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# On the automorphism group of a toral variety
Anton Shafarevich and Anton Trushin Moscow Center of Fundamental and Applied
Mathematics, Moscow, Russia; and Faculty of Computer Science, HSE University,
Pokrovsky Boulevard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>Moscow
Center of Fundamental and Applied Mathematics, Moscow, Russia; and Faculty of
Computer Science, HSE University, Pokrovsky Boulevard 11, Moscow, 109028
Russia<EMAIL_ADDRESS>
###### Abstract.
Let ${\mathbb{K}}$ be an algebraically closed field of characteristic zero. An
affine algebraic variety $X$ over ${\mathbb{K}}$ is toral if it is isomorphic
to a closed subvariety of a torus $({\mathbb{K}}^{*})^{d}$. We study the group
$\operatorname{Aut}(X)$ of regular automorpshims of a toral variety $X$. We
prove that if $T$ is a maximal torus in $\operatorname{Aut}(X)$, then $X$ is a
direct product $Y\times T$, where $Y$ is a toral variety with a trivial
maximal torus in the automorphism group. We show that knowing
$\operatorname{Aut}(Y)$, one can compute $\operatorname{Aut}(X)$. In the case
when the rank of the group ${\mathbb{K}}[Y]^{*}/{\mathbb{K}}^{*}$ is $\dim
Y+1$, the group $\operatorname{Aut}(Y)$ can be described explicitly.
###### Key words and phrases:
Affine variety, invertible function, algebraic torus, automorphism, rigid
variety
###### 2010 Mathematics Subject Classification:
Primary 14M25, 14L30; Secondary 14R20, 13N15, 14M17
The work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”
## 1\. Introduction
Let ${\mathbb{K}}$ be an algebraically closed field of characteristic zero.
The set of solutions of a system of polynomial equations in affine space has
been studied for a very long time. But some interesting properties may appear
when we consider the set of solutions inside a torus $({\mathbb{K}}^{*})^{d}$.
In other words we consider only solutions with non-zero coordinates. One of
examples of this approach is the Bernstein-Kushnirenko Theorem; see [2, 6].
Vladimir Popov in [9] proposed the following definition.
###### Definition 1.
An irreducible affine algebraic variety $X$ is called _toral_ if it is
isomorphic to a closed subvariety of a torus $({\mathbb{K}}^{*})^{d}$.
Some authors also use the term a ”very affine variety”; see [11, 3]. It can be
seen that $X$ is toral if and only if the algebra of regular functions on $X$
is generated by invertible functions; see [9, Lemma 1.14]. One of the reasons
why toral varieties are interesting is that they are rigid varieties; see [9,
Lemma 1.14].
###### Definition 2.
An affine algebraic variety $X$ is called _rigid_ if there is no non-trivial
action of the additive group $({\mathbb{K}},+)$ on $X.$
Despite the fact that the automorphism group of an affine algebraic variety
has a complicated structure sometimes it is possible to describe it for rigid
varieties. It was proven in [1] that the group of regular automorphisms
$\operatorname{Aut}(X)$ of a rigid variety $X$ contains a unique maximal torus
$T$. Moreover if we denote by $C(T)$ the centralizer of $T$ in
$\operatorname{Aut}(X)$ then the quotient group $\operatorname{Aut}(X)/C(T)$
is a discrete group. One can find examples of computation of
$\operatorname{Aut}(X)$ for rigid varieties in [1, 7, 8].
In this paper we study the automorphism group $\operatorname{Aut}(X)$ of a
toral variety $X$. We denote by ${\mathbb{K}}[X]$ the algebra of regular
functions on $X$ and by ${\mathbb{K}}[X]^{*}$ the multiplicative group of
invertible regular functions on $X$. Let $E(X)$ be the quotient group
${\mathbb{K}}[X]^{*}/{\mathbb{K}}^{*}.$ By [10], the group $E(X)$ is a free
finitely generated abelian group. For a toral variety $X$ the rank of $E(X)$
is not less than $\dim X.$
Any automorphism of $X$ induces an automorphism of $E(X).$ So we obtain a
homomorphism from $\operatorname{Aut}(X)$ to $E(X).$ We denote by $H(X)$ the
kernel of this homomorphism.
Suppose that $X$ is a closed subvariety of a torus
$T_{d}=({\mathbb{K}}^{*})^{d}$. In Proposition 1 we show that the group $H(X)$
is naturally isomorphic to a subgroup in $T_{d}$ which consists of elements
that preserve $X$ under the action by multiplication. In Proposition 2 we
propose a way to compute the subgroup $H(X)$.
In Theorem 1 we show that if $T$ is a maximal torus in $\operatorname{Aut}(X)$
then $X$ is isomorphic to a direct product $T\times Y$ where $Y$ is a toral
variety with a discrete automorphism group. Theorem 3 gives a way to find
$\operatorname{Aut}(X)$ knowing $\operatorname{Aut}(Y).$ If the rank of $E(Y)$
is $\dim Y+1$ then one can describe $\operatorname{Aut}(Y)$ (Theorem 3).
Also we consider the case when the rank of $E(X)$ is equal to $\dim X.$ By
Proposition 3 in this case $X$ is a torus. Moreover, it is the only case when
$\operatorname{Aut}(X)$ acts on $X$ with an open orbit.
We use the following notation. If $\varphi$ is a regular automorphism of an
affine variety $X$ then by $\varphi^{*}$ we mean an automorphism of
${\mathbb{K}}[X]$ dual to $\varphi$. If $A$ is a group and $B$ is a normal
subgroup in $A$ then by $[a]$ we denote the image of an element $a\in A$ in
the quotient group $A/B.$ If $X$ is a closed subvariety of an affine variety
$Z$ then by $I(X)$ we mean the ideal of regular functions on $Z$ which are
equal to zero on $X$.
The authors are grateful to Segrey Gaifullin for useful discussions. Also we
would like to thank Ivan Arzhantsev for helpful remarks and comments.
## 2\. General facts on toral varieties
Here we prove some initial properties of toral varieties and propose a way to
compute the group $H(X)$ for a toral variety $X$
Let $T_{d}$ be a torus of dimension $d$. We recall that the group
$\operatorname{Aut}(T_{d})$ is isomorphic to
$T_{d}\rtimes\mathrm{GL}_{d}({\mathbb{Z}})$; see[1, Example 2.3]. Here the
left factor $T_{d}$ acts on itself by multiplications and a matrix
$(a_{ij})\in\mathrm{GL}_{d}({\mathbb{Z}})$ defines an automorphism of $T_{d}$
which is given by the formula
$t_{i}\to t_{1}^{a_{i1}}\ldots t_{d}^{a_{id}},$
where $t_{1},\ldots,t_{d}$ are coordinate functions on $T_{d}.$
###### Proposition 1.
1. (1)
Let $X$ be a closed irreducible subvariety of the torus $T_{d}.$ Then $H(X)$
is a quasitorus isomorphic to a closed subgroup in $T_{d}$ which consists of
all elements that preserve $X$.
2. (2)
Let $X$ be a toral variety. Suppose that the rank of the group $E(X)$ is equal
to $r$. Then $X$ is isomorphic to a closed subvariety of
$T_{r}=({\mathbb{K}}^{*})^{r}$ and the group $\operatorname{Aut}(X)$ is
isomorphic to a subgroup of
$\operatorname{Aut}(T_{r})=T_{r}\rtimes\mathrm{GL}_{r}({\mathbb{Z}})$ which
consists of all automorphisms of $T_{r}$ that preserve $X.$
###### Proof.
1) Let $H$ be a subgroup in $T_{d}$ which consists of all elements which
preserve $X$. Any element of $H$ defines an automorphism of $X$. So we obtain
a homomorphism
$\psi:H\to\operatorname{Aut}(X).$
We will show that $\psi$ is injective and the image of $\psi$ is $H(X)$.
Let $t_{1},\ldots,t_{d}$ be coordinate functions on $T_{d}.$ Then their
restrictions to $X$ define invertible functions $f_{1},\ldots,f_{d}$ which
generate ${\mathbb{K}}[X].$ Let $\varphi$ be an automorphism in $H(X)$. Then
the automorphism $\varphi^{*}$ defines a trivial automorphism of $E(X).$ So
$\varphi^{*}(f_{i})=\lambda_{i}f_{i},$ for some
$\lambda_{1},\ldots,\lambda_{d}\in{\mathbb{K}}^{*}.$ Then the element
$(\lambda_{1},\ldots,\lambda_{d})\in T_{d}$ preserves $X$ and its restriction
to $X$ is $\varphi.$ So the image of $\psi$ contains $H(X).$
Conversely, consider an element $t=(\lambda_{1},\ldots,\lambda_{d})\in H$.
Then $(\psi(t))^{*}$ maps $f_{i}$ to $\lambda_{i}f_{i}.$ So it acts trivially
on $E(X).$ Therefore, the image of $\psi$ is contained in $H(X).$
The group $T_{d}$ acts on itself freely. So only the identity can act on $X$
trivially. Hence $\psi$ is injective.
2) Since rank $E(X)$ is equal to $r$ one can choose invertible functions
$f_{1},\ldots,f_{r}\in{\mathbb{K}}[X]^{*}$ such that $[f_{1}],\ldots,[f_{r}]$
form a basis of the group $E(X)$. Then $f_{1},\ldots,f_{r}$ generate the
algebra ${\mathbb{K}}[X]$ and there is a closed embedding of $X$ into $T_{r}.$
We denote by $t_{1},\ldots,t_{r}$ coordinate functions on $T_{r}$ which
corresponds to $f_{1},\ldots,f_{r}.$
Let $P$ be the subgroup in $T_{r}\rtimes\mathrm{GL}_{r}({\mathbb{Z}})$ of all
automorphisms that preserve $X$. Again, there is a homomorphism
$\psi:P\to\operatorname{Aut}(X)$ and we will show that $\psi$ is a bijection.
Each automorphism $\varphi\in\operatorname{Aut}(X)$ defines an automorphism of
the group $E(X).$ Since $E(X)$ is a free abelian group of the rank $r$ the
group $\operatorname{Aut}E(X)$ is isomorphic to
$\mathrm{GL}_{r}({\mathbb{Z}}).$
So the automorphism $\varphi^{*}$ has a form
$\varphi^{*}(f_{i})=\lambda_{i}f_{1}^{a_{i1}}\ldots f_{r}^{a_{ir}},$ where
$(a_{ij})\in\mathrm{GL}_{r}({\mathbb{Z}})$ and
$\lambda_{i}\in{\mathbb{K}}^{*}.$ But then the automorphism $\varphi$ is the
restriction of an automorphism
$\widetilde{\varphi}\in\operatorname{Aut}(T_{r})$ which is given by the
formula
$\widetilde{\varphi}(t_{i})=\lambda_{i}t_{1}^{a_{i1}}\ldots t_{r}^{a_{ir}}.$
So $\psi$ is surjective.
Now let $\phi$ be an automorphism of $T_{r}$ which acts trivially on $X$. Then
$\phi$ is given by the formula
$\phi(t_{i})=\lambda_{i}t_{1}^{a_{i1}}\ldots t_{r}^{a_{ir}}$
where $\lambda_{i}\in{\mathbb{K}}^{*}$ and the matrix $(a_{ij})$ belongs to
$\mathrm{GL}_{r}({\mathbb{Z}}).$ The corresponding automorphism of
${\mathbb{K}}[X]$ is given by the formula
$f_{i}\to\lambda_{i}f_{1}^{a_{i1}}\ldots f_{r}^{a_{ir}}$
but it is a trivial automorphism so we have
$f_{i}=\lambda_{i}f_{1}^{a_{i1}}\ldots f_{r}^{a_{ir}}.$
But the elements $[f_{1}],\ldots[f_{r}]\in E(X)$ are linearly independent. The
matrix $(a_{ij})$ is the identity matrix. So $\phi(t_{i})=\lambda_{i}t_{i}$
and $\phi$ belongs to $T_{r}\subseteq\operatorname{Aut}(T_{r})$. But the group
$T_{r}$ acts freely on itself. Hence $\phi$ is a trivial automorphism. So
$\psi$ is injective.
∎
###### Remark 1.
It follows from Proposition 1 that a toral variety $X$ can be embedded in a
torus $T_{r}$ in such a way that any automorphism $X$ can be extended to an
automorphism of $T_{r}.$ If $X$ is a subvariety of $Z$ it is always natural to
ask whether an automorphism of $X$ can be extended to an automorphism of $Z$.
Some results concerning this problem can be found in [4, 5].
Now let $X$ be a closed irreducible subvariety in $T_{d}.$ Then one can
describe the subgroup $H(X).$ Let $M\simeq{\mathbb{Z}}^{d}$ be the lattice of
characters of $T_{d}$. For $m=(m_{1},\ldots,m_{d})\in M$ by $\chi^{m}$ we mean
the character $t\to t_{1}^{m_{1}}\ldots t_{d}^{m_{d}}.$ Then each function in
${\mathbb{K}}[t_{1}^{\pm 1},\ldots,t_{d}^{\pm 1}]$ is a linear combinations of
characters. For a function
$f=\sum_{i}\alpha_{m_{i}}\chi^{m_{i}}\in{\mathbb{K}}[t_{1}^{\pm
1},\ldots,t_{d}^{\pm 1}]$ by _support of $f$_ we mean the subset
$\operatorname{Supp}f=\\{m_{i}\in M|\ \alpha_{m_{i}}\neq 0\\}\subseteq M.$
Let $I(X)$ be the ideal of functions in ${\mathbb{K}}[t_{1},\ldots,t_{d}]$
which are equal to zero on $X$. We say that $f\in I(X)$ is _minimal_ if there
is no non-zero $g\in I(X)$ such that
$\operatorname{Supp}g\subsetneq\operatorname{Supp}f.$
###### Lemma 1.
The ideal $I(X)$ is generated by minimal polynomials.
###### Proof.
If $f\in I(X)$ is not minimal then there is a $g\in I(X)$ with
$\operatorname{Supp}g\subsetneq\operatorname{Supp}f$. One can choose a
constant $\alpha$ such that $\operatorname{Supp}(f-\alpha
g)\subsetneq\operatorname{Supp}f.$ Applying induction by cardinality of
$\operatorname{Supp}f$ we see that $g$ and $f-\alpha g$ can be represented as
a sum of minimal polynomials. Then $f$ is also a sum of minimal polynomials. ∎
###### Remark 2.
We denote by $M(X)$ the subgroup of $M$ which is generated by all elements of
the form $m_{a}-m_{b}$ such that there is a minimal polynomial
$f=\sum\alpha_{m_{i}}\chi^{m_{i}}\in I(X)$ with non-zero $\alpha_{m_{a}}$ and
$\alpha_{m_{b}}$.
###### Proposition 2.
The subgroup $H(X)$ is isomorphic to a subgroup in $T_{d}$ which is given by
equations $\chi^{m}(t)=1$ for all $m\in M(X).$
###### Proof.
Denote by $H$ the subgroup in $T_{d}$ consisting of all elements which
preserve $X$. By Proposition 1 the group $H$ is isomorphic to $H(X).$ We will
show that $H$ is given in $T_{d}$ by equations $\chi^{m}(t)=1$ for all $m\in
M(X).$
Let $h\in H$ and $f=\sum_{i}\alpha_{m_{i}}\chi^{m_{i}}$ be a minimal
polynomial in $I(X).$ Then $h\circ
f=\sum_{i}\alpha_{i}\chi^{m_{i}}(h)\chi^{m_{i}}.$ The ideal $I(X)$ is
invariant under the action of $H$. So $h\circ f\in I(X)$. Suppose that there
are $a,b$ such that $\alpha_{a},\alpha_{b}\neq 0$ and
$\chi^{m_{a}}(h)\neq\chi^{m_{b}}(h).$ Then $g=\chi^{m_{a}}(h)f-h\circ f$ is a
non-zero function in $I(X)$ and
$\operatorname{Supp}g\subsetneq\operatorname{Supp}f.$ But $f$ is minimal. So
$\chi^{m_{a}}(h)=\chi^{m_{b}}(h)$. Therefore, $\chi^{m_{a}-m_{b}}(h)=1$ and
this implies that $\chi^{m}(h)=1$ for all $m\in M(X).$
Now consider the subgroup $H^{\prime}=\\{t\in T_{d}|\chi^{m}(t)=1,\ \forall\
m\in M(X)\\}.$ Then every minimal polynomial in $I(X)$ is semi-invariant with
respect to $H^{\prime}$. But $I(X)$ is generated by minimal polynomials. So
$I(X)$ is invariant under the action of $H^{\prime}$. Therefore,
$H^{\prime}\subseteq H.$
∎
At the end of this section, we note that toral varieties satisfy the following
conjecture formulated by Alexander Perepechko and Mikhail Zaidenberg.
###### Conjecture 1 (Conjecture 1.0.1 in [8]).
If $Y$ is a rigid affine algebraic variety over ${\mathbb{K}}$, then the
connected component $\operatorname{Aut}^{0}(Y)$ is an algebraic torus of rank
not greater than $\dim Y$.
Indeed, if $Y$ is a toral variety then the group $\operatorname{Aut}(Y)/H(Y)$
is isomorphic to a subgroup in
$\operatorname{Aut}(E(Y))\simeq\mathrm{GL}_{r}({\mathbb{Z}})$, where $r$ is a
rank of $E(Y).$ Then $\operatorname{Aut}(Y)/H(Y)$ is a discrete group. So
$\operatorname{Aut}^{0}(Y)$ is contained in $H(Y).$ But $H(Y)$ is a
quasitorus. Therefore, $\operatorname{Aut}^{0}(Y)$ is a torus.
## 3\. The structure of the automorphism group
It follows from [9, Lemma 1.14] that toral varieties are rigid. By [1, Theorem
2.1], there is a unique maximal torus in the automorphism group of an
irreducible rigid variety.
###### Theorem 1.
Let $X$ be a toral variety and $T$ be the maximal torus in
$\operatorname{Aut}(X).$ Then $X\simeq Y\times T$ where $Y$ is a toral variety
with a discrete automorphism group.
###### Proof.
Let $r$ by the rank of the group $E(X).$ By Proposition 1, the variety $X$ can
be embedded into the torus $T_{r}.$ We denote by $M$ the lattice of characters
of $T_{r}$ and by $M(X)$ the sublattice in $M$ which corresponds to $X$; see
Remark 2. One can choose a basis $e_{1},\ldots,e_{r}\in M$ such that
$b_{1}e_{1},\ldots b_{l}e_{l}$ is a basis of $M(X)$ for some
$b_{1},\ldots,b_{l}\in{\mathbb{N}}$ and $l\leq r$. Denote by
${t_{1}},\ldots{t_{r}}$ coordinates on $T_{r}$ corresponding to
$e_{1},\ldots,e_{r}$.
Then the equations $\chi^{m}(t)=1$ for all $m\in M(X)$ define a subgroup $H$
in $T_{r}$ which consists of elements of the form
$(\epsilon_{1},\ldots,\epsilon_{l},t_{l+1},\ldots,t_{r}),$
where $\epsilon_{1},\ldots,\epsilon_{l}$ are the roots of unity of degrees
$b_{1},\ldots,b_{l}$ respectively and $t_{l+1},\ldots
t_{r}\in{\mathbb{K}}^{*}.$
Then all minimal polynomials in $I(X)$ are semi-invariant with respect to $H$.
This means that minimal polynomials in $I(X)$ are homogeneous with respect to
variables ${t}_{l+1},\ldots,{t}_{r}.$ Since functions ${t}_{i}$ are invertible
one can choose a set of minimal generators of $I(X)$ which do not depend on
${t}_{l+1},\ldots,{t}_{r}.$ It implies that $X\simeq Y\times T_{r-l}$ where
$Y$ is a subvariety of $T_{l}.$
We see that $T_{r-l}$ coincides with the maximal torus in $H\simeq H(X)$. The
variety $Y$ is also a toral variety which is given by the ideal
$I(X)\cap{\mathbb{K}}[t_{1},\ldots t_{l}].$ Since the maximal torus in
$\operatorname{Aut}(X)$ is isomorphic to $T_{l}$ the maximal torus in
$\operatorname{Aut}(Y)$ is trivial. It follows from Proposition 1 that in this
case $\operatorname{Aut}(Y)$ is a discrete group.
∎
Let $X$ be a toral variety and suppose that $X\simeq T_{s}\times Y$ where $Y$
is a toral variety with a discrete automorphism group and $T_{s}$ is the torus
$({\mathbb{K}}^{*})^{s}.$ One can see that $\operatorname{Aut}(X)$ contains
the following subgroups.
There is a subgroup which is isomorphic to $\operatorname{Aut}(Y)$. This
subgroup acts in a natural way on $Y$ and acts trivially on $T_{s}.$ The
subgroup $\mathrm{GL_{s}}({\mathbb{Z}})$ acts naturally on $T_{s}$ and
trivially on $Y$. Moreover there is a subgroup which is isomorphic to
$({\mathbb{K}}[Y]^{*})^{s}\simeq(E(Y)\times{\mathbb{K}}^{*})^{s}.$ This
subgroup acts in the following way. If
$f_{1},\ldots,f_{s}\in{\mathbb{K}}[Y]^{*}$ then we can define an automorphism
of $T_{s}\times Y$ as follows
$(t_{1},\ldots,t_{s},y)\to(f_{1}(y)t_{1},\ldots,f_{s}(y)t_{s},y).$
The following theorem was proposed to the authors by Sergey Gaifullin.
###### Theorem 2.
Let $X\simeq T_{s}\times Y$ be a toral variety, where $Y$ is a toral variety
with a discrete automorphism. Then
$\operatorname{Aut}(X)\cong\operatorname{Aut}(Y)\ltimes(\mathrm{GL}_{s}({\mathbb{Z}})\ltimes(E(Y)\times{\mathbb{K}}^{*})^{s}).$
###### Proof.
There is a natural action of $T_{s}$ on $X$. We see that ${\mathbb{K}}[Y]$ is
the algebra of invariants of this action. Since $T_{s}$ is a maximal torus in
$\operatorname{Aut}(X)$ each automorphism of $T_{s}\times Y$ preserves
${\mathbb{K}}[Y].$ So we obtain a homomorphism
$\Phi:\operatorname{Aut}(X)\to\operatorname{Aut}(Y).$ Let $B$ be the kernel of
$\Phi$. The group $\operatorname{Aut}(Y)$ is naturally embedded into
$\operatorname{Aut}(T_{s}\times Y)$ and intersects trivially with $B$. At the
same time $\operatorname{Aut}(Y)$ maps isomorphically to the image of $\Phi$.
It implies that
$\operatorname{Aut}(T_{s}\times Y)=\operatorname{Aut}(Y)\ltimes B.$
We denote by $t_{1},\ldots,t_{s}$ coordinate functions on $T_{s}.$ Then
${\mathbb{K}}[T_{s}\times
Y]\simeq{\mathbb{K}}[T_{s}]\otimes{\mathbb{K}}[Y]={\mathbb{K}}[Y][t_{1}^{\pm
1},\ldots t_{s}^{\pm 1}].$
Let $\phi\in B.$ The algebra ${\mathbb{K}}[Y]$ is invariant with respect to
$\phi^{*}$. So for all $t\in T_{s}$ and $y\in Y$ we have
$\phi((t,y))=(t^{\prime},y),$
for some $t^{\prime}\in T_{s}.$ Therefore, for each $y\in Y$ the automorphism
$\phi$ defines an automorphism $\phi_{y}:T_{s}\to T_{s}.$ Hence for each $y\in
Y$ we have
$\phi^{*}(t_{i})(t,y)=t_{i}((\phi(t),\phi(y)))=t_{i}((\phi_{y}(t),y))=f_{i}(y)t_{1}^{a_{i1}(y)}\ldots
t_{s}^{a_{is}(y)}$
for some non-zero constant $f_{i}(y)$ and a matrix
$A(y)=(a_{ij}(y))\in\mathrm{GL}_{s}({\mathbb{Z}}).$ In reasons of continuity
the matrix $A(y)$ is the same for all $y\in Y$ and $f_{i}:Y\to Y$ are regular
functions on $Y$. Since $f_{i}(y)\neq 0$ for all $y\in Y$ the functions
$f_{i}$ are invertible. So we have
$\phi^{*}(t_{i})=f_{i}t_{1}^{a_{i1}}\ldots t_{s}^{a_{is}}$
for some $f_{i}\in{\mathbb{K}}[Y]^{*}$ and
$A\in\mathrm{GL}_{s}({\mathbb{Z}}).$
Then we have a homomorphism
$\overline{\Phi}:B\to\mathrm{GL}_{s}({\mathbb{Z}}),\ \phi\to A.$ Again, it is
easy to see that $\mathrm{GL}_{s}({\mathbb{Z}})$ is naturally embedded into
$B$ and maps isomorphically to $\mathrm{GL}_{s}({\mathbb{Z}})$ with respect to
$\overline{\Phi}.$ So
$B=\mathrm{GL}_{s}({\mathbb{Z}})\ltimes\mathrm{Ker}\ \overline{\Phi}.$
The kernel of $\overline{\Phi}$ consists of automorphisms
$\varphi\in\operatorname{Aut}(T_{s}\times Y)$ which have a form
$\varphi(t_{1},\ldots,t_{s},y)=(f_{1}(y)t_{1},\ldots,f_{s}(y)t_{s},y).$
for some $f_{1},\ldots,f_{s}\in{\mathbb{K}}[Y]^{*}.$ We see that for all
$f_{1},\ldots,f_{s}\in{\mathbb{K}}[Y]^{*}$ this formula defines an
automorphism of $T_{s}\times Y,$ so $\mathrm{{\rm Ker}\,}\
\overline{\Phi}=({\mathbb{K}}[Y]^{*})^{s}=(E(Y)\times{\mathbb{K}}^{*})^{s}.$ ∎
## 4\. The case $\mathrm{rk}\ E(X)=\dim X$
Let $X$ be a toral variety. Then $\mathrm{rk}\ E(X)\geq\dim X.$ Indeed,
suppose that $f_{1},\ldots,f_{r}$ are invertible functions and
$[f_{1}],\ldots,[f_{r}]$ is a basis in $E(X).$ Then $f_{1},\ldots,f_{r}$
generate ${\mathbb{K}}[X].$ So $r\geq\dim X$.
The following result shows that if $\mathrm{rk}\ E(X)=\dim X$ then $X$ is a
torus. Moreover this is the only case when $\operatorname{Aut}(X)$ acts with
an open orbit on $X$.
###### Proposition 3.
Let $X$ be a toral variety. Then the following conditions are equivalent.
1. (1)
$X$ is a torus;
2. (2)
$\mathrm{rk}\ E(X)=\dim X.$
Suppose also that the field ${\mathbb{K}}$ is uncountable. Then $X$ is a torus
if and only if $\operatorname{Aut}(X)$ acts on $X$ with an open orbit.
###### Proof.
The implication $1)\Rightarrow 2)$ is trivial.
Suppose that $\mathrm{rk}\ E(X)=\dim X.$ Then one can choose invertible
functions $f_{1},\ldots,f_{n}$ such that $[f_{1}],\ldots,[f_{n}]$ is a basis
of $E(X)$. Then ${\mathbb{K}}[X]$ is generated by
$f_{1},f_{1}^{-1},\ldots,f_{n},f_{n}^{-1}$. But $f_{1},\ldots,f_{n}$ are
algebraically independent, otherwise $\dim X<\mathrm{rk}\ E(X)$. So
${\mathbb{K}}[X]$ is isomoprhic to the algebra of Laurent polynomials. So we
obtain implication $2)\Rightarrow 1).$
Now we prove the second statement. If $X$ is a torus then
$\operatorname{Aut}(X)$ acts on $X$ transitively. So suppose $X$ is a toral
variety over an uncountable field ${\mathbb{K}}$ and $\operatorname{Aut}(X)$
acts on $X$ with an open orbit $U$.
Let $T$ be the maximal torus in $\operatorname{Aut}(X).$ Sine the quotient
group $\operatorname{Aut}(X)/T$ is a discrete group the set $U$ is a countable
union of orbits of $T.$ Since ${\mathbb{K}}$ is uncountable it implies that
one of the orbits of $T$ is open in $X.$ Then $\dim X=\dim T.$ By Theorem 1 we
have $X\simeq T\times Y$ for some toral variety $Y$. But since $\dim T=\dim
T\times Y$ we obtain that $Y$ is a point and $X\simeq T.$
∎
## 5\. The case $\mathrm{rk}\ E(X)=\dim X+1$
By Theorem 1 any toral variety is a direct product $T\times Y$ where $T$ is a
torus and $Y$ is a toral variety with a discrete automorphism group. By
Theorem 2 one can find $\operatorname{Aut}(X)$ knowing
$\operatorname{Aut}(Y).$ In this section we provide a way to find
$\operatorname{Aut}(Y)$ when $\mathrm{rk}\ E(Y)=\dim\ Y+1.$
Let $Y$ be a toral variety with a trivial maximal torus in
$\operatorname{Aut}(Y)$. Let $r$ be the rank of $E(Y)$. We suppose that
$r=\dim(Y)+1.$
By Proposition 1 the variety $Y$ can be embedded into $T_{r}$ as a
hypersurface. Then there is an irreducible polynomial
$h\in{\mathbb{K}}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$ such that $I(Y)=(h).$
The polynomial $h$ has a form
$h=\sum_{m\ \in\ \mathrm{Supp}\ h}\alpha_{m}\chi^{m}.$
Let $M$ be the lattice of characters of $T_{r}$ and $M(Y)$ be a sublattice in
$M$ which corresponds to $Y$; see Definition 2. Since the maximal torus in
$\operatorname{Aut}(Y)$ is trivial, the rank of the lattice $M(Y)$ is equal to
$r$. It means that the elements $m_{a}-m_{b}$ with
$m_{a},m_{b}\in\mathrm{Supp}\ h$ generate a sublattice of full rank in $M$.
We denote by $\mathrm{GAff}(M,h)$ the group of all degenerate integer affine
transformations $\varphi$ of $M$ which preserve $\mathrm{Supp}\ h$ and for any
linear combination
$\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})=0,\ a_{i}\in\mathbb{Z},\
m_{i,1},m_{i,2}\in\mathrm{Supp}\ h$
$\varphi$ satisfies
$\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{a_{i}}=\prod_{i}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{a_{i}}.$
(1)
###### Remark 3.
Let us fix a point $m_{0}\in\mathrm{Supp}\ h.$ Then any equation
$\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})=0,\ a_{i}\in\mathbb{Z},\
m_{i,1},m_{i,2}\in\mathrm{Supp}\ h$
can be written as
$\sum_{i}{b_{i}}(m_{i}-m_{0})=0,\ b_{i}\in{\mathbb{Z}},\
m_{i}\in\mathrm{Supp}\ h.$ (2)
Therefore, it is enough to check equations (1) for linear combinations of the
form (2). Let $N$ be the number of points in $\mathrm{Supp}\ h.$ Then the set
$\\{(b_{1},\ldots,b_{N-1})\in\mathbb{Z}^{N-1}|\sum_{i}{b_{i}}(m_{i}-m_{0})=0\\}$
is a sublattice in ${\mathbb{Z}}^{N-1}$ and it is enough to check the
equations in (1) for some basis in this sublattice.
###### Theorem 3.
Let $Y$ be a toral variety with trivial maximal torus in
$\operatorname{Aut}(Y).$ Suppose that $\mathrm{rk}\ E(Y)=\dim Y+1.$ Then
$\operatorname{Aut}(Y)/H(Y)\simeq\mathrm{GAff}(M,h).$
###### Proof.
Let $\psi$ be an automorphism of $Y$. By Proposition 1 the automorphism $\psi$
can be extended to an automorphism of $T_{r}$. We denote by $\psi^{*}$ the
respective automorphism of ${\mathbb{K}}[t^{\pm 1}_{1},\ldots,t^{\pm 1}_{r}]$.
Then $\psi^{*}$ has the form
$\psi^{*}(t_{i})=\lambda_{i}t_{1}^{a_{i1}}\ldots t_{r}^{a_{ir}},$
where $\lambda_{i}\in{\mathbb{K}}^{*}$ and
$(a_{ij})\in\mathrm{GL}_{r}({\mathbb{Z}}).$ We denote by $\lambda$ the element
$(\lambda_{1},\ldots,\lambda_{r})\in T_{r}$ and by $\overline{\psi}$ the
automorphism of $M$ which respects to the matrix $(a_{ij}).$ Then
$\psi^{*}(\chi^{m})=\chi^{m}(\lambda)\chi^{\overline{\psi}(m)}$
for all $m\in M.$
The polynomial $\psi^{*}(h)$ also generates $I(Y).$ So it differs from $h$ by
an invertible element of ${\mathbb{K}}[t^{\pm 1}_{1},\ldots,t^{\pm 1}_{r}].$
Then
$\psi^{*}(h)=\alpha\chi^{v}h$
for some $\alpha\in{\mathbb{K}}^{*}$ and $v\in M.$ Therefore, we have an
equation
$\psi^{*}(h)=\sum_{m\ \in\ \mathrm{Supp}\
h}\alpha_{m}\chi^{m}(\lambda)\chi^{\overline{\psi}(m)}=\alpha\sum_{m\ \in\
\mathrm{Supp}\ h}\alpha_{m}\chi^{m+v}.$ (3)
We define the map $\varphi:M\to M$ by the following formula:
$\varphi(m)=\overline{\psi}(m)-v.$
Then $\psi$ is an affine transformation of $M$ which preserves $\mathrm{Supp}\
h.$
We consider a linear combination
$\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})=0,\ a_{i}\in\mathbb{Z},\
m_{i,1},m_{i,2}\in\mathrm{Supp}\ h.$
Equation (3) can be written as
$\sum_{m\ \in\ \mathrm{Supp}\
h}\alpha_{m}\chi^{m}(\lambda)\chi^{\varphi(m)}=\alpha\sum_{m\ \in\
\mathrm{Supp}\ h}\alpha_{m}\chi^{m}=\alpha\sum_{m\ \in\ \mathrm{Supp}\
h}\alpha_{\varphi(m)}\chi^{\varphi(m)}.$
and it implies
$\frac{\alpha_{m_{i,1}}\chi^{m_{i,1}}(\lambda)}{\alpha_{m_{i,2}}\chi^{m_{i,2}}(\lambda)}=\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\chi^{m_{i,1}-m_{i,2}}(\lambda)=\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}$
for all $i.$
Then we have
$\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\chi^{m_{i,1}-m_{i,2}}(\lambda)\right)^{a_{i}}=\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{a_{i}}\left(\chi^{\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})}(\lambda)\right)=\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{a_{i}}$
$=\prod_{i}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{a_{i}}.$
So $\varphi\in\mathrm{Gaff}(M,h).$ Then we obtain a homomorphism
$\eta:\operatorname{Aut}(Y)\to\mathrm{Gaff}(M,h).$ Moreover we see that the
kernel of $\eta$ is $H(Y).$ Now we will show that $\eta$ is surjective.
Let $\varphi\in\mathrm{Gaff}(M,h)$ and $f_{1},\ldots,f_{r}$ be a basis in
$M(Y).$ Then there are $m_{i,1},m_{i,2}\in\ \mathrm{Supp}(h),$ and
$a_{ij}\in{\mathbb{Z}}$ such that
$f_{j}=\sum_{i}a_{ij}(m_{i,1}-m_{i,2}).$
There is a $\lambda\in T_{r}$ such that
$\chi^{f_{j}}(\lambda)=\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{-a_{ij}}\prod_{i}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{a_{ij}}.$
Let $d\varphi$ be the differential of $\varphi.$ We define an automorphism
$\psi^{*}$ of ${\mathbb{K}}[t_{1}^{\pm 1}\ldots t_{r}^{\pm 1}]$ by the
following rule:
$\psi^{*}(\chi^{m})=\chi^{m}(\lambda)\chi^{d\varphi(m)}.$
Let us check that $\psi^{*}$ preserves $I(Y).$ We have
$\psi^{*}(h)=\sum\alpha_{m}\chi^{m}(\lambda)\chi^{d\varphi(m)}.$
There is $v\in M$ such that
$\varphi(m)=d\varphi(m)+v.$
Then
$\chi^{v}\psi^{*}(h)=\sum\alpha_{m}\chi^{m}(\lambda)\chi^{\varphi(m)}.$
We see that $\mathrm{Supp}\ \chi^{v}\psi^{*}(h)=\mathrm{Supp}\ h.$ We will
show that there is $\alpha\in{\mathbb{K}}$ such that
$\chi^{v}\psi^{*}(h)=\alpha h.$
For any $b,c\in\mathrm{Supp}\ h$ there are numbers $d_{j}\in{\mathbb{Z}}$ such
that
$b-c=\sum_{j}d_{j}f_{j}.$
So
$\frac{\alpha_{b}\chi^{b}(\lambda)}{\alpha_{c}\chi^{c}(\lambda)}=\frac{\alpha_{c}}{\alpha_{b}}\chi^{b-c}(\lambda)=\frac{\alpha_{b}}{\alpha_{c}}\chi^{\sum_{j}d_{j}f_{j}}(\lambda)=$
$\frac{\alpha_{b}}{\alpha_{c}}(\prod_{j}\chi^{f_{j}}(\lambda))^{d_{j}}=\frac{\alpha_{b}}{\alpha_{c}}\prod_{i,j}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{-d_{j}a_{ij}}\prod_{i,j}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{d_{j}a_{ij}}.$
We have a combination
$b-c-\sum_{j}d_{j}f_{j}=b-c-\sum_{j,i}d_{j}a_{ij}(m_{i,1}-m_{i,2})=0.$
Since $\varphi\in\mathrm{Gaff}(M,h)$ we obtain
$\frac{\alpha_{b}}{\alpha_{c}}\prod_{i,j}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{-d_{j}a_{ij}}=\frac{\alpha_{\varphi(b)}}{\alpha_{\varphi(c)}}\prod_{i,j}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{-d_{j}a_{ij}}.$
It implies that
$\frac{\alpha_{b}\chi^{b}(\lambda)}{\alpha_{c}\chi^{c}(\lambda)}=\frac{\alpha_{\varphi(b)}}{\alpha_{\varphi(c)}}.$
So the coefficients of the polynomials $\chi^{v}\psi^{*}(h)$ and $h$ are
proportional. Then there is an $\alpha\in{\mathbb{K}}$ such that
$\chi^{v}\psi^{*}(h)=\alpha h.$ Hence $\psi^{*}(h)=\alpha\chi^{-v}h\in I(Y)$.
Therefore, $\psi^{*}$ preserves $I(Y)$ and defines an automorphism $\psi$. It
is a direct check that $\eta(\psi)=\varphi$. So $\varphi$ is surjective.
∎
###### Corollary 1.
Let $Y$ be a toral variety with trivial maximal torus in
$\operatorname{Aut}(Y).$ Suppose that $\mathrm{{\mathrm{rk}}}\ E(Y)=\dim Y+1.$
Then $\operatorname{Aut}(Y)$ is a finite group.
It is natural to formulate the following question.
###### Question 1.
Let $Y$ be a toral variety with trivial maximal torus in
$\operatorname{Aut}(Y).$ Is $\operatorname{Aut}(Y)$ a finite group?
Note that this is not true for a general rigid variety. One can find a
counterexample in [7].
At the end, we give two examples illustrating Theorem 3.
###### Example 1.
Let $Y$ be the affine line $\mathbb{A}^{1}$ without two points. The variety
$Y$ can be given in $({\mathbb{K}}^{*})^{2}$ as the set of solutions of the
equation
$h=x+y-1=0,\ (x,y)\in({\mathbb{K}}^{*})^{2}.$
So $Y$ is a toral variety. Let $M\simeq{\mathbb{Z}}^{2}$ be the lattice of
characetrs of $({\mathbb{K}}^{*})^{2}$. We see that the lattice $M(Y)$
contains elements $(1,0),(0,1),(1,-1)$, so $M(Y)=M.$ Therefore, $H(Y)$ is a
trivial group. The group $E(Y)$ is generated by $[x]$ and $[y]$. So we can
apply Theorem 3.
Now we find the group $\mathrm{Gaff}(M,h)$. The set $\mathrm{Supp}\ h$
consists of points $m_{0}=(0,0),\ m_{1}=(1,0),m_{2}=(0,1).$
$m_{2}$$m_{0}$$m_{1}$$M$ Figure 1. $\mathrm{Supp}(x+y-1)$
A linear combination
$a_{1}(m_{1}-m_{0})+a_{2}(m_{2}-m_{0})+a_{3}(m_{2}-m_{1})=a_{1}(1,0)+a_{2}(0,1)+a_{3}(-1,1)$
is equal to zero if and only if $a_{1}=a_{3}=-a_{2}.$ But then equations (1)
are trivial. By affine transformations of $M$ we can permute all points in
$\mathrm{Supp}\ h.$ Therefore,
$\operatorname{Aut}(Y)\simeq\mathrm{GAff}(M,h)\simeq S_{3}.$
The answer looks natural since the affine line without two points is the
projective line without three points.
###### Example 2.
Let $a,b,c\in{\mathbb{K}}^{*}$ be non-zero constants and $Y$ be the set of
solutions of the equation
$h=1+ax+by+cxy=0,\ (x,y)\in({\mathbb{K}}^{*})^{2}.$
Let $M\simeq{\mathbb{Z}}^{2}$ be the lattice of characters of
$({\mathbb{K}}^{*})^{2}$. The lattice $M(Y)$ contains $(1,0)$ and $(0,1)$ so
$M(Y)=M.$ Then $H(Y)$ is a trivial group. The group $E(Y)$ is generated by
$[x]$ and $[y]$. Again, we can apply Theorem 3.
The set $\mathrm{Supp}\ h$ consists of points $m_{0}=(0,0),\
m_{1}=(1,0),m_{2}=(0,1),m_{3}=(1,1).$
$m_{3}$$m_{2}$$m_{0}$$m_{1}$$M$ Figure 2. $\mathrm{Supp}(1+ax+by+cxy)$
The set of affine transformation which preserve $\mathrm{Supp}\ h$ is the
dihedral group $D_{4}.$
To understand what elements of $D_{4}$ belong to $\mathrm{GAff}(M,h)$ we need
to check the equations (3). To simplify notations we will denote $m_{0}$ by
$1,$ $m_{1}$ by $x,$ $m_{2}$ by $y$ and $m_{3}$ by $xy$. By $\alpha_{m}$ we
denote the coefficient in $h$ of monomial $\chi^{m}$. So we have
$\alpha_{1}=1,\ \alpha_{x}=a,\ \alpha_{y}=b,\ \alpha_{xy}=c.$
By Remark 3 it is enough to check the equations (3) only for linear
combinations of the form
$a_{1}(m_{1}-m_{0})+a_{2}(m_{2}-m_{0})+a_{3}(m_{3}-m_{0})=a_{1}(1,0)+a_{2}(0,1)+a_{3}(1,1)=0,$
where $a_{1},a_{2},a_{3}\in{\mathbb{Z}}.$ So we have $a_{1}=a_{2}=-a_{3}.$
Again, by Remark 3 it is enough to check the equation
$\left(\frac{\alpha_{x}}{\alpha_{1}}\right)^{1}\left(\frac{\alpha_{y}}{\alpha_{1}}\right)^{1}\left(\frac{\alpha_{xy}}{\alpha_{1}}\right)^{-1}=\left(\frac{\alpha_{\varphi(x)}}{\alpha_{\varphi(1)}}\right)^{1}\left(\frac{\alpha_{\varphi(y)}}{\alpha_{\varphi(1)}}\right)^{1}\left(\frac{\alpha_{\varphi(xy)}}{\alpha_{\varphi(1)}}\right)^{-1}.$
So an element $\varphi\in D_{4}$ belongs to $\mathrm{GAff}(M,h)$ if and only
if
$\frac{ab}{c}=\frac{\alpha_{\varphi(x)}\alpha_{\varphi(y)}}{\alpha_{\varphi(1)}\alpha_{\varphi(xy)}}.$
Substituting various elements of the dihedral group into this equality, we
obtain that the rotation by $\frac{\pi}{2}$ and the symmetries with respect to
the lines passing through the midpoints of the opposite sides belong to
$\mathrm{GAff}(M,h)$ if and only if
$\frac{ab}{c}=\frac{c}{ab}\iff(ab)^{2}=c^{2}.$
The rotation by $\pi$ and the symmetries with respect to the diagonals belong
to $\mathrm{GAff}(M,h)$ for any $a,b,c.$
So we obtain that if $(ab)^{2}=c^{2}$ then $\operatorname{Aut}(Y)\simeq D_{4}$
and if $(ab)^{2}\neq c^{2}$ then
$\operatorname{Aut}(Y)\simeq({\mathbb{Z}}/2{\mathbb{Z}})\oplus({\mathbb{Z}}/2{\mathbb{Z}}).$
## References
* [1] Ivan Arzhantsev and Sergey Gaifullin. The automorphism group of a rigid affine variety. Math. Nachr., 290:5, 662–671, 2017
* [2] David Bernstein. The number of roots of a system of equations. Funct. Anal. Appl., 9:3, 183–185, 1975
* [3] June Huh. The maximum likelihood degree of a very affine variety. Compos. Math., 149:8, 1245–1266, 2013
* [4] Shulim Kaliman. Extensions of isomorphisms between affine algebraic subvarieties of $k^{n}$ to automorphisms of $k^{n}$. Proc. Amer. Math. Soc. 113:2, 325–334, 1991
* [5] Shulim Kaliman and David Udumyan. On automorphisms of flexible varieties. Adv. Math., 396, Article no. 108112, 2022
* [6] Anatoly Koushnirenko. Newton polytopes and the Bezout theorem. Funct. Anal. Appl., 10:3, 82–83, 1976
* [7] Alexander Perepechko. Automorphisms of surfaces of Markov type. Math. Notes, 110:5, 732–737, 2021
* [8] Alexander Perepechko and Michael Zaidenberg. Automorphism group of affine rigid surfaces: the identity component. arXiv:2208.09738, 2022
* [9] Vladimir Popov. On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties. CRM Proc. Lect. Notes, 54:17, 289–311, 2011
* [10] Maxwell Rosenlicht. Some rationality questions on algebraic groups. Ann. Mat. Pura Appl., 43, 25–50, 1957
* [11] Jenia Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129:4, 1087–1104, 2007
|
Omm$#2\text{\,}#\mathrm{3}$
HIKE, High Intensity Kaon Experiments
at the CERN SPS
Letter of Intent
The HIKE Collaboration***Address all correspondence to<EMAIL_ADDRESS>
###### Abstract
A timely and long-term programme of kaon decay measurements at a new level of
precision is presented, leveraging the capabilities of the CERN Super Proton
Synchrotron (SPS). The proposed programme is firmly anchored on the experience
built up studying kaon decays at the SPS over the past four decades, and
includes rare processes, CP violation, dark sectors, symmetry tests and other
tests of the Standard Model. The experimental programme is based on a staged
approach involving experiments with charged and neutral kaon beams, as well as
operation in beam-dump mode. The various phases will rely on a common
infrastructure and set of detectors.
The HIKE collaboration
E. Cortina Gil1, J. Jerhot1, N. Lurkin1, T. Numao2, B. Velghe2, V. W. S.
Wong2, D. Bryman3, L. Bician4, Z. Hives4, T. Husek4, K. Kampf4, M. Koval4, A.
T. Akmete5, R. Aliberti5, V. Büscher5, L. Di Lella5, N. Doble5, L. Peruzzo5,
M. Schott5, H. Wahl5, R. Wanke5, B. Döbrich6, L. Montalto7, D. Rinaldi7, F.
Dettori8,9, A. Cardini9, A. Lai9, L. Bomben10, S. Carsi10, M. Prest10, A.
Selmi10, G. Lezzani11, P. Monti-Guarnieri11, L. Perna11, P. Dalpiaz12,13, V.
Guidi,12,13, A. Mazzolari12,13, I. Neri12,13, F. Petrucci12,13, M.
Soldani12,13, L. Bandiera13, A. Cotta Ramusino13, A. Gianoli13, M.
Romagnoni13, A. Sytov13, M. Lenti14,15, I. Panichi14,15, G. Ruggiero14,15, A.
Bizzeti15, F. Bucci15, A. Antonelli16, E. Di Meco16,30, G. Lanfranchi16, S.
Martellotti16, M. Martini16,31, M. Moulson16, D. Paesani16,30, I. Sarra16, T.
Spadaro16, G. Tinti16, E. Vallazza17, F. Ambrosino18,20, R. Giordano18,20, P.
Massarotti18,20, M. Napolitano18,20, G. Saracino18,20, C. Di Donato19,20, G.
D’Ambrosio20, M. D’Errico20, M. Mirra20, S. Neshatpour20, R. Fiorenza21, I.
Rosa21, D. De Salvador22, F. Sgarbossa22, G. Anzivino23,24, S. Germani23,24,
R. Volpe23,24, P. Cenci24, S. Cutini24, V. Duk24, P. Lubrano24, M. Pepe24, M.
Piccini24, F. Costantini25,26, S. Donati25,26, M. Giorgi25,26, S.
Giudici25,26, G. Lamanna25,26, E. Pedreschi25,26, J. Pinzino25,26, M.
Sozzi25,26, R. Fantechi26, V. Giusti26, F. Spinella26, I. Mannelli27, M.
Raggi28,29, A. Biagioni29, P. Cretaro29, O. Frezza29, F. Lo Cicero29, A.
Lonardo29, M. Turisini29, P. Vicini29, R. Ammendola30, V. Bonaiuto30, A.
Fucci30, A. Salamon30, F. Sargeni30, R. Arcidiacono32,33, B. Bloch-Devaux32,
E. Menichetti32,33, E. Migliore32,33, C. Biino33, F. Marchetto33, D.
Baigarashev34, Y. Kambar34, D. Kereibay34, Y. Mukhamejanov34, S. Sakhiyev34,
A. Briano Olvera35, J. Engelfried35, N. Estrada-Tristan35, R. Piandani35, M.
A. Reyes Santos35, K. A. Rodriguez Rivera35, P. C. Boboc36, A. M.
Bragadireanu36, S. A. Ghinescu36, O. E. Hutanu36, T. Blazek37, V. Cerny37, A.
Kleimenova37, Z. Kucerova37, D. Martinez Santos38, C. Prouve39, M. Boretto40,
F. Brizioli40, A. Ceccucci40, M. Corvino40, H. Danielsson40, F. Duval40, E.
Gamberini40, R. Guida40, E. B. Holzer40, B. Jenninger40, G. Lehmann Miotto40,
P. Lichard40, K. Massri40, E. Minucci40, M. Perrin-Terrin40, V. Ryjov40, J.
Swallow40, M. Van Dijk40, M. Zamkovsky40, R. Marchevski41, A. Gerbershagen42,
J. R. Fry43, F. Gonnella43, E. Goudzovski43, J. Henshaw43, C. Kenworthy43, C.
Lazzeroni43, C. Parkinson43, A. Romano43, J. Sanders43, A. Shaikhiev43, A.
Tomczak43, H. Heath44, D. Britton45, A. Norton45, D. Protopopescu45, J. B.
Dainton46, R. W. L. Jones46, A. De Santo47, F. Salvatore47, P. Cooper48, D.
Coward48, P. Rubin48
1 Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium
2 TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada
3 University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada
4 Charles University, 116 36 Prague 1, Czech Republic
5 Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany
6 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), D-80805
München, Germany
7 Dipartimento di Scienze e Ingegneria della Materia, dell’Ambiente ed
Urbanistica, Università Politecnica delle Marche, I-60131 Ancona, Italy
8 Dipartimento di Fisica dell’Università, I-09042 Cagliari, Italy
9 INFN, Sezione di Cagliari, I-09042 Cagliari, Italy
10 Università degli Studi dell’Insubria, 22100, Como and INFN Sezione Milano
Bicocca, I-20126 Milano, Italy
11 Dipartimento di Scienza e Alta Tecnologia, Università degli Studi
dell’Insubria, I-22100, Como, Italy
12 Dipartimento di Fisica e Scienze della Terra dell’Università, I-44122
Ferrara, Italy
13 INFN, Sezione di Ferrara, I-44122 Ferrara, Italy
14 Dipartimento di Fisica e Astronomia dell’Università, I-50019 Sesto
Fiorentino, Italy
15 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Italy
16 INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
17 INFN Sezione di Milano Bicocca, I-20126 Milano, Italy
18 Dipartimento di Fisica “Ettore Pancini” dell’Università degli Studi di
Napoli Federico II, I-80126 Napoli, Italy
19 Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope,
I-80143 Napoli
20 INFN, Sezione di Napoli, I-80126 Napoli, Italy
21 Scuola Superiore Meridionale e INFN, Sezione di Napoli, I-80138 Napoli,
Italy
22 Dipartimento di Fisica e Astronomia dell’ Università degli Studi di Padova,
I-35131 Padova e INFN Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy
23 Dipartimento di Fisica e Geologia dell’Università, I-06100 Perugia, Italy
24 INFN, Sezione di Perugia, I-06100 Perugia, Italy
25 Dipartimento di Fisica dell’Università, I-56100 Pisa, Italy
26 INFN, Sezione di Pisa, I-56100 Pisa, Italy
27 Scuola Normale Superiore e INFN, Sezione di Pisa, I-56100 Pisa, Italy
28 Dipartimento di Fisica, Sapienza Università di Roma, I-00185 Roma, Italy
29 INFN, Sezione di Roma I, I-00185 Roma, Italy
30 Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, I-00133
Roma, Italy
31 Dipartimento di Scienze Ingegneristiche, Università degli Studi Guglielmo
Marconi, I-00193 Roma, Italy
32 Dipartimento di Fisica dell’Università, I-10125 Torino, Italy
33 INFN, Sezione di Torino, I-10125 Torino, Italy
34 Institute of Nuclear Physics, 050032 Almaty, Kazakhstan
35 Instituto de Física, Universidad Autónoma de San Luis Potosí, 78240 San
Luis Potosí, Mexico
36 Horia Hulubei National Institute for R&D in Physics and Nuclear
Engineering, 077125 Bucharest-Magurele, Romania
37 Faculty of Mathematics, Physics and Informatics, Comenius University, 842
48, Bratislava, Slovakia
38 Axencia Galega de Innovacion, Conselleria de Economia e Industria, Xunta de
Galicia, 15704 Santiago de Compostela, Spain
39 Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, 15782 Santiago de Compostela, Spain
40 CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland
41 Ecole polytechnique fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland
42 PARTREC, UMCG, University of Groningen, 9747 AA Groningen, The Netherlands
43 School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham, B15 2TT, UK
44 School of Physics, University of Bristol, Bristol, BS8 1TH, UK
45 School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ,
UK
46 Faculty of Science and Technology, University of Lancaster, Lancaster, LA1
4YW, UK
47 Department of Physics and Astronomy, University of Sussex, Brighton, BN1
4GE, UK
48 Physics and Astronomy Department, George Mason University, Fairfax, VA
22030, USA
###### Contents
1. 1 Overview of HIKE
1. 1.1 The scientific goals of HIKE
2. 2 Scientific context
1. 2.1 Experimental context for high-scale BSM physics
2. 2.2 Experimental context for feebly-interacting particles
3. 2.3 The role of HIKE
3. 3 Physics with high-intensity kaon beams
1. 3.1 Flavour-changing neutral currents
2. 3.2 Lepton flavour universality tests
3. 3.3 Lepton flavour and lepton number violation
4. 3.4 Tests of low-energy QCD
5. 3.5 Test of first-row CKM unitarity
6. 3.6 Production of feebly-interacting particles in kaon decays
4. 4 Physics in the beam-dump mode
5. 5 High-intensity beams
1. 5.1 Beam delivery to the kaon production target
2. 5.2 Charged kaon beamline
3. 5.3 Neutral kaon beams
6. 6 Phase 1: a multi-purpose $K^{+}$ decay experiment
1. 6.1 Experimental layout
2. 6.2 Physics sensitivity
7. 7 Phase 2: a multi-purpose $K_{L}$ decay experiment
1. 7.1 Experimental layout
2. 7.2 Physics sensitivity
8. 8 Phase 3 (KLEVER): measurement of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay
1. 8.1 Experimental layout
2. 8.2 Rates and timing performance
3. 8.3 Expected performance for $K_{L}\to\pi^{0}\nu\bar{\nu}$
9. 9 Operation in beam-dump mode
1. 9.1 Experimental layout
2. 9.2 Physics sensitivity
10. 10 The HIKE detectors
1. 10.1 Detectors upstream of the fiducial decay volume
2. 10.2 Fiducial decay volume and its detectors
3. 10.3 Detectors downstream of the fiducial decay volume
4. 10.4 Detectors specific to KLEVER
11. 11 Data acquisition and high-level trigger
1. 11.1 Readout boards
2. 11.2 Streaming readout
3. 11.3 Triggered readout
4. 11.4 Event-building farm and event filter
12. 12 Online and offline computing
1. 12.1 Online monitoring
2. 12.2 Calibration and data quality
3. 12.3 Data processing model
4. 12.4 Distributed computing model
5. 12.5 Data reduction model
13. 13 Infrastructure and safety
14. 14 Summary and conclusions
## 1 Overview of HIKE
Over the past 75 years, experimental studies of kaon decays have played a
singular role in propelling the development of the Standard Model. As in other
branches of flavour physics, the continuing experimental interest in the kaon
sector derives from the possibility of conducting precision measurements,
particularly of suppressed rare processes, that may reveal the effects of new
physics with mass-scale sensitivity exceeding that which can be explored
directly, e.g., at the LHC or next-generation colliders. Because of the
relatively small number of kaon decay modes and the relatively simple final
states, combined with the relative ease of producing intense kaon beams, kaon
decay experiments are in many ways the quintessential intensity-frontier
experiments.
Over the past four decades, the CERN North Area has been the host to a
successful series of precision kaon decay experiments. Among the many results
obtained by these experiments is the discovery of direct CP violation, widely
quoted to be among the major discoveries made at CERN.
Continuation of high-intensity kaon experiments at CERN, including future
upgrades (providing a unique probe into BSM physics complementary to the
$B$-sector) has been identified as an essential scientific activity in the
2020 Update of the European Strategy for particle physics, and is strongly
supported in the national roadmaps across Europe.
The High Intensity Kaon Experiments (HIKE) project represents a broad, long-
term programme at CERN after LS3, based in the North Area ECN3 experimental
hall, covering all the main aspects of rare kaon decays and searches
accessible via kaon physics, from ultra-rare kaon decays to precision
measurements and searches for new phenomena. HIKE is intended to continue the
very successful tradition of kaon experiments at CERN in ECN3, the latest of
which is the currently operating NA62 experiment. HIKE will profit from a beam
intensity increase by a factor between four and six with respect to NA62, and
cutting-edge detector technologies. This will allow HIKE to play a pivotal
role in the quest for New Physics (NP) at the sensitivity required by the
present experimental limits and theoretical models, in a wide range of
possible masses and interaction couplings. The programme will consist of
multiple phases, first with charged and then neutral kaon beams, as well as
periods in beam dump mode. The long decay volume and detector characteristics
needed for kaon physics make HIKE suitable to search for new feebly-
interacting, long-lived particles, providing unique sensitivity to forward
processes. The detector design is challenging but at least one technological
solution exists for each subsystem, thanks also to the synergy with HL-LHC.
The various phases allow for insertion, repositioning or removal of specific
detector elements depending on the physics requirements, while the overall
setup remains broadly the same.
### 1.1 The scientific goals of HIKE
The HIKE programme consists of several phases using shared detectors and
infrastructure: a charged kaon phase between LS3 and LS4, and neutral kaon
experiments commencing after LS4 including a phase with tracking, and a phase
optimised specifically for the $K_{L}\to\pi^{0}\nu\bar{\nu}$ measurement
(KLEVER).
Phase 1: a multi-purpose $K^{+}$ decay experiment
The physics of $K^{+}$ decays will be scrutinised with the highest precision,
expanding and improving the physics programme of the NA62 experiment presently
running in ECN3. New detectors will replace those of NA62 with the goal of
improving the performance and sustaining higher rates.
The principal goals of this phase are:
* •
Measurement of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ branching ratio to 5%
relative precision, matching the SM theoretical uncertainty.
* •
Precision measurements of $K^{+}\to\pi^{+}e^{+}e^{-}$ and
$K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decays, and a precision lepton universality
test.
* •
Searches for lepton flavour/number violating decays $K^{+}\to\pi(\pi^{0})\mu
e$, $K^{+}\to\pi^{-}(\pi^{0})\ell^{+}\ell^{+}$,
$K^{+}\to\ell_{1}^{-}\nu\ell_{2}^{+}\ell_{2}^{+}$ and
$\pi^{0}\to\mu^{\pm}e^{\mp}$ at the ${\cal O}(10^{-12})$ sensitivity.
* •
Measurement of the quantity $R_{K}=\Gamma(K^{+}\to
e^{+}\nu)/\Gamma(K^{+}\to\mu^{+}\nu)$ to 0.1% precision.
* •
Measurement of the ratios of the branching ratios of the main decay modes
$K^{+}\to\pi^{0}\ell^{+}\nu$, $K^{+}\to\pi^{+}\pi^{0}$,
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ to a relative precision of a few per mille.
* •
Improvements of existing measurements of rare decays, including
$K^{+}\to\pi^{+}\gamma\gamma$, $K^{+}\to\pi^{+}\gamma\ell^{+}\ell^{-}$,
$K^{+}\to\pi^{+}\pi^{0}\gamma$, $K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$.
* •
Searches for production of feebly-interacting particles in $K^{+}$ decays.
* •
Collection of a dataset in the beam-dump mode (with appropriate time sharing
with the kaon mode), aiming for a factor 10 improvement in sensitivity to
decays of feebly-interacting particles with respect to what NA62 can achieve
by LS3.
Phase 2: a multi-purpose $K_{L}$ decay experiment
The physics of the $K_{L}$ decays will be addressed, including $K_{L}$ decays
into charged particles in the final state. This mode of operation will require
some modifications of the detector layout with respect to the charged mode.
The main objectives of this phase are:
* •
Observation of the ultra-rare decays $K_{L}\to\pi^{0}e^{+}e^{-}$ and
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$, or establishment of stringent upper limits on
the branching ratios of these decays at the ${\cal O}(10^{-11})$ level.
* •
Measurement of the $K_{L}\to\mu^{+}\mu^{-}$ decay branching ratio to 1%
relative precision.
* •
Searches for lepton flavour violating decays including
$K_{L}\to(\pi^{0})(\pi^{0})\mu^{\pm}e^{\mp}$ and $K_{L}\to
e^{\pm}e^{\pm}\mu^{\mp}\mu^{\mp}$ with ${\cal O}(10^{-12})$ sensitivity.
* •
Measurement of the ratios of the branching ratios of the main decay modes
$K_{L}\to\pi^{\pm}\ell^{\mp}\nu$, $K_{L}\to\pi^{+}\pi^{-}(\pi^{0})$,
$K_{L}\to\pi^{0}\pi^{0}(\pi^{0})$ to a relative precision of a few per mille.
* •
Collection of a further dataset in the beam-dump mode (with appropriate time
sharing with the kaon mode), and characterisation of the neutral beam
necessary to proceed to the third phase of HIKE.
Phase 3 (KLEVER): measurement of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay
The experimental apparatus will be modified to specifically address the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay, and to measure its branching ratio to a
20% relative precision. To optimise the measurement, the tracking systems will
be removed from the fiducial volume and additional photon veto detectors
installed. Although this phase will focus on $K_{L}\to\pi^{0}\nu\bar{\nu}$,
the setup will allow study of additional topics:
* •
Searches for additional flavour-changing neutral current decays, such as
$K_{L}\to\pi^{0}\pi^{0}\nu\bar{\nu}$.
* •
Limits on the production of a long-lived, feebly interacting particle $X$ in
$K_{L}\to\pi^{0}X$ decays, as a by-product of the measurement of
$K_{L}\to\pi^{0}\nu\bar{\nu}$.
* •
Searches for $K_{L}$ decays to feebly-interacting particles $X$ such as
$K_{L}\to XX$, with $X\to\gamma\gamma$.
* •
Searches for forbidden decays such as $K_{L}\to\pi^{0}\gamma$.
## 2 Scientific context
The Standard Model (SM) of particle physics describes the experimental results
obtained so far by experiments with exceptional precision. The culmination of
this success was the discovery of the Higgs boson by the ATLAS and CMS
collaborations at the LHC in 2012 [1, 2]. Thus the SM provides a solid
foundation for the present understanding of elementary particles and their
interactions. Despite this success, cosmological observations, experimental
tensions, and theoretical motivations strongly suggest that the SM is an
approximation of a more fundamental theory. One paradigm assumes that this
theory lies above the electroweak (EW) mass scale and manifests itself in
terms of new particles with masses well above the Higgs boson mass and having
sizeable interactions with the SM fields. Another paradigm accounts for
extensions of the SM that predict particles with masses below the EW scale,
feebly interacting with the SM fields. Such a low-energy NP paradigm is
usually referred to as a Feebly Interacting Particles (FIPs) scenario. Both
high-scale BSM and FIPs models provide explanations for the main open
questions in modern physics: the nature of dark matter, the baryon asymmetry
of the universe, cosmological inflation, the origin of neutrino masses and
oscillations, the strong CP problem, the hierarchical structure of the Yukawa
couplings, the hierarchy problem, and the cosmological constant.
HIKE intends to study NP in both high-scale BSM and FIPs directions. In this
respect, HIKE is perfectly aligned with the recommendation of the 2020 Update
of the European Strategy for Particle Physics [3], where a whole chapter is
devoted to flavour physics research with the statement: “Experimental hints
for deviations from SM predictions in flavour processes are one of our best
hopes to direct research towards the right energy scale where NP can be
found”.
### 2.1 Experimental context for high-scale BSM physics
The experimental techniques to investigate the presence of BSM physics at high
mass scales are direct searches for massive particles or processes forbidden
by the SM, and precision measurements of observables precisely known within
the SM. The first method is usually referred to as direct searches; the second
as indirect searches.
Direct searches allow unambiguous identification of NP already in single
processes. After the observation and the study of the Higgs boson, direct
searches for BSM particles were a primary goal of the LHC experiments ATLAS
and CMS. Still, the LHC centre-of-mass energy limits the sensitivity of this
technique to the TeV scale, or slightly above. Besides, a direct observation
alone gives little information on the BSM structure. The absence of any
significant direct observations of NP so far sets strong bounds on several
types of BSM models up to the TeV scale. The bounds are still statistically
limited in the TeV region, and the LHC high-luminosity programme will allow
them to be extended to the kinematic limits, if no observations emerge in the
meantime. Another type of direct searches focuses on the study of processes
forbidden by the SM due to accidental SM symmetries like such as charged
lepton flavour (CLF) or lepton number (LN) conservation. These searches
typically involve the study of hadron or lepton decays forbidden by the SM,
for example, $\tau\to\mu\mu\mu$ and $K_{L}\to\mu^{\pm}e^{\mp}$. Presently,
these searches are statistically limited, and experiments at the high-
intensity frontier are planned with increased sensitivity.
Indirect searches exploit the possibility that NP particles affect low-energy
observables virtually, via loop-level corrections to the SM prediction. The
golden modes of indirect searches are theoretically clean observables of rare
processes in the SM, i.e. occurring at loop-level at the lowest order, because
NP here may favorably compete with SM. Indirect searches allow sensitivities
to NP up to mass scales well above the TeV, but may require measurements of
several observables to provide conclusive evidence of NP and to determine its
nature. Given the limits to NP at TeV-scale already set by LHC through direct
searches, indirect searches represent a promising tool to further boost BSM
searches. Experimental flavour physics, measurements of Higgs couplings, and
measurements in the gauge sectors of the SM ($W$, $Z$, top masses and gauge
couplings) are typical examples of indirect searches. The existing
measurements are broadly consistent with the SM and already suggest that NP
may occur well beyond the TeV scale. Nevertheless, tensions between
experiments and the SM exist mainly in flavour physics observables, and
statistical limitations or the lack of measurements of rare processes
demonstrate the need for additional efforts and motivate future directions in
experimental particle physics. Presently, the most significant experimental
tensions with the SM are:
* •
the violation of lepton flavour universality in $b\to s$ and $b\to c$
transitions reported by LHCb and $B$-factories [4, 5, 6, 7, 8, 9, 10], with a
global significance of about 3–4$\sigma$ [11, 12, 13, 14];
* •
the violation of CKM unitarity in the first row, known as the “Cabibbo angle
anomaly”, with a significance of about 2–3$\sigma$ [15, 16, 17];
* •
the inconsistency between the exclusive and inclusive measurements of the CKM
matrix element $|V_{cb}|$ at the level above $3\sigma$ [18];
* •
the inconsistency between the experiment and the SM prediction for the
anomalous magnetic dipole moment of the muon at the level of 4.2$\sigma$ [19].
The sensitivity of the indirect searches to BSM physics scales with the size
of the datasets, which is particularly important in case of rare processes.
Moving to higher intensities is the way to increase statistics, however this
requires improvements of the detector performance to cope with harsher
experimental environments. Upgraded detectors also allow the improvement of
the accuracy of already precise measurements, overcoming systematic
uncertainties whenever they are the main limiting factor. The indirect search
approach is at the heart of the LHC intensity frontier programme, the
motivation for the upgrade of the LHCb experiment and for the continuation of
the $B$-factory programme with the Belle II experiment.
### 2.2 Experimental context for feebly-interacting particles
Direct searches for new low-mass particles can provide experimental insight
into the models involving dark sectors and predicting the existence of FIPs
[20, 21]. FIPs can be produced in meson decays or hadronic interactions of
protons, and can decay to SM particles. They are produced in rare processes
and are expected to have long lifetimes with respect to typical experimental
scales as a result of their feeble couplings to the SM fields.
Experiments may search for FIP production or decays. The first method focuses
on the associated production of FIPs with SM particles in meson decays; see
Ref. [22] for a review of possible production processes in kaon decays. The
meson mass ($\pi$, $K$, $D$, $B$) determines the kinematic region that can be
explored. Some examples are production of heavy neutral leptons in
$K^{+}\to\ell^{+}N$, dark photons in $\pi^{0}\to\gamma X$, and dark scalars in
$B\to KS$ or $K\to\pi S$. These kinds of searches can be pursued by flavour
experiments, like LHCb and NA62, but also at ATLAS and CMS. The second method
is based on FIP production, mainly in proton-nucleon interactions, and the
detection of the subsequent FIP decays. Forward configurations and long
experimental setups are best suited for this type of search, as those of
fixed-target experiments, but also LHCb. The solid angle covered by the
detector defines the kinematic range of the FIPs that can be studied. Examples
of FIP decays include dark scalars or dark photon decays
$S/A^{\prime}\to\mu^{+}\mu^{-}$ and heavy neutral lepton decays
$N\to\pi^{\pm}\mu^{\mp}$. Searches for FIPs performed so far have not reported
any hints of NP, and only limits in the plane of the coupling versus FIP mass
have been set within the framework of various NP models. Nevertheless, it is
worth continuing to explore the parameter space for FIP models. High beam
intensities and experiments with appropriate geometries are necessary for the
pursuit of these studies.
### 2.3 The role of HIKE
The HIKE experiment is primarily a flavour physics experiment, with a fixed
target configuration appropriate for the study of decays of kaons at high
energies, on the order of 100 GeV. The study of the kaon sector makes HIKE
complementary to LHCb and Belle II: kaons provide different, and in some cases
higher, sensitivity to NP than $B$ and $D$ mesons. Measurements of kaon
observables and correlations among them can help to find NP through the
comparison with corresponding SM expectations and investigation of the present
experimental inconsistencies with SM. More generally, the comparison between
the flavour picture emerging from kaons with that from $B$ mesons is a
powerful tool to investigate the indirect effects of NP: to shed light on
existing experimental anomalies, push the sensitivity to mass scales beyond
those attainable by $B$ and $D$ only, and provide insights to the flavour
structure of possible NP models.
Several models of BSM physics predict effects on kaon observables, ranging
from supersymmetric to non-supersymmetric models, NP involving heavy
$Z^{\prime}$ bosons, vector-like couplings, leptoquarks, and extra dimensions,
in both minimally flavour violating (MFV) scenarios or with new sources of
flavour violation. A common feature is that bounds on the parameter space of
these models coming from direct searches at LHC and $B$ and $D$ physics have
only a marginal impact on the possible effects on kaon observables, even more
so if the models are non-MFV and non-supersymmetric. In contrast, the main
constraints on the effects of NP on kaon observables come from the precisely
measured parameters $\varepsilon_{K}$ and $\Delta M_{K}$. Table 1 gives a
selected list of the most relevant NP models that may affect significantly
kaon observables [23].
Table 1: Compendium of new-physics models relevant for kaon processes [23]. NP scenarios | Process | References
---|---|---
Z-FCNC | $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon^{\prime}/\varepsilon$ | [24, 25, 26, 27, 28]
Z′ | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon^{\prime}/\varepsilon$, $\Delta M_{K}$ | [29, 24, 25, 30, 31]
Simplified models | $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon^{\prime}/\varepsilon$ | [32]
LHT | All $K$ decays | [33, 34, 35]
331 models | Small effects in $K\to\pi\nu\bar{\nu}$ | [30]
Vector-like quarks | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\Delta M_{K}$ | [36]
Supersymmetry | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [37, 38, 39, 40, 41, 42, 43, 44, 45]
2HDM | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [46, 47]
Universal extra dimensions | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [48, 49]
Randall-Sundrum models | All rare $K$ decays | [50, 51, 52, 53, 54]
Leptoquarks | All rare $K$ decays | [55, 56, 57]
SMEFT | Several processes in $K$ system | [25, 58]
SU(8) | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [59]
Diquarks | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon_{K}$ | [60, 61]
Vector-like compositeness | $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon_{K}$ | [62]
Presently, the main limitation to the investigation of the above models, and
more generally to the quest for NP with kaons, comes from the experimental
precision of the measurements of the kaon observables. To a lesser extent, the
limitation is due to theoretical uncertainties and the precision of the
measurements of the SM parameters, which will eventually be improved in the
upcoming years. The primary goal of HIKE is to improve the accuracy of the
measurements, wherever they exist, in order to match and possibly challenge
the theory precision, to study and measure for the first time channels not yet
observed, and to search with unprecedented sensitivity for kaon decays
forbidden by the SM.
A natural by-product of HIKE is the possibility to search for kaon decays to
FIPs, a line of inquiry that is already being pursued with success by NA62. In
addition, the fixed-target configuration and the high beam intensity
requirement make HIKE suitable to search for decays of FIPs, exploring regions
mainly below the $D$ mass, but with unprecedented sensitivity.
## 3 Physics with high-intensity kaon beams
Kaons offer a unique opportunity to investigate BSM physics in a way
complementary to $B$ or $D$ mesons, lepton physics, and EDM searches. The
strategy is to perform multiple experimental studies ranging from measurements
of branching ratios and form factors of kaon decays to searches for processes
forbidden or not existing within the SM. A global fit to the experimental
inputs provides the best sensitivity to NP, although single measurements also
put significant constraints. Evidence for NP can also show up as
inconsistencies between the observed flavour pattern and that expected from
the SM, violation of accidental SM symmetries like lepton flavour universality
(LFU), lepton flavour or number (LFV or LNV), and direct production of new
low-mass particles in kaon decays.
Because the BSM paradigm assumes NP at high masses, NP is expected to affect
kaon processes at short-distance (SD) scales. The link between kaon-physics
observables and BSM physics relies on the SM theoretical knowledge of these
observables. The development of the theoretical understanding of the kaon
sector is focused exclusively on calculations of observables relevant to pin
down SD contributions from experimental results. Examples are: the SM
calculation of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay branching ratios at NNLO leading to a few
percent precision on these quantities; the recent calculation of the CP
violation parameter $\varepsilon^{\prime}/\varepsilon$; computations of the
branching ratios for the rare decays $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$,
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$, $K_{L,S}\to\mu^{+}\mu^{-}$.
It is convenient to group kaon decays into the following classes.
* •
Flavour-changing neutral currents (Section 3.1). The $K\to\pi\nu\bar{\nu}$,
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$, $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ and
$K_{L}\to\mu^{+}\mu^{-}$ decays are suppressed in the SM, occurring at loop-
level at the lowest order. The corresponding SD contributions are related to
CKM matrix elements and are generally calculated with high precision within
the SM. Together with $\varepsilon_{K}$, rare decays allow the unitary
triangle to be over-constrained, independently from $B$ processes. With the
exception of $K\to\pi\nu\bar{\nu}$ decays, the above decays receive important
contributions from long distance (LD) physics. The theoretical framework used
to study these processes involves low-energy effective theories, like Chiral
Perturbation Theory (ChPT), which require experimental inputs (chiral
parameters) to provide and validate the predictions, as well as lattice QCD.
* •
Lepton flavour universality tests (Section 3.2). The ratios ${\cal
B}(K^{+}\to\pi^{+}e^{+}e^{-})/{\cal B}(K^{+}\to\pi^{+}\mu^{+}\mu^{-})$ and
${\cal B}(K^{+}\to e^{+}\nu)/{\cal B}(K^{+}\to\mu^{+}\nu)$ are extremely
sensitive probes of lepton flavour universality. Another sensitive test is
based on the ratio ${\cal B}(K\to\pi e\nu)/{\cal B}(K\to\pi\mu\nu)$, where
both neutral and charged kaon decays ($K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ and
$K^{+}\to\pi^{0}\ell^{+}\nu$) can be used.
* •
Lepton flavour and number violating decays (Section 3.3). Decay modes
forbidden by the SM by either lepton flavour or number conservation include
$K^{+}\to\pi^{-}(\pi^{0})\ell_{1}^{+}\ell_{2}^{+}$,
$K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$,
$K^{+}\to\ell_{1}^{-}\nu\ell_{2}^{+}\ell_{2}^{+}$,
$K_{L}\to(\pi^{0})(\pi^{0})\mu^{\pm}e^{\mp}$ and $\pi^{0}\to\mu^{\pm}e^{\mp}$.
Any experimental signal of these processes would constitute direct evidence
for NP. It should be noted that kaon beams represent sources of tagged,
Lorentz-boosted $\pi^{0}$ mesons.
* •
Tests of low-energy QCD (Section 3.4). The abundant kaon decays $K\to\pi\pi$
and $K\to\pi\pi\pi$ occur at the tree level and are dominated by LD physics.
The decays are described by ChPT which is the low-energy effective field
theory of QCD. Global fits to the measured branching ratios and kaon lifetimes
of the $K\to\pi\pi$ and $K\to\pi\pi\pi$ decays play a crucial role in the
extraction of the ChPT low-energy constants. Rare and radiative decays
including $K\to\pi\gamma\gamma$, $K\to\ell\nu\gamma$, $K\to\pi\ell\nu\gamma$,
$K\to\pi\pi\gamma$, $K\to\pi\pi\pi\gamma$, $K\to\pi\pi e^{+}e^{-}$, and
$K\to\pi\pi\ell\nu$ are also dominated by LD physics and similarly provide
access to the ChPT parameters.
* •
CKM first-row unitarity tests (Section 3.5). Branching ratios and form factors
of the tree-level $K^{+}\to(\pi^{0})\ell^{+}\nu$ and
$K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ decays represent the principal experimental
inputs for the determination of CKM matrix element $V_{us}$, with input from
lattice QCD.
* •
Production of feebly-interacting particles in kaon decays (Section 3.6) is
predicted by FIP models involving new particles below the kaon mass scale.
Examples include production of a heavy neutral lepton ($K^{+}\to\ell^{+}N$),
or a dark scalar ($K^{+}\to\pi^{+}S$).
HIKE intends to address experimentally all these processes at a new level of
sensitivity, improving the existing experimental picture to fully exploit the
complementarity of kaon with respect to heavy-quark physics in the quest for
NP.
### 3.1 Flavour-changing neutral currents
Kaon decays proceeding at loop level in the SM are denoted “rare”, and
typically have branching ratios below $10^{-7}$. They provide a tool
independent from $B$ and $D$ physics to test the flavour structure of the SM
by constraining the unitary triangle via loop-level observables.
The link between rare kaon decays and flavour physics in the $\rho-\eta$ plane
is schematically shown in Fig. 1. CP-violating short distance (SD) physics
contributes to the $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_{L}\to\pi^{0}e^{+}e^{-}$,
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$, and $K_{S}\to\mu^{+}\mu^{-}$ decays. The
amplitudes of these decays thus depend on the height, $\eta$, of the unitary
triangle. The SD physics contributing to the decay $K_{L}\to\mu^{+}\mu^{-}$ is
CP-conserving, and its amplitude depends on the base, $\rho$, of the unitary
triangle. The amplitude of the rare decay $K^{+}\to\pi^{+}\nu\bar{\nu}$
contains terms depending on both $\eta$ and $\rho$.
Figure 1: Relation between kaon rare decay modes and the parameters $\rho$ and
$\eta$ of the unitary triangle (UT). The direct link between decay modes and
the UT indicates short distance terms dependent on $\rho$ or $\eta$
contributing to the corresponding decay amplitudes. Decays not directly
connected to the UT are relevant to interpret the experimental results of the
decay modes to which they are related.
The $K_{L}\to\pi^{0}e^{+}e^{-}$, $K_{L}\to\pi^{0}\mu^{+}\mu^{-}$ and
$K_{L}\to\mu^{+}\mu^{-}$ decays are LD dominated. The extraction of their SD
components from experimental data proceeds through the study of ancillary
decay modes, listed in Fig. 1 in association with the rare decays mentioned
above. In contrast, LD contributions are sub-dominant to the amplitudes of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay, and negligible for the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay. As a matter of fact, these two decay
modes belong to the theoretically cleanest probes of the flavour structure of
the SM among all the kaon and $B$ meson decays.
#### 3.1.1 $K\to\pi\nu\bar{\nu}$ decays
The $K\to\pi\nu\bar{\nu}$ decays involve $s\to d$ quark transitions. They
proceed through box and penguin diagrams, which are SD dominated because of
the leading contribution of the virtual $t$ quark exchange. The quadratic GIM-
mechanism and the $t\to d$ Cabibbo suppression make place these processes
among the rarest meson decays in the SM. Their decay amplitudes can be
parameterised in terms of the precisely measured $K^{+}\to\pi^{0}e^{+}\nu$
decay, allowing theoretical computation free from hadronic uncertainties. The
SM predictions of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\to\pi^{0}\nu\bar{\nu}$ branching ratios can be written in terms of CKM
parameters as [63]
$\displaystyle{\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ $\displaystyle=(8.39\pm
0.30)\times
10^{-11}\left(\frac{|V_{cb}|}{0.0407}\right)^{2.8}\left(\frac{\gamma}{73.2^{\circ}}\right)^{0.74},$
(1) $\displaystyle{\cal B}(K_{L}\to\pi^{0}\nu\bar{\nu})$
$\displaystyle=(3.36\pm 0.05)\times 10^{-11}\left(\frac{|V_{ub}|}{3.88\times
10^{-3}}\right)^{2}\left(\frac{|V_{cb}|}{0.0407}\right)^{2}\left(\frac{\sin{\gamma}}{\sin{73.2^{\circ}}}\right)^{2}.$
(2)
Here $V_{cb}$ and $V_{ub}$ are elements of the CKM matrix, and $\gamma$ is the
angle of the unitary triangle defined as
$\arg[(-V_{ud}V_{ub}^{*})/(V_{cd}V_{cb}^{*})]$. The theoretical uncertainty
depends on the NNLO approximation of the computation of the Feynmann diagrams,
on the radiative corrections, and on the estimates of the long distance
contributions due to the exchange of the $u,d,c$ quarks. The prediction for
the neutral mode prediction is more precise than that for the charged mode
because the neutral amplitude is purely imaginary, allowing cancellation of
the long distance corrections. In either case, the main contribution to the
uncertainties for the branching ratio calculation comes from the precision of
the measurements of the CKM matrix elements, known as the parametric
uncertainty, and could be as large as 9% if the current measurements of
$V_{cb}$, $V_{ub}$ and $\gamma$ from the CKM fit are considered. A suitable
combination of external parameters nearly independent of new physics
contributions allows the reduction of the parametric uncertainty, reducing the
overall relative uncertainties in the branching ratios to 5% [64]. Other
recent calculations reach a similar quantitative conclusion [65]. A global fit
to the CKM unitary triangle, instead, allows the parametric uncertainty to be
factored out, and offers a powerful tool to compare the SM flavour structure
arising from kaon and $B$ physics. Equations 1 and 2 also indicate that
precise measurements of $K\to\pi\nu\bar{\nu}$ decays would shed light on CKM
parameters historically measured with $B$ mesons and suffering from long-
standing experimental tensions like $|V_{cb}|$.
The extreme SM suppression makes these decays particularly sensitive to NP.
From a model-independent point of view, the $K\to\pi\nu\bar{\nu}$ decays probe
NP at the highest mass scales [63], of the order of hundreds of TeV. Existing
experimental constraints on NP affect the $K\to\pi\nu\bar{\nu}$ branching
ratio weakly. Model-dependent scenarios predict sizeable deviations of the
branching ratios from the SM, as well as correlations between the branching
ratios for the charged and neutral modes, depending on the model [46, 55, 36,
28, 45, 44, 35, 66] (see also Table 1).
The NA62 experiment at CERN has observed $20$ candidate events for the decay
$K^{+}\to\pi^{+}\nu\bar{\nu}$ with 7 expected background events and 10
expected SM signal events [67], leading to the measurement
$\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})=(10.6^{+4.0}_{-3.4}|_{\text{stat}}\pm
0.9_{\text{syst}})\times 10^{-11}$ at $68\%$ confidence level. This represents
the most precise measurement to date of this process, providing first evidence
for its existence and falsifying the background-only hypothesis with
$3.4\sigma$ significance. The experiment is currently taking data in Run 2
(2021–LS3) with the aim of reaching a ${\cal O}(10\%)$ uncertainty
measurement, and has demonstrated its ability to sustain nominal beam
intensity. The NA62 experiment has therefore shown that the decay-in-flight
technique works well and is scalable to larger data samples.
Beyond the $\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ measurement, it is
important to establish if the decay has a purely vector nature as expected
within the SM. In BSM scenarios, an additional scalar contribution to the
decay is predicted [68, 69], leading to possibly LNV/LFV
$K^{+}\to\pi^{+}\nu\nu$ contributions, and since the neutrinos are not
detected, a $\nu\bar{\nu}$ pair cannot be directly distinguished from a
$\nu\nu$ pair, and the measurement of the branching ratio alone is
insufficient to understand the nature of the process. Because of the
suppression from the small neutrino mass, there is a negligible interference
between vector and scalar contributions, and the experimentally measured
branching ratio is given by [68, 23]:
$\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\overset{\textbf{(---)}}{\nu})=\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\rm
SM}+\sum_{i\leq j}^{3}\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu_{i}\nu_{j})_{\rm
LNV}.$ (3)
To fully test the SM prediction of a purely vector, lepton-number-conserving
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay, the momentum-transfer spectrum
must be studied to rule out the additional scalar BSM
$K^{+}\rightarrow\pi^{+}\nu\nu$ contribution to the overall experimentally
measured rate. Such a measurement requires a much larger dataset, as offered
within the HIKE programme.
The current upper limit on the branching ratio of the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay is $3\times 10^{-9}$ at 90% CL, set by the
KOTO experiment at J-PARC [70]. The KOTO experiment is currently taking data
with the goal of reaching ${\cal O}(10^{-11})$ sensitivity in the next 5
years. In the longer term, an ambitious upgrade, KOTO Step-2, is planned to
begin construction after 2025 in a proposed extension of the Hadron
Experimental Facility, with the goal of measuring the branching ratio to
within $\sim$20% in about three years of data taking [71].
#### 3.1.2 $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays
The ultra-rare $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays represent another set
of theoretically clean “golden modes” in kaon physics, second only to
$K\to\pi\nu\bar{\nu}$ in terms of their importance, and allowing for the
direct exploration of new physics contributions in $s\to d\ell\ell$
transitions (to be compared to $b\to s\ell\ell$ transitions). In the SM, they
are dominated by the indirect (mixing-induced) CP violation and its
interference with the direct CP-violating contribution. The decay rates can be
enhanced significantly in the presence of large new CP-violating phases, in a
correlated way with the effects in $K_{L}\to\pi^{0}\nu\bar{\nu}$ and
$\varepsilon^{\prime}/\varepsilon$ [23]. The SM expectation for the decay
branching ratios is [72, 73, 74]
$\displaystyle{\cal B}(K_{L}\to\pi^{0}e^{+}e^{-})$ $\displaystyle=$
$\displaystyle 3.54^{+0.98}_{-0.85}~{}\left(1.56^{+0.62}_{-0.49}\right)\times
10^{-11},$ $\displaystyle{\cal B}(K_{L}\to\pi^{0}\mu^{+}\mu^{-})$
$\displaystyle=$ $\displaystyle
1.41^{+0.28}_{-0.26}~{}\left(0.95^{+0.22}_{-0.21}\right)\times 10^{-11},$
with the two sets of values corresponding to constructive (destructive)
interference between direct and indirect CP-violating contributions.
Experimentally, the most stringent upper limits (at 90% CL) of the decay
branching ratios have been obtained by the KTeV experiment [75, 76]:
${\cal B}(K_{L}\to\pi^{0}e^{+}e^{-})<28\times 10^{-11},\quad{\cal
B}(K_{L}\to\pi^{0}\mu^{+}\mu^{-})<38\times 10^{-11}.$
The irreducible $K^{+}\to\gamma\gamma\ell^{+}\ell^{-}$ background, whose rate
vastly exceeds that of the signal decays $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$,
represents a significant problem in achieving experimental sensitivity at the
SM level, especially for $K_{L}\to\pi^{0}e^{+}e^{-}$ [77].
#### 3.1.3 $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ decays
The amplitudes of the $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ decays ($\ell=e,\mu$)
are LD-dominated and their branching ratios are $\mathcal{O}(10^{-7})$. The
differential decay amplitude of these decays with respect to the di-lepton
invariant mass depends on two form factor parameters, denoted $a_{+}$ and
$b_{+}$, that lepton universality predicts to be independent from the flavour
of the leptons. Differences between $a_{+}$ and $b_{+}$ between the electron
and muon channels can be correlated to LUV effects that can explain the
possible observed lepton anomalies in $B$ physics [78, 79].
The NA62 experiment has recently reported a new measurement of the form
factors of the $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decay using data collected in
2017–2018 [80]. The excellent kinematic resolution allowed a nearly
background-free selection of $3\times 10^{4}$ decays. The decay
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ was used for normalisation, and the parameter
values $a_{+}=-0.575\pm 0.013$ and $b_{+}=-0.722\pm 0.043$ were extracted from
a fit to the di-lepton mass spectrum. This improves significantly on the
previous measurements in the muon channel while also consistent within
uncertainty with measurements of the same parameters performed both in the
muon and electron channels [81, 82, 83, 84]. The NA62 experiment is expected
to improve the precision of the LFU test in the next years, addressing also
the measurement of the $K^{+}\to\pi^{+}e^{+}e^{-}$ form factors. However a
significant increase in precision is required to perform a LFU test with
comparable sensitivity to NP models as in current results from $B$ physics
[78, 85]. HIKE will offer a significantly larger data sample, further reducing
statistical uncertainties, which remain a large fraction of the total
uncertainty in the most recent NA62 measurement [80].
#### 3.1.4 $K_{L}\to\mu^{+}\mu^{-}$ decays
The $K_{L}\to\mu^{+}\mu^{-}$ is a rare and helicity suppressed decay. Its
branching ratio is driven by LD contributions, with a sub-dominant CP-
conserving SD component [86]. The SM prediction exhibits large uncertainties
due to the sign ambiguity of the interference between the LD and SD terms
contributing to the decay amplitude. The sign ambiguity leads to two SM
predictions:
$\text{[LD$+$]:}~{}~{}\left(6.82^{+0.77}_{-0.24}\pm 0.04\right)\times
10^{-9},~{}~{}~{}\text{[LD$-$]:}~{}~{}\left(8.04^{+1.66}_{-0.97}\pm
0.04\right)\times 10^{-9}.$ (4)
The first uncertainty comes from the calculation of the LD terms, dominates
the overall precision, and makes the uncertainty quite asymmetrical. This
asymmetry is reflected in the computation of the $K_{L}\to\mu^{+}\mu^{-}$ rate
with the inclusion of NP. This decay has been studied experimentally and the
measured branching ratio is $(6.84\pm 0.11)\times 10^{-9}$ [87]. The sign
ambiguity and the LD uncertainties prevent a clean theoretical interpretation
of this result. Theoretical efforts are foreseen in the next few years to
resolve the sign ambiguity through an appropriate matching between LD and SD
contributions and to reduce the LD uncertainty by at least a factor of two.
This could allow a sizeable reduction of the uncertainty of the SM prediction,
opening the possibility to exploit the high sensitivity of this decay to BSM
physics models.
The $K_{S}\to\mu^{+}\mu^{-}$ decay provides a complementary, theoretically
clean observable [88, 89]. While the total $K\to\mu^{+}\mu^{-}$ rate is
dominated by long-distance (LD) physics, the measurement (in a dedicated high-
intensity $K_{S}$ experiment) of CP violation in the interference of mixing
and decay via the time-dependent rate enables the extraction of the purely CP
odd short-distance amplitude, which is predicted within the SM with an ${\cal
O}(1\%)$ uncertainty.
The study of the $K_{S}\to\mu^{+}\mu^{-}$ decay is the main goal of the kaon
physics programme of the LHCb experiment. The high production rate of kaons in
proton-proton collisions at 13 TeV partly compensates the small LHCb
acceptance for kaon decays due to the long lifetime. LHCb has recently
reported an upper limit on the $K_{S}\to\mu^{+}\mu^{-}$ branching fraction
using the combined Run 1+2 dataset, ${\cal
B}(K_{S}\to\mu^{+}\mu^{-})<2.1\times 10^{-10}$ 90% CL [90], to be compared to
the SM prediction, ${\cal B}_{\rm SM}(K_{S}\to\mu^{+}\mu^{-})=(5.18\pm
1.50)\times 10^{-12}$ [91, 88]. The ultimate expected LHCb sensitivity for
this decay is close to the SM branching ratio and is expected to be limited by
the statistical uncertainty on the background subtraction and the signal yield
[92].
#### 3.1.5 Global sensitivity to New Physics
The rare decays discussed above offer the possibility to look for NP using a
global fit technique. As an example, Fig. 2 shows the result of this technique
using a low energy effective theory approach [85]. In this approximation, the
physics is described in terms of an effective Hamiltonian, which is an
expansion series of fermion current operators. The coefficient of this
expansion is the Wilson coefficient $C_{k}$, where $k$ is an index running
over the expansion terms. The assumption is that NP modifies $C_{k}$ with
respect to the SM values. The Wilson coefficients are then fitted using
experimental values of observables from rare kaon decays as input variables.
The results shown in Fig. 2 assume NP scenarios compatible with the LFUV
anomalies observed in $B$ decays.
Figure 2: Global fit to current data (purple, labelled as “current fit” in the
legend) and the results of two projections discussed in Ref. [85]: A, light-
red, where central values of observables currently only constrained with upper
bounds are set to the SM predictions; B, dark-red, where central values for
all of the observables are best-fit projections from existing data. The SM and
best-fit point with the current data are indicated with black and purple
crosses, respectively. The two figures represent the two possible signs of the
long-distance contribution to the $K_{L}\to\mu^{+}\mu^{-}$ process; the sign
ambiguity may be resolved with future data.
### 3.2 Lepton flavour universality tests
Lepton flavour universality (LFU) is a cornerstone of the SM postulating that
the lepton coupling to gauge bosons is independent of lepton type (“flavour”),
in contrast with the flavour-dependence of quark interactions. The origin of
the observed LFU, which is not associated with any known symmetry and is thus
not a fundamental conservation law, is a major question in modern physics.
Recently observed tensions between the SM predictions and experimental results
such as the measurement of the anomalous magnetic moment of the muon [93], the
Cabibbo angle anomaly [94], and the $B$-meson anomalies [8, 95, 6], suggest
possible LFU violation and motivate further the quest for improved
experimental LFU measurements.
The kaon sector offers opportunities for precision lepton flavour universality
tests by measuring the ratio ${\cal B}(K^{+}\to\pi^{+}e^{+}e^{-})/{\cal
B}(K^{+}\to\pi^{+}\mu^{+}\mu^{-})$ as discussed in the framework of a global
fit in a low-energy effective theory approach in Sections 3.1.3 and 3.1.5, as
well as ${\cal B}(K^{+}\to e^{+}\nu)/{\cal B}(K^{+}\to\mu^{+}\nu)$ and ${\cal
B}(K\to\pi e\nu)/{\cal B}(K\to\pi\mu\nu)$, exploiting cancellation of hadronic
effects [17, 78].
The ratio of purely leptonic decay rates of the charged kaon
$R_{K}=\Gamma(K^{+}\to e^{+}\nu)/\Gamma(K^{+}\to\mu^{+}\nu)$ is strongly
suppressed in the SM by conservation of angular momentum and an extremely
sensitive probe of LFU. Its SM expectation $R^{\rm SM}_{K}=(2.477\pm
0.001)\times 10^{-5}$ [96] is known to excellent (0.4‰) precision and the
currently most precise (4‰) experimental result is $R_{K}=(2.488\pm 0.007_{\rm
stat}\pm 0.007_{\rm syst})\times 10^{-5}$ [97]. The ratio $R_{K}$ is highly
sensitive to the possible violation of LFU naturally arising in new physics
scenarios involving sterile neutrinos [98, 99], leptoquarks [100], massive
gauge bosons [101], or an extended Higgs sector [102, 103]. Variations of
$R_{K}$ up to a few per mille from its SM expectation are predicted by these
models, without contradicting any present experimental constraints.
An analogous LFU test is based on the ratio of semileptonic kaon decay rates
$R_{K}^{(\pi)}=\Gamma(K\to\pi e\nu)/\Gamma(K\to\pi\mu\nu)$, where both neutral
and charged kaon decays ($K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ and
$K^{+}\to\pi^{0}\ell^{+}\nu$) can be used. For a given neutral or charged
initial state kaon, the Fermi constant, $V_{us}$, short-distance radiative
corrections, and the hadronic form factor at zero momentum transfer cancel out
when taking the ratio $R_{K}^{(\pi)}$ [17]. Therefore, in the SM this ratio is
entirely determined by phase space factors and long-distance radiative
corrections [104, 105, 106, 107, 16].
### 3.3 Lepton flavour and lepton number violation
In the SM, lepton number and lepton flavour are conserved due to accidental
global symmetries. Additionally, the SM requires neutrinos to be strictly
massless due to the absence of right-handed chiral states. However the
discovery of neutrino oscillations has demonstrated non-zero neutrino mass and
non-conservation of lepton flavour in the neutrino sector. On the other hand,
no evidence has yet been found for lepton flavour violation (LFV) in the
charged lepton sector, or lepton number violating (LNV) processes. Any such
observation would be a clear indication of NP.
Mixing of active neutrinos can, in principle, mediate LFV decays in the SM.
However the branching ratios are vanishingly small, and therefore any
observation of LFV kaon decays would be clear evidence of new physics. Such
LFV kaon decays are predicted in BSM scenarios with light pseudoscalar bosons
(ALPs) [108], a $Z^{\prime}$ boson [109, 110] or leptoquarks [111, 112]. As
discussed in Section 2.1, several hints of violations of lepton flavour
universality (LFUV) have been reported. Models which predict LFUV and explain
these potential anomalies in general naturally predict LFV processes [113,
114, 115]. Therefore search for LFV processes are of strong interest.
In BSM scenarios, LNV kaon decays are predicted for example in models with
Majorana neutrinos. In the minimal Type-I seesaw model [116, 117, 118], the
neutrino is a Majorana fermion with a mass term that violates lepton number by
two units. Non-zero neutrino masses make it possible, in principle, to
distinguish experimentally between the possible Dirac and Majorana natures of
the neutrino. Strong evidence for the Majorana nature of the neutrino would be
provided by the observation of lepton number violating (LNV) processes,
including kaon decays.
The recent results of LFV/LNV searches with the NA62 Run 1 (2016–2018) dataset
[119, 120, 121] include upper limits on the
$K^{+}\to\pi^{-}\ell^{+}_{1}\ell^{+}_{2}$ decay branching ratios at the ${\cal
O}(10^{-11})$ level, already leading to stringent constraints on active-
sterile mixing angles between Majorana neutrinos. Below the kaon mass, these
constraints are competitive with those obtained from neutrinoless double beta
decay [122, 123, 124, 125]. The NA62 searches are not limited by backgrounds,
and indicate that HIKE Phase 1 would make significant improvements in
sensitivity to LFV and LNV $K^{+}$ decays, extending the limits to the ${\cal
O}(10^{-12})$ level. On the other hand, the existing limit on the LFV
$K_{L}\to\mu^{\pm}e^{\mp}$ decay [126], which represents a stringent
constraint on several BSM scenarios, would be improved significantly by the
HIKE Phase 2; further details are provided in Section 7.2.
### 3.4 Tests of low-energy QCD
Most kaon decays are governed by physics at long distances. These decays can
be described by chiral perturbation theory (ChPT), which is the low-energy
effective field theory of QCD. The ChPT framework determines the kaon decay
amplitudes in terms of the so-called low-energy constants, which are
determined using experimental data. Therefore, measurements of kaon decay
rates and form factor parameters of various decay channels represent essential
tests of the ChPT predictions and crucial inputs to the theory at the same
time. A comprehensive overview of kaon decays and their relation to the ChPT
can be found in Ref. [127].
The HIKE dataset will provide a unique opportunity to perform precision
measurements of rare and radiative decays including $K\to\pi\gamma\gamma$,
$K\to\ell\nu\gamma$, $K\to\pi\ell\nu\gamma$, $K\to\pi\pi\gamma$,
$K\to\pi\pi\pi\gamma$, $K\to\pi\pi e^{+}e^{-}$, and $K\to\pi\pi\ell\nu$ of
both $K^{+}$ and $K_{L}$ mesons. Possible studies of the abundant decay modes
$K\to\pi\pi$ and $K\to\pi\pi\pi$ will also provide important inputs to future
ChPT parameter fits.
### 3.5 Test of first-row CKM unitarity
Measurements of semileptonic kaon decays $K\to\pi\ell\nu$ provide the
principal input for the extraction of the CKM parameter $V_{us}$, while the
ratios of (semi)leptonic $K^{+}$ and $\pi^{+}$ decay rates are used to extract
the ratio $V_{us}/V_{ud}$, with inputs provided from lattice QCD [128].
Determination of $V_{us}$ from kaon, pion, and $\tau$ decays, combined with
$V_{ud}$ measurement from super-allowed beta decays [129] and neutron decays
[130, 131], gives rise to a $3\sigma$ deficit in first-row CKM unitarity
relation known as the Cabibbo angle anomaly; a tension of similar significance
is observed between $K\to\ell\nu$ and $K\to\pi\ell\nu$ rates [15, 17]. The
uncertainty in $V_{us}$ comes in equal parts from the experimental errors and
theoretical uncertainties in the ratio of decay constants, $f_{K}/f_{\pi}$,
and the $K\to\pi\ell\nu$ form factor, $f_{+}(0)$. Substantial improvements in
the lattice QCD calculations of these hadronic factors are expected in the
next five years, thanks to decreased lattice spacing and accurate evaluation
of electromagnetic effects [132, 133, 134, 135]. Significant progress on the
calculation of radiative corrections has been achieved recently, reducing the
uncertainties on the long-distance electromagnetic corrections to a negligible
level [136, 107, 137, 16]. In order to shed light on the Cabibbo angle
anomaly, improved measurements of the principal $K^{+}$ and $K_{L,S}$
branching ratios are essential (and it should be noted that no $K_{L}$ decay
measurements have been made in the past decade). In particular a precision
measurement of the ratio of $K^{+}\to\pi^{0}\mu^{+}\nu$ and
$K^{+}\to\mu^{+}\nu$ rates is well-motivated [15]. In case the Cabibbo angle
anomaly persists, HIKE would be able to make precision measurements of
(semi)leptonic $K^{+}$ and $K_{L}$ decays based on special datasets collected
with minimum-bias triggers at low beam intensity.
It should be noted that the Belle II experiment is planning to extract
$V_{us}$ at an improved precision from a suite of inclusive and exclusive
measurements of $\tau$ decays, including ${\cal B}(\tau^{-}\to K^{-}\nu)/{\cal
B}(\tau^{-}\to\pi^{-}\nu)$, combined with theory improvements [138]. This
would provide a possibility to cross-check the kaon results.
### 3.6 Production of feebly-interacting particles in kaon decays
An alternative paradigm to the searches for NP at high energy is the search
for feebly-interacting particles (FIPs), which would be the manifestations of
weakly-interacting NP potentially at much lower energy scales. Interactions of
FIPs with SM fields are possible through BSM extensions involving low-
dimension operators, the so-called portals [20, 21], the simplest of which are
summarised in Table 2. For further discussion of the benchmark scenarios, see
also Section 9.2.
Table 2: Summary of generic portal models with FIPs [20, 21]. Portal | FIPs
---|---
Vector | Dark photon, $A^{\prime}$
Scalar | Dark Higgs/scalar, $S$
Fermion | Heavy neutral lepton (HNL), $N$
Pseudoscalar | Axion/axion-like particle (ALP), $a$
Due to the availability of large datasets and the suppression of the total
kaon decay width, kaon decays represent uniquely sensitive probes of light
hidden sectors via FIP production searches. The possible search strategies
have been reviewed recently [22], and the following have been identified as
the most promising. Searches for the $K\to\pi X_{\rm inv}$ decay by extension
of the $K\to\pi\nu\bar{\nu}$ measurements, where $X_{\rm inv}$ represents an
invisible particle, represent a unique probe into the dark-scalar and ALP
parameter space. Searches for heavy neutral lepton ($N$) production in
$K^{+}\to\ell^{+}N$ decays are approaching the seesaw neutrino mass models
with ${\cal O}(100~{}{\rm MeV})$ sterile neutrinos [139]. Searches for
resonances in the $K\to\pi\ell^{+}\ell^{-}$ and $K\to\pi\gamma\gamma$ decay
spectra are complementary to searches at beam-dump experiments for a
significant ALP mass range. Finally, searches for a leptonic force mediator
($X$) in $K^{+}\to\mu^{+}\nu X$ decays can probe a region of parameter space
providing an explanation for the muon $g-2$ anomaly [140].
Experimentally, the NA62 Run 1 dataset has been used to establish the analysis
methods for searches for dark-scalar and HNL production in $K^{+}$ decays,
producing world-leading limits on the dark-scalar coupling [67, 141, 142] and
HNL mixing parameters [143, 144] below the kaon mass. Further details on the
experimental aspects and HIKE sensitivity projections are provided in Sections
6.2.3 and 6.2.4.
## 4 Physics in the beam-dump mode
Thanks to the high-intensity beam and high-performance detectors (redundant
particle identification capability, highly efficient veto systems and high
resolution measurements of momentum, time, and energy), HIKE can achieve the
sensitivities required for competitive searches for production and decay of
long-lived light mediators in a variety of new-physics scenarios.
A number of possible Standard Model (SM) extensions aimed at explaining the
abundance of dark matter in our universe predict a new “hidden sector” with
mediator fields or matter fields in the MeV–GeV mass range. The realisations
of such scenarios are usually classified in terms of the Lagrangian operators
(the so-called “portals” listed in Table 2) connecting SM particles to the new
mediators. The related coupling constants are small, thus evading the existing
exclusion bounds and justifying the label of “feebly interacting” given to
these new-physics models [20, 21]. The mass of the mediator and hidden-sector
matter fields and the couplings are free parameters of the models. The
relevant features of the phenomenology of such models are:
* •
The mediators can be produced in proton-nucleus interactions through a number
of mechanisms, the most important being direct Primakoff production and meson-
mediated tertiary production. The specific production mechanism for each
mediator type differs in terms of production cross section and momentum-angle
spectra of the mediator emitted.
* •
At the 400-GeV energy of SPS protons, mediators with momenta above 10 GeV have
decay lengths ranging from tens of meters to tens of kilometres in a suitable,
interesting interval of the feeble coupling.
* •
Due to the feeble interaction with the SM particles, the emitted mediators can
reach the decay volume after punching through tens of meters of traversed
material before decaying.
As demonstrated by the NA62 Run 2 operation [145], the intense P42 400 GeV
proton beam extracted from the CERN SPS and the HIKE setup can be exploited to
search for production and decay of the emitted mediators. For this purpose,
HIKE can be operated in the so-called beam-dump mode. Considering that HIKE
will have the same type of forward geometry as NA62, the HIKE beam-dump
operation will be particularly sensitive to specific physics cases in which
the hidden sector mediator is produced by Primakoff scattering or light-meson
mediated decays. Prominent examples are the vector and some axion-like
particle scenarios. We address a number of new physics scenarios in Section 9,
tentatively assuming a total integrated statistics of $10^{19}$ POT for HIKE
Phase 1, and $5\times 10^{19}$ POT for the complete HIKE programme.
## 5 High-intensity beams
As the acronym HIKE suggests, the beam line must provide significantly higher
beam intensities than those that are now delivered to the NA62 experiment, on
the order of 4 to 6 times of that currently provided, which would correspond
to between 1.2 and $2\times 10^{13}$ protons on the T10 target per SPS
extraction. Here, we assume that a typical running year has 200 days with 3000
spills each, including an SPS uptime of about 80%, and a flat top length of
4.8 seconds.
The proposed HIKE program includes both a charged beam ($K^{+}$) phase and a
neutral beam ($K_{L}$) phase. For both parts of the program, protons at much
higher intensity have to be transported via the P42 beam line from the T4
target to the kaon production target T10. Conceptually, the $K^{+}$ beam is
proposed to be the same as the present beam line, but consolidation and
upgrades are needed to handle the higher beam intensities. Improved beam
instrumentation will keep the beam losses better under control. A more precise
calibration of the primary beam intensity measurements by the secondary
emission monitors at T4 and T10 is important for beam optimisation, and for
normalisation purposes in dump mode. Efforts in this respect have already been
started by the BI and EA groups at CERN. The neutral beamline is an entirely
new design and requires minor local modifications to the last section of the
P42 line.
### 5.1 Beam delivery to the kaon production target
The P42 beam line started operation in 1980 and hardly has been modified since
then [146, Chapter 5]. Most of the equipment dates from then and is showing
strong effects of ageing. Many issues and suggestions for mitigation have been
described in the report from the PBC Conventional Beams working group [147],
which has recently been updated. The first phase of a vast consolidation
program for the CERN North Area, titled NACONS, has recently been approved and
funded, and the second phase has been prepared [148, 149, 150]. This will
allow restoration of the equipment and beam infrastructure to a reliable
state. Some upgrades are required to cope with the up to 6 times higher beam
intensity, including operational requirements from the rest of the North Area
complex and with stricter safety and radiation protection requirements
compared to the original installation. Many studies have been performed by the
SPS Losses and Activation working group (SLAWG) [151, 152, 153], the Physics
Beyond Collider (PBC) Proton Delivery working group [154], the PBC
Conventional Beams working group [147], and more recently, by the PBC ECN3
beam delivery task force [155], focusing on the possibility to extract and
deliver such high intensities towards TCC8 and ECN3.
The present optics of the P42 beam is shown in Fig. 4. In order to reduce the
attenuation of the primary beam in the T4 target, several options are
considered. One is to increase the vertical beam size so that most of the beam
bypasses the target without interacting. Another option is to run dedicated
cycles for ECN3 in which the beam is bumped to pass above or below the target
plate. This option could allow in principle even higher intensities than
$2\times 10^{13}$ ppp on the T10 target. At present, the beam is provided in
spills of 4.8 seconds, typically twice per super-cycle of about 40 to 45
seconds during day time, and two equally-spaced spills during 33.6 seconds
during nights. One spill every 14.4 seconds is also technically possible and
will be the standard cycle adopted during the night from 2023 (Fig. 3).
Figure 3: The spill configuration used on 22 October 2022, as seen from SPS
Page 1.
In recent years, on average 3000 spills per day are delivered, including the
typical availability of SPS beams of about 80%. The figure of merit is the
time on flat top per year, which is proportional to the duty cycle and the
running time. It is also important that the beam intensity is delivered as
uniformly as possible. At the time of the NA62 proposal one assumed an
effective spill of 3.3 seconds, but thanks to recent improvements we
anticipate that it can be significantly better, ideally better than 4 seconds
for a 4.8 second spill. The spill length can be modified to some extent, but
it is important to maintain the duty cycle as well as the instantaneous
intensity, respecting both the RMS power limitation of the SPS and transfer
lines as well as the maximum intensity that the SPS is able to accelerate. The
latter implies that the intensity per spill shall be kept proportional to the
spill length.
Studies by the SY-STI group in the PBC context suggest that the TAXs in P42
and K12 are already running close to the maximum intensity allowed [147]. An
upgrade is being studied at the technical level in the framework of the ECN3
task force. This implies a substantial upgrade of the T10 target-TAX complex
to withstand the higher beam power and to be compatible with RP and
maintenance requirements. A new target design could be inspired by the
concepts of the CNGS target. The target material might be changed from
beryllium to carbon for better handling. However, also more dense target
materials can be considered as an option outside of the present baseline, as
they could be advantageous in particular for the neutral beamline. The new K12
TAX will be based on the concept that is developed for the P42 TAX. TAX holes
will be defined to ensure compatibility with the charged beam as well as the
neutral beam. It will probably require a different choice of material in the
TAX blocks and cooling closer to the beam impact point. The present
interlocks, P0-Survey and Dump Control, protect both the beam line equipment
and the NA62 experiment against incorrect magnet currents, T10 target and K12
TAX cooling problems and closed vacuum valves in the K12 beam line. For the
higher intensities in the future, this system must be sped up. This is planned
within the NACONS program baseline, independently of the proposed high
intensity upgrade for ECN3, profiting from a new generation of power
converters. On the shorter time scale, some improvements are already under
consideration, compatible with the presented HIKE timeline. Many studies are
on-going concerning radiation dose levels to equipment and on surface, which
must remain under control also at the higher intensities. All these issues
will be addressed in the forthcoming report from the ECN3 beam delivery task
force.
For a $K_{L}\to\pi^{0}\nu\bar{\nu}$ experiment, the production angle of the
kaon beam has to be increased to 8 mrad instead of the zero production angle
used for the charged mode. This can be achieved by a minor modification of the
P42 beam line, i.e. realigning the three last dipoles as documented in the
Conventional Beams Working Group Report [147], shown in Fig. 5. The high-
intensity proton beam will ultimately impinge on the kaon production target,
followed by the TAX beam dumps.
### 5.2 Charged kaon beamline
The HIKE charged kaon beamline will remain conceptually as it is now, with
essentially the same beam properties apart from the beam intensity (Fig. 6).
Consolidation of some equipment is required in the framework of NACONS, such
as the K12 machine protection system in synergy with the P42 interlock
upgrade. The main upgrades concern the T10 production target and the K12 TAX,
including the needed shielding. A pre-study was presented in the Conventional
Beams Working Group report [147] and is being followed up further in the
working group.
The K12 beam is a mixed beam containing 6% of kaons at 75 GeV/$c$. A study has
been performed to evaluate if the beam can be enriched by RF separation. RF
separation requires a much longer beamline, implying that more kaons decay
before the fiducial volume. This decreases not only the kaon flux but also the
initial kaon fraction in the beam before enrichment. An RF-separated beam
cannot fulfil the HIKE requirements, even with state-of-the-art RF systems
[156].
Figure 4: Beam optics for the P42 primary beam line, delivering protons from
the T4 production target to the T10 kaon production target. The red line
depicts the local values of the R12 (horizontal) and R34 (vertical) beam
transport matrix terms. The dotted blue line shows the local dispersion terms
R16 (horizontal) and R36 (vertical). Figure 5: Modifications of the T10 target
steering dipoles for increasing the production angle to 8 mrad, as required
for the KLEVER phase of the HIKE programme.
Figure 6: Beam optics for the K12 beamline. The red line depicts the local
values of the R12 (horizontal) and R34 (vertical) beam transport matrix terms.
The dotted black line shows the local dispersion terms R16 (horizontal) and
R36 (vertical).
### 5.3 Neutral kaon beams
The original design and simulations for the KLEVER beamline were performed by
the Conventional Beams Working Group [147]. The design is based on the
experience with the $K_{L}$ beam lines for NA31 and NA48, following guidelines
laid out in Ref. [157], and is shown schematically in Fig. 7. It forms the
basis for the beamline to be used also for HIKE Phase 2, and will be extended
in length for the precise measurement of $K_{L}\to\pi^{0}\nu\bar{\nu}$
(KLEVER).
Figure 7: Baseline design of neutral beamline for HIKE Phases 2 and 3.
The primary proton beam impinges on the T10 target with a downward vertical
angle of 8 mrad for KLEVER, and possibly 2.4 mrad for the $K_{L}$ experiment
with tracking. These choices balance several factors for the measurements of
$K_{L}\to\pi^{0}\nu\bar{\nu}$ and $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$,
respectively. As seen from Figs. 8 and 9, a smaller production angle increases
$K_{L}$ flux and hardens the $K_{L}$ spectrum, facilitating the task of
vetoing background channels with extra photons, such as
$K_{L}\to\pi^{0}\pi^{0}$. On the other hand, with harder input momentum
spectra, a smaller $K_{L}$ fraction and a larger $\Lambda$ faction decay
inside the fiducial volume. For the $K_{L}\to\pi^{0}\nu\bar{\nu}$ measurement,
$\Lambda\to n\pi^{0}$ decays constitute a potentially dangerous background.
Neutron interactions on residual gas in the vacuum tank and from halo neutrons
striking the detector constitute another dangerous background, which is
reduced by going to larger angle. The production angle of 8 mrad softens the
$\Lambda$ spectrum, so that almost all $\Lambda$ hyperons decay before the
start of the fiducial volume. For the $K_{L}\to\pi^{0}\nu\bar{\nu}$
measurement, the reduced $K_{L}$ flux is compensated to some extent by an
increase in the fraction of $K_{L}$ mesons that decay in the fiducial volume.
However, this is not true for $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$, and moreover,
background from $\Lambda$ decays and neutron interactions is not a concern in
this case. A production angle as small as possible is therefore preferred for
HIKE Phase 2, subject to limitations from the neutron flux in the beam at very
small angles. A production angle of 2.4 mrad (the same as for the $K_{L}$ beam
in NA48) is a likely choice. The $K_{L}$ and $\Lambda$ momentum spectra for
production angles of 2.4 and 8 mrad are shown in Fig. 10, and the fluxes and
other parameters are tabulated in Table 3.
Figure 8: Mean (left) and peak (right) momentum for $K_{L}$ (red) and
$\Lambda$ (blue) components vs beam production angle, from FLUKA (squares) and
Geant4 (circles), with solid (dashed) curves from fits to the form in Ref.
[158].
Figure 9: Left: neutral beam fluxes per POT per $\mu$sr of acceptance for
$K_{L}$ (blue) and $\Lambda$ (red) vs beam production angle. Right: ratios of
$\gamma$ (green), $n$ (red), and $\Lambda$ (blue) to $K_{L}$ fluxes vs beam
production angle. FLUKA (Geant4) results are shown with squares (circles),
with solid (dashed) curves from fits.
Figure 10: $K_{L}$ (blue) and $\Lambda$ (red) beam momentum spectrum (number of particles in beam per POT per $\mu$sr of neutral beam acceptance) for production angles of 2.4 mrad (left) and 8 mrad (right). The solid curves show a parameterisation [158] of FLUKA results at production; dashed curves show the momentum spectra for particles decaying in an indicative fiducial volume of the experiment extending from 135 m to 195 m downstream of the target. Table 3: Parameters of $K_{L}$, $n$, $\Lambda$ production at angles of 2.4 and 8.0 mrad, obtained from the FLUKA simulation of the beamline. An indicative fiducial volume (FV) extending from 135 m to 195 m from the production target is considered. | 2.4 mrad | 8.0 mrad
---|---|---
Mean $K_{L}$ momentum (GeV/$c$) | |
at production | 79 | 39
in FV | 46 | 26.4
Mean $\Lambda$ momentum (GeV/$c$) | |
at production | 151 | 66
in FV | 285 | 251
$K_{L}$ rate ($10^{-6}$/pot/$\mu$sr) | |
at production | 187 | 73
in FV | 6.8 | 4.1
$\Lambda$ rate ($10^{-6}$/pot/$\mu$sr) | |
at production | 310 | 37
in FV | 0.053 | 0.000132
Table 4: Particle fluxes in the neutral beam, obtained from the FLUKA simulation. The rates in MHz/GHz assume a primary beam intensity of $6.7\times 10^{12}$ pot/s. Beam component | Before converter | After converter | After final collimator
---|---|---|---
| pot-1 | GHz | pot-1 | MHz | pot-1 | MHz
$\gamma$ | | | | | |
2em $E>1$ GeV | $4.2\times 10^{-3}$ | 27.9 | $9.1\times 10^{-4}$ | 6100 | $2.97\times 10^{-5}$ | 198
2em $E>5$ GeV | $2.19\times 10^{-3}$ | 14.6 | $7.1\times 10^{-5}$ | 470 | $7.9\times 10^{-6}$ | 53
2em $E>10$ GeV | $1.55\times 10^{-3}$ | 10.3 | $1.81\times 10^{-5}$ | 121 | $3.15\times 10^{-6}$ | 21
2em $E>30$ GeV | $6.3\times 10^{-4}$ | 4.2 | $2.06\times 10^{-6}$ | 13.7 | $6.2\times 10^{-7}$ | 4.1
$n$ | | | | | |
2em $E>1$ GeV | $4.3\times 10^{-4}$ | 2.88 | $4.2\times 10^{-4}$ | 2820 | $6.7\times 10^{-5}$ | 440
$K_{L}$ | | | | | |
2em $E>1$ GeV | $1.37\times 10^{-4}$ | 0.91 | $1.29\times 10^{-4}$ | 870 | $2.11\times 10^{-5}$ | 140
The target is immediately followed by a first collimator that stops hadrons
outside the beam acceptance before they decay into muons. The non-interacting
protons are swept further downward by a strong dipole magnet. Protons and
charged secondaries are dumped in the TAX, which has apertures that allow the
beam to pass. A photon converter consisting of 9$X_{0}$ of high-$Z$ material
is positioned at the centre of the TAX between the two modules and reduces the
flux of high-energy photons ($E>5$ GeV) in the neutral beam by two orders of
magnitude. A thinner oriented crystal converter is an optional possibility
under study. Shower products emerging from the TAX are swept horizontally by a
dipole magnet. Three collimators define the beam acceptance in a clean way.
The defining collimator is located at $1/3$ of the distance to the final
collimator and defines the beam angular acceptance to not more than $\pm$0.4
mrad, which is necessary to measure the transverse momentum of the $\pi^{0}$
to sufficient precision in order to reject background from
$K_{L}\to\pi^{0}\pi^{0}$ decays. A cleaning collimator stops debris from
scatterings in the jaws of the defining collimator, and a final collimator
stops scattering products from the cleaning collimator. The defining and
cleaning collimator are each followed by strong sweeping magnets. The final
collimator will be active, and will define the start of the fiducial volume.
Sufficient space is available to add more collimation stages, depending on
future design iterations within the Conventional Beams Working Group.
#### 5.3.1 Simulation of the $K_{L}$ beamline
A detailed FLUKA simulation of the entire beamline has been developed by the
Conventional Beams Working Group. Fig. 11 shows the momentum distributions for
$K_{L}$, photons, and neutrons in the beam at various stages of the beamline
simulation: at generation (at the exit from the target), after the converter
and each of the collimators, and at the end of the beamline, at the front face
of the small-angle calorimeter (SAC). The normalisation of these distributions
provides the particle fluxes in the neutral beam per incident proton. Of
particular interest are the photon and neutron fluxes after the final
collimator. The fluxes of photons, neutrons and $K_{L}$ in the beam are
summarised in Table 4. For the purposes of the sensitivity estimates for the
KLEVER phase, $2.1\times 10^{-5}$ $K_{L}$ mesons enter the detector per proton
incident on the target. The beam rates in the table assume a primary beam
intensity of $2\times 10^{13}$ ppp with a (pessimistic) effective spill length
of 3 seconds. In addition to the 140 MHz of $K_{L}$ mesons entering the
detector, there are 440 MHz of neutrons: the $n/K_{L}$ ratio of about 3
observed for particle production at 8 mrad is not significantly changed during
transport of the neutral beam. In addition, there are about 50 MHz of photons
with $E>5$ GeV entering the detector, most of which are incident on the SAC at
the downstream end of the detector (the condition $E<5$ GeV corresponds to the
SAC threshold for KLEVER). The FLUKA simulation also contains an idealised
representation of the KLEVER experimental setup, for the purposes of
evaluating rates on the detectors from beam halo. These rates, with specific
reference to the KLEVER experimental geometry, are discussed in Section 8.2.
As seen from Fig. 11 and Table 4, the photon converter in the TAX dramatically
reduces the photon flux in the beam, especially for high energy photons. This
is critical to avoid blinding the SAC at the downstream end of the beamline.
In the baseline design, the converter is a tungsten prism of 32.9 mm
thickness, corresponding to 9.4$X_{0}$, or 7.3 photon conversion lengths. This
thickness is chosen to keep the rate of photons with $E>5$ GeV below 40 MHz at
the entrance to the SAC. On the other hand, this thickness corresponds to 58%
of a nuclear collision length and 33% of an interaction length, so that about
35% of $K_{L}$ mesons and 40% of neutrons interact in or are scattered out of
the beam by the converter.
Figure 11: Momentum distributions for $K_{L}$ mesons (top), photons (bottom
left), and neutrons (bottom right) in the neutral beam at various points along
the neutral beamline. Figure 12: Photon fluxes (for $E>5$ GeV) at the SAC as
functions of photon converter thickness for beryllium, copper, and lead
targets. The dotted line indicates the flux corresponding to a rate of 40 MHz,
considered to be the maximum tolerable rate for high-energy photons on the
SAC.
The use of a high-$Z$ material for the target is an interesting alternative to
the current baseline. Most of the prompt photons come from $\pi^{0}$ decays
within the target, and a high-$Z$ material causes many of these photons to be
converted immediately, while the overall thickness of the target in nuclear
interaction lengths is unchanged. No large differences are observed in terms
of hadronic production for targets of identical thickness in nuclear
interaction lengths [159]. Photon rates at the SAC for beryllium, copper and
tungsten targets (each with a nuclear interaction length equivalent to 400 mm
Be), determined with the FLUKA simulation, are shown in Fig. 12. The required
converter thicknesses for beryllium, copper, and tungsten targets of
equivalent thickness are 7.3, 5.2, and 3.8 photon conversion lengths,
respectively, resulting an effective increase in the $K_{L}$ flux into the
detector of 15% (28%) for a copper (tungsten) target relative to a beryllium
target. Changing the target material strongly impacts the target design, as
the energy deposit per unit of volume is much higher in a high-$Z$ target than
in beryllium, adding to cost and complexity of the new target design.
Another promising technique to reduce the photon content in the beam is to use
a crystal metal photon converter. If the crystal axis is aligned with the
incoming photon direction, the coherent effect of the crystal lattice promotes
pair production, leading to an effective decrease in the photon conversion
length [160, 161, 162]. The effects of coherent interactions increase with
photon energy and for decreasing angle of incidence. A series of exploratory
tests was performed with a set of tungsten crystals at the CERN SPS in summer
2018, together with the AXIAL collaboration [163]. In particular, a commercial
quality tungsten crystal of 10-mm thickness was targeted with a tagged photon
beam. When the <111> crystal axis was aligned with the beam to within 2.5
mrad, the multiplicity of charged particles was found to be enhanced by a
factor 1.6–2.3 for photon energies over the range of 30–100 GeV. Ref. [163]
also reports simulations validated by this result that suggest that a crystal
of this type could be used to reduce the thickness of the photon converter by
15–20% at no cost in effectiveness. Such a solution appears to be relatively
easy to implement.
#### 5.3.2 Beam for Phase 2 (a multi-purpose $K_{L}$ experiment)
The proposed HIKE Phase 2 is based on the $K_{L}$ beam, but a detector very
similar to the Phase 1 experiment, including a tracking system. This phase of
the programme will make use of the neutral beamline described above with a
production angle chosen to optimise the statistics for the measurement of
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays, likely 2.4 mrad. Alternative
beamline options can be studied. The full FLUKA simulation will be performed
when the choice of production angle is finalised. For the sensitivity
estimates in Section 7, the entries in Table 4 are scaled to account for
production angle of 2.4 mrad (Table 3).
#### 5.3.3 Beam for Phase 3 (KLEVER: measurement of
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay)
After the initial neutral beam studies for the KLEVER phase of HIKE were
completed, it was discovered that the background from $\Lambda\to\pi^{0}n$
decays is more difficult to suppress than previously thought. Therefore, the
amount of $\Lambda$ decays in the fiducial region has to be reduced further by
several orders of magnitude, whilst preserving as much as possible the kaon
flux. Various options were considered, such as lengthening the distance
between the target and the start of the fiducial volume, reducing the primary
proton beam energy (for example, from 400 to 300 GeV), and increasing the
production angle (for example, from 8 mrad to 20 mrad). Lengthening the beam
line by 150 m seems to be the best option, and several possibilities are under
consideration. Moving the T10 target further upstream is one of them, but this
may have complex radiation protection implications and might require the whole
experiment to be installed on a slope, depending on the exact target location.
The more straightforward option might be to prolong the ECN3 cavern by 150 m,
implying civil engineering works. Simulations show that this brings the
background from $\Lambda$ decays to an acceptable level (Section 8.3). The
technical details and costs of the different options are under discussion with
the Civil Engineering group of CERN.
An extension of the ECN3 cavern implies that the detector will be installed
150 m downstream of the present NA62 location. At the moment, we do not
contemplate changing the transverse dimensions of the setup. The positions of
the beamline elements would change as noted in Table 5. The beam solid angle
and hence the $K_{L}$ flux decreases by a factor of $(0.256/0.400)^{2}\approx
0.41$. Since the $p_{\perp}$ reconstruction at the downstream end of the FV
depends on the beam spot size, it is not trivial to recover the sensitivity of
the experiment by enlarging its transverse dimensions, and we favour the other
approaches discussed in Section 8.3. The full FLUKA simulation of the extended
configuration will be performed as soon as the design is finalised. For the
sensitivity estimates, the rates in Table 4 can be scaled to account for the
reduction in solid angle from the extension.
In case resources allow to advance the construction of the ECN3 extension,
HIKE Phase 2 can be performed with the extended beamline and the detector in
the new location. This would eliminate the need for infrastructural work to
change beamline configurations between Phase 2 and KLEVER and allow precise
information on the beam composition and halo to be obtained during Phase 2,
thereby improving the understanding of beam-related backgrounds in KLEVER.
Table 5: Collimator positions and other parameters for standard and extended KLEVER beamlines. Configuration | Standard | Extended
---|---|---
Target position [m] | 0 (T10) | 0 (T10)
Defining collimator position [m] | 40 | 90
Cleaning collimator position [m] | 80 | 180
Final collimator position [m] | 120 | 270
Beam opening angle [mrad] | 0.400 | 0.256
## 6 Phase 1: a multi-purpose $K^{+}$ decay experiment
### 6.1 Experimental layout
While the success of the NA62 experiment has proven that its layout is
suitable for a precision ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ measurement,
new or upgraded detectors will replace those of NA62 with the goal of
sustaining secondary-beam rates four times higher than those of NA62, to
substantially boost the statistical sensitivity. To this end, new technologies
will be used, in synergy with the upgrades of the LHC experiments. While some
detectors will be renovated for the start of the HIKE $K^{+}$ phase (Phase 1),
others are already intrinsically fast in terms of detector technologies but
will need readout upgrades. Therefore a staged approach will be used, where
renovated or new detectors are inserted as soon as they are needed and ready,
while maintaining the general principle that changes must serve the remaining
phases of the programme once they are applied.
In the baseline configuration, the $K^{+}$ beam is produced by $1.2\times
10^{13}$ $400~{}{\rm GeV}/c$ protons/spill at zero angle on target. The mean
$K^{+}$ momentum at the entrance to the decay volume is 75 GeV/$c$. The layout
(Fig. 13) includes a beam tracker, a kaon-identification detector, and a veto
counter upstream of the decay volume; a main tracking system with the MNP33
magnet in the decay volume; calorimeters, vetoes and particle identification
detectors downstream of the decay volume; and a large-angle veto system
surrounding the decay volume and part of the region downstream.
HIKE Phase 1 ($K^{+}$)
Figure 13: HIKE Phase 1 layout, with an aspect ratio of 1:10.
### 6.2 Physics sensitivity
#### 6.2.1 $K^{+}\to\pi^{+}\nu\bar{\nu}$ measurement
The first phase of HIKE will be dedicated to a detailed measurement of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay, with a target of reaching a branching
ratio measurement with $\mathcal{O}(5\%)$ precision, competitive with the
theoretical uncertainty. The approach towards this goal will continue, and
build upon, the successful strategy of the NA62 experiment. The keystones of
the measurement, required to suppress backgrounds at the
$\mathcal{O}(10^{11})$ level, are the following:
* •
High efficiency and high-precision tracking of both the $K^{+}$ upstream and
$\pi^{+}$ downstream. This, coupled with a careful choice of signal regions,
will allow kinematic suppression of backgrounds by a factor of
$\mathcal{O}(10^{3})$.
* •
High precision time measurements, allowing time-matching between upstream and
downstream detectors with $\mathcal{O}(20~{}\text{ps})$ precision. Simulations
show that with this time-matching performance, the effect of the higher
intensity can be totally compensated in the matching between upstream and
downstream tracks, without losing efficiency for the signal selection and
without increasing the relative contamination of background coming from
upstream decays and interactions with respect to NA62.
* •
Comprehensive and hermetic veto systems:
* –
Photon veto detectors with hermetic coverage of the 0–50 mrad range for
photons from $\pi^{0}\to\gamma\gamma$ decays, allowing photon suppression by a
factor of $\mathcal{O}(10^{8})$.
* –
Veto detectors to cover downstream regions not covered by the principal
detectors.
* –
Rejection of upstream decays and interactions, for suppressing possible
upstream backgrounds by an additional factor of $\mathcal{O}(10)$, with
changes to the beam line setup and new systems dedicated to the detection of
the upstream background events.
* •
High-performance particle identification system (for $\pi$/$\mu$
discrimination), suppressing backgrounds with muons by a factor
$\mathcal{O}(10^{7})$.
Crucial to the study of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay at HIKE will
be the exploitation of the high-intensity environment, taking advantage of the
high $K^{+}$ flux while mitigating detector pileup effects. A critical
performance indicator is the “random veto efficiency”, i.e., the fraction of
events passing signal selection criteria that are sensitive to accidental
activity in the detector systems. The random veto efficiency measured within
the NA62 selection for the $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay at the current
intensity working-point is $\varepsilon_{\rm RV}\approx 65\%$. This is
dependent on selection criteria that are limited by the timing precision of
the detectors. The random veto efficiency is approximately linear as a
function of the instantaneous beam intensity. At HIKE, this random veto
efficiency must be maintained or improved, requiring an improvement in time
resolution by the same factor as the intensity increase. With this
performance, a projection for the random veto efficiency as a function of
intensity is shown in Fig. 14. Individual contributions from photon veto
subsystems (SAV, LAV, ECAL) are indicated which in total give the ‘photon
rejection’ curve. The curve labelled ‘multiplicity rejection’ relates to the
selection criteria required to reject additional activity in an event (outside
the photon veto detectors). Finally, the total random veto efficiency,
combining photon and multiplicity rejection, is shown. At a secondary beam
intensity 4 times higher than NA62, with the detector updates and
corresponding selection changes shrinking veto windows by a factor of 4, the
random veto efficiency is maintained at the same level as currently achievable
at NA62.
Figure 14: Expected HIKE Phase 1 random veto efficiency $\varepsilon_{\rm RV}$
for the $K^{+}\to\pi^{+}\nu\bar{\nu}$ analysis as a function of instantaneous
beam intensity: the total effect (grey band) and the contributions from the
individual photon/multiplicity veto conditions. The NA62 graph is also shown
for comparison: the lower $\varepsilon_{\rm RV}$ values with respect to HIKE
Phase 1 are due to the worse time resolution.
The HIKE sensitivity for the ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$
measurement is estimated as follows.
* •
The effective number of collected $K^{+}$ decays per year is $N_{K}/{\rm
year}\simeq 2\times 10^{13}$. This can be extrapolated from the data already
collected by the NA62 experiment, scaling for an intensity four times larger
than the NA62 nominal intensity and 200 days of data-taking per year. This
automatically includes effects already present in the current setup, such as
duty cycle, SPS beam availability, secondary beamline downtime, and detector
and DAQ efficiencies.
* •
Acceptance of the signal selection: $\varepsilon_{A}\simeq 0.1$. The
improvements on the time-matching performance and on the resolution of the
charged-particle identification system will lead to an increase of about 50%
relative to that at the current NA62 working point.
* •
Random veto efficiency ($\varepsilon_{\rm RV}$) and trigger efficiency
($\varepsilon_{\rm trig}$), which according to the projections already
described should remain the same as for the current NA62 working point even at
higher intensity: $\varepsilon_{\rm RV}\simeq 0.7$, $\varepsilon_{\rm
trig}\simeq 0.9$.
Under these conditions, the single event sensitivity attainable in one year of
data-taking is found to be ${\cal B}_{\rm SES}(1\;{\rm year})=(N_{K}/{\rm
year}\cdot\varepsilon_{A}\cdot\varepsilon_{\rm RV}\cdot\varepsilon_{\rm
trig})^{-1}\simeq 8\times 10^{-13}$, leading to the number of expected SM
signal events per year:
$N_{\pi\nu\bar{\nu}}/{\rm year}\simeq 100.$
With an $O(10\%)$ relative background contamination in the signal sample and
systematic uncertainty sources under control to $O(1\%)$ or better, a
measurement of the branching ratio $\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})$
to $\mathcal{O}(5\%)$ precision can be reached by HIKE in 4 years of data-
taking. Assuming that the number of observed events is equal to the SM
expectation, values of ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ higher than 10%
with respect to the SM prediction will be excluded at 95% CL.
#### 6.2.2 Test for scalar amplitudes in the $K^{+}\to\pi^{+}\nu\bar{\nu}$
decay
As introduced in Section 3.1.1, in BSM scenarios the measurable branching
ratio of $K^{+}\to\pi^{+}\nu\bar{\nu}$ is formed from two components: the SM
process $K^{+}\to\pi^{+}\nu\bar{\nu}$ with a purely vector nature and a
possible BSM process $K^{+}\to\pi^{+}\nu\nu$ with a scalar nature [68, 69].
Therefore the experimentally measured branching ratio is given by [23, 68]:
$\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})=\mathcal{B}_{\rm
SM}(K^{+}\to\pi^{+}\nu\bar{\nu})+\sum_{i\leq j}^{3}\mathcal{B}_{\rm
LNV}(K^{+}\to\pi^{+}\nu_{i}\nu_{j}).$ (5)
The SM and BSM contributions are predicted to have different kinematic
distributions (Fig. 15). To identify the nature of the decay, an investigation
of the shape of the distribution of selected signal candidates as a function
of kinematic variables will be performed. In absolute terms, within the signal
regions (as defined in the current NA62 analysis), the acceptance for the BSM
mode is approximately 75% of the acceptance for the SM mode. With a sample of
several hundred $K^{+}\to\pi^{+}\nu\bar{\nu}$ candidates, HIKE will be able to
study the shape of the kinematic distributions as well as compare sets of sub-
categories of phase-space to distinguish between, or constrain the relative
contributions of, the possible vector and scalar natures of the process.
Figure 15: Simulated distributions of the squared missing mass $m_{\rm
miss}^{2}=(P_{K^{+}}-P_{\pi^{+}})^{2}$ vs pion momentum in the laboratory
frame for the SM $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay with a vector nature
(left) and the LNV $K^{+}\to\pi^{+}\nu\nu$ decay with a scalar nature (right).
Black boxes indicate the signal regions for which experimental sensitivity is
highest.
#### 6.2.3 Feebly-interacting particle production in $K^{+}\to\pi^{+}X_{\rm
inv}$ decays
As described in Section 3.6, searches for the $K^{+}\to\pi^{+}X_{\rm inv}$
decay, where $X_{\rm inv}$ is a FIP (either a dark scalar or ALP), can be
performed within the analysis framework developed for the study of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay. This is possible because the signature of
these two decays is identical except that the $K^{+}\to\pi^{+}X_{\rm inv}$
decay produces a peak in the $m_{\rm miss}^{2}=(P_{K^{+}}-P_{\pi^{+}})^{2}$
variable at $m_{X}^{2}$, where $m_{X}$ is the mass of the FIP. Searches are
therefore performed by looking for evidence of a peaking signal on top of the
background dominated by the SM $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay. The search
strategy is described in Ref. [141], which furthermore presents results based
on the analysis of 2017 data, which has been updated for the full Run 1
(2016–2018) dataset in Ref. [67]. Based on this experience, a sensitivity
projection for HIKE Phase 1 has been performed assuming a 40-fold increase in
the size of the data sample with respect to NA62 Run 1.
The principal background comes from the SM $K^{+}\to\pi^{+}\nu\bar{\nu}$
decay, and other backgrounds are negligible in comparison. The selection
acceptance and efficiency are assumed to be consistent with the results
obtained for the NA62 2018 data [67]. In this way realistic projections are
performed, accounting for all first-order effects. The results are shown in
Fig. 16 for the excluded regions in the interpretations of $X_{\rm inv}$ as
either a dark scalar mixing with the Higgs boson or an ALP with fermionic
couplings. Additional specialisation of the selection used for the
$K^{+}\to\pi^{+}X_{\rm inv}$ decays might be possible, further enhancing
sensitivity. HIKE Phase 3 sensitivity projections for the search for the
$K_{L}\to\pi^{0}X_{\rm inv}$ decay, refined with respect to Ref. [20], are
also shown in Fig. 16. HIKE Phases 1 and 3 provide complementary
sensitivities, allowing for exclusion of, or discovery within, new phase-space
regions which are otherwise difficult to cover.
Figure 16: Left: excluded regions at 90% CL of the $(m_{S},\sin^{2}\theta)$
parameter space for a dark scalar, $S$, of the BC4 model [20]. Right: excluded
regions of the parameter space $(m_{a},g_{Y})$ for an ALP, $a$, of the BC10
model [20]. The exclusion bound from analysis of NA62 Run 1 data [141, 142,
67] is shown in purple; the projected exclusion for HIKE Phase 1 and Phase 3
(KLEVER, assuming invisible $S$ or $a$ and updated from Ref. [20]) is shown in
red and light blue, respectively. Other exclusion bounds are shown in grey,
including results from E949 [164], CHARM [165, 166], NA48/2 [167], LHCb [168,
169], Belle [170], $K_{\mu 2}$ experiment [171], CLEO [172], KTEV [76],
MicroBooNE [173].
#### 6.2.4 Heavy neutral lepton production in $K^{+}\to\ell^{+}N$ decays
Searches for heavy neutral lepton (HNL) production in $K^{+}\to\ell^{+}N$
decays have been well established by the NA62 experiment using the main
$K^{+}\to\pi^{+}\nu\bar{\nu}$ trigger chain for the $K^{+}\to e^{+}N$ case,
and a downscaled control trigger chain for the $K^{+}\to\mu^{+}N$ case. NA62
has published world-leading exclusion on the HNL mixing parameters $|U_{\ell
4}|^{2}$ over much of the accessible mass range of 144–462 MeV/$c^{2}$ with
the Run 1 dataset [143, 144]. Both searches are limited by background. In
particular, the $K^{+}\to\mu^{+}\nu$ decay followed by $\mu^{+}\to
e^{+}\nu\bar{\nu}$ decay in flight, and the $\pi^{+}\to e^{+}\nu$ decay of the
pions in the unseparated beam, represent irreducible backgrounds to the
$K^{+}\to e^{+}N$ process. The peaking nature of the $K^{+}\to\ell^{+}N$
signal in terms of the reconstructed missing mass allows for data-driven
background evaluation, reducing the systematic uncertainties in the background
estimates.
HIKE Phase 1 sensitivity projections for HNL production searches in
$K^{+}\to\ell^{+}N$ decays, obtained from a detailed analysis assuming
NA62-like trigger chains and conditions, are shown in Fig. 17. HIKE Phase 1
offers world-leading sensitivity to $|U_{e4}|^{2}$ in the region
$m_{N}>140~{}{\rm MeV}/c^{2}$, approaching the seesaw bound and complementing
future long-baseline neutrino experiments [139], and is expected to improve on
the state-of-the-art for lower HNL masses via searches for the
$K^{+}\to\pi^{0}e^{+}N$ [174] and $\pi^{+}\to e^{+}N$ decays. HIKE Phase 1
also provides competitive sensitivity to $|U_{\mu 4}|^{2}$. The projection for
$|U_{\mu 4}|^{2}$ assumes data collection with a highly-downscaled control
trigger, and may improve by an order of magnitude in case a software trigger
based on the streaming readout (Section 11.2) is employed.
Figure 17: Summary of the upper limits at 90% CL of the HNL mixing parameters
$|U_{e4}|^{2}$ and $|U_{\mu 4}|^{2}$ obtained from production searches, and
HIKE Phase 1 sensitivity with $K^{+}\to\ell^{+}N$ decays. HIKE projection for
$|U_{\mu 4}|^{2}$ assumes data collection with a highly-downscaled control
trigger, and may improve by an order of magnitude in case a software trigger
is employed.
#### 6.2.5 Other measurements
HIKE Phase 1 is expected to increase the world samples of many rare $K^{+}$
decays by an order of magnitude, developing upon the trigger strategies
developed for the NA62 experiment. This would bring many rare decay
measurements to a new level of precision. In particular, we expect to collect
background-free samples of several times $10^{5}$ events of both
$K^{+}\to\pi^{+}e^{+}e^{-}$ and $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decays,
leading to a powerful test of lepton universality (LFU) by comparing the form-
factor parameters ($a_{+}$, $b_{+}$) of the two decay modes. This is of
particular interest in the framework of the global NP fit [85] to the kaon
decay data (Section 3.1.5, Fig. 2). Considering that the precision of the
recent NA62 measurement in the muon channel [80] is limited by the size of the
dataset, HIKE Phase 1 is expected to measure the difference of the form-factor
parameter $a_{+}$ between the two decay modes, $\Delta a_{+}^{e\mu}$, to a
precision of $\pm 0.007$, and the corresponding difference $\Delta
b_{+}^{e\mu}$ to a precision of $\pm 0.015$. Exploitation of the correlations
between the measured $a_{+}$ and $b_{+}$ parameters in each mode will provide
additional power in terms of LFU tests.
In terms of searches for $K^{+}$ and $\pi^{0}$ decays violating lepton number
or flavour conservation, including
$K^{+}\to\pi^{-}(\pi^{0})\ell_{1}^{+}\ell_{2}^{+}$,
$K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$,
$K^{+}\to\ell_{1}^{-}\nu\ell_{2}^{+}\ell_{2}^{+}$ and
$\pi^{0}\to\mu^{\pm}e^{\mp}$, the experimental technique based on dedicated
di-lepton trigger chains has been firmly established by the NA62 experiment,
leading to world-leading upper limits of ${\cal O}(10^{-11})$ on the branching
ratios of a number of processes with the NA62 Run 1 dataset [119, 120, 121].
These searches are not limited by background, and HIKE Phase 1 sensitivity is
expected to improve in the future almost linearly with the size of the dataset
to the ${\cal O}(10^{-12})$ level.
## 7 Phase 2: a multi-purpose $K_{L}$ decay experiment
### 7.1 Experimental layout
The baseline design of the multi-purpose $K_{L}$ decay experiment includes a
120 m long neutral beamline with the secondary beam opening angle of 0.4 mrad,
proposed and studied extensively for the KLEVER phase of the HIKE programme.
The beamline involves four stages of collimation, and provides adequate
suppression of the short-lived $K_{S}$ and $\Lambda$ components. A detailed
description of the beamline, including the expected $K_{L}$ yields and
momentum spectra, is provided in Section 5.3 Unlike the KLEVER phase, HIKE
Phase 2 detector will be equipped with a tracker, which offers the opportunity
to perform characterisation of the KLEVER beam. For the forthcoming proposal,
alternative beamline configurations can be studied.
The proposed $K_{L}$ production angle is 2.4 mrad (to be compared to 8 mrad
for the KLEVER phase), which improves the $K_{L}$ yield and increases the mean
$K_{L}$ momentum (Figs. 8, 10) therefore improving the acceptances for
$K_{L}\to(\pi^{0})\ell^{+}\ell^{-}$ decays. Thanks to the relatively compact
detector, HIKE Phase 2 allows for a 90 m long fiducial decay volume to be
accommodated in the present ECN3 experimental hall, and no major civil
engineering work is required in preparation for this phase of the programme.
It is proposed to use the HIKE Phase 1 experimental setup with minimal
modifications (Fig. 18). The GTK, KTAG, RICH and SAC detectors will be
removed, the STRAW spectrometer will be shortened to a total length of 25 m,
and central holes of the STRAW chambers will be realigned on the neutral beam
axis (Section 10.2.3). Reduction of the magnetic field of the spectrometer
dipole magnet by about 20%, leading to a momentum kick of 210 MeV/$c$, is
possible without significant degradation of the mass resolution.
HIKE Phase 2 ($K_{L}$)
Figure 18: HIKE Phase 2 layout, with an aspect ratio of 1:10.
### 7.2 Physics sensitivity
The expected $K_{L}$ yield in the beam is $5.4\times 10^{-5}$ per proton on
target, and the assumed integrated proton flux of $1.2\times 10^{19}$ pot/year
leads to the number of $K_{L}$ decays in the decay volume of $3.8\times
10^{13}$/year. The mean momentum of $K_{L}$ mesons entering the decay volume
is 79 GeV/$c$, while the mean momentum of decaying $K_{L}$ mesons is 46
GeV/$c$ (Table 3, Fig. 8). The expected signal yields for several principal
rare and forbidden $K_{L}$ decays have been evaluated using a full
Geant4-based simulation, reconstruction and analysis chain within the flexible
HIKE offline software platform being developed on the basis of the NA62
software (Section 12). The results are summarised in Table 6, and acceptances
in bins of the longitudinal coordinate of the decay vertex and $K_{L}$
momentum are shown in Fig. 19.
Table 6: HIKE Phase 2 sensitivity estimate for the principal rare and forbidden $K_{L}$ decays: assumed branching ratios (see Section 3.1.2 for details), acceptances and expected signal yields in five years of data taking. Mode | Assumed branching ratio | Acceptance | Signal yield in five years
---|---|---|---
$K_{L}\to\pi^{0}e^{+}e^{-}$ | $3.5\times 10^{-11}$ | 2.1% | 140
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$ | $1.4\times 10^{-11}$ | 6.0% | 160
$K_{L}\to\mu^{+}\mu^{-}$ | $7\times 10^{-9}$ | 17% | $2.3\times 10^{5}$
$K_{L}\to\mu^{\pm}e^{\mp}$ | – | 16% | –
Figure 19: Acceptances for rare $K_{L}$ decays for HIKE Phase 2, in bins of
the decay vertex longitudinal coordinate (left) and $K_{L}$ momentum (right).
Momentum distribution of the $K_{L}$ mesons decaying in the decay volume is
shown by a continuous curve in the right panel.
For the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays, the expected SM Dalitz plot
distributions [72] are used in the simulation, and a selection condition
$m_{ee}>140~{}{\rm MeV}/c^{2}$ is applied in the $K_{L}\to\pi^{0}e^{+}e^{-}$
case as required to suppress backgrounds from major $K_{L}$ decays into
neutral pions followed by $\pi^{0}\to\gamma e^{+}e^{-}$ decays. The resulting
single-event sensitivities for $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays (Table
6) improve by more than two orders of magnitude on the previous searches at
the kTeV experiment [75, 76]. Suppression of the
$K_{L}\to\gamma\gamma\ell^{+}\ell^{-}$ background, which represents a
significant limitation especially in the $K_{L}\to\pi^{0}e^{+}e^{-}$ case
[77], relies on the good photon energy resolution provided by the EM
calorimeter. The current NA62 LKr calorimeter [175] would provide a 2.2 MeV
di-photon mass resolution for the proposed HIKE Phase 2 setup. Further
potential backgrounds to $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays come from
major $K_{L}$ decays such as $K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ with $\pi^{\pm}$
misidentification or decay in flight, and from pileup effects.
Background estimation for the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays is
currently in progress, however the extensive experience of the NA62 lepton
flavour/number violation programme in terms of background reduction [119, 120,
121] suggests that the proposed experimental technique allows for background
suppression to the level of ${\cal O}(10^{-11})$ for rare and forbidden decays
into multiple tracks and photons, for a well-collimated beam. Considering the
expected signal yields in excess of 100 events (Table 6), we conclude that the
first observation of the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays is likely,
and a sensitivity at the SM branching ratio level of ${\cal O}(10^{-11})$ can
certainly be reached. Detailed HIKE Phase 2 sensitivity studies are in
progress for the forthcoming proposal.
The expected $K_{L}\to\mu^{+}\mu^{-}$ signal yield suggests a 0.2% statistical
precision on the measurement of the decay branching ratio. We expect that
overall a precision better than 1% is attainable when normalising to the
$K_{L}\to\pi^{+}\pi^{-}$ decay, which improves on the previous measurement by
the BNL-E871 experiment [87].
HIKE Phase 2 offers sensitivities of ${\cal O}(10^{-12})$ for branching ratios
of a broad range of rare and forbidden $K_{L}$ decay modes. As an example, the
acceptance for the lepton flavour violating $K_{L}\to\mu^{\pm}e^{\mp}$ decays
(Table 6) leads to expected upper limits of both decay branching ratios of
$8\times 10^{-14}$ at 90% CL in five years of operation (the null hypothesis
and assuming low background), improving the limit by a factor of 60 on the
previous search by the BNL-E871 experiment [126].
## 8 Phase 3 (KLEVER): measurement of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay
Exploratory work on an experiment to measure ${\cal
B}(K_{L}\to\pi^{0}\nu\bar{\nu})$ in a future stage of the ECN3 high-intensity
kaon physics programme began even before the start of NA62 data taking. The
basic design work for the KLEVER experiment ($K_{L}$ Experiment for VEry Rare
events) was carried out as a part of the first Physics Beyond Colliders
initiative at CERN, in preparation for the 2020 update of the European
Strategy for Particle Physics [176].
As in the other phases of the programme, KLEVER makes use of the 400-GeV
primary proton beam from the SPS, slow extracted and transported to the T10
target (Section 5.1) at an intensity of $2\times 10^{13}$ protons per pulse
(ppp), corresponding to about six times the nominal NA62 intensity. The
neutral secondary beam for KLEVER is derived at an angle of 8 mrad; the
beamline design is described in Section 5.3. In practice, the choice of
production angle has been optimised together with the limits of the fiducial
volume (FV) to maximise the signal sensitivity and minimise
$K_{L}\to\pi^{0}\pi^{0}$ and $\Lambda\to n\pi^{0}$ backgrounds. For a
production angle of 8 mrad, the neutral beam has a mean $K_{L}$ momentum of 40
GeV, so that 5% of $K_{L}$ decay inside an FV extending from 130 m to 190 m
downstream of the target. The $K_{L}$ mesons decaying inside the FV have a
mean momentum of 27 GeV/$c$. The boost from the high-energy beam facilitates
the rejection of background channels such as $K_{L}\to\pi^{0}\pi^{0}$ by
detection of the additional photons in the final state. On the other hand, the
layout poses particular challenges for the design of the small-angle vetoes,
which must reject photons from $K_{L}$ decays escaping through the beam exit
amidst an intense background from soft photons and neutrons in the beam.
Background from $\Lambda\to n\pi^{0}$ decays in the beam must also be kept
under control.
The KLEVER goal is to achieve a sensitivity of about 60 events for the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay at the SM BR with an $S/B=1$. At the SM
BR, this would correspond to a relative uncertainty of about 20%. We would
expect to be able to observe a discrepancy with SM predictions with $5\sigma$
significance if the true BR is a bit more than twice or less than one-quarter
of the SM BR, or with $3\sigma$ significance if the true BR is less than half
of the SM rate. As noted in Section 5.3, with an opening angle of 0.4 mrad,
the $K_{L}$ yield in the beam is $2.1\times 10^{-5}$ per proton on target
(pot). With a fiducial-volume acceptance of 5% and a selection efficiency of
3%, collection of 60 SM events would require a total primary flux of $6\times
10^{19}$ pot, corresponding to five years of running at an intensity of
$2\times 10^{13}$ ppp under NA62-like slow-extraction conditions with 3000
spills per day for 200 days per year.
The scheduling of the KLEVER phase is influenced by two considerations.
Firstly, to provide maximum protection from $\Lambda\to n\pi^{0}$ background,
the beamline would need to be lengthened by 150 m (Section 5.3.3). Secondly,
despite the efforts to benchmark the KLEVER beam simulation by comparison with
existing data on inclusive particle production by protons on lightweight
targets at SPS energies (Section 5.3.1), confidence in the simulations and
background estimates would be greatly increased by the possibility to acquire
particle production data and measure inclusive rates in the KLEVER beamline
setup with a tracking experiment in place. This would be possible in HIKE
Phase 2 (Section 7). Therefore, although in principle KLEVER could aim to
start data taking after LS4 (for which injector physics is currently foreseen
to begin in early 2034), the most natural timescale might envisage KLEVER as
running after Phase 2. We are studying scheduling options to allow KLEVER data
taking to begin before LS5 (foreseen to start in 2038).
### 8.1 Experimental layout
The KLEVER setup largely consists of a collection of high-efficiency photon
detectors arranged around a 160-m-long vacuum volume to guarantee hermetic
coverage for photons from $K_{L}$ decays emitted at polar angles out to 100
mrad and to provide a nearly free path through vacuum up to the main
electromagnetic calorimeter (MEC) for photons emitted into a cone of at least
7.5 mrad. The fiducial volume (FV) spans about 60 m just downstream of the
active final collimator (AFC), but the photon veto coverage extends along the
entire length up to the MEC. The layout of the detector elements is
schematically illustrated in Fig. 20.
HIKE Phase 3 (KLEVER)
Figure 20: HIKE Phase 3 (KLEVER) layout, with an aspect ratio of 1:10. The
baseline configuration, without the extension of the beamline by 150 m is
shown.
The largest elements are about 3 m in diameter. The beginning of the vacuum
volume is immediately downstream of the cleaning collimator at $z=80$ m, i.e.,
80 m downstream of the T10 target. A 40-m vacuum decay region allows the
upstream veto calorimeter (UV) surrounding the AFC to have an unobstructed
view for the rejection of $K_{L}$ decays occurring upstream of the detector
volume. The UV and AFC at $z=120$ m define the start of the detector volume,
which is lined with 25 large-angle photon and charged-particle veto stations
(LAV) in five different sizes, placed at intervals of 4 to 6 m to guarantee
hermeticity for decay particles with polar angles out to 100 mrad. The MEC, at
the downstream end of the vacuum volume, replaces the NA48 LKr calorimeter
used in NA62: it reconstructs the $\pi^{0}$ vertex for signal events and helps
to reject events with extra photons. A charged-particle veto detector (CPV) in
front of the MEC rejects $K_{L}\to\pi^{\pm}e^{\mp}\nu$ and
$K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ backgrounds, and a preshower detector (PSD)
allows reconstruction of the angles of incidence for photons, providing
additional constraints on signal candidates even if only one photon converts.
The small-angle vetoes on the downstream side of the MEC intercept photons
from $K_{L}$ decays that pass through the beam pipe. The small-angle
calorimeter (SAC) intercepts the neutral beam; its angular coverage as seen
from the downstream end of the FV extends to $\pm 2$ mrad. Because of the high
neutron and photon rate in the beam, the SAC design is one of the most
challenging aspects of the experiment. The intermediate-ring calorimeter (IRC)
is a ring-shaped detector between the SAC and MEC and intercepts photons from
downstream decays that make it through the calorimeter bore at slightly larger
angles. In addition to the photon vetoes, the experiment makes use of hadronic
calorimeters downstream of the MEC, to help reject background from hadron
interactions and the copious $K_{L}$ decays into charged particles such as
muons and pions.
Because of the experimental challenges involved in the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ measurement, and in particular, the very high
efficiency required for the photon veto systems, the performance requirements
for KLEVER drive the specifications for many of the new detectors for HIKE. In
this case, the LAVs, MEC, IRC, SAC, and hadronic calorimeters built for Phases
1 and 2 would all be able to be used for KLEVER in Phase 3. The UV, CPV, and
PSD would have to be built new for KLEVER, possibly with recycled components
from the analogous detectors for the earlier phases.
In the baseline design with the original 120 m beamline, the FV covers the
region $130~{}{\rm m}<z<190$ m. The positioning of the FV significantly
upstream of the calorimeter, together with the relatively high $K_{L}$
momentum, is key to obtaining sufficient $K_{L}\to\pi^{0}\pi^{0}$ background
rejection with the ring-shaped LAV geometry covering polar angles out to 100
mrad. This comes at a cost in acceptance for signal decays, which increases
significantly as the FV is moved closer to the calorimeter. Extending the
downstream limit of the FV would increase the sensitivity of the experiment
but will require improvements in background rejection; further optimisation
along these lines is under study.
Fig. 20 shows the experimental configuration without a lengthened beamline to
suppress background from $\Lambda$ decays. Lengthening the beamline would
change the distance from the cleaning collimator to the AFC, so that the
cleaning collimator would be upstream of the start of the vacuum tank instead
of right at its entrance. The detector configuration would be otherwise
unchanged.
### 8.2 Rates and timing performance
Table 7: Rates for events with hits on KLEVER detectors, by detector system (left) and for certain event classes (right), evaluated for the 120 m beamline configuration. Detector | Event rate (MHz) | Event class | Rate (MHz)
---|---|---|---
AFC | 2.3 | Exactly 2 hits on MEC | 4.8
UV | 7.1 | Exactly 2 photons on MEC | 1.0
LAV | 14 | 2 hits on MEC with UV, LAV veto | 3.1
MEC | 18 | 2 hits on MEC, no other hits | 0.007
IRC | 22 | |
SAC | 95 | |
The FLUKA simulation of the beamline performed by the Conventional Beams
Working Group (Section 5.3) contains an idealised representation of the
experimental setup for the purposes of evaluating rates on the detectors, both
from the decays of $K_{L}$ mesons and from the beam halo. The estimated rates
by detector system are listed in Table 7, assuming a primary intensity of
$2\times 10^{13}$ ppp and a (pessimistic) 3 s effective spill. The simulation
was performed for the standard, 120 m beamline (without extension). The rates
are obtained with a geometrical representation of the detector volumes and
assumed probabilities for hadrons to register signals. For the AFC, UV, LAV,
and MEC, the probability for registering signals is assumed to be 25% for
neutrons and $K_{L}$ mesons, and 100% for all other particles. For the IRC and
SAC, the assumed probability is 10% for neutral hadrons, and 100% for all
other particles. The particle tracking threshold is 1 GeV. Since a particle
detected in any of the detectors would effectively constitute a veto (the
rates of signal candidates being negligible for these purposes), this provides
an estimate of the total veto rate: 134 MHz, of which 104 MHz from events with
hits on a single detector and 30 MHz from events with hits on multiple
detectors. The total rate is dominated by the rate of interactions of beam
particles in the SAC, consisting of 13 MHz of hits from $K_{L}$ mesons, 44 MHz
of hits from neutrons, and 40 MHz of hits from beamline photons with $E>5$
GeV. Hadronic interactions can be efficiently recognised offline and we assume
only 10% of the corresponding rate contributes to the inefficiency from
accidental coincidence. The remaining veto rate from accidentals is then at
most 100 MHz, which we take as a figure of merit. The event time is obtained
from the $\pi^{0}$ candidate reconstructed in the MEC, and the accidental
coincidence rate is dominated by events on the SAC. Assuming a $\pm
5\sigma_{t}$ coincidence window, to limit the inefficiency from accidental
coincidence to below 25%, the time resolution $\sigma_{t}$ on the coincidence
must be better than 250 ps, and the detector design must minimise any
contributions from non-gaussian fluctuations to the timing distribution. These
considerations lead to the specification that the nominal time resolution for
the most critical detectors (the MEC and the SAC) must be better than 100 ps.
As noted above, the full FLUKA simulation was performed with the original, 120
m beamline configuration. Many of the rates in Table 7 would be expected to
decrease by the ratio of beam solid angle, i.e., by a factor of 0.41, when the
beamline is extended. This scaling may not be exact for the upstream detectors
(AFC, UV, LAV), as the rates on those detectors are determined in part by the
details of the collimation and sweeping scheme, but is is certainly true for
the downstream detectors, and in particular, for the SAC, which drives the
random veto rate for the experiment. Thus, the downstream extension not only
provides a significant general decrease in the rates on the detectors, but
would lower the estimated random veto rate from 25% to 10%.
The rates for certain classes of events to be acquired are also listed in
Table 7: for events with exactly two hits and with exactly two photons on the
calorimeter, independently of the other detectors; for events with two hits on
the calorimeter without hits on the UV or LAV (presumed to be in online veto);
and for events with two hits on the calorimeter and no hits on any of the
other detectors. These are estimates of the rates for the physical
events—there is no simulation of the detector response apart from the
efficiency assumptions outlined above—but these results do suggest that the
total rates for event classes that would approximately correspond to level-0
trigger conditions in NA62 are in fact of about the same order of magnitude as
in NA62.
Figure 21: Radial distribution of particle fluxes at the MEC for KLEVER, from
FLUKA beamline simulation.
Fig. 21 shows the radial distribution of the particle fluxes at the front face
of the MEC, as estimated with the FLUKA simulation of the neutral beam. The
beam halo from photons and neutral hadrons drops off rapidly for $10<r<20$ cm,
with $r$ the radial distance from the beam axis. For neutrons, the halo-to-
core ratio decreases from $2.5\times 10^{-2}$ to $8.8\times 10^{-5}$ over this
interval, corresponding to a decrease in the total neutron rate in the halo
from 12 MHz to 43 kHz. For $r>20$ cm, the neutral hadron fluxes are on the
order of a few Hz/cm2. At $r=20$ cm, there are 1 kHz/cm2 of photons and
charged pions and a few hundred Hz/cm2 of muons, dropping off more or less
linearly with radius.
Once the extended beamline layout is finalised, it will be necessary to re-run
the FLUKA-based simulation to obtain definitive estimates of the beam flux,
halo, and veto rates in the detectors.
### 8.3 Expected performance for $K_{L}\to\pi^{0}\nu\bar{\nu}$
Simulations of the experiment in the standard configuration carried out with
fast-simulation techniques (idealised geometry, parameterised detector
response, etc.) suggest that the target sensitivity (60 SM events with
$S/B=1$) is achievable. However, due to the the reduction in the beam solid
angle, the $K_{L}$ flux is decreased by 60% when the beamline is extended by
150 m. We are exploring options to increase the acceptance for signal decays.
The acceptance for $K_{L}\to\pi^{0}\nu\bar{\nu}$ decays in the fiducial volume
is only a few percent due to the cuts used to reject background, primarily
from $K_{L}\to\pi^{0}\pi^{0}$ decays, so increasing the signal acceptance is
closely related to optimising the background rejection.
#### 8.3.1 Signal selection and rejection of $K_{L}$ background
The basic criteria for signal selection are as follows:
* •
Events with exactly two photon clusters on the MEC and no other activity in
the detector, including in any of the photon vetoes, are selected.
* •
Assuming that the two photons have invariant mass equal to $m_{\pi^{0}}$ and
that the $K_{L}$ decays on the nominal beam axis ($r=0$), the position of the
decay vertex $z_{\rm rec}$ is calculated from the distance between the photon
clusters on the MEC. The vertex is required to lie inside the FV.
* •
The requirement that both photons must have radius $r>r_{\rm min}=35$ cm from
the beam axis helps to increase the rejection for events with overlapping
clusters, provides a safeguard against background from
$K_{L}\to\pi^{0}\pi^{0}$ decays upstream of the AFC, and helps to reject
residual $\Lambda\to n\pi^{0}$ decays.
* •
A specific background topology with two photons at large angle from an odd-
paired $K_{L}\to\pi^{0}\pi^{0}$ decaying just upstream of the MEC but falsely
reconstructing in the FV is rejected by the requirement that the photon of
lower energy have $E_{\rm min}>2~{}{\rm GeV}/r_{E_{\rm\min}}$, where
$r_{E_{\rm min}}$ refers to the photon with $E=E_{\rm min}$.
* •
Signal events are required to have $p_{\perp\,{\rm min}}>140$ MeV, where
$p_{\perp}$ refers to the $\pi^{0}$ as reconstructed by the MEC.
* •
At least one photon is required to convert in the PSD, allowing reconstruction
of the vertex position and transverse momentum with relaxed assumptions on the
radial position of the decaying $K_{L}$. The vertex position $z_{\rm PSD}$
must be upstream of the downstream edge of the FV and the transverse momentum
$p_{\perp\,{\rm PSD}}$ must be greater than $p_{\perp\,{\rm min}}$.
Table 8 shows the signal acceptance from the fast simulation. The numbers of
events at each stage of selection when the above cuts are sequentially applied
are listed, both for the original 120 m and extended 270 m beamline
configurations, assuming ${\cal B}(K_{L}\to\pi^{0}\nu\bar{\nu})=3\times
10^{-11}$. Note that for the extended beamline configuration, the FV extends
10 m further downstream than it does for the original configuration. Plots of
the distributions of the events satisfying these minimal criteria in the plane
of $p_{\perp}$ vs $z_{\rm rec}$ are shown in Fig. 22, for the extended
beamline configuration only. The two panels show the distributions for all
events with two photons on the MEC, and with the addition of the $r_{\rm
min}$, $E_{\rm min}$, and PSD requirements. Two features are apparent.
Firstly, the acceptance for signal events increases dramatically in proximity
to the calorimeter: the majority of the events with two photons on the MEC
actually lie downstream of the FV. Secondly, the $p_{\perp}$ resolution
degrades rapidly downstream of the FV, due to the error in $p_{\perp}$
reconstruction resulting from ignoring the radial displacement of the decaying
$K_{L}$. Note finally that the PSD cuts partially enforce the definition of
the signal box, which explains the shape of the distribution in Fig. 22
(right). For events passing the FV and cluster reconstruction cuts, the
$p_{\perp}<140$ MeV selection retains 60% of signal events.
Table 8: Estimated sensitivity for $K_{L}\to\pi^{0}\nu\bar{\nu}$. | Stage | Acceptance | Cumulative acceptance | Total events
---|---|---|---|---
120 m beamline, 60 m fiducial volume (130–190 m)
| Produced $K_{L}\to\pi^{0}\nu\bar{\nu}$ | 1 | 1 | 37800
| Decay in FV | 5.93% | 5.93% | 2240
| $2\gamma$ in MEC | 2.34% | 2.34% | 884
| \+ reconstructed in FV | 0.205 | $7.13\times 10^{-3}$ | 269
| \+ $r_{\rm min}>35$ cm | 0.504 | $3.60\times 10^{-3}$ | 136
| \+ $E_{\rm min}$ cut | 0.911 | $3.28\times 10^{-3}$ | 124
| \+ $p_{\perp}>140$ MeV | 0.568 | $1.86\times 10^{-3}$ | $70.4$
| \+ $(z,p_{\perp})_{\rm PSD}$ cuts with $0.5X_{0}$ | 0.469 | $0.874\times 10^{-3}$ | $33.0$
| or with $1.0X_{0}$ | 0.684 | $1.28\times 10^{-3}$ | $48.2$
270 m beamline, 70 m fiducial volume (280–350 m)
| Produced $K_{L}\to\pi^{0}\nu\bar{\nu}$ | 1 | 1 | 15483
| Decay in FV | 5.07% | 5.07% | 785
| $2\gamma$ in MEC | 1.95% | 1.95% | 302
| \+ reconstructed in FV | 0.429 | $8.38\times 10^{-3}$ | 130
| \+ $r_{\rm min}>35$ cm | 0.474 | $3.97\times 10^{-3}$ | 61.5
| \+ $E_{\rm min}$ cut | 0.867 | $3.44\times 10^{-3}$ | 53.3
| \+ $p_{\perp}>140$ MeV | 0.595 | $2.05\times 10^{-3}$ | 31.7
| \+ $(z,p_{\perp})_{\rm PSD}$ cuts with $0.5X_{0}$ | 0.479 | $0.982\times 10^{-3}$ | 15.2
| or with $1.0X_{0}$ | 0.699 | $1.43\times 10^{-3}$ | 22.2
Figure 22: Distribution of $K_{L}\to\pi^{0}\nu\bar{\nu}$ events in the
$(z_{\rm rec},p_{\perp})$ plane, from fast MC simulation of the configuration
with beamline extension, for all events with two photons on the MEC (left),
and after $r_{\rm min}$, $E_{\rm min}$, and preshower cuts (right).
In analogy to Table 8, Table 9 shows the number of $K_{L}\to\pi^{0}\pi^{0}$
events surviving the various signal selection cuts. Here, it is useful to
distinguish between events with the two detected photons from the same
$\pi^{0}$ (even pairings), with the two photons from different $\pi^{0}$’s
(odd pairings), and with at least one cluster containing overlapping photons
(fused clusters). As is readily seen from the left panels of Fig. 23, the
background from $\pi^{0}\pi^{0}$ events from all three categories is
concentrated in the proximity of the MEC, even more so than for signal events.
This arises from veto inefficiencies for high-angle, low-energy photons from
$K_{L}$ decays close to the MEC, in addition to the geometrical effect also
present for signal events. The background for odd events extends further
upstream into the FV than for the other categories, because of the incorrect
assumption that the two photon clusters come from a single $\pi^{0}$. These
events are rejected effectively by the PSD. Differently to Fig. 22, the right
panels of Fig. 23 show the distributions without the application of the
preshower cuts. This is because, in the case of the background, the cuts on
$(z,p_{\perp})_{\rm PSD}$ are almost as effective as those on $(z_{\rm
rec},p_{\perp})$ from the calorimeter.
Table 9: Estimated background from $K_{L}\to\pi^{0}\pi^{0}$. | Stage | Total | Even | Odd | Fused
---|---|---|---|---|---
120 m beamline, 60 m fiducial volume (130–190 m)
| Produced $K_{L}\to\pi^{0}\pi^{0}$ | $1.089\times 10^{12}$
| Decay in FV | $5.93\%=64.5\times 10^{9}$
| $2\gamma$ in MEC | $3.83\times 10^{8}$ | $0.94\times 10^{8}$ | $2.64\times 10^{8}$ | $0.26\times 10^{8}$
| \+ reconstructed in FV | $8.40\times 10^{6}$ | $0.70\times 10^{6}$ | $6.70\times 10^{6}$ | $1.00\times 10^{6}$
| \+ $r_{\rm min},E_{\rm min}$ cuts | $4.34\times 10^{6}$ | $0.33\times 10^{6}$ | $3.84\times 10^{6}$ | $0.18\times 10^{6}$
| \+ $p_{\perp}>140$ MeV | $261\pm 16$ | $154\pm 12$ | $86\pm 9$ | $21\pm 5$
| \+ $(z,p_{\perp})_{\rm PSD}$ cuts with $0.5X_{0}$ | $80\pm 9$ | $61\pm 8$ | $11\pm 3$ | $8\pm 3$
| or with $1.0X_{0}$ | $117\pm 13$ | $89\pm 11$ | $16\pm 5$ | $12\pm 4$
270 m beamline, 70 m fiducial volume (280–350 m)
| Produced $K_{L}\to\pi^{0}\pi^{0}$ | $0.446\times 10^{12}$
| Decay in FV | $5.07\%=22.6\times 10^{9}$
| $2\gamma$ in MEC | $9.45\times 10^{7}$ | $2.02\times 10^{7}$ | $6.71\times 10^{7}$ | $0.72\times 10^{7}$
| \+ reconstructed in FV | $5.12\times 10^{6}$ | $0.48\times 10^{6}$ | $4.07\times 10^{6}$ | $0.58\times 10^{6}$
| \+ $r_{\rm min},E_{\rm min}$ cuts | $21.3\times 10^{5}$ | $2.31\times 10^{5}$ | $18.3\times 10^{5}$ | $0.71\times 10^{5}$
| \+ $p_{\perp}>140$ MeV | $234\pm 15$ | $175\pm 13$ | $48\pm 7$ | $11\pm 3$
| \+ $(z,p_{\perp})_{\rm PSD}$ cuts with $0.5X_{0}$ | $70\pm 8$ | $59\pm 8$ | $7\pm 3$ | $4\pm 2$
| or with $1.0X_{0}$ | $102\pm 12$ | $86\pm 11$ | $10\pm 4$ | $6\pm 3$
Figure 23: Distribution of $K_{L}\to\pi^{0}\pi^{0}$ events in the $(z_{\rm
rec},p_{\perp})$ plane, from fast MC simulation of the configuration with
beamline extension, for all events with two photons on the MEC (left column),
and after $r_{\rm min}$ and $E_{\rm min}$ cuts (right column). The preshower
cuts are not applied. From top to bottom, rows show events with the two
photons from the same $\pi^{0}$ (even), with the two photos from different
$\pi^{0}$’s (odd), and with at least one cluster containing overlapping
photons (fused).
To ensure that it does not interfere with the detection of low-energy photons
by the MEC, the PSD radiator is thin, and the PSD cuts, although effective at
suppressing background, penalize the signal acceptance because of the
requirement for at least one photon to convert. With a $1X_{0}$ radiator, the
acceptance is increased by about 50% with respect to that obtained with a
$0.5X_{0}$ radiator, but the optimum thickness will require more detailed
simulations to determine. Tables 8 and 9 thus give the final expected numbers
of signal and background events in both hypotheses.
For the original 120 m beamline configuration, assuming a $1X_{0}$ radiator,
Tables 8 and 9 give 48.2 expected signal events and $117\pm 13$ background
events from $K_{L}\to\pi^{0}\pi^{0}$. For the case of the extended beamline,
22.2 signal events are expected after accounting for the reduction of the
$K_{L}$ flux, and $102\pm 12$ background events, where the increased
background results mainly from extending the fiducial volume by an additional
10 m in the downstream direction to partially make up for the reduced $K_{L}$
flux. This scheme is far from final. We are continuing to investigate
potential improvements to both the experiment and the analysis to increase the
sensitivity, reduce the background, and increase redundancy, including the
following:
* •
Recovering sensitivity for signal events by further extending the FV in the
downstream direction. As noted above, this will require improved background
rejection.
* •
Making better use of information from the PSD. In particular, a kinematic fit
to combine information from calorimeter and PSD shows promise for improving
the $(z,p_{\perp})$ reconstruction in the downstream region, where the angles
of incidence are greatest and the $p_{\perp}$ reconstruction is poorest, but
where the potential for recovery of signal events is highest.
* •
Continuing to improve the effectiveness of the photon veto systems, especially
in the downstream region. At the moment, we assume that the MEC gives no
response for photons with $E<100$ MeV (Section 10.3.3.2). If this threshold
can be lowered, which is likely with the design proposed for the MEC, the
populations of events in the downstream regions seen in Fig. 23 will be
significantly decreased.
* •
Use of multivariate analysis techniques to efficiently combine the event
selection criteria. While efforts have been made in this direction, current
attention is on development of the full simulation, which is prerequisite to
establishing a definitive search strategy.
#### 8.3.2 $\Lambda\to n\pi^{0}$ background and beamline extension
As noted in Section 5.3.3, additional measures must be taken to ensure
sufficient suppression of background from $\Lambda\to n\pi^{0}$ decays. The
most attractive option is to maintain the 8 mrad production angle and increase
the length of the beamline from target to AFC by 150 m.
Table 10: Estimated background from $\Lambda\to n\pi^{0}$. | Stage | Acceptance | Cumulative acceptance | Total events
---|---|---|---|---
120 m beamline, 60 m fiducial volume (130–190 m)
| Produced $\Lambda\to\ n\pi^{0}$ | 1 | 1 | $2.00\times 10^{14}$
| Decay in FV | $4.70\times 10^{-6}$ | $4.70\times 10^{-6}$ | $9.42\times 10^{8}$
| $2\gamma$ in MEC | $7.86\times 10^{-6}$ | $7.86\times 10^{-6}$ | $1.58\times 10^{9}$
| \+ reconstructed in FV | 0.351 | $2.76\times 10^{-6}$ | $5.53\times 10^{8}$
| \+ $r_{\rm min},E_{\rm min}$ cuts | 0.316 | $8.71\times 10^{-7}$ | $1.75\times 10^{8}$
| \+ $p_{\perp}>140$ MeV | $6.80\times 10^{-3}$ | $5.92\times 10^{-9}$ | ($1.19\pm 0.04)\times 10^{6}$
| \+ $(z,p_{\perp})_{\rm PSD}$ cuts with $0.5X_{0}$ | 0.132 | $7.81\times 10^{-10}$ | $(1.57\pm 0.14)\times 10^{5}$
| or with $1.0X_{0}$ | 0.193 | $1.14\times 10^{-9}$ | $(2.29\pm 0.21)\times 10^{5}$
270 m beamline, 70 m fiducial volume (280–350 m)
| Produced $\Lambda\to n\pi^{0}$ | 1 | 1 | $8.22\times 10^{13}$
| Decay in FV | $2.07\times 10^{-9}$ | $2.07\times 10^{-9}$ | $1.70\times 10^{6}$
| $2\gamma$ in MEC | $2.79\times 10^{-9}$ | $2.79\times 10^{-9}$ | $2.29\times 10^{5}$
| \+ reconstructed in FV | 0.445 | $1.24\times 10^{-9}$ | $1.02\times 10^{5}$
| \+ $r_{\rm min},E_{\min}$ cuts | 0.222 | $2.75\times 10^{-10}$ | $2.26\times 10^{4}$
| \+ $p_{\perp}>140$ MeV | 0.0207 | $5.70\times 10^{-12}$ | $468\pm 54$
| \+ $(z,p_{\perp})_{\rm PSD}$ cuts with $0.5X_{0}$ | 0.0760 | $4.33\times 10^{-13}$ | $35.6\pm 14.0$
| or with $1.0X_{0}$ | 0.111 | $6.32\times 10^{-13}$ | $52.0\pm 20.4$
Figure 24: Distribution of $\Lambda\to n\pi^{0}$ events in the $(z_{\rm
rec},p_{\perp})$ plane from fast MC simulation of the configuration with
beamline extension, for all events with two photons on the MEC (left), and
after $r_{\rm min}$, $E_{\rm min}$, and preshower cuts (right).
The effectiveness of this solution is demonstrated by Table 10, from which it
is seen that the beamline extension decreases the number of $\Lambda$ decays
with two photons on the MEC by a factor of nearly 7000 and the number passing
all cuts by a factor of more than 4000, from 229 000 to $52\pm 14$. The
centre-of-mass decay momentum of the $\pi^{0}$ in $\Lambda\to n\pi^{0}$ decay
is only 104 MeV, and even with the contribution from the neutral beam
divergence, the true value of $p_{\perp}$ is always less than 120 MeV, so the
$p_{\perp}$ and preshower cuts are particularly effective. The distributions
in $(z_{\rm rec},p_{\perp})$ are shown in Fig. 24. Before the $r_{\rm min}$,
$E_{\rm min}$, and preshower cuts, most of the $\Lambda$ decays occur at the
most upstream end of the FV: the 70 m FV spans more than 3 average decay
lengths. The distribution for the 52 surviving events is seen in the right
panel. Most of these events can be eliminated using a hadronic calorimeter at
small angle backing up the SAC to veto the high-energy neutron from the
$\Lambda\to n\pi^{0}$ decay. Because of the mass asymmetry in the decay, the
neutrons from decays in the FV that pass analysis cuts are always emitted into
the SAC acceptance and have an energy distribution with a mean of 220 GeV; 98%
of these neutrons have $E>150$ GeV. Relatively few of the beam neutrons have
energies this high (Fig. 11); a hadronic calorimeter with a threshold at
$E=150$ GeV would add less than 10 MHz to the total ${\cal O}$(100 MHz) veto
rate for the extended configuration. The concept requires full simulation, but
it seems reasonable to assume that the residual $\Lambda$ background could be
reduced by an order of magnitude or more.
#### 8.3.3 Other backgrounds
In addition to the backgrounds from $K_{L}\to\pi^{0}\pi^{0}$ and $\Lambda\to
n\pi^{0}$, there are a number other background channels that require further
advances in the simulation to properly study, and in particular, the full
incorporation of the detailed FLUKA beamline simulation into the HIKE Monte
Carlo. While mitigation of potential background contributions from these
sources might ultimately require specific modifications to the experimental
setup, we expect this task to be less complicated than dealing with the
primary challenges from $K_{L}\to\pi^{0}\pi^{0}$ and $\Lambda\to n\pi^{0}$.
##### $K_{L}\to 3\pi^{0}$
Background from $K_{L}\to 3\pi^{0}$ is not expected to be significant due to
the relative improbability of losing four photons. A preliminary study was
performed with the fast simulation for an earlier configuration of the
experiment. Essentially all of the selected events have the $K_{L}$ vertex
very near the calorimeter; the detected photons have low energy and are
concentrated at very small radius. For combinatoric reasons, there are
significantly more events reconstructed in the FV with one or more photons
lost to cluster fusion than in the case of $K_{L}\to\pi^{0}\pi^{0}$. The
fiducial volume cut and $r_{\rm min}$ cuts are highly effective at eliminating
these events, and the residual background has very low $p_{\perp}$. This study
will be repeated with the updated experimental configuration at a level of
statistics sufficient to rule out or evaluate residual contributions to the
background.
##### $K_{L}\to\gamma\gamma$
This channel has also been studied by fast simulation. Because the transverse
momentum of the $\gamma\gamma$ pair is zero in the rest frame, the $p_{\perp}$
cut used to reject $K_{L}\to\pi^{0}\pi^{0}$ background is extremely effective,
and no events are expected from $K_{L}\to\gamma\gamma$ decays in the core of
the neutral beam. Any background from this channel must come from the decays
of $K_{L}$’s scattering from the collimator surfaces. The KOTO experiment has
reported background from $K_{L}\to\gamma\gamma$ at a level corresponding to
about 6 events at SM sensitivity [177]. Cluster shape cuts to constrain the
angles of photon incidence were helpful in reducing this background. KLEVER
has numerous advantages, including a long beamline with four stages of
collimation, resulting in beam with an excellent core-to-halo ratio, and
reconstruction constraints from the PSD.
##### Decays to charged particles
The role of the CPV in rejecting decays with charged particles in the final
state is discussed in Section 10.4. An important decay against which the CPV
not completely effective is $K_{L}\to\pi^{+}\pi^{-}\pi^{0}$: because of the
low decay momentum, it is possible for the $\pi^{+}$ and/or $\pi^{-}$ to
escape down the beam pipe. KOTO has resolved this problem by installing
additional charged particle vetoes along the walls of the downstream beam
pipe, and it is straightforward to adopt the same solution for KLEVER.
Radiative semileptonic decays and $K_{e4}$ decays will also require simulation
with the full HIKE MC.
##### Beam interactions on collimator surfaces
There are a number of potential backgrounds from the inelastic scattering of
beam particles from the collimator surfaces. Hyperons and $K_{S}$ mesons can
produced in these interactions. Compared to the $\Lambda$ and $K_{S}$
components of the neutral beam, these secondaries have much lower momentum and
should decay away if produced on the upstream collimators. Scattering in the
AFC is more dangerous, so further beamline optimisation, including the
possibility of introducing additional stages of collimation, may be necessary.
Based on KOTO’s experience, $K^{0}p\to K^{+}n$ charge exchange is possibly the
most important process [177], because the $K^{+}$ can undergo $K_{e3}$ decay
inside the FV with the $\pi^{0}$ reconstructed and the $e^{+}$ emitted at very
low energy and at very high angle. KOTO is exploring the use of very thin
upstream charged particle detectors to reduce this background. In KLEVER, the
first three stages of collimation are followed by magnetic sweepers, so this
background can only arise from scattering in the AFC, which for this process
leaves a signal to veto.
##### Neutron interactions in the MEC
Neutron interactions in the calorimeter can create background due to the
possibility for hadronic shower fragments to travel a substantial distance in
the calorimeter before reinteracting, creating two clusters. The principal
strategy for abatement of this background is careful beamline design to reduce
the neutral beam halo. This is also the logic behind widening the calorimeter
bore so that the beam penumbra intercepts the IRC downstream of the MEC. In
addition, the particle-identification capabilties of the calorimeter play an
important role. KOTO, which uses a CsI calorimeter, identifies neutron
interactions via cluster shape analysis and double-sided light readout. The
MEC design permits optimisation of the transverse segmentation to allow
cluster shape analysis, while longitudinal shower information is obtained from
the use of spy tiles (Section 10.3.3.2). Since the halo is concentrated on the
inner regions of the calorimeter, the $r_{\rm min}$ cut is expected to be
effective. KLEVER will also make use of information from the HIKE hadron
calorimeters to reject hadronic showers.
##### Beam-gas interactions
The background from single $\pi^{0}$ production in interactions of beam
neutrons on residual gas has been estimated with the FLUKA-based simulation
(Section 5.3) to be at most a few percent of the expected signal for a
residual gas pressure of $10^{-7}$ mbar.
#### 8.3.4 Outlook
As discussed in the previous sections, the current simulations give 22.2
expected signal events, $102\pm 12$ expected background events from
$K_{L}\to\pi^{0}\pi^{0}$, and $52\pm 20$ expected background events from
$\Lambda\to n\pi^{0}$, which can potentially be reduced to the low single
digits or eliminated by the introduction of a hadronic calorimeter to back up
the SAC. We have identified in-principle solutions for the mitigation of the
remaining backgrounds that require further study. The designs of the beamline
and experiment may require relatively minor modifications, such as the
introduction of additional collimation stages or of small detectors to
accomplish specific tasks such as upstream or downstream charged-particle
veto, but these are well-defined problems amenable to solution. We also
consider the following factors in interpreting the signal and background
estimates:
* •
Random veto will reduce both signal and background by about 10% (Section 8.2).
* •
The use of an oriented crystal metal photon converter in the TAX collimator
will reduce the scattering of the neutral beam by about 20%, leading to an
effective increase in $K_{L}$ flux of 10% (Section 5.3.1). Since this involves
no significant technical difficulty, we assume that this will cancel the loss
from random veto.
* •
Alternatively, the use of a high-$Z$ target could increase the effective
$K_{L}$ flux by up to 30% due to the dramatic reduction in the thickness
required for the photon converter. However, we do not take this into account
for the moment, as the associated costs and technical difficulties have not
been evaluated.
With these factors taken into consideration, the signal and background
estimates from the simulation remain unchanged.
Relative to the sensitivity target of 60 SM events, the apparent shortfall
arises principally because of the reduction of the the $K_{L}$ flux a factor
of 2.4 with the extended beamline. With the performance gains expected from
reoptimisation of the analysis, with particular attention to extension of the
FV, better use of the information from the PSD, and continued improvements to
the effectiveness of the photon veto systems with emphasis on the calorimeter
design, this loss can largely be recovered, so that the original sensitivity
goal remains within reach.
An effort is underway to develop a comprehensive simulation and to use it to
validate the results obtained so far. This work is being carried out within
the framework of the flexible HIKE Monte Carlo platform, allowing new
configurations of the existing and proposed detectors to be simulated on the
basis of the NA62 software.
## 9 Operation in beam-dump mode
### 9.1 Experimental layout
The HIKE beam dump operation will build upon the experience accumulated in
NA62 with beam-dump data taking. In 2021, $1.4\times 10^{17}$ protons were
collected in 10 days of data taking at NA62 in beam-dump mode, with the T10
target used to generate the standard NA62 secondary hadron beam removed from
the beamline. Proton beam was made to interact in the NA62 movable
collimators, called TAX, located 23 m downstream of T10 within two pair of
dipoles, and 80 m upstream of the decay volume. An ad-hoc setting of the
dipoles allows a substantial reduction of the rate of muons emitted by pion
decays in the proton-induced hadronic showers in the TAX (the so called “halo”
muons). Secondary interactions of halo muons can induce background to the
searches proposed, and optimisation of the beamline reduces this background by
at least one order of magnitude [147]. On-going and completed studies have
shown that the residual background is negligible, in particular when searching
for two-body decays of new-physics mediators. In this case, the background
rejection is improved by requiring that the trajectory of the mediator points
to the beam dump. Based on the NA62 experience, it is assumed that the
background will remain negligible at HIKE after an increase of 10 to 50 times
in statistics with respect to the NA62 beam-dump operation. Therefore
sensitivity projections are performed assuming no background limitation.
Sensitivity projections are produced assuming $10^{19}$ and $5\times 10^{19}$
POT accumulated in beam-dump mode. Operation at $4\times 10^{13}$ POT for 4.8
s spills is assumed. When compared to the NA62 2021 beam-dump period, this
corresponds to an intensity increase by a factor of 8. Assuming the same
efficiencies as for the 2021 NA62 data taking, HIKE will be able to collect
$10^{19}$ POT in three to four months. The trigger rate foreseen during beam-
dump operation is modest, amounting to less than 200 kHz overall.
Using a movable dump allows for a quick switch between a kaon-beam and a beam-
dump operation, making it possible to perform specific online calibration
procedures in kaon-beam mode during beam-dump operation periods. The
importance of this aspect was demonstrated in 2021, significantly improving
the overall data quality. A careful consideration concerning the cooling of
the dump is needed: the intensity foreseen demands a cooling power exceeding
the maximum presently available for the NA62 TAX collimators by a factor of
four. Cooling and radio-protection considerations are discussed in Section 5.
As demonstrated by the 2021 NA62 beam-dump data, frequent and accurate
calibration of the beam secondary-emission intensity monitors (BSI) is
necessary, to ensure an overall uncertainty in the integrated proton flux at
the level of 5%. Monitors based on the present BSI-technology are adequate to
withstand the projected beam intensity. Exploratory experimental studies by
the CERN beam-group experts show that the goal of achieving a 5% uncertainty
in the proton flux measurement is within reach.
### 9.2 Physics sensitivity
The projected sensitivity curves of the HIKE experiments in beam-dump mode are
evaluated as exclusion plots at 90% CL, obtained in the assumption of zero
observed events and negligible expected background. Note that some of the
scenarios are also accessible by HIKE operating in kaon mode (Section 3.6).
With reference to the portal classification discussed in Ref. [20], the models
relevant to the HIKE sensitivity projections are:
* BC1
A new single vector state $A^{\prime}$, that interacts with the
electromagnetic current through a small coupling constant, $\varepsilon$. In
the NA62 beam-dump setup, such dark photon can be produced by proton strahlung
and through the decays of secondary mesons ($\pi^{0}$, $\eta^{(\prime)}$,
etc.). The dark photon decays to SM particles, as final states with dark
matter fields are assumed to be kinematically forbidden. Below 700 MeV, di-
lepton decays dominate the $A^{\prime}$ width. The free parameters of the
model are $M_{A^{\prime}}$ and $\varepsilon$. The expected sensitivity of HIKE
to the dark photon production and decay is shown in Fig. 25.
* BC4
A new scalar state $S$, that interacts with the SM Higgs through a small
mixing constant, $\theta$. In the beam-dump setup, such dark scalar is mostly
produced through flavor-changing neutral-current decays of $B$ mesons ($B\to
K^{(\ast)}S$). The effective coupling constant is proportional to the fermion
mass. Hence, when scanning the $S$ mass from MeV to GeV and above, each new
decay mode kinematically accessible ($e^{+}e^{-}$, $\mu^{+}\mu^{-}$,
$\pi^{+}\pi^{-}$, etc.) tends to either saturate or significantly increase the
$S$ width. The free parameters of the model are $m_{S}$ and $\theta$. The
expected HIKE sensitivity is shown in Fig. 26 (left), where we have used the
scalar hadronic decay widths from [165].
* BC6, 7, 8
New fermions $N$, that interact with the SM lepton doublets. These heavy-
neutral leptons (HNL) mix with the SM neutrinos through a matrix called $U$.
Models BC6, BC7 and BC8 assume the presence of a single HNL of Majorana type
and the dominance of electron, $\mu$, or $\tau$ neutrino couplings,
respectively. In the beam-dump setup, the production of HNL is dominated by
leptonic and semileptonic decays of charmed mesons or $\tau$ leptons, each
model favouring the corresponding lepton flavour. The decay width is dominated
by meson-lepton or meson-neutrino two-body final states. The free parameters
of the BC6,7,8 models are $m_{N}$ and $U_{e,\mu,\tau}$, respectively. The
expected HIKE sensitivity curves are obtained with the same assumptions of
[178] and are shown in Fig. 27.
* BC9, 10, 11
A new pseudoscalar state $a$ (axion-like particle, ALP), that interacts with
the SM fields. Models BC9, BC10 and BC11 assume a dominant coupling of an ALP
with SM photons, fermions, and gluons, respectively. In BC9, the production in
a proton beam dump is dominated by the interaction of photons from the decays
of secondary mesons ($\pi^{0}$, $\eta^{(\prime)}$) with the EM fields of the
TAX nuclei. The decay width is dominated by the ALP decay to two photons in
the whole $m_{a}$ range. The free parameters of the BC9 model are $m_{a}$ and
the coupling $C_{\gamma\gamma}$. The expected HIKE sensitivity is shown in
Fig. 26 (right). In BC10, the production is dominated by decays of secondary
$B$ mesons ($B\to K^{(\ast)}a$). For $m_{a}$ below 700 MeV, the decay width is
dominated by decays to di-lepton final states. The free parameters of the BC10
model are $m_{a}$ and the Yukawa coupling $g_{Y}$ to the SM fermionic fields.
The expected HIKE sensitivity is shown in Fig. 28 (left). The broad
phenomenology of BC11 can reproduce either of the two former models, depending
on the ALP mass: one-loop corrections lead to an effective coupling to
photons, and an effective coupling to quarks is generated. The former case
leads to the ALP di-photon decay; the latter to hadronic decays, e.g.
$a\to\pi^{+}\pi^{-}\gamma$, but not to leptonic decays. The expected HIKE
sensitivity is shown in Fig. 28 (right).
Figure 25: Expected HIKE 90% CL exclusion limits for the BC1 scenario
($A^{\prime}\to\ell^{+}\ell^{-}$). The currently excluded areas of parameter
space are shown in grey.
Figure 26: 90% CL exclusion limits obtained in simulations using the ALPINIST
framework with the selection criteria described in [179]. Current exclusion
limits are shown with filled and future projections with empty contours. Left:
Search for a dark scalar $S$ in the BC4 scenario. For HIKE, production from
$B\to K^{(\ast)}S$ decays and sensitivity to $S\to ee,\mu\mu,\pi\pi,KK$ decays
are considered. The $S$ hadronic decay widths derived in [165] have been used.
Exclusions from LHCb [168, 169], NA62 Run 1 [67, 142], E137, and CHARM
measurements are separately shown. The E137 exclusion is obtained using the
ALPINIST framework with data provided in [180]. Right: search for an axion-
like particle $a$ in the BC9 scenario. HIKE sensitivity to $a\to\gamma\gamma$
decays is considered.
Figure 27: HIKE 90% CL exclusion limits for BC6 (top-left), BC7 (top-centre)
and BC8 (bottom) scenarios in the beam-dump mode, assuming sensitivity to two-
track final states and NA62 geometrical acceptance. The currently excluded
areas of parameter space are shown in grey.
Figure 28: Exclusion limits at 90% CL obtained with the ALPINIST framework
for an axion-like particle $a$. Current exclusion limits are shown with filled
and future projections with empty contours. In the BC10 scenario (left), we
assume HIKE sensitivity to ALP production from $B$ meson decays and to the ALP
decays $a\to ee$ and $a\to\mu\mu$. In the BC11 scenario (right), we assume
HIKE sensitivity to $a\to\gamma\gamma$ decays (dashed contour) and also to the
hadronic decays $a\to\pi^{+}\pi^{-}\gamma$, $3\pi$, $2\pi\eta$, etc. (full
contour).
## 10 The HIKE detectors
The aim is to achieve the same detector performances of NA62 at a factor 4–5
higher beam intensity, with suitable time resolution and rate capability.
Describe which detector modifications and readout we need.
### 10.1 Detectors upstream of the fiducial decay volume
#### 10.1.1 Cherenkov kaon tagger (KTAG)
A differential ring-focusing Cherenkov detector (KTAG), used for $K^{+}$
tagging in the NA62 experiment at a kaon rate of about 40 MHzprovides tagging
efficiency above 95$\%$ [175, 181]. The KTAG uses nitrogen radiator gas (to be
replaced with hydrogen before LS3) and ring-imaging optics to focus Cherenkov
photons onto eight spherical mirrors reflecting them onto eight photodetector
arrays. A diaphragm is used so that only Cherenkov light emitted by kaons
reaches the photodetectors. Each array is equipped with a matrix of 48
Hamamatsu R7400-U03 and R9880-210 single-anode phototubes (PMTs) with peak
quantum efficiency (QE) of 20–40$\%$. The average rate of detected photons per
PMT depends on the QE and PMT position and varies between 3–5 MHz, for an
effective area of 2.5 cm2. A single-photon time resolution of 300 ps and the
mean number of detected photons per kaon of about 20 lead to a 70 ps overall
kaon time resolution.
For HIKE Phase 1, the KTAG is expected to provide a time resolution of 15–20
ps and a tagging efficiency above 95%. Simulations of the KTAG with hydrogen
radiator gas show that a mean number of detected photons per kaon of about 30
is expected; the improvement in the light yield with respect to nitrogen
radiator is due to newly built ring-imaging optical components optimised for
hydrogen. The expected $K^{+}$ rate in the HIKE beam is about 200 MHz,
corresponding to a 10 MHz/cm2 maximum rate of detected photons (including a
safety margin) assuming the NA62 values of the QE and acceptance of the
photodetection system. The KTAG should be upgraded with new photodetection,
front-end and readout systems to satisfy these requirements. Photodetection
devices under consideration for the HIKE KTAG detector are micro-channel plate
photomultipliers (MCP-PMTs) which are compact devices providing a single-
photon time resolution of 50–70 ps with a low dark count rate (below 1
kHz/cm2), able to operate at the expected maximum photon rate of 10 MHz/cm2.
Assuming a gain of $10^{6}$, 300 days of runtime and a 50% duty cycle, the
above photon rate leads to an integrated anode charge (IAC) of 16 C/cm2.
Figure 29: Schematics of a Photonis Planacon square-shaped MCP-PMT array under
investigation.
Figure 30: Sketches of a single MCP-PMT array (left) and a matrix of four MCP-
PMTs (right) to be used to detect photons in each of the eight octants of the
HIKE KTAG detector. Figure 31: QE of the Photonis Planacon MCP-PMTs as
functions of wavelength for the two photocathode types considered (standard
bialkali and Hi-QE Blue). A cutoff at 180 nm corresponds to the minimum
wavelength of the Cherenkov photons reaching the photodetection system.
A limitation of standard MCP-PMTs is their limited lifetime, caused by a rapid
decrease in the QE of the photocathode (PC) with increasing IAC. This ageing
of the PC is caused by heavy feedback ions from the residual gas. For standard
MCP-PMTs the QE was observed to drop by 50% for IAC of 0.2 C/cm2 [182]. MCP-
PMT treatment with atomic layer deposition (ALD) coating increases the
lifetime dramatically [183]. The Planacon XP85112 from Photonis, a square-
shaped MCP-PMT treated with an improved ALD technique involving two layers,
has been demonstrated to reach a lifetime of 33 C/cm2 without any sign of
ageing [184]. This represents an improvement in lifetime by a factor of more
than 150 with respect to standard MCP-PMTs, and a factor of at least 10
compared to earlier lifetime optimisation techniques.
A Photonis Planacon MCP-PMT of 2”$\times$2” size consisting of $8\times 8$
pixel sensors (Fig. 29), treated with two-layer ALD coating, is a viable
solution for the photodetection system of the KTAG in high intensity
applications. The Cherenkov photons at each photodetector plane can be
detected with a matrix of four MCP-PMTs (Fig. 30). The MCP-PMT is available
with standard bialkali and Hi-QE Blue photocathode types; the two QE
parameterisations provided by the manufacturer are shown in Fig. 31.
Simulation results obtained with a geometrical filling factor of 75% and a
collection efficiency of 60% show that the 15–20 ps kaon time resolution is
achievable. The linearity of MCP-PMTs with rate at high gain is under
investigation. However the rate stability can be adjusted by modifying the
resistance and capacitance of the MCP coating inside the MCP-PMT.
The above results for the kaon time resolution do not include contributions
from front-end and readout systems. The fastIC [185] and picoTDC [186]
electronics (Section 11.1) being currently developed for other applications
are viable options. The FastIC technology provides a 20 ps time resolution and
better linearity for signals from SiPM and MCP-PMTs. PicoTDC chips with 64
channels and 12 ps binning can provide a 4 ps time resolution. Both time
contributions will only have a marginal effect on the KTAG timing capability.
#### 10.1.2 Beam tracker
The GigaTracker [187] is the beam tracker used by the NA62 experiment to
measure the momentum and time of the beam particles with high momentum,
angular and time resolutions. The detector is made by planes of hybrid pixel
silicon detectors, with a dipole magnets between the planes to allow the
momentum measurement. For NA62 Run 1 (2016–2018) three silicon planes were
used, then for NA62 Run 2 (2021–LS3) a fourth plane was added. Each plane is
made of a n-in-p planar sensor, with a thickness of
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a sensitive area of $60\times
27~{}\rm{mm^{2}}$, bump-bonded to 10 readout ASICs (TDCPix). The ASICs need to
operate with a cooling system: the NA62 GigaTracker used a silicon
microchannel cooling plate that represented the first use of this technology
in high-energy experiments. In order to minimise the probability of
interactions between the beam particles and the detector, the material budget
must be kept as low as possible: the total thickness of a single plane is
around $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$, namely $0.5\%X_{0}$. The
NA62 GigaTracker has achieved very good performances, namely a track time
resolution of ${\cal O}(100~{}{\rm ps})$, an angular resolution of 16 $\mu$rad
and a momentum resolution of 0.2%.
Among all the above characteristics of the detector, the time resolution
represents a limiting parameter for its operations in the context of HIKE.
With more than four-fold instantaneous intensity increase, the time resolution
should scale accordingly, namely from less than 200 ps to below 50 ps for a
single hit. The radiation hardness of the new detector needs to be improved by
at least a factor four if a replacement approach is to be preserved, or more
in case of a single installation for the entire HIKE data taking campaign.
Table 11 summarises the comparison between the current NA62 GigaTracker and
the new detector needed for HIKE.
Table 11: Comparison between the NA62 Gigatracker main characteristics and the requirements for the HIKE beam tracker. One year corresponds to 200 days of beam. | NA62 GigaTracker | New beam tracker
---|---|---
Single hit time resolution | < 200 ps | < 50 ps
Track time resolution | < 100 ps | < 25 ps
Peak hit rate | $2~{}\rm{MHz/mm^{2}}$ | $8~{}\rm{MHz/mm^{2}}$
Pixel efficiency | > 99 % | > 99 %
Peak fluence / 1 year [$10^{14}~{}1~{}\rm{MeV~{}n_{eq}/cm^{2}}$] | 4 | 16
The interest for silicon detectors with fast timing information capable to
operate in a high-radiation environment is shared among different particle-
physics collaborations, including the LHC experiments for the high luminosity
phase of the collider. Following the 2020 Update to the European Strategy for
Particle Physics, 4D tracking at high fluences was identified as one of the
most urgently needed technologies for future detectors (2021 European
Collaboration for Future Accelerators; Detector Research and Development
Roadmap DRDT 3.2 and DRDT 3.3). Therefore several promising R&D projects are
ongoing and their outcome could be adopted for HIKE.
A strong option is the timeSPOT project [188, 189]: which develops a
technology called hybrid 3D-trenched pixels in which the pixel electrode
geometry is optimised for timing performance. A representation of one cell of
a 3D-trench sensor is reported in Fig. 32. As for standard 3D-pixels, the
sensors are able to withstand very large irradiation. The project has
experimentally demonstrated that sensors with a pitch of
$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$ provide a time resolution of
$10\text{\,}\mathrm{p}\mathrm{s}$ up to fluence of
$2.5\text{\times}{10}^{16}\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}~{}\mathrm{n}_{eq}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$
[190]. The timeSPOT collaboration is planning to extend these tests up to
fluences of
$1\text{\times}{10}^{17}\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}~{}\mathrm{n}_{eq}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
that has never been achieved so far. Excellent detection efficiencies were
also measured [190] by operating the sensor inclined by an angle of
$20\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the beam incidence.
As the electrode itself is not sensitive, the inclination ensures that
particle always cross some active pixel region. Sensor with a size of $2\times
2$ cm2 can be produced and technical solution like stitching are being
explored to produce larger devices.
A first prototype of the ASIC, TIMESPOT1, has been realised [191]. The chip
contains a pixel matrix of $32\times 32$ elements, with a pixel pitch of
$55~{}\rm{\mu m}$. Each pixel in the matrix contains an analogue and a digital
circuit, with its TDC to measure the time of the arrival of the signals from
the sensor. Digitised signals are then sent to the periphery of the matrix
where 8 multiplexed output links send the data out at 1.28 Gb/s speed. Efforts
are on-going to scale the size of the ASIC. The above considerations make
timeSPOT a viable option for the HIKE beam tracker.
Figure 32: Elementary cell of a 3D-trench silicon detector, as designed by the
TIMESPOT project [192]. Doping profiles are shown (n++ in red, p– in green,
p++ in blue). A) 3D rendering; B) pixel section; C) pixel layout.
Other projects involving different technologies, such as monolithic pixel
sensors [193] or LGADs [194, 195], are also taken in consideration. Monolithic
detectors can be made with very small thickness and excellent time
resolutions, however they have a lower radiation tolerance compared to the
3D-trench sensors.
#### 10.1.3 Veto counter
The introduction of the veto counter detector (VC) surrounding the beamline
has helped reduce the upstream background for the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ analysis. However, a part of the background
remain undetected, passing at a very small angle inside the holes
accommodating the beam pipe. The random veto due to the usage of the veto
counter is less than 2% at NA62 nominal intensity, but is expected to increase
with the beam intensity. When more than one particle is present in the event,
the granularity of the current setup does not allow to reliably separate them
and distinguish between accidental activity due to halo muons and genuine
background candidates.
An improved design based on scintillating fibre (SciFi) technology, developed
for the SciFi tracking detector at LHCb [196], is proposed to tackle these
limitations. The detector will consist of three detection stations: two
located in front of the main collimator, separated by a layer of lead acting
as a photon converter, and a third one located immediately behind the
collimator, see figure 33. Each station needs to extend at least up to
$10\text{\,}\mathrm{cm}$ above the beamline and $30\text{\,}\mathrm{cm}$ below
the beamline to cover the full range where upstream decays are expected to
cross the detection planes. For the same reason, the horizontal coverage needs
to extend up to $6\text{\,}\mathrm{cm}$ on either side of the beam centre. A
central hole will accommodate the reduced-size beam pipe, allowing it to reach
only $2\text{\,}\mathrm{cm}$ from the beam centre. The thin SciFi modules
would then be attached to the beam pipe allowing for an increased detection
efficiency for interactions of photons and charged particles with the beam
pipe. Following these specifications, the active area of the new veto counter
will cover a rectangular surface area of
$26\times$$78\text{\,}\mathrm{c}\mathrm{m}^{2}$ with a hole at the centre to
accommodate the passing beam. With the present state of the technology as used
in LHCb, each station is made of scintillating fibres of diameter
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$, read out by silicon photo-
multipliers (SiPMs). The fibres are arranged in mats of multiple fibre layers
forming a detection plane. Two orthogonal SciFi planes form a single veto
counter station. The described detector scheme will provide X-Y reading of
charged particles crossing a detection station with a spatial resolution
$\sigma_{x,y}\sim$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, a time
resolution below $200\text{\,}\mathrm{ps}$, and detection efficiency larger
than 99$\%$. The number of readout channels for the above design will be about
4000 channels per station. If needed, several fibres could be coupled together
into a single SiPM to reduce the number of channels at the expense of some
spatial resolution. The thickness of the detection planes can also be modified
to increase detection efficiency and improve time resolution. The small
granularity and high spatial resolution of each detection plane will provide
tracking capabilities by combining information from consecutive stations,
further improving the time resolution.
Figure 33: Zoomed view of upstream region of HIKE Phase 1 (see Fig. 13),
highlighting the location of the veto counter stations.
The expected hit rate in the detector is extrapolated from the rates at the
nominal NA62 beam intensity, assuming it grows linearly with the beam
intensity. The hit rate at four times the NA62 beam intensity is expected to
be 30 MHz per detection plane, which is too high for the existing readout
electronics of the SciFi detector but can be addressed with the TDC-Felix
electronics commissioned in NA62 (Section 11). The front-end electronics must
also be adapted for HIKE. The present front-end ASIC used for the LHCb SciFi
receives signals from the SiPMs, which are then shaped and digitised by a
2-bit ADC with a sampling rate of 40 MHz. Due to the time resolution
requirements for HIKE, new front-end electronics will be used, based on ToT or
FADC, similar to the solutions for other detector systems (see for example the
KTAG and LAV sections).
A different technology option is an update on the design currently used in
NA62 using tiles of plastic scintillator. In this case, the use of tiles with
smaller height (4 cm in NA62) and the usage of a double plane for each station
(a plane with tiles placed horizontally and a plane with tiles placed
vertically) will allow to achieve a smaller granularity and separate signals
left by multiple particles. An increased acceptance coverage will be achieved
by adding tiles also on the left and right of the beam. A rectangular hole of
$42\times 78$ mm2 size will allow the beam pipe to pass through the detector.
In addition, some tiles could be placed along the beam pipe to increase the
detection of particles escaping the beam pipe between stations.
Further improvements in upstream background rejection power could be achieved
by rearranging the beamline elements in the veto counter region. Combined with
the proposed hardware improvements, the new veto counter detector will reduce
the upstream background at least three times more than the present system.
#### 10.1.4 ANTI-0
The baseline option for the charged particle veto upstream of the decay volume
(very upstream ANTI station called ANTI-0) is a cell structure hodoscope
covering area of $\oslash$2.2 m around the beampipe at the entrance of the
fiducial decay volume. The NA62 ANTI-0 hodoscope design is based on
rectangular scintillating tiles with transverse size of $120\times 120$ mm2
read by SiPMs through short lightguides arranged in a chessboard style at both
sides of the central foil. The general design of ANTI-0 for HIKE follows that
of the NA62 ANTI-0 hodoscope [197] with the two major changes: finer
granularity to sustain up to x6 intensity, and two active layers to ensure
higher efficiency.
An additional improvement, currently under investigation, is the possibility
to use reabsorption-free fast nanocrystal-based plastic scintillators [198].
The new scintillator already now can sustain a higher radiation dose, having
an emission peak shifted to $\lambda>550$ nm, tolerating transparency losses
in UV and blue regions. Some perovskite-based scintillators show very fast
decay time, with the first time component of the order of $\sim 0.3$ ns [199],
which could be crucial for building fast future detectors. These technological
possibilities will be further investigated for the HIKE Proposal. More details
can be found in the description of the MEC.
### 10.2 Fiducial decay volume and its detectors
#### 10.2.1 Charged anti-coincidence detector
The Charged Anti-coincidence detector (CHANTI) provides veto for events with
particles scattered inelastically off the last station of the beam tracker and
halo particles originating from upstream decays entering the fiducial decay
region close to the beam.
The NA62 CHANTI comprises six stations, each station covering an area of
$30\times 30$ cm2 with a $9\times$5\text{\,}\mathrm{c}\mathrm{m}^{2}$$ inner
opening for the beam. The CHANTI covers hermetically the angular region
between $34\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ and
$1.38\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$ with respect to the beam axis.
The CHANTI, with its time resolution of about
$800\text{\,}\mathrm{p}\mathrm{s}$, induces a signal loss due to accidental
activity of about 5$\%$ at the nominal intensity but is expected to become
more significant with increasing intensity.
The design of the HIKE CHANTI stations is based on the usage of thick plastic
scintillating fibres (from $\oslash 2$ up to $\oslash 3$ mm) read by SiPMs
from both ends. The expected particle rate has been estimated using the NA62
beamline simulation, assuming an hadron beam intensity six times higher than
the nominal 750MHz of NA62. The expected rate per channel in kHz is shown in
Fig. 34 for the option with $\oslash 2$ mm thick fibres overlapping by 0.5 mm.
The option with $\oslash 2$ mm fibres, that is the maximum diameter currently
widely available on the market, assumes 464 fibres per station (208
forvertically oriented X station, and 256 for horizontally oriented Y station)
and about 1000 SiPMs/station. The required number of electronic channels could
be reduced by connecting signals from the peripheral fibres into analogue OR
before digitization.
##### 10.2.1.1 Alternative detector design using SciFi technology
A more ambitious option would be to pursue the SciFi detector technology
rather than scintillator technology. The new detector would consists of six
stations with $39\times$$39\text{\,}\mathrm{c}\mathrm{m}^{2}$ transverse
dimensions, positioned to hermetically cover the required angular region with
respect to the beam axis. Each station is made of scintillating fibres
($\oslash$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) arranged in two planes
that provide X-Y information about the incoming charged particles. Each plane
is made of mats of several fibre layers, which are read out by SiPMs. The
layout allows a time resolution below $200\text{\,}\mathrm{p}\mathrm{s}$ per
station and a spatial resolution
$\sigma_{x,y}\sim$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The time
resolution would be at least a four-fold improvement, and the exceptional
spatial resolution would provide tracking capabilities. The stations would be
thicker than the ones currently used at LHCb [196], bringing the detection
efficiency to above 99$\%$.
The new CHANTI detector could also be used in conjunction with the new
VetoCounter (Section 10.1.3), both made of SciFi technology, to separate halo
muons from charged particles produced in interactions in the beam-tracker.
Owing to their identical spatial and time resolutions and largely overlapping
sensitive regions, a combination of the information between the two detectors
can reduce significantly the accidental veto due to halo-muon activity.
Using a SciFi detector for the new CHANTI presents a technological challenge.
At present detectors using SciFi technology are operated in air and a
dedicated R$\&$D program is required to enable their operation in vacuum. A
solution to this challenge will allow cooling of the SiPM to liquid Nitrogen
temperatures, eliminating the noise completely and improving radiation
tolerance. This is an important improvement over the present CHANTI, which
uses SiPMs without cooling. The SiPMs currently in operation at NA62 are
heavily impacted by the effect of accumulated radiation dose; such issue would
not be present for a SciFi detector with cooled SiPMs in vacuum.
The number of readout channels for the above design would be about 3000
channels per station, but it can be reduced if several fibres are coupled
together and fed into a single SiPM. This would be at the expense of some
spatial resolution, and would not be an issue. The readout is planned to be
based on the TDC-Felix electronics 11, that is able to sustain the hit rate
presented in Fig. 34.
Figure 34: CHANTI expected rate per fibre for the vertical and horizontal
planes.
#### 10.2.2 Large angle vetos
As demonstrated by the successful $K^{+}\to\pi^{+}\nu\bar{\nu}$ measurements
at NA62, the efficiency of the existing photon veto systems, including the
existing Large Angle Vetoes (LAV) [175] based on the OPAL lead glass [200], is
adequate for the HIKE $K^{+}\to\pi^{+}\nu\bar{\nu}$ measurement. In the
analysis of NA62 Run 1 data, the overall $\pi^{0}$ detection inefficiency was
about $1\times 10^{-8}$ [201], and the expected number of background events
from $K^{+}\to\pi^{+}\pi^{0}$ (the only significant background channel with
photons) was about 7% of the expected number of signal events [67]. The
$\pi^{0}$ veto efficiency estimate was validated by studies of the single-
photon detection efficiency for each veto subsystem [142]. The inefficiency of
the NA62 LAVs for single photons was found to rapidly decrease with photon
energy to a value of about $3\times 10^{-3}$ for photons with $E=300$ MeV, and
thereafter to slowly decrease with energy to values of a few $10^{-4}$ for
photons of $4$–$6$ GeV.111This refers to the LAV system in operation,
including the effects of the non-hermeticity of the retrofitted arrangement of
lead-glass blocks. The actual detection efficiency for the lead-glass
detectors themselves was found with electrons to be an order of magnitude
higher [202]. In the decay-in-flight technique for the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ measurement, the $K^{+}$/$\pi^{+}$ vertex is
accurately reconstructed and required to be in the fiducial volume, and the
fiducial momentum cuts on the secondary $\pi^{+}$ guarantee that the $\pi^{0}$
in $K^{+}\to\pi^{+}\pi^{0}$ decays has at least 40 GeV of energy. As a result,
the LAV system only has to cover out to 50 mrad in the polar angle as seen
from the fiducial volume. Moreover, the two photons from
$K^{+}\to\pi^{+}\pi^{0}$ decays have an anticorrelation such that for events
with one low-energy photon heading into the LAVs, the second photon has high
energy and is seen in the LKr. These conditions allow the very low detection
inefficiency for $\pi^{0}$’s to be obtained even with LAVs with single-photon
inefficiencies on the order of $10^{-3}$.
On the other hand, the time resolution obtained with the existing LAVs is
problematic for HIKE. The Cherenkov light produced in the large lead-glass
blocks is highly directional, and the light propagates to the PMT
photocathodes via complicated paths with multiple reflections. There is
considerable time spread arising from the angle of particle incidence, and
perhaps for different particle species, the Cherenkov characteristics of mips
and electromagnetic showers being quite different. Above all, the light yield
of the lead-glass blocks for electromagnetic showers is just 0.2 p.e./MeV. As
a result, the LAVs have a time resolution of about 1 ns with significant
extra-gaussian tails [175]. In order to maintain the random veto rate constant
in HIKE, the time resolution needs to be improved by at least a factor of 4,
and the tails substantially reduced or eliminated.
The LAVs are potentially the most resource-intensive subsystem to build for
HIKE, so it is essential that they be reusable for all phases of the program.
The design of the LAVs for HIKE is therefore driven by the efficiency
requirements for the $K_{L}$ program, and in particular, for the measurement
of $K_{L}\to\pi^{0}\nu\bar{\nu}$. The time resolution requirements for the
$K_{L}$ phase are similar to those for the $K^{+}$ phase. The preliminary
estimate of the total hit rate on the LAVs in KLEVER is 14 MHz (Table 7),
implying a 3.5% random veto rate for a $\pm 5\sigma$ coincidence window of 2.5
ns. The efficiency requirements for the $K_{L}$ phase, on the other hand, are
much more stringent than for the $K^{+}$ phase. Unlike for the case of
$K^{+}\to\pi^{+}\nu\bar{\nu}$, for $K_{L}\to\pi^{0}\nu\bar{\nu}$ decays, the
reconstruction of the $K_{L}$ decay vertex is uncertain, so the photon vetoes
need to cover the entire length of the experiment from the upstream end of the
fiducial volume to the calorimeter. Moreover, the kinematic anticorrelation
between the energies (or angles) of missed photons from
$K_{L}\to\pi^{0}\pi^{0}$ decays is largely washed out. Nevertheless, because
of the boost from the high-energy beam, the KLEVER simulations show that it is
sufficient for the large-angle photon vetoes (LAVs) to cover polar angles out
to 100 mrad, as long as the fiducial volume is located well upstream of the
calorimeter and the LAV detectors themselves satisfy the efficiency
requirements discussed below: indicatively, the photon detection inefficiency
must be less than 5% at 10 MeV, less than $2.5\times 10^{-4}$ at 100 MeV, and
less than $2.5\times 10^{-6}$ for energies above 2.5 GeV.
##### 10.2.2.1 Layout for the $K^{+}$ phase
Table 12: Parameters of the existing NA62 Large Angle Vetoes. Stations | Diameter [mm] | Block radius [mm] | Layers | Blocks
---|---|---|---|---
| Outer wall | Inner | Outer | |
LAV1–LAV5 | 2168 | 537 | 907 | 5 | 160
LAV6–LAV8 | 2662 | 767 | 1137 | 5 | 240
LAV9–LAV11 | 3060 | 980 | 1350 | 4 | 240
LAV12 | 3320 | 1070 | 1440 | 4 | 256
The positions and radii of the LAV stations for the HIKE $K^{+}$ phase are
expected to be quite similar to those for NA62. In NA62, there are 12 LAV
stations: 11 incorporated in the vacuum tank, and one in air just upstream of
the calorimeter. Parameters of the NA62 LAV stations are listed in Table 12.
##### 10.2.2.2 Layout for KLEVER
For KLEVER, a total of 25 LAV stations in five different sizes, operated in
vacuum and placed at intervals of 4 to 6 m, are needed to guarantee coverage
out to $\theta=100$ mrad. The most downstream LAV station (LAV25) leaves as
much of the front face of the Main Electromagnetic Calorimeter (MEC) uncovered
as possible, and the dimensions and positions of the other LAVs have been
chosen so that LAV25 defines the most restrictive aperture for photons from
the fiducial volume. The dimensions and segmentation of the 25 LAV stations
are specified in Table 13.
Table 13: Dimensions, segmentation, number of readout channels, and total quantity of plastic scintillator for the five different types of LAV stations. LAVs | $r_{\rm int}$ (m) | $r_{\rm ext}$ (m) | Sectors | Total channels | Tot. scint. (kg)
---|---|---|---|---|---
1–11 | 0.44 | 0.85 | 40 | 3520 | 9690
12–15 | 0.58 | 0.99 | 48 | 1536 | 4290
16–18 | 0.72 | 1.23 | 56 | 1344 | 4970
19–21 | 0.86 | 1.37 | 64 | 1536 | 5680
22–25 | 1.00 | 1.51 | 72 | 2304 | 8525
As noted above, 11 of the LAV stations will be constructed for the HIKE Phase
1. The 12th station, operated in air rather than in vacuum, might be used for
Phase 1 only. The remaining 14 stations in vacuum will be constructed for the
HIKE Phase 2, as needed, and for KLEVER. The sizes and positions of the LAVs
were implemented in the KLEVER simulation before the HIKE $K^{+}$ phase was
proposed. Because the polar angle coverage was extended by increasing the
number of LAV stations rather than their dimensions, the dimensional
parameters of the KLEVER LAVs, though not the same as for NA62, are quite
similar. Although the LAVs must be positioned carefully in KLEVER to guarantee
the needed coverage, with some careful engineering work, it should be possible
to reconcile the LAV designs for the HIKE $K^{+}$ and $K_{L}$ phases. For
example, from the comparison of Tables 12 and 13, it seems likely that LAVs
1–5 for the $K^{+}$ program could be recycled as LAVs 11–15 for $K_{L}$, LAVs
6–8 for $K^{+}$ would become LAVs 16–18 for $K_{L}$, and LAVs 9–11 for $K^{+}$
would become LAVs 19–21 for $K_{L}$. Various sections of the NA62 vacuum tank
would have to be modified or remade.
##### 10.2.2.3 Design and expected performance
As a reference for what low-energy photon detection efficiencies can be
achieved for the new LAVs, Fig. 35 shows the inefficiency parameterization
used for the KOPIO proposal [203].
Figure 35: Photon detection inefficiency parameterization from KOPIO [203],
broken down by source. Measured inefficiencies for the E949 barrel veto [204]
are also plotted.
For the energy range $50~{}{\rm MeV}<E<170~{}{\rm MeV}$, the overall
parameterization (black circles) is based on detection inefficiencies measured
for the E787/949 barrel photon veto [204] using $K^{+}\to\pi^{+}\pi^{0}$
events; these data are also shown in the figure. Outside of this range, the
parameterization is guided by FLUKA simulations with different detector
designs, and the overall result is (slightly) adjusted to reflect the
segmentation of the KOPIO shashlyk calorimeter, but for most of the interval
$E<200$ MeV, the results do not differ much from the E949 measurements. The
contributions to the inefficiency from photonuclear interactions, sampling
fluctuations, and punch through were estimated from known cross sections,
statistical considerations, and mass attenuation coefficients.
One possible design for the HIKE LAVs would be similar to the Vacuum Veto
System (VVS) detectors planned for the CKM experiment at Fermilab [205]. The
CKM VVS is a lead/scintillator-tile detector with a segmentation of 1 mm Pb +
5 mm scintillator, for an electromagnetic sampling fraction of 36%. This
segmentation is the same as for the E787/949 barrel photon veto, so the same
low-energy efficiencies might be expected. The wedge-shaped tiles are stacked
into modules and arranged to form a ring-shaped detector. The scintillation
light is collected and transported by 1-mm-diameter WLS fibers in radial
grooves, as seen in Fig. 36.
Figure 36: Prototype tile with WLS fiber readout for the CKM VVS detector
[205].
In the approximate geometry for HIKE, the LAV modules would consist of 96
layers, for a total thickness of $\sim$60 cm, corresponding to $\sim$18
$X_{0}$. There are 40–72 modules per detector, each with 20 fibers per tile.
In the original VVS design, the fibers brought the light to optical windows
for readout by PMTs outside of the vacuum. Readout by SiPMs inside the vacuum
would make for shorter fibers and would facilitate the mechanical design—the
availability of economical SiPM arrays with large effective area makes this an
attractive option. In the HIKE geometry, the fibers in a module would be
bundled for readout by eight SiPM arrays. As in the VVS design, alternating
fibers from each tile would be read by different SiPMs to provide redundancy,
and there would be four readout layers in depth.
Figure 37: Measurements of detection inefficiency for tagged electrons with
203, 350, 483 MeV energies for three veto prototypes, made at the Frascati BTF
in 2007. The red squares are for the CKM VVS prototype.
In the original NA62 proposal [206], before the OPAL lead glass became
available, very similar detectors were the baseline solution for the existing
LAVs, and in 2007, the efficiency of the CKM VVS prototype in the energy range
200–500 MeV was measured using a tagged electron beam at the Frascati Beam-
Test Facility (BTF) [202]. The results are shown in Fig. 37. Earlier, the
efficiency of the same prototype was measured at the Jefferson National
Laboratory with tagged electrons in the interval 500–1200 MeV. An inefficiency
of $3\times 10^{-6}$ was found at 1200 MeV, even with a high threshold (80
MeV, or 1 mip) [207]. The efficiency parameterization for the LAVs used in our
simulations is based on the KOPIO efficiencies up to the point at 129 MeV, and
then extrapolated through the three points measured at the BTF, to $2.5\times
10^{-6}$ for photons with $E>2.5$ GeV. This is not unreasonable, considering
that the Jefferson Lab measurement shows that nearly this inefficiency is
already obtained at 1.2 GeV. The tests of this prototype at the BTF were not
optimized for the measurement of the time resolution, but indicated a time
resolution of better than 250 ps for 500 MeV electrons, which should be
sufficient for HIKE, including KLEVER.
##### 10.2.2.4 Readout
A leading contribution to the random veto inefficiency for the LAVs in NA62 is
from halo muons. If the interactions of halo muons could be reliably
distinguished from photon showers, the random veto inefficiency for the LAVs
would be decreased, potentially relaxing the requirements on the LAV time
resolution. In NA62, the LAV signals are discriminated against two thresholds,
and signal amplitudes are obtained using the time over threshold technique.
While some use of this information has been made in NA62 to partially recover
the LAV random veto inefficiency and efforts are continuing in this direction,
in practice, the separation between mips and low-energy photon showers has not
so far proved to be reliable. To improve upon this situation, we are
investigating the gains to be had with a fully digitizing FADC readout for the
LAVs, as is foreseen for all of the photon veto detectors in KLEVER.
#### 10.2.3 Spectrometer
A spectrometer similar to the NA62 straw tracker [175], comprising four straw
chambers and a dipole magnet, is planned to reconstruct the momentum and
direction of charged particles in the final state. The straw tubes are
expected to remain in the vacuum tank containing the decay region, profiting
from the successful technology developed for NA62. Straws with diameter less
than 5 mm are necessary to handle the expected particle rates at the higher
beam intensity. Straw diameter reduction by a factor $\sim 2$ with respect to
the NA62 straws will lead to shorter drift times and an improvement in the
resolution of the trailing edge time from the current ${\sim}30$ ns to
${\sim}6$ ns. A smaller diameter of the straw also requires a change in the
geometric placement of the straws in a single view. Design work based on Monte
Carlo simulations was performed, and the straw layout was optimised taking
into account realistic spacing and dimension requirements, resulting in a
choice of eight straw layers per view, shown in Fig. 38.
The material used to make new straws using the ultrasonic welding technique
will be the same as in the current spectrometer, namely Mylar coated with 50
nm of copper and 20 nm of gold on the inside. To reduce the detector material
budget, the thickness of the mylar will be reduced from $36~{}\mu$m to either
$12~{}\mu$m or $19~{}\mu$m. The diameter of the gold-plated tungsten anode
wires might be reduced from $30~{}\mu$m to $20~{}\mu$m. The final decision on
the mylar thickness and the wire diameter will be made based on mechanical
stability tests. The development of small-diameter thin-walled straws has
synergies with R&D work for COMET phase II at J-PARC [208], and is included in
the ECFA detector R&D roadmap [209].
Based on the results of the design study, a Geant4-based simulation of the new
spectrometer was developed using the same dimensions and positions of the
straw chambers, the number and orientation of views in the chamber, the gas
composition (Ar+CO2 with 70:30 ratio) and the properties of the dipole magnets
as in the current NA62 layout. A comparison between the two straw detectors is
given in Table 14, and a Geant4 visualisation of the new spectrometer is shown
in Fig. 39. The new spectrometer is planned to have the capability of aligning
the central holes of the straw chambers on the beam axis for both $K^{+}$
(Section 6) and $K_{L}$ (Section 7) modes of operation.
The NA62 track reconstruction algorithm was adapted for the new detector, and
a preliminary resolution comparison indicates that the new spectrometer could
improve the resolution for the reconstructed track angles and momenta by
10–20% with respect to the existing NA62 spectrometer while maintaining the
high track reconstruction efficiency.
Investigations of possible technological solutions for the straw connectivity,
design of a new high-voltage board, and a pre-production of straw tubes with a
diameter of 4.82 mm have already been started (Figs. 42, 43, 44).
Figure 38: Optimised layout of straw tubes with a diameter of 4.82 mm in a view. Table 14: Comparison of the NA62 spectrometer and the new straw spectrometer. | Current NA62 spectrometer | New straw spectrometer
---|---|---
Straw diameter | 9.82 mm | 4.82 mm
Straw length | 2100 mm | 2100 mm
Planes per view | 4 | 8
Straws per plane | 112 | ${\sim}160$
Straws per chamber | 1792 | ${\sim}5200$
Mylar thickness | $36~{}\upmu$m | (12 or 19) $\upmu$m
Anode wire diameter | 30 $\upmu$m | (20 or 30) $\upmu$m
Total material budget | 1.7% $X_{0}$ | (1.0 – 1.5)% $X_{0}$
Maximum drift time | ${\sim}150~{}$ns | ${\sim}80~{}$ns
Hit leading time resolution | (3 – 4) ns | (1 – 4) ns
Hit trailing time resolution | ${\sim}30~{}$ns | ${\sim}6~{}$ns
Average number of hits hits per view | 2.2 | 3.1
Figure 39: Geant4 visualisation of the new straw spectrometer.
Figure 40: Geant4 visualisation of a new straw chamber: (left) front view;
(right) tilted back view.
Figure 41: Preliminary comparison of resolutions of track momentum (left) and
track $\theta_{X}$ angle (right) between the existing NA62 spectrometer (blue)
and the new spectrometer with $12~{}\mu$m mylar thickness (red). A similar
improvement is observed in the reconstructed $\theta_{Y}$ angle.
Figure 42: Left: detail of the connectivity elements (coaxial) in order to
minimise the signal path to the front-end electronics. Right: cross-section of
the prototype to validate connectivity and basic performance of a 4.82 mm
diameter straw using new front-end electronics with the capability to measure
the trailing edge.
Figure 43: Left: pre-production of straws with a diameter of 4.82 mm and a
wall thickness of 19 $\mu$m. Right: test of signal connectivity and high-
voltage stability of individual components. Figure 44: Prototype of a HV
board for the new straw spectrometer.
### 10.3 Detectors downstream of the fiducial decay volume
#### 10.3.1 The RICH detector
The NA62 RICH detector, which uses neon at atmospheric pressure as the
radiator, is well suited for operation within the HIKE programme in terms of
the mechanical structure (vessel, mirror support, end-caps). Major changes
should only concern the Cherenkov light sensors and the two flanges hosting
them. Improvement of the geometrical acceptance for negative particles is also
being considered.
##### 10.3.1.1 New photodetectors
The present NA62 RICH is equipped with Hamamatsu R7400-U03 phototubes with a
time resolution of 240 ps for single photons, quantum efficiency (QE) of
$\sim$20$\%$ and with a distance between the centres of adjacent sensors of 18
mm [210]. This distance, constrained by the sensor size, gives the main
contribution to the resolution on the single hit position ($\sim$4.7 mm) and
consequently to the overall resolution on the ring radius ($\sim$1.5 mm), the
main parameter driving the performance on the particle identification (PID) of
the RICH detector [211]. The single hit time resolution together with the
number of hits associated to each ring (which depends on the QE) determines
the overall time resolution of the RICH for positive tracks of 80 ps on
average.
The main requirement for RICH operation at HIKE Phase 1 is a ring time
resolution of 20–30 ps, in order to reduce the coincidence window with the
upstream detectors (beam tracker and KTAG). The RICH is the only downstream
detector that could reach such resolution for the kaon decay products.
Besides, a reduction of the sensor size would improve the particle
identification performance. Silicon photomultiplier (SiPMs) meet these
requirements: SiPMs with 100 ps time resolution and QE of 40% or above are
already available from the main manufacturers.
Considering a similar active area geometrical filling factor (80%) and a
similar average number of hits per ring to the NA62 case, and taking into
account a factor 2 improvement in the QE, we have evaluated the number of hits
per ring and the track time resolution with the new configuration. The time
resolution for pion momenta of 15 and 45 GeV/$c$ (the limits of the RICH
working region) for the NA62 RICH, and those expected for the future RICH
instrumented with SiPMs, are listed in Table 15. The latter meets the HIKE
requirements.
Table 15: Comparison of the time resolution of the NA62 RICH and the new RICH. | NA62 RICH | Future RICH
---|---|---
Sensor type | PMT | SiPM
Sensor time resolution | 240 ps | 100 ps
Sensor quantum efficiency | 20% | 40%
Number of hit for $\pi^{+}$ at 15 GeV/$c$ | 7 | 14
Number of hit for $\pi^{+}$ at 45 GeV/$c$ | 12 | 24
Time resolution for $\pi^{+}$ at 15 GeV/$c$ | 90 ps | 27 ps
Time resolution for $\pi^{+}$ at 45 GeV/$c$ | 70 ps | 20 ps
Replacement of the light sensors, necessary to improve the time resolution,
also represents an opportunity to improve the RICH performance in terms of
particle identification. A smaller sensor size and an improved QE will allow
to establish a more optimal RICH working point, improving both the muon
rejection and pion identification efficiency in the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ analysis. Fig. 45 illustrates the improvement in
the ring radius resolution $\sigma_{\rm Radius}$ achieved by sensor size
reduction and QE improvement. For a small SiPM size, the contributions to the
single hit resolution coming from misalignment of the reflecting mirrors and
neon (radiator) dispersion, 0.6 mm and 2.1 mm respectively, become dominant.
Figure 45: Layout of the present RICH NA62 sensors compared to SiPM candidates
(in scale) and the corresponding performance in terms of the ring radius
resolution.
In considering SiPMs as photodetector candidates for the HIKE RICH, the
effects of the main drawbacks related to these sensors must be evaluated:
* -
Dark count: for a 800 ps coincidence window, and an annulus area of $7\times
10^{4}~{}{\rm mm}^{2}$ considered for evaluation of the possible particle
identification hypotheses, a dark count rate of several kHz/mm2 would produce
a non-negligible number of spurious hits. To lower the contamination to the
level of few percent, the SiPM should be operated at low temperature. A
possible layout of the flanges housing the SiPM and the cooling system is
discussed in Section 10.3.1.2.
* -
Cross-talk is strongly dependent on the SiPM type, can be reduced by cooling
the SiPM, and will be considered for the choice of the final sensor. Cross-
talk leads to an extra contribution (for a small fraction of events) to the
hit space resolution, and consequently to the ring radius resolution.
* -
Ageing/radiation hardness: the RICH flanges are located at a distance of 1.5 m
from the beam pipe and are not traversed by the bulk of the particle flux.
Nevertheless the radiation level in the high intensity environment should be
investigated. Cooling of the SiPMs would reduce the effects of radiation
damage.
Alternative photodetectors are the MCP-PMTs considered for the HIKE KTAG
detector and described in Section 10.1.1.
##### 10.3.1.2 New mechanics
The NA62 RICH vessel, end-caps and the mirror support panel will not be
changed in the HIKE configuration, apart for minor modifications. Concerning
the mirrors, a replacement or a re-aluminisation should be considered to
recover from the deterioration of the coating layer already observed in 2014
during their installation. The new reflecting surface must match, in terms of
reflectivity as a function of light wavelength, the quantum efficiency of the
photodetectors, so both the reflective and coating layers will be chosen
accordingly.
The change of sensors will request modification of the two flanges hosting
them. This represents an opportunity to increase the RICH acceptance for
negatively-charged particles (note that the NA62 RICH is optimised for the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay). Simulations show that increasing the
instrumented area from 5700 cm2 to 7600 cm2 would lead to a good acceptance
for negative tracks. The end-cap region that can be instrumented without
compromising the vacuum-proof mechanics is shown in Fig. 46. In the redesign
of the new flanges, a photosensor cooling system will be introduced, which
would avoid inducing a temperature gradient in the RICH radiator gas. An
adequate system to guarantee thermal insulation between the sensor flanges and
the vessel must be considered.
Figure 46: The upstream RICH end-cap after its construction, in 2013. The
circular hole hosts the sensor flange in the NA62 RICH. The red contour
delimits the region that can be instrumented with new photo-sensors.
#### 10.3.2 The timing detector
The timing detector will allow to have a redundant and complementary
measurement of the transit time of charged particles in the downstream
detector region with respect to the RICH. The information of the timing plane
can be used for efficiency studies, trigger purposes and to veto accidental
activity. The timing detector can also be used to have more than one time
measurement for pions and muons with momentum below the Cherenkov thresholds
in the RICH of 12 and 8 GeV/$c$, respectively. The requested time resolution
is about 100 ps.
For the construction of this detector HIKE will profit from the expertise
gained by NA62 with the NA48-CHOD and NA62-CHOD detectors. The NA48-CHOD,
consisting of two planes of scintillator slabs of approximately $100\times 6$
cm2 size, cannot be used in the HIKE environment; the high particle rate (and
the high probability of more than one track hitting the same counter) will not
allow to correct the measured time for the light propagation time in the
scintillator to the photo-sensors depending on the track impact point. On the
contrary, the NA62-CHOD layout (Fig. 47), consisting of tiles of approximately
$10\times 13$ cm2 (in the central region) will be suitable for HIKE. The
expected maximum rate per tile of 2.8 MHz (inferred from experience with NA62
data taking) is affordable and can be lowered further by reducing the
dimension of the tiles near the beam pipe, where the particle rate is higher.
Figure 47: Layout of the present NA62 CHOD in the $xy$ and $xz$ planes.
The NA62-CHOD time resolution (500 ps) is limited by the light transportation
outside the detector acceptance via wavelength shifting fibers. In the HIKE
setup, the photosensors will be attached directly to the tiles to avoid the
degradation. In this layout the sensors will be irradiated by the downstream
particle flux. The photosensor type will be chosen profiting from the studies
foreseen for other detectors (KTAG, RICH, HCAL, MUV). To further improve the
time resolution, an additional detector plane can be considered.
#### 10.3.3 The HIKE electromagnetic calorimeter
The forward electromagnetic calorimeter is crucial for any kaon program. In
particular, in the HIKE $K^{+}$ phase this detector serves as the principal
photon veto for the measurement of $K^{+}\to\pi^{+}\nu\bar{\nu}$ and is
essential for the reconstruction of the final states for decays like $K_{\ell
2}$, $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$, and $K^{+}\to\pi^{+}\gamma\gamma$. For
the neutral beam configurations, the electromagnetic calorimeter reconstructs
the $\pi^{0}$ vertex for $K_{L}\to\pi^{0}\nu\bar{\nu}$ events and helps to
reject events with extra photons. Many of the same issues arise in the design
of the electromagnetic calorimeter for the $K^{+}$ and $K_{L}$ phases. We
therefore seek a design for a fast calorimeter with excellent photon detection
efficiency and energy resolution to be used in all phases of the HIKE program.
While the new calorimeter is investigated and built, the existing NA62 LKr
calorimeter is presented as a possible solution for the $K^{+}$ program since
it could meet the performance requirements.
##### Performance requirements
The principal performance requirements for the electromagnetic calorimeter are
excellent energy resolution and intrinsic detection efficiency for high-energy
photons, good two-cluster separation for photons, and excellent time
resolution.
It is natural to inquire as to whether the liquid-krypton calorimeter (LKr)
[212] currently used in NA62 can be reused for HIKE. The energy, position, and
time resolution of the LKr calorimeter were measured in NA48 to be
$\displaystyle\frac{\sigma_{E}}{E}$
$\displaystyle=0.0042\oplus\frac{0.032}{\sqrt{E{\rm(GeV)}}}\oplus\frac{0.09}{E{\rm(GeV)}},$
(6) $\displaystyle\sigma_{x,y}$ $\displaystyle=0.06~{}{\rm
cm}\oplus\frac{0.42~{}{\rm cm}}{\sqrt{E{\rm(GeV)}}},$ (7)
$\displaystyle\sigma_{t}$ $\displaystyle=\frac{2.5~{}{\rm
ns}}{\sqrt{E{\rm(GeV)}}}.$ (8)
Indeed, the efficiency and energy resolution of the LKr appear to be
satisfactory for all phases of HIKE. Studies of $K^{+}\to\pi^{+}\pi^{0}$
decays in NA48 data and tests conducted in 2006 with tagged photons from an
electron beam confirmed that the LKr has an inefficiency of less than
$10^{-5}$ for photons with $E>10$ GeV, providing the needed rejection for
forward photons [213]. These studies were fully confirmed in NA62.
Notwithstanding the presence of a much larger amount of material upstream of
the LKr calorimeter in NA62 than in NA48, a study of single-photon efficiency
underpinning the NA62 measurement of ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$
(and used to obtain a limit on ${\rm BR}(\pi^{0}\to{\rm invisible})$) found an
inefficiency of about $10^{-5}$ at $E=20$ GeV, slightly decreasing at higher
energies [142]. The LKr time resolution, however, is a significant issue. The
necessary upgrade for the LKr calorimeter to be used in the $K^{+}$ phase is
discussed below (Section 10.3.3.1). For the $K_{L}$ phases, and for KLEVER in
particular, the calorimeter provides the measurement of the event time, and
must have a time resolution of 100 ps or better for the reconstruction of
$\pi^{0}$’s with energies of a few GeV. Additionally, the size of the LKr
inner bore would limit the beam solid angle and hence the kaon flux during the
$K_{L}$ phases. A new calorimeter that would meet these requirements is
described in Section 10.3.3.2.
##### 10.3.3.1 The NA62 LKr calorimeter
The LKr energy resolution meets the HIKE requirements, while the time
resolution must be substantially improved. This can be achieved by performing
a major upgrade of the LKr readout electronics, as discussed below.
##### Major upgrade of the LKr readout electronics
The readout of the NA62 liquid krypton calorimeter is based on the measurement
of the initial current induced by the charge generated by the showers, to have
a fast response and to separate pile-up pulses. The electronics chain for each
cell is composed of three elements:
* •
a preamplifier, sitting in the cold liquid, to collect the charge with an
integration time of about 150 ns;
* •
a so-called transceiver module, mounted immediately after the cryostat feed-
throughs, whose function is to restore the fast signal, removing the pole of
the integration in the preamplifier, and to prepare a differential signal to
drive long cables to the next stage;
* •
a shaper, to have a narrow signal (70 ns), followed by an FADC to digitise the
signal at a rate of 40 MHz.
The increase in intensity by a factor 4 will produce more pile-up events,
whose detection could be difficult with the existing chain. To cope with this,
the baseline for a new readout structure will be the reduction of the shaping
time to the minimum possible of about 28 ns (the transceiver output has a
risetime of about 20 ns which adds to the time constants of the shaper) with a
reduction of the amplitude of about 40%, and subsequent digitization at 160
MHz, which is above the Nyquist limit, but which could help in identifying
superposition of pulses looking at the width of the pulse.
##### Space charge
The increase of intensity will also worsen the effect of the space charge:
already in the current operation we observe a loss of response in the hottest
cells, due to the build up of the space charge during the burst. This effect
is inversely proportional to the square of the value of the high voltage
across the cell and directly proportional to the energy density created by
ions. The effect of the space charge is, at low densities, to reduce the
electric field at the anode and to increase it at the cathode. There is then a
threshold value after which the field at the anode is zero and above this
value an increasing fraction of the cell is unresponsive.
With the current NA62 beam intensity we see already that in the hottest cells
we are near or just above the critical value, clearly seen both from the
decrease in the response across the burst and from a modulation of the
response across the $x$ coordinate of the cells. An increase in intensity will
worsen this behaviour for the hottest cells, but it will also extend the area
affected by this problem. To mitigate this issue, an increase of the high
voltage is possible: going from the actual 3.5 kV to 5 kV will reduce by two
the value of the critical parameter and the performance will be the same as
now, which is not a problem for veto operations. An additional improvement for
all analyses that do not use the calorimeter as a veto will be to increase the
size of the actual Intermediate Ring Counter (IRC, Section 10.3.4), to cover
the area most affected by the space charge.
##### 10.3.3.2 The new HIKE electromagnetic calorimeter
We are investigating the possibility of replacing the LKr with a shashlyk
calorimeter patterned on the PANDA FS calorimeter, in turn based on the
calorimeter designed for the KOPIO experiment [214]. This design featured
modules $110\times 110$ mm2 in cross section made of alternating layers of
0.275-mm-thick lead absorber and 1.5-mm-thick injection-moulded polystyrene
scintillator. This composition has a radiation length of 3.80 cm and a
sampling fraction of 39%. The scintillator layers were optically divided into
four $55\times 55$ mm2 segments; the scintillation light was collected by WLS
fibres traversing the stack longitudinally and read out at the back by
avalanche photodiodes (APDs). KOPIO was able to obtain an energy and time
resolution of 3.3% and 73 ps at 1 GeV with this design, establishing that it
is capable of providing the same energy resolution as the LKr while meeting
the time resolution requirements for HIKE.
For HIKE, the design would be updated to use silicon photomultipliers (SiPMs)
instead of APDs. The final choice of module size and readout granularity has
yet to be determined, but on the basis of KLEVER simulations that assume that
clusters are resolved if more than 6 cm apart, readout cells of $5\times 5$
cm2 seem reasonable. The Molière radius for the fine-sampling shashlyk design
described above is 3.39 cm, so smaller cells can be used if needed. For
comparison, the Molière radius of liquid krypton is 5.86 cm, and the NA48 LKr
calorimeter features $2\times 2$ cm2 cells.
Figure 48: Wireframe drawing of the MEC as implemented in Geant4 for the NA62
Monte Carlo.
The new calorimeter, referred to as the main electromagnetic calorimeter (MEC)
in KLEVER, would have an inner bore of at least 12 cm in radius to allow the
passage of the neutral beam. The bore could be widened to as much as 15 cm to
allow the penumbra of beam photons and neutral hadrons to pass through and be
intercepted by the small-angle vetoes. For the $K^{+}$ phase, a smaller inner
aperture would be required; in NA62, this angular region is covered by the IRC
(Section 10.3.4), and a new IRC based on the same shashlyk technology would be
used together with the MEC during the $K^{+}$ phase. The sensitive area has an
outer radius of 125 cm. Figure 48 shows a wireframe drawing of the calorimeter
as implemented in the NA62 Geant4-based Monte Carlo for HIKE.
Figure 49: Geant4 simulation of a 5 GeV photon showering in a MEC module.
Figure 50: Detection efficiency as a function of incident particle energy, for
various thresholds on visible energy, from the Geant4 simulation.
For the KLEVER sensitivity studies of Section 8.3, the inefficiency for photon
detection assumed for the MEC was assumed to be 1 for energies below 100 MeV,
falling exponentially to $10^{-3}$ at 1 GeV, $10^{-4}$ at 5 GeV, and thence to
$10^{-5}$ at 15 GeV, in accordance with experience with the LKr as discussed
above. The model of the shashlyk calorimeter developed for the HIKE Monte
Carlo does not include the light and signal readout, which are still being
defined, but does contain a detailed representation of the module structure,
allowing studies of the efficiency as a function of visible energy deposition.
Figure 49 shows the simulated interaction of a 5 GeV photon in a module of the
calorimeter. Figure 50 shows results obtained from the simulation for the
photon detection inefficiency as a function of incident energy for various
thresholds on visible energy. The simulation, which confirms the visible
energy fraction of 39%, demonstrates that the fine-samplng shashlyk design
satisfies the efficiency requirements for KLEVER, as expected.
The radiation resistance of the scintillator is a potential concern,
especially for the $K_{L}$ phase. In KLEVER, the dose rate is dominated by
photons from $K_{L}$ decays. The rate is most intense on the innermost layers
of the calorimeter, for which it is about 2 kHz/cm2. Precise dose rate
calculations have yet to be performed, but an estimate suggests a dose of 4
kGy/yr to the scintillator for the innermost layers. This estimate would
suggest that radiation damage, while a concern, is likely manageable.
Radiation robustness may be a factor in the final choice of scintillator.
Figure 51: Shashlyk tiles made of perovskite nanocomposite scintillator tested
as an alternative to conventional scintillator, in ambient light (left) and
under ultraviolet light (right).
Although current information suggests that optimized formulations of
conventional polystyrene scintillator are sufficiently luminous, fast, and
radiation resistant, in synergy with the AIDAinnova project NanoCal, we are
evaluating the advantages that can be obtained with less conventional choices
for the light emitter (e.g., perovskite [198, 215] or chalcogenide [216]
quantum dots) or matrix material (e.g., polysiloxane [217]). In particular, we
have participated in a recent head-to-head experimental comparison of small
shashlyk prototypes made from conventional scintillator, specifically, the
extrusion-moulded polystyrene scintillator formulated at IHEP Protvino for
KOPIO [214], with 1.5% PTP and 0.04% POPOP, and a nanocomposite scintillator
consisting of 0.2% lead halide bromide (CsPbBr3) nanocrystals in PMMA222Glass
To Power SpA, Rovereto TV, Italy.. The latter, which emits at 520 nm (Figure
51), is expected to be a very fast and bright alternative to conventional
scintillators; its comparatively long wavelength emission and use of PMMA as a
matrix material is expected to confer good radiation hardness.
Figure 52: Small shashlyk prototypes tested in fall 2022, during construction
(left) and on the H2 beam line for testing (right).
Identical prototypes consisting of 12 layers of 0.6-mm lead and 3-mm
scintillator were constructed out of components from the PANDA/KOPIO
calorimeter (Figure 52). For the conventional prototype, the original
scintillating tiles were used with Kuraray Y-11(200) green-emitting WLS fibre.
The nanocomposite prototype used Kuraray O-2(100) orange-emitting WLS fibre.
Each was read out with a single Hamamatsu 13360-6050 SiPM ($6\times 6$ mm2, 50
$\mu$m pixel size) and fast amplifier with a gain of 4. Both protoypes worked
well; the conventional prototype showed significantly more light output than
the nanocomposite prototype, but we have reason to believe that the luminosity
and attenuation length of the orange fibres plays a significant role. A series
of tests are planned to explore different fibre-scintillator pairings
(including the use of custom-produced fibres) as well as successively
optimised nanocomposite scintillator formulations.
In addition to the basic criteria on energy resolution, efficiency, time
resolution, and two-cluster separation, ideally, the MEC would provide
information useful for particle identification. For example, in KLEVER,
identification of pion interactions would provide additional suppression of
background for decays with charged particles in the final state, and, as the
experience with KOTO suggests, it is crucial to have as much information as
possible to assist with $\gamma/n$ discrimination. The fine transverse
segmentation of the MEC will play an important role: a simple cut on cluster
RMS with the existing LKr can suppress up to 95% of pion interactions in NA62
data. Fast digitisation of the signals from the MEC is expected to provide
additional $\gamma/n$ discrimination. For both the $K^{+}$ and $K_{L}$ phases,
the MEC will be backed up with hadronic veto calorimeters as discussed in
Section 10.3.5.
Figure 53: Top: Geant4 model of small prototype for romashka calorimeter,
featuring spy tiles placed at key points in the shashlyk stack, together with
a photograph of the module constructed at Protvino. Bottom: Fiber routing
scheme for independent readout of spy tiles, giving rise to the name romashka
(chamomile).
We are also experimenting with concepts to obtain information on the
longitudinal shower development from the shashlyk design. One possible concept
makes use of “spy tiles”, 10-mm thick scintillator bricks incorporated into
the shashlyk stack but optically isolated from it and read out by separate WLS
fibres. The spy tiles are located at key points in the longitudinal shower
development: near the front of the stack, near shower maximum, and in the
shower tail, as illustrated in Fig. 53. This provides longitudinal sampling of
the shower development, resulting in additional information for $\gamma/n$
separation. The prototype shown in Fig. 53 was constructed at Protvino in
early 2018. Its basic functionality was tested in the OKA beamline in April
2018, and more comprehensive tests were carried out in September 2019 at DESY
in collaboration with LHCb. Simulations suggest (and preliminary test beam
data validate, to a certain extent) that the romashka design with spy tiles
can give at least an order of magnitude of additional neutron rejection
relative to what can be obtained from the transverse segmentation of the
calorimeter alone, providing an overall suppression of up to 99.9% for neutron
interactions. The small prototype has a cross sectional area of only $55\times
55$ mm2 (one readout cell) and a depth of $14X_{0}$ (60% of the design depth),
and both transverse and longitudinal leakage significantly complicated
attempts to measure the time resolution. For electrons with $1<E<5$ GeV, the
measured time resolution was about 200 ps and virtually constant; we expect
that this can be significantly improved. The time resolution for hits on the
spy tiles (independently of information from the shashlyk stack) was on the
order of 500–600 ps, which may be difficult to improve. This is not expected
to be a problem, however: the main shashlyk signal establishes the event time
and the association of the PID information from the shashlyk tiles is based on
the segmentation, with occupancies per cell of at most a few tens of kHz on
the innermost layers.
Although promising, the romashka design is not the only solution under
investigation for obtaining information on the longitudinal shower
development. Other concepts being explored, besides variations on the spy tile
readout such as on-board SiPMs, include alternatives such as two-sided
front/back readout and explicit segmentation into two or more modules in
depth.
We plan to continue R&D work on improvements to the shashlyk design such as
from the use of innovative scintillators and longitudinal segmentation schemes
through 2023, while at the same time, we are designing a full-scale prototype
of the baseline solution with conventional scintillator and a uniform shashlyk
stack for beam testing next year.
#### 10.3.4 Intermediate-ring calorimeter (IRC)
In NA62, the IRC is a small, ring-shaped calorimeter that sits just upstream
of the LKr to provide photon veto coverage for the angular interval
corresponding to the inactive region near the inner surface of the LKr
cryostat. The NA62 IRC has an inner radius of 60 mm and an outer radius of 145
mm. The inner and outer radii are not concentric—the inner bore is offset in
the horizontal direction by 10 mm, so that it is centered on the beam axis. In
NA62, the IRC is a shashlyk calorimeter with layers consisting of 1.5 mm of
lead and 1.5 mm of scintillator. It is segmented into quadrants and read out
with PMTs.
In HIKE, as discussed in Section 10.3.3, the IRC will be used to extend the
coverage of the calorimeter to small radii, and performance specifications
similar to those for the calorimeter will be necessary. If the LKr used is
during the $K^{+}$ phase, the IRC will cover the region of highest rates on
the LKr, where space-charge effects are important. The new MEC, not needing a
cryostat, will have minimum dead space at its inner radius. However, it could
be convenient to have a bore of 15 cm or more to allow passage of the neutral
beam halo during the $K_{L}$ phase of the experiment. In that case, during the
$K^{+}$ phase, the IRC would be used in the same way with the MEC as it would
with the LKr. For the HIKE $K_{L}$ phases, the IRC will then be moved
downstream of the calorimeter and used to veto photons at the small radii
occupied by the penumbra of the neutral beam.
Ideally, the same instrument would be used for both phases, but due to the
horizontal displacement of the $K^{+}$ beam at the location of the IRC,
changes to the geometry may be necessary. If two different IRCs are needed,
they can certainly be built with identical technology, namely, as shashlyk
calorimeters with geometry similar to that for the NA62 IRC and sampling and
readout granularity as for the HIKE MEC, providing more light for better time
resolution and higher readout granularity for better rate resistance.
#### 10.3.5 Hadron Calorimeter (HCAL) and Muon Veto Detector (MUV)
##### 10.3.5.1 Hadron calorimeter requirements
As in NA62, the hadron calorimeter (HCAL) is the main detector for $\pi/\mu$
identification and separation. While minimum ionizing muons can be
distinguished from pion showers without difficulty, so-called catastrophically
interacting muons, which deposit all or a large fraction of their energy in
the calorimeter, are of more concern. The $\pi/\mu$ separation is therefore
achieved both by longitudinal tracking through the whole calorimeter and by
examining the transversal shower shapes of electromagnetic muon versus
hadronic pion showers. Similar to the NA62 MUV1-3 detectors, an average muon
misidentification probability of ${\cal O}(10^{-6})$ over the momentum range
from 10 to about 50 GeV is necessary (see Section 6). The HCAL therefore needs
segmentation in all three dimensions.
The 4–6 times higher intensity in HIKE with respect to NA62 leads to a total
rate of charged tracks from $K^{+}$ decays of ${\cal O}(10~{}\text{MHz})$ on
the calorimeters. This converts into a sub-nanosecond time resolution which is
required to avoid dead-time from random veto. Such a time resolution cannot be
achieved with scintillating strips as in the NA62 MUV1/MUV2, since the long
light paths cause a resolution of 1 ns or worse. In addition, a strip read-out
would suffer from double hits on the same strip at high rates.
The HIKE HCAL therefore needs to have a cellular layout to reduce both the
rate on each channel and the time resolution. With an iron absorber with a
Molière radius of 1.72 cm, a cell size of $6\times 6$ cm2 allows a sufficient
distinction between hadronic pion showers and electromagnetic showers from
catastrophically interacting muons. A corresponding layout of an HCAL with an
octagonal shape with an inner radius of 126 cm is shown in Fig. 54. It
consists of a grid of $42\times 42$ cells. The subtraction of the beam pipe
region and the corners to obtain an octagon results in 1440 cells in the
transverse plane.
Figure 54: Transverse HCAL layout with an inner radius of 126 cm and readout
cells of $6\times 6$ cm2 cross section. The plane comprises in total 1440
cells.
We consider two options for an iron-scintillator based calorimeter, which are
described in the following. The decision between the two options will be taken
in the near future, after more detailed simulation of the particle ID
capabilities of the two designs.
##### 10.3.5.2 High-granularity hadron calorimeter
For the ILD detector at the proposed International Linear Collider (ILC) the
CALICE Collaboration has developed a highly granular analogue hadron
calorimeter (AHCAL) [218] and a similar detector is being constructed for the
CMS High Granularity Calorimeter Upgrade (HGCAL) [219]. In both these cases
the high granularity is needed to improve jet energy resolution with particle-
flow methods. While jet reconstruction is not necessary in HIKE, high
granularity is nevertheless required for optimum $\pi/\mu$ separation and sub-
nanosecond time resolution in a high- rate environment as described above.
The basic design of a high-granularity HCAL is an iron-scintillator sampling
calorimeter with 3 cm iron absorber plates interleaved with 40 layers of
scintillating tiles of $6\times 6$ cm2 cross section and 1 cm thickness, which
gives enough light yield to detect minimum ionizing particles. The total HCAL
therefore comprises 7.2 nuclear interaction lengths and has a length of about
2 m, when considering additional space for electronics and air. Each
scintillating tile is readout separately by a single SiPM, which is connected
to a common readout board (see Fig. 55 for the Calice AHCAL layout). The SiPM
needs a relatively large dynamic range, i.e. a large number of pixels, for a
linear energy measurement for both minimum ionizing particles and
electromagnetic and hadronic showers. Candidates for such SiPMs are e.g. the
existing Hamamatsu types S13360 or S14160 with more than 14000 pixels and a
sensitive area of $3\times 3$ or $6\times 6$ mm2.
Figure 55: Left: $3\times 3$ cm2 scintillator tiles of the CALICE AHCAL,
wrapped and unwrapped, mounted on a common readout board with SiPMs [218].
Centre and right: back and front sides of a readout board with $6\times 6$ cm2
scintillator tiles [220].
Each readout board of $36\times 36$ cm2 size carries $6\times 6=36$
scintillating tiles, wrapped in reflective foil and glued to the PCB. The
SiPMs are surface-mounted on the PCB and are housed in small cavities on the
bottom side of the tiles (Fig. 55). The front-end electronics together with a
digitizing ASIC occupy the back side of the readout board. The digitization
therefore takes place directly next to the photo detectors, ensuring an
excellent time resolution.
Assuming the general design shown in Fig. 54 with 1440 cells in each layer and
in total 40 scintillating layers, the number of readout boards is 1800 (not
all fully equipped) and the total number of channels is 57600. The possibility
of a reduced channel count by e.g. using larger tiles in the outer region or
less active layers is being studied at the moment.
As described above, the digitization of the SiPM signals is done with an on-
board ASIC. A suitable such chip is the HGCROC [221], which has been developed
for the high-rate environment of the CMS HGCAL.
##### 10.3.5.3 Shashlik hadron calorimeter
As second option we consider a shashlik calorimeter with the same $6\times 6$
cm2 cell size as described above. Each module (cell) consists of 80 absorber
layers of 1.5 cm iron and 80 active layers of 0.5 cm plastic scintillator
(Fig. 56). The scintillator light is read out with wavelength-shifting (WLS)
fibres to either photomultiplier tubes or large SiPMs or SiPM arrays. For
calibration, the scintillators can be activated by light input through an
additional clear fibre.
In the simplest set-up the number of channels is just 1440, greatly reduced
with respect to the Calice-like design. However, such a design would not
provide any information about the longitudinal shower development. In NA62,
the two separate modules MUV1 and MUV2 are of great importance for $\pi/\mu$
separation, therefore a shashlik design should also at least contain two
modules with separate readout or, alternatively, having light readout both on
the front and the back side of the detector.
Figure 56: A shashlik HCAL module. The beam comes from the left. The central
fibre serves for injection of monitoring light.
##### 10.3.5.4 Muon veto detector (MUV)
The muon veto detector (MUV) vetoes muons and contributes to the muon
identification and suppression at a event-building-farm or the analysis level.
It is placed following the HCAL behind a 80 cm thick iron wall (corresponding
to about 5 additional nuclear interaction lengths). The MUV needs a time
resolution of the order of 100–150 ps, to keep the loss of signal due to
random activity at few percent level, as it is presently achieved by NA62 with
a time resolution of about 500 ps.
The MUV will be built from scintillating tiles with direct photo detector
readout. Two feasible options exist. The first scenario is applicable with a
high-granularity HCAL, where the MUV detector would be just another
scintillating layer, but behind the iron wall and with scintillating tiles of
5 cm thickness to ensure high light yields for an optimum time resolution. The
second MUV option is a copy of the NA62 MUV3 detector, also consisting of 5 cm
thick scintillating tiles, but read out by PMTs from the back side. In
contrast to NA62, where the tile front faces measure $22\times 22$ cm2, the
tile size would need to be reduced to about $8\times 8$ cm2 to be able to cope
with the higher rate in HIKE.
#### 10.3.6 Small-angle electromagnetic calorimeter (SAC)
For the measurements of $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\to\pi^{0}\nu\bar{\nu}$, as well as other measurements of rare decays
requiring hermetic photon vetoes, the coverage of the veto system must extend
down to zero in the polar angle, to intercept photons that would otherwise
escape the detector through the downstream beam pipe. Vetoing photons at small
angle is easier for the $K^{+}$ phase than for the $K_{L}$ phases, because the
$K^{+}$ beam can be diverted and dumped outside the acceptance of the small-
angle calorimeter. For the $K_{L}$ phase, instead, the SAC sits directly in
the neutral beam; it must reject photons from $K_{L}$ decays that would
otherwise escape via the downstream beam exit while remaining relatively
insensitive to the very high flux of neutral hadrons in the beam, so that the
experiment is not blinded by the random vetoes from these hadrons. The design
and construction of the SAC for the $K_{L}$ phase is thus a unique challenge.
To the extent that R&D for the KLEVER SAC is well underway, its construction
should be complete before LS3, allowing it to be validated for the $K^{+}$
phase before the end of the present run. The operational experience gained
during the $K^{+}$ phase will then be used to validate the SAC with respect to
the more stringent requirements for KLEVER, which form the basis for the
following discussion. Note that for much of HIKE Phase 2 (specifically, for
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ measurements) a production angle of 2–4 mrad
is preferred. For running at the smallest production angles, the SAC might not
be installed because of the extremely high total neutral beam rate.
Figure 57: Maximum tolerable inefficiency of the KLEVER SAC vs incident photon
energy, for photons from $K_{L}\to\pi^{0}\pi^{0}$ events, from simulation.
The maximum tolerable inefficiency for photon detection for the KLEVER SAC is
illustrated as a function of incident photon energy in Fig. 57. This plot was
obtained from the fast simulation described in Section 8.3 by cumulating the
energy distribution in the SAC for $K_{L}\to\pi^{0}\pi^{0}$ events that evade
all other vetoes and pass all other cuts used to select
$K_{L}\to\pi^{0}\nu\bar{\nu}$ events. The following, indicative efficiency
requirements can be identified:
* •
For $E<5~{}{\rm GeV}$, the SAC can be blind. For the $K\to\pi\nu\bar{\nu}$
studies in both the $K^{+}$ and $K_{L}$ experiments, background processes that
are not otherwise efficiently vetoed do not have photons on the SAC.
* •
For photons with $5~{}{\rm GeV}<E<30~{}{\rm GeV}$, the SAC inefficiency must
be less than 1%. Most photons on the SAC with energies in this range from
$K_{L}$ decays that are otherwise accepted as signal candidates are from
events in which there are two photons on the SAC, so that the efficiency
requirement is relatively relaxed.
* •
Only for photons with $E>30~{}{\rm GeV}$ must the inefficiency be very low
($<10^{-4}$). For $K_{L}$ decays passing analysis cuts with a photon on the
SAC in this energy range, the other photon is emitted at large angle and has
low energy, so the SAC veto is important.
Although these SAC efficiency requirements are not intrinsically challenging,
from the simulations of the beamline without extension (Section 5.3), there
are about 130 MHz of $K_{L}$ mesons, 440 MHz of neutrons, and 40 MHz of high-
energy ($E>5$ GeV) beam photons incident on the SAC, and the required
efficiencies must be attained while maintaining insensitivity to the nearly
600 MHz of neutral hadrons in the beam. In order to keep the false veto rate
to an acceptable level, the hadronic component must be reduced to at most a
few tens of MHz, so that the total accidental rate is dominated by the beam
photons and in any case significantly less than the 100 MHz target cited in
Section 8.2. These requirements lead to the following considerations:
* •
The SAC must be as transparent as possible to the interactions of neutral
hadrons. In practice, this means that the nuclear interaction length
$\lambda_{\rm int}$ of the SAC must be much greater than its radiation length
$X_{0}$.
* •
The SAC must have good transverse segmentation to provide $\gamma/n$
discrimination.
* •
It would be desirable for the SAC to provide additional information useful for
offline $\gamma/n$ discrimination, for example, from longitudinal
segmentation, from pulse-shape analysis, or both.
* •
The SAC must have a time resolution of 100 ps or less.
* •
The SAC must have double-pulse resolution capability at the level of a few ns.
In addition, in five years of operation, the SAC will be exposed to a neutral
hadron fluence of about $10^{14}$ $n$/cm2, as well as a dose of up to several
MGy from photons.
Figure 58: Dimensional sketch of a SAC for KLEVER based on dense, high-$Z$
crystals with both transverse and longitudinal segmentation.
One possible design that is well-matched to the KLEVER requirements would be
to use a highly segmented, homogeneous calorimeter with dense, high-$Z$
crystals providing very fast light output. As an example, the small-angle
calorimeter for the PADME experiment used an array of 25 lead fluoride (PbF2)
crystals of $30\times 30\times 140~{}{\rm mm}^{3}$ dimensions. PbF2 is a
Cherenkov radiator and provides very fast signals. For single crystals read
out with PMTs, a time resolution of 81 ps and double-pulse separation of 1.8
ns were obtained for 100–400 MeV electrons [222], satisfying the KLEVER timing
performance requirements. The PADME performance was obtained with fully
digitising waveform readout at 5 GS/s; waveform digitisation would also be
required for KLEVER (Section 11).
At the doses expected at KLEVER, radiation-induced loss of transparency to
Cherenkov light could be significant for PbF2, as suggested by existing
studies with ionising doses of up to ${\cal O}(10~{}{\rm kGy})$ [223, 224,
225]. However, these studies also found significant annealing and dose-rate
effects in PbF2, as well as the effectiveness of bleaching with UV light. If
the effects of radiation damage to PbF2 prove to be a significant problem, a
good, radiation-hard alternative could be recently developed, optimised lead
tungstate (PbWO4, PWO) [226, 227, 228]. In particular, ultrafast PWO (PWO-UF)
with a decay time constant of 640 ps, good light yield, and high radiation
tolerance has recently been developed [229].
For the KLEVER SAC, a design with longitudinal segmentation is under study, as
shown in Fig. 58. This design would feature four layers of $20\times 20\times
40$ mm3 PbF2 or PWO-UF crystals (each $\sim$4$X_{0}$ in depth). To minimise
leakage, the gaps between layers would be kept as small as possible. Compact
PMTs such as Hamamatsu’s R14755U-100 could fit into a gap of as little as 12
mm. Readout with SiPMs would facilitate a compact SAC design even further, but
may require advances in SiPM radiation resistance and timing performance.
Figure 59: Geant4 simulation of the KLEVER SAC showing a 1-GeV photon incident
from the right. White tracks show photons, cyan tracks show Cherenkov photons,
and the red track shows a negatively charged particle.
A Geant4 simulation of the KLEVER SAC has been developed for the HIKE Monte
Carlo. This simulation can be used to study the performance with different
crystals and geometries (the readout response is not yet simulated). Fig. 59
shows the simulation of an interaction of a 1-GeV photon in the KLEVER SAC
with PbF2 crystals and a 5-mm gap width between layers for readout with SiPMs.
The Cherenkov photon yield is 85,000 per GeV of incident photon energy. There
is a $\sim$10% loss of shower containment at the highest photon energies,
which is not seen to depend on the size of the gaps between layers over the
interval of 0–20 mm. This is considered to be an acceptable trade-off for the
purpose of maintaining the detector relatively transparent to hadrons. The
response to the main components of the neutral beam has been studied.
Figure 60: SAC inefficiency for photons (top) and efficiency for neutrons and
$K_{L}$ mesons (bottom left and right) vs. incident particle energy, for
various thresholds on observed incident particle energy as measured from the
number of Cherenkov photons produced.
Fig. 60 shows the inefficiency of the SAC for photon detection and the
efficiency of the SAC for neutron and $K_{L}$ detection as a function of
incident particle energy, for various thresholds set on the measured incident
particle energy from the number of Cherenkov photons produced. It is seen that
with a threshold set at the number of Cherenkov photons produced by a photon
of 4–5 GeV, the SAC efficiency requirements are satisfied, while about 30% of
the neutral hadrons leave signals above threshold, contributing to the random
veto rate.
R&D work on the KLEVER SAC is currently being carried out in synergy with
other collaborations, facilitated in part by participation in the AIDAinnova
research network. CRILIN, an electromagnetic calorimeter under development for
the International Muon Collider Collaboration, is an independently proposed,
highly granular, longitudinally segmented, fast crystal calorimeter with SiPM
readout and performance requirements similar to those for the KLEVER SAC
[230], and much development work for KLEVER is being carried out in
collaboration with CRILIN, with particular emphasis on the SiPMs, front-end
electronics, and signal readout, as well as on solutions for detector
mechanics and SiPM cooling. The first test beam measurements with individual
PbF2 and PWO crystals were performed in summer 2021 at the Frascati BTF and
the SPS North Area, followed by additional tests in fall 2022 in which some of
the first commercially available samples of PWO-UF were also tested. These
tests were focused on understanding the best possible time resolution that can
be obtained, studying the systematics of light collection in the small
crystals, and validating the CRILIN choices of SiPMs and the design of the
readout amplifier. In autumn 2022, single $10\times 10\times 40$ mm3 crystals
of PbF2 (4.3$X_{0}$) and PWO-UF (4.5$X_{0}$) were exposed to high-energy
(60–120 GeV) electron beams at the SPS H2 beamline. Each crystal was viewed by
a matrix of four Hamamatsu 14160-4010 SiPMs ($4\times 4$ mm2, 10 $\mu$m pixel
size), which were read out in pairs, providing independent readout channels
for the left and right sides of the crystal (Fig. 61).
Figure 61: Left: Single crystal read out with $2\times 2$ matrix of four
$4\times 4$ mm2 SiPMs. Right: Installation of crystal on H2 beam line during
fall 2022 KLEVER/CRILIN test beam.
The SiPM was chosen to represent a plausible choice in the current state of
the art for high-speed response, short pulse width, and good radiation
resistance. The signals from the SiPMs were amplified with the CRILIN
electronics and digitized at 5 GS/s. The data collected are currently being
analyzed, but some preliminary conclusions are already apparent:
* •
The time resolution obtainable from the combination of either crystal, PbF2 or
PWO-UF, with the chosen SiPM and CRILIN electronics is excellent, with the
final time resolution expected to be significantly better than required for
the KLEVER SAC.
* •
The light yield for PWO-UF is approximately twice that for PbF2; the time
resolution for PbF2 is slightly better.
* •
For either crystal, the light produced is highly localized on the rear face of
the short crystal, requiring some care with the segmented readout.
* •
The CRILIN electronics performed very well; KLEVER should evaluate the
possibility of faster shaping to obtain better double pulse discrimination.
Figure 62: Two-layer $3\times 3$ crystal CRILIN prototype: mechancial drawing
(left), assembled front module (center), and PCB with SiPMs for one layer
(right).
The next step of CRILIN development is to test a two-layer, $3\times 3$
crystal array. The mechanical prototype (Fig. 62) features a scheme for SiPM
cooling. The light is read out in the same manner as for the single crystals
tested in 2021–2022, with two channels per crystal, for a total of 36
channels. Modular electronics for power distribution, SiPM signal processing,
and control have also been developed. This prototype will be populated with
both PbF2 and PWO-UF crystals and tested in the coming months to validate the
segmentation scheme, test the time resolution, study various optimisations
such as the crystal surface preparation, and evaluate the performance of the
engineering solutions adopted.
As noted above, the full SAC design with 4 layers of 4-cm crystals suffers
slightly from shower leakage, while the interaction probability for hadrons in
the beam is about 30%. A possibility to make the SAC more hermetic for photon
showers and less sensitive to hadrons is to exploit the effects of the
coherent interactions of high-energy photons in oriented crystals to induce
prompt electromagnetic showering. This is the same technique to be used for
optimisation of the photon converter in the neutral beam (Section 5.3.1).
Because of the relative ease in producing high-$Z$ optical crystals of high
quality, there are good prospects for using this technique to construct a
compact calorimeter with a very small radiation length, referred to the
primary interaction [231]. A decrease by a factor of 5 in the effective
radiation length for a 4-mm thick PWO crystal has been observed for 120 GeV
electrons incident to within 1 mrad of the [001] axis [232]. Considering that
the SAC acceptance for photons from $K_{L}\to\pi^{0}\pi^{0}$ decays in the
fiducial volume extends at most to $\pm 2$ mrad, it should be possible to
orient the crystals to enhance the probability for photon conversion. This
would allow the thickness of the SAC layers to be reduced, reducing the rate
of neutral hadron interactions while maintaining shower containment and photon
detection efficiency.
The potential gains from aligning the crystals, as well as the procedures and
mechanics needed, are under study. In summer 2021, KLEVER and CRILIN, together
with the STORM collaboration, used a tagged photon beam to measure the
enhancement of shower processes in thicker PbF2 and PWO crystals as a function
of angular alignment, correlating variables such as the energy dispersed in
the crystal and multiplicity of charged particles produced in the shower with
the production of Cherenkov and scintillation light in the crystals. STORM
collected similar data with electron beams in summer 2021 and 2022. The data
are still under analysis, but as a preliminary generalisation, the effective
radiation length for crystals of 2–4 cm thickness is reduced by 20–30% by
alignment, with an angular acceptance of about 1 mrad [233]. Further studies
are planned with the CRILIN prototypes, and the OREO collaboration, continuing
the work of STORM, has plans to build a prototype PWO-UF calorimeter in which
the [100] axes of the crystals on the front layer are aligned. In addition to
allowing studies of the variation of the energy deposition, radiation length
and Molière radius with beam energy and crystal alignment, the OREO prototype
will serve as a platform for developing the procedures needed to build a
calorimeter with aligned crystals.
#### 10.3.7 The hadron sampling calorimeter (HASC)
The original NA62 setup was improved by the addition of a hadron sampling
calorimeter (HASC-1) adjacent to the beampipe, downstream of the muon
detector. The primary purpose of this detector is to reduce the
$K\to\pi^{+}\pi^{+}\pi^{-}$ background to the $K^{+}\to\pi^{+}\nu\bar{\nu}$
decay, by vetoing the topology in which the $\pi^{-}$ undergoes hadronic
interaction in the first STRAW chamber, while an energetic $\pi^{+}$ travels
in the beampipe undetected by the IRC and emerges downstream. Analysis of the
NA62 Run 1 dataset has revealed that HASC-1 is also efficient as a photon
veto, providing a 30% reduction of the $K^{+}\to\pi^{+}\pi^{0}$ background to
the $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay. Consequently, a second calorimeter
(HASC-2) was installed in 2021 at the HASC-1 longitudinal position,
symmetrically with respect to the beam axis, which improves further the
$K^{+}\to\pi^{+}\pi^{0}$ rejection.
Each of the HASC-1 and HASC-2 stations consists of 9 identical modules. Each
module is a sandwich of 120 lead/scintillator alternating tiles, with a total
volume of $10\times 10\times 120~{}\mathrm{cm}^{3}$ (W x H x L). The sampling
ratio is 4:1, the scintillator tiles having a dimension of $100\times
100\times 4~{}{\rm mm}^{3}$ while the lead thickness is 16 mm. Each module is
organised in 10 longitudinal readout sections, each scintillator tile of every
section being optically coupled with a wavelength shifting (WLS) optical fiber
of 1 mm2 cross-section. At the rear side of each module there are 10 optical
connectors, originally designed to be coupled with 3$\times$3 mm2 green-
sensitive micro-pixel avalanche photodiodes (MAPD) (currently the S12572-015C
Hamamatsu SiPM sensors are used).
The FE electronics and SiPMs installed on the HASC-2 station are cooled down
to 21ºC with a custom-made system consisting of three Peltier thermoelectric
coolers / module and a water-air heat exchanger used to blow cold air in the
modules end-cap cases. The temperature is maintained constant by a temperature
controller – MCU based – which regulates the Peltier supply voltage via a PID
routine. For HIKE, this scheme will be extended to HASC-1, improving the
quality of SiPM signals and reducing ageing due to radiation to an acceptable
level.
Signal rates observed in the HASC during the NA62 data taking in 2022 at
nominal beam intensity are summarised in Table 16. The rates fall rapidly with
the distance from the beampipe. The modules next to the beampipe contain
$35\%$ and $25\%$ of the total activity in the HASC-1 and HASC-2 stations,
respectively.
Table 16: Signal rates observed in the HASC during the NA62 data taking in 2022. Detector part | Rates in NA62 setup at nominal intensity
---|---
HASC-1 | 1 MHz
HASC-1, most active module | 0.35 MHz
HASC-1, most active channel | 0.05 MHz
HASC-2 | 0.21 MHz
HASC-2, most active module | 0.05 MHz
HASC-2, most active channel | 6 kHz
Total (HASC-1 and HASC-2) | 1.21 MHz
The NA62 HASC provides near-optimal geometric coverage for the relevant
topologies of the $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ and $K^{+}\to\pi^{+}\pi^{0}$
backgrounds to the $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay, and is suitable also
for HIKE Phase 1. Assuming the HIKE beam profile to be similar to the NA62
one, we expect at most a rate of 200 kHz in the most active channel for the
HIKE HASC. The main issue in the operation of the current HASC at the HIKE
beam intensity is the random veto. It is determined chiefly by the time
resolution of the detector, presently at the level of 370 ps FWHM for HASC-2
and 430 ps for HASC-1, in agreement with HAMAMATSU specifications [234]. For
HIKE, the resolution should be improved to $\mathcal{O}(100~{}\mathrm{ps})$.
The HAMAMATSU S13362-3050DG SiPMs achieving this value are available on the
market [235], and represent a promising candidate for the upgrade. Table 17
shows a comparison between the current HASC SiPMs and the candidate for the
upgrade. Another appealing feature of the new SiPMs is the embedded two-stage
thermoelectric cooling which greatly simplifies the current cooling scheme.
Table 17: Summary of SIPM characteristics for the model currently employed in the NA62 HASC and the one proposed for the HIKE upgrade. SIPM model | S12572-015C (current) | S13362-3050DG (upgrade)
---|---|---
Photon detection efficiency | 25% | 40%
Typical dark count | 1000 kcps | 25 kcps
Gain | $2.3\times 10^{5}$ | $1.7\times 10^{6}$
Time resolution FWHM | 400 ps | 110 ps [236]
An alternative option for the HASC upgrade, which is currently under study,
involves PMTs. A particularly interesting example is HAMAMATSU H14220A, which
could be used to read out individual scintillator plates (instead of groups of
six, as done presently), eliminating pileup due to detecting a convoluted
signal hence offering lower rise time ($\tau_{\mathrm{r}}$) and fall time
($\tau_{\mathrm{f}}$). The main drawback of reading single scintillators is
the lower number of collected photons ($N_{\mathrm{ph}}$), compared with SiPM
readout, which could have a negative impact on the time resolution which is
known to be proportional with
$\sqrt{\tau_{\mathrm{r}}\tau_{\mathrm{f}}/N_{\mathrm{ph}}}$ [237].
### 10.4 Detectors specific to KLEVER
#### 10.4.1 Upstream veto (UV) and active final collimator (AFC)
The upstream veto (UV) rejects $K_{L}\to\pi^{0}\pi^{0}$ decays in the 40 m
upstream of the fiducial volume where there are no large-angle photon vetoes.
The UV is a shashlyk calorimeter with the same basic structure as the MEC
(without the spy tiles). Because the UV does not participate in event
reconstruction, the readout granularity can be somewhat coarser than for the
MEC. The sensitive area of the UV has inner and outer radii of 10 cm and 100
cm.
The active final collimator (AFC) is inserted into the hole in centre of the
UV. The AFC is a LYSO collar counter with angled inner surfaces to provide the
last stage of beam collimation while vetoing photons from $K_{L}$’s that decay
in transit through the collimator itself. The collar is made of 24 crystals of
trapezoidal cross section, forming a detector with an inner radius of 60 mm
and an outer radius of 100 mm. The UV and AFC are both 800 mm in depth. The
maximum crystal length for a practical AFC design is about 250 mm, so the
detector consists of 3 or 4 longitudinal segments. Each crystal is read out on
the downstream side with two avalanche photodiodes (APDs). These devices
couple well with LYSO and offer high quantum efficiency and simple signal and
HV management. Studies indicate that a light yield in excess of 4000 p.e./MeV
should be easy to achieve.
#### 10.4.2 Charged-particle rejection
For the rejection of charged particles, $K_{L}\to\pi^{+}e^{-}\nu$ is a
benchmark channel because of its large branching ratio and because the final
state electron can be mistaken for a photon. Simulations indicate that the
needed rejection can be achieved with two planes of charged-particle veto
(CPV) each providing 99.5% detection efficiency, supplemented by the
$\mu^{\pm}$ and $\pi^{\pm}$ recognition capabilities of the MEC (assumed in
this case to be equal to those of the LKr) and the HIKE hadronic calorimeters,
which could be reused in KLEVER. The CPVs are positioned $\sim$3 m upstream of
the MEC and are assumed to be constructed out of thin scintillator tiles. In
thicker scintillation hodoscopes, the detection inefficiency arises mainly
from the gaps between scintillators. For KLEVER, the scintillators will be
only $\sim$5 mm thick ($1.2\%X_{0}$), and the design will be carefully
optimized to avoid insensitive gaps.
#### 10.4.3 Preshower detector
The PSD measures the directions for photons incident on the MEC. Without the
PSD, the $z$-position of the $\pi^{0}$ decay vertex can only be reconstructed
by assuming that two clusters on the MEC are indeed photons from the decay of
a single $\pi^{0}$. With the PSD, a vertex can be reconstructed by projecting
the photon trajectories to the beamline. The invariant mass is then an
independent quantity, and $K_{L}\to\pi^{0}\pi^{0}$ decays with mispaired
photons can be efficiently rejected. The vertex can be reconstructed using a
single photon and the constraint from the nominal beam axis. Simulations show
that with $0.5X_{0}$ of converter (corresponding to a probability of at least
one conversion of 50%) and two tracking planes with a spatial resolution of
100 $\mu$m, placed 50 cm apart, the mass resolution is about 20 MeV and the
vertex position resolution is about 10 m. The tracking detectors must cover a
surface of about 5 m2 with minimal material. Micropattern gas detector (MPGD)
technology seems perfectly suited for the PSD. Information from the PSD will
be used for bifurcation studies of the background and for the selection of
control samples, as well as in signal selection.
## 11 Data acquisition and high-level trigger
Particle rate in HIKE Phase 1 is 3000 MHz in the beam tracker and 200 MHz for
$K^{+}$ identification. Single-particle rates in the downstream detectors is
about 50 MHz, including both $K^{+}$ decay products and the charged beam’s
muon halo. The neutral beam of HIKE Phases 2 and 3 is 50% more intense that
the Phase 1 charged beam. The small-angle calorimeter (SAC) in the neutral
beam will encounter rates of about 100 MHz, and the other detectors encounter
rates of about 20 MHz. The high particle rates will give rise to a harsh
radiation environment in the HIKE experimental cavern. To limit the impact of
radiation on the DAQ system, we aim to minimise the amount of electronics
located in the cavern, particularly electronics related to data processing and
storage, and use radiation-tolerant optical links and error-correcting data
formats such as (lp)GBT.
### 11.1 Readout boards
The readout boards serve two purposes: to host devices that timestamp digital
signals produced by the detectors, and to encode and transmit these data away.
The readout boards are located in the high-radiation environment close to the
detector, so radiation-hardened technologies will be used. The time of signals
from the detectors will be recorded either with time-digital converters (TDC),
analogue-to-digital converters (ADC), or dedicated ASIC devices. The
64-channel PicoTDC is under development at CERN and is anticipated to be ready
before HIKE Phase 1. The PicoTDC can timestamp signals at 12 ps precision
[238] and outputs data in the GBT format, allowing simple readout boards to be
used that only convert electric signals from the PicoTDC into optical signals
that are sent to the rest of the DAQ system. Possible commercial solutions for
fast FADC-based readout boards, to be adapted to the requirements of the DAQ
in synergy with the manufacturer, are under investigation. The HIKE Phase 1
beam tracker data will be handled by dedicated ASICs.
The number of detector channels read by TDC or ADC in HIKE Phase 1 is
estimated to be about 50000. Approximately half of the detector channels will
be read by TDC devices, the other half by ADCs. Therefore an estimated 400 TDC
devices with 64 channels each, and 1600 ADC devices with 16 channels each,
will be required for HIKE Phase 1. For HIKE Phase 2, the 10240 channels of the
LAV, $\mathcal{O}(2000)$ channels of the MEC, and 400 channels of the SAC will
be read using ADCs, requiring about 800 ADC devices with 16 channels each. Spy
tiles in the MEC would contribute an additional 8000 channels that can be read
using TDCs, in which case about 125 TDC devices with 64 channels each will be
needed.
### 11.2 Streaming readout
Moving the bulk of the DAQ hardware away from the high-radiation environment
suggests implementing a streaming readout system, where detector signals are
collected by radiation-tolerant readout boards and then transferred
immediately, without an external trigger signal and without complicated
processing or storage. The transferred data are received and buffered in the
RAM of a set of servers, then transmitted to a computing farm for selection
and filtering.
The first stage of data readout may be implemented using a FELIX-based system,
developed at CERN and exploited in several HEP use cases. The current version
of the FELIX PCIe card has 24 optical links with a 9.6 Gb/s line rate and
supports several data formats. When operating with the radiation-tolerant GBT
format the effective data bandwidth is 3.2 Gb/s per link. New versions of
FELIX, currently under development and anticipated on the timescale of HIKE,
can provide faster links (e.g. 25 Gb/s) or a larger number of links (e.g. 48).
The FELIX system also takes care of distributing clock, timing and commands to
the front-ends, as well as receiving monitoring data for detector control
purposes. FELIX cards are hosted in servers, and all data received by the
FELIX cards are transferred to the host and buffered in memory. The servers
then provide data to the computing farm as needed, over a high-speed ethernet
network. As modern servers can be equipped with 1 TB of RAM, the data recorded
during a whole SPS spill (4.8 s) may be buffered, greatly relaxing any latency
requirement on the data processing as the inter-spill period (at least 9.6 s)
can be exploited. Other options for interfacing the detector data with the
computing farm are presently available, such as the PCI40 board developed for
the LHCb and ALICE experiments, and the NaNet board, used by the KM3Net
experiment.
The main challenges of a streaming readout system are the transfer and
processing of large quantities of data. The highest data rates are expected
during the KLEVER phase, in particular from the SAC, MEC, and LAV detectors.
The raw data rates are summarised in Table 18 assuming the use of 14-bit ADCs.
The largest rate is 72 Tb/s from the LAV. This can be considerably reduced by
only reading channels that contain non-zero energy, by implementing a zero-
suppression (ZS) algorithm in the readout boards. Assuming that reading data
covering 16 ns is enough to contain the data from one event, the zero-
suppressed data rates are given in Table 19. In this case, the SAC dominates
the data rate. Taking the FELIX system with GBT data format as the benchmark
solution, the zero-suppressed SAC data could be collected by 9 FELIX cards.
Reading the non-suppressed data is unfeasible, and the streaming readout
design depends on the successful implementation of zero-suppression in the
ADC-based readout boards.
Table 18: Raw data rates expected from the MEC, SAC, and LAV in HIKE Phase 3. Detector | Channels | Sampling rate (GHz) | Raw data rate (Tb/s)
---|---|---|---
LAV | 10240 | 0.5 | 72
MEC | 2000 | 1 | 26
SAC | 400 | 1 | 6
Table 19: Zero-suppressed data rates expected from the MEC, SAC, and LAV in HIKE Phase 3. Detector | Event rate (MHz) | Average N channels hit per event | ZS data rate (Gb/s)
---|---|---|---
LAV | 14 | 5 | 8
MEC | 18 | 10 | 40
SAC | 95 | 30 | 640
### 11.3 Triggered readout
In case zero suppression of the calorimeter data proves unfeasible, or data
transmission at the required speed cannot be achieved on the calorimeter
readout boards, a trigger may still be necessary to reduce the required data
throughput. The key parameters of the trigger are latency, which dictates how
much data will have to be stored on the readout boards, and selectivity, which
dictates how much data will have to be transferred to the computing farm. The
level of data reduction that the trigger can achieve is a compromise between
the sophistication of the algorithms, with more sophisticated algorithms
providing better selectivity, and the time taken to form the trigger decision.
We envisage a software trigger, with the trigger decision prepared using
dedicated trigger algorithms running on the computing farm, most likely on a
subset of the event data. Recalling the anticipated LAV data rate is 72 Tb/s
(9 TB/s), assuming a software-trigger latency of 100 ms and 100 LAV readout
boards, each LAV readout board must be able to cache about 9 GB of data. If
the trigger can achieve a reduction factor of 100 – the same as the combined
hardware and software trigger of NA62 – the transfer rate would be 720 Gb/s,
which can be handled by 10 FELIX cards.
### 11.4 Event-building farm and event filter
Regardless of whether a streaming or triggered DAQ is implemented, an event
filter will be used to select candidate events, reducing the amount of data
sent to permanent storage. The first stage of the event-filter is to combine
fragments of events coming from the different detectors and reconstruct basic
objects in each detector to identify physics events. Taking
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decays as an example, the event-building code
will identify coincident signals in the RICH, STRAW, and KTAG as a potential
$K^{+}\to\pi^{+}\nu\bar{\nu}$ event. This stage will be performed in an event-
building computing farm located in the surface building of the experimental
cavern. The backbone of the event-building farm will be standard server
computers running C++ algorithms. However, the event-building process centres
around combinatorial algorithms that are ideally suited for hardware
acceleration using FPGA or GPU devices. We envisage the use of heterogeneous
computing architectures to handle the demanding processing task given to the
event-building farm. Subsequently, the event filter will examine signals in
the events to identify the small fraction that should be kept for detailed
analysis offline. This stage will be performed using dedicated servers in the
Computing Centre on the CERN Prévessin site. A number of machine-learning-
based algorithms have been developed at NA62 for particle identification and
other tasks, and these can be the basis for evaluating events in the HIKE
event filter.
## 12 Online and offline computing
The baseline computing model is a scaled version of the current NA62 system. A
flexible HIKE software platform is currently under development, on the basis
of the NA62 software. HIKE will profit from the use of dedicated machines in
Prévessin for many, if not all, of the described functions below, provided
enough band capability is present between the experiment site and the
computing centre.
### 12.1 Online monitoring
The HIKE online monitoring system will be based on an updated model of the
current one used in NA62. At present in NA62, a subset of the data are sent
from farm nodes running the event-building processes to dedicated four
monitoring computers; each event is then processed by the standard
reconstruction software. The reconstructed events are then forwarded to a last
machine where the different streams are recombined for displaying by dedicated
processes. A set of low-level monitoring histograms produced by the
reconstruction processes are also combined and displayed, for the purpose of
quick data quality checks.
For HIKE, a multi-level analysis pipeline, where each level performs more
complex and time consuming tasks, should be implemented to shorten the latency
between the acquisition of the data and the first feedback. Low level (data
corruption) and high level (data quality) algorithms will perform error
detection and alert the shift crew and the run control software. As often as
possible automatic recovery procedures will be implemented to minimise data
losses. A set of low-level calibration constants will also be computed during
this first reconstruction and used both for monitoring purposes and to be fed
back into the acquisition system. Depending on the details of the trigger
scheme, the computing farm could also perform some of data filtering tasks.
Reconstructed events will be processed with data quality algorithms, also used
during full offline data processing (see next section), yielding output
information collected over a whole run (about 1500 bursts) to be reviewed by
the detector experts.
### 12.2 Calibration and data quality
Calibration and data quality monitoring strategy will be built on the NA62
systems. Calibration will be performed in several stages. First, geometrical
detector alignments will be initially obtained from special “muon” runs, where
TAXes are closed and the spectrometer magnet is switched off to obtain
straight penetrating tracks. Later, these geometrical alignments will be
refined and monitored using kaon decays in standard data-taking conditions.
Raw timing alignments within individual detectors, like slewing corrections,
will be obtained from data and assumed to be constant over a certain data
taking period, although they are permanently monitored. A similar procedure to
that of NA62 for relative timing alignment between different detectors (“T0s”)
will be developed, to achieve an average precision at the picosecond level,
with a width corresponding to the individual timing resolutions of each
detector. This will be done on a per-burst basis. During processing
information on detector-specific calibrations will also be obtained, on a per-
burst basis, like energy conversions, dead/noisy channels etc, which will be
fed back to the analysis software.
Data quality will be additionally monitored during full offline processing by
software specific to each detector, but also including more common properties
like efficiencies for different (online or offline) trigger algorithms.
Graphical and numerical outputs will be produced for every detector, together
with a database containing the monitored (and calibrated) quantities and
alarms raised if certain thresholds are passed for one burst.
### 12.3 Data processing model
Based on experience, the NA62 processing model, with some improvements and
changes, can be adapted for HIKE. In NA62, raw data processing and user
analysis is performed only on CERN computing clusters, while Monte Carlo
production is performed using the international GRID, including CERN Grid
resources.
Processing of raw data for HIKE will be performed at CERN and using GRID
resources (with a split still to be defined depending on the details of
available computing and data sizes at the time of data taking), due to the
increased amount of data. The splitting of the processed data according to
trigger and/or physics analysis interests (“filtering”) must be refined to
obtain stream sizes that are smaller and easier to handle. Stronger selection
criteria will be applied, and the amount of information written for every
selected event (slimming) will be reduced.
### 12.4 Distributed computing model
Large-scale generation of Monte Carlo simulations will be performed
exclusively on the Grid. The existing NA62 Grid framework is a one-of-its-kind
system, created specifically for NA62 and providing an easy-to-use and fully
automated production system [239]. With an uptime of 99.99% over the last ten
years, it is a proven system that we envisage to extend to HIKE. Besides its
simulation and reconstruction capabilities, our Grid framework can be extended
to handle user analysis, in case this becomes a necessity for HIKE.
### 12.5 Data reduction model
Due to the expected increase in the amount of data (due to the increased
intensity) and in the size of data (due to the increase of signal channels),
both filtering and slimming are foreseen for HIKE. For filtering, overlap
between different filtering streams must be minimised, while several filter
outputs may be combined at analysis level. For slimming, after reconstruction
no low-level hit information will be stored, discarding as much as possible
any information that can be:
1. 1.
Recomputed from stored information.
2. 2.
Likely not needed at analysis level.
In specific cases of detector studies, or when an analysis requires access to
low-level information, a selected list of events/bursts will be reconstructed
again. The low-level information will be kept for those small dedicated
samples for the duration of the study.
Generally, most analyses make use of standard high-level objects (downstream
tracks, vertices) without the need to access most of the information stored in
the underlying objects from which they were built (clusters, tracks,
candidates). Those high level object could therefore contain only the subset
of information commonly used (time and position at associated sub-detectors)
and stored in high-level analysis files which would be the basis for most user
analyses. Again, some dedicated studies occasionally will require access to
the full reconstructed objects, but these are generally aiming at deeper
understanding of detector effects and to develop procedures to deal with them.
In most cases the final procedure will not rely on the full reconstruction
information, but only on that already available in the high-level objects. In
addition, such procedures can generally be standardised and applied to several
analyses without further study. It will therefore be sufficient to reconstruct
a subset of data with full information for the duration of the study.
## 13 Infrastructure and safety
HIKE will be housed in TCC8 and ECN3 where NA62 is at present. An evaluation
of the necessary modifications to the experimental area has been carried out
in collaboration with the ECN3 Task Force [155], the North Area Consolidation
Project (NA-CONS) [148, 149, 150] and the Conventional Beams Working Group
[147], where topics such as requirements for beam infrastructure, vacuum,
cooling and ventilation, electrical distribution, handling and transport of
detectors, access to the cavern, IT infrastructure, gas distribution,
cryogenic systems, civil engineering and radiation protection have been
addressed. The full list of requirements is provided in Ref. [240]. In order
to ensure full compatibility with the North Area consolidation and as a
central access point to all CERN service groups, discussion for the
implementation of all requirements is steered through the NA-CONS Technical
Coordination Committee (TCC) meetings, while integration aspects are handled
in the Integration Committee for Experimental Areas (ICEA) as part of NA-CONS.
The best match between the HIKE requirements and the infrastructure studies
will be further investigated for the Proposal. We would like to highlight that
the experimental layout of HIKE does not require any civil engineering work
before at least LS4. The minimal changes and updates in the infrastructure
will have a positive impact on the foreseen costs. Furthermore, the
requirements for the electrical distribution and the detector cooling will be
similar to that of NA62. The vacuum system with seven cryo-pumps went through
thorough maintenance in 2021 and will be suitable to provide a good vacuum at
the $10^{-6}$ mbar level for HIKE.
To keep the NA62 LKr calorimeter operational for HIKE Phase 1, continuous
maintenance of the cryogenics is needed. In addition, besides the installation
of a new readout back end, the legacy components of the readout chain should
be reviewed. Actions in this direction will be started already in the next
YETS and procedures for regular review intervention will be defined.
The human, technical and financial resources needed for HIKE are being
evaluated. Table 20 shows a tentative indication of the cost of the main
detectors for HIKE. The total for Phase 1 is 22.3 MCHF. The Phase 2 will see
the addition of the Main Electromagnetic Calorimeter (for an extra 5 MCHF).
The remaining costs for Phase 3 (KLEVER) will be mainly the increase in the
number of Large Angle Vetoes from 12 to 25 units (for an extra 9 MCHF). The
numbers are only intended to give an idea of the financial extent of the
project. A refined estimate will be prepared for the forthcoming proposal.
Table 20: Tentative indication of detector costs for HIKE. Detector | Cost (MCHF) | Comments
---|---|---
Kaon ID (KTAG) | 0.5 | Using MCP-PMTs
Beam Tracker | 2.5 | Development, and production of 4 planes
Charged particle veto (CHANTI) | 0.4 | 6 stations, with SiPMs
VetoCounter (VC) | 0.2 | 3 stations (SciFi technology)
Anti0 | 0.1 | 1 plane, same technology as TimingPlanes
Large Angle Vetos (LAV) | 8 | 12 modules (Phase 1)
STRAW | 3.5 | 4 Straw chambers and associated readout
LKr upgrade OR MEC | 2.5 OR 5 | Readout upgrade OR new MEC
Small Angle Calorimeter (SAC) | 2 | High-Z crystals
Pion ID (RICH) | 0.8 | Using SiPMs
Timing Detector | 0.2 | 2 planes, scint. tiles and SiPMs
Hadronic Calorimeter + Muon plane | 1.5 | Shashlyk technology and SiPMs
HASC | 0.1 | Using SiPMs
MEC (if not already in Phase 1) | 5 | Shashlyk technology and SiPMs (Phase 2)
Large Angle Vetoes (LAV) | 9 | Additional 13 modules (Phase 3)
## 14 Summary and conclusions
The HIKE project presented in this document constitutes a multi-phase
experimental programme taking full advantage of the ECN3 experimental hall to
provide a comprehensive study of flavour physics in the kaon sector. This
physics programme is complementary to, and out of reach of, the LHC
experiments. Importantly, HIKE will be sensitive to new physics up to the
highest mass scales. The HIKE facility will, in addition, search for feebly-
interacting particles with unprecedented sensitivity. A summary of the HIKE
phases and related intensity requirements is presented in Table 21.
The experimental apparatus changes over time with a staged approach, which
allows HIKE to evolve and adapt the physics scope in its three phases, an
important feature for a project that embraces a time scale of more than decade
during which the physics landscape could change. Thanks to the successful
experience of NA62 and its predecessor NA48, the experimental technique is
well established and robust expectations of sensitivity can be provided driven
by the analysis of the existing data. In this way the expected sensitivity for
benchmark studies have been presented above. The HIKE facility allows other
users to run in parallel, even in the same area. This facilitates a high
degree of diversity in the North-Area physics programme which, we believe, is
crucial for the future of particle physics.
Table 21: Summary of intensity requirements for the HIKE Phases. The number of kaon decays refers to those inside a fiducial decay volume (see Sections about physics sensitivity). For the $K^{+}$ phase, the maximum required intensity is specified; possibilities of reducing the intensity and increasing the momentum bite will be studied for the full proposal. An intensity higher than $2\times 10^{13}$ can be used by HIKE when operating in dump mode. Phase | Protons on target/spill | $K$ decays/year | Protons on TAX
---|---|---|---
Phase 1 ($K^{+}$) | $1.2\times 10^{13}$ | $2\times 10^{13}$ | -
Phase 2 ($K_{L}$+tracking) | $2\times 10^{13}$ | $3.8\times 10^{13}$ | -
Dump Mode | - | - | (2–4)$\times 10^{13}$
Phase 3 ($K_{L}$, KLEVER) | $2\times 10^{13}$ | $1.3\times 10^{13}$ | -
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# Rigorous Assessment of Model Inference Accuracy using Language Cardinality
Donato Clun 0000-0001-5190-8957 Imperial College LondonUK
<EMAIL_ADDRESS>, Donghwan Shin 0000-0002-0840-6449 University of
SheffieldUK<EMAIL_ADDRESS>, Antonio Filieri 0000-0001-9646-646X
Imperial College LondonUK<EMAIL_ADDRESS>and Domenico Bianculli
0000-0002-4854-685X University of LuxembourgLuxembourg
<EMAIL_ADDRESS>
###### Abstract.
Models such as finite state automata are widely used to abstract the behavior
of software systems by capturing the sequences of events observable during
their execution. Nevertheless, models rarely exist in practice and, when they
do, get easily outdated; moreover, manually building and maintaining models is
costly and error-prone. As a result, a variety of model inference methods that
automatically construct models from execution traces have been proposed to
address these issues.
However, performing a systematic and reliable accuracy assessment of inferred
models remains an open problem. Even when a reference model is given, most
existing model accuracy assessment methods may return misleading and biased
results. This is mainly due to their reliance on statistical estimators over a
finite number of randomly generated traces, introducing avoidable uncertainty
about the estimation and being sensitive to the parameters of the random trace
generative process.
This paper addresses this problem by developing a systematic approach based on
analytic combinatorics that minimizes bias and uncertainty in model accuracy
assessment by replacing statistical estimation with deterministic accuracy
measures. We experimentally demonstrate the consistency and applicability of
our approach by assessing the accuracy of models inferred by state-of-the-art
inference tools against reference models from established specification mining
benchmarks.
## 1\. Introduction
Software system models, typically in the form of Finite State Automata (FSA),
have been widely used to abstract the behavior of software components and
their interactions. Such models are important in many applications, including
test data generation (Fraser and Walkinshaw, 2012), model checking (Clarke Jr
et al., 2018), and program comprehension (Cook and Wolf, 1998). Nevertheless,
they are rarely available during software development and, if they do, get
easily outdated. This is mainly because manually building and maintaining such
models is both time-consuming and error-prone. To address this problem, a
variety of _model inference_ algorithms have been proposed (Biermann and
Feldman, 1972; Beschastnikh et al., 2011; Walkinshaw et al., 2016; Emam and
Miller, 2018; Mariani et al., 2017). These algorithms automatically extract a
system’s behavior model by capturing visible events generated during the
system’s execution.
_Model assessment_ , i.e., assessing the accuracy of inferred models, is an
essential task to evaluate and compare different model inference algorithms.
When reference models are given as ground truth, model assessment seems
straightforward; for example, one can compute the similarity between the
languages defined by the inferred and reference models. However, the languages
are often infinite, making it complex to measure the degree of their
similarity.
To address this issue, it is common to rely on statistical accuracy
estimation. For example, the idea behind a popular method, called _trace
similarity_ (Lo and Khoo, 2006a), is to generate sampled traces111A _trace_ is
the sequence of events generated during the system’s execution. Also known as
_run_ , _sequence_ , _execution trace._ by randomly traversing both reference
and inferred models, and estimate the accuracy by checking how many of the
traces generated using one model are accepted by the other. The idea is that a
high number of accepted sampled traces is an indicator of a high degree of
similarity. Although this idea is very intuitive, the random traverse can
introduce a large degree of uncertainty about the estimation, depending on the
parameters of the random trace generative process (e.g., the probability of
ending the traverse in an accepting state rather than continuing through one
of its outgoing transitions), and the topology of the model (e.g., a model may
have features that are hard to exercise through random exploration). While
there is no intrinsic probability distribution over the traces accepted by a
finite state model, the random trace generation does induce such a
distribution, therefore introducing an evaluation bias: the measured accuracy
will reflect how well the inferred model can classify traces drawn according
to that distribution.
In an attempt to reduce the uncertainty due to the random traversal, an
alternative consists in using deterministic trace generation methods
(Walkinshaw et al., 2008). These generate sets of traces that are guaranteed
to cover all the models’ behaviors without favoring any specific one. Although
this increases the reproducibility and the chances of revealing discrepancies,
there are still remaining issues. First, each discrepancy (a trace upon which
the reference and the inferred model disagree) may be representative of a
larger or smaller class of traces, thus making it difficult to quantify its
impact. Second, these methods often generate sets of traces containing a
disproportionately large amount of traces not accepted by the reference model,
which may lead to skewed accuracy results. Third, the generated set of traces
can become exceedingly large if the difference between the number of states in
the reference and the inferred model is large, which is a common occurrence in
model inference, hindering its practical usability.
In this paper we propose a method to rigorously measure the accuracy of an
inferred model against a reference one (both in the form of FSA) by
considering all the possible traces up to an arbitrary finite maximum trace
length. The maximum trace length considered is a parameter set by the user,
which can be set to a value relevant for the application under analysis, to a
value large enough to guarantee that all the behaviors of the reference and
inferred models are exercised, or to an arbitrarily large value, effectively
computing the asymptotic accuracy.
Our method is deterministic (i.e., for a given pair of reference and inferred
models it always returns the same result), does not depend on the model
structure (i.e., assessing against the same reference model different models
accepting the same language will generate the same accuracy measurement), and
does not introduce any evaluation bias other than the maximum trace length
(i.e., the computed metrics accurately describe how well the inferred model
can classify traces drawn with uniform distribution from the set of all the
traces up to the maximum length).
Within this paper we also highlight how an inferred model can show a variable
level of accuracy depending on the length of the traces used in the
assessment. This is an aspect that is generally not considered by current
assessment methodologies. Following this observation, we propose an additional
assessment method that computes a pair of precision and recall values for each
trace length, within a range specified by the user. Each value considers all
the possible traces of a given length.
Central to our solution is the use of analytic combinatorics to count the
number of traces in a (possibly infinite) regular language, up to an arbitrary
maximum trace length, without explicitly enumerating them. In this regard, to
increase the practical applicability of our approach, we improved the
scalability of the analytic combinatorics approaches currently used to compute
the cardinality of regular languages (Flajolet and Sedgewick, 2009).
We have implemented the proposed method in a prototype and experimentally
evaluated it using reference models previously used in the model inference
literature, and inferred models generated using well known model inference
methods. The experimental results obtained when assessing the applicability of
our method on the real-world model show that it is scalable enough to be used
in practice. We have also compared the generated accuracy measurements with
the results obtained using other popular assessment methods. The results
indicate differences caused by the evaluation bias introduced by current
assessment methods, and show that our proposed approach can be used to address
this issue.
In summary, this paper makes the following contributions:
* •
A novel model assessment method that measures the precision and recall values
of an inferred model, with respect to a reference ground truth model,
considering all the traces up to a finite maximum trace length. This method
is:
* –
Deterministic: repeated executions return the same result.
* –
Comprehensive: it considers all the traces up to the maximum length.
* –
Unbiased: all the traces considered in the evaluation have the same weight on
the result.
* –
Model-independent: assessing different models accepting the same language
against the same reference model generates the same result.
* •
A further development of the assessment method, measuring the model accuracy
separately for each trace length, over a given range.
* •
An empirical comparison of the accuracy measurement obtained using our methods
with measurements generated using other popular assessment methods.
* •
An experimental evaluation of the applicability of our method.
* •
An improvement of currently used analytic combinatorics approaches to compute
the cardinality of regular languages, up to a finite maximum trace length.
The rest of the paper is organized as follows. In Section 2, we present a
brief background on the theory of formal languages, model assessment, and
analytic combinatorics as a tool to compute the cardinality of regular
languages. In Section 3, we discuss two prominent classes of model assessment
methods from the literature: statistical estimation (the most commonly used
accuracy assessment method) and model-based (developed to address shortcomings
of the first), reflecting on the open issues that motivated this work. Section
4 presents our main contribution: a novel assessment method, based on
analytical measures for the cardinality of languages, that addresses the
issues highlighted in the preceding section, and a further development that
allows to evaluate how the model accuracy changes over a range of trace
lengths. To increase the practical applicability of our method, in Section 5,
we present an improvement of currently used analytic combinatorics approaches
to compute the cardinality of regular languages. In Section 6 we evaluate our
method experimentally, focusing on whether it is suitable to replace the
assessment methods discussed in Section 3, and how the different methods’
assessment results compare. Section 7 presents relevant related work, and
Section 8 concludes the paper with future work directions.
## 2\. Background
### 2.1. Models and Languages
In this paper, we consider models in the form of Deterministic Finite-state
Automata (DFA). A DFA is a tuple $\mathcal{A}=(\Sigma,Q,q_{0},F,\delta)$,
where $\Sigma$ is a finite alphabet, $Q$ is the set of states, $q_{0}\in Q$ is
the initial state, $F\subseteq Q$ is the set of accepting states, and
$\delta:Q\times\Sigma\rightarrow Q$ is the transition function. A trace is a
finite sequence $t=\langle\sigma_{1}\sigma_{2}\dots\sigma_{n}\rangle$ of
elements $\sigma_{i}\in\Sigma$ for $i=1,\dots,n$. A trace
$t=\langle\sigma_{1}\sigma_{2}\dots\sigma_{n}\rangle$ is _accepted_ by
$\mathcal{A}$ if there exists a sequence of states $\langle
q_{0},q_{1},\dots,q_{n}\rangle$ such that (1) $q_{i}\in Q$ for $i=1,\dots,n$,
(2) $\delta(q_{i-1},\sigma_{i})=q_{i}$ for $i=1,\dots,n$, (3) $q_{0}$ is the
initial state, and (4) $q_{n}\in F$. A state $q\in Q$ is an _error state_ if
no accepting state is reachable from $q$. Let $\Sigma^{*}$ be the set of all
possible traces over $\Sigma$ (including the empty trace). The language
accepted by $\mathcal{A}$, denoted by
$\mathfrak{L}(\mathcal{A})\subseteq\Sigma^{*}$, is the set of all traces
accepted by $\mathcal{A}$. Two DFAs are equivalent if they accept the same
language.
DFAs accept regular languages, which are closed under union, intersection, and
complement. These operations on regular languages correspond to analogous
operations on the automata accepting the languages. Therefore, with abuse of
notation, we will use the union ($\cdot\cup\cdot$), intersection
($\cdot\cap\cdot$), and complement ($\overline{\cdot}$) operators on both
automata and the languages they accept; for example,
$\mathcal{A}\cup\overline{\mathcal{B}}$ represents the DFA accepting any trace
that is accepted by $\mathcal{A}$ or not accepted by $\mathcal{B}$.
### 2.2. Model Assessment as Language Comparison
Given a reference model $\mathcal{R}$ and an inferred model $\mathcal{H}$ over
the same alphabet $\Sigma$, to assess the accuracy of the inferred model we
need a measure of how well the language accepted by $\mathcal{H}$ approximates
the language accepted by $\mathcal{R}$. Drawing from established theory on
classification assessment methods (Tharwat, 2020), $\Sigma^{*}$ can be
partitioned into four subsets of traces: True Positives
($\Sigma^{*}_{\mathit{TP}}=\mathfrak{L}(\mathcal{R}\cap\mathcal{H})$), True
Negatives
($\Sigma^{*}_{\mathit{TN}}=\mathfrak{L}(\overline{\mathcal{R}}\cap\overline{\mathcal{H}})$),
False Positives
($\Sigma^{*}_{\mathit{FP}}=\mathfrak{L}(\overline{\mathcal{R}}\cap\mathcal{H})$),
and False Negatives
($\Sigma^{*}_{\mathit{FN}}=\mathfrak{L}(\mathcal{R}\cap\overline{\mathcal{H}})$).
By generalization, any set of traces $E\subseteq\Sigma^{*}$ (which we call
evaluation set) can be partitioned into four subsets $E_{\mathit{TP}}$,
$E_{\mathit{TN}}$, $E_{\mathit{FP}}$, and $E_{\mathit{FN}}$, containing true
positives, true negatives, false positives and false negatives, respectively.
If the evaluation set is finite, several accuracy metrics can be defined based
on the _relative cardinality_ ($|\cdot|$) of these four languages. In this
paper, we will focus on precision and recall:
(1)
$\mathit{precision}=\frac{|E_{\mathit{TP}}|}{|E_{\mathit{TP}}|+|E_{\mathit{FP}}|};\quad\textit{recall}=\frac{|E_{\mathit{TP}}|}{|E_{\mathit{TP}}|+|E_{\mathit{FN}}|}$
which are widely used in model inference literature (Krka et al., 2014; Lo et
al., 2012; Lo and Khoo, 2006a).
In the rest of this paper we will continue to use $\mathcal{R}$ and
$\mathcal{H}$ to denote the reference and the inferred model respectively,
both assumed to be defined over the same alphabet $\Sigma$.
### 2.3. Analytic Combinatorics and Cardinality of Regular Languages
Analytic combinatorics (Flajolet and Sedgewick, 2009) is a theory used to
build, manipulate and analyze exact enumerative descriptions of combinatorial
structures, typically focusing on structures whose realizations are too many
for explicit enumeration. _Generating functions_ are a core tool in analytic
combinatorics, as they enable to concisely define and operate on certain
infinite sequences. Let us consider a discrete, possibly infinite sequence of
real values $a_{n}$, with $n=0,1,2,\cdots$. An _ordinary generating function_
(OGF) is a mathematical object encoding the sequence as the coefficients of a
formal power series:
$f(z)=a_{0}+a_{1}z+a_{2}z^{2}+a_{3}z^{3}+\cdots=\sum_{n=0}^{\infty}a_{n}z^{n}$.
Given an OGF $f(z)$, the $n$-th coefficient $a_{n}$ can be retrieved as the
$n$-th Taylor coefficient of $f(z)$, i.e.,
$a_{n}=\frac{1}{n!}\frac{\partial^{n}}{\partial z^{n}}f(z)\big{|}_{z=0}$.
A regular language $L$, accepted by a DFA $\mathcal{A}$, is a combinatorial
structure representing all the traces belonging to the language. The _sequence
of cardinalities_ of $L$ is the sequence of number of traces $a_{n}$ of length
$n=0,1,2,\cdots$ accepted by $\mathcal{A}$, i.e., the cardinality of the
language accepted by $\mathcal{A}$ restricted to traces of length $n$,
$\mathfrak{L}(\mathcal{A})\cap\Sigma^{n}$. For our purpose, we aim at
constructing an OGF $\mathfrak{G}_{L}(z)$ encoding in a compact way the
sequence of cardinalities of $L$.
The OGF for the sequence of cardinalities of a regular language is always a
rational function (Flajolet and Sedgewick, 2009). For example, let us consider
the language $L=\Sigma^{*}$, with $\Sigma=\\{0,1\\}$. The empty trace, with
length 0, is accepted ($a_{0}=1$); two traces of length 1 are accepted
($a_{1}=2$); four of length 2 ($a_{2}=4$); and so on. The infinite sequence
$a_{n}=\\{1,2,4,8,16,\cdots\\}$ can be compactly encoded by the OGF
$\mathfrak{G}_{L}(z)=\frac{1}{1-2z}$.
Given a regular language $L$, the construction of the OGF
$\mathfrak{G}_{L}(z)$ is typically reduced to the problem of counting the
number of accepting paths of length $n$ of the minimal automaton accepting
$L$, via the established _transfer matrix method_ (Flajolet and Sedgewick,
2009; Stanley, 2011). This algorithm relies on linear algebraic operations on
the matrix representing the transition relation of the automaton (Flajolet and
Sedgewick, 2009, Proposition I.3). While the transfer matrix method works in
general, in Section 5 we will present an alternative, equivalent algorithm to
obtain the OGF of a regular language that significantly outperformed the
transfer matrix method in our tests.
Once the OGF $\mathfrak{G}_{L}(z)$ is known, the values of the sequence
$a_{i},i\geq 0$ can be recovered as the Taylor coefficients of
$\mathfrak{G}_{L}(z)$. Since the OGF for the sequence of cardinalities of a
regular language is a rational function, however, it is possible (and faster)
to derive a recurrence relation that allows us to compute the values of the
sequence without performing symbolic differentiation, using the following
method. Let the OGF $A(z)$ be defined as $A(z)=N(z)/D(z)$, where $N(z)$ and
$D(z)$ are polynomials of maximum degree $m$. Then
(2)
$\frac{N(z)}{D(z)}=\frac{b_{0}+b_{1}z+\cdots+b_{m}z^{m}}{c_{0}+c_{1}z+\cdots+c_{m}z^{m}}=\sum_{i=0}^{\infty}a_{i}z^{i}$
By multiplying by $D(z)$ and expanding the sum, one obtains:
(3)
$b_{0}+b_{1}z+\cdots+b_{m}z^{m}=(c_{0}+c_{1}z+\cdots+c_{m}z^{m})(a_{0}+a_{1}z+a_{2}z^{2}+\cdots)$
and by equating the coefficients one obtains a set of equations that
recursively defines the sequence
(4)
$a_{n}=\begin{cases}\frac{b_{n}-\sum_{i=1}^{n}c_{i}a_{n-i}}{c_{0}}&\text{if
$n\leq m$}\\\ \frac{-\sum_{i=1}^{m}c_{i}a_{n-i}}{c_{0}}&\text{if
$n>m$}\end{cases}$
## 3\. Existing methods for Model Assessment
Following Equation (1), when a finite evaluation set of traces $E$ is
available, measuring the accuracy of $\mathcal{H}$ against $\mathcal{R}$ using
precision and recall can be achieved by comparing the acceptance of each trace
in $E$. However, obtaining a finite and representative evaluation set over
$\Sigma^{*}$ is a challenging problem. In this section, we discuss the two
prominent classes of methods from the literature to generate evaluation sets
for model assessment. Methods in the first class — _Statistical Estimation_ —
generate the evaluation set of traces by means of a random walk over the
automata’s paths. Methods in the second class — _Model-Based_ — use a
deterministic traversal of the automata’s transition relation to generate an
evaluation set that is likely to reveal divergences between the reference
model and the inferred one.
### 3.1. Statistical Estimation using Random Walks
Statistical estimation (Busany et al., 2019; Lo et al., 2012, 2009; Walkinshaw
and Bogdanov, 2013; Walkinshaw et al., 2013; Mariani et al., 2010) represents
a class of model assessment methods based on finite evaluation multisets
(denoted by $E$) of traces randomly generated (sampled) over $\Sigma^{*}$. The
random process used to generate $E$ is the characterizing element of each
method. The most popular statistical estimation method is _trace similarity_ ,
proposed by Lo and Khoo (2006a), which generates $E$ by performing _random
walks_ on the reference and inferred models, rather than generating random
sequences from $\Sigma^{*}$.
The core random walk algorithm is defined as follows, starting from the
initial state. If the current state is accepting, the procedure randomly
decides whether to terminate the walk or to continue; if the walk is not
terminated, it randomly selects one among the outgoing transitions from the
current state, adds the transition symbol to the trace, updates the current
state to the target of the transition, and then the random walk continues from
the new current state. If an error state222As defined in Section 2.1, an error
state is a state from which no accepting state can be reached. is reached, the
current trace is discarded and the random walk restarts from the initial
state.
To compute the _precision_ using trace similarity, an evaluation multiset $E$
is generated by performing random walks on the inferred model $\mathcal{H}$,
therefore generating traces that are either true positives or false positives.
Precision is then defined as the ratio of traces in $E$ accepted also by the
reference model $\mathcal{R}$, i.e. the proportion of true positives333The
rationale for this process stems from the observation that
$\frac{|TP|}{|TP|+|FP|}=\frac{|\mathfrak{L}(\mathcal{R}\cap\mathcal{H})|}{|\mathfrak{L}(\mathcal{R}\cap\mathcal{H})|+|\mathfrak{L}(\overline{\mathcal{R}}\cap\mathcal{H})|}=\frac{|\mathfrak{L}(\mathcal{R}\cap\mathcal{H})|}{|\mathfrak{L}(\mathcal{H})|}$,
which on the surface reminds of the conditional probability of a trace being
accepted by $\mathcal{R}$ given that the trace is accepted by $\mathcal{H}$.
We shall demonstrate in the remaining of this section that this conditional
probability interpretation is itself conditional on the random walk process,
which may render the computed precision meaningless.. Similarly, to compute
the _recall_ , the evaluation multiset $E$ is generated by performing random
walks on the reference model $\mathcal{R}$, generating traces that are either
true positives or false negatives. Recall is then defined as the proportion of
traces in $E$ accepted also by the inferred model $\mathcal{H}$. In both
cases, model coverage can be achieved by repeated, independent random walks,
e.g., until each transition is traversed a minimum number of times.
It is important to note that, when Lo and Khoo (2006a) originally proposed the
trace similarity method, it was assumed either (a) that the models are
probabilistic and the random walk is performed according to the transition
probabilities of the models, or (b) that the models are deterministic and the
user specifies how the random choices necessary to generate traces are made
(defaulting to a uniform distribution over the outgoing transitions if not
otherwise specified). The transition probability distribution is an important
aspect of the assessment because, as we will soon discuss, it affects the
results. In the model inference literature, the use of trace similarity on
deterministic models, i.e., whose transition probabilities are not specified
by the model itself, is predominant. This is because the inference methods
that are most well-known (e.g., K-tail (Chow, 1978)) or considered state-of-
the-art (e.g., MINT (Walkinshaw et al., 2016)) generate deterministic models.
Moreover, the reference models used in the assessment are also most often
deterministic, because they are either manually created (for example, from API
documentation and reference books (Pradel et al., 2010)) or they come from a
formal specification of the target language, thus with no information about
the probability distribution of the traces.
On the other hand, a random walk algorithm does require a probability
distribution to select which transition to traverse at each step, and to
decide whether to continue or to terminate the random walk when an accepting
state is reached.
To decide which transition to follow, it is common practice to select one of
the possible alternatives with uniform probability distribution (Lo and Khoo,
2006a). For the decision of whether to continue or terminate the random walk
once a final state is reached, a number of strategies are commonly used; e.g.,
Walkinshaw et al. (2013) define a termination probability that is inversely
proportional to the number of outgoing transitions.
A factor that is often overlooked in the statistical estimation of the model
accuracy is how the (arbitrary) randomness of the trace generation impacts the
model assessment. Inferred and reference models merely represent languages,
i.e., (possibly infinite) sets of traces over a finite alphabet. There is no
intrinsic probability distribution over the traces of a finite state model. On
the other hand, the random trace generation process used to produce the
evaluation multiset (e.g., a random walk, in the case of trace similarity)
induces a probability distribution over the accepted language. This
distribution may be non-uniform (i.e., different traces may be generated with
different probabilities) therefore introducing a _sampling bias_ in the model
assessment: the precision and recall values obtained using this random
sampling will reflect how well the inferred model can classify traces drawn
according to that distribution. Unless this distribution reflects domain
knowledge about the application being analyzed (i.e., if the models are
purposefully probabilistic, and they describe the probability that different
traces have of being generated by the system under analysis), this sampling
bias is not desirable.
For example, let us consider a random walk using a fixed termination
probability $p_{a}$ to decide the trace termination, and uniform sampling
among the available alternatives to decide which transition to follow.
Intuitively, the termination probability has an impact on the length of the
sampled traces: increasing $p_{a}$ makes shorter traces more likely to be
sampled (correspondingly, longer traces less likely). However, it is hard to
predict the actual length distribution without taking into account the
specific topology of the model from which the traces are sampled. Moreover,
the topology of the model also induces a non-uniform distribution among
accepted traces of the same length: the likelihood of a trace depends on the
probability of selecting each of its transitions in a given order, with each
of these selections generally depending on the number of successors of the
source state.
We have thus identified two sampling biases in the random walk:
* •
shorter traces are more likely to be sampled (but exactly how likely is
topology-dependent);
* •
for a given trace length, some traces are more likely to be sampled than
others.
When a random walk is used to generate the evaluation multiset for trace
similarity, both sampling biases become systematic faults of the method, and
their impact on the model assessment is difficult to quantify a priori since
both depend on specific random walk parameters and the topology of the model
to which the random walk is applied. Furthermore, since for trace similarity
methods precision and recall are measured using random walks on different
models (i.e., $\mathcal{H}$ for precision, $\mathcal{R}$ for recall), the
assessment mixes two _different sampling biases_ , making the two measures
incomparable and generally impossible to aggregate.
(a) Reference model $\mathcal{R}$.
(b) Inferred model $\mathcal{H}$.
Figure 1. Example of reference and inferred models
Let us show with an example how sensitive the model’s accuracy calculated by
trace similarity is to the choice of random walk parameters. We take a
concrete example based on the reference model Signature, adapted from the
benchmark models provided in Krka et al. (2014). For readability, we used
single symbols on transition labels and omitted transitions to error states.
The reference model $\mathcal{R}$ is shown in Figure 1(a), while Figure 1(b)
shows the inferred model $\mathcal{H}$, obtained with MINT (Walkinshaw et al.,
2016) — a state-of-the-art model inference tool — with its default
configuration, on a set of 100 traces accepted by $\mathcal{R}$. To see the
impact of different parameters of random walks on the precision of
$\mathcal{H}$, we vary the termination probability (denoted with $p_{a}$)
between $0.01$ and $1$, while the choice of the outgoing transition is always
done uniformly among the available alternatives. For each value of $p_{a}$,
50,000 random traces are generated to assess the precision of $\mathcal{H}$.
Figure 2 shows the result. When $p_{a}=1$, the random walk always generates a
trace of length zero (since the empty string is accepted by $\mathcal{H}$),
which is a true positive, and therefore the precision value converges to 1. On
the contrary, as $p_{a}$ decreases, longer traces are generated and the
precision value converges to 0.2. In this simple example, this range of values
could have been predicted analytically because, for traces of length $l$
($l\geq 2$), the number of true positives is $5^{(l-2)}$ (the set of true
positives is the language determined by the regular expression $a\cdot
b\cdot\\{b,c,d,e,f\\}^{*}$) while the number of false positives is $4\cdot
5^{(l-2)}$ (the set of false positives is the language determined by the
regular expression $a\cdot\\{c,d,e,f\\}\cdot\\{b,c,d,e,f\\}^{*}$), and
therefore $\lim_{l\to\infty}\frac{5^{(l-2)}}{5^{(l-2)}+4\cdot 5^{(l-2)}}=0.2$.
As a result, we can see that the precision value varies between 0.2 and 1.0
depending on the value of $p_{a}$, implying that the result of trace
similarity is extremely sensitive to the choice of the random walk
parametrization. Unlike in this simple example, however, the effect of the
random walk on the assessment result is in general difficult to predict
analytically.
Figure 2. Sensitivity of statistical estimation to changes in the random walk
for the models in Figure 1
#### 3.1.1. Statistical Estimation by Sampling $\Sigma^{*}$
The sampling bias caused by the random trace generation is fundamentally
unavoidable, because the sampled languages are in general infinite and there
exists no uniform distribution over a discrete infinite set. There are,
however, alternative methods to generate a random evaluation multiset $E$ (and
then compute precision and recall using equation (1)) that allow for more
control over the sampling bias.
The simplest method consists in generating traces by combining symbols
randomly selected from the alphabet $\Sigma$, without considering the models
under analysis, thus making the sampling bias model-independent. This can be
done, for example, by firstly selecting a random trace length $l$, and then
concatenating $l$ symbols, each one randomly selected from $\Sigma$. The
generated trace is then added to the evaluation multiset $E$, and the process
is repeated until a stopping condition is met (e.g., a certain number of
traces has been generated). Note that, when the maximum length of a sample
trace $n$ is fixed, it is possible to perform this trace generation in a way
that ensures uniform probability distribution over $\Sigma^{\leq n}$: each
trace length $l\leq n$ must be chosen with probability proportional to
$|\Sigma^{l}|$, and each symbol must be chosen with uniform distribution.
Despite its simplicity and its ability to reduce the sampling bias, compared
to methods based on random walks like trace similarity, sampling from
$\Sigma^{n}$ is rarely used in practice due to its computational cost: for
most practical problems and large values of $n$, the proportion of
$\Sigma^{n}$ accepted by either $\mathcal{H}$ or $\mathcal{R}$ tends to
decrease with $n$, leading to most samples being true negatives, whose number
does not contribute to the computation of either precision or recall. We will
show instances of this problem experimentally in Section 6.
### 3.2. Model-Based Assessment
Walkinshaw et al. (Walkinshaw et al., 2008) proposed a new method based on
methodologies from model-based testing (MBT), aimed at mitigating the sampling
bias introduced by the random sampling and increasing the reproducibility of
the results, as it is deterministic and thus not affected by the statistical
uncertainty that comes from using a finite randomly generated evaluation set.
The intuition is that MBT methods can deterministically generate a set of
traces that is _comprehensive_ (in the sense that any erroneous behavior of
the inferred model would be detected) that can be used as evaluation set.
There are multiple MBT methods for DFAs (Chow, 1978; Bonifácio et al., 2008)
that, given a reference model $\mathcal{R}$ and an upper bound on the number
of states the inferred model ($\mathcal{H}$ in our case) is allowed to
contain, can check the equivalence of $\mathcal{R}$ and $\mathcal{H}$ by
testing them on a finite set of traces, i.e., it can generate a set of traces
$T$ such that, ideally,
$\mathfrak{L}(\mathcal{R})=\mathfrak{L}(\mathcal{H})\Leftrightarrow\forall
t\in T:\mathcal{R}(t)=\mathcal{H}(t)$. The idea proposed by Walkinshaw et al.
(Walkinshaw et al., 2008) is to compute precision and recall using the set of
traces $T$ (generated using the MBT method of choice) as evaluation set $E$.
As an example, let us describe how the W-method (Chow, 1978), a widely-used
MBT method for DFA, generates a set of tests $T$. Given the reference model
$\mathcal{R}=(\Sigma,Q_{\mathcal{R}},q_{i_{\mathcal{R}}},F_{\mathcal{R}},\delta_{\mathcal{R}})$
and an upper bound to the number of states of $\mathcal{H}$ denoted with $m$,
it constructs two sets of traces: a state cover (denoted with
$C_{\mathcal{R}}$) and a characterization set (denoted with
$D_{\mathcal{R}}$). The state cover $C_{\mathcal{R}}$ is a prefix-closed
subset of $\Sigma^{*}$ containing traces reaching each state of $\mathcal{R}$,
i.e., $\forall q\in Q_{\mathcal{R}}\;\exists t\in
C_{\mathcal{R}}:\delta_{\mathcal{R}}(q_{i_{\mathcal{R}}},t)=q$. The
characterization set $D_{\mathcal{R}}$ is a subset of $\Sigma^{*}$ such that,
for any pair of states $q_{a},q_{b}\in Q_{\mathcal{R}}$ with $q_{a}\neq
q_{b}$, it contains a distinguishing trace $t$ such that
$\delta_{\mathcal{R}}(q_{a},t)$ is an accepting state but
$\delta_{\mathcal{R}}(q_{b},t)$ is not, or vice versa. The test set $T$
generated by the W-method is then defined as
$T=C_{\mathcal{R}}(\\{\epsilon\\}\cup\Sigma\cup\Sigma^{2}\cup\dots\cup\Sigma^{k+1})D_{\mathcal{R}}$,
where $k=m-|Q_{\mathcal{R}}|$ and $T_{a}T_{b}=\\{t_{a}t_{b}\mid t_{a}\in
T_{a},t_{b}\in T_{b}\\}$ for two sets of traces $T_{a}$ and $T_{b}$.
The comprehensiveness of $T$ ensures that any erroneous behavior in the
inferred model will affect the accuracy metrics, and the way in which $T$ is
generated does not favor specific parts of the model under analysis, leading
to less skewed accuracy results when compared to statistical estimation.
Nevertheless, using $T$ to measure the accuracy of an inferred model has three
major shortcomings.
First, there are multiple approaches in the area of MBT for DFA with different
aims. For example, the Wp-method (Fujiwara et al., 1991) is a further
development of the W-method aimed at reducing the size of $T$ while providing
the same guarantees. Since different approaches will generate different sets
of tests $T$, usually with different number of tests accepted or rejected by
either of the models, they will also lead to different accuracy results.
Second, as already noted by Walkinshaw et al. (2008), MBT methods often
generate a set $T$ containing a disproportionately large amount of traces not
accepted by the reference model, which may lead to skewed accuracy results.
Furthermore, $T$ can become exceedingly large if the difference between the
number of states of $\mathcal{R}$ and the number of states of $\mathcal{H}$ is
large. For example, in the W-method described above, the cardinality of $T$
grows with $|\Sigma^{k+1}|$, where $k$ is the difference between the number of
states. This is a significant issue especially in the context of model
inference from positive examples only, where it is not unusual to obtain
inferred models having a number of states that is one order of magnitude
larger than the number of states of the reference model, making this type of
assessment infeasible. This will also be shown experimentally in Section 6.
Third, although MBT guarantees that the generated $T$ will certainly contain a
counterexample trace highlighting any incorrect behavior of the inferred
model, each counterexample trace has the same “weight” on the computed
accuracy metrics, even though it could represent a smaller or larger class of
errors. Let us consider the example shown in Figure 3; it contains a reference
model $\mathcal{R}$ (Figure 3(a)) and two incorrect models $\mathcal{H}_{1}$
(Figure 3(b)) and $\mathcal{H}_{2}$ (figure 3(c)), which were manually created
introducing the erroneous transitions highlighted in red. Both
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ have only false negative errors.
(a) Reference model $\mathcal{R}$
(b) An incorrect model $\mathcal{H}_{1}$
(c) Another incorrect model $\mathcal{H}_{2}$
Figure 3. A example case where the assessment based on the W-method would give
misleading results.
By enumerating all traces we verified that, for any trace length,
$\mathcal{H}_{1}$ has fewer false negatives than $\mathcal{H}_{2}$ (e.g., for
traces of length less than or equal to 10, $\mathcal{H}_{1}$ has 11 false
negative traces, while $\mathcal{H}_{2}$ has 64 false negative traces).
However, when the model accuracy is evaluated using the set of traces $T$
generated by the W-method, $\mathcal{H}_{1}$ has 96% recall while
$\mathcal{H}_{2}$ has 98% recall (precision is 100% in both cases). This
happens because individual false negative traces in $T$ do not represent an
equal number of false negative traces in the inferred languages.
## 4\. Measuring Accuracy with Trace Counting
To overcome the limitations of statistical and model-based accuracy
evaluations, we propose to compute precision and recall, using analytical
measures for the cardinality of the languages of true positives, false
positives and false negatives — normally, restricted up to a finite maximum
trace length to handle languages with unbounded traces.
In particular, the proposed cardinality measures should be: 1) _deterministic_
, making the process repeatable and avoiding the convergence limitations of
statistical methods; 2) _comprehensive_ , accounting for every trace; 3)
_unbiased_ , giving each trace the same weight on the result; 4) _model-
independent_ , generating the same accuracy measurement on all models
accepting the same language.
Notably, a statistical estimation in which all the traces (up to the
prescribed maximum length) are sampled uniformly with the same probability
(such as via the model-independent random sampling from $\Sigma^{*}$ discussed
in Section 3.1.1) would converge to the same values we propose to compute
analytically. This will be shown experimentally in Section 6.
### 4.1. Language Cardinality Measures
We propose to compute two classes of cardinality-based measures to evaluate
the accuracy of an inferred model (the hypothesis $\mathcal{H}$) against a
reference model $\mathcal{R}$.
The first step is the construction of the languages corresponding to the
definitions of true positive, false positive, and false negative, as required
for the computation of precision and recall. These languages will be accepted,
respectively, by the automata $\mathcal{A}_{TP}$, $\mathcal{A}_{FP}$, and
$\mathcal{A}_{FN}$ defined as:
(5)
$\mathcal{A}_{TP}=\mathcal{R}\cap\mathcal{H};\quad\mathcal{A}_{FP}=\overline{\mathcal{R}}\cap\mathcal{H};\quad\mathcal{A}_{FN}=\mathcal{R}\cap\overline{\mathcal{H}}$
We can then use analytic combinatorics methods, as described in Section 2.3,
to obtain the sequences $\mathit{tp}_{n}$, $\mathit{fp}_{n}$,
$\mathit{fn}_{n}$ of the number of traces of length $n=0,1,2,\cdots$ accepted
by $\mathcal{A}_{TP}$, $\mathcal{A}_{FP}$, and $\mathcal{A}_{FN}$ respectively
— the _cardinality of the languages_ accepted by the three automata
intersected with the evaluation set $E=\Sigma^{n}$.
Finally, we compose these elementary cardinality measures to compute the
derived precision and recall metrics for assessing an inferred hypothesis
model: _cumulative-length_ and _single-length_.
Cumulative-Length. This type of assessment considers all the traces of length
_up to a provided value_ $n$ (i.e., the evaluation set $E$ is $\Sigma^{\leq
n}$). We compute the cardinalities
$C_{TP}=\sum_{i=0}^{n}\mathit{tp}_{i};\quad
C_{FP}=\sum_{i=0}^{n}\mathit{fp}_{i};\quad
C_{FN}=\sum_{i=0}^{n}\mathit{fn}_{i}$
using the sequences of cardinalities of the languages accepted by the automata
in Equation 5, and then compute precision and recall:
(6) $\textit{precision}_{\leq
n}=\frac{C_{TP}}{C_{TP}+C_{FP}};\quad\textit{recall}_{\leq
n}=\frac{C_{TP}}{C_{TP}+C_{FN}}$
Our analysis does not mandate for a specific value of the maximum trace length
$n$. If any domain knowledge for the application under analysis is available
and suggests the use of a particular value, this should be used. In the
absence of this information, the user should consider that the choice of the
maximum trace length may affect the assessment result — as we will investigate
experimentally in Section 6.4.
Resolving to using very large values of $n$ so to approximate the _asymptotic_
values of $\textit{precision}_{\leq n}$ and $\textit{recall}_{\leq n}$ (i.e.,
the limit of such measures for $n\to\infty$) is appealing, but should be done
with caution, since models with the same asymptotic values can exhibit
different accuracy for shorter trace lengths. Nonetheless, the asymptotic
values of precision and recall carry useful information. If the precision
eventually converges to zero, the hypothesis language contains more false
positives than true positives (in order of magnitude); if it converges to one,
there are more true positives than false positives, while a value in between
these extremes is achievable if the numbers of true and false positive have
the same order of magnitude (including when they are both finite). Analogous
considerations can be formulated for the asymptotic convergence of the recall
measure in Equation (6) by comparing the orders of magnitude of the true
positives and false negatives languages. We will further investigate
experimentally the behavior of our cardinality-based accuracy metrics for
large values of $n$ in Section 6.4444The analytical computation of such
asymptotic values is usually non-trivial and will not be considered in this
work. The interested reader may refer to “Part B: Complex Asymptotics” of
(Flajolet and Sedgewick, 2009) for an extensive treatment of the subject..
Single-length Assessment. A second pair of cardinality-based assessment
metrics can be obtained by computing precision and recall on the sublanguages
of $\mathcal{A}_{TP}$, $\mathcal{A}_{FP}$, and $\mathcal{A}_{FN}$ restricted
to traces of exactly length $n$, i.e., the traces at the intersection of
$\Sigma^{n}$ and the corresponding automaton.
Given a trace length, computing precision and recall is straightforward, using
directly values from the sequences of cardinalities previously defined:
(7)
$\textit{precision}_{=n}=\frac{\mathit{tp}_{n}}{\mathit{tp}_{n}+\mathit{fp}_{n}};\quad\textit{recall}_{=n}=\frac{\mathit{tp}_{n}}{\mathit{tp}_{n}+\mathit{fn}_{n}}$
This assessment type can be used to build a more comprehensive picture of the
model accuracy by repeating the assessment for every trace length within a
range specified by the user. The result has multiple desirable features.
First, is not sensitive to the parameter selection: the choice of the range of
trace lengths over which the single-length assessment is repeated does not
affect the result, but just defines the scope of the assessment. Second, it
makes clear how the model accuracy changes across the trace lengths considered
— a characteristic that current popular assessment methods do not highlight
despite the fact that, in general, models _do_ have variable accuracy
depending on the trace length, as it will be observed in our experimental
evaluation (Section 6). Third, it allows computing derived accuracy metrics
that consider different trace lengths with different weight. This may be used,
for example, to weight precision and recall on the frequency of trace lengths
observed in a specific deployment of the system.
Although our method does not mandate for a specific range, it should be wide
enough to cover all the possible behaviors of the models under analysis, for
example ensuring that the upper bound of the range is at least equal to the
number of states of the largest automaton under analysis.
## 5\. Fast Computation of OGFs for Model Assessment
In our model assessment method we compute the values of precision and recall
using the counts of true positives, false positives and false negatives, up to
a finite maximum trace length. To obtain these counts without explicitly
enumerating all the possible traces, we use analytic combinatorics (see
Section 2.3), to count the number of different accepting paths on the automata
accepting these languages.
To do so, the first step is to obtain the _ordinary generating function_ (OGF)
for the cardinality sequence of the language under analysis. This is generally
done using the _transfer matrix method_ (Flajolet and Sedgewick, 2009;
Stanley, 2011), which relies on linear algebraic operations on the matrix
representing the transition relation of the automaton (Flajolet and Sedgewick,
2009), resulting in its worst-case complexity being cubic in the state count.
This section describes an alternative state elimination algorithm that,
despite having the same worst case complexity, in practice in our preliminary
evaluation performed significantly better than the transfer matrix method, by
exploiting the sparsity of the transition matrix.
Our method is analogous to the state elimination algorithm by Brzozowski and
McCluskey (Brzozowski and McCluskey, 1963) to generate a regular expression
given a finite state automaton. While in the Brzozowski and McCluskey’s
algorithm each transition is labeled with a regular expression indicating the
language that causes traversing the transition, in our method it is labeled
with the OGF of the cardinality sequence of that same language. The reduction
rules used when a state is eliminated then allow us to progressively build the
OGF through operations between rational functions.
Our method is presented in algorithm 1
Data: DFA $\mathcal{A}=(\Sigma,Q,q_{i},F,\delta)$
Result: The OGF of the sequence of cardinalities of the language accepted by
$\mathcal{A}$
1 $(\mathcal{G}=\langle
N,E\rangle,\mathit{initial},\mathit{final})\leftarrow\texttt{digraphConstruction}(\mathcal{A})$
2 while
_$\texttt{hasNonInitialOrFinalNode}(\mathcal{G},\mathit{initial},\mathit{final})$_
do
3
$n\leftarrow\texttt{chooseNonInitialOrFinalNode}(\mathcal{G},\mathit{initial},\mathit{final})$
4 $\texttt{eliminateNode}(\mathcal{G},n)$
5
6 end while
return $E(\mathit{initial},\mathit{final}\,)$
Algorithm 1 Fast computation of the OGF of the sequence of cardinalities of
the language accepted by a DFA.
and works as follows.
Given an input DFA, we construct (line 1) a directed graph $\mathcal{G}$,
using digraphConstruction (algorithm 2):
Data: $\mathcal{A}=(\Sigma,Q,q_{i},F,\delta)$ deterministic finite-state
automaton over the alphabet $\Sigma$.
Result: $\mathcal{G}=\langle N,E\rangle$ digraph with edges labeled with
rational functions, where $N$ is the set of nodes, and $E:N\times N\rightarrow
RF$ are the edges labeled with rational functions (0 if no edge is present).
1 $N\leftarrow Q\cup\\{\mathit{initial},\mathit{final}\\}$
2 foreach _$(a,b)\in N\times N$_ do
3 $E(a,b)\leftarrow 0$
4
5 end foreach
6foreach _$q\in Q$_ do
7 foreach _$s\in\Sigma$_ do
8 $t\leftarrow\delta(q,s)$
9 $E(q,t)\leftarrow E(q,t)+z$
10
11 end foreach
12
13 end foreach
14$E(\mathit{initial},q_{i})\leftarrow 1$
15 foreach _$q\in F$_ do
16 $E(q,\mathit{final})\leftarrow 1$
17
18 end foreach
19$\mathcal{G}=\langle N,E\rangle$
return $(\mathcal{G},\mathit{initial},\mathit{final}\,)$
Algorithm 2 digraphConstruction. Generates the digraph corresponding to the
given DFA.
each state of the DFA corresponds to one node of $\mathcal{G}$ (distinct
states correspond to distinct nodes), and the edge between any ordered pair of
nodes of $\mathcal{G}$ is labeled with the generating function of the sequence
of cardinalities of the language of words of length one, containing the
symbols causing the transition between the two corresponding states of the
DFA. If the cardinality of this language is $n$ (i.e., there are $n$ symbols
causing the transition from the source state to the destination state), the
cardinality sequence is $\\{0,n,0,0,0,\dots\\}$, thus the generating function
on the corresponding edge of $\mathcal{G}$ is $\mathfrak{G}(z)=nz$. In
addition, we add to $\mathcal{G}$ a node called initial, an edge from initial
to the node corresponding to the initial state of the DFA, a node called
final, and an edge from each node corresponding to an accepting state of the
DFA to final. All these additional edges are labeled with the generating
function $\mathfrak{G}(z)=1$, which is the generating function of the
cardinality sequence $\\{1,0,0,\dots\\}$ (i.e., of the language containing
only the empty string). Adding these edges does not change the final
generating function, but it is a simple way of dealing with multiple final
states and transitions back to the initial state. Figure 4(a)
(a) Initial DFA
(b) Corresponding digraph $\mathcal{G}$
Figure 4. Example of digraph construction
shows the example DFA taken from (Aydin et al., 2015) (which computes the OGF
using the transfer matrix method), whereas Figure 4(b) depicts the graph
$\mathcal{G}$ constructed with these rules described above.
Then, at lines 1–1 of algorithm 1, we eliminate, one by one and using
eliminateNode (Algorithm 3), all the nodes of $\mathcal{G}$ except initial and
final, as follows.
Data: $\mathcal{G}=\langle N,E\rangle$ labeled directed graph, where $N$ is
the set of nodes, and $E:N\times N\rightarrow RF$ are the edges labeled with
rational functions (0 if no edge is present); $n\in N$ the node to be
eliminated.
Result: The node $n$ is removed from $\mathcal{G}$ and the OGFs on the
remaining edges are updated.
1
2$\mathfrak{G}_{\mathit{loop}}\leftarrow\frac{1}{1-E(n,n)}$
$E(n,n)\leftarrow 0$
/* remove self loop if present */
3
$P\leftarrow\\{p\in N:E(p,n)\neq 0\\}$
/* predecessors of $n$ */
$S\leftarrow\\{s\in N:E(n,s)\neq 0\\}$
/* successors of $n$ */
4
5foreach _$p\in P$_ do
6 foreach _$s\in S$_ do
7 $E(p,s)\leftarrow E(p,s)+E(p,n)\,E(n,s)\,\mathfrak{G}_{\mathit{loop}}$
8
9 end foreach
10 $E(p,n)\leftarrow 0$
11
12 end foreach
13foreach _$s\in S$_ do
14 $E(n,s)\leftarrow 0$
15
16 end foreach
$N\leftarrow N\setminus n$
Algorithm 3 eliminateNode. Eliminates a node while maintaining the correct OGF
on the remaining digraph edges.
Let $n$ be the node to be eliminated; algorithm eliminateNode works by
replacing every pair composed by one edge entering $n$ from a predecessor node
$p$ and one edge exiting $n$ to a successor node $s$, with a single edge from
$p$ to $s$, keeping into account the loop edge from $n$ to $n$ if present. The
procedure computes the OGF on the new edges using the fact that union,
concatenation, and Kleene star of regular languages translate to algebraic
operations between the OGF of the cardinality sequences of the operand
languages. Specifically: if $A$ and $B$ are regular languages, their
concatenation $C=A\cdot B$ is a regular language with
$\mathfrak{G}_{C}(z)=\mathfrak{G}_{A}(z)\mathfrak{G}_{B}(z)$; if $A$ and $B$
are regular languages and $A\cap B=\emptyset$ (which in our case is a
consequence of the automaton determinism), then their union $C=A\cup B$ is a
regular language with
$\mathfrak{G}_{C}(z)=\mathfrak{G}_{A}(z)+\mathfrak{G}_{B}(z)$; if $A$ is a
regular language with $\mathfrak{G}_{A}(z)=A(z)$ then $A^{*}$ (Kleene star) is
a regular language with $\mathfrak{G}_{A^{*}}(z)=\frac{1}{1-A(z)}$. As a
result, Algorithm 3 computes the OGF of the Kleene star of the language that
would be on the loop of the node to be eliminated (line 3), eliminates the
loop (line 2) — otherwise the set of predecessors and successors subsequently
defined would contain $n$ — and then iterates through the edges entering $n$
and exiting $n$, updating the OGF on the edge from the predecessor node to the
successor node as illustrated in Figure 5.
(a) Before eliminating $q_{2}$
(b) After eliminating $q_{2}$
Figure 5. Node elimination
When initial and final are the only nodes left in $\mathcal{G}$, the OGF on
the edge from initial to final is the OGF of the cardinality sequence of the
language accepted by the DFA.
Figure 6 shows one possible sequence of state eliminations to obtain the
generating functions of the DFA shown in Figure 4(a).
(a) After eliminating $q_{3}$
(b) After eliminating $q_{2}$
(c) After eliminating $q_{1}$
Figure 6. Example sequence of state eliminations for the DFA shown in Figure
4(a)
We remark the order with which the nodes are eliminated is not relevant for
the correctness of the result. However, it does significantly affect the
performance due to the cost of the operations between rational functions
required to compute the generating functions on the edges involved in each
elimination. A simple heuristic that proved to be beneficial in our
experiments is choosing first the node with the lowest OGF degree555The OGF
degree is defined as the maximum of the degrees of its constituent numerator
and denominator polynomials. on the self loop (thus the nodes with no self
loop are eliminated first).
## 6\. Experimental Evaluation
In this section, we report on the experimental evaluation of our model
assessment method in terms of the following aspects. First, since the
cumulative-length assessment requires, as an input parameter, a maximum trace
length, we investigate the impact of changing its value on the model
assessment results. We also analyze whether the asymptotic values (i.e., the
results obtained when the maximum trace length go to infinity) are useful to
characterize the model accuracy. Second, to better understand to what extent
the model assessment results differ depending on the methods used, we compare
the results obtained using our method with the results obtained using existing
methods, specifically the statistical estimation and MBT-based methods. Third,
in Section 4, we claimed that a statistical estimation in which traces having
the same length have the same probability of being sampled would converge to
the same results obtained by our single-length assessment method. We will
verify this experimentally by comparing the single-length assessment and a
model-independent sampling of $\Sigma^{*}$ in terms of the precision and
recall values. Finally, we evaluate the applicability of our method on models
inferred using well known model inference approaches.
To summarize, we address the following research questions:
1. RQ1:
How does the choice of the maximum trace length affect the model assessment
results?
2. RQ2:
How do the values of precision and recall obtained using our method compare
with the results obtained using other evaluation methods?
3. RQ3:
Is the single-length assessment over a range of lengths a usable alternative
to the statistical evaluation of model accuracy using model-independent
sampling of $\Sigma^{*}$?
4. RQ4:
Is our method applicable to the assessment of inferred models representing
aspects of actual software systems?
### 6.1. Evaluation Subjects
To evaluate model assessment methods, we need various pairs of reference and
inferred models. We describe how we select reference models and generate
inferred models from them below.
#### Reference Models
We selected 41 publicly available reference models, taken from existing
studies (Pradel et al., 2010; Krka et al., 2014); all of them have a well-
documented origin and were previously used in the model inference literature.
Pradel et al. (2010) selected 32 commonly used classes from the Java SDK API
and identified their method ordering constraints using the API documentation
and well-known reference books. These constraints were then translated into
reference models representing all possible valid traces of method calls. The
resulting reference models are publicly available on the authors’
website666http://mp.binaervarianz.de/icsm2010/index.html. These models were
used also in previous work on model inference (Busany et al., 2019).
Krka et al. (2014) selected 9 open-source libraries, found the corresponding
reference models (manually specified in previous work), and checked them
manually for inconsistencies. These models were checked against execution
traces collected from actual executions of software using those libraries, and
the transitions on methods that were never invoked in the collected traces
were eliminated. The resulting reference models are publicly available on the
authors’ website777https://softarch.usc.edu/wiki/doku.php?id=inference:start.
These models have been used also in previous work (Walkinshaw et al., 2016; Le
et al., 2015)
#### Inferred Models
Ideally, the inferred models should be generated using model inference engines
on traces produced by executions of actual software systems. Unfortunately, we
were not able to find and execute the exact versions of the software systems
represented by the 41 reference models, to collect their execution traces. As
an alternative, we used the same type of random walk described in Section 3.1
to generate a set of random traces for each reference model, and then we
processed each set of traces using different model inference engines. As for
the random walk parameters, we used the termination probability $p_{a}=0.1$
and the uniform probability for selecting an outgoing transition among
available transitions following Walkinshaw et al. (2013). For each reference
model, we repeated the random walk until at least 100 traces were generated
_and_ each state of the model had been visited at least four times, as
suggested by Busany et al. (2019). From each set of traces we inferred two
models, using two different model inference algorithms: K-tail (Chow, 1978)
(the most well-known model inference algorithm) and MINT (Walkinshaw et al.,
2016) (a state-of-the-art model inference technique).
It is worth noting that, in the area of model inference, it is common practice
to use randomly generated traces when traces coming from actual software
executions are not available (e.g., (Busany et al., 2019; Lo et al., 2012;
Walkinshaw and Bogdanov, 2013; Walkinshaw et al., 2013)). Moreover, model
assessment is independent of how inferred models are generated, thus is not a
major issue in our evaluation. Nevertheless, we will discuss this as a
potential threat to validity in Section 6.8.
To summarize, our experimental evaluation is based on 82 test subjects (pairs
of reference and inferred models): 41 reference models, with two inferred
models each. Table 1 shows the size of the reference and inferred models in
terms of the minimum, the average, and the maximum number of states and
transitions.
Table 1. Characteristics of reference and inferred models | State count | | Transition count
---|---|---|---
| Min | Avg | Max | | Min | Avg | Max
Reference | 2 | 9.2 | 41 | | 7 | 77.8 | 465
Inferred | 2 | 349 | 2060 | | 5 | 752 | 4994
### 6.2. Evaluation Settings
We performed the assessment of all the 82 test subjects using our
implementation of trace similarity, MBT-based assessment, and our method with
trace counting, all developed in Java and publicly available (see Section
6.3).
All the experiments were executed on an AMD EPYC 7401P (24 cores, 48 threads)
with 448 GB of RAM. Since the implementation of our evaluation method is
single-thread, 24 evaluations were executed concurrently.
The evaluation runtime of every model assessment performed in our evaluation
was subject to a timeout of two days. It is worth pointing out that all the
single-length and cumulative-length assessments on the same test subject
require the same evaluation runtime. The reason is that this evaluation
runtime is dominated by the time required to compute the generating functions,
which are the same for all assessments of the same test subject.
### 6.3. Data availability
To support open science and enhance the reproducibility of our evaluation, we
provide a replication package for the reviewers, including all the artifacts:
the source code, the reference models and the inferred models used. Upon
acceptance, the replication package will be released to the public.
### 6.4. RQ1: Input Parameter Sensitivity
##### Methodology
As described in Section 4, our method can be applied in two ways: _cumulative-
length_ and _single-length_ assessments. The single-length assessment repeated
over a range of trace lengths is not sensitive to parameter changes: the
_range_ changes just the _scope_ of the assessment, i.e., for which lengths
the values of precision and recall are computed, without affecting the output
values themselves. To evaluate the parameter sensitivity of the cumulative-
length assessment, we computed the precision and recall values varying the
only parameter (i.e., the maximum trace length) from 0 to 200 in steps of 1.
##### Results
All the assessments of 61 of the 82 test subjects terminated within the 2-day
timeout. Figure 7 shows how the precision and recall values vary depending on
the maximum trace length parameter; each line is a result for one test
subject.
(a)
(b)
Figure 7. Values of $\textit{precision}_{\leq n}$ and $\textit{recall}_{\leq
n}$ measured using the cumulative-length assessment, varying the maximum trace
length parameter $n$ in the range [0, 200]. One plot per test subject. Five
test subjects are highlighted, and their characteristics are summarized in
Table 2.
The foremost takeaway of the evaluation results is that the choice of the
maximum trace length parameter does indeed affect the computed accuracy
values, and it does so differently, depending on the inferred model. The
results highlighted the prevalence of cases in which precision and/or recall
values cover a large part of the range $[0,1]$, depending on the parameter
value. Specifically, among the 61 test subjects that finished within the
timeout, 20 have a precision value (and 54 have a recall value) that covers
the entire range $[0,1]$, depending on the maximum trace length parameter.
This emphasizes how — for any choice of a single value of the maximum trace
length — some relevant information about the model accuracy is inevitably
lost, leading to possibly misleading results.
Figure 7 also shows the variety of trends that the precision and recall values
can have, as the maximum trace length is increased: e.g., convergence to 0
with and without a peak, constant equal to 1, and convergence to $0<k<1$. In
the figure we have highlighted (with colors and special markers) five test
subjects that, together, cover the most common trends we observed in the
results of this experiment. Appendix A gives further details on how we
selected these test subjects, and Appendix B describes the selected test
subjects in more detail. Table 2 summarizes the characteristics of these
representative test subjects.
Table 2. Characteristics of the selected test subjects | Reference model | Inference approach | Inferred model
---|---|---|---
| Name | States | Trans. | States | Trans.
Subject 1 (S1) | java.net.URL (Pradel et al., 2010) | 5 | 57 | MINT | 718 | 993
Subject 2 (S2) | SMTPProtocol (Krka et al., 2014) | 3 | 14 | MINT | 24 | 110
Subject 3 (S3) | StringTokenizer (Pradel et al., 2010) | 4 | 14 | 2-Tails | 83 | 175
Subject 4 (S4) | Signature (Krka et al., 2014) | 3 | 7 | MINT | 15 | 63
Subject 5 (S5) | StringTokenizer (Krka et al., 2014) | 4 | 7 | 2-Tails | 30 | 61
Furthermore, Figure 7 shows that the asymptotic value888 We determined the
asymptotic value by manually checking to which value the accuracy stabilizes.
There were some cases (seven for the precision value, four for recall value)
in which we were not able to determine the asymptotic value because, within
the observed range, it was not clear around which value it stabilized. can be
reached through different paths, with fast or slow convergence, and with or
without local minima or maxima. This confirms that judging or comparing model
accuracies according to the asymptotic value of precision and recall can be
misleading, since models with the same asymptotic accuracy generally can have
different accuracies for shorter traces.
Although investigating the asymptotic accuracy of models obtained with current
inference methods is outside the scope of this paper, an interesting
observation is that the asymptotic values of the accuracy metrics are
generally either zero or one. This is because the precision and recall values
are quotients between language cardinalities that tend to infinity as the
maximum trace length considered is increased, therefore the quotient often
goes to zero or one depending on the relative growth rate of the languages in
Equation 1. In our evaluation, the most common asymptotic value is zero: 45
subjects have a precision value converging to zero, while 55 have a recall
value converging to zero. The number of subjects having accuracy converging to
a value greater than zero and smaller than one is only three for precision and
two for recall. This is an observation that is outside the scope of this
paper, and may not generalize to models obtained with different inference
methods; it will be further investigated in future work.
#### Answer to RQ1
The maximum trace length parameter does affect the computed cumulative-length
precision and recall values, and it does so differently, depending on the
inferred model. This parameter should be tuned, case by case, to a value that
is relevant for the application under analysis. If no domain knowledge is
available, it is preferable to consider how precision and recall values change
considering different trace lengths, using the single-length metrics.
### 6.5. RQ2: Comparison of Model Assessment Methods
This section discusses how the results generated by the assessment methods
described and discussed in the paper compare with each other. In particular,
we will examine trace similarity, MBT-based assessment, cumulative-length
assessment, and single-length assessment.
Note that the model assessment methods discussed in this paper can be
classified into two types depending on their output types: trace similarity,
MBT-based assessment, and the cumulative-length assessment are all methods
that return one pair of precision and recall values, while the single-length
assessment instead returns one pair of precision and recall values _per trace
length_ in the given range (which is a parameter of the method). These two
types cannot be directly compared. For this reason the comparison proposed in
this section is divided in two parts.
In the first part, we will compare trace similarity, MBT-based, and
cumulative-length, with the goal of understanding the consistency and the
differences between the computed precision and recall values across the
different assessment methods and different parameter values.
In the second part, we will discuss the single-length assessment results in
relation with the results generated with the other currently used methods. In
this case a direct comparison is neither possible nor meaningful, due to the
different output type. To make them comparable, we will _condition the other
assessment methods on the trace length_ : the evaluation sets generated by
these methods will be partitioned according to the trace length, and the
subsets will be used to compute one pair of precision and recall values per
trace length, which can then be directly compared with the results from the
single-length assessment.
#### Comparison of Trace Similarity, MBT-based, and Cumulative-Length
Assessments
As discussed above, we first compare trace similarity, MBT-based, and
cumulative-length.
##### Methodology
To understand the consistency and the differences between the computed
precision and recall values across the different assessment methods, we run
the methods on the 81 test subjects, with the following parameters settings.
Trace similarity requires defining how to make the nondeterministic choices
needed to conduct the random walk (i.e., whether to terminate the walk, and
which available transition should be traversed — see Section 3.1). This has a
direct effect on the sampling bias, thus affecting the assessment results. In
our setup, the selection of the outgoing transition to be traversed was done
uniformly among the available alternatives, as it is often done in practice
(e.g., (Walkinshaw et al., 2013; Lo and Khoo, 2006a)). Regarding the
termination of the trace when a final state is reached, we found different
approaches in the model inference literature. Sometimes the termination
probability is a function of the outdegree of the accepting state (e.g., as in
reference (Walkinshaw et al., 2013)), sometimes is not (e.g., as in reference
(Lo et al., 2012)). To highlight how this choice may affect the assessment
result, we used a fixed termination probability $p_{a}$, and we executed the
trace similarity assessment setting $p_{a}$ to three values (0.02, 0.1 and
0.5), thus obtaining three pairs of precision and recall values for each test
subject. Trace similarity also requires specifying the target size for the
evaluation set $E$, which in our experiments was set to 100,000 traces. To
further ensure adequate model coverage, we continued adding traces to $E$
beyond the target size, until each transition of the model was followed at
least 10 times (as proposed in reference (Lo and Khoo, 2006a)), or a 30-minute
time limit was reached. As a result, in some cases we generated more than
100,000 traces.
The MBT-based assessment requires a model-based testing method for finite
state automata, to generate the evaluation set $E$. We use the W-method, as
done by the original proponents of the method.
The cumulative-length assessment requires a parameter indicating the maximum
length of the traces to be considered. Since there is no “correct” value to
choose in the absence of domain knowledge for a specific test subject, we
evaluate the results when varying the parameter from 0 to 200 in steps of 1.
##### Results
All the model assessments performed using trace similarity returned a result
within the timeout. The cumulative-length assessments terminated within the
timeout for 61 of the 82 test subjects (for all parameter values), while for
the remaining 21 subjects a timeout occurred during the computation of the
generating function. The MBT-based assessment terminated within the timeout
only in two of the 82 cases under analysis. The reason is that the cardinality
of the evaluation set generated with the W-method grows with $|\Sigma^{d+1}|$,
where $d$ is the difference in the number of states between the inferred and
reference model, therefore the method is usable only when the reference and
inferred models have a similar number of states. This is rarely the case in
our set of test subjects: the difference $d$ is on average 247 ($\sigma=265$).
Due to the lack of a sufficient number of results from the MBT-based
assessment, we omit this method in the rest of the comparison.
Before we compare the trace similarity and cumulative-length assessment
results, it is worth focusing on how trace similarity is affected by the
termination probability $p_{a}$ of the random walk. Table 3 shows the effect
of $p_{a}$ on the average length of the traces generated in our experimental
evaluation. These are consistent with an exponential distribution, however we
remind that the relationship between termination probability and distribution
of trace lengths is model dependent, and our experiments lead to this
distribution only because most of the non-error states of the models under
analysis are accepting states.
Table 3. Mean trace length generated by the random walk, depending on the $p_{a}$ parameter. $p_{a}$ | Mean trace length
---|---
0.5 | 1.99
0.1 | 11.28
0.02 | 49.45
Table 4 summarizes the absolute difference between the model accuracy
evaluated using trace similarity with different random walk termination
probability $p_{a}$. For example the mean absolute difference between the
precision value computed using trace similarity with a random walk with
$p_{a}=0.5$, and the precision value computed using trace similarity with a
random walk with $p_{a}=0.02$, is $36\text{\,}\mathrm{\textup{pp}}$
($\sigma=$29\text{\,}\mathrm{\textup{pp}}$$), with $\mathrm{\textup{pp}}$
indicating percentage points. This difference indicates that the choice of the
random walk parameters values can affect the trace similarity result in a way
that does not allow us to reliably determine the model accuracy. Note that a
larger difference in the $p_{a}$ parameter leads to a larger difference in the
measured accuracy (e.g., line 1 vs line 3 of Table 4). This is explained
intuitively in terms of the sampling bias discussed in Section 3.1: a random
walk with higher termination probability $p_{a}$ will generally produce
shorter traces, and we know, from the RQ1 results in Section 6.4, that in most
cases the model accuracy is higher on shorter traces.
Table 4. Mean and standard deviation of the absolute difference between the model accuracy evaluated using trace similarity with different random walk termination probability $p_{a}$. Values in percentage points ($\mathrm{\textup{pp}}$). | Mean | $\sigma$
---|---|---
$|\textit{Precision}_{\textit{TS}_{0.5}}-\textit{Precision}_{\textit{TS}_{0.1}}|$ | $19\text{\,}\mathrm{\textup{pp}}$ | $14\text{\,}\mathrm{\textup{pp}}$
$|\textit{Precision}_{\textit{TS}_{0.1}}-\textit{Precision}_{\textit{TS}_{0.02}}|$ | $19\text{\,}\mathrm{\textup{pp}}$ | $17\text{\,}\mathrm{\textup{pp}}$
$|\textit{Precision}_{\textit{TS}_{0.5}}-\textit{Precision}_{\textit{TS}_{0.02}}|$ | $36\text{\,}\mathrm{\textup{pp}}$ | $29\text{\,}\mathrm{\textup{pp}}$
$|\textit{Recall}_{\textit{TS}_{0.5}}-\textit{Recall}_{\textit{TS}_{0.1}}|$ | $37\text{\,}\mathrm{\textup{pp}}$ | $20\text{\,}\mathrm{\textup{pp}}$
$|\textit{Recall}_{\textit{TS}_{0.1}}-\textit{Recall}_{\textit{TS}_{0.02}}|$ | $15\text{\,}\mathrm{\textup{pp}}$ | $10\text{\,}\mathrm{\textup{pp}}$
$|\textit{Recall}_{\textit{TS}_{0.5}}-\textit{Recall}_{\textit{TS}_{0.02}}|$ | $52\text{\,}\mathrm{\textup{pp}}$ | $27\text{\,}\mathrm{\textup{pp}}$
Figure 8. Comparison of trace similarity with three different random walks
($\textit{TS}_{\textit{0.5}}$, $\textit{TS}_{\textit{0.1}}$ and
$\textit{TS}_{\textit{0.02}}$), and the cumulative evaluation proposed in
Section 4 (black plot) over five representative test subjects.
Figure 8 shows the comparison of the precision and recall values computed
using trace similarity and our cumulative-length assessment, for the five
representative test subjects selected in RQ1. To improve the readability, the
range of the trace length parameter for the cumulative-length assessment shown
in the figure is from 0 to 100. The same information is available for all test
subjects in the supporting material (see Section 6.3). Note the three
different trace similarity assessments per test subject (using different
values of the $p_{a}$ parameter of the random walk).
The most important observation is that both trace similarity and cumulative-
length results values are different in most cases, and vary over a large range
of values depending on the methods’ parameters.
A closer look reveals a consistency between the trend of the accuracy results
computed using trace similarity as $p_{a}$ is decreased, and the trend of the
accuracy results computed by the cumulative-length assessment for longer
traces: in all subjects for which the cumulative-length assessment shows
decreasing model accuracy for longer traces, also the trace similarity
assessment shows decreasing accuracy as $p_{a}$ is decreased from 0.5 to 0.02.
Analogously, the cumulative-length assessment for subject S1 shows an
(initially) increasing precision, and the corresponding trace similarity
result is also consistent with this trend. The reason behind this consistency
is the effect of the random walk termination probability $p_{a}$ on the length
of the traces generated by the random walk, as discussed before. However, it
is important to stress that the distribution of trace lengths generated by the
random walk is not enough to fully capture how the sampling bias affects the
trace similarity result, because also different traces of the same length may
have different probabilities of being generated. The nondeterministic choices
required to conduct the random walk can be performed in a variety of ways, and
their effect on the result is model-dependent. On the other hand, the
cumulative-length assessment has only the maximum trace length parameter,
whose effect can be intuitively understood, making it easier to tune. Section
6.5 will include a scenario in which trace similarity and cumulative-length
assessment diverge due to this effect.
Furthermore, in Figure 8 we can observe that for some test subjects all the
model assessment methods compute the same value (100%) for precision. We
manually verified that in these cases the inferred language is a subset of the
reference language, hence the evaluation set cannot contain false positive
traces, regardless how it is generated, and thus the computed precision value
is always 100%.
#### Comparison of Statistical Estimation and Single-Length Assessment
We now turn our attention to the results of the single-length assessment over
a range of trace lengths, and how it compares with the results of trace
similarity.
##### Methodology
As discussed above, a direct comparison is not possible, since the single-
length assessment returns one pair of precision and recall values _per trace
length_ , while trace similarity returns one pair of precision and recall
values overall.
To obtain comparable results, we will _condition the trace similarity on the
trace length_ , by partitioning each evaluation set $E$ according to the trace
length. This will allow us to use the resulting subsets to compute one pair of
precision and recall values per trace length, and it will also allow us to
look at the distribution of the traces in $E$ across different trace lengths.
In fact, we will initially focus on this distribution, which is the first
effect of the sampling bias induced by the random walk mentioned in Section
3.1. Then, we will compare the values of precision and recall for different
trace lengths (from 1 to 100 in steps of 1). Since our single-length
assessment has no sampling bias, any difference in precision or recall value
for a specific trace length implies statistical noise and/or the non-
uniformity of the sampling induced by the random walk (i.e., the second effect
of the sampling bias as mentioned in Section 3.1).
##### Results
The results for the five representative test subjects previously used in RQ1
are shown in Figure 9.
In each plot, the red line and the black line with markers represent how the
accuracy metric values (left y-axis) computed by trace similarity conditioned
on the trace length and the single-length assessment, respectively, vary
depending on the trace length (x-axis). In addition, the solid area filled in
blue indicates the number of samples (right y-axis) in the evaluation set $E$
for each trace length (x-axis). Note that, in some of the plots, the red line
and the blue area do not exist after a certain trace length, meaning that the
random walk generated an empty $E$ and no precision and recall values were
computed by trace similarity.
(a) Trace similarity using a random walk with $p_{a}=0.5$
(b) Trace similarity using a random walk with $p_{a}=0.1$
(c) Trace similarity using a random walk with $p_{a}=0.02$
Figure 9. Comparison of the single-length assessment and trace similarity
conditioned on the trace length, with three different random walk parameters
Based on the results shown in Figure 9, we can make the following
observations.
* •
The distribution of samples highlights how shorter traces are more likely to
be sampled, and how the distribution of samples depends both on the random
walk parameters and on the model upon which the random walk is performed. For
example with $p_{a}=0.5$ (Figure 9(a)) the distribution of samples for subject
S1 contains longer traces, compared to the other test subjects. Conversely,
with $p_{a}=0.02$ (Figure 9(c)) the distribution of samples for the precision
assessment of subject S1 contains shorter traces, compared to the other test
subjects.
* •
The random walk can introduce a sampling bias between traces having the same
length. This is particularly visible in Figure 9(c), in the recall plots of
subject S5, which show a difference between trace similarity conditioned on
the length (red line, which converges to 0.8) and the single-length evaluation
(black line, which converges to 0.3). This observation explains why in Figure
8, in the case of subject S5, the recall values obtained using standard trace
similarity consistently exceed the recall values obtained using our
cumulative-length assessment, despite the fact that the inferred model
consistently shows a 0.3 recall value for traces having a length greater than
20. Note that in the other test subjects, the plots of trace similarity
conditioned on the trace length, and those of the single-length assessment
show the same trend, highlighting how this effect is model-dependent.
#### Answer to RQ2
The sampling bias affects the trace similarity result in a way that is hard to
predict a priori, because it depends on both the topology of the model and the
parameters of the random walk. When trace similarity is compared to the
cumulative-length assessment, the conclusion is that the choice of the
parameter values of both assessment methods affect the result to an extent
that does not allow us to reliably determine the model accuracy, thus
preventing a meaningful comparison among different models in terms of
accuracy, or among the accuracy values of the same model measured with
different methods.
On the other hand, the single-length assessment is not affected by sampling
bias because it measures the model accuracy for each trace length in the given
range, considering (for each length) all the possible traces. Leveraging this
characteristic, we compared the single-length assessment with trace similarity
conditioned on the trace length, analyzing the effect of the non-uniform
sampling among traces of the same length: this effect is model-dependent. We
also examined the distribution of trace length generated by the various random
walk in trace similarity, noticing how shorter traces are more likely to be
sampled, and how this effect depends on the model topology and the parameters
of the random walk.
### 6.6. RQ3: Single-length assessment and model-independent sampling of
$\Sigma^{*}$
The goal of RQ3 is to compare, in terms of precision and recall, the single-
length assessment we proposed and a model-independent sampling of $\Sigma^{*}$
inspired by the observation made in Sections 3.1.1 and 4. It is possible to
statistically assess precision and recall using a sampling method that
generates traces by combining symbols randomly chosen from the alphabet,
rather than through random walks on models. In practice, however, using this
method is feasible only for short trace lengths, because to evaluate precision
(respectively, recall) it is necessary to generate traces that are accepted by
the inferred (respectively, reference) model. This generation process becomes
infeasible with real-world models and long traces, due to the gap between the
exponential growth of $|\Sigma^{n}|$ and the slower growth of the size of the
accepted language of length $n$, as the length $n$ is increased. Nonetheless,
this approach has the advantage of controlling the sampling bias (which
becomes model independent), enabling sampling with uniform distribution over a
finite subset of $\Sigma^{*}$.
Answering this research question will let us check experimentally whether a
statistical estimation of the model accuracy (in which all the traces having
the same length have the same probability of being sampled) generates the same
results as our single-length assessment. If the results show consensus between
the two assessment methods, they will confirm that our method is a usable
alternative to the statistical evaluation of model accuracy using model-
independent sampling of $\Sigma^{*}$.
##### Methodology
For each test subject, we computed precision and recall for each trace length
starting at 0 and increasing it until a one-hour timeout was reached. This was
done using Algorithm 4, which takes as input the reference model
$\mathcal{R}$, the inferred model $\mathcal{H}$, the accuracy metric $m$ to be
computed (precision or recall) and the trace length $n$ for which to compute
it. The algorithm generates traces without considering the reference or the
inferred model, by concatenating $n$ symbols randomly selected from the
alphabet with uniform distribution. Each generated trace is then tested
against the models: if $m$ is _precision_ (respectively, _recall_), the trace
is tested against $\mathcal{H}$ (respectively, $\mathcal{R}$), to determine
whether it is part of the inferred (respectively, reference) language. If this
is the case, the trace is further tested against $\mathcal{R}$ (respectively,
$\mathcal{H}$) to determine whether it is a true positive. This is repeated
until 1000 “useful” traces are generated, i.e., traces that are accepted by
$\mathcal{H}$ (respectively $\mathcal{R}$) and therefore contribute to
computing the metric. The target number of useful traces is a compromise
between the scalability and the noise of the evaluation. It was chosen
empirically, and deemed appropriate because it gives a 99% chance that the
real precision/recall value is within ±4.08% of the measured value. Note that
the maximum trace length reached with this model assessment method is model-
dependent, since it depends on the proportion of traces in $\Sigma^{*}$
accepted by the models under analysis. In fact, it may differ even between
precision and recall on the same test subject, since the proportion may be
different between reference and inferred model.
Finally, we compared the resulting precision and recall values with the
single-length assessment results.
Data: Reference model $\mathcal{R}$, inferred model $\mathcal{H}$, alphabet
$\Sigma$, trace length $l$, target number of samples $n$, metric $m$ (either
precision or recall)
Result: Accuracy value
1 $\mathit{truePositives}\leftarrow 0$
2 $\mathit{acceptedTraces}\leftarrow 0$
3 while _$\mathit{acceptedTraces} <n$_ do
$t\leftarrow\text{randomTrace}(\Sigma,l)$
// $l$ symbols chosen from $\Sigma$ with uniform distribution
4 if _m = precision $\wedge$ $t\in\mathfrak{L}(\mathcal{H})$_ then
5 $\mathit{acceptedTraces}\leftarrow\mathit{acceptedTraces}+1$
6 else if _m = recall $\wedge$ $t\in\mathfrak{L}(\mathcal{R})$_ then
7 $\mathit{acceptedTraces}\leftarrow\mathit{acceptedTraces}+1$
8 end if
9 if _$t\in\mathfrak{L}(\mathcal{R})\wedge t\in\mathfrak{L}(\mathcal{H})$_
then
10 $\mathit{truePositives}\leftarrow\mathit{truePositives}+1$
11 end if
12
13 end while
return _$\frac{\mathit{truePositives}}{\mathit{acceptedTraces}}$_
Algorithm 4 Statistical assessment of the model accuracy for traces of length
$l$, using a model-independent sampling of $\Sigma^{l}$
##### Results
The differences between the model accuracies obtained using the single-length
assessment and the model-independent sampling of $\Sigma^{*}$ (computed using
all the trace lengths for which both results are available) are summarized in
Table 5.
Table 5. Mean and standard deviation of the absolute difference in precision and recall evaluated using the single-length assessment and the model-independent sampling of $\Sigma^{*}$ (values in percentage points). | $\textit{precision}_{=n}$ | $\textit{recall}_{=n}$
---|---|---
Mean | $0.44\text{\,}\mathrm{\textup{pp}}$ | $0.22\text{\,}\mathrm{\textup{pp}}$
$\sigma$ | $0.70\text{\,}\mathrm{\textup{pp}}$ | $0.55\text{\,}\mathrm{\textup{pp}}$
The small differences between the methods’ results are caused by the sampling
error intrinsic in any accuracy measure performed using randomly generated
samples, and are within the margin of error expected for the used number of
samples. This confirms that a statistical estimation in which all the traces
of the same length have the same probability of being sampled generates the
same results as our single-length assessment, in line with your discussion in
Section 4.
To have a closer look at the results for individual test subjects, Figure 10
shows the results of the same five representative test subjects previously
used in RQ1 and RQ2. The results obtained using the model-independent sampling
of $\Sigma^{*}$ are represented by red markers (note the different ranges of
trace lengths for which the results are available, due to the timeout), while
the single-length assessment is represented by the black line. We remark the
two methods exhibit a similar trend also on test subject 5, for which instead
the trace similarity method returned different results due to the non-uniform
probability distribution of traces having the same length, induced by the
random walk.
Figure 10. Partitioned comparison of sampling $\Sigma^{*}$
#### Answer to RQ3
The precision and recall values obtained using the single-length assessment we
proposed are consistent with those obtained using a model-independent sampling
of $\Sigma^{*}$. This implies that our method is preferable, because it is not
limited to short trace lengths, and because, being deterministic, is not
affected by statistical noise.
### 6.7. RQ4: Practical applicability on inferred models
This section discusses the practical applicability of our approach by
evaluating its scalability with respect to the complexity of the models under
analysis, measuring the execution time of each assessment on each test
subject.
##### Methodology
Model complexity is widely studied, and can be measured in a variety of ways,
with different complexity metrics appropriate in different scenarios. We will
evaluate the scalability of our method with respect to two different model
complexity metrics: _deterministic state complexity_ (Yu, 2001) and _star
height_ (Eggan, 1963). Specifically, given a regular language $L$, these
metrics are defined as follows. The _deterministic state complexity_ is
defined as the number of states of the minimal DFA accepting $L$. We will
refer to this measure simply as the “number of states”, since all the automata
used in this work are in their minimal form. The number of states is a
commonly used model complexity metric in the field of model inference
(Walkinshaw et al., 2016; Lo and Khoo, 2006b). The _star height_ of $L$ is the
minimum star height among all the regular expressions representing $L$, where
the star height of a regular expression is the maximum nesting depth of Kleene
star operators in the expression. Though the start height is less common, it
is a better predictor of the execution time of our assessment method since
this is dominated by the computation of the OGFs. Referring to the OGF
computation algorithm described in Section 5, it can be seen that an automaton
without closed cycles leads to intermediate OGFs that are easy to compute
since they are polynomials (instead of rational functions) generated using
only sums and products of polynomials. Conversely, all cycles present in the
automaton under analysis must eventually be eliminated during the OGF
computation by composing the intermediate OGFs using sums, products, and
quotients of rational functions. Intuitively, languages with higher star
heights are likely to have more closed cycles in the corresponding DFA
representation, increasing the complexity of the OGF computation, whereas a
star height of zero indicates a finite language (thus with a corresponding DFA
with no loops) of which the OGF is a polynomial of finite degree that is easy
to compute.
We should note that the star height is by no means a complete metric; the cost
of each state elimination operation in the OGF computation of Section 5 is
hard to predict a priori, because it depends on the complexity of the rational
functions involved. Nonetheless, developing a new ad-hoc structural complexity
metric for the OGF computation is outside the scope of this work.
Finding the minimum star height of a language is a notoriously hard problem
(Kirsten, 2005). Therefore, in our evaluation, we use an approximated value
obtained by transforming the inferred model in a regular expression using the
Brzozowski and McCluskey algorithm (Brzozowski and McCluskey, 1963) and then
counting the maximum nesting depth of Kleene star operators in this
expression.
Note that both reference and inferred models influence the execution time.
However, in our experiments the size of the inferred model is one or two
orders of magnitude larger than the size of the reference model: therefore we
consider the effect of the latter negligible.
Since the differences in execution times between different assessments
(single-length and cumulative-length) of the same subject are negligible, we
give one execution time _per test subject_ rather than _per assessment_. This
is because the execution time is dominated by the computation of the OGFs,
which is the same for all the assessments on the same test subject.
##### Results
Figure 11(a) shows the number of subject assessments completed over time. On
one hand, 21 test subjects assessments did not finish within the timeout, and
when the timeout occurred all of them were in the OGF computation phase. This
confirms that the execution time of our method is dominated by the computation
of the OGFs. Among the subject assessments that did not finish within the
timeout, the minimum state count was 68 and the maximum was 2061, while the
minimum approximated star height was 13 and the maximum was 304. On the other
hand, 41 of 82 subjects had an assessment execution time of less than one
minute, showing the practical suitability of the method. Additionally, a
benefit of our method is that the OGFs, once generated, can be reused to
perform further analyses with different parameters with negligible overhead.
Figure 11(b) shows the correlation between number of states of the inferred
model, and execution time. The largest model successfully assessed had 1465
states, and was assessed in $9935\text{\,}\mathrm{s}$. The Pearson correlation
coefficient between execution time and number of states was 0.07, highlighting
how inadequate is this complexity metric to capture the computational
complexity of the assessment.
Figure 11(c) shows the correlation between star height and execution time. The
most complex inferred model successfully assessed had an approximated star
height of 38 (i.e., the maximum nesting depth of the Kleene star operator in a
regular expression accepting the inferred language was 38), and was assessed
in $151\,000\text{\,}\mathrm{s}$ — the longest execution time encountered
among the test subjects that finished within the timeout. The Pearson
correlation coefficient between star height and execution time was 0.51,
implying that the start height can be a rough indicator of the execution time
of our method.
$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$0$$20$$40$$60$$80$Elapsed
time (s)Completed subject assessments (a) Number of subject assessments
completed over time.
$10^{1}$$10^{2}$$10^{3}$$10^{-2}$$10^{0}$$10^{2}$$10^{4}$Number of
statesExecution time (s) (b) Assessment time, depending on the number of
states.
$10^{0}$$10^{1}$$10^{-2}$$10^{0}$$10^{2}$$10^{4}$Star heightExecution time (s)
(c) Assessment time, depending on the star height of the inferred model.
Figure 11. Scalability of our method
Although the experiments were executed on a multicore system, our current
implementation is still sequential, thus the performance metrics in this
section are to be considered for a single core.
#### Answer to RQ4
Despite the inadequacy of widely adopted model complexity metrics, we report
that our approach was able to assess inferred models representing aspects of
actual software systems, using commodity hardware and despite being at a
prototype stage.
### 6.8. Threats to validity
There are a number of factors that could impact the validity of our
experimental results.
_Choice of metrics._ The choice of the accuracy metrics used in this
evaluation could threaten the validity of our results if the metrics fail to
capture adequately the model accuracy. To mitigate this threat we used
precision and recall (Tharwat, 2020), which are well known metrics and have
often been used in the area of model inference (Lo and Khoo, 2006a; Pradel et
al., 2010; Walkinshaw et al., 2016; Krka et al., 2014; Polyvyanyy et al.,
2020; Walkinshaw et al., 2008). Moreover, this risk is also partially
mitigated by the fact that the proposed assessment method can be adapted to
compute additional metrics (e.g., specificity or F-measure) without affecting
the scalability.
The choice of the metric used to capture the model complexity in the
evaluation of our method scalability could also pose a construct validity
threat. To minimize this risk, we provided results using two different
complexity metrics: model state count and language star height.
_Reference models used._ Using certain reference models could limit the
generalizability of our results. To mitigate this issue, we considered a
variety of reference models (41 in total), obtained from two different
publicly available sources we found in the model inference literature (Pradel
et al., 2010; Krka et al., 2014).
_Training set generation and model inference setup._ Due to the lack of a
suitable and widely accepted model inference benchmark suite containing traces
of real software executions, we used random walks on the reference models to
generate the traces fed to a model inference method. This could introduce a
bias in the training set, which may become a bias in the corresponding
inferred models. Similarly, also the choice of the inference method (and its
parameter values) could introduce a bias in the inferred model. Nevertheless,
the model assessment method proposed in this paper and considered in our
evaluation is independent of the method used for generating inferred models.
Furthermore, we used the same evaluation subjects (i.e., pairs of reference
and inferred models) for the model assessment methods used in our comparative
evaluation, yielding fair comparison results. Moreover, testing inference
methods using traces generated from random walks is common practice (Busany et
al., 2019; Lo et al., 2012; Walkinshaw and Bogdanov, 2013; Walkinshaw et al.,
2013).
_Input parameters of other assessment methods._ The comparison discussed in
RQ2 is affected by the parameters of the considered assessment methods, e.g.,
the characteristics of the random walk used for trace similarity. To obtain
results that are representative of how the assessment methods are is used in
practice, our choice of parameter values was guided by what is frequently used
in the literature. Nonetheless, different method parameters would in general
lead to different results.
_Statistical noise_. Although single-length, cumulative-length, and MBT-based
assessment methods are deterministic, trace similarity involves randomness,
which could affect the evaluation results. However, due to the low
computational cost for generating traces through random walks and checking
them against a model, we were able to perform the trace similarity assessment
using at least $100\,000$ samples, ensuring low statistical noise.
_Errors in the implementation._ We used a prototype implementation of our
method, which may contain faults. Moreover, in order to develop a self-
contained solution that does not rely on external algebra systems such as
Wolfram Mathematica (Inc., 2022) or Maple (Waterloo Maple, Inc.., 2022), the
algebraic system used to perform operations on rational functions was
implemented by us from scratch: it may also contain faults. To mitigate the
issue, we performed systematic sanity checks by comparing the language
cardinalities obtained using different counting methods.
## 7\. Related Work
Our method is related to the work done in the area of _model assessment_ and
_trace (model) counting_ 999This area of research is generally called _model
counting_. In our context, this name could be confusing, since we have already
used the term _model_ to indicate a representation of an aspect of a system.
To avoid any confusion, hereafter we refer to it as _trace (model) counting_..
### 7.1. Model Assessment
As discussed in Section 3.1, Lo and Khoo (2006a) proposed a method called
_trace similarity_ for empirically assessing the accuracy of inferred models
against reference models, both in the form of probabilistic or non-
probabilistic finite-state automata. The method measures the accuracy in terms
of _precision_ and _recall_ using traces randomly generated from the models.
Specifically, for a reference model $R$ and an inferred model $H$, they
defined _recall_ as the percentage of traces generated by $R$ that are
accepted by $H$ and _precision_ as the percentage of traces generated by $H$
that are accepted by $R$. They also proposed an algorithm called TraceGen that
generates a set of random traces from a model with the aim of covering every
transition in the model at least $n$ times, where $n$ is a coverage parameter.
Lo et al. (2012) later extended the trace similarity method to additionally
assess the accuracy of inferred models in terms of _specificity_ , which is a
metric indicating the ability to correctly reject illegal behaviors. One
common issue of these methods, as discussed in Section 3.1, is that the
accuracy of inferred models depends on the traces drawn according to a certain
probability distribution imposed by the random trace generation. Our method
resolves this issue by computing precision and recall values that consider all
the traces, up to a user-defined arbitrarily large maximum length.
As discussed in Section 3.2, the aforementioned issue was also acknowledged by
Walkinshaw et al. (2008), which addressed it by adapting the W-Method (Chow,
1978) (originally developed for model-based software testing to generate a set
of traces covering all distinguishable runs of the model under test) to
generate a “representative” trace set that covers all the model behaviors
without privileging any specific one. Although this solution guarantees to
cover all behaviors, it could still return misleading results (e.g., an
inferred model having fewer counterexamples could obtain a lower accuracy
value) since it does not consider the number of accepted traces affected by
each behavior, which is something that our method does — up to a finite
maximum trace length set by the user. Furthermore, as shown in our evaluation
results, the MBT-based method is not scalable due to the exponential growth of
the cardinality of the representative set with the number of different states
between reference and inferred models.
Recently, Polyvyanyy et al. (2020) have proposed a framework that compares
reference and inferred models in terms of their languages (i.e., the languages
accepted by the models). While they defined precision and recall similarly to
ours (i.e., Equation 6), they overlooked the usage of language cardinality,
arguing that it is useful only for finite languages. Instead, they suggested
framework instantiations using _topological entropy_ as a language measure
which, intuitively, measures the increase in the variability of the words of
the language as their length goes to infinity. It is not clear whether, using
this measure, an inferred model with fewer counterexamples would always obtain
a higher accuracy value, even when the counterexample languages of the two
inferred models are not in a subset-of relationship. In our method, all the
traces up to the maximum trace length specified by the user are considered,
and they all have the same weight on the result. Moreover, the maximum trace
length parameter can be set either to a value that is relevant for the
application under analysis, or to a very large value, resulting in effectively
computing the asymptotic accuracy.
Other than the _language_ perspective, where traces (and thus languages)
generated by models matter in assessing the accuracy of inferred models, model
assessment can also take a _structural_ perspective, where the model
structure, i.e., how states and transitions are arranged, is considered.
Walkinshaw and Bogdanov (2013) proposed an algorithm called LTSDiff that
compares the structure of two models, identifying missing or superfluous
states and transitions, and computing precision and recall based on these.
Pradel et al. (2010) noticed that structural differences can have a variable
impact on the size of the language difference. To address the issue, they
proposed an approach that counts the number of transitions that are in common
between reference and inferred models after abstracting the models using a
variant of the k-tail algorithm. This allows one to identify states that are
_similar_ , even when they are not exactly equivalent, with the parameter $k$
controlling how much imprecision and incompleteness the metrics accept. We
remark that the language perspective and the structural one are complementary:
a structural approach is unable to consider the full extent of the impact
(i.e., number of traces affected) of each missing or erroneous state or
transition, while a language approach cannot provide insights about the
structural similarity of the models.
### 7.2. Trace (Model) Counting
A number of approaches perform trace (model) counting using the _transfer
matrix method_ (Flajolet and Sedgewick, 2009; Stanley, 2011), which relies on
linear algebraic operations on the matrix representing the transition relation
of the automaton. For example, Aydin et al. (2015) used it to perform
quantitative information flow and probabilistic analysis of software, and
Aydin et al. (2018) later extended the work including also parametric
constraints, still using the same trace (model) counting approach. While the
transfer matrix method generates the same OGF as our method, as discussed in
Section 5, our method is more scalable, enabling the assessment of larger
models.
Luu et al. (2014) proposed an approach to count the number of strings
satisfying a given set of constraints, using _analytic combinatorics_.
Constraints are individually translated to OGF, and later composed. However,
both constraint translation and composition are not precise, returning an
upper and lower bound instead.
Trinh et al. (2017) developed a trace (model) counting approach for a class of
string constraints based on S3P (an SMT solver for strings constraints) (Trinh
et al., 2016). While S3P works by building a reduction tree, reducing the
original formula into simpler formulas until a satisfying assignment or a
contradiction is found, trace (model) counting is performed by exhaustively
building the entire reduction tree, in which each node is associated with the
OGF representing the count for that node of the tree, and the counts are
propagated bottom-up, from the leaf nodes.
## 8\. Conclusions
In the evaluation of newly developed model inference systems, rigorously
assessing the accuracy of the generated inferred models against ground truth
reference models is crucial. In this paper we have highlighted how commonly
used assessment methods provide results that may not reflect the actual
accuracy of the inferred model under analysis: statistical methods are often
affected by a systematic bias caused by overlooking the impact of the random
trace generation, while deterministic methods that guarantee to discover any
error in the model may not correctly capture the number of traces it affects.
To tackle these shortcomings, we have proposed a method to rigorously measure
the accuracy of an inferred model against a ground truth reference model.
First, our method is comprehensive: it considers all the possible traces up to
an arbitrary finite maximum trace length defined by the user. This is in
contrast with other language-based methods that perform the assessment based
on a much smaller subset of traces.
Second, it is deterministic: our method does not rely on random sampling, and
gives exact results. While current statistical methods may provide a more
flexible time/accuracy trade-off, our approach avoids the convergence
limitations of statistical methods,
Third, it is unbiased: it considers all the traces up to the maximum length,
and each trace has the same weight on the results. This is an advantage over
assessment methods using a random trace generation process to generate a
finite set of traces on which the assessment result is based: the random
sampling induces a probability distribution over the accepted language
introducing a bias in the result, which is not desirable unless it reflects
the domain knowledge about the application being analyzed.
Fourth, it is model-independent: assessing against the same reference model
different models accepting the same language will generate the same accuracy
measurement. This is in contrast with trace similarity, where the topology of
the model affects the sampling, and thus the result.
We have also highlighted how characterizing model accuracy using only a pair
of precision and recall values may be inadequate because an inferred model can
have a variable level of accuracy depending on the _length of the traces_ used
in the assessment. As a result, we have proposed an additional assessment
method, measuring the accuracy separately for each trace length, over a given
range.
We have evaluated our assessment methods experimentally and compared ours with
the currently popular assessment methods, on reference models previously used
in the model inference literature and inferred models generated using well
known model inference methods. The results highlighted the shortcomings of
current assessment methods that are addressed by our solution. The results
also show that our approach is scalable enough to be used in practice.
As part of future work, we plan to further evaluate our approach on test
subjects from industry, investigate how the accuracy provided by modern model
inference tools changes with the length of the traces considered, and further
discuss the asymptotic analysis of precision and recall and its usefulness in
assessing the model accuracy.
###### Acknowledgements.
The support of the EPSRC Centre for Doctoral Training in High Performance
Embedded and Distributed Systems (HiPEDS, Grant Reference EP/L016796/1) is
gratefully acknowledged.
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## Appendix A Appendix: test subjects categorization and selection
In this section we discuss how we categorized the results for the test
subjects in the experimental evaluation, depending on the characteristics of
the trends of the accuracy metric (precision and recall) values over the range
of trace length, and how we used this categorization to choose the five test
subjects that are given as examples in Section 6.
To differentiate the trends, we looked at the value of each accuracy metric
for the shortest and the longest available trace length, with the goal of
obtaining a rough indication whether the metric is increasing, decreasing, or
constant.
The values were aggregated in three categories: greater than 0.999 (below
represented with $\mathit{1}$), smaller than 0.001 (below represented with
$\mathit{0}$), and any other value (below represented with $k$). This is
motivated by the observation that the precision and recall values at the
extremes of the trace length range are generally close to either zero or one,
and when this is not the case it is worth considering the subject separately
because it indicates that in Equation 6 numerator and denominator have the
same order of magnitude, which is a relevant characteristic of the inferred
language.
Based on this categorization, we counted the number of test subjects in each
category, obtaining Table 6 (e.g. $\mathit{1}\rightarrow k$ indicates that the
value of the accuracy metric value for short traces is greater than 0.999 and
for long traces is between 0.001 and 0.999).
Table 6. Number of test subjects for each metric trend | | Recall
---|---|---
| | $\mathit{0}\rightarrow\mathit{0}$ | $\mathit{0}\rightarrow k$ | $\mathit{0}\rightarrow\mathit{1}$ | $k\rightarrow\mathit{0}$ | $k\rightarrow k$ | $k\rightarrow\mathit{1}$ | $\mathit{1}\rightarrow\mathit{0}$ | $\mathit{1}\rightarrow k$ | $\mathit{1}\rightarrow\mathit{1}$
Precision | $\mathit{0}\rightarrow\mathit{0}$ | 0 | 0 | 0 | 1 | 0 | 0 | 20 | 2 | 0
$\mathit{0}\rightarrow k$ | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0
$\mathit{0}\rightarrow\mathit{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0
$k\rightarrow\mathit{0}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$k\rightarrow k$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$k\rightarrow\mathit{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\mathit{1}\rightarrow\mathit{0}$ | 0 | 0 | 0 | 0 | 0 | 0 | 14 | 2 | 2
$\mathit{1}\rightarrow k$ | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 1 | 0
$\mathit{1}\rightarrow\mathit{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 1 | 0
Looking at table 6, we observed that all the most common accuracy trends have
a recall value starting at 1 and ending at 0. Within these we selected four
subjects, each one having one of the most common precision value trend.
Specifically:
* •
Subject 1 was randomly selected among the 20 subjects having precision
starting at 0 and ending at 0. It is worth noting that in all 20 subjects we
observed that the precision increases to a maximum value, and then decreases.
* •
Subject 2 was randomly selected among the 14 subjects having both precision
and recall starting at 1 and ending at 0.
* •
Subject 3 was randomly selected among the five subjects having precision
starting at 1 and ending at 1. It is worth noting that in all 5 subjects we
observed that the precision is actually constant throughout the range.
* •
Subject 4 was randomly selected among the eight subjects having precision
starting at 1 and ending at a value between 0 and 1.
These trend categories cover 47 of the 61 available test subjects.
In addition, we selected one more subject (Subject 5) showing a recall value
ending at a value greater than zero. This is motivated by the fact that
although this trend is a rare occurrence, it highlights a relevant
characteristic of the inferred model.
## Appendix B Appendix: description of the selected test subjects
In test subject 1 the inferred model precision starts at zero, reaches a
maximum for length 10 (47.5%), and then converges to zero. A possible
motivation for this behavior is suggested by the large size of the inferred
model: it is possible that the inference approach could not find any good
generalization of the training data, and as a result the accuracy of the model
follows the distribution of lengths of the traces contained in the training
set. In fact, we verified that the most frequent trace length in the training
data is indeed 10.
In test subject 2 the precision and recall values converge to zero, indicating
that the growth of the size of the language of true positives is slower than
the growth of the size of the language of both inferred positives and
reference positives.
For test subject 3 the constant precision equal to 1 is explained by the fact
that the language of false positives is empty (which we also manually
verified), indicating that the inferred language is a (non-empty) subset of
the reference language.
In test subject 4 the precision converges to a value greater than zero,
indicating that as the trace length is increased, the cardinalities of
numerator and denominator of Equation 6 have the same order of growth. The
same is true for the recall in test subject 5.
Notably, test subject 5 has also an empty language of false positives, as
suggested by the constant precision equal to 1. Consequently, the inferred
language is a subset of the reference language, and thanks to the asymptotic
value of the recall we can determine that it covers 30% of the reference
language — an exceptionally accurate inference result.
|
# CMS RPC data taking during the LHC Run-2 and activities during Long Shutdown
2
Kevin Mota Amarilo on behalf of CMS Collaboration Rio de Janeiro State
University, Institute of Physics, R. São Francisco Xavier, 524, Rio de Janeiro
- RJ, 20550-013, Brasil<EMAIL_ADDRESS>
###### Abstract
The CMS experiment collected around 150 fb-1 of proton-proton collision data
at $\sqrt{s}$ = 13 TeV during the Run-2 data taking period of LHC. The CMS RPC
system provided redundant information for robust muon triggering,
reconstruction and identification. To ensure stable data taking, the CMS RPC
collaboration has performed detector operation, calibration and performance
studies. After the end of Run-2, it was started the second LHC long shutdown
period (LS2), an important opportunity for maintenance and preparation for the
next data taking period (Run-3) and the installation of services in
preparation for the Phase-II upgrade. The activities included maintenance of
power, gas and online systems. In this presentation, the overall performance
of the CMS RPC system during the Run-2 period is summarized as well as all the
activities done in preparation for future data taking periods.
## 1 Introduction
One of the main actors of the CMS (Compact Muon Solenoid) experiment [1] is
its Muon System [2], which uses gaseous detector technologies to accomplish
muon triggering, identification, transverse momentum and charge measurement.
Through the end of Run 2 in 2018, the components of the CMS Muon system were:
drift tubes (DT) at barrel region ($|\eta|<1.2$), cathode strip chambers (CSC)
at endcap region ($1.2<|\eta|<2.4$) and Resistive Plate Chambers (RPC), in
barrel and endcap regions ($|\eta|<1.9$). During LS2 a new type of detector
was installed, the gas electron multiplier (GEM), as part of the Muon System
Phase-2 Upgrades to complement measurements in the higher $\eta$ region for
Phase-II upgrade [3], which will also include new RPC chambers, the so called
improved-RPC (iRPC).
The CMS RPC system is composed of double gap phenolic resin (called bakelite)
chambers, operating in avalanche mode with gas mixture 95.2%
$C_{2}H_{2}F_{4}$, to enhance the ionization caused by incident particles,
4.5% $iC_{4}H_{10}$ as a quencher gas to reduce streamer formation and 0.3%
$SF_{6}$ to control secondary ionization. The RPCs are designed and calibrated
to have a time resolution of around 2 ns and a number of adjacent strips fired
in a single muon hit (Cluster size) between 2 and 3.
## 2 CMS RPC Run-2 data taking
Between 2015 to 2018 during Run-2, CMS recorded proton-proton collisions with
$\sqrt{s}$ = 13 TeV from LHC, with total integrated luminosity of 150.26 fb-1,
out of which only 0.15 % was lost due to RPC problems. The total accumulated
charge in Run-1 and Run-2 of the RPC system was 2.3 mC/cm2 in barrel and 7.5
mC/cm2 in endcap. Studies at the CERN Gamma Irradiation Facility ++ (GIF++)
have showed no drop in efficiency of RPCs up to an integrated charge of 153
mC/cm2 [8]
### 2.1 RPC efficiency and cluster size stability
Hit efficiency and cluster size (CLS) are important parameters for RPC
performance. An RPC hit’s geometrical coordinates are calculated as the
geometrical center of the cluster formed by the fired adjacent strips. The CLS
of RPC hits should be kept less than 4 strips to avoid fake muon triggers,
therefore, proper calibration is very important to keep good performance.
The calibration is done by analyzing the efficiency and CLS dependence on the
effective high voltage (HV), which is the high voltage applied corrected by
the environmental pressure and temperature variations. HV working point scans
were performed once or twice a year in dedicated collision runs, effective
voltage is applied in values between 8600 and 9800 V. The collected data is
analyzed and the proper working points are selected. More information on the
calibration can be found in [4].
Figure 1: RPC average efficiency vs integrated luminosity during Run-2 for
barrel stations. Red vertical lines show the planned technical stops (TS) and
the grey ones — Year-End-Technical stops (YETS). Figure 2: RPC average cluster
size vs integrated luminosity during Run-2 for barrel stations. Red vertical
lines show the planned technical stops (TS) and the grey ones — Year-End-
Technical stops (YETS).
The Figures 1 and 2 shows the Run-2 efficiency and CLS history. Each point
corresponds to the average efficiency or CLS per barrel station. Each change
in the trends is related to the deployment of a new working point. With
exception to a drop in efficiency and CLS between in August 2018 caused by a
configuration setting issue, it is possible to see that the RPC system had a
stable performance with efficiency greater than 95 % and CLS between 2 and 3,
within the CMS Muon Trigger requirements.
### 2.2 RPC contribution to the CMS Muon Trigger
The main contribution of the RPC system is to the CMS Level-1 (L1) Muon
Trigger [5], which is divided into 3 $\eta$-regions: the barrel muon track
finder (BMTF) for $|\eta|<0.83$, the overlap muon track finder (OMTF) for
$0.83<|\eta|<1.24$ and the endcap muon track finder (EMTF) for $|\eta|>1.24$.
Each one of the track finders can access information of the detectors
concurrently, building tracks and assigning pT and exploiting the redundancies
of the muon system.
In the BMTF, RPC timing information is used to improve DT trigger primitives
and bunch crossing assignment. In the OMTF, RPC-only segments are build from
the 8 chambers available, resulting in a substantial gain in in efficiency.
Figure 3 demonstrates that using RPC information in the OMTF increases
efficiency by 15 %. Finally, in the EMTF, RPC hit position is used in case of
CSC trigger primitive absence. More information on this subject can be found
in [6].
Figure 3: Trigger efficiency versus pT for the OMTF derived from algorithm
emulation applied on real data. Using (Red) and not using (Blue) RPC
information. Ref. [6].
### 2.3 RPC ohmic current monitoring
The ohmic current is defined as the current with no beam, up to around 7000 V,
where the gas amplification contribution is negligible and the currents follow
the Ohm’s law.
Figure 4 shows the history of ohmic current measured in four RPC stations. An
increase in the currents was observed in all stations, correlated to the
background, the RE+4 and RE-4 background rate is about 40 Hz/cm2 and W+0 and
RE-1 is less than 10 Hz/cm2. In November of 2018, at the start of the Heavy
Ion period where the luminosity is very low and the background is effectively
zero, the currents started to decrease. In RE-4 the decrease was faster with
the increase of the gas flux.
The increase was correlated with the production of flourine ions (F-) in the
gas gap due to the electrical discharge inside the gap. The flourine can
combine with the water in the gas and form HF, that can damage the gap. This
explains the velocity of decrease of currents correlation with gas flux, as
the gas can flush out the F-ions before the combination with water. More
information on the fluorine formation inside RPC gas gap can be found in [7].
Figure 4: RPC ohmic current monitoring history for the stations W+0, RE+1,
RE+4 and RE-4.
## 3 LS2 Activities
During LS2, CMS is undergoing an intensive upgrade and maintenance program
during the second two-year-long shutdown period (LS2). To ensure an excellent
performance of the detector in the subsequent physics program, the RPC group
pursued in the present shutdown a thorough detector consolidation program to
repair most of the hardware problems. In preparation for future installation
of new iRPC chambers, cooling and cable services for the new detectors were
installed. This includes thousands of kilometres of high voltage and low
voltage cables, stainless steel gas pipes between predistribution gas racks in
the Service cavern (USC) and gas distribution racks in the experimental cavern
(UXC), copper pipes between distribution racks and chambers, gas impedance
boxes, support equipment, and optical fibers for the iRPCs.
In order to keep the optimal performance of the system, an extensive HV and
low voltage (LV) maintenance campaign was performed. The goal of HV
maintenance was to identify the problematic parts of the HV power system and
to fix it in the best possible way, recuperating the performance of the
chambers. A total of 65 HV channels were repaired. The LV maintenance aim was
to ensure a proper operation and configuration of the detector electronics and
ensure a good functionality of the LV power boards and communication buses. A
total of 12 LV problems were fixed.
A very important activity was the extraction of the chambers from the two RE4
stations to allow the CSC ME4/1 chamber extraction for electronics
refurbishment. The chambers were brought to the surface, and accommodated to a
new laboratory with controlled environmental conditions and new gas lines to
provide the standard RPC gas mixture. All needed reparation and re-validation
was done in the laboratory before the re-installation.
The activity with most priority during LS2, was gas system consolidation. The
aim was to minimize the environmental impact of the RPC system, as the
standard gas mixture is mainly composed of fluorine composed gases (F-gases)
with high global warming potential (GWP). The actions taking place during LS2
are:
* •
Gas Leak identification and repair, where 49 out of 99 leaky chambers in
barrel where repaired.
* •
Recuperation of the Exhaust, which was not working during Run-2 for the
installation of the first $C_{2}H_{2}F_{4}$ recuperation system with
efficiency of 80 %, which have been developed by CERN EP-DT Gas team.
* •
By end of 2021, CERN EP-DT Group is going to install automatic pressure
regulation valves on the redistribution gas racks in USC to minimize pressure
variations in the chambers, which can be a possible source of new leaks.
* •
Turn off the remaining leaky chambers which was not possible to repair (about
3.5 % of RPC system), to keep the amount of fresh gas added to the system at a
minimum.
## 4 RPC commissioning with cosmic rays data during LS2
In Figure 5, the efficiency of the barrel chambers is compared using cosmic
rays runs of 2018 (End of Run-2) and in 2021 (after all the repairs of LS2).
The average efficiency of the chambers with efficiency greater than 70 % is
compared and is about the same (gain of 0.6 %). The fraction of chambers with
efficiency greater than 70 % increased in around 6 % as a result of all the
repairs, which is very important for the muon system redundancy.
Figure 5: RPC barrel efficiency comparison between cosmic runs of 2018 (Red)
and 2021 (Blue).
## 5 Conclusion
The CMS RPC system showed a stable performance after operation in Run-1 and
Run-2 ($\sim$185 fb-1) with average efficiency greater than 95 % and average
CLS between 2 and 3 strips/hit. We observed reversible ohmic current increase
in high background regions correlated to the F- formation inside the gaps.
During LS2 RPC did a massive maintenance campaign to repair gas leaks, HV and
LV in order to keep the optimal performance and is planning to keep the
greenhouse gases emissions at a minimum by keeping all the remaining leaky
chambers off and using the new $C_{2}H_{2}F_{4}$ recuperation system. The
performance of the RPC System is improved with respect to the end of Run-2 and
the system is ready and commissioned for Run-3 data taking.
## References
## References
* [1] CMS Collaboration 2008 The CMS experiment at the CERN LHC JINST 3 S08004 doi:10.1088/1748- 0221/3/08/S08004
* [2] CMS Collaboration 1997 The CMS muon project: Technical Design Report (Geneva: CERN) URL `https://cds.cern.ch/record/343814`
* [3] CMS Collaboration 2017 The Phase-2 Upgrade of the CMS Muon Detectors (Geneva) URL `https://cds.cern.ch/record/2283189`
* [4] Abbrescia M et al. 2005 Cosmic ray tests of double-gap resistive plate chambers for the CMS experiment, Nucl. Instrum. Meth. A 550 116 URL `https://www.sciencedirect.com/science/article/pii/S0168900205011915`
* [5] CMS Collaboration 2013 CMS Technical Design Report for the Level-1 Trigger Upgrade (Geneva) URL `https://cds.cern.ch/record/1556311`
* [6] A. Samalan et al 2020 RPC system in the CMS Level-1 Muon Trigger JINST 15 C10007
* [7] A. Gelmi, R. Guida and B. Mandelli on behalf of CMS Muon 2021 Gas mixture quality studies for the CMS RPC detectors during LHC Run 2 JINST 16 C04004
* [8] J. Eysermans and M. I. Pedraza Morales 2017 Aging studies for the CMS RPC system. Proceedings of Science. URL `https://cds.cern.ch/record/2286783/files/EPS-HEP2017_785.pdf`
|
ifaamas [AAMAS ’23]Proc. of the 22nd International Conference on Autonomous
Agents and Multiagent Systems (AAMAS 2023)May 29 – June 2, 2023 London, United
KingdomA. Ricci, W. Yeoh, N. Agmon, B. An (eds.) 2023 2023 ???
# Artificial prediction markets present a novel opportunity for human-AI
collaboration
Anonymous submission
###### Abstract
Despite high-profile successes in the field of Artificial Intelligence,
machine-driven technologies still suffer important limitations, particularly
for complex tasks where creativity, planning, common sense, intuition, or
learning from limited data is required. These limitations motivate effective
methods for human-machine collaboration. Our work makes two primary
contributions. We thoroughly experiment with an artificial prediction market
model to understand the effects of market parameters on model performance for
benchmark classification tasks. We then demonstrate, through simulation, the
impact of exogenous agents in the market, where these exogenous agents
represent primitive human behaviors. This work lays the foundation for a novel
set of hybrid human-AI machine learning algorithms.
###### Key Words.:
prediction markets; machine learning; artificial intelligence; human-AI
collaboration
## 1 Introduction
A body of work on artificial prediction markets is emerging. These are
numerically simulated markets, populated by artificial agents (bot-traders)
for the purpose of supervised learning of probability estimators
barbu2012introduction. While nascent, this literature has demonstrated the
plausibility of using a trained market as a supervised learning algorithm,
achieving comparable performance to standard approaches on simple
classification tasks barbu2012introduction; barbu2013artificial;
jahedpari2014artificial; nakshatri2021design. In fact, these results are
sensible given the deep mathematical connections between prediction markets
and learning chen2008complexity; chen2010new; abernethy2011optimization.
Like other machine learning algorithms, functioning of an artificial
prediction market depends on several researcher-determined parameters: number
of agents; liquidity; initial cash; alongside parameters related to training
processes. Scenarios in which performance is robust or brittle to these
settings is yet unclear. Prior work has observed that artificial markets may
suffer from lack of participation rajtmajer2022synthetic. That is, like their
human counterparts in traditional prediction markets, agents may not invest in
the market if they do not have sufficient information Arrow:2008;
tetlock2008liquidity; rothschild2014extent; in practice, this occurs when an
asset representing test data point is too dissimilar to training examples.
In our view, the most promising opportunity afforded by artificial prediction
markets is eventual human-AI collaboration – a market framework should
theoretically support human traders participating alongside agents to evaluate
outcomes. Whether and how artificial prediction markets might benefit from
this hybrid scenario is an open question. The work we undertake here provides,
through simulation, initial support for this opportunity in the context of a
simple artificial market and primitive human behaviors.
Our work is framed by two primary research questions.
RQ1: How does performance of a simple artificial prediction market depend on
hyper-parameter selection?
RQ2: What impact does the inclusion of exogenous agents representing simple
(human-like) behaviors have on market performance?
Our findings support those of prior recent work indicating the promise of
artificial prediction markets for classification tasks. We demonstrate the
sensitivity of this approach to hyper-parameter selection and highlight, in
particular, the role of liquidity in moderating performance. Finally, we
demonstrate the exciting opportunity for _hybrid prediction markets_ to serve
as a framework for human-AI collaboration. We suggest that this approach may
be particularly valuable in contexts where machine learning falls short (e.g.,
lack of training data, complex tasks) and potential for human-only approaches
is either undesirable or infeasible.
## 2 Related Work
Our work builds upon and contributes to two primary literatures, namely, work
on artificial prediction markets and work in collaborative human-AI
technologies.
### 2.1 Artificial Prediction Markets
Prediction markets are simple futures markets used to aggregate disperse
information into efficient forecasts of uncertain future events
wolfers2004prediction; hanson2006information; manski2006interpreting;
wolfers2006interpreting. Specifically, market participants buy and sell
contracts that pay out based on the outcomes of future events. Market prices
generated from these contracts can be understood as a collective prediction
among market participants. Prediction markets have been successfully used,
e.g., for forecasting election outcomes berg2008prediction, sports betting
spann2009sports, forecasting infectious disease activity polgreen2007use, and
aggregating employee wisdom in corporate settings cowgill2009using;
gillen2012information.
Artificial prediction markets are a variation on this idea, wherein
numerically simulated markets populated by trained agents (bot-traders) are
used for the purpose of supervised learning of probability estimators
barbu2012introduction; barbu2013artificial. In initial formulations by Barbu
and Lay lay2010supervised; barbu2012introduction; lay2012artificial, each
agent is represented as a budget and a simple betting function. During
training, each agent’s budget is updated based on the accuracies of its
predictions over a training dataset. Authors found that these markets
outperformed standard approaches on benchmark classification and regression
tasks. Later, Storkey and colleagues storkey2011machine; storkey2012isoelastic
developed the so-called machine learning market, also for the purpose of
classification. In their formulation, each agent purchases contracts in order
to maximize a utility function. Most recently, Nakshatri et al.
nakshatri2021design proposed an artificial prediction market wherein agent
purchase logic is defined geometrically, in particular, by a convex semi-
algebraic set in feature space. Time varying asset prices affect the structure
of the semi-algebraic sets leading to time-varying agent purchase rules. Agent
parameters are trained using an evolutionary algorithm. Authors show that
their approach has desirable properties, e.g., the market satisfies certain
universal approximation properties, and there exist sufficient conditions for
convergence. Our work builds on this approach.
Like their human-populated counterparts, artificial prediction markets have
found a number of real-world applications barbu2013artificial;
jahedpari2014artificial. Ongoing theoretical work has offered support for
these promising experimental findings, highlighting the mathematical
connections between artificial markets and machine learning
chen2008complexity; ChenPennock:2010; abernethy2011optimization; hu2014multi.
### 2.2 Human-AI Collaboration
Despite high-profile successes in the field of Artificial intelligence (AI)
he2015delving; brown2019superhuman; kleinberg2018human; zhu2018human, machine-
driven solutions still suffer important limitations particularly for complex
tasks where creativity, common sense, intuition or learning from limited data
is required jarrahi2018artificial; lai2019human; green2019principles;
li2016crowdsourced; kamar2016directions; amershi2019guidelines;
muller2017organic. Both the promises and challenges of AI have motivated work
exploring frameworks for human-machine collaboration dellermann2021future;
wang2019human; nunes2015survey; puig2020watch; wu2022survey. The hope is that
we can eventually develop hybrid systems that bring together human intuition
and machine rationality to effectively and efficiently tackle today’s grand
challenges.
Recent work in hybrid intelligence systems has demonstrated the feasibility
and highlighted the potential of integrating human input into AI systems
kamar2016directions, or even, of human-AI collaboration wang2020human. The
spectrum of these efforts range from accounting for human factors in
technology design bansal2019beyond; canonico2019wisdom; harper2019role to
efficiently utilizing human inputs for training data amershi2014power in
applications as diverse as business nagar2011making; sowa2021cobots, civic
welfare fogliato2022case, criminal justice travaini2022machine, and healthcare
tschandl2020human; lee2021human; rajpurkar2022ai.
The work we describe here brings together the bodies of prior work on
artificial prediction markets and hybrid intelligence, proposing hybrid
prediction markets for direct integration of human wisdom into the deployment
of a machine learning algorithm.
## 3 Data
We consider three classification tasks. The first two are benchmark tasks used
broadly to compare performance of machine learning algorithms. The third is
the task of classifying scientific research outcomes as replicable or not
replicable – a challenging, complex task on which both machine learning
algorithms altmejd2019predicting; yang2020estimating; pawel2020probabilistic;
wu2021predicting and human assessment dreber2015using; camerer2016evaluating;
camerer2018evaluating; forsell2019predicting; gordon2020replication;
gordon2021predicting have achieved respectable but not excellent performance.
The replication prediction task, we suggest, is an example of the type of
problem well-suited to hybrid human-AI approaches and, specifically, the
hybrid prediction markets we propose.
### 3.1 Benchmark Machine Learning Datasets
The Iris dataset misc_iris_53 is one of the best known datasets in statistics
and machine learning, and one the earliest datasets used for evaluation of
classification methodologies. The dataset contains three classes of 50
instances each, where each class refers to a type of iris plant. One class is
linearly separable from the other; the latter are not linearly separable from
each other. Approaches based on support vector classification
mohan2020support, random forest classification mishina2015boosted;
chicho2021machine, and logisitc regression pinto2018iris have been reported to
work very well on this task, achieving $100\%$ or near-$100\%$ accuracy.
The Heart Disease dataset misc_heart_disease_45 is also a multivariate dataset
used for benchmark classification algorithms. Fourteen patient attributes are
used to predict presence or absence of heart disease. Random forest
singh2016heart, Xgboost rajadevi2021feature, and logistic regression
desai2019back achieve performance just under $90\%$ accuracy. While, support
vector classification achieves $86\%$ rajadevi2021feature.
### 3.2 Replication Studies Outcomes
In the last decade, several large-scale replication projects have been
undertaken across psychology, economics, political science, cancer biology and
other domains open2015estimating; camerer2016evaluating;
camerer2018evaluating; klein2014investigating; klein2018many;
cova2021estimating; errington2014open. Amongst their important impacts, these
studies have created small ground-truth datasets of replication studies
outcomes that can be used for train and test of automated approaches for
replication prediction. Specifically, we use the dataset and extracted
features considered by rajtmajer2022synthetic for ease of comparison. The
dataset containes 192 findings in the social and behavioral sciences, each
labeled either Replicable or Not Replicable, and a set of 41 features
extracted from each associated paper representing biblometric, venue-related,
author-related, statistical and semantic information. See wu2021predicting for
further detail on feature extraction processes.
Of note, authors in rajtmajer2022synthetic achieve $89.4\%$ accuracy,
remarkable for the task of replication prediction. However, accuracy is
calculated based on the approximately one-third of the test data that gets
evaluated by the market. Because agent participation is voluntary and agents
do not participate if they do not have sufficient information about a test
point, some (or much) of the data can be left unclassified. Our work uses the
same data and market structure described in rajtmajer2022synthetic. This
allows us to explore the effects of hyper-parameters (RQ1) and the inclusion
of exogenous agents (RQ2) on these performance/participation trade-offs.
## 4 Prediction Market Model
We use as a base model the artificial binary prediction market described in
nakshatri2021design. The state of the prediction market is defined by a pair
of integers $\mathbf{q}_{t}=(q^{0}_{t},q^{1}_{t})\in\mathbb{Z}_{+}^{2}$ giving
the number of units of the two asset classes that have been sold. For
simplicity we refer to the assets as $0$ and $1$. Traders are agents
$\mathcal{A}=\\{a_{1},\dots,a_{n}\\}$ who buy assets $0$ and $1$ using
policies $\\{\gamma_{1},\dots,\gamma_{n}\\}$. Also following
nakshatri2021design, we assume for simplicity that agents cannot sell. If
agent purchase policy $\gamma_{i}$ is conditioned on exogenous information
$\mathbf{x}\in D\subseteq\mathbb{R}^{n}$ then,
$\gamma_{i}:(\mathbf{q}_{t},\mathbf{x})\mapsto(r^{0},r^{1})$ and agent $i$
purchases $r^{0}$ units of $A_{0}$ and $r^{1}$ units of $A_{1}$, thus causing
a state update. In what follows, we assume that agents specialize in the
purchase of either Asset $0$ or Asset $1$ so that if $r^{0}>0$, then
$r^{1}=0$.
Asset prices are computed using a logarithmic market scoring rule (LMSR):
$\displaystyle p^{0}_{t}=\frac{\exp{(\beta q^{0}_{t})}}{\exp{(\beta
q^{0}_{t})}+\exp{(\beta q^{1}_{t})}}$ $\displaystyle
p^{1}_{t}=\frac{\exp{(\beta q^{1}_{t})}}{\exp{(\beta q^{0}_{t})}+\exp{(\beta
q^{1}_{t})}}.$
This is the softmax function of $(q^{0}_{t},q^{1}_{t})$. Liquidity $\beta$
adjusts the price change given a change in asset quantities
lekwijit2018optimizing. The fact that prices vary as a function of
$\mathbf{q}_{t}$ ensures that the policy need not take spot price into
consideration explicitly. It is often more convenient to work in units of
$1/\beta$ as $\beta$ can become arbitrarily close to zero. Experimental
results are therefore reported for this quantity as the liquidity factor.
To start the market, all agents may purchase assets at time $t=0$. After this,
we assume that agents arrive at the market with arrival rate $\lambda$ and
inter-arrival time governed by an exponential distribution. This allows us to
avoid scenarios in the hybrid setting where the synthetic traders swamp the
market.
The LMSR imposes a market maker price, so that actual trade costs are given
by:
$\displaystyle\kappa_{t}^{0}(\Delta
q^{0})=\frac{1}{\beta}\log\left\\{\frac{\exp[\beta(q^{0}_{t}+\Delta
q^{0})]+\exp[\beta q^{1}_{t}]}{\exp[\beta q^{0}_{t}]+\exp[\beta
q^{1}_{t}]}\right\\}$ $\displaystyle\kappa_{t}^{1}(\Delta
q^{1})=\frac{1}{\beta}\log\left\\{\frac{\exp[\beta
q^{0}_{t}]+\exp[\beta(q^{1}_{t}+\Delta q^{1})]}{\exp[\beta
q^{1}_{t}]+\exp[\beta q^{0}_{t}]}\right\\}.$
Here $\kappa^{i}_{t}(\Delta q^{i})$ is the cost to a trader for purchasing
$\Delta q^{i}$ units of Asset $i$ (with $i\in\\{0,1\\})$ at time $t$. For
small values of $\beta$ (large values of $1/\beta$) the cost of purchase
approaches the spot-price nakshatri2021design.
Agent purchase logic is governed by a time-varying bank value $B_{i}$ and a
characteristic function
$\psi_{i}:\mathbb{R}^{n}\times\mathbb{R}^{2}\times\mathbb{R}^{m}\to\mathbb{R}$
to reason about information $\mathbf{x}$ and its decision to buy an asset in
class $y_{i}$ is governed by:
$\Delta
q^{y_{i}}_{i}=H\left\\{\sigma[\psi_{i}(\mathbf{x},\mathbf{q};\bm{\theta})]-\kappa^{y_{i}}\right\\}\cdot
H\left(B_{i}-\kappa^{y_{i}}\right).$ (1)
Here $\sigma:\mathbb{R}\to[0,1]$ is a sigmoid function and $H(x)$ is the unit
step function defined as $0$ at $x=0$. The expression
$\sigma[\psi_{i}(\mathbf{x},\mathbf{q})]$ defines the value Agent $i$ places
on Asset $y_{i}$ as a function of the market state (and hence spot-prices) and
the information in the external information $\mathbf{x}$. If Agent $i$ places
more value on Asset $y_{i}$ than its present price $\kappa^{y_{i}}$, then
$H\left\\{\sigma[\psi_{i}(\mathbf{x},\mathbf{p})]-\kappa^{y_{i}}\right\\}=1$
and $\Delta q^{y_{i}}_{i}=1$ just in case the agent has sufficient funds given
by $H\left[B_{i}-\kappa^{y_{i}}\right]$. That is, Agent $i$ purchases a share
of Asset $i$. Notice we are assume that agents may buy one share of an asset
at a time. This both simplifies the agent logic and also would prevent the
agents from out-competing humans in the market in the hybrid scenario. The
vector $\bm{\theta}$ is a set of parameters that define the specific outputs
of $\psi_{i}$ and thus affect the agent purchase logic.
Let $\bm{\Theta}$ be the (matrix) of all parameter vectors for the agents.
After running for $T$ time units with input information $\mathbf{x}$, the spot
price for Asset $1$ is $p^{1}_{T}(\mathbf{x};\bm{\Theta})$. If we are given
input information $\\{\mathbf{x}_{1},\dots,\mathbf{x}_{N}\\}$ with class
information $\\{y_{1},\dots,y_{N}\\}$, then training the market is the process
of solving:
$\min_{\bm{\Theta}}\;\;\frac{1}{N}\sum_{j=1}^{N}\left\lvert
p^{1}_{T}(\mathbf{x}_{j};\bm{\Theta})-y_{j}\right\rVert^{2}.$
This problem is solved in nakshatri2021design using a genetic algorithm to
obtain a market that can classify external information $\mathbf{x}\in D$.
At the close of the market, the price of a each asset is taken as a proxy for
the market’s confidence in the corresponding outcome. In our binary market
model, there are two mutually exclusive possible outcomes and so the
(normalized) prices should sum to $1$. In this way, the market can be used for
regression or classification. In the three examples we consider here, the
market is used for classification. A separate market is run for each point in
the test set and the asset with the higher price is considered the market’s
classification decision for that test point.
We note, critically, that based on this model, agent participation is
voluntary and decision to participate is driven by $\Delta q^{y_{i}}_{i}=1$
from Equation 1. If this condition is not met during the course of the market
for any agents, there will be no market activity and thus no classification
decision for that test point. Authors in rajtmajer2022synthetic have noted
that this may occur frequently, particularly in cases where the training data
set is small or points in the test set are significant different from training
the data. Accordingly, we calculate accuracy and F1 based on the scored subset
of the data, while also reporting participation as a performance metric.
The artificial prediction market model includes five hyper-parameters that are
not optimized by the genetic algorithm discussed in nakshatri2021design:
1. 1.
Agent inter-arrival rate ($\lambda$);
2. 2.
Agent initial bank value ($B_{i}(0)$);
3. 3.
Market liquidity ($1/\beta$);
4. 4.
Simulation running time ($T$) or duration;
5. 5.
Number of generations in the genetic (training) algorithm.
As such, these parameters are researcher-determined and warrant further study
(RQ1). Our first set of experiments, described below, explore the specific
roles of agent inter-arrival rate ($\lambda$), agent initial bank value
($B_{i}(0)$) (or, “cash”), and market liquidity ($1/\beta$) on performance. We
explore the robustness of performance to selection of these hyper-parameters,
highlighting accuracy and F1 score but also trade-offs with agent
participation.
In experiments that follow, the genetic algorithm is trained over five
generation. The objective function of the genetic algorithm maximizes root
mean square error of the estimated score. Agent performance is evaluated based
on profit; nonprofitable agents are deleted from the pool. The ten most
profitable agents are retained and, amongst them, the seven most profitable
agents are selected for mutation and crossover.
## 5 Experimental Design
The experiments we describe support the two primary research questions we have
laid out, respectively. First, we capture the effects of different
combinations of hyper-parameters on market performance (RQ1). Second, we
explore the impact of exogenous agents not trained through the evolutionary
training process, but rather who adopt one of a set of three simple purchasing
rules meant to represent primitive human inputs (RQ2).
### 5.1 Market robustness to hyper-parameters
We study the effects of inter-arrival rate $\lambda$, agent initial bank value
$B_{i}(0)$ (or, “cash”), and market liquidity $1/\beta$ on artificial market
performance. As mentioned, number of generations is fixed at five during
training; while, market duration is fixed at 20. These parameters were fixed
(vs. manipulated) to avoid combinatorial complexity during this initial study;
however, they should be further studied in future work. In practice, we have
found these values to be sufficient for market behavior to converge while also
offering reasonable run time. Liquidity is tested for the set of values
$\\{5,10,20,50,75,100,150,200,300\\}$. Initial cash is tested for
$\\{1,2,3,4,5,10,20\\}$; $\lambda$ is tested for
$\\{0.1,0.25,0.5,0.01,0.025,0.05,1\\}$. Our experiments consider all
combinations of these hyper-parameter values, $441$ total, and measure
corresponding performance in terms of accuracy, F1 score, and agent
participation (i.e., total number of scored test points). Performance for each
hyper-parameter set is determined based on 5-fold cross validation with 80/20
train/test splits. From these outcomes, we select best and worst-performing
hyper-parameter sets to be used for downstream analyses. This process is
outlined in Figure 10.
### 5.2 Market behavior with exogenous agents
We introduce three classes of exogenous agents representing simple,
fundamental behaviors which operate fully separate from the agent logic and
feature-based training protocol used for the other agents in the market. These
classes of behavior are intended to represent behavioral primitives that, in
combination, would underlie the actions of human participants in a hybrid
scenario. The first, _ground truth_ agents (GT) have perfect knowledge of the
correct outcome and always buy contracts corresponding to the correct outcome
whenever they have the opportunity to participate (which is moderated by their
arrival rate, $\lambda$). The second are _ground truth inverse_ agents
(GTinv). These agents also know the correct outcome but always buy contracts
corresponding to the incorrect outcome whenever they have an opportunity to
participate. This scenario is equivalent to the case where agents are simply
certain but incorrect in their forecast. Finally, our third class of agents
are _random_ agents which purchase contracts corresponding to one or the other
outcome randomly. Understanding that the decisions of human participants in
the hybrid prediction market would likely not exactly fall into any of these
three categories, we consider our simulations of markets with agents drawn
from these classes as drawing initial boundaries around the type of impact
human participants might have on the performance of an artificial market
depending on the complexity of the forecasting task, e.g., there may be a
classification task which is very easy for humans but difficult for algorithms
wherein we would expect near-perfect performance from human participants.
Our experiments measure impact of exogenous agents on market performance
measured, as before, by accuracy, F1 score, and agent participation. Because
exogenous agents are not trained, they are not subject to the genetic
algorithm. Rather, exogenous agents are added directly to the agent pool
during test. Our experiments test the impact of adding varying number of
agents from each class. We specify this number based on percentage of the
total agent pool. Specifically, we test hybrid market performance with the
inclusion of $\\{0.1\%,0.5\%,1\%\\}$ GT and GTinv agents. We test hybrid
market performance with the inclusion of random agents accounting for
$\\{1\%,5\%,10\%,50\%\\}$ of the total agent pool. Random agents are included
at a higher rate given the comparatively lesser impact they have on asset
prices.
## 6 Experimental Results
This section discusses the results obtained from all the experiments executed
on the three datasets mentioned above. We have analyzed the performance of the
artificial prediction market using different metrics to capture the degree of
coherence between observations and predictions. Owing to the size of these
datasets, we chose to go for 5-fold cross validation to ensure that we make
best use of the entire data and gather more accurate metrics in terms of the
performance of our markets across various hyper-parameter combinations. Our
market has been generalised to work like a binary classifier, therefore,
accuracy and F1 score were extracted as the performance evaluation metrics.
Accuracy is the fraction of correct predictions made by our model. For binary
classification, accuracy can also be calculated in terms of two positives
(True Positives(TP) and False Positives(FP)) and two negatives (True
Negatives(TN) and False Negatives(FN))as follows.
$\displaystyle Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$
Apart from the accuracy of predictions, we also wanted to understand how well
our model classifies across the two classes. Hence, we chose F1-Score as our
evaluation metric since it combines the Precision and Recall of a classifier
(which means that it accounts for both FPs and FNs) into a single metric by
taking a harmonic mean as shown below.
$\displaystyle
F1-Score=2*\frac{\text{Precision}*\text{Recall}}{\text{Precision}+\text{Recall}}$
If a model produces no false positives then the precision of the model is 1.
F1-score ranges between 0 and 1. Higher the F1, better the model.
### 6.1 Analysis using IRIS Data
We started off our tests with the standard IRIS dataset which contains total
150 datapoints of 50 classes each as mentioned in Section 3.1. As shown in
Figure 10, we capture the average accuracy and average f1 for all
experimentally generated hyper-parameter combinations. Figure 1 highlights the
average F1 score with different combinations of $\lambda$, initial cash and
liquidity. From the colour datapoints in the figure, we can infer that the
best F1 scores are obtained when initial cash ranges from 1-4 and when
liquidity is greater than 100. We obtained a best case accuracy of 94.1% with
a F1 score 0.91 for the combination (liquidity: 300, lambda: 1, initial
cash:1). Hence, we a small subset was taken for deeper understanding to find
clearer patterns across combinations by fixing a pair from this set and
varying the third parameter.
From Table 1, 2, 3, we can clearly observe that initial cash and liquidity
tends to show a pattern in terms of the affect on performance of the market
while it’s relatively difficult to assess the impact of $\lambda$ on the
overall changes in captured metrics. The market performance increases as
liquidity increases (Table 3) but decreases when initial cash increases(Table
1). For example, the accuracy is 94% when the initial cash is 1 but it
decreases to 75% when the cash increased to 20.
Figure 2 showcases the average accuracy vs F1 score for the IRIS data with
different values of initial cash. In most of the cases, F1 scores are inline
with the accuracy and the best numbers are seen with an initial cash of 1
(highlighted with blue markers).
Table 1: Average F1 Score and Accuracy with different values of initial cash for best hyper-parameters of IRIS data Initial Cash | Liquidity | $\lambda$ | Accuracy | F1 Score
---|---|---|---|---
1.0 | 300.0 | 1.0 | 0.94 | 0.91
2.0 | 300.0 | 1.0 | 0.81 | 0.58
3.0 | 300.0 | 1.0 | 0.81 | 0.64
4.0 | 300.0 | 1.0 | 0.87 | 0.76
5.0 | 300.0 | 1.0 | 0.76 | 0.42
10.0 | 300.0 | 1.0 | 0.75 | 0.35
20.0 | 300.0 | 1.0 | 0.75 | 0.37
| | | |
Table 2: Average F1 Score and Accuracy with different values of $\lambda$ for best hyper-parameters of IRIS data $\lambda$ | Liquidity | Initial Cash | Accuracy | F1 Score
---|---|---|---|---
0.100 | 300.0 | 1.0 | 0.87 | 0.79
0.250 | 300.0 | 1.0 | 0.87 | 0.80
0.500 | 300.0 | 1.0 | 0.87 | 0.78
0.010 | 300.0 | 1.0 | 0.87 | 0.79
0.025 | 300.0 | 1.0 | 0.91 | 0.86
0.050 | 300.0 | 1.0 | 0.88 | 0.82
1.000 | 300.0 | 1.0 | 0.94 | 0.91
| | | |
Table 3: Average F1 Score and Accuracy with different values of Liquidity for best hyper-parameters of IRIS data Liquidity | $\lambda$ | Initial Cash | Accuracy | F1 Score
---|---|---|---|---
5.0 | 1.0 | 1.0 | 0.67 | 0.00
10.0 | 1.0 | 1.0 | 0.67 | 0.00
20.0 | 1.0 | 1.0 | 0.67 | 0.03
50.0 | 1.0 | 1.0 | 0.75 | 0.33
75.0 | 1.0 | 1.0 | 0.85 | 0.72
100.0 | 1.0 | 1.0 | 0.86 | 0.74
150.0 | 1.0 | 1.0 | 0.86 | 0.76
200.0 | 1.0 | 1.0 | 0.88 | 0.81
300.0 | 1.0 | 1.0 | 0.94 | 0.91
| | | |
Figure 1: Average F1 with different values of $\lambda$, liquidity and initial
cash for Iris Data Figure 2: Average accuracy vs average F1 with different
values of Initial Cash for Iris Data
### 6.2 Analysis using Heart Disease Data
The complete set of 441 combinations were generated for heart disease dataset
as mentioned in Section 3.1. Figure 3 shows the average F1 score for all 441
combinations of hyper-parameters where the yellow colored datapoints represent
better F1 score than the magenda ones. These yellow colored datapoints are
spreaded all over the combinations of hyper-parameters set. Figure 4 describe
the obtained average accuracy with the average F1 score with different values
of $\lambda$. There is no effect of lambda on market performance as we can see
it is not possible to infer any pattern form the figure 4. The main intention
to plot the figure is to show there is no proper best/worse value of $\lambda$
regarding market performance. The best performance was obtained with accuracy
66% and F1 score 0.71 when only one datapoint was not scored by the model.
Now, to have more clear insight we took a small subset from the 441
combinations to explore more with these parameters.
The best market performance was obtained when the liquidity is in the range
(50-75), but there’s no defined pattern in the performance with change in
values for $\lambda$ and initial cash in case of Heart data. From table 4 we
can infer that with Only in case of increase in liquidity, effect the
performance of the model. For heart data when we have liquidity 50 the
obtained accuracy is 66% but it was reduced to 57% when liquidity is 200 which
means higher values of liquidity are not good for this dataset.
Table 4: Average F1 and Accuracy with different Liquidity for Heart Disease Data Liquidity | $\lambda$ | Initial cash | Accuracy | F1 score
---|---|---|---|---
50.0 | 0.05 | 20.0 | 0.66 | 0.71
20.0 | 0.05 | 20.0 | 0.60 | 0.66
75.0 | 0.05 | 20.0 | 0.61 | 0.64
200.0 | 0.05 | 20.0 | 0.57 | 0.62
100.0 | 0.05 | 20.0 | 0.58 | 0.62
300.0 | 0.05 | 20.0 | 0.59 | 0.61
150.0 | 0.05 | 20.0 | 0.58 | 0.59
10.0 | 0.05 | 20.0 | 0.54 | 0.56
5.0 | 0.05 | 20.0 | 0.59 | 0.54
| | | |
Figure 3: Average F1 with different values of $\lambda$, liquidity and initial
cash for Heart Data Figure 4: Average accuracy vs average F1 with different
values of $\lambda$ for Heart Data
### 6.3 Analysis using Replication Data
We further repeated the phase 1 of the experiments on replication data to
understand how the market parameters change across datasets of varying
complexities and to capture similarities, dissimilarities and patterns across
datasets. This generalization would be helpful for future research in this
area. The average F1 score for all hyper-parameter combinations can be seen in
Figure 5. Figure 6 is the density plot for the replication data across the
accuracy and F1 score for all those 441 combinations.
Figure 8 shows the average accuracy with average F1 score for the replication
data with different values of initial cash. We observed that the market
performs well with initial cash of 1 and 20 respectively in some of the cases.
From these plots, we can further infer that there’s no clear best/worst
performing range of initial cash and lambda for the replication data as well.
Table 5: Agent participation with different values of Liquidity for Replication data Liquidity | Mean of agents participation
---|---
5 | 24.49
10 | 26.15
20 | 29.13
50 | 35.43
75 | 37.48
100 | 38.88
150 | 40.57
200 | 41.51
300 | 42.45
|
Table 6: Mean of Agent participation with different values of Initial Cash for Replication data Initial Cash | Mean of agents participation
---|---
1 | 35.39
2 | 35.87
3 | 35.22
4 | 34.96
5 | 34.79
10 | 34.90
20 | 34.73
|
Table 7: Mean of Agents participation with different values of $\lambda$ for Replication data $\lambda$ | Mean of agents participation
---|---
0.01 | 35.08
0.025 | 34.93
0.05 | 35.06
0.1 | 35.13
0.25 | 35.18
0.5 | 35.06
1 | 35.41
|
Table 8: First five best and worst Hyper-parameters for Replication Data Liquidity | $\lambda$ | Initial Cash | F1 | Accuracy | Scored%
---|---|---|---|---|---
5 | 0.05 | 1 | 0.84 | 0.79 | 36
10 | 0.05 | 10 | 0.84 | 0.76 | 35
5 | 1 | 4 | 0.83 | 0.75 | 37
5 | 0.1 | 1 | 0.83 | 0.77 | 36
5 | 0.1 | 2 | 0.83 | 0.75 | 37
150 | 0.25 | 4 | 0.69 | 0.58 | 35
100 | 0.05 | 2 | 0.68 | 0.0.57 | 35
150 | 0.5 | 20 | 0.68 | 0.58 | 35
10 | 0.5 | 3 | 0.65 | 0.66 | 55
75 | 0.1 | 2 | 0.64 | 0.65 | 52
| | | | |
Table 9: F1 score for different Exogenous Agents No-Ex | GT0.1% | GT0.5% | GT1% | GTinv0.1% | GTinv0.5% | GTinv1% | Random1% | Random5% | Random10% | Random50%
---|---|---|---|---|---|---|---|---|---|---
0.84 | 0.93 | 1 | 1 | 0.34 | 0.23 | 0.09 | 0.79 | 0.81 | 0.80 | 0.82
0.84 | 0.91 | 0.96 | 0.97 | 0.34 | 0.31 | 0.28 | 0.74 | 0.76 | 0.75 | 0.75
0.83 | 0.94 | 1 | 1 | 0.32 | 0.25 | 0.06 | 0.79 | 0.81 | 0.83 | 0.78
0.83 | 0.90 | 0.95 | 0.99 | 0.33 | 0.28 | 0.24 | 0.76 | 0.74 | 0.82 | 0.77
0.83 | 0.88 | 0.91 | 0.94 | 0.33 | 0.32 | 0.29 | 0.75 | 0.77 | 0.79 | 0.73
0.69 | 0.89 | 0.94 | 0.96 | 0.34 | 0.31 | 0.28 | 0.76 | 0.77 | 0.81 | 0.77
0.69 | 0.91 | 0.97 | 0.96 | 0.34 | 0.29 | 0.29 | 0.78 | 0.77 | 0.77 | 0.78
0.68 | 0.90 | 0.95 | 0.96 | 0.33 | 0.29 | 0.29 | 0.78 | 0.78 | 0.79 | 0.76
0.66 | 0.89 | 0.90 | 0.94 | 0.33 | 0.33 | 0.29 | 0.76 | 0.76 | 0.79 | 0.74
0.65 | 0.90 | 0.91 | 0.94 | 0.34 | 0.32 | 0.30 | 0.77 | 0.75 | 0.81 | 0.74
| | | | | | | | | |
Table 10: Accuracy score for different Exogenous Agents No-Ex | GT0.1% | GT0.5% | GT1% | GTinv0.1% | GTinv0.5% | GTinv1% | Random1% | Random5% | Random10% | Random50%
---|---|---|---|---|---|---|---|---|---|---
0.79 | 0.92 | 1 | 1 | 0.27 | 0.14 | 0.05 | 0.76 | 0.77 | 0.77 | 0.78
0.76 | 0.90 | 0.95 | 0.97 | 0.27 | 0.21 | 0.17 | 0.67 | 0.68 | 0.67 | 0.70
0.75 | 0.94 | 1 | 1 | 0.26 | 0.14 | 0.03 | 0.76 | 0.77 | 0.80 | 0.77
0.77 | 0.89 | 0.95 | 0.99 | 0.26 | 0.18 | 0.14 | 0.70 | 0.66 | 0.77 | 0.73
0.75 | 0.87 | 0.90 | 0.94 | 0.27 | 0.23 | 0.18 | 0.67 | 0.70 | 0.74 | 0.68
0.58 | 0.88 | 0.93 | 0.95 | 0.26 | 0.21 | 0.17 | 0.70 | 0.70 | 0.76 | 0.71
0.57 | 0.90 | 0.96 | 0.95 | 0.25 | 0.18 | 0.18 | 0.73 | 0.70 | 0.71 | 0.72
0.58 | 0.89 | 0.94 | 0.95 | 0.25 | 0.19 | 0.17 | 0.73 | 0.71 | 0.73 | 0.74
0.65 | 0.88 | 0.89 | 0.93 | 0.25 | 0.26 | 0.18 | 0.71 | 0.68 | 0.74 | 0.69
0.64 | 0.89 | 0.90 | 0.94 | 0.27 | 0.23 | 0.19 | 0.72 | 0.68 | 0.76 | 0.70
| | | | | | | | | |
Figure 5: Average F1 with different values of Lambda, liquidity and initial cash for Replication Data Figure 6: Density plot for replication data across Accuracy and F1 score Figure 7: Average accuracy vs average F1 with different values of Liquidity for Replication Data Figure 8: Average accuracy vs average F1 with different values of Initial Cash for Replication Data Figure 9: F1 with different percentage of added GT inverse agents for replication data Figure 10: Phase 1 Architecture Figure 11: Phase 2 Architecture Figure 12: F1 with different percentage of added Random agents for replication data Figure 13: F1 with different percentage of added GT agents for replication data Table 11: Agent participation with different percentages of GT agents No-Ex | GT0.1% | GT0.5% | GT1%
---|---|---|---
4112 | 4543 | 4671 | 5352
3280 | 3766 | 4266 | 5339
3431 | 3325 | 4057 | 5132
4142 | 4442 | 5244 | 5496
3581 | 3989 | 4638 | 5023
5963 | 6462 | 6934 | 7544
5802 | 5784 | 6356 | 7028
6076 | 6100 | 6790 | 7475
3906 | 4063 | 4286 | 5180
6448 | 5436 | 5873 | 6766
| | |
The most impactful hyper-parameter we found after doing the complete study is
the liquidity for all three datasets. Figure 7 highlights the average accuracy
vs average F1 score with different values of liquidity for the replication
data. For all these three datasets the observation is same and it is evident
that the performance of the market doesn’t depend on the lambda. The best
performance was observed with accuracy 79% and F1 score 0.84 when the initial
cash was 1 and liquidity was 5. The total number of papers were 145 out of
which 93 were not scored and 52 were scored.
Figure 14: Agent Participation with different percentage of added GT agents for replication data Table 12: Agent participation with different percentages of GTinv agents No-Ex | GTinv0.1% | GTinv0.5% | GTinv1%
---|---|---|---
4112 | 4284 | 5374 | 6785
3280 | 3636 | 4512 | 5484
3431 | 3928 | 4664 | 5299
4142 | 4085 | 5169 | 7010
3581 | 4067 | 4883 | 6409
5963 | 6207 | 6414 | 7803
5802 | 5985 | 6174 | 7448
6076 | 6218 | 6529 | 7430
3906 | 4045 | 5042 | 6048
6448 | 5607 | 6120 | 7228
| | |
Figure 15: Agent Participation with different percentage of added GT inverse
agents for replication data
Another very interesting observation is the mean of agent participation with
different values of liquidity which is mentioned in Table 5. The agent
participation clearly increases with increase in the value of liquidity.
Liquidity constant directly impacts the shifts in the price.The agents tend to
participate more as the price shifts tends to increase. Hence, this behavior
is inline with our expectations. We captured similar data for different values
of lambda and Initial cash as shown in Table 7 and Table 6 respectively. The
mean of the agent participation tends to show no significant variation with
change in lambda and initial cash and hence this supports our previous
observation that these parameters don’t produce a consistent decipherable
effect on the performance of the market.
At the end of our experiments, with the hyper-parameter combinations we
understood that there is no fixed best hyper-parameter set for all kinds of
datasets. Every researcher needs to experiment and set the best hyper-
parameter combination for their application to get the best performance from
the prediction market. For the next set of experiments, we decided to take the
best and worst 5 combinations of hyper-parameters for the replication dataset
which is listed in Table 8. The prediction market classifies a subset of the
input papers with good accuracy and F1 score. The last column shows the total
percentage of papers scored for each of these combinations. The highest number
of scored papers is seen with a (liquidity, $\lambda$, initial cash)
combination of (10, 0.5, 3), when the market was able to get 55% scored papers
with 65% accuracy and with an F1 of 0.66.
#### 6.3.1 Discussion RQ1
Bring together.
### 6.4 Analysis using Exogenous Agents
In the second phase of the study, we have introduced simulated exogenous
agents to analyse their effect on the market performance. For this purpose, we
chose to inject incremental percentages of simulated agents to track the
shifts in accuracy, F1-score and eventually the overall agent participation.
Our agents are divided into three categories - Ground truth (GT), Ground Truth
Inverse(GT Inv) and Random agents respectively. The accuracy and F1-score for
the top 5 best and worst combinations after the introduction of these agents
has been listed down in Table 10 and Table 9 respectively.
#### 6.4.1 Introduction of Ground Truth Agents in the Agent Pool
Agents with Ground Truth information have the potential to causes sudden
changes on how the market behaves. Hence, we selected very small percentages
of such agents in our tests.
$\%\>of\>GT\>agents=\\{0.1,0.5,1\\}$
In the set above, 0.1% implies that we would inject 1 GT agent for every 1000
agents currently in the pool.
From Table 9 and Table 10, it is evident that even a small amount of GT agents
can improve the market performance substantially. Figure 13 clearly charts
down this incremental improvement. On injecting 0.1% of GT agents, we see a
spike in F1 score from 0.84 to 0.93 and the accuracy shoots from 79% to 92% as
shown in the tables above. Further gradual addition of such agents ensures
100% accuracy and F1 for the best combinations. The impact of such agents
becomes more evident in the cases where the previously worst performing set of
hyper-parameters now showed an improvement in accuracy from 64% to 89% with
just 0.1% of GT agents.
#### 6.4.2 Introduction of Ground Truth Inverse Agents in the Agent Pool
Similar to GT agents, GTinv agents can intuitively cause sudden drops in
performance. These drops were clearly seen in our tests and can be tracked
from Table 9 and Table 10 respectively. Just like GT Agents, we chose the
following set of incremental percentages for GTinv agents since they drive the
market in a defined direction.
$\%\>of\>GTInv\>Agents=\\{0.1,0.5,1\\}$
Figure 9 plots the F1 score for different percentage of GTinv agents. The
black bar represents the case where there are no GTinv agents in the market.
It is clearly visible from the bar plot that market performance deteriorates
with the increase in amount of GTinv agents. Addition of just 0.1% of GTinv
agents dropped the accuracy from 79% to 27%. The accuracy further drops down
to 14% after adding 0.5% of agents and any further addition pushes the
accuracy down to zero as per our tests. Similar behavior is seen in the F1
scores as well. The best F1 score is 0.84 prior to the introduction of GTinv
agents but it drops down to 0.34 with just 0.1% of such agents.
#### 6.4.3 Random Agents
We further created a set of simulated agents which weren’t trained on the
default feature set to simulate real-time human experiments. We found that
it’s infeasible to assess a pattern in the market performance owing to the
randomness introduced by these agents. Figure 12 shows how the F1 score
changes when these Random agents are injected to the training pool. We have
used a different set of percentages for these agents as shown below.
$\%\>of\>Random\>Agents=\\{1,5,10,50\\}$
As expected, the F1 scores don’t showcase a clear shift in any direction as
can be seen from Figure 12, but these agents have the potential to cause an
overall change in the market performance. The impact of these agents varies
across best to worst choices of markets. Adding 10% Random agents drops the
F1-score from 0.84 to 0.80 for the best case but it also improves the score
from 0.65 to 0.81 for the previously worst performing combination. Similar
behavior is seen if we add 5% random agents as it changes F1-score from 0.84
to 0.81 for the best case but it improves from 0.65 to 0.75 for the worst case
scenario. On increasing the percentage to 50%, we see a further dip in
performance compared to the case with just 10% random agents But if you
increase it to 50% then the overall performance will go down from the results
obtained for 10% random agents.
#### 6.4.4 Agent Participation varies with added Exogenous agents
The addition of these agents provides an opportunity to derive various
intuitions. One such analysis was how the agent participation changes with
added GT and GTinv agents. Agent participation increases with the increase in
percentage of these agents in the market as they have the potential to drive
other agents by bidding in defined directions. Table 11 and Table 12 shows
total agent participation for different percentages of GT agents and GTinv
agents respectively. From Figure 14, we can observe that participation was
highest for 1% of newly added GT agents and lowest for no exogenous agents for
all 10 combinations. If we take the first combination for example, when there
are no exogenous agents the participation was 4112, for 0.1% GT agents
participation increases to 4543 and then it became 5352 for 1% added GT agents
during market run. Similarly, Figure 15 shows total agent participation for
different percentage of added GTinv agents. It can be seen that incremental
addition of GTinv agents drives increased agent participation for all 10
combinations. For example, if we chose the last set of hyper-parameter
combination the agent participation was 6448 for no exogenous agents but it
increased to 7228 on adding 1% of GTinv agents.
#### 6.4.5 Discussion RQ2
## 7 Expected Contributions
This study provides the research community with a new approach to the
prediction market that has not been explored yet. The analysis of the
simulated humans with trained AI agents and how they behave together in the
field of research reproducibility is new. We have tried to analyze the
scenario when there will be real human participants in the prediction market
and will buy/sell alongside the trained AI agents and how their participation
will affect the market performance as well as the agent participation. The
comparative study of the artificial prediction market and the market with
exogenous agents (also known as simulated humans) point out the need for
human-AI collaboration in the prediction market which should motivate future
researchers to explore more in the field of Human-AI study. The overall study
with the hyper-parameters will be very helpful for future research because it
will save time and effort. They don’t have to run multiple experiments to
understand the behavior of these parameters. They will have a clear idea about
the experiments needed for their dataset to choose the best set.
The proposed research could have an impact on research society in broad ways
because the prediction market has been massively used in almost every research
domain. Scientific research reproducibility is one of the most sensitive areas
for all researchers. The proposed research will produce a clear vision of the
future hybrid prediction market. A lot of money and time will be saved.
Previously it was observed that there was a lot of loss in the market due to
replication failures. People can use that money and time for other productive
research which can be used in different ways for the betterment of society.
## 8 Conclusions
Previous research suggested that artificial prediction markets work wonders in
various applications. In many cases, they perform better than human
predictions. But some of the applications are very complex and they need human
intelligence for better understanding. In these situations, the combined
prediction of humans and AI agents can work wonders as per our hypothesis.
Therefore, we have experimented with simulated humans in the artificial
prediction market to explore how the participation of different exogenous
agents will affect the market performance. The artificial prediction market
with random agents, GT agents, and GT inverse agents was analyzed to
understand the bounds of the behavior of the market in a hybrid setting. After
adding a different percentage of exogenous agents, the performance of both
markets (artificial prediction market and market with simulated humans) was
compared using different evaluation matrices. Trained market agents will not
participate if they don’t have enough similar training features. In those
cases, human participation can drive agent participation in the market. The
final hope of this research study is that it will encourage other researchers
to explore more in Human-AI collaboration more. Also, we provided an important
understanding of different hyper-parameters and how they differ for different
datasets. Future researchers will have a complete understanding of how to use
these parameters to improve the performance of the prediction market. For
future work, we will introduce human participants in our prediction market
with AI agents to investigate the real scenario.
|
# RPC based tracking system at CERN GIF++ facility
K. Mota Amarilo<EMAIL_ADDRESS>A. Samalan M. Tytgat M. El Sawy G.A.
Alves F. Marujo E.A. Coelho E.M. Da Costa H. Nogima A. Santoro S.
Fonseca De Souza D. De Jesus Damiao M. Thiel M. Barroso Ferreira Filho A.
Aleksandrov R. Hadjiiska P. Iaydjiev M. Rodozov M. Shopova G. Soultanov
A. Dimitrov L. Litov B. Pavlov P. Petkov A. Petrov E. Shumka S.J. Qian
H. Kou Z.-A. Liu J. Zhao J. Song Q. Hou W. Diao P. Cao C. Avila D.
Barbosa A. Cabrera A. Florez J. Fraga J. Reyes Y. Assran M.A. Mahmoud
Y. Mohammed I. Crotty I. Laktineh G. Grenier M. Gouzevitch L. Mirabito
K. Shchablo I. Bagaturia I. Lomidze Z. Tsamalaidze V. Amoozegar B.
Boghrati M. Ebraimi M. Mohammadi Najafabadi E. Zareian M. Abbrescia G.
Iaselli G. Pugliese F. Loddo N. De Filippis R. Aly D. Ramos W.
Elmetenawee S. Leszki I. Margjeka D. Paesani L. Benussi S. Bianco D.
Piccolo S. Meola S. Buontempo F. Carnevali L. Lista P. Paolucci F.
Fienga A. Braghieri P. Salvini P. Montagna C. Riccardi P. Vitulo E.
Asilar J. Choi T.J. Kim S.Y. Choi B. Hong K.S. Lee H.Y. Oh J. Goh I.
Yu C. Uribe Estrada I. Pedraza H. Castilla-Valdez A. Sanchez-Hernandez R.
L. Fernandez M. Ramirez-Garcia E. Vazquez M. A. Shah N. Zaganidis A. Radi
H. Hoorani S. Muhammad A. Ahmad I. Asghar W.A. Khan J. Eysermans F.
Torres Da Silva De Araujo
on behalf of the CMS Muon Group
###### Abstract
With the HL-LHC upgrade of the LHC machine, an increase of the instantaneous
luminosity by a factor of five is expected and the current detection systems
need to be validated for such working conditions to ensure stable data taking.
At the CERN Gamma Irradiation Facility (GIF++) many muon detectors undergo
such studies, but the high gamma background can pose a challenge to the muon
trigger system which is exposed to many fake hits from the gamma background. A
tracking system using RPCs is implemented to clean the fake hits, taking
profit of the high muon efficiency of these chambers. This work will present
the tracking system configuration, used detector analysis algorithm and
results.
###### keywords:
Resistive plate chambers , CMS Experiment , gaseous detectors , HL-LHC
###### PACS:
0000 , 1111
###### MSC:
0000 , 1111
††journal: Nuclear Instruments and Methods in Physics Research A
[a]Ghent University, Dept. of Physics and Astronomy, Proeftuinstraat 86,
B-9000 Ghent, Belgium. [aa]Université Libre de Bruxelles, Avenue Franklin
Roosevelt 50-1050 Bruxelles, Belgium. [b]Centro Brasileiro Pesquisas Fisicas,
R. Dr. Xavier Sigaud, 150 - Urca, Rio de Janeiro - RJ, 22290-180, Brazil.
[c]Dep. de Fisica Nuclear e Altas Energias, Instituto de Fisica, Universidade
do Estado do Rio de Janeiro, Rua Sao Francisco Xavier, 524, BR - Rio de
Janeiro 20559-900, RJ, Brazil. [d]Bulgarian Academy of Sciences, Inst. for
Nucl. Res. and Nucl. Energy, Tzarigradsko shaussee Boulevard 72, BG-1784
Sofia, Bulgaria. [e]Faculty of Physics, University of Sofia,5 James Bourchier
Boulevard, BG-1164 Sofia, Bulgaria. [f]School of Physics, Peking University,
Beijing 100871, China. [g]State Key Laboratory of Particle Detection and
Electronics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China. [gg]University of Chinese Academy of Sciences, No.19(A)
Yuquan Road, Shijingshan District, Beijing 100049, China. [h]Universidad de
Los Andes, Carrera 1, no. 18A - 12, Bogotá, Colombia [i]Egyptian Network for
High Energy Physics, Academy of Scientific Research and Technology, 101 Kasr
El-Einy St. Cairo Egypt. [ii]Suez University, Elsalam City, Suez - Cairo Road,
Suez 43522, Egypt. [j]Center for High Energy Physics(CHEP-FU), Faculty of
Science, Fayoum University, 63514 El-Fayoum, Egypt. [jj]Department of Physics,
Faculty of Science, Ain Shams University, Cairo, Egypt. [jjj] Physics
Department Faculty of Science Helwan University Ain Helwan 11795 Cairo Egypt.
[k]Univ Lyon, Univ Claude Bernard Lyon 1, CNRS-IN2P3, IP2I Lyon, UMR
5822,F-69622, Villeurbanne, France. [l]Georgian Technical University, 77
Kostava Str., Tbilisi 0175, Georgia. [m]School of Particles and Accelerators,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531,
Tehran, Iran. [mm]School of Engineering, Damghan University, Damghan,
3671641167, Iran. [n]INFN, Sezione di Bari, Via Orabona 4, IT-70126 Bari,
Italy. [o]INFN, Laboratori Nazionali di Frascati (LNF), Via Enrico Fermi 40,
IT-00044 Frascati, Italy. [p]INFN, Sezione di Napoli, Complesso Univ. Monte S.
Angelo, Via Cintia, IT-80126 Napoli, Italy. [pp]Dipartimento di Ingegneria
Elettrica e delle Tecnologie dell’Informazione - Università Degli Studi di
Napoli Federico II, IT-80126 Napoli, Italy. [q]INFN, Sezione di Pavia, Via
Bassi 6, IT-Pavia, Italy. [qq]INFN, Sezione di Pavia and University of Pavia,
Via Bassi 6, IT-Pavia, Italy. [r]Hanyang University, 222 Wangsimni-ro, Sageun-
dong, Seongdong-gu, Seoul, Republic of Korea. [s]Korea University, Department
of Physics, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea. [t]Kyung
Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of
Korea. [u]Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-
do 16419, Republic of Korea. [v]Benemerita Universidad Autonoma de Puebla,
Puebla, Mexico. [w]Cinvestav, Av. Instituto Politécnico Nacional No. 2508,
Colonia San Pedro Zacatenco, CP 07360, Ciudad de Mexico D.F., Mexico.
[x]Universidad Iberoamericana, Mexico City, Mexico. [y]Sultan Qaboos
University, Al Khoudh,Muscat 123, Oman. [z]National Centre for Physics, Quaid-
i-Azam University, Islamabad, Pakistan. [za]Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA 02139, United States. [zc]III.
Physikalisches Institut (A), RWTH Aachen University, Sommerfeldstrasse
D-52056, Aachen, Germany.
## 1 Introduction
The Resistive Plate Chambers (RPC) act as muon triggering detector in the Muon
System of the Compact Muon Solenoid (CMS) experiment [1]. With the coming
upgrade of the Large Hadron Collider (LHC) machine, the High Luminosity LHC
(HL-LHC) [2], many detector systems are preparing to improve their
capabilities. In particular, the CMS RPC project has two important planned
activities – the replacement of the off-detector electronics (called link
system) and the extension of the RPC coverage from $|\eta|=$ 1.9 to 2.4 with
the new improved RPC (iRPC) detectors [3].
To study the performance and stability of RPCs in the HL-LHC conditions, tests
are taking place in the CERN Gamma Irradiation Facility (GIF++), where a high
energy particle beam (normally muons) and photons from a gamma source (13 TBq,
137Cs) are combined [4]. The maximum expected background rate for the RPCs in
the current CMS Muon system is 600 Hz/cm2 and for the new iRPCs it goes up to
2 kHz/cm2 (including the safety factor of three) [3]. Such high rates can pose
a challenge to the measurement, as the fake hits can bias the data-taking.
All results shown in this work were recorded during GIF++ test beam in October
2021.
## 2 Experimental Setup
Figure 1 shows a representation of the setup in GIF++. There are two trolleys
where the RPC detectors are placed for irradiation. In addition to the RPCs,
two scintillators are placed on each trolley to trigger on the muon beam. The
data acquisition is performed using a CAEN Time-to-Digital Converter (TDC)
module of type V1190A where the frontend electronics of the chambers are
connected. A V1718 VME controller module is responsible for the communication
between the TDC and the computer where the data are stored. To host the VME
controller and the TDC a 6U VME 6021 crate is used.
Figure 1: Experimental setup at GIF++.
In the trolley farther from the gamma source, there are two tracking RPC
chambers with their strip planes oriented perpendicular to each other to allow
measurement in two directions. The chambers used are the following:
* 1.
Tracking chambers: Referred to as GT1 and GT2, these are double gap chambers
made of 2 mm thick High Pressure Laminate (HPL, also known as bakelite). They
are equipped with a strip plane with 32 strips of 1.45 cm pitch. The signals
are read with the CMS FrontEnd electronics with threshold set to 150 fC.
* 2.
Test Chamber: Referred to as KODEL-C, this is also a double gap chamber made
of HPL with 1.4 mm thickness. It is equipped with a strip plane with 32 strips
of 1.95 cm pitch and custom FrontEnd electronics with threshold set to 75 fC.
All the chambers are flushed with the CMS RPC standard gas mixture (95.2%
freon ($C_{2}H_{2}F_{4}$), 4.5% isobutane ($iC_{4}H_{10}$), and 0.3% sulphur
hexafluoride ($SF_{6}$)) with a relative humidity of $\approx$40%. The
tracking chambers are always powered on with the high voltage (HV) working
point (WP) applied to their electrodes.
## 3 Tracking algorithm
To enable the tracking analysis, at least one hit is required in both tracking
chambers inside the muon time window defined by a Gaussian fit, as indicated
in Fig. 2. A 2D hit profile is made to check their alignment with the muon
beam, as shown in Fig. 3. The hit profile is also determined for the test
chamber and it is used for alignment between the chambers.
Figure 2: Hits time profile of the tracking chamber GT2. The time window where
the hits are accepted in the analysis is represented in the red hatched box.
Figure 3: 2D hit profile of the strips signals for the hits in the muon time
window.
The tracking algorithm relies on the assumption that the beam is perpendicular
to the strip plane, therefore it is only needed to extrapolate the position of
the hit from the tracking to the test chamber and check for a matching hit.
For every event, the following steps are taken:
1. 1.
Perform the clusterization of the hits in the tracking chambers, where events
with more than one cluster are rejected. The cluster barycentres are defined
as the mean position in the cluster;
2. 2.
Perform the clusterization for the test chambers and calculate the clusters’
barycentres;
3. 3.
Form a perpendicular track starting from the tracking cluster barycentre;
4. 4.
Check for a match in any cluster on the test chamber.
Figure 4 shows an example of event where the analysis was performed and a
matching hit was found.
Figure 4: Example of one event in which the hit on tracking chamber was
matched on the test chamber. The strips in red represent the hits and the
point in black is the cluster barycentre.
## 4 Performance of the Tracking Algorithm
Efficiency curves of the test chambers were used to evaluate the validity of
the tracking in rejecting the fake hits caused by the gammas. On Fig. 5 the
efficiency curves are compared for three gamma background conditions. There
are three curves in each plot. The black curve is calculated without the
tracking, taking into consideration all the hits in the muon time window. For
the blue one, only the hits that passed the tracking criteria were considered.
Finally, for the red curve, the HV was recalculated to remove the voltage drop
caused by the resistance of the electrodes - this correction decouples the
shift to the right of the curve on higher cluster rates, caused by the
increase of the current. Therefore, HVgas is the effective HV applied to the
gas volume.
As expected, in Fig. 5(a), where the data were taken in a run with source OFF,
the three curves are equivalent since the cluster rate (CLR) and the currents
are low. in Fig. 5(b) with CLR $\approx$ 648 Hz/cm2, a shift to the right on
the tracking corrected curve is observed, and it is much more evident in Fig.
5(c) with CLR $\approx$ 1.8 kHz/cm2. The increase of the maximum efficiency on
the raw efficiency curve from the Fig. 5(b) to 5(c) indicates high gamma
contamination.
(a)
(b)
(c)
Figure 5: Efficiencies and their sigmoid fits measured at three photon fluxes
with measured at the working-point high voltages gamma cluster rates of 1.804
kHz (a), 0.645 kHz (b), and 1.48 Hz (c), respectively. The curve in black is
the efficiency calculated with all hits inside the muon window, the blue one
is with applied tracking correction and the one in red – with tracking and
resistance correction.
In Fig. 6(a), it is possible to see the curves with tracking and resistance
correction. The shift to the right is caused by the reduction of the gas gain
at higher rates, while the drop in the maximum efficiency is caused by the
dead-time of the electronics. In Fig. 6(b), no resistance correction is
applied, so the main reason for the shift is the voltage drop over the
electrode’s resistance, coupled with the reasons mentioned before.
(a)
(b)
Figure 6: Efficiency curves for various gamma background rates. In (a) the HV
applied is corrected only by pressure and temperature (HVeff), while in (b)
the HV is also corrected by the resistance of the gaps (HVgas)
.
## 5 Conclusion
The implementation of the tracking system was motivated by the necessity to
test the CMS RPC chambers at the conditions of the HL-LHC. The results show
that the tracking system performs very well to remove fake hits from the gamma
background, even at rates as high as 2 kHz/cm2. The test chamber performed
very well and showed increase on the working point of $\approx$650 V with
efficiency loss of $\approx$ 7.5%, using the custom electronics with threshold
of 75 fC. This system is currently being used for the ageing studies of the
CMS RPC system.
## Acknowledgements
We would like to thank our colleagues from CERN Gamma Irradiation facility. We
would also like to acknowledge the enduring support for the Upgrade of the CMS
detector and the supporting computing infrastructure provided by the following
funding agencies: FWO (Belgium); CNPq, CAPES and FAPERJ (Brazil); MES and BNSF
(Bulgaria); CERN; CAS, MoST, and NSFC (China); MINCIENCIAS (Colombia); CEA and
CNRS/IN2P3 (France); SRNSFG (Georgia); DAE and DST (India); IPM (Iran); INFN
(Italy); MSIP and NRF (Republic of Korea); BUAP, CINVESTAV, CONACYT, LNS, SEP,
and UASLP-FAI (Mexico); PAEC (Pakistan); DOE and NSF (USA).
## References
* [1] CMS Collaboration, The CMS Experiment at the CERN LHC, JINST 3 (2008) S08004. doi:10.1088/1748-0221/3/08/S08004.
* [2] O. Aberle, et al., High-Luminosity Large Hadron Collider (HL-LHC): Technical design report, CERN Yellow Reports: Monographs, CERN, Geneva, 2020. doi:10.23731/CYRM-2020-0010.
URL https://cds.cern.ch/record/2749422
* [3] CMS Collaboration, The Phase-2 Upgrade of the CMS Muon Detectors, Tech. rep., CERN, Geneva (2017).
URL https://cds.cern.ch/record/2283189
* [4] R. Guida, GIF++: The new CERN irradiation facility to test large-area detectors for HL-LHC, in: 2015 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), IEEE, 2015, pp. 1–4.
|
# On the number of subrings of $\mathbb{Z}^{n}$ of prime power index
Hrishabh Mishra Chennai Mathematical Institute, H1, SIPCOT IT Park,
Kelambakkam, Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>and
Anwesh Ray Centre de recherches mathématiques, Université de Montréal,
Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal (Québec) H3T 1J4,
Canada<EMAIL_ADDRESS>
###### Abstract.
Let $n$ and $k$ be positive integers, and $f_{n}(k)$ (resp. $g_{n}(k)$) be the
number of unital subrings (resp. unital irreducible subrings) of
$\mathbb{Z}^{n}$ of index $k$. The numbers $f_{n}(k)$ are coefficients of
certain zeta functions of natural interest. The function $k\mapsto f_{n}(k)$
is multiplicative, and the study of the numbers $f_{n}(k)$ reduces to
computing the values at prime powers $k=p^{e}$. Given a composition
$\alpha=(\alpha_{1},\dots,\alpha_{n-1})$ of $e$ into $n-1$ positive integers,
let $g_{\alpha}(p)$ denote the number of irreducible subrings of
$\mathbb{Z}^{n}$ for which the associated upper triangular matrix in Hermite
normal form has diagonal $(p^{\alpha_{1}},\dots,p^{\alpha_{n-1}},1)$. Via
combinatorial analysis, the computation of $f_{n}(p^{e})$ reduces to the
computation of $g_{\alpha}(p)$ for all compositions of $i$ into $j$ parts,
where $i\leq e$ and $j\leq n-1$. We extend results of Liu and Atanasov-Kaplan-
Krakoff-Menzel, who explicitly compute $f_{n}(p^{e})$ for $e\leq 8$. The case
$e=9$ proves to be significantly more involved. We evaluate $f_{n}(e^{9})$
explicitly in terms of a polynomial in n and p up to a single term which is
conjecturally a polynomial. Our results provide further evidence for a
conjecture, which states that for any fixed pair $(n,e)$, the function
$p\mapsto f_{n}(p^{e})$ is a polynomial in $p$. A conjecture of Bhargava on
the asymptotics for $f_{n}(k)$ as a function of $k$ motivates the study of the
asymptotics for $g_{\alpha}(p)$ for certain infinite families of compositions
$\alpha$, for which we are able to obtain general estimates using techniques
from the geometry of numbers.
###### Key words and phrases:
counting subrings of prescribed index, combinatorics of subrings of a fixed
ring, zeta functions of groups and rings
###### 2020 Mathematics Subject Classification:
20E07, 11H06, 11A25, 11M41
## 1\. Introduction
Let $G$ be an infinite group. Given a finite index subgroup $H$, we set
$[G\mathrel{\mathop{\mathchar 58\relax}}H]$ to denote the index of $H$ in $G$.
For $k\in\mathbb{Z}_{\geq 1}$, let $a_{k}(G)$ be the number of finite-index
subgroups $H$ of $G$ such that $[G\mathrel{\mathop{\mathchar 58\relax}}H]=k$.
In [GSS88], Grunewald, Segal and Smith introduced the _zeta function_ of $G$,
defined as follows
$\zeta_{G}(s)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{H}[G\mathrel{\mathop{\mathchar
58\relax}}H]^{-s}=\sum_{k=1}^{\infty}a_{k}(G)k^{-s},$
where the above sum runs over all finite index subgroups $H$ of $G$. The
properties of such zeta functions are described in [DSWP08]. In this context,
zeta functions measure _subgroup growth_. For a comprehensive account of this
theme, we refer to [LS03]. Throughout, $n>1$ is an integer and
$\mathbb{Z}^{n}$ is the ring consisting of integer $n$-tuples with
componentwise addition and multiplication. It is well known that the zeta
function $\zeta_{\mathbb{Z}^{n}}(s)$ is given by
$\zeta_{\mathbb{Z}^{n}}(s)=\zeta(s)\zeta(s-1)\dots\zeta(s-(n-1))$
(cf. _loc. cit._ for five different proofs of the above result).
We study distribution questions for the number of finite-index unital subrings
of $\mathbb{Z}^{n}$ of prescribed index. This problem is closely related to
the distributions of orders in a fixed number ring $\mathcal{O}_{K}$, a
problem studied by Brackenhoff [Bra09], Kaplan Marcinek and Takloo-Bighash
(cf. [KMTB15]). Given $k\in\mathbb{Z}_{\geq 1}$, following [Liu07], let
$f_{n}(k)$ be the number of commutative unital subrings $S$ of
$\mathbb{Z}^{n}$ such that $[\mathbb{Z}^{n}\mathrel{\mathop{\mathchar
58\relax}}S]=k$. The _subring zeta function_ is given by
$\zeta_{\mathbb{Z}^{n}}^{R}(s)=\sum_{k=1}^{\infty}f_{n}(k)k^{-s},$
which decomposes into an Euler product
$\zeta_{\mathbb{Z}^{n}}^{R}(s)=\prod_{p}\zeta_{\mathbb{Z}^{n},p}^{R}(s),$
where the product is over all primes $p$. The local Euler factor at $p$ is
given by
$\zeta_{\mathbb{Z}^{n},p}^{R}(s)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{e=0}^{\infty}f_{n}(p^{e})p^{-es}.$
It is not hard to show that $\zeta_{\mathbb{Z}^{2}}^{R}(s)=\zeta(s)$, and that
$\zeta_{\mathbb{Z}^{3}}^{R}(s)=\frac{\zeta(3s-1)\zeta(s)^{3}}{\zeta(2s)^{2}},$
cf. [Liu07, Proposition 4.1]. The formula for $n=3$ was first obtained by
Datskovsky and Wright [DW88]. A formula for $n=4$ was obtained by Nakagawa cf.
[Nak96]. Recently, there has been interest in the following question (cf.
[Ish22b, Question 1.2]).
###### Question 1.1.
Let $n,e$ be a pair of natural numbers. What can be said about $f_{n}(p^{e})$
as a function of $p$?
For $e\leq 5$, Liu [Liu07] gives explicit formulae for $f_{n}(p^{e})$.
Subsequently, these formulae were generalized by Atanasov, Kaplan, Krakoff and
Menzel [AKKM21] for $e=6,7,8$. These computations indicate that for a fixed
pair $(n,e)$, the function $p\mapsto f_{n}(p^{e})$ is a polynomial in $p$ (cf.
[AKKM21, Question 1.13]), and this has been further studied by Isham in
[Ish22a]. The zeta function $\zeta_{\mathbb{Z}^{n}}^{R}(s)$ is said to be
_uniform_ if there is a rational function $W(X,Y)\in\mathbb{Q}(X,Y)$ such that
for every prime $p$, $\zeta_{\mathbb{Z}^{n},p}^{R}(s)=W(p,p^{-s})$. It is not
hard to show that the zeta function $\zeta_{\mathbb{Z}^{n}}(s)$ is uniform if
for all $e$, $f_{n}(p^{e})$ is a polynomial in $p$. For further details, we
refer to [AKKM21, section 5.1]. Below we summarize the known values of
$f_{n}(p^{e})$ for $e\leq 5$.
###### Theorem 1.2 (Liu [Liu07]).
With respect to notation above,
$\begin{split}f_{n}(1)=&1,f_{n}(p)=\binom{n}{2},f_{n}(p^{2})=\binom{n}{2}+\binom{n}{3}+3\binom{n}{4},\\\
f_{n}(p^{3})=&\binom{n}{2}+(p+1)\binom{n}{3}+7\binom{n}{4}+10\binom{n}{5}+15\binom{n}{6},\\\
f_{n}(p^{4})=&\binom{n}{2}+(3p+1)\binom{n}{3}+(p^{2}+p+10)\binom{n}{4}+(10p+21)\binom{n}{5}+70\binom{n}{6}\\\
+&105\binom{n}{7}+105\binom{n}{8},\\\ f_{n}(p^{5})=&{n\choose
2}+(4p+1){n\choose 3}+(7p^{2}+p+13){n\choose 4}+(p^{3}+p^{2}+41p+31){n\choose
5}\\\ +&(15p^{2}+35p+141){n\choose 6}.\\\ \end{split}$
For the values for $e$ in the range $6\leq e\leq 8$, we refer to [KMTB15].
In this manuscript, we extend the above mentioned results to $e=9$. First, we
introduce some further notation. A subring of prime power index decomposes
into a direct product of irreducible subrings, which we now proceed to
describe.
###### Definition 1.3.
A subring $L$ of index $p^{e}$ is said to be _irreducible_ if for each vector
$x=(x_{1},\dots,x_{n})^{t}\in L$, we have that
$x_{1}\equiv x_{2}\equiv\dots\equiv x_{n}\left(\operatorname{mod}p\right).$
An irreducible subring matrix is a matrix $A$ in Hermite normal form
(1.1)
$A=\begin{pmatrix}p^{\alpha_{1}}&pa_{1,2}&pa_{1,3}&\cdots&pa_{1,n-1}&1\\\
&p^{\alpha_{2}}&pa_{2,3}&\cdots&pa_{2,n-1}&1\\\
&&p^{\alpha_{3}}&\cdots&pa_{3,n-1}&1\\\ &&&\ddots&\vdots&\vdots\\\
&&&&p^{\alpha_{n-1}}&1\\\ &&&&&1\end{pmatrix},$
with $\alpha_{i}\geq 1$ for all $i$. The subring $L$ of index $p^{e}$ is
irreducible if and only if $A_{L}$ is an irreducible subring matrix. Let
$g_{n}(p^{e})$ be the number of irreducible subrings of $\mathbb{Z}^{n}$ of
index $p^{e}$. It is shown that every subring of $p$-power power index is a
direct sum of irreducible subrings, and as a consequence, we have the
following recursive formula (cf. Proposition 4.4 of [Liu07])
(1.2) $f_{n}(p^{e})=\sum_{i=0}^{e}\sum_{j=1}^{n}{n-1\choose
j-1}f_{n-j}(p^{e-i})g_{j}(p^{i}).$
In order to compute $f_{n}(p^{9})$ it thus suffices to compute $g_{j}(p^{e})$
for all values $e\leq 9$, $1\leq j\leq n$. We remark that the convention for
the definition of $g_{n}(p^{e})$ used here is that of [AKKM21], and differs
from that in [Liu07].
A composition of $m$ into $r$ parts consists of a tuple
$\alpha=(\alpha_{1},\dots,\alpha_{r})$ of positive integers such that
$\sum_{i=1}^{r}\alpha_{i}=m$. Let $\mathcal{C}_{n,e}$ be set of all
compositions of $e$ into $(n-1)$ parts. Given $\alpha\in\mathcal{C}_{n,e}$,
let $g_{\alpha}(p)$ be the number of irreducible subring matrices (cf.
Definition 2.3) of the form (1.1). We find that
$g_{n}(p^{e})=\sum_{\alpha\in\mathcal{C}_{n,e}}g_{\alpha}(p)$, and thus to
compute $g_{n}(p^{e})$, we need to compute $g_{\alpha}(p)$ for all
$\alpha\in\mathcal{C}_{n,e}$. Therefore, the numbers $g_{\alpha}(p)$ can be
thought of as the basic building blocks for computing the numbers $f_{n}(k)$.
In practice, it is possible to compute $g_{\alpha}(p)$ in many cases via
combinatorial case by case analysis. For larger values of $e$, the computation
of $f_{n}(p^{e})$ becomes significantly more involved. The total number of new
compositions that one must consider does grow exponentially, and even though
one may compute the value of $g_{\alpha}(p)$ for various types of compositons
that fit into a general framework, there are many exceptional compositions
that do not fit into such a framework. Furthermore, the combinatorial (case by
case) analysis does get increasing more challenging for exceptional
compositions of longer length. With this in mind, we state the main result
below.
###### Theorem 1.4.
With respect to notation above, we have that
$\displaystyle f_{n}(p^{9})-\gamma(n,p)={n\choose
2}+(p^{3}+4p^{2}+4p+1){n\choose 3}+(11p^{4}+30p^{3}+9p^{2}+p+25){n\choose 4}$
$\displaystyle+(11p^{6}+14p^{5}+137p^{4}-16p^{3}+81p^{2}+202p+73){n\choose
5}+(4p^{7}+76p^{6}+128p^{5}+71p^{4}$
$\displaystyle+464p^{3}+1088p^{2}+386p+571){n\choose
6}+(p^{9}+22p^{8}+23p^{7}+59p^{6}+31p^{5}+2032p^{4}$
$\displaystyle+2152p^{3}+2467p^{2}+4949p+2485){n\choose
7}+(p^{10}+p^{9}+2p^{8}+2p^{7}+507p^{6}+955p^{5}$
$\displaystyle+3293p^{4}+5980p^{3}+23410p^{2}+17011p+13707){n\choose
8}+(36p^{8}+37p^{7}+157p^{6}+1159p^{5}$
$\displaystyle+8545p^{4}+34997p^{3}+59371p^{2}+93649p+64019){n\choose
9}+(675p^{6}+795p^{5}+18960p^{4}$
$\displaystyle+48927p^{3}+250632p^{2}+330657p+297103){n\choose
10}+(990p^{5}+12540p^{4}+148830p^{3}$
$\displaystyle+497640p^{2}+1157145p+1245992){n\choose
11}+(13860p^{4}+97020p^{3}+1049895p^{2}+2961805p$
$\displaystyle+4727041){n\choose
12}+(135135p^{3}+1036035p^{2}+5900895p+15346045){n\choose 13}+(945945p^{2}$
$\displaystyle+7252245p+40500460){n\choose 14}+(4729725p+80615535){n\choose
15}+110810700{n\choose 16}$ $\displaystyle+91891800{n\choose
17}+34459425{n\choose 18},$
where $\gamma(n,p)\mathrel{\mathop{\mathchar
58\relax}}=\left(\sum_{k=2}^{n}{k-1\choose 5}\right)g_{(3,2,2,1,1)}(p)$.
It proves to be difficult to compute the exact value of $g_{(3,2,2,1,1)}(p)$
since it reduces to the explicit computation of the number of solutions to the
following system of polynomial equations over
$\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$
$\begin{split}&(x_{3}^{2}-x_{3})-x_{2}(x_{7}^{2}-x_{7})-x_{1}(x_{5}^{2}-x_{5})=0\\\
&(x_{4}^{2}-x_{4})-x_{2}(x_{8}^{2}-x_{8})-x_{1}(x_{6}^{2}-x_{6})=0\\\
&x_{3}x_{4}-x_{2}x_{7}x_{8}-x_{1}x_{5}x_{6}=0.\end{split}$
There are $3$ equations in $8$ variables. Computations done for primes $p\leq
19$ suggest that the number of solutions for the above equations is given by
$N_{p}=p^{5}+12p^{4}-20p^{3}+30p^{2}-10p$. Our computations then suggest that
$g_{(3,2,2,1,1)}(p)=p^{7}+24p^{6}-29p^{5}+21p^{4}-4p^{3}.$
We are unable to verify this claim for primes $p>19$, since it seems to be a
difficult problem in general to compute the number of points on varieties over
finite fields in a large number of variables. We note that in very specific
cases, exact formulae are known. For instance, in the case of hypersurfaces
(cf. [Dwo62]), or in the case when the system consists of diagonal equations
(cf. [Spa79]) such questions are studied. One may however bound the size of
$g_{(3,2,2,1,1)}(p)$, and it follows from Proposition 7.5 that
$g_{(3,2,2,1,1)}(p)\geq p^{4}$.
We describe the method of proof in more detail. It follows from results in
[Liu07, AKKM21] that $g_{\alpha}(p)$ can be computed for compositions $\alpha$
of one of the following types
* •
$\alpha=(\beta,1,\dots,1)$ and $n\geq 2$,
* •
$\alpha=(2,1\dots,1,\beta,1,\dots,1)$ and $n\geq 3$.
We refer to results in section 2 for further details. In section 3, we prove
some general results which allow us to compute $g_{\alpha}(p)$, for
compositions $\alpha$ of one of the following types
* •
$\alpha=(\beta,1,\dots,1,\gamma)$ is a composition of length $(n-1)\geq 3$
such that $\beta>2$ and $\gamma\geq\beta-1$,
* •
$\alpha$ is of the form $\alpha=(2,1,\dots,1,2,1,\dots,1,\beta)$, where
$\beta>1$,
* •
$\alpha$ is of the form $\alpha=(2,1,\dots,1,2,1,\dots,1,\beta,1)$, where
$\beta>1$,
* •
$\alpha$ is of the form $\alpha=(2,1,\dots,1,3,1,\dots,1,2)$.
We refer to a composition that does not fit into any of the above families as
an exceptional composition. In section 4 (resp. section 5), we compute
$g_{\alpha}(p)$ for all relevant compositions beginning with $2$ (resp. $3$).
In section 6, we compute $g_{\alpha}(p)$ for all relevant compositions
beginning with $4,5$ or $6$. The proof of Theorem 1.4 is provided in section
7.1. All known computations indicate that for any composition $\alpha$, the
function $g_{\alpha}(p)$ is a polynomial in $p$. The conditions for a matrix
$A$ give rise to polynomial conditions on the entries $a_{i,j}$, and
$g_{\alpha}(p)$ in many cases is the number of $\mathbb{F}_{p}$-points on a
scheme defined by integral polynomial equations. It is certainly of interest
to note that in all the examples considered, these schemes have polynomial
point count in the sense of [HRV08, p.616, ll. -6 to -2].
We now come to describing the general asymptotic results proved in this
manuscript. The analysis of the numbers $f_{n}(p^{e})$ can be translated into
properties of the associated subring zeta functions. For instance, Isham in
[Ish22b] proves lower bounds for $g_{n}(p^{e})$ and deduces that the subring
zeta function $\zeta_{\mathbb{Z}^{n},p}^{R}(s)$ diverges for all $s$ such that
$\operatorname{Re}s\leq c_{7}(n)$, where $c_{7}(n)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}_{0\leq d\leq n-1}\frac{d(n-1-d)}{(n-1+d)}$. This
comes as a consequence of proving lower bounds for $f_{n}(p^{e})$. Some
related results are also proven by Brackenhoff in [Bra09]. Liu attributes the
following conjecture regarding the asymptotics for the numbers $f_{n}(k)$ to
Bhargava (cf. [Liu07, p.298]). The conjecture was communicated to Liu via
personal communication, cf. _loc. cit._
###### Conjecture 1.5.
For $n$ odd, $f_{n}(k)=O(k^{\frac{n-1}{6}+\epsilon})$, and for $n$ even,
$f_{n}(k)=O(k^{\frac{n^{2}-2n}{6n-8}+\epsilon})$.
In particular, for a fixed prime number $p$ and a fixed value of $n$, the
conjecture predicts that
$f_{n}(p^{e})=\begin{cases}&O(p^{\frac{e(n-1)}{6}})\text{ if }n\text{ is
odd,}\\\ &O(p^{\frac{e(n^{2}-2n)}{6n-8}})\text{ if }n\text{ is odd.}\\\
\end{cases}$
Conjecture 1.5 motivates our results in section 7.2, where we investigate the
asymptotics for $g_{\alpha}(p)$ for certain natural families of compositions
$\alpha$. Let $n>1$ be a natural number and $t$ be in the range $1\leq t\leq
n-1$. Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n-1})$ be a composition
of length $(n-1)$ and for $k\in\mathbb{Z}_{\geq 1}$, set
$\alpha_{/k,t}\mathrel{\mathop{\mathchar
58\relax}}=(\alpha_{1},\alpha_{2},\dots
k\alpha_{t}\dots,\alpha_{n-2},\alpha_{n-1})$, i.e., the composition obtained
upon multiplying the $t$-th coordinate of $\alpha$ by $k$. Note that
$g_{\alpha_{/k,t}}(p)$ contributes to $g_{n}(p^{e_{k}})$, where
$e_{k}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i=1}^{n-1}\alpha_{i}+(k-1)\alpha_{t}$. Therefore, the
asymptotic growth of $g_{\alpha_{/k,t}}(p)$ as a function of $k$ provides
insight into that of $g_{n}(p^{e_{k}})$. The following result is proven via a
combination of techniques from the geometry of numbers and the combinatorial
analysis of large subring matrices.
###### Theorem 1.6.
Let $n>1$ be a natural number and
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n-1})$ a composition of length
$(n-1)$ such that $\alpha_{i}>1$ are integers for each
$i\in\\{1,2,\dots,n-1\\}$ and $1\leq t\leq n-1$. Then,
$g_{\alpha_{/k,t}}(p)=O(p^{\gamma k(n-2)})\text{ as }k\to\infty,$
where $\gamma=\max\\{\alpha_{1},\dots,\alpha_{t}\\}$. The implied constant
depends on $p$ and $n$, and not on $k$.
Although the above result is weaker than the prediction of Conjecture 1.5, the
authors expect that such arguments have the potential to lead to stronger
results in the future.
### Acknowledgment
The second author is supported by the CRM-Simons postdoctoral fellowship.
## 2\. Preliminaries
Let $n\geq 1$ be an integer and $M_{n}(\mathbb{Z})$ denote the ring of
$n\times n$-matrices with integer entries. By definition, a _lattice_ in
$\mathbb{Z}^{n}$ is a subgroup $L$ of finite index in $\mathbb{Z}^{n}$. This
index is denoted $[\mathbb{Z}^{n}\mathrel{\mathop{\mathchar 58\relax}}L]$. Set
$u_{1},\dots,u_{n}$ to be the standard basis of $\mathbb{Z}^{n}$, with
$u_{i}=(0,0,\dots,0,1,0,\dots)$, with $1$ in the $i$-th position and $0$s in
all other positions. Let $v_{1},\dots,v_{n}$ be a basis of $L$, i.e., a set of
vectors such that every element $v\in L$ is uniquely represented as an
integral linear combination $v=\sum_{i=1}^{n}b_{i}v_{i}$. Expressing
$v_{i}=\sum_{j=1}^{n}a_{i,j}u_{j}$, consider the integer matrix
$A=\left(a_{i,j}\right)\in M_{n}(\mathbb{Z})$. We may choose
$v_{1},\dots,v_{n}$ such that the associated matrix $A$ is in Hermite normal
form, i.e.,
$A=\begin{pmatrix}a_{1,1}&a_{1,2}&a_{1,3}&\cdots&a_{1,n-1}&a_{1,n}\\\
&a_{2,2}&a_{2,3}&\cdots&a_{2,n-1}&a_{2,n}\\\
&&a_{3,3}&\cdots&a_{3,n-1}&a_{3,n}\\\ &&&\ddots&\vdots&\vdots\\\
&&&&a_{n-1,n-1}&a_{n-1,n}\\\ &&&&&a_{n,n}\end{pmatrix}$
with $0\leq a_{i,j}<a_{i,i}$ for all tuples $(i,j)$ such that $1\leq i<j\leq
n$. The following result is used to reinterpret counting problems for subrings
of $\mathbb{Z}^{n}$ in terms of intgral matrices satisfying precribed
conditions.
###### Lemma 2.1.
There is a bijection between lattices $L\subset\mathbb{Z}^{n}$ of index $k>0$
and integer $n\times n$ matrices $A$ in Hermite normal form with determinant
$k$.
###### Proof.
The reader is referred to the proof of [Liu07, Proposition 2.1]. ∎
Thus, to a lattice $L$ we associate the matrix $A_{L}$ in Hermite normal form,
and to a matrix $A$ in Hermite normal form, we associate a unique lattice
$L_{A}$. Given two integral vectors $u=(u_{1},\dots,u_{n})^{t}$ and
$v=(v_{1},\dots,v_{n})^{t}$, denote the _composite_ by $u\circ
v\mathrel{\mathop{\mathchar 58\relax}}=(u_{1}v_{1},\dots,u_{n}v_{n})^{t}$. A
lattice $L$ is multiplicatively closed if $u\circ v\in L$ for all elements
$u,v\in L$. A _subring_ of $\mathbb{Z}^{n}$ shall in this paper be taken to
mean a multiplicatively closed lattice which contains the identity element
$\mathbf{1}\mathrel{\mathop{\mathchar 58\relax}}=(1,1,\dots,1)^{t}$. In
particular, the index of a subring is finite. Given a positive integer $k$,
let $f_{n}(k)$ be the number of subrings of $\mathbb{Z}^{n}$ with index equal
to $k$. Define the subring zeta function as follows
$\zeta_{\mathbb{Z}^{n}}^{R}(s)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{k=1}^{\infty}f_{n}(k)k^{-s}.$
The function $f_{n}(k)$ is multiplicative (cf. [Liu07, Proposition 2.7]),
i.e., given two coprime integers $k_{1}>0$ and $k_{2}>0$, we have that
$f_{n}(k_{1}k_{2})=f_{n}(k_{1})f_{n}(k_{2})$. The zeta function exhibits an
Euler product
$\zeta_{\mathbb{Z}^{n}}^{R}(s)=\prod_{p}\zeta_{\mathbb{Z}^{n},p}^{R}(s),$
where
$\zeta_{\mathbb{Z}^{n},p}^{R}(s)=\sum_{e=0}^{\infty}f_{n}(p^{e})p^{-es}$.
Thus, the study of the function $f_{n}(k)$ and the subring zeta function,
comes down to the determination of $f_{n}(p^{e})$, where $n>0,e\geq 0$ and $p$
is a prime number. More precisely, given $n$ and $e$, we would like to
determine $f_{n}(p^{e})$ as a function of $p$.
We recall Liu’s bijection between subrings of $\mathbb{Z}^{n}$ and integral
matrices in Hermite normal form.
###### Proposition 2.2.
Let $n,k>0$ be integers. The association $L\mapsto A_{L}$ gives a bijection
between subrings $L$ of $\mathbb{Z}^{n}$ of index $k$ and matrices
$A\in\operatorname{M}_{n}(\mathbb{Z})$ in Hermite normal form with $\det(A)=k$
and columns $v_{1},\dots,v_{n}$ such that
1. (1)
$\mathbf{1}$ is in the column span of $A$,
2. (2)
for all $i,j$ in the range $1\leq i,j\leq n$, $v_{i}\circ v_{j}$ is in the
column span of $A$.
###### Proof.
The reader is referred to [Liu07, Proposition 2.1, 2.2]. ∎
###### Definition 2.3.
A matrix $A$ satisfying the conditions of Proposition 2.2 is referred to as a
_subring_ matrix. The column span of $A$ is denoted $\operatorname{Col}(A)$.
Let $\alpha=(\alpha_{1},\dots,\alpha_{n-1})$ be a composition of length
$(n-1)$. Recall from the introduction that $g_{\alpha}(p)$ is the number of
irreducible subring matrices with diagonal
$(p^{\alpha_{1}},\dots,p^{\alpha_{n-1}},1)$. We recall some useful results
from [AKKM21, Liu07] which shall allow us to deduce the values of
$g_{\alpha}(p)$ for specific choices of $\alpha$.
###### Lemma 2.4.
Let $n\geq 2$ and $\alpha=(\beta,1,\dots,1)$ be a composition of length
$(n-1)$. Then, the following assertions hold.
1. (1)
If $\beta=2$, then $g_{\alpha}(p)=p^{n-2}$.
2. (2)
If $\beta\geq 3$, then $g_{\alpha}(p)=(n-1)p^{n-2}$.
###### Proof.
The above result is [AKKM21, Lemma 3.5]. ∎
###### Lemma 2.5.
Let $n\geq 3$ and let $\alpha=(2,1\dots,1,\beta,1,\dots,1)$ be a composition
of length $(n-1)$ where $\beta$ is in the $k$-th position. Let $r=n-1-k$,
i.e., the number of $1$s after $\beta$. Then, the following assertions hold.
1. (1)
If $\beta=2$, then,
$g_{\alpha}(p)=p^{n-3+r}+(r+1)p^{n-3}(p-1).$
2. (2)
If $\beta\geq 3$, then
$g_{\alpha}(p)=(r+1)\left(p^{n-3+r}+p^{n-3}(p-1)\right).$
###### Proof.
The above result is [AKKM21, Lemma 3.6]. ∎
## 3\. General results for computing the value of $g_{\alpha}(p)$
In this section, we prove a number of general results computing
$g_{\alpha}(p)$ for various choices of $\alpha$. Given an irreducible subring
with associated matrix $A\in M_{n}(\mathbb{Z})$, recall that
$v_{1},\dots,v_{n}$ shall denote the columns of $A$, where $v_{i}$ is the
$i$-th column.
###### Lemma 3.1.
Let $n\geq 4$, $\alpha=(\beta,1,\dots,1,\gamma)$ be a composition of length
$(n-1)$ such that $\beta>2$ and $\gamma\geq\beta-1$. Then, we have
$g_{\alpha}(p)=p^{n-3+\lfloor\frac{\beta}{2}\rfloor}+(n-3)p^{n-2}.$
###### Proof.
Consider a matrix $A\in M_{n}(\mathbb{Z})$ in Hermite normal form
$A=\begin{pmatrix}p^{\beta}&pa_{1}&pa_{2}&\cdots&pa_{n-2}&1\\\
&p&0&\cdots&0&1\\\ &&p&\cdots&0&1\\\ &&&\ddots&\vdots&\vdots\\\
&&&&p^{\gamma}&1\\\ &&&&&1,\end{pmatrix}$
where the entries $a_{1},\dots,a_{n-2}$ satisfy $0\leq a_{i}\leq
p^{\beta-1}-1$. First we write down conditions for
$v_{j}^{2}\in\operatorname{Col}(A)$. Clearly, this condition is automatically
satisfied for $j=1$ and $j=n$. Consider the values of $j$ that lie in the
range $2\leq j\leq n-1$. We may write $j=i+1$, where $i$ lies in the range
$1\leq i\leq n-2$. First, we consider the case when $i\leq n-3$. Observe that
$v_{i+1}^{2}=(a_{i}^{2}p^{2},0,\dots,0,p^{2},0,\dots,0)^{t}.$
Note that $v_{i+1}^{2}$ is contained in $\operatorname{Col}(A)$ if and only if
$v_{i+1}^{2}-pv_{i+1}$ is contained in $\operatorname{Col}(A)$. We find that
$v_{i+1}^{2}-pv_{i+1}=\left((a_{i}^{2}-a_{i})p^{2},0,\dots,0\right)^{t}.$
Therefore, $v_{i+1}^{2}$ is contained in $\operatorname{Col}(A)$ if and only
if
(3.1) $a_{i}^{2}-a_{i}\equiv 0\mod{p^{\beta-2}}.$
We find that $v_{n-1}^{2}=(a_{n-2}^{2}p^{2},0,\dots,0,p^{2\gamma},0)^{t}$ is
in $\operatorname{Col}(A)$ if and only if
$v_{n-1}^{2}-p^{\gamma}v_{n-1}=\left(a_{n-2}^{2}p^{2}-a_{n-2}p^{\gamma+1},0,\dots,0\right)^{t}$
is contained in $\operatorname{Col}(A)$. Therefore, we deduce that
$v_{n-1}^{2}$ is in $\operatorname{Col}(A)$ if and only if
(3.2) $a_{n-2}^{2}p^{2}-a_{n-2}p^{\gamma+1}\equiv 0\mod{p^{\beta}}.$
It is assumed that $\gamma\geq\beta-1$ and hence the above condition is
equivalent to
$a_{n-2}^{2}\equiv 0\mod{p^{\beta-2}},$
i.e.,
(3.3) $a_{n-2}\equiv 0\mod{p^{\lceil\frac{\beta-2}{2}\rceil}}.$
For $1\leq i<j\leq n-2$, we find that $v_{i+1}\circ v_{j+1}$ is contained in
$\operatorname{Col}(A)$ if and only if
(3.4) $a_{i}a_{j}\equiv 0\mod{p^{\beta-2}}.$
Putting it all together, we find that $A$ is a subring matrix if and only if
the following conditions are satisfied
1. (1)
$a_{i}(a_{i}-1)\equiv 0\mod{p^{\beta-2}}$ for all $i$ in the range $1\leq
i\leq n-3$,
2. (2)
$a_{n-2}\equiv 0\mod{p^{\lceil\beta/2\rceil-1}}$,
3. (3)
$a_{i}a_{j}\equiv 0\mod{p^{\beta-2}}$ for all $i,j$ in the range $1\leq
i<j\leq n-2$.
From condition (1) above, we find that at most one of $a_{i}\equiv
1\mod{p^{\beta-2}}$ for $i=1,\dots,n-3$. We thus are led to the following
cases.
Case 1: First, we consider the case when $a_{i}\equiv 0\mod{p^{\beta-2}}$ for
all $i$ in the range $1\leq i\leq n-3$. Since $a_{i}$ is in the range $0\leq
a_{i}<p^{\beta-1}$, we find that there are $p$ choices for each $a_{i}$ and
$p^{\beta-\lceil\beta/2\rceil}=p^{\lfloor\beta/2\rfloor}$ choices for
$a_{n-2}$. In total, we find that there are $p^{n-3+\lfloor\beta/2\rfloor}$
matrices in this case.
Case 2: Consider the case when one of the $a_{i}$ satisfies $a_{i}\equiv
1\mod{p^{\beta-2}}$. Then, from (3) we find that
$a_{n-2}\equiv a_{i}a_{n-2}\equiv 0\mod{p^{\beta-2}}.$
Thus, for each index $i$ in the range $1\leq i\leq n-2$, we have $p^{n-2}$
choices for which $a_{i}\equiv 1\mod{p^{\beta-2}}$. There are $(n-3)$ values
taken by $i$ for which $a_{i}\equiv 1\mod{p^{\beta-2}}$. Thus, in this case,
there are $(n-3)p^{n-2}$ choices.
Putting together the calculations from cases 1 and 2, we find that
$g_{\alpha}(p)=p^{n-3+\lfloor\beta/2\rfloor}+(n-3)p^{n-2}$. ∎
###### Lemma 3.2.
Let $\alpha$ be of the form $\alpha=(2,1,\dots,1,2,1,\dots,1,\beta)$, where
$\beta>1$ and the second $2$ occurs at position $k\geq 2$, then
$g_{\alpha}(p)=p^{n-3+r}+p^{n-3}(p-1)r,$
where $r=n-1-k$.
###### Proof.
Let $A$ be an integral matrix in Hermite normal form
$A=\begin{pmatrix}p^{2}&pa_{1}&pa_{2}&\cdots&pa_{k-1}&\cdots&\cdots&pa_{n-2}&1\\\
&p&0&\cdots&0&\cdots&\cdots&0&1\\\ &&p&\cdots&0&\cdots&\cdots&0&1\\\
&&&\ddots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
&&&&p^{2}&pb_{1}&\cdots&pb_{r}&1\\\ &&&&&p&\cdots&0&1\\\
&&&&&&\ddots&\vdots&1\\\ &&&&&&&p^{\beta}&1\\\ &&&&&&&&1.\end{pmatrix}$
Since the first entry of $\alpha$ is equal to $2$, it is easy to see that
$v_{i}^{2}$ is in $\operatorname{Col}(A)$ for $i=1,\dots,k-1$. We find that
$v_{k}^{2}$ is in $\operatorname{Col}(A)$ if and only if
$v_{k}^{2}-p^{2}v_{k}$ is in $\operatorname{Col}(A)$. Since $p^{2}$ divides
$a_{k-1}^{2}p^{2}-p^{3}a_{k-1}$, this condition is seen to be satisfied. For
$i=1,\dots,r$, we observe that
$v_{k+i}^{2}=(a_{k-1+i}^{2}p^{2},\cdots,b_{i}^{2}p^{2},\cdots,p^{2})^{t}\text{
is in }\operatorname{Col}(A)$
if and only if
$v_{k+i}^{2}-pv_{i+k}=\left((a_{k-1+i}^{2}-a_{k-1+i})p^{2},\dots,(b_{i}^{2}-b_{i})p^{2},0,\dots,0\right)^{t}$
is in $\operatorname{Col}(A)$. Subtracting $(b_{i}^{2}-b_{i})v_{k}$ from the
above, we get
$v_{k+i}^{2}-pv_{i+k}-(b_{i}^{2}-b_{i})v_{k}=\left((a_{k-1+i}^{2}-a_{k-1+i})p^{2}-(b_{i}^{2}-b_{i})a_{k-1}p,0,\dots,0\right)^{t},$
which is in $\operatorname{Col}(A)$ if and only if first entry is divisible by
$p^{2}$. We have thus shown that $v_{k+i}^{2}$ is in $\operatorname{Col}(A)$
if and only if
(3.5) $(b_{i}^{2}-b_{i})a_{k-1}\equiv 0\mod{p}.$
Now,
$v_{n-1}^{2}=\left(a_{n-2}^{2}p^{2},\dots,b_{r}^{2}p^{2},0,\dots,0,p^{2\beta},0,\dots\right)^{t}$
is in $\operatorname{Col}(A)$ if and only if $v_{n-1}^{2}-p^{\beta}v_{n-1}$ is
in $\operatorname{Col}(A)$. Note that therefore, $v_{n-1}^{2}$ is in
$\operatorname{Col}(A)$ if and only if $b_{r}a_{k-1}\equiv 0\mod{p}$.
It is clear that for all $1\leq i\leq k$ and all values of $j$, $v_{i}\circ
v_{j}\in\operatorname{Col}(A)$. Now consider $v_{k+i}\circ v_{k+j}$ for $1\leq
i<j\leq r$, it is equal to
$\left(a_{k-1+i}a_{k-1+j}p^{2},0,\dots,0,b_{i}b_{j}p^{2},0,\dots,0\right)^{t}.$
We find that $v_{k+i}\circ v_{k+j}$ is contained in $\operatorname{Col}(A)$ if
and only if
$v_{k+i}\circ v_{k+j}-b_{i}b_{j}v_{k-1}\in\operatorname{Col}(A).$
Therefore, $v_{k+i}\circ v_{k+j}$ is contained in $\operatorname{Col}(A)$ if
and only if $b_{i}b_{j}a_{k-1}\equiv 0\mod{p}$. Therefore, $A$ is a subring
matrix if and only if
1. (1)
$(b_{i}^{2}-b_{i})a_{k-1}\equiv 0\mod{p}$ for $i=1,\dots,r-1$,
2. (2)
$b_{r}a_{k-1}\equiv 0\mod{p}$,
3. (3)
$b_{i}b_{j}a_{k-1}\equiv 0\mod{p}$, for all $i,j$ such that $1\leq i<j\leq r$.
We consider two cases.
Case 1: Consider the case when $a_{k-1}=0$. In this case, the total number of
choices is the total number of choices of
$(a_{1},\dots,a_{k-2},a_{k},\dots,a_{n-2})$ and $(b_{1},\dots,b_{r})$. Thus,
the total number of choices are $p^{n-3+r}$.
Case 2: Consider the other case, i.e., when $a_{k-1}\neq 0$. Since
$a_{k-1}<p$, we find that $p\nmid a_{k-1}$. Therefore, the conditions are as
follows
1. (1)
$b_{i}^{2}-b_{i}\equiv 0\mod{p}$,
2. (2)
$b_{r}\equiv 0\mod{p}$,
3. (3)
$b_{i}b_{j}\equiv 0\mod{p}$, $1\leq i<j\leq r$.
Note that all elements $b_{i}$ satisfy the bounds $0\leq b_{i}<p$. Therefore,
at most one of the $b_{i}$ satisfies $b_{i}=1$. Subdivide into two cases,
first consider the case when all the $b_{i}$ are equal to $0$ and then
consider the case when at most one of the $b_{i}$ is equal to $1$. The total
number of choices is $p^{n-3}(p-1)r$. Upon adding up all the conclusions from
the case decomposition above, we find that
$g_{\alpha}(p)=p^{n-3+r}+p^{n-3}(p-1)r.$
∎
###### Lemma 3.3.
Let $\alpha$ be of the form $\alpha=(2,1,\dots,1,2,1,\dots,1,\beta,1)$, where
$\beta>1$, the second $2$ occurs at position $k\geq 2$ and set
$r\mathrel{\mathop{\mathchar 58\relax}}=n-1-k$. Then, the following assertions
hold.
1. (1)
If $\beta=2$, then we find that
$g_{\alpha}(p)=2p^{n-3+r}+p^{n-5+r}(p-2)+2rp^{n-3}(p-1)+p^{n-4}(p-1)(p-2)(p+r-2).$
2. (2)
If $\beta\geq 3$, then,
$g_{\alpha}(p)=2p^{n-3+r}+2p^{n-5+r}(p-1)+2rp^{n-3}(p-1)+2p^{n-4}(p-1)^{2}(p+r-2).$
###### Proof.
Let $A$ be an integral matrix in Hermite normal form
$A=\begin{pmatrix}p^{2}&pa_{1}&pa_{2}&\cdots&pa_{k-1}&\cdots&\cdots&\cdots&pa_{n-2}&1\\\
&p&0&\cdots&0&\cdots&\cdots&\cdots&0&1\\\
&&p&\cdots&0&\cdots&\cdots&\cdots&0&1\\\
&&&\ddots&\vdots&\vdots&\cdots&\cdots&\vdots&\vdots\\\
&&&&p^{2}&pb_{1}&\cdots&\cdots&pb_{r}&1\\\ &&&&&p&\cdots&\cdots&0&1\\\
&&&&&&\ddots&\cdots&\vdots&1\\\ &&&&&&&p^{\beta}&pc_{1}&1\\\ &&&&&&&&p&1\\\
&&&&&&&&&1.\end{pmatrix}$
Since the first entry is $p^{2}$, it is clear that $v_{i}^{2}$ is contained in
$\operatorname{Col}(A)$ for $i=1,2,\dots,k$. For $i=1,2,\dots,r-2$, we find
that $v_{k+i}^{2}$ is contained in $\operatorname{Col}(A)$ if and only if
$v_{k+i}^{2}-pv_{k+i}$ is contained in $\operatorname{Col}(A)$. So,
$(a_{k-1+i}^{2}p^{2}-a_{k-1}p^{2},\dots,b_{i}^{2}p^{2}-b_{i}p^{2},\dots)^{t}$
should be in $\operatorname{Col}(A)$. It is clear that this is true if and
only if
(3.6) $a_{k-1}(b_{i}^{2}-b_{i})\equiv 0\mod{p}$
Similarly we see that $v_{n-2}^{2}\in\operatorname{Col}(A)$ if and only if
$v_{n-2}^{2}-p^{\beta}v_{n-2}\in\operatorname{Col}(A)$. Now,
$(a_{n-3}^{2}p^{2}-a_{n-3}p^{\beta+1},\dots,b_{r-1}^{2}p^{2}-b_{r-1}p^{\beta+1},\dots)^{t}$
should be in $\operatorname{Col}(A)$. It is again clear that this is true if
and only if
(3.7) $a_{k-1}b_{r-1}\equiv 0\mod p$
as $\beta>1$. Again, $v_{n-1}^{2}\in\operatorname{Col}(A)$ if and only if
$v_{n-1}^{2}-pv_{n-1}\in\operatorname{Col}(A)$, so
$(a_{n-2}^{2}p^{2}-a_{n-2}p^{2},\dots,(b_{r}^{2}-b_{r})p^{2},\dots,(c_{1}^{2}-c_{1})^{2},0,0)^{t}$
should be in $\operatorname{Col}(A)$. Then we have $c_{1}^{2}-c_{1}\equiv
0\mod p^{\beta-2}$, and
(3.8) $b_{r-1}(c_{1}^{2}-c_{1})\equiv 0\mod p^{\beta-1}$ (3.9)
$a_{k-1}\left(b_{r}^{2}-b_{r}-\frac{b_{r-1}(c_{1}^{2}-c_{1})}{p^{\beta-1}}\right)+a_{n-3}\frac{c_{1}^{2}-c_{1}}{p^{\beta-2}}\equiv
0\mod p$
The requirement that $v_{i}v_{j}$ belongs to $\operatorname{Col}(A)$ for
$i\neq j$ translates the following conditions on entries of the matrix (for
all $1\leq i<j\leq r$, and $\\{i,j\\}\neq\\{r-1,r\\}$)
$a_{k-1}b_{i}b_{j}\equiv 0\mod p$
First, we suppose that $\beta\geq 3$, then we have conditions,
1. (1)
$a_{k-1}(b_{i}^{2}-b_{i})\equiv 0\mod{p}$, $i=1,\dots,r-2$,
2. (2)
$b_{r-1}a_{k-1}\equiv 0\mod p$,
3. (3)
$c_{1}^{2}-c_{1}\equiv 0\mod p^{\beta-2}$,
4. (4)
$b_{r-1}(c_{1}^{2}-c_{1})\equiv 0\mod p^{\beta-1}$,
5. (5)
$a_{k-1}\left(b_{r}^{2}-b_{r}-\frac{b_{r-1}(c_{1}^{2}-c_{1})}{p^{\beta-1}}\right)+a_{n-3}\frac{c_{1}^{2}-c_{1}}{p^{\beta-2}}\equiv
0\mod p$,
6. (6)
$a_{k-1}b_{i}b_{j}\equiv 0\mod p$, $1\leq i<j\leq r$, and
$\\{i,j\\}\neq\\{r-1,r\\}$.
We consider two cases.
Case 1 : First, we consider the case when $a_{k-1}=0$. Then, the above
equations reduce to the following:
1. (1)
$c_{1}^{2}-c_{1}\equiv 0\mod p^{\beta-2}$,
2. (2)
$b_{r-1}(c_{1}^{2}-c_{1})\equiv 0\mod p^{\beta-1}$,
3. (3)
$\frac{a_{n-3}(c_{1}^{2}-c_{1})}{p^{\beta-2}}\equiv 0\mod p$.
If $c_{1}\in\\{0,1\\}$, then the number of such matrices is $2p^{n-3+r}$. On
the other hand, if $c_{1}\notin\\{0,1\\}$, then, there are $2p-2$ choices for
$c_{1}$. Since $b_{r-1}=0,a_{n-3}=0$, we deduce that there are
$2(p-1)p^{n+r-5}$ more matrices.
Case 2 : Suppose $a_{k-1}\neq 0$, then equations reduce to
1. (1)
$b_{i}^{2}-b_{i}\equiv 0\mod p$, $i=1,\dots,r-2$,
2. (2)
$b_{r-1}=0$,
3. (3)
$c_{1}^{2}-c_{1}\equiv 0\mod p^{\beta-2}$,
4. (4)
$a_{k-1}\left(b_{r}^{2}-b_{r}\right)+a_{n-3}\frac{c_{1}^{2}-c_{1}}{p^{\beta-2}}\equiv
0\mod p$,
5. (5)
$b_{i}b_{j}\equiv 0\mod p$, $1\leq i<j\leq r$, and $\\{i,j\\}\neq\\{r-1,r\\}$
We divide into two sub-cases.
* •
Case 2A : Suppose that $c_{1}\in\\{0,1\\}$, then we get that
$b_{r}\in\\{0,1\\}$ and as in Lemma 3.2, we get that at most one of the
$b_{i}$ satisfies $b_{i}=1$. Subdivide into two cases, first consider the case
when all the $b_{i}$ are equal to $0$ and then consider the case when at most
one of the $b_{i}$ is equal to $1$. Therefore, the total number of choices is
$2rp^{n-3}(p-1)$.
* •
Case 2B : Consider the other subcase, i.e., $c_{1}\notin\\{0,1\\}$. Then,
there is a unique value of $a_{n-3}$ for each value of $b_{r},a_{k-1},c_{1}$,
so we get $2p^{n-3}(p-1)^{2}(p+r-2)$ matrices
Putting it all together, the result is proven. The case $\beta=2$ is similar,
and the number of matrices change only in Case 1, with $c_{1}\notin\\{0,1\\}$
and Case 2B. ∎
###### Lemma 3.4.
Let $\alpha$ be of the form $\alpha=(2,1,\dots,1,3,1,\dots,1,2)$, where $3$
occurs at position $k\geq 2$. Then, we have that
$g_{\alpha}(p)=rp^{n-3+r}+p^{n-3}(p-1)(p+r-1),$
where $r\mathrel{\mathop{\mathchar 58\relax}}=n-1-k$.
###### Proof.
Let $A$ be an integral matrix in Hermite normal form
$A=\begin{pmatrix}p^{2}&pa_{1}&pa_{2}&\cdots&pa_{k-1}&\cdots&\cdots&pa_{n-2}&1\\\
&p&0&\cdots&0&\cdots&\cdots&0&1\\\ &&p&\cdots&0&\cdots&\cdots&0&1\\\
&&&\ddots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
&&&&p^{3}&pb_{1}&\cdots&pb_{r}&1\\\ &&&&&p&\cdots&0&1\\\
&&&&&&\ddots&\vdots&1\\\ &&&&&&&p^{2}&1\\\ &&&&&&&&1.\end{pmatrix}$
Since the first diagonal entry is $p^{2}$, we see that $v_{i}^{2}$ is
contained in $\operatorname{Col}(A)$ for all $i=1,\dots,k$. Note that
$v_{k+i}^{2}\in\operatorname{Col}(A)$ if and only if
$v_{k+i}^{2}-pv_{k+i}\in\operatorname{Col}(A)$ for $i=1,2,\dots,r-1$. Hence,
we find that
$\left((a_{k-1+i}^{2}-a_{k-1+i})p^{2},0,\dots,0,(b_{i}^{2}-b_{i})p^{2},0,\dots,0\right)^{t}$
should be in $\operatorname{Col}(A)$. We deduce that that this is the case if
and only if the following conditions are satisfied
(3.10) $\begin{split}&b_{i}^{2}-b_{i}\equiv 0\mod p,\\\
&a_{k-1}(b_{i}^{2}-b_{i})\equiv 0\mod p^{2}.\end{split}$
It is also required that $v_{n-1}^{2}$ is contained in
$\operatorname{Col}(A)$, which is the case if and only if
$v_{n-1}^{2}-p^{2}v_{n-1}$ is contained in $\operatorname{Col}(A)$. Therefore,
we find that
$\left((a_{n-2}^{2}-a_{n-2}p)p^{2},0,\dots,0,(b_{r}^{2}-b_{r}p)p^{2},0,\dots,0\right)^{t}$
should be in $\operatorname{Col}(A)$. We get that $b_{r}^{2}-b_{r}p\equiv
0\mod p$ and $a_{k-1}(b_{r}^{2}-b_{r}p)\equiv 0\mod p^{2}$, therefore, we find
that $p\mid b_{r}$. Using similar arguments we deduce that
$v_{k+i}v_{k+j}\in\operatorname{Col}(A)$ for $1\leq i<j\leq r$ if and only if
the following conditions are satisfied
(3.11) $\begin{split}&b_{i}b_{j}\equiv 0\mod p,\\\ &a_{k-1}b_{i}b_{j}\equiv
0\mod p^{2}.\end{split}$
It is clear that $v_{i}v_{j}$ is contained in $\operatorname{Col}(A)$ for
$1\leq i<j\leq k$. Therefore, we have following conditions on the entries of
$A$
1. (1)
$b_{i}^{2}-b_{i}\equiv 0\mod p$ and $a_{k-1}(b-i^{2}-b_{i})\equiv 0\mod p^{2}$
for $i=1,\dots,r-1$,
2. (2)
$b_{r}=b_{r}^{\prime}p$, where $0\leq b_{r}^{\prime}\leq p-1$,
3. (3)
$b_{i}b_{j}\equiv 0\mod p$ and $a_{k-1}b_{i}b_{j}\equiv 0\mod p^{2}$ for all
values $1\leq i<j\leq r$.
We consider two cases as follows.
Case 1 : First, we consider the case when $a_{k-1}=0$. In this case, the
conditions on $A$ reduce to the following
1. (1)
$b_{i}^{2}-b_{i}\equiv 0\mod p$ for all $i=1,\dots,r-1$,
2. (2)
$b_{r}=b_{r}^{\prime}p$, where $0\leq b_{r}^{\prime}\leq p-1$,
3. (3)
$b_{i}b_{j}\equiv 0\mod p$ for $1\leq i<j\leq r$.
As in Lemma 3.3, either all the values of $b_{i}$ for $1\leq i\leq r-1$ are
$0\mod p$, or at most one of these values is $1\mod p$. Further dividing into
cases, it is easy to see that the number of matrices is $rp^{n-3+r}$.
Case 2 : Next, we consider the other case, namely assume that $a_{k-1}\neq 0$.
The conditions on $A$ then reduce to the following
1. (1)
$b_{i}^{2}-b_{i}\equiv 0\mod p^{2}$ for all $i=1,\dots,r-1$,
2. (2)
$b_{r}=b_{r}^{\prime}p$, where $0\leq b_{r}^{\prime}\leq p-1$,
3. (3)
$b_{i}b_{j}\equiv 0\mod p^{2}$ for all $(i,j)$ satisfying $1\leq i<j\leq r$.
Consequently, it follows that either $b_{i}=0$ for all $i=1,\dots,r-1$, or at
most one of them is $1$. Therefore, we find that the number of matrices in
this case is equal to $p^{n-3}(p-1)(p+r-1)$ matrices. Adding up our
conclusions, we prove the assertion of the lemma. ∎
## 4\. Calculating the values of $g_{\alpha}(p)$ for compositions beginning
with $2$
### 4.1. Compositions of length $4$
We consider the compositions $\alpha$ of length $4$. In all, there are $15$ of
them, listed below
$(2,5,1,1)$ | $(2,1,2,4)$
---|---
$(2,1,5,1)$ | $(2,3,3,1)$
$(2,1,1,5)$ | $(2,3,1,3)$
$(2,4,2,1)$ | $(2,1,3,3)$
$(2,4,1,2)$ | $(2,3,2,2)$
$(2,2,4,1)$ | $(2,2,3,2)$
$(2,2,1,4)$ | $(2,2,2,3)$.
$(2,1,4,2)$ |
It follows directly from Lemmas 2.5, 3.2 and 3.3 that
$g_{(2,5,1,1)}(p)$ | $3p^{4}+3p^{3}-3p^{2}$
---|---
$g_{(2,1,5,1)}(p)$ | $4p^{3}-2p^{2}$
$g_{(2,1,1,5)}(p)$ | $p^{3}$
$g_{(2,2,4,1)}(p)$ | $4p^{4}+2p^{3}-4p^{2}$
$g_{(2,2,1,4)}(p)$ | $p^{4}+2p^{3}-2p^{2}$
$g_{(2,1,2,4)}(p)$ | $2p^{3}-p^{2}$.
We compute $g_{\alpha}(p)$ for the rest of the compositions. We note that many
arguments are similar, and we summarize the arguments that tend to repeat.
###### Lemma 4.1.
With respect to notation above, we have that $g_{(2,1,4,2)}(p)=3p^{4}-2p^{3}$.
###### Proof.
Let $A$ be the matrix,
$\begin{pmatrix}p^{2}&a_{1}p&a_{2}p&a_{3}p&1\\\ 0&p&0&0&1\\\
0&0&p^{4}&b_{1}p&1\\\ 0&0&0&p^{2}&1\\\ 0&0&0&0&1\end{pmatrix}$
where $0\leq a_{1}\leq p-1$ and $0\leq b_{1}\leq p^{3}-1$. We arrive at
conditions for $A$ to be a subring matrix. It is easy to see that
* •
$v_{2}^{2}$,$v_{3}^{2}$,$v_{2}v_{3}$, $v_{2}v_{4}$, $v_{3}v_{4}$ are in
$\operatorname{Col}(A)$.
* •
We find that $v_{4}^{2}$ is in $\operatorname{Col}(A)$ if and only if
$v_{4}^{2}-p^{2}v_{4}$ is contained in $\operatorname{Col}(A)$. In other
words,
$\begin{split}&(a_{3}^{2}p^{2},0,b_{1}^{2}p^{2},p^{4},0)^{t}-p^{2}(a_{3}p,0,b_{1}p,p^{2},0)^{t}\\\
=&\left((a_{3}^{2}p^{2}-a_{3}p^{3}),0,(b_{1}^{2}p^{2}-b_{1}p^{3}),0,0\right)^{t}\end{split}$
is in $\operatorname{Col}(A)$. The expression
$(b_{1}^{2}p^{2}-b_{1}p^{3})=b_{1}(b_{1}-p)p^{2}$ must be divisible by $p^{4}$
and moreover, we find that $v_{4}^{2}-p^{2}v_{4}$ is contained in
$\operatorname{Col}(A)$ if and only if
$v_{4}^{2}-p^{2}v_{4}-\frac{b_{1}(b_{1}-p)}{p^{2}}v_{3}$ is in
$\operatorname{Col}(A)$. This holds if and only if in addition, we have that
$b_{1}(b_{1}-p)a_{2}\equiv 0\mod p^{3}$.
We deduce from above that the necessary conditions are as follows
1. (1)
$b_{1}(b_{1}-p)\equiv 0\mod p^{2}$,
2. (2)
$b_{1}(b_{1}-p)a_{2}\equiv 0\mod p^{3}$.
From equation (1), we get that $b_{1}=b_{1}^{\prime}p$, where $0\leq
b_{1}^{\prime}\leq p^{2}-1$. Equation (2) asserts that
$b_{1}^{\prime}(b_{1}^{\prime}-1)a_{2}\equiv 0\mod p$. Consider the following
case decomposition.
* •
Case 1 : Assume that $a_{2}=0$. In this case, the number of such matrices is
$p^{4}$.
* •
Case 2 : Consider the other case, i.e., $a_{2}\neq 0$. Then, we have
$b_{1}^{\prime}(b_{1}^{\prime}-1)\equiv 0\mod p$. Hence, the total number of
such matrices is easily seen to be $2p^{3}(p-1)$.
We conclude from the above that
$g_{(2,1,4,2)}(p)=p^{4}+2p^{3}(p-1)=3p^{4}-2p^{3}$. ∎
###### Lemma 4.2.
With respect to notation above, we find that
$g_{(2,4,1,2)}(p)=p^{5}+3p^{4}-p^{3}-p^{2}$.
###### Proof.
Let $A$ be the matrix
$\begin{pmatrix}p^{2}&a_{1}p&a_{2}p&a_{3}p&1\\\ 0&p^{4}&b_{1}p&b_{2}p&1\\\
0&0&p&0&1\\\ 0&0&0&p^{2}&1\\\ 0&0&0&0&1\end{pmatrix}$
where $0\leq a_{i}\leq p-1$ and $0\leq b_{j}\leq p^{3}-1$. We obtain
conditions for $A$ to be a subring matrix. By the same arguments as in Lemma
4.1, we find that
* •
$v_{2}^{2}$, $v_{2}v_{3}$, $v_{2}v_{4}$ are in $\operatorname{Col}(A)$.
* •
We find that $v_{3}^{2}$ is in $\operatorname{Col}(A)$ if and only if
$b_{1}(b_{1}-1)\equiv 0\mod p^{2}$ and $b_{1}(b_{1}-1)a_{1}\equiv 0\mod
p^{3}$.
* •
We find that $v_{4}^{2}$ is in $\operatorname{Col}(A)$ if and only if
$b_{2}=b_{2}^{\prime}p$, $0\leq b_{2}^{\prime}\leq p^{2}-1$ and
$b_{2}^{\prime}(b_{2}^{\prime}-1)a_{1}\equiv 0\mod p$.
* •
We find that $v_{3}v_{4}$ is in $\operatorname{Col}(A)$ if and only if
$b_{1}b_{2}^{\prime}\equiv 0\mod p$ and $a_{1}b_{1}b_{2}^{\prime}\equiv 0\mod
p^{2}$.
Hence, $A$ is a subring matrix if and only if
1. (1)
$b_{1}(b_{1}-1)\equiv 0\mod p^{2}$,
2. (2)
$a_{1}b_{1}(b_{1}-1)\equiv 0\mod p^{3}$,
3. (3)
$b_{2}=b_{2}^{\prime}p$ for $0\leq b_{2}^{\prime}\leq p^{2}-1$,
4. (4)
$a_{1}b_{2}^{\prime}(b_{2}^{\prime}-1)\equiv 0\mod p$,
5. (5)
$b_{1}b_{2}^{\prime}\equiv 0\mod p$,
6. (6)
$a_{1}b_{1}b_{2}^{\prime}\equiv 0\mod p^{2}$.
In order to compute the total number of matrices satisfying all of the above
conditions, we consider the following cases.
* •
Case 1 : Assume that $a_{1}=0$. If $b_{1}\equiv 0\mod p^{2}$ then number of
such matrices is $p^{5}$ otherwise $b_{1}\equiv 1\mod p^{2}$, number of such
matrices is $p^{4}$.
* •
Case 2 : Consider the case when $a_{1}\neq 0$. We get $b_{1}=0$ or $b_{1}=1$.
If $b_{1}=0$, number of such matrices is $2p^{3}(p-1)$. Otherwise $b_{1}=1$
and number of such matrices is $p^{2}(p-1)$ (as $b_{2}=0$ in this case).
Therefore, we find that
$g_{(2,4,1,2)}(p)=p^{5}+p^{4}+2p^{3}(p-1)+p^{2}(p-1)=p^{5}+3p^{4}-p^{3}-p^{2}$.
∎
###### Lemma 4.3.
We have that
$\begin{split}&g_{(2,3,1,3)}(p)=3p^{4}-p^{2},\\\ &g_{(2,1,3,3)}(p)=p^{4},\\\
&g_{(2,2,3,2)}=p^{5}+p^{4}-p^{3}.\end{split}$
###### Proof.
The proof is similar to Lemma 4.2, and we omit it. ∎
###### Lemma 4.4.
We have that $g_{(2,3,2,2)}(p)=p^{5}+4p^{4}-9p^{3}+4p$.
###### Proof.
Let $A$ be the matrix,
$\begin{pmatrix}p^{2}&a_{1}p&a_{2}p&a_{3}p&1\\\ 0&p^{3}&b_{1}p&b_{2}p&1\\\
0&0&p^{2}&c_{1}p&1\\\ 0&0&0&p^{2}&1\\\ 0&0&0&0&1\end{pmatrix}$
where $0\leq a_{i}\leq p-1$, $0\leq b_{j}\leq p^{2}-1$ and $0\leq c_{1}\leq
p-1$. Below are the conditions for $A$ to be a subring matrix.
* •
It is easy to see that $v_{2}^{2}$ is in $\operatorname{Col}(A)$.
* •
We find that $v_{3}^{2}$ is contained in $\operatorname{Col}(A)$ if and only
if $v_{3}^{2}-p^{2}v_{3}$ is contained in $\operatorname{Col}(A)$. Hence, we
find that $v_{3}^{2}\in\operatorname{Col}(A)$ if and only if
$b_{1}=b_{1}^{\prime}p$, where $0\leq b_{1}^{\prime}\leq p-1$.
* •
Similar reasoning shows that $v_{4}^{2}\in\operatorname{Col}(A)$ if and only
if $b_{2}^{2}-(c_{1}^{2}-c_{1})b_{1}^{\prime}\equiv 0\mod p$ and
$\left(\frac{b_{2}^{2}-(c_{1}^{2}-c_{1})b_{1}^{\prime}}{p}-b_{2}\right)a_{1}+c_{1}(c_{1}-1)a_{2}\equiv
0\mod p$.
* •
We next consider the requirement that $v_{i}v_{j}$ is in
$\operatorname{Col}(A)$ for distinct values of $i$ and $j$. It is easy to see
that $v_{1}v_{2},v_{1}v_{3},v_{1}v_{4},v_{2}v_{3}$ and $v_{2}v_{4}$ are in
$\operatorname{Col}(A)$ and that $v_{3}v_{4}$ is in $\operatorname{Col}(A)$ if
and only if $b_{1}^{\prime}a_{1}(b_{2}-c_{1})\equiv 0\mod p$.
Summarizing the above, the conditions we arrive at are as follows
1. (1)
$b_{1}=b_{1}^{\prime}p$, where $0\leq b_{1}^{\prime}\leq p-1$,
2. (2)
$b_{2}^{2}-(c_{1}^{2}-c_{1})b_{1}^{\prime}\equiv 0\mod p$,
3. (3)
$\left(\dfrac{b_{2}^{2}-(c_{1}^{2}-c_{1})b_{1}^{\prime}}{p}-b_{2}\right)a_{1}+c_{1}(c_{1}-1)a_{2}\equiv
0\mod p$,
4. (4)
$b_{1}^{\prime}a_{1}(b_{2}-c_{1})\equiv 0\mod p$.
Consider the following cases.
* •
Case 1 : First consider the case when $a_{1}=0$. We further subdivide our
argument into the following subcases.
* $\diamond$
Case 1A : Assume that $c_{1}\in\\{0,1\\}$. Then, we find that $b_{2}\equiv
0\mod p$. There are $2p^{4}$ such matrices.
* $\diamond$
Case 1B : Assume that $c_{1}\notin\\{0,1\\}$. In this case, we get $a_{2}=0$,
and thus, there are $p^{3}(p-2)$ such matrices.
* •
Case 2 : Next consider the case when $a_{1}\neq 0$. Then
$b_{1}^{\prime}(b_{2}-c_{1})\equiv 0\mod p$. We have the following subcases.
* $\diamond$
Case 2A : Assume that $b_{1}^{\prime}=0$, then we find that $b_{2}\equiv 0\mod
p$. Now if $c_{1}\in\\{0,1\\}$ then we get $2p^{3}(p-1)$ such matrices. If
$c_{1}\notin\\{0,1\\}$ we get $p^{2}(p-1)(p-2)$ such matrices.
* $\diamond$
Case 2B : Assume that $b_{1}^{\prime}\neq 0$, then we get $b_{2}\equiv
c_{1}\mod p$. If $c_{1}=0$ we get $p^{3}(p-1)^{2}$ such matrices. If $c_{1}=1$
we get no subring matrices as we get $b_{2}\equiv 1\mod p$ and $b_{2}\equiv
0\mod p$, but no such integer exists. If $c_{1}\notin\\{0,1\\}$, we get
$p^{2}(p-1)(p-2)$ subring matrices.
Adding up the numbers obtained in the above case decomposition, we find that
$g_{(2,3,2,2)}(p)=p^{5}+4p^{4}-9p^{3}+4p$. ∎
###### Lemma 4.5.
We have that
$\begin{split}&g_{(2,4,2,1)}(p)=2p^{5}+8p^{4}-15p^{3}+6p^{2},\\\
&g_{(2,3,3,1)}(p)=12p^{4}-10p^{3}+2p^{2}.\end{split}$
###### Proof.
The proof is similar to that of Lemma 4.4, and is omitted. ∎
### 4.2. Compositions of length $5$
In this subsection we consider the compositions of length $5$ that begin with
$2$. We need to compute $g_{\alpha}(p)$ for $20$ compositions of this form.
They are listed below,
$(2,4,1,1,1)$ | $(2,1,4,1,1)$
---|---
$(2,1,1,4,1)$ | $(2,1,1,1,4)$
$(2,3,2,1,1)$ | $(2,3,1,2,1)$
$(2,3,1,1,2)$ | $(2,2,3,1,1)$
$(2,2,1,3,1)$ | $(2,2,1,1,3)$
$(2,1,3,1,2)$ | $(2,1,3,2,1)$
$(2,1,2,3,1)$ | $(2,1,2,1,3)$
$(2,1,1,3,2)$ | $(2,1,1,2,3)$
$(2,2,2,2,1)$ | $(2,2,2,1,2)$
$(2,2,1,2,2)$ | $(2,1,2,2,2)$
Table 1. Compositions of length $5$ that begin with $2$
In the next two lemmas we obtain values of $g_{\alpha}(p)$ for all the
compositions listed above.
###### Lemma 4.6.
We have following values
$g_{(2,4,1,1,1)}(p)$ | $p^{6}+4p^{4}-4p^{3}$
---|---
$g_{(2,1,4,1,1)}(p)$ | $p^{5}+3p^{4}-3p^{3}$
$g_{(2,1,1,4,1)}(p)$ | $3p^{4}-2p^{3}$
$g_{(2,1,1,1,4)}(p)$ | $p^{4}$
$g_{(2,3,1,1,2)}(p)$ | $3p^{6}+p^{5}+p^{4}-2p^{3}$
$g_{(2,1,3,1,2)}(p)$ | $3p^{5}-p^{3}$
$g_{(2,1,1,3,2)}(p)$ | $p^{5}$
$g_{(2,2,1,1,3)}(p)$ | $p^{6}+3p^{4}-3p^{3}$
$g_{(2,1,2,1,3)}(p)$ | $p^{5}+2p^{4}-2p^{3}$
$g_{(2,1,1,2,3)}(p)$ | $2p^{4}-p^{3}$
$g_{(2,2,1,3,1)}(p)$ | $2p^{6}+4p^{5}+2p^{4}-8p^{3}+2p^{2}$
$g_{(2,1,2,3,1)}(p)$ | $4p^{5}+4p^{4}-4p^{3}$
###### Proof.
The result follows immediately from Lemmas 2.5, 3.2, 3.3 and 3.4. ∎
Next, we obtain values of $g_{\alpha}(p)$ for remaining $8$ compositions.
###### Lemma 4.7.
We have following values,
$g_{(2,3,2,1,1)}(p)$ | $3p^{7}+9p^{6}+7p^{5}-18p^{4}-3p^{2}$
---|---
$g_{(2,2,3,1,1)}(p)$ | $p^{7}+5p^{6}+8p^{5}-11p^{4}$
$g_{(2,1,3,2,1)}(p)$ | $10p^{5}-11p^{4}+4p^{3}$
$g_{(2,3,1,2,1)}(p)$ | $7p^{6}+2p^{5}-2p^{4}-5p^{3}+2p^{2}$
$g_{(2,2,2,2,1)}(p)$ | $9p^{6}+2p^{5}-18p^{4}+12p^{3}-4p^{2}$
$g_{(2,2,2,1,2)}(p)$ | $2p^{6}+9p^{5}-16p^{4}+8p^{3}-2p^{2}$
$g_{(2,2,1,2,2)}(p)$ | $p^{6}+3p^{5}-3p^{4}$
$g_{(2,1,2,2,2)}(p)$ | $2p^{5}-p^{3}$
###### Proof.
We will prove this for the composition $(2,1,2,2,2)$, expression for other
compositions can be obtained in a similar way. Let $A$ be a matrix in Hermite
normal form of following type,
$\begin{pmatrix}p^{2}&a_{1}p&a_{2}p&a_{3}p&a_{4}p&1\\\ &p&0&0&0&1\\\
&&p^{2}&b_{1}p&b_{2}p&1\\\ &&&p&c_{1}p&1\\\ &&&&p^{2}&1\\\ &&&&&1\\\
\end{pmatrix}$
First, we determine the conditions on entries of the matrix which make $A$ a
subring matrix.
1. (1)
First, we note that $v_{2}^{2}\in\mathrm{Col}(A)$ and
$v_{3}^{2}\in\mathrm{Col}(A)$.
2. (2)
Next, we know that $v_{4}^{2}\in\mathrm{Col}(A)$ if and only
$v_{4}^{2}-p^{2}v_{4}\in\mathrm{Col}(A)$. Now, this is the case if and only if
$b_{1}a_{2}\equiv 0\mod p$.
3. (3)
Similarly, $v_{5}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{5}^{2}-p^{2}v_{5}\in\mathrm{Col}(A)$ and this is true if and only if
$b_{1}c_{1}\equiv 0\mod p$ and $b_{2}^{2}a_{2}-c_{1}^{2}a_{3}\equiv 0\mod p$.
4. (4)
We also note that if above conditions are satisfied then
$v_{2}v_{3},v_{2}v_{4},v_{2}v_{5},v_{3}v_{4},v_{3}v_{5}$ and $v_{4}v_{5}$ are
in $\mathrm{Col}(A)$.
The conditions we get are,
* •
$b_{1}a_{2}=0$,
* •
$b_{1}c_{1}=0$,
* •
$b_{2}^{2}a_{2}-c_{1}^{2}a_{3}\equiv 0\mod p$.
First we assume that $b_{1}=0$. Now further if $c_{1}=0$ then we get
$p^{3}(2p-1)$ matrices. If $c_{1}\neq 0$ then we get $p^{4}(p-1)$ matrices.
Next, we assume that $b_{1}\neq 0$ then $a_{2}=0$ and $c_{1}=0$, therefore we
get $p^{4}(p-1)$ matrices. Therefore, we get that
$g_{(2,1,2,2,2)}(p)=2p^{5}-p^{3}$. ∎
## 5\. Calculating the values of $g_{\alpha}(p)$ for compositions beginning
with $3$
### 5.1. Compositions of length $4$
In this section, we compute $g_{\alpha}(p)$ for $10$ compositions $\alpha$
beginning with $3$ and of length $4$. They are listed as follows
$(3,4,1,1)$ | $(3,3,1,2)$
---|---
$(3,1,4,1)$ | $(3,2,3,1)$
$(3,1,1,4)$ | $(3,1,2,3)$
$(3,2,1,3)$ | $(3,3,2,1)$
$(3,1,3,2)$ | $(3,2,2,2)$.
We make note of some known computations.
###### Lemma 5.1.
The following values of $g_{\alpha}(p)$ are known
$g_{(3,4,1,1)}(p)$ | $12p^{4}-9p^{3}+p^{2}$
---|---
$g_{(3,1,4,1)}(p)$ | $2p^{4}+6p^{3}-2p^{2}$
$g_{(3,1,1,4)}(p)$ | $3p^{3}$.
###### Proof.
The above result follows from [Ish22a, Lemma 4.6]. ∎
###### Lemma 5.2.
We have that $g_{(3,2,1,3)}(p)=5p^{4}-4p^{3}+2p^{2}.$
###### Proof.
Let $A$ be any integer matrix in Hermite normal form of the type
$\begin{pmatrix}p^{3}&a_{1}p&a_{2}p&a_{3}p&1\\\ &p^{2}&b_{1}p&b_{2}p&1\\\
&&p&0&1\\\ &&&p^{3}&1\\\ &&&&1.\\\ \end{pmatrix}$
We determine the conditions so that $A$ is a subring matrix.
1. (1)
It is required that $v_{2}^{2}\in\mathrm{Col}(A)$ and this is the case if and
only if
$(a_{1}^{2}p^{2},p^{4},0,0,0)^{t}-p^{2}(a_{1}p,p^{2},0,0,0)^{t}$
is in $\mathrm{Col}(A)$. We deduce that $a_{1}\equiv 0\mod p$, hence
$a_{1}=a_{1}^{\prime}p$ where $0\leq a_{1}^{\prime}\leq p-1$.
2. (2)
It is required that $v_{3}^{2}\in\mathrm{Col}(A)$; clearly this is equivalent
to the statement that $v_{3}^{2}-pv_{3}\in\mathrm{Col}(A)$. Now, it is easy to
see that latter condition is true if and only if
(5.1) $(a_{2}^{2}-a_{2})-a_{1}^{\prime}(b_{1}^{2}-b_{1})\equiv 0\mod p.$
3. (3)
Similarly, it is clear that $v_{4}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{4}^{2}-p^{3}v_{4}\in\mathrm{Col}(A)$ and latter condition is equivalent to
(5.2) $a_{3}^{2}-b_{2}^{2}a_{1}^{\prime}\equiv 0\mod p.$
4. (4)
One can easily check that $v_{2}v_{3}$ and $v_{2}v_{4}$ are both in
$\mathrm{Col}(A)$ and $v_{3}v_{4}\in\mathrm{Col}(A)$ if and only
(5.3) $a_{2}a_{3}-a_{1}^{\prime}b_{1}b_{2}\equiv 0\mod p.$
Thus, to summarize, we arrive at the following conditions
1. (1)
$a_{1}=a_{1}^{\prime}p$ where $0\leq a_{1}^{\prime}\leq p-1$,
2. (2)
$(a_{2}^{2}-a_{2})-a_{1}^{\prime}(b_{1}^{2}-b_{1})\equiv 0\mod p$,
3. (3)
$a_{3}^{2}-b_{2}^{2}a_{1}^{\prime}\equiv 0\mod p$,
4. (4)
$a_{2}a_{3}-a_{1}^{\prime}b_{1}b_{2}\equiv 0\mod p.$
We count matrices $A$ satisfying these conditions by dividing into cases.
Case 1 : First consider the case when $a_{3}\equiv\mod p$. From equation (5.2)
we deduce that $a_{1}^{\prime}b_{2}\equiv 0\mod p$. We consider two subcases
below.
* •
Case 1A : We consider the case when $b_{2}=0$. If $b_{1}\in\\{0,1\\}$, then,
there are $4p^{3}$ matrices. If $b_{1}\notin\\{0,1\\}$, then, there are
$p^{3}(p-2)$ matrices. Thus, there are $p^{3}(p+2)$ in this case in total.
* •
Case 1B : Next consider the case when $b_{2}\neq 0$. From equation (5.2) we
find that $a_{1}^{\prime}=0$. From (5.1), we deduce $a_{2}^{2}-a_{2}\equiv
0\mod p$. Hence, there are $2p^{3}(p-1)$ matrices in this case.
Case 2 : In this case we count matrices such that $a_{3}\not\equiv 0\mod p$.
From (5.2), we deduce that $b_{2}\neq 0$. Consider following subcases.
* •
Case 2A : Suppose $b_{1}=0$, then from equation (5.3) we get that $a_{2}\equiv
0\mod p$. Hence, there are $p^{2}(p-1)^{2}$ matrices in this case.
* •
Case 2B : Assume that $b_{1}=1$, then from equation (5.1) we find that
$a_{2}(a_{2}-1)\equiv 0\mod p$. However, from (5.2) we cannot have
$a_{2}\equiv 0\mod p$, therefore $a_{2}\equiv 1\mod p$. Hence, we find that
$a_{3}\equiv b_{2}\mod p$. As a result, there are $p^{2}(p-1)$ matrices in
this case.
* •
Case 2C : In the last case we consider the case when $b_{1}\notin\\{0,1\\}$
then from equations (5.1), (5.2), (5.3) we get that $a_{2}\not\equiv 0\mod p$.
Now, we get that from the three equations (5.1), (5.2), (5.3) that,
$a_{1}^{\prime}\equiv\frac{a_{2}(a_{2}-1)}{b_{1}(b_{1}-1)}\mod p,$
$a_{1}^{\prime}\equiv\frac{a_{3}^{2}}{b_{2}^{2}}\mod p,$
$a_{1}^{\prime}\equiv\frac{a_{2}a_{3}}{b_{1}b_{2}}\mod p.$
From these we find that $a_{2}\equiv b_{1}\mod p$ and $a_{3}\equiv b_{2}\mod
p$. Hence, there are $(p-2)(p-1)p^{2}$ matrices in this case.
Adding up the values in each case, we prove the assertion of the lemma. ∎
###### Lemma 5.3.
We have that $g_{\alpha}(p)=2p^{4}$ where $\alpha=(3,1,3,2).$
###### Proof.
Let $A$ be any integer matrix in Hermite normal form of the type
$\begin{pmatrix}p^{3}&a_{1}p&a_{2}p&a_{3}p&1\\\ &p&0&0&1\\\
&&p^{3}&c_{1}p&1\\\ &&&p^{2}&1\\\ &&&&1\\\ \end{pmatrix}$
We derive the conditions so that $A$ is a subring matrix and use them to
evaluate $g_{(3,1,3,2)}(p)$.
1. (1)
It is clear that $v_{2}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{2}^{2}-pv_{2}\in\mathrm{Col}(A)$, the second condition translates to
(5.4) $a_{1}^{2}-a_{1}\equiv 0\mod p.$
2. (2)
Similarly, $v_{3}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{3}^{2}-p^{3}v_{3}\in\mathrm{Col}(A)$, and therefore, $a_{2}\equiv 0\mod
p$. In other words, $a_{2}=a_{2}^{\prime}p$, where $0\leq a_{2}^{\prime}\leq
p-1$.
3. (3)
Arguing as in previous two cases, we see that $v_{4}^{2}\in\mathrm{Col}(A)$ if
and only if $c_{1}=c_{1}^{\prime}p$, where $0\leq c_{1}^{\prime}\leq p-1$ and
$a_{3}=a_{3}^{\prime}p$ with $0\leq a_{3}^{\prime}\leq p-1$.
4. (4)
It is easy to see that if the above conditions are satisfied, then
$v_{2}v_{3},v_{2}v_{4}$ and $v_{3}v_{4}$ are in $\mathrm{Col}(A)$.
To summarize, the conditions are as follows
1. (1)
$a_{1}^{2}-a_{1}\equiv 0\mod p$,
2. (2)
$a_{2}=a_{2}^{\prime}p$ where $0\leq a_{2}^{\prime}\leq p-1$,
3. (3)
$c_{1}=c_{1}^{\prime}p$ where $0\leq c_{1}^{\prime}\leq p-1$,
4. (4)
$a_{3}=a_{3}^{\prime}p$ with $0\leq a_{3}^{\prime}\leq p-1$.
From the above, it is clear that $g_{(3,1,3,2)}(p)=2p^{4}$. ∎
###### Lemma 5.4.
We have that
(5.5) $g_{(3,3,1,2)}(p)=8p^{4}-6p^{3}+2p^{2}.$
###### Proof.
Let $A$ be an integer matrix in Hermite normal form
$\begin{pmatrix}p^{3}&a_{1}p&a_{2}p&a_{3}p&1\\\ &p^{3}&b_{1}p&b_{2}p&1\\\
&&p&0&1\\\ &&&p^{2}&1\\\ &&&&1.\\\ \end{pmatrix}$
We determine the conditions so that $A$ is a subring matrix.
1. (1)
Arguing as in Lemma 5.3, we find that the condition
$v_{2}^{2}\in\mathrm{Col}(A)$ translates to the condition that
$a_{1}=a_{1}^{\prime}p$, with $0\leq a_{1}^{\prime}\leq p-1$.
2. (2)
Next, we see that $v_{3}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{3}^{2}-pv_{3}\in\mathrm{Col}(A)$. Therefore we deduce that
$b_{1}^{2}-b_{1}\equiv 0\mod p$ and
(5.6) $(a_{2}^{2}-a_{2})p-a_{1}^{\prime}(b_{1}^{2}-b_{1})\equiv 0\mod p.$
3. (3)
It is again clear that $v_{4}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{4}^{2}-p^{2}v_{4}\in\mathrm{Col}(A)$. We deduce that this is true if and
only if $b_{2}=b_{2}^{\prime}p$, where $0\leq b_{2}^{\prime}\leq p-1$, and
$a_{3}=a_{3}^{\prime}p$, where $0\leq a_{3}^{\prime}\leq p-1$.
4. (4)
It is clear that $v_{2}v_{3}$ and $v_{2}v_{4}$ are in $\mathrm{Col}(A)$
whenever above conditions are satisfied. By similar arguments as above we see
that $v_{3}v_{4}\in\mathrm{Col}(A)$ if and only if
$a_{1}^{\prime}b_{1}b_{2}^{\prime}\equiv 0\mod p$.
We have the following conditions on $A$.
1. (1)
$a_{1}=a_{1}^{\prime}p$ with $0\leq a_{1}^{\prime}\leq p-1$,
2. (2)
$b_{1}^{2}-b_{1}\equiv 0\mod p$ and
$(a_{2}^{2}-a_{2})p-a_{1}^{\prime}(b_{1}^{2}-b_{1})\equiv 0\mod p$,
3. (3)
$b_{2}=b_{2}^{\prime}p$ where $0\leq b_{2}^{\prime}\leq p-1$,
4. (4)
$a_{3}=a_{3}^{\prime}p$ where $0\leq a_{3}^{\prime}\leq p-1$,
5. (5)
$a_{1}^{\prime}b_{1}b_{2}^{\prime}\equiv 0\mod p$.
We consider three cases depending upon whether $b_{1}$ is $0$, $1$ or not in
$\\{0,1\\}$, and count number of matrices in each case.
Case 1 : First consider the case when $b_{1}=0$. Clearly there are $2p^{4}$
matrices in this case.
Case 2 : Consider the case in which $b_{1}=1$. It is easy to see that there
are $2p^{2}(2p-1)$ matrices in this case.
Case 3 : Assume that $b_{1}\notin\\{0,1\\}$. Then if $b_{2}^{\prime}=0$ we get
$2p^{3}(p-1)$ matrices. On the other hand, if $b_{2}\neq 0$ then
$a_{1}^{\prime}=0$ follows from $a_{1}^{\prime}b_{2}^{\prime}\equiv 0\mod p$.
Therefore, we find that $a_{2}^{2}-a_{2}\equiv 0\mod p$, and there are
$4p^{2}(p-1)^{2}$ matrices. Adding up the number of matrices from each case we
arrive at the assertion. ∎
###### Lemma 5.5.
The following equality holds
(5.7) $g_{(3,2,3,1)}(p)=12p^{4}-2p^{3}+4p^{2}.$
###### Proof.
Let $A$ be any integer matrix in Hermite normal form of the type
$\begin{pmatrix}p^{3}&a_{1}p&a_{2}p&a_{3}p&1\\\ &p^{2}&b_{1}p&b_{2}p&1\\\
&&p^{3}&c_{1}p&1\\\ &&&p&1\\\ &&&&1.\\\ \end{pmatrix}$
We determine the conditions so that $A$ is a subring matrix.
1. (1)
It is easy to see that $v_{2}^{2}\in\mathrm{Col}(A)$ if and only if
$a_{1}=a_{1}^{\prime}p$ where $0\leq a_{1}^{\prime}\leq p-1$.
2. (2)
We note that $v_{3}^{2}$ is in $\mathrm{Col}(A)$ if and only if
$v_{3}^{3}-p^{3}v_{3}\in\mathrm{Col}(A)$. Therefore, we require that
$(a_{2}^{2}p^{2},b_{1}^{2}p^{2},p^{6},0,0)^{t}-p^{3}(a_{2}p,b_{1}p,p^{3},0,0)^{t}$
is in $\mathrm{Col}(A)$. Clearly, this is equivalent to the congruence
(5.8) $a_{2}^{2}-a_{1}^{\prime}b_{1}^{2}\equiv 0\mod p.$
3. (3)
It is required that $v_{4}^{2}\in\mathrm{Col}(A)$. It is easy to see that this
condition is equivalent to the condition that
$v_{4}^{2}-pv_{4}\in\mathrm{Col}(A)$. The last entry in the column
$v_{4}^{2}-pv_{4}$ is $(c_{1}^{2}-c_{1})p^{2}$ therefore, first condition is
$c_{1}^{2}-c_{1}\equiv 0\mod p$. As in previous lemmas we obtain following
second condition, $b_{1}(c_{1}^{2}-c_{1})\equiv 0\mod p^{2}$ and from this we
deduce that $b_{1}(c_{1}^{2}-c_{1})=0$ and the third condition is,
(5.9)
$(a_{3}^{2}-a_{3})p^{2}-a_{2}(c_{1}^{2}-c_{1})-a_{1}^{\prime}(b_{2}^{2}-b_{2})p^{2}\equiv
0\mod p^{3}.$
4. (4)
If entries of matrix $A$ satisfy above conditions then it is easy to see that
$v_{2}v_{3}$, $v_{2}v_{4}$ are in $\mathrm{Col}(A)$ and
$v_{3}v_{4}\in\mathrm{Col}(A)$ if and only if the following congruence holds
(5.10) $a_{2}(a_{3}-c_{1})-a_{1}^{\prime}b_{1}(b_{2}-c_{1})\equiv 0\mod p.$
Thus, the conditions so that $A$ is a subring matrix are as follows
1. (1)
$a_{1}=a_{1}^{\prime}p$ with $0\leq a_{1}^{\prime}\leq p-1$,
2. (2)
$c_{1}^{2}-c_{1}\equiv 0\mod p$ and $b_{1}(c_{1}^{2}-c-1)=0$,
3. (3)
$(a_{3}^{2}-a_{3})p^{2}-a_{2}(c_{1}^{2}-c_{1})-a_{1}^{\prime}(b_{2}^{2}-b_{2})p^{2}\equiv
0\mod p^{3}$,
4. (4)
$a_{2}(a_{3}-c_{1})-a_{1}^{\prime}b_{1}(b_{2}-c_{1})\equiv 0\mod p$.
There are three cases to consider, depending upon the value of $c_{1}$.
Case 1 : First suppose that $c_{1}=0$. Then by symmetry, the set of matrices
is in bijection with the set of irreducible subring matrices with diagonal
$(3,2,1,3)$. Therefore, we deduce that $g_{(3,2,1,3)}(p)=5p^{4}-4p^{3}+2p^{2}$
(from the argument in Lemma 5.2).
Case 2 : Next, consider the case when $c_{1}=1$. The equation (5.9) reduces to
the following
(5.11) $(a_{3}^{2}-a_{3})-a_{1}^{\prime}(b_{2}^{2}-b_{2})\equiv 0\mod p.$
First we suppose that $b_{1}=0$. Then we find that $a_{2}\equiv 0\mod p$.
Again considering two cases when $b_{2}\in\\{0,1\\}$ or $\notin\\{0,1\\}$ it
is clear that there are $p^{3}(p+2)$ matrices in this case with $b_{1}=0$. If,
$b_{1}\neq 0$, then we further divide into cases such that $b_{2}=0$, $1$ or
not in $\\{0,1\\}$. It is easy to see that $2p^{2}(p-1)(2p-1)$ in this case
with $b_{1}\neq 0$.
Case 3 : Finally, we consider the case when $c_{1}\notin\\{0,1\\}$. From
conditions above we find that $b_{1}=0$; from equation 5.9 we deduce that
$a_{2}=0$ and equation 5.9 gets reduced to the equation 5.11. We consider sub-
cases depending upon whether $b_{2}\in\\{0,1\\}$ or not. There are $8p^{3}$
matrices in the case when $b_{2}\in\\{0,1\\}$. On the other hand, there are
$2p^{3}(p-2)$ matrices for which $b_{2}\notin\\{0,1\\}$.
Adding up the number of matrices in each case we prove the result. ∎
###### Lemma 5.6.
The following equality holds
$\begin{split}&g_{(3,1,2,3)}(p)=p^{4}+2p^{3}-p^{2},\\\
&g_{(3,3,2,1)}(p)=p^{6}+2p^{5}-13p^{4}+9p^{3},\\\
&g_{(3,2,2,2)}(p)=p^{6}-2p^{5}+6p^{4}-4p^{3}+3p^{2}-5p+2.\end{split}$
###### Proof.
The proof is very similar to previous results, and we omit it. ∎
### 5.2. Compositions of length $5$
We consider the composition of length $5$ that begin with $3$ we need to
evaluate $g_{\alpha}(p)$ for $10$ compositions of this form. We list all
compositions of this form in the table below
$(3,3,1,1,1)$ | $(3,2,1,1,2)$
---|---
$(3,1,3,1,1)$ | $(3,1,2,1,2)$
$(3,1,1,3,1)$ | $(3,1,2,2,1)$
$(3,1,1,1,3)$ | $(3,1,1,2,2)$
$(3,2,1,2,1)$ | $(3,2,2,1,1)$.
Table 2. Compositions of length $5$ that begin with $3$.
For some of these compositions, we can use results from [Ish22a] to deduce
value of $g_{\alpha}(p)$. The next result follows from the results in _loc.
cit._
###### Lemma 5.7.
We have following values
$g_{(3,3,1,1,1)}(p)$ | $16p^{6}+12p^{5}-20p^{4}+8p^{3}$
---|---
$g_{(3,1,3,1,1)}(p)$ | $18p^{5}-6p^{4}$
$g_{(3,1,1,3,1)}(p)$ | $2p^{5}+10p^{4}-4p^{3}$
$g_{(3,1,1,1,3)}(p)$ | $4p^{4}$.
###### Proof.
The above results are direct consequence of [Ish22a, Lemma 4.6]. ∎
Next we evaluate the value of $g_{\alpha}(p)$ for the remaining compositions.
###### Lemma 5.8.
The following table of relations holds
$g_{(3,1,2,2,1)}(p)$ | $13p^{5}-8p^{4}+8p^{3}-2p^{2}$
---|---
$g_{(3,1,2,1,2)}(p)$ | $5p^{5}-4p^{4}+2p^{3}$
$g_{(3,1,1,2,2)}(p)$ | $p^{5}+3p^{4}-2p^{3}$
$g_{(3,2,1,1,2)}(p)$ | $4p^{6}-8p^{5}+35p^{4}-31p^{3}+4p^{2}$
$g_{(3,2,1,2,1)}(p)$ | $9p^{6}-4p^{5}+28p^{4}-39p^{3}+20p^{2}-10p$.
###### Proof.
We will prove this result only for the composition $(3,1,1,2,2)$, and the
proof is similar for the other compositions listed above. Therefore, we will
count the number of integer subring matrices $A$ in Hermite normal form of the
type
$\begin{pmatrix}p^{3}&a_{1}p&a_{2}p&a_{3}p&a_{4}p&1\\\ &p&0&0&0&1\\\
&&p&0&0&1\\\ &&&p^{2}&b_{1}p&1\\\ &&&&p^{2}&1\\\ &&&&&1.\\\ \end{pmatrix}$
We determine the conditions on entries of the matrix which make $A$ a subring
matrix.
1. (1)
We note that $v_{2}^{2}$ and $v_{3}^{2}\in\mathrm{Col}(A)$ if and only if
$a_{1}^{2}-a_{1}\equiv 0\mod p$ and $a_{2}^{2}-a_{2}\equiv\mod p$.
2. (2)
We also want $v_{4}^{2}$ to be in $\mathrm{Col}(A)$ and this is true if and
only
$(a_{3}^{2}p^{2},0,0,p^{4},0,0)^{t}-p^{2}(a_{3}p,0,0,p^{2},0,0)^{t}$
is in $\mathrm{Col}(A)$. Therefore, $v_{4}^{2}\in\operatorname{Col}A$ if and
only if $a_{3}=a_{3}^{\prime}p$ with $0\leq a_{3}^{\prime}\leq p-1$.
3. (3)
Similarly, $v_{5}^{2}\in\mathrm{Col}(A)$ if and only if
$v_{5}^{2}-p^{2}v_{5}\in\mathrm{Col}(A)$. This condition is equivalent to
(5.12) $(a_{4}^{2}-a_{4})-a_{3}^{\prime}b_{1}^{2}\equiv 0\mod p.$
4. (4)
Similar arguments show that $v_{2}v_{4},v_{3}v_{4},v_{4}v_{5}$ are in
$\mathrm{Col}(A)$ if above conditions are satisfied. Furthermore,
$v_{2}v_{3},v_{2}v_{5},v_{3}v_{5}\in\mathrm{Col}(A)$ if and only if
$a_{1}a_{2},a_{1}a_{4},a_{2}a_{4}$ are all divisible by $p$.
We count the number of matrices satisfying the above conditions. It benefits
us to consider two cases.
Case 1: First, we consider the case when $b_{1}=0$. Note that in this case, at
most one of $a_{1},a_{2},a_{4}$ is $1$ $\mod p$, and the others need to be $0$
$\mod p$. Hence, our count leads to $4p^{4}$ matrices for which $b_{1}=0$.
Case 2: Next, we consider the case when $b_{1}\neq 0$. Note that in this case
there are $p^{4}(p-1)$ matrices for which $a_{1},a_{2}$ are both divisible by
$p$. On the other hand, either $a_{1}$ or $a_{2}$ is not divisible by $p$,
then, $a_{4}$ is divisible by $p$. The equation 5.12 implies that
$a_{3}^{\prime}=0$. Thus, there are $2p^{3}(p-1)$ matrices for which either
$a_{1}$ or $a_{2}$ is divisible by $p$. Thus, the total number of matrices for
this case is $p^{3}(p+2)(p-1)$.
Putting together the number of matrices from both cases we deduce that
$g_{(3,1,1,2,2)}(p)$ $=p^{5}+3p^{4}-2p^{3}$. ∎
There is one composition $\alpha$ for which we are not able to compute
$g_{\alpha}(p)$ using the above arguments. This is the composition
$\alpha=(3,2,2,1,1)$. Let us consider this case in some further detail. Let
$A$ be an integer subring matrix of the form
$A=\begin{pmatrix}p^{3}&a_{1}p&a_{2}p&a_{3}p&a_{4}p&1\\\
&p^{2}&b_{1}p&b_{2}p&b_{3}p&1\\\ &&p^{2}&c_{1}p&c_{2}p&1\\\ &&&p&0&1\\\
&&&&p&1\\\ &&&&&1.\\\ \end{pmatrix}$
Since the above matrix is in Hermite normal form, $1\leq a_{i}\leq p^{2}-1$,
and $1\leq b_{i},c_{j}\leq p-1$. Below, we list the conditions for $A$ to be a
subring matrix
* •
$a_{2}^{2}-a_{1}^{\prime}b_{1}^{2}\equiv 0\mod p$,
* •
$b_{1}c_{1}(c_{1}-1)=0$, $b_{1}c_{2}(c_{2}-1)=0$, $b_{1}c_{1}c_{2}=0$,
* •
$(a_{3}^{2}-a_{3})-a_{2}(c_{1}^{2}-c_{1})/p-a_{1}^{\prime}(b_{2}^{2}-b_{2})\equiv
0\mod p$,
* •
$(a_{4}^{2}-a_{4})-a_{2}(c_{2}^{2}-c_{2})/p-a_{1}^{\prime}(b_{3}^{2}-b_{3})\equiv
0\mod p$,
* •
$a_{2}(a_{3}-c_{1})-a_{1}^{\prime}b_{1}(b_{2}-c_{1})\equiv 0\mod p$,
* •
and $a_{2}(a_{4}-c_{2})-a_{1}^{\prime}b_{1}(b_{3}-c_{2})\equiv 0\mod p$,
* •
$a_{3}a_{4}-a_{2}c_{1}c_{2}/p-a_{1}^{\prime}b_{2}b_{3}\equiv 0\mod p$.
In the subcase when $b_{1}=0$ counting solutions to the equations is reduced
to counting number of solutions in $\mathbb{F}_{p}^{8}$ to following system of
polynomial equations:
$(x_{3}^{2}-x_{3})-x_{2}(x_{7}^{2}-x_{7})-x_{1}(x_{5}^{2}-x_{5})=0$
$(x_{4}^{2}-x_{4})-x_{2}(x_{8}^{2}-x_{8})-x_{1}(x_{6}^{2}-x_{6})=0$
$x_{3}x_{4}-x_{2}x_{7}x_{8}-x_{1}x_{5}x_{6}=0$
Let $N_{p}$ denote the number of solutions to this system in
$\mathbb{F}^{8}_{p}$. Using SageMath we calculated $N_{p}$ for
$p=2,3,5,7,11,13,17,19$ and these computations suggest that
$N_{p}=p^{5}+12p^{4}-20p^{3}+30p^{2}-10p$. Using this and arguments similar to
those used in proving previous results we deduce that
$g_{(3,2,2,1,1)}(p)=p^{7}+24p^{6}-29p^{5}+21p^{4}-4p^{3}$. In summary, our
computations lead us to make the following conjecture.
###### Conjecture 5.9.
We have that $g_{(3,2,2,1,1)}(p)=p^{7}+24p^{6}-29p^{5}+21p^{4}-4p^{3}$.
## 6\. Calculating the values of $g_{\alpha}(p)$ for compositions beginning
with $4,5$ or $6$
### 6.1. Values of $g_{\alpha}(p)$ for compositions beginning with $4$
#### 6.1.1. Compositions of length $4$
We will evaluate $g_{\alpha}(p)$ for $6$ compositions of length $4$ that begin
with $4$. They are listed below
$(4,2,2,1)$ | $(4,2,1,2)$
---|---
$(4,1,2,2)$ | $(4,3,1,1)$
$(4,1,3,1)$ | $(4,1,1,3)$.
Table 3. Compositions of length $4$ that begin with $4$.
###### Lemma 6.1.
We have following values,
$g_{(4,2,2,1)}(p)$ | $6p^{6}+12p^{5}-10p^{4}-7p^{3}+4p^{2}$
---|---
$g_{(4,2,1,2)}(p)$ | $p^{5}+9p^{4}-5p^{3}-4p^{2}$
$g_{(4,1,2,2)}(p)$ | $p^{5}+2p^{4}-p^{2}$
$g_{(4,3,1,1)}(p)$ | $2p^{6}-2p^{5}+17p^{4}-6p^{3}-2p^{2}$
$g_{(4,1,3,1)}(p)$ | $2p^{4}+6p^{3}$
$g_{(4,1,1,3)}(p)$ | $p^{4}+2p^{3}$.
###### Proof.
The proof is omitted. ∎
#### 6.1.2. Compositions of length $5$
We will evaluate $g_{\alpha}(p)$ for $4$ compositions of length $5$ that begin
with $4$. We list them below
$(4,2,1,1,1)$ | $(4,1,2,1,1)$
---|---
$(4,1,1,2,1)$ | $(4,1,1,1,2)$.
Table 4. Compositions of length $5$ that begin with $4$.
###### Lemma 6.2.
We have following values,
$g_{(4,2,1,1,1)}(p)$ | $5p^{6}+16p^{5}-17p^{4}-27p^{3}+12p^{2}$
---|---
$g_{(4,1,2,1,1)}(p)$ | $p^{6}+13p^{5}-8p^{4}+3p^{3}+2p^{2}$
$g_{(4,1,1,2,1)}(p)$ | $p^{6}+8p^{4}-4p^{3}$
$g_{(4,1,1,1,2)}(p)$ | $p^{5}+3p^{4}$.
###### Proof.
We will explicitly obtain the expression for $g_{(4,1,1,1,2)}(p)$. The other
computations are similar, and thus omitted. Let $A$ be a matrix in Hermite
normal form of following type
$\begin{pmatrix}p^{4}&a_{1}p&a_{2}p&a_{3}p&a_{4}p&1\\\ &p&0&0&0&1\\\
&&p&0&0&1\\\ &&&p&0&1\\\ &&&&p^{2}&1\\\ &&&&&1.\\\ \end{pmatrix}$
First, we determine the conditions on entries of the matrix which make $A$ a
subring matrix. Note that for $i=2,3,4$, $v_{i}^{2}\in\mathrm{Col}(A)$ if and
only if $a_{i-1}(a_{i-1}-1)\equiv 0\mod p^{2}$ and
$v_{5}^{2}\in\mathrm{Col}(A)$ if and only $a_{4}\equiv 0\mod p$. We also note
that for $1<i<j$, $v_{i}v_{j}\in\mathrm{Col}(A)$ if and only if
$a_{i-1}a_{j-1}\equiv 0\mod p^{2}$. First we suppose that for $i=2,3,4$, we
have that $a_{i-1}\equiv 0\mod p^{2}$. There are $p^{5}$ matrices of this
form. Next, suppose for exactly one of $a_{1},a_{2},a_{3}$ is congruent to $1$
modulo $p^{2}$ and others are divisible by $p^{2}$ then we get $3p^{4}$
matrices of this form. Therefore, $g_{(4,1,1,1,2)}(p)$ = $p^{5}+3p^{4}$. ∎
### 6.2. Values of $g_{\alpha}(p)$ for compositions beginning with $5$
#### 6.2.1. Compositions of length $4$
We calculate $g_{\alpha}(p)$ for $3$ compositions of length $4$ that begin
with $5$. The compositions are listed below
$(5,2,1,1)$
---
$(5,1,2,1)$
$(5,1,1,2)$.
Table 5. Compositions of length $4$ that begin with $5$.
###### Lemma 6.3.
The following table of relations hold
$g_{(5,2,1,1)}(p)$ | $p^{6}-3p^{5}+17p^{4}-8p^{3}-5p^{2}+2p$
---|---
$g_{(5,1,2,1)}(p)$ | $7p^{4}-p^{3}-2p^{2}$
$g_{(5,1,1,2)}(p)$ | $p^{4}+2p^{3}$.
###### Proof.
We will prove the result only for the composition $(5,1,2,1)$. The other cases
are omitted since the arguments are similar to this case. Let $A$ be a matrix
in Hermite normal form of the following type
$\begin{pmatrix}p^{5}&a_{1}p&a_{2}p&a_{3}p&1\\\ &p&0&0&1\\\
&&p^{2}&b_{1}p&1\\\ &&&p&1\\\ &&&&1.\\\ \end{pmatrix}$
We obtain the conditions on the entries of $A$ below.
1. (1)
First, we must have that $v_{2}^{2}\in\mathrm{Col}(A)$ and this is true if and
only if $v_{2}^{2}-pv_{2}\in\mathrm{Col}(A)$. The latter condition is
equivalent to the condition $a_{1}(a_{1}-1)\equiv 0\mod p$.
2. (2)
We also want that $v_{3}^{2}\in\mathrm{Col}(A)$ and we see that this is
equivalent to the condition $v_{3}^{2}-p^{2}v_{3}\in\mathrm{Col}(A)$, which is
true if and only if $a_{2}(a_{2}-p)\equiv\mod p$. Therefore we get that
$a_{2}=a_{2}^{\prime}p$ with $0\leq a_{2}^{\prime}\leq p^{3}-1$ and
$a_{2}^{\prime}(a_{2}^{\prime}-1)\equiv 0\mod p$.
3. (3)
Using similar arguments we obtain that $v_{4}^{2}\in\mathrm{Col}(A)$ if and
only if $a_{3}(a_{3}-1)-b_{1}(b_{1}-1)a_{2}^{\prime}\equiv 0\mod p^{3}$. We
also deduce that $v_{2}v_{3},v_{2}v_{4}$ and $v_{3}v_{4}$ are in
$\mathrm{Col}(A)$ if and only if $a_{1}a_{2}\equiv 0\mod p$, $a_{1}a_{3}\equiv
0\mod p$ and $a_{2}(a_{3}-b_{1})\equiv 0\mod p$.
The conditions we get are as follows
* •
$a_{1}(a_{1}-1)\equiv 0\mod p$,
* •
$a_{2}=a_{2}^{\prime}p$ with $0\leq a_{2}^{\prime}\leq p^{3}-1$ and
$a_{2}^{\prime}(a_{2}^{\prime}-1)\equiv 0\mod p$,
* •
$a_{3}(a_{3}-1)-b_{1}(b_{1}-1)a_{2}^{\prime}\equiv 0\mod p^{3}$,
* •
$a_{1}a_{2}\equiv 0\mod p$, $a_{1}a_{3}\equiv 0\mod p$ and
$a_{2}(a_{3}-b_{1})\equiv 0\mod p$.
Next, we count the number of matrices by dividing in two cases depending upon
if $a_{1}\equiv 0\mod p^{3}$ or $a_{1}\equiv 0\mod p^{3}$.
Case 1 : First consider the case when $a_{1}\equiv 1\mod p^{3}$, as we must
have $a_{1}a_{2}\mod p^{3}$ we deduce that
$a_{2}=a_{2}^{\prime\prime\prime}p^{3}$ and $a_{3}\equiv 0\mod p$. Therefore,
conditions reduce to $b_{1}(b_{1}-1)a_{2}^{\prime\prime\prime}\equiv 0\mod p$.
Counting we see that there are $3p^{3}-2p^{2}$ matrices in this case.
Case 2 : Now, we consider that case when $a_{1}\equiv 0\mod p^{3}$. We further
consider two separate cases depending upon whether $a_{2}^{\prime}$ is
divisible by $p$ or not.
* •
Case 2A : Consider the case when $a_{2}^{\prime}$ is divisible by $p$. we have
two possibilities, either $p^{3}\nmid a_{2}$ or $p^{3}\mid a_{2}$, if
$p^{3}\nmid a_{2}$ then we deduce that $a_{3}\equiv b_{1}\mod p$, therefore
$b_{1}=0$ or $1$. We get $2p^{3}(p-1)$ matrices. if $p^{3}\mid a_{2}$, again
considering two cases depending upon whether $b_{1}\in\\{0,1\\}$ or not and
counting we see that we get $2p^{4}$ matrices.
* •
Case 2B : We consider the case when $a_{2}^{\prime}\equiv 1\mod p$, we deduce
that $a_{3}\equiv b_{1}\mod p^{3}$. Conditions reduce to
$b_{1}(b_{1}-1)(1-a_{2}^{\prime})\equiv 0\mod p^{3}$. Now, suppose
$b_{1}\in\\{0,1\\}$, then we get $2p^{4}$ matrices. Otherwise,
$a_{2}^{\prime}=1$ and we get $p^{2}(p-2)$ matrices.
It follows from the above observations that the number of matrices in each
case we deduce that $g_{(5,1,2,1)}(p)=7p^{4}-p^{3}-2p^{2}$. ∎
#### 6.2.2. Compositions of length $5$
In this section we evaluate $g_{\alpha}(p)$ for the composition
$\alpha=(5,1,1,1,1)$. This is the only composition of length $5$ which starts
with $5$. We have the following result.
###### Lemma 6.4.
We have that $g_{\alpha}(p)=5p^{4}$, where $\alpha=(5,1,1,1,1)$.
###### Proof.
The result follows directly from [AKKM21, Lemma 3.5]. ∎
### 6.3. Values of $g_{\alpha}(p)$ for compositions beginning with $6$
We evaluate $g_{\alpha}(p)$ for the only composition $\alpha=(6,1,1,1)$ of
length $4$ which begins with $6$.
###### Lemma 6.5.
We have that $g_{(6,1,1,1)}(p)=4p^{3}$.
###### Proof.
The above result is an immediate consequence [AKKM21, Lemma 3.5]. ∎
## 7\. Main results
### 7.1. Proof of the main result
In this section we prove Theorem 1.4. We note that for $n>1$ and $e\geq n-1$
the following relation holds
(7.1)
$g_{n}(p^{e})=g_{n-1}(p^{e-1})+\sum_{\alpha\in\mathcal{C}^{\prime}_{n,e}}g_{\alpha}(p),$
where $\mathcal{C}^{\prime}_{n,e}$ denotes the set of compositions in
$\mathcal{C}_{n,e}$ whose first coordinate is greater than $1$.
###### Proposition 7.1.
We have that,
$g_{5}(p^{9})=11p^{6}+14p^{5}+137p^{4}-16p^{3}+p^{2}+2p+3,$
and
$g_{6}(p^{9})=4p^{7}+76p^{6}+128p^{5}+56p^{4}-111p^{3}+43p^{2}-9p+1+g_{(3,2,2,1,1)}(p).$
###### Proof.
The result follows immediately from (7.1) and computations in previous
sections. ∎
Liu computes the values of $g_{n}(p^{e})$ for $n=3,4$ (cf. [Liu07, Proposition
6.1 and 6.2]). We record these values for $e=9$ and refer to [AKKM21, p. 231]
the values of $e$ which lie in the range $4\leq e\leq 8$. We find that
$g_{3}(p^{9})=p^{3}+4p^{2}+4p+1,$
and that
$g_{4}(p^{9})=11p^{4}+30p^{3}+9p^{2}+p+1.$
###### Proposition 7.2.
For $n>1$, the following relation holds
$\displaystyle f_{n}(p^{9})-f_{n-1}(p^{9})$ $\displaystyle=$ $\displaystyle
Q(n)(276480P_{10}(n)p^{10}+276480P_{9}(n)p^{9}+138240P_{8}(n)p^{8}$
$\displaystyle+34560P_{7}(n)p^{7}+34560P_{6}(n)p^{6}+11520P_{5}(n)p^{5}$
$\displaystyle+11520P_{4}(n)p^{4}+1440P_{3}(n)p^{3}+1440P_{2}(n)p^{2}$
$\displaystyle+240P_{1}(n)p+P_{0}(n))+{n-1\choose 5}g_{(3,2,2,1,1)}(p),$
where,
$\displaystyle Q(n)$ $\displaystyle=$ $\displaystyle\frac{n-1}{1393459200},$
$\displaystyle P_{10}(n)$ $\displaystyle=$
$\displaystyle(n-2)(n-3)(n-4)(n-5)(n-6)(n-7),$ $\displaystyle P_{9}(n)$
$\displaystyle=$ $\displaystyle n(n-2)(n-3)(n-4)(n-5)(n-6),$ $\displaystyle
P_{8}(n)$ $\displaystyle=$
$\displaystyle(n-2)(n-3)(n-4)(n-5)(n-6)(9n^{2}-131n+784),$ $\displaystyle
P_{7}(n)$ $\displaystyle=$
$\displaystyle(n-2)(n-3)(n-4)(n-5)(37n^{3}-761n^{2}+6482n-18144),$
$\displaystyle P_{6}(n)$ $\displaystyle=$
$\displaystyle(n-2)(n-3)(n-4)(75n^{5}-2468n^{4}+36349n^{3}-279672n^{2}$
$\displaystyle+1101372n-1732080).$ $\displaystyle P_{5}(n)$ $\displaystyle=$
$\displaystyle(n-2)(n-3)(n-4)(33n^{6}-1220n^{5}+21757n^{4}-208732n^{3}$
$\displaystyle+1053542n^{2}-2414076n+1592640),$ $\displaystyle P_{4}(n)$
$\displaystyle=$
$\displaystyle(n-2)(n-3)(42n^{8}-2102n^{7}+51106n^{6}-749135n^{5}$
$\displaystyle+7010050n^{4}-41819711n^{3}+152633830n^{2}$
$\displaystyle-307862664n+260890560),$ $\displaystyle P_{3}(n)$
$\displaystyle=$
$\displaystyle(n-2)(273n^{10}-18123n^{9}+571282n^{8}-10999886n^{7}$
$\displaystyle+140978185n^{6}-1241293667n^{5}+7531038196n^{4}$
$\displaystyle-30849468932n^{3}+81151302432n^{2}-123180801984n$
$\displaystyle+81613163520),$ $\displaystyle P_{2}(n)$ $\displaystyle=$
$\displaystyle(n-2)(147n^{11}-10843n^{10}+377832n^{9}-8095642n^{8}$
$\displaystyle+117343063n^{7}-1198590955n^{6}+8745674590n^{5}$
$\displaystyle-45324367680n^{4}+162663439888n^{3}-383226514464n^{2}$
$\displaystyle+531138427776n-326776343040),$ $\displaystyle P_{1}(n)$
$\displaystyle=$
$\displaystyle(n-2)(315n^{12}-25368n^{11}+956025n^{10}-22147762n^{9}$
$\displaystyle+349611873n^{8}-3946086924n^{7}+32542876455n^{6}$
$\displaystyle-196940866290n^{5}+865325849028n^{4}-2683887407672n^{3}$
$\displaystyle+5560471449216n^{2}-6887555482752n+3844971970560),$
$\displaystyle P_{0}(n)$ $\displaystyle=$ $\displaystyle
135n^{16}-14400n^{15}+730980n^{14}-23334360n^{13}+522513250n^{12}$
$\displaystyle-8678453720n^{11}+110319301164n^{10}-1092302298312n^{9}$
$\displaystyle+8494128305343n^{8}-51930963880392n^{7}+248190356069720n^{6}$
$\displaystyle-915171074718208n^{5}+2545385435375472n^{4}$
$\displaystyle-5146605000021888n^{3}+7110255039457536n^{2}$
$\displaystyle-5973926003435520n+2289569122713600.$
###### Proof.
Recall the recurrence relation (1.2)
$f_{n}(p^{9})=\sum_{i=0}^{9}\sum_{j=1}^{n}{n-1\choose
j-1}f_{n-j}(p^{9-i})g_{j}(p^{i}).$
Note that since $i\leq 9$, we find that $j\leq 10$ in the above recurrence
relation. We compute the values of $g_{j}(p^{i})$ and then recursively, use
the above to compute the value of $f_{n}(p^{9})$. In greater detail, we use
values from [AKKM21, p. 231], the Theorem 1.2, and the values computed in
previous sections, to obtain the above recurrence relation. ∎
Using the recurrence relation above and noting that
$f_{n}(p^{9})=\sum_{k=2}^{n}\left(f_{k}(p^{9})-f_{k-1}(p^{9})\right)$
we obtain Theorem 1.4. We performed all the computations in SageMath.
### 7.2. Some bounds on $g_{\alpha}(p)$
In this section we obtain upper bounds on $g_{\alpha}(p)$ for certain
compositions $\alpha$. Let $u_{1},\dots,u_{n}$ be the standard basis of
$\mathbb{Z}^{n}$. The vector $u_{i}=(0,0,\dots,0,1,0,\dots,0)$ consists of $0$
in all entries except for $1$ in the $i$-th entry. Let $L$ be an irreducible
subring of $\mathbb{Z}^{n}$ and let $\mathfrak{m}_{L}$ be the ideal in $L$
consisting of vectors all of whose coordinates are divisible by $p$. Setting
$\rho_{L}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{dim}_{\mathbb{F}_{p}}\mathfrak{m}_{L}/\mathfrak{m}_{L}^{2}$.
Following [Liu07, p. 292, l.-1], an irreducible subring is _full_ if
$\rho_{L}=n-1$. Assume that the matrix for $L$ is in Hermite normal form with
respect to the basis $u_{1},\dots,u_{n}$. Let $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{n}\rightarrow\mathbb{Z}^{n-1}$ be the projection onto
the last $(n-1)$ coordinates $u_{2},\dots,u_{n}$. Suppose that the basis is
chosen so that with respect to the ordered basis $(u_{1},\dots,u_{n})$, the
matrix $A$ associated with $L$ is in the Hermite normal form. Let
$v_{1},\dots,v_{n}$ be the columns of $A$, and let $\tau(L)$ be the lattice
generated by $L$ and by $v_{1}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{p}v_{1}$. It is easy to see that $\tau(L)$ is a subring
(cf. [Liu07, p.291, l.-2]).
###### Proposition 7.3.
Let $L$ be an irreducible subring of index $p^{e+n}$ such that $\pi(L)$ is
full and $pu_{1}\notin L$. Then the map
$L\mapsto\frac{1}{p}\mathfrak{m}_{\tau(L)}$ is a $p^{n-2}$-to-one surjection
onto subrings of index $p^{e}$.
###### Proof.
The above result is [Liu07, Proposition 5.6]. ∎
Let $\beta\mathrel{\mathop{\mathchar
58\relax}}=(\beta_{1},\beta_{2},\dots,\beta_{n-1})$ be a composition of length
$n-1$ such that $\beta_{1}>1$, $\beta_{i}>0$ for $i\in\\{2,\dots,n-1\\}$. By
abuse of notation, we say that an irreducible subring $L$ has diagonal $\beta$
if the entries on the diagonal of the matrix $A_{L}$ are
$p^{\beta_{1}},\dots,p^{\beta_{n-1}},1$. We set $\mathcal{R_{\beta}}$ to
denote the set of irreducible subrings with diagonal $\beta$. Let
$\beta^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=(\beta_{1}-2,\beta_{2}-1,\dots,\beta_{n-1}-1)$, let
$\mathcal{S}_{\beta^{\prime}}$ denote the set of subring matrices with
diagonal $\beta^{\prime}$. Let $\mathcal{R}^{\prime}_{\beta}$ be the subset of
$\mathcal{R}_{\beta}$ consisting of irreducible subrings with diagonal $\beta$
such that $\pi(L)$ is full and $pu_{1}\notin L$, with respect to the basis
$u_{1},\dots,u_{n}$ for which $A_{L}$ is in Hermite normal form. Given a
finite set $S$, set $\\#S$ to denote the cardinality of $S$.
###### Lemma 7.4.
Let $\beta\mathrel{\mathop{\mathchar
58\relax}}=(\beta_{1},\beta_{2},\dots,\beta_{n-1})$ be a composition of length
$n-1$ such that the following conditions are satisfied
1. (1)
$\beta_{1}>1$,
2. (2)
$\beta_{i}>0$ for $i\in\\{2,\dots,n-1\\}$,
3. (3)
$\sum_{i=1}^{n-1}\beta_{i}>n$.
Then, there is a $p^{n-2}$-to-one surjection from
$\mathcal{R}_{\beta}^{\prime}$ to $\mathcal{S}_{\beta^{\prime}}$. In
particular, we have that
$\\#\mathcal{R}_{\beta}\geq\\#\mathcal{R}^{\prime}_{\beta}\geq
p^{n-2}\\#\mathcal{S}_{\beta^{\prime}}.$
###### Proof.
It follows from Proposition 7.3 that the map
$L\mapsto\frac{1}{p}\mathfrak{m}_{\tau(L)}$ defines a $p^{n-2}$ to one
surjective map $\Phi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{R}_{\beta}^{\prime}\rightarrow\mathcal{S}_{\beta^{\prime}}$.
The inequality follows immediately from this. ∎
The following result comes as a consequence of the above Lemma.
###### Proposition 7.5.
Let $\beta\mathrel{\mathop{\mathchar
58\relax}}=(\beta_{1},\beta_{2},\dots,\beta_{n-1})$ be a composition of length
$n-1$ such that $\beta_{1}>1$, $\beta_{i}>0$ for $i\in\\{2,\dots,n-1\\}$ and
$\sum_{i=1}^{n-1}\beta_{i}>n$. Setting $m_{\beta}\mathrel{\mathop{\mathchar
58\relax}}=\min\\{\lfloor\frac{\beta_{1}}{2}\rfloor,\beta_{2},\dots,\beta_{n-1}\\}$,
we have the following lower bound for $g_{\beta}(p)$
$g_{\beta}(p)\geq p^{(n-2)m_{\beta}}.$
###### Proof.
The result follows upon repeatedly applying the Lemma 7.4. ∎
In particular we see that if $\alpha\mathrel{\mathop{\mathchar
58\relax}}=(k,\ell,\dots,\ell)$ of length $n-1$, then $g_{\alpha}(p)\geq
p^{{\lfloor\frac{k}{2}\rfloor}(n-2)}$. We note that this is a special case of
[Ish22b, Corollary 4.6].
###### Theorem 7.6.
We have the following polynomial lower bound
$g_{n}(p^{e})\geq\sum_{j=1}^{\lfloor
e/n\rfloor}p^{(n-2)j}\left(\binom{e-1-nj}{n-2}-\binom{e-1-n(j+1)}{n-2}\right).$
###### Proof.
Let $\mathcal{C}_{n,e}^{j}$ be the subset of $\mathcal{C}_{n,e}$ consisting of
all compositions $\beta=(\beta_{1},\dots,\beta_{n-1})$ such that
$m_{\beta}=j$. It is easy to see that
$\\#\mathcal{C}_{n,e}^{j}=\\#\mathcal{C}_{n,e-nj}-\\#\mathcal{C}_{n,e-n(j+1)}=\binom{e-1-nj}{n-2}-\binom{e-1-n(j+1)}{n-2}.$
It follows from Proposition 7.5 that
$g_{n}(p^{e})\geq\sum_{j=1}^{\lfloor
e/n\rfloor}p^{(n-2)j}\\#\mathcal{C}_{n,e}^{j},$
and the result follows. ∎
We now prove Theorem 1.6.
###### Proof of Theorem 1.6.
Let us assume for simplicity that $t=n-1$. The proof in generality is
identical to this case. Given an $n\times n$ matrix $A$, let $A_{[i,j]}$ be
the $i\times j$ matrix obtained upon deleting the last $(n-i)$ rows and
$(n-j)$ columns. Thus, $A_{[i,j]}$ is the upper left $i\times j$ submatrix of
$A$. Let $S$ denote the set of subring matrices of the form
$A=\begin{pmatrix}p^{\alpha_{1}}&\dots&\dots&\dots&a_{1}p&1\\\
&p^{\alpha_{2}}&\dots&\dots&a_{2}p&1\\\ &&\ddots&&\vdots&\vdots\\\
&&&&\vdots&\vdots\\\ &&&&p^{k\alpha_{n-1}}&1\\\ &&&&&1\end{pmatrix}$
that are in Hermite normal form. We note that the following conditions hold
1. (1)
$0\leq a_{i}\leq p^{\gamma-1}-1$,
2. (2)
setting $\hat{A}\mathrel{\mathop{\mathchar 58\relax}}=A_{[n-2,n-2]}$, we note
that
$(a_{1}^{2}p^{2}-a_{1}p^{k\alpha_{n-1}+1},\dots,a_{n-2}^{2}p^{2}-a_{n-2}p^{k\alpha_{n-1}+1})^{t}\in\operatorname{Col}\hat{A}=\hat{A}(\mathbb{Z}^{n-2}).$
We note in passing that $g_{\alpha_{/k,t}}(p)=\\#S$. We denote the set of all
irreducible subring matrices of size $(n-1)$ and with diagonal
$(p^{\alpha_{1}},p^{\alpha_{2}},\dots,p^{\alpha_{n-2}},1)$ by $\mathcal{A}$.
We note that for each index $i$ in the range $1\leq i\leq n-2$ we have that
$|a_{i}^{2}p^{2}-a_{i}p^{k\alpha_{i}+1}|\leq|a_{i}^{2}|p^{2}+|a_{i}p^{k\alpha_{i}+1}|\leq
2p^{(k+1)\gamma}$.
In order to simplify notation, for $B\in\mathcal{A}$, set
$B^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=B_{[n-2,n-2]}$. Setting
$\mathcal{C}\mathrel{\mathop{\mathchar
58\relax}}=[-2p^{(k+1)\gamma},2p^{(k+1)\gamma}]^{n-2}$, we find that
(7.2)
$\\#S\leq\sum_{B\in\mathcal{A}}\\#\left(B^{\prime}(\mathbb{Z}^{n-2})\cap\mathcal{C}\right).$
For $B\in\mathcal{A}$, the number of points in $v\in\mathbb{Z}^{n-2}$ such
that $B^{\prime}v\in\mathcal{C}$ is same as the number of points
$w\in\mathcal{C}$ such that
$\left(B^{\prime}\right)^{-1}(w)\in\mathbb{Z}^{n-2}$. Therefore using
inequality 7.2 we deduce that
(7.3)
$\\#S\leq\sum_{B\in\mathcal{A}}\\#(\left(B^{\prime}\right)^{-1}(\mathcal{C})\cap\mathbb{Z}^{n-2}).$
For $w=(w_{1},\dots,w_{n-2})\in\mathbb{Z}^{n-2}$, set $\lVert w\rVert$ to be
the Euclidean norm $\sqrt{\sum_{i}w_{i}^{2}}$. For any $w\in\mathcal{C}$ we
have that
$\lVert(B^{\prime})^{-1}w\rVert\leq\lVert(B^{\prime})^{-1}\rVert\lVert
w\rVert$. It is easy to see that $\lVert w\rVert\leq
2p^{(k+1)\gamma}\sqrt{n-2}$. On the other hand, since $\mathcal{A}$ is a
finite set which is defined independent of $k$, we find that for any
$B\in\mathcal{A}$ and $w\in\mathcal{C}$, $\lVert(B^{\prime})^{-1}w\rVert\leq
M\sqrt{n-2}p^{(k+1)\gamma}$ where $M\in\mathbb{R}_{>0}$ is a suitably large
constant (not depending on $k$). Using the inequality (7.3), we deduce that
(7.4)
$\\#S\leq\sum_{L\in\mathcal{A}}\\#([-M\sqrt{n-2}p^{(k+1)\gamma},M\sqrt{n-2}p^{(k+1)\gamma}]^{n-2}\cap\mathbb{Z}^{n-2})$
Fxing an integer $N>M\sqrt{n-2}$, we deduce that
(7.5)
$\\#S\leq\sum_{L\in\mathcal{A}}\\#([-Np^{(k+1)\gamma},Np^{(k+1)\gamma}]^{n-2}\cap\mathbb{Z}^{n-2})$
Therefore, $\\#S\leq\\#(\mathcal{A})(2N)^{n-2}p^{(k+1)\gamma{(n-2)}}$. We have
proved that
$g_{\alpha_{/k,t}}(p)=\\#S\leq\\#(\mathcal{A})(2N)^{n-2}p^{\gamma{(n-2)}}p^{\gamma
k{(n-2)}}$
Therefore, $g_{\alpha_{/k,t}}(p)=O(p^{\gamma k(n-2)})\text{ as }k\to\infty.$ ∎
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|
# A Component of the Smith High Velocity Cloud Now Crossing the Galactic Plane
Felix J. Lockman Green Bank Observatory, Green Bank, WV 24944, USA Robert A.
Benjamin Dept. of Physics, University of Wisconsin-Whitewater, Whitewater, WI
53190 Nicolas Pichette Dept. of Physics, Montana State University, Bozeman,
MT Christopher Thibodeau Current Address: 236 Braeden Brooke Dr. San Marcos,
TX 78666 Department of Physics, Astronomy & Geosciences, Towson University,
Towson, MD.
###### Abstract
We have identified a new structure in the Milky Way: a leading component of
the Smith high velocity cloud that is now crossing the Galactic plane near
longitude $25\arcdeg$. Using new 21cm Hi data from the Green Bank Telescope
(GBT) we measured the properties of several dozen clouds that are part of this
structure. Their kinematics is consistent with that of the Smith Cloud with a
$V_{\mathrm{LSR}}$ exceeding that permitted by circular rotation in their
direction. Most of the clouds in the Leading Component show evidence that they
are interacting with disk gas allowing the location of the interaction to be
estimated. The Leading Component crosses the Galactic plane at a distance from
the Sun of 9.5 kpc, about 4.5 kpc from the Galactic Center. Its Hi mass may be
as high as $10^{6}$ $M_{\mathrm{\odot}}$, comparable to the mass of the
neutral component of the Smith Cloud, but only a fraction of this is contained
in clouds that are resolved in the GBT data. Like the Smith Cloud, the Leading
Component appears to be adding mass and angular momentum to the ISM in the
inner Galaxy. We suggest that the Smith Cloud is not an isolated object, but
rather part of a structure that stretches more than $40\arcdeg$ ($\sim 7$ kpc)
across the sky, in two pieces separated by a gap of $\sim 1$ kpc.
ISM: clouds – ISM: individual objects (Smith Cloud) – Galaxy: structure –
Galaxy: evolution
††facilities: GBT
## 1 Introduction
The old puzzle of the Milky Way’s high velocity clouds – those aggregations of
neutral and ionized gas whose velocities deviate significantly from that
allowed by Galactic rotation – continues to confound and delight. The origin
of most high velocity clouds (HVCs) is uncertain. Because the category is
defined solely on kinematics (e.g., Wakker & van Woerden, 1991), it can
encompass everything from gas stripped from satellite galaxies, accretion from
the intergalactic and circumgalactic medium, and material recycled from the
Galactic disk (Oort, 1969; Wakker & van Woerden, 1997; Marinacci et al., 2010;
Binney & Fraternali, 2012; Putman et al., 2012). HVCs cover a large fraction
of the sky (Lockman et al., 2002; Lehner et al., 2022). The large HVC
complexes lie in the Galactic halo and are typically found within 10 kpc of
the Galactic plane (Putman, 2017). HVCs have now been detected in other
galaxies, e.g., M31 (Braun & Thilker, 2004; Thilker et al., 2004; Westmeier et
al., 2008; Wolfe et al., 2016), though not with the sensitivity and richness
of detail available to studies of the Milky Way.
HVCs are important in understanding the evolution of galaxies as they can be
manifestations of both gas accretion and recycling from a galactic fountain
(Putman, 2017; Richter, 2017; Fraternali, 2017; Li & Tonnesen, 2020). A number
of HVCs have a cometary or head-tail morphology or other indications that they
are interacting with an external medium (Putman et al., 2011; Barger et al.,
2020). There is also evidence for the interaction of infalling HVCs with
neutral gas in the Galactic plane (Lockman, 2003; McClure-Griffiths et al.,
2008; Park et al., 2016). These clouds must have survived their passage
through the halo to merge with the Milky Way disk.
The Smith Cloud (catalog ) is one of the most prominent high-velocity clouds;
it was discovered even before the identification of HVCs as a class of
interstellar object (Smith, 1963; Muller et al., 1963). In neutral hydrogen
emission the Smith Cloud extends over more than $10^{\circ}$ on the sky with a
highly-organized cometary shape. It has a magnetic field draped around it
(Hill et al., 2013; Betti et al., 2019). The Cloud contains at least $10^{6}$
$M_{\mathrm{\odot}}$ of Hi and probably an equal mass in ionized gas but has
no prominent stellar counterpart (Lockman et al., 2008; Hill et al., 2009;
Stark et al., 2015). Its metallicity has been determined through optical
emission and UV absorption spectroscopy to be approximately half Solar (Hill
et al., 2009; Fox et al., 2016), but this may reflect more the medium with
which it has exchanged material than any initial property of the cloud
(Gritton et al., 2014; Henley et al., 2017; Heitsch et al., 2022). The
distance to the Smith Cloud derived from three different methods is 12.4 kpc,
placing it 2.9 kpc below the Galactic Plane (Putman et al., 2003; Wakker et
al., 2008; Lockman et al., 2008). The kinematics of the cloud has been
analyzed by Lockman et al. (2008, hereafter L08) who suggest that it is on a
course to collide with the Milky Way disk in $\approx 30$ Myr, adding mass and
angular momentum to the Galaxy, and possibly triggering a burst of star
formation (Alig et al., 2018). The Smith Cloud has been modelled as the gas
remnant of a dwarf galaxy merging with the Milky Way (Bland-Hawthorn et al.,
1998), the baryonic component of a dark matter subhalo (Nichols & Bland-
Hawthorn, 2009; Nichols et al., 2014; Tepper-García & Bland-Hawthorn, 2018),
and the product of supernove or the Galactic fountain (Sofue et al., 2004;
Marasco & Fraternali, 2017).
This is the first in a series of papers exploring aspects of the Smith Cloud.
Future papers will report on studies of its structure, interaction with the
circumgalactic medium, molecular content, and trajectory. Here we describe a
major interstellar feature associated with the Smith Cloud: a leading
component that is now passing through the Galactic disk.
HVCs are easily detected at high Galactic latitude where their discrepant
velocities stand out most clearly from Galactic disk gas. There is no reason
to believe, however, that the phenomenon is not widespread, and that a
significant fraction of the HVC population has been missed through confusion
with gas at permitted velocities (Zheng et al., 2015). Indeed, distinctions
based solely on velocity certainly give a biased view of phenomena in the
Galactic halo (Bish et al., 2021; Marasco et al., 2022).
In a search for potential HVCs at low Galactic latitude, we constructed a
slice through the LAB Hi survey (Kalberla et al., 2005) in the first Galactic
quadrant, integrating over a velocity interval 15-35 $\rm km\,s^{-1}$ greater
than expected for circular Galactic rotation for a reasonable Galactic
rotation curve. In the inner Galaxy, the maximum value of
$|{V_{\mathrm{LSR}}}|$ from circular rotation occurs at the tangent point,
where the distance from the Sun, r, is $R_{0}cos(\ell)$, and $R_{0}$ is the
distance to the Galactic Center. By sampling emission from velocities slightly
greater than allowed by circular rotation we are measuring gas in the line
wing, which in the absence of non-circular motions should arise from regions
close to the tangent point. In this instance we used the rotation curve
function given by Brand & Blitz (1993) with coefficients from Burton (1988),
although as we will show, the basic findings are not sensitive to details of
the adopted rotation curve. The results over a $40^{\circ}\times 40^{\circ}$
region are shown in Figure 1.
The Hi emission visible in this figure arises from three separate sources.
First, because of temperature and turbulence there will always be some
emission in the line wings at velocities extending beyond that allowed by
Galactic rotation. The amount of emission in the line wings depends on the
velocity dispersion of the gas, but also on $|d{V_{\mathrm{LSR}}}/dr|^{-1}$ –
often called velocity crowding – which is the degree to which the velocity of
gas at different distances from the Sun, r, projects to a difference in
$V_{\mathrm{LSR}}$ (Burton, 1971; Celnik et al., 1979). The sense of this
geometric effect is that $V_{\mathrm{LSR}}$ changes more slowly with distance
at high longitudes than at low longitudes, so more emission appears in line
wings at a forbidden velocity at $\ell=50\arcdeg$ than at $\ell=30\arcdeg$.
The dashed yellow lines above and below latitude $0\arcdeg$ in Figure 1 mark a
distance $\pm 0.25$ kpc from the plane at the tangent points and outline
expectations for the contribution from line wings and velocity crowding.
A second factor affecting the appearance of Figure 1 is large-scale streaming
motions that arise from, for example, gas response to spiral arms (Burton,
1971). Although these are easily detected in the data (e.g., Fig. 8 in
McClure-Griffiths & Dickey (2007) and Figs. 6 and 7 in Levine et al. (2008)),
they are not included in rotation curves that describe only the symmetric
component. Large-scale streaming contributes to the emission observed in
Figure 1 around longitude $32\arcdeg$ and $50\arcdeg$. Note that both
turbulence and streaming motions are expected to be approximately symmetric
around the Galactic Plane, and that is what is observed.
But Figure 1 also shows emission not related to Galactic rotation in any
sense. The Smith HVC dominates latitudes $b\lesssim-10{{}^{\circ}}$ between
longitudes $35^{\circ}\leq\ell\leq 50^{\circ}$, and there is a component in Hi
emission almost as extensive as the Smith Cloud that lies along its axis but
extends through the Galactic plane, crossing it around longitude
$\ell=25\arcdeg$. A dashed cyan line connects these two features, which are
clearly not symmetric about the Galactic Plane. The existence of a band of
emission at forbidden velocity, aligned with the Smith Cloud, suggests that
both may be part of a larger structure that shares a common kinematic anomaly.
Because the morphology of the Smith Cloud indicates that its space motion is
towards lower longitude and more positive latitude (L08), i.e., to the upper
right in Figure 1, we call the component that extends through the Galactic
plane the “Leading Component”. To understand its properties more fully we
analyze new 21cm Hi observations from the Green Bank Telescope (GBT) made over
a large region for another purpose, and here present those data and the
results.
Figure 1: Map of the integrated Hi brightness temperature along a slice
through the LAB survey that includes only emission between 15 and 35 $\rm
km\,s^{-1}$ in excess of that allowed by circular Galactic rotation. Dashed
yellow lines mark a distance $z=\pm 0.25$ kpc from the plane at the tangent
point distance of each longitude. Normal thermal and turbulent motions, along
with velocity crowding, will produce forbidden-velocity emission in this range
whose brightness is expected to increase to higher longitude over areas
approximately bound by curves of constant $|z|$. The Smith Cloud is in the
lower left between longitudes $40\arcdeg\lesssim\ell\lesssim 50\arcdeg$
centered around latitude $b\approx-15\arcdeg$. This Figure shows that there is
also emission that extends through the Galactic plane along the axis of the
Cloud, marked by the diagonal dashed cyan line. We call this the Leading
Component. The grey scale is proportional to the square root of the integrated
Hi brightness temperature.
In section 2 the new GBT Hi survey is described, as well as an existing study
that provides high resolution images of several clouds in the Leading
Component. A catalog of Hi clouds found in the Leading Component in presented
in section 3 along with a discussion of their general properties. Section 4
discusses evidence that some of these leading clouds are interacting with
normally rotating disk gas, allowing derivation of a kinematic distance. Their
overall space velocity is analyzed in section 5, which is followed by a
discussion of the implications of the present work.
## 2 Observations
### 2.1 The GBT Hi Survey
Figure 2: Longitude-latitude maps from the new GBT Hi survey. Coverage is
almost complete over $13\arcdeg\leq\ell\leq 35\arcdeg$ and $-10\arcdeg\leq
b\leq+10\arcdeg$. The new data show the tip of the Smith Cloud in the lower
left of the left panel. The Leading Component stands out as a band of emission
with the same position angle as the Smith Cloud, extending from
$(\ell,b)=(32\arcdeg,-7\arcdeg)$ to $(22\arcdeg,+4\arcdeg)$. Black rectangles
mark areas blanked in the GBT data. Left Panel: The integrated Hi emission
over $+226\leq{V_{\mathrm{GSR}}}\ \leq+275\ {\rm km\,s^{-1}}$. Right Panel:
Channel map at ${V_{\mathrm{GSR}}}=243.8\ {\rm km\,s^{-1}}$ over the central
region of the GBT survey. Here we see that some of the Leading Component is
resolved into individual clouds.
Measurements of the 21cm line of Hi were made with the Green Bank Telescope
(GBT) using the standard L band receiver and the GBT spectrometer (Prestage et
al., 2009). Data were taken ”on-the-fly” using in-band frequency switching and
were calibrated as described in Boothroyd et al. (2011). A low order
polynomial was fit to emission-free regions of the spectra after they were
gridded into a cube with a channel spacing of 0.80 $\rm km\,s^{-1}$ over a
velocity range of $-200\leq{\rm V_{LSR}}\leq+275$ $\rm km\,s^{-1}$. The data
cover $13^{\circ}\leq\ell\ \leq 35^{\circ}$ over a latitude range $\pm
10^{\circ}$ with a partial extension to latitude $-14^{\circ}$ at $\ell\geq
30^{\circ}$ to include the tip of the Smith Cloud. The effective angular
resolution is $10\arcmin$. The median rms noise in brightness temperature for
a 1 $\rm km\,s^{-1}$ channel is 0.12 K. The noise varies somewhat across the
mapped area, being greatest within $2^{\circ}$ of the Galactic plane where
there was often a significant increase in the system temperature from
continuum emission. A small fraction of the mapped area had to be blanked over
regions of strong continuum emission or because of occasional radio frequency
interference.
For study of the Leading Component, the new data have a distinct advantage
over the premier existing Hi survey, HI4PI (HI4PI Collaboration et al., 2016).
Emission from the leading component is not particularly faint, so although
HI4PI has about one-third lower noise level than the new GBT data, noise is
not a critical factor. The $10\arcmin$ angular resolution of the GBT data
compared with the $16\farcm 2$ of HI4PI is, however, important in sorting out
confusion between the leading component and other Hi emission. Virtually all
of the Leading Component lies at J2000 declination $\delta<-5{{}^{\circ}}$,
which is outside the declination limit of any survey using the Arecibo
telescope.
The new GBT survey is shown in Figure 2 as emission integrated over forbidden
velocities (left), i.e., those velocities not permitted by circular Galactic
rotation, and as a map of emission in a single velocity channel (right). The
left panel shows the tip of the Smith Cloud at the lower left and the Leading
Component in the center of the Figure. The velocity with respect to the Local
Standard of Rest, $V_{\mathrm{LSR}}$, is not appropriate for the study of
objects that cover a large angle on the sky, as the changing projection of the
LSR across the structure becomes significant. A more suitable measure is the
velocity with respect to the Galactic Standard of Rest defined as
${V_{\mathrm{GSR}}}={V_{\mathrm{LSR}}}+|V_{0}|sin(\ell)cos(b)$, where V0 is
the circular velocity of Galactic Rotation at the Sun’s distance from the
Galactic Center, ${\rm R_{0}}$. Throughout this paper we use ${\rm R_{0}}=8.1$
kpc and ${\rm V_{0}=230}\ {\rm km\,s^{-1}}$, consistent with the results of
recent studies (Gravity Collaboration et al., 2018; Eilers et al., 2019). The
right panel of Figure 2 is a channel map over the central part of the new GBT
survey at a constant $V_{\mathrm{GSR}}$ of 243.8 $\rm km\,s^{-1}$, showing
that the Leading Component contains discrete clouds.
### 2.2 High Angular Resolution Measurements
Pidopryhora et al. (2015, hereafter P2015) combined GBT and VLA data to
produce Hi maps of 10 interstellar clouds at $\sim 1\arcmin$ angular
resolution, corresponding to a linear resolution of 3 pc at a distance of 10
kpc. The clouds cover a range of longitude ($16\fdg 0\leq\ell\ \leq\ 44\fdg
8$) and latitude ($-8\fdg 7\ \leq b\leq+4\fdg 5$). The sample was selected to
have little confusion with other neutral Hi, so these clouds are somewhat
outside of the Galactic plane (median $|b|=4\fdg 9)$ and have a velocity
beyond the terminal velocity (${V_{\mathrm{LSR}}}\geq\ V_{t}$), where the
terminal velocity is the maximum permitted by Galactic rotation in their
direction. Three of the P2015 clouds turn out to be clouds in the Leading
Component. Although the noise in the P2015 data is about three times worse
than the GBT data (approximately 0.36 K in a 1 $\rm km\,s^{-1}$ channel) its
much higher angular resolution reveals important structure within the clouds.
## 3 Leading Clouds
Figure 3: GBT Hi spectra showing examples of the clouds detected in this
survey associated with the Leading Component of the Smith Cloud identified by
their $\ell,b$ coordinates. The leading cloud is always the highest velocity
spectral feature, marked with the solid arrow. All these clouds appear to be
interacting with a component of disk gas at a lower (permitted) velocity,
marked with a dashed arrow. This is discussed in more detail in section 4.
Figure 4: GBT Hi spectra of Leading Component clouds that are confused with
lower velocity emission (arrows). The longitude and latitude is indicated in
each panel. These spectral features are distinct clouds in channel maps and
have a $V_{\mathrm{LSR}}$ exceeding that permitted by Galactic rotation, but
cannot be separated from lower velocity emission and, like others with similar
properties, are therefore not included in Table 1.
### 3.1 Cloud Properties
At the angular resolution of the GBT data the Leading Component contains a
number of individual clouds many of which are isolated in $\ell$, b, and
velocity, so that their properties can be measured accurately. The right panel
of Figure 2 shows a region of the survey centered near
$(\ell,b)=(27\arcdeg,-2\arcdeg)$ in which a number of individual clouds can be
identified. The clouds lie in a swath that begins a few degrees above the top
of the Smith Cloud around $(\ell,b)=(35\arcdeg,-8\arcdeg)$ and extends at
least to $b>+4{{}^{\circ}}$ at $\ell<24{{}^{\circ}}$. Discrete clouds
associated with the leading component were identified using three criteria: 1)
a location on or near the main region of the leading component; 2) a
$V_{\mathrm{LSR}}$ that was close to or beyond that permitted by circular
Galactic rotation in their direction; and, 3) for most of the clouds, evidence
that they were interacting with gas at a lower, permitted velocity. Spectra
towards 4 of the leading clouds are shown in Figure 3. They are detected well
above the noise in the GBT data.
There are also cloud-like structures visible in channel maps whose spectral
components cannot be separated from lower velocity emission. This is
especially common close to the Galactic plane. Examples are shown in Figure 4.
These are not included in Table 1.
Clouds were identified in channel maps and a Gaussian was fit to the spectrum
at the location of the peak $N_{\mathrm{{\rm HI}}}$. A 2-d Gaussian model was
used to derive an angular size and position angle. Hi masses were measured by
integrating over an area of the channel map encompassing the measurable extent
of the cloud at the velocity of the Hi peak. Background Hi emission was
estimated from measurements around the edge of the cloud and subtracted. The
result was then multiplied by the FWHM of the line at the brightest location
to obtain a line power integrated over area and velocity. Some clouds were so
confused with other emission that it was not possible to measure their size.
Properties of 38 clouds are given in Table 1. We emphasize that the sample of
clouds presented here does not completely account for all the clouds or all
the Hi mass in the leading component; it represents only the more easily-
identifiable and less-confused clouds in the GBT data. The longitude and
latitude of the peak Hi brightness is in the first two columns, and quantities
in cols. 3 through 6 refer to that location. Errors in line parameters were
derived from the Gaussian fit to the spectrum. The median peak $T_{b}$ of the
clouds is 2.5 K which is a detection at the $20\sigma$ level above the noise.
The faintest cloud is detected at $6.5\sigma$. The angular size in col. 7 is
not corrected for beam smearing. The position angle is measured counter-
clockwise with reference to the Galactic North Pole such that a cloud
elongated parallel to the Galactic plane would have PA $=90\arcdeg$. For the
purposes of calculating general cloud properties we assume that they are all
at a distance $d_{10}=10$ kpc. This is comparable to the distance of the Smith
Cloud, 12.4 kpc (L08), and results can be scaled easily. The distance of the
Leading Component is discussed further in section 4.
Figure 5: Properties of the leading clouds. Top left: $N_{\mathrm{{\rm HI}}}$
measured at the position of peak line brightness assuming optically thin
emission. Top right: Cloud diameter deconvolved for beam smearing assuming a
10 kpc distance. Center left: Hi mass of the leading clouds assuming a
distance of 10 kpc. The median $M_{{\rm HI}}=1.6\times 10^{3}\ \ (d_{10}^{2})\
{M_{\mathrm{\odot}}}$. Center right: Estimate of the average volume density in
the clouds from the ratio of the peak column density to the diameter. The
diameter is corrected for beam convolution and scaled to a distance
$d_{10}=10$ kpc. The median $n_{HI}=0.43\ {\rm cm^{-3}}$. Bottom: Distribution
of position angles for the leading clouds. Each cloud is displayed twice, once
at its PA and once at its PA+180∘. The two vertical arrows mark the PA of the
Smith Cloud, $130^{\circ}$. The data suggest that many individual clouds share
the same elongation as the Smith Cloud.
Properties of the leading cloud population are summarized in Table 2.
The measured angular size of the clouds, $\Theta_{obs}$, was corrected for
beam convolution by subtracting the $10\arcmin$ resolution of the data cube in
quadrature from the measured $\Theta_{maj}$ and $\Theta_{min}$. Cloud
diameters were derived from the square root of the product of the major and
minor deconvolved angular sizes, at a distance of 10 kpc. They are shown in
the upper right panel of Figure 5.
The cloud Hi mass distribution is shown in Figure 5 for an assumed distance of
10 kpc. The sum of the Hi mass of the tabulated clouds is $1.1\times 10^{5}$
$M_{\mathrm{\odot}}$. The Hi mass in the entire leading component was
estimated by summing over a large region encompassing the leading component,
and integrating over all ${V_{\mathrm{GSR}}}\geq 226$ $\rm km\,s^{-1}$. The
result, $8\times 10^{5}$ $M_{\mathrm{\odot}}$ for a distance of 10 kpc, has a
large uncertainly, but implies that the tabulated clouds contain only a small
fraction ($\sim 15\%$) of the total Hi associated with the system.
A lower limit to the volume density of the clouds can be derived from the
ratio of the peak column density, $N_{\mathrm{{\rm HI}}}$, to the cloud
diameter. This is a lower limit because the observed $N_{\mathrm{{\rm HI}}}$
is certainly reduced to some extent by beam convolution, while this effect was
taken out, at least approximately, in calculating the diameter.
The distribution of cloud position angles is shown in the bottom panel of
Figure 5, where to avoid artificial discontinuities at $180^{\circ}$ each
cloud is counted twice, once at its PA and once at its PA+180∘. The
distribution peaks around the PA of the Smith Cloud, which is also the PA of
the Leading Component, $130^{\circ}$, suggesting that some individual clouds
are being shaped by the same interaction with their environment as the Smith
Cloud itself.
### 3.2 Comparison with higher angular resolution observations
Three of the leading clouds from Table 1 were observed in the P2015 survey and
we can compare properties derived from GBT data with those from the higher
angular resolution data. The results are given in Table 3, where P2015
diameters have been scaled to a distance of 10 kpc. Velocities are in good
agreement, while the higher angular resolution of P2015 gives larger peak
values of $N_{\mathrm{{\rm HI}}}$ by a factor of 2 - 3. The higher resolution
data often reveal cloud cores with a diameter less than one-third that given
by the GBT data. The increase in peak $N_{\mathrm{{\rm HI}}}$ and smaller
cloud core size combine to increase the central cloud density by factors of
more than six. It is interesting that these clouds are elongated along the
direction of the Leading Component axis, with position angles clustered around
the $130\arcdeg$ PA of the Smith Cloud.
### 3.3 Leading Cloud Kinematics
Values of $V_{\mathrm{LSR}}$ and $V_{\mathrm{GSR}}$ for the clouds are shown
in a position-velocity diagram in Figure 6. The solid line is at the median
$V_{\mathrm{GSR}}$ of the sample, 237 $\rm km\,s^{-1}$, while the mean
$V_{\mathrm{GSR}}$ is identical at $237.1\pm 6.8$ $\rm km\,s^{-1}$($1\sigma)$.
The $V_{\mathrm{GSR}}$ of the central region of the Smith Cloud, marked by the
dashed line, is essentially identical to that of the leading clouds at 241
$\rm km\,s^{-1}$. The dotted line at 226 $\rm km\,s^{-1}$ marks the lower
velocity limit of the integrated emission displayed in the left panel of
Figure 2, which was used to estimate the Hi mass of the entire Leading
Component system.
As we will discuss in section 5, the quantity $V_{\mathrm{GSR}}$ is directly
related to the total space velocity of an object. The combination of similar
$V_{\mathrm{GSR}}$ and an apparent association on the sky strongly supports a
physical connection between the Leading Component and the Smith Cloud.
From a rotation curve which gives $V_{\theta}(R)$, a terminal (maximum)
velocity can be calculated for every longitude $|\ell|<90\arcdeg$: $Vt\equiv
cos(b)\ [V_{\theta}(R_{0})sin(\ell)-V_{\theta}(R_{t})]$, where $R_{t}=R_{0}\
sin(\ell).$ Each cloud can then be assigned a ”forbidden” velocity,
${V_{\mathrm{for}}}\equiv{V_{\mathrm{LSR}}}-V_{t}$, which gives the amount by
which the cloud’s $V_{\mathrm{LSR}}$ is in excess of that allowed by circular
Galactic rotation at any location along that line of sight.
We use two rotation curves: a flat rotation curve with $V_{\theta}(R)\equiv
V_{0}=230\ {\rm km\,s^{-1}}$, which is very close to the curve measured by
Eilers et al. (2019) at $R\geq 5.27$ kpc, and a ”Universal” rotation curve
(Persic et al., 1996) with coefficients from Mróz et al. (2019), that we call
the PM model. The PM model differs from that of Reid et al. (2014) by only a
few percent over the region of interest.
Figure 6: The leading clouds in position and velocity. The $V_{\mathrm{LSR}}$
(blue dots below) and $V_{\mathrm{GSR}}$ (red stars above) of the leading
cloud sample is plotted vs. longitude. The solid horizontal line is the median
$V_{\mathrm{GSR}}$ of the clouds, 237 $\rm km\,s^{-1}$, while the dashed line
is the $V_{\mathrm{GSR}}$ of the Smith Cloud, 241 $\rm km\,s^{-1}$. There is
no strong trend in $V_{\mathrm{GSR}}$ with longitude implying that the entire
complex is consistent with having a single velocity vector projected to
different values of $V_{\mathrm{LSR}}$ at different locations. The dotted line
shows the lower limit to the $V_{\mathrm{GSR}}$ used in integrating over
velocity to create the moment-zero map in the left panel of Figure 2.
Figure 7 shows the ”forbidden velocities” $V_{\mathrm{for}}$ for the cloud
sample plotted against longitude from the PM rotation curve on the left and
the flat rotation curve on the right. The median value of $V_{\mathrm{for}}$
is 17.3 and 8.1 $\rm km\,s^{-1}$, respectively. Forbidden velocities are
larger in the PM model because, unlike the flat model, the PM rotational
velocities decrease toward the Galactic Center reducing the maximum permitted
$V_{\mathrm{LSR}}$ at lower longitudes. When analyzed with a flat rotation
curve there are five clouds with ${V_{\mathrm{for}}}\lesssim 0\ {\rm
km\,s^{-1}}$, but these all show signs of interaction with lower-velocity gas
implying that they have a high non-circular velocity component (discussed in
section 4). In Figure 7 clouds with evidence of interaction are marked with a
red star.
Figure 7: The ”forbidden velocity” of the cloud sample, $V_{\mathrm{for}}$,
defined as the difference between a cloud’s $V_{\mathrm{LSR}}$ and the maximum
permitted by circular Galactic rotation in their direction (blue dots). This
is a lower limit to the peculiar line-of-sight velocity of a cloud. A more
realistic estimate of a cloud’s deviation from circular rotation is given by
the quantity $V_{\mathrm{dif}}$, discussed in section 4. If the clouds show
signs of interaction with lower velocity gas the points are overplotted with
red stars. Left panel: Forbidden velocities evaluated for the Persic
”Universal” rotation curve (Persic et al., 1996) with coefficients from Mróz
et al. (2019). Right Panel: Forbidden velocities for a flat rotation curve at
a circular velocity of 230 $\rm km\,s^{-1}$. While there are 5 clouds with
${V_{\mathrm{for}}}\lesssim 0$ for a flat rotation curve, values for the more
realistic Persic curve, and evidence for interaction, suggests that they have
anomalous velocities compared to their location in the Galaxy and are properly
included in the leading cloud sample.
## 4 Interaction with Disk Gas
Most of the clouds appear to be interacting with gas at lower
${V_{\mathrm{LSR}}}$, i.e., with regularly rotating material in the Milky Way
disc. In some cases the spectra show continuous emission between the velocity
of the leading cloud and a lower-velocity component, overlapping in position
with the cloud. The spectra in Figure 3 show four examples, where the lower
velocity gas is marked with a dashed arrow. In many cases the connection is
especially convincing, as illustrated in Figure 8. Here, in a display of
latitude vs. $V_{\mathrm{LSR}}$, we can see one or more edges of leading
clouds forming a continuous bridge of Hi emission to lower velocities,
consistent with the outer regions of the clouds being stripped and
decelerated. We compare Figure 8 to Figure 9 (from L08), which is a velocity-
position cut through the minor axis of the Smith Cloud showing the same
kinematic bridge between the anomalous velocity of the Smith Cloud and gas in
the Milky Way disk.
Figure 8: Examples of leading clouds showing their interaction with the Milky
Way disk gas. In these latitude-$V_{\mathrm{LSR}}$ plots the leading cloud is
always the feature at the highest velocity, marked with an arrow. There are
”bridges” to lower velocity Hi emission connecting these clouds to gas at a
velocity consistent with normal Galactic rotation. This is often accompanied
by an enhancement in the lower velocity emission. We interpret these figures
as showing that these leading clouds are interacting with normal disk gas,
with some cloud material being stripped away and decellerating to the lower,
permitted velocity. Figure 9: From Lockman et al. (2008), a position-velocity
cut through the minor axis of the Smith Cloud, showing the same transition at
the cloud edge between anomalous velocity gas and normally-rotating disk gas
as is seen at the edges of the leading clouds of Figure 8. The arrow marks a
small cloud that we interpret as having been detached from the Smith Cloud and
decellerated by its interaction with the circumgalactic medium of the Milky
Way.
The high-resolution P2015 data give additional information on the interaction
for two leading clouds: G22.74+4.30 and G26.84-6.38 (we refer to these clouds
by their GBT positions in Table 1, which deviate somewhat from the location of
the brightest $N_{\mathrm{{\rm HI}}}$ in the P2015 data). A map of total
$N_{\mathrm{{\rm HI}}}$ and a position-velocity cut through each cloud is
shown in Figure 10. Unlike the clouds in Figure 8, these clouds do not exhibit
kinematic bridges of material that join the cloud with lower-velocity gas, but
in both cases there is a bright Hi emission component centered approximately
at the position of the leading cloud, but at a much lower velocity. The
interaction between cloud and disk gas for G25.20+4.42 is shown more clearly
in the GBT data of Figure 8 than in the P2015 data because the P2015 data do
not have the sensitivity to detect the extended bridge of emission that
connects this cloud kinematically to the disk.
It is interesting and perhaps significant that of the 10 clouds in the P2015
sample, only those three associated with the Leading Component show
unambiguous evidence of interaction with gas at lower velocities.
In Table 1 the $V_{\mathrm{LSR}}$ of gas interacting with leading clouds is
given as $V_{MW}$. This quantity is blank for clouds without identifiable
interaction. The quantity $V_{\mathrm{dif}}$, which is the difference in line-
of-sight velocity between the cloud and the disk gas that it is encountering,
${V_{\mathrm{dif}}}\equiv{V_{\mathrm{LSR}}}-V_{MW}$, is shown in Figure 11.
The evidence for interaction provides two important pieces of information:
1. 1.
the leading clouds are in fact kinematically anomalous by a significant
amount;
2. 2.
the velocity of the interacting gas might be used to derive a distance to the
interaction.
Figure 10: Top panels: High angular resolution images of the total
$N_{\mathrm{{\rm HI}}}$ from the P2015 data for two of the leading clouds
observed in that survey: 22.74+4.30 (left) and 26.84-6.38 (right). Bottom:
Cuts through the clouds in longitude-velocity (22.74+4.30, left) and latitude-
velocity (26.84-6.38, right) along the directions given by dotted lines in the
upper panels. The leading clouds are marked with a solid arrow and a dashed
arrow identifies the Milky Way disk gas with which they are interacting.
Unlike the examples in Figure 8, these data do not show kinematic bridges
between the clouds and Milky Way gas. Instead, we derive evidence of
interaction from the high degree of spatial coincidence between the gas at
forbidden and permitted velocities, coincidence which is not seen in P2015
clouds except for those that are part of the Leading Component. The velocity
of the interacting gas is used to derive a kinematic distance to the
interaction. Figure 11: The line-of-sight velocity difference
$V_{\mathrm{dif}}$ between the leading clouds and interacting gas, plotted
against longitude. The median value is 44 $\rm km\,s^{-1}$, and this is the
best estimate of the line-of-sight deviation of the leading clouds from
circular rotation at their location.
### 4.1 Location of the Interaction
Figure 12: Location of regions where the leading clouds appear to be
interacting with disk gas (small blue circles). These were derived for the PM
rotation curve with error bars showing the effect of a $\pm 5\ {\rm
km\,s^{-1}}$ change in $V_{\mathrm{GSR}}$. Red star and crosses are components
of the Smith Cloud (SC) assuming a constant distance from the Sun for this
object of 12.4 kpc. The large blue circle is at the position of the Sun and
the ”x” shows the Galactic center. (a) Projection of the cloud interaction
locations on the x-z plane where x is in the direction of the Galactic Center
(which is at 0,0), and z is distance from the Galactic plane. (b) Projection
of cloud interaction locations on the z-y plane, where y is perpendicular to
the line of sight from the Sun to the Galactic Center. To some extent the
correlated structure in this panel simply reflects the alignment of the
Leading Component with the Smith Cloud in longitude and latitude. (c) Cloud
interaction locations in the x-y plane, which lies in the plane of the Milky
Way. The black star shows the position of the Smith Cloud where it is
projected to cross the Galactic plane in 27 Myr from the trajectory given in
L08. The sense of Galactic rotation is indicated by the arrow extending
upwards from the symbol for the Sun.
We assume that the velocity of material apparently interacting with a leading
cloud is in fact disk gas in circular rotation and thus can be used to
identify the location of the interaction within the Galaxy. An identical line
of argument was used for the Smith Cloud and produced a distance consistent
with that derived from independent methods (L08). While there are many
uncertainties in this approach, and the distance to any individual cloud may
have a large error, this kinematic analysis has the potential to provide
information on the location of the leading cloud population that is not
otherwise available. In respect of the uncertainties in derived distances we
do not discuss the location of individual clouds but rather limit our comments
to properties of the ensemble. These do not vary greatly between the two
rotation curves that were used.
We analyze the cloud interactions with the two rotation curves described
earlier: a flat rotation curve with $V_{\theta}(R)=230\ {\rm km\,s^{-1}}$, and
the PM ”Universal” rotation curve (Persic et al., 1996; Mróz et al., 2019).
For objects at the longitudes of the leading clouds, a given positive value of
$V_{\mathrm{LSR}}$ occurs at a single value of R, the distance from the
Galactic Center, but at two distances from the Sun. For the leading cloud
sample the derived near distance is around 4 kpc, while the far distance is
$>9$ kpc. Given that the Smith Cloud is at a distance of 12.2 kpc we always
choose the far kinematic distance for the leading clouds. The velocity of the
interacting gas is rarely determined with high precision: to estimate an
effect of this uncertainly on our results we evaluated the kinematic distances
at the $V_{\mathrm{LSR}}$ of the interaction $\pm 5\ {\rm km\,s^{-1}}$. In the
few cases where a velocity is greater than permitted by the Galactic rotation
model the tangent point distance is adopted as the most likely.
Figure 12 shows derived distances to the interacting gas and by inference, to
the leading clouds themselves, using the PM rotation curve where the Sun is at
$x=-R_{0}=-8.1$ kpc. The interaction sites of the leading clouds cluster
around median values of $x,y,z=0.2,4.5,-0.6$ kpc, at a median distance from
the Sun of 9.5 kpc. This location is more than 8 kpc away from the spot where
the Smith Cloud is expected to cross the Galactic disk according to the
trajectory of L08, marked with a black star on panel (c) of Figure 12.
Figure 13: Derived values of $V_{\theta}$ as a function of $cos(\ell-\theta)$
for $V_{R}=0$, assuming that so close to the Galactic plane $V_{z}sin(b)=0$.
For very low values of $cos(\ell-\theta)$ any $V_{R}$ component will be
projected across the line of sight and $V_{\mathrm{GSR}}$ will be a function
of $V_{\theta}$ alone (Equation 1). The horizontal line shows the adopted
$V_{\theta}=240$ $\rm km\,s^{-1}$. Figure 14: Derived values of $V_{R}$ for
$V_{\theta}=240\ {\rm km\,s^{-1}}$ assuming that so close to the Galactic
Plane $V_{z}sin(b)=0$. The horizontal line shows the median value, 65 $\rm
km\,s^{-1}$. Error bars show the effect of changing $V_{\mathrm{\theta}}$ by
$\pm 10$ $\rm km\,s^{-1}$, and the dashed lines show the median values of
$V_{R}$ that result.
### 4.2 Infall or Outflow?
Throughout this work we have assumed that the Smith Cloud and the Leading
Component are manifestations of infall into the Milky Way. There have been
suggestions, however, that the Smith Cloud results from a fountain process and
is thus now moving away from the Galactic plane (Sofue et al., 2004; Marasco &
Fraternali, 2017). We believe that the properties of clouds in the Leading
Component resolve this issue. For one, the Leading Component is found both
above and below the Galactic plane at the same angle with respect to the plane
as the Smith Cloud. This seems a very unlikely geometry for a supernova-driven
superbubble originating in the Milky Way disk, but is unremarkable for an
object passing through the Galactic plane. Second, some leading clouds have
components that connect them to lower velocity emission, emission that is
allowed by Galactic rotation (Figure 8). Were these clouds being accelerated
outwards by a fast wind, the presumably stripped gas would be at a higher
velocity. These considerations lead to the same conclusion when applied to the
Smith Cloud itself (Figure 9). This discussion has no bearing on the origin of
these objects, but the case for their infall at the current time is
persuasive.
## 5 Space Velocity
In a right-handed cylindrical coordinate system centered on the Galactic
center where the angle $\theta$ is measured from the Sun-center line such that
$\ell\approx\theta$ at large distances, $V_{\mathrm{GSR}}$ can be written as
the sum of three components, $V_{\theta},{V_{\mathrm{R}}},$ and
$V_{\mathrm{z}}$
${V_{\mathrm{GSR}}}=[-R_{0}\ sin(\ell)\ V_{\theta}/R\ \ +V_{R}\
cos(\ell-\theta)]cos(b)\ +V_{z}sin(b)$ (1)
where in this system the circular velocity $V_{\theta}<0$.
Figure 15: Derived values of $V_{z}sin(b)$ for $V_{\theta}=240\ {\rm
km\,s^{-1}}$ and ${V_{\mathrm{R}}}=65\ {\rm km\,s^{-1}}$. The horizontal line
shows the median value, 0.2 $\rm km\,s^{-1}$. Error bars show the effect of
changing the adopted $V_{\mathrm{\theta}}$ by $\pm 10$ $\rm km\,s^{-1}$. The
lines formed with dashes and plus signs show values expected for $V_{z}=-100$
$\rm km\,s^{-1}$ and $V_{z}=+100$ $\rm km\,s^{-1}$, respectively. The small
values of the derived $V_{z}sin(b)$ imply that $V_{z}$ cannot be determined
accurately from these data.
There are two limiting cases of interest: first, near the Galactic Plane
$sin(b)\approx 0$, so vertical velocities do not contribute significantly to
$V_{\mathrm{GSR}}$. Second, at locations near the tangent points the quantity
$cos(\ell-\theta)\approx 0$, and the radial component of motion, $V_{R}$, does
not contribute to $V_{\mathrm{GSR}}$. At these locations $V_{\mathrm{GSR}}$ is
a direct measure of $V_{\theta}$. Figure 13 shows $V_{\theta}$ derived from
these assumptions plotted against $cos(\ell-\theta)$. The five clouds with the
lowest values of $cos(\ell-\theta)$ give similar results for $V_{\theta}$
which we will adopt as its most likely value, 240 $\rm km\,s^{-1}$, indicated
by a horizontal line in Figure 13.
For a constant $V_{\theta}=240\ {\rm km\,s^{-1}}$, and, assuming that
$V_{z}sin(b)$ is small close to the Galactic plane, Equation 1 can be solved
for the $V_{R}$ of the leading clouds. The results are shown in Figure 14
where the median value, 65 $\rm km\,s^{-1}$, is indicated by a horizontal
line. The error bars show the effect of a 10 $\rm km\,s^{-1}$ change in
$V_{\mathrm{\theta}}$.
In principle, having estimates for $V_{\mathrm{\theta}}$ and $V_{\mathrm{R}}$
should allow estimation of $V_{\mathrm{z}}$ for each cloud, but the
determination is not robust, as sin(b) for these clouds is so small. Figure 15
shows $V_{\mathrm{z}}$sin(b) vs. latitude with expectations for $V_{z}=\pm
100$ $\rm km\,s^{-1}$. The small values and large uncertainties suggest that
$V_{\mathrm{z}}$ cannot be determined reliably so close to the Galactic Plane.
The morphology and kinematics of the Smith Cloud indicate that its motion is
to higher Galactic latitude, which requires ${V_{\mathrm{z}}}>0$. If the
leading clouds share this motion they should have ${V_{\mathrm{z}}}\ sin(b)<0$
at $b<0$ and lie along a curve like the one marked by the plus signs.
The estimated values for the space velocity of the Leading Component clouds
are compared with these for the Smith Cloud from L08 in Table 4. They are
generally consistent.
The total space velocity of the Leading Component clouds is $V_{tot}>250\ {\rm
km\,s^{-1}}$ where the inequality indicates that we are unable to establish a
reliable value for $V_{z}$. The Leading Clouds have a median line-of-sight
velocity excess of 44 $\rm km\,s^{-1}$ with respect to the disk gas at their
location. This is in the direction of Galactic rotation. As gas is stripped
from these clouds (Figure 8) it adds not only mass but angular momentum to the
disk.
## 6 What is the Leading Component?
In this paper we have presented arguments for the existence of an organized
interstellar structure whose orientation and kinematics are not consistent
with Galactic rotation, but which has a connection to the Smith high velocity
cloud. The principle evidence for these claims is
1. 1.
A band of Hi emission crossing the Galactic plane a few degrees ahead of the
Smith Cloud with an orientation identical to that of the Smith Cloud.
2. 2.
The clouds in this structure have velocities larger than allowed by Galactic
rotation (Figure 7) but the velocities are virtually identical to the
$V_{\mathrm{GSR}}$ of the Smith Cloud, indicating that there is a kinematic as
well as a spatial connection between the structures.
3. 3.
Many of the leading clouds are interacting with material at permitted
velocities, confirming their peculiar kinematics (Figure 8 and Figure 11).
We thus believe that the Leading Component is an anomalous feature of the ISM
related to the Smith Cloud, and despite its location in the Galactic plane
this material had its origin as a high velocity cloud.
Evidence of interaction with normal disk gas allows us to estimate that where
the Leading Component crosses the Galactic plane near longitude $25\arcdeg$ it
is 9.5 kpc from the Sun and 4.5 kpc from the Galactic Center. This is a region
of the Galaxy with significant star formation activity as indicated by the
presence of numerous HII regions (Anderson et al., 2012), some of which have a
parallactic distance identical to the median kinematic distance to the Leading
Component (Sato et al., 2014). The Leading Component has a neutral hydrogen
mass comparable to that of the Smith Cloud, and it extends $\approx 17\arcdeg$
across the sky which, for our derived distance, makes its linear size $\sim 3$
kpc, identical to that of the Smith Cloud.
From estimates of interaction locations we derive the velocity components for
the leading clouds and find that they are similar to those of the Smith Cloud
(Table 4). Because the two objects have a similar $V_{\mathrm{GSR}}$ (Figure
6) the similarity of the individual velocity components is not surprising, but
is nonetheless reassuring that we are observing a single phenomenon. There is
a gap between the Leading Component and the Smith Cloud of about $4\arcdeg$ in
which there is no strong emission that can be associated with either object
(see the left panel of Figure 2). This would correspond to a gap of 0.6 - 1.6
kpc depending on the projection and assumed distance of the clouds.
The total space velocity of the Leading Component is $>250$ $\rm km\,s^{-1}$,
where the limit reflects our inability to measure $V_{z}$ accurately. This is
comparable to the L08 determination for the Smith Cloud of $V_{tot}=296\pm 20$
$\rm km\,s^{-1}$. There is no information to suggest that either object has a
velocity capable of escaping the Galaxy. The estimates of cloud space velocity
from section 5 required several approximations; a more complete study of the
entire Smith Cloud and Leading Component taking into account the Galactic
potential and dynamical effects is in progress.
The Smith Cloud has been analyzed as an isolated system, perhaps the baryonic
component of a dark matter subhalo (Nichols & Bland-Hawthorn, 2009; Nichols et
al., 2014; Tepper-García & Bland-Hawthorn, 2018), and the question of the
survival of its gaseous component given its strong interaction with the Milky
Way’s circumgalactic medium has received considerable attention (Gritton et
al., 2014; Galyardt & Shelton, 2016). The discovery of the Leading Component
suggests that the Smith Cloud may not be an isolated object but instead the
more prominent manifestation of a system that stretches over more than 7 kpc
on the sky with a gap between its two parts. It is interesting that “chains”
of infalling gas occur naturally in some simulations that include simultaneous
infall and outflow – cloud chains that may survive to pass through a galaxy’s
disk (Melso et al., 2019). Perhaps that is what we are observing here. If the
metallicity of the Leading Component is similar to that of the Smith Cloud at
$\approx 0.5$ solar, it will lower the average metallicity of disk gas over
the region of interaction.
In Lockman et al. (2008) and subsequent publications there was speculation
about the future of the Smith Cloud, as its trajectory indicated that it would
enter and merge with the Milky Way disk in a few tens of Myr perhaps
triggering a burst of star formation (Alig et al., 2018). Given the existence
of the Leading Component it appears that the merger is already underway.
Acknowledgments The Green Bank Observatory is a facility of the National
Science Foundation, operated by Associated Universities, Inc. The observations
were made under GBT proposal codes 06C${\\_}$038, 08A${\\_}$014, and
09A${\\_}$007\. We thank E.C. Kornacki for assistance with the data reduction
and the anonymous referee for useful comments. ECK, NP, and CT were supported
by the NSF REU programs at the National Radio Astronomy Observatory and the
Green Bank Observatory. FJL acknowledges the influence of the classic work by
Toomre (1977) on the opening sentence of this paper.
Table 1: Leading clouds detected in the GBT Observations. Columns as follow: (1) Galactic longitude; (2) Galactic latitude; (3) Maximum line brightness temperature; (4) LSR velocity; (5) Line width FWHM; (6) Maximum ${\rm HI}$ column density; (7) observed angular size FWHM; (8) position angle with respect to the Galactic pole; (9) ${\rm HI}$ mass assuming a distance of 10 kpc. Columns (3)-(5) are from a Gaussian fit. Column (7) is from a 2D Gaussian fit to the channel map at the velocity of peak emission and is not corrected for beam smearing. Column (9) is calculated assuming a distance from the Sun of 10 kpc. Uncertainties are propagated from the error associated with the Gaussian fits. If a cloud shows evidence that it is interacting with normal gas in the Milky Way disk the velocity of that gas is given as $V_{MW}$ in column (10). $\ell$ | $b$ | $T_{\mathrm{pk}}$ | ${V_{\mathrm{LSR}}}$ | FWHM | $N_{\mathrm{{\rm HI}}}$ | $\Theta_{obs}$ | PA | $M_{{\rm HI}}\ $ (10 kpc) | $V_{MW}$
---|---|---|---|---|---|---|---|---|---
($\arcdeg$) | ($\arcdeg$) | (K) | ($\rm km\,s^{-1}$) | ($\rm km\,s^{-1}$) | ($10^{19}\ {\rm cm^{-2}}$) | ($\arcmin$) | ($\arcdeg$) | ($M_{\mathrm{\odot}}$) | ($\rm km\,s^{-1}$)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
21.67 | +2.63 | 2.41$\pm$0.05 | 149.6$\pm$0.3 | 18.9$\pm$0.8 | 8.8 | — | — | 5.6e+03 | 119
22.74 | +4.30 | 3.48$\pm$0.04 | 137.3$\pm$0.1 | 11.0$\pm$0.2 | 7.5 | 17.9$\times$15.0 | 125$\pm$6 | 1.2e+03 | 78
23.45 | -2.00 | 1.84$\pm$0.04 | 146.2$\pm$0.1 | 12.3$\pm$0.3 | 4.4 | — | — | 2.4e+03 | 112
23.86 | -3.57 | 1.83$\pm$0.03 | 136.6$\pm$0.2 | 19.7$\pm$0.6 | 7.0 | 22.6$\times$16.0 | 84$\pm$3 | 1.3e+03 | 103
25.15 | -0.55 | 4.71$\pm$0.09 | 143.1$\pm$0.3 | 17.3$\pm$0.6 | 15.8 | 71.4$\times$57.0 | 27$\pm$6 | 1.0e+04 | —
25.20 | +4.42 | 2.76$\pm$0.04 | 147.7$\pm$0.1 | 12.8$\pm$0.2 | 6.9 | 16.7$\times$14.0 | 134$\pm$6 | 1.4e+03 | 87
25.36 | -3.41 | 2.63$\pm$0.05 | 132.2$\pm$0.2 | 16.9$\pm$0.5 | 8.6 | 72.0$\times$45.6 | 94$\pm$3 | 3.8e+03 | 91
25.60 | -2.71 | 1.72$\pm$0.05 | 145.0$\pm$0.2 | 13.5$\pm$0.6 | 4.5 | 37.0$\times$13.6 | 168$\pm$1 | 1.1e+03 | 126
25.72 | -1.42 | 1.70$\pm$0.10 | 146.7$\pm$0.3 | 9.1$\pm$0.9 | 3.0 | 17.8$\times$12.4 | 126$\pm$4 | 6.6e+02 | 105
25.74 | +4.78 | 1.06$\pm$0.03 | 142.6$\pm$0.3 | 24.8$\pm$0.8 | 5.1 | 17.2$\times$13.7 | 84$\pm$6 | 7.7e+02 | —
26.05 | -2.90 | 2.13$\pm$0.06 | 140.8$\pm$0.2 | 13.3$\pm$0.5 | 5.5 | 41.3$\times$25.0 | 126$\pm$2 | 1.3e+03 | 114
26.36 | +3.49 | 7.62$\pm$0.03 | 120.2$\pm$0.0 | 15.9$\pm$0.1 | 23.5 | 84.6$\times$45.6 | 1$\pm$2 | 1.1e+04 | 66
26.77 | -3.28 | 3.16$\pm$0.03 | 127.1$\pm$0.1 | 15.6$\pm$0.2 | 9.6 | — | — | 6.9e+03 | 81
26.84 | -2.24 | 1.44$\pm$0.06 | 136.8$\pm$0.4 | 12.4$\pm$1.0 | 3.4 | 36.3$\times$27.9 | 129$\pm$5 | 9.5e+02 | 116
26.84 | -6.38 | 3.78$\pm$0.04 | 123.0$\pm$0.1 | 10.0$\pm$0.1 | 7.4 | 29.2$\times$14.7 | 132$\pm$1 | 1.9e+03 | 94
27.94 | -4.11 | 2.90$\pm$0.04 | 132.7$\pm$0.1 | 12.5$\pm$0.2 | 7.0 | 47.9$\times$21.8 | 9$\pm$1 | 3.2e+03 | 87
28.02 | -2.75 | 4.03$\pm$0.05 | 124.6$\pm$0.2 | 18.4$\pm$0.5 | 14.4 | 77.4$\times$37.8 | 127$\pm$1 | 8.6e+03 | 110
28.76 | -1.83 | 11.35$\pm$0.08 | 121.8$\pm$0.1 | 12.8$\pm$0.1 | 28.2 | 37.4$\times$20.3 | 89$\pm$2 | 1.1e+04 | 96
29.29 | -4.34 | 3.46$\pm$0.04 | 135.8$\pm$0.1 | 9.0$\pm$0.1 | 6.1 | 19.9$\times$15.3 | 124$\pm$7 | 1.0e+03 | —
29.31 | -7.30 | 1.36$\pm$0.04 | 122.8$\pm$0.1 | 11.1$\pm$0.4 | 2.9 | 19.7$\times$11.0 | 51$\pm$2 | 4.3e+02 | —
29.36 | -5.03 | 1.71$\pm$0.05 | 130.9$\pm$0.1 | 6.8$\pm$0.2 | 2.3 | 15.3$\times$11.8 | 163$\pm$3 | 2.8e+02 | —
29.92 | -7.92 | 0.79$\pm$0.03 | 132.3$\pm$0.4 | 18.1$\pm$1.1 | 2.8 | 27.2$\times$18.0 | 16$\pm$4 | 1.1e+03 | 74
29.95 | -6.14 | 3.49$\pm$0.03 | 128.2$\pm$0.1 | 13.5$\pm$0.2 | 9.1 | 22.0$\times$15.8 | 149$\pm$3 | 1.9e+03 | 63
30.02 | -7.42 | 3.78$\pm$0.03 | 117.7$\pm$0.1 | 19.6$\pm$0.2 | 14.4 | 22.0$\times$16.8 | 98$\pm$3 | 1.3e+03 | 71
30.07 | -3.92 | 5.30$\pm$0.11 | 121.3$\pm$0.2 | 15.2$\pm$0.6 | 15.6 | — | — | 1.7e+03 | —
30.37 | -6.31 | 1.33$\pm$0.03 | 119.2$\pm$0.2 | 21.5$\pm$0.7 | 5.5 | 46.6$\times$38.0 | 91$\pm$7 | 2.0e+03 | —
30.64 | -4.16 | 5.35$\pm$0.04 | 128.2$\pm$0.0 | 12.5$\pm$0.1 | 13.0 | 28.3$\times$15.4 | 137$\pm$2 | 2.5e+03 | 82
30.98 | -6.85 | 1.01$\pm$0.02 | 119.2$\pm$0.5 | 29.6$\pm$1.2 | 5.8 | — | — | 2.3e+03 | 68
31.35 | -4.80 | 2.40$\pm$0.03 | 121.7$\pm$0.1 | 15.2$\pm$0.3 | 7.1 | 54.6$\times$20.2 | 129$\pm$1 | 6.2e+02 | 76
31.47 | -6.53 | 3.57$\pm$0.03 | 111.8$\pm$0.1 | 17.8$\pm$0.2 | 12.3 | 38.9$\times$22.1 | 162$\pm$1 | 3.1e+03 | 51
31.54 | -7.55 | 3.86$\pm$0.03 | 105.2$\pm$0.1 | 19.4$\pm$0.2 | 14.5 | 32.1$\times$18.2 | 48$\pm$2 | 1.5e+03 | 65
31.57 | -3.46 | 2.14$\pm$0.05 | 120.7$\pm$0.2 | 15.3$\pm$0.6 | 6.4 | — | — | 1.2e+03 | 95
31.64 | -5.73 | 4.39$\pm$0.05 | 112.2$\pm$0.0 | 8.0$\pm$0.1 | 6.8 | — | — | 1.1e+03 | 85
32.04 | -4.10 | 1.79$\pm$0.03 | 120.6$\pm$0.2 | 19.6$\pm$0.5 | 6.8 | — | — | 1.0e+03 | —
32.06 | -5.38 | 5.22$\pm$0.03 | 112.9$\pm$0.1 | 15.2$\pm$0.1 | 15.4 | — | — | 3.2e+03 | 76
32.15 | -7.35 | 2.73$\pm$0.03 | 110.2$\pm$0.2 | 18.8$\pm$0.4 | 10.0 | 30.0$\times$24.6 | 107$\pm$4 | 3.8e+03 | 50
32.28 | -4.97 | 3.15$\pm$0.04 | 121.1$\pm$0.1 | 13.7$\pm$0.2 | 8.3 | 21.8$\times$17.1 | 78$\pm$7 | 1.2e+03 | 56
32.46 | -6.55 | 2.46$\pm$0.03 | 123.4$\pm$0.1 | 14.0$\pm$0.2 | 6.7 | 44.1$\times$17.7 | 147$\pm$1 | 2.3e+03 | 43
Table 2: Leading cloud population properties. Distant-dependent quantities were calculated assuming a distance from the Sun, $d_{10}=10$ kpc. Spectral properties were derived from a Gaussian fit at the location of the peak 21cm brightness. The quantity $V_{\mathrm{GSR}}$ was calculated using a Galactic circular velocity of 230 $\rm km\,s^{-1}$ at the location of the Sun. $V_{\mathrm{dif}}$ is the difference in velocity between the cloud and the material it appears to be interacting with. Angular sizes were derived from a Gaussian fit at the velocity of peak emission corrected by subtracting in quadrature the $10\arcmin$ effective angular resolution of the survey. Column (4) gives the number of clouds used to establish each quantity. Property | Median | Range | n clouds
---|---|---|---
(1) | (2) | (3) | (4)
$T_{pk}\ $ (K) | 2.7 | 0.79 – 11.35 | 38
FWHM ($\rm km\,s^{-1}$) | 15.2 | 6.8 – 29.6 | 38
Peak $N_{\mathrm{{\rm HI}}}$ $(10^{19}\ {\rm cm^{-2}})$ | 7.0 | 2.3 – 28.2 | 38
$V_{\mathrm{GSR}}$ ($\rm km\,s^{-1}$) | 237 | 222 – 248 | 38
$\Theta_{maj}\ (\arcmin)$ | 28.3 | 11.6 – 84 | 29
$\Theta_{min}\ (\arcmin)$ | 15.0 | 4.6 – 56 | 29
PA $(\arcdeg)$ | 124 | 1 – 168 | 29
Diam ($d_{10}$ pc) | 57 | 25 – 183 | 29
$n\ \ ({\rm cm^{-3}}\ \ d_{10}^{-1})$ | 0.43 | 0.13 – 1.24 | 29
$M_{{\rm HI}}\ (10^{3}\ d_{10}^{2}$ $M_{\mathrm{\odot}}$) | 1.6 | 0.28 – 11.0 | 38
${V_{\mathrm{dif}}}\ $ ($\rm km\,s^{-1}$) | 44 | 15 – 81 | 30
Table 3: Properties of three leading component clouds observed by the GBT at $10\arcmin$ resolution and by P2015 at $1\arcmin$ resolution. The P2015 diameters refer to the cloud core and were scaled to a distance of 10 kpc. Property | G22.74+4.30 | G25.20+4.42 | G26.84-6.38
---|---|---|---
| GBT, P2015 | GBT, P2015 | GBT, P2015
(1) | (2) | (3) | (4)
Peak $N_{\mathrm{{\rm HI}}}$ $(10^{20}\ {\rm cm^{-2}})$ | 0.75, 1.7 | 0.69, 1.6 | 0.74, 2.2
FWHM ($\rm km\,s^{-1}$) aaG22.74+4.30 and G26.84-6.38 have two spectral components in the P2015 data, one broad and one narrow. | 11.0, 5.8/24.3 | 12.8, 13.4 | 10.0, 3.0/14.5
Diam ($d_{10}$ pc) | 37.5, 12.6 | 33.3, 13.3 | 50.0, 11.0
$n\ \ ({\rm cm^{-3}}\ \ d_{10}^{-1})$ | 0.65, 4.4 | 0.67, 3.9 | 0.48, 6.4
PA $(\arcdeg)$ | 125, 122 | 134, 156 | 132, 128
Table 4: Space velocities of the Smith Cloud and the Leading Component. Smith Cloud entries are from the model in L08. Leading cloud entries are from the discussion in section 5 where the the uncertainty in $V_{R}$ shows the effect of a 10 $\rm km\,s^{-1}$ change in the adopted $V_{\theta}$, and $V_{z}$ could not be determined. Object | $V_{\theta}$ | $V_{R}$ | $V_{z}$
---|---|---|---
| $\rm km\,s^{-1}$ | $\rm km\,s^{-1}$ | $\rm km\,s^{-1}$
(1) | (2) | (3) | (4)
Smith Cloud | $270\pm 21$ | $94\pm 18$ | $73\pm 26$
Leading Component | $240$ | $65\pm 35$ | —
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|
# Structural evolution of the kagome superconductors $A$V3Sb5 ($A$ = K, Rb,
and Cs) through charge density wave order
Linus Kautzsch Materials Department, University of California Santa Barbara,
Santa Barbara, CA, 93106, United States Brenden R. Ortiz Materials
Department, University of California Santa Barbara, Santa Barbara, CA, 93106,
United States Krishnanand Mallayya Department of Physics, Cornell
University, Ithaca, NY, 14853, United States Jayden Plumb Materials
Department, University of California Santa Barbara, Santa Barbara, CA, 93106,
United States Ganesh Pokharel Materials Department, University of California
Santa Barbara, Santa Barbara, CA, 93106, United States Jacob P. C. Ruff
CHESS, Cornell University, Ithaca, NY, 14853, United States Zahirul Islam
Advanced Photon Source, Argonne National Laboratory, Lemont, IL, 60439, United
States Eun-Ah Kim Department of Physics, Cornell University, Ithaca, NY,
14853, United States Ram Seshadri Materials Department, University of
California Santa Barbara, Santa Barbara, CA, 93106, United States Stephen D.
Wilson<EMAIL_ADDRESS>Materials Department, University of California
Santa Barbara, Santa Barbara, CA, 93106, United States
###### Abstract
The kagome superconductors KV3Sb5, RbV3Sb5, and CsV3Sb5 are known to display
charge density wave (CDW) order which impacts the topological characteristics
of their electronic structure. Details of their structural ground states and
how they evolve with temperature are revealed here using single crystal X-ray
crystallographic refinements as a function of temperature, carried out with
synchrotron radiation. The compounds KV3Sb5 and RbV3Sb5 present
2$\times$2$\times$2 superstructures in the $Fmmm$ space group with a staggered
tri-hexagonal deformation of vanadium layers. CsV3Sb5 displays more complex
structural evolution, whose details have been unravelled by applying machine
learning methods to the scattering data. Upon cooling through the CDW
transition, CsV3Sb5 displays a staged progression of ordering from a
2$\times$2$\times$1 supercell and a 2$\times$2$\times$2 supercell into a final
2$\times$2$\times$4 supercell that persists to $T$ = 11 K and exhibits an
average structure where vanadium layers display both tri-hexagonal and Star of
David patterns of deformations. Diffraction from CsV3Sb5 under pulsed magnetic
fields up to $\mu_{0}H$ = 28 T suggest the real component of the CDW state is
insensitive to external magnetic fields.
††preprint: APS/123-QED
## I Introduction
The charge density wave (CDW) instability is central to many of the
unconventional properties reported in the $A$V3Sb5 ($A=$ K, Rb, and Cs) family
of kagome superconductors [1, 2, 3]. Within the kagome lattice of these
materials, the CDW is hypothesized to harbor both real and imaginary
components within the resulting order parameter, with the former corresponding
to a real space charge inhomogeneity and the latter corresponding to a
“chiral” flux phase that breaks time reversal symmetry [4]. The real component
of the CDW results in a superstructure deformation of the crystal lattice.
Precise details of the structural pattern of this deformation has been an
active area of recent investigation.
Understanding the pattern of the real space structural deformation stands to
inform a number of theoretical models of the anomalous CDW transtion in AV3Sb5
compounds. For instance, models of the band folding through the transition and
calculations of the resulting Berry curvature may help inform the origin of
the large anomalous Hall effect reported in the CDW state [5, 6]. At high
temperatures, the structure of all three variants assumes an ideal
(undistorted) kagome network of vanadium ions in the space group $P6/mmm$ as
shown in Fig. 1 [1]. The lattice deformation that results upon cooling through
the CDW transition primarily arises from the displacement of the kagome
network of vanadium atoms [7]. The most energetically favored possibilities
are predicted to be one of two breathing modes: either the Star of David-type
(SoD) deformation or its inverse, a tri-hexagonal (TrH) deformation [8, 9,
10]. Both are triple-q distortion modes, suggestive of the influence of
nesting effects between the saddle-points ($M$-points) close to the Fermi
level in AV3Sb5.
Figure 1: (a) and (b) show the refined room temperature ($T=290$ K) structure
of the $A$V3Sb5 ($A=$ K, Rb, and Cs) family in the hexagonal space group
$P6/mmm$.
An added complexity in solving the pattern of CDW order in AV3Sb5 is that
interlayer interactions are strong enough to promote a fully three-dimensional
CDW. This has been observed in scanning tunneling microscopy (STM)
measurements as a half unit cell phase-shift between layers resolved across
step-edges [11] as well as direct observation of superlattice reflections with
a finite $q_{z}$ propagation wave vector in x-ray diffraction measurements [7,
12]. The out-of-plane $q_{z}$ periodicity varies between AV3Sb5 variants with
the unit cell doubled along the c-axis in A= K, Rb variants [8, 12] and the
unit cell at least partially quadrupled in A= Cs [7, 13, 14]. The increased
interlayer spacing driven by the larger Cs interstitial layer seemingly
modifies the interactions between kagome planes and drives the formation of a
delicate three-dimensional CDW phase whose modulation between kagome planes
potentially hosts phase coexistence and metastable states.
Metastability in the CDW phase of CsV3Sb5 has been primarily observed with
x-ray diffraction measurements. Initial reports found conflicting observations
of $2\times 2\times 2$ [12] and $2\times 2\times 4$ CDW superlattices [7],
suggesting sensitivity to growth conditions. Furthermore, recent x-ray
diffraction studies have reported the coexistence of $2\times 2\times 2$ and
$2\times 2\times 4$ structures with different onset temperatures and thermal
quenching/annealing behaviors [13, 14]. Conclusions regarding which state is
stabilized by annealing also vary, likely reflecting the added variable of
disorder introduced via the growth process in the ground state selection. For
instance, recent studies have shown that small amounts of kagome-plane dopants
can quench the $2\times 2\times 4$ state, leaving only a quasi-two-dimensional
$2\times 2\times 2$ phase remaining [15].
The above complexity of the CDW state in CsV3Sb5 is relevant as it is the most
easily grown and widely studied AV3Sb5 variant. In our prior synchrotron
study, an average structural solution for CsV3Sb5 was presented presuming a
minimal model where the system maintains hexagonal symmetry ($P\bar{3}$). This
average structure suggested a superstructure comprised of modulated TrH and
SoD distortions in a $2\times 2\times 4$ cell [7]. This prior solution could
also be indexed as an orthorhombic cell; however twinning/domain formation was
not directly resolved in the diffraction data. Recently, however, STM [16] and
optical measurements [17] have resolved three sets of orthorhombic twin
domains in the CDW state of all three $A$V3Sb5 compounds. This provides an
excellent basis for developing twinned orthorhombic structural models of these
systems via synchrotron x-ray diffraction measurements.
Here, we report the structural deformations resulting from CDW order in all
three AV3Sb5 compounds determined via single crystal synchrotron x-ray
diffraction measurements. Structural refinements using an orthorhombic, three
domain, twinned model are presented for all three compounds. For KV3Sb5 and
RbV3Sb5, a 2$\times$2$\times$2 superstructure is observed and structural
refinement, combined with recent NMR analysis [18, 19, 20], allows for a
staggered TrH deformation of the vanadium layers to be resolved and the atomic
positions determined. We also revisit the average structure of CsV3Sb5 in an
orthorhombic cell using the same three domain structure, where we identify a
2$\times$2$\times$4 superlattice in the ground state. The average low-
temperature structural solution remains a modulation of SoD and TrH-type
deformations, consistent with prior reports [7, 21, 22]. Curiously, we also
resolve a cross-over regime below $T=94$ K where a 2$\times$2$\times$1 and a
2$\times$2$\times$2 lattice form and then diminish at lower temperatures.
Finally, as a test for potential time reversal symmetry breaking in the charge
density wave order parameter, we explore the field dependence of the
superstructure in CsV3Sb5 under the application of a $\mu_{0}H=$ 28 T pulsed
magnetic field.
## II Methods
### II.1 Structural solutions
Synchrotron x-ray diffraction experiments were carried out at the QM2 beam
line at CHESS. The incident x-ray wavelength of $\lambda=0.41328$ $\rm\AA$ was
selected using a double-bounce diamond monochromator. A stream of cold flowing
helium gas was used to cool the sample. The diffraction experiment was
conducted in transmission geometry using a 6-megapixel photon-counting pixel-
array detector with a silicon sensor layer. Data was collected in full 360∘
sample rotations with a step size of 0.1∘. Scattering planes in reciprocal
space were visualized using the NeXpy software package. The diffraction data
was indexed and integrated using the APEX3 software package including
absorption and extinction correction. Crystallographic structural solutions
were determined using the SHELX software package [23]. An unsupervised machine
learning tool using Gaussian Mixture Models called X-ray Temperature
Clustering (X-TEC) [24] was used to identify classes of peaks in reciprocal
space with distinct temperature dependencies.
### II.2 Pulsed magnetic field experiments
Pulsed magnetic field experiments were carried out at the beam line 6 ID-C of
the Advanced Photon Source at Argonne National Laboratory using a dual
cryostat, single solenoid pulsed-magnet system described in Ref. [25]. X-rays
with a wavelength of $\lambda=0.56356$ $\rm\AA$ were used, and the
crystallographic $c$-axis of CsV3Sb5 was aligned parallel to the direction of
the incident x-ray beam and the direction of the magnetic field. A low-noise
Lambda 750K detector was used with $55\times 55$ $\mu$ m2 CdTe pixels
providing near 100% efficiency. The detector was run in 12-bit mode at a 1 kHz
repetition rate with XRD pattern collected every 1 ms. Multiple frames were
collected over the course of repeated 9 ms pulses with a maximum magnetic
field of $H_{\mathrm{max}}=28$ T.
Figure 2: Refined structural models for KV3Sb5 and RbV3Sb5. (a) shows the z =
0.5 layer comprised of vanadium and antimony atoms of the refined room
temperature structure of KV3Sb5 and RbV3Sb5 in the hexagonal space group
$P6/mmm$. (b) and (c) show the suggested TrH model for the 2$\times$2$\times$2
superstructure in the orthogonal space group $Fmmm$. The vanadium planes at
$z=0$ and $z=0.5$ in the $Fmmm$ structure are displayed.
## III Experimental Results
### III.1 Structure of KV3Sb5 and RbV3Sb5
We begin by discussing diffraction experiments on KV3Sb5 at $T=10$ K.
Scattering data indicate a 2$\times$2$\times$2 superstructure, and, similar to
prior measurements [7], the area detector resolution does not allow for the
momentum-space resolution of twinning in the distorted state. Recent scanning
optical studies have shown the presence of three structural twins that form
below the CDW transition [17]. These domains have their principle axes rotated
by 120∘ in the ab-plane, consistent with an underlying orthorhombic lattice.
This would imply a pseudohexagonal twinning with three orthorhombic twins that
are rotated 120∘ with respect to each other in the $ab$-plane, which in turn
raises the apparent symmetry of the diffraction pattern to hexagonal.
Theoretical investigations identified the space groups $Cmcm$ [26] and $Fmmm$
[10] as possible starting orthorhombic solutions to the twinned
pseudohexagonal structure.
As reflection intensities indicate a Laue class of $mmm$, model refinements
were performed in both the face-centered orthorhombic space group $Fmmm$ and
the base-centered space group $Cmmm$. Refinement in both space groups yields
identical deformations of the vanadium layers with the distinction being that
$Cmmm$ allows for unique distortions within neighboring vanadium layers.
Refinement in $Cmmm$ also implicitly allows for any intensity at positions
forbidden in $Fmmm$ to be accounted for outside of twinning models. Analysis
in these two possible groups gave comparable or slightly better $R_{1}$ values
for the higher symmetry $Fmmm$ space group, and this group was used for the
final low-temperature structural determination.
KV3Sb5 data refined in $Fmmm$ is well-represented by two structural models
(Tab. 1) consisting of either a staggered TrH ($R_{1}=6.22$, Fig. 2) or a
staggered SoD distortion ($R_{1}=6.31$) of the vanadium layers. We note here
that the assignments of TrH or SoD-type distortions in this paper are based on
the dominant displacement type of V atoms, and that in all solutions there is
detectable variance amongst V-V distances within a single plane (ie. there is
not a single V-V distance in a given distorted kagome plane). This is further
parameterized in Discussion section of this manuscript. The pattern of a given
distortion type is shifted by half of a lattice constant ($0.5a$) with respect
to neighboring layers, and the two solutions differ in the sign of the
displacement vectors of the vanadium atoms in the $ab$-plane.
X-ray refinement alone is unable to discriminate between the distortion types,
which is a result of the previously discussed twinning. However, recent NMR
studies identify a TrH-type distortion and allow us to break the degeneracy
between these solutions [18, 19]. The resulting low-temperature structure
generated from the staggered TrH solution of KV3Sb5 is plotted in Fig. 2 (b).
This is consistent with recent studies [13, 27, 9], and crystallographic
information files (CIF) for both TrH and SoD distortion types can be found in
the supplemental information. After collecting data at 10 K, the sample was
warmed to 290 K and the high temperature structure measured. This undistorted
state was refined in $P6/mmm$ at 290 K with bond lengths illustrated in Fig. 2
(a).
Similar analysis was employed for the x-ray diffraction data of RbV3Sb5, where
RbV3Sb5 also forms a 2$\times$2$\times$2 superstructure in the CDW state. At
$T=10$ K, the data is again best represented when indexed in the space group
$Fmmm$, and our refinements show superior solutions using a staggered TrH
model ($R_{1}=6.40$) versus a staggered SoD model ($R_{1}=7.41$). This is
consistent with NMR results [18], and the TrH solution is selected as the
correct structure for presentation and the resulting bond lengths in the
vanadium network are shown in Fig. 2 (b). For reference, CIF files for both
TrH and SoD distortion types can also be found in the supplemental information
[28]. After measurement at 10 K, the 290 K structure was determined at 290 K
in $P6/mmm$ with bond lengths illustrated in Fig. 2 (a).
### III.2 Structure of CsV3Sb5
Figure 3: Temperature dependence of the unit cell parameters of CsV3Sb5 upon
traversing the CDW transition. The evolution of the $a$-, $b$-, and $c$-axes
are plotted in panel (a). (b) shows $a$ and $b$ where $a_{\mathrm{hex}}=a/2$
for $T\leq 90$ K and $b_{\mathrm{hex}}=\sqrt{b^{2}+4a_{hex}^{2}}/4$ for $T\leq
90$ K and (c) shows $c$ where $c_{\mathrm{hex}}=c/2$ for $T=90$ K and
$c_{\mathrm{hex}}=c/4$ for $T<90$ K. The subcell parameters undergo no
significant changes. Figure 4: X-TEC analysis of CsV3Sb5 diffraction data. (a)
X-TEC identifies three distinct temperature trajectories (blue, green, and
yellow clusters) whose intensities exhibit a sharp onset near the CDW
transition temperature. The lines denote the mean and shading denotes the
standard deviation of the re-scaled (z-scored) intensities $\tilde{I}(T)$
within each cluster. The grey clusters identify the diffuse background
scattering. (b) The average intensity of the blue, green, and yellow cluster
reveals the different onset temperatures for the $2\times 2\times 4$ CDW
($T_{2\times 2\times 4}\approx 85K$) and $2\times 2\times 2$ CDW ($T_{2\times
2\times 2}\approx 90K$) state and the $2\times 2\times 1$ state emerging
between 95 K and 80 K. $\Delta I(T)$ is the average intensity after
subtracting its minimum value. (c) A section of the (H, 2.5, L) plane, in
reciprocal lattice units (r.l.u.) and indexed in P6/mmm, showing the pixels
colored by the cluster assignments of their intensities. (d,e,f) Distribution
of the out-of-plane momentum $L$ of the peaks in the respective clusters.
Now turning to analysis of the structure of CsV3Sb5, temperature dependent
synchrotron x-ray diffraction experiments were carried out on a single crystal
of CsV3Sb5. The final data set was collected at $T=290$ K where data were
indexed in the hexagonal space group $P6/mmm$. The Bragg peaks of the CsV3Sb5
crystal studied are narrow with minimal stacking disorder, though the
crystallinity is anisotropic. Quantifying this with Gaussian fits to
representative Bragg peaks shows minimum correlation lengths in the $ab$-plane
of $\xi_{min}=890$ $\rm\AA$ or $\xi_{min}=1000$ $\rm\AA$ (depending on the
pixel-limited cut direction) and correlation lengths of $\xi=400$ $\rm\AA$
along the $c$-axis, indicating excellent crystallinity of the sample [28]. The
lower correlation length along the $c$-axis (interlayer direction) is typical
in layered materials and beyond the resolution of experiments using relaxed
collimation.
Figure 5: Refined structural model for CsV3Sb5. (a) shows the kagome network
(z = 0.5) of the 290 K room temperature structure in the hexagonal space group
$P6/mmm$. (b), (c), (d) and (e) represent a single-phase model of the
2$\times$2$\times$4 superstructure in the orthorhombic space group $Cmmm$.
Three kagome layers form with TrH deformations that are capped by a layer with
a predominant SoD deformation.
Temperature-dependent diffraction data were collected on warming after using
an initial cooling rate of approximately 10 K/min to cool to 10 K. Data were
then collected by warming at $\approx 5$ K/min with pauses for measurement
scans (12 min/scan) performed at select temperatures. Collecting a data series
using this cooling/warming profile show that at temperatures below $T=94$ K,
superstructure reflections appear as expected. Just below the CDW transition,
a clear staging behavior is observed. At 90 K, superlattice reflections
consistent with a 2$\times$2$\times$2 supercell are observed due to the
appearance of half-integer $L$ peaks as well as half-integer peaks in the
($H$, $K$)-plane. Below this temperature, at 85 K, weak quarter-integer $L$
peaks also appear, signifying the onset of an enlarged 2$\times$2$\times$4
supercell. The cell parameters through the CDW transition are plotted in Fig.
3
Figure 6: Magnetic field dependence of x-ray diffraction of the
Q=(0.5,0.5,0.25) CDW superlattice reflection in CsV3Sb5 in a pulsed magnetic
field experiment with $H_{\mathrm{max}}=28$ T at $T=11$ K and $T=82.5$ K. (a)
and (b) CDW reflection at $T=82.5$ K and $t=5$ ms from 1 sweep (1 ms exposure
in (a)) and integrated over 61 sweeps (61 ms exposure in (b)). (c) Magnetic
field $H$ as a function of the experiment time in milliseconds. (d) Sum of the
integrated intensity of the (0.5,0.5,0.25) superlattice reflection from a
total of 45 pulsed field sweeps at $T=11$ K and 61 sweeps at $T=82.5$ K.
It is clear that the nature of the charge density wave and the potential
coexistence/competition between different states involves subtle interactions
and changes in the diffraction patterns. In addition, the large temperature-
dependent and Q-dependent data sets contain substantial quantities of
information that can be overlooked during standard data reduction and
integration processes. Recently, an unsupervised machine learning tool called
X-ray Temperature Clustering (X-TEC) was developed to enable analysis of
parametric, complex data sets. X-TEC uses a Gaussian Mixture Model to identify
peaks in reciprocal space with distinct temperature dependencies. In order to
distinguish the functional form of the intensity-temperature trajectory rather
than their magnitudes, the intensity of each momenta $\vec{q}$ is re-scaled
(z-scored) as
$\tilde{I}_{\vec{q}}(T)=\left(I_{\vec{q}}(T)-\mu_{\vec{q}}\right)/\sigma_{\vec{q}}$
where $\mu_{\vec{q}}$ is the mean over temperature $T$, and $\sigma_{\vec{q}}$
is the standard deviation in $T$ [24].
By applying X-TEC on the CsV3Sb5 data, $\sim 15,000$ peaks were identified
with three distinct temperature trajectories color-coded as yellow, green and
blue clusters. These clusters exhibit an onset behavior around the CDW
transition temperature as shown in Figs. 4(a) and (b). By looking at the
locations of these clusters in reciprocal space (Fig. 4(c)), we find that all
three clusters have the same in-plane momenta at $(H,K)\equiv(0.5,0.5)$, but
differ in their out-of-plane momentum $L$. While the green clusters are
located at $L=1/4$ (Fig. 4(e)) corresponding to the $2\times 2\times 4$
structure, the blue clusters derive from the $L=1/2$ peaks (Fig. 4(f)) of the
$2\times 2\times 2$ structure, and the yellow clusters to an additional class
of $L=0$ peaks (Fig. 4 (d)) representing a $2\times 2\times 1$ structure. From
their intensity-temperature trajectories (Fig. 4(b)), we find that the onset
of CDW proceeds in two stages. First, the $2\times 2\times 1$ and $2\times
2\times 2$ cells appear below temperature $\approx 95$ K, followed by the
onset of a $2\times 2\times 4$ state at $\approx 85$ K. Remarkably, with the
onset of the $2\times 2\times 4$ peaks, the $2\times 2\times 1$ peaks are
strongly suppressed and vanish below 80 K, and, additionally, the $2\times
2\times 2$ peaks initially decrease with the onset of $2\times 2\times 4$
order and then recover their intensity at lower temperatures. This suggests an
exchange of scattering weight between order parameters.
Table 1: Parameters of structural models for the room temperature and the low-temperature structures of KV3Sb5, RbV3Sb5, and CsV3Sb5. All .cif files from the refinements can be found in the supplemental information [28]. | KV3Sb5 | RbV3Sb5 | CsV3Sb5
---|---|---|---
Wavelength ($\rm\AA$) | 0.41328
Temperature (K) | 290 | 10 | 290 | 10 | 290 | 90 | 11
Space group | $P6/mmm$ | $Fmmm$ | $P6/mmm$ | $Fmmm$ | $P6/mmm$ | $Fmmm$ | $Cmmm$
Type | | $2\times 2\times 2$ | | $2\times 2\times 2$ | | $2\times 2\times 2$ | $2\times 2\times 4$
Distortion | | TrH | SoD | | TrH | SoD | | TrH | SoD | TrH/SoD
$N_{\mathrm{measured}}$ | 2751 | 17014 | 2239 | 16168 | 2888 | 6788 | 24048
$N_{\mathrm{independent}}$ | 232 | 2401 | 201 | 2444 | 238 | 2159 | 5002
$N_{\mathrm{sig.}}$ [$I>2\sigma(I)$] | 231 | 1617 | 201 | 1906 | 238 | 641 | 3098
$a$ ($\rm\AA$) | 5.4260(9) | 10.9576(9) | 5.4941(6) | 11.0065(9) | 5.5236(6) | 11.0275(9) | 11.005(2)
$b$ ($\rm\AA$) | 5.4260(9) | 18.9813(16) | 5.4941(6) | 19.0575(16) | 5.5236(6) | 19.1082(16) | 19.046(4)
$c$ ($\rm\AA$) | 8.845(2) | 17.8552(15) | 9.1071(14) | 18.1010(15) | 9.3623(15) | 18.6213(15) | 37.176(7)
$\rho$ (g/cm3) | 5.90 | 5.73 | 5.91 | 5.93 | 6.00 | 6.06 | 7.22
$R_{1}>4\sigma$ (%) | 8.62 | 6.22 | 6.31 | 10.42 | 6.40 | 7.41 | 5.91 | 7.71 | 7.73 | 10.0
$R_{1}$ all (%) | 8.63 | 6.44 | 6.56 | 10.42 | 6.74 | 7.82 | 5.91 | 9.61 | 9.62 | 11.55
Twin fraction 1 | - | 0.35 | 0.36 | - | 0.32 | 0.32 | - | 0.31 | 0.30 | 0.31
Twin fraction 2 | - | 0.33 | 0.33 | - | 0.33 | 0.33 | - | 0.36 | 0.36 | 0.34
Twin fraction 3 | - | 0.32 | 0.31 | - | 0.34 | 0.34 | - | 0.33 | 0.34 | 0.35
The 2$\times$2$\times$4 superstructure persists down to temperatures of 11 K.
We note here that the in-plane and out-of-plane correlation lengths of the CDW
superlattice peaks here are identical to the primary structural peaks. This
indicates that the 2$\times$2$\times$4 CDW state is long-range with
correlation lengths constrained only by the sample’s underlying crystallinity.
Notably, our diffraction data using this cooling/warming profile do not
resolve anomalies at $T=60$ K as seen in magnetotransport experiments [29, 30]
or a crossover to a 2$\times$2$\times$2 state similar to Ref. [13]. This
difference potentially arises from subtle effects attributed to the different
cooling profiles during measurement and potentially from sample dependent
effects imparted during the crystal growth process.
We now discuss the structural refinement in the 2$\times$2$\times$2 regime at
$T=90$ K and the 2$\times$2$\times$4 superstructure at lower temperatures.
Using the same twinning approach described earlier, the 2$\times$2$\times$2
state at 90 K is best fit in the $Fmmm$ group rendering either a staggered SoD
or TrH distortion (CIFs provided in the supplemental material [28]). This is
consistent with the low temperature state recently reported by Stahl et al.
[13] where only a 2$\times$2$\times$2 superstructure was observed. Upon
cooling into the 2$\times$2$\times$4 state, the data at 10 K are instead best
refined in the lower symmetry, base-centered $Cmmm$ group. Fig. 5 shows the
resulting model of the superstructure consisting of vanadium layers with both
SoD and TrH deformations. The three TrH layers are staggered along the
$c$-axis by $0.5a$ (Fig. 5 (d),(e),(f)) and are then capped by a SoD layer
(Fig. 5 (c)). The model assumes a uniform phase at this temperature and
represents the measured diffraction data well, especially considering the
large number of significant reflections (3098 shown in Tab. 1). This mixed
distortion-type result is consistent with previous refinements in a
$P\overline{3}$ average structure containing both SoD and TrH layers [3] as
well as recent angle-resolved photoemission data [21, 22]. Mixed reports from
NMR measurements [19, 20] provide a picture interpreted as either SoD or TrH
distortion types, and it is unclear whether a mixed distortion-type structure
can also be modeled via the data. Moreover, our X-TEC analysis indicates that
a 2$\times$2$\times$2 phase competes with and potentially coexists with the
2$\times$2$\times$4 phase at lowest temperatures and that the stacking
sequence of the TrH- and SoD-like layers may be influenced by this
coexistence.
### III.3 Pulsed magnetic field experiment
Scanning tunneling microscopy measurements have reported that a magnetic field
couples to the apparent Fourier weight attributed to CDW order at the surface
in $A$V3Sb5 compounds [8], suggesting the presence of a chiral CDW state that
breaks time reversal symmetry. As means of testing whether a magnetic field
couples to the real component of the CDW order parameter in the bulk, the
magnetic field dependence of the superlattice reflections was investigated. A
pulsed magnetic field of $H_{\mathrm{pulse}}=28$ T was used, and the geometry
of the inner bore of the magnet coil allows for the measurement of the
Q$=(0.5,0.5,0.25)$ superlattice reflection in transmission geometry with the
field applied parallel to the $c$-axis. Scattering data were collected at
$T=11$ K and at $T=82.5$ K after cooling the sample into the CDW state under
zero field conditions. Magnetic field pulses were applied that last for
approximately 10 ms (half-sine, start to finish), and where the maximum
magnetic field of 28 T is reached. Multiple pulses were used to improve
measurement statistics.
The Q$=(0.5,0.5,0.25)$ peak was measured at $T=11$ K for a total of 45 pulses
and at $T=82.5$ K for a total of 61 pulses. For illustration, Fig. 6 (a) shows
the result of 1 ms of data collection during a single field pulse on the
Q$=(0.5,0.5,0.25)$ superlattice peak on a pixel array. Fig. 6 (b) then shows
the same peak after a cumulative sum of 61 pulses. Each time bin corresponds
to a field value shown in Fig. 6 (c).
The integrated intensities of the Q$=(0.5,0.5,0.25)$ peak as a function of
time were summed over the number of individual pulses to increase the signal-
to-noise ratio with the results plotted in Fig. 6 (d). No significant change
of the integrated intensity outside of the experimental error could be
observed at either of the two temperatures explored. The pre-pulse
fluctuations and sharp post-pulse decrease in the integrated intensity are
attributed to mechanical vibrations. Within our experimental resolution, we do
not observe any evidence of magnetic-field dependence of the superlattice peak
intensity, consistent with a recent report [13] and signifying that the real
component of the CDW order parameter is unchanged.
## IV Discussion and Conclusions
Tab. 1 summarizes the parameters of the crystallographic refinements carried
out on KV3Sb5, RbV3Sb5, and CsV3Sb5 crystals at key temperatures discussed in
the preceding sections. All refined models represent the collected data well
and the $R$-values are low considering the high number of significant
reflections collected in synchrotron experiments. The CDW distortion results
in an average V-V bond length that is nearly the same in all three AV3Sb5
variants with (V-V)avg=2.7022$\pm 0.022$ in A = K, (V-V)avg=2.6992$\pm 0.028$
in A = Rb, and (V-V)avg=2.6979$\pm 0.036$ in A = Cs. The standard deviations
in these V-V distances highlight the fact that the assignments of TrH or SoD
distortions are only effective classifications and that the V-V bond lengths
have substantial variance within a given layer.
The increase in the standard deviation of distorted bond lengths about a fixed
average value likely indicates a progressively increasing instability of the
TrH state as the A-site changes from K to Rb to Cs. Notably, the Cs system has
the largest room temperature V-V distance, rendering its low-temperature
reconstruction of the kagome network the most dramatic of the series. This is
likely reflected in the larger heat capacity anomaly in the Cs system relative
to the other two as well as its accommodation of a mixed TrH and SoD
distortion.
The mixed SoD and TrH state refined in the CDW state of CsV3Sb5 should also be
viewed through the lens of the experimental parameters used in this study.
Slower cooling or faster quenching may modify the resulting structure in the
CDW state given the seeming competition with the $2\times 2\times 2$ TrH state
and a newly observed $2\times 2\times 1$ state that both appear just below
$T_{CDW}$. It is possible that materials with different levels of strain
frozen during the growth process or dilute impurity content may modify this
behavior as well. Subtle disorder along the interlayer direction in the form
of variable crystallinity between samples does not seem to play a dominant
role as we have screened many crystals with variable quality and they all
exhibit the same $2\times 2\times 4$ low-temperature superlattice using this
same cooling profile.
In summary, we studied the structural deformations related to the CDW
transitions in the systems CsV3Sb5, KV3Sb5, and RbV3Sb5 using synchrotron
x-ray diffraction. Our diffraction data allow for twinned models of
orthorhombic superstructures in the CDW state. We find 2$\times$2$\times$2
superstructures in KV3Sb5 and RbV3Sb5 and our crystallographic refinements
allow us to parameterize the distortion of vanadium-based kagome layers in a
staggered TrH state. Temperature dependent diffraction data from the CsV3Sb5
system indicate a complex evolution of the structural superlattice. Initially,
a 2$\times$2$\times$1 state appears along with a 2$\times$2$\times$2 state at
temperatures below 90 K, and this is followed by the onset of a
2$\times$2$\times$4 superlattice below 85 K. Structural refinement of this
larger superstructure reveals a three-layer staggered TrH deformation capped
by a layer with a SoD type distortion. Scattering data from CsV3Sb5 measured
under a 28 T pulsed magnetic field fails to resolve a change in the CDW
superlattice intensity, setting bounds on the response of the real component
of the CDW state to the application of a magnetic field.
## V Acknowledgments
SDW acknowledges helpful discussions with Turan Birol, Rafael Fernandes, and
Binhai Yan. This work was supported by the National Science Foundation (NSF)
through Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for
Quantum Materials Science, Engineering and Information (Q-AMASE-i): Quantum
Foundry at UC Santa Barbara (DMR-1906325). The research made use of the shared
facilities of the NSF Materials Research Science and Engineering Center at UC
Santa Barbara (DMR- 1720256). The UC Santa Barbara MRSEC is a member of the
Materials Research Facilities Network. (www.mrfn.org). This work is based upon
research conducted at the Center for High Energy X-ray Sciences (CHEXS) which
is supported by the National Science Foundation under award DMR-1829070. The
high-field pulsed magnet and a choke coil were installed at the Advanced
Photon Source through a partnership with International Collaboration Center at
the Institute for Materials Research (ICC-IMR) and Global Institute for
Materials Research Tohoku (GIMRT) at Tohoku University. The development and
use of X-TEC by K.M. and E.-A.K. was supported by the U.S. Department of
Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering. KM and E-AK are supported in part by the Gordon and Betty Moore
Foundation’s EPiQS Initiative, Grant GBMF10436.
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|
# Orbits of linear series on the projective line
Anand Deopurkar Anand Deopurkar, Mathematical Sciences Institute, Australian
National University<EMAIL_ADDRESS>and Anand Patel Anand Patel,
Department of Mathematics, Oklahoma State University<EMAIL_ADDRESS>
###### Abstract.
We compute the equivariant fundamental class of the orbit closure of a linear
series on the projective line. We also describe the boundary of the orbit
closure and how the orbits specialise in one parameter families.
## 1\. Introduction
Let $M$ be a variety with the action of an algebraic group $G$. Given a point
$m\in M$, let $\operatorname{Orb}(m)\subset M$ be the closure of the $G$-orbit
of $m$. The fundamental class of $\operatorname{Orb}(m)$ in the equivariant
Chow ring of $M$ encodes fascinating geometric information. The class is
particularly meaningful when the $G$-orbits have geometric significance. The
main result of this paper (Theorem 1.1) is a complete and explicit expression
for these classes for
$M=\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
with the natural action of
$G=\operatorname{GL}(r+1)\times\operatorname{GL}(2)$. Here $\mathbf{k}$ is an
algebraically closed field in which $d!\neq 0$.
The orbits of the $\operatorname{GL}(r+1)\times\operatorname{GL}(2)$ action on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
represent isomorphism classes of linear series of rank $r$ and degree $d$ on
the projective line. Their fundamental class turns out to depend only on the
ramification profile. For example, when $r=1$ and the linear series is base-
point free, the class is a function of the multiplicities in the ramification
divisor. Likewise, when $r=2$ and the linear series is base-point free, the
class depends on the multiplicities of cusps and hyperflexes.
The group $\operatorname{GL}(r+1)$ acts freely on the open subset of
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
consisting of injective maps. The quotient is the Grassmannian ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. So we have a surjective
homomorphism
(1)
$A^{*}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))\to
A^{*}_{\operatorname{GL}(2)}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})).$
Taking images under this map gives classes of $\operatorname{GL}(2)$-orbits in
the Grassmannian. Forgetting the group action gives a further homomorphism
(2) $A^{*}_{\operatorname{GL}(2)}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}))\to A^{*}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})).$
Taking images under this map gives non-equivariant classes of the
$\operatorname{GL}(2)$-orbits. All of these, including the non-equivariant
ones, were unknown. Table 1 shows the weighted (multiplied by the order of the
stabiliser) orbit classes for all base-point free pencils of quartics.
Ramification profile | Weighted orbit class in the Schubert basis
---|---
$2,2,2,2,2,2$ | $24\cdot\Omega_{3}+48\cdot\Omega_{2,1}$
$3,2,2,2,2$ | $16\cdot\Omega_{3}+40\cdot\Omega_{2,1}$
$4,2,2,2$ | $12\cdot\Omega_{3}+24\cdot\Omega_{2,1}$
$3,3,2,2$ | $8\cdot\Omega_{3}+32\cdot\Omega_{2,1}$
$3,3,3$ | $24\cdot\Omega_{2,1}$
$4,3,2$ | $4\cdot\Omega_{3}+16\cdot\Omega_{2,1}$
Table 1. The weighted orbit classes of base-point free pencils of quartics in
${\bf Gr}(2,\operatorname{Sym}^{4}\mathbf{k}^{2})$ as a function of the
ramification profile.
Many mathematicians have studied equivariant and non-equivariant orbit
classes. We were inspired mainly by the work of Aluffi and Faber [3, 2, 1] on
the (non-equivariant) classes of orbits of points on ${\bf P}^{1}$ and curves
in ${\bf P}^{2}$. There has been exciting recent progress on extending their
work to the equivariant setting; see [13] for points on ${\bf P}^{1}$ and
quartic curves in ${\bf P}^{2}$, [12] for hyperplane arrangements in ${\bf
P}^{n}$, and [6] for generic cubics in ${\bf P}^{3}$. There is also
significant work in a more representation theoretic context. See, for example,
[11, 5] for classes of orbits of quiver representations and [4] for orbits of
matrices.
### 1.1. Main theorem
To state the main result, we need some notation. We have an identification
(3)
$A^{*}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))=\mathbf{Z}[\mu_{0},\dots,\mu_{r},\omega_{1},\omega_{2}]^{S_{r+1}\times
S_{2}},$
where the $\mu_{i}$’s and the $\omega_{j}$’s are the “Chern roots” of the
tautological representations of $\operatorname{GL}(r+1)$ and
$\operatorname{GL}(2)$, respectively. Given integers $a_{0}<\dots<a_{r}$ in
$\\{0,\dots,d\\}$, set
$\phi_{a_{0},\dots,a_{r}}(\mu;\omega)=\prod_{j=0}^{r}\prod_{i}((d-i)\omega_{1}+i\omega_{2}-\mu_{j}),$
where the index $i$ ranges over $\\{0,\dots,d\\}-\\{a_{0},\dots,a_{r}\\}$. Set
$\psi_{a_{0},\dots,a_{r}}(\mu;\omega)=\phi_{a_{0},\dots,a_{r}}(\mu;\omega_{1},\omega_{2})-\phi_{a_{0},\dots,a_{r}}(\mu;\omega_{2},\omega_{1}).$
Note that $\psi$ is symmetric in the $\mu$’s and anti-symmetric in the
$\omega$’s.
Given an injective
$s\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$,
let $S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$ be its image. Associated to
$S$ and a point $p\in{\bf P}^{1}$, we have the vanishing sequence
$a_{0}(p)<\dots<a_{r}(p)$
characterised by the property that for each $i$, the space $S$ contains a
section vanishing to order exactly $a_{i}(p)$ at $p$. For all but finitely
many points $p\in{\bf P}^{1}$, the vanishing sequence is given by
$a_{i}(p)=i$. Let $B\subset{\bf P}^{1}$ be the points where it differs. The
ramification profile of $S$ is the multi-set
$\\{(a_{0}(b),\dots,a_{r}(b))\mid b\in B\\}.$
###### Theorem 1.1 (Theorem 5.5).
Let $\mathbf{k}$ be an algebraically closed field in which $d!\neq 0$. Let
$s\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be an injective map with image $S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$.
Let $\\{(a_{i}(b))\mid b\in B\\}$ be the ramification profile of $S$. Let
$\Gamma\subset\operatorname{PGL}(2)$ be the stabiliser of
$S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$, and assume that it is finite.
Let $\operatorname{Orb}(s)$ be the closure of the
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$ orbit of $s$. Then the
class
$|\Gamma|\cdot[\operatorname{Orb}(s)]\in
A^{(r+1)(d-r)-3}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))$
is given by
$|\Gamma|\cdot[\operatorname{Orb}(s)]=\frac{1}{(\omega_{1}-\omega_{2})^{3}}\sum_{b\in
B}\psi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega)+\frac{2-|B|}{(\omega_{1}-\omega_{2})^{3}}\cdot\psi_{0,\dots,r}(\mu;\omega).$
Despite the apparent denominators, the expression must be a polynomial
(although this is not obvious).
To get the $\operatorname{GL}(2)$-equivariant class of
$\operatorname{Orb}(S)\subset{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$, we apply the homomorphism (1),
which sends the $\mu$’s to the Chern roots of the universal sub-bundle. To get
the non-equivariant class of $\operatorname{Orb}(S)\subset{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$, we further substitute
$\omega_{1}=\omega_{2}=0$.
###### Remark 1.2.
Although Theorem 1.1 gives the non-equivariant classes by the series of
substitutions mentioned above, it does not give a simple expression for it.
Indeed, the substitution $\omega_{1}=\omega_{2}=0$ can only be performed after
dividing by the denominator in Theorem 1.1, and the result of the division is
not obvious.
###### Remark 1.3.
Given $[S]\in{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$, we can consider $[S]$ as
a point of the projective space ${\bf
P}\left(\wedge^{r+1}\operatorname{Sym}^{d}W\right)$ by applying the Plucker
embedding. The class of $\operatorname{Orb}([S])$ in ${\bf
P}\left(\wedge^{r+1}\operatorname{Sym}^{d}W\right)$ depends only on the
multiplicities in the ramification divisor of $S$. The class of
$\operatorname{Orb}([S])$ in ${\bf Gr}(r+1,\operatorname{Sym}^{d}W)$ is a
richer invariant, sensitive to the full ramification profile.
### 1.2. Orbit closures
The method of proof of the main theorem also yields a description of the
boundary points of the orbit closure.
###### Theorem 1.4 (Theorem 4.5).
The orbit closure $\operatorname{Orb}(S)\subset{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ is a union of finitely many
orbits. In addition to the orbit of $S$, these are the orbits of subspaces of
the form
$\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle,$
where $a_{0}<\dots<a_{r}$ is the vanishing sequence of $S$ at some point of
${\bf P}^{1}$ and $w_{1},w_{2}$ is a basis of $\mathbf{k}^{2}$.
At a generic point of ${\bf P}^{1}$, the vanishing sequence is given by
$a_{i}=i$. The orbit of the corresponding linear series $\langle
w_{1}^{d}w_{2}^{0},\dots,w_{1}^{d-r}w_{2}^{r}\rangle$ is $1$-dimensional. At a
ramification point, the vanishing sequence is different, and the orbit of the
corresponding linear series $\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle$ is
$2$-dimensional.
### 1.3. Orbit specialisation
Consider a one-parameter family of linear series in which some ramification
points collide. Then the flat limit of the orbit closure contains the orbit
closure of the limiting linear series. In a suitable family, the limit of the
orbit closure contains the orbit closure of a unique other linear series,
which we now describe. Let $A$ be the multi-set of vanishing sequences of the
points that collide and $B$ the multi-set of vanishing sequences of the
remaining points. Let $c=(c_{0},\dots,c_{r})$ be the vanishing sequence of the
limiting linear series at the point of collision. Set
$c^{\prime}=(d-c_{r},\dots,d-c_{0})$. Denote by $\operatorname{WOrb}$ the
weighted orbit closure, namely
$\operatorname{WOrb}=|\Gamma|\cdot\operatorname{Orb}.$
Recall that we can treat the class $[\operatorname{WOrb}]$ as a function of
the ramification profile.
###### Theorem 1.5 (Proposition 6.8).
In the notation above, we have the equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-equivariant classes
$[\operatorname{WOrb}(A\cup
B)]=[\operatorname{WOrb}(A\cup\\{c^{\prime}\\})]+[\operatorname{WOrb}(B\cup\\{c\\})].$
The statement above is a special case of a large family of linear relations
among the classes $[\operatorname{WOrb}]$; see Section 6 for more.
By repeated collisions of ramification points, we can express an arbitrary
orbit class as a non-negative linear combination of orbit classes of linear
series with 3 ramification points (the expression is not unique). For example,
in the case of quartic pencils, Figure 1 shows a sequence of collisions and
the resulting complementary orbits. As a result, we get the following equality
$\begin{split}\operatorname{WOrb}((0,2)\times 6)&=\\\
\operatorname{WOrb}((0,3)&\times 2,(1,2))+2\operatorname{WOrb}((0,2)\times
2,(1,4))+\operatorname{WOrb}((0,2)\times 2,(2,3)).\end{split}$
In the non-equivariant Chow ring of the Grassmannian, the above evaluates to
the identity
$24\cdot\Omega_{3}+48\cdot\Omega_{2,1}=\left(8\Omega_{3}+20\cdot\Omega_{2,1}\right)+2\cdot\left(8\cdot\Omega_{3}+8\cdot\Omega_{2,1}\right)+12\cdot\Omega_{2,1}.$
$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,3)$$+$$(0,3)$$+$$(1,2)$$+$$(0,2)$$(0,2)$$(1,4)$$(0,2)$$(0,2)$$(1,4)$$(0,2)$$(0,2)$$(2,3)$
Figure 1. An example of orbit specialisation for a generic quartic pencil.
Collisions of ramification points create a new ramification profile and a
unique complementary orbit shown underneath.
### 1.4. Ideas in the proof
To find the class of the closure of the $G$-orbit of $m\in M$, we use a
complete orbit parametrisation for $m$. A complete orbit parametrisation
consists of a proper $G$-variety $X$ with a dense orbit and an equivariant map
$\phi\colon X\to M$ whose image contains $m$. Then, up to a scalar, the class
we seek is $\phi_{*}[X]$.
We begin with ${\bf P}^{3}={\bf
P}\operatorname{Hom}(\mathbf{k}^{2},\mathbf{k}^{2})$ with the natural
$\operatorname{GL}(2)$ action on the target $\mathbf{k}^{2}$ and the
equivariant rational map
$\phi\colon{\bf P}^{3}\dashrightarrow{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$
defined by $m\mapsto m(S)$. The map is only rational because when $m$ is
degenerate, the image $m(S)\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$ need
not be $(r+1)$-dimensional. We prove (Theorem 4.1) that $\phi$ extends to a
regular map on a simple blow up:
(4) $\widetilde{{\bf P}}^{3}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}).$
This is the same blow-up that Aluffi and Faber use in [1] for $r=0$. We
compute the push-forward using equivariant localisation as in [13, Appendix
B]. We upgrade from the orbit class in ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ to the orbit class in
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
using general principles, outlined in Section 2.
### 1.5. Organisation
In Section 2, we discuss equivariant orbit classes in general and classes on
Grassmannians in particular. We also discuss general results about one
parameter specialisations of orbits. In Section 3, we recall vanishing
sequences, the Wronskian, and the ramification divisor of a linear series. In
Section 4, we begin the proof of the main theorem by constructing a complete
parametrisation for a linear series. In Section 5, we finish the proof of the
main theorem with a localisation computation. In Section 6, we discuss one-
parameter specialisations of orbits of linear series, and the resulting
additive decomposition of the orbit class.
### 1.6. Conventions
Throughout, $\mathbf{k}$ denotes an algebraically closed field of
characteristic 0 or of characteristic $>d$. All schemes are of finite type
over $\mathbf{k}$. A _variety_ is a reduced scheme, separated over
$\mathbf{k}$. Given a vector space $V$, the notation $\underline{V}$ denotes
the trivial vector bundle with fiber $V$ on the scheme that is clear from the
context. Given a vector bundle $V$, the _projectivisation_ ${\bf P}V$ is the
space of one-dimensional sub-bundles. Unless mentioned otherwise, _point_
means a closed point. Given a variety $X$ with a $G$-action, $A^{i}_{G}(X)$
denotes the $G$-equivariant Chow group of $X$ of cycles of co-dimension $i$,
as defined in [7].
### 1.7. Acknowledgements
We started working on the project when Anand P. visited Anand D. at the
Institute for Computational and Experimental Research in Mathematics (ICERM)
at Brown University during the semester program _Braids_ in 2022. We thank
ICERM and the organisers of _Braids_. Anand D. thanks the Australian Research
Council for the grant DE180101360 that supported a part of this project.
## 2\. Equivariant orbit classes
Let $G$ be an algebraic group and $M$ a variety with a $G$ action. A _complete
parametrisation_ of the $G$ orbit of $m\in M$ is a $G$-variety $X$ together
with a proper $G$-equivariant map $\phi\colon X\to M$ such that
1. (1)
there exists an $x\in X$ such that $\phi(x)=m$,
2. (2)
the $G$-orbit of $x$ is dense in $X$, and
3. (3)
the stabiliser of $x\in X$ is trivial.
Let $\Gamma\subset G$ be the stabiliser of $m\in M$ and let $\phi\colon X\to
M$ be a complete parametrisation of the $G$ orbit of $m$. Then we have
(5) $\phi_{*}[X]=\begin{cases}|\Gamma|\operatorname{Orb}(m)&\text{ if $\Gamma$
is finite}\\\ 0&\text{otherwise}.\end{cases}$
We now specialise to two particular settings.
In the first setting, we take $M=W$, where $W$ is a non-zero
$G$-representation and the closely related setting of $M={\bf P}W$. Assume
that the image of $G\to\operatorname{GL}(W)$ contains the scalar matrices.
Suppose the map $G\to\operatorname{PGL}(W)$ factors through
$G\to\overline{G}$. (In applications, $G$ will be a general linear group and
$\overline{G}$ will be the corresponding projective general linear group.)
Let
$0\to O(-1)\to\underline{W}\to Q\to 0$
be the universal sequence on ${\bf P}W$.
###### Proposition 2.1.
Let $w\in W$ be non-zero. Let $\Gamma\subset\overline{G}$ be the stabiliser of
$[w]\in{\bf P}W$ and assume that it is finite. Let $\phi\colon X\to{\bf P}W$
be a complete parametrisation of the $\overline{G}$ orbit of $[w]\in{\bf P}W$.
Then, in $A^{*}_{G}(W)=A^{*}_{G}(\bullet)$, we have
$|\Gamma|[\operatorname{Orb}(w)]=\int_{X}c_{\rm top}(\phi^{*}Q).$
###### Proof.
Let $\pi\colon W^{*}\to{\bf P}W$ be the projection. Since the image of $G$
contains the scalars, we have
$\operatorname{Orb}(w)=\pi^{*}\operatorname{Orb}([w]),$
and hence
(6)
$|\Gamma|\cdot\operatorname{Orb}(w)=|\Gamma|\cdot\pi^{*}\operatorname{Orb}([w]).$
Since $\phi$ is a complete parametrisation of the $\overline{G}$ orbit of
$[w]$, we have
$|\Gamma|\cdot\operatorname{Orb}([w])=\phi_{*}[X].$
We now compute the right hand side.
On $X\times{\bf P}W$, let $\pi_{i}$ for $i=1,2$ be the two projections.
Consider the composite
$\pi_{2}^{*}O(-1)\to\underline{W}\to\pi_{1}^{*}\phi^{*}Q.$
Its vanishing locus is precisely the graph of $\phi\colon X\to{\bf P}W$.
Therefore, we have
(7) $\phi_{*}[X]={\pi_{2}}_{*}c_{\rm
top}\operatorname{Hom}(\pi_{2}^{*}O(-1),\pi_{1}^{*}\phi^{*}Q).$
In (6), we substitute $\phi_{*}[X]$ from (7) and use the push-pull formula on
the fiber square
${X\times W^{*}}$${W^{*}}$${X\times{\bf P}W}$${{\bf P}W.}$$\scriptstyle{\pi}$
Let $\widetilde{\pi}_{i}$ for $i=1,2$ be the two projections on $X\times
W^{*}$. Noting that the pull-back of $O(-1)$ to $W^{*}$ is trivial, in
$A^{*}_{G}(W^{*})$ we have
(8)
$\begin{split}|\Gamma|[\operatorname{Orb}(w)]&={\widetilde{\pi_{2}}}_{*}c_{\rm
top}\operatorname{Hom}(O,\widetilde{\pi}_{1}^{*}\phi^{*}Q)\\\ &=\int_{X}c_{\rm
top}(\phi^{*}Q).\end{split}$
Since $A^{*}_{G}(W^{*})=A^{*}_{G}(W)$ in co-dimension less than $\dim W$,
equation (8) also holds in $A^{*}_{G}(W)$. ∎
In the next setting, we assume that $G=\operatorname{GL}(V)\times H$ and
$M=\operatorname{Hom}(V,W)$, where $W$ is an $H$-representation. Let $\dim
V=r+1$. A closely related setting is that of $M={\bf Gr}(r+1,W)$ with the
induced action of $H$. Suppose the map $H\to\operatorname{PGL}(W)$ factors
through $H\to\overline{H}$. (In applications, $H$ will be a general linear
group and $\overline{H}$ will be the corresponding projective general linear
group.)
Let
$0\to E\to\underline{W}\to Q\to 0$
be the universal sequence on the Grassmannian.
###### Proposition 2.2.
Let $s\in\operatorname{Hom}(V,W)$ be an injective homomorphism with image $S$.
Let $\Gamma\subset\overline{H}$ be the stabiliser of $[S]\in{\bf Gr}(r+1,W)$
and assume that it is finite. Let $\phi\colon X\to{\bf Gr}(r+1,W)$ be a
complete parametrisation of the $\overline{H}$-orbit of $[S]$. Then, in
$A^{*}_{G}(\operatorname{Hom}(V,W))=A^{*}_{G}(\bullet)$, we have
$|\Gamma|[\operatorname{Orb}(s)]=\int_{X}c_{\rm
top}\operatorname{Hom}(\underline{V},\phi^{*}Q).$
###### Proof.
Consider the $H$-equivariant bundle $\phi^{*}E$ on $X$. Set
$F=\operatorname{Hom}(\underline{V},\phi^{*}E).$
Then $F$ admits an action of $G=\operatorname{GL}V\times H$. Set $Y={\bf P}F$;
let $\pi\colon Y\to X$ be the projection; and let $\psi=\pi\circ\phi$. The
composite
(9)
$O_{Y}(-1)\xrightarrow{i}\operatorname{Hom}(\underline{V},\psi^{*}E)\xrightarrow{j}\operatorname{Hom}(\underline{V},\underline{W})$
yields a $G$-equivariant map
$Y\to{\bf P}\operatorname{Hom}(V,W).$
This map is a complete parametrisation of the
$\operatorname{PGL}V\times\overline{H}$-orbit of $[s]$. It is easy to check
that the stabiliser of $[s]$ in $\operatorname{PGL}V\times\overline{H}$ maps
isomorphically to $\Gamma\subset\overline{H}$ under the second projection. By
Proposition 2.1, in $A^{*}_{G}(\operatorname{Hom}(V,W))$ we have
$|\Gamma|[\operatorname{Orb}(s)]=\int_{Y}c_{\rm
top}(\operatorname{coker}(j\circ i)).$
From the sequence in (9), we have an exact sequence
$0\to\operatorname{coker}i\to\operatorname{coker}(j\circ
i)\to\operatorname{coker}j\to 0.$
and hence the equality of top Chern classes
$c_{\rm top}(\operatorname{coker}j\circ i)=c_{\rm
top}(\operatorname{coker}i)\cdot c_{\rm top}(\operatorname{coker}j).$
Observe that
$\operatorname{coker}j=\pi^{*}\operatorname{Hom}(\underline{V},\phi^{*}Q).$
Since $Y\to X$ is a projective space bundle and $i$ is its universal sub-
bundle, we have
$\int_{Y/X}c_{\rm top}(\operatorname{coker}i)=[X].$
Now, the push-pull formula gives
$\int_{Y/X}c_{\rm top}(\operatorname{coker}i)\cdot c_{\rm
top}(\operatorname{coker}j)=c_{\rm
top}(\operatorname{Hom}(\underline{V},\phi^{*}Q)).$
Integrating the two sides along $X$ gives the result. ∎
We now study specialisations of orbits in families. The results are
essentially from [13, § 2.5], but we state and prove them more generally.
Fix an action of an algebraic group $G$ on a variety $M$. For simplicity,
assume that $G$ is irreducible. Given $m\in M$, define the $G$-equivariant
cycle
$\operatorname{WOrb}(m)=\begin{cases}|\operatorname{Stab}(m)|\cdot\operatorname{Orb}(m)&\text{
if $\operatorname{Stab}(m)$ is finite,}\\\ 0&\text{otherwise}.\end{cases}$
Let $\Delta$ be the spectrum of a DVR with general point $\eta$ and special
point $0$. Fix a section $m_{\eta}{\colon}\eta\to M_{\eta}$. Let $\Sigma$ be
the cycle on $M$ obtained as the limit of the cycle
$\operatorname{WOrb}(m_{\eta})$ on $M_{\eta}$. Our goal is to understand
$\Sigma$. We begin with an easy observation.
###### Proposition 2.3.
Suppose $m_{\eta}$ has a limit $\overline{m}\in M$. Then the cycle
$\Sigma-\operatorname{WOrb}(\overline{m})$ is effective.
###### Proof.
Let $X\supset G$ be an equivariant compactification. Consider the map
$\alpha\colon G_{\eta}\to M_{\eta}$ defined by $g\mapsto gm_{\eta}$, and let
$Y\subset X\times M\times\Delta$ be the closure of the graph of $\alpha$. Then
$Y$ is irreducible of dimension $\dim G+1$. Let $\pi\colon Y\to M$ be the
projection. Observe that we have
${\pi}_{*}Y_{\eta}=\operatorname{WOrb}(m_{\eta})\text{ and
}{\pi}_{*}Y_{0}=\Sigma.$
By construction, $Y$ is invariant under the $G$-action on $X\times
M\times\Delta$. Since $y=(1,\overline{m},0)\in Y_{0}$, we have a copy of
$G\subset Y_{0}$ given by $G\cdot y$. Let $Z\subset Y_{0}$ be its closure.
Note that ${\pi}_{*}Z=\operatorname{WOrb}(\overline{m})$. Furthermore, since
$\dim Z=\dim Y_{0}$, we see that $Z\subset Y_{0}$ is an irreducible component.
Then $\Sigma-\operatorname{WOrb}(\overline{m})$ is the push-forward of the
effective cycle $[Y_{0}]-[Z]$. ∎
We have the following strengthening of Proposition 2.3 (see [13, Proposition
2.13]).
###### Proposition 2.4.
Consider sections $g_{1},\dots,g_{n}{\colon}\eta\to G_{\eta}$ such that for
all $i\neq j$, the section $g_{i}g_{j}^{-1}$ does not extend to a section
$\Delta\to G_{\Delta}$. Suppose $\overline{m}_{1},\dots,\overline{m}_{n}\in M$
are limits of $g_{1}m_{\eta},\dots,g_{n}m_{\eta}$. Then the cycle
$\Sigma-\operatorname{WOrb}(\overline{m}_{1})-\dots-\operatorname{WOrb}(\overline{m}_{n})$
is effective.
###### Proof.
Let $X\supset G$ be an equivariant compactification. Consider the map
$\alpha\colon G_{\eta}\to G_{\eta}^{n}\times M_{\eta}$ given by
$\alpha\colon g\mapsto(gg_{1}^{-1},\dots,gg_{n}^{-1},gm_{\eta})$
and let $Y\subset X^{n}\times M\times\Delta$ be the closure of its image.
Denoting by $\pi$ the projection onto the $M$ factor, we have
${\pi}_{*}Y_{\eta}=\operatorname{WOrb}(m_{\eta})\text{ and
}{\pi}_{*}Y_{0}=\Sigma.$
Let $\overline{g}_{i,j}\in X$ be the limit of $g_{i}g_{j}^{-1}$. Then
$\overline{g}_{i,j}\not\in G$ for $i\neq j$. Then the limit of $\alpha(g_{i})$
is the point
$y_{i}=(\overline{g}_{i,1},\dots,\overline{g}_{i,i-1},1,\overline{g}_{i,i+1},\overline{g}_{i,n},\overline{m}_{i})\in
Y_{0}.$
Each $y_{i}$ gives a copy of $G\subset Y_{0}$, say $G_{i}$, defined by
$G_{i}=G\cdot y_{i}$. Since $\overline{g}_{i,j}\not\in G$ for $i\neq j$, the
$j$-th components of the points of $G_{i}$ do not lie in $G$ for $i\neq j$ and
lie in $G$ for $i=j$. In particular, for $i\neq j$, the $G_{i}$ and $G_{j}$
are disjoint. Let $Z_{i}$ be the closure of $G_{i}$. Then the $Z_{i}$ are
distinct irreducible components of $Y_{0}$. The cycle
$\Sigma-\operatorname{WOrb}(\overline{m}_{1})-\dots-\operatorname{WOrb}(\overline{m}_{n})$
is the $\pi$-pushforward of the effective cycle $Y_{0}-Z_{1}-\dots-Z_{n}$. ∎
## 3\. Vanishing sequences and the Wronskian
Let $C$ be a smooth curve, $L$ a line bundle of degree $d$ on $C$, and
$S\subset H^{0}(C,L)$ a subspace of dimension $r+1$. The pair $(S,L)$ is
called a _linear series of rank $r$ and degree $d$_ or a $g^{r}_{d}$ on $C$.
Given a point $p\in C$, there exists a unique sequence of integers
$a_{0}(p)<\dots<a_{r}(p)$ in $\\{0,\dots,d\\}$ such that $S$ contains a
section $s_{i}$ that vanishes to order exactly $a_{i}(p)$ at $p$. The sequence
$a_{i}(p)$ is called the _vanishing sequence_ of $S$ at $p$. The sections
$s_{0},\dots,s_{r}$ form a basis of $S$.
Denote by $j_{r+1}L$ the bundle of jets of order $r+1$ of $L$; that is
$j_{r+1}L={\pi_{2}}_{*}\left(\pi_{1}^{*}L\otimes O_{(r+1)\Delta}\right).$
Here $\Delta\to C\times C$ is the diagonal and the $\pi_{i}$ are the two
projections. The linear series $S\subset H^{0}(C,L)$ induces an evaluation map
$\underline{S}\to L,$
and also a map
(10) $\underline{S}\to j_{r+1}L.$
The _Wronskian_ of $S$ is the determinant of (10).
###### Proposition 3.1.
The Wronskian is non-zero and vanishes to order $\sum a_{i}(p)-i$ at $p$.
###### Proof.
The proof is a part of [9, Proposition 1.1]. We give it for completeness.
Let $s_{i}\in S$ vanish to order $a_{i}=a_{i}(p)$ at $p$. Denote by
$\widehat{M}$ the completion of an $O_{C}$-module $M$ at $p$. Choose an
isomorphism of $\mathbf{k}\llbracket t\rrbracket$-modules
$\widehat{L}\cong\mathbf{k}\llbracket t\rrbracket$. Then we have an
isomorphism
$\widehat{j_{r+1}L}\cong\mathbf{k}\llbracket t,u\rrbracket/(u-t)^{r+1}.$
If the section $s\in\widehat{L}\cong\mathbf{k}\llbracket t\rrbracket$ is given
by the power series $s(t)$, then the induced section $j_{r+1}(s)$ of
$\widehat{j_{r+1}L}$ is given by
$j_{r+1}(s)=s(t)+s^{\prime}(t)(u-t)+\dots+s^{(r)}(t)\frac{1}{r!}(u-t)^{r}.$
Let $s_{i}$ correspond to the power series
$s_{i}(t)=t^{a_{i}}z_{i}\in\mathbf{k}\llbracket t\rrbracket$, where
$z_{i}\in\mathbf{k}\llbracket t\rrbracket$ is a unit. With respect to the
bases $s_{0},\dots,s_{r+1}$ of $S$ and $1,(u-t),\dots,(u-t)^{r}/r!$ of
$\widehat{j_{r+1}L}$, the matrix of the map (10) is given by
$\begin{pmatrix}t^{a_{0}}z_{0}&t^{a_{1}}z_{1}&\dots&t^{a_{r}}z_{r}\\\
(t^{a_{0}}z_{0})^{\prime}&(t^{a_{1}}z_{1})^{\prime}&\dots&(t^{a_{r}}z_{r})^{\prime}\\\
\vdots&\vdots&\ddots&\vdots\\\
(t^{a_{0}}z_{0})^{(r)}&(t^{a_{1}}z_{1})^{(r)}&\dots&(t^{a_{r}}z_{r})^{(r)}.\end{pmatrix}$
The lowest degree term in its determinant is $t^{\sum a_{i}-i}$ and its
coefficient is the determinant of the $(r+1)\times(r+1)$ matrix $C$ with
$C_{ij}=a_{i}(a_{i}-1)\cdots(a_{i}-j+1)$. It remains to show that the
coefficient is non-zero. By a sequence of invertible row transformations, the
matrix $C$ reduces to the Vandermonde matrix $V$ with $V_{ij}=a_{i}^{j}$.
Since the $a_{i}$’s are distinct integers in $\\{0,\dots,d\\}$, their images
are distinct in $\mathbf{k}$, which has characteristic 0 or characteristic
$>d$. Hence the determinant of $V$, and hence of $C$, is non-zero. ∎
Let $\rho(S)\subset C$ be the vanishing locus of the Wronskian of $S$. We call
$\rho(S)$ the _ramification divisor_ of $S$ and its points the _ramification
points_.
###### Corollary 3.2.
For all $p\in C-\rho(S)$, we have $a_{i}(p)=i$. In particular, for all but
finitely many $p$, we have $a_{i}(p)=i$.
###### Proof.
Follows immediately from Proposition 3.1. ∎
Note that $\rho(S)$ is the zero-locus of a section of $\det j_{r+1}L$. Recall
that we have a sequence of surjective maps
$j_{r+1}L\to j_{r}L\to\dots\to j_{2}L\to j_{1}L=L\to 0$
with
$\ker(j_{i+1}L\to j_{i}L)=L\otimes\Omega_{C}^{i}.$
As a result, we get
(11) $\det j_{r+1}L=L^{r+1}\otimes\Omega_{C}^{r(r+1)/2}.$
Consider the Grassmannian ${\bf Gr}={\bf Gr}(r+1,H^{0}(C,L))$ with the
universal sub-bundle
$S\subset\underline{H^{0}(C,L)}.$
On ${\bf Gr}\times C$, we have the universal evaluation map
$\pi_{1}^{*}S\to\pi_{2}^{*}L,$
which induces a map
$\pi_{1}^{*}S\to\pi_{2}^{*}j_{r+1}L.$
Taking the determinant of the last map and applying ${\pi_{1}}_{*}$ gives the
following map on ${\bf Gr}$:
$\det S\to\underline{H^{0}(C,\det j_{r+1}L)}.$
By Proposition 3.1, the map above is non-zero at every point of ${\bf Gr}$,
and hence yields a morphism
(12) $\rho\colon{\bf Gr}(r+1,H^{0}(C,L))\to{\bf P}H^{0}(C,\det j_{r+1}L).$
###### Proposition 3.3.
The map $\rho$ in (12) is finite. For $C={\bf P}^{1}$, it is also surjective.
###### Proof.
Assume that $\dim H^{0}(C,L)>r+1$; otherwise, the statement is vacuous.
Since the Grassmannian has Picard rank one, any morphism from it is either
finite or constant. It is easy to see that $\rho$ is not constant; so it must
be finite.
For $C={\bf P}^{1}$ and $L=O(d)$, we have by (11)
$\det j_{r+1}L\cong O((r+1)d)\otimes O(-r(r+1))=O((r+1)(d-r)).$
Hence, both the source and target of $\rho$ have the same dimension, namely
$(r+1)(d-r)$. ∎
## 4\. Complete parametrisation of the orbit of a linear series
Fix a $2$-dimensional vector space $V$ and an $(r+1)$-dimensional subspace
$S\subset\operatorname{Sym}^{d}V$.
On ${\bf P}V\cong{\bf P}^{1}$, let $O(-1)\subset\underline{V}$ be the
universal sub-bundle and $Q$ the quotient $\underline{V}/O(-1)$. Then we have
a canonical identification $\operatorname{Sym}^{d}V=H^{0}({\bf P}V,Q^{d})$. We
regard $S$ as a linear series associated to the line bundle $Q^{d}$.
Let $W$ be another $2$-dimensional vector space. Consider the
${\bf P}^{3}={\bf P}\operatorname{Hom}(V,W).$
We have a rational map
$\phi\colon\mathbf{P}^{3}\dashrightarrow{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$
defined by
(13) $m\mapsto[m(S)].$
Note that the map is equivariant with respect to the obvious
$\operatorname{GL}W$ action. The goal of this section is to understand a
$\operatorname{GL}W$-equivariant resolution of $\phi$.
Let $B\subset\mathbf{P}V$ be the ramification divisor of $S$. For every $b\in
B$, let $L_{b}\subset\mathbf{P}^{3}$ be the linear subspace
(14) $L_{b}=\\{[m]\mid m(b)=0\\}.$
Then $L_{b}$ is a copy of ${\bf P}^{1}$ linearly embedded in ${\bf P}^{3}$,
and for $b_{1}\neq b_{2}$, the lines $L_{b_{1}}$ and $L_{b_{2}}$ are disjoint.
Let
$\beta\colon\widetilde{{\bf P}}^{3}\to{\bf P}^{3}$
be the blow-up at $\bigsqcup_{b\in B}L_{b}$. Note that the lines $L_{b}$ are
$\operatorname{GL}W$-invariant and hence the blow-up $\widetilde{{\bf P}}^{3}$
inherits an action by $\operatorname{GL}W$. See Figure 2 for a sketch of ${\bf
P}^{3}$ and the lines $L_{b}$.
Figure 2. The map $\phi\colon{\bf P}^{3}\dashrightarrow{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$ becomes regular after blowing up the lines
$L_{b}$ (red) on the discriminant quadric $\Delta$. Along the lines of the
opposite ruling (blue, dashed), the map is constant.
The key observation is the following consequence of a result of Aluffi and
Faber [1].
###### Theorem 4.1.
The rational map
$\phi\colon{\bf P}^{3}\dashrightarrow{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$
extends to a regular map
$\widetilde{\phi}\colon\widetilde{{\bf P}}^{3}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W).$
For the proof, we need a simple lemma.
###### Lemma 4.2.
Let $\rho\colon X\to Y$ be a finite map between varieties. Let $P$ be an
irreducible normal variety with a rational map $P\dashrightarrow X$. If the
composite $P\dashrightarrow Y$ extends to a regular map $P\to Y$, then so does
$P\dashrightarrow X$.
###### Proof.
Let $\Gamma_{X}\subset P\times X$ and $\Gamma_{Y}\subset P\times Y$ be the
closures of the graphs of $P\dashrightarrow X$ and $P\dashrightarrow Y$,
respectively. Then the projections $\Gamma_{X}\to P$ and $\Gamma_{Y}\to P$ are
birational. In fact, by the hypothesis on $P\dashrightarrow Y$, the projection
$\Gamma_{Y}\to P$ is an isomorphism. The map $\Gamma_{X}\to\Gamma_{Y}$ induced
by $\rho$ is finite and birational. Since the target $\Gamma_{Y}\cong P$ is
normal, we conclude that $\Gamma_{X}\to\Gamma_{Y}$ is an isomorphism. Hence
the projection $\Gamma_{X}\to P$ is also an isomorphism. We deduce that
$P\dashrightarrow X$ extends to a regular map. ∎
###### Proof of Theorem 4.1.
Set
${\bf Gr}={\bf Gr}(r+1,\operatorname{Sym}^{d}W)$
and
${\bf P}={\bf P}H^{0}({\bf P}^{1},\det j_{r+1}O(d))={\bf
P}\operatorname{Sym}^{(r+1)(d-r)}W.$
Recall that we have the finite map
$\rho\colon{\bf Gr}\to{\bf P}$
defined by the Wronskian. By definition, $\rho$ is
$\operatorname{GL}W$-equivariant.
Let $R$ be the Wronksian of $S$. Consider the rational map
${\bf P}^{3}\dashrightarrow{\bf P}$
defined by
$m\mapsto[m(R)].$
Then we have the commutative diagram
${\widetilde{\bf P}^{3}}$${{\bf Gr}}$${\widetilde{\bf P}^{3}}$${{\bf
P}.}$$\scriptstyle{\rho}$
By [1, Proposition 1.2], the map $\widetilde{\bf P}^{3}\dashrightarrow{\bf P}$
extends to a regular map $\widetilde{\bf P}^{3}\to{\bf P}$. By Lemma 4.2, we
conclude the same about $\widetilde{\bf P}^{3}\dashrightarrow{\bf Gr}.$ (In
[1], the base field is assumed to have characteristic 0, but Proposition 1.2
and its proof hold in any characteristic.) ∎
Let $\Delta\subset{\bf P}^{3}={\bf P}\operatorname{Hom}(V,W)$ be the locus of
degenerate maps. Then $\Delta$ is a quadric hypersurface, defined by the
vanishing of the determinant. We have an isomorphism
$\Delta\xrightarrow{\sim}{\bf P}V\times{\bf P}W$
given by
(15) $m\mapsto(\ker m,\operatorname{im}m).$
Under this isomorphism, the line $L_{b}$ defined in (14) corresponds to the
line $\\{b\\}\times{\bf P}W$.
We now do a simple calculation that allows us to understand the map
$\widetilde{\phi}\colon\widetilde{\bf P}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W).$
Choose bases $\langle v_{1},v_{2}\rangle$ for $V$ and $\langle
w_{1},w_{2}\rangle$ for $W$. Let us write elements of
$\operatorname{Hom}(V,W)$ as $2\times 2$-matrices with respect to these bases.
Consider the one-parameter family of homomorphisms given by
(16) $m_{t}=\begin{pmatrix}1&0\\\ 0&t\end{pmatrix},\quad t\in{\bf A}^{1}.$
For $t\neq 0$, let
$S_{t}=m_{t}(S)\subset\operatorname{Sym}^{d}W,$
and
$S_{0}=\lim_{t\to 0}S_{t}\subset\operatorname{Sym}^{d}W.$
The following proposition describes $S_{0}$.
###### Proposition 4.3.
Let $S_{0}$ be defined as above. Let $b\in{\bf P}V$ be the point corresponding
to $[v_{2}]\subset V$ and let $a_{0}<\dots<a_{r}$ be the vanishing sequence of
$S$ at $b$. Then $S_{0}\subset\operatorname{Sym}^{d}W$ is given by
$S_{0}=\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$
###### Proof.
Let $s_{i}\in S$ vanish to order $a_{i}$ at $b$. Then there exists a non-zero
$c_{i}\in\mathbf{k}$ and a homogeneous polynomial $F_{i}(v_{1},v_{2})$ of
degree $d-a_{i}-1$ such that
$s_{i}=c_{i}v_{1}^{d-a_{i}}v_{2}^{a_{i}}+v_{2}^{a_{i}+1}F_{i}(v_{1},v_{2}).$
We see that
$m_{t}(s_{i})=c_{i}t^{a_{i}}w_{1}^{d-a_{i}}w_{2}^{a_{i}}+t^{a_{i}+1}w_{2}^{a_{i}+1}F_{i}(w_{1},tw_{2}).$
Therefore, we get
$\lim_{t\to 0}\langle m_{t}(s_{i})\rangle=\langle
w_{1}^{d-a_{i}}w_{2}^{a_{i}}\rangle,$
and hence
$\lim_{t\to 0}m_{t}(S)=\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$
∎
###### Corollary 4.4.
If $b\in{\bf P}V$ is not a ramification point of $S$, then $S_{0}$ depends
only on the subspace $\operatorname{im}m_{0}\subset W$.
###### Proof.
In this case, the vanishing sequence at $b$ is $0<\dots<r$. From Proposition
4.3, we see that $S_{0}\subset\operatorname{Sym}^{d}W$ is the subspace of
sections that vanish to order at least $d-r$ at
$\operatorname{im}m_{0}=[w_{1}]\in{\bf P}W$. ∎
Corollary 4.4 implies that the map $\widetilde{\phi}$ is constant on the
proper transforms of the lines ${\bf P}V\times\\{w\\}\subset\Delta$ (see
Figure 2).
###### Theorem 4.5.
Let $S^{\prime}\subset\operatorname{Sym}^{d}W$ be the image of
$S\subset\operatorname{Sym}^{d}V$ under an isomorphism $V\to W$. The
$\operatorname{GL}W$-orbit of $[S^{\prime}]\in{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$ contains finitely many orbits in its
closure. In addition to the orbit of $[S^{\prime}]$, these are the orbits of
$\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle$ where
$a_{0}<\dots<a_{r}$ is the vanishing sequence of $S$ at some $p\in{\bf P}V$.
###### Proof.
The closure of the $\operatorname{GL}W$-orbit of $[S^{\prime}]$ is precisely
the image of the morphism
$\widetilde{\phi}\colon\widetilde{\bf P}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W).$
Let $\widetilde{\Delta}\subset\widetilde{\bf P}$ be the pre-image of
$\Delta\subset{\bf P}^{3}$. Then $\widetilde{\bf P}-\widetilde{\Delta}$ is a
copy of $\operatorname{PGL}_{3}$, and its image is the orbit of
$[S^{\prime}]$.
Consider $\widetilde{m}\in\widetilde{\Delta}$ and let $m\in\Delta$ be its
image. Suppose $m$ does not lie on any line $L_{b}$ and let
$\operatorname{im}m=[w_{1}]\in{\bf P}W$. Then by Corollary 4.4, the image
$\widetilde{\phi}(\widetilde{m})\in{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$
corresponds to
(17) $\langle w_{1}^{d}w_{2}^{0},\dots,w_{1}^{d-r}w_{2}^{r}\rangle,$
where $w_{2}\in W$ is any vector linearly independent from $w_{1}$.
Suppose $m$ lies on $L_{b}$. Let $w=\operatorname{im}m\in{\bf P}W$, so that
$m\in\Delta$ corresponds to $(b,w)$ in the isomorphism (15). Suppose $v_{2}\in
V$ is such that $b=[v_{2}]$ and $w_{1}\in W$ such that $w=[w_{1}]$. The point
$\widetilde{m}$ corresponds to a direction in the normal bundle $N_{L_{b}/{\bf
P}^{3}}$ at $p$. Consider the direction in $N_{L_{b}/{\bf P}^{3}}|_{p}$
corresponding to the line ${\bf P}V\times\\{w\\}\subset\Delta$ through $m$. By
Corollary 4.4, the map $\widetilde{\phi}$ is constant on this line with value
given by the subspace (17). If $\widetilde{m}$ corresponds to this direction,
then $\widetilde{\phi}(\widetilde{m})$ corresponds to the same subspace. All
the other directions in $N_{L_{b}/{\bf P}^{3}}|_{p}$ can be realised by
tangent directions of families $m_{t}$ for suitable choices of $v_{1}\in V$
and $w_{2}\in W$. If $\widetilde{m}$ corresponds to the tangent direction
given by $m_{t}$, then Proposition 4.3 implies that
$\widetilde{\phi}(\widetilde{m})$ corresponds to the subspace
(18) $\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$
In conclusion, we see that the image of every point of $\widetilde{\Delta}$ is
of the form (17), which corresponds to the generic vanishing sequence, or
(18), which corresponds to the vanishing sequence of a ramification point. ∎
## 5\. Localisation
The goal of this section is to compute the equivariant orbit class of a linear
series using localisation on $\widetilde{{\bf P}}^{3}$. The calculation is
similar to the one in [13, Appendix B].
Choose a basis $\langle w_{1},w_{2}\rangle$ of $W$ and let
$T\subset\operatorname{GL}(W)$ be the diagonal torus with respect to this
basis. Choose coordinates $t_{1},t_{2}$ on $T$ so that the action on $W$ is
$(t_{1},t_{2})\colon(w_{1},w_{2})\mapsto(t_{1}w_{1},t_{2}w_{2}).$
Then the $T$-fixed locus of ${\bf P}^{3}={\bf P}\operatorname{Hom}(V,W)$
consists of two skew lines $\Lambda_{1}$ and $\Lambda_{2}$ given by
$\Lambda_{i}={\bf P}\operatorname{Hom}(V,\langle w_{i}\rangle).$
Note that the lines $\Lambda_{i}\subset{\bf P}^{3}$ lie in the discriminant
quadric $\Delta$, and are in the opposite ruling as the lines $L_{b}$ blown up
to get $\widetilde{{\bf P}}^{3}$. The $T$-fixed locus of $\widetilde{{\bf
P}}^{3}$ consists of the proper transforms $\widetilde{\Lambda}_{i}$ of
$\Lambda_{i}$ and points $p_{i,b}$ for $i=1,2$ and $b\in B$. The point
$p_{i,b}$ maps under the blow-down to the point $\Lambda_{i}\cap L_{b}$. We
now describe it in local coordinates.
Choose a basis $v_{1},v_{2}$ of $V$ such that $[v_{2}]=b\in{\bf P}V$. In the
bases $v_{1},v_{2}$ of $V$ and $w_{1},w_{2}$ of $W$, we represent
$m\in\operatorname{Hom}(V,W)$ by the matrix
$m=\begin{pmatrix}X_{11}&X_{12}\\\ X_{21}&X_{22}\end{pmatrix}.$
The matrix entries $X_{ij}$ are functionals on $\operatorname{Hom}(V,W)$. The
torus $T$-acts by
$(t_{1},t_{2})\colon X_{ij}\mapsto t_{i}^{-1}X_{ij}.$
The line $\Lambda_{1}\subset{\bf P}^{3}$ is defined by $X_{21}=X_{22}=0$ and
the line $L_{b}\subset{\bf P}^{3}$ by $X_{12}=X_{22}=0.$ Their intersection is
the point $X_{12}=X_{21}=X_{22}=0$. Choose affine coordinates
$x_{ij}=X_{ij}/X_{11}$ around this point. Then the blow-up of ${\bf P}^{3}$ in
$L_{b}$ has the coordinate charts
$(x_{21},x_{12},x_{22},u\mid x_{12}=ux_{22})\text{ and
}(x_{21},x_{12},x_{22},v\mid x_{22}=vx_{12}).$
The point $p_{1,b}$ is contained in the second chart, and is given by
(19) $p_{1,b}:x_{21}=x_{22}=v=0.$
Given $(m_{1},m_{2})\in\mathbf{Z}$, let $\chi(m_{1},m_{2})$ be the character
of $T$ given by
$\chi(m_{1},m_{2})\colon(t_{1},t_{2})\mapsto t_{1}^{m_{1}}\cdot
t_{2}^{m_{2}}.$
The next two propositions give the normal bundles of the $T$-fixed loci.
###### Proposition 5.1.
Let $N_{1,b}$ be the tangent space to $p_{1,b}\in\widetilde{{\bf P}}^{3}$. As
a $T$-representation, we have
$N_{1,b}\cong\chi(-1,1)\oplus\chi(-1,1)\oplus\chi(1,-1).$
###### Proof.
From (19), we see that the co-tangent space at $p_{1,b}$ is generated by
$x_{21},x_{22},v$, on which $T$ acts by
$(t_{1},t_{2})\colon x_{21}\mapsto t_{1}t_{2}^{-1}x_{21}\quad x_{22}\mapsto
t_{1}t_{2}^{-1}x_{22}\quad v\mapsto t_{1}^{-1}t_{2}v.$
We now take the dual. ∎
###### Proposition 5.2.
Let $\widetilde{N}_{1}$ be the normal bundle of
$\widetilde{\Lambda}_{1}\subset\widetilde{{\bf P}}^{3}$. In the
$T$-equivariant Grothendieck group of $\widetilde{\Lambda}_{1}={\bf P}^{1}$
(with the trivial $T$-action), we have
$[N_{1}]=\left([O]+[O\left(2-|B|\right)]\right)\otimes\chi(-1,1).$
###### Proof.
Let $N_{1}$ be the normal bundle of $\Lambda_{1}\subset{\bf P}^{3}$. The
defining equations show that $\Lambda_{1}$ is the zero locus of sections of
$O_{{\bf P}^{3}}(1)\otimes\chi(0,1)$ and the restriction of $O_{{\bf
P}^{3}}(1)$ to $\Lambda_{1}$ is $O(1)\otimes\chi(-1,0)$. Therefore, we get
$N_{1}\cong\left(O(1)\oplus O(1)\right)\otimes\chi(-1,1).$
Recall that $\beta\colon\widetilde{\bf P}^{3}\to{\bf P}^{3}$ is the blow-up
along $\bigsqcup_{b}L_{b}$. We have the exact sequence
$0\to\widetilde{N}_{1}\to\beta^{*}N_{1}\to Q\to 0,$
where $Q\cong O_{\operatorname{supp}B}\otimes\chi(-1,1)$. The statement
follows. ∎
Let $\widetilde{S}$ be the pull-back to $\widetilde{\bf P}^{3}$ of the
universal sub-bundle of ${\bf Gr}(r+1,\operatorname{Sym}^{d}W)$. The following
two propositions describe $\widetilde{S}$ on the $T$-fixed locus.
###### Proposition 5.3.
Let $p=p_{1,b}$ and let $a_{0}<\dots<a_{r}$ be the vanishing sequence of $S$
at $b$. As a $T$-representation, we have
$\widetilde{S}|_{p}\cong\bigoplus_{i=0}^{r}\chi(d-a_{i},a_{i}).$
###### Proof.
Consider the family $m_{t}$ as in (16), where $[v_{2}]=b$. Then, in
$\widetilde{\bf P}$ we have
$\lim_{t\to 0}m_{t}=p.$
Therefore, we have
$\widetilde{S}|_{p}=\lim_{t\to 0}m_{t}(S).$
Proposition 4.3 gives the limit and yields
$\widetilde{S}|_{p}=\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$
The statement follows. ∎
###### Proposition 5.4.
As a $T$-equivariant sheaf on $\widetilde{\Lambda}_{1}$, we have
$[\widetilde{S}]|_{\widetilde{\Lambda}_{1}}=\bigoplus_{i=0}^{r}\chi(d-i,i).$
###### Proof.
By Corollary 4.4, the map $\widetilde{\phi}$ is constant on
$\widetilde{\Lambda}_{1}$, and its value corresponds to the subspace
$\langle
w_{1}^{d}w_{2}^{0},\dots,w_{1}^{d-r}w_{2}^{r}\rangle\subset\operatorname{Sym}^{d}W.$
The statement follows. ∎
We are now ready to prove the main theorem of the paper, which we restate.
###### Theorem 5.5.
Let
$s\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be an injective map with image $S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$.
Let $\\{(a_{i}(b))\mid b\in B\\}$ be the ramification profile of $S$. Let
$\Gamma\subset\operatorname{PGL}(2)$ be the stabiliser of
$S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$, and assume that it is finite.
Let $\operatorname{Orb}(s)$ be the closure of the
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$ orbit of $s$. Then the
class
$|\Gamma|\cdot[\operatorname{Orb}(s)]\in
A^{(r+1)(d-r)-3}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))$
is given by
$|\Gamma|\cdot[\operatorname{Orb}(s)]=\frac{1}{(\omega_{1}-\omega_{2})^{3}}\sum_{b\in
B}\psi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega)+\frac{2-|B|}{(\omega_{1}-\omega_{2})^{3}}\cdot\psi_{0,\dots,r}(\mu;\omega).$
###### Proof.
Set $V=W=\mathbf{k}^{2}$ and consider the rational map
${\bf P}^{3}={\bf P}\operatorname{Hom}(V,W)\dashrightarrow{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$
given by
$m\mapsto m(S).$
Let $\phi\colon\widetilde{{\bf P}}^{3}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$ be its resolution provided by Theorem 4.1.
Then $\phi$ is a complete parametrisation of the $\operatorname{PGL}W$-orbit
of $[S]$. By Proposition 2.2, we have
$|\Gamma|\cdot[\operatorname{Orb}(s)]=\int_{\widetilde{{\bf
P}}^{3}}c_{(r+1)(d-r)}\operatorname{Hom}(\underline{\mathbf{k}}^{r+1},\phi^{*}Q),$
where $Q$ is the universal quotient bundle on the Grassmannian. In our case,
we have
$\phi^{*}Q=\operatorname{Sym}^{d}\underline{W}/\widetilde{S}.$
Let $T\subset\operatorname{GL}(2)$ be a maximal torus. By the equivariant
localisation formula [8, Corollary 1], we have
(20) $\int_{\widetilde{{\bf
P}}^{3}}c_{(r+1)(d-r)}\operatorname{Hom}(\underline{\mathbf{k}}^{r+1},\phi^{*}Q)=\sum_{F}\int_{F}\frac{c_{(r+1)(d-r)}\operatorname{Hom}(\underline{\mathbf{k}}^{r+1},\phi^{*}Q)}{e(N_{F})},$
where the sum is over the $T$-fixed loci $F$ and $e(N_{F})$ is the top Chern
class of the equivariant normal bundle of $F\subset\widetilde{{\bf P}}^{3}$.
In our case, the fixed loci consist of the lines $\widetilde{\Lambda}_{i}$ for
$i=1,2$ and the points $p_{i,b}$ for $i=1,2$ and $b\in B$.
Let us compute the contribution from $F=\widetilde{\Lambda}_{1}$. Proposition
5.2 give us the numerator of the integrand:
$\prod_{j=0}^{r}\prod_{i=r+1}^{d}((d-i)\omega_{1}+i\omega_{2}-\mu_{j})=\phi_{0,\dots,r}(\mu;\omega).$
Proposition 5.4 gives us the denominator of the integrand:
$(\omega_{2}-\omega_{1})(\omega_{2}-\omega_{1}+(2-|B|)h),$
where $h=c_{1}(O(1))$. The integral is the coefficient of $h$ in the power
series expansion of the quotient. We may compute it by formally
differentiating with respect to $h$ and setting $h=0$. The result is
(21)
$\int_{\widetilde{\Lambda}_{1}}\frac{c_{(r+1)(d-r)}\operatorname{Hom}(\mathbf{k}^{r+1},\phi^{*}Q)}{e(N_{\widetilde{\Lambda}_{1}})}=\frac{2-|B|}{(\omega_{1}-\omega_{2})^{3}}\cdot\phi_{0,\dots,r}(\mu;\omega).$
The contribution from $F=\widetilde{\Lambda}_{2}$ is the same as above with
$\omega_{1}$ and $\omega_{2}$ swapped.
Let us compute the contribution from $F=p_{1,b}$. Proposition 5.3 gives us the
numerator:
$\prod_{j=0}^{r}\prod_{i}((d-i)\omega_{1}+i\omega_{2}-\mu_{j})=\phi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega).$
On the left hand side above, the index $i$ ranges in the set
$\\{0,\dots,d\\}-\\{a_{0}(b),\dots,a_{r}(b)\\}$. Proposition 5.1 gives us the
denominator:
$(\omega_{1}-\omega_{2})^{3}.$
Taking the integral is trivial in this case, so the result is
(22)
$\int_{p_{1,b}}\frac{c_{(r+1)(d-r)}\operatorname{Hom}(\mathbf{k}^{r+1},\phi^{*}Q)}{e(N_{p_{1,b}})}=\frac{1}{(\omega_{1}-\omega_{2})^{3}}\cdot\phi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega).$
The contribution from $F=p_{2,b}$ is the same as above with $\omega_{1}$ and
$\omega_{2}$ swapped.
By combining (21), (22), and their analogues for $\widetilde{\Lambda}_{2}$ and
$p_{2,b}$ we see that the integral in (20) is as claimed. ∎
## 6\. Orbit specialisation
The goal of this section is to understand limits of the orbit closure of a
linear series under one-parameter specialisations. Recall that
$\operatorname{Orb}$ denotes the orbit closure and $\operatorname{WOrb}$ the
weighted orbit closure, namely the cycle obtained by multiplying the orbit
closure by the order of the stabiliser group in $\operatorname{PGL}(2)$.
Fix a positive integer $m$ and for each $j=1,\dots,m$, fix an increasing
sequence of non-negative integers $a_{0}(j)<\dots<a_{r}(j)$. Assume that
(23) $\sum_{j,i}(a_{i}(j)-i)=(r+1)(d-r).$
We consider one-parameter degenerations of linear series on ${\bf P}^{1}$ with
ramification profile $\\{(a_{0}(j),\dots,a_{r}(j))\mid j=1,\dots,m\\}$. We use
limit linear series, for which we refer the reader to [10]. Although [10]
assumes characteristic 0, the theory holds without modification for
characteristic $>d$; see [14].
Let $\Delta$ be the spectrum of a DVR with special point $0$ and generic point
$\eta$. Let $p_{1,\eta},\dots,p_{m,\eta}$ be distinct sections $\eta\to{\bf
P}^{1}_{\eta}$ and let $(S_{\eta},O(d))$ be a linear series that has vanishing
sequence $a_{0}(j)<\dots<a_{r}(j)$ at $p_{j,\eta}$. Let
$(\mathcal{X}\to\Delta,p_{1},\dots,p_{m})$ be a semi-stable extension of
$({\bf P}^{1}_{\eta}\to\eta,p_{1,\eta},\dots,p_{m,\eta})$ such that
$\mathcal{X}$ is non-singular. Then the central fiber
$(X,\overline{p}_{1},\dots,\overline{p}_{m})$ is a semi-stable pointed
rational curve. The linear series $S_{\eta}$ yields a refined limit linear
series $\mathcal{S}$ on $\mathcal{X}$ (use the implication $3\implies 1$ of
[10, Proposition 2.5]). Let $S$ be the reduction of $\mathcal{S}$ to the
central fiber.
We record the ramification profile of the limit linear series $S$ using a
decorated graph. We begin with the dual graph $\Gamma$ of the pointed curve
$(X,\overline{p}_{1},\dots,\overline{p}_{m})$. Recall that $\Gamma$ is
obtained by drawing a vertex $\nu$ for every irreducible component $X_{\nu}$
of $X$, an edge $\mu$ to $\nu$ for every node of $X$ on $X_{\mu}$ and
$X_{\nu}$, and a half-edge $i$ on $\nu$ if the marked point $\overline{p}_{i}$
lies on $X_{\nu}$. We think of the full-edge from $\mu$ to $\nu$ as a
combination of two half-edges, one at $\mu$ and one at $\nu$. We use the term
_dangling half-edge_ when we refer to a true half-edge, and not a part of a
full-edge. On $\Gamma$ we record the ramification information of $S$ as
follows. We label a half-edge $e$ incident to a vertex $\nu$ by the vanishing
sequence of the $X_{\nu}$-aspect of $S$ at the point of $X_{\nu}$ represented
by $e$. If $e$ is a full edge, joining $\mu$ and $\nu$ say, then it gets two
labels, one at $\mu$, say $(a_{0},\dots,a_{r})$, and one at $\nu$, say
$(b_{0},\dots,b_{r})$. Since $S$ is a refined series, the two labels are
complementary; that is, they satisfy $a_{i}+b_{r-i}=d$. Figure 3 shows an
example of a labelled dual graph $\Gamma$ arising from a limit linear series
of rank $2$ and degree $4$.
$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,3)$$(1,4)$$(0,3)$$(1,4)$$(1,2)$$(2,3)$
Figure 3. We represent the ramification information of a refined limit
$g^{r}_{d}$ by a labelled dual graph as above, where $r=1$ and $d=4$. The
labels on the half-edges are vanishing sequences, and the two labels on a
full-edge are complementary.
For a vertex $\nu$ of $\Gamma$, let $S(\nu)$ be the $X_{\nu}$-aspect of the
limit linear series $S$.
###### Theorem 6.1.
In the notation above, let $\Sigma$ be the cycle on ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ obtained as the limit of the
cycle $\operatorname{WOrb}(S_{\eta})$. Then we have the equality of
$\operatorname{GL}(2)$-invariant cycles
$\Sigma=\sum_{\nu}\operatorname{WOrb}(S(\nu)).$
###### Proof.
We first show that the cycle
$D=\Sigma-\sum_{\nu\in\Gamma}\operatorname{WOrb}(S(\nu))$
is effective using Proposition 2.4.
For every vertex $\nu$ of $\Gamma$, let $\mathcal{X}_{\nu}$ be the surface
obtained by blowing down all components of $X$ except $X_{\nu}$. Then we have
an isomorphism $\mathcal{X}_{\nu}\cong{\bf P}^{1}\times\Delta$; fix such an
isomorphism and identify the two. Choose an arbitrary vertex of $\Gamma$ and
call it $0$. For every $\nu$, let $g_{\nu}\in\operatorname{PGL}_{2}(\eta)$ be
the unique element such that
$\mathcal{X}_{0,\eta}={\bf P}^{1}\times\eta\xrightarrow{g_{\nu}}{\bf
P}^{1}\times\eta=\mathcal{X}_{\nu,\eta}$
represents the identity map
$\mathcal{X}_{0,\eta}=\mathcal{X}_{\eta}\to\mathcal{X}_{\nu,\eta}=\mathcal{X}_{\eta}.$
For $\mu\neq\nu$, the identity map on the generic fiber does not extend to an
isomorphism $\mathcal{X}_{\mu}\to\mathcal{X}_{\nu}$. Therefore,
$g_{\mu}g_{\nu}^{-1}\in\operatorname{PGL}_{2}(\eta)$ does not extend to an
element of $\operatorname{PGL}_{2}(\Delta)$. On $\mathcal{X}_{\nu}={\bf
P}^{1}\times\Delta$, the limit of $[S_{\eta}]{\colon}\eta\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ is $[S(\nu)]\in{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}).$ Now, Proposition 2.4 applied
to $M={\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ and
$G=\operatorname{PGL}_{2}$ implies that
$D=\Sigma-\sum_{\nu}\operatorname{WOrb}(S(\nu))$
is effective.
Let $N=(d-r)(r+1)$. We apply the Wronskian map
$\rho{\colon}{\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})\to{\bf
P}\operatorname{Sym}^{N}\mathbf{k}^{2}$
and see that the cycle
$\rho_{*}D=\rho_{*}\Sigma-\sum_{\nu}\rho_{*}\operatorname{WOrb}(S(\nu))$
is effective. Let $R_{\eta}$ be the ramification divisor of $S_{\eta}$ and
$R(\nu)$ the ramification divisor of $S(\nu)$. Then the cycle $\rho_{*}\Sigma$
on ${\bf P}\operatorname{Sym}^{N}\mathbf{k}^{2}$ is the limit of the cycle
$\operatorname{WOrb}(R_{\eta})$ and $\rho_{*}\operatorname{WOrb}(S(\nu))$ is
the cycle $\operatorname{WOrb}(R(\nu))$. Aluffi and Faber compute the degree
of the weighted orbit closure for any divisor on ${\bf P}^{1}$ (see [1,
Proposition 1.3] and note that in their paper this degree is called the “pre-
degree”). Using their formula, it is straightforward to check that
$\deg\rho_{*}D=0$ (see Remark 6.2). Since $\rho_{*}D$ is effective, we
conclude that $\rho_{*}D=0$ and since $\rho$ is finite, that $D=0$. ∎
###### Remark 6.2.
We indicate how to check that the pre-degree of the orbit of $R_{\eta}$ is the
sum of the pre-degrees of the orbits of $R(\nu)$ using [1, Proposition 1.3].
The pre-degree of the orbit closure of a divisor depends only on the
multiplicities of the points in the divisor. More precisely, by [1,
Proposition 1.3], the pre-degree for a divisor of degree $N$ with
multiplicities $m_{1},\dots,m_{s}$ is given by
(24) $\begin{split}p(m_{1},\dots,m_{s})&=N^{3}-3N\sum m_{i}^{2}+2\sum
m_{i}^{3}\\\ &=\sum_{i\neq j\neq k}m_{i}m_{j}m_{k}.\end{split}$
To understand the multiplicities in the divisors $R(\nu)$, consider the
labelled dual graph $\Gamma$ and replace the vanishing sequences by their
weights, where the weight of the sequence $(a_{0},\dots,a_{r})$ is
$\sum_{i}(a_{i}-i)$. Then the multiplicities in $R_{\eta}$ are the labels of
the dangling half-edges and the multiplicities of $R(\nu)$ are the labels of
the half-edges incident to $\nu$. The sum of the labels of all the half-edges
incident to a vertex is $N=(d-r)(r+1)$, and the sum of the two labels of a
full-edge is also $N$. Consider the operation of contracting an edge of
$\Gamma$ as shown below
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Since
$N=v+w=m_{1}+\dots+m_{\ell}+v=m_{\ell+1}+\dots+m_{s}+w,$
we have $v=m_{\ell+1}+\dots+m_{s}$ and $w=m_{1}+\dots+m_{\ell}$. From (24) it
follows that
$p(m_{1},\dots,m_{\ell},v)+p(m_{\ell+1},\dots,m_{s},w)=p(m_{1},\dots,m_{s}).$
By contracting the edges one by one, we reduce the sum of the pre-degrees for
$R(\nu)$ to a single sum, which is the pre-degree for $R_{\eta}$.
###### Corollary 6.3.
In the setup of Theorem 6.1, in $A^{*}_{\operatorname{GL}(2)}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}))$ we have
$[\operatorname{WOrb}(S_{\eta})]=\sum_{\nu}[\operatorname{WOrb}(S(\nu))].$
###### Proof.
Since $[\operatorname{WOrb}(S_{\eta})]=[\Sigma]$, the statement follows from
Theorem 6.1. ∎
Recall that the class $\operatorname{WOrb}(S)$ depends only on the
ramification profile of $S$. The ramification profile of $S_{\eta}$ is
recorded by the labels on the dangling half-edges of $\Gamma$. The
ramification profile of $S(\nu)$ is recorded by the labels on the half-edges
incident to $\nu$. Thus, the equality in Corollary 6.3 can be read off purely
from the labelled graph $\Gamma$.
A natural question is: which labelled graphs $\Gamma$ arise from degenerations
of linear series? Here, the obvious necessary conditions are also sufficient.
Let $\Gamma$ be a tree with $m$ dangling half-edges. Suppose all the half-
edges of $\Gamma$, including the full-edges considered as two half-edges, are
labelled with increasing $(r+1)$-tuples of integers in $\\{0,\dots,d\\}$.
Assume that the two labels on a full-edge are complementary.
###### Proposition 6.4.
Let $\Gamma$ be a labelled graph as above. Suppose for every vertex $\nu$ of
$\Gamma$, there is a linear series $S(\nu)$ of rank $r$ and degree $d$ on
${\bf P}^{1}$ whose ramification profile agrees with the multi-set of labels
of half-edges incident to $\nu$. Then $\Gamma$ arises from a degeneration.
That is, there exists a $\Delta$, a semi-stable $m$-pointed curve
$(\mathcal{X}\to\Delta,p_{1},\dots,p_{m})$ and a refined linear series
$\mathcal{S}$ on $\mathcal{X}$ such that the labelled dual graph of its
central fiber is $\Gamma$.
###### Proof.
The proof is a simple application of the smoothing theorem for $g^{r}_{d}$’s
[10, Theorem 3.4].
Let $(X,\overline{p}_{1},\dots,\overline{p}_{m})$ be a semi-stable curve with
dual graph $\Gamma$. We assemble the linear series $S(\nu)$ together to get a
refined limit linear series $S$ on $X$. Note that the expected dimension of
the space $G^{r}_{d}$ on a rational curve with a prescribed ramification
divisor is $0$. Since the Wronskian map is finite and the ramification
divisors of all aspects of $S$ are prescribed, $S$ is an isolated point of
$G^{r}_{d}(X,\overline{p}_{1},\dots,\overline{p}_{m})$. By [10, Theorem 3.4],
there exists a $\Delta$ and a smoothing
$(\mathcal{X}\to\Delta,p_{1},\dots,p_{m},\mathcal{S})$ of
$(X,\overline{p}_{1},\dots,\overline{p}_{m},S)$. ∎
###### Remark 6.5.
The existence of a $g^{r}_{d}$ on ${\bf P}^{1}$ with prescribed ramification
profile is a Schubert theoretic condition on ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. Such a series exists if and
only if the product of certain Schubert classes on the Grassmannian is non-
zero. See the remark after [9, Theorem 2.3] for a precise statement.
Theorem 6.1 asserts an equality of $\operatorname{GL}(2)$-invariant cycles on
${\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. It can be easily
upgraded to an equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-invariant cycles on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$.
###### Proposition 6.6.
In the notation of Theorem 6.1, let
$s_{\eta}\colon\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})_{\eta}$
be a homomorphism with image $S_{\eta}$. Let
$s_{\nu}\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be homomorphisms with images $S(\nu)$. Let $\Sigma$ be the flat limit of
$\operatorname{WOrb}(s_{\eta})$. Then, on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$,
we have the equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-invariant cycles
$\Sigma=\sum_{\nu}\operatorname{WOrb}(s_{\nu}).$
###### Proof.
Let
$U\subset\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be the open subset of injective homomorphisms. The action of
$\operatorname{GL}(r+1)$ on $U$ is free and the map
$\operatorname{im}\colon U\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$
taking a homomorphism to its image is the quotient by
$\operatorname{GL}(r+1)$. On $U$, the assertions follow from Theorem 6.1.
It suffices to prove that $\Sigma$ does contain any $(r+1)^{2}+3$-dimensional
components in the complement of $U$. We claim that $\Sigma$ is the closure of
$\Sigma\cap U$, from which the statement follows.
To see this, take $x\in\Sigma$. Then, possibly after a base change, we can
choose
$x_{\eta}\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})_{\eta}$
in the $\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-orbit of $s_{\eta}$
such that the flat limit of $x_{\eta}$ is $x$. Let
$S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$ be the $(r+1)$-dimensional
subspace corresponding to the flat limit of $\operatorname{im}(x_{\eta})$ in
${\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. Then it is easy to see
that $S$ contains the image of $x$. Let $\ell$ be the dimension of the image
of $x$. By making a change of coordinates on $\mathbf{k}^{r+1}$ if necessary,
assume that $x$ sends the last $r+1-\ell$ standard basis vectors to 0. Let
$s^{\prime}\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be a map that agrees with $x$ on the first $\ell$ basis vectors and whose
image is $S$. Then $s^{\prime}\in\Sigma\cap U$. Consider the diagonal one
parameter subgroup $\lambda_{t}\to\operatorname{GL}(r+1)$ with diagonal
entries
$(\underbrace{1,\dots,1}_{\ell},\underbrace{t,\dots,t}_{r+1-\ell}).$
Then, by construction, $\lim_{t\to 0}\lambda_{t}(s^{\prime})=x$. Since
$\Sigma\cap U$ is $\operatorname{GL}(r+1)$-invariant, and
$s^{\prime}\in\Sigma\cap U$, we get that $\lambda_{t}(s^{\prime})\in\Sigma\cap
U$ for $t\neq 0$. We conclude that $x$ lies in the closure of $\Sigma\cap U$.
∎
###### Corollary 6.7.
In the setup of Proposition 6.6, in
$A^{*}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))$,
we have
$[\operatorname{WOrb}(s_{\eta})]=\sum_{\nu}[\operatorname{WOrb}(s_{\nu})].$
###### Proof.
Since $[\operatorname{WOrb}(s_{\eta})]=[\Sigma]$, the statement follows from
Proposition 6.6. ∎
Finally, we discuss the specialisation mentioned in the introduction (Theorem
1.5). Let $p_{1},\dots,p_{m}$ be sections $\Delta\to{\bf P}^{1}\times\Delta$
that are distinct on the generic fiber but $p_{1}=\dots=p_{\ell}$ on the
central fiber (with no other coincidences). Let $S_{\eta}$ be a linear series
on the generic fiber whose ramification points are $p_{1},\dots,p_{m}$ and
whose vanishing sequence at $p_{i}$ is $a(i)=(a_{0}(i),\dots,a_{r}(i))$. Let
$S$ be the flat limit of $S_{\eta}$. It is easy to see that the vanishing
sequence of $S$ at $p_{i}$ for $i>\ell$ is $a(i)$. Let
$c=(c_{0},\dots,c_{r+1})$ be the vanishing sequence of $S$ at
$p=p_{1}=\dots=p_{\ell}$. Let $c^{\prime}=(d-c_{r},\dots,d-c_{0})$ be the
complementary sequence.
###### Proposition 6.8.
In the setup above, there exists a linear series of rank $r$ and degree $d$ on
${\bf P}^{1}$ with ramification profile $\\{a(1),\dots,a(\ell),c^{\prime}\\}$.
Furthermore, on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$ we
have the equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-equivariant classes
$[\operatorname{WOrb}(a(1),\dots,a(m))]=[\operatorname{WOrb}(a(1),\dots,a(\ell),c^{\prime})]+[\operatorname{WOrb}(a(\ell+1),\dots,a(m),c)].$
###### Proof.
The existence of the linear series can be deduced using Schubert calculus. We
give an alternate geometric proof using the smoothing theorem.
Consider the pointed stable limit of $({\bf P}^{1}_{\eta},p_{1},\dots,p_{m})$.
The dual graph of limit, together with the labelling given by the limit linear
series, has the form
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{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\scriptsize$a(\ell+1)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
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}\hbox{{\scriptsize$c^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
In this diagram, $T$ is a labelled tree with dangling half-edges labelled
$a(1),\dots,a(\ell)$, and one of its vertices, say $t$, is connected to the
remaining vertex, say $o$, by an edge that has the label $c^{\prime}$ near $t$
and $c$ near $o$. We apply Proposition 6.4 to $T$ together with the dangling
edges labelled $a(1),\dots,a(\ell)$ and $c^{\prime}$. A geometric general
fiber of the asserted degeneration gives a linear series of rank $r$ and
degree $d$ on ${\bf P}^{1}$ with ramification profile
$\\{a(1),\dots,a(\ell),c^{\prime}\\}$.
By applying Proposition 6.4 to the two vertex graph
$\leavevmode\hbox to155.87pt{\vbox
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.71223pt}{-3.21947pt}\pgfsys@lineto{10.96613pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
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{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{99.02391pt}{-3.0686pt}\pgfsys@lineto{107.08055pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}{}\pgfsys@moveto{4.91386pt}{0.0pt}\pgfsys@lineto{90.27214pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$c^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{71.68611pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$c$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$
we see that the graph above arises from a degeneration. We now apply Corollary
6.7 to this degeneration. ∎
## References
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* [5] A. S. Buch and R. Rimányi. A formula for non-equioriented quiver orbits of type $A$. J. Algebr. Geom., 16(3):531–546, 2007.
* [6] A. Deopurkar, A. Patel, and D. Tseng. A universal formula for counting cubic surfaces, 2021. Pre-print, arXiv:2109.12672.
* [7] D. Edidin and W. Graham. Equivariant intersection theory (With an appendix by Angelo Vistoli: The Chow ring of ${\mathcal{M}}_{2}$). Invent. Math., 131(3):595–644, 1998.
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* [11] L. M. Fehér, R. Rimányi, and A. Weber. Characteristic classes of orbit stratifications, the axiomatic approach. In Schubert calculus and its applications in combinatorics and representation theory. Selected papers presented at the “International Festival in Schubert Calculus”, Guangzhou, China, November 6–10, 2017, pages 223–249. Singapore: Springer, 2020.
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|
# Top-Down Synthesis for Library Learning
Matthew Bowers Massachusetts Institute of TechnologyCambridgeMAUSA
<EMAIL_ADDRESS>, Theo X. Olausson Massachusetts Institute of
TechnologyCambridgeMAUSA<EMAIL_ADDRESS>, Lionel Wong Massachusetts
Institute of TechnologyCambridgeMAUSA<EMAIL_ADDRESS>, Gabriel Grand
Massachusetts Institute of TechnologyCambridgeMAUSA<EMAIL_ADDRESS>, Joshua
B. Tenenbaum Massachusetts Institute of TechnologyCambridgeMAUSA , Kevin
Ellis Cornell UniversityIthacaNYUSA and Armando Solar-Lezama Massachusetts
Institute of TechnologyCambridgeMAUSA
(2023; 2022-07-07; 2022-11-07)
###### Abstract.
This paper introduces corpus-guided top-down synthesis as a mechanism for
synthesizing library functions that capture common functionality from a corpus
of programs in a domain specific language (DSL). The algorithm builds
abstractions directly from initial DSL primitives, using syntactic pattern
matching of intermediate abstractions to intelligently prune the search space
and guide the algorithm towards abstractions that maximally capture shared
structures in the corpus. We present an implementation of the approach in a
tool called Stitch and evaluate it against the state-of-the-art deductive
library learning algorithm from DreamCoder. Our evaluation shows that Stitch
is 3-4 orders of magnitude faster and uses 2 orders of magnitude less memory
while maintaining comparable or better library quality (as measured by
compressivity). We also demonstrate Stitch’s scalability on corpora containing
hundreds of complex programs that are intractable with prior deductive
approaches and show empirically that it is robust to terminating the search
procedure early—further allowing it to scale to challenging datasets by means
of early stopping.
Program Synthesis, Library Learning, Abstraction Learning
††copyright: rightsretained††doi: 10.1145/3571234††journalyear:
2023††submissionid: popl23main-p278-p††journal: PACMPL††journalvolume:
7††journalnumber: POPL††article: 41††publicationmonth: 1††ccs: Software and
its engineering Automatic programming
## 1\. Introduction
Figure 1. (A) Given an initial DSL of lower-level primitives and a corpus of
programs written in the initial DSL, (B) Stitch uses a fast and memory
efficient corpus-guided top-down search algorithm to construct function
abstractions which maximally capture shared structure across the corpus of
programs. (C) Stitch automatically rewrites the initial corpus using the
learned library functions.
One way programmers manage complexity is by hiding functionality behind
functional abstractions. For example, consider the graphics programs at the
bottom of Fig. 1A. Each uses a generic set of drawing primitives and renders a
technical schematic of a hardware component, shown to the left of each
program. Faced with the task of writing more of these rendering programs, an
experienced human programmer likely would not continue using these low-level
primitives. Instead, they would introduce new functional abstractions, like
the one at the bottom of Fig. 1B which renders a regular polygon given a size
and number of sides. Useful abstractions like these allow more concise and
legible programs to render the existing schematics. More importantly, well-
written abstractions should generalize, making it easier to write new graphics
programs for similar graphics tasks.
Recently, the program synthesis community has introduced new approaches that
can mimic this process, automatically building a library of functional
abstractions in order to tackle more complex synthesis problems (Ellis et al.,
2021, 2020; Shin et al., 2019; Lázaro-Gredilla et al., 2019; Dechter et al.,
2013). One popular approach to library learning is to search for common tree
fragments across a corpus of programs, which can be introduced as new
abstractions (Shin et al., 2019; Lázaro-Gredilla et al., 2019; Dechter et al.,
2013). Ellis et al. 2021, however, introduces an algorithm that reasons about
variable bindings to abstract out well-formed functions instead of just tree
fragments. While it produces impressive results, the system in Ellis et al.
2021 takes a deductive approach to library learning that is difficult to scale
to larger datasets of longer and deeper input programs. This approach is
deductive in that it uses semantics-preserving rewrite rules to attempt to
refactor existing programs to expose shared structure. This requires
representing and evaluating an exponentially large space of proposed
refactorings to identify common functionality across the corpus. Prior work,
such as Ellis et al. 2021, 2020, approaches this challenge by combining a
dynamic bottom-up approach to refactoring with version spaces to more
efficiently search over the refactored programs. However, these deductive
approaches face daunting memory and search requirements as the corpus scales
in size and complexity.
This paper introduces an alternate approach to library learning, while
preserving the focus on well-formed function abstractions from Ellis et al.
2021. Instead of taking a deductive approach based on refactoring the corpus
with rewrite rules, we directly synthesize abstractions. We call this approach
corpus-guided top-down synthesis, and it is based on the insight that when
applied to the task of synthesizing abstractions, top-down search can be
guided precisely towards discovering shared abstractions over a set of
existing training programs. At every step of the search, syntactic comparisons
between a partially constructed abstraction and the set of training programs
can be used to strongly constrain the search space and direct the search
towards abstractions that capture the greatest degree of shared syntactic
structure.
We implement this approach in Stitch, a corpus-guided top-down library
learning tool written in Rust (Fig. 1). Stitch is open-source and available
both as a Python package and a Rust crate; the code, installation
instructions, and tutorials are available at the
GitHub111https://github.com/mlb2251/stitch. We evaluate Stitch through a
series of experiments (Section 6), and find that: Stitch learns libraries of
comparable quality to those found by the algorithm of Ellis et al. 2021 on
their iterative bootstrapped library learning task, while being 3-4 orders of
magnitude faster and using 2 orders of magnitude less memory (Section 6.1);
Stitch learns high quality libraries within seconds to single-digit minutes
when run on corpora containing a few hundred programs with mean lengths
between 76–189 symbols (sourced from Wong et al. 2022), while even the
simplest of these corpora lies beyond the reach of the algorithm of Ellis et
al. 2021 (Section 6.2); and that Stitch degrades gracefully when resources are
constrained (Section 6.3). We also perform ablation studies to expose the
relative impact of different optimizations in Stitch (Section 6.4). Finally,
we show that Stitch is complementary to deductive rewrite approaches (Section
6.5).
In summary, our paper makes the following contributions:
1. (1)
Corpus-guided top-down synthesis (CTS): A novel, strongly-guided branch-and-
bound algorithm for synthesizing function abstractions (Sec. 3).
2. (2)
CTS for program compression: An instantiation of the CTS framework for utility
functions favoring abstractions that compress a corpus of programs (Sec. 4).
3. (3)
Stitch: A parallel Rust implementation of CTS for compression that achieves
3-4 orders of magnitude of speed and memory improvements over prior work, and
the analysis of its performance, scaling, and optimizations through several
experiments (Sec. 6).
## 2\. Overview
Figure 2. (A) Schematic of corpus-guided top-down search (CTS). Partial
abstractions can contain holes indicating unfinished subtrees, denoted ??,
while complete abstractions do not contain holes. (B) Partial and complete
abstractions match at locations in the corpus. The blue partial abstraction
can be expanded into the green complete abstraction so the green abstraction
matches in a subset of places that the blue abstraction does. For complete
abstractions, matching indicates that the subtree can be rewritten to use the
abstraction.
In this section, we build intuition for the algorithmic insights that power
Stitch. As a running example, we focus on learning a single abstraction from
the following corpus of programs:
(1) $\displaystyle\lambda x.\
\texttt{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+ 3}
({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}*} (+ 2 4)
2)}$ $\displaystyle\lambda xs.\ \texttt{map ($\lambda x.\
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+ 3}
({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}*} 4 (+ 3
x))) xs}$ $\displaystyle\lambda x.\ \texttt{* 2
({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+ 3}
({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}*} x (+ 2
1)))}$
The notion of what a good abstraction is depends on the application, so our
algorithm is generic over the utility function that we seek to maximize.
Following prior work (Ellis et al., 2021; Shin et al., 2019; Dechter et al.,
2013) we focus on compression as a utility function: a good abstraction is one
which minimizes the size of the corpus when rewritten with the abstraction.
The utility function used by Stitch is detailed in Section 4 and corresponds
exactly to the compression objective, but at a high level the function seeks
to maximize the product of the size of the abstraction and the number of
locations where the abstraction can be used. This product balances two key
features of a highly compressive abstraction: the abstraction should be
general enough that it applies in many locations, but specific enough that it
captures a lot of structure at each location.
The optimal abstraction that maximizes our utility in this example is:
(2)
$\texttt{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}fn0}}=\lambda\alpha.\
\lambda\beta.\texttt{(+ 3 (* $\alpha$ $\beta$))}$
And the shared structure that this abstraction captures is highlighted in blue
in Eq. 1. When rewritten to use this abstraction, the size of the resulting
programs is minimized:
(3) $\displaystyle\lambda x.\
\texttt{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}fn0}
(+ 2 4) 2}$ $\displaystyle\lambda xs.\ \texttt{map ($\lambda x.\
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}fn0} 4 (+ 3
x)) xs}$ $\displaystyle\lambda x.\ \texttt{* 2
({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}fn0} x (+ 2
1))}$
Stitch synthesizes the optimal abstraction directly through top-down search.
Top-down search methods construct program syntax trees by iteratively refining
partially-completed programs that have unfinished holes, until a complete
program that meets the specification is produced (Feser et al., 2015;
Polikarpova et al., 2016; Balog et al., 2016; Ellis et al., 2021; Nye et al.,
2021; Shah et al., 2020). This kind of search is often made efficient by
identifying branches of the search tree that can be pruned away because the
algorithm can efficiently determine that none of the programs in that branch
can be correct. We aim to apply a similar idea to the problem of synthesizing
good abstractions; the idea is to explore the space of functions in the same
top-down way, but in search of a function that maximizes the utility measure.
The key observation is that this new objective affords even more aggressive
pruning opportunities than the traditional correctness objective, allowing us
to synthesize optimal abstractions very efficiently. We call this approach
corpus-guided top-down search (CTS).
##### Corpus-Guided Top-Down Search.
Like other top-down synthesis approaches, our algorithm explores the space of
abstractions by repeatedly refining abstractions with holes, as illustrated in
Fig. 2A. We call abstractions with holes partial abstractions in contrast to
complete abstractions which have no holes. Our top-down algorithm searches
over abstraction bodies, so to synthesize the optimal abstraction fn0 in our
running example, we synthesize its body: (+ 3 (* $\alpha$ $\beta$)).
We say that a partial abstraction matches at a subtree in the corpus if
there’s a possible assignment to the holes and arguments that yields the
subtree. For example, consider the subtree (+ 3 (* (+ 2 4) 2) from the first
program in our running example, shown as a syntax tree at the bottom of figure
2B. The partial abstraction (+ 3 ??) shown in blue matches here with the hole
$\texttt{??}=\texttt{(* (+ 2 4) 2)}$, and the complete abstraction (+ 3 (*
$\alpha$ $\beta$)) matches here with $\alpha=\texttt{(+ 2 4)}$ and
$\beta=\texttt{2}$. For complete abstractions, matching corresponds to being
able to use the abstraction to rewrite this subtree, resulting in compression.
We refer to the set of subtrees at which a complete or partial abstraction
matches as the set of match locations. For example, the three match locations
of (+ 3 (* $\alpha$ $\beta$)) are the subtree (+ 3 (* (+ 2 4) 2)) in the first
program, (+ 3 (* 4 (+ 3 x))) in the second program, and (+ 3 (* x (+ 2 1))) in
the third program.
In traditional top-down synthesis, a branch of search can be safely pruned by
proving that a program satisfying the specification cannot exist in that
branch. In CTS, we can safely prune a branch of search if we can prove that it
cannot contain the optimal abstraction. One way to prove this is by computing
an upper bound on the utility of abstractions in the branch, and discarding
the branch if we have previously found an abstraction with a utility that is
higher than this bound. An efficient and conservative way to compute this
upper bound is to overapproximate the set of match locations, then upper bound
the compressive utility gain from each match location.
Our key observation to compute this bound is that during search, expanding a
hole in a partial abstraction yields an abstraction that is more precise and
thus matches at a subset of the locations that the original matched at. This
subset observation is depicted in Fig. 2 where the refinement of the partial
abstraction (+ 3 ??) (blue) to the goal (+ 3 (* $\alpha$ $\beta$)) (green)
results in a larger abstraction that matches at a subset of the locations
(match locations shown at the top of Fig. 2B). An important consequence of
this is that the match locations of a partial abstraction serves as an over-
approximation of the match locations of any abstraction in this branch of top-
down search.
To upper bound the compressive utility gained by rewriting at a match
location, notice that the most compressive abstraction that matches at a
subtree is the constant abstraction corresponding to the subtree itself. For
example, the largest possible abstraction at (+ 3 (* (+ 2 4) 2) is just (+ 3
(* (+ 2 4) 2). In other words, the size of the subtree is a strict upper bound
on how much compression can be achieved at a match location. Thus we get the
upper bound for some partial abstraction $A_{\texttt{??}}$:
(4)
$U_{\text{upperbound}}(A_{\texttt{??}})=\sum_{e\in\text{matches}(A_{\texttt{??}})}{\text{size}(e)}$
Equipped with this bound, our algorithm performs a branch-and-bound style top-
down search, where at each step of search it discards all partial abstractions
that have utility upper bounds that are less than the utility of the best
complete abstraction found so far.
Our full algorithm presented in Section 3 has some additional complexity. It
handles rewriting in the presence of variables soundly, it uses an exact
utility function that accounts for additional compression gained by using the
same variable in multiple places, and it handles situations where match
locations overlap and preclude one another. We also introduce two other
important forms of pruning while maintaining the optimality of the algorithm,
and we use the upper bound as a heuristic to prioritize more promising
branches of search first.
Figure 3. Visualization of Stitch library learning on the nuts-bolts
subdomain from Wong et al. 2022. (A) From the base DSL primitives (top row),
Stitch iteratively discovers a series of abstractions that compress programs
in the domain. Arrows demonstrate how abstractions from selected iterations
build on one another to achieve increasingly higher-level behaviors. (B)
Rewriting a single item from the domain with the cumulative benefit of
discovered abstractions yields increasingly compact expressions. Colors
indicate correspondence between object parts and program fragments: orange =
outer octagon, green = ring of six circles, purple = inner circle.
##### Building Up Abstraction Libraries.
The top-down search algorithm described above yields a single abstraction.
However, we can easily run this algorithm for multiple iterations on a corpus
of programs to build up an entire library of abstractions. Fig. 3 illustrates
the power of this kind of iterative library learning, which interleaves
compression and rewriting. At each iteration, Stitch discovers a single
abstraction that is used to rewrite the entire corpus of programs. Successive
iterations therefore yield abstractions that build hierarchically on one
another, achieving increasingly higher-level behaviors. As the library grows
to contain richer and more complex abstractions, individual programs shrink
into compact expressions.
##### Structure of the Paper.
In the subsequent sections, we formalize our corpus-guided top-down search
algorithm (Section 3), its application to the problem of compression (Section
4), and how it may be layered on top of data structures such as version spaces
(Section 5). We then report experimental results (Section 6) showcasing both
diverse library learning settings as well as ablations of Stitch’s search
mechanisms. Finally, we conclude by situating Stitch within the landscape of
related work (Section 7) and future work (Section 8) in the areas of library
learning and program synthesis.
## 3\. Corpus-Guided Top-Down Search
### 3.1. Problem Setup
In this section we provide the definitions necessary to understand our
problem.
##### Grammar
Our algorithm operates on lambda-calculus expressions with variables
represented through de Bruijn indices (de Bruijn, 1972); expressions come from
a context-free grammar of the form $e\Coloneqq\lambda.\ e^{\prime}\mid
e^{\prime}\ e^{\prime\prime}\mid\$i\mid t$, where
$t\in\mathcal{G}_{\texttt{sym}}$ refers to the set of built-in primitives in
the domain-specific language. For example, in an arithmetic domain
$\mathcal{G}_{\texttt{sym}}$ would include operators like + and constants like
3. We say that $e^{\prime}$ is a subexpression of $e$ if $e=C[e^{\prime}]$,
where $C$ is a context as defined in the contextual semantics of Felleisen and
Hieb 1992. An expression is closed if it has no free variables, in which case
the expression is also a program. A set of programs $\mathcal{P}$ is a corpus.
When representing variables through de Bruijn indices, $\$i$ refers to the
variable bound by the $i$th closest lambda above it. Thus $\lambda x.\ \lambda
y.\ (x\ y)$ is represented as $\lambda.\ \lambda.\ (\$1\ \$0)$. Beta reduction
with de Bruijn indices requires upshifting: incrementing free variables in the
argument when substitution recurses into a lambda, since the free variables
must now point past one more lambda. Similarly, inverse beta reduction
requires downshifting.
##### Abstraction
Given a grammar $\mathcal{G}$, we define the abstraction grammar
$\mathcal{G}_{A}$ as $\mathcal{G}$ extended to include abstraction variables,
denoted by Greek letters $\alpha$, $\beta$, etc. Formally,
$A\Coloneqq\lambda.\ A^{\prime}\mid A^{\prime}\ A^{\prime\prime}\mid\$i\mid
t\mid\alpha$ where $t\in\mathcal{G}_{\texttt{sym}}$. A term $A$ from this
grammar represents the body of an abstraction; e.g. the abstraction
$\lambda\alpha.\ \lambda\beta.\ (\texttt{+}\ \alpha\ \beta)$ is simply
represented by the term $(\texttt{+}\ \alpha\ \beta)$ from the language of
$\mathcal{G}_{A}$.
##### Partial Abstraction
A partial abstraction $A_{\texttt{??}}$ is an abstraction that can
additionally include holes. A hole, denoted by ??, represents an unfinished
subtree of the abstraction. Thus, $A_{\texttt{??}}\Coloneqq
A\mid\texttt{??}\mid\lambda.\ A_{\texttt{??}}^{\prime}\mid
A_{\texttt{??}}^{\prime}\ A_{\texttt{??}}^{\prime\prime}$. Any abstraction is
thus also a partial abstraction. Given a grammar $\mathcal{G}$, the grammar of
partial abstractions is denoted by $\mathcal{G}_{A_{\texttt{??}}}$. Each hole
in a partial abstraction $A$ can be referred to by a unique index, which can
be explicitly written as $\texttt{??}_{i}$.
##### Lambda-Aware Unification
We introduce lambda-aware unification as a modification of traditional
unification adapted to our algorithm. $\textsc{LambdaUnify}(A,e)$ returns a
mapping (if one exists) from abstraction variables and holes to expressions
$[\alpha_{i}\to e_{i}^{\prime},\ \ldots,\ \texttt{??}_{j}\to u_{j}^{\prime},\
\ldots]$ such that
(5) $(\lambda\alpha_{i}.\ \ldots\ \lambda\texttt{??}_{j}.\ \ldots A)\
e_{i}^{\prime}\ \ldots\ u_{j}^{\prime}\ \ldots=e$
through beta reduction. A key difference from traditional unification is that
the expression $e$ may be deep inside a program written using de Bruijn
indices, so LambdaUnify must perform some index arithmetic in order to
generate its output mapping; in particular, raising a subtree out of a lambda
during this inverse beta reduction requires downshifting variables in it.
The definition of LambdaUnify is presented in Fig. 4 (left). In U-App,
$\textsc{merge}(l_{1},l_{2})$ merges two $[\alpha_{i}\to e_{i}]$ mappings to
create a new mapping that includes all bindings from $l_{1}$ and $l_{2}$, and
fails if the same abstraction variable $\alpha$ maps to different expressions
in $l_{1}$ and $l_{2}$. The rule U-Same applies only when the abstraction
argument to LambdaUnify is an expression (i.e. it contains no holes or
abstraction variables) and is syntactically identical to the expression it is
being unified with. In U-Lam, DownshiftAll returns a new mapping with all
abstracted expressions $e$ and hole expressions $u$ downshifted using the
$\downarrow$ operator presented in Fig. 4 (right). The $\downarrow$ operator
has been modified from traditional downshifting to allow for partial
abstractions that contain holes $\texttt{??}_{j}$ to break the rules of
variable binding, because holes represent unfinished subtrees of the
abstraction.
In particular, a new syntactic form ”$\&i$” is used to represent a $\$i$
variable that has been downshifted further than traditional downshifting would
permit. $\&i$ variables are created by $\downarrow$ when a traditional
downshift would otherwise convert a free variable to a (different, incorrect)
bound variable. These variables represent references to lambdas present within
the body of the abstraction and are allowed in expressions $u_{j}^{\prime}$
bound to holes but not expressions $e_{i}^{\prime}$ bound to abstraction
variables.
To account for $\&i$ variables, the beta reduction used in Eq. 5, which is
defined in Fig. 5, uses a modified upshift operator $\uparrow$ defined to be
an inverse to the $\downarrow$ operator. With this modification of beta
reduction, any $\&i$ variables with negative indices will ultimately be
shifted back to positive indices during reduction. Furthermore, the beta
reduction procedure is equivalent to traditional beta reduction when there are
no holes and thus no $\&i$ variables.
*[left=U-AbsVar] LambdaUnify(α, e) ↝[ α→e ] *[left=U-Hole] LambdaUnify(??_i, e) ↝[ ??_i →e ] *[left=U-App] LambdaUnify(A_1,e_1) ↝l_1
LambdaUnify(A_2,e_2) ↝l_2
l = merge(l_1,l_2) LambdaUnify((A_1 A_2),(e_1 e_2)) ↝l *[left=U-Lam]
LambdaUnify(A, e) ↝l’
l = DownshiftAll(l’) LambdaUnify((λ. A),(λ. e)) ↝l *[left=U-Same]
LambdaUnify(e,e) ↝[ ]
$\displaystyle\textsc{DownshiftAll}([\alpha_{i}\to e_{i}^{\prime},\
\texttt{??}_{j}\to u_{j}^{\prime},\ ...])$ $\displaystyle\ \ \ \
=[\alpha_{i}\to\ \downarrow_{0}e_{i}^{\prime},\ \texttt{??}_{j}\to\
\downarrow_{0}u_{j}^{\prime},\ ...]$
$\displaystyle\downarrow_{d}\lambda.b=\lambda.\downarrow_{d+1}b$
$\displaystyle\downarrow_{d}(fx)=(\downarrow_{d}f\ \downarrow_{d}x)$
$\displaystyle\downarrow_{d}\$i=\begin{cases}\$i,&\text{if }i<d\\\
\$(i-1),&\text{if }i>d\\\ \&(i-1),&\text{if }i=d\\\ \end{cases}$
$\displaystyle\downarrow_{d}\&i=\&(i-1),$ $\displaystyle\downarrow_{d}t=t,\
\text{if $t$ is a primitive (i.e. $t\in\mathcal{G}_{\texttt{sym}}$) }$
Figure 4. (Left) Inference rules for lambda-aware unification. (Right)
Definition of DownshiftAll.
$\displaystyle(\lambda\alpha_{i}.\ \ldots\ \lambda\texttt{??}_{j}.\ \ldots A)\
e_{i}^{\prime}\ \ldots\ u_{j}^{\prime}\ \ldots$ $\displaystyle\ \ \ \
=[\alpha_{i}\to e_{i}^{\prime},\ \texttt{??}_{j}\to u_{j}^{\prime},\ ...]\circ
A$ $\displaystyle l\circ\lambda.b=\lambda.\textsc{UpshiftAll}(l)\circ b$
$\displaystyle l\circ(fx)=(l\circ f)(l\circ x)$ $\displaystyle l\circ\$i=\$i$
$\displaystyle l\circ\alpha_{i}=l[\alpha_{i}]$ $\displaystyle
l\circ\texttt{??}_{i}=l[\texttt{??}_{i}]$ $\displaystyle l\circ t=t,\
\text{for $t\in\mathcal{G}_{\texttt{sym}}$}$
$\displaystyle\textsc{UpshiftAll}([\alpha_{i}\to e_{i}^{\prime},\
\texttt{??}_{j}\to u_{j}^{\prime},\ ...])$ $\displaystyle\ \ \ \
=[\alpha_{i}\to\ \uparrow_{0}e_{i}^{\prime},\ \texttt{??}_{j}\to\
\uparrow_{0}u_{j}^{\prime},\ ...]$
$\displaystyle\uparrow_{d}\lambda.b=\lambda.\uparrow_{d+1}b$
$\displaystyle\uparrow_{d}(fx)=(\uparrow_{d}f\ \uparrow_{d}x)$
$\displaystyle\uparrow_{d}\$i=\begin{cases}\$i,&\text{if }i<d\\\
\$(i+1),&\text{if }i\geq d\\\ \end{cases}$
$\displaystyle\uparrow\&i=\begin{cases}\&(i+1),&\text{if }i+1\neq d\\\
\$(i+1),&\text{if }i+1=d\\\ \end{cases}$ $\displaystyle\uparrow_{d}t=t,\
\text{for $t\in\mathcal{G}_{\texttt{sym}}$}$
Figure 5. (Left) Definition of modified beta reduction and substitution
($\circ$). (Right) Definition of UpshiftAll.
We provide a proof of the correctness of LambdaUnify with respect to Eq. 5 in
Appendix B, as well as a Coq proof of the correctness in stitch.v in the
supplemental material. However, to understand the key idea behind the proof we
focus here on an example illustrating the different cases involving the
$\downarrow$ operator and $\&i$ indices. Consider what happens when you run
(6) $\textsc{LambdaUnify}(\lambda.\ f\ \texttt{??}_{0},\ \lambda.\ e)$
By Eq. 5, the goal is to produce a mapping $[\texttt{??}_{0}\rightarrow
u^{\prime}_{0}]$ such than when replacing $\texttt{??}_{0}$ with
$u^{\prime}_{0}$ it produces $(\lambda.\ e)$. In other words, $(\lambda.\
\lambda.\ f\ \$1)\ u^{\prime}_{0}=\lambda.\ e$. Now, suppose
(7) $\textsc{LambdaUnify}(f~{}\texttt{??}_{0},\
e)\rightsquigarrow[\texttt{??}_{0}\rightarrow\lambda.\ \$i]$
This means that $(\lambda.\ f\ \$0)\ (\lambda.\ \$i)=e$ so $e=f\ (\lambda.\
\$i)$. There are three possibilities for $i$ that need to be considered,
corresponding to the 3 cases in the definition of $\downarrow_{d}\$i$. In the
first case, when $i=0$ and thus $e=f\ (\lambda.\ \$0)$, it means that $\$i$ in
Eq. 7 is bound to the lambda in the return mapping. In this case, DownshiftAll
in U-Lam should not do anything because if we can replace $\texttt{??}_{0}$
with $(\lambda.\ \$0)$ in $(f\ \texttt{??}_{0})$ to produce $e$, the same
replacement in $(\lambda.\ f\ \texttt{??}_{0})$ will produce $(\lambda.\ e)$.
The second case is when $i\geq 2$; this means that $e$ has some variables that
were defined outside of it and they have been captured by $\texttt{??}_{0}$.
For example, if $i=2$ and thus $e=f\ (\lambda.\ \$2)$, the solution to Eq. 6
is $[\texttt{??}_{0}\rightarrow\lambda.\ \$1]$, since $(\lambda.\ \lambda.\ f\
\$1)(\lambda.\ \$1)=\lambda.\ e$; in other words, when computing
$\textsc{LambdaUnify}(\lambda.~{}f~{}\texttt{??}_{0},\ \lambda.\ e)$ from Eq.
7, the $\$i$ in the return mapping has to be downshifted to account for the
fact that it will be substituted inside an additional lambda.
The third case, when $i=1$, is more problematic. In this case, we have that
$e=f\ (\lambda.\ \$1)$. This creates a problem, since there is no $i$ such
that $(\lambda.\ \lambda.\ f\ \$1)(\lambda.\ \$i)=\lambda.\ e$; the $\$i$
needs to refer to the lambda surrounding $e$. The downshift operator addresses
this through the special treatment of the $\&i$ index.
LambdaUnify relates to prior work on unification modulo binders (Huet, 1975;
Miller, 1991, 1992; Dowek et al., 1995, 1996) but focuses on a more
efficiently solvable syntax-driven subset of the more general unification
problems tackled in this prior work. We give a more detailed comparison to
this prior work in Section 7.
##### Match Locations
The set of match locations of a partial abstraction $A_{\texttt{??}}$ in a
corpus $\mathcal{P}$, denoted $\textsc{Matches}(\mathcal{P},A_{\texttt{??}})$,
is the maximal set of context-expression pairs
$\\{(C_{1},e_{1}),(C_{2},e_{2}),\ \ldots,(C_{k},e_{k})\\}$ such that $\forall
k.\ p=C_{k}[e_{k}]$ and $p\in\mathcal{P}$ and
$\textsc{LambdaUnify}(e_{k},A_{\texttt{??}})$ succeeds. We discard locations
where the mapping produced by LambdaUnify has &i indices in expressions bound
by abstraction variables, while we allow them in expressions bound by holes as
holes are allowed to violate variable binding rules. In Section 3.2 we explain
how we maintain the set of matches incrementally.
##### Rewrite Strategies
A corpus $\mathcal{P}$ can be rewritten to use a complete abstraction $A$ as
follows. We introduce a new terminal symbol $t_{A}$ into the symbol grammar
$\mathcal{G}_{\texttt{sym}}$ to represent the abstraction, and consider it
semantically equivalent to $(\lambda\alpha_{0}.\ ...\ \lambda\alpha_{k}.\ A)$.
We then re-express $\mathcal{P}$ in terms of $t_{A}$ as follows. We can
replace the match location $(C,e)\in\textsc{Matches}(\mathcal{P},A)$ with
$t_{A}$ applied with the argument assignments $[\alpha_{i}\to
e_{i}]=\textsc{LambdaUnify}(A,e)$ to yield a semantically equivalent
expression, $C[(t_{A}\ e_{0}\ ...\ e_{k})]$. Note that since this is a
complete abstraction, it contains no holes.
One complication is that rewriting at one match location may preclude
rewriting at another match location if they overlap with each other. A rewrite
strategy $\mathcal{R}$ is a procedure for selecting a subset of
$\textsc{Matches}(\mathcal{P},A)$ to rewrite at, in which no match precludes
another match. We refer to this set as
$\textsc{RewriteLocations}_{\mathcal{R}}(\mathcal{P},A)$. We refer to
rewriting a corpus $\mathcal{P}$ under an abstraction $A$ with rewrite
strategy $\mathcal{R}$ by $\textsc{Rewrite}_{\mathcal{R}}(\mathcal{P},A)$. We
refer to rewriting at only a single particular match location $m$ with
abstraction $A$ by $\textsc{RewriteOne}(m,A)$.
##### Utility
CTS optimizes a user-defined utility function $U_{\mathcal{P},\mathcal{R}}(A)$
which scores an abstraction $A$ given a corpus $\mathcal{P}$ and rewrite
strategy $\mathcal{R}$. There are no strict constraints on the form or
properties of the utility function. While in Section 4 we will focus on
compressive utility functions, our framework does not generally require this.
We say that a rewrite strategy is optimal with respect to a utility function
if the rewrite strategy chooses locations to rewrite at such that the utility
is maximized. A naïve optimal rewrite strategy exhaustively checks the utility
of rewriting with each of the $2^{|\textsc{Matches}(\mathcal{P},A)|}$ possible
subsets of $\textsc{Matches}(\mathcal{P},A)$, however depending on the
specific utility function computing an optimal or approximately optimal
strategy may be significantly more computationally tractable. In Section 4 we
detail the rewrite strategy used by Stitch, which is an optimal, linear-time
rewrite strategy for compressive utility functions based on dynamic
programming.
### 3.2. Algorithm
Given a corpus $\mathcal{P}$, rewrite strategy $\mathcal{R}$, and utility
function $U_{\mathcal{P},\mathcal{R}}(A)$, the objective of CTS is to find the
abstraction $A$ that maximizes the utility $U_{\mathcal{P},\mathcal{R}}(A)$.
CTS takes a branch-and-bound approach (Land and Doig, 1960; Morrison et al.,
2016) to this problem, as described in this section.
##### Expansion
We construct the body of a partial abstraction $A_{\texttt{??}}$ through a
series of top-down expansions starting from the trivial partial abstraction
??. Given a partial abstraction $A_{\texttt{??}}$ we can expand a hole in this
partial abstraction using any production rule from the partial abstraction
grammar $\mathcal{G}_{A_{\texttt{??}}}$, yielding a new abstraction
$A_{\texttt{??}}^{\prime}$. We denote this expansion by $A_{\texttt{??}}\to
A_{\texttt{??}}^{\prime}$. In Figure 2A our overall top-down search is
depicted as a series of expansions. We say that a complete abstraction $A$ can
be derived from a partial abstraction $A_{\texttt{??}}$, denoted
$A_{\texttt{??}}\to^{*}A$, if $A$ is in the transitive reflexive closure of
the expansion operation, i.e. there exists a sequence of expansions from
$A_{\texttt{??}}$ to $A$. Any abstraction can be derived from ??.
##### A Naïve Approach
The goal of our algorithm is to find the maximum-utility abstraction. One
simple, inefficient approach to this is to simply enumerate the entire space
of abstractions through top down synthesis and return the one with the highest
utility. This naïve approach maintains a queue of partial abstractions
initialized with just ??. At each step of the algorithm it pops an abstraction
from the queue, chooses a hole in it and expands that hole using each possible
production rule. Whenever an expansion produces a new partial abstraction it
pushes it to the queue, and when it produces a complete abstraction it
calculates its utility and updates the best abstraction found so far. Since an
abstraction cannot match a program that is smaller than it, the algorithm can
stop expanding when all expansions would lead to abstractions larger than the
largest program in the corpus. This algorithm will enumerate all possible
abstractions exactly once each.
##### Introducing Strict Dominance Pruning
While the naïve approach will find the optimal abstraction, it is extremely
inefficient. We can improve on this using an idea core to branch-and-bound
algorithms: pruning. We allow the algorithm to choose to prune a partial
abstraction instead of expanding it, in which case it simply discards the
abstraction and chooses another from the queue. Of course, to maintain
optimality we must be certain that we never prune the branch of search
containing the optimal abstraction. To formalize safe pruning we will use the
notion of strict dominance from branch-and-bound literature (Morrison et al.,
2016; Ibaraki, 1977; Chu and Stuckey, 2015), which we explain below in terms
of a notion of covering.
A complete abstraction $A^{\prime\prime}$ covers another complete abstraction
$A^{\prime}$, written $\text{covers}(A^{\prime\prime},A^{\prime})$, if the
utility of $A^{\prime\prime}$ is greater than that of $A^{\prime}$. A partial
abstraction $A_{\texttt{??}}^{\prime}$ is strictly dominated by another
partial abstraction $A_{\texttt{??}}^{\prime\prime}$ if and only if every
complete abstraction that is derivable from $A_{\texttt{??}}^{\prime}$ is
covered by a complete abstraction that is derivable from
$A_{\texttt{??}}^{\prime\prime}$.
Formally, $A_{\texttt{??}}^{\prime\prime}$ strictly dominates
$A_{\texttt{??}}^{\prime}$ iff $\forall A^{\prime}.\
A_{\texttt{??}}^{\prime}\to^{*}A^{\prime}\implies\exists A^{\prime\prime}.\
A_{\texttt{??}}^{\prime\prime}\to^{*}A^{\prime\prime}\land\text{covers}(A^{\prime\prime},A^{\prime})$.
Equipped with this formalism, we claim that it is safe to prune an abstraction
$A_{\texttt{??}}^{\prime}$ if we know that there exists some
$A_{\texttt{??}}^{\prime\prime}$ which strictly dominates it. Note that
$A_{\texttt{??}}^{\prime\prime}$ does not necessarily contain the optimum and
may even be pruned in the search if it is strictly dominated by another
abstraction. Knowing the existence of $A_{\texttt{??}}^{\prime\prime}$ is
enough to enable pruning, regardless of whether it has been pruned or has not
yet been enumerated during search.
###### Lemma 1.
Naïve search augmented with strict dominance pruning finds the optimal
abstraction.
###### Proof.
We proceed with a proof by induction. We seek to prove the equivalent
statement that the optimum is never pruned and thus it will be enumerated by
the search.
Our inductive hypothesis is that the optimum has not yet been pruned. In the
base case this is trivially true, since no pruning has taken place yet. In the
inductive step we must prove that a step of pruning maintains the validity of
the inductive hypothesis. Recall that a step of pruning will discard some
partial abstraction $A_{\texttt{??}}^{\prime}$ which is strictly dominated by
another abstraction $A_{\texttt{??}}^{\prime\prime}$. Suppose then for the
sake of contradiction that we did prune the optimum in this step; then the
optimum must have been derivable from $A_{\texttt{??}}^{\prime}$. By strict
dominance we know that all abstractions derivable from
$A_{\texttt{??}}^{\prime}$ must be covered by some abstraction derivable from
$A_{\texttt{??}}^{\prime\prime}$. Hence, there must exist some abstraction
that covers the optimum. However, since the utility of the optimum is greater
than or equal to that of all other abstractions, there is by definition no
such abstraction; we have thus arrived at a contradiction. ∎
This lemma ensures the safety of composing together pruning strategies, as
long as they all are forms of strict dominance pruning, since no instance of
strict dominance pruning will remove the optimum. Determining which partial
abstractions are strictly dominated by which others is specific to the utility
function being used, and in Section 4 we identify two instances of strict
dominance used when instantiating this framework for a compression-based
utility.
##### Upper Bound Pruning
We further improve this algorithm to employ upper bound based pruning, in
which we bound the maximum utility that can be obtained in a branch of search,
and discard the branch if we have previously found a complete abstraction with
higher utility than this bound. This is the most common form of pruning used
in branch-and-bound algorithms (see Morrison et al. 2016 Section 5.1 for a
review). Our algorithm is generic over the upper bound function
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})$ which upper bounds the
utility of any complete abstraction $A$ that can be derived from
$A_{\texttt{??}}$.
Formally, the bound must satisfy $\forall A.\ A_{\texttt{??}}\to^{*}A\implies
U_{\mathcal{P},\mathcal{R}}(A)\leq\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})$.
While this could trivially be satisfied with
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})=\infty$ for any choice of
a utility function, a tighter bound will allow for more pruning. The soundness
of upper bound pruning in maintaining the optimality of the solution trivially
follows from the fact that the pruned branch only contains complete
abstractions that are at most as high-utility as the best abstraction found so
far.
While we defer to Section 4 to construct the upper bound used by Stitch, it is
worth mentioning a key insight that helps in constructing tight bounds for
many utility functions. When bounding the utility of $A_{\texttt{??}}$, it is
useful to bound the set of possible locations where rewrites can occur for any
abstraction $A$ derived from $A_{\texttt{??}}$. The following lemma provides
such a bound:
###### Lemma 2.
$\textsc{Matches}(\mathcal{P},A_{\texttt{??}})$ is an upper bound on the set
of locations where rewrites can occur in any $A$ derived from
$A_{\texttt{??}}$.
This follows from the fact that as partial abstractions are expanded they
become more precise and thus match at a subset of the locations. Since
rewriting only happens at a subset of match locations,
$\textsc{Matches}(\mathcal{P},A_{\texttt{??}})$ serves as a bound on the set
of locations where rewrites can occur.
An important consequence of Lemma 2 is that if a partial abstraction matches
at zero locations, then all abstractions derived from it will have zero
rewriting locations, so no rewriting can occur. Such branches can therefore be
safely pruned.
##### Improving the Search Order
Finally, without sacrificing the optimality of this algorithm we can
heuristically guide the order in which the space of abstractions is explored.
The queue of partial abstractions used in top-down search can be replaced with
a priority queue ordered by the upper bound. This way, the algorithm will
first explore more promising, higher-bound branches of search, narrowing in on
the optimal abstraction more quickly. In branch-and-bound literature this
choice of a search order that uses a sound upper bound is sometimes referred
to as best-bound search and makes the algorithm a form of A∗ search (Hart et
al., 1968) since the upper bound is an admissible heuristic (Morrison et al.,
2016).
In preliminary experiments on a subset of benchmarks we found that search
order had little effect on the overall runtime of the algorithm, but the best-
bound ordering was moderately helpful in more quickly narrowing in on the
optimal solution (i.e. useful when running the algorithm with a limited time
budget), so this is used for all experiments.
##### Efficient Incremental Matching
When expanding an abstraction $A_{\texttt{??}}$ to a new abstraction
$A_{\texttt{??}}^{\prime}$, it is easy to compute
$\textsc{Matches}(\mathcal{P},A_{\texttt{??}}^{\prime})$ since we know it is a
subset of $\textsc{Matches}(\mathcal{P},A_{\texttt{??}})$. Thus, there is no
need to perform matching from scratch against every subtree in the corpus to
compute the match locations. Instead, we can simply inspect the relevant
subtree at each match location of the original abstraction to see which match
locations will be preserved by a given expansion. In fact, except when
expanding into an abstraction variable $\alpha$, the sets of match locations
obtained by different expansions of a hole will be disjoint.
When expanding to an abstraction variable $\alpha$, if $\alpha$ is an existing
abstraction variable then this is a situation where the same variable is being
used in more than one place, as in the square abstraction $(\lambda\alpha.\
\texttt{*}\ \alpha\ \alpha)$. In this case we restrict the match locations to
the subset of locations where within the location all instances of $\alpha$
are bound to syntactically identical subtrees.
Additionally, if the user provides a maximum arity limit, then an expansion
that causes the abstraction to exceed this limit can be discarded.
##### Algorithm Summary and Key Points
The CTS algorithm combines all of the aforementioned optimizations in a top-
down search algorithm with pruning. In summary, CTS takes as input a corpus
$\mathcal{P}$, a rewrite strategy $\mathcal{R}$, a utility function
$U_{\mathcal{P},\mathcal{R}}(A)$, and an upper bound function
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})$, and returns the optimal
abstraction with respect to the utility function. CTS performs a top-down
search over partial abstractions and prunes branches of search when partial
abstractions match at zero locations or are eliminated by upper bound pruning
or strict dominance pruning. CTS is made further efficient through the
aforementioned search-order heuristic and efficient incremental matching.
Importantly, none of these optimizations sacrifice the optimality of the
abstraction found — all pruning is done soundly, as discussed in the prior
sections on upper bound pruning and strict dominance pruning.
We also note that CTS is amenable to parallelization without losing
optimality. This can be implemented using a shared priority queue that is
safely accessed by different worker threads through a locking primitive.
Strict dominance pruning trivially remains sound as it doesn’t depend on any
global information about the state of the search. Upper bound pruning also
remains sound even if workers only occasionally synchronize their best-
abstraction-so-far, as this will just mean that they occasionally have weaker
upper bounds which are still sound.
Since the algorithm maintains a best abstraction so-far, it can also be
terminated early, making it an anytime algorithm. Finally, to learn a library
of abstractions, CTS can be run repeatedly (much like DreamCoder), adding one
abstraction at a time to the library and rewriting the corpus with each
abstraction as it is learned before running CTS again. A listing of the full
algorithm is provided in Appendix A.
## 4\. Applying Corpus-Guided Top-Down Search to Compression
Having presented the general framework and algorithm of corpus-guided top-down
search, we now instantiate this framework for optimizing a compression metric.
### 4.1. Utility
In compression we seek to minimize some measure of the size, or more generally
the cost, of a corpus of programs after rewriting them with a new abstraction.
As in prior work (Ellis et al., 2021) we penalize large abstractions by
including abstraction size in the utility. The compressive utility function
for a corpus $\mathcal{P}$ and abstraction $A$ is given below.
(8)
$U_{\mathcal{P},\mathcal{R}}(A)\triangleq-\text{cost}(A)+\text{cost}(\mathcal{P})-\text{cost}(\textsc{Rewrite}_{\mathcal{R}}(\mathcal{P},A))$
Here, $\text{cost}(\cdot)$ is a cost function of the following form:
(9) $\displaystyle\text{cost}(\lambda.\
e^{\prime})=\text{cost}_{\lambda}+\text{cost}(e^{\prime})$
$\displaystyle\text{cost}(e^{\prime}e^{\prime\prime})=\text{cost}_{\texttt{app}}+\text{cost}(e^{\prime})+\text{cost}(e^{\prime\prime})$
$\displaystyle\text{cost}(\$i)=\text{cost}_{\texttt{\$i}}$
$\displaystyle\text{cost}(t)=\text{cost}_{t}(t),\text{ for
}t\in\mathcal{G}_{\texttt{sym}}$
$\displaystyle\text{cost}(\alpha)=\text{cost}_{\alpha}$
where $\text{cost}_{\lambda}$, $\text{cost}_{\texttt{app}}$,
$\text{cost}_{\texttt{\$i}}$, and $\text{cost}_{\alpha}$ are non-negative
constants, and $\text{cost}_{t}(t)$ is a mapping from grammar primitives to
their (non-negative) costs. Finally, we introduce
$\text{cost}_{\alpha=0}(\cdot)$ as a version of $\text{cost}(\cdot)$ where
$\text{cost}_{\alpha}=0$.
We can equivalently construct the utility in Eq. 8 by summing over the
compression gained from performing each rewrite separately, as given below.
(10)
$U_{\mathcal{P},\mathcal{R}}(A)=-\text{cost}(A)+\sum_{e\in\textsc{RewriteLocations}_{\mathcal{R}}(\mathcal{P},A)}{\text{cost}(e)-\text{cost}(\textsc{RewriteOne}(e,A))}$
By reasoning about the way that rewriting transforms a program, this utility
can be broken down even further:
(11)
$U_{\mathcal{P},\mathcal{R}}(A)=-\text{cost}(A)+\sum_{e\in\textsc{RewriteLocations}_{\mathcal{R}}(\mathcal{P},A)}{U_{local}(A,e)}$
(12)
$U_{local}(A,e)=\\!\\!\underbrace{\text{cost}_{\alpha=0}(A)\vphantom{\sum_{\alpha\in\text{a}(A)}}}_{\text{abstraction
size}}-\underbrace{(\text{cost}_{t}(t_{A})+\text{cost}_{\texttt{app}}\cdot\text{arity}(A))\vphantom{\sum_{\alpha\in\text{a}(A)}}}_{\text{application
utility (negative)}}+\underbrace{\sum_{[\alpha\to
e^{\prime}]\in\text{args}(A,e)}{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\text{usages}(\alpha)-1)\cdot\text{cost}(e^{\prime})}}_{\text{multiuse
utility}}$
where $t_{A}$ is the new primitive corresponding to abstraction $A$. This form
of the utility function can be efficiently computed without explicitly
performing any rewrites; and it sheds light on the different sources of
compression. The three main terms in this expression are (1) the abstraction
size that comes from the shared structure that is removed, (2) the negative
application utility that comes from introducing the new primitive and lambda
calculus app nonterminals to apply it to each argument, and (3) the multiuse
utility which comes from only needing to pass in a single copy of an argument
that might be use in more than one place in the body. We emphasize that this
form of the utility function is equivalent to the original definition based on
rewriting given in Eq. 8.
### 4.2. Upper Bounding the Utility
We seek an upper bound function
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})$ such that for any $A$
derived from $A_{\texttt{??}}$,
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})\geq
U_{\mathcal{P},\mathcal{R}}(A)$. We begin from the decomposition of the
utility function given in Eq. 10. Since costs are always non-negative, we can
upper bound this by dropping the $-\text{cost}(A)$ term as well as the
negative term within the sum:
(13)
$U_{\mathcal{P},\mathcal{R}}(A)\leq\sum_{e\in\textsc{RewriteLocations}_{\mathcal{R}}(\mathcal{P},A)}{\text{cost}(e)}$
Intuitively, dropping the negative term from the sum is equivalent to assuming
we compressed the cost of this location all the way down to cost 0. We can
also bound $\textsc{RewriteLocations}_{\mathcal{R}}(\mathcal{P},A)$ as
$\textsc{Matches}(\mathcal{P},A_{\texttt{??}})$ using Lemma 2, yielding our
final upper bound in terms of $A_{\texttt{??}}$:
(14)
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})\triangleq\sum_{e\in\textsc{Matches}(\mathcal{P},A_{\texttt{??}})}{\text{cost}(e)}$
### 4.3. Strict Dominance Pruning
We identify two forms of strict dominance pruning compatible with a
compressive utility. The first is redundant argument elimination: a partial
abstraction can be dropped if it has two abstraction variables that always
take the same argument as each other across all match locations, for example
if it had variables $\alpha$ and $\beta$ and $\alpha=\beta=\texttt{(+ 3 5)}$
at one location, $\alpha=\beta=\texttt{2}$ at another location, and so on for
all match locations. This abstraction would then be strictly dominated by the
abstraction that doesn’t take $\beta$ as an argument and instead reuses
$\alpha$ in place of $\beta$, and can hence be eliminated. Thus an abstraction
like (+ (* $\alpha$ $\beta$) ??) is strictly dominated by an abstraction like
(+ (* $\alpha$ $\alpha$) ??) if $\alpha=\beta$ across all match locations in
the former abstraction.
The second is argument capture: when a partial abstraction takes the same
argument for at least one abstraction variable across all match locations,
this abstraction can be discarded. This is because every abstraction derivable
is covered by another abstraction which is identical except it has the
argument inlined into the body. For example, if in the abstraction (+ (*
$\alpha$ $\beta$) ??) the abstraction variable $\alpha$ takes the same
argument $\alpha=$(+ 3 5) across every match location, there is a strictly
dominating partial abstraction that simply has (+ 3 5) inlined in its body: (+
(* (+ 3 5) $\beta$) ??). Note that argument capture does not apply when the
argument contains a free variable, as inlining would result in an invalid
abstraction.
A close reader might notice that if the number of times $\alpha$ appears in
$A$ is greater than the number of rewrite locations, the abstraction with
argument capture applied, $A^{\prime}$, will have slightly lower utility.
Employing this pruning rule in our search means that we will find the optimal
abstraction subject to the constraint that all possible argument captures have
occurred. This utility difference from the optimal abstraction without
argument capture is bounded by at most $\text{cost}(A^{\prime})$, since the
difference comes from the contribution of the size of the abstraction body
itself to the utility. 222 Specifically, for some $[\alpha\to e]$, the
abstraction without argument capture is higher in utility by
$-(\text{cost}(e)+\text{cost}_{\texttt{app}})*|\textsc{RewriteLocations}_{\mathcal{R}}(\mathcal{P},A)|+(\text{cost}(e)-\text{cost}_{\alpha})*\text{usages}(\alpha)$.
For a cost function where $\text{cost}_{\texttt{app}}<\text{cost}_{\alpha}$,
this is positive when the argument is used more times than the abstraction
itself is used. In preliminary experiments across all of our experimental
datasets, we never find this edge case to change which abstraction is chosen
as optimal.
### 4.4. Rewrite Strategy
Stitch employs a linear-time, optimal rewrite strategy for a compression
objective. The goal of an optimal rewrite strategy is to efficiently select
the optimal subset of $\textsc{Matches}(\mathcal{P},A)$ to perform rewrites at
in order to maximize the utility.
The main challenge is that when match locations overlap, the strategy must
decide which of the two to accept. For example, consider the program (foo (foo
(foo bar)) and the abstraction $t_{A}=\lambda\alpha.\ $(foo (foo $\alpha$)).
This abstraction matches at the root of the program with $\alpha=$ (foo bar),
resulting in the rewritten program ($t_{A}$ (foo bar)). However, though the
abstraction also matched at the subtree (foo (foo bar) with $\alpha=$ bar in
the original program, this match location is no longer present in the
rewritten program, so only one of the two locations can be chosen by the
rewrite strategy.
Our approach is a bottom-up dynamic programming algorithm, which begins at the
leaves of the program and proceeds upwards. At each node $e$, we compute the
cumulative utility so far if we reject a rewrite here ($\text{util}_{R}$), if
we accept a rewrite here ($\text{util}_{A}$), or if we choose the better of
the two options ($\text{util}^{*}$):
(15) $\displaystyle\text{util}_{R}[e]=\sum_{e^{\prime}\in
children(e)}{\text{util}^{*}[e^{\prime}]}$
$\displaystyle\text{util}_{A}[e]=\left\\{\begin{array}[]{ll}0&\text{if
}e\notin\textsc{Matches}(\mathcal{P},A)\\\
U_{local}(A,e)+\sum\limits_{e^{\prime}\in\text{args}(A,e)}{\text{util}^{*}[e^{\prime}]}&\text{otherwise}\\\
\end{array}\right.$
$\displaystyle\text{util}^{*}[e]=\text{max}(\text{util}_{R}[e],\text{util}_{A}[e])$
where $U_{local}(A,e)$ is the utility gained from a single rewrite as defined
in Eq. 12. After calculating these quantities the rewrite strategy can start
from the program root and recurse down the tree, rewriting at each node where
rewriting is optimal (i.e. $\text{util}_{A}>\text{util}_{R}$) and then
recursing into its arguments after rewriting. Since all arguments were
originally subtrees in the program, these quantities will have been calculated
for all of them, with the caveat that their de Bruijn indices may have been
shifted. Shifting indices of a subtree does not change whether an abstraction
can match there nor does it change the utility gained from using that
abstraction, so this simply requires some extra bookkeeping to track.
This algorithm is optimal by a simple inductive argument: at each step of the
dynamic programming problem, we can assume that we know the cumulative utility
of all children and potential arguments at this node, so we can use Eq. 15 to
calculate the cumulative utility of either rejecting or accepting the rewrite
at this node.
## 5\. Combining corpus-guided top-down search with deductive approaches
In complex domains, assembling good libraries may involve more than just
finding matching code-templates; sometimes, some initial refactoring is
necessary to expose common structure. For example, consider learning the
abstraction $\lambda\alpha.\ \texttt{(* 2 }\alpha\texttt{)}$ for doubling
integers, given the expressions (* 2 8), (* 7 2), and (right-shift 3 1). This
is only possible if the system can use the commutativity of multiplication to
rewrite (* 7 2) into (* 2 7), and bitvector properties to rewrite (right-shift
3 1) into (* 2 3); such rewrites are not natively supported by CTS.
In this section, we discuss how CTS can be combined with refactoring systems
based on deductive rewrites to increase its expressivity further. The core
idea is simple: run a rewrite system on the corpus to produce a set of
refactorings of the corpus in a version space, then run CTS over the resulting
data structure.333Note that while we have previously presented CTS as
operating over syntax trees, the core notions of upper bounds and matching
that CTS operates on are not restricted to program trees. Intuitively, this
will lead to improved performance compared with a purely deductive approach
since the cost of rewriting grows exponentially with the number iterations of
rewrites that are applied. This is a problem for fully deductive approaches
like Ellis et al. 2021 because extracting the abstractions often requires a
long chain of rewrites, especially for higher-arity abstractions. However, a
small number of rewrites is typically sufficient to expose the underlying
commonalities, as it was in the example above; performing only a handful of
rewrites and then using CTS to actually extract the library thus avoids the
exponential blow-up of computing several rewrites in sequence, while still
benefiting from the increase in expressivity afforded by the rewrites.
##### Version Spaces
To illustrate this hybrid approach, we combine CTS with _version spaces_ (Lau
et al., 2003; Polozov and Gulwani, 2015) representing sets of programs
semantically equivalent under the $\beta$-inversion rewrite. Version spaces
are represented as terms from a grammar obtained by extending the grammar of
expressions with the union ($\uplus$) operator that represents a set of
equivalent expressions: $v\Coloneqq\varnothing\mid v^{\prime}\ \uplus\
v^{\prime\prime}\mid\lambda.\ v^{\prime}\mid v^{\prime}\
v^{\prime\prime}\mid\$i\mid t$. We define a denotation operator,
$\llbracket{v}\rrbracket$, mapping a version space $v$ to a set of terms: for
the union operator, $\llbracket{v\uplus
v^{\prime}}\rrbracket=\llbracket{v}\rrbracket\cup\llbracket{v^{\prime}}\rrbracket$;
for applications, $\llbracket{v\ v^{\prime}}\rrbracket=\left\\{e\
e^{\prime}\,:\,\forall
e,e^{\prime}\in\llbracket{v}\rrbracket\times\llbracket{v^{\prime}}\rrbracket\right\\}$;
for lambda abstractions, $\llbracket{\lambda v}\rrbracket=\left\\{\lambda e\
\,:\,\forall e\in\llbracket{v}\rrbracket\right\\}$; for the empty set,
$\llbracket{\varnothing}\rrbracket=\varnothing$; and for de Bruijn indices and
terminals, $\llbracket{v}\rrbracket=v$. Ellis et al. 2021 gives a procedure
called $I\beta(e)$, which take as input an expression $e$ and outputs a
version space $v$ inverting one step of $\beta$-reduction: that is,
$e^{\prime}\to^{\beta}e$ iff $e^{\prime}\in\llbracket{I\beta(e)}\rrbracket$.
*[left=RU-AbsVar] RewriteUnify(α,v) ↝[ α→v ] *[left=RU-Hole] RewriteUnify(??_i,v) ↝[ ??_i →v ] *[left=RU-App] RewriteUnify(A_1,v_1) ↝l_1
RewriteUnify(A_2,v_2) ↝l_2
l = vmerge(l_1,l_2) RewriteUnify((A_1 A_2),(v_1 v_2)) ↝l *[left=RU-Lam]
RewriteUnify(A,v) ↝l’
l = DownshiftAll(l’) RewriteUnify((λ. A),(λ. v)) ↝l *[left=RU-Same]
RewriteUnify(v,v) ↝[ ] *[left=RU-Union] RewriteUnify(A,v_1) ↝l
RewriteUnify(A,v_1⊎v_2) ↝l
*[left=VM-Same] [α→v_1]∈l_1 [α→v_2]∈l_2 [α→(v_1∩v_2)]∈vmerge(l_1,l_2) *[left=VM-Diff] [α→v]∈l_1
α／∈args(l_2) [α→v]∈vmerge(l_1,l_2)
Figure 6. Defining RewriteUnify, which takes as input an abstraction and a
version space of possible refactorings, and yields multiple substitutions
corresponding to all the ways that the abstraction’s holes and variables can
match with programs encoded by the version space.
To run CTS on top of this rewriting system requires a generalization of
LambdaUnify: instead of unifying an abstraction with a term, yielding a single
substitution, we unify against a _set of terms_ (a version space), yielding a
set of candidate substitutions. The relation RewriteUnify (Fig. 6)
accomplishes this, and equipped with this relation we can express the size of
a subtree $e$ after expanding it into the version space $I\beta(e)$ and
rewriting it with abstraction $A$ as:
(16)
$\text{cost}(\textsc{Rewrite}(A,e))=\min_{l:\;\textsc{RewriteUnify}(A,I\beta(e))\rightsquigarrow
l}\text{cost}_{t}(t_{A})+\text{cost}_{\texttt{app}}\cdot\text{arity}(A)+\sum_{v\in\text{args}(l)}\min_{e^{\prime}\in\llbracket{v}\rrbracket}\text{cost}(e^{\prime})$
We then can approximate the utility of rewriting a corpus (based on Eq. 10) as
follows
(17)
$U_{\mathcal{P},\mathcal{R}}(A)\approx-\text{cost}(A)+\sum_{p\in\mathcal{P}}{\;\max_{e\in\text{subtrees}(p)}{\text{cost}(e)-\text{cost}(\textsc{Rewrite}(A,e))}}$
This utility is exact when the optimal way of rewriting the corpus using $A$
has at most one rewrite location per program. This is because it considers the
utility of the single best site at which to perform the rewrite instead of
considering multiple simultaneous rewrites at different locations within a
single program. We can bound this approximate utility for a partial
abstraction by computing the approximate utility directly while treating any
version space bound to a hole in $\text{args}(l)$ as having zero cost.
##### Example: Learning map from fold.
Consider learning the higher-order function map $=\lambda\alpha.$(fold
($\lambda x.\lambda\ell.$ (cons ($\alpha$ $x$) $\ell$))) from two example
programs: doubling a list of numbers, expressed as (fold ($\lambda
x.\lambda\ell.$ (cons (+ $x$ $x$) $\ell$))), and decrementing a list of
numbers, expressed as (fold ($\lambda x.\lambda\ell.$ (cons (- $x$ 1)
$\ell$))). Matching these programs with the map function requires re-
expressing them as (fold ($\lambda x.\lambda\ell.$ (cons (($\lambda z.$ (+ z
z)) $x$) $\ell$))) and (fold ($\lambda x.\lambda\ell.$ (cons (($\lambda z.$ (-
z 1)) $x$) $\ell$))), which matches the map function with the substitutions
$\alpha=\texttt{($\lambda z.$ (+ z z))}$ and $\alpha=\texttt{($\lambda z.$ (-
z 1))}$ respectively. These two re-expressions are performed by running a
single round of $\beta$-inversion as a deductive rewriting step.
While the majority of our experiments will focus on evaluating the approach
laid out in Section 3 and Section 4, we implement and evaluate a prototype of
integrating version spaces in this way in Section 6.5.
## 6\. Experiments
In this section, we evaluate corpus-guided top-down search for library
learning. Specifically, our evaluation focuses on five hypotheses about the
performance of Stitch:
1. (1)
Stitch learns libraries of comparable quality to those found by existing
deductive library learning algorithms in prior work, while requiring
significantly less resources. In Section 6.1 we run Stitch on the library
learning tasks from (Ellis et al., 2021) and directly compare Stitch to
DreamCoder, the deductive algorithm introduced in that work. We find that
Stitch learns libraries which usually match or exceed the baseline in quality
(measured via a compression metric), while improving the resource efficiency
in terms of memory usage and runtime by 2 and 3-4 orders of magnitude compared
to the baseline (respectively).
2. (2)
Stitch scales to corpora of programs that contain more and longer programs
than would be tractable with prior work. In Section 6.2, we evaluate Stitch’s
ability to learn libraries within eight graphics domains from Wong et al.
2022, which are considerably larger and more complex than have been considered
in previous work. We find that Stitch on average obtains a test set
compression ratio of 2.55x-11.57x in 0.19s-60.16s, with a peak memory usage of
11.21MB-714.55MB in these domains. The problems are large enough to time out
with the DreamCoder baseline.
3. (3)
Stitch degrades gracefully when resource-constrained. In Section 6.3 we
investigate Stitch’s performance when run as an anytime algorithm; i.e., one
that can be terminated early for a best-effort result if a corpus is too large
or there are limits on time or memory. We reuse the eight graphics domains
from Wong et al. 2022 and find that with its heuristic guidance Stitch
converges upon a set of high quality abstractions very early in search, doing
so within 1% of the total search time in 3 out of 8 domains and within 10% in
all except one.
4. (4)
All the elements of Stitch matter. In Section 6.4, we carry out an ablation
study on Stitch and find the argument capturing and upper bound pruning
methods are essential to its performance, while redundant argument elimination
also proves useful in certain domains. With all optimizations disabled, we
find that Stitch cannot run in $\leq 90$ minutes and $\leq 50$GB of RAM on any
of the domains from Wong et al. 2022.
5. (5)
Stitch is complementary to deductive rewrite-based approaches to library
learning. These prior experiments show the superior runtime performance of
Stitch relative to deductive rewrite systems, but deductive systems have an
important advantage over Stitch: the ability to incorporate arbitrary rewrite
rules to expose more commonality among different programs and in that way
discover better libraries. Such deductive approaches are especially more apt
at learning higher-order abstractions. In Section 6.5, we give evidence that
this expressivity gap can be reduced by running Stitch on top of a deductive
rewrite system, allowing it to learn many new abstractions while still using
$<$2% of the compute time.
For all experiments, we parameterize Stitch’s $\text{cost}(e)$ function (as
defined in Section 4.1) as follows:
$\text{cost}_{\texttt{app}}=\text{cost}_{\lambda}=1$,
$\text{cost}_{\texttt{\$i}}=\text{cost}_{\alpha}=\text{cost}_{t}(t)=100$. To
avoid overfitting, DreamCoder prunes the abstractions that are only useful in
programs from a single task. We add this to Stitch as well, treating each
program as a separate task for datasets that don’t divide programs into tasks.
We run all experiments on a machine with two AMD EPYC 7302 processors, 64
CPUs, and 256GB of RAM. We note however that Stitch itself runs exceptionally
well on a more average machine. For example, on one author’s laptop (ThinkPad
X1 Carbon Gen 8), the experiments from Section 6.2 can be replicated with
nearly identical runtimes, and these are the most computationally intensive
experiments outside of the ablation study.
### 6.1. Iterative Bootstrapped Library Learning
#### Experimental setup
Our first experiment is designed to replicate the experiments in DreamCoder,
which is the state-of-the-art in deductive library learning. DreamCoder learns
libraries iteratively: the system is initialized with a low-level DSL, and
then alternates between synthesizing programs (via a neurally-guided
enumerative search) that solve a training corpus of inductive tasks and
updating the library of abstractions available to the synthesizer. Traces from
the experiments carried out by Ellis et al. 2021 are publicly
available444https://github.com/mlb2251/compression_benchmark and include all
of the intermediate programs that were synthesized as well as the libraries
learned from those programs.
In this experiment, we take these traces and evaluate Stitch on each instance
where library learning was performed, comparing the quality of the resulting
library to the original one found by DreamCoder. We also re-run DreamCoder on
these same benchmarks in order to evaluate its resource usage, capturing its
runtime and memory usage in the same environment as Stitch.
The library learning algorithm in Ellis et al. 2021 implements a stopping
criterion to determine how many abstractions to retain on any given set of
training programs. In our comparative experiments, we run the DreamCoder
baseline first, and then match the number of abstractions learned by Stitch at
each iteration to those learned by the baseline under its stopping criterion
so that timing comparisons are fair. We record the total time spent both
performing abstraction learning and rewriting for both Stitch and DreamCoder.
We replicate experiments on five distinct domains from Ellis et al. 2021:
* •
Lists: A functional programming domain consisting of 108 total inductive
tasks.
* •
Text: A string editing domain in the style of FlashFill (Gulwani et al., 2015)
consisting of 128 total inductive tasks.
* •
LOGO: A graphics domain consisting of 80 total inductive tasks.
* •
Towers: A block-tower construction domain consisting of 56 total inductive
tasks.
* •
Physics: A domain for learning equations corresponding to physical laws from
observations of simulated data, consisting of 60 total inductive tasks.
#### Assessing library quality with a compression metric
The standard Stitch configuration optimizes a compression metric that
minimizes the size of the programs after being rewritten to use the
abstraction. This is a standard metric in program synthesis, since shorter
programs are frequently easier to synthesize. Optimizing against this metric
is equivalent to maximizing the likelihood of the rewritten programs under a
uniform PCFG.
The DreamCoder synthesizer is more sophisticated than simple enumeration; it
takes as input a learned typed bigram PCFG and leverages it to synthesize
programs more efficiently. When performing compression, it optimizes against
this given PCFG in order to find abstractions that will be more profitable for
its specific synthesizer. For the purpose of this evaluation, however, we
restrict ourselves to the uniform PCFG because the one used by DreamCoder
requires programs to be in a particular normal form. Another aspect of
DreamCoder relevant to its compression metric is that DreamCoder synthesizes
multiple programs that solve the same task and then selects the abstraction
that works best on _some_ program for a given task. This is expressed formally
in the equation below.
(18)
$\text{cost}(A)+\sum\limits_{\text{task}}\min\limits_{p\in\text{task}}\text{cost}\left(\textsc{Rewrite}(p,A)\right)$
It is trivial to implement this best-of-task metric in Stitch, so we use this
for the comparisons with DreamCoder.
Figure 7. Compression rates obtained when running Stitch on the five domains
considered in 6.1 relative to those of DreamCoder. Higher is better for
Stitch: a ratio above 1.0 indicates that Stitch achieves greater compression
than DreamCoder. The specific compression metric used in the ratio is given in
Eq. 18.
Figure 8. Peak memory usage of Stitch and DreamCoder while running on the five
domains considered in 6.1, averaged over all benchmarks. Lower is better;
black lines indicate $\pm$ one standard deviation. Note the logarithmic
y-axis.
Figure 9. Wall-clock time required to find (and rewrite under) one abstraction
in each of the five domains from 6.1, averaged over all benchmarks. Lower is
better; black lines indicate $\pm$ one standard deviation. Note the
logarithmic y-axis.
#### Results
We compare DreamCoder and Stitch for library quality and resource efficiency.
Library quality. We first examine the quality of the libraries learned by both
Stitch and DreamCoder using the compression metric in Eq. 18: Fig. 7 shows the
ratio between them across all of the benchmarks for each domain. A ratio of
1.0 indicates that programs are the exact same length under the libraries
learned by DreamCoder and Stitch, a ratio greater than 1.0 indicates that
Stitch learns more compressive libraries, and a ratio less than 1.0 indicates
that Stitch learns libraries which are less compressive. For example, a ratio
of 1.1 indicates that DreamCoder rewrote to produce a corpus 10% larger than
that of Stitch, so it achieved less compression.
These results show that Stitch generally learns libraries of comparable and
often greater quality than DreamCoder when matching the number of abstractions
learned by the latter. In the logo, towers, text, and physics domains, Stitch
always finds abstractions that are of equal—and often considerably
greater—compressive quality than DreamCoder does; in the list domain, Stitch
more often than not still obtains better compression, but sometimes loses out
to DreamCoder. This is a result of the fact that Stitch cannot learn higher-
order abstractions, which are useful in this domain; although we emphasize our
focus is on scalability, we will later present an extension of Stitch capable
of handling most higher-order functions in Section 6.5, based on the formalism
developed in Section 5. Nonetheless, we conclude that Stitch learns libraries
whose quality is comparable to and often better than those found by
DreamCoder.
Resource efficiency. In addition to the quality of the libraries found, we are
interested in how the two methods compare in terms of time and space
requirements. Since we a priori believe Stitch to be significantly faster, for
this evaluation we allow DreamCoder to use 8 CPUs but limit Stitch to single
threading555While this may seem unfair to Stitch, it is worth noting that it
would be unlikely to benefit from multithreading when running on the order of
milliseconds anyway; DreamCoder, on the other hand, would struggle greatly in
these domains without the aid of parallelism.. The results are shown in Fig. 9
and Fig. 9; in summary, we find that Stitch takes tens of milliseconds to
discover abstractions across all five domains—achieving a 3-4 order of
magnitude speed-up over DreamCoder—while also requiring more than 2 orders of
magnitude less memory. We thus conclude that Stitch is dramatically more
efficient than the state-of-the-art deductive baseline when replicating the
iterative library learning experiments of Ellis et al. 2021.
### 6.2. Large-Scale Corpus Library Learning
#### Experimental setup.
While the previous experiment allowed us to benchmark Stitch directly against
a state-of-the-art deductive baseline, the iterated learning setting
considered by Ellis et al. 2021 only evaluates library learning on relatively
small corpora of short programs discovered by the synthesizer. Our second
experiment is instead designed to evaluate Stitch in a more traditional
learning setting, in which we aim to learn libraries of abstractions from a
large corpus of existing programs all at once.
We source our larger-scale program datasets from Wong et al. 2022, which
present a series of datasets designed as a benchmark for comparing human-level
abstraction learning and graphics program writing against automated synthesis
and library learning models. These datasets are divided into two distinct
high-level domains (technical drawings and block-tower planning tasks), each
consisting of four distinct subdomains containing 250 programs:
* •
Technical drawing domains: nuts and bolts; vehicles; gadgets; furniture: CAD-
like graphics programs that render technical drawings of common objects,
written in an initial DSL consisting of looped transformations (scaling,
translation, rotation) over simple geometric curves (lines and arcs).
* •
Tower construction domains: bridges; cities; houses; castles: Planning
programs that construct complex architectures by placing blocks, written in an
initial DSL that moves a virtual hand over a canvas and places horizontal and
vertical bricks.
We choose these datasets not only for their size and scale (full dataset
statistics in Table 1), but also for the complexity of their potential
abstractions: Wong et al. 2022 explicitly design their corpora to contain
complex hierarchical structures throughout the programs, making them an
interesting setting for library learning.
Domain | #Programs | Average program length | Average program depth
---|---|---|---
nuts & bolts | 250 | $76.03\pm 24.22$ | $15.18\pm 2.13$
gadgets | 250 | $142.85\pm 87.32$ | $20.88\pm 2.37$
furniture | 250 | $171.74\pm 48.41$ | $31.83\pm 5.33$
vehicles | 250 | $141.70\pm 40.23$ | $21.22\pm 1.35$
bridges | 250 | $137.03\pm 59.71$ | $92.35\pm 39.80$
cities | 250 | $161.70\pm 55.56$ | $109.80\pm 37.66$
castles | 250 | $189.09\pm 60.18$ | $128.27\pm 40.77$
houses | 250 | $168.13\pm 55.75$ | $114.85\pm 37.60$
Table 1. Summary statistics about the domains from Wong et al. 2022. Program
length is the number of terminal symbols in the program; program depth is the
length of the longest path from root to leaf in the program tree. Both are
reported as the mean over the entire dataset $\pm$ one standard deviation.
When performing library learning in a synthesis setting by compressing a
corpus of solutions, it’s desirable to find abstractions that would be useful
for solving new tasks, as opposed to abstractions that overfit to the existing
solutions. To evaluate how well the abstractions we learn apply to heldout
programs in the domain, we split the corpora into train and test sets, running
Stitch on the train set and evaluating its compression on the test set. Since
Wong et al. 2022 do not present a train/test split of their datasets, we use
stochastic cross validation to evaluate the generalization of the libraries
found by Stitch. For each domain, we randomly sample 80% of the dataset to
train on and reserve the last 20% as a held out test set; we repeat this
procedure 50 times. We then ask Stitch to learn a library consisting of $\leq
10$ abstractions with a maximum arity of 3, and average the results across the
different random seeds.
To obtain a baseline to compare its performance against, we once again turned
to DreamCoder (Ellis et al., 2021). However, we found that DreamCoder was
unable to discover even a single abstraction when run directly on any of the
datasets from Wong et al. 2022, despite being given hours of runtime and 256GB
of RAM. We also experimented with heavily sub-sampling the training dataset
before passing it to DreamCoder, but failed to find a configuration under
which DreamCoder finds any interesting abstractions at all due to the fact
that it immediately blows up on programs as long as these. As a result, we
resort to presenting Stitch’s performance metrics without any baseline to
compare against; we stress that this is a direct result of the fact that
Stitch is the first library learning tool that scales to such a challenging
setting.
#### Results
Domain | Compression Ratio | Runtime (s) | Peak mem. usage (MB)
---|---|---|---
Training set | Test set
nuts & bolts | $12.00\pm 0.25$ | $11.57\pm 0.49$ | $0.24\pm 0.04$ | $11.10\pm 0.17$
gadgets | $4.03\pm 0.15$ | $3.91\pm 0.31$ | $1.83\pm 0.35$ | $30.11\pm 0.85$
furniture | $4.95\pm 0.07$ | $4.85\pm 0.26$ | $2.83\pm 0.49$ | $30.88\pm 0.62$
vehicles | $4.28\pm 0.14$ | $4.14\pm 0.25$ | $1.52\pm 0.29$ | $26.98\pm 0.88$
bridges | $4.36\pm 0.06$ | $3.78\pm 0.14$ | $17.58\pm 1.26$ | $189.08\pm 10.10$
cities | $3.15\pm 0.05$ | $3.06\pm 0.14$ | $50.74\pm 3.96$ | $413.63\pm 9.51$
castles | $2.57\pm 0.07$ | $2.55\pm 0.08$ | $77.26\pm 6.98$ | $683.89\pm 32.96$
houses | $8.92\pm 0.21$ | $8.85\pm 0.57$ | $15.54\pm 1.37$ | $241.17\pm 2.86$
Table 2. Results for the large-scale library learning experiment in Sec. 6.2.
The compression ratio refers to how many times smaller the corpus is after
rewriting under the learned library compared to the original corpus; higher is
thus better. All results are given as the mean $\pm$ one standard deviation
over 50 runs with different random seeds for the dataset splitting.
The results are summarized in Table 2. We find that Stitch scales up to even
the most complex sub-domains, running in 77 seconds with a peak memory usage
below 1GB on castles. On four out of eight of the domains, Stitch finishes in
single-digit seconds and consumes only tens of megabytes. This stands in stark
contrast to DreamCoder, which we were unable to run on the very simple nuts &
bolts domain even with 256GB RAM and several hours worth of compute budget.
These results thus support our claim that Stitch scales to corpora of programs
that would be intractable with prior library learning approaches.
We hope that by providing our results on these datasets in full, future work
in this field will benefit from having a directly comparable baseline.
### 6.3. Robustness to Early Search Termination
#### Experimental setup
This experiment is designed to evaluate how early into the search procedure
Stitch finds what will eventually prove to be the optimal abstraction. This is
highly relevant in settings where the set of training programs is too large to
run the search to completion. The experiment showcases one of Stitch’s more
subtle strengths: corpus-guided top-down abstraction search is an _anytime_
algorithm, and thus does not need to be run to completion to give useful
results.
We re-use the domains from Wong et al. 2022 and once again evaluate the
quality of the library learned (measured in program compression), similarly to
what was done in the previous experiment. However, since we are interested in
how quickly Stitch finds _what it perceives to be_ the optimal abstraction, we
measure compressivity of the training dataset itself (rather than a held-out
test set) and capture the compression ratio obtained by each _candidate_
abstraction found during search (rather than just the compression ratio
obtained when search has been run to completion). Thus, we are able to
investigate how early on during the search procedure Stitch converges on a
chosen library. We restrict Stitch to learning a single abstraction with a
maximum arity of 3.
#### Results
Figure 10. Reduction in size obtained on the training set when rewritten under
the best abstraction found thus far vs. the number of nodes expanded during
search. The y-axis is normalized with respect to the optimal abstraction; the
x-axis is normalized with respect to the total number of nodes explored by
Stitch. Lines thus end earlier the quicker Stitch finds the optimal
abstraction; however in all runs Stitch continues to search until reaching
100% on the x-axis, exploring the rest of the abstraction space without
finding a new best abstraction. Note the logarithmic x-axis.
The results are shown in Fig. 10. These results validate our hypothesis that
Stitch is empirically robust to terminating the search procedure early: in
every sub-domain except for _nuts & bolts_ Stitch converges to the optimal
abstraction very early on, having only completed a tiny fraction of the total
search.666Given that significant attention has already been given to wall-
clock run-times of Stitch on similar workloads in 6.2, we here use the number
of nodes explored instead of wall-clock time to ensure deterministic and
easily reproducible results. We believe that this has great importance for
Stitch’s applicability in data-rich settings since it suggests that a nearly-
optimal abstraction can be found even if the search must be terminated early
(e.g. after a fixed amount of time has passed), making early stopping an
empirically useful way of speeding up the library learning process.
### 6.4. Ablation Study
#### Experimental setup
Stitch implements several different optimizations, which we have argued hasten
the search for abstractions. To verify that this claim holds in practice, we
now carry out a brief ablation study.
Since the space of every possible combination of optimizations is too large to
present succinctly, we focus our attention on four ablations:
* •
no-arg-capture (from Section 4.3), which disables the pruning of abstractions
which are only ever used with the exact same set of arguments (and these
arguments could therefore just be in-lined for greater compression).
* •
no-upper-bound, which disables the upper bound based pruning.
* •
no-redundant-args (from Section 4.3), which disables the pruning of multi-
argument abstractions that have a redundant argument that could be removed
because it is always the same as another argument.
* •
no-opts, which disables all of Stitch’s optimizations.
To isolate the impact of disabling optimizations, we run a single iteration of
abstraction learning on each of the 8 domains from 6.2 and collect the number
of nodes explored during the search. The first iteration is generally the most
challenging as the corpus is large and has not yet been compressed at all.
Focusing on the number of nodes explored (rather than for example runtime)
allows for deterministic results. We also fix the maximum arity of
abstractions to 3 in all runs, aiming to strike a good balance between compute
requirements and how much the optimization will be exposed. We limit each run
to 50GB of virtual address space, as well as 90 minutes of compute.
System Domain | bridges | castles | cities | gadgets | furniture | houses | nuts & bolts | vehicles
---|---|---|---|---|---|---|---|---
no-arg-capture | RAM | RAM | RAM | 16714.28 | RAM | RAM | 30.21 | 26.67
no-upper-bound | 12.15 | 23.71 | 27.90 | 207.76 | 119.00 | 36.18 | 179.12 | 179.99
no-redundant-args | 1.95 | 1.42 | 1.37 | 1.01 | 1.00 | 1.01 | 1.09 | 1.00
no-opt | RAM | RAM | RAM | | RAM | RAM | | RAM
Table 3. Results from the ablation study. Each cell contains the ratio between
the number of nodes explored during search by that particular system and the
number of nodes explored by the baseline on the same domain. Lower is better;
1.00 means performance is identical to the baseline. Cells labeled RAM crashed
due to reaching the 50GB virtual address space limit, while those labeled
reached the 90 minute time restriction.
#### Results
The results are shown in Table 3. We first note as a sanity check that each
ablation does indeed lead to reduced performance in general (i.e. explores a
search space larger than the baseline does). Furthermore, the results suggest
that upper bound based pruning is the most important in the nuts & bolts and
vehicles domains, while pruning out argument capture abstractions is the most
important in the other six domains; it is noteworthy that this latter ablation
by itself causes Stitch to reach the memory limit on more than half of the
domains. On the other hand, disabling the pruning of redundant arguments has a
relatively modest impact on the size of the search space, but still leads to
an almost 2x improvement in the bridge domain.
Perhaps the most important takeaway from this ablation study is that when all
optimizations are disabled, Stitch fails to find an abstraction within the
resource budget on any of the domains. This verifies our hypothesis that
corpus-guided pruning of the search space is the key factor involved in making
top-down synthesis of abstractions tractable.
### 6.5. Learning Libraries of Higher-Order Functions
#### Experimental setup
Our experiments up till now have focused on performant and scalable library
learning. This comes at the expense of some expressivity: deductive rewrite
systems can, in principle, express broader spaces of refactorings. For
example, a rewrite based on inverting $\beta$-reduction allows inventing
auxiliary $\lambda$-abstractions, which helps with learning higher-order
functions: in Ellis et al. 2021, DreamCoder is shown to recover higher-order
functions such as map, fold, unfold, filter, and zip_with, starting from the
Y-combinator. This works by constructing a version space which encodes every
refactoring that is equivalent up to $\beta$-inversion rewrites. But
DreamCoder’s coverage comes at a steep cost as inverting $\beta$-reduction is
expensive.
In this experiment, we seek to give evidence that it is possible to make
Stitch recover all of these higher-order functions by layering it on top of
the version space obtained after a single step of DreamCoder’s
$\beta$-inversion, following the formalism outlined in Section 5. We then
compare this modified version of Stitch with DreamCoder on its ability to
learn these higher-order functions from programs generated by intermediate
DreamCoder iterations, and measure the runtime of each approach.
#### Probabilistic Re-ranking
While the approach outlined in Section 5 should suffice to layer Stitch on top
of deductive rewrite systems based on version spaces, some extra care needs to
be taken to combine it with DreamCoder. This is because DreamCoder implements
a probabilistic Bayesian objective for judging candidate abstractions
(exploiting the connection between compression and probability (Shannon,
1948)), seeking the library $L$ which maximizes
$P(L)\prod_{p\in\mathcal{P}}P(p|L)$ for a given or learned prior $P(L)$ and
program likelihood $P(p|L)$. Stitch, on the other hand, effectively judges
compression quality via a cost function capturing the (weighted) size of the
programs as detailed in Section 4.
To implement this probabilistic heuristic in Stitch, we simply run Stitch on
the version spaces as-is but then re-score each complete abstraction popped
off of the priority queue under the Bayesian objective, using DreamCoder’s
models of the prior and likelihood. To make this integration easier, we re-
implement Stitch in Python, giving a prototype version called pyStitch which
is only used for this experiment. Our implementation employs strict dominance
pruning, and prunes using the bound on the approximate utility given in
Section 5.
System | Fold | Unfold | Map | Filter | ZipWith | Time (s)
---|---|---|---|---|---|---
| Base | ✓ | $\times$ | ✓ | $\times$ | ✓ | 503
| +Bayes | ✓ | $\times$ | ✓ | $\times$ | ✓ | 817
| +VS | ✓ | ✓ | ✓ | $\times$ | ✓ | 3231
pyStitch | +Bayes+VS | ✓ | ✓ | ✓ | ✓ | ✓ | 3042
DreamCoder | Step 1 | $\times$ | $\times$ | ✓ | $\times$ | ✓ | 67
Step 2 | $\times$ | $\times$ | ✓ | ✓ | $\times$ | 116
Step 3 | $\times$ | $\times$ | ✓ | ✓ | ✓ | 2254
Step 4 | ✓ | ✓ | ✓ | ✓ | ✓ | 228048
Table 4. Comparing library learners on functional programming exercises.
DreamCoder, $n$-steps: deductive baseline rewriting $n$ steps of
$\beta$-reduction. pyStitch: Python reimplementation of Stitch, which enables
better interoperability with DreamCoder’s version space algebra (+VS) and
probabilistic models (+Bayes). pyStitch+VS+Bayes learns all of the same
higher-order functions, using $<2\%$ of the compute.
#### Results
The results are shown in Table 4. We note first the large discrepancy in
runtimes; running DreamCoder with 4 steps of rewriting takes roughly 2.5 days
of compute on this domain, while even the slowest version of pyStitch still
finishes in less than an hour. In terms of the number of higher-order
abstractions found, DreamCoder only finds all five when run in its most
expensive configuration; reducing the computation cost quickly decreases its
expressivity. For pyStitch, the Bayesian re-ranking alone (pyStitch+Bayes)
does not yield any improvements, while running it on the version spaces
(pyStitch+VS) yields 4 out of 5 functions. However, it is only when these two
adaptations are used in conjunction (pyStitch+Bayes+VS) that the Stitch-based
method is able to find all of the higher-order functions.
In summary, we find that running CTS after a single step of version space
rewriting and then probabilistically re-ranking the results suffices to
recover the core higher-order functions that DreamCoder learns, while using
$<$2% of its compute. We thus conclude that running Stitch on top of a
deductive rewrite system reduces the expressivity gap, while retaining
superior performance.
## 7\. Related Work
Stitch is related to two core ideas from prior work: deductive refactoring and
library learning systems, which introduce the idea of learning abstractions
that capture common structure across a set of programs, but have largely been
driven by deductive algorithms; and guided top-down program synthesis systems,
which use cost functions to guide top-down enumerative search over a space of
programs, but have largely been used to synthesize whole programs for
individual tasks in prior work.
### 7.1. Deductive Refactoring and Library Learning
Recent work shares Stitch’s goal of learning libraries of program abstractions
which capture reusable structure across a corpus of programs (Ellis et al.,
2018, 2021; Dechter et al., 2013; Cropper, 2019; Shin et al., 2019; Allamanis
and Sutton, 2014; Iyer et al., 2019; Wong et al., 2021; Jones et al., 2021).
Several of these prior approaches also introduce a utility metric based on
program compression in order to determine the most useful candidate
abstractions to retain (Dechter et al., 2013; Ellis et al., 2021; Wong et al.,
2021; Lázaro-Gredilla et al., 2019; Iyer et al., 2019).
Much of this this prior work follows a bottom-up approach to abstraction
learning, combining a bottom-up traversal across individual training programs
with a second stage to extract shared abstractions from across the training
corpus. This approach includes systems that work through direct memoization of
subtrees across a corpus of programs (Dechter et al., 2013; Lin et al., 2014;
Lázaro-Gredilla et al., 2019); antiunification (caching tree templates that
can be unified with training program syntax trees) (Ellis et al., 2018;
Henderson, 2013; Hwang et al., 2011; Iyer et al., 2019); or by more
sophisticated refactoring using one or more rewrite rules to expose additional
shared structure across training programs (Ellis et al., 2018, 2021; Chlipala
et al., 2017; Liang et al., 2010).
Many of the bottom up algorithms draw more generally on deductive synthesis
approaches that apply local rewrite rules in a bottom-up fashion to program
trees in order to refactor them — historically, to synthesize programs from a
declarative specification of desired function (Burstall and Darlington, 1977;
Manna and Waldinger, 1980). Deductive approaches to library learning, however,
confront fundamental memory and search-time scaling challenges as the corpus
size and depth of the training programs increases; prior deductive approaches
such as (Ellis et al., 2018, 2021) use version spaces (Lau et al., 2003;
Mitchell, 1977) to mitigate the memory usage during bottom-up abstraction
proposal. Still, deductive approaches are challenging to bound and prune
(unlike the top-down approach we take in Stitch), as they generally traverse
individual program trees locally and must store possible abstraction
candidates in memory before the extraction step.
Some prior work (Shin et al., 2019; Allamanis and Sutton, 2014) takes an MCMC-
based approach with better scaling behavior than deductive rewrites, but
differs from the goals of Stitch as they don’t address binders and focus on
common syntactic fragments instead of well-formed functions.
### 7.2. Guided Top-Down Program Synthesis
Stitch uses a corpus-guided top-down approach to learning library abstractions
that is closely related to recent guided enumerative synthesis techniques.
This includes methods that leverage type-based constraints on holes (Feser et
al., 2015; Polikarpova et al., 2016), over- and under-approximations of the
behaviors of holes (Lee et al., 2016; Chen et al., 2020), and probabilistic
techniques to heuristically guide the search (Balog et al., 2016; Ellis et
al., 2020, 2021; Nye et al., 2021; Shah et al., 2020). These approaches have
largely been applied in to synthesize entire programs based on input/output
examples or another form of specification, in contrast to the abstraction-
learning goal in our work (Allamanis et al., 2018; Balog et al., 2016; Chen et
al., 2018; Ellis et al., 2018; Ganin et al., 2018; Koukoutos et al., 2017).
Like Stitch however, these approaches sometimes use cost functions (such as
the likelihood of a partially enumerated program under a hand-crafted or
learned probabilistic generative model over programs) in order to direct
search towards more desirable program trees. However, Stitch’s cost function
leverages a more direct relationship between partially-enumerated candidate
functions and the existing training corpus, unlike the cost functions
typically applied in inductive synthesis, which must be estimated from
input/output examples.
### 7.3. Lambda-Aware Unification
The LambdaUnify procedure presented in Section 3 relates to prior work on
unification modulo binders (Huet, 1975; Miller, 1991, 1992; Dowek et al.,
1995, 1996).
Notion of beta-equivalence. This prior work is concerned with more general
notions of equivalence modulo beta-reduction, while LambdaUnify is based on a
restricted but fast syntax-driven equivalence. For example, in Dowek et al.
1996 one might try to unify $(\alpha\ \texttt{foo})$ with $(\texttt{foo})$ and
get the two solutions: $[\alpha\to(\lambda.\ \$0)]$ and $[\alpha\to(\lambda.\
\texttt{foo})]$. In contrast, LambdaUnify will never introduce a
$\lambda-$abstraction to create a higher order argument and thus $(\alpha\
\texttt{foo})$ and $(\texttt{foo})$ would not unify at all, since one is an
application while the other is a primitive. Instead, higher order abstraction
arguments are handled by combining the algorithm with a deductive approach
when desirable as in Section 5. We also note that in
$\textsc{LambdaUnify}(A,e)$ it is assumed that $e$ is an expression and thus
does not contain abstraction variables, which further simplifies the approach
compared to this prior work.
Handling of binders. Dowek et al. 1995 and Dowek et al. 1996 both employ de
Bruijn indexed variables and therefore must similarly account for the shifting
of variables in arguments when inverting beta reduction. While LambdaUnify
handles this through DownshiftAll, Dowek et al. 1995 and Dowek et al. 1996
both converting their terms to the $\lambda\sigma-$calculus of explicit
substitutions (Abadi et al., 1989) which allows them to insert upshifting
operators at each abstraction variable location. This alternate handling of
shifting is useful given the more general notion of equivalence they are
considering, but is excessive for our simpler syntax-guided task.
Handling of holes. In $\textsc{LambdaUnify}(A,e)$ we allow $A$ to contain
holes ?? which are allowed to violate index shifting rules during unification
as they are considered unfinished subtrees of the abstraction. This is handled
through the use of $\&i$ indices. While the special-case handling of holes in
expressions is not directly part of this prior work, there are similarities
between it and the Skolemization (Skolem, 1920) done in Miller 1992.
Skolemization allows for lifting an existential quantifier (i.e. an
abstraction variable or hole) above a universal quantifier (i.e. a lambda) by
turning the abstraction variable or hole into a function of its local context
— in essence piping the local context into the hole. In the context of the
lambda calculus this is essentially a form of lambda lifting (Johnsson, 1985),
which is the process of lifting a local function that contains free variables
by binding the free variables as additional arguments to the function and
passing them in at each call site. In Miller 1992 this is used for abstraction
variables (as there are no holes) and aids in the more general notion of beta-
equivalence they are dealing with, while for our purposes the $\&i$ variables
suffice and don’t require extra manipulations of lambdas to thread in the
local context.
### 7.4. Upper Bounds in Network Motifs
There is also an interesting connection between this work and prior work on
finding network motifs (Milo et al., 2002), which are frequently-occurring
subgraphs within a corpus of graphs. One approach to mining high-frequency
motifs is based on growing a motif one edge at a time, much like we grow
abstractions one node at a time during our search (Schreiber and
Schwöbbermeyer, 2005; Kuramochi and Karypis, 2004, 2001). This approach makes
use of the insight that the frequency of a motif will only decrease as it is
grown, giving an upper bound on the frequency of any motif derived from it.
While we employ a more complex compressive utility function than their
frequency-based utility, our upper bound is based on a similar insight that
larger abstractions will match at a subset of the locations.
### 7.5. Comparison to babble
babble (Cao et al., 2023) is concurrent work in library learning that likewise
adopts the compression objective from DreamCoder. While babble focuses
primarily on expressive library learning through an algorithm that can reason
over semantic equivalences represented through e-graphs, Stitch focuses
primarily on efficient library learning through a parallel anytime branch-and-
bound algorithm.
Stitch synthesizes the maximally-compressive abstraction for a corpus of
programs through a branch-and-bound top-down search, computing an upper bound
on the compression of any partially-constructed abstraction to guide and prune
the search. In contrast, babble operates on an e-graph representing many
equivalent corpora and uses anti-unification to propose candidate abstractions
and a custom e-graph beam-search extraction algorithm to select a maximally-
compressive set of abstractions from the proposal.
At a high level, Stitch aims to be efficient while babble aims to be
expressive. The key differences between the algorithms are:
* •
Efficiency: On DreamCoder’s benchmarks, babble is 10-100x faster than
DreamCoder while Stitch is 1,000-10,000x faster than DreamCoder. Unlike
babble, Stitch is an anytime algorithm that can be stopped early for a best-
so-far result.
* •
Expressivity: babble can learn libraries modulo equational theories, allowing
it to find common semantic abstractions despite syntactic differences in the
programs. Stitch only explores non-syntactic equivalences preliminarily in the
PyStitch prototype with the DreamCoder beta-inversion rewrite rule (Section 5
and Section 6.5).
* •
Jointly learning a library: babble can learn multiple abstractions at once
that jointly provide compression, while DreamCoder and Stitch repeatedly learn
a single abstraction at a time. Stitch provides an optimality guarantee on
each abstraction learned while babble approximates the joint objective through
a beam search.
These two approaches has advantages and disadvantages, and we believe that
there is strong potential in combining the two.
## 8\. Conclusion and Future Work
We have presented corpus-guided top-down synthesis (CTS)—an efficient new
algorithm for synthesizing libraries of functional abstractions capturing
common functionality within a corpus of programs. CTS directly synthesizes the
abstractions, rather than exposing them through a series of rewrites as is
done by deductive systems. Key to its performance is the usage of a guiding
utility function, which allows CTS to effectively search (and prune out large
portions of) the space of possible abstractions.
We implement this algorithm in Stitch, an open-source library learning tool
with an associated Python library and Rust
crate.777https://github.com/mlb2251/stitch We evaluate Stitch across five
experimental settings, demonstrating that it learns comparably compressive
libraries with 2 orders of magnitude less memory and 3-4 orders of magnitude
less time compared to the state-of-the-art deductive algorithm of Ellis et al.
2021. We also find that Stitch scales to learning libraries of abstractions
from much larger datasets of deeper program trees than is possible with prior
work, and that the anytime property of corpus-guided top-down search —
abstractions discovered via top-down search are already compressive early in
search and improve as it continues — offers the opportunity for high-quality
library learning even in complex domains through early search termination.
There remain many open problems and exciting directions in abstraction
learning. One major direction is extending the CTS approach to handle
reasoning over more general deductive rewrite systems, like those encoded in
egg (Willsey et al., 2021) as babble does (Cao et al., 2023) (also see Section
7.5). The PyStitch prototype presented in Section 5 only works with a single
rewrite rule (beta-inversion) and uses an approximate utility without an
optimality guarantee. This is because it is challenging to adapt the upper
bound used by Stitch to version spaces or e-graphs due to the trade-off
between the compression gained from equational rewrites and compression gained
from abstraction learning.
The speedup afforded by Stitch may allow for new applications of abstraction
learning as well, as it can now be used as an inexpensive subroutine that can
be called much more frequently than prior algorithms. To further extend its
range of applications, CTS could also be adapted to finding abstractions in
data structures beyond program trees, such as dataflow DAGs and more general
graph structures like molecules, if the bounds and matching can be adapted.
Finally, while we work with fairly large lambda calculus programs in this
work, there is a clear gap in scale between these programs and those in an
actual codebase, so exploring applications of CTS to real-world code is an
exciting direction.
## Data Availability Statement
All source code can be found at https://github.com/mlb2251/stitch. An artifact
for reproducing results in this work is available at
https://github.com/mlb2251/stitch-artifact or alternatively a static version
is available at https://zenodo.org/record/7151663 (Bowers et al., 2022).
###### Acknowledgements.
We thank A. Lew and J. Andreas for helpful discussions, and J. Feser and I.
Kuraj for feedback on the manuscript. M.B. and G.G are supported by the
Sponsor National Science Foundation (NSF) Graduate Research Fellowship under
Grant No. Grant #2141064. M.B. is also supported by the Sponsor Defense
Advanced Research Projects Agency (DARPA) under the SDCPS Contract Grant
#FA8750-20-C-0542. T.X.O. is supported by Herbert E. Grier (1933) and Dorothy
J. Grier Fund Fellowship. L.W. and J.B.T. are supported by Sponsor AFOSR
under grant number Grant #FA9550-19-1-0269, the MIT Quest for Intelligence,
the MIT-IBM Watson AI Lab, ONR Science of AI, and DARPA Machine Common Sense.
G.G. is also supported by the MIT Presidential Fellowship. A.S. is supported
by the NSF under Grant No. Grant #1918839. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of sponsors.
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## Appendix A Corpus-guided top-down search: full algorithm
In this appendix, we give an expansive description of the full corpus-guided
top-down search (CTS) algorithm to ease re-implementation.
The full algorithm is given in Algorithm 1. It takes as input a corpus of
programs $\mathcal{P}$, a utility function $U_{\mathcal{P},\mathcal{R}}(A)$,
and utility upper bound function
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})$. The algorithm searches
for and returns the complete abstraction $A_{best}$ that maximizes the utility
function. The algorithm maintains a priority queue $Q$ of partial abstractions
ordered by their utility upper bound (or alternatively a custom priority
function), a best complete abstraction so-far $A_{best}$, and a corresponding
best utility so far $U_{best}$ for that abstraction. The priority queue is
initialized to hold the single-hole partial abstraction ??, from which any
abstraction can be derived. $A_{best}$ and $U_{best}$ are initialized as the
best arity-zero abstraction and corresponding utility (line 4), since arity-
zero abstractions can be quickly and completely enumerated (each unique,
closed subexpression in the corpus is an arity-zero abstraction).
We then proceed to the core loop of the algorithm from lines 6-31. At each
step of this loop we pop the highest-priority partial abstraction
$A_{\texttt{??}}$ off of the priority queue and process it. We discard
$A_{\texttt{??}}$ if its utility upper bound doesn’t exceed our best utility
found so far (lines 8-10). We then choose a hole $h$ in $A_{\texttt{??}}$ to
expand with the procedure Choose-Hole. Choose-Hole can be a custom function;
we find from preliminary experiments on a subset of the datasets that choosing
the most recently introduced hole is effective in practice.
The algorithm then uses the procedure Expansions to iterate over all possible
single step expansions of the hole $h$ in $A_{\texttt{??}}$, such as replacing
the hole with + or with (app ?? ??). For each expanded abstraction
$A_{\texttt{??}}^{\prime}$, it is easy to compute its set of match locations
$\textsc{Matches}(\mathcal{P},A_{\texttt{??}}^{\prime})$ since we know this
will be a subset of the match locations of $A_{\texttt{??}}$ (and these will
be disjoint subsets, except when expanding into an abstraction variable
$\alpha$). We can easily inspect the relevant subtree at each match location
of the original abstraction to see which expansions are valid and which match
locations will be preserved by a given expansion. Note that an expansion to a
free variable can be pruned immediately, as any resulting abstraction will not
be well-formed.
When expanding to an abstraction variable $\alpha$, if $\alpha$ is a new
variable not already present in the partial abstraction, the set of match
locations is unchanged. If $\alpha$ is an existing abstraction variable then
this is a situation where the same variable is being used in more than one
place, as in the square abstraction $(\lambda\alpha.\ \texttt{*}\ \alpha\
\alpha)$. In this case we restrict the match locations to the subset of
locations where within a location all instances of $\alpha$ match against the
same subtree. Additionally, if a maximum arity is provided then an expansion
that causes the abstraction to exceed this limit is not considered.
When considering each possible expanded abstraction
$A_{\texttt{??}}^{\prime}$, there is room for strong corpus-guidance. First,
we don’t need to consider any expansions that would result in zero match
locations since all abstractions in this branch of search will have zero
rewrite locations per Lemma 2 (lines 13-15). Furthermore, we use
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}}^{\prime})$ to upper bound
the utility achievable in this branch of search and discard it if it is less
than our best utility so far (lines 16-18). Since each
$A_{\texttt{??}}^{\prime}$ covers a disjoint set of match locations (except in
the case of expanding into an abstraction variable), the set of match
locations often drops rapidly and can allow for the calculation of a tight
upper bound depending on the utility function. As a final step of pruning, if
we can identify that $A_{\texttt{??}}^{\prime}$ is strictly dominated by some
other abstraction $A_{\texttt{??}}^{\prime\prime}$, we may discard
$A_{\texttt{??}}^{\prime}$ (lines 19-21). Note additionally that arity-zero
abstractions were precomputed, then abstractions that match at a single
location are safe to prune as well (as long as they don’t have free variables)
as arity-zero abstractions are always superior for single match locations.
For any partial abstraction $A_{\texttt{??}}^{\prime}$ that has not been
pruned, we then check whether it is a complete abstraction (there are no
remaining holes) or whether it is still a partial abstraction. Partial
abstractions get pushed to the priority queue ranked by their utility upper
bound, and complete abstractions are used to update $A_{best}$ and $U_{best}$
if they have a higher utility than any prior complete abstractions.
Once there are no more partial abstractions remaining in the priority queue,
the algorithm terminates. Note that since the algorithm maintains a best
abstraction so-far, it can also be terminated early, making it an any-time
algorithm. We also note that this algorithm is amenable to parallelization as
it can be easily parallelized over the while loop on lines 6-31, especially
since the algorithm remains sound even when the upper bound $U_{best}$ used
pruning is not always up to date.
This algorithm can be iterated to build up a library of abstractions.
Finally note that in our implementation, we employ structural hashing to
simplify equality checks between subtrees and avoid re-doing work on multiple
identical copies of a subtree. Our complete implementation of CTS for
compression is provided at https://github.com/mlb2251/stitch.
Algorithm 1 Corpus-guided top-down abstraction synthesis. Color-coded: Upper
bound pruning, Zero-usage pruning, Strict dominance pruning
1:Corpus of input programs $\mathcal{P}$, utility function
$U_{\mathcal{P},\mathcal{R}}(A)$, and utility upper bound function
$\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})$
2:The maximally compressive abstraction $A_{best}$
3:$Q\leftarrow$ Priority-Queue { ?? } $\triangleright$ New priority queue with
the single partial abstraction ??
4:$A_{best}\leftarrow\textsc{Best-Arity-Zero-Abstraction}(\mathcal{P})$
$\triangleright$ Initialize best abstraction so far
5:$U_{best}\leftarrow U_{\mathcal{P},\mathcal{R}}(A_{best})$
6:while $\text{non-empty}(Q)$ do
7: $A_{\texttt{??}}\leftarrow\text{pop-max}(Q)$ $\triangleright$ Next partial
abstraction to expand
8: if $\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}})\leq U_{best}$ then
9: continue $\triangleright$ Upper bound pruning
10: end if
11: $h\leftarrow\textsc{Choose-Hole}(A_{\texttt{??}})$ $\triangleright$ Choose
a hole to expand
12: for
$(A_{\texttt{??}}^{\prime},M^{\prime})\in\textsc{Expansions}(A_{\texttt{??}},h,\mathcal{P})$
do $\triangleright$ abstraction $A_{\texttt{??}}^{\prime}$ and match locations
$M^{\prime}$
13: if $\text{length}(M^{\prime})==0$ then
14: continue $\triangleright$ No match locations in corpus
15: end if
16: if $\bar{U}_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}}^{\prime})\leq
U_{best}$ then
17: continue $\triangleright$ Upper bound pruning
18: end if
19: if $\text{strictly-dominated}(A_{\texttt{??}}^{\prime},\mathcal{P})$ then
20: continue $\triangleright$ Strict dominance pruning
21: end if
22: if $\text{has-holes}(A_{e})$ then
23: $Q\leftarrow Q\cup A_{\texttt{??}}^{\prime}$ $\triangleright$ add partial
abstraction to heap
24: else
25: if $U_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}}^{\prime})>U_{best}$ then
26: $U_{best}\leftarrow U_{\mathcal{P},\mathcal{R}}(A_{\texttt{??}}^{\prime})$
$\triangleright$ new best complete abstraction
27: $A_{best}\leftarrow A_{\texttt{??}}^{\prime}$
28: end if
29: end if
30: end for
31:end while
## Appendix B proof of Correctness of LambdaUnify
As introduced in Section 3, $\textsc{LambdaUnify}(A,e)$ returns a mapping from
abstraction variables and holes to expressions $[\alpha_{i}\to
e_{i}^{\prime},\ \ldots,\ \texttt{??}_{j}\to u_{j}^{\prime},\ \ldots]$ such
that
(19) $(\lambda\alpha_{i}.\ \ldots\ \lambda\texttt{??}_{j}.\ \ldots A)\
e_{i}^{\prime}\ \ldots\ u_{j}^{\prime}\ \ldots=e$
through beta reduction. The LambdaUnify procedure is given in Fig. 4 and the
definition of beta reduction with a modified upshifting operator is given in
Fig. 5.
We can write Eq. 19 in terms of substitution as
(20) $[\alpha_{i}\to e_{i}^{\prime},\ \ldots,\ \texttt{??}_{j}\to
u_{j}^{\prime},\ \ldots]\ \circ A=e$
Which can in turn be written in terms of LambdaUnify as
(21) $\textsc{LambdaUnify}(A,e)\circ A=e$
To prove the correctness of LambdaUnify, we must show that Eq. 21 holds for
all $A$ and $e$. We do this in two ways: a written proof (below) of our
correctness theorem (Theorem B.1), and a machine-verifiable proof in Coq. The
Coq proof is given in stitch.v in the supplementary material and was proven
with CoqIDE 8.15.2. The meanings of the definitions and lemmas in the Coq
proof are commented. The Coq proof differs from the written proof in places
because encoding the proof in Coq required coming up with some representations
that worked well for Coq, though these differences are limited. For example,
the Coq proof requires fully formalizing the behavior of Merge and proofs
related to it which is more simply explained in the written proof. When lemmas
in the written proof below correspond directly to lemmas in the Coq proof, the
corresponding Coq lemma is indicated in the written proof below.
### B.1. Well-formedness
We formalize a notion of well-formedness to capture the set of expressions
that are accessible starting from an expression with no $\&i$ variables then
applying any series of upshifts and downshifts. We define $\text{WF}_{d}(e)$,
the well-formedness of an expression $e$ at a depth $d$, as follows:
$\displaystyle\text{WF}_{d}\ \lambda.b=\text{WF}_{d+1}\ b$
$\displaystyle\text{WF}_{d}\ (fx)=\text{WF}_{d}\ f\land\text{WF}_{d}\ x$
$\displaystyle\text{WF}_{d}\ \$i=\top$ $\displaystyle\text{WF}_{d}\
\&i=\begin{cases}\top,&\text{if }i<d\\\ \bot,&\text{if }i\geq d\\\
\end{cases}$ $\displaystyle\text{WF}_{d}\ t=\top,\ \text{for
$t\in\mathcal{G}_{\texttt{sym}}$}$
Note that trivially any expression $e$ without any $\&i$ variables is well
formed, as $\&i$ variables are the only source of $\bot$ in this definition.
If every expression in a mapping $l$ is well-formed, we say that the mapping
itself is well-formed, written $(\text{WFMap}\ l)$.
### B.2. Relevant Lemmas
The following two lemmas show that upshifting and downshifting preserve well-
formedness. In Coq these lemmas are upshift_wf and downshift_wf.
###### Lemma 3.
$\forall e.\ \forall d.\ \text{WF}_{d}\ e\implies\text{WF}_{d}\ \uparrow_{d}e$
###### Proof.
We proceed by induction on e.
Case $e=\lambda.\ b$. Our goal is
$\text{WF}_{d}\ \uparrow_{d}\lambda.b$
We can move $\uparrow_{d}$ and $\text{WF}_{d}$ into the lambda by unfolding
their definitions to get
$\text{WF}_{d+1}\ \uparrow_{d+1}b$
We also have the premise $\text{WF}_{d}\ \lambda.b$, where we can likewise
move the $\text{WF}_{d}$ into the lambda to get $\text{WF}_{d+1}\ b$. The
inductive hypothesis $\forall d.\ \text{WF}_{d}\ b\implies\text{WF}_{d}\
\uparrow_{d}b$ can be instantiated with $d$ as $d+1$ to get $\text{WF}_{d+1}\
b\implies\text{WF}_{d+1}\ \uparrow_{d+1}b$, the conclusion of which is
precisely our goal and the premise of which is precisely our other premise.
Case $e=f\ x$. Our goal is
$\text{WF}_{d}\ \uparrow_{d}(f\ x)$
We can move $\uparrow_{d}$ and $\text{WF}_{d}$ into the application by
unfolding their definitions to get
$(\text{WF}_{d}\ \uparrow_{d}f)\ \land(\text{WF}_{d}\ \uparrow_{d}x)$
We can use our inductive hypotheses $\text{WF}_{d}\ f\implies\text{WF}_{d}\
\uparrow_{d}f$ to prove the left side of this conjunction if we can prove
$\text{WF}_{d}\ f$. We have the premise that $\text{WF}_{d}\ (f\ x)$ from this
overall induction case, which by definition of $\text{WF}_{d}$ can only be
true if $\text{WF}_{d}\ f\land\text{WF}_{d}\ d$. Thus we can prove the left
side of the conjunction, and the proof of the right side is identical with $f$
replaced by $x$ using the other inductive hypothesis.
Case $e=\$i$. Our goal is
$\text{WF}_{d}\ \uparrow_{d}\$i$
We will handle this in 2 cases:
* •
Case $i<d$.
$\text{WF}_{d}\ \uparrow_{d}\$i$ $=\text{WF}_{d}\ \$i$
This is true because $i indices are always well-formed.
* •
Case $i\geq d$.
$\text{WF}_{d}\ \uparrow_{d}\$i$ $=\text{WF}_{d}\ \$(i+1)$
This is true because $i indices are always well-formed.
Case $e=\&i$. Our goal is
$\text{WF}_{d}\ \uparrow_{d}\&i$
We will handle our goal in 2 cases:
* •
Case $i+1\neq d$.
$\text{WF}_{d}\ \uparrow_{d}\$i$ $=\text{WF}_{d}\ \&(i+1)$
We know from our premise that $\text{WF}_{d}\ \&i$ which means $i<d$, so
$i+1\leq d$. Combining this with our knowledge from the case branch that
$i+1\neq d$, we know that $i+1<d$ which means that $\&(i+1)$ is well formed.
* •
Case $i+1=d$.
$\text{WF}_{d}\ \uparrow_{d}\&i$ $=\text{WF}_{d}\ \$(i+1)$
This is true because $i indices are always well-formed.
Case $e=t$. Our goal is $\text{WF}_{d}\ \uparrow_{d}t$ which simplifies by
definition of $\uparrow_{d}$ to $\text{WF}_{d}\ t$ which is true by definition
of $\text{WF}_{d}$.
∎
###### Lemma 4.
$\forall e.\ \forall d.\ \text{WF}_{d}\ e\implies\text{WF}_{d}\
\downarrow_{d}e$
###### Proof.
We proceed by induction on e. The cases $e=t$, $e=(f\ x)$ and $e=\lambda.b$
are identical to those in Lemma 3 but with $\uparrow$ replaced with
$\downarrow$ so they will be omitted.
Case $e=\$i$. Our goal is
$\text{WF}_{d}\ \downarrow_{d}\$i$
We will handle this in 3 cases:
* •
Case $i<d$.
$\text{WF}_{d}\ \downarrow_{d}\$i$ $=\text{WF}_{d}\ \$i$
This is true because $i indices are always well-formed.
* •
Case $i>d$.
$\text{WF}_{d}\ \downarrow_{d}\$i$ $=\text{WF}_{d}\ \$(i-1)$
This is true because $i indices are always well-formed.
* •
Case $i=d$.
$\text{WF}_{d}\ \downarrow_{d}\$i$ $=\text{WF}_{d}\ \&(i-1)$
Since $i=d$ we know that $i-1<d$ which means $\&(i-1)$ is well-formed.
Case $e=\&i$. Our goal is
$\text{WF}_{d}\ \downarrow_{d}\&i$
which simplifies to
$\text{WF}_{d}\ \&(i-1)$
We know from our premise that $\text{WF}_{d}\ \&i$ so $i<d$ which means
$i-1<d$ so $\&(i-1)$ is well-formed.
∎
The following lemma shows that upshifting is an inverse of downshifting. In
Coq this lemma is upshift_downshift.
###### Lemma 5.
$\forall e.\ \forall d.\ \text{WF}_{d}\
e\implies\uparrow_{d}\downarrow_{d}e=e$
###### Proof.
We proceed by induction on $e$.
Case $e=\lambda.\ b$. Our goal is
$\uparrow_{d}\downarrow_{d}\lambda.\ b=\lambda.\ b$
By the definitions of $\uparrow_{d}$ and $\downarrow_{d}$ we can propagate
them into the body of a lambda while incrementing $d$:
$\lambda.\ \uparrow_{d+1}\downarrow_{d+1}b=\lambda.\ b$
We also have the additional premise $\text{WF}_{d}\ \lambda.b$ which
simplifies to $\text{WF}_{d+1}\ b$.
Since our inductive hypothesis $\forall d.\ \text{WF}_{d}\
b\implies\uparrow_{d}\downarrow_{d}b=b$ is universally quantified over $d$ we
can instantiate it with $d+1$ in place of $d$ and discharge its precondition
with our other premise $\text{WF}_{d+1}\ b$ to get
$\uparrow_{d+1}\downarrow_{d+1}b=b$ and apply this to our goal to get
$\lambda.\ b=\lambda.\ b$
which is trivially true.
Case $e=f\ x$. Our goal is
$\uparrow_{d}\downarrow_{d}(f\ x)=f\ x$
By the definitions of $\uparrow_{d}$ and $\downarrow_{d}$ we can propagate
them into both sides of the application:
$(\uparrow_{d}\downarrow_{d}f)(\uparrow_{d}\downarrow_{d}x)=f\ x$
We also have the additional premise $\text{WF}_{d}\ (f\ x)$ can be broken into
the two premises $\text{WF}_{d}\ f$ and $\text{WF}_{d}\ x$.
We can then apply our inductive hypotheses $\forall d.\ \text{WF}_{d}\
f\implies\uparrow_{d}\downarrow_{d}f=f$ and $\forall d.\ \text{WF}_{d}\
x\implies\uparrow_{d}\downarrow_{d}x=x$ with these premises to rewrite our
goal to
$f\ x=f\ x$
which is trivially true.
Case $e=\$i$. Our goal is
$\uparrow_{d}\downarrow_{d}\$i=\$i$
We will handle this in 3 cases:
* •
Case $i<d$.
$\uparrow_{d}\downarrow_{d}\$i=\$i$
Unfold definition of $\downarrow_{d}$
$\uparrow_{d}\$i=\$i$
Unfold definition of $\uparrow_{d}$
$\$i=\$i$
* •
Case $i>d$.
$\uparrow_{d}\downarrow_{d}\$i=\$i$
Unfold definition of $\downarrow_{d}$
$\uparrow_{d}\$(i-1)=\$i$
Unfold definition of $\uparrow_{d}$ noting that $(i-1)\geq d$
$\$((i-1)+1)=\$i$
Simplify
$\$i=\$i$
* •
Case $i=d$.
$\uparrow_{d}\downarrow_{d}\$i=\$i$
Unfold definition of $\downarrow_{d}$
$\uparrow_{d}\&(i-1)=\$i$
Unfold definition of $\uparrow_{d}$ noting that $(i-1)+1=d$
$\$((i-1)+1)=\$i$
Simplify
$\$i=\$i$
Case $e=\&i$. Our goal is
$\uparrow_{d}\downarrow_{d}\&i=\&i$
Unfolding the definition of $\downarrow_{d}$ yields
$\uparrow_{d}\&(i-1)=\&i$
Since we know that $\text{WF}_{d}\ \&i$ by our premise, we know that $i<d$ so
$(i-1)+1<d$ so $(i-1)+1\neq d$ so $\uparrow_{d}$ unfolds to:
$\&((i-1)+1)=\&i$
which simplifies to
$\&i=\&i$
which is trivially true.
Case $e=t$. Our goal is
$\uparrow_{d}\downarrow_{d}t=t$
Neither $\uparrow_{d}$ nor $\downarrow_{d}$ have any effect on a grammar
primitive $t$, so this is trivially true.
∎
The following lemma shows that any result of LambdaUnify is a well-formed
mapping. In Coq this lemma is called lu_wf.
###### Lemma 6.
$\forall A.\ \forall e.\text{WFMap}\ \textsc{LambdaUnify}(A,e)$
###### Proof.
This is a straightforward proof by induction. In the $\alpha_{i}$ and and
$\texttt{??}_{i}$ cases LambdaUnify adds expressions from the original
expression to the mapping, which are well-formed as they can’t contain $\&i$
variables. In the lambda case DownshiftAll shifts variables using $\downarrow$
which is guaranteed to preserve well-formedness by Lemma 4. In the application
case merge simply combines mappings from the function and argument calls to
LambdaUnify which are well-formed by induction, resulting in an overall well-
formed mapping. ∎
The following lemma follows from Lemma 5 and shows that UpshiftAll is an
inverse of DownshiftAll. In Coq this lemma is upshift_all_downshift_all.
###### Lemma 7.
$\text{WFMap}\ l\implies\textsc{UpshiftAll}(\textsc{DownshiftAll}(l))=l$
###### Proof.
Writing $l$ as $[\alpha_{i}\to\ e_{i}^{\prime},\ \texttt{??}_{j}\to\
u_{j}^{\prime},\ ...]$ and unfolding the definitions of UpshiftAll
DownshiftAll we get:
$[\alpha_{i}\to\ \uparrow_{0}\downarrow_{0}e_{i}^{\prime},\
\texttt{??}_{j}\to\ \uparrow_{0}\downarrow_{0}u_{j}^{\prime},\
...]=[\alpha_{i}\to\ e_{i}^{\prime},\ \texttt{??}_{j}\to\ u_{j}^{\prime},\
...]$
This equality will hold if
$\uparrow_{0}\downarrow_{0}u_{j}^{\prime}=u_{j}^{\prime}$ and
$\uparrow_{0}\downarrow_{0}e_{i}^{\prime}=e_{i}^{\prime}$ for all
$u_{j}^{\prime}$ and $e_{i}^{\prime}$. $\text{WFMap}\ l$ tells us that all
expressions $u_{j}^{\prime}$ and $e_{i}^{\prime}$ are well-formed and thus
satisfy the precondition of Lemma 5, which proves these equalities true.
∎
The following two lemmas allow us to ”forget” one side of a merge when it
won’t have any effect on the substitution. In Coq the first lemma is
merge_forget3 and the second is captured by the definition Compatible.
###### Lemma 8.
If $l_{1}$ is a mapping that binds every abstraction variable and hole in $A$,
then $\textsc{merge}(l_{1},l_{2})\circ A=l_{1}\circ A$.
###### Proof.
$l_{1}$ already assigns to every abstraction variable and hole, and merging in
$l_{2}$ can’t overwrite these assignments since merge fails when the same
variable or hole is assigned to two different expressions. Thus merging in
$l_{2}$ has no effect on the resulting expression from substitution. ∎
###### Lemma 9.
If $l_{2}$ is a mapping that binds every abstraction variable and hole in $A$,
then $\textsc{merge}(l_{1},l_{2})\circ A=l_{2}\circ A$.
###### Proof.
This proof is identical in structure to Lemma 8. $l_{2}$ already assigns to
every abstraction variable and hole, and merging in $l_{1}$ can’t overwrite
these assignments since merge fails when the same variable or hole is assigned
to two different expressions. Thus merging in $l_{1}$ has no effect on the
resulting expression from substitution. ∎
The following lemma shows that any result of LambdaUnify satisfies the
precondition of Lemma 9 and Lemma 8.
###### Lemma 10.
$\textsc{LambdaUnify}(A,e)$ returns a mapping that binds all abstraction
variables $\alpha_{i}$ and all holes $\texttt{??}_{i}$ in $A$.
###### Proof.
This is a straightforward proof by induction. In the $\alpha_{i}$ and and
$\texttt{??}_{i}$ cases LambdaUnify binds the variable and hole respectively,
in the lambda case DownshiftAll has no effect on which abstraction variables
are bound (and it binds all the variables in the body by induction), and in
the application case merge results in combining the function and argument
mappings (which bind all variables by induction) to result in a mapping which
binds all variables that appear in both the function and argument. ∎
### B.3. Correctness Theorem
In Coq this theorem is called correctness.
###### Theorem B.1.
$\forall A.\ \forall e.\ \textsc{LambdaUnify}(A,e)\circ A=e$
###### Proof.
We proceed by induction over the proof tree of $\textsc{LambdaUnify}(A,e)$.
Case U-AbsVar. Our goal is $\textsc{LambdaUnify}(A,\alpha)\circ\alpha=e$ which
simplifies by definition of LambdaUnify to $[\alpha\to e]\circ\alpha=e$ which
is true by the definition of substitution.
Case U-Hole. Our goal is
$\textsc{LambdaUnify}(A,\texttt{??})\circ\texttt{??}=e$ which simplifies by
definition of LambdaUnify to $[\texttt{??}\to e]\circ\texttt{??}=e$ which is
true by the definition of substitution.
Case U-App. Our goal is
$\textsc{LambdaUnify}(A_{1}\ A_{2},\ e_{1}\ e_{2})\circ(A_{1}\ A_{2})=e_{1}\
e_{2}$
This simplifies by definition of LambdaUnify to
$\textsc{merge}(\textsc{LambdaUnify}(A_{1},e_{1}),\textsc{LambdaUnify}(A_{2},e_{2}))\circ(A_{1}\
A_{2})=e_{1}\ e_{2}$
We can distribute the substitution over the application (by definition of
substitution in Fig. 5 (Left)) to get
$\displaystyle($
$\displaystyle(\textsc{merge}(\textsc{LambdaUnify}(A_{1},e_{1}),\textsc{LambdaUnify}(A_{2},e_{2}))\circ
A_{1})$
$\displaystyle(\textsc{merge}(\textsc{LambdaUnify}(A_{1},e_{1}),\textsc{LambdaUnify}(A_{2},e_{2}))\circ
A_{2}))$ $\displaystyle=e_{1}\ e_{2}$
We can then forget the irrelevant part of each merge using Lemma 9 and Lemma 8
where the precondition (binding all abstraction variables and holes) is
satisfied by Lemma 10:
$\displaystyle($ $\displaystyle(\textsc{LambdaUnify}(A_{1},e_{1})\circ A_{1})$
$\displaystyle(\textsc{LambdaUnify}(A_{2},e_{2})\circ A_{2}))$
$\displaystyle=e_{1}\ e_{2}$
We can then finish the proof by directly applying our inductive hypotheses
$\textsc{LambdaUnify}(A_{1},e_{1})\circ A_{1}=e_{1}$ and
$\textsc{LambdaUnify}(A_{2},e_{2})\circ A_{2}=e_{2}$.
Case U-Lam. Our goal is
$\textsc{LambdaUnify}(\lambda.\ A,\lambda.\ e)\circ\lambda.\ A=\lambda.\ e$
This simplifies by definition of LambdaUnify to
$\textsc{DownshiftAll}(\textsc{LambdaUnify}(A,e))\circ\lambda.\ A=\lambda.\ e$
We can move the substitution under the lambda while inserting UpshiftAll as
per the definition of substitution
$\lambda.\
\textsc{UpshiftAll}(\textsc{DownshiftAll}(\textsc{LambdaUnify}(A,e)))\circ
A=\lambda.\ e$
Since we know that any result of LambdaUnify is well-formed by Lemma 6 we can
apply Lemma 7 to get
$\lambda.\ \textsc{LambdaUnify}(A,e)\circ A=\lambda.\ e$
We can then apply our inductive hypothesis $\textsc{LambdaUnify}(A,e)\circ
A=e$ to get
$\lambda.\ e=\lambda.\ e$
and we are done.
Case U-Same. Our goal is $\textsc{LambdaUnify}(e,e)\circ e=e$ which simplifies
by definition of LambdaUnify to $[]\circ e=e$ which is trivially true as the
empty substitution has no effect.
∎
|
# Statistical Nonlocality of Dynamically Coherent Structures
Andre N. Souza1<EMAIL_ADDRESS>Tyler Lutz2 Glenn R. Flierl1
1Massachusetts Institute of Technology, Cambridge, MA, United States 2 Yale
University, New Haven, CT, United States
###### Abstract
We introduce a class of stochastic advection problems amenable to analysis of
turbulent transport. The statistics of the flow field are represented as a
continuous time Markov process, a choice that captures the intuitive notion of
turbulence as moving from one coherent structure to another. We obtain closed
form expressions for the turbulent transport operator without invoking
approximations. We recover the classical estimate of turbulent transport as a
diffusivity tensor, the components of which are the integrated auto-
correlation of the velocity field, in the limit that the operator becomes
local in space and time.
## 1 Introduction
The study of passive scalar transport is of fundamental importance in
characterizing turbulence. Given that predicting a chaotic dynamical
trajectory over long times is infeasible (Lorenz, 1963) one must switch to a
statistical perspective to make headway on transport properties. From the
analysis of anomalous dispersion by Taylor (1922), operator notions of mixing
from Knobloch (1977), computations of “effective diffusivity” by Avellaneda &
Majda (1991), simplified models of turbulence by Pope (2011), rigorous notions
of mixing in terms of Sobolev norms by Thiffeault (2012), or upper bounds on
transport as in Hassanzadeh et al. (2014), different approaches elucidate
fundamental properties of turbulence.
In addition to furthering our understanding of turbulence, there are practical
applications for turbulence closures. In particular, Earth Systems Models
require closure relations for the transport of unresolved motions Schneider et
al. (2017); however, the closure relations are marred by structural and
parametric uncertainty, requiring ad-hoc tuning to compensate for biases.
There are structural biases associated with scaling laws and closure
assumptions between turbulent fluxes and gradients. Modern studies are
bridging the gap by incorporating more complex physics and novel scaling laws,
Tan et al. (2018); Gallet & Ferrari (2020), but the chosen functional forms to
represent fluxes remain unknown.
The multi-scale nature of turbulent flows, the presence of coherent
structures, as well as the interconnection of reacting chemical species and
prognostic fields, instead suggests that fluxes are perhaps better modeled
using nonlinear nonlocal (in space, time, and state) operators. Data-driven
methods relying on flexible interpolants can significantly reduce structural
bias, but often at the expense of interpretability, generalizability, or
efficiency. Thus, understanding scaling laws and functional forms of
turbulence closures is still necessary to physically constrain data-driven
methods and decrease their computational expense. A promising avenue for
significant progress, lying at the intersection of theory and practice, is the
calculation of closure relations for passive scalars.
The present work aims to characterize the transport of passive scalars by flow
fields with known statistics. Our notion of transport is the ensemble mean
flux, which we express as an operator that acts on the ensemble mean tracers
in terms of the statistics of the flow field. We make arguments akin to those
in Kraichnan (1968) to motivate the operator approach, but our method of
calculation is fundamentally different. Given that the goal is to construct an
operator, rather than estimating a diffusivity tensor acting on the ensemble
mean gradients or deriving upper bounds, we take a field-theoretic perspective
(Hopf, 1952). Doing so allows us to derive a coupled set of partial
differential equations representing conditional mean tracers where the
conditional averages are with respect to different flow states. The turbulent
transport operator is then a Schur complement of the resulting linear system
with respect to statistical “perturbation” variables. If the flow statistics
are given by a continuous-time Markov process with a small finite state space,
the Schur complement becomes tractable to compute analytically. Obtaining a
closed form functional for turbulent fluxes allows for a direct statistical
simulation approach similar to those of Venturi et al. (2013); Allawala &
Marston (2016); Farrell & Ioannou (2019).
The paper is organized as follows. In section 2 we formulate the closure
problem and recast it as one of solving coupled partial differential
equations. In section 3 we show how to explicitly solve the closure problem
for a few flows with simple temporal structure but arbitrarily complex spatial
structure. In section 4 we outline the general theory. Appendices supplement
the body of the manuscript. Appendix A provides a direct field-theoretic
derivation of arguments in section 2 and Appendix B provides a heuristic
overview of obtaining continuous-time Markov processes and their statistics
from deterministic or stochastic dynamical systems.
## 2 Problem Formulation
We consider the advection and diffusion of an ensemble of passive scalars
$\theta_{\omega}$
$\displaystyle\partial_{t}\theta_{\omega}+\nabla\cdot\left(\bm{u}_{\omega}\theta_{\omega}-\kappa\nabla\theta_{\omega}\right)$
$\displaystyle=s(\bm{x})$ (1)
by a stochastic flow field $\bm{u}_{\omega}(\bm{x},t)$ where $\omega$ labels
the ensemble member. Here $s$ is a deterministic mean zero source term and
$\kappa$ is a diffusivity constant111For laboratory flows $\kappa$ would be
the molecular diffusivity; for larger-scale problems, we rely on the fact that
away from boundaries, the ensemble mean advective flux (but not necessarily
other statistics) may still be much larger than the diffusive flux and thus
the exact value of $\kappa$ will not matter.. Our target is to obtain a
meaningful equation for the ensemble mean,
$\displaystyle\partial_{t}\langle\theta\rangle+\nabla\cdot\left(\langle\bm{u}\theta\rangle-\kappa\nabla\langle\theta\rangle\right)$
$\displaystyle=s(\bm{x})$ (2)
which requires a computationally amenable expression for the mean advective
flux, $\langle\bm{u}\theta\rangle$, in terms of the statistics of flow field,
$\bm{u}(\bm{x},t)$, and the ensemble average of the tracer,
$\langle\theta\rangle$. Thus the closure problem is to find an operator
$\mathcal{O}$ that relates the ensemble mean, $\langle\theta\rangle$, to the
ensemble mean advective-flux, $\langle\bm{u}\theta\rangle$, i.e.,
$\displaystyle\mathcal{O}[\langle\theta\rangle]=\langle\bm{u}\theta\rangle.$
(3)
We show how to define (and solve for) the operator $\mathcal{O}$. The operator
will be linear with respect to its argument and depend on the statistics of
the flow field.
We assume all tracer ensemble members to have the same initial condition and
thus the ensemble average here is with respect to different flow realizations.
The only source of randomness comes from different flow realizations.
Throughout the manuscript we assume homogeneous Neumann boundary conditions
for the tracer and zero wall-normal flow for the velocity field when
boundaries are present. These restrictions, combined with the assumption that
the source term is mean zero, imply that the tracer average is conserved.
For the statistics of the flow field, we consider a continuous time Markov
process with $N$ states corresponding to steady flow fields
$\bm{u}_{n}(\bm{x})$ where $n$ is the associated state index. We start with
finitely many states for simplicity rather than necessity. Physically, we
think of these states as representing coherent structures in a turbulent
flow222This is viewed as a finite volume discretization in function space
where the states are the “cell averages” of a control volume in function
space.. A turbulent flow, by its very nature, chaotic and unpredictable over
long time horizons, is modelled as being semi-unpredictable through our
choice. Over short horizons, the probability of remaining in a given state is
large. On the medium term, the flow is limited to moving to a subset of likely
places in phase space. Over long time horizons, the most one can say about the
flow is related to the likelihood of being found in the appropriate subset of
phase space associated with the statistically steady state.
Thus we proceed by characterizing the probability, $\mathbb{P}$, of
transitioning from state $n$ to state $m$ by a transition matrix
$\mathscr{P}(\tau)$,
$\displaystyle\mathbb{P}\\{\bm{u}(\bm{x},t+\tau)=\bm{u}_{m}(\bm{x})|\bm{u}(\bm{x},t)=\bm{u}_{n}(\bm{x})\\}$
$\displaystyle=[\mathscr{P}(\tau)]_{mn}.$ (4)
The transition probability is defined through its relation to the generator
$\mathcal{Q}$,
$\displaystyle\mathscr{P}(\tau)\equiv\exp(\mathcal{Q}\tau)$ (5)
where $\exp(\mathcal{Q}\tau)$ is a matrix exponential. Each entry of
$\mathscr{P}(\tau)$ must be positive. Furthermore, the column sum of
$\mathscr{P}(\tau)$ for each $\tau$, sum to one since the total probability
must sum to one. Similarly $\mathcal{Q}$’s off-diagonal terms must be
positive333Indeed, to first order
$\exp(\mathcal{Q}dt)=\mathbb{I}+\mathcal{Q}dt$. The positivity requirement of
the transition probability $\mathscr{P}(dt)=\exp(\mathcal{Q}dt)$ necessitates
the positivity of $\mathcal{Q}$’s off-diagonal terms as well as the negativity
of the diagonal terms. and the column sum of $\mathcal{Q}$ must be zero.
We denote the probability of being found at state $m$ at time $t$ by
$\mathcal{P}_{m}(t)$,
$\displaystyle\mathcal{P}_{m}(t)=\mathbb{P}\\{\bm{u}(\bm{x},t)=\bm{u}_{m}(\bm{x})\\}.$
(6)
The evolution equation for $\mathcal{P}_{m}(t)$ is the master equation,
$\displaystyle\frac{d}{dt}\mathcal{P}_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n}.$ (7)
We assume that Equation 7 has a unique steady state and denote the components
of the steady state by $P_{m}$.
We have used several ”P”s at this stage and their relation are:
1. 1.
$\mathbb{P}$ denotes a probability.
2. 2.
$\mathscr{P}(\tau)$ denotes the transition probability matrix for a time
$\tau$ in the future.
3. 3.
$\mathcal{P}_{m}(t)$ denotes the probability of being in state $m$ at time
$t$. The algebraic relation
$\sum_{m}P_{m}(t+\tau)\bm{\hat{e}}_{m}=\mathscr{P}(\tau)\sum_{n}P_{n}(t)\bm{\hat{e}}_{n}$
holds.
4. 4.
$P_{m}$ is the statistically steady probability of being found in state $m$.
In the limit
$\lim_{t\rightarrow\infty}\mathcal{P}_{m}(t)=P_{m}.$
We exploit the given information about the flow field to infer the mean
statistics of the passive tracer $\theta_{\omega}$. We do so by conditionally
averaging the tracer field $\theta_{\omega}$ with respect to a given flow
state $\bm{u}_{n}$. More precisely, given the stochastic partial differential
equation,
$\displaystyle\mathbb{P}\\{\bm{u}_{\omega}(\bm{x},t+\tau)=\bm{u}_{m}(\bm{x})|\bm{u}_{\omega}(\bm{x},t)=\bm{u}_{n}(\bm{x})\\}$
$\displaystyle=[\exp(\mathcal{Q}\tau)]_{mn},$ (8)
$\displaystyle\partial_{t}\theta_{\omega}+\nabla\cdot\left(\bm{u}_{\omega}\theta_{\omega}-\kappa\nabla\theta_{\omega}\right)$
$\displaystyle=s(\bm{x}),$ (9)
we shall obtain equations for probability weighted conditional means of
$\theta_{\omega}$ defined by
$\displaystyle\Theta_{m}(\bm{x},t)$
$\displaystyle\equiv\langle\theta_{\omega}\rangle_{\bm{u}(\bm{x},t)=\bm{u}_{m}(\bm{x})}\mathcal{P}_{m}(t).$
(10)
We will show that the evolution equation for $\Theta_{m}$ is
$\displaystyle\frac{d}{dt}\mathcal{P}_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n},$ (11)
$\displaystyle\partial_{t}\Theta_{m}+\nabla\cdot\left(\bm{u}_{m}\Theta_{m}-\kappa\nabla\Theta_{m}\right)$
$\displaystyle=s(\bm{x})\mathcal{P}_{m}+\sum_{n}\mathcal{Q}_{mn}\Theta_{n}.$
(12)
The explicit dependence on the generator in Equation 12, as we shall see,
yields considerable information. We recover the equation for the tracer
ensemble mean, Equation 2, by summing Equation 12 over the index $m$, using
$\langle\theta\rangle=\sum_{m}\Theta_{m}$, $\sum_{m}\mathcal{Q}_{mn}=\bm{0}$,
and $\sum_{m}\mathcal{P}_{m}=1$,
$\displaystyle\partial_{t}\sum_{m}\Theta_{m}+\nabla\cdot\left(\sum_{m}\bm{u}_{m}\Theta_{m}-\kappa\nabla\sum_{m}\Theta_{m}\right)$
$\displaystyle=s(\bm{x})$ (13) $\displaystyle\Leftrightarrow$
$\displaystyle\partial_{t}\langle\theta\rangle+\nabla\cdot\left(\langle\bm{u}\theta\rangle-\kappa\nabla\langle\theta\rangle\right)$
$\displaystyle=s(\bm{x}).$ (14)
We comment that the presence of the generator when taking conditional averages
is similar to the entrainment hypothesis in the atmospheric literature. See,
for example, Tan et al. (2018) for its use in motivating a turbulence closure;
however, here we derive the result from the direct statistical representation
as opposed to hypothesize its presence from a dynamical argument.
Most of the terms in Equation 12 are obtained by applying a conditional
average to Equation 9, commuting with spatial derivatives when necessary, and
then multiplying through by $\mathcal{P}_{m}$; however, the primary difficulty
lies in proper treatment of the conditional average of the temporal
derivative. We circumvent the problem in a roundabout manner: The strategy is
to discretize the advection-diffusion equation, write down the resulting
master equation, compute moments of the probability distribution, and then
take limits to restore the continuum nature of the advection-diffusion
equation. For an alternative derivation where we forego discretization see
Appendix A and for a brief overview of the connection between the discrete,
continuous, and mixed master equation see Appendix B or, in a simpler context,
Hagan et al. (1989).
A generic discretization (in any number of dimensions) of Equation 9 is of the
form
$\displaystyle\frac{d}{dt}\theta^{i}+\sum_{jkc}A_{ijk}^{c}u^{k,c}_{\omega}\theta^{j}-\sum_{j}D_{ij}\theta^{j}$
$\displaystyle=s^{i}$ (15)
for some tensor $A_{ijk}^{c}$, representing advection, and matrix $D_{ij}$,
representing diffusion. Here each $i,j$ and $k$ corresponds to a spatial
location, and the index $c$ corresponds to a component of the velocity field
$\bm{u}$. The variable $\theta^{i}$ is the value of the tracer at grid
location $i$ and $u^{k,c}$ is the value of the $c$’th velocity component and
grid location $k$. The master equation for the joint probability density for
each component $\theta^{i}$ and Markov state $m$, $\rho_{m}(\bm{\theta})$,
where $\bm{\theta}=(\theta^{1},\theta^{2},....)$ and the $m$-index denotes a
particular Markov state, is a combination of the Liouville equation for 15 and
the transition rate equation for 8,
$\displaystyle\partial_{t}\rho_{m}$
$\displaystyle=\sum_{i}\frac{\partial}{\partial\theta^{i}}\left[\left(\sum_{jkc}A_{ijk}^{c}u^{k,c}_{m}\theta^{j}-\sum_{j}D_{ij}\theta^{j}-s^{i}\right)\rho_{m}\right]+\sum_{n}\mathcal{Q}_{mn}\rho_{n}.$
(16)
Define the following moments,
$\displaystyle\mathcal{P}_{m}$ $\displaystyle=\int d\bm{\theta}\rho_{m}\text{
and }\Theta_{m}^{j}=\int d\bm{\theta}\theta^{j}\rho_{m}.$ (17)
We obtain an equation for $\mathcal{P}_{m}$ by integrating 130 by
$d\bm{\theta}$ to yield
$\displaystyle\frac{d}{dt}\mathcal{P}_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n}$ (18)
as expected from Equation 7. The equation for $\Theta_{m}^{\ell}$ is obtained
by multiplying 130 by $\theta^{\ell}$ and then integrating with respect to
$d\bm{\theta}$,
$\displaystyle\frac{d}{dt}\Theta_{m}^{\ell}$ $\displaystyle=-\sum_{jkc}A_{\ell
jk}^{c}u^{k,c}_{m}\Theta^{j}_{m}+\sum_{j}D_{\ell
j}\Theta^{j}_{m}+s^{\ell}\mathcal{P}_{m}+\sum_{n}\mathcal{Q}_{mn}\Theta_{n}^{\ell},$
(19)
where we integrated by parts on the $\int
d\bm{\theta}\theta^{\ell}\partial_{\theta^{i}}\bullet$ term. Upon taking
limits of 19 we have the following equations
$\displaystyle\frac{d}{dt}\mathcal{P}_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n}$ (20)
$\displaystyle\partial_{t}\Theta_{m}+\nabla\cdot\left(\bm{u}_{m}\Theta_{m}-\kappa\nabla\Theta_{m}\right)$
$\displaystyle=s(\bm{x})\mathcal{P}_{m}+\sum_{n}\mathcal{Q}_{mn}\Theta_{n}.$
(21)
We compare Equation 21 to the direct application of the conditional average to
Equation 9 followed by multiplication with $\mathcal{P}_{m}$ to infer,
$\displaystyle\langle\partial_{t}\theta_{\omega}\rangle_{\bm{u}(\bm{x},t)=\bm{u}_{m}(\bm{x})}\mathcal{P}_{m}$
$\displaystyle=\partial_{t}\Theta_{m}-\sum_{n}\mathcal{Q}_{mn}\Theta_{n}.$
(22)
In summary, for an m-dimensional advection diffusion equation and $N$ Markov
states, Equations 11-12 are a set of $N$-coupled m-dimensional advection
diffusion equations with $N$ different steady velocities. When the statistics
of the flow field are described by $c$ continuous variables, the resulting
equation set becomes an $m+c$ dimensional system. Stated differently, if the
statistics of $\bm{u}_{\omega}$ are characterized by a transitions between a
continuum of states associated with a linear operator
$\mathcal{F}_{\bm{\omega}}$ with variables $\bm{\omega}\in\mathbb{R}^{c}$,
then
$\displaystyle\partial_{t}\mathcal{P}$
$\displaystyle=\mathcal{F}_{\bm{\omega}}[\mathcal{P}]$ (23)
$\displaystyle\partial_{t}\Theta+\nabla\cdot\left(\bm{u}\Theta-\kappa\nabla\Theta\right)$
$\displaystyle=s(\bm{x})\mathcal{P}+\mathcal{F}_{\bm{\omega}}[\Theta],$ (24)
where $\mathcal{P}=\mathcal{P}(\bm{\omega},t)$,
$\Theta=\Theta(\bm{x},\bm{\omega},t)$, and
$\bm{u}=\bm{u}(\bm{x},\bm{\omega})$. Equations 20-21 are thought of as finite
volume discretizations of flow statistics in Equations 23-24.
Our primary concern in this work is to use Equations 11-12 to calculate
meaningful expressions for $\langle\bm{u}\theta\rangle$; however, we shall
first take a broader view to understand the general structure of the turbulent
fluxes. The following argument is attributed to Weinstock (1969), but we use
our own notation and make additional simplifications.
Applying the Reynolds decomposition
$\displaystyle\theta_{\omega}=\langle\theta\rangle+\theta^{\prime}_{\omega}\text{
and }\bm{u}_{\omega}=\langle\bm{u}\rangle+\bm{u}^{\prime}_{\omega}$ (25)
yields
$\displaystyle\partial_{t}\langle\theta\rangle+\nabla\cdot\left(\langle\bm{u}\rangle\langle\theta\rangle+\langle\bm{u}^{\prime}\theta^{\prime}\rangle-\kappa\nabla\langle\theta\rangle\right)$
$\displaystyle=s$ (26)
$\displaystyle\partial_{t}\theta^{\prime}_{\omega}+\nabla\cdot\left(\langle\bm{u}\rangle\langle\theta\rangle-\langle\bm{u}^{\prime}\theta^{\prime}\rangle+\bm{u}_{\omega}\theta_{\omega}-\kappa\nabla\theta^{\prime}_{\omega}\right)$
$\displaystyle=0$ (27)
The perturbation equation is rewritten as
$\displaystyle\partial_{t}\theta^{\prime}_{\omega}+\nabla\cdot\left(\bm{u}_{\omega}^{\prime}\theta_{\omega}^{\prime}-\langle\bm{u}^{\prime}\theta^{\prime}\rangle+\langle\bm{u}\rangle\theta_{\omega}^{\prime}-\kappa\nabla\theta^{\prime}_{\omega}\right)$
$\displaystyle=-\nabla\cdot\left(\bm{u}_{\omega}^{\prime}\langle\theta\rangle\right)$
(28)
This is an infinite system (or finite depending on the number of ensemble
members) of coupled pde’s between the different ensemble members. The ensemble
members are coupled due to the presence of the turbulent flux,
$\langle\bm{u}^{\prime}\theta^{\prime}\rangle$. The key observation is to
notice the terms on the left hand side involve the perturbation variables and
not the ensemble mean of the gradients. Assuming it is possible to find the
inverse, the Green’s function for the large linear system is used to yield
$\displaystyle\theta^{\prime}_{\omega}(\bm{x},t)$ $\displaystyle=-\int
d\bm{x}^{\prime}dt^{\prime}d\mu_{\alpha}\mathcal{G}_{\alpha\omega}(\bm{x},t|\bm{x}^{\prime},t^{\prime})\nabla\cdot\left(\bm{u}_{\alpha}^{\prime}\langle\theta\rangle\right)$
(29)
where we also have to integrate with respect to the measure defining the
different ensemble members through $d\mu_{\alpha}$. Notation wise this would
means $\langle\theta\rangle=\int d\mu_{\omega}\theta_{\omega}$. We use this
expression to rewrite the turbulent flux as
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle$
$\displaystyle=-\int
d\bm{x}^{\prime}dt^{\prime}d\mu_{\omega}d\mu_{\alpha}\bm{u}^{\prime}_{\omega}(\bm{x},t)\mathcal{G}_{\alpha\omega}(\bm{x},t|\bm{x}^{\prime},t^{\prime})\left[\nabla\cdot\left(\bm{u}_{\alpha}^{\prime}(\bm{x}^{\prime},t^{\prime})\langle\theta\rangle(\bm{x}^{\prime},t^{\prime})\right)\right]$
(30)
We make two simplifications for illustrative purposes.
1. 1.
All ensemble averages are independent of time.
2. 2.
The flow is incompressible, i.e., $\nabla\cdot\bm{u}=0$.
Equation 30 becomes
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle$
$\displaystyle=-\int
d\bm{x}^{\prime}dt^{\prime}d\mu_{\omega}d\mu_{\alpha}\left[\bm{u}^{\prime}_{\omega}(\bm{x},t)\mathcal{G}_{\alpha\omega}(\bm{x},t|\bm{x}^{\prime},t^{\prime})\bm{u}_{\alpha}^{\prime}(\bm{x}^{\prime},t^{\prime})\right]\cdot\nabla\langle\theta\rangle(\bm{x}^{\prime})$
(31)
We perform the $t^{\prime},\alpha,\omega$ integrals first to define the
turbulent-diffusivity tensor kernel as
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle$
$\displaystyle=-\int d\bm{x}^{\prime}\underbrace{\int
dt^{\prime}d\mu_{\omega}d\mu_{\alpha}\left[\bm{u}^{\prime}_{\omega}(\bm{x},t)\otimes\bm{u}_{\alpha}^{\prime}(\bm{x}^{\prime},t^{\prime})\mathcal{G}_{\alpha\omega}(\bm{x},t|\bm{x}^{\prime},t^{\prime})\right]}_{\bm{\mathcal{K}}(\bm{x}|\bm{x}^{\prime})}\cdot\nabla\langle\theta\rangle(\bm{x}^{\prime})$
(32) $\displaystyle=-\int
d\bm{x}^{\prime}\bm{\mathcal{K}}(\bm{x}|\bm{x}^{\prime})\cdot\nabla\langle\theta\rangle(\bm{x}^{\prime})$
(33)
The independence of $\bm{\mathcal{K}}$ with respect to $t$ follows from the
time-independence of $\langle\bm{u}^{\prime}\theta^{\prime}\rangle$ and
$\langle\theta\rangle$. In total we see
$\displaystyle\langle\bm{u}\theta\rangle=\langle\bm{u}\rangle\langle\theta\rangle-\int
d\bm{x}^{\prime}\bm{\mathcal{K}}(\bm{x}|\bm{x}^{\prime})\cdot\nabla\langle\theta\rangle(\bm{x}^{\prime}).$
(34)
An insight from Equation 34 is the dependence of turbulent fluxes
$\langle\bm{u}^{\prime}\theta^{\prime}\rangle$ at location $\bm{x}$ as a
weighted sum of gradients of the mean variable $\langle\theta\rangle$ at
locations $\bm{x}^{\prime}$. The operator is linear and amenable to
computation, even in turbulent flows, Bhamidipati et al. (2020).
We consider the spectrum for the turbulent diffusivity operator $\int
d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet$ as a
characterization of turbulent-mixing by the flow field $\bm{u}(\bm{x},t)$. We
comment that the operator $\int
d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet$ is a mapping from
vector fields to vector fields whereas the kernel
$\mathcal{K}(\bm{x}|\bm{x}^{\prime})$ is a mapping from two positions to a
tensor.
For example, consider a one-dimensional problem in a periodic domain
$x\in[0,2\pi)$. If $\mathcal{K}(x|x^{\prime})=\kappa_{e}\delta(x-x^{\prime})$
for some positive constant $\kappa_{e}$, the spectrum of the operator is flat
and turbulent-mixing remains the same on every length scale. If
$\mathcal{K}(x|x^{\prime})=-\kappa_{e}\partial_{xx}^{2}\delta(x-x^{\prime})$
then the rate of mixing increases with increasing wavenumber, one gets
hyperdiffusion. And lastly, if $\int
dx^{\prime}\mathcal{K}(x|x^{\prime})\bullet=(\kappa_{e}-\partial_{xx})^{-1}$,
then the kernel is nonlocal and the rate of mixing decreases at smaller length
scales.
In the following section we calculate $\int
d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet$ directly from the
conditional equations and then discuss the general structure in Section 4.
## 3 Examples
We now go through three examples to understand the implications of Equations
11-12. The three examples follow sequentially in increasing complexity. The
first example considers transitions between two Markov states. There we
introduce a generalizable approach to computing the turbulent diffusivity. We
then apply the same approach to a slightly more complex problem, transitions
between three Markov states. And finally, we conclude with a calculation
involving transitions between four Markov states where we delve into details
with cellular flow states. The generator for the two and three state systems
are derived from a finite volume discretization of an Ornstein-Uhlenbeck
process, see Appendix B.2.
Although we present the general form of the kernels here, in Sections 3.1 and
3.2 one can consider the case where the advection-diffusion equation is one-
dimensional, periodic, and with a flow field that is constant in space. In
this case one can decompose the advection-diffusion equation into Fourier
modes that are decoupled from one another and allows for alternative
computations of the same result using more standard techniques.
### 3.1 Two State
To start we consider the simplest time-dependent mean-zero incompressible
stochastic flow field: the transition between two incompressible states
$\bm{u}_{1}(\bm{x})=\bm{u}(\bm{x})$ and $\bm{u}_{2}(\bm{x})=-\bm{u}(\bm{x})$
where each state is equally likely in the statistically steady state. For this
we use the generator
$\displaystyle\mathcal{Q}$ $\displaystyle=\gamma\begin{bmatrix}-1&1\\\
1&-1\end{bmatrix}$ (35)
where $\gamma>0$. The eigenvector and eigenvalues of the generator are
$\displaystyle\bm{v}^{1}=\begin{bmatrix}1/2\\\ 1/2\end{bmatrix}\text{ and
}\bm{v}^{2}=\begin{bmatrix}1/2\\\ -1/2\end{bmatrix}$ (36)
with respective eigenvalues $\lambda^{1}=0$ and $\lambda^{2}=-2\gamma$. The
first eigenvector is the steady state probability, which we see has
probability $1/2$ for each state, as expected.
In what follows we denote the Laplacian by $\Delta$. Equations 11-12 for the
two-state system are
$\displaystyle\partial_{t}\mathcal{P}_{1}$
$\displaystyle=-\gamma\mathcal{P}_{1}+\gamma\mathcal{P}_{2}$ (37)
$\displaystyle\partial_{t}\mathcal{P}_{2}$
$\displaystyle=-\gamma\mathcal{P}_{2}+\gamma\mathcal{P}_{1}$ (38)
$\displaystyle\partial_{t}\Theta_{1}+\nabla\cdot\left(\bm{u}\Theta_{1}\right)$
$\displaystyle=\kappa\Delta\Theta_{1}+s(\bm{x})\mathcal{P}_{1}-\gamma\Theta_{1}+\gamma\Theta_{2}$
(39)
$\displaystyle\partial_{t}\Theta_{2}-\nabla\cdot\left(\bm{u}\Theta_{2}\right)$
$\displaystyle=\kappa\Delta\Theta_{2}+s(\bm{x})\mathcal{P}_{2}-\gamma\Theta_{2}+\gamma\Theta_{1}.$
(40)
We assume a statistically steady state so that the temporal derivatives
vanish. Consequently, $\mathcal{P}_{1}=\mathcal{P}_{2}=1/2$ and the equations
reduce to
$\displaystyle\nabla\cdot\left(\bm{u}\Theta_{1}\right)$
$\displaystyle=\kappa\Delta\Theta_{1}+s(\bm{x})/2-\gamma\Theta_{1}+\gamma\Theta_{2}$
(41) $\displaystyle-\nabla\cdot\left(\bm{u}\Theta_{2}\right)$
$\displaystyle=\kappa\Delta\Theta_{2}+s(\bm{x})/2-\gamma\Theta_{2}+\gamma\Theta_{1}$
(42)
We rewrite the above equation set into a mean and perturbation by changing
basis to $\varphi_{i}$ variables according to
$\displaystyle\begin{bmatrix}1/2&1/2\\\
1/2&-1/2\end{bmatrix}\begin{bmatrix}\varphi_{1}\\\
\varphi_{2}\end{bmatrix}=\begin{bmatrix}\Theta_{1}\\\
\Theta_{2}\end{bmatrix}\Leftrightarrow\begin{bmatrix}\varphi_{1}\\\
\varphi_{2}\end{bmatrix}=\begin{bmatrix}1&1\\\
1&-1\end{bmatrix}\begin{bmatrix}\Theta_{1}\\\ \Theta_{2}\end{bmatrix}$ (43)
yielding,
$\displaystyle\nabla\cdot\left(\bm{u}\varphi_{2}\right)$
$\displaystyle=\kappa\Delta\varphi_{1}+s(\bm{x})$ (44)
$\displaystyle\bm{u}\cdot\nabla\varphi_{1}$
$\displaystyle=\kappa\Delta\varphi_{2}-2\gamma\varphi_{2}.$ (45)
We used the incompressibility condition to yield the representation in
Equation 45. Our choice of basis is no accident, we used the eigenvectors of
the generator, $\mathcal{Q}$, to define the transformation in Equation 43. We
recognize the variable $\varphi_{1}$ as the ensemble mean
$\varphi_{1}=\langle\theta\rangle$ and $\varphi_{2}$ as a ”perturbation”
variable. The turbulent flux term is
$\langle\bm{u}^{\prime}\theta^{\prime}\rangle=\bm{u}\varphi_{2}$. We eliminate
the dependence on the perturbation variable by inverting the Helmholtz
operator in Equation 45. In total we have the following representation of the
mean equation
$\displaystyle\nabla\cdot\left(\bm{u}(\kappa\Delta-2\gamma)^{-1}[\bm{u}\cdot\nabla\langle\theta\rangle]\right)$
$\displaystyle=\kappa\Delta\langle\theta\rangle+s(\bm{x})$ (46)
from whence we extract the turbulent diffusivity operator
$\displaystyle\int
d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet=\bm{u}(2\gamma-\kappa\Delta)^{-1}\bm{u}.$
(47)
We point out a few salient features of Equation 47. The inverse Helmholtz
operator, $\left(2\gamma-\kappa\Delta\right)^{-1}$, damps high spatial
frequency components of ensemble mean gradients. Thus, the operator’s
eigenvalues decrease as one examines increasingly fine-scale structure.
Intuitively, as one examines a small-scale structure, the presence of
diffusivity leads to lower turbulent fluxes, expressing the notion that it is
difficult to transport something that immediately diffuses. The second
observation pertains to the presence of the eigenvalue of the generator in the
Helmholtz operator. If the flow field changes rapidly, transitioning between
the disparate states, then $\gamma$ is large, and one can expect the
turbulent-diffusivity to be local. In other words, the flow does not stay
sufficiently long time near a coherent structure.
In this example, the non-locality of the turbulent diffusivity is enabled by
the presence of the regular diffusion operator. However, in the following
example, we show that this need not be the case.
### 3.2 Three State
For this example we consider a Markov process that transitions between three
incompressible states $\bm{u}_{1}(\bm{x})=\bm{u}(\bm{x})$,
$\bm{u}_{2}(\bm{x})=0$, and $\bm{u}_{3}(\bm{x})=-\bm{u}(\bm{x})$. For this let
the generator be
$\displaystyle\mathcal{Q}$ $\displaystyle=\gamma\begin{bmatrix}-1&1/2&0\\\
1&-1&1\\\ 0&1/2&-1\end{bmatrix}$ (48)
where $\gamma>0$. The eigenvectors of the generator are
$\displaystyle\bm{v}^{1}=\begin{bmatrix}1/4\\\ 1/2\\\ 1/4\end{bmatrix}\text{ ,
}\bm{v}^{2}=\begin{bmatrix}1/2\\\ 0\\\ -1/2\end{bmatrix}\text{ and
}\bm{v}^{3}=\begin{bmatrix}1/4\\\ -1/2\\\ 1/4\end{bmatrix}$ (49)
with respective eigenvalues $\lambda^{1}=0$, $\lambda^{2}=-\gamma$, and
$\lambda^{3}=-2\gamma$.
The statistically steady three-state manifestation of Equations 11-12 are
$\displaystyle\nabla\cdot\left(\bm{u}\Theta_{1}\right)$
$\displaystyle=\kappa\Delta\Theta_{1}+s(\bm{x})/4-\gamma\Theta_{1}+\gamma\Theta_{2}/2$
(50) $\displaystyle 0$
$\displaystyle=\kappa\Delta\Theta_{2}+s(\bm{x})/2-\gamma\Theta_{2}+\gamma\Theta_{1}+\gamma\Theta_{3}$
(51) $\displaystyle-\nabla\cdot\left(\bm{u}\Theta_{3}\right)$
$\displaystyle=\kappa\Delta\Theta_{3}+s(\bm{x})/4-\gamma\Theta_{3}+\gamma\Theta_{2}/2$
(52)
Similar to before we define a transformation using the eigenvectors of the
generator $\mathcal{Q}$,
$\displaystyle\begin{bmatrix}1/4&1/2&1/4\\\ 1/2&0&-1/2\\\
1/4&-1/2&1/4\end{bmatrix}\begin{bmatrix}\varphi_{1}\\\ \varphi_{2}\\\
\varphi_{3}\end{bmatrix}=\begin{bmatrix}\Theta_{1}\\\ \Theta_{2}\\\
\Theta_{3}\end{bmatrix}\Leftrightarrow\begin{bmatrix}\varphi_{1}\\\
\varphi_{2}\\\ \varphi_{3}\end{bmatrix}=\begin{bmatrix}1&1&1\\\ 1&0&-1\\\
1&-1&1\end{bmatrix}\begin{bmatrix}\Theta_{1}\\\ \Theta_{2}\\\
\Theta_{3}\end{bmatrix}$ (53)
The resulting equations are
$\displaystyle\nabla\cdot\left(\bm{u}\varphi_{2}\right)$
$\displaystyle=\kappa\Delta\varphi_{1}+s(\bm{x})$ (54)
$\displaystyle\frac{1}{2}\bm{u}\cdot\nabla\varphi_{1}+\frac{1}{2}\bm{u}\cdot\nabla\varphi_{3}$
$\displaystyle=\kappa\Delta\varphi_{2}-\gamma\varphi_{2}$ (55)
$\displaystyle\bm{u}\cdot\nabla\varphi_{2}$
$\displaystyle=\kappa\Delta\varphi_{3}-2\gamma\varphi_{3}.$ (56)
We again comment that $\varphi_{1}=\langle\theta\rangle$ and that
$\varphi_{2}$ and $\varphi_{3}$ are thought of as perturbation variables.
Furthermore the turbulent flux is
$\langle\bm{u}^{\prime}\theta^{\prime}\rangle=\bm{u}\varphi_{2}$. We eliminate
dependence on the perturbation variables $\varphi_{2}$ and $\varphi_{3}$ by
first solving for $\varphi_{3}$ in terms of $\varphi_{2}$,
$\displaystyle\varphi_{3}$
$\displaystyle=\left(\kappa\Delta-2\gamma\right)^{-1}\bm{u}\cdot\nabla\varphi_{2}$
(57)
and then solving for $\varphi_{2}$ in terms of $\varphi_{1}$,
$\displaystyle\varphi_{2}$
$\displaystyle=\left(\kappa\Delta-\gamma-\frac{1}{2}\bm{u}\cdot\nabla\left(\kappa\Delta-2\gamma\right)^{-1}\bm{u}\cdot\nabla\right)^{-1}\frac{1}{2}\bm{u}\cdot\nabla\varphi_{1}$
(58)
And finally we write our equation for the ensemble mean as
$\displaystyle\nabla\cdot\left(\bm{u}\left(\kappa\Delta-\gamma-\frac{1}{2}\bm{u}\cdot\nabla\left(\kappa\Delta-2\gamma\right)^{-1}\bm{u}\cdot\nabla\right)^{-1}\frac{1}{2}\bm{u}\cdot\nabla\langle\theta\rangle\right)$
$\displaystyle=\kappa\Delta\langle\theta\rangle+s(\bm{x})$ (59)
from whence we extract the turbulent diffusivity operator
$\displaystyle\int d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet$
$\displaystyle=\bm{u}\left(\gamma-\kappa\Delta+\frac{1}{2}\bm{u}\cdot\nabla\left(\kappa\Delta-2\gamma\right)^{-1}\bm{u}\cdot\nabla\right)^{-1}\frac{1}{2}\bm{u}.$
(60)
We see that, unlike the two-state system, the $\kappa\rightarrow 0$ limit
retains a nonlocal feature since
$\displaystyle\int
d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet\rightarrow\bm{u}\left(\gamma-\frac{1}{4\gamma}(\bm{u}\cdot\nabla)(\bm{u}\cdot\nabla)\right)^{-1}\frac{1}{2}\bm{u}$
(61)
and the operator $(\bm{u}\cdot\nabla)(\bm{u}\cdot\nabla)$ can have a
significant spatial structure. Similar to before, larger transition rates
imply increasing local structures. In the $\kappa\rightarrow 0$ limit the non-
dimensional parameter of interest is the characteristic timescale of the
steady flow field, $LU^{-1}$, as it compares to the characteristic timescale
for transitioning, $\gamma^{-1}$.
For the last case, we emphasize the algebraic structure of Equations 11-12 by
working in detail through a four-state system.
### 3.3 Four State
Here we will consider a two-dimensional velocity field motivated by Flierl &
McGillicuddy (2002). The flow is two-dimensional, periodic with
$x\in[0,2\pi)$, and wall-bounded with $z\in[-1,1]$. Our Markov velocity states
are defined through the stream-functions
$\displaystyle\psi_{1}$
$\displaystyle=\sin(x)\cos\left(\frac{\pi}{2}z\right)\text{,
}\psi_{2}=\cos(x)\cos\left(\frac{\pi}{2}z\right)\text{, }$ (62)
$\displaystyle\psi_{3}$
$\displaystyle=-\sin(x)\cos\left(\frac{\pi}{2}z\right)\text{, and
}\psi_{4}=-\cos(x)\cos\left(\frac{\pi}{2}z\right),$ (63)
and the corresponding velocity states are
$\bm{u}_{m}=(\partial_{z}\psi_{m},-\partial_{x}\psi_{m})$. The states are
simply $\pi/2$ phase shifts of one another along the periodic $x$ direction
and are thought of as describing a cellular flow that randomly propagates
through a channel. The generator is given by
$\displaystyle\mathcal{Q}$ $\displaystyle=\gamma\begin{bmatrix}-1&1/2&0&1/2\\\
1/2&-1&1/2&0\\\ 0&1/2&-1&1/2\\\
1/2&0&1/2&-1\end{bmatrix}+\omega\begin{bmatrix}0&1/2&0&-1/2\\\ -1/2&0&1/2&0\\\
0&-1/2&0&1/2\\\ 1/2&0&-1/2&0\end{bmatrix}.$ (64)
The parameter $\gamma$ is, as in the previous examples, related to the amount
of time spent in a given state. When $\omega=0$ the cells move with equal
likelihood to the left or the right, whereas $\omega\neq 0$ biases the cells
to move in a particular direction. We require $|\omega|\leq\gamma$ for a
probabilistic interpretation of results.
We assume that the system is in a statistically steady state. Thus we start by
observing that the normalized right eigenvectors of $\mathcal{Q}$ are
$\displaystyle\bm{v}_{1}$ $\displaystyle=\begin{bmatrix}1/4\\\ 1/4\\\ 1/4\\\
1/4\end{bmatrix}\text{ , }\bm{v}_{2}=\begin{bmatrix}1/2\\\ \iota/2\\\ -1/2\\\
-\iota/2\end{bmatrix}\text{ , }\bm{v}_{3}=\begin{bmatrix}1/2\\\ -\iota/2\\\
-1/2\\\ \iota/2\end{bmatrix}\text{ , and }\bm{v}_{4}=\begin{bmatrix}1/4\\\
-1/4\\\ 1/4\\\ -1/4\end{bmatrix}$ (65)
with corresponding eigenvectors
$\lambda_{1}=0,\lambda_{2}=-\gamma-\omega\iota,\lambda_{3}=-\gamma+\omega\iota,\lambda_{4}=-2\gamma$,
respectively. Inspection of $\bm{v}^{1}$, the vector associated with the
statistically steady state, reveals that each state is equally likely in the
steady state.
The steady-state equations are
$\displaystyle\bm{u}_{1}\cdot\nabla\Theta_{1}$
$\displaystyle=\kappa\Delta\Theta_{1}+s/4-\gamma\Theta_{1}+\frac{\gamma+\omega}{2}\Theta_{2}+\frac{\gamma-\omega}{2}\Theta_{4}$
(66) $\displaystyle\bm{u}_{2}\cdot\nabla\Theta_{2}$
$\displaystyle=\kappa\Delta\Theta_{2}+s/4-\gamma\Theta_{2}+\frac{\gamma+\omega}{2}\Theta_{3}+\frac{\gamma-\omega}{2}\Theta_{1}$
(67) $\displaystyle\bm{u}_{3}\cdot\nabla\Theta_{3}$
$\displaystyle=\kappa\Delta\Theta_{3}+s/4-\gamma\Theta_{3}+\frac{\gamma+\omega}{2}\Theta_{4}+\frac{\gamma-\omega}{2}\Theta_{2}$
(68) $\displaystyle\bm{u}_{4}\cdot\nabla\Theta_{4}$
$\displaystyle=\kappa\Delta\Theta_{4}+s/4-\gamma\Theta_{4}+\frac{\gamma+\omega}{2}\Theta_{1}+\frac{\gamma-\omega}{2}\Theta_{3}.$
(69)
As was done in prior sections we will change basis by utilizing the
eigenvectors of $\mathcal{Q}$. We avoid the use of imaginary eigenvectors by
instead using linear combinations $(v^{2}+v^{3})/2$ and
$(v^{2}-v^{3})/(2\iota)$ at the cost of a non-diagonal generator in the
resulting basis. Explicitly we use the similarity transformation defined by
the matrices
$\displaystyle S=\begin{bmatrix}1/4&1/2&0&1/4\\\ 1/4&0&1/2&-1/4\\\
1/4&-1/2&0&1/4\\\ 1/4&0&-1/2&-1/4\end{bmatrix}\text{ and
}S^{-1}=\begin{bmatrix}1.0&1.0&1.0&1.0\\\ 1.0&0.0&-1.0&0.0\\\
0.0&1.0&0.0&-1.0\\\ 1.0&-1.0&1.0&-1.0\end{bmatrix}.$ (70)
Thus we make the change of variables
$\displaystyle\begin{bmatrix}1.0&1.0&1.0&1.0\\\ 1.0&0.0&-1.0&0.0\\\
0.0&1.0&0.0&-1.0\\\ 1.0&-1.0&1.0&-1.0\end{bmatrix}\begin{bmatrix}\Theta_{1}\\\
\Theta_{2}\\\ \Theta_{3}\\\
\Theta_{4}\end{bmatrix}=\begin{bmatrix}\varphi_{1}\\\ \varphi_{2}\\\
\varphi_{3}\\\ \varphi_{4}\end{bmatrix}.$ (71)
We observe $\varphi_{1}=\langle\theta\rangle$. With this change of variable
and using $\bm{u}_{1}=-\bm{u}_{3}$ with $\bm{u}_{2}=-\bm{u}_{4}$, the
resulting system of equations is written in block operator form as,
$\displaystyle\begin{bmatrix}-\kappa\Delta&\nabla\cdot\left(\bm{u}_{1}\bullet\right)&\nabla\cdot\left(\bm{u}_{2}\bullet\right)&0\\\
0&\gamma-\kappa\Delta&-\omega&\frac{1}{2}\bm{u}_{1}\cdot\nabla\\\
0&\omega&\gamma-\kappa\Delta&-\frac{1}{2}\bm{u}_{2}\cdot\nabla\\\
0&\bm{u}_{1}\cdot\nabla&-\bm{u}_{2}\cdot\nabla&2\gamma-\kappa\Delta\end{bmatrix}\begin{bmatrix}\varphi_{1}\\\
\varphi_{2}\\\ \varphi_{3}\\\ \varphi_{4}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}s(x,z)\\\
-\frac{1}{2}\bm{u}_{1}\cdot\nabla\varphi_{1}\\\
-\frac{1}{2}\bm{u}_{2}\cdot\nabla\varphi_{1}\\\ 0\end{bmatrix}$ (72)
where the dependence of the perturbation variables on $\varphi_{1}$ is
included as a source rather than as a part of the block operator. The inverse
of the $3\times 3$ lower right submatrix matrix of Equation 72,
$\displaystyle\mathcal{G}=\begin{bmatrix}\gamma-\kappa\Delta&-\omega&\frac{1}{2}\bm{u}_{1}\cdot\nabla\\\
\omega&\gamma-\kappa\Delta&-\frac{1}{2}\bm{u}_{2}\cdot\nabla\\\
\bm{u}_{1}\cdot\nabla&-\bm{u}_{2}\cdot\nabla&2\gamma-\kappa\Delta\end{bmatrix}^{-1}$
(73)
is the Green’s function associated with the perturbation variables
$\varphi_{2}$, $\varphi_{3}$, $\varphi_{4}$. The perturbation Green’s function
is used to represent the turbulent diffusivity operator as
$\displaystyle\int d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet$
$\displaystyle=\begin{bmatrix}\bm{u}_{1}&\bm{u}_{2}&0\end{bmatrix}\begin{bmatrix}\gamma-\kappa\Delta&-\omega&\frac{1}{2}\bm{u}_{1}\cdot\nabla\\\
\omega&\gamma-\kappa\Delta&-\frac{1}{2}\bm{u}_{2}\cdot\nabla\\\
\bm{u}_{1}\cdot\nabla&-\bm{u}_{2}\cdot\nabla&2\gamma-\kappa\Delta\end{bmatrix}^{-1}\begin{bmatrix}\bm{u}_{1}/2\\\
\bm{u}_{2}/2\\\ 0\\\ \end{bmatrix}$ (74)
Written this way, we emphasize the Schur-complement structure of the ensemble
mean equations when eliminating dependence on the perturbation variables.
To better understand the turbulent diffusivity operator for this example, we
consider two local approximations. Assuming a scale separation we approximate
$\displaystyle\int\mathcal{K}(\bm{x}|\bm{x}^{\prime})\cdot\nabla\langle\theta\rangle(\bm{x}^{\prime})d\bm{x}^{\prime}\approx\int\mathcal{K}(\bm{x}|\bm{x}^{\prime})d\bm{x}^{\prime}\cdot\nabla\langle\theta\rangle(\bm{x})=\bm{D}_{1}\cdot\nabla\langle\theta\rangle(\bm{x})$
(75)
where we integrate the turbulent diffusivity kernel with respect to
$\bm{x}^{\prime}$
$\displaystyle\bm{D}_{1}$
$\displaystyle\equiv\int\mathcal{K}(\bm{x}|\bm{x}^{\prime})d\bm{x}^{\prime}.$
(76)
We comment that, upon discretization, the linear operator $\int
d\bm{x}^{\prime}\mathcal{K}(\bm{x}|\bm{x}^{\prime})\bullet$ is represented as
a matrix. Performing the integration with respect to $d\bm{x}^{\prime}$
amounts to performing a row sum on the matrix.
A second local estimate of the turbulent diffusivity is obtained by neglecting
the dissipation terms and perturbation gradients, i.e.
$\displaystyle\begin{bmatrix}\gamma-\kappa\Delta&-\omega&\frac{1}{2}\bm{u}_{1}\cdot\nabla\\\
\omega&\gamma-\kappa\Delta&-\frac{1}{2}\bm{u}_{2}\cdot\nabla\\\
\bm{u}_{1}\cdot\nabla&-\bm{u}_{2}\cdot\nabla&2\gamma-\kappa\Delta\end{bmatrix}^{-1}\approx\begin{bmatrix}\gamma&-\omega&0\\\
\omega&\gamma&0\\\ 0&0&2\gamma\end{bmatrix}^{-1}.$ (77)
Thus the local eddy diffusivity is
$\displaystyle\bm{D}_{2}$
$\displaystyle=\begin{bmatrix}\bm{u}_{1}&\bm{u}_{2}&0\end{bmatrix}\begin{bmatrix}\gamma&-\omega&0\\\
\omega&\gamma&0\\\ 0&0&2\gamma\end{bmatrix}^{-1}\begin{bmatrix}\bm{u}_{1}/2\\\
\bm{u}_{2}/2\\\ 0\\\ \end{bmatrix}$ (78)
$\displaystyle=\frac{1}{2(\gamma^{2}+\omega^{2})}\left[\gamma\left(\bm{u}_{1}\otimes\bm{u}_{1}+\bm{u}_{2}\otimes\bm{u}_{2}\right)+\omega\left(\bm{u}_{2}\otimes\bm{u}_{1}-\bm{u}_{1}\otimes\bm{u}_{2}\right)\right]$
(79)
where we have interpreted products of vector fields as outer products. Using
the velocity fields,
$\displaystyle\bm{u}_{1}$
$\displaystyle=-\frac{\pi}{2}\sin(x)\sin\left(\frac{\pi}{2}z\right)\hat{x}-\cos(x)\cos\left(\frac{\pi}{2}z\right)\hat{z}$
(80) $\displaystyle\bm{u}_{2}$
$\displaystyle=-\frac{\pi}{2}\cos(x)\sin\left(\frac{\pi}{2}z\right)\hat{x}+\sin(x)\cos\left(\frac{\pi}{2}z\right)\hat{z},$
(81)
we compute each outer product to obtain the components of the local turbulent
diffusivity,
$\displaystyle[\bm{D}_{2}]_{\hat{x}\otimes\hat{x}}$
$\displaystyle=\frac{\gamma\pi^{2}}{8(\gamma^{2}+\omega^{2})}\sin^{2}\left(\frac{\pi}{2}z\right)$
(82) $\displaystyle[\bm{D}_{2}]_{\hat{x}\otimes\hat{z}}$
$\displaystyle=-[\bm{D}_{2}]_{\hat{z}\otimes\hat{x}}=\frac{\omega\pi}{4(\gamma^{2}+\omega^{2})}\sin\left(\frac{\pi}{2}z\right)\cos\left(\frac{\pi}{2}z\right)$
(83) $\displaystyle[\bm{D}_{2}]_{\hat{z}\otimes\hat{z}}$
$\displaystyle=\frac{\gamma}{2(\gamma^{2}+\omega^{2})}\cos^{2}\left(\frac{\pi}{2}z\right)$
(84)
In Section 4 we show, in general, the equivalence of neglecting dissipation
terms and perturbation gradients, as was done in Equation 77, and estimating
the diffusivity by computing the integrated auto-correlation of the
statistically steady velocity field,
$\displaystyle\bm{D}_{2}=\int_{0}^{\infty}\langle\bm{u}(\bm{x},t+\tau)\otimes\bm{u}(\bm{x},t)\rangle
d\tau.$ (85)
Figure 1: A comparison of two local diffusivity estimates. The analytic
diffusivity estimate is in red and numerically computed diffusivity estimate
uses blue dots. The z-axis is depth and the x-axis is the diffusivity
amplitude. Here $\gamma=\omega=100$ and $\kappa=1/100$.
Figure 2: A comparison of two local diffusivity estimates. The analytic
diffusivity estimate is in red and numerically computed diffusivity estimate
uses blue dots. The z-axis is depth and the x-axis is the diffusivity
amplitude. Here $\gamma=\omega=1$ and $\kappa=1$.
Figure 3: A representation of the turbulent diffusivity operator corresponding
to the case in Figure 2. If the diffusivity was indeed local then each of the
form matrices would only have a diagonal component. The x-z coordinate axis
have been collapsed to a single index, hence the banded structure of the
output. Thus the rows are the ”output” axis corresponding to the $(x,z)$ of
$\int dx^{\prime}dz^{\prime}K(x,z|x^{\prime},z^{\prime})\bullet$ and the
columns are the $(x^{\prime},z^{\prime})$ corresponding to input. The ordering
is chosen such that each block-diagonal structure corresponds to a fixed
$x_{i},x_{j}^{\prime}$ grid location. The similarity between different block
rows follow from the periodicity of the $x$ coordinate. Here $\gamma=\omega=1$
and $\kappa=1$.
We illustrate the local diffusivity calculation in two scenarios:
1. 1.
$\gamma=\omega=100$ and $\kappa=1/100$
2. 2.
$\gamma=\omega=\kappa=1$
In the first case, we expect the local diffusivity estimate to work well, and
the two diffusivity estimates to correspond to one another. In the latter
case, there is no scale separation between transition rates, diffusive
timescales, and advective timescales; we expect nonlocal effects to play a
significant role. We discretize all operators using a collocation method as
described by Trefethen (2000) to explore the nonlocality of the full turbulent
flux operator, Equation 74. We use 65 Chebyshev modes in the wall-bounded
direction and 8 Fourier modes in the periodic direction to approximate each
operator, leading to a $1560\times 1560$ sized matrix representation of
Equation 73. There are no significant changes upon halving or doubling the
resolution in each direction. Furthermore, given the independence of the
periodic direction in $\bm{D}_{2}$, we use the local estimate
$\displaystyle\overline{\bm{D}}_{1}(z)$
$\displaystyle=\frac{1}{2\pi}\int\mathcal{K}(x,z|x^{\prime},z^{\prime})dx^{\prime}dz^{\prime}dx$
(86)
for comparison between $\bm{D}_{1}$ and $\bm{D}_{2}$.
For the first case, we compute the local diffusivities and display their
result in Figure 1. We see that there is excellent agreement between the two
approaches, except near the boundary in the $K^{xz}$ component of the
diffusivity tensor. Here we note the influence of the homogenous Neumann
boundary conditions in the diffusivity estimate.
For the second case, we compute the local diffusivities and display their
result in Figure 1. We see that there is poor agreement between the two
approaches in each component of the diffusivity tensor. Given the lack of
scale separation between the timescales of the problem, this comes as no
surprise.
To further explore the discrepancy, we show the full operator, $\int
dx^{\prime}dz^{\prime}\mathcal{K}(x,z|x^{\prime},z^{\prime})\bullet$, in
Figure 3. The current turbulent diffusivity operator is a four-dimensional
object, which we represent with a two-dimensional heatmap. To do so, we
flatten both the $x$ and $z$ dimensions into a single index and order them so
that the $z$ values are sequential and $x$ values are separated upon the
completion of a $z$-range. The structured pattern of the heatmap in Figure 3
is a consequence of the periodicity of the $x$ direction. There is a block
structure in each of the components of the operator. These blocks correspond
to the diffusivity operator at a fixed $x,x^{\prime}$ location and represent
the variation in $z,z^{\prime}$. Thus the diagonal block component corresponds
to the $x=x^{\prime}$ part of the operator. Each block row corresponds to a
fixed $x$ value, and each block column corresponds to a fixed $x^{\prime}$
value. One can count 8 blocks appearing on a given row (and column), which
corresponds to our choice of using 8 Fourier modes in the periodic direction.
Each block row seems to be a periodic translation of one another; however, we
emphasize that this is merely in appearance rather than actuality. The four
states are not sufficient to guarantee translation invariance.
The rich structure of the current operator stands in stark contrast to a local
diffusivity operator. A local diffusivity operator is a diagonal matrix. We
emphasize that the significant off-diagonal components imply that turbulent
fluxes are not related solely to local gradients but must incorporate a
weighted sum of gradients in a neighborhood of a given location.
In the following section, we gather the approach used in the examples and
generalize.
## 4 General Approach
We have seen three examples that all follow a similar pattern:
1. 1.
Compute the eigenvectors of the generator $\mathcal{Q}$.
2. 2.
Transform the equations into a basis that diagonalizes $\mathcal{Q}$.
3. 3.
Separate the mean equation from the perturbation equations.
4. 4.
Solve for the perturbation variables in terms of the mean variable.
Here we aim to gather the above procedure in the general case where we have
access to the eigenvectors of $\mathcal{Q}$. Furthermore, in the last example,
we claimed that the local turbulent diffusivity approximation as calculated by
neglecting the effects of diffusion and perturbation gradients is equivalent
to calculating the integrated auto-correlation of the Markov process. We
justify that claim in Section 4.3.
### 4.1 Notation
Let us establish a notation for the general procedure. We again let
$\mathcal{Q}$ denote the generator with corresponding transition probability
matrix $\mathscr{P}(\tau)$ given by the matrix exponential
$\displaystyle\mathscr{P}(\tau)$ $\displaystyle=\exp(\tau\mathcal{Q}).$ (87)
The entries of the matrix $[\mathscr{P}(\tau)]_{mn}$ denotes the transition
probability of state $n$ to the state $m$. In each column of the transition
matrix the sum of the entries is one. We assume a unique zero eigenvalue for
$\mathcal{Q}$ with all other eigenvalues negative. We also assume that the
eigenvalues can be ordered in such a way that they are decreasing, i.e.
$\lambda_{1}=0$, $\lambda_{2}<0$ and $\lambda_{i}\leq\lambda_{j}$ for $i>j$
with $j\geq 2$. These choices result in a unique statistical steady state
which we denote by the vector $\bm{v}_{1}$ with the property
$\displaystyle\mathcal{Q}\bm{v}_{1}$ $\displaystyle=0\bm{v}_{1}\text{ and
}\mathscr{P}(\tau)\bm{v}_{1}=\bm{v}_{1}\text{ for all }\tau.$ (88)
and similarly for the left eigenvector, $\bm{w}_{1}$. We denote the entries of
$\bm{v}_{1}$ and $\bm{w}_{1}$ by column vectors
$\displaystyle\bm{v}_{1}$ $\displaystyle=\begin{bmatrix}P_{1}\\\ P_{2}\\\
\vdots\\\ P_{M}\end{bmatrix}\text{ and }\bm{w}_{1}=\begin{bmatrix}1\\\ 1\\\
\vdots\\\ 1\end{bmatrix}$ (89)
where $M$ is the number of states. We assume that the eigenvector $\bm{v}_{1}$
is normalized such that $\sum_{m}P_{m}=1$. Consequently,
$\bm{w}_{1}\cdot\bm{v}_{1}=\bm{w}_{1}^{T}\bm{v}_{1}=1$. We introduce unit
vectors $\hat{\bm{e}}_{m}$ whose $m^{\prime}th$ entry is zero and all other
entries are zero, e.g.
$\displaystyle\hat{\bm{e}}_{1}$ $\displaystyle=\begin{bmatrix}1\\\ 0\\\
\vdots\\\ 0\end{bmatrix}\text{ , }\hat{\bm{e}}_{2}=\begin{bmatrix}0\\\ 1\\\
\vdots\\\ 0\end{bmatrix}\text{ and }\hat{\bm{e}}_{M}=\begin{bmatrix}0\\\ 0\\\
\vdots\\\ 1\end{bmatrix}.$ (90)
Thus, $\bm{v}_{1}=\sum_{m}P_{m}\bm{\hat{e}}_{m}$,
$\bm{w}_{1}=\bm{\hat{e}}_{m}$. Furthermore,
$\bm{\hat{e}}_{m}\cdot\bm{v}_{1}=P_{m}$ and
$\bm{\hat{e}}_{m}\cdot\bm{w}_{1}=1$ for each $m$.
For the discussion that follows we will assume that the matrix $\mathcal{Q}$
has an eigenvalue decomposition. In general we denote the right eigenvectors
of $\mathcal{Q}$ by $\bm{v}_{i}$ for $i=1,..,M$ and the left eigenvectors by
$\bm{w}_{i}$ for $i=1,...,M$. These vectors are all associated with
eigenvalues $\lambda_{i}$ for $i=1,...,M$ where $i=1$ denotes the unique
eigenvalue $\lambda_{1}=0$. We recall that the left eigenvectors can be
constructed from the right eigenvectors by stacking all the left eigenvectors
in a matrix $V$, computing the inverse $V^{-1}$, and extracting the rows of
the inverse. The aforementioned procedure guarantees the normalization
$\bm{w}_{j}\cdot\bm{v}_{i}=\bm{w}_{i}^{T}\bm{v}_{j}=\delta_{ij}$. Thus we have
the relations
$\displaystyle\mathcal{Q}\bm{v}_{n}=\lambda_{n}\bm{v}_{n}\text{ and
}\bm{w}^{T}_{n}\mathcal{Q}=\lambda_{n}\bm{w}^{T}_{n}.$ (91)
With notation now in place, we observe that the operators $\mathcal{Q}$ and
$\mathscr{P}(\tau)$ are characterized by their spectral decomposition
$\displaystyle\mathcal{Q}$
$\displaystyle=\sum_{i}\lambda_{i}\bm{v}_{i}\bm{w}_{i}^{T}\text{ and
}\mathscr{P}(\tau)=\sum_{i}e^{\tau\lambda_{i}}\bm{v}_{i}\bm{w}_{i}^{T}.$ (92)
We remind the reader of the various use of ”P”s and their relation:
1. 1.
$\mathbb{P}$ denotes a probability.
2. 2.
$\mathscr{P}(\tau)$ denotes the transition probability matrix for a time
$\tau$ in the future.
3. 3.
$\mathcal{P}_{m}(t)$ denotes the probability of being in state $m$ at time
$t$. The algebraic relation
$\sum_{m}P_{m}(t+\tau)\bm{\hat{e}}_{m}=\mathscr{P}(\tau)\sum_{n}P_{n}(t)\bm{\hat{e}}_{n}$
holds.
4. 4.
$P_{m}$ is the statistically steady probability of being found in state $m$.
In the limit
$\lim_{t\rightarrow\infty}\mathcal{P}_{m}(t)=P_{m}.$
We now introduce our Markov states as steady vector fields. The use of several
vector spaces imposes a burden on notation: The vector spaces associated with
Markov states, ensemble members, and the vector field $\bm{u}$. Instead of
using overly decorated notation with an excessive number of indices, we
introduce the convention that $\bm{u}$ will always belong to the vector space
associated with the vector field, and all other vectors are associated with
the vector space of Markov states. Effectively we let elements of our vector
space associated with Markov states belong to a different algebra than real
numbers.
### 4.2 The Spectral Representation
With this notation now in place, the statistically steady equations
$\displaystyle\nabla\cdot\left(\bm{u}_{m}\Theta_{m}\right)$
$\displaystyle=\kappa\Delta\Theta_{m}+P_{m}s+\sum_{n}\mathcal{Q}_{mn}\Theta_{n}$
(93)
are represented as the matrix system
$\displaystyle\sum_{m}\hat{\bm{e}}_{m}\nabla\cdot\left(\bm{u}_{m}\Theta_{m}\right)$
$\displaystyle=\sum_{m}\hat{\bm{e}}_{m}\kappa\Delta\Theta_{m}+s\bm{v}_{1}+\mathcal{Q}\left(\sum_{m}\hat{\bm{e}}_{m}\Theta_{m}\right)$
(94)
where we made use of $\sum_{m}\bm{\hat{e}}_{m}P_{m}=\bm{v}_{1}$. We now re-
express Equation 97 in terms of a basis that uses the eigenvectors of the
transition matrix. Define components $\varphi_{n}$ by the change of basis
formula
$\displaystyle\sum_{m}\Theta_{m}\bm{\hat{e}}_{m}$
$\displaystyle=\sum_{n}\varphi_{n}\bm{v}_{n}\Leftrightarrow\sum_{n}\varphi_{n}\bm{\hat{e}}_{n}=\sum_{mn}(\bm{w}_{n}\cdot\bm{\hat{e}}_{m})\bm{\hat{e}}_{n}\Theta_{m}$
(95)
We make the observation $\varphi_{1}=\langle\theta\rangle$. We have the
following relations based on the general definitions of the left eigenvectors
$\bm{w}_{n}$ and right eigenvectors $\bm{v}_{n}$,
$\displaystyle\Theta_{n}$
$\displaystyle=\sum_{i}(\bm{\hat{e}}_{n}\cdot\bm{v}_{i})\varphi_{i}\text{ and
}\varphi_{n}=\sum_{m}(\bm{w}_{n}\cdot\bm{\hat{e}}_{m})\Theta_{m}.$ (96)
Multiplying Equation 97 by $\bm{w}^{T}_{j}$ and making use of Equation 96 we
get
$\displaystyle\sum_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\nabla\cdot\left(\bm{u}_{n}\Theta_{n}\right)$
$\displaystyle=\kappa\Delta\varphi_{j}+\delta_{1j}s+\lambda_{j}\varphi_{j}$
(97) $\displaystyle\Rightarrow$
$\displaystyle\sum_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\nabla\cdot\left(\bm{u}_{n}\left[\sum_{i}\bm{\hat{e}}_{n}\cdot\bm{v}_{i}\varphi_{i}\right]\right)$
$\displaystyle=\kappa\Delta\varphi_{j}+\delta_{1j}s+\lambda_{j}\varphi_{j}$
(98) $\displaystyle\Rightarrow$
$\displaystyle\sum_{in}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})(\bm{\hat{e}}_{n}\cdot\bm{v}_{i})\nabla\cdot\left(\bm{u}_{n}\varphi_{i}\right)$
$\displaystyle=\kappa\Delta\varphi_{j}+\delta_{1j}s+\lambda_{j}\varphi_{j}$
(99)
We now wish to decompose Equation 99 into a mean equation, index $j=1$, and
perturbation equations $j>1$. For the mean equation, we make use of the
properties
$\displaystyle\lambda_{1}=0\text{ , }\bm{w}_{1}\cdot\hat{\bm{e}}_{n}=1\text{ ,
and
}\sum_{(i=1)n}(\bm{w}_{1}\cdot\hat{\bm{e}}_{n})(\bm{\hat{e}}_{n}\cdot\bm{v}_{i})\nabla\cdot\left(\bm{u}_{n}\varphi_{i}\right)=\nabla\cdot\langle\bm{u}\varphi_{1}\rangle$
(100)
to arrive at (after changing summation index from $n$ to $m$),
$\displaystyle\nabla\cdot\left(\langle\bm{u}\rangle\varphi_{1}\right)+\nabla\cdot\left[\sum_{(i\neq
1)m}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}\varphi_{i}\right]$
$\displaystyle=\kappa\Delta\varphi_{1}+s$ (101)
We make the observation that the turbulent flux is
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle$
$\displaystyle=\sum_{(i\neq
1)m}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}\varphi_{i}.$ (102)
The perturbation equations, indices $j>1$, are
$\displaystyle\sum_{in}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})(\bm{\hat{e}}_{n}\cdot\bm{v}_{i})\nabla\cdot\left(\bm{u}_{n}\varphi_{i}\right)$
$\displaystyle=\kappa\Delta\varphi_{j}+\lambda_{j}\varphi_{j}\text{ for }j>1.$
(103)
We isolate the dependence on the mean gradients by rearranging the above
expression as follows for $j>1$
$\displaystyle\sum_{(i\neq
1)n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})(\bm{\hat{e}}_{n}\cdot\bm{v}_{i})\nabla\cdot\left(\bm{u}_{n}\varphi_{i}\right)-\kappa\Delta\varphi_{j}-\lambda_{j}\varphi_{j}$
$\displaystyle=-\sum_{n}P_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\nabla\cdot\left(\bm{u}_{n}\varphi_{1}\right)$
(104)
where we used $\bm{\hat{e}}_{n}\cdot\bm{v}_{i}=P_{n}$. Assuming that the
operator on the left-hand side of Equation 104 is invertible, we introduce the
Green’s function, $\mathcal{G}_{ij}$ to yield
$\displaystyle\varphi_{i}=-\int d\bm{x}^{\prime}\sum_{(j\neq
1)n}\mathcal{G}_{ij}(\bm{x}|\bm{x}^{\prime})P_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\nabla_{\bm{x}^{\prime}}\cdot\left(\bm{u}_{n}(\bm{x}^{\prime})\varphi_{1}(\bm{x}^{\prime})\right)\text{
for }i\neq 1$ (105)
Thus we represent our turbulent flux as
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle$
$\displaystyle=-\int d\bm{x}^{\prime}\sum_{(i\neq 1)(j\neq
1)mn}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}(\bm{x})\mathcal{G}_{ij}(\bm{x}|\bm{x}^{\prime})P_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\nabla_{\bm{x}^{\prime}}\cdot\left(\bm{u}_{n}(\bm{x}^{\prime})\varphi_{1}(\bm{x}^{\prime})\right).$
(106)
For compressible flow, the eddy-flux depends on both the ensemble mean
gradients and the ensemble mean value; otherwise, when each Markov state is
incompressible,
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle$
$\displaystyle=-\int d\bm{x}^{\prime}\sum_{(i\neq 1)(j\neq
1)mn}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}(\bm{x})\mathcal{G}_{ij}(\bm{x}|\bm{x}^{\prime})P_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\bm{u}_{n}(\bm{x}^{\prime})\cdot\nabla_{\bm{x}^{\prime}}\varphi_{1}(\bm{x}^{\prime}),$
(107)
in which case the turbulent diffusivity kernel is
$\displaystyle\mathcal{K}(\bm{x}|\bm{x}^{\prime})$ $\displaystyle=\sum_{(i\neq
1)(j\neq
1)mn}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}(\bm{x})\mathcal{G}_{ij}(\bm{x}|\bm{x}^{\prime})P_{n}(\bm{w}_{j}\cdot\hat{\bm{e}}_{n})\bm{u}_{n}(\bm{x}^{\prime}).$
(108)
The above expression completes the procedure that we enacted for the examples
in Section 3.
We now discuss local approximations to the turbulent diffusivity operator.
### 4.3 Local Approximation
We start with the same local diffusivity approximation of Section 3.3 but
using the spectral representation of Equations 11-12. In the perturbation
equations, neglect the dissipation operator and perturbation gradients, e.g.
only include index $i=1$, to yield the following reduction of Equation 103,
$\displaystyle\sum_{n}(\bm{w}_{i}\cdot\hat{\bm{e}}_{n})P_{n}\left(\bm{u}_{n}\cdot\nabla\varphi_{1}\right)$
$\displaystyle=\lambda_{i}\varphi_{i}\text{ for }i>1$ (109)
where we used $(\bm{\hat{e}}_{n}\cdot\bm{v}_{1})=P_{n}$ and have changed
indices from $j$ to $i$. We solve for $\varphi_{i}$ for $i>1$ and focus on the
perturbation flux term in Equation 101
$\displaystyle\langle\bm{u}^{\prime}\theta^{\prime}\rangle=\sum_{(i\neq
1)m}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}\varphi_{i}$ (110)
to get the local turbulent-diffusivity estimate,
$\displaystyle\sum_{(i\neq
1)m}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}\varphi_{i}$
$\displaystyle=\sum_{(i\neq
1)m}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})\bm{u}_{m}\left[\frac{1}{\lambda_{i}}\sum_{n}(\bm{w}_{i}\cdot\hat{\bm{e}}_{n})P_{n}\left(\bm{u}_{n}\cdot\nabla\varphi_{1}\right)\right]$
(111) $\displaystyle=\underbrace{\left[\sum_{(i\neq
1)mn}\frac{-1}{\lambda_{i}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})(\bm{w}_{i}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}\otimes
P_{n}\bm{u}_{n}\right]}_{\bm{D}}\cdot(-\nabla\varphi_{1}).$ (112)
We aim to show that the turbulent diffusivity from Equation 112
$\displaystyle\bm{D}$ $\displaystyle=\sum_{(i\neq
1)mn}\frac{-1}{\lambda_{i}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})(\bm{w}_{i}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}\otimes
P_{n}\bm{u}_{n}$ (113)
is equivalent to estimating the diffusivity by calculating the integral of the
velocity perturbation autocorrelation in a statistically steady state,
$\displaystyle\bm{D}=\int_{0}^{\infty}\langle\bm{u}^{\prime}(\bm{x},t+\tau)\otimes\bm{u}^{\prime}(\bm{x},t)\rangle
d\tau$ (114)
The above turbulent diffusivity is expected to work well in the limit that
diffusive effects can be neglected and the velocity field transitions rapidly
with respect to the advective timescale. Under such circumstances it is not
unreasonable to think of velocity fluctuations as analogous to white noise
with a given covariance structure. For example, letting $\bm{\xi}$ be a white
noise process and $\bm{\sigma}$ be a variance vector, if
$\displaystyle\bm{u}^{\prime}(\bm{x},t)\approx\bm{\sigma}(\bm{x})\xi\text{
where }\langle\xi(t+\tau)\xi(t)\rangle=\delta(\tau)$ (115)
then a diffusivity is given by
$\displaystyle\bm{D}(\bm{x})=\int_{0}^{\infty}\langle\bm{u}^{\prime}(\bm{x},t+\tau)\otimes\bm{u}^{\prime}(\bm{x},t)\rangle
d\tau$ $\displaystyle=\bm{\sigma}(\bm{x})\otimes\bm{\sigma}(\bm{x}).$ (116)
Indeed, we will show that the intuitive estimate,
$\displaystyle\bm{D}(\bm{x})=\int_{0}^{\infty}\langle\bm{u}^{\prime}(\bm{x},t+\tau)\otimes\bm{u}^{\prime}(\bm{x},t)\rangle
d\tau$ (117)
does correspond to Equation 113.
We begin with two observations. First, the statistically steady velocity field
satisfies
$\displaystyle\langle\bm{u}(\bm{x},t)\rangle=\sum_{m}P_{m}\bm{u}_{m}(\bm{x}),$
(118)
where $\bm{u}_{m}(\bm{x})$ for each $m$ are the states of the Markov process.
Second, recall that the vector $\mathscr{P}(\tau)\bm{\hat{e}}_{n}$ is a column
vector of probabilities whose entries denote the probability of being found in
state $m$ given that at time $\tau=0$ the probability of being found in state
$n$ is one. Thus, the conditional expectation of $\bm{u}(\bm{x},t+\tau)$ given
$\bm{u}(\bm{x},t)=\bm{u}_{n}(\bm{x})$ is
$\displaystyle\langle\bm{u}(\bm{x},t+\tau)\rangle_{\bm{u}(\bm{x},t)=\bm{u}_{n}(\bm{x})}$
$\displaystyle=\left(\sum_{m}\bm{u}_{m}(\bm{x})\hat{\bm{e}}_{m}\right)^{T}\mathscr{P}(\tau)\bm{\hat{e}}_{n}$
(119)
$\displaystyle=\sum_{im}e^{\tau\lambda_{i}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})(\bm{w}_{i}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}(\bm{x}).$
(120)
Equation 119 expresses the conditional expectation as a weighted sum of Markov
states $\bm{u}_{m}(\bm{x})$.
We are now in a position to characterize the local turbulent-diffusivity
estimate. The local turbulent-diffusivity is computed by taking the long time
integral of a statistically steady flow field’s autocorrelation function, i.e.
$\displaystyle\bm{D}(\bm{x})=\int_{0}^{\infty}\bm{R}(\bm{x},\tau)d\tau$ (121)
where
$\displaystyle\bm{R}(\bm{x},\tau)\equiv\langle\bm{u}(\bm{x},t+\tau)\otimes\bm{u}(\bm{x},t)\rangle-\langle\bm{u}(\bm{x},t+\tau)\rangle\otimes\langle\bm{u}(\bm{x},t)\rangle.$
(122)
We calculate the second term under the statistically steady assumption of
Equation 122,
$\displaystyle\langle\bm{u}(\bm{x},t+\tau)\rangle\otimes\langle\bm{u}(\bm{x},t)\rangle=\left(\sum_{m}P_{m}\bm{u}_{m}(\bm{x})\right)\otimes\left(\sum_{n}P_{n}\bm{u}_{n}(\bm{x})\right).$
(123)
For the first term of Equation 122 we decompose the expectation into
conditional expectations,
$\displaystyle\langle\bm{u}(\bm{x},t+\tau)\otimes\bm{u}(\bm{x},t)\rangle=\sum_{n}\langle\bm{u}(\bm{x},t+\tau)\otimes\bm{u}(\bm{x},t)\rangle_{\bm{u}(\bm{x},t)=\bm{u}_{n}(\bm{x})}P_{n}$
(124)
Given that we are in a statistically steady state, we use Equation 119 to
establish
$\displaystyle\langle\bm{u}(\bm{x},t+\tau)\otimes\bm{u}(\bm{x},t)\rangle$
$\displaystyle=\sum_{n}\langle\bm{u}(\bm{x},t+\tau)\otimes\bm{u}(\bm{x},t)\rangle_{\bm{u}(\bm{x},t)=\bm{u}_{n}(\bm{x})}P_{n}$
(125)
$\displaystyle=\sum_{imn}e^{\tau\lambda_{i}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})(\bm{w}_{i}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}\otimes
P_{n}\bm{u}_{n}.$ (126)
We isolate the $i=1$ index and use $\lambda_{1}=0$,
$\bm{\hat{e}}_{m}\cdot\bm{v}_{1}=P_{m}$, and
$\bm{w}_{1}\cdot\bm{\hat{e}}_{n}=1$ to arrive at
$\displaystyle\sum_{mn}e^{\tau\lambda_{1}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{1})(\bm{w}_{1}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}\otimes
P_{n}\bm{u}_{n}=\left(\sum_{m}P_{m}\bm{u}_{m}\right)\otimes\left(\sum_{n}P_{n}\bm{u}_{n}\right).$
(127)
Equation 127 cancels with 123 so that in total we have the following
characterization of Equation 122
$\displaystyle\bm{R}(\bm{x},\tau)$ $\displaystyle=\sum_{(i\neq
1)mn}e^{\tau\lambda_{i}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})(\bm{w}_{i}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}\otimes
P_{n}\bm{u}_{n}.$ (128)
Equation 128 is integrated to yield the local turbulent-diffusivity
$\displaystyle\bm{D}(\bm{x})=\int_{0}^{\infty}\bm{R}(\bm{x},\tau)d\tau=\sum_{(i\neq
1)mn}\frac{-1}{\lambda_{i}}(\bm{\hat{e}}_{m}\cdot\bm{v}_{i})(\bm{w}_{i}\cdot\bm{\hat{e}}_{n})\bm{u}_{m}\otimes
P_{n}\bm{u}_{n}$ (129)
where we used $\lambda_{i}<0$ for $i>1$. A comparison of Equation 129 to
Equation 113 reveals the correspondence. Thus we see that estimating the
diffusivity through the velocity autocorrelation integral is equivalent to
neglecting diffusive effects and perturbation gradients.
## 5 Conclusions
We have introduced a class of stochastic partial differential equations
amenable to analysis in this work. The class of problems falls under the
umbrella of stochastic advection, where the flow state is modeled as a
continuous time Markov process. We reformulated the problem of finding a
turbulence closure for passive scalars advected by a stochastic flow field
into solving a set of partial differential equations by conditionally
averaging the passive scalar equation with respect to the flow state.
The resulting dimensionality of the equations depended on the number of
variables required to describe flow statistics and the dimensionality of the
flow. A flow characterized by $m$ discrete variables leads to a set of
$m$-coupled equations of the same dimensionality as the original. A flow
characterized by a continuum of statistical variables can be discretized and
reduced to the former. Eliminating the system’s dependence on all but the
ensemble mean leads to an operator characterization of the turbulence closure,
allowing for an exploration of closures that don’t invoke a scale separation
hypothesis.
We explored three examples of increasing complexity–Markov states
characterized by two, three, and four states–and outlined a general approach
to obtaining a closure based on the spectrum of the transition probability
operator. In the examples, we examined the role of non-locality in determining
a statistically steady turbulence closure. We calculated closures for all
three systems and numerically evaluated a Green’s function for the four-state
system. We also found the small velocity amplitude, weak scalar diffusivity,
and fast transition rate limit reduce the closure to a spatially heterogeneous
tensor acting on ensemble mean gradients. Furthermore, we related this tensor
to the time-integrated auto-correlation of the stochastic flow field.
We have not exhausted the number of examples offered by the formulation nor
simplifications leading to analytically tractable results. Interesting future
directions include using Markov states estimated directly from turbulence
simulations, analyzing scale-separated flows, generalizing the advection-
diffusion equation to reaction-advection-diffusion equations, and formulating
optimal mixing problems. When the number of Markov states increases, the
computational burden of estimating turbulent diffusivity operators becomes
demanding; thus, there is a need to develop methods that exploit the structure
of the problem as much as possible.
Mathematically there are many challenges as well. All the arguments provided
here are formal calculations, and the necessity of rigorous proofs remains.
For example, a direct proof of the conditional averaging procedure is
necessary. Ultimately, the goal is to reduce the stochastic-advection
turbulence closure problem to one that can leverage theory from partial
differential equations.
[Supplementary data]Supplementary material and movies are available at
ZENODO
https://github.com/sandreza/StatisticalNonlocality
[Acknowledgements]We would like to thank the 2018 Geophysical Fluid Dynamics
summer school where much of this work was completed. We would also like to
thank Tobias Bischoff, Simon Byrne, and Raffaele Ferrari for their
encouragement and discussion with regards to the present manuscript.
[Funding]Our work is supported by the generosity of Eric and Wendy Schmidt by
recommendation of the Schmidt Futures program, and by the National Science
Foundation under grant AGS-6939393.
[Declaration of interests]The authors report no conflict of interest.
## Appendix A An Alternative Formal Derivation
We wish to show that one can work directly with the continuous formulation of
the advection-diffusion equations for the derivation of the conditional mean
equations. Although we consider a finite (but arbitrarily large) number of
Markov states here, considering a continuum follows mutatis mutandi. In
Section 2 we wrote down the master equation for the discretized stochastic
system as
$\displaystyle\partial_{t}\rho_{m}$
$\displaystyle=\sum_{i}\frac{\partial}{\partial\theta^{i}}\left[\left(\sum_{jkc}A_{ijk}^{c}u^{k,c}_{m}\theta^{j}-\sum_{j}D_{ij}\theta^{j}-s^{i}\right)\rho_{m}\right]+\sum_{n}\mathcal{Q}_{mn}\rho_{n}.$
(130)
We introduce the (spatial) volume element $\Delta\bm{x}_{i}$ to rewrite
Equation 130 in the evocative manner,
$\displaystyle\partial_{t}\rho_{m}$
$\displaystyle=\sum_{i}\Delta\bm{x}_{i}\frac{1}{\Delta\bm{x}_{i}}\frac{\partial}{\partial\theta^{i}}\left[\left(\sum_{jkc}A_{ijk}^{c}u^{k,c}_{m}\theta^{j}-\sum_{j}D_{ij}\theta^{j}-s^{i}\right)\rho_{m}\right]+\sum_{n}\mathcal{Q}_{mn}\rho_{n}.$
(131)
We now take limits
$\displaystyle\sum_{i}\Delta\bm{x}_{i}$
$\displaystyle\overset{``\text{lim}"}{=}\int d\bm{x},$ (132)
$\displaystyle\frac{1}{\Delta\bm{x}_{i}}\frac{\partial}{\partial\theta^{i}}$
$\displaystyle\overset{``\text{lim}"}{=}\frac{\delta}{\delta\theta(\bm{x})},$
(133)
$\displaystyle\sum_{jkc}A_{ijk}^{c}u^{k,c}_{m}\theta^{j}+\sum_{j}D_{ij}\theta^{j}-s^{i}$
$\displaystyle\overset{``\text{lim}"}{=}\bm{u}_{m}\cdot\nabla\theta-\kappa\Delta\theta-s,$
(134)
to get the functional evolution equation for the probability density,
$\displaystyle\partial_{t}\rho_{m}$ $\displaystyle=\int
d\bm{x}\frac{\delta}{\delta\theta(\bm{x})}\left(\left[\bm{u}_{m}\cdot\nabla\theta-\kappa\Delta\theta-s\right]\rho_{m}\right)+\sum_{n}\mathcal{Q}_{mn}\rho_{n}$
(135)
where $\bm{x}$ is a continuous index. As before we can derive the CM equations
directly from the above equations. To do so we make the additional
correspondence
$\displaystyle d\bm{\theta}$
$\displaystyle\overset{``\text{lim}"}{=}\mathcal{D}[\theta].$ (136)
We now define the same quantities as before, but using the field integral
$\displaystyle\mathcal{P}_{m}$
$\displaystyle\equiv\int\mathcal{D}[\theta]\rho_{m}\text{ and
}\Theta_{m}(\bm{y})\equiv\int\mathcal{D}[\theta]\theta(\bm{y})\rho_{m}.$ (137)
We mention that the discrete indices $i,j,k$ from Equations 2 before get
replaced by the continuous labels such as $\bm{x}$ and $\bm{y}$. We only make
use of a few formal properties of the field integral, with direct
correspondence the the $n$-dimensional integrals. We use linearity, i.e. for
two mappings with compatible ranges $\mathcal{F}[\theta]$ and
$\mathcal{H}[\theta]$,
$\displaystyle\int\mathcal{D}[\theta]\left(\mathcal{F}[\theta]+\mathcal{H}[\theta]\right)$
$\displaystyle=\int\mathcal{D}[\theta]\mathcal{F}[\theta]+\int\mathcal{D}[\theta]\mathcal{H}[\theta]$
(138)
We use the analogue to the divergence theorem,
$\displaystyle\int\mathcal{D}[\theta]\int
d\bm{x}\frac{\delta}{\delta\theta(\bm{x})}\left(\left[\bm{u}_{m}\cdot\nabla\theta-\kappa\Delta\theta-s\right]\rho_{m}\right)$
$\displaystyle=0$ (139) $\displaystyle\Leftrightarrow$ $\displaystyle\int
d\bm{\theta}\nabla_{\bm{\theta}}\cdot(\bm{f}\rho)$ $\displaystyle=0$ (140)
since the integral of a divergence should be zero if the probabilities vanish
at infinity (i.e. that our tracer cannot have infinite values at a given point
in space). We also make use of the integration by parts, i.e. for some
functionals $\mathcal{F}$ and $\mathcal{H}$,
$\displaystyle\int\mathcal{D}[\theta]\mathcal{H}\int
d\bm{x}\frac{\delta}{\delta\theta(\bm{x})}\mathcal{F}$
$\displaystyle=-\int\mathcal{D}[\theta]\int
d\bm{x}\frac{\delta\mathcal{H}}{\delta\theta(\bm{x})}\mathcal{F}$ (141)
$\displaystyle\Leftrightarrow$ $\displaystyle\int
d\bm{\theta}h\nabla_{\bm{\theta}}\cdot\bm{f}$ $\displaystyle=-\int
d\bm{\theta}\left(\nabla_{\bm{\theta}}h\right)\cdot\bm{f}$ (142)
And finally we also interchange sums and integrals,
$\displaystyle\int\mathcal{D}[\theta](\Delta\theta)\rho_{m}$
$\displaystyle=\Delta\int\mathcal{D}[\theta]\theta\rho_{m}=\Delta\Theta_{m}$
(143) $\displaystyle\Leftrightarrow$ $\displaystyle\int
d\bm{\theta}\left(\sum_{j}D_{\ell j}\theta^{j}\rho_{m}\right)$
$\displaystyle=\sum_{j}D_{\ell j}\int
d\bm{\theta}\theta^{j}\rho_{m}=\sum_{j}D_{\ell j}\Theta^{j}_{m}.$ (144)
We proceed similarly for the $\bm{u}_{m}\cdot\nabla$ term. We also use
properties of the variational derivative such as,
$\displaystyle\frac{\delta\theta(\bm{y})}{\delta\theta(\bm{x})}=\delta(\bm{x}-\bm{y})\Leftrightarrow\frac{\partial\theta^{\ell}}{\partial\theta^{i}}=\delta_{\ell
i}.$ (145)
Taken together one can directly obtain Equations 11 and 12 by first
integrating Equation 135 with respect to $\mathcal{D}[\theta]$ to get
$\displaystyle\partial_{t}\mathcal{P}_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n}$ (146)
and multiplying Equation 135 by $\theta(\bm{y})$ and then integrating with
respect to $\mathcal{D}[\theta]$ to get
$\displaystyle\partial_{t}\Theta_{m}(\bm{y},t)+\nabla_{\bm{y}}\cdot\left(\bm{u}_{m}(\bm{y})\Theta_{m}(\bm{y},t)-\kappa\nabla_{\bm{y}}\Theta_{m}(\bm{y},t)\right)$
$\displaystyle=s(\bm{y})\mathcal{P}_{m}+\sum_{n}\mathcal{Q}_{mn}\Theta_{n}(\bm{y},t).$
(147)
In the above expression, removing explicit dependence of the position variable
yields,
$\displaystyle\partial_{t}\Theta_{m}+\nabla\cdot\left(\bm{u}_{m}\Theta_{m}-\kappa\nabla\Theta_{m}\right)$
$\displaystyle=sP_{m}+\sum_{n}\mathcal{Q}_{mn}\Theta_{n}.$ (148)
Our reason for mentioning the above methodology is that it allows for
expedited computations. There is no need to explicitly discretize, perform
usual $n$-dimensional integral manipulations, and then take limits afterward.
For example, computing the conditional two-moment equations defined by the
variable
$\displaystyle C_{m}(\bm{y},\bm{z},t)$
$\displaystyle\equiv\int\mathcal{D}[\theta]\theta(\bm{y})\theta(\bm{z})\rho_{m},$
(149)
is obtained by multiplying Equation 135 by $\theta(\bm{y})$ and
$\theta(\bm{z})$ and integrating with respect to $\mathcal{D}[\theta]$,
$\displaystyle\partial_{t}C_{m}+\nabla_{\bm{y}}\cdot\left(\bm{u}_{m}(\bm{y})C_{m}-\kappa\nabla_{\bm{y}}C_{m}\right)+\nabla_{\bm{z}}\cdot\left(\bm{u}_{m}(\bm{z})C_{m}-\kappa\nabla_{\bm{z}}C_{m}\right)$
(150)
$\displaystyle=s(\bm{z})\Theta_{m}(\bm{y})+s(\bm{y})\Theta_{m}(\bm{z})+\sum_{n}\mathcal{Q}_{mn}C_{n}.$
(151)
In particular we note the source term on the right hand side and the
appearance of the first conditional moment. In the derivation we used the
product rule
$\displaystyle\frac{\delta(\theta(\bm{y})\theta(\bm{z}))}{\delta\theta(\bm{x})}$
$\displaystyle=\delta(\bm{x}-\bm{y})\theta(\bm{z})+\delta(\bm{x}-\bm{z})\theta(\bm{y}).$
(152)
If the advection-diffusion equation is $m$-dimensional and we have $N$ Markov
states, the above equation is $2m+N$ dimensional. Indeed the equation for the
$M^{\prime}th$ moment is a $M\times m+N$ dimensional partial differential
equation.
## Appendix B A Heuristic Overview of the Master Equation and Discretizations
In this section we provide an argument for the form of the master equation in
the main text, Equation 130 in Section 2. Our starting point is Section B.1
where we use the Liouville equation for two continuous variables. We then
apply the finite volume method to the Fokker-Planck equation of an Ornstein-
Uhlenbeck process to derive the transition matrices used in the two-state and
three-state systems in Section 3. We conclude with a formal argument for the
use of discrete Markov states as an approximation to the compressible Euler
equations in B.3.
### B.1 Two Variable System
Suppose that we have an two variables $x,y\in\mathbb{R}$ governed by the
equations,
$\displaystyle\frac{dx}{dt}$ $\displaystyle=f(x)+\sqrt{2}\sigma\xi$ (153)
$\displaystyle\frac{dy}{dt}$ $\displaystyle=g(x,y).$ (154)
where $\xi$ is white noise. In this context we think of $x$ as being our flow
field $\bm{u}$ and $y$ as the tracer $\theta$. The master equation implied by
the dynamics is
$\displaystyle\partial_{t}\rho$
$\displaystyle=-\partial_{x}\left(f(x)\rho-\sigma^{2}\partial_{x}\rho\right)-\partial_{y}\left(g(x,y)\rho\right).$
(155)
We now discretize the equation with respect to the $x-$variable by
partitioning space into non-overlapping cells, characterized by domains
$\Omega_{m}$. First we start with the Fokker-Planck equation for $x$, which is
independent of the $y-$variable,
$\displaystyle\partial_{t}P$
$\displaystyle=-\partial_{x}\left(f(x)P-\sigma^{2}\partial_{x}P\right).$ (156)
Observe the relation $\int\rho(x,y,t)dy=P(x,t)$. Define our coarse grained
variable $\mathcal{P}_{m}$ as
$\displaystyle\mathcal{P}_{m}$ $\displaystyle\equiv\int_{\Omega_{m}}P(x)dx$
(157)
which is a probability. Thus the discretization of Equation 156 becomes
$\displaystyle\partial_{t}\int_{\Omega_{m}}Pdx$
$\displaystyle=-\int_{\Omega_{m}}\partial_{x}\left(f(x)\rho-\sigma^{2}\partial_{x}\rho\right)dx$
(158) $\displaystyle\approx$ $\displaystyle\partial_{t}\mathcal{P}_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n}$ (159)
for a generator $\mathcal{Q}$ which we derive in B.2 with respect to a chosen
numerical flux. Heuristically, going from Equation 158-159 is accomplished by
observing that $\mathcal{P}_{m}$ is a probability and the operator
$\mathcal{L}\equiv\partial_{x}\left(f(x)\bullet-\sigma^{2}\partial_{x}\bullet\right)$
is linear; thus, upon discretization, the operator is represented a
matrix444It is, of course, possible to approximate using a nonlinear operator,
but for simplicity we only consider the linear case. acting on the chosen
coarse grained variables $\mathcal{P}_{n}$. The property
$\sum_{m}\mathcal{Q}_{mn}=\bm{0}$ is the discrete conservation of probability.
Going back to Equation 155, defining
$\displaystyle\rho_{m}(y)$ $\displaystyle\equiv\int_{\Omega_{m}}\rho(x,y)dx,$
(160)
introducing $x_{m}\in\Omega_{m}$, and performing the same discretization for
the joint Markov system yields,
$\displaystyle\partial_{t}\rho_{m}$
$\displaystyle=\sum_{n}\mathcal{Q}_{mn}\rho_{n}-\partial_{y}\left(g(x_{m},y)\rho_{m}\right),$
(161)
where we used the approximation
$\displaystyle\int_{\Omega_{m}}g(x,y)\rho(x,y)dx$ $\displaystyle\approx
g(x_{m},y)\int_{\Omega_{m}}\rho(x,y)dx=g(x_{m},y)\rho_{m}(y).$ (162)
The $x_{m}$ are the Markov states and the $\mathcal{Q}_{mn}$ serves as the
specification for transitioning between different states. We also observe that
one can simply start with the discrete states for $x$ and continuous variables
for $y$ to directly obtain Equation 161 as was done in the main text.
In what follows we give a concrete example of deriving a transition matrix
$\mathcal{Q}$ from a finite-volume discretization of an Ornstein-Uhlenbeck
(OU) process. We explicitly mention the kind of discretization that we use
since retaining mimetic properties of the transition matrix $\mathcal{Q}$ is
not guaranteed with other discretizations. Furthermore, using a finite volume
discretization allows for the resulting discretization to be interpreted as a
continuous time Markov process with a finite state space.
### B.2 Example Discretization
Consider an Ornstein-Uhlenbeck process and the resulting Fokker-Planck
equation,
$\displaystyle\partial_{t}\rho$
$\displaystyle=-\partial_{x}\left(-x\rho-\partial_{x}\rho\right).$ (163)
We discretize the above equation with $N+1$ cells, where $N=1$ and $N=2$
correspond to the two and three state systems, respectively. Using a finite
volume discretization, we take our cells to be
$\displaystyle\Omega_{m}$ $\displaystyle=[\Delta
x\left(m-1/2-N/2\right),\Delta x\left(m+1/2-N/2\right)]$ (164)
$\displaystyle\Delta x$ $\displaystyle=\frac{2}{\sqrt{N}}$ (165)
for $m=0,1,...,N$. Our choice implies that cell centers (the discrete Markov
states) are
$\displaystyle x_{m}$ $\displaystyle=\Delta x(m-N/2)$ (166)
for $m=0,...,N$ and cell faces are
$\displaystyle x_{m}^{f}$ $\displaystyle=\Delta x(m-1/2-N/2)$ (167)
for $m=0,...,N+1$. We define
$\displaystyle\mathcal{P}_{m}$ $\displaystyle=\int_{\Omega_{m}}\rho dx\text{
and }\overline{\rho}_{m}\Delta x=\mathcal{P}_{m}.$ (168)
Upon integrating with respect to the control volume we obtain,
$\displaystyle\frac{d}{dt}\mathcal{P}_{m}$
$\displaystyle=-\left.\left(-x\rho-\partial_{x}\rho\right)\right|_{x=x_{m}^{f}}+\left.\left(-x\rho-\partial_{x}\rho\right)\right|_{x=x_{m+1}^{f}}$
(169)
The numerical flux is chosen as follows,
$\displaystyle\left.\left(-x\rho-\partial_{x}\rho\right)\right|_{x=x_{m}^{f}}$
$\displaystyle\approx-\frac{x_{m-1}\overline{\rho}_{m-1}+x_{m}\overline{\rho}_{m}}{2}-\frac{\overline{\rho}_{m}-\overline{\rho}_{m-1}}{\Delta
x}$ (170)
$\displaystyle=-\frac{x_{m-1}\mathcal{P}_{m-1}+x_{m}\mathcal{P}_{m}}{2\Delta
x}-\frac{\mathcal{P}_{m}-\mathcal{P}_{m-1}}{(\Delta x)^{2}}$ (171)
$\displaystyle=\frac{1}{2}\left((N-m+1)\mathcal{P}_{m-1}-m\mathcal{P}_{m}\right)$
(172)
where we use the convention $\mathcal{P}_{-1}=\mathcal{P}_{N+1}=0$ so that
boundaries, corresponding to indices $m=0$ and $m=N+1$, imply no flux
conditions. Combining the flux estimates for both cell boundaries, the
evolution equation for the probabilities $\mathcal{P}_{m}$ becomes
$\displaystyle\partial_{t}\mathcal{P}_{m}$
$\displaystyle=\frac{1}{2}\left[(N-m+1)\mathcal{P}_{m-1}-N\mathcal{P}_{m}+(m+1)\mathcal{P}_{m+1}\right],$
(173)
which implies the transition matrix
$\displaystyle\mathcal{Q}_{mn}$
$\displaystyle=\frac{1}{2}\left(-N\delta_{mn}+n\delta_{(m+1)n}+(N-n)\delta_{(m-1)n}\right).$
(174)
Equation 173 emphasize the row structure of the matrix whereas Equation 174
emphasizes the column structure. The steady state probability distribution is
the binomial distribution555The continuous steady state distribution is a
Normal distribution $\rho(x)=(2\pi)^{-1/2}\exp\left(-x^{2}/2\right)$.
$\displaystyle P_{m}$ $\displaystyle=2^{-N}\binom{N}{m}.$ (175)
Furthermore, the eigenvectors and eigenvalues of the matrix are in
correspondence with the eigenfunctions and eigenvalues of the OU process as
noted by Hagan et al. (1989). In particular, the cell centers, Equation 166,
is a left eigenvector of $\mathcal{Q}_{mn}$ with eigenvalue $\lambda=-1$. This
relation is useful for calculating the auto-correlation of the Markov process
since Equation 119 only involves one eigenvalue. We used the transition
matrix, Equation 174, in the construction of the two and three state systems.
Similar to the construction in this section, four-state system transition
matrix is obtained from discretizing a random walk in a periodic domain with a
drift. The term proportional to $\gamma$ is attributed to diffusion and the
term proportional to $\omega$ is attributed to drift. The resulting cell
centers are then taken as the phase in the periodic direction of a fixed
stream function.
### B.3 A Finite Volume Approximation in Function Space
We start with the compressible Euler-Equations
$\displaystyle\partial_{t}\rho+\nabla\cdot\left(\rho\bm{u}\right)$
$\displaystyle=0,$ (176)
$\displaystyle\partial_{t}\rho\bm{u}+\nabla\cdot\left(\rho\bm{u}\otimes\bm{u}\right)+\nabla
p$ $\displaystyle=0,$ (177) $\displaystyle\partial_{t}\rho
e+\nabla\cdot\left(\bm{u}\left[\rho e+p\right]\right)$ $\displaystyle=0,$
(178) $\displaystyle p(\rho,\rho\bm{u},\rho e)$ $\displaystyle=p$ (179)
where $\rho$ is density, $\rho\bm{u}$ is the momentum,
$\bm{u}=\rho\bm{u}/\rho$ is the velocity, $\rho e$ is the total energy
density, and $p$ is a thermodynamic pressure666For example, one could use the
pressure for an ideal gas $p=(\gamma-1)(\rho e-\rho|\bm{u}|^{2}/2)$ with
$\gamma=7/5$.. Here we introduce $Z$ as a probability density in function
space for the state variables $S\equiv(\rho,\rho\bm{u},\rho e)$. In the
notation of A, the evolution equation for the statistics $Z$ are
$\displaystyle\partial_{t}Z$ $\displaystyle=\int
d\bm{x}\left[\frac{\delta}{\delta\rho}\left(\nabla\cdot\left[\rho\bm{u}\right]Z\right)+\frac{\delta}{\delta\rho\bm{u}}\left(\nabla\cdot\left[\rho\bm{u}\otimes\bm{u}\right]Z+\nabla
pZ\right)+\frac{\delta}{\delta\rho e}\left(\nabla\cdot\left(\bm{u}\left[\rho
e+p\right]\right)Z\right)\right],$ (180)
where we have suppressed the index $\bm{x}$ in the variational derivatives.
Now consider a partition in function space into domains $\Omega_{m}$ and let
$S_{m}$ denote a value of a state within the set $S_{m}$. In this case we
define the probability as
$\displaystyle\mathcal{P}_{m}$
$\displaystyle\equiv\int_{\Omega_{m}}\mathcal{D}[\rho]\mathcal{D}[\rho\bm{u}]\mathcal{D}[\rho
e]Z.$ (181)
In analogy with the calculations in Section B.2, integrating equation 180 with
respect to a control volume $\Omega_{m}$ would result in an equation of the
form
$\displaystyle\partial_{t}\mathcal{P}_{m}=\sum_{n}\mathcal{Q}_{mn}\mathcal{P}_{n}$
(182)
for some generator $\mathcal{Q}_{mn}$. The entries of the generator are
functionals of the states $S_{m}\in\Omega_{m}$. Performing the necessary
integrals and re-expressing it in this finite form is done indirectly through
data-driven methods with time-series as in Klus et al. (2016), Fernex et al.
(2021), or Maity et al. (2022). The difficulty of performing a discretization
from first principles comes from choosing the subsets of function space to
partition and carrying out the integrals in function space. Periodic orbits
and fixed points of a flow serve as a natural skeleton for function space, but
are typically burdensome to compute. We offer no solution, but hope that in
the future such direct calculations are rendered tractable. In the meanwhile,
indirect data-driven methods are the most promising avenue for the calculation
of the generator $\mathcal{Q}$.
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|
∎ 11institutetext: 1 Theoretical and Applied Mechanics, Northwestern
University, 2145 Sheridan Rd, Evanston, 60208, Illinois, USA.
2 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan
Rd, Evanston, 60208, Illinois, USA.
3 Department of Mathematics, Princeton University, Washington Road, Princeton,
08540, New Jersey, USA.
†Corresponding Author: Wing Kam Liu, 11email<EMAIL_ADDRESS>
# Deep Learning Discrete Calculus (DLDC): A Family of Discrete Numerical
Methods by Universal Approximation for STEM Education to Frontier Research
Sourav Saha1 Chanwook Park2 Stefan Knapik2 Jiachen Guo1 Owen Huang3 Wing Kam
Liu1,2
###### Abstract
The article proposes formulating and codifying a set of applied numerical
methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the
knowledge from discrete numerical methods to interpret the deep learning
algorithms through the lens of applied mathematics. The DLDC methods aim to
leverage the flexibility and ever increasing resources of deep learning and
rich literature of numerical analysis to formulate a general class of
numerical method that can directly use data with uncertainty to predict the
behavior of an unknown system as well as elevate the speed and accuracy of
numerical solution of the governing equations for known systems. The article
is structured in two major sections. In the first section, the building blocks
of the DLDC methods are presented and deep learning structures analogous to
traditional numerical methods such as finite difference and finite element
methods are constructed with a view to incorporate these techniques in
Science, Technology, Engineering, Mathematics (STEM) syllabus for K-12
students. The second section builds upon the building blocks of the previous
discussion, and proposes new solution schemes for differential and integral
equations pertinent to multiscale mechanics. Each section is accompanied with
mathematical formulation of the numerical methods, analogous DLDC formulation,
and suitable examples.
###### Keywords:
Numerical Methods Discrete Calculus Kernel Learning Partial Differential
Equation Convolution
## 1 Introduction
The problems of engineering and physical science can be categorized into three
types saha2021hierarchical , type 1 problems are the problems with limited
physical understanding and a lot of experimental data, type 2 problems are
those with incomplete physical understanding and some experimental data, and
type 3 problems are those for which there is sufficient knowledge about the
system but solving for the system response is computationally challenging. An
example of type 1 problem is relating the spatio-temporal variation of
temperature to the resulting mechanical properties in metal additive
manufacturing xie2021mechanistic , a type 2 problem would be calibrating the
heat source and other relevant models for computational fluid dynamics
simulation for metal additive manufacturing process gan2021benchmark , and a
type 3 problem would be relating the microstructure and fatigue life of a 3D
printed metallic part xie2021mechanistic ; kafka2021image . Although the
classification is not a strict one, it points to two facts: a) even with
numerical modeling, aid of data is needed at some level, and b) by combining
data science and numerical method one can cover the spectrum of solving for
completely unknown phenomenon to challenging problems. Hence, the question
naturally arises, is there a way we can come up with a new method that gives
us the best of both worlds?
Establishing this connection between numerical methods and deep learning is
the most sought after prize in the literature nowadays. The quest started with
solving the dynamic system directly using data science and deep learning
methods sirignano2018dgm ; raissi2018deep ; han2017deep . The deep learning
methods primarily include deep neural network yu2018deep , recurrent neural
network xiao2018nonlinear , convolutional neural network zhu2018convolutional
, and residual neural network kani2017dr . Other techniques from data science
such as unsupervised learning are also heavily used to analyze the data and
extract meaningful features that may or may not have explicit physical meaning
(sometimes called the latent variables) karl2016deep ;
chakraverty2017artificial . However, it was apparent that using directly deep
learning methods results in lack of generalization magill2018neural . The
neural networks are essentially highly non-linear interpolation functions with
parameters being trained on observations liu2021mechanistic . Hence, to
predict the behaviour of a system outside the training range becomes
challenging. Moreover, the neural networks in such methods may become very
complex and it is often impossible to interpret the inner working of the
neural network in use. This results in a ”black box” method which we can just
use without proper know-how. Finally, these methods are data-hungry i.e., it
requires a lot of data to implement these.
As a solution to the lack of generalization, combining the knowledge of
mechanics with data science is proposed as Mechanistic Data Science (MDS)
liu2021mechanistic . The knowledge of mechanics can be incorporated in
selection of features and curing of data to be used as features, or, governing
principles such as conservation of energy can be directly used as an
optimization constraint. The later method is also termed as the Physics
Informed Neural Network (PINN) raissi2019physics ; karniadakis2021physics ;
pang2019fpinns . These methods have become extremely popular as these serve
perfectly to solve type 3 problems krishnapriyan2021characterizing ;
li2021physics ; leung2021nh . Moreover, these methods only partially solve the
issue of the requirement of large datasets. However, neither of these theories
solves the problem of lack of interpretation as the neural network is still
there. Finally, for type 1 problems (where we only have data) the idea of
adding governing principle does not work.
Data science techniques to discover the underlying governing equation from
data have been proposed brunton2016discovering ; kaiser2018sparse ;
kaheman2020sindy . Another example is Dimension-Net which discovers the
underlying non-dimensional numbers directly from data gan2021universal ;
xie2021data . While these techniques are exciting, one needs to have some idea
about the system behavior to use these techniques as these are primarily
regression techniques working on a library of candidate functions. Therefore,
there is still a need for a data science framework that can directly use the
experimental data to solve for the dynamics of the system and at the same time
interpret the technique.
Efforts towards merging numerical methods with deep learning have gained
increased attention in the last few years. The initial efforts were to
establish the parallels between a subset of numerical methods with a subset of
deep learning techniques chen2018neural ; zhang2021hierarchical . For example,
how a recurrent neural network behaves like an ordinary differential equation
niu2019recurrent such as wave equation hughes2019wave , or, how some finite
difference techniques or multi-grid methods have parallels with convolutional
neural network he2019mgnet . While these works are important as an
interpretation of the deep learning techniques, they do not suggest how to
improve the numerical solvers using data science or vice-versa. One of many
early efforts that tries to leverage deep learning to improve existing
numerical methods is called Hierarchical Deep Learning Finite Element Method
(HiDeNN-FEM) zhang2021hierarchical where using custom neural networks
different linear and non-linear interpolation shape functions are obtained. A
recent breakthrough in interpretable general deep learning methods for solving
partial differential equation came in the form of operator learning
kovachki2021neural . Two major methods were proposed concurrently for operator
learning called Fourier Neural Operator (FNO) li2020fourier , and DeepONet
lu2019deeponet . There have been studies showing comparisons between these two
techniques lu2022comprehensive . Keeping the similarities and differences
aside, the philosophy of both of these methods are same: mapping functions
from input space to output space instead of mapping data. Based on the same
philosophy, several other structures are being proposed including graph kernel
network and non-local kernel network anandkumar2020neural ; you2022nonlocal .
Figure 1: The key features of the Deep Learning Discrete Calculus (DLDC)
method.
Aside from solving challenging problems in scientific and engineering
research, contemporary deep learning methods have a huge pedagogical potential
as well. In order to develop a competent workforce, big tech companies and
entrepreneurs are now focusing on integrating artificial intelligence (AI) in
high school curriculum tucker2020exploring . There is increasing need for
skilled workforce over outstanding academic achievers in the industries.
Naturally, the question arises how to train the young students in AI and at
the same time, make the curriculum interesting for them. If the AI is
introduced in a traditional way through teaching computer science, the
diversity in the student body will become very difficult to attain. Due to the
socioeconomic background of the students, direct training in machine learning
algorithms may become very hard to attain. Thus untested introduction to AI
for middle and high school students will create another bias. On the other end
of the spectrum, the way mathematics, especially, calculus is taught in the
high schools may become too abstract for some students to grasp. It is natural
for the young students to be interested to learn a concept with a clear idea
about how to implement it. Usually, in the school curriculum the calculus is
taught through some textbook examples which often make the students
disinterested in the topic wang2022perspective . This lack of interest results
in reduced number of students choosing Science, Technology, Engineering, and
Mathematics (STEM) majors. There have been a lot research
touretzky2022artificial ; touretzky2022machine ; yin2022ai4all on how to
incorporate data science and AI into high school education. However, most of
these studies are at strategic level and do not provide a clear pathway to
merge two broad fields of calculus and AI. It is the authors’ opinion that
introducing AI through deep learning and interpretation of deep learning via
elementary calculus is the most optimized way to teach calculus at the high
school level. This has two-way benefit: one, implementing calculus through
advanced computational tools with real-life data will be possible for young
students. It will increase their interest and understanding on the basic
calculus. Two, interpretability of deep learning algorithms can be improved if
the algorithms are thought from the first principle.
With the broad scope in mind, in this article, the authors propose an applied
numerical method that aims to combine the mechanistic knowledge, classical
numerical methods, and deep learning algorithms called Deep Learning Discrete
Calculus (DLDC) (see, Figure 1). A formal definition of Deep Learning Discrete
Calculus can be given as:
The Deep Learning Discrete Calculus is a discipline to integrate fundamental
definitions of calculus with numerical methods using deep learning neural
network.
Deep learning is an extremely powerful tool for feature extraction and
prediction, but it requires massive and cleansed data sets to perform well.
For many engineering applications, the collection of so much data is
prohibitively expensive or even impossible, making traditional deep learning
approaches nonviable. Incorporating discrete calculus and numerical method
tools into neural network architectures can empower deep learning to address
engineering problems where datasets may be small and noisy. Additionally,
numerical method-informed deep learning frameworks sometimes have the
potential to extrapolate beyond available observations, which is something
traditional deep learning implementations are infamously incapable of. The
synthesis of numerical analysis ideas with machine learning can reduce the
expenses of collecting data, pre-processing data, and training models, while
improving the interpretability, accuracy, and generalizability of deep neural
networks. Figure 1 shows the key features of the DLDC method. The DLDC aims to
become a general method for solving the three different types of problems
discussed before. The article has two major parts. The first part aims to
introduce the concept of DLDC through building blocks of calculus, such as,
differential and integral calculus for STEM education. In later sections, the
article discusses how to apply the DLDC methods to solve differential and
integral equations, and how it has advantages over the traditional numerical
methods. Each demonstration is accompanied by an example to further reinforce
the ideas presented in the article.
## 2 Formulating DLDC for STEM Education
Figure 2: A neural network model with input, hidden, and output layers.
In this section, the basic mathematical foundations of the DLDC methods are
presented. This section will be particularly useful for implementing in STEM
education. Before going into too much detail, a brief discussion on neural
network is required. For extensive discussion, the readers can refer to
liu2021mechanistic . A neural network has three types of layers, input,
hidden, and output. The input variables of interest go into input layer, and
the output variables are obtained through the output layer. The hidden
layer(s) takes input variable and pass those through a non-linear function,
called activation function, with optimization parameters. Mathematically, if
$x$ is the input variable, $\mathcal{A}$ is the activation function, $y$ is
the output variable, the equation for a neural network can be written as,
$\centering y=\mathcal{A}(Wx+b)\@add@centering$ (1)
where $W$ is the weight, and $b$ is the bias. By varying the weight and bias
to minimize the error between predicted output and training data, one can
achieve an approximation of any linear and non-linear functional relationship.
A schematic relationship showing the details of a neural network with one
input, one hidden, and one output layer is shown in Figure 2. The convention
of super and sub-scripts for weights and biases are shown in Figure 2.
Following this convention, the output can be written as a function of the
trainable weights, biases, and given input as
$\centering
y=\mathcal{A}\left[W_{1,1}^{2,3}(\mathcal{A}(W_{1,1}^{1,2}x+b_{1}^{2}))+W_{2,1}^{2,3}\mathcal{A}(W_{1,2}^{1,2}x+b_{2}^{2})\right]\@add@centering$
(2)
The trainable $\bm{W}$ and $\bm{b}$ parameters are varied to minimize the
following loss function:
$Lossfunction=\frac{1}{P}\sum_{n=1}^{P}(y-y^{*})^{2}$ (3)
### 2.1 Differential Calculus
Based on this basic definition of neural network, we proceed to show how to
build a neural network from scratch to mimic the finite difference methods.
Let us consider a function $f(x)$ as shown in Figure 2(a). If we want to have
an approximation of first derivative of $f(x)$ at A, the forward difference
method will give us chapra2011numerical ,
$\centering\frac{dy}{dx}=\frac{f(x_{j+1})-f(x_{j})}{x_{j+1}-x_{j}}\@add@centering$
(4)
Here, the index $j$ indicates point A, and $j+1$ indicates the next point on
the function. The central difference method gives,
$\centering\frac{dy}{dx}=\frac{f(x_{j+1})-f(x_{j-1})}{x_{j+1}-x_{j-1}}\@add@centering$
(5)
A higher order approximation of the first derivative involving three points
looks like,
$\centering\frac{dy}{dx}=\frac{-3f(x_{j})+4f(x_{j+1})-f(x_{j+2})}{x_{j+1}-x_{j-1}}\@add@centering$
(6)
Therefore, from observation, we can simply write the general form of these
equations as,
$\centering\frac{dy}{dx}=\sum_{j=1}^{n}w_{j}f(x_{j})\@add@centering$ (7)
Figure 3: (a) A schematic diagram showing a continuous function $f(x)$ and
discrete sampling data, (b) The DLDC structure for the first derivative
(forward difference method).
This form for numerical differentiation is famously known as differential
quadrature method in numerical analysis bellman1971differential ;
bert1996differential ; shu2012differential . If we compare the form of Eqn. 7
with Eqn. 1, a clear parallel can be observed. The finite difference formulas
can be obtained using linear activation functions with zero bias. Inspired
from this observation, a first-order derivative equivalent neural network is
proposed in Figure 2(b). In the figure, the weights and biases are expressed
in index notation for convenience. In $W_{i,j}^{l-1,l}$, the subscripts $i$
and $j$ denotes the incoming neuron and outgoing neuron, respectively. The
superscript $l$ identifies the layers. The index $l-1$ indicates the incoming
layer and $l$ indicates the outgoing layer. The network has one hidden layer,
rectified linear unit activation function, and zero bias. The weights between
input and hidden layer are constrained to be 1, and the weights between the
hidden and output layer (red marked) are optimized. All the activation
functions are linear and biases are set to zero. The cost function can be
written as,
$\centering\frac{1}{N}\sum_{i=1}^{N}\left(\frac{dy}{dx}^{true}-\frac{dy}{dx}^{prediction}\right)^{2}_{i}\@add@centering$
(8)
Here, $N$ is the sample size. It is interesting to note that after
optimization, the trainable weights becomes
$W_{1,1}^{2,3}=\frac{1}{x_{j+1}-x{j}}$ and
$W_{2,3}^{2,1}=\frac{-1}{x_{j+1}-x{j}}$. This value resembles the forward
difference method. If a neural network like this is trained once against data
generated by a known function and it’s derivative, the trained network can be
used to predict the first derivative of any function provided the sampling
remains the same.
This concept has been applied to formulate the first order derivative of
different types of known function. Some of the results are presented in Figure
4. In case of Figure 4(a), the neural network is trained for a simple
sinusoidal function in a range between $0-360$ degrees. With the trained
neural network, we tried to predict the first-order derivative of cosine
function which gives very good prediction. In the next example, see, Figure
4(b), a simple polynomial was used to generate training data for the neural
network. Once the network is trained, the same network is applied to predict
the derivative of a higher-order polynomial (Figure 4(c)) and a relatively
complicated function (Figure 4(d)).
Figure 4: The DLDC results for first-order derivative. (a) Performance for
predicting the first derivative of cosine function. The training was done with
sine function. (b) Training performance for simple polynomial, and (c), (d)
performance of the network to predict derivatives of more complicated
functions.
### 2.2 Integral Calculus
The next building block is to solve an integration problem. Numerically, an
integral of a function with respect to a variable is an estimation of area
under the curve representing that function when plotted against the said
variable. Based on this simple concept, numerical methods such as trapezoidal
or Simpson’s method are in use. However, engineers are mostly interested to
solve time dependent ordinary or partial differential equation. The first
method that comes to mind for such computation is the Euler’s method. Let us
suppose we want to solve an ordinary differential equation.
$\centering\frac{dy}{dt}=f(y,t)\@add@centering$ (9)
where $y$ is the dependent variable and $t$ is time. A DLDC equivalent
structure for explicit Euler method and associated prediction is shown in
Figure 5. The generalized alpha method for Euler’s method is:
$\centering y^{n+1}=y^{n}+\Delta
t\left([1-\alpha].f^{n}+[\alpha].f^{n+1}\right),\@add@centering$ (10)
where, $n$ is the time step, $\Delta t$ is step size, and $\alpha$ is a factor
that indicates an explicit method if the value is 0 and implicit method if the
value is 1. The structure of the DLDC for Euler integration is exactly the
same as forward difference method, except the inputs are the values of the
integrand and dependent variable at the previous step and the output is the
value of the dependent variable in the next time step. In the neural network
shown in Fig. 5 (a), the biases are set to zero and activation functions are
linear. The black weights are fixed to a value of 1 and the red weights are
only allowed to train. The values of the red marked weights in the figure are
the final values of these weights after training. It turns out that after the
training, the weights finally get to a value that makes the output mimic the
equation Eq. 10.
Figure 5: (a) The DLDC structure for the explicit Euler formulation, (b) A
comparison between the analytical and the DLDC prediction for the time
integration.
The numerical prediction in the example was made by solving
$\frac{dy}{dt}=15t^{2}+8t$ with $y=0$ initial condition. The predicted and
analytical solutions diverge slightly near 1 second. But this error is coming
from the Euler method itself as the weights of the trained DLDC is exactly
same as the explicit Euler method. Similar network can be constructed for
implicit Euler method as well.
Figure 6: The DLDC structure for 2-point Gauss quadrature method. The red
marked terms are to be optimized during training.
The next attempt was made to construct a DLDC method for Gauss quadrature
integration method chapra2011numerical . In brief, the Gauss quadrature method
can perform integration by converting integral to fixed limit between -1 to 1.
The equations look like,
$\centering\int_{a}^{b}f(x)dx=\int_{-1}^{1}f\left(\frac{b-a}{2}\xi+\frac{a+b}{2}\right)\frac{dx}{d\xi}d\xi,\@add@centering$
(11)
$\centering\int_{-1}^{1}g(x)dx=\sum_{i=1}^{n}w_{i}g(x_{i}).\@add@centering$
(12)
Here, $n$ is the number of quadrature points and $w_{i}$ are weights assigned
to each point. $g(x_{i})$ is the value of the converted function at each
quadrature points. The DLDC structure for the 2-point Gauss quadrature is
shown in Figure 6. In order to train the network, a third order polynomial
$g(x,\textbf{a})=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$ is assumed and the
corresponding Gauss integral
$\int_{-1}^{1}g(x,\textbf{a})dx=2a_{0}+\frac{2}{3}a_{2}$. The training data is
generated for different values of a and the integral. In the DLDC neural
network, the activation function is rectified linear unit between input and
first hidden layer and at the last layer, and linear for the second hidden
layer. If we train the neural network with respect to
$x_{1},x_{2},w_{11}^{34},w_{21}^{34}$ with following cost function,
$\centering\frac{1}{N}\sum_{n=1}^{N}\left(\sigma_{1}^{4}(x_{1},x_{2},w_{11}^{34},w_{21}^{34};\textbf{a}^{n})-\int_{-1}^{1}g(x,\textbf{a})dx\right)^{2},\@add@centering$
(13)
where
$\centering\sigma_{1}^{4}(x_{1},x_{2},w_{11}^{34},w_{21}^{34};\textbf{a}^{n})=\sum_{j=1}^{N_{i-1}}w_{jk}^{i-1,i}\sigma_{j}^{i-1}+b_{k}^{i}=c_{1}g(x_{1};\textbf{a}^{n})+c_{2}g(x_{2},\textbf{a}^{n}),\@add@centering$
(14)
after the training the quadrature coordinates $x_{1}$, $x_{2}$, and the
weights $c_{1}$ and $c_{2}$ have exact same value as theoretical Gauss
quadrature method. The structure was extended to 3 and 4-point rules and for
all the cases the trained values for the co-ordinates and weights match well
with the analytical value (see Figure 7). Just like the Euler method and
finite difference approximation, the Gauss quadrature can be trained with a
known integration can be used for any other integration without retraining.
Figure 7: A table showing the comparison between different quadrature methods
and optimized values for the DLDC.
## 3 DLDC Formulation for Frontier Research
This section builds upon the ideas of the DLDC formulation for STEM education
and proposes discrete calculus-inspired deep learning methods to solve partial
differential equations and integral equations which are at the core of applied
mechanics.
### 3.1 Solving Partial Differential Equations
#### 3.1.1 Convolution Hierarchical Deep-learning Neural Network (C-HiDeNN)
In this section, we use DLDC to formulate a new interpolation theory based on
convolution, called Convolution Hierarchical Deep-learning Neural Network
(C-HiDeNN). C-HiDeNN interpolants are smoother and more accurate than those of
FEM. They are designed to have arbitrary reproducing polynomial orders,
resulting in superior convergence behavior without using higher order
elements.
To understand C-HiDeNN, we should review HiDeNN-FEM first saha2021hierarchical
; zhang2021hierarchical . HiDeNN-FEM is a neural network representation of
FEM. HiDeNN-FEM shape functions are small neural networks that are designed to
be mathematically equivalent to FEM shape functions. Therefore, the weight and
bias of HiDeNN-FEM shape functions are represented as a function of nodal
coordinates and nodal field variables. The element-wise shape functions are
then gathered to form a global neural network and finally, a loss function
based on the principle of minimum potential energy is minimized.
In HiDeNN-, the solution is obtained by optimizing nodal coordinates and nodal
field variables with respect to the potential energy loss function. There are
two solution schemes for the HiDeNN-FEM. First, the nodal coordinates are kept
fixed and only the nodal variables are updated. This makes the HiDeNN-FEM
mathematically equivalent to FEM, thus the solution accuracy and computation
time of HiDeNN-FEM is on the same order as FEM. Second, both the nodal
coordinates and nodal variables are updated, thus making it equivalent to
r-adaptive FEM. For detailed discussions, readers may refer to
zhang2021hierarchical .
Figure 8: C-HiDeNN formulation for 1D Poisson problem. Light blue and orange
terms are weights and biases of the neural network, respectively. If there is
no weight or bias assigned for a neuron, it will have fixed negligible
weight=1 and bias=0. Functions inside neurons with blue edges represent
activation functions while those with black edges represent inputs (green
color) and outputs (white color) of the neuron. (a) shows nodal coordinates in
both physical and parametric space, focusing on the element of interest
$e_{I}$. Nodal patch domains and other terminologies are defined below. (b)
represents the C-HiDeNN shape function of element $e_{I}$, which constitutes
the hierarchical DeNN layer of the global neural network (d). (c) is the
convolution patch function that can be found at the greed dotted box in (b).
This figure is taken from park2023convolution .
Since HiDeNN-FEM is still based on finite element interpolations, the
improvement of global solution accuracy is insignificant even with the
automatic r-adaptivity. This prompted the idea of convolution to answer the
question: ”Can we incorporate convolution filters to HiDeNN-FEM shape
functions to achieve highly smooth and accurate interpolants?”. This idea is
written as (in 1D):
$u^{h,e}(\xi)=\sum_{i\in A^{e}}N_{i}(\xi)\sum_{j\in
A^{i}_{s}}W^{x_{i}}_{a,p,j}\left(x^{h,e}(\xi)\right)u_{j}=\sum_{k\in
A^{e}_{s}}\tilde{N}_{k}(\xi)u_{k},$ (15)
with elementwise mapping between natural coordianates and physical
coordinates:
$x^{h,e}(\xi)=\sum_{i\in A^{e}}N_{i}(\xi)x_{i}.$ (16)
In Eq. 15, two interpolants appear: 1) general polynomial interpolants,
$N_{i}(\xi)$, defined over element $A^{e}$; 2) convolution patch functions,
$W^{x_{i}}_{a,p,j}\left(x^{h,e}(\xi)\right)$, defined over nodal patch domain
$A^{i}_{s}$. The elemental patch domain is visualized in Fig.8(a) where the
integer-valued patch size $s$ refers to the number of element layers
surrounding node $i$. That is, $A^{i=x_{I}}_{s=2}$ contains node $x_{I-2}$
through node $x_{I+2}$ while $A^{i=x_{I+1}}_{s=2}$ contains node $x_{I-1}$
through node $x_{I+3}$.
The general polynomial interpolants $N_{i}(\xi)$ can be thought as any finite
element shape functions that are compactly supported and satisfy Kronecker
delta and partition of unity. The convolution patch functions
$W^{x_{i}}_{a,p,j}\left(x^{h,e}(\xi)\right)$ are also compactly supported
supported interpolation functions that satisfy Kronecker delta and reproducing
conditions. Here, the dilation parameter $a$ determines the size of support
domain and $p$ is the reproducing polynomial order. For the convolution patch
functions, we borrow from well-developed meshfree techniques. In this study,
the radial point interpolation method is adopted because this method returns
stable interpolants that satisfy Kronecker delta property and reproducing
conditions. For details, readers may refer to liu2001local ;
liu2005introduction .
Finally, the double summation in Eq. 15 is combined into a single summation
over the elemental patch domain $A^{e}_{s}=\cup_{i\in A^{e}}A^{i}_{s}$, and
the resulting convolution interpolants are $\tilde{N}_{k}(\xi)$ where $k\in
A^{e}_{s}$. The convolution interpolants therefore satisfy compact
supportness, partition of unity, Kronecker delta, and reproducing conditions,
making it easy to apply boundary conditions.
In terms of DLDC, the C-HiDeNN shape function can be written as a neural
network illustrated in Fig.8(b). The first two hidden layers refer to the
finite element shape functions, and the third hidden layer is for the
convolution patch functions. Note that the number of neurons is determined by
the number of nodes in nodal patch domains. That is, larger patch size $s$
leads to larger number of neurons or higher connectivity. This is analogous to
the convolutional neural network (CNN) kernels.
In C-HiDeNN, higher reproducing polynomial order $p$ is achieved by setting
large patch size $s$. Based on our preliminary study, $s\geq p$ to get stable
convolution patch functions. In other words, C-HiDeNN can achieve higher order
$p$ by increasing the connectivity of nodes while using the same linear
elements. This is completely different in FEM where higher order elements must
be used to build higher order shape functions. Therefore, the global degrees
of freedom (DOFs) of C-HiDeNN are the same as linear FEM, but the bandwidth of
the global stiffness matrix becomes larger due to increased nodal
connectivity.
The weights of the third hidden layer in Fig.8(b) are built form another sub-
neural network shown in Fig.8(c). This follows the radial point interpolation
theory and details are discussed in park2023convolution . Finally, the
elementwise shape functions are assembled to form a global neural network and
potential energy loss function shown in Fig.8.
To demonstrate superior accuracy of C-HiDeNN compared to FEM, a 2-D Poisson
problem has been solved:
$\begin{split}\nabla\cdot(\nabla
u(\textbf{x}))+b_{f}(\textbf{x})&=0\quad\textrm{in}\ \Omega\\\
u&=0\quad\textrm{on}\ \Gamma\end{split}$ (17)
where $b_{f}(\textbf{x})$ is the body force and a square domain $\Omega$ whose
lower left corner is at $(0,0)$ and upper right cornder is at (10,10) is used.
$\Gamma$ is the boundary of the domain $\Omega$. We set the analytical field
variable as:
$u(\textbf{x})=\frac{1}{625}(x^{2}-10x)(y^{2}-10y)\left(2e^{-2((x-3)^{2}+(y-3)^{2})}+e^{-2((x-7)^{2}+(y-7)^{2})}\right).$
(18)
L2 and H1 error estimators are used:
$\begin{split}\lVert e\rVert_{L_{2}}&=\lVert
u-u^{h}\rVert_{L_{2}}=\frac{\left(\int_{\Omega}(u-u^{h})^{2}dx\right)^{1/2}}{\left(\int_{\Omega}u^{2}dx\right)^{1/2}}\\\
\lVert e\rVert_{H_{1}}&=\lVert
u-u^{h}\rVert_{H_{1}}=\frac{\left(\int_{\Omega}(u-u^{h})^{2}dx+\int_{\Omega}\lVert\nabla
u-\nabla
u^{h}\rVert^{2}_{2}dx\right)^{1/2}}{\left(\int_{\Omega}u^{2}dx+\int_{\Omega}\lVert\nabla
u\rVert^{2}_{2}dx\right)^{1/2}}\end{split}$ (19)
Figure 9: Convergence plot. (a, b) are for $L_{2}$ norm error and (c, d) are
for $H_{1}$ norm error estimation. (a, c) are the error vs. mesh size plots
and (b, d) are the error vs. degrees of freedom plots. Numbers on the graph
are the convergence rates (italic font for C-HiDeNN). For C-HiDeNN, patch size
$s=3$ and dilation parameter $a=30$. The graphs are colored by the reproducing
polynomial order $p$. FEM uses dashed lines and C-HiDeNN uses solid lines.
Fig.9 shows C-HiDeNN-FEM can have arbitrary convergence rates depending on the
reproducing polynomial order $p$. That is, for the same $p$, the convergence
rates (slope of Fig.9(a,c)) of FEM and C-HiDeNN are asymptotically the same.
However, their y-intercepts are different. When $p=1$, C-HiDeNN is around two
orders of magnitude more accurate than FEM for a given mesh size. The gap
decreases as $p$ goes up and the FEM turns around when $p=3$ (i.e., blue
dotted curves are lower than blue solid curves in Fig.9(a,c)). However, this
is not a fair comparison as higher order FEM uses higher order elements that
has more DOFs while C-HiDeNN still uses the linear elements.
The errors are plotted with respect to DOFs in Fig.9(b,d). These two plots
show that C-HiDeNN curves are always lower than those of FEM for the same $p$.
That is, for the same DOFs, C-HiDeNN is always more accurate than FEM. In
other words, to achieve the same level of error, FEM needs more DOFs than
C-HiDeNN for the same $p$. When $p=3$, for example, FEM requires 5 to 10 times
more DOFs than C-HiDeNN-FEM for a given accuracy (see Fig.9(b,d)).
It is important to note that the convolution operations over the local patch
domains are the same as those performed in convolutional neural network (CNN).
The larger the patch size $s$, the more neurons connectivity. The dilation
parameter $a$ acts as a convolution operator, dictating the feature extraction
of the data. Thus, many of the well-developed high-performance computing
algorithms utilizing parallel architectures (such as GPU and TPU) in CNN can
be integrated with C-HiDeNN. park2023convolution discusses in detail on how
the parallel programming can accelerate C-HiDeNN computation and demonstrates
that it can be as fast as commercial FEM software running on CPU.
#### 3.1.2 Space Time Finite Element Method with DLDC
This section proposes a DLDC strategy for solving partial differential
equations based on a space-time finite element formulation hughes1988space ;
hulbert1990space ; wang2017time . Space-time finite element methods are
numerical discretization schemes to predict the spatiotemporal responses of
dynamic systems. When used to approximate solutions to partial differential
equations, they require assumed values of the underlying governing parameters.
For structural problems, these parameters may correspond to material
properties such as density, stiffness, and damping capacity. For many
applications these assumptions are reasonable, but for many others the
distribution of properties throughout a material domain is unknown. For the
latter, experimental observation is required for the accurate calibration of
model parameters. Consider an elastic one-dimensional bar governed by the
following equation.
$\centering EA\frac{\partial^{2}u}{\partial x^{2}}-\rho
A\frac{\partial^{2}u}{\partial t^{2}}+f=0.\@add@centering$ (20)
Here, $t$ is time, $x$ is location, $u$ is displacement, $E$ is elastic
stiffness, $A$ is cross-sectional area, $\rho$ is density of the material, and
$f$ is external excitation. The discretization of the bar must be done through
the space and time dimensions simultaneously, and here it is done with linear
quadrilateral elements. Details of the Galerkin formulation are included in
the Appendix. The nodes are numbered counting upwards through space in the
same order for each time instant, beginning with the first time instant. This
results in a highly structured space-time stiffness matrix that will be
exploited in the developed DLDC method. For this problem with one spatial
dimension, $N$ nodes through space and $T$ nodes through time will result in a
total of $N\times T$ degrees of freedom. Besides the enforcement of initial
and boundary conditions, the stiffness matrix retains a diagonal structure
such that the $N$ equations corresponding to a given time instant only have
$3N$ non-zero coefficients. This corresponds to the displacement field at time
$t$ being only directly related to the displacement fields at times $t-1$ and
$t+1$. For a system discretized with linear quadrilateral elements, a piece of
the matrix equation’s larger diagonal structure is depicted below with
subscripts denoting sub-matrix size and superscripts denoting time step.
$\centering\begin{bmatrix}\mathbf{A}_{N\times N}&\mathbf{B}_{N\times
N}&\mathbf{C}_{N\times N}&0&0\\\ 0&\mathbf{A}_{N\times N}&\mathbf{B}_{N\times
N}&\mathbf{C}_{N\times N}&0\\\ 0&0&\mathbf{A}_{N\times N}&\mathbf{B}_{N\times
N}&\mathbf{C}_{N\times N}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{N\times
1}^{t-2}\\\ \mathbf{u}_{N\times 1}^{t-1}\\\ \mathbf{u}_{N\times 1}^{t}\\\
\mathbf{u}_{N\times 1}^{t+1}\\\ \mathbf{u}_{N\times 1}^{t+2}\\\
\end{bmatrix}=\begin{bmatrix}\mathbf{f}_{N\times 1}^{t-1}\\\
\mathbf{f}_{N\times 1}^{t}\\\ \mathbf{f}_{N\times 1}^{t+1}\\\
\end{bmatrix}\@add@centering$ (21)
Here, $\bf{A}$, $\bf{B}$, and $\bf{C}$ represent repeated sub-matrices within
the global stiffness matrix. For the elastic bar governed by an equation with
no damping term, it is noted that $\bf{A}$ and $\bf{C}$ are identical. The
solution of this matrix equation can be done sequentially, solving only $N$
simultaneous equations at a time before progressing to the next time instant.
These computations can be formulated as an autoregressive neural network.
Autoregressive neural networks are sequential feed-forward models used to
predict sequence values based on prior observations of the sequence
taskaya2005comparative ; triebe2019ar-net . Generally, this can be represented
by the following equation with $ARNN$ denoting the autoregressive neural
network as a function of prior observation data, where $\mathbf{X}^{t}$ and
$\bm{\varepsilon}^{t}$ represent the observation and prediction error
respectively at step $t$ in a sequence of data.
$\centering\mathbf{X}^{t}=ARNN(\mathbf{X}^{t-1},\mathbf{X}^{t-2},\mathbf{X}^{t-3},...,\mathbf{X}^{t-p})+\bm{\varepsilon}^{t}\@add@centering$
(22)
Here, the autoregressive model would be considered of order $p$, since the
previous $p$ observations are used to calculate a prediction. Predictions
spanning multiple time steps may be achieved by using predicted sequence
values as inputs for the prediction of further steps in the sequence. By
inspection of the space-time finite element matrix equation 21, the following
equivalent second order autoregressive neural network can be constructed.
$\centering\mathbf{u}^{t+1}=ARNN(\mathbf{f}^{t},\mathbf{u}^{t},\mathbf{u}^{t-1})=\mathbf{C}^{-1}\left[\mathbf{f}^{t}-\begin{bmatrix}\mathbf{A}&\mathbf{B}\end{bmatrix}\begin{bmatrix}\mathbf{u}^{t-1}\\\
\mathbf{u}^{t}\end{bmatrix}\right]\@add@centering$ (23)
At first glance, this re-framing of the space-time finite element formulation
as a neural network provides no apparent advantage. The computations required
to solve the partial differential equation remain unchanged. However,
recalling that the space-time stiffness matrix elements depend on the
coefficients of the partial differential equation, it follows that the neural
network provides a functional mapping of these coefficients to predicted
sequence values. Sub-matrices, $\bf{A}$, $\bf{B}$, and $\bf{C}$, of the space-
time stiffness matrix are functions of the differential equation coefficients
according to the finite element assembly. In the context of the elastic bar
example, given force and prior displacement fields, the constructed neural
network maps the elastic stiffness, cross-sectional area, and density of the
bar to the displacement field at the next time step.
$\centering\mathbf{u}^{t+1}=ARNN(E,A,\rho:\mathbf{f}^{t},\mathbf{u}^{t},\mathbf{u}^{t-1})+\bm{\varepsilon}^{t+1}=\mathbf{C}^{-1}(E,A,\rho)\left[\mathbf{f}^{t}-\begin{bmatrix}\mathbf{A}(E,A,\rho)&\mathbf{B}(E,A,\rho)\end{bmatrix}\begin{bmatrix}\mathbf{u}^{t-1}\\\
\mathbf{u}^{t}\end{bmatrix}\right]+\bm{\varepsilon}^{t+1}\@add@centering$ (24)
This empowers the automatic learning of the differential equation coefficients
from observed spatiotemporal data. If $E$, $A$, and $\rho$ are taken as the
trainable parameters of the autoregressive neural network, they can be
optimized such that the error term, $\bm{\varepsilon}$, is minimized over all
time steps. By embedding the space-time finite element method in the
architecture of the neural network, variable time step and spatial mesh sizes
can be readily accommodated unlike with black-box neural network
implementation. Furthermore, while traditional dense or deep neural network
models are prone to over-fitting, much less data is needed to accurately
predict system responses with this FEM-informed auto-regressive neural network
since the number of trainable parameters is limited to the number of equation
coefficients. Additionally, unlike traditional deep learning implementations,
extrapolation beyond the training data set to new initial and boundary
conditions can be expected since the computations performed during a forward
pass retain the form of the underlying numerical method.
To numerically demonstrate the proposed finite element-informed auto-
regressive neural network, we consider a spring-mass-damper system governed by
the following differential equation.
$m\frac{\partial^{2}u}{{\partial t}^{2}}+c\frac{\partial u}{\partial
t}+ku=f_{0}sin(\omega t)$ (25)
(a) Exact Training Data and Predicted Solution
(b) Convergence of the Training Procedure
Figure 10: Training the STFEM Auto-Regressive Neural Network with Exact Data.
Here, $t$ is time, $u$ is displacement, $m$ is mass, $c$ is damping
coefficient, $k$ is spring stiffness, and $f_{0}$ and $\omega$ are the maximum
magnitude and frequency of an external sinusoidal force. Training data is
generated by solving the equation using 150 finite elements spanning 3 time
units for the following parameter values and initial conditions:
$m=1,c=10,k=100,f_{0}=10,\omega=2\pi,u(t=0)=1$, and $\frac{\partial
u}{\partial t}(t=0)=1$. A STFEM autoregressive neural network is implemented
with $m$, $c$, and $k$ as the trainable parameters. The sum of the absolute
differences, also known as the L1 loss function, between the true
displacements, $u^{t}$, and their predicted values are taken as the objective
function to be minimized. The Adam optimization algorithm is used to
iteratively update the learnable parameters. A convergence plot of the
training procedure is depicted in Fig. 10(b), which shows that the learnable
parameters in the neural network approach the true values of the equation
coefficients as the sum of the absolute errors, $|{\varepsilon}^{t}|$, in the
prediction of ${u}^{t}$ is minimized, resulting in accurate prediction of the
equation solution visualized in Fig. 10(a). The learned values of these
coefficients within the neural network can then be used to make predictions
for problems with new initial conditions and boundary conditions, generalizing
and extrapolating beyond training data like a traditional numerical method
while remaining a data-driven machine learning framework. As shown in Fig. 11,
the model is able to accurately predict the behavior of the spring-mass-damper
system for a variety of problems with different boundary and initial
conditions even though it was only trained on a single case.
Figure 11: Prediction for New Initial Conditions and Boundary Conditions from
Exact Training Data.
Gaussian noise of mean 0 and variance 0.001 is added to a new set of training
data for the same problem, which is now solved with a coarser 50 element mesh
spanning 3 time units. The resulting time series is shown in Fig. 12, and Fig.
13 illustrates that even when provided with a very small and noisy training
data set the STFEM structured auto-regressive neural network can accurately
capture the system behavior across a variety of new conditions. The ability of
the STFEM neural network to produce accurate predictions for cases beyond the
training data set is not typical of traditional neural networks, which
typically only perform well when applied to cases similar to those seen during
training.
Figure 12: True Solution and Generated Noisy Training Data.
Figure 13: Prediction for New Initial Conditions and Boundary Conditions from
Noisy Training Data.
Dense and deep neural networks offer great function representation capacity,
which make them powerful tools for predicting complex phenomena if they are
provided with sufficiently large and diverse datasets to learn from. With
limited data, neural networks exhibit overfitting due to their large number of
trainable parameters leading to poor interpolation capability. Additionally,
the extrapolation capability of neural networks to make good predictions
beyond the range of their training data is generally very poor with “black
box” implementation. These qualities preclude neural networks from solving
engineering problems where the collection of data is expensive or cannot span
the space of desired prediction capability. By designing an autoregressive
neural network architecture consistent with the space-time finite element
method, the generalizability and interpretability of the numerical method can
be preserved while harnessing the advantage of data-driven machine learning to
reduce the need to make potentially incorrect parameter assumptions. The STFEM
neural network may be trained on video data for only a single loading case to
produce reasonably accurate parameter values. This intelligently designed
neural network, which captures mesh connectivity and spatiotemporal
discretization, both reduces the cost of model training and increases
robustness to data noise. The simple neural network architecture, with limited
learnable parameters, eliminates the need for collecting a massive quantity of
data. In fact, with exact data, only a few time observations are required to
eventually converge to the true system parameters, which could then be used to
make predictions across a wide variety of boundary conditions in the same
manner as finite element analysis. Since the functional structure of the
neural network is specified as the finite element method, the collection of
additional data only serves to reduce inaccuracies arising due to noise in the
observations. This contrasts traditional deep-learning models which seek to
learn a functional structure from scratch, hence their massive data
requirement.
### 3.2 Solving Integral Equations
The section will discuss how the DLDC methods can be used to solve the
integral equations. For brevity and the underlying context of the article,
this section will focus on how to solve the Lippmann-Schwinger equation
lippmann1950variational efficiently with the understanding of the numerical
methods through DLDC. This form of integral equation has myriads of
applications including scattering gopal2022accelerated and multiscale
mechanics of materials zecevic2022new ; moulinec1998numerical . A general
integral equation appears as,
$f(x)=\int_{a}^{b}\mathcal{K}(x,y)u(y)dy,$ (26)
where $u$ is an unknown function, $f$ is a known function, and $\mathcal{K}$
is the kernel function. Solution of the integral equation depends on
determining the kernel function. The importance of this kernel function will
be clear if discussed in the context of solving a differential equation
through Green’s function. For example, consider the following equation,
$\mathcal{L}u(x)=f(x),\forall x\in\mathcal{\mathbf{R}}$ (27)
where, $\mathcal{L}$ is a linear differential operator, $u(x)$ is the unknown
function, and $f(x)$ is the forcing function. The solution of this
differential equation is given by,
$u(x)=\int\mathcal{G}(x-y)f(y)dy,$ (28)
Here, $\mathcal{G}(\cdot)$ is the Green’s function, and the following
relationship holds true,
$\mathcal{L}\mathcal{G}(x,y)=\delta(x-y),$ (29)
where $\delta(\cdot)$ is the Delta function. If one compares Eq. 26 and Eq.
28, it will be apparent that for linear differential operator
$\mathcal{L}(\cdot)$, the Green’s function is the kernel function. If one
solves for this Green’s function, the differential equation will be solved
automatically. It is very hard to analytically solve for this Green’s
function. Usually, this Green’s function is determined numerically.
In recent operator learning paradigm kovachki2021neural ; li2020fourier , this
kernel function is replaced by a convolutional neural network. This neural
network or kernel is trained by adding subsequent convolutional layers. In
order to achieve resolution independence, in Fourier Neural Network (FNO) the
input space of the function variables is taken to Fourier domain and trained
li2020fourier . Another approach is using graph kernels li2020neural where a
subset of the domain points from input and the output functional space are
considered to construct a graph and convolutional kernel is employed for
training. While these networks are shown to solve complex problems including
Navier-Stokes equation li2020fourier , when the domain to solve becomes very
large for example large-scale simulation of 3D microstructure as shown in
saha2021macroscale ; saha2021microscale , the number of training parameters
may explode and a lot of offline training efforts are required. In this
section, using the concepts of mechanics and DLDC, the article gives
formulations for two methods that can reduce the computational burden when
solving large-scale engineering problems through convolutional kernels.
Figure 14: The idea of solving a problem with SCA is explained at the top row
of the figure. The 2D representative volume element with $100\times 100$ mesh
is discretized into 8 clusters. In DLDC method, each cluster is assumed to be
a node of the graph and mechanistic functional spaces are taken as nodal
attributes. The interaction matrix is similar to nodal interaction among the
clusters.
As a demonstration, the article chooses to solve resort to the self-consistent
clustering analysis liu2016self . The SCA algorithm has demonstrated
considerable success in modeling damage, failure, and mechanical response of
multiscale materials system yu2019self ; kafka2021image . The idea of SCA is
to group similar material points together in a structure based on their
elastic response, and solve the Lippmann-Schwinger equation only for the group
or cluster of material points instead of the entire domain. More details can
be found here liu2016self . The following section will touch the basics of SCA
for convenience of discussion.
The homogenization problem for a representative volume element (RVE) can be
modeled as the following integral equation.
$\bm{\epsilon(\mathbf{x})}=\bm{\epsilon^{0}(x)}-\int_{\Omega}\mathcal{G}(x,y):\left[\bm{\sigma(y)}-\bm{C^{0}(y):\epsilon(y)}\right]dy,$
(30)
where $\bm{\epsilon(x)}$ is local strain at material point $x$, $\Omega$ is
the physical domain (RVE), $\bm{\epsilon^{0}(x)}$ is background strain,
$\bm{\sigma(y)}$ is the local stress, $C^{0}$ is the reference material
stiffness. With SCA, the Eq. 30 is not solved for all the material points
rather on clustered domain. The clustering is done by K-means clustering
hartigan1979algorithm . At first, a small, elastic load is applied to the RVE
and the resulting strain concentration matrix is stored. Based on this strain
concentration matrix, the RVE is clustered (see Figure 14). On this clustered
domain the Eq. 30 becomes
$\Delta\bm{\epsilon^{I}}=\Delta\bm{\epsilon^{0}}-\sum_{j=1}^{k}\left[\frac{1}{c^{I}|\Omega|}\int_{\Omega}\int_{\Omega}\chi^{I}(x)\chi^{J}(y)\mathcal{G}(x,y)dxdy\right]:\left[\Delta\sigma^{I}(y)-C^{0}(y):\Delta\epsilon^{J}(y)\right].$
(31)
In the equation, $c$ is cluster identifier, $k$ is the number of clusters,
$I,J$ are the cluster indices, and $\chi^{I}(x)$ is the cluster variables
which assumes value 1 when $x$ is in cluster $I$ and zero otherwise. If one
compares Eqn. 30 and 31, it will be apparent that the Green’s function is
replaced by the convolution operation also known as the interaction tensor.
$\bm{D}^{IJ}=\frac{1}{c^{I}|\Omega|}\int_{\Omega}\int_{\Omega}\chi^{I}(x)\chi^{J}(y)\mathcal{G}(x,y)dxdy.$
(32)
In conventional SCA method, this interaction tensor is pre-computed as so-
called ”offline” database. Based on this pre-computed kernel/interaction
tensor, the strain on each cluster is computed from Eq. 32. With the DLDC
method, the article will show how to determine this kernel on the cluster
centroids so that the number of parameters to be trained can be reduced.
Using graph kernel networks, the Eq. 31 can be solved on reduced functional
space. The justification comes from the mathematical form of the graph kernel
neural networks. The graph kernel network has the following form for variable
update:
$v_{t+1}(x)=\sigma\left(Wv_{t}(x)+\int_{\Omega}\mathcal{K}(x,y,a(x),a(y))v_{t}(y)dy\right).$
(33)
Here, $v_{t+1}$ is the output variable in a transformed functional space at
$(t+1)$-th time step, $\sigma$ is the activation function, $W$ is weight,
$(x,y,a(x),a(y)$ is the edge index in the input function space where $x$ and
$y$ are coordinates, and $a(x)$ and $a(y)$ are values of input function. More
details can be found here []. The discretized form of the Eq. 33 is:
$v_{t+1}(x)=\sigma\left(Wv_{t}(x)+\frac{1}{|N(x)|}\sum_{y\in
N}\mathcal{K}(x,y,a(x),a(y))v_{t}(y)dy\right).$ (34)
Figure 15: Problem statement for the one-dimensional self-consistent
clustering analysis. a) The one-dimensional problem domain for training the
network, b) The cluster centroids after k-means clustering, c) Comparison
between analytical and SCA solution for 33 clusters, d) Evolution of error
during training.
In this equation, $N(x)$ defines a neighborhood inside radius $r$. Drawing
similarities with Eq. 31, it is apparent that the interaction tensor and the
summation term in Eq. 34 have similarities. The only difference is that, in
graph kernel network the neighborhood nodes can be selected by changing the
radius of whereas the SCA is inherently global when computing the interaction
tensor. As an example, let us consider a one-dimensional domain of length 10
as shown in Figure 15 (a) discretized into 1000 points in equal intervals. The
material is a linear elastic one with stiffness varying as
$C(x)=\frac{1}{1+x^{2}}$. On this domain, the overall applied strain
$\epsilon$ is varied from 0.05 to 0.5 and the domain is clustered starting
from 2 to 128 clusters. For each such discretization, the cluster centroids
are taken as the input samples. Essentially, 1000 material points are reduced
2-128 material points with the assumption that the local strain has a constant
value inside each of these clusters. A distribution of possible cluster
centroids with 33 clusters is shown in Figure 15(b). To generate the data, the
solution with SCA is validated against the analytical solution (Figure 15(c)).
The training is performed with a modified version of the graph kernel network
with 6 iterative layers and a radius of 2 (i.e., the interaction is
established between only 2 neighbor clusters). The training performance of the
network is shown in Figure 15. The figure suggests that the training
performance is quite good as the normalized mean squared error is going down
with number of epochs. A more thorough analysis on the efficacy of such
construction is shown in a companion paper owensahagao2022 where it is shown
how varying the hyperparameters of the neural network such as the radius of
influence $r$ or number of layers for training. Apart from the reduction of
training parameters and domain points, the method can also extrapolate to some
extent. It is often identified as the ”resolution independence”. An example of
such extrapolation is shown in Figure 16.The figure shows the prediction of
the kernel learning algorithm and SCA for 300 clusters and 0.2 applied strain.
This particular case is outside the domain of training and testing datasets.
In spite of that, we can see that the kernel learning algorithm is as good as
SCA. However, further testing is required before claiming that kernel learning
method has resolution independence. Nevertheless, the results are promising
and shows a way how we can overcome the limitations of the current graph
kernel networks while solving three-dimensional engineering problems.
Figure 16: Comparison between kernel learning prediction and SCA for 300
clusters.
## 4 Future Developments
While the structure of the neural networks proposed in DLDC shows the
potential of a general framework of solving fundamental calculus and applied
science problems, there are still some limitations that need to be overcome.
For example, the forward difference or central difference neural networks
presented in the differential calculus section works well only same
discretization or sampling intervals are used. This is apparent from the
construction of neural networks as well. However, in real-life applications,
unstructured data or uneven sampling is quite common. The next step is to
extend the DLDC structures similar to differential quadrature method to obtain
the derivative of unstructured data. The weights can be learned via either
convolutional kernel (similar to C-HiDeNN) or fully-connected neural network
or its variant (such as autoencoder). While the DLDC structure gives the exact
reproduction of the Gauss quadrature method, it is to be explored if the
flexibility of deep neural network can be used to see if it can solve
integration that hard to solve with Gauss methods, such as integration around
singularities. The authors are currently working to extend C-HiDeNN into a
more general form and to solve multiresolution problems such as
mcveigh2006multiresolution ; mcveigh2009multiresolution . While the proposed
integral equation solver can significantly reduce the number of training
parameters for graph kernels, we still need to train the method with some
data. The authors are currently working to extend C-HiDeNN method to solve the
integral Lippmann-Schwinger equation so that the convolutional patch function
can capture interaction among the cluster centroids.
## 5 Conclusions
The proposed DLDC method aims to combine the rich knowledge of numerical
methods with modern data science techniques and create a new perspective on
solving challenging problems in engineering and physical sciences. The
building blocks of the DLDC are still in the making. The purpose of building
the tools is to make a general and easy to follow process of making a deep
neural network to solve any governing differential equation. The vision is to
combine all the benefits from different numerical methods and propose a
unified approach to solve engineering problems. At the same time, the DLDC
methods can be used as a teaching tool for calculus in K-12 education system.
These tools will simultaneously introduce the students with deep learning and
applied calculus. This is expected to increase participation in STEM programs
from students. Moreover, the DLDC methods try to avoid multiple training and
can be more flexible using the deep neural network. The DLDC version of finite
element method i.e., C-HiDeNN shows higher accuracy compared to FEM with same
degrees of freedom. The proposed integral equation solver potentially avoids
large number of training parameters that was the bottleneck for using graph
kernel network. However, further research is required to establish the DLDC as
viable method to solve for general problems in science and engineering.
###### Acknowledgements.
The authors would like to acknowledge the support of National Science
Foundation (NSF, USA) grants CMMI-1762035 and CMMI-1934367 and AFOSR, USA
grant FA9550-18-1-0381. C. Park and S. Saha would like to thank the Division
of Orthopaedic Surgery and Sports Medicine at Ann and Robert H. Lurie
Children’s Hospital for their philanthropic grant. The authors would like to
acknowledge the contribution of Alberto Ciampaglia, Department of Mechanical
and Aerospace Engineering, Politecnico di Torino, Italy and Visiting
Researcher, Department of Mechanical Engineering, Northwestern University to
this article.
## 6 Appendix 1: Space-Time Finite Element Formulation for a 1D Elastic Bar
The governing equation is:
$EA\frac{\partial^{2}u}{\partial x^{2}}-\rho A\frac{\partial^{2}u}{\partial
t^{2}}=0.$ (35)
For a bar with left end at $x=L$ and right end at $x=R$, vibrating from time
$t=0$ to time $t=T$, the above equation can be converted to a weak form by
applying the principle of virtual work.
$\int\limits^{T}_{0}\int\limits^{R}_{L}\tilde{v}\left(EA\frac{\partial^{2}u}{\partial
x^{2}}-\rho A\frac{\partial^{2}u}{\partial t^{2}}\right)\partial x\partial
t=0$ (36)
$\int\limits^{T}_{0}\int\limits^{R}_{L}\left(EA\frac{\partial\tilde{v}}{\partial
x}\frac{\partial^{2}u}{\partial x^{2}}-EA\frac{\partial\tilde{v}}{\partial
x}\frac{\partial u}{\partial x}-\rho A\frac{\partial\tilde{v}}{\partial
t}\frac{\partial^{2}u}{\partial t^{2}}+\rho A\frac{\partial\tilde{v}}{\partial
t}\frac{\partial u}{\partial t}\right)\partial x\partial t=0$ (37)
$\left[\int_{0}^{T}EA\tilde{v}\frac{\partial u}{\partial x}\,\partial
t\right]_{x=L}^{x=R}-\int\limits^{T}_{0}\int\limits^{R}_{L}EA\frac{\partial\tilde{v}}{\partial
x}\frac{\partial u}{\partial x}\partial x\partial t-\left[\int_{L}^{R}\rho
A\tilde{v}\frac{\partial u}{\partial t}\,\partial
x\right]_{t=0}^{t=T}+\int\limits^{T}_{0}\int\limits^{R}_{L}\rho
A\frac{\partial\tilde{v}}{\partial t}\frac{\partial u}{\partial t}\partial
x\partial t=0$ (38)
Finite element shape function matrices can be used to discretize the
simplified weak form. For an element spanning $x=x_{0}^{e}$ to $x=x_{1}^{e}$
through space and $t=t_{0}^{e}$ to $t=t_{1}^{e}$ through time, the element
matrices correspond to the following expressions.
$\text{Element Stiffness Matrix
}[K]^{e}=\int\limits^{t_{1}^{e}}_{t_{0}^{e}}\int\limits^{x_{1}^{e}}_{x_{0}^{e}}EA\frac{\partial\tilde{v}}{\partial
x}\frac{\partial u}{\partial x}\partial x\partial t$ (39)
$\text{Element Mass Matrix
}[M]^{e}=\int\limits^{t_{1}^{e}}_{t_{0}^{e}}\int\limits^{x_{1}^{e}}_{x_{0}^{e}}\rho
A\frac{\partial\tilde{v}}{\partial t}\frac{\partial u}{\partial t}\partial
x\partial t$ (40)
The global boundary conditions are prescribed via the other terms in Eq. 38.
$\text{Space Boundary Conditions }=\left[\int_{0}^{T}EA\tilde{v}\frac{\partial
u}{\partial x}\,\partial
t\right]_{x=L},\left[\int_{0}^{T}EA\tilde{v}\frac{\partial u}{\partial
x}\,\partial t\right]_{x=R}$ (41)
$\text{Time Boundary Conditions }=\left[\int_{L}^{R}\rho
A\tilde{v}\frac{\partial u}{\partial t}\,\partial
x\right]_{t=0},\left[\int_{L}^{R}\rho A\tilde{v}\frac{\partial u}{\partial
t}\,\partial x\right]_{t=T}$ (42)
The time boundary condition at $t=0$ corresponds to the enforcement of initial
velocity. The time boundary condition at the final time, $t=T$, can be
eliminated from the global matrix equation in favor of equations that enforce
initial displacement at $t=0$.
## 7 Appendix 2: Time Finite Element Formulation for a Spring Mass Damper
System
The governing equation is:
$m\frac{d^{2}u}{dt^{2}}+c\frac{du}{dt}+ku=f_{0}sin(\omega t).$ (43)
For a mass oscillating from time $t=0$ to time $t=T$, the above equation can
be converted to a weak form by applying the principle of virtual work.
$\int_{0}^{T}\tilde{v}\left(m\frac{d^{2}u}{dt^{2}}+c\frac{du}{dt}+ku-
f_{0}sin(\omega t)\right)\,dt=0$ (44)
$\left[m\tilde{v}\frac{du}{dt}\right]_{t=0}^{t=T}-\int_{0}^{T}m\frac{d\tilde{v}}{dt}\frac{du}{dt}\,dt+\int_{0}^{T}c\tilde{v}\frac{du}{dt}\,dt+\int_{0}^{T}k\tilde{v}u\,dt-\int_{0}^{T}\tilde{v}f_{0}sin(\omega
t)\,dt=0$ (45)
Finite element shape function matrices can be used to discretize the
simplified weak form. For an element spanning $t=t_{0}^{e}$ to $t=t_{1}^{e}$
through time, the element matrices correspond to the following expressions.
$\text{Element Mass Matrix
}[M]^{e}=\int_{t_{0}^{e}}^{t_{1}^{e}}m\frac{d\tilde{v}}{dt}\frac{du}{dt}\,dt$
(46)
$\text{Element Damping Matrix
}[C]^{e}=\int_{t_{0}^{e}}^{t_{1}^{e}}c\tilde{v}\frac{du}{dt}\,dt$ (47)
$\text{Element Stiffness Matrix
}[K]^{e}=\int_{t_{0}^{e}}^{t_{1}^{e}}k\tilde{v}u\,dt$ (48)
$\text{Element Force Matrix
}[F]^{e}=\int_{t_{0}^{e}}^{t_{1}^{e}}\tilde{v}f_{0}sin(\omega t)\,dt$ (49)
The element force matrices account for the force boundary condition due to the
external sinusoidal excitation of the sprung mass. The temporal boundary
conditions are prescribed via the other term in Eq. 45.
$\text{Time Boundary Conditions
}=\left[m\tilde{v}\frac{du}{dt}\right]_{0},\left[m\tilde{v}\frac{du}{dt}\right]_{T}$
(50)
The time boundary condition at $t=0$ corresponds to the enforcement of initial
velocity. The time boundary condition at the final time, $t=T$, can be
eliminated from the global matrix equation in favor of equations that enforce
initial displacement at $t=0$.
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|
# Holonomic Control of Arbitrary Configurations of Docked Modboats
Zhijie Qiao, Gedaliah Knizhnik, and Mark Yim The authors are with the GRASP
Laboratory, University of Pensylvannia, Philadelphia, PA 19104.
<EMAIL_ADDRESS>
###### Abstract
The Modboat is a low-cost, underactuated, modular robot capable of surface
swimming, docking to other modules, and undocking from them using only a
single motor and two passive flippers. Undocking is achieved by causing
intentional self-collision between the tails of neighboring modules in certain
configurations; this becomes a challenge, however, when collective swimming as
one connected component is desirable. Prior work has developed controllers
that turn arbitrary configurations of docked Modboats into steerable vehicles,
but they cannot counteract lateral forces and disturbances. In this work we
present a centralized control strategy to create holonomic vehicles out of
arbitrary configurations of docked Modboats using an iterative potential-field
based search. We experimentally demonstrate that our controller performs well
and can control surge and sway velocities and yaw angle simultaneously.
## I Introduction
Aquatic modular self-reconfigurable robotic systems (MSRRs) are of great
interest to researchers and industry; they can be used for monitoring ocean
environments, performing exploration tasks, and collecting flow
information[1], while adapting to changing conditions and scales of interest.
Conventional wisdom has been that such MSRRs must be built from modules
capable of holonomic motion [2, 3, 4, 5], which has limited development due to
increased complexity. Recent work by the authors, however, has shown that
effective aquatic MSRRs can be built from underactuated surface-swimming
modules [6, 7, 8, 9, 10].
The underactuated modules used in this prior work — the Modboats — use passive
flippers and an inertial rotor powered by a single motor to generate thrust
and steering [6]. They are capable of docking and reconfiguration through
permanent-magnet based docks and can undock from one another using mechanical
self-collision of protruding tails (see Fig. 1), all while using only one
motor [7]. While this passive docking setup and mechanical undocking method
greatly reduces actuation complexity for the system, it also introduces a
major constraint when attempting to swim collectively: any collective behavior
must constantly avoid self-collisions between neighboring tails, which would
cause the docked configuration to disintegrate.
Prior work addressed this concern by introducing restrictions on the phase and
thrust direction allowed for the Modboat modules [8, 9], which are technically
capable of thrusting in any direction [10]. By limiting the thrust direction
to the surge axis of the configuration and requiring in-phase swimming, these
approaches were able to guarantee no unintentional self-collisions and
maintain steerability for any arbitrary structure [8, 9]. But the resulting
controllers could not produce force along the configurations’ sway axes, which
made them highly susceptible to noise and external disturbances. It is also
reasonable to suspect that a structure of $N\geq 3$ modules should be
holonomic in the plane, given a reasonable control law, but thus far such a
controller has not been developed due to the complexity of the collision
constraint [9].
In this work, we propose a holonomic collective control approach for an
arbitrary configuration of docked Modboats, by using an iterative potential
field search to find collision-free desirable movements. While iterative path
planning has been widely adopted in autonomous robot navigation [11, 12, 13,
14], little work has been done to explore its use in the avoidance of internal
collision constraints. Our controller uses such an approach to relax the
assumption in [8, 9] of thrust only along the surge axis, while using a
collision checker to avoid any unintended undocking.
The rest of this work is organized as follows: in Section II we formulate the
problem and present an approach for finding collision-free motions that
generate desired forces. Section III presents the control approach for
determining desired forces for given motions, and Section IV presents the
experimental evaluation of our approach. The results are discussed in Section
V.
Figure 1: Example structure of three docked Modoats in a parallel
configuration, with the structure frame given by $S$ and the surge ($y$) and
sway ($x$) axes marked. For each boat, the top body is shown in black, and the
tail is shown in color, each pointing at
$-\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$$. The passive flippers are not
shown but connect to the tails and do not protrude from the top body
footprint. More details of the Modboat structure can be found in [6].
## II Methodology
Consider a configuration of docked Modboat modules labeled $i\in[1,N]$. To
move, the modules collectively perform a series of swim cycles, in which each
module executes the waveform given in (1), where $\phi_{i}(t)$ is the motor
angle of module $i$ and $\phi_{0,i}$, $A_{i}$, and $T$ are the centerline,
amplitude, and period of the oscillation, respectively. The period is constant
for all modules for concurrency111In this work we use $T=1.5~{}$\mathrm{s}$$,
which has been empirically determined to be an effective period for the system
[15, 9], although the methodology is applicable to any period., while the
centerline and amplitude vary for each module for control. Parameters are
determined at the beginning of each cycle and executed for a single period.
Since the final pose of one cycle and the initial pose of the next cycle may
not be coincident, an additional transition cycle is allowed between swim
cycles to prepare, after which point the process repeats.
$\phi_{i}(t)=\phi_{0,i}+A_{i}\cos{\left(2\pi t/T\right)},\quad t\in[0,T)$ (1)
$F(A)=0.022\mathinner{\\!\left\lvert
A\right\rvert}-0.019,\quad\mathinner{\\!\left\lvert
A\right\rvert}\in[0.9,2.6]$ (2)
In prior work we have shown that the use of waveform (1) results in a linear
relationship between force and amplitude, given in (2) for a period of
$1.5~{}$\mathrm{s}$$[8, 9]. Unlike in prior work [8, 9], however, in this work
we allow $\phi_{0}$ to take on any value. We also allow amplitude to take on
both positive and negative values for numerical flexibility; negative
amplitudes result in a phase shift of
$\pi~{}$\mathrm{r}\mathrm{a}\mathrm{d}$$, but do not affect the generated
force, as per (2).
Figure 2: Numerically simulated phase-space with collision region, with
$\phi_{1}$ being the left boat and $\phi_{2}$ being the right boat. Collided
configurations are shown in red, and free space in green, and top-body
neighbors display an offset version of this relationship. The blue line
represents a sample motion, with boat $1$ having $(\phi_{0},A)=(-1.5,0.5)$,
and boat $2$ having $(\phi_{0},A)=(2.0,1.0)$
###### Problem 1 (Valid Swim Cycles).
Given a configuration of docked Modboats $i\in[1,N]$ and a set of desired
forces
$\vec{F}_{des}=\begin{bmatrix}F_{x,des}&F_{y,des}&\tau_{des}\end{bmatrix}^{T}$
find a set $\Phi$ of pairs $(\phi_{0},A)_{i}$ for $i\in[1,N]$ such that the
generated forces
$\vec{F}=\begin{bmatrix}F_{x}&&F_{y}&&\tau\end{bmatrix}^{T}$
equal (as closely as possible) the desired forces, i.e.
$\vec{F}=\vec{F}_{des}$, while avoiding tail collisions between neighboring
modules.
###### Problem 2 (Valid Transition Cycles).
Given multiple sets of motion pairs $\Phi_{k}$, $k\in\mathbb{Z}^{+}$, where
$k$ represents the swim cycle for which the solution is used, find a set of
motions $\Psi$ to transition from the final pose in $\Phi_{k}$ to the initial
pose in $\Phi_{k+1}$ while avoid tail collisions between neighboring modules.
The goal of this work, then is to find solutions to 1 and 2 under these
conditions, i.e. to find a valid set of swim and transition cycles that
generate a desired set of forces without causing internal collisions. This
will allow the configuration of docked Modboats to function as a holonomic
vehicle.
### II-A Collision Region
The first step in solving 1 is to identify what set of poses causes a
collision between neighbors. We can construct a graph of this collision region
by numerically solving for the intersection of the tails for any pair of
boats, as described in [9]. The resulting graph is shown in Fig. 2 for a
single pair of boats; in this phase space any motion of the two neighboring
boats is a straight line, as proven in [9].
Figure 3: Distance to Collision (DoC) analysis for four different intersecting
cases, using the collision region shown in Fig. 2. $AB$ is the phase-space
trajectory and $CDEF$ is the collision region. The DoC is shown in red in each
case.
The critical question is how to define the distance to collision (DoC) given
the phase-space representation in Fig. 2 and linear trajectories. For any pair
of boats, we can define the DoC as $DoC=D(\phi_{i},A_{i},\phi_{j},A_{j})$.
Alternatively, given a line $AB$ representing a trajectory in phase-space and
a polygon with boundary $CDEF$ representing the collision region, we propose
to define DoC as follows and as illustrated in Fig. 3:
1. 1.
If $AB$ is completely outside $CDEF$, the DoC is the minimum distance from
$AB$ to $CDEF$.
2. 2.
If $AB$ intersects $CDEF$ on point $G$ with $B$ interior to $CDEF$, the DoC is
$-|BG|$.
3. 3.
If $AB$ is inside $CDEF$ with extensions intersecting $CDEF$ on $G$ and $H$,
the DoC is $-\left(|AB|+\min[|AG|,|BH|]\right)$.
4. 4.
If $AB$ intersects $CDEF$ on $G$ and $H$, the DoC is
$-\left(|GH|+\min[|AG|,|BH|]\right)$.
In summary, the DoC captures the distance along the phase-space trajectory to
move into or out of the collision region. While more optimal strategies (i.e.
moving sideways) exist, this strategy is physically meaningful in the context
of the Modboats. Computing the DoC is also expensive, since the calculation
needs to be performed for each pair of boats at every step in the solution
process, so we precompute a DoC table with a discretization of
$0.1~{}$\mathrm{r}\mathrm{a}\mathrm{d}$$ for the centerline and amplitude.
Note that in practice Fig. 2 must be extended to $[-2\pi,2\pi]$ to account for
angle wrapping, which creates several identical but shifted collision regions.
In this case the DoC must be calculated for every collision region and the
most conservative value is used.
### II-B Attractive Field
Given the collision-space representation developed in Section II-A, we begin
to solve 1 by creating an attractive potential to drive the values for
generated structural forces $\vec{F}$ to their desired values $\vec{F}_{des}$,
which can be computed based on the gradient of an error term. Let our error
vector be given by $\vec{e}$ in (3), and our weight vector be given by
$\vec{w}$ in (4), which accounts for the relative unit scale of force and
torque (we have heuristically determined that
$[\begin{matrix}1&1&10\end{matrix}]^{T}$ is effective). Then the error in
generated forces is given by $E=\vec{e}\cdot\vec{w}$.
$\vec{e}=\vec{F}_{des}-\vec{F}$ (3)
$\vec{w}=\begin{bmatrix}w_{x}&w_{y}&w_{\tau}\end{bmatrix}^{T}$ (4)
Eq. (1) is parameterized by $\phi_{0,i}$ and $A_{i}$ for each boat, or
$\vec{\phi}_{0}$ and $\vec{A}$ for all boats. Then (5) and (6) give the
gradient in terms of those quantities.
$\displaystyle\nabla_{\vec{\phi}_{0}}E$
$\displaystyle=-\left(\nabla_{\vec{\phi}_{0}}\vec{F}\right)\vec{w}$ (5)
$\displaystyle\nabla_{\vec{A}\hphantom{{}_{|}}}E$
$\displaystyle=-\left(\nabla_{\vec{A}\hphantom{{}_{|}}}\vec{F}\right)\vec{w}$
(6)
From (2) and Fig. 1, we can define the forces produced by each boat as in (7),
which combine to form the configuration’s generated forces as
$\vec{F}=\sum_{i}\vec{F}_{i}$. Note that the generated force for each boat
points opposite the centerline direction of the tail tip, and $F_{i}=F(A_{i})$
from (2).
$\vec{F}_{i}=\begin{bmatrix}F_{x}\\\ F_{y}\\\
\tau\end{bmatrix}_{i}=\begin{bmatrix}-F_{i}\cos(\phi_{0,i})\\\
-F_{i}\sin(\phi_{0,i})\\\
-F_{i}\sin(\phi_{0,i})x_{i}+F_{i}\cos(\phi_{0,i})y_{i}\end{bmatrix}$ (7)
Figure 4: Repulsive field strength segmented function vs.distance to
collision. This stepped function determines the strength of the repulsive
field and prioritizes the worst collisions.
The gradient of the force produced by each boat can be taken as in (5) with
respect to its centerline, and (6) with respect to its amplitude. These
gradients then form the rows of the gradients in (5) and (6) with respect to
$\vec{\phi}_{0}$ and $\vec{A}$, respectively.
$\displaystyle\nabla_{\phi_{0,i}}\vec{F}_{i}$
$\displaystyle=\begin{bmatrix}\hphantom{-}F_{i}\sin(\phi_{0,i})\\\
-F_{i}\cos(\phi_{0,i})\\\
-F_{i}\cos(\phi_{0,i})x_{i}-F_{i}\sin(\phi_{0,i})y_{i}\end{bmatrix}^{T}$ (8)
$\displaystyle\nabla_{A_{i}}\vec{F}_{i}$
$\displaystyle=\begin{bmatrix}-0.022\cos(\phi_{0,i})\\\
-0.022\sin(\phi_{0,i})\\\
-0.022\left[\sin(\phi_{0,i})x_{i}-\cos(\phi_{0,i})y_{i}\right]\end{bmatrix}^{T}$
(9)
Eqs. (5) and (6) can then be used — with values obtained from (8) and (9) — to
obtain the desired step direction from the attractive field as in (10).
However, since the size of this step may be arbitrarily small, we take a fixed
size step in the direction given by the gradient, where we use the
$\operatorname{sgn}$ function rather than taking the unit vector to account
for the discretized phase space.
$\displaystyle\begin{split}\vec{d}_{\vec{\phi}_{0}}&=\Delta
d\hphantom{|}\operatorname{sgn}{(\nabla_{\vec{\phi}_{0}}E)}\\\
\vec{d}_{\vec{A}\hphantom{{}_{|}}}&=\Delta
d\hphantom{|}\operatorname{sgn}{(\nabla_{\vec{A}\hphantom{{}_{|}}}E)}\end{split}$
(10)
### II-C Repulsive Field
To solve 1 we also need a repulsive field to drive each Modboat’s tail away
from collisions with any of its neighbors, which occur at motor angles given
by Fig. 2. Because of the non-standard nature of distance to collision in this
scenario, we take the following approach: for each boat, let $D_{0}$ represent
the DoC at the current state, while $D_{\phi}$ and $D_{A}$ represent the DoC
after updating $\phi_{0,i}$ and $A_{i}$ while neighboring parameters
(subscript $j$) remain constant, as in (11).
$\begin{bmatrix}D_{0}\\\ D_{\phi}\\\
D_{A}\end{bmatrix}_{i,j}=\begin{bmatrix}D(\phi_{0,i}\hphantom{xxd_{\phi_{0,i}}},A_{i}\hphantom{xxd_{A,i}},\phi_{0,j},A_{j})\\\
D(\phi_{0,i}+d_{\phi_{0,i}},A_{i}\hphantom{xxd_{A,i}},\phi_{0,j},A_{j})\\\
D(\phi_{0,i}\hphantom{xxd_{\phi_{0,i}}},A_{i}+d_{A,i},\phi_{0,j},A_{j})\end{bmatrix}$
(11)
The repulsive field strength $U_{i}$ is constructed in (12) for each boat
using a segmented expression based on the DoC, as shown in Fig. 4, which has
been shown to work well in our experiments, and its sign is adjusted based on
whether the attractive field step results in “more” or “less” collision.
$\begin{bmatrix}U_{\phi}\\\
U_{A}\end{bmatrix}_{i}=\sum_{j}\left(\begin{bmatrix}\operatorname{sgn}{(D_{\phi}-D_{0})}\\\
\operatorname{sgn}{(D_{A}-D_{0})}\end{bmatrix}S(D_{0})\right)_{i,j}$ (12)
The repulsive field outputs in (12) are calculated by summing over all
occupied neighbor sites $j$ (maximum 4). If $U_{i}$ is positive, the update in
(10) is performed for boat $i$; otherwise, no update is performed. This allows
each boat to prioritize the worst collision case and reach a balance among all
its neighbors.
### II-D Overall Procedure
for _$n_{epoch}\in[1,N]$_ do
for _$i\in[1,N_{1}]$_ do
Apply attractive field;
for _$i\in[N_{1},N_{2}]$_ do
Apply attractive field if repulsive field $>$ 0;
for _$i\in[N_{2},N_{3}]$_ do
Apply -attractive field if repulsive field $<$ 0;
Algorithm 1 Potential field algorithm for generating collision free solutions
to 1.
The attractive field in Section II-B and repulsive field in Section II-C
together should be sufficient to solve 1. The repulsive field formulation in
Section II-C has two issues, however: (a) it does not guarantee the final
solution will be collision free, and (b) the Modboats may be unable to
traverse through collided phase-space to reach a desired solution. To address
these issues, we apply a three-stage iterative approach to solving 1 as
presented in Algorithm 1:
1. 1.
Applying the attractive field alone prioritizes finding a solution to the
desired forces.
2. 2.
Applying the attractive field and repulsive field together searches for a
valid solution.
3. 3.
Applying only the repulsive field alone prioritizes ensuring the solution is
valid.
This approach is quite effective, as shown in Section IV, although it still
does not guarantee the final solution is fully collision free or globally
optimal. Since collision avoidance is critical, the final solution used is the
most optimal collision-free solution found along the way.
When applied to all modules $i\in[1,N]$ concurrently, Algorithm 1 produces a
collision-free set $\Phi$ of $(\phi_{0},A)_{i}$ $\forall i\in[1,N]$, solving
1. Each Modboat then executes (1) with those parameters for a single swim
cycle (i.e. a single period), before repeating the process for the next cycle
and set of desired forces.
### II-E Transition Solver
Figure 5: To make finding valid transition sets more tractable, we decompose
the structure into horizontal and vertical sub-problems. For each sub-problem,
only interactions in the given direction are considered, and solution sets are
found via AC [16] for each boat. The full collision-free solution is the
intersection of the solution sets for each boat in the sub-problems.
Although the procedure in Section II-D solves 1 and produces collision free
movements, when used to generate a sequence of movements $\Phi_{k}$ for
$k\in\mathbb{Z}^{+}$ it does not guarantee that the transition from the last
position of one movement to the start of the next is itself collision free. An
additional transition solver is needed to solve 2 and find a collision free
set of transition paths $\Psi$. We assume, for ease of computation, that
transitions take the entirety of $t_{trans}$ for all boats, and occur at a
constant speed.
Any Modboat can take either a clockwise or counterclockwise path from the last
position of its previous cycle to the first position of its next. For any pair
of modules, then, there are four possible transitions, each of which is a line
in the phase-space in Fig. 2 and can be simply checked for collision. Of
these, at least one is likely to be collision-free, but to increase the number
of available options we also consider negating the amplitude of all boats in
the next cycle, which shifts the start locations of the next cycle but does
not otherwise affect the solution. This set of possible transitions is then
run through the arc consistency (AC) algorithm [16] to find a valid collision-
free transition for each boat pair.
Because Modboat configurations are two-dimensional there many interactions
between neighbors, since diagonal neighbors — while not directly neighbors —
influence each other through shared connections. To make the search space more
tractable, we split the spatial constraint by considering horizontal
interactions and vertical interactions as two separate problems, as in Fig. 5.
When AC is run on each row (column) of the horizontal (vertical) sub-problem,
it generates a set of valid transitions for each boat. The full solution is
then generated by intersecting the solution sets for each boat from the
horizontal and vertical sub-solutions, which recovers the full spatial
constraint.
In our testing with simulated random inputs and prior locations, a valid
transition set exists for all boats in over $99\%$ of cases for square
configurations of Modboats with 2–5 boats to a side. However, since the number
of possible sets to evaluate scales exponentially with the number of boats,
this approach becomes intractable for larger structures; a more efficient
strategy will need to be developed for larger configurations, but is left to
future work.
A critical thing to note is that the transition requires a finite $t_{trans}$
between cycles of (1), which means that decisions about generated forces are
made at intervals of $T+t_{trans}$. Using a small $t_{trans}$ maintains the
responsiveness of the overall controller, but the fast transitions introduce
unwanted dynamic disturbances. Using a large $t_{trans}$ minimizes the dynamic
disturbances, but slows the response of the overall controller. We minimize
this impact by selecting the solution set that results in the minimum overall
distance travelled, but we also expect most transitions to be small during
normal operation, since the control generated in Section III should be
relatively continuous unless sharp maneuvers are needed.
## III Control
The methodology of Section II provides the parameters for (1) — namely
$\phi_{0}$ and $A$ — for each boat $i$ in a structure given its shape and a
set of desired values $\vec{F}_{des}$. To determine these desired values and
implement holonomic control for the configuration as a whole, PID control is
applied to the equations of motion as derived in [8].
Control for a desired yaw angle $\Theta$ is given in Eqs. 13, 14 and 15, where
$\Omega$ is the observed angular velocity of the structure, $T$ is the period
of (1), $I$ is the Modboat configuration’s moment of inertia, and $C_{R}$ is a
drag coefficient [9]. Note that (13) includes a prediction of the yaw at the
end of the current cycle, which accounts for the delay introduced by discrete
control during sharp yaw maneuvers.
$e_{\Theta}=\Theta_{des}-\left(\Theta_{obs}+\Omega T\right)$ (13)
$\alpha=K_{p,\Theta}e_{\Theta}+K_{d\Theta}\frac{de_{\Theta}}{dt}$ (14)
$\tau_{des}=I\alpha+C_{R}|\Omega|\Omega$ (15)
Velocity control in [8, 9] — which presented steerable vehicles — considered
the desired velocity as a surge (i.e. body-fixed frame) velocity value. Since
the method presented in this paper results in a holonomic vehicle, we consider
the desired values as
$\vec{v}_{des}=[\begin{matrix}v_{x,des}&v_{y,des}\end{matrix}]^{T}$ expressed
in the world frame. Just as in [8] an artificial linear acceleration is
computed based on the observed error and then integrated into the commanded
velocity $\vec{v}_{c}$, as in Eqs. 16, 17 and 18.
$\vec{e}_{v}=\vec{v}_{des}-\vec{v}_{obs}$ (16)
$\vec{a}_{y}=K_{p,v}\vec{e}_{v}+K_{d,v}\frac{d\vec{e}_{v}}{dt}$ (17)
$\vec{v}_{c}=\vec{v}_{des}+(\gamma^{n-1}\sum_{i=0}^{n-1}\vec{a}_{y}+\vec{a}_{y})T$
(18)
A diminishing coefficient $\gamma<1$ is added to (17) to prevent control lag
due to excessive error accumulation. The commanded velocity is then converted
to desired force values using the quadratic drag relationship [8] and
converted to the boat frame using (19). $\vec{F}_{des}$ can then be
constructed from (19) and (15) and decomposed into $\vec{\phi}_{0}$ and
$\vec{A}$ using the methodology in Section II.
$\begin{bmatrix}F_{x,des}\\\
F_{y,des}\end{bmatrix}=C_{L}\begin{bmatrix}\hphantom{-}\sin(\Theta_{obs})&\cos(\Theta_{obs})\\\
-\cos(\Theta_{obs})&\sin(\Theta_{obs})\end{bmatrix}\left(\vec{v}_{c}\odot\vec{v}_{c}\right)$
(19)
## IV Experiments
Experiments were conducted in a $4.5~{}$\mathrm{m}$\times
3.0~{}$\mathrm{m}$\times 1.2~{}$\mathrm{m}$$ tank of still water equipped with
an OptiTrack motion capture system that provides the real-time position,
velocity, and orientation data for each boat at
$120~{}$\mathrm{H}\mathrm{z}$$. The control methodology in Sections II and III
was computed in Python on an offboard PC and, the resulting parameters for (1)
and transitions were sent to each Modboat via WiFi at the beginning of each
cycle.
Experimental evaluation of the methodology presented in this work is ongoing,
but preliminary results show great potential. The following four evaluations
have so far been conducted using three boats in a parallel configuration, and
the results shown in Fig. 6:
1. 1.
$[\begin{matrix}v_{x}&v_{y}&\Theta\end{matrix}]_{des}=[\begin{matrix}0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
for a duration of $90~{}$\mathrm{s}$$.
2. 2.
$[\begin{matrix}v_{x}&v_{y}&\Theta\end{matrix}]_{des}=[\begin{matrix}0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
for a duration of $90~{}$\mathrm{s}$$.
3. 3.
$[\begin{matrix}v_{x}&v_{y}&\Theta\end{matrix}]_{des}=[\begin{matrix}0.03~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0.01~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
for a duration of $90~{}$\mathrm{s}$$.
4. 4.
$[\begin{matrix}v_{x}&v_{y}&\Theta\end{matrix}]_{des}=[\begin{matrix}0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
for a duration of $60~{}$\mathrm{s}$$, then
$[\begin{matrix}v_{x}&v_{y}&\Theta\end{matrix}]_{des}=[\begin{matrix}0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
for another $60~{}$\mathrm{s}$$.
(a) (b) (c) (d)
Figure 6: Experimental trajectories for the four preliminary tests.
$[\begin{matrix}v_{x}&v_{y}&\Theta\end{matrix}]_{des}=$ (a)
$[\begin{matrix}0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
(b)
$[\begin{matrix}0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$
(c)
$[\begin{matrix}0.03~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0.01~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$.
(d)
$[\begin{matrix}0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$,
then
$[\begin{matrix}0~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&0.04~{}$\mathrm{m}\mathrm{/}\mathrm{s}$&\pi/2~{}$\mathrm{r}\mathrm{a}\mathrm{d}$\end{matrix}]$.
The COM pose is shown in orange.
## V Discussion
The results in Fig. 6 show excellent preliminary performance. The Modboat
configuration is able to travel in a desired direction in either a head-on
(Fig. 6a) or sideways (Fig. 6b) orientation, and even mix velocities in the
world frame (Fig. 6c). In a significant stress test, the controller is also
able to provide excellent 90 degree turning performance, as evidenced by the
trajectory in Fig. 6d, which shows little overshoot and good direction
tracking during both legs of the trajectory. Notably, our controller has a
difficult time maintaining a steady orientation (all tests in Fig. 6), but
nevertheless maintains reasonable directional swimming.
## VI Conclusion
In this work we have presented a potential-field based control approach to
allow a group of three or more docked Modboats to function as a holonomic
vehicle, improving on prior work [8, 9] that could only create a steerable
vehicle. This method works for arbitrary structures of docked modules and has
the potential to be scaled to large structures containing many modules.
Preliminary experimental results have shown that this approach is effective in
controlling the velocity and somewhat effective at controlling orientation in
a few different testing scenarios. The major limitation of this strategy is
the need for a fixed transition time outside the main swim cycle, which is
necessary to avoid collisions but creates undesirable dynamics and slows down
the overall controller response time. Future work will consider a transition
strategy that minimizes this impact, as well as solution strategies for the
swim cycle that minimize the need for transitions.
Future work will also consider testing with larger numbers of modules and
expanding our evaluation to non-controlled environments, like lakes or rivers.
This will stress the ability of our controller to reject temporary
disturbances. Developing this approach to work with larger configurations will
also necessitate developing a more efficient transition solver that displays
better scaling.
## Acknowledgment
We thank Dr. M. Ani Hsieh for the use of her instrumented water basin in
obtaining all of the testing data.
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|
# }
[
###### keywords:
Finite Elements, MFEM library, Lagrange, Raviart-Thomas, Taylor-Hood, Laplace
Equation, Navier-Stokes Equations.
###### keywords:
Elementos Finitos, Librería MFEM, Lagrange, Raviart-Thomas, Taylor-Hood,
Ecuación de Laplace, Ecuaciones de Navier-Stokes.
[firstpage = 1, volume = 0, number = 0, month = 00, year = 1900, day = 00,
monthreceived = 0, yearreceived = 1900, monthaccepted = 0, yearaccepted =
1900]
authors[] authors department = Departamento de Matemáticas, institution =
Universidad Nacional de Colombia, city = Bogotá D.C., country = Colombia ]
We revise the finite element formulation for Lagrange, Raviart-Thomas, and
Taylor-Hood finite element spaces. We solve Laplace equation in first and
second order formulation, and compare the solutions obtained with Lagrange and
Raviart-Thomas finite element spaces by changing the order of the shape
functions and the refinement level of the mesh. Finally, we solve Navier-
Stokes equations in a two dimensional domain, where the solution is a steady
state, and in a three dimensional domain, where the system presents a
turbulent behaviour. All numerical experiments are computed using MFEM
library, which is also studied.
Revisamos la formulación de elementos finitos para los espacios de elementos
finitos de Lagrange, Raviart-Thomas y Taylor-Hood. Solucionamos la ecuación de
Laplace en su formulación de primer y segundo orden, y comparamos las
soluciones obtenidas con los espacios de elementos finitos de Lagrange y
Raviart-Thomas al cambiar el orden de las funciones base y el nivel de
refinamiento de la malla. Finalmente, resolvemos las ecuaciones de Navier-
Stokes en un dominio bidimensional, donde la solución es un estado estable, y
en un dominio tridimensional, donde el sistema presenta un comportamiento
turbulento. Todos los experimentos numéricos se realizan utilizando la
librería MFEM, la cual es también estudiada.
###### Contents
1. 1 Preliminaries
1. 1.1 Partial Differential Equations
1. 1.1.1 Laplace equation
2. 1.1.2 Navier-Stokes equations
2. 1.2 The Finite Element Method
1. 1.2.1 FEM for elliptic problems
2. 1.2.2 Lagrange finite elements
3. 1.2.3 Lagrange spaces of higher order
4. 1.2.4 Raviart-Thomas finite elements
5. 1.2.5 Taylor-Hood finite elements
3. 1.3 MFEM Library
1. 1.3.1 Information about the library
2. 1.3.2 Overview
3. 1.3.3 Code structure
2. 2 Lagrange vs. Raviart-Thomas finite elements
1. 2.1 Problem
2. 2.2 Code
3. 2.3 Tests
4. 2.4 Results
5. 2.5 Analysis
6. 2.6 Some other examples
3. 3 Numerical Experiments with NS
1. 3.1 2D Experiment: Steady State
2. 3.2 3D Experiment: Turbulence
4. 4 Conclusion and Perspectives
5. 5 Appendices
1. 5.1 Appendix A : Code for comparison
2. 5.2 Appendix B : Numerical values of the comparison
3. 5.3 Appendix C : MiniApp Code for Navier-Stokes
## 1 Preliminaries
In this section we are going to recall the theoretical background needed in
the rest of the paper. First, we are going to review the finite element
methods used for the Laplace equation in second and first order form. We write
the strong and weak form of the problem and introduce the Lagrange and mixed
finite element spaces. We also introduce the MFEM library by giving an
overview of its main characteristics and the general structure of a finite
element code in MFEM.
### 1.1 Partial Differential Equations
For the scope of this work, partial differential equations are of the form
$F\left(u,p,\frac{\partial p}{\partial x_{i}},\frac{\partial u}{\partial
x_{i}},\frac{\partial^{2}p}{\partial x_{i}^{2}},\frac{\partial^{2}u}{\partial
x_{i}^{2}},f\right)=0;\ \ i=1,2,3,4$
where $u:\Omega\subseteq\mathbb{R}^{3}\rightarrow U\subseteq\mathbb{R}^{3}$,
$p:\Omega\subseteq\mathbb{R}^{3}\rightarrow P\subseteq\mathbb{R}$, $f$ is a
restriction parameter and $F$ is any mathematical expression within its
variables.
It is well known that PDEs have multiple solutions, in fact, there is a
vectorial space consisting of all the solutions for a given equation. For this
reason, the equation is usually presented with a boundary condition that
forces a unique solution for the equation [2]. If some problem is being solved
in a given domain $\Omega\subseteq\mathbb{R}^{3}$, whose boundary is $\Gamma$,
then the problem has the form
$\left\\{\begin{split}&F\left(u,p,\frac{\partial p}{\partial
x_{i}},\frac{\partial u}{\partial x_{i}},\frac{\partial^{2}p}{\partial
x_{i}^{2}},\frac{\partial^{2}u}{\partial x_{i}^{2}},f\right)=0\text{ in
}\Omega,\\\ &F_{0}\left(u,p,\frac{\partial p}{\partial x_{i}},\frac{\partial
u}{\partial x_{i}},\frac{\partial^{2}p}{\partial
x_{i}^{2}},\frac{\partial^{2}u}{\partial x_{i}^{2}},g\right)=0\text{ in
}\Gamma.\end{split}\right.$
The two main equations that we treat are the Laplace equation and the Navier-
Stokes equation, whose represent a fluids phenomenon in real life. Note that
$x_{1}=x,x_{2}=y,x_{3}=z,x_{4}=t$ (in time-space interpretation) and that, $p$
is the fluid’s pressure and $u$ is the fluid’s velocity.
#### 1.1.1 Laplace equation
Laplace equation consists on finding
$p:\Omega\subseteq\mathbb{R}^{3}\rightarrow\mathbb{R}$ such that
$-\Delta p=f,\text{ in }\Omega$ (1)
where $f:\Omega\rightarrow\mathbb{R}$ is a given function and $\Delta
p=\nabla\cdot\nabla p=\frac{\partial^{2}p}{\partial
x^{2}}+\frac{\partial^{2}p}{\partial y^{2}}+\frac{\partial^{2}p}{\partial
z^{2}}$ [4]. It is clear that the equation is a partial differential equation
of the form presented before in Section 1.1 because it only depends on the
second order derivatives of $p$.
Now, we present two common ways of imposing a boundary condition to the
problem: Dirichlet and Neumann. Let $\Gamma$ be the boundary of $\Omega$.
Dirichlet boundary condition
The Dirichlet boundary condition is
$p=g\text{ in }\Gamma$ (2)
where $g:\Gamma\rightarrow\mathbb{R}$ is a given function [4]. If $g=0$, the
condition is called homogeneous Dirichlet boundary condition.
Neumann boundary conditions
The Neumann boundary condition is
$-\nabla p\cdot\eta=h\text{ in }\Gamma$ (3)
where $h:\Gamma\rightarrow\mathbb{R}$ is a given function and $\eta$ is the
boundary’s normal vector [4].
As seen on [4], problem (1) can be stated with the two types of boundary
condition (2) and (3), by imposing each condition on different parts of
$\Gamma$. However, the version of the problem presented later on Section 1.2.2
has the homogeneous Dirichlet boundary condition.
Finally, the Laplace equation (1) can be formulated on two different ways:
first order formulation and second order formulation. The second order
formulation is the one presented already on (1), which involves second order
derivatives. Now, if we set $u=\nabla p$, the problem can be stated [4] as
$\left\\{\begin{split}\mbox{div}(u)=-f\text{ in }\Omega,\\\ u=\nabla p\text{
in }\Omega,\end{split}\right.$ (4)
which is the first order formulation for the problem and, notice that it only
involves first order derivatives of $u$ and $p$.
On Section 2, we compare both formulations of Laplace equation (1) and (4). We
use Lagrange finite elements to solve (1) and Raviart-Thomas finite elements
to solve (4).
#### 1.1.2 Navier-Stokes equations
In this section we revise incompressible Stokes and Navier-Stokes equations
with Dirichlet boundary condition. First, incompressible Navier-Stokes
equations consist on finding the velocity $u:\Omega\rightarrow\mathbb{R}^{3}$
and the pressure $p\rightarrow\mathbb{R}$ that solve the system of equations
(5) [8].
$\begin{split}\frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\nabla
p=f,\text{ in }\Omega,\\\ \nabla\cdot u=0,\text{ in }\Omega,\\\ u=g,\text{ in
}\Gamma,\end{split}$ (5)
where $\Omega\subseteq\mathbb{R}^{3}$ is the spatial domain,
$\Gamma=\partial\Omega$ the boundary of the domain, $\nu$ is called the
kinematic viscosity (more information below), $f$ is the forcing term (given
function), and $g$ is the specified Dirichlet boundary condition.
On the other hand, when removing the non-linear term $(u\cdot\nabla)u$ from
the first equation, we get Stokes linear equations which is system of
equations (6) [8].
$\begin{split}\frac{\partial u}{\partial t}-\nu\Delta u+\nabla p=f,\text{ in
}\Omega,\\\ \nabla\cdot u=0,\text{ in }\Omega,\\\ u=g,\text{ in
}\Gamma.\end{split}$ (6)
Take into account that $u(x,y,z,t)=(u_{x},u_{y},u_{z})$ is a vector for each
point in time-space and that $p(x,y,z,t)=p$ is a scalar for each point in
time-space. For real life fluid problems modeled by these equations, $(x,y,z)$
is the position of a fluid’s particle in space and $t$ is the time. Also, the
equation $\nabla\cdot u=0$ is the one that establishes the incompressible
condition for the fluid.
Kinematic Viscosity
On the modeling of fluid flow, as presented on [7], a very important parameter
appears and its called Reynolds number, $Re$. On [7], it is defined as
$Re=\frac{\rho LU}{\mu},$
where $\rho$ is the density of the fluid, $L$ is the characteristic linear
dimension of the domain of the flow, $U$ is some characteristic velocity, and
$\mu$ is the fluid’s viscosity. For example, as seen on [7], $L$ is defined
supposing that $\Omega=[0,L]^{n}$, i.e., $L$ is the length of the domain on
each direction, and $U$ can be taken as the square root of the average initial
kinetic energy in $\Omega$.
However, for the purpose of this work we neglect the parameters $L$ and $U$ in
such way that $Re=\frac{\rho}{\mu}$. Therefore, the kinematic viscosity of a
fluid is, according to [7],
$\nu:=\frac{\mu}{\rho}=\frac{1}{Re}.$ (7)
Notice that if the viscosity of the fluid, $\mu$, is higher and the density of
the fluid, $\rho$, is lower, then, the kinematic viscosity of the fluid,
$\nu$, is higher. This parameter quantifies the resistance that a fluid
imposes to movement due to an external force, like gravity [7].
On Section 3, we do some numerical experiments in 2D and 3D using the Navier
Miniapp of MFEM library, which solves (5), by using the formulation presented
on Section 1.2.5.
### 1.2 The Finite Element Method
First, on Section 1.2.1 we show how the solution for a differential equation
is also a solution for a minimization problem and a variational problem. This
serves as a basic case for showing that partial differential equations are, in
fact, solved via minimization or variational problems; which are the ones
solved with finite element methods.
Then, in Sections 1.2.2 and 1.2.4 we study two finite element methods. On both
of them, the following procedure was applied:
1. 1.
Consider the problem of solving Laplace equation (23) with homogeneous
Dirichlet boundary condition.
2. 2.
Multiply by some function (test function) and integrate by parts. Apply
boundary conditions.
3. 3.
Discretize the domain and select finite-dimensional function spaces for the
solution and the test functions.
4. 4.
Produce a matrix system to solve for solution weights in the linear
combination representation of the approximated solution.
The basis functions that form part of the finite-dimensional spaces are called
shape functions. In Lagrange formulation, those are the functions in $V_{h}$,
and in mixed formulation, those are the functions in $H_{h}^{k}$ and
$L_{h}^{k}$, where the parameter $h$ denotes the size of the elements in the
triangulation of the domain. Moreover, in Lagrange formulation the boundary
condition is essential, and in mixed formulation, it is natural.
Furthermore, on Section 1.2.3 we show high order shape functions used in
Lagrange finite element method and, on Section 1.2.5 we study the most common
finite elements used to approximate the solution for Navier-Stokes equations.
#### 1.2.1 FEM for elliptic problems
Let $\mathbb{D}$ be the two-point boundary value problem (8) taken from [2].
$\begin{split}&-u^{\prime\prime}(x)=f(x),\ x\in(0,1),\\\
&u(0)=u(1)=0,\end{split}$ (8)
where $f$ is a given continuous function and
$u^{\prime\prime}(x)=\frac{d^{2}u(x)}{dx^{2}}$. Notice that this problem is a
partial differential equation of the form presented on section 1.1 but with
all the functions being of the form $\mathbb{R}\rightarrow\mathbb{R}$ and
$\Omega=(0,1)$ being a 1D domain, with homogeneous boundary condition ($0$ on
$\Gamma=\\{0,1\\}$).
By integrating $-u^{\prime\prime}(x)=f(x)$ twice, it is clear that the problem
(8) has a unique solution $u$ [2]. For example, if $f(x)=e^{x}$
$\begin{split}&-u^{\prime\prime}(x)=e^{x}\\\
\implies&-u^{\prime}(x)=e^{x}+c_{1}\\\
\implies&-u(x)=e^{x}+c_{1}x+c_{2}\end{split}$
and by applying boundary conditions,
$\left\\{\begin{split}&u(0)=0\implies c_{2}=-1,\\\ &u(1)=0\implies
c_{1}=1-e.\end{split}\right.$
It follows that $u(x)=-e^{x}+(e-1)x+1$ is the unique solution. Recall that the
boundary conditions force the problem to have a unique solution, as mentioned
previously on the work.
Now, following [2], define the linear space $V$ of all continuous functions on
$[0,1]$ that vanish at $\\{0,1\\}$, whose derivative is piecewise continuous
and bounded on $[0,1]$. Also, define the linear functional
$F:V\rightarrow\mathbb{R}$, by
$F(v)=\frac{1}{2}\int_{0}^{1}\left[v^{\prime}(x)\right]^{2}dx-\int_{0}^{1}f(x)v(x)dx.$
With this settled, let $\mathbb{M}$ be the optimization problem of finding
$u\in V$ such that
$F(u)\leq F(v)$ (9)
for all $v\in V$, and let $\mathbb{V}$ be the variational problem of finding
$u\in V$ such that
$\int_{0}^{1}u^{\prime}(x)v^{\prime}(x)dx=\int_{0}^{1}f(x)v(x)dx$ (10)
for all $v\in V$.
Remark that if $u,v\in V$ then $w=u+\alpha v\in V$ for any
$\alpha\in\mathbb{R}$, because $w$ is continuous in $[0,1]$;
$w(1)=w(0)=u(0)+\alpha v(0)=0$, i.e. vanishes on $\\{0,1\\}$; and
$w^{\prime}=u^{\prime}+\alpha v^{\prime}$ is piecewise continuous and bounded
on $[0,1]$. The rest of this section is focused on showing that $\mathbb{D}$,
$\mathbb{M}$ and $\mathbb{V}$ are equivalent problems [2].
Equivalence $\mathbb{D}\iff\mathbb{V}$
Let $u_{\mathbb{D}}$ be the solution for $\mathbb{D}$. Then, multiply
$-u_{\mathbb{D}}^{\prime\prime}(x)=f(x)$ on both sides by some $v\in V$ (this
$v$ is called a test function) and integrate to obtain
$-\int_{0}^{1}u_{\mathbb{D}}^{\prime\prime}(x)v(x)dx=\int_{0}^{1}f(x)v(x)dx.$
(11)
Using the formula for integration by parts,
$\int_{0}^{1}a(x)b^{\prime}(x)dx=a(1)b(1)-a(0)b(0)-\int_{0}^{1}a^{\prime}(x)b(x)dx,$
(12)
with $a(x)=v(x)$ and $b(x)=u_{\mathbb{D}}^{\prime}(x)$, and applying the fact
that $v(0)=v(1)=0$ we get
$\int_{0}^{1}u_{\mathbb{D}}^{\prime\prime}(x)v(x)dx=-\int_{0}^{1}v^{\prime}(x)u_{\mathbb{D}}^{\prime}(x)dx.$
(13)
Then, replacing (13) on equation (11) we get that
$\int_{0}^{1}v^{\prime}(x)u_{\mathbb{D}}^{\prime}(x)dx=\int_{0}^{1}f(x)v(x)dx.$
(14)
Notice that (14) is the equation associated to the variational problem
$\mathbb{V}$ (see (10)), and as $v\in V$ was arbitrary, we have that
$u_{\mathbb{D}}$ satisfies (14) (and so, (10)) for all $v\in V$. Therefore,
$u_{\mathbb{D}}$ is also a solution for $\mathbb{V}$.✓
On the other hand, let $u_{\mathbb{V}}\in V$ be the solution for $\mathbb{V}$
(on [2], it is shown that the solution for $\mathbb{V}$ is unique). Then, we
have by (10) that, for all $v\in V$,
$\int_{0}^{1}u_{\mathbb{V}}^{\prime}(x)v^{\prime}(x)dx-\int_{0}^{1}f(x)v(x)dx=0.$
(15)
Now, applying integration by parts (12) in the same way as before, we obtain
(13). Replacing (13) on (15) and unifying the integral, we get
$-\int_{0}^{1}[u_{\mathbb{V}}^{\prime\prime}(x)+f(x)]v(x)dx=0$ (16)
for all $v\in V$. Then, by (18) we have that
$u_{\mathbb{V}}^{\prime\prime}(x)+f(x)=0$ for $x\in(0,1)$. In other words,
$-u_{\mathbb{V}}^{\prime\prime}(x)=f(x),\ x\in(0,1).$ (17)
Notice that (17), along with the fact that
$u_{\mathbb{V}}(0)=u_{\mathbb{V}}(1)=0$, is the equation associated to the
differential problem $\mathbb{D}$ (see (8)). Therefore, $u_{\mathbb{V}}$ is
also a solution for $\mathbb{V}$ as long as $u^{\prime\prime}(x)$ exists and
is continuous (regularity assumption). But, this last condition for
$u_{\mathbb{V}}$ holds, as seen on [2], so, the result holds.✓
To complete the proof, we have to prove (18), which is exercise 1.1 from [2]:
$\begin{split}\text{If $w$ is continuous on $[0,1]$ and}\\\
\int_{0}^{1}w(x)v(x)dx=0\text{ for all }v\in V,\\\ \text{then }w(x)=0\text{
for all }x\in(0,1).\end{split}$ (18)
Proof: By contradiction, suppose that $w(x)\not=0$ for some $x\in(0,1)$. Then,
$w(x_{0})=c\in\mathbb{R}$ for $x_{0}\in(0,1)$. Without loss of generality,
suppose that $c>0$ (for $c<0$ the argument is analogous). As $w$ is
continuous, there is an interval centered at $x_{0}$,
$I=(x_{0}-\delta,x_{0}+\delta)$, such that $f(x)>0$ for $x\in I$. Now, define
$v(x)=\left\\{\begin{split}0,\ if\ &x\not\in I,\\\ \frac{c}{\delta}[x+(\delta-
x_{0})],\ if\ &x\in(x_{0}-\delta,x_{0})\\\
-\frac{c}{\delta}[x-(\delta+x_{0})],\ if\
&x\in(x_{0},x_{0}+\delta),\end{split}\right.$
which is a function sketched on Figure 1.
Figure 1: Sketch of the proof for (18) involving an arbitrary function $w(x)$
(solid black line) and the built function $v\in V$ (solid red line).
Notice that $v(0)=v(1)=0$ because there is always a $\delta>0$ such that
$0<x_{0}-\delta$ and $x_{0}+\delta<1$, and $v(x)=0$ for $x\not\in I$. Also,
$v(x)$ is continuous on $[0,1]$ by construction (it was built with 4 lines
that have connections in $(x_{0}-\delta,0),(x_{0},c)$ and $(x_{0}+\delta,0)$).
Moreover,
$v^{\prime}(x)=\left\\{\begin{split}0,\ if\ &x\not\in I,\\\ \frac{c}{\delta},\
if\ &x\in(x_{0}-\delta,x_{0}),\\\ -\frac{c}{\delta},\ if\
&x\in(x_{0},x_{0}+\delta),\end{split}\right.$
is clearly piecewise continuous and bounded on $[0,1]$. Therefore, $v\in V$.
However, as $w(x)v(x)>0$ for $x\in I$ and $w(x)v(x)=0$ for $x\not\in I$, then
$\int_{0}^{1}w(x)v(x)dx=\int_{x_{0}-\delta}^{x_{0}+\delta}w(x)v(x)dx>0.$
In other words, if $w(x)\not=0$ for some $x\in(0,1)$, then we found $v\in V$
such that $\int_{0}^{1}w(x)v(x)dx\not=0$, which contradicts the hypothesis
that $\int_{0}^{1}w(x)v(x)dx=0$ for all $v\in V$. Therefore, $w(x)=0$ for all
$x\in(0,1)$.∎
Equivalence $\mathbb{V}\iff\mathbb{M}$
Let $u_{\mathbb{V}}\in V$ be the solution for $\mathbb{V}$. Take some $v\in V$
and set $w=v-u_{\mathbb{V}}\in V$. Then
$\begin{split}&F(v)=F(u+w)\\\
=&\frac{1}{2}\int_{0}^{1}\left[u_{\mathbb{V}}^{\prime}(x)+w^{\prime}(x)\right]^{2}dx-\int_{0}^{1}f(x)[u_{\mathbb{V}}(x)+w(x)]dx\\\
=&\frac{1}{2}\left(\int_{0}^{1}[u_{\mathbb{V}}^{\prime}(x)]^{2}dx+\int_{0}^{1}2u_{\mathbb{V}}^{\prime}(x)w^{\prime}(x)dx+\int_{0}^{1}[w^{\prime}(x)]^{2}dx\right)\\\
&-\int_{0}^{1}f(x)u_{\mathbb{V}}(x)dx-\int_{0}^{1}f(x)w(x)dx\\\
=&\frac{1}{2}\int_{0}^{1}[u_{\mathbb{V}}^{\prime}(x)]^{2}dx-\int_{0}^{1}f(x)u_{\mathbb{V}}(x)dx+\int_{0}^{1}u_{\mathbb{V}}^{\prime}(x)w^{\prime}(x)dx\\\
&-\int_{0}^{1}f(x)w(x)dx+\frac{1}{2}\int_{0}^{1}[w^{\prime}(x)]^{2}dx\\\
\stackrel{{\scriptstyle\eqref{equiv1:eq4}}}{{=}}&\frac{1}{2}\int_{0}^{1}[u_{\mathbb{V}}^{\prime}(x)]^{2}dx-\int_{0}^{1}f(x)u_{\mathbb{V}}(x)dx+\frac{1}{2}\int_{0}^{1}[w^{\prime}(x)]^{2}dx\\\
=&F(u_{\mathbb{V}})+\frac{1}{2}\int_{0}^{1}[w^{\prime}(x)]^{2}dx\geq
F(u_{\mathbb{V}}).\end{split}$
In other words, we have that
$F(v)\geq F(u_{\mathbb{V}}).$ (19)
Notice that (19) is the equation associated to the optimization problem
$\mathbb{M}$ (see (9)), and as $v\in V$ was arbitrary, we have that
$u_{\mathbb{V}}$ satisfies (19) (and so, (9)) for all $v\in V$. Therefore,
$u_{\mathbb{V}}$ is also a solution for $\mathbb{M}$.✓
On the other hand, let $u_{\mathbb{M}}\in V$ be a solution for $\mathbb{M}$.
Then, for any $v\in V$ and any $\alpha\in\mathbb{R}$ we have that
$u_{\mathbb{M}}+\alpha v\in V$, and so $F(u_{\mathbb{M}})\leq
F(u_{\mathbb{M}}+\alpha v)$ by (9). Therefore, the minimum for
$F(u_{\mathbb{M}}+\alpha v)$ is achieved at $\alpha=0$.
Now, define $g(\alpha)=F(u_{\mathbb{M}}+\alpha v)$, which is a differentiable
function [2]. We have that
$\begin{split}&g(\alpha)=F(u_{\mathbb{M}}+\alpha v)\\\
=&\frac{1}{2}\int_{0}^{1}\left[u_{\mathbb{M}}^{\prime}(x)+\alpha
v^{\prime}(x)\right]^{2}dx-\int_{0}^{1}f(x)[u_{\mathbb{M}}(x)+\alpha
v(x)]dx\\\
=&\frac{1}{2}\int_{0}^{1}[u_{\mathbb{M}}^{\prime}(x)]^{2}dx+\alpha\int_{0}^{1}u_{\mathbb{M}}^{\prime}(x)v^{\prime}(x)dx+\frac{\alpha^{2}}{2}\int_{0}^{1}[v^{\prime}(x)]^{2}dx\\\
&-\int_{0}^{1}f(x)u_{\mathbb{M}}(x)dx-\alpha\int_{0}^{1}f(x)v(x)dx.\end{split}$
(20)
Since $g(\alpha)=F(u_{\mathbb{M}}+\alpha v)$ has a minimum at $\alpha=0$, then
$g^{\prime}(0)=0$. Therefore, taking the derivative on (20) and replacing
$\alpha=0$, we get
$\begin{split}&g^{\prime}(\alpha)=\int_{0}^{1}u_{\mathbb{M}}^{\prime}(x)v^{\prime}(x)dx+\alpha\int_{0}^{1}[v^{\prime}(x)]^{2}dx-\int_{0}^{1}f(x)v(x)dx\\\
\implies&g^{\prime}(0)=\int_{0}^{1}u_{\mathbb{M}}^{\prime}(x)v^{\prime}(x)dx-\int_{0}^{1}f(x)v(x)dx.\end{split}$
(21)
Finally, as $g^{\prime}(0)=0$, we get from (21), that
$\int_{0}^{1}u_{\mathbb{M}}^{\prime}(x)v^{\prime}(x)dx=\int_{0}^{1}f(x)v(x)dx.$
(22)
Notice that (22) is the equation associated to the variational problem
$\mathbb{V}$ (see (10)), and as $v\in V$ was arbitrary, we have that
$u_{\mathbb{M}}$ satisfies (22) (and so, (10)) for all $v\in V$. Therefore,
$u_{\mathbb{M}}$ is also a solution for $\mathbb{V}$.✓
As mentioned before, on [2], it is shown that the solution for $\mathbb{V}$ is
unique. Therefore, as $\mathbb{V}$ and $\mathbb{M}$ are equivalent, then the
solution for $\mathbb{M}$ is also unique.
#### 1.2.2 Lagrange finite elements
In this section we consider the Laplace equation [2],
$\begin{split}-\Delta&p=f\text{ in }\Omega,\\\ &p=0\text{ in
}\Gamma,\end{split}$ (23)
where $\Omega\subseteq\mathbb{R}^{2}$ is an open-bounded domain with boundary
$\Gamma$, $f$ is a given function and $\Delta p=\frac{\partial^{2}p}{\partial
x^{2}}+\frac{\partial^{2}p}{\partial y^{2}}$. Following [2], consider the
space
$V=\\{v:v\text{ continuous on }\Omega,\frac{\partial v}{\partial
x},\frac{\partial v}{\partial y}\text{ piecewise continuous on }\Omega\text{
and }v=0\text{ on }\Gamma\\}.$
Alternatively we can work in the Sobolev space (see [2, 4, 3])
$H^{1}(\Omega)=\\{v\in L^{2}(\Omega)\ \Big{|}\ \frac{\partial v}{\partial
x},\frac{\partial v}{\partial y}\in L^{2}(\Omega)\\}.$
Here $L^{2}(\Omega)=\\{v:\Omega\rightarrow\mathbb{R}\
\Big{|}\int_{\Omega}v^{2}<\infty\\}$.
We multiply the first equation of (23) by some $v\in V$ (referred to as test
function) and integrate over $\Omega$ to obtain
$-\int_{\Omega}\Delta p\ v=\int_{\Omega}f\ v.$ (24)
Applying divergence theorem we obtain the Green’s formula ([2]),
$-\int_{\Omega}\Delta p\ v=\int_{\Omega}\nabla v\cdot\nabla p-\int_{\Gamma}v\
\nabla p\cdot\eta,$ (25)
where $\eta$ is the outward unit normal to $\Gamma$. Since $v=0$ on $\Gamma$,
the third integral equals $0$. Note that the boundary integral does not depend
on $p$’s value on $\Gamma$ but rather on the normal derivative of $p$ in
$\Gamma$. Due to this fact the boundary condition $p=0$ on $\Gamma$ is know as
an essential boundary condition.
Then, replacing (25) on (24), we get,
$\int_{\Omega}\nabla v\cdot\nabla p=\int_{\Omega}f\ v.$ (26)
This holds for all $v\in V$. This is called weak formulation of the Laplace
equation (23). We remark that, according to [2], if $p\in V$ satisfies (26)
for all $v\in V$ and is sufficiently regular, then $p$ also satisfies (23),
i.e., it’s a (classical) solution for our problem. For more details see [2]
and references therein.
In order to set the problem for a computer to solve it, we are going to
discretize it and encode it into a linear system.
First, consider a triangulation $T_{h}$ of the domain $\Omega$. This is,
$T_{h}=\\{K_{1},\dots,K_{m}\\}$ a set of non-overlapping triangles such that
$\Omega=K_{1}\cup\dots\cup K_{m}$ and no vertex ($N_{i}$) of one triangle lies
on the edge of another triangle, as seen on Figure 2.
Figure 2: A triangulation for a given domain $\Omega$ showing a node $N_{i}$,
and formed by some triangles $K_{j}$. Note: Triangles have been separated in
the edges to take a better look, but the triangulation has no empty spaces.
Visualization: [6].
The $h$ in the notation $T_{h}$ is a measure of the size of mesh, it usually
refers to a typical element diameter or perhaps to the largest element
diameter in the triangulation. In this manuscript $h$ is defined by
$h=\max\\{\mbox{diam}(K):K\in T_{h}\\}$ where $\mbox{diam}(K)=\text{longest
side of }K$.
Now, let $V_{h}=\\{v:v\text{ continuous on }\Omega,v|_{K}\text{ linear for
}K\in T_{h},\ v=0\text{ on }\Gamma\\}$.
We consider the nodes ($N_{1},\dots,N_{M}$) of the triangulation that are not
on the boundary, because $p=0$ there, and we define some functions
$\varphi_{j}\in V_{h}$ in such way that
$\varphi_{j}(N_{i})=\left\\{\begin{array}[]{lcc}1&,\ i=j\\\ \\\ 0&,\
i\not=j\\\ \end{array}\right.$
for $i,j=1,\dots,M$. See Figure 3 for an illustration of $\varphi_{j}\in
V_{h}$.
Figure 3: Illustration of the function $\varphi_{j}$ produced with MFEM
library. On the left picture we plot the function $\varphi_{j}$. On the right
picture we show the same plot depicting the elements of the underlying
triangulation. Visualization: [6].
With this, $V_{h}=\mbox{span}\\{\varphi_{i}:i=1,\dots,M\\}$ and for any given
$v\in V_{h}$ we have $v(x)=\sum_{j=1}^{M}\xi_{j}\varphi_{j}(x),$ with
$\xi_{j}=v(N_{j})$ and $x\in\Omega\cup\Gamma$. So, $V_{h}$ is a finite-
dimensional subspace of $V$. See [2] for details.
Then, if $p_{h}\in V_{h}$ satisfies (26) for all $v\in V_{h}$, in particular,
$\int_{\Omega}\nabla p_{h}\cdot\nabla\varphi_{j}=\int_{\Omega}f\ \varphi_{j},\
\ j=1,\dots,M.$ (27)
Since $\nabla p_{h}=\sum_{i=1}^{M}\xi_{i}\nabla\varphi_{i}$ with
$\xi_{i}=p_{h}(N_{i})$, replacing on (27) we get,
$\sum_{i=1}^{M}\xi_{i}\int_{\Omega}\nabla\varphi_{i}\cdot\nabla\varphi_{j}=\int_{\Omega}f\
\varphi_{j},\ \ j=1,\dots,M.$ (28)
Finally, (28) is a linear system of $M$ equations and $M$ unknowns
($\xi_{1},\dots,\xi_{M}$), which can be written as,
$A\xi=b,$ (29)
where $A[i,j]=\int_{\Omega}\nabla\varphi_{i}\cdot\nabla\varphi_{j}$,
$\xi[i]=p_{h}(N_{i})$ and $b[i]=\int_{\Omega}f\ \varphi_{i}$.
We can solve (29) with MFEM library as done on Section 2 (Example#1). Before
continuing with the next section, let us show some theorems regarding the
error between the solution $p$ for problem (23) and its approximation $p_{h}$.
The theorems are presented on the general form but, after the proof, we show
how is it used on our particular problem.
For the following theorem, $A$ is a bilinear form on $V\times V$ and $L$ is a
linear form on $V$ such that
1. 1.
$A$ is continuous ($\mathcal{C}$)
There is a constant $\gamma>0$ such that
$|A(v,w)|\leq\gamma||v||_{V}||w||_{V},\ \forall v,w\in V.$
2. 2.
$A$ is $V$-elliptic ($V_{\epsilon}$)
There is a constant $\alpha>0$ such that
$\alpha||v||_{V}^{2}\leq A(v,v).$
###### Theorem 1.1.
[2] Céa Lemma
If $p\in V$ is the solution for
$A(p,v)=L(v),\ \forall v\in V$
and $p_{h}\in V_{h}\subset V$ is the solution for
$A(p_{h},v_{h})=L(v_{h}),\ \forall v_{h}\in V_{h}$
then,
$||p-p_{h}||_{V}\leq\frac{\gamma}{\alpha}||p-v_{h}||_{V},\ \forall v_{h}\in
V_{h}.$
###### Proof 1.2.
Using the hypothesis that $A(p,v)=L(v)$ for all $v\in V$, along with the fact
that $V_{h}\subset V$, we have that $A(p,v_{h})=L(v_{h})$ for all $v_{h}\in
V_{h}$. Now, we subtract the last equation with the one given as hypothesis,
$A(p_{h},v_{h})=L(v_{h})$, to get, $A(p-p_{h},v_{h})=L(v_{h})-L(v_{h})=0$ for
all $v_{h}\in V_{h}$.
For an arbitrary $w\in V_{h}$, let $v_{h}=p_{h}-w\in V_{h}$. Then,
$\begin{split}&\alpha||p-p_{h}||^{2}_{V}\\\
\leq&A(p-p_{h},p-p_{h})+0\quad(\text{See }V_{\epsilon})\\\
=&A(p-p_{h},p-p_{h})+A(p-p_{h},w)\\\ =&A(p-p_{h},p-p_{h}+w)\\\
=&A(p-p_{h},p-p_{h}+p_{h}-v_{h})\\\ =&A(p-p_{h},p-v_{h})\\\
\leq&\gamma||p-p_{h}||_{V}||p-v_{h}||_{V}\quad(\text{See
}\mathcal{C}).\end{split}$
In other words, we have that
$\alpha||p-p_{h}||^{2}_{V}\leq\gamma||p-p_{h}||_{V}||p-v_{h}||_{V}$ for all
$v_{h}\in V_{h}$. Dividing by $\alpha||p-p_{h}||_{V}$ on both sides, we get,
$||p-p_{h}||_{V}\leq\frac{\gamma}{\alpha}||p-v_{h}||_{V}$ for all $v_{h}\in
V_{h}.$ Note that $||p-p_{h}||_{V}\not=0$ because $p_{h}$ is supposed to be an
approximation for $p$, and not the exact solution.
On the particular case of (26), we have that $A(p,v)=\int_{\Omega}\nabla
v\cdot\nabla p$ and $L(v)=\int_{\Omega}f\ v.$ Using Cauchy-Schwarz Inequality
for Integrals we have that
$A(p,v)=\int_{\Omega}\nabla v\cdot\nabla p\leq\sqrt{\int_{\Omega}|\nabla
p|^{2}}\cdot\sqrt{\int_{\Omega}|\nabla v|^{2}}=1\cdot||p||_{V}||v||_{V}.$
In other words, the parameter for the continuity of the bilinear form of our
particular case is $\gamma=1$. Also, notice that
$1\cdot||v||_{V}^{2}=\left(\sqrt{\int_{\Omega}|\nabla
v|^{2}}\right)^{2}=\int_{\Omega}|\nabla v|^{2}=\int_{\Omega}\nabla v\nabla
v=A(v,v).$
That is, the parameter for the $V$-ellipticity of the bilinear form for our
particular case is $\alpha=1$. Therefore, Céa Lemma ensures that
$\sqrt{\int_{\Omega}|p-p_{h}|^{2}}\leq\sqrt{\int_{\Omega}|p-v_{h}|^{2}}$ for
all $v_{h}\in V_{h}$.
Now, before presenting the second theorem, define the operator
$\mathcal{I}^{h}:\mathcal{C}(\Omega)\rightarrow V_{h}$ that associates every
continuous function whose domain is $\Omega$, $f\in\mathcal{C}(\Omega)$, with
a function $\mathcal{I}^{h}f\in V_{h}$ [4]. This operator is an interpolation
operator defined by the nodes of the triangulation of $\Omega$: if
$T_{h}=\\{K_{1},\dots,K_{m}\\}$ is a triangulation of $\Omega$, then
$\mathcal{I}^{h}f(K_{i})=f(K_{i}),\ i=1,\dots,m$.
###### Theorem 1.3.
[4] Let $\Omega\subseteq\mathbb{R}^{2}$ be a polygonal domain. Let
$\\{T_{h_{i}}\\}$ be a family of triangulations of $\Omega$, with $T_{h_{i}}$
being a quasi-uniform triangulation. Then,
$||\mathcal{I}^{h}p-p||_{1}\leq c\cdot h\cdot||p||_{2},$
where
$||f||_{2}=\left(\int_{\Omega}f(x)^{2}+|\nabla
f(x)|^{2}+\sum_{ij}(\partial_{ij}f)^{2}dx\right)^{1/2}.$
The proof of this theorem is out of the scope for this work. However, it can
be checked on [4]. In summary, for our case, Céa Lemma states that Laplace
problem can be approximated by the space $V_{h}$, and Theorem 1.3 states that
the approximation is a good one.
#### 1.2.3 Lagrange spaces of higher order
This short section has the purpose of explaining Lagrange finite element
spaces of higher order. Previously, on Section 1.2.2, when introducing
Lagrangian elements, the shape function’s degree was set to one. Better
approximations can be obtained by using polynomials of higher order. One can
define, for a fixed order $k$,
$\begin{split}V^{k}_{h}=\\{v:&v\text{ continuous on }\Omega,\\\ &v|_{K}\text{
polynomial of order at most }k,K\in T_{h},\ v=0\text{ on
}\Gamma\\}.\end{split}$
For example, as seen in [4], the space of Bell triangular finite elements for
a given triangulation $T_{h}$ is the space of functions that are polynomials
of order 5 when restricted to every triangle $K\in T_{h}$. That is, if $v$ is
in this space, then,
$v|_{K}(x,y)=a_{1}x^{5}+a_{2}y^{5}+a_{3}x^{4}y+a_{4}xy^{4}+\dots+a_{16}x+a_{17}y+a_{18}$
for all $K\in T_{h}$. Here, the constants $a_{i},\ i=1,\dots,18$ correspond to
$v$’s DOF (degrees of freedom).
On Figures 4 and 5, we present a visualization of some shape functions of
different orders. We encourage the reader to compare them with Figure 3 and
notice the degree of the polynomial in the nonzero part of the shape
functions.
Figure 4: Illustration of finite element basis (shape) functions of order 2.
On the left picture we show one continuous basis function. On the right we
also show the underlying triangulation. Visualization: [6].
Figure 5: Illustration of finite elements basis function of orders 5 (left)
and 10 (right). Visualization: [6].
#### 1.2.4 Raviart-Thomas finite elements
First, let’s define some important spaces, where $\Omega$ is a bounded domain
in $\mathbb{R}^{2}$ and $\Gamma$ its boundary. See [2, 3, 4] and references
therein for details. The space of all square integrable functions,
$L^{2}(\Omega)=\\{v:\Omega\rightarrow\mathbb{R}\
\Big{|}\int_{\Omega}v^{2}<\infty\\}.$
We also use the first order Sobolev space,
$H^{1}(\Omega)=\\{v\in L^{2}(\Omega)\ \Big{|}\ \frac{\partial v}{\partial
x},\frac{\partial v}{\partial y}\in L^{2}(\Omega)\\}$
and the subspace of $H^{1}(\Omega)$ of functions with vanishing value on the
boundary,
$H_{0}^{1}(\Omega)=\\{v\in H^{1}(\Omega)\ |\ v=0\ on\ \Gamma\\}.$
We also introduce the space of square integrable vector functions with square
integrable divergence,
$H(\mbox{div};\Omega)=\\{\mathbf{v}\in L^{2}(\Omega)\times L^{2}(\Omega)\ |\
\mbox{div}(\mathbf{v})\in L^{2}(\Omega)\\}.$
As above, let $\Omega\in\mathbb{R}^{2}$ be a bounded domain with boundary
$\Gamma$ and consider problem (23). This time we require explicitly that $f\in
L^{2}(\Omega)$. Recall (from Section 1.2.2) that this problem can be reduced
to
$\int_{\Omega}\nabla v\cdot\nabla p=\int_{\Omega}f\ v,\text{ for all $v\in
V$},$
where Dirichlet boundary condition ($p=0\ in\ \Gamma$) is essential. Recall
that we can take $V=H_{0}^{1}(\Omega)$ as seen in [2, 3].
Let $u=\nabla p$ in $\Omega$. Then, problem (23) can be written as the
following system of fist order partial differential equations,
$\begin{split}u&=\nabla p\text{ in }\Omega\\\ \mbox{div}(u)&=-f\text{ in
}\Omega\\\ p&=0\text{ in }\Gamma,\end{split}$ (30)
because $\Delta p=\mbox{div}(\nabla p)$. Now, following a similar procedure as
in Section 1.2.2, multiply the first equation of (30) by some $\mathbf{v}\in
H(\mbox{div};\Omega)$ and integrate both sides to obtain,
$\int_{\Omega}u\ \mathbf{v}=\int_{\Omega}\nabla p\cdot\mathbf{v}.$ (31)
Consider Green’s identity [3],
$\int_{\Omega}\mathbf{v}\cdot\nabla p+\int_{\Omega}p\
\mbox{div}(\mathbf{v})=\int_{\Gamma}(\mathbf{v}\cdot\eta)p,$ (32)
where $\eta$ is the normal vector exterior to $\Gamma$.
Replacing (32) in (31), and considering the third equation of (30), we get,
$\int_{\Omega}u\ \mathbf{v}+\int_{\Omega}p\
\mbox{div}(\mathbf{v})=\int_{\Gamma}(\mathbf{v}\cdot\eta)p,$ (33)
On the other hand, we can multiply the second equation of problem (30) by some
$w\in L^{2}(\Omega)$, integrate and obtain,
$\int_{\Omega}w\ \mbox{div}(u)=-\int_{\Omega}f\ w.$ (34)
Note that the boundary integral depends directly on the value of $p$ in
$\Gamma$. And, this is referred to as the case of a natural boundary
condition. Observe that the same boundary condition appeared as an essential
boundary condition in the second order formulation considered before (Section
1.2.2). In this first order formulation it showed up as a natural boundary
condition.
Finally, applying boundary condition $p=0\ \text{in }\Gamma$ into (33), and
joining (33) and (34). We get the following problem deduced from (30),
$\begin{split}&\int_{\Omega}u\ \mathbf{v}+\int_{\Omega}p\
\mbox{div}(\mathbf{v})=0\\\ &\int_{\Omega}w\ \mbox{div}(u)=-\int_{\Omega}f\
w.\end{split}$ (35)
For this problem, which is a variational formulation of (30), the objective is
to find $(u,p)\in H(\mbox{div};\Omega)\times L^{2}(\Omega)$ such that it is
satisfied for all $\mathbf{v}\in H(\mbox{div};\Omega)$ and all $w\in
L^{2}(\Omega)$.
For the discretized problem related to (35), in [3] the following spaces are
defined for a triangulation $T_{h}$ of the domain $\Omega$ and a fixed integer
$k\geq 0$,
$\begin{split}&H_{h}^{k}:=\\{\mathbf{v_{h}}\in H(\mbox{div};\Omega)\ |\
\mathbf{v_{h}}|_{K}\in RT_{k}(K)\text{ for all }K\in T_{h}\\},\text{ and}\\\
&L_{h}^{k}:=\\{w_{h}\in L^{2}(\Omega)\ |\ w_{h}|_{K}\in\mathbb{P}_{k}(K)\text{
for all }K\in T_{h}\\},\end{split}$
where
$\begin{split}&\mathbb{P}_{k}(K)=\\{p:K\rightarrow\mathbb{R}\ |\ p\text{ is a
polynomial of degree }\leq k\\},\text{ and}\\\
&RT_{k}(K)=[\mathbb{P}_{k}(K)\times\mathbb{P}_{k}(K)]+\mathbb{P}_{k}(K)x.\end{split}$
Note that $\mathbf{p}\in RT_{k}(K)$ if and only if there are some
$p_{0},p_{1},p_{2}\in\mathbb{P}_{k}(K)$ such that
$\mathbf{p}(x)=\begin{pmatrix}p_{1}(x)\\\
p_{2}(x)\end{pmatrix}+p_{0}(x)\begin{pmatrix}x\\\ y\end{pmatrix}\text{ for all
}\begin{pmatrix}x\\\ y\end{pmatrix}\in K,$
and, also note that $\mathbf{p}$ has a degree of $k+1$.
Then, (35) gives the following discrete problem: find $(u_{h},p_{h})\in
H_{h}^{k}\times L_{h}^{k}$ such that
$\begin{split}&\int_{\Omega}u_{h}\ \mathbf{v}_{h}+\int_{\Omega}p_{h}\
\mbox{div}(\mathbf{v}_{h})=0\\\ &\int_{\Omega}w_{h}\
\mbox{div}(u_{h})=-\int_{\Omega}f\ w_{h},\end{split}$ (36)
for all $\mathbf{v}_{h}\in H_{h}^{k}$ and all $w_{h}\in L_{h}^{k}$.
As spaces $H_{h}^{k}$ and $L_{h}^{k}$ are finite dimensional, they have a
finite basis. That is, $H_{h}^{k}=\mbox{span}\\{\varphi_{i}:i=1,\dots,M\\}$
and $L_{h}^{k}=\mbox{span}\\{\psi_{j}:j=1,\dots,N\\}$. Then,
$u_{h}=\sum_{i=1}^{M}u_{i}\varphi_{i}$ and
$p_{h}=\sum_{j=1}^{N}p_{j}\psi_{j}$, where $u_{i}$ and $p_{j}$ are scalars.
In particular, as $\varphi_{k}\in H_{h}^{k}$ and $\psi_{l}\in L_{h}^{k}$, we
have that problem (36) can be written as,
$\begin{split}&\int_{\Omega}\left(\sum_{i=1}^{M}u_{i}\varphi_{i}\right)\varphi_{k}+\int_{\Omega}\left(\sum_{j=1}^{N}p_{j}\psi_{j}\right)\mbox{div}(\varphi_{k})=0\\\
&\int_{\Omega}\psi_{l}\mbox{div}\left(\sum_{i=1}^{M}u_{i}\varphi_{i}\right)=\int_{\Omega}f\psi_{l},\end{split}$
(37)
for $k=1,\dots,M$ and $l=1,\dots,N$. Which is equivalent to the following, by
rearranging scalars,
$\begin{split}&\sum_{i=1}^{M}u_{i}\int_{\Omega}\varphi_{i}\cdot\varphi_{k}+\sum_{j=1}^{N}p_{j}\int_{\Omega}\psi_{j}\mbox{div}(\varphi_{k})=0\\\
&\sum_{i=1}^{M}u_{i}\int_{\Omega}\psi_{l}\mbox{div}(\varphi_{i})=\int_{\Omega}f\psi_{l},\end{split}$
(38)
for $k=1,\dots,M$ and $l=1,\dots,N$. The problem (38) can be formulated into
the following matrix system
$\begin{pmatrix}A&B\\\ B^{t}&0\end{pmatrix}\begin{pmatrix}U\\\
P\end{pmatrix}=\begin{pmatrix}0\\\ F\end{pmatrix},$ (39)
where $A$ is a $N\times N$ matrix, $B$ is a $M\times N$ matrix with $B^{t}$
it’s transpose, $U$ is a $M$-dimensional column vector and $P,F$ are
$N$-dimensional column vectors.
The entries of these arrays are
$A[i,j]=\int_{\Omega}\varphi_{i}\cdot\varphi_{j}$,
$B[i,j]=\int_{\Omega}\psi_{j}\mbox{div}(\varphi_{i})$, $U[i]=u_{i}$,
$P[i]=p_{i}$ and $F[i]=\int_{\Omega}f\psi_{i}$.
The linear system (39) can be solved for $(U,P)$ with a computer using MFEM
library. Note that with the entries of $U$ and $P$, the solution
$(u_{h},p_{h})$ of (36) can be computed by their basis representation.
The spaces defined to discretize the problem are called Raviart-Thomas finite
element spaces. The fixed integer $k$ is also called the order of the shape
functions or the order of the finite element space. The parameter $h$ is the
same as in Section 1.2.2, which is a measure of size for $T_{h}$. See [3] for
details.
#### 1.2.5 Taylor-Hood finite elements
In this section, we show the spatial discretization done in [8], for Stokes
equations (40),
$\begin{split}\frac{\partial u}{\partial t}-\nu\Delta u+\nabla p=f,\text{ in
}\Omega,\\\ \nabla\cdot u=0,\text{ in }\Omega,\\\ u=g,\text{ in
}\Gamma,\end{split}$ (40)
and then mention the corresponding spatial discretization for Navier-Stokes
equations (41), as an extension of the previous one,
$\begin{split}\frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\nabla
p=f,\text{ in }\Omega,\\\ \nabla\cdot u=0,\text{ in }\Omega,\\\ u=g,\text{ in
}\Gamma.\end{split}$ (41)
When applying a numerical method to solve both systems of equations, (40) and
(41), time has to be discretized too. However, time discretization is out of
the scope of this paper (it can be found on section 4 of [8]).
First of all, let $T_{h}=\\{K_{1},\dots,K_{m}\\}$ be a discretization of the
domain $\Omega$. That is, $\Omega=K_{1}\cup\dots\cup K_{m}$, and
$K_{1},\dots,K_{m}$ don’t overlap between them, and no vertex of one of them
lies on the edge of another (check triangulation on Section 1.2.2). If
$\Omega$ is in 2D, $K_{i}$ is a quadrilateral, and if $\Omega$ is in 3D,
$K_{i}$ is a a hexahedron. Then, define the following finite element function
spaces on $T_{h}$, where $d\in\\{2,3\\}$ is the dimension of $\Omega$ [8].
$\begin{split}&U_{h}^{k}=\\{v\in(H^{1}(\Omega))^{d}\ :\
v(K)\in(\mathcal{Q}_{k}(K))^{d}\text{ for all }K\in T_{h}\\},\\\
&P_{h}^{k}=\\{s\in H^{1}(\Omega)\ :\ s(K)\in\mathcal{Q}_{k}(K)\text{ for all
}K\in T_{h}\\},\end{split}$
where $\mathcal{Q}_{k}(K)$ is the set of all polynomials whose degree on each
of their variables is less or equal than $k$, with domain $K$. For example, if
$K$ is two dimensional,
$\mathcal{Q}_{2}(K)=\\{a_{0}+a_{1}x+a_{2}y+a_{3}x^{2}+a_{4}y^{2}+a_{5}xy+a_{6}x^{2}y+a_{7}xy^{2}+a_{8}x^{2}y^{2}:a_{i}\in\mathbb{R},\
i=1,\dots,8\\}$. Notice that the total degree of $a_{8}x^{2}y^{2}$ is $4$, but
the degree on each variable ($x$ or $y$) is just $2$, as desired.
As done on previous sections, multiply the equations of (40) by some test
function $v\in U_{h}^{k}$ and $s\in P_{h}^{k}$, respectively, to obtain
$\begin{split}&\int_{\Omega}\frac{\partial u}{\partial
t}v-\nu\int_{\Omega}\Delta u\ v+\int_{\Omega}\nabla p\cdot
v=\int_{\Omega}fv,\\\ &\int_{\Omega}\left(\nabla\cdot u\right)s=0.\end{split}$
(42)
Now, applying Green’s formula (25) with homogeneous boundary condition
($u=g=0$ in $\Gamma$), as done on Section 1.2.2, the term
$-\nu\int_{\Omega}\Delta u\ v$ becomes $\nu\int_{\Omega}\nabla u\cdot\nabla
v$. Also, it is usual to multiply the second equation by $-1$, therefore, (42)
becomes the finite element formulation (43) found on [8]. The idea is to find
$(u,p)\in(U_{h}^{k},P_{h}^{k})$, for all $(v,s)\in(U_{h}^{k},P_{h}^{k})$, such
that
$\begin{split}&\int_{\Omega}\frac{\partial u}{\partial
t}v-\nu\int_{\Omega}\nabla u\cdot\nabla v+\int_{\Omega}\nabla p\cdot
v=\int_{\Omega}fv,\\\ &-\int_{\Omega}\left(\nabla\cdot
u\right)s=0.\end{split}$ (43)
Let $\\{\phi_{i}:i=1,\dots,n\\}$ be a basis for $U_{h}^{k}$ and
$\\{\psi_{j}:j=1,\dots,m\\}$ be a basis for $P_{h}^{k}$. Therefore,
$u(x,t)=\sum_{i=1}^{n}u_{i}(t)\phi_{i}(x)$
and
$p(x,t)=\sum_{j=1}^{m}p_{j}(t)\psi_{j}(x)$
where $u_{i}$ and $p_{j}$ are functions depending only on $t$. Replacing these
representations on (43) and noticing that
$(\phi_{i},\psi_{j})\in(U_{h}^{k},P_{h}^{k})$, we get the system of equations
(44), with $I=1,\dots,n$ and $J=1,\dots,m$.
$\begin{split}&\int_{\Omega}\frac{\partial\left(\sum_{i=1}^{n}u_{i}\phi_{i}\right)}{\partial
t}\phi_{I}-\nu\int_{\Omega}\nabla\left(\sum_{i=1}^{n}u_{i}\phi_{i}\right)\cdot\nabla\phi_{I}\\\
+&\int_{\Omega}\nabla\left(\sum_{j=1}^{m}p_{j}\psi_{j}\right)\cdot\phi_{I}=\int_{\Omega}f\phi_{I},\\\
&-\int_{\Omega}\left(\nabla\cdot\left(\sum_{i=1}^{n}u_{i}\phi_{i}\right)\right)\psi_{J}=0.\end{split}$
(44)
After rearranging scalars, using properties of the dot product and the
operator $\nabla$, and noting that $\phi_{i}$ does not depend on $t$, (44) can
be formulated as (45).
$\begin{split}&\int_{\Omega}\left(\sum_{i=1}^{n}\phi_{i}\frac{\partial
u_{i}}{\partial
t}\right)\phi_{I}-\nu\int_{\Omega}\left(\sum_{i=1}^{n}u_{i}\nabla\phi_{i}\right)\cdot\nabla\phi_{I}\\\
+&\int_{\Omega}\left(\sum_{j=1}^{m}p_{j}\nabla\psi_{j}\right)\cdot\phi_{I}=\int_{\Omega}f\phi_{I},\
\ I=1,\dots,n\\\
&-\int_{\Omega}\left(\sum_{i=1}^{n}u_{i}\nabla\cdot\phi_{i}\right)\psi_{J}=0,\
J=1,\dots,m.\end{split}$ (45)
Finally, (45) can be formulated as (46) after swapping integrals with
summations and rearranging integration scalars.
$\begin{split}&\sum_{i=1}^{n}\frac{\partial u_{i}}{\partial
t}\int_{\Omega}\phi_{i}\phi_{I}-\sum_{i=1}^{n}u_{i}\int_{\Omega}\nu\nabla\phi_{i}\cdot\nabla\phi_{I}\\\
+&\sum_{j=1}^{m}p_{j}\int_{\Omega}\phi_{I}\cdot\nabla\psi_{j}=\int_{\Omega}f\phi_{I},\
\ I=1,\dots,n\\\
&-\sum_{i=1}^{n}u_{i}\int_{\Omega}\psi_{J}\nabla\cdot\phi_{i}=0,\
J=1,\dots,m.\end{split}$ (46)
As before, the problem can be reduced to the matrix system (47), which is the
semi-discrete Stokes problem [8].
$\begin{split}M\dot{u}+Lu+Gp=f,\\\ -Du=0,\end{split}$ (47)
where $M$ and $L$ are $n\times n$ matrices, $G$ is a $n\times m$ matrix, $D$
is a $m\times n$ matrix, $u$ and $f$ are $n$-dimensional vectors, $p$ is a
$m$-dimensional vector and $\dot{u}$ is the notation used for the partial
derivate of $u$ with respect to time $t$. The entries of these arrays are
$M[i,j]=\int_{\Omega}\phi_{i}\phi_{j}$,
$L[i,j]=\int_{\Omega}\nu\nabla\phi_{i}\cdot\nabla\phi_{j}$,
$G[i,j]=\int_{\Omega}\phi_{i}\cdot\nabla\psi_{j}$,
$D[i,j]=\int_{\Omega}\psi_{i}\nabla\cdot\phi_{j}$,
$f[i]=\int_{\Omega}f\phi_{i}$, $p[i]=p_{i}$, $u[i]=u_{i}$ and
$\dot{u}[i]=\frac{\partial u_{i}}{\partial t}$.
As mentioned on [8], for the steady Stokes problem, $\dot{u}=0$ is taken. In
such case, (47) becomes the linear matrix system (48).
$\begin{pmatrix}L&G\\\ -D&0\end{pmatrix}\begin{pmatrix}u\\\
p\end{pmatrix}=\begin{pmatrix}f\\\ 0\end{pmatrix}$ (48)
Furthermore, for the Navier-Stokes equations (41), the semi-discrete
formulation is [8]:
$\begin{split}M\dot{u}+Lu+\mathcal{N}(u)+Gp=f,\\\ -Du=0,\end{split}$ (49)
where
$\mathcal{N}(u)[i]=\bigintss_{\Omega}\begin{pmatrix}u_{1}&\dots&u_{n}\end{pmatrix}\begin{pmatrix}(\phi_{1}\cdot\nabla)\phi_{1}&\dots&(\phi_{1}\cdot\nabla)\phi_{n}\\\
\vdots&\ddots&\vdots\\\
(\phi_{n}\cdot\nabla)\phi_{1}&\dots&(\phi_{n}\cdot\nabla)\phi_{n}\\\
\end{pmatrix}\begin{pmatrix}u_{1}\\\ \vdots\\\ u_{n}\end{pmatrix}\phi_{i}$
is the discretized nonlinear vector-convection term, of size $n\times n$.
Finally, recall that the spaces $U_{h}^{k}$ and $P_{h}^{k}$ have a given order
$k$. As mentioned on [8], Taylor-Hood finite element space is the tuple
$(U_{h}^{k},P_{k-1})$, which is used to solve steady and unsteady Stokes
problem (convergence is optimal and stable for $k\geq 2$). And, for Navier-
Stokes equations, the finite element space used is $(U_{h}^{k},P_{h}^{k})$,
called $P_{N}P_{N}$ space.
According to MFEM documentation [5], the implementation for the solution of
Navier-Stokes equations is done following [8], which is the theory presented
on this section.
### 1.3 MFEM Library
In this manuscript, we worked with MFEM’s Example#1 and Example#5 which can be
found on [5]. Example#1 uses standard Lagrange finite elements and Example#5
uses Raviart-Thomas mixed finite elements. Further, in Section 2.1, we find
the parameters so that both problems are equivalent and then (Section 2.4), we
compare the solutions.
We finally mention that for a fair comparison between Lagrange and mixed
finite element’s approximation, Lagrange shape functions of order $k-1$ will
be compared to the corresponding (mixed) approximation obtained by using
$RT_{k}(K)$.
Afterwards, we worked with MFEM’s miniapp for solving Navier-Stokes equations,
which corresponds to the experiments done on Section 3. On Section 3.1 we
worked in a 2-dimensional domain, and on Section 3.2 we worked in a
3-dimensional domain.
#### 1.3.1 Information about the library
According to it’s official site [5], MFEM is a free, lightweight, scalable C++
library for finite element methods that can work with arbitrary high-order
finite element meshes and spaces.
MFEM has a serial and a parallel version. The serial version is the one
recommended for beginners, and is used in Section 2. On the other hand, the
parallel version provides more computational power and enables the use of some
MFEM mini-apps, like the Navier-Stokes mini app, which is used in Section 3.
Moreover, the Modular Finite Element Method (MFEM) library is developed by the
MFEM Team at the Center for Applied Scientific Computing (CASC), located in
the Lawrence Livermore National Laboratory (LLNL), under the BSD licence.
However, as it is open source, the public repository can be found at
github.com/mfem in order for anyone to contribute.
Also, since 2018, a wrapper for Python (PyMFEM) is being developed in order to
use MFEM library among with Python code, which demonstrates the wide
applicability that the library can achieve. And, in 2021, the first community
workshop was hosted by the MFEM Team, which encourages the use of the library
and enlarges the community of MFEM users.
Finally, take into account that the use of the library requires a good manage
of C++ code, which is a programming language that’s harder to use compared to
other languages, such as Python. This understanding of C++ code is important
because some parts of the library are not well documented yet, and, by
checking the source code, the user may find a way of implementing what is
required.
#### 1.3.2 Overview
The main classes (with a brief and superficial explanation of them) that we
are going to use in the code are:
* •
Mesh: domain with the partition.
* •
FiniteElementSpace: space of functions defined on the finite element mesh.
* •
GridFunction: mesh with values (solutions).
* •
$\\_$Coefficient: values of GridFunctions or constants.
* •
LinearForm: maps an input function to a vector for the rhs.
* •
BilinearForm: used to create a global sparse finite element matrix for the
lhs.
* •
$\\_$Vector: vector.
* •
$\\_$Solver: algorithm for solution calculation.
* •
$\\_$Integrator: evaluates the bilinear form on element’s level.
* •
NavierSolver: class associated to the navier mini-app, which is used to solve
Navier-Stokes equations.
The ones that have $\\_$ are various classes whose name ends up the same and
work similarly.
Note:
lhs: left hand side of the linear system.
rhs: right hand side of the linear system.
#### 1.3.3 Code structure
A MFEM general code has the following steps (directly related classes with the
step are written):
1. 1.
Receive an input file (.msh) with the mesh and establish the order for the
finite element spaces.
2. 2.
Create a mesh object, get the dimension, and refine the mesh (refinement is
optional). Mesh
3. 3.
Define the finite element spaces required. FiniteElementSpace
4. 4.
Define the coefficients, functions, and boundary conditions of the problem.
XCoefficient
5. 5.
Define the LinearForm for the rhs and assemble it. LinearForm, XIntegrator
6. 6.
Define the BilinearForm for the lhs and assemble it. BilinearForm, XIntegrator
7. 7.
Solve the linear system. XSolver, XVector
8. 8.
Recover solution. GridFunction
9. 9.
Show solution with a finite element visualization tool like GLVis [6]
(optional).
And, for the general code structure of the navier mini-app, we have the
following steps:
1. 1.
Receive an input file (.msh) with the mesh, create a parallel mesh object and
refine the mesh (refinement is optional). ParMesh
2. 2.
Create the flow solver by stating the order of the finite element spaces and
the parameter for kinematic viscosity $\nu$. NavierSolver
3. 3.
Establish the initial condition, the boundary conditions and the time step
$dt$.
4. 4.
Iterate through steps in time with the NavierSolver object and save the
solution for each iteration in a parallel GridFunction. ParGridFunction
5. 5.
Show the solution with a finite element visualization tool like ParaView [10]
(optional).
Notice that the Mesh and GridFunction classes used in the mini-app are for the
parallel version of MFEM. The reason for this, is that the navier mini-app is
available for the parallel version of MFEM only. Also, the mini-app is coded
in such way that the code is simple (see Appendix 5.3).
## 2 Lagrange vs. Raviart-Thomas finite elements
In this section, we take examples 1 and 5 from [5], define their problem
parameters in such way that they’re equivalent, create a code that implements
both of them at the same time and compares both solutions ($L_{2}$ norm), run
the code with different orders, and analyse the results.
Some considerations to have into account for a fair comparison are that, the
order for the Mixed method should be 1 less than the order for Lagrange
method, because, with this, both shape functions would have the same degree.
Also, we will compare pressures and velocities with respect to the order of
the shape functions and the size of the mesh ($h$ parameter). Furthermore, for
the problem, the exact solution is known, so, we will use it for comparison.
And, the maximum order and refinement level to be tested is determined by our
computational capacity (as long as solvers converge fast).
### 2.1 Problem
As mentioned before, we have to find the parameters for example 1 and 5 from
[5], in such way that both problems are equivalent. This step is important
because example # 1 is solved using Lagrange finite elements, while example #
5 is solved using mixed finite elements. Therefore, in order to make the
comparison, the problem must be the same for both methods.
Example#1 [5]: Compute $p$ such that
$\begin{split}-\Delta&p=1\text{ in }\Omega\\\ &p=0\text{ in
}\Gamma.\end{split}$ (50)
Example#5 [5]: Compute $p$ and $\mathbf{u}$ such that
$\begin{split}&k\mathbf{u}+\nabla p=f\text{ in }\Omega\\\
&-\mbox{div}(\mathbf{u})=g\text{ in }\Omega\\\ &-p=p_{0}\text{ in
}\Gamma.\end{split}$ (51)
From the first equation of (51),
$\mathbf{u}=\frac{f-\nabla p}{k}.$ (52)
Then, replacing (52) on the second equation of (51),
$-\mbox{div}\left(\frac{f-\nabla p}{k}\right)=g.$ (53)
If we set $k=1;\ f=0\ and\ g=-1$ in (53), we get
$-\Delta p=1,$ (54)
which is the first equation of (50).
So, setting ($*$) $p_{0}=0,\ k=1;\ f=0\ and\ g=-1$ in (51), we get,
$\begin{split}&\mathbf{u}+\nabla p=0\text{ in }\Omega\\\
&-\mbox{div}(\mathbf{u})=-1\text{ in }\Omega\\\ &-p=0\text{ in
}\Gamma.\end{split}$ (55)
Notice that from the first equation we get that $\mathbf{u}=-\nabla p$. This
is important because in problem (50) we don’t get the solution for
$\mathbf{u}$ from the method, so, we will have to find it from $p$’s
derivatives.
In the code, we will set the value of the parameters in the way shown here, so
that both problems are the same. As seen in (52)-(54), problem (51) is
equivalent to problem (50) with the values assigned for coefficients and
functions at ($*$).
### 2.2 Code
The first part of the code follows the structure mentioned in Section 1.3.3,
but implemented for two methods at the same time (and with some extra lines
for comparison purposes). Also, when defining boundary conditions, the
essential one is established different from the natural one. And, after
getting all the solutions, there’s a second part of the code where solutions
are compared between them and with the exact one.
Note:
The complete code with explanations can be found on the Appendix A.
However, before taking a look into it, the reader may have the following into
account. The following table shows the convention used for important variable
names along the code:
Variable Name | Object
---|---
X_space | Finite element space X
X_mixed | Variable assigned to a mixed method related object
u | Velocity solution
p | Pressure solution
X_ex | Variable assigned to an exact solution object
### 2.3 Tests
The tests of the two methods presented previously, were run on MFEM library on
the domain shown on Figure 6.
Figure 6: Illustration of the star domain used for the numerical tests.
Visualization: [6].
Each run test is determined by the order of the Lagrange shape functions and
the h parameter of the mesh. Remember that mixed shape functions have order
equal to $\textit{order}-1$. The parameter order is changed directly from the
command line, while the parameter h is changed via the number of times that
the mesh is refined ($h=h(\\#refinements)$). As we refine the mesh more times,
finite elements of the partition decrease their size, and so, the parameter
$h$ decreases.
Tests were run with: $order=1,\dots,N$ and $refinements=0,\dots,M$, where
$N,M$ depend on the computation capacity. The star domain was partitioned
using quads (instead of triangles), and such partition is shown on Figure 7.
Figure 7: Initial mesh used for numerical tests (no refinements).
Visualization: [6].
Results on Section 2.4 are presented in graphs. However, all the exact values
that were computed can be found in the Appendix B.
### 2.4 Results
In Figure 8 we show the computed solution when running the code with
$\textit{order}=2$ and $\\#Refinements=3$. We use the visualization tool [6].
We mention that, at the scale of the plot, Lagrange and Mixed solutions look
the same.
Figure 8: Illustration of computed pressure and velocities. GLVis ([6]) is
used for this visualization. Pressure (left) and $L^{2}$ norm of the vector
velocity (right). Visualization: [6].
In the following results, if $u=(u_{x},u_{y})$ is the solution obtained by the
mixed or Lagrange finite element method and $u_{ex}=(u_{x_{ex}},u_{y_{ex}})$
is the exact solution for the problem, then,
$U_{error}=\frac{\sqrt{\left(||u_{x}-u_{x_{ex}}||_{L^{2}}\right)^{2}+\left(||u_{y}-u_{y_{ex}}||_{L^{2}}\right)^{2}}}{||u_{ex}||_{L^{2}}}.$
Figure 9: Variation of error with respect to the refinement level for the
approximation of the solution of problem (50) with Lagrangian finite elements
of order 1 and problem (51) with mixed Raviart-Thomas finite elements of order
0. Figure 10: Variation of error with respect to the refinement level for the
approximation of the solution of problem (50) with Lagrangian finite elements
of order 2 and problem (51) with mixed Raviart-Thomas finite elements of order
1. Figure 11: Variation of error with respect to the refinement level for the
approximation of the solution of problem (50) with Lagrangian finite elements
of order 3 and problem (51) with mixed Raviart-Thomas finite elements of order
2. Figure 12: Variation of error with respect to the refinement level for the
approximation of the solution of problem (50) with Lagrangian finite elements
of order 4 and problem (51) with mixed Raviart-Thomas finite elements of order
3.
### 2.5 Analysis
In this section we comment and analyze the results in tables presented on the
Appendix B.
To understand the information presented, take into account that the exact
solution would have value $0$ in X err. Also, if the two solutions obtained
(Lagrange and Mixed) are exactly the same, the value in P comp and U comp
would be $0$. And, lower values of $h$ mean more mesh refinements, ie, smaller
partition elements.
As expected, computational time increases as order and refinements increase.
Here are the most relevant observations that can be obtained after analysing
the data corresponding to absolute errors.
For fixed order, absolute errors have little variation when reducing $h$ (max
variation is $4.722$e$-03$ in $Uerr$ order 1); $Perr$ increases as $h$
decreases, while $Pmx\ err$ decreases as $h$ decreases; and, $Uerr$ increases
as $h$ decreases, while $Umx\ err$ decreases as $h$ decreases.
Absolute errors variation (respect to refinement) is lower when order is
higher. For example; in order 2, $Perr$ is the same for each $h$ (up to three
decimal places); while in order 6, $Perr$ is the same for each $h$ (up to five
decimal places).
For fixed $h$, absolute errors remain almost constant between orders.
Moreover, $Perr$ (absolute error obtained for pressure with Lagrange) is
always lower than $Pmx\ err$ (absolute error obtained for pressure with mixed)
and $Uerr$ (absolute error obtained for velocity with Lagrange) is always
lower than $Umx\ err$ (absolute error obtained for velocity with mixed).
As order increases, pressure and velocity absolute errors tend to be the same.
In order 10, the difference between $Perr$ and $Pmx\ err$ is $0.000001$ and
the difference between $Uerr$ and $Umx\ err$ is $<0.0000009$.
However, notice that in all the cases, the absolute error was higher than $1$.
In $L_{2}$ norm, this value is pretty little, and shows that we are only
getting approximations of the exact solution.
Now, the most relevant observations that can be obtained after analysing the
data corresponding to comparison errors. First of all, comparison error tends
to $0$; and comparison errors $Ucomp$ and $Pcomp$ decrease as $h$ decreases.
For a fixed order, comparison error can be similar to a higher order
comparison error, as long as enough refinements are made. Moreover, when order
increases, comparison errors are lower for fixed $h$.
Pressure comparison error lowers faster than velocity comparison error.
Maximum comparison errors were found at order 1 with no refinements, where
$Pcomp\approx 7.5$e$-02$ and $Ucomp\approx 3.7$e$-02$, and minimum comparison
errors were found at order 10 with 1 refinement (higher refinement level
computed for order 10), where $Pcomp\approx 5.1$e$-06$ and $Ucomp\approx
9.8$e$-04$. It can be seen that $Pcomp$ improved in almost four decimal places
while $Ucomp$ improved in just 2.
### 2.6 Some other examples
In this section we show three of the examples that MFEM library provides [5].
We only show the problem, a brief verification of the exact solution and the
solution obtained using MFEM, without going into details of any type. The
purpose is to show the wide variety of applications that MFEM library can have
and let the reader familiarize with the visualization of some finite element
method solutions.
Example 1
This example is Example # 3 of [5] and consists on solving the second order
definite Maxwell equation
$\nabla\times\nabla\times E+E=f$
with Dirichlet boundary condition. In the example, the value for $f$ is given
by
$f\begin{pmatrix}x\\\ y\\\
z\end{pmatrix}=\begin{pmatrix}\left(1+\pi^{2}\right)\sin(\pi y)\\\
\left(1+\pi^{2}\right)\sin(\pi z)\\\ \left(1+\pi^{2}\right)\sin(\pi
x)\end{pmatrix}.$
The exact solution for $E$ is
$E\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}=\begin{pmatrix}\sin(\pi y)\\\
\sin(\pi z)\\\ \sin(\pi x)\end{pmatrix},$
and can be verified by computing
$\begin{split}&\nabla\times\nabla\times E+E\\\
=&\begin{pmatrix}\frac{\partial}{\partial x}\\\ \frac{\partial}{\partial y}\\\
\frac{\partial}{\partial
z}\end{pmatrix}\times\begin{pmatrix}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial y}\\\ \frac{\partial}{\partial
z}\end{pmatrix}\times\begin{pmatrix}\sin(\pi x)\\\ \sin(\pi z)\\\ \sin(\pi
y)\end{pmatrix}+\begin{pmatrix}\sin(\pi y)\\\ \sin(\pi z)\\\ \sin(\pi
x)\end{pmatrix}\\\ =&\begin{pmatrix}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial y}\\\ \frac{\partial}{\partial
z}\end{pmatrix}\times\begin{pmatrix}-\pi\cos(\pi z)\\\ -\pi\cos(\pi x)\\\
-\pi\cos(\pi y)\end{pmatrix}+\begin{pmatrix}\sin(\pi y)\\\ \sin(\pi z)\\\
\sin(\pi x)\end{pmatrix}\\\ =&\begin{pmatrix}\pi^{2}\sin(\pi y)\\\
\pi^{2}\sin(\pi z)\\\ \pi^{2}\sin(\pi x)\end{pmatrix}+\begin{pmatrix}\sin(\pi
y)\\\ \sin(\pi z)\\\ \sin(\pi x)\end{pmatrix}=f.\end{split}$
The solution for $E$, $E_{h}$, computed with MFEM library is presented on
Figure 13. The error for the approximation is
$\left|\left|E_{h}-E\right|\right|_{L^{2}}=0.39154.$
Figure 13: Visualization of the norm of the solution for the electromagnetic
diffusion problem corresponding to the second order definite Maxwell equation.
Solution for $E$, $E_{h}$, obtained using MFEM library. Visualization: [6].
Example 2
This example is Example # 4 of [5] and consists on solving the diffusion
problem corresponding to the second order definite equation
$-\nabla(\alpha\mbox{div}(F))+\beta F=f$
with Dirichlet boundary condition, and, with parameters $\alpha=1$ and
$\beta=3$. In the example, the value for $f$ is given by
$f\begin{pmatrix}x\\\ y\end{pmatrix}=\begin{pmatrix}(3+2\pi^{2})\cos(\pi
x)\sin(\pi y)\\\ (3+2\pi^{2})\cos(\pi y)\sin(\pi x)\end{pmatrix}.$
The exact solution for $F$ is
$F\begin{pmatrix}x\\\ y\end{pmatrix}=\begin{pmatrix}\cos(\pi x)\sin(\pi y)\\\
\cos(\pi y)\sin(\pi x)\end{pmatrix},$
and can be verified by computing
$\begin{split}&-\nabla(\alpha\mbox{div}(F))+\beta F\\\
=&-\nabla\left(\mbox{div}\begin{pmatrix}\cos(\pi x)\sin(\pi y)\\\ \cos(\pi
y)\sin(\pi x)\end{pmatrix}\right)+3\begin{pmatrix}\cos(\pi x)\sin(\pi y)\\\
\cos(\pi y)\sin(\pi x)\end{pmatrix}\\\ =&-\nabla\left(-2\pi\sin(\pi x)\sin(\pi
y)\right)+3\begin{pmatrix}\cos(\pi x)\sin(\pi y)\\\ \cos(\pi y)\sin(\pi
x)\end{pmatrix}\\\ =&-\begin{pmatrix}-2\pi^{2}\cos(\pi x)\sin(\pi y)\\\
-2\pi^{2}\sin(\pi x)\cos(\pi y)\end{pmatrix}+3\begin{pmatrix}\cos(\pi
x)\sin(\pi y)\\\ \cos(\pi y)\sin(\pi x)\end{pmatrix}\\\
=&\begin{pmatrix}2\pi^{2}\cos(\pi x)\sin(\pi y)\\\ 2\pi^{2}\cos(\pi y)\sin(\pi
x)\end{pmatrix}+\begin{pmatrix}3\cos(\pi x)\sin(\pi y)\\\ 3\cos(\pi y)\sin(\pi
x)\end{pmatrix}=f.\end{split}$
The solution for $F$, $F_{h}$, computed with MFEM library is presented on
Figure 14. The domain is a square with a circular hole in the middle. On the
visualization, the triangular elements of the mesh are shown. The error for
the approximation is
$\left|\left|F_{h}-F\right|\right|_{L^{2}}=5.55372\times 10^{-6}.$
Figure 14: Visualization of the norm of the solution for the diffusion problem
showing the mesh elements. Solution for $F$, $F_{h}$, obtained using MFEM
library. Visualization: [6].
Example 3
This example is Example # 7 of [5] and consists on solving the Laplace problem
with mass term corresponding to the equation
$-\Delta u+u=f.$
In the example, the value for $f$ is given by
$f\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}=\frac{7xy}{x^{2}+y^{2}+z^{2}}.$
The exact solution for $u$ is
$u\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}=\frac{xy}{x^{2}+y^{2}+z^{2}},$
and can be verified by computing
$\begin{split}&-\Delta u+u\\\
=&-\mbox{div}\left(\nabla\left(\frac{xy}{x^{2}+y^{2}+z^{2}}\right)\right)+\frac{xy}{x^{2}+y^{2}+z^{2}}\\\
=&-\mbox{div}\left(\frac{1}{(x^{2}+y^{2}+z^{2})^{2}}\begin{pmatrix}y(-x^{2}+y^{2}+z^{2})\\\
x(x^{2}-y^{2}+z^{2})\\\
-2xyz\end{pmatrix}\right)+\frac{xy}{x^{2}+y^{2}+z^{2}}\\\
=&-\left(-\frac{6xy}{x^{2}+y^{2}+z^{2}}\right)+\frac{xy}{x^{2}+y^{2}+z^{2}}\\\
=&\frac{7xy}{x^{2}+y^{2}+z^{2}}=f.\end{split}$
The solution for $u$, $u_{h}$, computed with MFEM library is presented on
Figure 15. The error for the approximation is
$\left|\left|u_{h}-u\right|\right|_{L^{2}}=0.00236119.$
Figure 15: Visualization of the solution for the Laplace problem with mass
term. Solution for $u$, $u_{h}$, obtained using MFEM library. Visualization:
[6].
These examples show that MFEM can solve several types of equations that can
include divergence, curl, gradient and Laplacian operators. Also, the given
parameter $f$ was a function of the form
$\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$,
$\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ and
$\mathbb{R}^{3}\rightarrow\mathbb{R}$, which shows that MFEM can work with
scalar and vectorial functions in a 2D or 3D domain. Although the shown
visualizations were norms of the solution, using another visualization tool,
such as [10], the vectors of the solution can be seen.
## 3 Numerical Experiments with NS
In this section we run some computational experiments solving Navier-Stokes
equations using MFEM’s Navier Stokes mini app. One of the experiments is in a
2D domain (Section 3.1) and the other one is in a 3D domain (Section 3.2). The
two dimensional experiment converges to a steady state when the initial
condition is the steady state with a small perturbation. In such experiment,
we compare graphically the pressure solution obtained when changing the order
and the refinement level. On the other hand, in the three dimensional
experiment, we revise a graphical solution for a dynamical system, where
turbulence is present.
### 3.1 2D Experiment: Steady State
The two dimensional domain is a $2\times 4$ rectangle whose vertex coordinates
are $(-0.5,1.5)$, $(-0.5,-0.5)$, $(1,1.5)$ and $(1,-0.5)$, as shown on Figure
16.
Figure 16: Two dimensional domain used in the 2D Navier-Stokes experiment. It
is a $2\times 4$ rectangle. Visualization: [10].
Also, the default mesh (with no refinements) is presented on figure 17.
Figure 17: Two dimensional domain mesh used in the 2D Navier-Stokes
experiment. Visualization: [10].
For this experiment, $Re=40$ and the velocity boundary condition was settled
to be
$u_{0}(x,y)=\begin{pmatrix}1-e^{2\left(10-\sqrt{100+\pi^{2}}\right)x}\cdot\cos\left(2\pi
y\right)\\\ \frac{2\left(10-\sqrt{100+\pi^{2}}\right)}{2\pi}\cdot
e^{2\left(10-\sqrt{100+\pi^{2}}\right)x}\cdot\sin\left(2\pi
y\right)\end{pmatrix}.$ (56)
If the initial condition is picked to be $u_{0}$, then the system is already
on a steady state. As we wanted the experiment to reach the steady state, then
initial condition was ensured to be
$u_{i}(x,y)=u_{0}(x,y)+\delta\begin{pmatrix}(x+0.5)(x-1)(y+0.5)(y-1.5)\\\
(x+0.5)(x-1)(y+0.5)(y-1.5)\end{pmatrix},$ (57)
where the $\delta$ parameter in the initial condition gives the magnitude of
the perturbation from the steady state. The parameter was picked to be
$\delta=0.001$. Notice that the term after $\delta$ vanishes in the boundary
of the domain.
The expected result in pressure (computed with order 6 and 5 refinements) is
presented on figure 18, which corresponds to the steady state reached by the
system.
Figure 18: Expected solution for the pressure in the 2D experiment, computed
with order 6 and 5 refinements. It is the steady state of the system.
Visualization: [10].
For illustration of how the system changed, the initial condition for the
pressure is shown in figure 19.
Figure 19: Initial condition for pressure in the 2D experiment, computed with
order 6 and 5 refinements. Visualization: [10].
Notice that the initial condition has a higher pressure on the upper-right
part of the domain and it’s not uniform along any of the axes. However, after
the system reaches the steady state pressure is constant for a fixed $x$
value.
Now, the experiment consists on iterating through different orders and
changing the refinement level for each of the orders, in order to check
differences in the graphical solution. The experiment was done using a time
step of $dt=0.001$ with a total time of $T=0.05$. It was computed with the
parallel version of MFEM library (with 4 cores), using the Navier Miniapp [5].
The system was solved 36 times, corresponding to $order=1,2,3,4,5,6$, and for
each of them, $\\#refinements=0,1,2,3,4,5$. After checking all of the results
in ParaView [10], we summarized the general behaviour of the steady state
solutions, as shown on figure 20. As notation, each of the results has a
corresponding $(k,r)$ value, where $k$ denotes the order and $r$ the amount of
refinements made.
Figure 20: Summary of the steady state solutions for the 2D experiment as
order and refinement level changes. The orders and refinement levels that
don’t appear are almost identical to the expected steady state. Visualization:
[10].
From the summary it is clear that with low order and low refinement levels,
the approximation for the solution is not a good one. It’s interesting to
notice that increasing $1$ in the order can be similar to increasing $1$ in
the number of refinements, as seen on the steady state of $(1,1)\sim(2,0)$,
$(1,2)\sim(2,1)$ and $(3,1)\sim(4,0)$.
Furthermore, with a high order, the solution converges as expected even if the
mesh is not refined. Finally, note that for the lowest order or for the lowest
refinement level, an increase of $5$ in the counterpart achieves a good
approximation (the steady states that are not shown are the ones that already
achieve a good approximation).
### 3.2 3D Experiment: Turbulence
The three dimensional domain is a $0.4\times 0.4\times 2.5$ parallelepiped
with a vertex on the origin $(0,0,0)$ and the other ones on the positive part
of the coordinate system. It has a cylindrical hole, parallel to the
$yx$-plane, whose cross section center is located at $(0.5,0.2)$ and has a
radius of $0.05$. The domain is shown on Figure 21.
Figure 21: Three dimensional domain used in the 3D Navier-Stokes experiments.
It is a $0.4\times 0.4\times 2.5$ parallelepiped with a cylindrical hole.
Visualization: [10].
For the experiment, both the initial condition and the boundary condition for
velocity were settled to be the same:
$u_{0}\begin{pmatrix}x\\\ y\\\ z\\\ t\end{pmatrix}=u_{i}\begin{pmatrix}x\\\
y\\\ z\\\ t\end{pmatrix}=\begin{pmatrix}u_{x}(x,y,z,t)\\\ 0\\\
0,\end{pmatrix}$ (58)
where
$u_{x}(x,y,z,t)=\left\\{\begin{split}\frac{36yz}{0.41^{4}}\sin\left(\frac{\pi
t}{8}\right)(0.41-y)(0.41-z),\text{ if }x\leq 10^{-8},\\\ 0,\text{ if
}x>10^{-8}.\end{split}\right.$ (59)
Notice that the velocity condition simulates a system where the fluid is
entering through the squared face of the domain near the hole. Also, the
boundary condition was forced only on the inlet and on the walls of the
domain.
Furthermore, the experiment was computed with $4$-th order elements, using a
time step of $dt=0.001$, a total time of $T=8$ and the parameter of kinematic
viscosity for the fluid being $\nu=0.001$.
For the visualization of the solutions, we used ParaView’s [10] stream tracer
functionality, which shows the stream lines of the system at a given moment.
Recall that the stream lines are tangent to the vector field of the velocity,
and they show the trajectory of particles through the field (at a given
instant of time). Also, there are 5 visualizations, corresponding to times
$t=0,2,4,6,8$, and have the value of pressure codified in a color scale.
At $t=0$, the stream lines show a laminar flow (all lines are almost parallel
and follow the same direction) that avoids the obstacle, and the pressure is
high on the left because the fluid is entering though that part of the domain,
as shown on figure 22.
Figure 22: Stream lines for the velocity of the 3D experiment at $t=0$.
Visualization: [10].
Then, at $t=2$, more fluid is coming in, therefore, the pressure on the left
increases. However, the value of pressure after the obstacle starts to have
some variations, which will cause the turbulence later.
Figure 23: Stream lines for the velocity of the 3D experiment at $t=2$.
Visualization: [10].
At $t=4$, the turbulence starts to show up. As seen on figure 24, after the
obstacle, some of the stream lines have spiral forms.
Figure 24: Stream lines for the velocity of the 3D experiment at $t=4$.
Visualization: [10].
Then, at $t=6$, the pressure on the left finally starts to lower (with a lot
of variation) and the pressure on the right starts to increase, because the
fluid is already passing through the obstacle and no more fluid is coming in..
More spiral-shaped stream lines show up after the obstacle.
Figure 25: Stream lines for the velocity of the 3D experiment at $t=6$.
Visualization: [10].
Finally, at $t=T=8$, the pressure on the left is low and on the right is high.
However, the high variation of pressure in previous time steps generated a lot
of turbulence near the obstacle. The fluid presents two states, a laminar one,
and a turbulent one. The turbulent state, characterized by spiral movement and
swirls, is presented near the obstacle; while the laminar state, characterized
by straight lines, is presented away from the obstacle. As seen on figure 26.
Figure 26: Stream lines for the velocity of the 3D experiment at $t=T=8$.
Visualization: [10].
## 4 Conclusion and Perspectives
The MFEM library allows us to approximate the solution of partial differential
equations in a versatile way. Moreover, the library has a lot of potential
because it can compute with high order elements without requiring a very
powerful computer, for example, we could run experiments with elements of
order 10 (which is a relatively high order). Also, the navier mini app of the
library provides a simple, precise and efficient way for simulating dynamical
systems that involve incompressible fluids. Furthermore, it is important to
use a good visualization tool, preferably one that allows the visualization of
vector fields and stream lines when working with fluid equations. Finally,
recall that finite element methods enable the study of systems that depend on
difficult partial differential equations.
Following this work, some study can be made on some of the following topics:
* •
Fluid modeling via partial differential equations.
* •
Error theorems for finite element methods.
* •
Effects of the mesh in the solution.
* •
Solutions for the equations when the parameters are not constants, but
functions.
* •
Discretization of time.
* •
Picking of the time step $dt$, in order to achieve an appropriate solution.
* •
Turbulence and laminar flow.
## References
* [1] Felipe Cruz. Comparing Lagrange and Mixed finite element methods using MFEM library. Beyond Research work at National University of Colombia. Arxiv: https://arxiv.org/submit/3724279/view
* [2] Claes Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. ISBN10 048646900X. Dover Publications Inc. 2009.
* [3] Gabriel N. Gatica. A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. ISBN 978-3-319-03694-6. Springer. 2014.
* [4] Juan Galvis & Henrique Versieux. Introdução à Aproximação Numérica de Equações Diferenciais Parciais Via o Método de Elementos Finitos. ISBN: 978-85-244-325-5. 28 Colóquio Brasileiro de Matemática. 2011.
* [5] R. Anderson and J. Andrej and A. Barker and J. Bramwell and J.-S. Camier and J. Cerveny V. Dobrev and Y. Dudouit and A. Fisher and Tz. Kolev and W. Pazner and M. Stowell and V. Tomov and I. Akkerman and J. Dahm and D. Medina and S. Zampini, MFEM: A modular finite element methods library, Computers & Mathematics with Applications 81 (2021), 42-74. Main Page: mfem.org. DOI 10.11578/dc.20171025.1248
* [6] GLVis - OpenGL Finite Element Visualization Tool. Main Page: glvis.org. DOI 10.11578/dc.20171025.1249
* [7] Kalita, P. & Łukaszewicz, G. (2016). Navier-Stokes Equations. An Introduction with Applications. Springer. DOI 10.1007/978-3-319-27760-8.
* [8] Michael Franco, Jean-Sylvain Camier, Julian Andrej, Will Pazner (2020) High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners (https://arxiv.org/abs/1910.03032)
* [9] Schneider, T., et al. (2018). Decoupling Simulation Accuracy from Mesh Quality. New York University, USA. 2018 Association for Computing Machinery. https://doi.org/10.1145/3272127.3275067.
* [10] Ayachit, Utkarsh, The ParaView Guide: A Parallel Visualization Application, Kitware, 2015, ISBN 978-1930934306.
## 5 Appendices
### 5.1 Appendix A : Code for comparison
Here, the code used for Section 2 (written in C++) is shown, with a brief
explanations of it’s functionality.
$\triangleright$Include the required libraries (including MFEM) and begin main
function.
⬇
#include "mfem.hpp"
#include <fstream>
#include <iostream>
using namespace std;
using namespace mfem;
int main(int argc, char *argv[]){
$\triangleright$Parse command-line options (in this project we only change
"order" option) and print them.
⬇
const char *mesh_file = "../data/star.mesh";
int order = 1;
bool visualization = true;
OptionsParser args(argc, argv);
args.AddOption(&mesh_file, "-m", "–mesh",
"Mesh␣file␣to␣use.");
args.AddOption(&order, "-o", "–order",
"Finite␣element␣order␣(polynomial␣degree).");
args.AddOption(&visualization, "-vis", "–visualization", "-no-vis", "–no-
visualization",
"Enable␣or␣disable␣GLVis␣visualization.");
args.Parse();
if (!args.Good()){
args.PrintUsage(cout);
return 1;
}
args.PrintOptions(cout);
$\triangleright$Create mesh object from the star.mesh file and get it’s
dimension.
⬇
Mesh *mesh = new Mesh(mesh_file,1,1);
int dim = mesh->Dimension();
$\triangleright$Refine the mesh a given number of times (UniformRefinement).
⬇
int ref_levels;
cout << "Refinements:␣";
cin >> ref_levels;
for (int l = 0; l < ref_levels; l++){
mesh->UniformRefinement();
}
$\triangleright$Get size indicator for mesh size (h_max) and print it.
⬇
double mesh_size, h = 0;
for (int i=0;i<mesh->GetNE();i++){
mesh_size = mesh->GetElementSize(i,2);
if(mesh_size>h){
h = mesh_size;
}
}
cout << "h:␣" << h << endl;
$\triangleright$Define finite element spaces. For mixed finite element method,
the order will be one less than for Lagrange finite element method. The last
one is a vector $L^{2}$ space that we will use later to get mixed velocity
components.
⬇
FiniteElementCollection *H1 = new H1_FECollection(order, dim);
FiniteElementSpace *H1_space = new FiniteElementSpace(mesh, H1);
FiniteElementCollection *hdiv_coll(new RT_FECollection(order-1, dim));
FiniteElementCollection *l2_coll(new L2_FECollection(order-1, dim));
FiniteElementSpace *R_space = new FiniteElementSpace(mesh, hdiv_coll);
FiniteElementSpace *W_space = new FiniteElementSpace(mesh, l2_coll);
FiniteElementSpace *V_space = new FiniteElementSpace(mesh, l2_coll, 2);
$\triangleright$Define the parameters of the mixed problem. C functions are
defined at the end. Boundary condition is natural.
⬇
ConstantCoefficient k(1.0);
void fFun(const Vector & x, Vector & f);
VectorFunctionCoefficient fcoeff(dim, fFun);
double gFun(const Vector & x);
FunctionCoefficient gcoeff(gFun);
double f_bound(const Vector & x);
FunctionCoefficient fbndcoeff(f_bound);
$\triangleright$Define the parameters of the Lagrange problem. Boundary
condition is essential.
⬇
ConstantCoefficient one(1.0);
Array<int> ess_tdof_list;
if (mesh->bdr_attributes.Size()){
Array<int> ess_bdr(mesh->bdr_attributes.Max());
ess_bdr = 1;
H1_space->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
}
$\triangleright$Define the exact solution. C functions are defined at the end.
⬇
void u_ex(const Vector & x, Vector & u);
double p_ex(const Vector & x);
double u_ex_x(const Vector & x);
double u_ex_y(const Vector & x);
$\triangleright$Get space dimensions and create vectors for the right hand
side.
⬇
Array<int> block_offsets(3);
block_offsets[0] = 0;
block_offsets[1] = R_space->GetVSize();
block_offsets[2] = W_space->GetVSize();
block_offsets.PartialSum();
BlockVector rhs_mixed(block_offsets);
Vector rhs(H1_space->GetVSize());
$\triangleright$Define the right hand side. These are LinearForm objects
associated to some finite element space and rhs vector. "f" and "g" are for
the mixed method and "b" is for the Lagrange method. Also, "rhs" vectors are
the variables that store the information of the right hand side.
⬇
LinearForm *fform(new LinearForm);
fform->Update(R_space, rhs_mixed.GetBlock(0), 0);
fform->AddDomainIntegrator(new VectorFEDomainLFIntegrator(fcoeff));
fform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(fbndcoeff));
fform->Assemble();
LinearForm *gform(new LinearForm);
gform->Update(W_space, rhs_mixed.GetBlock(1), 0);
gform->AddDomainIntegrator(new DomainLFIntegrator(gcoeff));
gform->Assemble();
LinearForm *b(new LinearForm);
b->Update(H1_space, rhs, 0);
b->AddDomainIntegrator(new DomainLFIntegrator(one));
b->Assemble();
$\triangleright$Create variables to store the solution. "x" is the Vector used
as input in the iterative method.
⬇
BlockVector x_mixed(block_offsets);
GridFunction u_mixed(R_space), p_mixed(W_space), ux_mixed(W_space),
uy_mixed(W_space), ue(V_space);
Vector x(H1_space->GetVSize());
GridFunction ux(W_space),uy(W_space),p(H1_space);
$\triangleright$Define the left hand side for mixed method. This is the
BilinearForm representing the Darcy matrix (39). VectorFEMMassIntegrator is
asociated to $k*u-\nabla p$ and VectorFEDDivergenceIntegrator is asociated to
$\mbox{div}(u)$.
⬇
BilinearForm *mVarf(new BilinearForm(R_space));
MixedBilinearForm *bVarf(new MixedBilinearForm(R_space, W_space));
mVarf->AddDomainIntegrator(new VectorFEMassIntegrator(k));
mVarf->Assemble();
mVarf->Finalize();
SparseMatrix &M(mVarf->SpMat());
bVarf->AddDomainIntegrator(new VectorFEDivergenceIntegrator);
bVarf->Assemble();
bVarf->Finalize();
SparseMatrix & B(bVarf->SpMat());
B *= -1.;
SparseMatrix *BT = Transpose(B);
BlockMatrix D(block_offsets);
D.SetBlock(0,0, &M);
D.SetBlock(0,1, BT);
D.SetBlock(1,0, &B);
$\triangleright$Define the left hand side for Lagrange method. This is the
BilinearForm asociated to the laplacian operator. DiffusionIntegrator is
asociated to $\Delta u$. The method FormLinearSystem is only used to establish
the essential boundary condition.
⬇
OperatorPtr A;
Vector XX,BB;
BilinearForm *a(new BilinearForm(H1_space));
a->AddDomainIntegrator(new DiffusionIntegrator(one));
a->Assemble();
a->FormLinearSystem(ess_tdof_list, p, *b, A, XX, BB);
$\triangleright$Solve linear systems with MINRES (for mixed) and CG (for
Lagrange). SetOperator method establishes the lhs. Mult method executes the
iterative algorithm and receives as input: the rhs and the vector to store the
solution.
⬇
int maxIter(10000);
double rtol(1.e-6);
double atol(1.e-10);
MINRESSolver Msolver;
Msolver.SetAbsTol(atol);
Msolver.SetRelTol(rtol);
Msolver.SetMaxIter(maxIter);
Msolver.SetPrintLevel(0);
Msolver.SetOperator(D);
x_mixed = 0.0;
Msolver.Mult(rhs_mixed, x_mixed);
if (Msolver.GetConverged())
std::cout << "MINRES␣converged␣in␣" << Msolver.GetNumIterations() <<
"␣iterations␣with␣a␣residual␣norm␣of␣" << Msolver.GetFinalNorm() << ".\n";
else
std::cout << "MINRES␣did␣not␣converge␣in␣" << Msolver.GetNumIterations() <<
"␣iterations.␣Residual␣norm␣is␣" << Msolver.GetFinalNorm() << ".\n";
CGSolver Lsolver;
Lsolver.SetAbsTol(atol);
Lsolver.SetRelTol(rtol);
Lsolver.SetMaxIter(maxIter);
Lsolver.SetPrintLevel(0);
Lsolver.SetOperator(*A);
x = 0.0;
Lsolver.Mult(rhs,x);
if (Lsolver.GetConverged())
std::cout << "CG␣converged␣in␣" << Lsolver.GetNumIterations() <<
"␣iterations␣with␣a␣residual␣norm␣of␣" << Lsolver.GetFinalNorm() << ".\n";
else
std::cout << "CG␣did␣not␣converge␣in␣" << Lsolver.GetNumIterations() <<
"␣iterations.␣Residual␣norm␣is␣" << Lsolver.GetFinalNorm() << ".\n";
$\triangleright$Save the solution into GridFunctions, which are used for error
computation and visualization.
⬇
u_mixed.MakeRef(R_space, x_mixed.GetBlock(0), 0);
p_mixed.MakeRef(W_space, x_mixed.GetBlock(1), 0);
p.MakeRef(H1_space,x,0);
$\triangleright$Get missing velocities from the solutions obtained.
Remember that $u=-\nabla p$. Mixed components are extracted using the
auxiliary variable "ue" defined before.
⬇
p.GetDerivative(1,0,ux);
p.GetDerivative(1,1,uy);
ux *= -1;
uy *= -1;
VectorGridFunctionCoefficient uc(&u_mixed);
ue.ProjectCoefficient(uc);
GridFunctionCoefficient ux_mixed_coeff(&ue,1);
GridFunctionCoefficient uy_mixed_coeff(&ue,2);
ux_mixed.ProjectCoefficient(ux_mixed_coeff);
uy_mixed.ProjectCoefficient(uy_mixed_coeff);
$\triangleright$Create the asociated Coefficient objects for error
computation.
⬇
GridFunction* pp = &p;
GridFunctionCoefficient p_coeff(pp);
GridFunction* uxp = &ux;
GridFunction* uyp = &uy;
GridFunctionCoefficient ux_coeff(uxp);
GridFunctionCoefficient uy_coeff(uyp);
FunctionCoefficient pex_coeff(p_ex);
VectorFunctionCoefficient uex_coeff(dim,u_ex);
FunctionCoefficient uex_x_coeff(u_ex_x);
FunctionCoefficient uex_y_coeff(u_ex_y);
$\triangleright$Define integration rule.
⬇
int order_quad = max(2, 2*order+1);
const IntegrationRule *irs[Geometry::NumGeom];
for (int i=0; i < Geometry::NumGeom; ++i){
irs[i] = &(IntRules.Get(i, order_quad));
}
$\triangleright$Compute exact solution norms. Here, the parameter $2$ in
ComputeLpNorm makes reference to the $L^{2}$ norm.
⬇
double norm_p = ComputeLpNorm(2., pex_coeff, *mesh, irs);
double norm_u = ComputeLpNorm(2., uex_coeff, *mesh, irs);
double norm_ux = ComputeLpNorm(2., uex_x_coeff, *mesh, irs);
double norm_uy = ComputeLpNorm(2., uex_y_coeff, *mesh, irs);
$\triangleright$Compute and print absolute errors.
⬇
double abs_err_u_mixed = u_mixed.ComputeL2Error(uex_coeff,irs);
printf("Velocity␣Mixed␣Absolute␣Error:␣%e\n", abs_err_u_mixed / norm_u);
double abs_err_p_mixed = p_mixed.ComputeL2Error(pex_coeff,irs);
printf("Pressure␣Mixed␣Absolute␣Error:␣%e\n", abs_err_p_mixed / norm_p);
double abs_err_p = p.ComputeL2Error(pex_coeff,irs);
printf("Pressure␣Absolute␣Error:␣%e\n", abs_err_p / norm_p);
double abs_err_ux = ux.ComputeL2Error(uex_x_coeff,irs);
double abs_err_uy = uy.ComputeL2Error(uex_y_coeff,irs);
double abs_err_u = pow(pow(abs_err_ux,2)+pow(abs_err_uy,2),0.5);
printf("Velocity␣Absolute␣Error:␣%e\n", abs_err_u / norm_u);
$\triangleright$Compute and print comparison errors.
⬇
double err_ux = ux_mixed.ComputeL2Error(ux_coeff,irs);
double err_uy = uy_mixed.ComputeL2Error(uy_coeff,irs);
double err_u = pow(pow(err_ux,2)+pow(err_uy,2),0.5);
printf("Velocity␣Comparison␣Error:␣%e\n", err_u / norm_u);
double err_p = p_mixed.ComputeL2Error(p_coeff, irs);
printf("Pressure␣Comparison␣Error:␣%e\n", err_p / norm_p);
$\triangleright$Visualize the solutions and the domain. GLVis visualization
tool uses port $19916$ to receive data.
⬇
char vishost[] = "localhost";
int visport = 19916;
if(visualization){
Vector x_domain(H1_space->GetVSize());
GridFunction domain(H1_space);
x_domain=0.0;
domain.MakeRef(H1_space,x_domain,0);
socketstream dom_sock(vishost, visport);
dom_sock.precision(8);
dom_sock << "solution\n" << *mesh << domain << "window_title␣’Domain’" <<
endl;
socketstream um_sock(vishost, visport);
um_sock.precision(8);
um_sock << "solution\n" << *mesh << u_mixed << "window_title␣’Velocity␣Mixed’"
<< endl;
socketstream pm_sock(vishost, visport);
pm_sock.precision(8);
pm_sock << "solution\n" << *mesh << p_mixed << "window_title␣’Pressure␣Mixed’"
<< endl;
socketstream uxm_sock(vishost, visport);
uxm_sock.precision(8);
uxm_sock << "solution\n" << *mesh << ux_mixed <<
"window_title␣’X␣Velocity␣Mixed’" << endl;
socketstream uym_sock(vishost, visport);
uym_sock.precision(8);
uym_sock << "solution\n" << *mesh << uy_mixed <<
"window_title␣’Y␣Velocity␣Mixed’" << endl;
socketstream p_sock(vishost, visport);
p_sock.precision(8);
p_sock << "solution\n" << *mesh << p << "window_title␣’Pressure’" << endl;
socketstream ux_sock(vishost, visport);
ux_sock.precision(8);
ux_sock << "solution\n" << *mesh << ux << "window_title␣’X␣Velocity’" << endl;
socketstream uy_sock(vishost, visport);
uy_sock.precision(8);
uy_sock << "solution\n" << *mesh << uy << "window_title␣’Y␣Velocity’" << endl;
}}
$\triangleright$Define C functions.
⬇
void fFun(const Vector & x, Vector & f){
f = 0.0;
}
double gFun(const Vector & x){
return -1.0;
}
double f_bound(const Vector & x){
return 0.0;
}
void u_ex(const Vector & x, Vector & u){
double xi(x(0));
double yi(x(1));
double zi(0.0);
u(0) = - exp(xi)*sin(yi)*cos(zi);
u(1) = - exp(xi)*cos(yi)*cos(zi);
}
double u_ex_x(const Vector & x){
double xi(x(0));
double yi(x(1));
double zi(0.0);
return -exp(xi)*sin(yi)*cos(zi);
}
double u_ex_y(const Vector & x){
double xi(x(0));
double yi(x(1));
double zi(0.0);
return -exp(xi)*cos(yi)*cos(zi);
}
double p_ex(const Vector & x){
double xi(x(0));
double yi(x(1));
double zi(0.0);
return exp(xi)*sin(yi)*cos(zi);
}
### 5.2 Appendix B : Numerical values of the comparison
The order parameter will be fixed for each table and $h$ parameter is shown in
the first column. To interpret the results take into account that P refers to
pressure, U refers to velocity, mx refers to mixed (from mixed finite element
method), err refers to absolute error (compared to the exact solution), and
comp refers to comparison (the error between the two solutions obtained by the
two different methods).
Order = 1
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 7.549479e-02 | 1.021287e+00 | 1.025477e+00 | 3.680827e-02 | 1.029378e+00 | 1.037635e+00
0.286032 | 3.627089e-02 | 1.022781e+00 | 1.023990e+00 | 1.727281e-02 | 1.032760e+00 | 1.035055e+00
0.143016 | 1.791509e-02 | 1.023236e+00 | 1.023596e+00 | 9.222996e-03 | 1.033725e+00 | 1.034369e+00
0.0715079 | 8.922939e-03 | 1.023372e+00 | 1.023480e+00 | 5.111295e-03 | 1.033999e+00 | 1.034182e+00
0.035754 | 4.455715e-03 | 1.023412e+00 | 1.023445e+00 | 2.859769e-03 | 1.034077e+00 | 1.034130e+00
0.017877 | 2.226845e-03 | 1.023424e+00 | 1.023435e+00 | 1.603788e-03 | 1.034100e+00 | 1.034115e+00
Order = 2
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 8.069013e-03 | 1.023329e+00 | 1.023554e+00 | 1.399079e-02 | 1.033924e+00 | 1.034255e+00
0.286032 | 2.138257e-03 | 1.023391e+00 | 1.023470e+00 | 7.845012e-03 | 1.034056e+00 | 1.034146e+00
0.143016 | 5.704347e-04 | 1.023417e+00 | 1.023442e+00 | 4.400448e-03 | 1.034093e+00 | 1.034120e+00
0.0715079 | 1.537926e-04 | 1.023426e+00 | 1.023434e+00 | 2.469526e-03 | 1.034104e+00 | 1.034112e+00
0.035754 | 4.194302e-05 | 1.023428e+00 | 1.023431e+00 | 1.385966e-03 | 1.034107e+00 | 1.034110e+00
Order = 3
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 8.691241e-04 | 1.023389e+00 | 1.023471e+00 | 8.745151e-03 | 1.034060e+00 | 1.034143e+00
0.286032 | 2.477673e-04 | 1.023417e+00 | 1.023443e+00 | 4.911967e-03 | 1.034094e+00 | 1.034120e+00
0.143016 | 7.316263e-05 | 1.023426e+00 | 1.023434e+00 | 2.756849e-03 | 1.034104e+00 | 1.034112e+00
0.0715079 | 2.178864e-05 | 1.023428e+00 | 1.023431e+00 | 1.547232e-03 | 1.034108e+00 | 1.034110e+00
Order = 4
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 3.199774e-04 | 1.023412e+00 | 1.023448e+00 | 6.119857e-03 | 1.034088e+00 | 1.034124e+00
0.286032 | 9.547574e-05 | 1.023424e+00 | 1.023435e+00 | 3.434952e-03 | 1.034103e+00 | 1.034114e+00
0.143016 | 2.862666e-05 | 1.023428e+00 | 1.023431e+00 | 1.927814e-03 | 1.034107e+00 | 1.034111e+00
Order = 5
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 1.552006e-04 | 1.023420e+00 | 1.023439e+00 | 4.578518e-03 | 1.034099e+00 | 1.034117e+00
0.286032 | 4.658038e-05 | 1.023427e+00 | 1.023433e+00 | 2.569749e-03 | 1.034106e+00 | 1.034112e+00
0.143016 | 1.406993e-05 | 1.023429e+00 | 1.023431e+00 | 1.442205e-03 | 1.034108e+00 | 1.034110e+00
Order = 6
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 8.612580e-05 | 1.023424e+00 | 1.023435e+00 | 3.584133e-03 | 1.034103e+00 | 1.034114e+00
0.286032 | 2.600417e-05 | 1.023428e+00 | 1.023431e+00 | 2.011608e-03 | 1.034107e+00 | 1.034111e+00
0.143016 | 7.897631e-06 | 1.023429e+00 | 1.023430e+00 | 1.128989e-03 | 1.034109e+00 | 1.034110e+00
Order = 7
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 5.243187e-05 | 1.023426e+00 | 1.023433e+00 | 2.899307e-03 | 1.034105e+00 | 1.034112e+00
0.286032 | 1.589631e-05 | 1.023429e+00 | 1.023431e+00 | 1.627221e-03 | 1.034108e+00 | 1.034110e+00
Order = 8
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 3.409225e-05 | 1.023427e+00 | 1.023432e+00 | 2.404311e-03 | 1.034107e+00 | 1.034111e+00
0.286032 | 1.037969e-05 | 1.023429e+00 | 1.023430e+00 | 1.349427e-03 | 1.034108e+00 | 1.034110e+00
Order = 9
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 2.328387e-05 | 1.023428e+00 | 1.023431e+00 | 2.033288e-03 | 1.034107e+00 | 1.034110e+00
0.286032 | 7.124397e-06 | 1.023429e+00 | 1.023430e+00 | 1.141177e-03 | 1.034109e+00 | 1.034110e+00
Order = 10
h | P comp | P err | Pmx err | U comp | U err | U mx err
---|---|---|---|---|---|---
0.572063 | 1.664200e-05 | 1.023429e+00 | 1.023431e+00 | 1.746755e-03 | 1.034108e+00 | 1.034110e+00
0.286032 | 5.085321e-06 | 1.023429e+00 | 1.023430e+00 | 9.803705e-04 | 1.034109e+00 | 1.034109e+00
### 5.3 Appendix C : MiniApp Code for Navier-Stokes
Here, the code used for Section 3 (written in C++) is shown, with a brief
explanations of it’s functionality.
$\triangleright$Include the required libraries (including navier miniapp).
⬇
#include "navier_solver.hpp"
#include <fstream>
using namespace mfem;
using namespace navier;
using namespace std;
$\triangleright$Define the context for the problem. In this case, we define
parameters for the 2D and 3D experiment. In general, the parameters to define
should be $\nu$, $dt$, $T$, $k$ and $\\#refinements$.
⬇
struct NavierContext{
//Parameters for 2D experiment
int max_order_steady = 6;
int max_refinements_steady = 5;
double kinvis_steady = 1.0 / 40.0;
double t_final_steady = 50 * 0.001;
double dt_steady = 0.001;
double delta = 0.001;
//Parameters for 3D experiment
double kinvis_3d = 0.001;
double t_final_3d = 8.0;
double dt_3d = 1e-3;
} ctx;
$\triangleright$Define a velocity as a C function. This function represents
the initial and boundary conditions for the velocity in the 3D experiment.
⬇
void vel_3d(const Vector &x, double t, Vector &u){
double xi = x(0);
double yi = x(1);
double zi = x(2);
double U = 2.25;
if(xi <= 1e-8){
u(0) = 16.0 * U * yi * zi * sin(M_PI * t / 8.0) * (0.41 - yi) * (0.41 - zi) /
pow(0.41, 4.0);
}else{ u(0) = 0.0; }
u(1) = 0.0;
u(2) = 0.0;}
$\triangleright$Define another velocity as a C function. This function
represents the boundary condition, $u_{0}$, for the 2D experiment.
⬇
void vel_steady(const Vector &x, double t, Vector &u){
double reynolds = 1.0 / ctx.kinvis_steady;
double lam = 0.5 * reynolds - sqrt(0.25 * reynolds * reynolds + 4.0 * M_PI *
M_PI);
double xi = x(0);
double yi = x(1);
u(0) = 1.0 - exp(lam * xi) * cos(2.0 * M_PI * yi);
u(1) = lam / (2.0 * M_PI) * exp(lam * xi)
* sin(2.0 * M_PI * yi);}
$\triangleright$Define a third velocity as a C function. This function
represents the initial condition, $u_{i}$, for the 2D experiment.
⬇
void vel(const Vector &x, double t, Vector &u){
double reynolds = 1.0 / ctx.kinvis_steady;
double lam = 0.5 * reynolds - sqrt(0.25 * reynolds * reynolds + 4.0 * M_PI *
M_PI);
double xi = x(0);
double yi = x(1);
double delta = ctx.delta;
u(0) = 1.0 - exp(lam * xi) * cos(2.0 * M_PI * yi);
u(1) = lam / (2.0 * M_PI) * exp(lam * xi) * sin(2.0 * M_PI * yi);
u(0) = u(0)+delta*(xi+0.5)*(xi-1)*(yi+0.5)*(yi-1.5);
u(1) = u(1)+delta*(xi+0.5)*(xi-1)*(yi+0.5)*(yi-1.5);
}
$\triangleright$Begin a function that receives an order and the amount of
refinements. This function will compute the 2D experiment. First, it defines
the mesh shown on figure 17.
⬇
void NS_steady(int order, int refinement){
Mesh *mesh = new Mesh(2, 4, Element::QUADRILATERAL, false, 1.5, 2.0);
mesh->EnsureNodes();
GridFunction *nodes = mesh->GetNodes();
*nodes -= 0.5;
$\triangleright$Refine the mesh and create the parallel version of the mesh.
⬇
for (int i = 0; i < refinement; ++i){
mesh->UniformRefinement();}
auto *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
delete mesh;
$\triangleright$Create the NavierSolver object (receives the mesh, $k$, and
$\nu$ as parameters). Set the initial condition, the function vel. And set the
boundary condition, the function vel_steady.
⬇
//Create the flow solver
NavierSolver flowsolver(pmesh, order, ctx.kinvis_steady);
flowsolver.EnablePA(true);
//Set the initial condition
ParGridFunction *u_ic = flowsolver.GetCurrentVelocity();
VectorFunctionCoefficient u_excoeff(pmesh->
Dimension(), vel);
u_ic->ProjectCoefficient(u_excoeff);
//Add Dirichlet boundary conditions to velocity space
Array<int> attr(pmesh->bdr_attributes.Max());
attr = 1;
flowsolver.AddVelDirichletBC(vel_steady, attr);
$\triangleright$Set up the problem ($dt$) and define the ParGridFunctions to
store the solution.
⬇
double t = 0.0;
bool last_step = false;
flowsolver.Setup(ctx.dt_steady);
ParGridFunction *u_gf = flowsolver.GetCurrentVelocity();
ParGridFunction *p_gf = flowsolver.GetCurrentPressure();
$\triangleright$Create the ParaView file and associate the variables to save
the solution.
⬇
ParaViewDataCollection
pvdc("STEADY"+to_string(order)+to_string(refinement),pmesh);
pvdc.SetDataFormat(VTKFormat::BINARY32);
pvdc.SetHighOrderOutput(true);
pvdc.SetLevelsOfDetail(order);
pvdc.SetCycle(0);
pvdc.SetTime(t);
pvdc.RegisterField("velocity",u_gf);
pvdc.RegisterField("pressure",p_gf);
pvdc.Save();
$\triangleright$Iterate from $t=0$ to $t=T$. For each iteration, the step is
taken and saved in the ParaView file.
⬇
for(int step = 0; !last_step; ++step){
//Check for final step
if (t + ctx.dt_steady >= ctx.t_final_steady - ctx.dt_steady / 2){last_step =
true;}
//Do the step
flowsolver.Step(t, ctx.dt_steady, step);
//Save paraview information
pvdc.SetCycle(step);
pvdc.SetTime(t);
pvdc.Save();}
delete pmesh;}
$\triangleright$Begin a function that receives an order and the amount of
refinements. This function will compute the 3D experiment. First, it defines
the mesh associated to the domain shown on figure 21. This part of the code
was planned to be used with different orders and refinement levels, however,
it was only run with one case.
⬇
void NS_3D(int order, int refinement){
Mesh *mesh = new Mesh("box-cylinder.mesh");
$\triangleright$Refine the mesh and create the parallel version of the mesh.
⬇
for (int i = 0; i < refinement; ++i){
mesh->UniformRefinement();}
auto *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
delete mesh;
$\triangleright$Create the NavierSolver object. Set the initial condition, the
function vel_3d. And set the boundary condition, the function vel_3d too.
Notice that the boundary condition is only applied to some parts of the mesh.
⬇
//Create the flow solver
NavierSolver flowsolver(pmesh, order, ctx.kinvis_3d);
flowsolver.EnablePA(true);
//Set the initial condition
ParGridFunction *u_ic = flowsolver.GetCurrentVelocity();
VectorFunctionCoefficient u_excoeff(pmesh->
Dimension(), vel_3d);
u_ic->ProjectCoefficient(u_excoeff);
//Add Dirichlet boundary conditions to velocity space restricted to selected
attributes on the mesh
Array<int> attr(pmesh->bdr_attributes.Max());
attr[0] = 1; //Inlet
attr[2] = 1; //Walls
flowsolver.AddVelDirichletBC(vel_3d, attr);
$\triangleright$Set up the problem ($dt$) and define the ParGridFunctions to
store the solution.
⬇
double t = 0.0;
bool last_step = false;
flowsolver.Setup(ctx.dt_3d);
ParGridFunction *u_gf = flowsolver.GetCurrentVelocity();
ParGridFunction *p_gf = flowsolver.GetCurrentPressure();
$\triangleright$Create the ParaView file and associate the variables to save
the solution.
⬇
ParaViewDataCollection pvdc("3D"+to_string(order)+to_string(refinement),
pmesh);
pvdc.SetDataFormat(VTKFormat::BINARY32);
pvdc.SetHighOrderOutput(true);
pvdc.SetLevelsOfDetail(order);
pvdc.SetCycle(0);
pvdc.SetTime(t);
pvdc.RegisterField("velocity", u_gf);
pvdc.RegisterField("pressure", p_gf);
pvdc.Save();
$\triangleright$Iterate from $t=0$ to $t=T$. For each iteration, the step is
taken and saved in the ParaView file.
⬇
for(int step = 0; !last_step; ++step){
//Check for final step
if(t + ctx.dt_3d >= ctx.t_final_3d - ctx.dt_3d / 2){last_step = true;}
//Do the step
flowsolver.Step(t, ctx.dt_3d, step);
//Save paraview information every 10 steps
if (step % 10 == 0){
pvdc.SetCycle(step);
pvdc.SetTime(t);
pvdc.Save();}}
delete pmesh;}
$\triangleright$Finally, define the main function with a MPI session (for
parallel computation). The 2D experiment corresponds to the use of the
function NS_steady while iterating through orders and refinement levels. And,
the 3D experiment is running the function NS_3D with order $4$. No refinements
were done because the original mesh (done by MFEM [5]) gives a good solution
already.
⬇
int main(int argc, char *argv[]){
//Define mpi sessions (parallel programming)
MPI_Session mpi(argc,argv);
//Steady experiment
for(int order=1;order<=ctx.max_order_steady;order++){
for(int refinements=0;refinements<=ctx.max_refinements_steady;refinements++){
NS_steady(order,refinements);
}
}
//3D experiment
NS_3D(4,0);
}
Note: This code was run using the command mpirun -n 4, which uses 4 cores of
the computer.
|
# UV/vis-to-IR Photonic Down Conversion Mediated by Excited State Vibrational
Polaritons
Connor K. Terry<EMAIL_ADDRESS>Department of Chemistry, Northwestern University, Evanston, Illinois
60208-3113, USA. Justin<EMAIL_ADDRESS>Department of Chemistry,
Northwestern University, Evanston, Illinois 60208-3113, USA. Emily A.
<EMAIL_ADDRESS>Department of Chemistry, Northwestern
University, Evanston, Illinois 60208-3113, USA. Roel
<EMAIL_ADDRESS>Department of Chemistry,
Northwestern University, Evanston, Illinois 60208-3113, USA.
###### Abstract
This work proposes a new photophysical phenomenon whereby UV/vis excitation of
a molecule involving a Franck-Condon (FC) active vibration yields infrared
(IR) emission by strong coupling to an optical cavity. The resulting UV/vis-
to-IR photonic down conversion process is mediated by vibrational polaritons
in the electronic excited state potential. It is shown that the formation of
such excited state vibrational polaritons (ESVP) via UV/vis excitation only
occurs with molecules having vibrational modes with both a non-zero FC
activity and IR activity in the excited state. Density functional theory
calculations are shown to effectively identify a candidate molecule,
1-Pyreneacetic acid (PAA), with this property and the dynamics of ESVP are
modeled using the truncated Wigner approximation. Overall, this work presents
a new avenue of polariton chemistry where excited state dynamics, driven by
photoexcitation, are influenced by the formation of vibrational polaritons.
Along with this, the photonic down conversion is potentially useful in both
the sensing of excited state vibrations and in quantum transduction schemes.
## 1 Introduction
Most light-matter interactions involved in spectroscopy are in the weak
coupling regime such that the intramolecular Coulombic fields are
significantly stronger than the interacting radiation fields.[1] As such,
perturbative theories accurately describe many spectroscopic phenomena.[2]
These theories break down in the “strong coupling” regime where light and
matter states hybridize, resulting in energy splittings that can be
spectroscopically observed.[3] The quanta of such hybrid light-matter states
are called “polaritons”, the formation of which is commonly realized by the
use of high finesse optical cavities with quantized radiation modes that can
be coupled to dipole carrying transitions within states of matter. Polaritons
formed between infrared (IR)-active molecular or condensed phase vibrations
and an optical cavity are, naturally, called “vibrational polaritons”.
In recent experiments, it has been demonstrated that product selectivity and
rates of ground state reactions may be modulated by the formation of
vibrational polaritons through coupling an optical cavity to the vibrational
coordinate on which a reaction proceeds.[4, 5, 6, 7, 8, 9, 10] Although there
has been significant experimental and theoretical work towards understanding
such ground state vibrational polaritons and their effect on reactions along
with other physiochemical phenomena,[11, 3, 12, 13, 14, 15, 16, 17, 18]
analogies on the electronic excited state potential have remained
underexplored.[19, 20] Here, we propose a novel photonic down conversion
process that results from the formation of excited state vibrational
polaritons (ESVP) via UV/vis electronic excitation of a molecule containing a
vibrational mode that is coupled to an IR cavity. Under vibronic coupling, the
UV/vis excitation of the molecule induces a wavepacket consisting of ESVP,
while Rabi oscillations occur between occupied vibrational and photonic states
as time evolves. These photons in turn may leak from the cavity, leading to IR
light emission.[21]
The UV/vis-to-IR photonic down conversion process proposed in this work has
potential applications in quantum information science, where the vibrational
wave-packet, formed by a UV/vis excitation of the cavity-embedded molecule,
can be directly transduced into a photonic wavepacket through coupling to the
optical cavity. Hence, molecules with different vibronic couplings and excited
state potential energy surfaces produce different coherences within the
photonic wavepacket emitted via the ESVP photonic down conversion process.
This could allow for the manipulation of coherent light by molecular design.
Furthermore, the UV/vis-to-IR photonic down conversion process implies that an
optical cavity can be used to enhance the emission of excited state
vibrations, analogous to the Purcell effect in the weak light-matter coupling
regime.[22, 23] The effective detection of excited state vibrations through
this process would be a powerful measurement tool for identifying vibrational
modes induced by electronic transitions—information that has been difficult to
obtain experimentally,[24] and that often requires a determination through
electronic structure calculations and molecular dynamics simulations.[25]
Detection of photonic emission from vibrations is typically impossible due to
their non-radiative relaxation being on the order of picoseconds,[26, 27]
which out-competes any non-enhanced emission that occurs on a $>$ 1 µs time
scale.[28, 29] It was experimentally indicated by Raschke and coworkers that
ground state vibrational emission could be enhanced by coupling molecules to
an optical resonator, yielding a 50% decrease in the vibrational dephasing
lifetime for poly(methylmethacrylate),[29] although photonic emission from
vibrations was not actually observed.
Another potential application of ESVP is in the modulation of photophysical
processes that occur via transitions between excited electronic states, and
which can be mediated by vibrational coherences. For example, vibrational
coherences have been shown to mediate singlet fission in certain molecular
systems,[30, 31, 32, 33, 34, 35] promote charge and energy transfer in
biological light harvesting systems,[36, 37, 38, 39] as well as drive the
photochemical reaction of vision.[40, 41, 42] The hybridization of cavity
photons with relevant vibrational modes could modulate such coherences as well
as change excited state potential energy surfaces (PES), altering reaction
pathways and rates similarly to what has been shown for ground state
reactions.
In this work, we utilize the truncated Wigner approximation (TWA)[43] to
theoretically predict the time evolution of quantum mechanical observables in
a combined cavity-molecule system, either without dissipation or when coupled
to dissipative harmonic baths. The TWA is exact for systems governed by linear
or quadratic potentials, i.e., systems consisting of harmonic oscillators.
Furthermore, the TWA benefits from linear scaling of complexity with
increasing number of degrees of freedom (DOF), as opposed to the exponential
increase in complexity of the Hilbert space of the system. Therefore, the TWA
enables exact modeling of the evolution of polariton wavepackets coupled to
baths of harmonic oscillators without requiring a truncated Hilbert space and
without tracing out the bath DOF, as is done in standard quantum master
equation techniques.[44, 45] We are therefore able to concurrently and
explicitly monitor the population transfer into both a radiative and non-
radiative bath, giving insights into how different interaction parameters
affect photonic emission from ESVP in the presence of non-radiative
vibrational relaxation.
## 2 Results
### 2.1 Photonic Down Conversion Mechanism
Figure 1: Schematic of the UV/vis-to-IR photonic down conversion process
mediated by ESVP. An impulsive excitation by UV/vis light (blue arrow) creates
ESVP due to vibronic interactions of the cavity coupled vibrational mode. The
ESVP then leak IR radiation from the cavity (red arrows).
Figure 1 shows a schematic of the ESVP mediated photonic down conversion
process. A molecule is positioned inside an optical cavity, initially in its
ground electronic state. A mode of the optical cavity is assumed to be near
resonant with a spectrally isolated vibrational mode of the molecule. We treat
the cavity as transparent to high optical frequencies allowing a UV/vis pulse
to promote the molecule into an electronically excited state. We also assume
the FC approximation where vibronic coupling is characterized by a linear
shift in the excited state nuclear PES, as shown in Figure 1, with
$Q_{0}^{(g)}$ and $Q_{0}^{(e)}$ being the ground and excited state equilibrium
nuclear configurations of the vibrational mode of interest, respectively.
Assuming a non-zero Huang-Rhys (HR) factor (vibronic coupling constant), a
vibrational wavepacket will be created upon electronic excitation of the
molecule. In the absence of the cavity, this wavepacket will have an expected
occupation number, $\langle\hat{N}_{\textrm{vib}}\rangle$, equal to the HR
factor ($S$),[46, 47] and its energy expectation value equals
$S\hbar\omega_{v}$, which is called the reorganization energy, where
$\omega_{v}$ is the vibrational frequency. When the cavity is present and the
excited state vibrational transition dipole moment
($\bm{\mu}_{\textrm{IR}}^{(e)}$) is non-zero for the cavity resonant
vibrational mode, the vibrational wavepacket will comprise a superposition of
polariton eigenstates and Rabi oscillations will occur between vibrations and
cavity photons as time evolves. Furthermore, given a finite quality factor
($\mathcal{Q}$) of the cavity, which is a measure of its radiative energy
loss,[48] the cavity photons involved in such ESVP will leak out of the cavity
resulting in IR emission that can be detected and/or harnessed.[21]
Altogether, these steps constitute a photonic down conversion process where a
single incident UV/vis photon is converted into one or more IR photons.
In this work, we only consider dissipation of cavity photons through the
cavity mirrors so that $\mathcal{Q}$ is dependent only on the amount of
detectable radiative emission. In general, other dissipation pathways should
also be accounted for such as absorption by the cavity mirrors and bound mode
photoluminescence,[21] which would also decrease $\mathcal{Q}$. Therefore, the
results presented herein give an idealized estimate of the amount of photonic
down conversion that can occur within a lossy cavity.
### 2.2 Theoretical Model
We use the Pauli-Fierz Hamiltonian as a basis for constructing a model of the
ESVP mediated photonic down conversion process.[49, 50, 51] Accordingly, the
cavity-molecule system is described as
$\hat{H}_{\textrm{system}}=\hat{H}_{\textrm{mol}}+\frac{\hat{p}^{2}}{2}+\frac{\omega_{c}^{2}}{2}(\hat{q}+\bm{A}\cdot\bm{\hat{\mu}})^{2},$
(1)
where $\hat{p}$ and $\hat{q}$ are the cavity mode momentum and position
operators, respectively, and $\omega_{c}$ is the frequency of the cavity mode.
Moreover, $\bm{A}=\sqrt{2/\hbar\omega_{c}}A_{0}\bm{\epsilon}_{c}$, where
$A_{0}=\sqrt{\hbar/2\omega_{c}\varepsilon V}$ is the amplitude of the cavity
mode vector potential, with $\varepsilon$ being the effective permittivity of
the cavity and $V$ being the cavity mode volume, and $\bm{\epsilon}_{c}$ is
the cavity mode polarization unit vector. Lastly, $\bm{\hat{\mu}}$ is the
molecular dipole operator and $\hat{H}_{\textrm{mol}}$ describes the molecule.
We express $\hat{H}_{\textrm{system}}$ in the molecular diabatic basis at the
ground state equilibrium, that is, the basis of electronic eigenstates,
$|\alpha\rangle$, that satisfy
$\hat{H}_{\textrm{mol}}^{\textrm{el}}(Q_{0}^{(g)})|\alpha\rangle=E_{\alpha}(Q_{0}^{(g)})|\alpha\rangle.$
(2)
Here, $\hat{H}_{\textrm{mol}}^{\textrm{el}}(Q_{0}^{(g)})$ is the component of
$\hat{H}_{\textrm{mol}}$ containing a dependence on the electronic DOF,
parameterized by $Q$, which is taken to be fixed at $Q=Q_{0}^{(g)}$. We
truncate this basis to only include the ground and a single excited diabatic
state denoted as $|g\rangle$ and $|e\rangle$, respectively. In this basis,
$\hat{H}_{\textrm{mol}}$, including a single harmonic vibrational mode, can be
written as
$\hat{H}_{\textrm{mol}}=\frac{\hat{P}^{2}}{2}+\frac{1}{2}\omega_{v}^{2}\hat{Q}^{2}+\left(E_{\textrm{vert}}-\omega_{v}^{2}Q_{0}^{(e)}\hat{Q}\right)|e\rangle\langle
e|,$ (3)
where $\hat{P}$ and $\hat{Q}$ are the mass-weighted momentum and position
operators of the vibrational mode, respectively,
$E_{\textrm{vert}}=\hbar\Omega_{e,g}+S\hbar\omega_{v}$ is the energy of the
vertical transition from the ground to excited electronic PES at the point of
the ground state nuclear equilibrium configuration (the FC point) with
$\Omega_{e,g}$ being the transition frequency between the ground and excited
electronic states at their respective nuclear equilibrium position.
Furthermore, in Eq. (3) we have taken the ground state nuclear equilibrium
configuration to be at the origin so that $Q^{(g)}_{0}=0$, and it is assumed
that there is no coupling between diabatic states. We have also assumed that
the vibrational frequency is independent of electronic state.
Eq. (1) and Eq. (3) effectively describe a system of coupled harmonic
oscillators: the vibrational mode and the cavity, which are coupled through an
interaction of the cavity field with the molecular dipole. Expanding the
quadratic term in Eq. (1) gives
$\hat{H}_{\textrm{system}}=\hat{H}_{\textrm{mol}}+\hat{H}_{\textrm{cav}}+\hat{H}_{\textrm{int}},$
(4)
where
$\hat{H}_{\textrm{cav}}=\frac{\hat{p}^{2}}{2}+\frac{\omega_{c}^{2}}{2}\hat{q}^{2}$
(5)
and
$\hat{H}_{\text{int}}=\omega_{c}^{2}\bm{A}\cdot\bm{\hat{\mu}}\,\hat{q}+\frac{\omega_{c}^{2}}{2}(\bm{A}\cdot\bm{\hat{\mu}})^{2}.$
(6)
As is commonly done with Jaynes-Cummings like models,[21, 52, 13] we ignore
the 2nd term of $\hat{H}_{\text{int}}$, which is the dipole self energy,
reserving a survey of its effects to a follow up study (as it may become
significant in ultra strong coupling[51]).
Once $\hat{H}_{\textrm{system}}$ is expressed in the diabatic basis, each
element of the dipole operator in this basis is expanded to the first order
with respect to $\hat{Q}$ about the ground state nuclear equilibrium position
($Q_{0}^{(g)}=0$) in order to account for linear coupling between the cavity
and the vibrational mode. This gives rise to two cavity-vibration interaction
terms,
$\hat{H}_{\textrm{int}}=\hat{H}_{\text{PFS}}+\hat{H}_{\text{cav-vib}},$ (7)
where
$\hat{H}_{\text{PFS}}=\omega_{c}^{2}\bm{A}\cdot\bm{\mu}^{0}_{e}\,\hat{q},$ (8)
and where
$\hat{H}_{\text{cav-
vib}}=\omega_{c}^{2}\bm{A}\cdot\bm{\mu}_{e}^{\prime}\hat{Q}\,\hat{q}.$ (9)
Here, $\bm{\mu}_{e}^{0}$ is the permanent dipole moment of the excited state
at $Q_{0}^{(g)}$ and $\bm{\mu}_{e}^{\prime}$ is the derivative of the nuclear
dipole with respect to $Q$ (see “Methods”). The term $\hat{H}_{\text{PFS}}$
polarizes the bare cavity Fock states (hence, denoted PFS for polarizes Fock
states),[53] and its effect on the ESVP mediated down conversion process is
discussed in section 3 of the SI. However, the term $\hat{H}_{\text{cav-vib}}$
is the dominant interaction term and mixes cavity and vibrational DOF.
Therefore, we only consider
$\hat{H}_{\textrm{system}}=\hat{H}_{\textrm{mol}}+\hat{H}_{\textrm{cav}}+\hat{H}_{\textrm{cav-
vib}}$.
For the application of the TWA we keep $\hat{H}_{\textrm{system}}$ in terms of
phase space operators; nonetheless, we believe it valuable to connect our
model with the coupling constant ($g$) used in Jaynes-Cummings like
models.[21, 52, 13] This is done by substituting
$\hat{Q}=\sqrt{\frac{\hbar}{2\omega_{v}}}(\hat{b}+\hat{b}^{\dagger})$ and
$\hat{q}=\sqrt{\frac{\hbar}{2\omega_{c}}}(\hat{a}+\hat{a}^{\dagger})$ into Eq.
(9), as well as taking the rotating wave approximation (RWA), giving rise to
the expression
$g=\omega_{c}A_{0}\bm{\epsilon}_{c}\cdot\,\bm{\mu_{\textrm{IR}}^{(e)}},$ (10)
where
$\bm{\mu}_{\textrm{IR}}^{(e)}=\bm{\mu}_{e}^{\prime}\sqrt{\frac{\hbar}{2\omega_{v}}}$
(11)
is the vibrational transition dipole moment, which can be found through
electronic structure calculations or from experimental IR spectra.[54, 55].
### 2.3 Relevant Molecular Parameters
Figure 2: Pictorial representation of the normal mode decomposition for an
optical cavity coupled to a molecular vibration. The cavity and vibration
harmonic oscillators are depicted as springs. The cavity, with position
coordinate $q$, and vibration, with position coordinate $Q$, are coupled by
$G$. The cavity and vibrational coordinates are then rotated into a coordinate
system of two uncoupled oscillators called the polariton basis, with
coordinates $R_{-}$ and $R_{+}$.
Using the model Hamiltonian described in section 2.2, we determine the
relevant molecular properties that are required for the photonic down
conversion to occur through ESVP. This is achieved by monitoring the cavity
photon occupancy ($\hat{N}_{\text{cav}}$), which is representative of the
amount of photons that can be harnessed as IR emission if the cavity is leaky
(finite $\mathcal{Q}$). Rather than representing the Hilbert space by a
truncated basis of cavity photon and vibrational modes, we apply the TWA to
arrive at an exact analytical expression for
$\langle\hat{N}_{\text{cav}}\rangle$ (see SI section 2 for more details). Note
that the RWA is not invoked here. To apply the TWA, we rotate
$\hat{H}_{\textrm{system}}$ into a basis of two uncoupled oscillators called
the polariton normal mode basis (see Figure 2 and “Methods”). Due to the
displacement of the excited state PES, upon a vertical excitation of the
molecule both the upper and lower polariton harmonic oscillators are displaced
along their respective coordinates, as depicted in Figure 2. This displacement
of excited state polariton oscillators creates a non-equilibrium initial
condition that drives the dynamics. At finite temperature the resulting photon
occupancy, following the UV/vis excitation, is given by
$\begin{split}\langle\hat{N}_{\text{cav}}(t)\rangle=\frac{S\omega_{v}^{3}}{2(C^{2}+1)}&\bigg{[}\frac{1}{2\omega_{c}}\left(\frac{\sin(\Omega_{-}t)}{\Omega_{-}}-\frac{\sin(\Omega_{+}t)}{\Omega_{+}}\right)^{2}\\\
&+2\omega_{c}\left(\frac{\sin^{2}(\Omega_{-}t/2)}{\Omega_{-}^{2}}-\frac{\sin^{2}(\Omega_{+}t/2)}{\Omega_{+}^{2}}\right)^{2}\bigg{]}+N_{\beta},\end{split}$
(12)
where
$C=\frac{\omega_{v}^{2}-\omega_{c}^{2}}{4\frac{g}{\hbar}\sqrt{\omega_{v}\omega_{c}}}$,
and where the polariton frequencies are given by $\Omega_{\pm}=\sqrt{A\pm B}$
with $A=\frac{1}{2}(\omega_{v}^{2}+\omega_{c}^{2})$ and
$B=\frac{1}{2}\sqrt{(\omega_{v}^{2}-\omega_{c}^{2})^{2}+16\frac{g^{2}}{\hbar^{2}}\omega_{v}\omega_{c}}$.
The effect of finite temperature is captured by
$N_{\beta}=\frac{1}{4}\sum_{\gamma}\left(c_{c}^{(\gamma)2}\left(\frac{\Omega_{\gamma}}{\omega_{c}}+\frac{\omega_{c}}{\Omega_{\gamma}}\right)\coth(\beta\hbar\Omega_{\gamma}/2)\right)-\frac{1}{2}$,
which describes the thermal distribution of vibrational polaritons on the
electronic ground state, before electronic excitation of the molecule, where
$\gamma$ denotes the upper ($\gamma=+$) or lower ($\gamma=-$) polariton mode,
$\beta$ is the inverse temperature, and $c_{c}^{(\gamma)}$ is the projection
of the cavity mode onto the $\gamma$ polariton mode. In the following
analysis, we take the zero temperature limit ($\beta\rightarrow\infty$) and
assume the cavity to be in perfect resonance with the vibration so that
$\omega_{c}=\omega_{v}$, yielding $c_{c}^{(\gamma)}=1/\sqrt{2}$ for $\gamma=+$
and $\gamma=-$. This results in
$N_{\beta}=\frac{1}{8}\sum_{\gamma}\left(\frac{\Omega_{\gamma}}{\omega_{c}}+\frac{\omega_{c}}{\Omega_{\gamma}}\right)-\frac{1}{2}$.
For weak coupling, where $\Omega_{\gamma}\approx\omega_{c}=\omega_{v}$, it is
apparent that $N_{\beta}\approx 0$. But, larger coupling strengths result in
$N_{\beta}>0$, which is concurrent with the breakdown of the RWA. In that
case, the polaritonic ground state within the RWA, which has no photons or
vibrational quanta, mixes with the state containing a single cavity photon and
single vibration, and therefore, even at zero temperature there will be a non-
zero number of cavity photons if the coupling is large.
A key result from Eq. (12) is that $\langle\hat{N}_{\text{cav}}(t)\rangle$ is
proportional to $S$. This can be understood by considering that $S$ is
proportional to the squared displacement of the excited state PES. If the
displacement is zero, all FC factors will be zero indicating that no
vibrational transitions will be induced upon the UV/vis excitation. Increasing
$S$ increases the displacement, and consequentially, the number of vibrational
quanta created upon electronic excitation. Given a non-zero $g$ value, these
vibrational quanta will be converted into cavity photons.
Figure 3: (a) Time evolution of the cavity photon occupancy,
$\langle\hat{N}_{\text{cav}}\rangle$, for a single intramolecular vibration
coupled to a cavity following an electronic excitation of the molecule, for
different values of $g$ and for a fixed HR factor ($S=0.75$). (b) The maximum
value of the $\langle\hat{N}_{\text{cav}}\rangle$ time trace
(Max$\\{\langle\hat{N}_{\text{cav}}\rangle\\}$) as a function of both $g$ and
$S$. The white dashed line represents $S=0.75$, with each point on the line
corresponding to the $S$ and $g$ values used for the traces in (a). The cavity
frequency was set in resonance with the vibration:
$\omega_{c}=\omega_{v}=1600\textrm{ cm}^{-1}$.
Figure 3a shows the time evolution of $\langle\hat{N}_{\text{cav}}\rangle$
given by Eq. (12) for a cavity mode in perfect resonance with the vibrational
mode, i.e., $\omega_{c}=\omega_{v}=7.3$ mHa or 1600 $\textrm{cm}^{-1}$.
Results are shown for five evenly spaced $g$ values between 0 and 0.9 mHa. At
time zero, $\langle\hat{N}_{\text{cav}}\rangle=0$ for all values of $g$ since
the UV/vis excitation acts exclusively on the electronic DOF, inducing
vibrations in the excited state. This vibrational wavepacket has zero
projection onto the cavity mode at time zero and is not an eigenstate of
$\hat{H}_{\textrm{system}}$. As time evolves, Rabi oscillations occur as the
wavepacket gains a non-zero projection onto the cavity mode giving rise to the
$\langle\hat{N}_{\text{cav}}\rangle$ traces observed. The multi-oscillatory
behavior observed in the $\langle\hat{N}_{\text{cav}}\rangle$ traces is due to
interference between the contributing polariton frequencies. It can be seen
that as $g$ increases so does the maximum value
$\langle\hat{N}_{\text{cav}}\rangle$ can have
(Max$\\{\langle\hat{N}_{\text{cav}}\rangle\\}$). For small $g$ values,
Max$\\{\langle\hat{N}_{\text{cav}}\rangle\\}$ is approximately equal to the
expected number of vibrational quanta produced upon electronic excitation,
$S$. However, the energy stored in the cavity-vibration coupling will also
contribute to $\langle\hat{N}_{\text{cav}}\rangle$, and as $g$ becomes less
negligible in comparison to the cavity and vibration energies,
$\langle\hat{N}_{\text{cav}}\rangle$ will become noticeably greater than $S$,
as observed in Figure 3a.
Figure 3b shows a heat map of Max$\\{\langle\hat{N}_{\text{cav}}\rangle\\}$ as
a function of $g$ and $S$. From this, it is clear that maximizing both $g$ and
$S$ will lead to the largest Max$\\{\langle\hat{N}_{\text{cav}}\rangle\\}$
value. Accordingly, a molecule that has a vibrational mode with a significant
$\bm{\mu}_{\textrm{IR}}^{(e)}$ value, and hence a large $g$ value (see Eq.
(10)), along with having significant vibronic coupling, i.e., large $S$, will
make a good candidate for use in ESVP mediated photonic down conversion.
### 2.4 Radiative and Non-Radiative Dissipation of ESVP
So far our analysis has assumed the molecule and cavity to be fully isolated,
apart from their mutual coupling, without including emission from the cavity,
which is an integral step of the ESVP mediated photonic down conversion
process. Moreover, practical implementations of this process will suffer from
non-radiative relaxation of the intramolecular vibration. It is therefore
necessary to consider photonic emission and non-radiative dissipation from the
cavity-vibration system. To this end, we are interested in how the photonic
emission can be maximized throughout the parameter space of the system. It is
evident that a large $S$ value will increase the probability of photonic down
conversion as this will produce more excited state vibrational quanta. Other
parameters, however, such as the non-radiative relaxation rate, the cavity-
vibration coupling, and the cavity emission rate (related to $\mathcal{Q}$)
have a non-trivial relationship to the percentage of incoming UV/vis photons
converted to emitted IR photons.
To study the influence of these parameters on the photonic down conversion
process, we add two harmonic baths to our model: 1) a bath representing an
external electromagnetic field that accounts for cavity emission, and 2) a
bath of (solvent) phonon modes that account for non-radiative vibrational
relaxation. Accordingly, the total system-bath Hamiltonian is given by
$H_{\textrm{total}}=H_{\textrm{system}}+\sum_{i=1}^{E+B}\left(\frac{\hat{p}_{i}^{2}}{2}+\omega_{i}^{2}\frac{\hat{r}_{i}^{2}}{2}\right)+\sum_{i=1}^{E}\kappa_{i}\,\hat{r}_{i}\hat{q}+\sum_{i=E+1}^{E+B}\chi_{i}\,\hat{r}_{i}\hat{Q},$
(13)
where $\hat{r}_{i}$ is the position operator for the $1\leq i\leq E$ external
field modes and $E<i\leq E+B$ non-radiative bath modes, $\kappa_{i}$ is the
coupling constant between the $i$th external field mode and the cavity, and
$\chi_{i}$ is the coupling constant between the $i$th non-radiative bath mode
and the vibration. It should be noted that, from Eq. (26) and Eq. (31) in
“Methods”,
$\kappa_{i}=\sqrt{\frac{\omega_{i}}{2\pi\mathcal{Q}D_{E}}},$ (14)
where $D_{E}$ is the density of states of the external electromagnetic field,
chosen to be constant as a function of $\omega_{i}$.[56] Hence, the cavity-
external field coupling, $\kappa_{i}$, and $\mathcal{Q}$ are inversely
related. This is an intuitive result because as $\mathcal{Q}$ decreases,
cavity photons are expected to dissipate more efficiently due to a stronger
coupling of the cavity to the external field.
Figure 4: (a) Schematic showing the total Hamiltonian,
$\hat{H}_{\textrm{total}}$, including the cavity-molecule system, the non-
radiative bath modes, and the external-field modes that constitute emission.
(b-d) Dynamics of different occupation numbers for different quality factor
values ($\mathcal{Q}$) at $g=0.04$ mHa: Black = cavity photons; blue =
intramolecular vibrations; orange = external field photons; green = non-
radiative bath. (e) A heat map of the % emission of IR photons as a function
of $g$ and $\mathcal{Q}^{1/2}$, where % emission is evaluated at 18 ps. (f) %
emission as a function of $\mathcal{Q}^{1/2}$ evaluated at different $g$
values. These traces correspond to % emission along the dashed lines in (e).
The labeled points along the $g=0.04$ mHa trace correspond to the
$\mathcal{Q}$ and $g$ values of the dynamics shown in (b-d). The cavity
frequency was set in resonance with the vibration:
$\omega_{c}=\omega_{v}=1600\textrm{ cm}^{-1}$.
The schematic in Figure 4a represents the total Hamiltonian given in Eq. (13).
All components of the ensemble are comprised of harmonic oscillators and the
red arrows show the coupling between components. Following a similar procedure
as was done for the calculation of $\langle\hat{N}_{\text{cav}}\rangle$ in the
isolated cavity-vibration case, we used the TWA to explicitly describe the
time evolution of the system, non-radiative bath, and external field. This
time, normal modes of the $\hat{H}_{\textrm{total}}$ Hessian matrix were
determined. The time evolution was monitored by means of the occupation
numbers of the different components.
Figures 4b-d show the resulting occupation numbers following a UV/vis
excitation of the molecule for different $\mathcal{Q}$ values and for a fixed
$g$ value of 0.04 mHa. These data show that by changing $\mathcal{Q}$, regimes
with quantitatively different behaviors are obtained. For the relatively small
value $\mathcal{Q}$ $=9$ (Figure 4b), the cavity occupancy is significantly
dampened due the strong coupling to the external field. Therefore, the cavity
dissipation lifetime is shorter than the Rabi oscillation period. It can also
be seen that the non-radiative relaxation (green) out-competes photonic
emission (orange) for this case. For the value $\mathcal{Q}$ $=100$ (Figure
4c), short lived Rabi-oscillations are observed. The presence of Rabi-
oscillations are indicative of strong coupling between the cavity and the
molecular vibrational mode. In this case, the photonic emission outcompetes
the non-radiative relaxation. For the value $\mathcal{Q}$ $=1600$ (Figure 4d),
longer-lived Rabi-oscillations are observed between the cavity and the
molecular vibration mode. In this case, similar to the $\mathcal{Q}$ $=9$
case, the non-radiative relaxation outcompetes the photonic emission. The
change in dynamic behavior with $\mathcal{Q}$ observed in Figures 4b-d are the
result of $\mathcal{Q}$ affecting the cavity decay lifetime (contingent on
both the radiative and non-radiative dissipation rates) relative to $g$. The
strong coupling regime, indicated by the presence of Rabi-oscilations between
the cavity and vibration as well as peak splitting in the emission spectrum,
occurs when the cavity decay lifetime is long compared to the cavity-vibration
Rabi oscillation period.[57]
Figure 4e shows a heat map of the percentage of vibrations induced by the
UV/vis excitation converted to photonic emission (% emission) as a function of
$g$ and $\mathcal{Q}$. The % emission is calculated as the ratio of external
field photons at equilibrium normalized to the initial number of vibrations
created by the UV/vis excitation, given by $S$, that is,
$\textrm{\%
emission}=\frac{1}{S}\sum_{i}^{E}\langle\hat{N}_{\textrm{ext}}^{\,i}(t_{\textrm{eq}})\rangle.$
(15)
Here, $\hat{N}_{\textrm{ext}}^{\,i}$ is the $i$th external field occupation
number operator and $t_{\textrm{eq}}$ is the time at which equilibrium is
reached, here taken to be $t_{\textrm{eq}}=18$ ps, which suffices for all
values of $g$ and $\mathcal{Q}$ used. By using this definition the % emission
is independent of $S$. For example, the results in Figure 4e were computed
with $S=0.5$ but would be identical for any $S$ value used. Determining the
net amount of photons produced in the photonic down conversion process simply
requires multiplying the % emission by $S$.
Figure 4f shows slices of the % emission heat map along $\mathcal{Q}$ for
different $g$ values given by the dashed lines in Figure 4e. The labeled
points in 4f indicate the related dynamics in Figure 4b-d that give rise to
those % emission values. It can be seen from Figure 4f that the % emission
increases with $g$ for all values of $\mathcal{Q}$, which indicates that
increasing $g$ will always increase the photonic down conversion yield.
Furthermore, all traces in Figure 4 have a maximum along $\mathcal{Q}$, which
we denote as $\mathcal{Q}^{\textrm{max}}$. This is due to the competition
between radiative and non-radiative decay. For $\mathcal{Q}$ values larger
than $\mathcal{Q}^{\textrm{max}}$, the probability of cavity photons leaking
from the cavity into the external field becomes small enough that non-
radiative dissipation outcompetes emission. On the other hand, for
$\mathcal{Q}$ values smaller than $\mathcal{Q}^{\textrm{max}}$, the cavity
mode is coupled strong enough to the external field modes that it begins to
decouple to the vibration, and again, non-radiative dissipation outcompetes
emission. In fact, in the limit where $\mathcal{Q}$ becomes very small, the
cavity mode mixes into the external field to the extent that the system
effectively becomes a vibration interacting with a free-space electromagnetic
field, and radiative emission from the vibration is not expected due to the
fast non-radiative relaxation.
$\mathcal{Q}^{\textrm{max}}$ should be applicable for any polariton system
where the matter component has a non-radiative relaxation pathway, implying
that a finite cavity $\mathcal{Q}$ value is needed to maximize the photonic
emission from such systems. This motivates cavity design principles for
measuring and utilizing ESVP mediated photonic down conversion, as well as for
any system that uses an optical resonator to enhance emission.
### 2.5 Calculation of Relevant Parameters for Pyrene and PAA
The identification of molecules that allow for the formation of ESVP through a
UV/vis excitation is critical for all potential applications of the ESVP
mediated photonic down conversion process. Following the results of Figures 3
and 4, molecules must contain vibrational modes that are both FC active (i.e.,
having non-zero HR factor) and IR active within the excited state for the ESVP
photonic down conversion to occur. According to the rule of mutual exclusion,
such modes exist only in molecules lacking inversion symmetry (vide infra).
Furthermore, the extent to which the photonic down conversion process can
occur in non-centrosymmetric molecules is dependent on the strength of their
vibronic couplings and excited state IR activity. This section focuses on
extracting these molecular properties in specific molecules while making a
connection to our theoretical model.
Figure 5: DFT calculations of the vibronic couplings and excited state IR
dipoles for pyrene and 1-pyreneacetic acid (PAA). Calculated absorption
spectra for (a) pyrene and (b) PAA were performed at different temperatures
(red = 300 K; blue = 0.1 K) and with different standard deviations ($\sigma$)
for the Gaussian distribution of electronic energies taken to represent
inhomogeneous broadening (red $=200\textrm{ cm}^{-1}$; blue $=10\textrm{
cm}^{-1}$). Experimental spectra (black) were taken in cyclohexane for pyrene
and toluene for PAA at room temperature ($\sim$ 293 K). Calculated HR factors
($S$) and excited state IR dipole moments ($\mu^{(e)}_{\textrm{IR}}$) for
vibrational modes of (c) pyrene and (d) PAA are plotted as a function of the
wavenumber of each mode. $\mu^{(e)}_{\textrm{IR}}$ is plotted “on top of” the
$S$ values, and are only shown for modes with $S>0$, while its value is
indicated above each bar.
We have performed DFT and time-dependent-DFT (TD-DFT) calculations for two
molecules: one with and one without inversion symmetry. Figure 5 shows results
for both pyrene and 1-pyreneacetic acid (PAA), which are rigid chromophores
with clear vibronic progressions due to vibronic coupling. Pyrene is
centrosymmetric and should not have vibrational modes that are both IR and
Raman active in accordance with the rule of mutual exclusion.[58] This is
relevant as Raman activity implies FC activity. In centrosymmetric molecules,
all electronic states contain an inversion symmetry such that electronically
exciting them will only induce symmetric vibrations, meaning the excited state
PES is displaced only along these symmetric modes. This displacement allows
ground state vibrational overtones to be reached in the second order Raman
process making them Raman active.[59] Importantly, since only symmetric modes
are induced upon electric excitation, which are IR inactive, no ESVP should
form in centrosymetric molecules through UV/vis excitation. PAA, on the other
hand, has a carboxylic acid functional group that breaks the inversion
symmetry of the pyrene moiety and should contain IR and FC active vibrational
modes. By comparing these two molecules we can test whether simple violations
of the rule of mutual exclusion can serve to design molecules for ESVP
mediated photonic down conversion.
In order to determine the accuracy of the calculations, absorption spectra
were calculated and compared to experimental spectra (Figure 5a and 5b). These
spectra were calculated using the TWA as detailed in other work,[60] and are
exact within the FC and harmonic approximation. Results were calculated at
finite temperature by taking a Boltzmann population on the ground state
vibrations and also include inhomogeneous broadening due to a Gaussian
distribution ($\sigma$ denoting the standard deviation) of the electronic
transition energy. At room temperature, calculated spectra (red) with
$\sigma=200\textrm{ cm}^{-1}$ show excellent qualitative agreement with the
experimental spectra (black) indicating that the calculations capture the
correct vibronic couplings and normal modes within the molecules. The
calculated spectra at 0.1 K and $\sigma=10\textrm{ cm}^{-1}$ remove the
majority of the inhomogenous broadening as well as broadening due to
transitions from hot ground state vibrations that occur at room temperature.
This gives “the skeleton” of the vibronic progression with peaks corresponding
only to transitions from ground vibrational states on the electronic ground
state. A renormalization of the DFT-calculated electronic transition energy
was applied for our calculations to match the experimental data.
Figure 5c and 5d show $S$ (blue) and $\bm{\mu}_{\textrm{IR}}^{(e)}$ (green)
for the different vibrational modes as a function of each mode’s wavenumber
for both pyrene and PAA. Although both molecules contain many modes with non-
zero $\bm{\mu}_{\textrm{IR}}^{(e)}$ values, we only show
$\bm{\mu}_{\textrm{IR}}^{(e)}$ for modes with non-zero $S$ values as these are
the only vibrational modes that are excited by electronic transitions. $S$
values are calculated for ground state vibrations and
$\bm{\mu}_{\textrm{IR}}^{(e)}$ values are calculated for excited state normal
modes. Because Duschinsky rotations can occur,[61] the overlap squared of each
ground state mode with each excited state mode was taken in order to match
them. In both molecules, there was generally minimal mixing of ground state
normal modes in the excited state, validating our application of the FC
approximation. Figure 5c shows that each FC active vibrational mode has a zero
$\bm{\mu}_{\textrm{IR}}^{(e)}$ value for pyrene, in accordance with the rule
of mutual exclusion. On the other hand, Figure 5d shows that each FC active
vibrational mode in PAA has a non-zero $\bm{\mu}_{\textrm{IR}}^{(e)}$ value
due to the broken inversion symmetry of the pyrene moiety. Therefore, an
optical cavity may be tuned to one of the FC active vibrational modes of PAA
shown in Figure 5d—a good choice being the highest frequency mode at 1680
$\textrm{cm}^{-1}$—and upon UV/vis excitation of the PAA molecule, ESVP would
form.
The results from the DFT calculations presented here can be connected to the
theoretical model presented in sections 2.3 and 2.4 through the calculated $S$
and $g$ values. However, for a single molecule coupled to a cavity mode, $g$
is typically very small when compared to values required for the UV/vis-to-IR
photonic down conversion process. For example, for the mode at 1680
$\textrm{cm}^{-1}$, $\bm{\mu}_{\textrm{IR}}^{(e)}=0.57$ Debye, which
corresponds to a $g$ value of $1.8\times 10^{-6}$ mHa. This is assuming a
cavity volume of $V=1.06\times 10^{-16}\textrm{ m}^{3}$ using
$V=\lambda^{3}/2$, where $\lambda$ is the wavelength of the cavity mode.
Comparing this $g$ value with Figure 4e, there would be negligible photonic
down conversion for all $\mathcal{Q}$ values. The small $g$ value and photonic
down conversion yield are due to having only a single molecule coupled to the
cavity. Experimental vibrational-polariton measurements rely on an ensemble of
molecules to be coupled to a cavity mode, and the Rabi splittings observed
correspond to larger coupling strengths than would be expected from each
individual molecule.[62] This is refereed to as collective strong coupling for
which it is known that the Rabi-splitting increases with the square root of
the number of cavity coupled molecules ($N$).[13, 63] As a rough estimate, an
effective $g$ value that scales with $\sqrt{N}$ can, therefore, be considered.
Using the concentration of molecules present in the optical cavity and knowing
the molecule’s absorption cross section, one could estimate the number of
excited molecules within the cavity and calculate an effective coupling
strength from the DFT results. This would allow the % emission from photonic
down conversion of the molecule to be determined as a function of the
$\mathcal{Q}$. It should be noted, however, that collective strong coupling
remains underexplored, especially in the presence of multiple quanta such as
involved in the ESVP mediated photonic down conversion process, the topic of
which we reserve for future work.
## 3 Discussion
In this work, a novel UV/vis-to-IR photonic down conversion process is shown
to be driven by ESVP for a molecule in an optical cavity. Accordingly, a
UV/vis excitation of the molecule induces vibrational polaritons on the
molecule’s electronic excited state potential. Such ESVP may then leak IR
light, resulting in a photonic down conversion that can potentially be
exploited for applications in sensing, quantum information, and in the
manipulation of photo-induced reactions. The mechanism and relevant molecular
parameters that allow for the down conversion process to occur are the HR
factor and the excited state IR dipole, which both need to be non-zero.
According to the rule of mutual exclusion, this implies that the photonic down
conversion process can only occur within molecules lacking inversion symmetry.
By studying the parameter space of the ESVP mediated photonic down conversion
while including both radiative and non-radiative dissipation, we determined
the optimal parameters to maximize the down conversion yield, facilitating
experimental and practical implementations. Due to the inclusion of non-
radiative dissipation, the photonic down conversion is maximized at a specific
cavity quality factor. With excess quality factors, the probability of cavity
photon leakage becomes small and non-radiative processes will out-compete
emission. However, with lower than optimal quality factors, the cavity mode
becomes highly delocalized through mixing with free-space electromagnetic
field modes outside of the cavity inhibiting strong coupling with the
molecular vibration. These two limiting behaviors give rise to an intermediate
quality factor, $\mathcal{Q}^{\textrm{max}}$, that maximizes the down
conversion process.
In the previous subsection, we demonstrated how to use DFT calculations to
predict if ESVP will form on a molecule when it is excited with UV/vis light
within a cavity. We believe this to be useful for designing molecules that
maximize the ESVP mediated photonic down conversion, allowing for applications
in quantum information and sensing to be realized. Such DFT calculations can
also determine if an optical cavity could be used to modulate a molecule’s
excited state dynamics following a UV/vis excitation with ramifications for
photo-induced reactions. The importance of the rule of mutual exclusion is
emphasized by our evaluation of both pyrene, containing and inversion
symmetry, and 1-pyreneacetic acid, lacking an inversion symmetry. Our results
show that only 1-pyreneacetic acid has both FC active (non-zero HR factor) and
excited state IR active vibrational modes, which permit the formation of ESVP.
We note that a potential exception to the requirement of molecular inversion
symmetry for the formation of ESVP induced by a UV/vis excitation would be the
occurrence of excited state symmetry breaking due to solvent interactions.[64]
This results in ground state vibrational modes of different symmetries mixing
in the excited state and could allow excited state IR active vibrations to be
produced on ground state centrosymmetric molecules following a UV/vis
excitation of the molecule. These vibrations could, therefore, couple to a
cavity, opening further ways to mediate the photonic down conversion.
## 4 Methods
### 4.1 Diabatic Expansion of the Dipole Operator
The dipole operator can be separated into components that depend on electronic
and nuclear DOF as
$\bm{\hat{\mu}}=\bm{\hat{\mu}_{\textrm{el}}}+\bm{\hat{\mu}_{\textrm{nu}}}.$
(16)
When expanding $\hat{H}_{\textrm{system}}$ in the diabatic basis, each element
of the dipole operator in this basis becomes
$\bm{\hat{\mu}_{\alpha\beta}}(\hat{Q})=\langle\alpha|\bm{\hat{\mu}_{\textrm{el}}}|\beta\rangle+\bm{\hat{\mu}_{\textrm{nu}}}(\hat{Q})\delta_{\alpha\beta},$
(17)
which act on the nuclear Hilbert space of $\hat{Q}$ and not on the electronic
Hilbert space. Note that $\bm{\hat{\mu}_{\textrm{nu}}}$ does not act on the
diabatic basis states as they do not depend on $Q$ (see Eq. (2)). The linear
expansion of each element of the dipole operator is given by
$\bm{\hat{\mu}}_{\alpha\beta}(\hat{Q})=\bm{\mu_{\alpha\beta}}(Q_{0}^{(g)})+\frac{\partial\bm{\mu_{\alpha\beta}}(Q)}{\partial
Q}\bigg{|}_{Q_{0}^{(g)}}(\hat{Q}-Q_{0}^{(g)}),$ (18)
with the first coefficient being
$\bm{\mu_{\alpha\beta}}(Q_{0}^{(g)})=\langle\alpha|\bm{\hat{\mu}_{\textrm{el}}}|\beta\rangle+\bm{\mu_{\textrm{nu}}}(Q_{0}^{(g)})\delta_{\alpha\beta},$
(19)
where $\langle\alpha|\bm{\hat{\mu}_{\textrm{el}}}|\beta\rangle$ is the
permanent electic dipole of state $\alpha$ when $\alpha=\beta$ and the
transition dipole between the states $\alpha$ and $\beta$ when
$\alpha\neq\beta$. Further, $\bm{\mu_{\textrm{nu}}}(Q_{0}^{(g)})$ is the
dipole moment of the nuclei at configuration $Q_{0}^{(g)}$. It follows that
the second coefficient in Eq. (18) is given by the derivative of the nuclear
dipole moment with respect to $Q$ at configuration $Q_{0}^{(g)}$,
$\frac{\partial\bm{\mu_{\alpha\beta}}(Q)}{{\partial
Q}}\bigg{|}_{Q_{0}^{(g)}}=\frac{\partial\bm{\mu_{\textrm{nu}}}(Q)}{{\partial
Q}}\bigg{|}_{Q_{0}^{(g)}}\delta_{\alpha\beta}.$ (20)
We ignored coupling between the cavity and the molecule’s electronic DOF as
the cavity mode is assumed to be near resonant with the vibrational mode of
interest making it significantly off-resonant from the electronic transition.
Accordingly, we set elements of the dipole operator where $\alpha\neq\beta$ to
zero. Because of this, the electronic ground and excited state contributions
to $\hat{H}_{\textrm{system}}$ are uncoupled and can be treated separately.
For the purpose of this study, we focused on the excited state contribution
where $\alpha=\beta=e$. To simplify notation we defined
$\bm{\mu}_{e}^{\prime}\equiv\frac{\partial\bm{\mu_{ee}}(Q)}{\partial
Q}\bigg{|}_{Q_{0}^{(g)}},$ (21)
and
$\bm{\mu}_{e}^{0}\equiv\bm{\mu_{ee}}(Q_{0}^{(g)})-\bm{\mu}_{e}^{\prime}Q_{0}^{(g)}.$
(22)
Note that $\bm{\mu}_{e}^{0}$ is the permanent dipole moment of the excited
state at the ground state equilibrium configuration, which was taken to be
$Q_{0}^{(g)}=0$.
### 4.2 Polariton Normal Mode Basis
$\hat{H}_{\textrm{system}}$ can be rotated into a basis of two uncoupled
oscillators called the polariton normal mode basis (see Figure 2). This basis
is found by determining the eigenvectors of the cavity-vibration Hessian
matrix,
$\bm{H}=\begin{pmatrix}\omega_{c}^{2}&G\\\ G^{*}&\omega_{v}^{2}\end{pmatrix}.$
(23)
The polariton harmonic oscillators have upper frequency ($\Omega_{+}$) and
lower frequency ($\Omega_{-}$) given by the square root of the eigenvalues of
$\bm{H}$. The off diagonal element of the Hessian matrix, $G$, is the cavity-
vibration bilinear coupling constant given by
$G=\frac{2}{\hbar}\sqrt{\omega_{c}\omega_{v}}g.$ (24)
The analytical expression for $\langle\hat{N}_{\text{cav}}(t)\rangle$ was
determined by rotating the cavity photon occupancy operator,
$\hat{N}_{\text{cav}}=\frac{1}{2\hbar\omega_{c}}(\omega_{c}^{2}\hat{q}^{2}+\hat{p}^{2})-1/2,$
(25)
into the polariton basis by expanding $\hat{q}$ and $\hat{p}$ in terms of
polariton phase space operators, $\hat{R}_{-}$ and $\hat{R}_{+}$. The Weyl
symbol of $\hat{N}_{\text{cav}}$, in the polariton basis, was determined by
replacing the polariton phase space operators with phase space variables that
can be evolved according to the classical equations of motion for two
uncoupled harmonic oscillators. From this, we calculate the expectation value
of the time evolved Weyl symbol of $\hat{N}_{\text{cav}}$ acting on the time
independent Wigner distribution of the initial position and momentum of the
system.[65, 43]
### 4.3 System-Bath Couplings
An ohmic spectral density was taken for both the non-radiative bath and the
external field, $J_{E}(\omega)=\eta_{E}\,\omega$ and
$J_{B}(\omega)=\eta_{B}\,\omega$, respectively, where $\eta_{E}$ ($\eta_{B}$)
is the cavity (vibration) damping constant due to coupling to the external
field (non-radiative bath). By taking a large but finite number of non-
radiative bath and external field modes (we used 500 modes for each bath),
$\hat{H}_{\textrm{total}}$ approximates the thermodynamic number of bath modes
under experimental conditions.[66] In our modeling, we used a constant density
of states ($D_{B}$ for the non-radiative bath; $D_{E}$ for the external field)
and frequency dependent coupling constants,
$\kappa_{i}=\sqrt{\frac{\eta_{E}\,\omega_{i}}{D_{E}}},$ (26)
and
$\chi_{i}=\sqrt{\frac{\eta_{B}\,\omega_{i}}{D_{B}}}.$ (27)
In all of our calculations, the vibrational damping constant, $\eta_{B}$, was
fixed such that an excited vibration would have a lifetime of 2 ps for $g=0$,
which is typical of condensed-phase molecular vibrations (we study different
non-radiative lifetimes in section 5 of the SI).[26, 27] Furthermore,
$\eta_{E}$ is related to $\mathcal{Q}$, which is defined as the ratio of the
cavity center frequency to its emission linewidth, $\Delta\omega_{c}$,[67]
$\mathcal{Q}=\frac{\omega_{c}}{\Delta\omega_{c}}.$ (28)
An expression for the emission profile linewidth of an optical cavity coupled
to a bath of external field modes can be extracted from the Lindblad master
equation as[45]
$\Delta\omega_{c}=2\pi D_{E}|\kappa_{\omega_{c}}|^{2},$ (29)
where $\kappa_{\omega_{c}}$ is the cavity-external field coupling at the
cavity frequency. Substituting equation 29 into equation 28 gives
$\mathcal{Q}=\frac{\omega_{c}}{2\pi D_{E}|\kappa_{\omega_{c}}|^{2}}.$ (30)
Setting $\kappa_{i}=\kappa_{\omega_{c}}$ and $\omega_{i}=\omega_{c}$ in
equation 26 and then comparing the result to equation 30, it follows that
$\eta_{E}=\frac{1}{2\pi\mathcal{Q}}.$ (31)
It should be noted that the Lindblad equation invokes the RWA and therefore
will give inaccurate $\mathcal{Q}$ values in the ultrastrong cavity-external
field coupling regime.
### 4.4 DFT Calculations of Huang-Rhys Factors and Excited State IR Dipoles
DFT calculations were performed using the Qchem software (version 5.3). All
DFT and TD-DFT calculations utilized the B3LYP functional and
$\textrm{6-31+G}^{*}$ basis set. In order to determine $S$ and
$\bm{\mu}_{\textrm{IR}}^{(e)}$ for pyrene and PAA, first the ground state
equilibrium geometry was calculated for each molecule in its singlet ground
state, followed by a normal mode analysis about this ground state minimum to
obtain the intramolecular vibrations, both done via DFT calculations. TD-DFT
calculations were then used to determine the forces felt by the nuclei due to
the molecule being excited into the first, bright excited singlet state. This
force is due to the electron density of the excited state acting on the ground
state nuclear configuration, shifting the equilibrium nuclear configuration.
Excited state force vectors from the TD-DFT calculations were projected onto
the ground state vibrational modes. These forces determine the shift in the
excited state PES and were used to calculate HR factors for each vibrational
mode. Finally, we performed TD-DFT excited state normal mode analyses at the
FC point to calculate the excited state intramolecular vibrations and their
corresponding $\bm{\mu}_{\textrm{IR}}^{(e)}$ values. No description of solvent
interactions was included in the computed spectra.
## 5 Acknowledgments
The authors thank Jonathan D. Schultz for the helpful discussions regarding
this work. This work was supported as part of the Center for Molecular Quantum
Transduction (CMQT), an Energy Frontier Research Center funded by the U.S.
Department of Energy, Office of Science, Basic Energy Sciences under Award No.
DE-SC0021314. This material is based upon work supported by the National
Science Foundation Graduate Research Fellowship under Grant No. DGE-2234667.
## Declarations
The authors declare no competing interests.
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|
# Analysis of UAV Radar and Communication Network Coexistence with Different
Multiple Access Protocols
Sung Joon Maeng, Jaehyun Park, , and İsmail Güvenç,
This work has been supported in part by the NSF award CNS-1910153. This work
has been also supported in part by National Research Foundation of Korea under
the framework of international cooperation program (2022K2A9A2A06035926).S. J.
Maeng, İ. Güvenç are with the Department of Electrical and Computer
Engineering, North Carolina State University, Raleigh, NC 27606 USA (e-mail:
<EMAIL_ADDRESS>iguvenc@ncsu.edu).J. Park with the Department of Electronic
Engineering, Pukyong National University, Busan 608-737, South Korea (e-mail:
jaehyun@pknu.ac.kr).
###### Abstract
Unmanned aerial vehicles (UAVs) are expected to be used extensively in the
future for various applications, either as user equipment (UEs) connected to a
cellular wireless network, or as an infrastructure extension of an existing
wireless network to serve other UEs. Next generation wireless networks will
consider the use of UAVs for joint communication and radar and/or as dedicated
radars for various sensing applications. Increasing number of UAVs will
naturally result in larger number of communication and/or radar links that may
cause interference to nearby networks, exacerbated further by the higher
likelihood of line-of-sight signal propagation from UAVs even to distant
receivers. With all these, it is critical to study network coexistence of UAV-
mounted base stations (BSs) and radar transceivers. In this paper, using
stochastic geometry, we derive closed-form expressions to characterize the
performance of coexisting UAV radar and communication networks for spectrum
overlay multiple access (SOMA) and time-division multiple access (TDMA). We
evaluate successful ranging probability (SRP) and the transmission capacity
(TC) and compare the performance of TDMA and SOMA. Our results show that SOMA
can outperform TDMA on both SRP and TC when the node density of active UAV-
radars is larger than the node density of UAV-comms.
###### Index Terms:
Coexistence, guard zone, HPPP, multiple access, sensing and communication,
stochastic geometry, UAV communication, UAV radar detection.
## I Introduction
Recently, various different applications of cellular-connected unmanned aerial
vehicles (UAVs) have been getting significant attention due to their cost-
efficient deployment and controllable mobility. UAVs are utilized in many
fields such as environmental monitoring and surveillance [1], public safety
[2], video broadcasting [3], and delivery [4]. Moreover, UAV-mounted base
stations (UAV-BS) and user equipment (UAV-UE), as well as UAV-mounted radars
(UAV-radar) are commonly considered in sensing and communication applications,
since the UAVs are available to quickly change the position to serve users at
the outage area, and/or surveil/track the location of detected moving targets.
Joint design of the radar and communication systems is considered as one of
the key research areas for wireless networks beyond 5G systems which can
benefit significantly from the use of autonomous UAVs.
In the meanwhile, as the demand for using wider bandwidths has been increasing
to support higher throughput and massive connectivity, the band of operation
for broadband wireless networks has been moving to higher frequencies such as
millimeter-wave (mmWave) and sub/THz bands that are also commonly used by
radar systems. Some traditional radar bands, including certain bands below 6
GHz, are also being opened for shared use with communication networks due to
the increasing congestion in the dedicated spectrum for cellular networks. All
these developments call for rigorously studying the coexistence scenarios for
radar and communication networks and coming up with strategies for effective
spectrum sharing [5].
Stochastic geometry-based techniques are commonly used in the literature for
obtaining closed-form expressions on the performance of wireless networks
where transmit sources are randomly deployed in the spatial domain [6]. For
example, the analysis of accumulated interference from multiple nodes
following a homogeneous Poisson point process (HPPP) is useful to evaluate the
capacity of wireless networks [6]. In this paper, we specifically investigate
UAV radar sensing and communication network coexistence scenarios. In
particular, we consider scenarios where radar transmission and data
transmission are coordinated by two different multiple-access protocols:
spectrum overlay multiple access (SOMA) and time-division multiple access
(TDMA). In SOMA, radar sensing and data communication share the same spectrum
so that the spectrum is overlapped. On the other hand, in TDMA, radar
detection and communication are separated by time. We utilize stochastic
geometry-based analysis where UAVs are randomly located in 3D space following
a two-dimensional homogeneous Poisson point process (HPPP). We individually
analyze the radar detection performance and the data communication performance
using the successful ranging probability (SRP) and the transmission capacity
(TC), respectively.
Table I: Literature review for stochastic geometry based wireless network performance analysis. Ref. | Analysis objective | Application | Radar | Networks
---|---|---|---|---
[7] | TC with different spatial diversity techniques | Terrestrial | ✗ | Ad hoc
[8] | Information and energy outage probability and area harvested energy | Terrestrial | ✗ | SWIPT in ad hoc networks
[9] | Channel outage and packet loss probability | Terrestrial | ✗ | DL URLLC communications
[10, 11] | SRP | Terrestrial | ✓ | Radar with road scenario
[12] | SRP | Terrestrial | ✓ | Radar networks
[13, 14] | Coverage probability with different antenna patterns and directivity | Terrestrial | ✗ | DL cellular
[15] | Connection and secrecy probability | Terrestrial | ✗ | DL secure communication cellular
[16] | Coverage probability with underlay and overlay protocols | UAV | ✗ | UAV-to-UAV and UL terrestrial
[17] | Coverage probability and spectral efficiency | UAV | ✗ | Two-tiers cellular
[18, 19] | Coverage probability and spectral efficiency | UAV | ✗ | DL communication
[20] | Successful transmission probability, energy and SINR coverage | UAV | ✗ | DL SWIPT and UL communication
[21] | Coverage probability | UAV | ✗ | UAV-aided DL and UL communication
[22] | LoS probability | UAV | ✗ | BS-to-UAV link
[23] | Coverage probability, motion energy, and flight time | UAV | ✗ | UAV path planning
[24] | Connection, secrecy, and energy-information outage probability | UAV | ✗ | Secure communication in SWIPT
This work | SRP and TC | UAV | ✓ | Radar and communication coexistence
Contributions of this paper can be summarized as follows:
* –
We derive closed-form expressions for SRP and TC on the UAV radar and
communication coexistence scenario where UAVs are placed following HPPP with a
guard zone.
* –
We analyze the performance of SRP and TC in SOMA and TDMA, respectively. We
also investigate behaviors of SRP and TC depending on the node density, radius
of the guard zone, power splitting factor in SOMA, and time division factor in
TDMA.
* –
We analytically compare TDMA and SOMA on SRP and TC and show that TDMA
outperforms SOMA on SRP while SOMA is better than TDMA on TC in the general
condition. Furthermore, we analyze the condition that SOMA can be superior to
TDMA on both SRP and TC metrics.
The rest of this paper is organized as follows. Section II presents the
literature review. In Section III, we describe the UAV radar and communication
network coexistence design. In Section IV, we provide the signal propagation
model when UAVs are distributed by HPPP. In Section V, we derive the closed-
form expressions of the SRP and the TC. In Section VI, we analyze SRP
depending on the system parameters and the multiple access protocols. In
Section VII, we analyze the TC depending on the system parameters and the
multiple access protocols. In Section VIII, we compare the SRP and the TC
performance of SOMA and TDMA. In Section IX, we show the simulation results to
verify the analysis in previous sections, and Section X provides concluding
remarks.
## II Literature Review
The operation of UAVs on BSs and radar detectors have been investigated in the
literature. A flying UAV-BS can maximize the capacity or minimize the outage
of networks by optimizing UAV trajectory [25, 26]. In [27, 28], the trajectory
and precoder of UAV-BS are optimized to maximize physical layer secrecy. In
[29], a UAV-radar is used in measuring the depth of the snow on the sea. Human
detection and classification by a UAV-radar have been studied in [30]. Target
detection using radar imaging from UAV-radar has been investigated in [31]. In
[32], the feasibility of a surveillance system using a UAV-radar has been
explored.
The study of coexistence networks has been explored in the literature. In
[33], a beamforming approach has been studied to facilitate the coexistence
between downlink (DL) multi-user-multiple-input-multiple-output (MU-MIMO)
communication and MIMO radar system. In [34], the joint design of the radar
and communication system for the coexistence of MIMO radar and MIMO
communication has been studied. Moreover, UAV communication and radar sensing
network coexistence that utilizes the spectrum for both purposes has been
investigated for an efficient and flexible system design [35]. In [36], joint
UAV communication and cooperative sensing network based on beam sharing scheme
has been explored.
Stochastic geometry based network analysis has been thoroughly investigated in
the literature. TC is analyzed in ad hoc networks with different spatial
diversity techniques where transmitting nodes are distributed by an HPPP [7],
and this work is extended to the wireless information and power transfer
(SWIPT)-based ad hoc networks in [8]. In [9], packet loss probability
depending on the packet size, packet duration, and SINR are derived in
downlink ultra-reliable and low-latency communications (URLLC) scenarios where
distributed antenna ports are randomly placed following an HPPP. In [10, 11,
12], the effect of radar interference on the radar detection performance is
analyzed and SRP is evaluated using stochastic geometry. More specifically,
the geometric layout of vehicles on a road where the locations of vehicles on
a certain lane are decided by unidimensional HPPP model is investigated in
[10, 11]. In [11], radar cross-section (RCS) characteristics are modeled and
analyzed using HPPPs for automotive radar network scenarios. In [13, 14], the
effects of different directional antenna patterns, node densities, and antenna
array sizes on coverage probability are studied for mmWave networks. The
locations of BSs and the eavesdroppers are randomly distributed by independent
HPPPs in [15], and closed-form expression of secrecy probability for secure
communications is explored.
Closed-form analysis of network performance using stochastic geometry
techniques have also been studied in UAV networks in [16, 17, 18, 19, 20, 21,
22, 23, 24]. In [16], coexisting UAV-to-UAV links and uplink (UL) ground-BS to
ground-user links are considered. Then, coverage of two different scenarios
are studied, where the spectrum for each link is either reused, or it is
allocated in a dedicated manner. The literature review with representative
works related to stochastic geometry-based wireless network performance
analysis is summarized in Table I. To the best of our knowledge, the study of
radar networks based on stochastic geometry has been limited, and UAV
communication and radar network coexistence scenario has not been investigated
yet.
Table II: Key symbols and notations used in this paper. Symbol | Definition
---|---
${\lambda_{\rm r}}^{\prime}$ | Node density of UAV-radars
${\lambda_{\rm d}}^{\prime}$ | Node density of UAV-comms
${\bar{\lambda}}_{\rm r}$ | Active node density of UAV-radars
${\lambda_{\rm r}}$ | Effective node density of active UAV-radars in HPPP
${\lambda_{\rm d}}$ | Effective node density of UAV-comms in HPPP
$\mathsf{h}_{\rm UAV}$ | UAVs height
$r_{0}$ | Radius of guard zone
$\phi$ | Power splitting factor
$\tau$ | Time division factor
$\delta$ | Duty circle
$\mathsf{P}_{\rm Tx}$ | Transmit power
$\mathsf{G}_{\rm t}$ | Tx antenna gain
$\alpha$ | Path-loss exponent
$\alpha_{\rm I}$ | Path-loss exponent from the interference
$\mathsf{G}_{\rm r}$ | Receiver antenna gain
$\mathsf{G}_{\rm rI}$ | Receiver antenna gain from the interference
$\bar{\sigma}$ | Average RCS
$\sigma$ | RCS
$S_{\rm e}$ | Effective aperture of radar receiver
$\mathsf{G}_{\rm p}$ | Processing gain of radar receiver
$f_{\rm c}$ | Carrier frequency
$R_{0}$ | Target distance
$r_{i}$ | Distance from the interferer
$c$ | Speed of light
$h_{0}$ | Small-scale fading
$h_{i}$ | Small-scale fading from the interference
$\beta_{\rm th}$ | Target SINR threshold for outage probability
$\gamma{\rm th}$ | Target SINR threshold for successful range probability
$\beta_{0}$ | SINR of the received data signal
$\gamma{0}$ | SINR of the received radar signal
$N_{0}$ | Noise power
## III System Model
We consider UAV networks where radar nodes and communication nodes coexist.
Radar-mounted UAVs (UAV-radars) detect and track a target on the ground by
transmitting radar signals and receiving the reflected signals from the
target. On the other hand, UAVs that are equipped with a BS (UAV-comms)
communicate with a ground user. We assume that UAV-radars and UAV-comms follow
a two-dimensional HPPP independently where the node densities are
${\lambda_{\rm r}}^{\prime}$ and ${\lambda_{\rm d}}^{\prime}$ respectively.
All UAVs fly at a fixed identical height $\mathsf{h}_{\rm UAV}$. Fig. 1
describes two different network representations of the radar detection
scenario and the communication scenario in the HPPP model. The guard zone with
radius $r_{0}$ is considered between two UAVs, or between a UAV and a user to
protect them from potential strong interference. The distance between a UAV-
radar and a target in the radar detection scenario and the distance between a
UAV-comm and a served user in the communication scenario is $R_{0}$. UAV-
radars are assigned to active UAV-radar by the random spectrum access with the
duty cycle $\delta$, and the rest of UAV-radars remain inactive UAV-radars in
the networks.
(a)
(b)
Figure 1: UAV radar and communication network coexistence. A blue UAV is
either: (a) a typical UAV-radar that detects a target using radar
transmission; or (b) a serving UAV that communicates with a typical user. The
green UAV-comms or the red (active) UAV-radars can interfere with radar
detection or communication signals. The black (inactive) UAV-radars do not
transmit any interference signals. To avoid strong interference, a guard zone
is with a radius $r_{0}$ established between UAVs, and between a UAV and a
user. Figure 2: Two different multiple access schemes: SOMA (left) and TDMA
(right).
In the radar and communication coexistence, UAV-radars and UAV-comms need to
coordinate the time and spectrum resources for radar and data transmissions.
We consider two different multiple access schemes: SOMA where radar signal and
data signal share the same spectrum during transmission time, and TDMA where
the time for radar and data transmission is scheduled at separate time slots.
Fig. 2 illustrates different radar and data allocations depending on multiple
access schemes. The power allocated to radar and communication is determined
by power splitting factor $\phi$ for SOMA, and the time duration assigned to
radar and communication is decided by time division factor $\tau$ for TDMA.
Note that the interference behavior in this network coexistence is dependent
on the multiple access. In this paper, we focus on the analysis and comparison
of SOMA and TDMA on data transmission and radar detection. The key parameters
are summarized in Table II.
## IV Signal Propagation Models in HPPP
In this section, we describe signal and interference models in HPPP when SOMA
and TDMA are adopted respectively. Throughout this paper, we denote SOMA and
TDMA as $\rm s.o.$ and $\rm t.d.$ at the superscript.
### IV-A Radar and Data Signal Models
#### IV-A1 SOMA
We place a typical UAV-radar with the origin $(0,\leavevmode\nobreak\
0,\leavevmode\nobreak\ \mathsf{h}_{\rm UAV})$ and distance from the target at
$(x_{\rm t},y_{\rm t},0)$ is $R_{0}=\sqrt{x_{\rm t}^{2}+y_{\rm
t}^{2}+(\mathsf{h}_{\rm UAV})^{2}}$ as in Fig. 1(a). We assume that the height
of a UAV-radar $\mathsf{h}_{\rm UAV}$ is sufficiently high so that line-of-
sight is secured to detect the target and the free-space path loss model can
be considered [37]. The power of the received signal that is reflected back
from the target can be expressed as [10],
$\displaystyle\mathsf{P}_{\rm r}^{\rm
s.o.}=\left(\frac{(1-\phi)\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}}{4\pi
R_{0}^{\alpha}}\right)\left(\frac{\sigma S_{\rm e}}{4\pi
R_{0}^{\alpha}}\right)\mathsf{G}_{\rm p},$ (1)
where $\mathsf{P}_{\rm Tx}$, $\mathsf{G}_{\rm t}$, $\alpha$ indicate transmit
power, Tx antenna gain, and path loss exponent, and $\sigma$, $S_{\rm e}$,
$\mathsf{G}_{\rm p}$ denote radar cross-section (RCS) of the target, the
effective aperture of radar receiver, and the processing gain. Swerling I
model is considered for the RCS and the RCS of the target follows the
exponential distribution,
$\sigma\sim\frac{1}{\bar{\sigma}}e^{-\frac{\sigma}{\bar{\sigma}}}$ [11]. The
effective area is given by
$\displaystyle S_{\rm e}=\frac{\mathsf{G}_{\rm r}c^{2}}{4\pi f_{\rm c}^{2}},$
(2)
where $c$, $f_{\rm c}$ denote the speed of light and the carrier frequency.
Then, the power of the reflected back radar signal in (1) can be rewritten as
$\displaystyle\mathsf{P}_{\rm r}^{\rm s.o.}=\frac{(1-\phi)\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}\sigma}{(4\pi)^{3}f_{\rm c}^{2}R_{0}^{2\alpha}}.$ (3)
In the communication scenario as in Fig. 1(b), a typcial user is located at
the origin and it receives the signal from the serving UAV-comm at $(x_{\rm
t},y_{\rm t},\mathsf{h}_{\rm UAV})$, which is at a distance of
$R_{0}=\sqrt{x_{\rm t}^{2}+y_{\rm t}^{2}+(\mathsf{h}_{\rm UAV})^{2}}$ from the
user. The power of the received signal of the user is given by
$\displaystyle\mathsf{P}_{\rm d}^{\rm s.o.}=\frac{\phi\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}c^{2}}{(4\pi)^{2}f_{\rm
c}^{2}R_{0}^{\alpha}}h_{0},$ (4)
where $h_{0}\sim\exp(1)$ represents Rayleigh small-scale fading. Note that the
allocated power of the radar and the data signals are split by the power
splitting factor $\phi$ in SOMA as illustrated in Fig 2.
#### IV-A2 TDMA
The received signal power of the radar signal and the data signal in TDMA are
expressed as
$\displaystyle\mathsf{P}_{\rm r}^{\rm t.d.}=\frac{\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}\sigma}{(4\pi)^{3}f_{\rm c}^{2}R_{0}^{2\alpha}},$ (5)
$\displaystyle\mathsf{P}_{\rm d}^{\rm t.d.}=\frac{\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}c^{2}}{(4\pi)^{2}f_{\rm
c}^{2}R_{0}^{\alpha}}h_{0}.$ (6)
Since the radar detection and communication are separately conducted in
different time slots in TDMA, transmit power is not adjusted as in SOMA.
### IV-B Effective Radar and Communication Node Densities
The PPP in this network introduces the guard zone and it can be modeled by the
Matérn hard-core point processes (MHCPP) type-II, which can be further
approximated by the HPPP model.
#### IV-B1 SOMA
The approximated effective node density of the radar and the communication
from MHCPP type-II can be written as [38]
$\displaystyle\lambda_{\rm r}^{\rm s.o.}=\frac{1-e^{-\bar{\lambda}_{\rm
r}^{\rm s.o.}\pi r_{0}^{2}}}{\pi r_{0}^{2}},\leavevmode\nobreak\ \lambda_{\rm
d}^{\rm s.o.}=\frac{1-e^{-\lambda^{\prime}_{\rm d}\pi r_{0}^{2}}}{\pi
r_{0}^{2}},$ (7)
where ${\bar{\lambda}}_{\rm r}^{\rm s.o.}=\delta\lambda^{\prime}_{\rm r}$ is
the active UAV-radar node density, and $\delta$ is the duty cycle.
#### IV-B2 TDMA
The effective node density of the radar and the communication from MHCPP type-
II can be written as
$\displaystyle\lambda_{\rm r}^{\rm t.d.}=\frac{1-e^{-\bar{\lambda}^{\rm
t.d.}_{\rm r}\pi r_{0}^{2}}}{\pi r_{0}^{2}},\leavevmode\nobreak\ \lambda_{\rm
d}^{\rm t.d.}=\frac{1-e^{-\lambda^{\prime}_{\rm d}\pi r_{0}^{2}}}{\pi
r_{0}^{2}},$ (8)
where ${\bar{\lambda}}_{\rm r}^{\rm
t.d.}=\frac{\delta}{1-\tau}\lambda^{\prime}_{\rm r}$ is the active UAV-radar
node density. Note that the duty cycle $\delta$ in SOMA increases to
$\frac{\delta}{1-\tau}$ in TDMA as much as the reduced radar transmission time
by $\tau$, since it is assumed that the total number of active UAV-radar nodes
during the time period is the same for both SOMA and TDMA. Since the effective
node density of UAV-comms for TDMA and SOMA is equal, we merge the notation of
the node density as $\lambda_{\rm d}=\lambda_{\rm d}^{\rm s.o.}=\lambda_{\rm
d}^{\rm t.d.}$.
### IV-C Interference Models
In this subsection, we obtain the power of interference coming from nearby
active UAV-radars and UAV-comms in the HPPP model.
#### IV-C1 SOMA
Since the radar detection and communication occupy the same spectrum band at
the same time, the aggregated interference power from UAV-radars and UAV-comms
can be expressed as
$\displaystyle\mathsf{I}^{\rm s.o.}$
$\displaystyle=\underbrace{\sum_{r_{i}\in\Phi(\lambda_{\rm d})\backslash
r_{0}}\frac{\phi\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
rI}c^{2}}{(4\pi)^{2}f_{\rm c}^{2}r_{i}^{\alpha_{\rm
I}}}h_{i}}_{\text{interfence from nearby UAV-comms }(\mathsf{I}_{1})}$
$\displaystyle+\underbrace{\sum_{r_{j}\in\Phi(\lambda_{\rm r}^{\rm
s.o.})\backslash r_{0}}\frac{(1-\phi)\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm
t}\mathsf{G}_{\rm rI}c^{2}}{(4\pi)^{2}f_{\rm c}^{2}r_{j}^{\alpha_{\rm
I}}}h_{j}}_{\text{interfence from nearby UAV-radars }(\mathsf{I}_{2})},$ (9)
where $r\in\Phi(\lambda)\backslash r_{0}$ means a two-dimensional HPPP with a
density $\lambda$ and $r>r_{0}$, $\mathsf{G}_{\rm rI}$ and $\alpha_{\rm I}$
are Rx antenna gain and path-loss exponent from the interfering signals,
respectively, $h_{i}$, $h_{j}$ represent small-scale fading from the
interference, and $r_{i}$, $r_{j}$ are distance between a typical UAV-radar or
user and an interferer. Since the radar detection and communication are
carried out together in SOMA, the interference power at both the typical UAV-
radar (Fig. 1(a)) and the typical information receiver (Fig. 1(b)) is the
same: $\mathsf{I}^{\rm s.o.}=\mathsf{I}^{\rm s.o.}_{\rm d}=\mathsf{I}^{\rm
s.o.}_{\rm r}$.
#### IV-C2 TDMA
The interfence comes only from the UAV-radars in the radar detection scenario.
Likewise, the interference comes only from UAV-comms in communication
scenario. Then the aggregated interference power for each case can be
expressed as
$\displaystyle\mathsf{I}^{\rm t.d.}_{\rm r}$
$\displaystyle=\sum_{r_{i}\in\Phi(\lambda_{\rm r}^{\rm t.d.})\backslash
r_{0}}\frac{\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
rI}c^{2}}{(4\pi)^{2}f_{\rm c}^{2}r_{i}^{\alpha_{\rm I}}}h_{i},$ (10)
$\displaystyle\mathsf{I}^{\rm t.d.}_{\rm d}$
$\displaystyle=\sum_{r_{i}\in\Phi(\lambda_{\rm d})\backslash
r_{0}}\frac{\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
rI}c^{2}}{(4\pi)^{2}f_{\rm c}^{2}r_{i}^{\alpha_{\rm I}}}h_{i}.$ (11)
## V Performance Analysis of Radar Detection and Data Communication
In this section, we discuss performance evaluation metrics for two different
scenarios. Specifically, we derive the SRP in the radar detection scenario and
the TC in the communication scenario.
### V-A Successful Ranging Probability
SRP is the probability that a UAV-radar succeeds in detecting the target,
which is decided by the signal-to-interference-plus-noise ratio (SINR). SRP is
defined by
$\displaystyle\mathsf{Pr_{s.r.}}(\gamma_{\rm
th})=\mathsf{Pr}(\gamma_{0}>\gamma_{\rm th}),$ (12)
where $\gamma_{0}=\frac{\mathsf{P}_{\rm r}}{\mathsf{I}_{\rm
r}+\mathsf{N}_{0}}$ and $\mathsf{N}_{0}$ are SINR of the received radar signal
and the noise power, respectively. In addition, $\gamma_{\rm th}$ denotes SINR
threshold where the target is successfully detected. In what follows, we
derive the closed-form expression of SRP in both SOMA and TDMA.
#### V-A1 SOMA
SRP in (12) can be derived from (3), (IV-C1) as follows:
$\displaystyle\mathsf{Pr_{s.r.}^{\rm s.o.}}(\gamma_{\rm
th})=\mathsf{Pr}\left(\frac{\mathsf{P}^{\rm s.o.}_{\rm r}}{\mathsf{I}^{\rm
s.o.}+\mathsf{N}_{0}}>\gamma_{\rm th}\right)$
$\displaystyle=\mathsf{Pr}\left(\frac{(1-\phi)\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}\sigma}{\mathsf{I}^{\rm s.o.}+\mathsf{N}_{0}}>(4\pi)^{3}f_{\rm
c}^{2}R_{0}^{2\alpha}\gamma_{\rm th}\right)$
$\displaystyle\approx\mathsf{Pr}\left(\sigma>\frac{(4\pi)^{3}f_{\rm
c}^{2}R_{0}^{2\alpha}\gamma_{\rm th}\mathsf{I}^{\rm
s.o.}}{(1-\phi)\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
r}\mathsf{G}_{\rm p}c^{2}}\right)$
$\displaystyle=\int_{0}^{\infty}\left\\{1-F_{\sigma}\left(\frac{(4\pi)^{3}f_{\rm
c}^{2}R_{0}^{2\alpha}\gamma_{\rm th}y}{(1-\phi)\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}}\right)\right\\}f_{\mathsf{I}^{\rm s.o.}}(y){\rm d}y$
$\displaystyle=\int_{0}^{\infty}e^{-\frac{(4\pi)^{3}f_{\rm
c}^{2}R_{0}^{2\alpha}\gamma_{\rm th}y}{(1-\phi)\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}\bar{\sigma}}}f_{\mathsf{I}^{\rm s.o.}}(y){\rm d}y$
$\displaystyle=\mathcal{L}_{\mathsf{I}^{\rm s.o.}}\left(\frac{(4\pi)^{3}f_{\rm
c}^{2}R_{0}^{2\alpha}\gamma_{\rm th}}{(1-\phi)\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}\bar{\sigma}}\right)$ (13)
where the approximation comes from the interference limit regime assumption,
$F_{\sigma}(X)=1-e^{-\frac{X}{\bar{\sigma}}}$ is the cumulative distribution
function (CDF) of $\sigma$, and $f_{\mathsf{I}^{\rm s.o.}}(\rm x)$ is the
probability density function (PDF) of $\mathsf{I}^{\rm s.o.}$,
$\mathcal{L}_{\mathsf{I}^{\rm s.o.}}(z)$ indicates Laplace transform of the
PDF of $\mathsf{I}^{\rm s.o.}$.
$\mathcal{L}_{\mathsf{I}^{\rm s.o.}}(z)$ can be derived as follows. The
interference term can be rewritten as $\mathsf{I}^{\rm
s.o.}=\mathsf{I}_{1}+\mathsf{I}_{2}$ where $\mathsf{I}_{1}$, $\mathsf{I}_{2}$
are the first and the second terms in (IV-C1) respectively. Then, we can
obtain [9]
$\displaystyle\mathcal{L}_{\mathsf{I}_{1}}(z)$
$\displaystyle=\exp{\left\\{-2\pi\lambda_{\rm d}{\rm A}_{1}(z)\frac{(z\phi{\rm
K}_{1})^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}},$
$\displaystyle\mathcal{L}_{\mathsf{I}_{2}}(z)$
$\displaystyle=\exp{\left\\{-2\pi\lambda_{\rm r}^{\rm s.o.}{\rm
A}_{2}(z)\frac{(z(1-\phi){\rm K}_{1})^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm
I}}\right\\}},$ (14)
where ${\rm A}_{1}(z)={\rm B}\left(\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm B}\left(\frac{1}{1+z\phi{\rm
K}_{1}r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm
I}}\right)$, ${\rm A}_{2}(z)={\rm B}\left(\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm B}\left(\frac{1}{1+z(1-\phi){\rm
K}_{1}r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm
I}}\right)$, ${\rm K_{1}}=\frac{\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm
t}\mathsf{G}_{\rm rI}c^{2}}{(4\pi)^{2}f_{\rm c}^{2}}$, ${\rm B}(a,b)$ is the
beta function, and ${\rm B}(x;a,b)=\int_{0}^{x}u^{a-1}(1-u)^{b-1}{\rm d}u$ is
the incomplete beta fucntion. Then, we can derive
$\displaystyle\mathcal{L}_{\mathsf{I}^{\rm
s.o.}}(z)=\mathcal{L}_{\mathsf{I}_{1}}(z)\mathcal{L}_{\mathsf{I}_{2}}(z)=$
$\displaystyle\exp{\left\\{-2\pi\left(\phi^{\frac{2}{\alpha_{\rm
I}}}\lambda_{\rm d}{\rm A}_{1}(z)+\left(1-\phi\right)^{\frac{2}{\alpha_{\rm
I}}}\lambda_{\rm r}^{\rm s.o.}{\rm A}_{2}(z)\right)\frac{(z{\rm
K}_{1})^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}}.$ (15)
Then, the closed-form expression of SRP in SOMA can be obtained as (V-A1) at
the top of the next page.
$\displaystyle\mathsf{Pr_{s.r.}^{\rm s.o.}}(\gamma_{\rm
th})=\exp\left\\{-2\pi\left(\left(\frac{\phi}{1-\phi}\right)^{\frac{2}{\alpha_{\rm
I}}}\lambda_{\rm d}\underbrace{\left\\{{\rm B}\left(\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm B}\left(\frac{1}{1+\left(\frac{\phi
4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm th}}{(1-\phi)\mathsf{G}_{\rm
r}\mathsf{G}_{\rm p}\bar{\sigma}}\right)r_{0}^{-\alpha_{\rm
I}}};\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm
I}}\right)\right\\}}_{C_{1}}+\right.\right.$
$\displaystyle\left.\left.\lambda_{\rm r}^{\rm s.o.}\underbrace{\left\\{{\rm
B}\left(\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm
B}\left(\frac{1}{1+\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm
I}}\right)\right\\}}_{C_{2}}\right)\frac{\left(\frac{4\pi{\rm G}_{\rm
rI}R_{0}^{2\alpha}\gamma_{\rm th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}$
(16)
$\displaystyle\mathsf{Pr_{s.r.}^{\rm t.d.}}(\gamma_{\rm
th})=\exp\left\\{-2\pi\lambda_{\rm r}^{\rm t.d.}\underbrace{\left\\{{\rm
B}\left(\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm
B}\left(\frac{1}{1+\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm
I}}\right)\right\\}}_{C_{2}}\frac{\left(\frac{4\pi{\rm G}_{\rm
rI}R_{0}^{2\alpha}\gamma_{\rm th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}$
(17)
$\displaystyle C^{\rm s.o.}={\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-2\pi\left(\lambda_{\rm d}\underbrace{\left\\{{\rm
B}\left(\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm
B}\left(\frac{1}{1+\left(\frac{{\rm G}_{\rm rI}R_{0}^{\alpha}\beta_{\rm
th}}{\mathsf{G}_{\rm r}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)\right\\}}_{C_{3}}+\right.\right.$
$\displaystyle\left.\left.\left(\frac{1-\phi}{\phi}\right)^{\frac{2}{\alpha_{\rm
I}}}\lambda_{\rm r}^{\rm s.o.}\underbrace{\left\\{{\rm
B}\left(\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm
B}\left(\frac{1}{1+\left(\frac{(1-\phi){\rm G}_{\rm
rI}R_{0}^{\alpha}\beta_{\rm th}}{\phi\mathsf{G}_{\rm
r}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm
I}}\right)\right\\}}_{C_{4}}\right)\frac{\left(\frac{{\rm G}_{\rm
rI}R_{0}^{\alpha}\beta_{\rm th}}{\mathsf{G}_{\rm
r}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}$ (18)
$\displaystyle C^{\rm t.d.}=\tau{\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-2\pi\lambda_{\rm d}\underbrace{\left\\{{\rm
B}\left(\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm I}}\right)-{\rm
B}\left(\frac{1}{1+\left(\frac{{\rm G}_{\rm rI}R_{0}^{\alpha}\beta_{\rm
th}}{\mathsf{G}_{\rm r}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)\right\\}}_{C_{3}}\frac{\left(\frac{{\rm
G}_{\rm rI}R_{0}^{\alpha}\beta_{\rm th}}{\mathsf{G}_{\rm
r}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}$ (19)
#### V-A2 TDMA
SRP can be derived from (5), (10) as follow:
$\displaystyle\mathsf{Pr_{s.r.}^{\rm t.d.}}(\gamma_{\rm th})$
$\displaystyle=\mathsf{Pr}\left(\frac{\mathsf{P}^{\rm t.d.}_{\rm
r}}{\mathsf{I}^{\rm t.d.}_{\rm r}+\mathsf{N}_{0}}>\gamma_{\rm th}\right),$
$\displaystyle=\mathcal{L}_{\mathsf{I}^{\rm t.d.}_{\rm
r}}\left(\frac{(4\pi)^{3}f_{\rm c}^{2}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}c^{2}\bar{\sigma}}\right),$ (20) $\displaystyle\mathcal{L}_{\mathsf{I}^{\rm
t.d.}_{\rm r}}(z)$ $\displaystyle=\exp{\left\\{-2\pi\lambda_{\rm r}^{\rm
t.d.}{\rm A}_{3}(z)\frac{(z{\rm K}_{1})^{\frac{2}{\alpha_{\rm
I}}}}{\alpha_{\rm I}}\right\\}},$ (21)
where ${\rm A}_{3}(z)={\rm B}(\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm
I}})-{\rm B}(\frac{1}{1+z{\rm K}_{1}r_{0}^{-\alpha_{\rm
I}}};\frac{2}{\alpha_{\rm I}},1-\frac{2}{\alpha_{\rm I}})$. Note that detailed
mathematical steps are skipped, since many steps are similar to (V-A1),
(V-A1). Then, the closed-form expression of SRP in TDMA can be expressed as
(V-A2) at the top of the next page.
### V-B Transmission Capacity
TC is defined by the achievable data rate given an outage constraint
multiplied by the spatial density and the data transmission time duration [7,
8]. At first, the outage probability can be expressed as
$\displaystyle\mathsf{Pr_{out}}(\beta_{\rm
th})=\mathsf{Pr}(\beta_{0}<\beta_{\rm th}),$ (22)
where $\beta_{0}=\frac{\mathsf{P}_{\rm d}}{\mathsf{I}_{\rm d}+\mathsf{N}_{0}}$
is SINR of the received data signal, and $\beta_{\rm th}$ is a target SINR.
Then, TC is given as
$\displaystyle C^{\rm s.o.}$ $\displaystyle={\lambda}_{\rm
d}(1-\mathsf{Pr_{out}}(\beta_{\rm th}))\log(1+\beta_{\rm th}),$ (23)
$\displaystyle C^{\rm t.d.}$ $\displaystyle=\tau{\lambda}_{\rm
d}(1-\mathsf{Pr_{out}}(\beta_{\rm th}))\log(1+\beta_{\rm th}),$ (24)
where $C^{\rm s.o.},\leavevmode\nobreak\ C^{\rm t.d.}$ denote transmission
capacity of SOMA and TDMA, respectively. Next, we derive the closed-form
expression of TC in both SOMA and TDMA.
#### V-B1 SOMA
Outage probability in (22) can be derived as
$\displaystyle\mathsf{Pr_{out}^{s.o.}}(\beta_{\rm
th})=1-\mathsf{Pr}\left(\frac{\mathsf{P}^{\rm s.o.}_{\rm d}}{\mathsf{I}^{\rm
s.o.}+\mathsf{N}_{0}}>\beta_{\rm th}\right)$
$\displaystyle=1-\int_{0}^{\infty}\left\\{1-F_{h_{0}}\left(\frac{(4\pi)^{2}f^{2}R_{0}^{\alpha}\beta_{\rm
th}y}{\phi\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
r}c^{2}}\right)\right\\}f_{\mathsf{I}^{\rm s.o.}}(y){\rm d}y$
$\displaystyle=1-\mathcal{L}_{\mathsf{I}^{\rm
s.o.}}\left(\frac{(4\pi)^{2}f^{2}R_{0}^{\alpha}\beta_{\rm
th}}{\phi\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
r}c^{2}}\right),$ (25)
where $F_{h_{0}}(X)=1-e^{-X}$. From (V-A1), (23), and (V-B1), the closed-from
expression of the TC in SOMA is given as (V-A1) at the top of the next page.
#### V-B2 TDMA
Outage probability in TDMA can be derived as
$\displaystyle\mathsf{Pr_{out}^{t.d.}}(\beta_{\rm th})$
$\displaystyle=1-\mathsf{Pr}\left(\frac{\mathsf{P}^{\rm t.d.}_{\rm
d}}{\mathsf{I}^{\rm t.d.}_{\rm d}+\mathsf{N}_{0}}>\beta_{\rm th}\right)$
$\displaystyle=1-\mathcal{L}_{\mathsf{I}^{\rm t.d.}_{\rm
d}}\left(\frac{(4\pi)^{2}f^{2}R_{0}^{\alpha}\beta_{\rm th}}{\mathsf{P}_{\rm
Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm r}c^{2}}\right).$ (26)
The Laplace transform of $\mathsf{I}^{\rm t.d.}_{\rm d}$ can be derived as
$\displaystyle\mathcal{L}_{\mathsf{I}^{\rm t.d.}_{\rm d}}(z)$
$\displaystyle=\exp{\left\\{-2\pi\lambda_{\rm
d}\int_{r_{0}}^{\infty}\mathbb{E}_{h}\left[1-e^{-z{\rm K}_{1}hr^{-\alpha_{\rm
I}}}\right]r{\rm d}r\right\\}}$ $\displaystyle=\exp{\left\\{-2\pi\lambda_{\rm
d}{\rm A}_{3}(z)\frac{(z{\rm K}_{1})^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm
I}}\right\\}}.$ (27)
From (23), (V-B2), (V-B2), the closed-form expression of the TC in TDMA can be
derived as (19) at the top of the page.
## VI Network Design Strategy for Successful Ranging Probability
In this section, we discuss how network parameters such as node densities,
radius of guard zone, power splitting factor, and time division factor are
determined from the analysis in Section V for a given SRP constraint.
### VI-A Node Densities
The density of the node in the networks affects the power of the interference
signal. Specifically, as the UAV-radar node density $\lambda_{\rm r}^{\prime}$
increases, the interference power at the typical UAV-radar increases and
therefore, the SINR of the received radar signal decreases, which results in
the lower SRP. In SOMA, SINR is also affected by the UAV-comm node density
$\lambda_{\rm d}^{\prime}$ due to the simultaneous transmission of data and
radar signals. Therefore, one can be interested in finding the maximum node
density given a target SRP ($\mathsf{\bar{Pr}_{s.r.}}$) and SINR threshold
$\gamma_{\rm th}$ of the SRP.
#### VI-A1 SOMA
When SINR threshold $\gamma_{\rm th}$ and the target SRP are given, we can
rearrange (V-A1) such that only the terms that are related to the node
densities $\lambda_{\rm d}$ and $\lambda_{\rm r}^{\rm s.o.}$ are placed to the
left side of the equation. Then, we obtain inequality as follows:
$\displaystyle\left(\frac{\phi}{1-\phi}\right)^{\frac{2}{\alpha_{\rm
I}}}C_{1}\lambda_{\rm d}+C_{2}\lambda_{\rm r}^{\rm
s.o.}\leq\frac{-\log\mathsf{\bar{Pr}_{s.r.}^{\rm s.o.}}\alpha_{\rm
I}}{2\pi\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}},$ (28)
where $C_{1}$ and $C_{2}$ are indicated in (V-A1). If we assume a condition
$\lambda_{\rm d}=\lambda_{\rm r}^{\rm s.o.}$, the above inequality can be
rewritten as
$\displaystyle\lambda_{\rm d}=\lambda_{\rm r}^{\rm
s.o.}\leq\frac{-\log\mathsf{\bar{Pr}_{s.r.}^{\rm s.o.}}\alpha_{\rm
I}}{2\pi\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm
I}}}\left(\left(\frac{\phi}{1-\phi}\right)^{\frac{2}{\alpha_{\rm
I}}}C_{1}+C_{2}\right)}.$ (29)
The maximum node densities $\lambda_{\rm d}^{\star}$ and $\lambda_{\rm r}^{\rm
s.o.\star}$ can be obtained when (29) goes to equality. Note that this
analysis can be easily extended to the condition that $\lambda_{\rm d}$ and
$\lambda_{\rm r}^{\rm s.o.}$ are given by the different ratio ($\lambda_{\rm
d}\propto\lambda_{\rm r}^{\rm s.o.}$) to find the maximum node densities.
#### VI-A2 TDMA
In the same manner of obtaining (29) for SOMA, the maximum node densities of
the UAV-radar $\lambda_{\rm r}^{\rm t.d.\star}$ with the given target SRP and
SINR threshold can be expressed from (V-A2) as
$\displaystyle\lambda_{\rm r}^{\rm
t.d.\star}=\frac{-\log\mathsf{\bar{Pr}_{s.r.}^{\rm s.o.}}\alpha_{\rm
I}}{2\pi\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}C_{2}}.$ (30)
### VI-B Radius of Guard Zone
Guard zone constrains the minimum distance between nodes to avoid strong
interference. As the minimum distance increases, the power of the interference
decreases. This implies that SRP is reduced as radius of guard zone $r_{0}$
increases. When we design networks with target SRP
($\mathsf{\bar{Pr}_{s.r.}}$) and the SINR threshold $\gamma_{\rm th}$, the
minimum radius of the guard zone $r_{0}$ that satisfies the target performance
can be obtain by solving (V-A1) in SOMA and (V-A2) in TDMA for $r_{0}$.
### VI-C Power Splitting Factor $\phi$ in SOMA
Power splitting factor $\phi$ determines the transmit power ratio between UAV-
comms and UAV-radars in SOMA where the radar signal power proportionally
decreases as $\phi$ increases. From the closed-form expression of the SRP in
(V-A1), the terms that are affected by $\phi$ are
$\left(\frac{\phi}{1-\phi}\right)^{\frac{2}{\alpha_{\rm I}}}$ and ${\rm
B}\left(\frac{1}{1+\left(\frac{\phi 4\pi{\rm G}_{\rm
rI}R_{0}^{2\alpha}\gamma_{\rm th}}{(1-\phi)\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)$. Since $\frac{\phi}{1-\phi}$ and
incomplete beta function are monotonic increasing functions, it is easily
proved that SRP is decreasing function with respect to the power splitting
factor $\phi$. This can be intuitively interpreted as higher transmit power of
UAV-comms increasing the power of the interference signal.
###### Proposition 1
When $0\leq\phi<0.5$, the impact of the UAV-comm node density $\lambda_{\rm
d}$ on SRP is less than the UAV-radar node density $\lambda_{\rm r}^{\rm
s.o.}$. When $\phi=0.5$, the impact of the communication node density
$\lambda_{\rm d}$ on SRP is equal to the radar node density $\lambda_{\rm
r}^{\rm s.o.}$, while when $0.5<\phi\leq 1$, the impact of the communication
node density $\lambda_{\rm d}$ on SRP is greater than the radar node density
$\lambda_{\rm r}^{\rm s.o.}$.
###### Proof:
From (V-A1), we can observe that varying $\phi$ only affects the node density
of UAV-comm $\lambda_{\rm d}$ term, not the node density of UAV-radar
$\lambda_{\rm r}^{\rm s.o.}$ term. Then, when $\phi=0.5$,
$\frac{\phi}{1-\phi}$ becomes 1, which leads to the result that the impact of
$\lambda_{\rm d}$ becomes the same as the impact of $\lambda_{\rm r}^{\rm
s.o.}$. On the other hand, when $\phi$ is greater than 0.5,
$\frac{\phi}{1-\phi}$ becomes greater than 1 as well, which makes the
multiplying term by $\lambda_{\rm d}$ becomes greater than the multiplying
term by $\lambda_{\rm r}^{\rm s.o.}$. In the same way, when $\phi$ is less
than 0.5, the multiplying term by $\lambda_{\rm d}$ becomes less than the
multiplying term by $\lambda_{\rm r}^{\rm s.o.}$. ∎
Propostion 1 implies that SRP is affected by the ratio between the node
density of UAV-comm ($\lambda_{\rm d}$) and the UAV-radar ($\lambda_{\rm
r}^{\rm s.o.}$) and when $\phi$ is given, a different ratio of UAV-comm and
UAV-radar node density can improve SRP, which is observed in Fig. 6 of Section
IX.
### VI-D Time Division Factor $\tau$ in TDMA
As we mention in Section IV-B, the increase in $\tau$ reduces radar
transmission time and increase the duty cycle, which results in higher node
density of the active UAV-radar $\bar{\lambda}^{\rm t.d.}_{\rm r}$. The
effective UAV-radar node density $\lambda_{\rm r}^{\rm t.d.}$ in the HPPP
approximation is proportionally increased by $\bar{\lambda}^{\rm t.d.}_{\rm
r}$ in (8).
## VII Network Design Strategy for Transmission Capacity
In this section, we analyze TC depending on network design parameters. We find
the node densities that maximize the TC and we investigate the impact of the
radius of guard zone. We also investigate the effect of the power splitting
factor and the time division factor on the TC.
### VII-A Node Densities
As the node density of the UAV-comm $\lambda_{\rm d}$ increases, SINR is
decreased by the larger number of interferers but the higher node density can
increase the capacity of the unit area. Because of this trade-off, we can find
the maximum node density $\lambda_{\rm d}$ that maximizes TC.
#### VII-A1 SOMA
When target SINR $\beta_{\rm th}$, the UAV-radar node density $\lambda_{\rm
r}^{\rm s.o.}$, radius of guard zone $r_{0}$, and the power splitting factor
$\phi$ are given, we can find the $\lambda_{\rm d}$ that maximizes the TC from
(V-A1). The term in (V-A1) that is affected by $\lambda_{\rm d}$ are written
as
$\displaystyle D_{1}={\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left(-2\pi{\lambda}_{\rm d}C_{3}\frac{\left(\frac{{\rm G}_{\rm
rI}R_{0}^{\alpha}\beta_{\rm th}}{\mathsf{G}_{\rm
r}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right),$ (31)
where $C_{3}$ is indicated in (V-A1). From (31), the first and the second
derivative of transmission capacity with respective to ${\lambda}_{\rm d}$ can
be expressed as
$\displaystyle(C^{\rm s.o.})^{{}^{\prime}}$ $\displaystyle=\log(1+\beta_{\rm
th})\exp\left(-2\pi{\lambda}_{\rm
d}C_{3}^{\prime}\right)\left(1-2\pi{\lambda}_{\rm d}C_{3}^{\prime}\right),$
(32) $\displaystyle(C^{\rm s.o.})^{{}^{\prime\prime}}$ $\displaystyle=4\pi
C_{3}^{\prime}\log(1+\beta_{\rm th})\exp\left(-2\pi{\lambda}_{\rm
d}C_{3}^{\prime}\right)\left(\pi{\lambda}_{\rm d}C_{3}^{\prime}-1\right),$
(33)
where $C_{3}^{\prime}=C_{3}\frac{\left(\frac{{\rm G}_{\rm
rI}R_{0}^{\alpha}\beta_{\rm th}}{\mathsf{G}_{\rm
r}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}$. Then, transmission
capacity is maximized at
$\displaystyle{\lambda}_{\rm d}^{\star}=\frac{1}{2\pi
C_{3}^{\prime}}\quad\quad\text{(SOMA)}.$ (34)
In SOMA, the node density of UAV-radar $\lambda_{\rm r}^{\rm s.o.}$ also
increases the power of interference, which reduces the TC. In
$\eqref{eq:tc_so_ana}$, the terms that include the radar node density
$\lambda_{\rm r}^{\rm s.o.}$ are given as
$\displaystyle D_{2}={\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left(-2\pi\left(\frac{1-\phi}{\phi}\right)^{\frac{2}{\alpha_{\rm
I}}}{\lambda}_{\rm r}^{\rm s.o.}C_{4}^{\prime}\right),$ (35)
where $C_{4}^{\prime}=C_{4}\frac{\left(\frac{{\rm G}_{\rm
rI}R_{0}^{\alpha}\beta_{\rm th}}{\mathsf{G}_{\rm
r}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}$. From the above
equation, it can be found that TC is a decreasing function of the
${\lambda}_{\rm r}^{\rm s.o.}$.
#### VII-A2 TDMA
Similarly to SOMA, we can optimize the UAV-comm node density $\lambda_{\rm d}$
in TDMA. From $\eqref{eq:tc_td_ana}$, the TC can be rewritten as
$\displaystyle C^{\rm t.d.}=\tau{\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left(-2\pi{\lambda}_{\rm d}C_{3}^{\prime}\right).$ (36)
Then, the optimal UAV-comm node density that maximizes TC can be derived as
$\displaystyle{\lambda}_{\rm d}^{\star}=\frac{1}{2\pi
C_{3}^{\prime}}\quad\quad\text{(TDMA)}.$ (37)
In addition, the TC in TDMA is not affected by $\lambda_{\rm r}^{\rm t.d.}$.
###### Remark 1
From the above analysis, TC is maximized at ${\lambda}_{\rm
d}^{\star}=\frac{1}{2\pi C_{3}^{\prime}}$ for both SOMA and TDMA. On the other
hand, the TC in SOMA decreases as $\lambda_{\rm r}^{\rm s.o.}$ increases,
while TC in TDMA is independent of $\lambda_{\rm r}^{\rm t.d.}$.
### VII-B Radius of Guard Zone
Guard zone improves SINR and it reduces the effective node density
$\lambda_{\rm d}$ from (7). Therefore, as radius of guard zone, $r_{0}$,
increases TC would be either improved by higher SINR or degraded by the lower
node density. Since it is mathematically intractable to obtain the first and
the second derivatives of TC with respect to $r_{0}$ in (V-A1) and (19), we
observe the effect of $r_{0}$ by simulations. From simulation results in Fig.
3(b), it is observed that the maximum TC decreases as the $r_{0}$ increases
from 5 m to 25 m, which implies that the TC is a decreasing function of
$r_{0}$ in a typical parameter setup.
### VII-C Power Splitting Factor $\phi$ and Time Division Factor $\tau$
In SOMA, TC is improved as $\phi$ increases since the transmit power of UAV-
comm becomes higher, which improves SINR. The terms in (V-A1) that are
affected by $\phi$ are $\left(\frac{1-\phi}{\phi}\right)^{\frac{2}{\alpha_{\rm
I}}}$ and ${\rm B}\left(\frac{1}{1+\left(\frac{(1-\phi){\rm G}_{\rm
rI}R_{0}^{\alpha}\beta_{\rm th}}{\phi\mathsf{G}_{\rm
r}}\right)r_{0}^{-\alpha_{\rm I}}};\frac{2}{\alpha_{\rm
I}},1-\frac{2}{\alpha_{\rm I}}\right)$. Since $\frac{1-\phi}{\phi}$ is a
decreasing function and an incomplete beta function is a monotonic increasing
function, it is easily proved that TC is an increasing function in terms of
$\phi$.
In TDMA, $\tau$ decides the time duration of the data transmission, and larger
$\tau$ increases TC. From (19), it is observed that TC is linearly increasing
with respect to $\tau$.
## VIII Performance Comparison of SOMA and TDMA
We compare the performance of SRP and transmission capacity between two
different multiple access strategies to give intuition in the design of UAV
radar sensing and communication network coexistence. We consider two different
scenarios when $\phi=\tau=0.5$: case 1 and case 2. In case 1, we analyze the
condition where the node density of UAV-comms and active UAV-radars is equal,
and we compare SOMA with TDMA by SRP and TC. In case 2, we analyze the
condition that the node density of UAV-radars is greater than that of UAV-
comms, and we find the condition that both SRP and TC of SOMA are higher than
those of TDMA.
### VIII-A Case 1: $\lambda_{\rm d}^{\prime}=\bar{\lambda}_{\rm r}\neq 0$
We first analyze a special case that $\lambda_{\rm
d}^{\prime}=\bar{\lambda}_{\rm r}\neq 0$ and $\phi=\tau=0.5$ where the active
UAV-radar and the UAV-comm node density are equal and the resources allocation
of the data transmission and the radar detection are the same. In this
condition, SRP of SOMA and TDMA can be rewritten from (V-A1) and (V-A2) as
$\displaystyle\mathsf{Pr_{s.r.}^{\rm s.o.}}(\gamma_{\rm th})$
$\displaystyle=\exp\left\\{-4\pi\lambda_{\rm r}^{\rm
s.o.}C_{2}\frac{\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\},$
$\displaystyle\mathsf{Pr_{s.r.}^{\rm t.d.}}(\gamma_{\rm th})$
$\displaystyle=\exp\left\\{-2\pi\lambda_{\rm r}^{\rm
t.d.}C_{2}\frac{\left(\frac{4\pi{\rm G}_{\rm rI}R_{0}^{2\alpha}\gamma_{\rm
th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}.$
(38)
Then, we can derive the following proposition.
###### Proposition 2
In case 1, SRP of TDMA is always greater than SOMA: $\mathsf{Pr_{s.r.}^{\rm
s.o.}}(\gamma_{\rm th})<\mathsf{Pr_{s.r.}^{\rm t.d.}}(\gamma_{\rm th})$.
###### Proof:
From (VIII-A), proposition 2 is proved if $2\lambda_{\rm r}^{\rm
s.o.}>\lambda_{\rm r}^{\rm t.d.}$ holds. From (7), (8), the statement can be
derived as
$\displaystyle 2-2e^{-\bar{\lambda}_{\rm r}^{\rm s.o.}\pi
r_{0}^{2}}>1-e^{-\bar{\lambda}_{\rm r}^{\rm t.d.}\pi r_{0}^{2}}$
$\displaystyle\to$ $\displaystyle 2-2e^{-\lambda_{\rm r}^{\prime}\delta\pi
r_{0}^{2}}>1-e^{-2\lambda_{\rm r}^{\prime}\delta\pi r_{0}^{2}}$
$\displaystyle\to$ $\displaystyle\left(e^{-\lambda_{\rm r}^{\prime}\delta\pi
r_{0}^{2}}\right)^{2}-2e^{-\lambda_{\rm r}^{\prime}\delta\pi r_{0}^{2}}+1>0$
$\displaystyle\to$ $\displaystyle\left(e^{-\lambda_{\rm r}^{\prime}\delta\pi
r_{0}^{2}}-1\right)^{2}>0.$ (39)
∎
Next, in this special case, TC can be rewritten from (V-A1), (19) as
$\displaystyle C^{\rm s.o.}$ $\displaystyle={\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-4\pi\lambda_{\rm d}C_{3}^{\prime}\right\\},$ $\displaystyle
C^{\rm t.d.}$ $\displaystyle=\frac{1}{2}{\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-2\pi\lambda_{\rm d}C_{3}^{\prime}\right\\}.$ (40)
Then, we can derive the following proposition.
###### Proposition 3
In case 1, transmission capacity of SOMA is greater than TDMA, when outage
probability $\mathsf{Pr_{out}^{t.d.}}(\beta_{\rm th})<\frac{1}{2}$,
$\mathsf{Pr_{out}^{s.o.}}(\beta_{\rm th})<\frac{3}{4}$.
###### Proof:
From (VIII-A), $C^{\rm s.o.}>C^{\rm t.d.}$, if the following inequality holds:
$\displaystyle{\lambda}_{\rm d}\log(1+\beta_{\rm
th})\left(\exp\left\\{-2\pi\lambda_{\rm d}C_{3}^{\prime}\right\\}\right)^{2}$
$\displaystyle>\frac{1}{2}{\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-2\pi\lambda_{\rm d}C_{3}^{\prime}\right\\}$
$\displaystyle\to$ $\displaystyle\quad\quad\exp\left\\{-2\pi\lambda_{\rm
d}C_{3}^{\prime}\right\\}>\frac{1}{2},$ $\displaystyle\to$
$\displaystyle\quad\quad 1-\mathsf{Pr_{out}^{t.d.}}(\beta_{\rm
th})>\frac{1}{2},$ $\displaystyle\to$
$\displaystyle\quad\quad\mathsf{Pr_{out}^{t.d.}}(\beta_{\rm th})<\frac{1}{2},$
$\displaystyle\to$ $\displaystyle\quad\quad\mathsf{Pr_{out}^{s.o.}}(\beta_{\rm
th})=1-\left(\exp\left\\{-2\pi\lambda_{\rm
d}C_{3}^{\prime}\right\\}\right)^{2}<\frac{3}{4}.$ (41)
∎
Note that the condition that outage probability is greater than $\frac{3}{4}$
is a generally desirable condition. Therefore, in case 1 ($\lambda_{\rm
d}^{\prime}=\bar{\lambda}_{\rm r}\neq 0$ and $\phi=\tau=0.5$), TDMA
outperforms SOMA for SRP, but SOMA is better than TDMA for TC.
### VIII-B Case 2: $\lambda_{\rm d}^{\prime}<\bar{\lambda}_{\rm r}$
We can also analyze another special case where $\lambda_{\rm
d}^{\prime}<\bar{\lambda}_{\rm r}$, $\phi=\tau=0.5$, and an additional
condition that the UAV-radar node density $\lambda_{\rm r}^{\prime}$ is
sufficiently small. Then, the effective UAV-radar node densities $\lambda_{\rm
r}^{\rm s.o.}$, $\lambda_{\rm r}^{\rm t.d.}$ in (7), (8) can be approximated
by the first order Taylor expansion at $\lambda_{\rm r}^{\prime}=0$ as
$\lambda_{\rm r}^{\rm s.o.}\approx\delta\lambda_{\rm r}^{\prime}$ and
$\lambda_{\rm r}^{\rm t.d.}\approx 2\delta\lambda_{\rm r}^{\prime}$. In this
condition, we can have the following proposition.
###### Proposition 4
In case 2 where $\lambda_{\rm d}^{\prime}<\bar{\lambda}_{\rm r}$, the radar
node density $\lambda_{\rm r}^{\prime}$ is sufficient small, and
$\phi=\tau=0.5$, SRP of SOMA is greater than TDMA.
###### Proof:
From (V-A1) and (V-A2), we can obtain SRP in case 2 as follows:
$\displaystyle\mathsf{Pr_{s.r.}^{\rm s.o.}}(\gamma_{\rm th})$
$\displaystyle=\exp\left\\{-2\pi(\lambda_{\rm d}+\delta\lambda_{\rm
r}^{\prime})C_{2}\frac{\left(\frac{4\pi{\rm G}_{\rm
rI}R_{0}^{2\alpha}\gamma_{\rm th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\},$
$\displaystyle\mathsf{Pr_{s.r.}^{\rm t.d.}}(\gamma_{\rm th})$
$\displaystyle=\exp\left\\{-2\pi(2\delta\lambda_{\rm
r}^{\prime})C_{2}\frac{\left(\frac{4\pi{\rm G}_{\rm
rI}R_{0}^{2\alpha}\gamma_{\rm th}}{\mathsf{G}_{\rm r}\mathsf{G}_{\rm
p}\bar{\sigma}}\right)^{\frac{2}{\alpha_{\rm I}}}}{\alpha_{\rm I}}\right\\}.$
(42)
Then, we can easily prove that $\mathsf{Pr_{s.r.}^{\rm
s.o.}}>\mathsf{Pr_{s.r.}^{\rm t.d.}}(\gamma_{\rm th})$ if $\lambda_{\rm
d}^{\prime}<\delta\lambda_{\rm r}^{\prime}=\bar{\lambda}_{\rm r}$. ∎
Next, we can also obtain the following proposition regarding TC.
###### Proposition 5
In the case that $\phi=\tau=0.5$, TC of SOMA is greater than TDMA, when
$\mathsf{Pr}(\beta_{1}<\beta_{\rm th})<\frac{1}{2}$, where
$\beta_{1}=\frac{\mathsf{P}^{\rm s.o.}_{\rm r}}{\mathsf{I}_{1}}$ denotes SIR
in SOMA considering the interference only comes from the active UAV-radar
nodes ( $\mathsf{I}_{1}$ in (IV-C1)).
###### Proof:
In the case that $\phi=\tau=0.5$, transmission capacity can be rewritten from
(V-A1), (19) as
$\displaystyle C^{\rm s.o.}$ $\displaystyle={\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-2\pi\lambda_{\rm
d}C_{3}^{\prime}\right\\}\exp\left\\{-2\pi\lambda_{\rm r}^{\rm
s.o.}C_{3}^{\prime}\right\\},$ $\displaystyle C^{\rm t.d.}$
$\displaystyle=\frac{1}{2}{\lambda}_{\rm d}\log(1+\beta_{\rm
th})\exp\left\\{-2\pi\lambda_{\rm d}C_{3}^{\prime}\right\\}.$ (43)
Then, $C^{\rm s.o.}>C^{\rm t.d.}$ holds, when
$\displaystyle 1-\exp\left\\{-2\pi\lambda_{\rm r}^{\rm
s.o.}C_{3}^{\prime}\right\\}<\frac{1}{2}$
$\displaystyle\to\leavevmode\nobreak\ $
$\displaystyle\mathsf{Pr}(\beta_{1}<\beta_{\rm th})<\frac{1}{2},$ (44)
where $\mathsf{I}_{1}=\sum_{r_{j}\in\Phi(\lambda_{\rm r}^{\rm s.o.})\backslash
r_{0}}\frac{\frac{1}{2}\mathsf{P}_{\rm Tx}\mathsf{G}_{\rm t}\mathsf{G}_{\rm
rI}c^{2}}{(4\pi)^{2}f_{\rm c}^{2}r_{j}^{\alpha_{\rm I}}}h_{j}$. ∎
###### Remark 2
From Proposition 4 and Proposition 5, SOMA can outperform TDMA in both SRP and
TC, if the conditions in Proposition 4 and Proposition 5 are satisfied. This
implies that the active UAV-radar node density is greater than the UAV-comm
node density ($\lambda_{\rm d}^{\prime}<\bar{\lambda}_{\rm r}$) while the
outage probability considering the interference only from the active UAV-
radars is less than 0.5. Moreover, when the first condition holds, the second
condition is generally desirable since the target outage probability is mostly
less than 0.5 and the interference coming from the UAV-comms is smaller than
the active UAV-radars.
Table III: Parameter settings for UAV radar and communication network coexistence analysis. Parameter | Value
---|---
Transmit power ($\mathsf{P}_{\mathsf{Tx}}$) | $20$ dBm
Transmitter antenna gain ($\mathsf{G}_{\rm t}$) | 10 dBi
Receiver antenna gain ($\mathsf{G}_{\rm r}$) | 10 dBi
Receiver antenna gain from the interference ($\mathsf{G}_{\rm rI}$) | -10 dBi
Target distance ($R_{0}$) | 50 m
Average RCS ($\bar{\sigma}$) | 30 dBsm
Path-loss exponent ($\alpha$) | 2.0
Path-loss exponent from the interference ($\alpha_{\rm I}$) | 2.5
Processing gain ($\mathsf{G}_{\rm p}$) | 10 dBi
Duty cycle ($\delta$) | 0.1
Carrier frequency ($f_{\rm c}$) | $35$ GHz
(a)
(b)
Figure 3: SRP and TC depending on SINR threshold and radius of guard zone
($r_{0}$) where $\phi=0.5$, $\tau=0.5$, $\lambda^{\prime}_{\rm d}=0.01$,
$\lambda^{\prime}_{\rm r}=0.1$. In case 1, TDMA outperforms SOMA on SRP while
SOMA is superior to TDMA on TC.
## IX Simulation Results
In this section, we evaluate the performance of UAV radar and communication
network coexistence based on simulation and analysis. SRP and TC with SOMA and
TDMA are presented with the change of the different parameters. We consider 35
GHz carrier frequency for mmWave communication and Ka-band radar. The key
parameters are listed in Table III.
### IX-A SRP and TC Dependence SINR Threshold and Radius of Guard Zone
(a)
(b)
Figure 4: The SRP and the TC depending on SINR threshold where
$\lambda^{\prime}_{\rm d}=0.00025$, $\lambda^{\prime}_{\rm r}=0.005$,
$r_{0}=5$ m, $\phi=0.5$, $\tau=0.5$. In case 2, SOMA outperforms TDMA on both
the SRP and the TC.
In this subsection, we compare SOMA and TDMA by SRP and TC depending on radius
of guard zone and SINR threshold. In Fig. 3, we show SRP and TC of both SOMA
and TDMA with $\lambda^{\prime}_{\rm d}$, $\lambda^{\prime}_{\rm r}$ and
$\phi$, $\tau$ by case 1 in Section VIII-A. As we discuss in Proposition 2 and
Proposition 3, TDMA outperforms SOMA in SRP while SOMA is superior to TDMA in
the TC. We also observe that as radius of guard zone $r_{0}$ increases, SRP
improves but TC degrades, which is matched to the analysis in Section VI-B and
Section VII-B.
Fig. 4 shows SRP and TC with a system configuration in case 2 in Section
VIII-B. It is observed that both SRP and TC could be better in SOMA if we
consider case 2, which is mentioned in Remark 2. Note that in a general system
parameter setting, we obtain SRP and the TC performance of case 1.
### IX-B SRP and TC Dependence Power Splitting Factor and Time Division
Factor
(a)
(b)
Figure 5: Change of SRP and TC as $\phi$ in SOMA and $\tau$ in TDMA increases
where $\lambda^{\prime}_{\rm d}=0.01$, $\lambda^{\prime}_{\rm r}=0.1$,
$\beta_{\rm th}=-5$ dB, $\gamma_{\rm th}=-10$ dB. Figure 6: Change of SRP as
$\phi$ increases with different ratio of the node density of the UAV-radar and
the UAV-comm where $\gamma_{\rm th}=-10$ dB, which is analyzed in Propostion
1.
In this subsection, we evaluate SRP and TC depending on $\phi$ in SOMA and
$\tau$ in TDMA. Fig. 5(a) shows that as $\phi$ increases TC improves but SRP
decreases as we discuss in Section VI-C and in Section VII-C, which represents
the impact of the different power ratio between the radar signal and the data
signal on SRP and TC. In, Fig. 5(b), it is observed that TC increases as
$\tau$ becomes large. On the other hand, the SRP slowly decreases as $\tau$
increases when we compare it with $\phi$ in SOMA in Fig. 5(a).
Fig. 6 show the effect of the different radio of the active UAV-radar node
density $\bar{\lambda}_{\rm r}^{\rm s.o.}$ and the UAV-comm node density
$\lambda_{\rm d}^{\prime}$ on the SRP in SOMA. It is observed that when
$0.1<\phi<0.5$, higher UAV-comm node density achieves higher SRP while when
$0.5<\phi<1$, higher active UAV-radar node density achieves higher SRP, which
can be interpreted by Proposition 1.
### IX-C SRP and TC Dependence Node Density of UAV-radar and UAV-comm
(a)
(b)
Figure 7: Change of SRP and TC depending on the UAV-radar node density where
$\lambda^{\prime}_{\rm d}=0.01$, $r_{0}=5$ m, $\beta_{\rm th}=0$ dB,
$\gamma_{\rm th}=-10$ dB, $\phi=0.5$, $\tau=0.5$.
(a)
(b)
Figure 8: Change of SRP and TC depending on the UAV-comm node density where
$\lambda^{\prime}_{\rm r}=0.01$, $r_{0}=5$ m, $\beta_{\rm th}=0$ dB,
$\gamma_{\rm th}=-10$ dB, $\phi=0.5$, $\tau=0.5$.
In this subsection, we simulate the dependence of SRP and TC on the node
density of the UAV-radar ($\lambda_{\rm r}^{\prime}$) and the UAV-comm
($\lambda_{\rm d}^{\prime}$). As we discuss in Section VI-A and Section VII-A,
Fig. 7(a) shows that SRP is a decreasing function of $\lambda_{\rm
r}^{\prime}$ for both SOMA and TDMA. In addition, in Fig. 7(b), it is observed
that TC decreases as $\lambda_{\rm r}^{\prime}$ increases in SOMA, however,
the TC is not affected by $\lambda_{\rm r}^{\prime}$ in TDMA.
In Fig. 8(a), we observe that SRP is a decreasing function of $\lambda_{\rm
d}^{\prime}$ in SOMA, while SRP is not affected by $\lambda_{\rm r}^{\prime}$
in TDMA. Fig. 8 shows that TC is maximized at ${\lambda}_{\rm
d}^{\prime\star}=0.0115$ for both SOMA and TDMA, which can be derived from
Remark 1 and (7).
## X Conclusion
In this paper, we investigate the coexistence of UAV radar and communication
network. We deploy UAV-radars and UAV-comms by using HPPP where UAV-radars
detect and track targets and UAV-comms communicate with their serving users in
the same frequency band. We take into account two different multiple-access
protocols, SOMA, and TDMA, to operate both radar signals and data signals
simultanously. We analyze the performance of SRP in the radar detection
scenario and TC in the data communication scenario. We show that in general,
TDMA outperforms SOMA on SRP, while SOMA outperforms TDMA on TC. However, SOMA
can achieve higher SRP as well as higher TC when the node density of UAV-
radars is higher than that of UAV-comms (i.e., $\lambda_{\rm
d}^{\prime}<\bar{\lambda}_{\rm r}$), for $\phi=\tau=0.5$, and
$\mathsf{Pr}(\beta_{1}<\beta_{\rm th})<\frac{1}{2}$. We also find the UAV-comm
node density that maximizes TC by derving the first and the second derivative
of its analytic form and analyze the behavior of SRP and TC depending on the
node density, the radius of the guard zone, the power splitting factor, and
time division factor.
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|
# Strong two-meson decays of light and charmed vector mesons
Roberto Correa da Silveira LFTC, Universidade Cidade de São Paulo, Rua Galvão
Bueno 868, São Paulo, SP 01506-000, Brazil Fernando E. Serna LFTC,
Universidade Cidade de São Paulo, Rua Galvão Bueno 868, São Paulo, SP
01506-000, Brazil Departamento de Física, Universidad de Sucre, Carrera 28
No. 5-267, Barrio Puerta Roja, Sincelejo 700001, Colombia Bruno El-Bennich
LFTC, Universidade Cidade de São Paulo, Rua Galvão Bueno 868, São Paulo, SP
01506-000, Brazil
###### Abstract
We calculate the strong decay couplings for $\rho\to\pi\pi$, $\phi\to KK$,
$K^{*}\to K\pi$ and $D^{*}\to D\pi$ in a unified and consistent approach based
on the impulse approximation, nonperturbative solutions of the quark-gap
equation and the Poincaré invariant Bethe-Salpeter amplitudes of vector and
pseudoscalar mesons. In particular, we obtain the coupling
$g_{D^{*}\\!D\pi}=17.24^{+3.06}_{-2.30}$ in very good agreement with the
experimental value by CLEO, which corresponds to a strong effective coupling
between heavy vector and pseudoscalar mesons to the pion of
$\hat{g}=0.58^{+0.10}_{-0.08}$.
## I Introduction
One of the challenges in hadron physics is to understand the spectrum,
constituent composition and momentum distribution of quarks and gluons within
the hadrons. To obtain deeper insight into the hadron’s structure, their
excitations have been intensively investigated in the past decades. This
includes radial excitations, higher angular momentum states and exotic states
containing constituent gluons that contribute to the total angular momentum of
the hadron.
The vector mesons, being the lowest spin excitations of the pseudoscalars,
offer a first glimpse into an electromagnetic excitation of a $\bar{q}q$ pair.
This is because neutral vector mesons can directly couple to the photon via an
electromagnetic current since their quantum numbers, $J^{PC}=1^{--}$, are
those of the photon. Naturally, they have been much studied and from the
viewpoint of functional approaches to Quantum Chromodynamics (QCD) they were
helpful to establish the ladder truncation of the Bethe-Salpeter equation
(BSE) Maris:1999nt , at least for the ground states of lighter vector mesons.
Of course, beyond the masses of the pseudoscalar and vector mesons, their
electromagnetic and electroweak properties are of fundamental interest and
there is no lack of studies dedicated to weak decay constants, elastic and
transition form factors Maris:2000sk ; deMelo:1997hh ; Bakker:2002mt ;
deMelo:2012hj ; DeMelo:2018bim ; deMelo:2014gea ; Chang:2013nia ; Raya:2015gva
; Ding:2018xwy ; Xu:2019ilh ; daSilva:2012gf ; El-Bennich:2012mkr ; El-
Bennich:2008dhc ; Ivanov:2007cw ; El-Bennich:2009gbu .
Beyond these observables, the strong decays of vector mesons into two
light(er) mesons provide another source of information on the nonperturbative
dynamics complementary to electromagnetic interactions and weak decays. They
are the simplest possible decays that proceed via strong interactions and
since the vector meson decays via a $P$-wave interaction, the Bethe-Salpeter
amplitude (BSA) is probed differently than in the electroweak sector. In here,
our main object of interest is the reaction $D^{*}\to D\pi$ which we studied
in Refs. El-Bennich:2010uqs ; El-Bennich:2011tme ; El-Bennich:2012hom ; El-
Bennich:2016bno within the limitations of not having the BSA of charmed
mesons at hand but motivated by the first measurement of the $D^{*}$ width,
$\Gamma(D^{*+})=96\pm 4\pm 22$ keV CLEO:2001sxb . This result is of great
interest, as it is one of the few quantities in flavor physics that does not
probe electroweak properties of heavy mesons and which opens a window on
nonperturbative QCD in mesons with two very distinct mass scales. Moreover,
the strong coupling $g_{D^{*}\\!D\pi}$ one can extract from the decay width is
related to a putative universal coupling $\hat{g}$ between heavy-light mesons
and a low-momentum pion in the heavy-meson chiral Lagrangian Casalbuoni:1996pg
; El-Bennich:2021ldv ; Braghin:2021qmu . At leading order in the $1/m_{D}$
expansion this relation is
$g_{D^{*}\\!D\pi}=2\sqrt{m_{D}^{*}m_{D}}\,\hat{g}/f_{\pi}$.
These calculations were based on one-covariant models of the $D$ and $D^{*}$
wave functions and were therefore not Poincaré invariant, so that the momentum
partition parameters had to be chosen according to some semi-classical
criterium El-Bennich:2010uqs . This, of course, was not satisfactory and the
motivation remained to compute the decay amplitude guided by Ref.
Jarecke:2002xd which dealt with the decays $\rho\to\pi\pi$, $\phi\to\bar{K}K$
and $K^{*}\to K\pi$. In this work we close this gap and compute the vector-
meson decay with the complete Poincaré covariant structure of the BSA for the
$D$, $D^{*}$ and pion. Along the way, we also obtain the strong couplings
considered in Ref. Jarecke:2002xd which we update.
The remainder of this paper is composed of five sections: in Section II we
explain the framework in which the strong couplings are calculated and define
the decay kinematics; in Section III we describe the functional approach to
QCD we use to calculate the quark propagators and BSA of the mesons within a
given truncation scheme in Euclidean space; in Section IV the numerical method
to solve the BSE is summarized and the meson’s masses and weak decay constants
are calculated. Finally, in Section V we present our results for the strong
couplings and wrap up with final remarks in Section VI.
## II Strong Decay Amplitude
In what follows, we limit ourselves to the _impulse approximation_ of the
strong decays depicted in Fig. 1. As argued in Ref. Jarecke:2002xd , since the
$\rho$ and $\phi$ appear as resonance poles in the timelike electromagnetic
form factors of the pion and kaon, the pole residues are proportional to the
respective coupling constants $g_{\rho\pi\pi}$ and $g_{\phi KK}$. Hence, if
these form factors are obtained in impulse approximation, so will be the
couplings. Now, the impulse approximation for the electromagnetic coupling to
mesons conserves the current as long as the meson’s BSA and quark-photon
vertex are calculated in the ladder and the quark propagators in rainbow
truncation, respectively, and the resulting electromagnetic form factors are
in excellent agreement with experiment Maris:2000sk ; Chang:2013nia ;
Raya:2015gva ; Ding:2018xwy .
In the time-like region, on the other hand, the ladder truncation of the BSE
fails to produce the $\rho$ pole in $e^{+}e^{-}\to\gamma^{*}\to\pi^{+}\pi^{-}$
and was amended to include effective pion degrees of freedom in the BSE
scattering kernel Williams:2018adr ; Miramontes:2019mco ; Miramontes:2021xgn .
Therefore, we expect that the impulse approximation for the strong decays of
lighter mesons misses some of the relevant physics, in particular in the case
of the $\rho$ meson whose decay width is almost 20% of its mass. Going beyond
this approximation, not merely in the BSE kernel but also in the decay
amplitude, is a technically and numerically challenging task. As the main aim
here is to improve on earlier calculations of $D^{*}\to D\pi$, we deliberately
ignore these corrections.
The strong decay coupling for a process $V\to PP$ is defined as,
$\langle
P(p_{2})P(q)|V(p_{1},\lambda)\rangle:=\,g_{V\\!PP}\,\epsilon^{\lambda}\\!\cdot
q\ ,$ (1)
where the initial state is a vector meson with transverse polarization
$\epsilon^{\lambda}_{\mu}$ and momentum $p_{1}^{2}=-m_{V}^{2}$, while the
light(er) mesons have on-shell momenta $p_{1}^{2}=-m_{P}^{2}$,
$q^{2}=-m_{P}^{2}$, with $q=p_{1}-p_{2}$, and can have different flavor
content. The decay amplitude in impulse approximation can be expressed by the
loop integral:
$\displaystyle g_{V\\!PP}\;\epsilon^{\lambda}\\!\cdot q\,$
$\displaystyle=\int^{\Lambda}\\!\frac{d^{4}k}{(2\pi)^{4}}\,\mathrm{Tr_{CD}}\,\big{[}\epsilon^{\lambda}\\!\cdot\Gamma_{V}(k_{V},p_{1})$
$\displaystyle S_{f}(k_{1})$
$\displaystyle\bar{\Gamma}_{P}(k_{P},-p_{2})S_{f}(k_{2})\bar{\Gamma}_{P}(k_{P}^{\prime},-q)S_{f}(k_{3})\big{]}\
.$ (2)
In here, $\Gamma(k,P)$ are the BSAs of the mesons, $S(k_{i})$ are the quarks
propagators and the trace is over color and Dirac indices. Following the
momentum flow in Fig. 1, the quark momenta are defined as,
$\displaystyle k_{1}$ $\displaystyle=k+w_{1}p_{1}\,,$ (3) $\displaystyle
k_{2}$ $\displaystyle=k+w_{1}p_{1}-p_{2}\,,$ (4) $\displaystyle k_{3}$
$\displaystyle=k-w_{2}p_{1}\,.$ (5)
with the constraint $w_{1}+w_{2}=1$ on the partition parameters due to
momentum conservation. The relative BSA momenta are given by,
$\displaystyle k_{V}$ $\displaystyle=k+\tfrac{1}{2}(w_{1}-w_{2})p_{1}\,,$ (6)
$\displaystyle k_{P}$ $\displaystyle=k+w_{1}p_{1}-\tfrac{1}{2}\,p_{2}\,,$ (7)
$\displaystyle k_{P}^{\prime}$
$\displaystyle=k+\tfrac{1}{2}(w_{1}-w_{2})p_{1}-\tfrac{1}{2}\,p_{2}\,.$ (8)
Note that the relative momentum of the vector meson is only real if
$w_{1}=w_{2}$, as in the meson’s rest frame $p_{1}=(\mathbf{0},im_{V})$ in the
Euclidean-space formulation we use. This will be discussed in more detail in
Section V.
Figure 1: Decay diagram depicting a generic strong decay $V\to PP$ in the
impulse approximation of Eq. (2). The shaded ovals represent vector ($V$) and
pseudoscalar ($P$) meson BSAs (15), while the dark-shaded circles symbolize
dressed quark propagators (34) and the double-lined arrows describe the
incoming vector-meson momentum and outgoing pseudoscalar-meson momenta.
We conclude this section by mentioning some definitions with respect to the
charge when one of the final mesons is an isovector state. We follow Ref.
Bracco:2011pg and define the generic $D^{*}D\pi$ coupling as the one
containing the neutral meson:
$\displaystyle g_{D^{*}\\!D\pi}$
$\displaystyle=\,g_{D^{*\pm}D\mp\pi^{0}}=g_{D^{*0}D^{0}\pi^{0}}$
$\displaystyle=\,\tfrac{1}{\sqrt{2}}\,g_{D^{*-}D^{0}\pi^{+}}=\tfrac{1}{\sqrt{2}}\,g_{D^{*+}D^{0}\pi^{-}}\
.$ (9)
Likewise, considering SU(3) flavor algebra one has,
$g_{K^{*}\\!K\pi}=\sqrt{3}\,g_{K^{*+}K^{+}\pi^{0}}=\sqrt{\tfrac{3}{2}}\,g_{K^{*+}K^{0}\pi^{+}}\
,$ (10)
and moreover $g_{\rho\pi\pi}=g_{\rho^{0}\pi^{+}\pi^{-}}$, $g_{\phi KK}=g_{\phi
K^{+}K^{-}}$.
In Section III we describe how the ingredients of the strong decay amplitude
(2), namely the quark propagators and BSAs, are obtained from solving the
quark-gap equation and the BSE.
## III Pseudoscalar and Vector Meson Bound States
### III.1 Bethe-Salpeter Equation
The relativistic initial and final bound states in the decay amplitude (2) are
described by Poincaré covariant BSAs, $\Gamma^{fg}_{P}(k,P)$ and
$\Gamma^{fg}_{V\nu}(k,P)$, which are the solutions of the homogeneous BSE in
the $J^{PC}=0^{-+}$ and $J^{PC}=1^{--}$ channels, respectively Bashir:2012fs :
$\displaystyle\Gamma^{fg}_{P}(k,P)=\int^{\Lambda}\\!\frac{d^{4}q}{(2\pi)^{4}}\,K^{fg}(k,q,P)\,\chi^{fg}_{P}(q,P)\
,$ (11)
$\displaystyle\Gamma^{fg}_{V\nu}(k,P)=\int^{\Lambda}\\!\frac{d^{4}q}{(2\pi)^{4}}\,K^{fg}(k,q,P)\,\chi^{fg}_{V\nu}(q,P)\
.$ (12)
In these BSEs, $k$ is the relative quark-antiquark momentum, $P$ is the meson
momentum and $K^{fg}(k,q,P)$ is the fully amputated scattering kernel which
sums up all possible quark-antiquark interactions. The Bethe-Salpeter wave
functions, $\chi^{fg}_{P}(k,p)$ and $\chi^{fg}_{V\nu}(k,P)$, are obtained by
attaching the quark propagators to the BSA,
$\displaystyle\chi^{fg}_{P}(k,P)$ $\displaystyle=\
S_{f}(k_{\eta})\,\Gamma^{fg}_{P}(k,P)S_{g}(k_{\bar{\eta}})\ ,$ (13)
$\displaystyle\chi^{fg}_{V\nu}(k,P)$ $\displaystyle=\
S_{f}(k_{\eta})\,\Gamma^{fg}_{V\nu}(k,P)S_{g}(k_{\bar{\eta}})\ ,$ (14)
with the shorthands, $k_{\eta}=k+\eta P$ and $k_{\bar{\eta}}=k-\bar{\eta}P$,
that define momentum-partition parameters: $\eta+\bar{\eta}=1$.
The BSA has the most general Poincaré covariant form that can be composed of
the Dirac matrices and the relative and total momenta consistent with the
quantum numbers $P$ and $C$ of a given meson,
$\Gamma_{M}^{fg}(k,P)=\sum_{i=1}^{N}\,T^{i}(k,P)\,\mathcal{F}_{i}^{fg}\big{(}k,P,z_{k}\big{)}\
,$ (15)
where $T^{i}(k,P)$ are Dirac covariants, $\mathcal{F}_{i}^{fg}$ are scalar
Lorentz-invariant amplitudes and $z_{k}=k\cdot P/|k\|P|$ is an angle between
$k$ and $P$. In case of pseudoscalar mesons, we choose the usual $N=4$
covariants,
$\displaystyle T^{1}(k,P)$ $\displaystyle=i\gamma_{5}\,$ (16) $\displaystyle
T^{2}(k,P)$ $\displaystyle=\gamma_{5}\,\gamma\cdot P\ ,$ (17) $\displaystyle
T^{3}(k,P)$ $\displaystyle=\gamma_{5}\,\gamma\cdot k\,k\cdot P\ ,$ (18)
$\displaystyle T^{4}(k,P)$
$\displaystyle=\gamma_{5}\,\sigma_{\mu\nu}\,k_{\mu}P_{\nu}\ ,$ (19)
and for a vector meson $N=8$ covariant vector components are required:
$\displaystyle T^{1}_{\nu}(k,P)$ $\displaystyle=\ i\gamma^{T}_{\nu}\ ,$ (20)
$\displaystyle T^{2}_{\nu}(k,P)$ $\displaystyle=\
i\left[3k^{T}_{\nu}\gamma\cdot
k^{T}-\gamma^{T}_{\nu}\big{(}k^{T}\big{)}^{\\!2}\right]\ ,$ (21)
$\displaystyle T^{3}_{\nu}(k,P)$ $\displaystyle=\ ik\cdot P\,\gamma\cdot
P\,k^{T}_{\nu}\ ,$ (22) $\displaystyle T^{4}_{\nu}(k,P)$ $\displaystyle=\
i\left[\gamma^{T}_{\nu}\gamma\cdot P\,\gamma\cdot
k^{T}+k^{T}_{\nu}\,\gamma\cdot P\right]\ ,$ (23) $\displaystyle
T^{5}_{\nu}(k,P)$ $\displaystyle=\ k^{T}_{\nu}\ ,$ (24) $\displaystyle
T^{6}_{\nu}(k,P)$ $\displaystyle=\ k\cdot P\left[\gamma^{T}_{\nu}\gamma\cdot
k^{T}-\gamma\cdot k^{T}\,\gamma^{T}_{\nu}\right]\ ,$ (25) $\displaystyle
T^{7}_{\nu}(k,P)$ $\displaystyle=\ \gamma^{T}_{\nu}\gamma\cdot P-\gamma\cdot
P\,\gamma^{T}_{\nu}-2\,T_{\nu}^{8}(k,P)\ ,$ (26) $\displaystyle
T_{\nu}^{8}(k,P)$ $\displaystyle=\
\hat{k}^{T}_{\nu}\,\gamma\cdot\hat{k}^{T}\,\gamma\cdot P\ .$ (27)
The transverse projections are $V^{T}_{\nu}=V_{\nu}-P_{\nu}(V\cdotp P)/P^{2}$
with $P\cdot V^{T}=0$ for any four-vector $V_{\nu}$ and
$\hat{k}^{T}\cdot\hat{k}^{T}=1$. Note that the $T^{i}_{\mu}(k,P)$ in Eqs. (20)
to (27) form an orthogonal basis Maris:1999nt ; Gao:2014bca with respect to
the Dirac trace.
In order to calculate the meson’s weak decay constant, one has to normalize
the meson’s BSA. We do so with the derivative of the eigenvalue trajectory,
$\lambda(P^{2})$, of the BSE Nakanishi:1965zz ; Nakanishi:1965zza :
$\displaystyle\left(\frac{\partial\ln\lambda}{\partial
P^{2}}\right)^{\\!-1}\\!\\!$ $\displaystyle=\
\operatorname{tr}_{\mathrm{CD}}\int\frac{d^{4}k}{(2\pi)^{4}}\,\bar{\Gamma}^{fg}_{M}(k;-P)$
$\displaystyle\times\
S_{f}(k_{\eta})\Gamma^{fg}_{M}(k;P)S_{g}(k_{\bar{\eta}})\,.$ (28)
With this we calculate the weak decay constant of the pseudoscalar meson
defined by
$f_{P}P_{\mu}=\langle
0\,|\bar{q}_{g}\gamma_{5}\gamma_{\mu}q_{f}|P(k,P)\rangle\ ,$ (29)
which can be expressed by the integral:
$f_{P}P_{\mu}=\frac{\mathcal{Z}_{2}N_{c}}{\sqrt{2}}\int^{\Lambda}\\!\frac{d^{4}k}{(2\pi)^{4}}\,\operatorname{Tr}_{\mathrm{D}}\left[\gamma_{5}\gamma_{\mu}\,\chi_{P}^{fg}(k,P)\right]\,.$
(30)
Likewise, the weak decay constant of a vector meson is defined by the
amplitude,
$f_{V}m_{V}\,\epsilon^{\lambda}_{\mu}=\langle
0\,|\bar{q}_{g}\gamma_{\mu}q_{f}|V(k,P,\lambda)\rangle\ ,$ (31)
where $m_{V}$ is the vector-meson mass and $\epsilon^{\lambda}_{\mu}(P)$ is
the polarization vector of the transverse vector meson of helicity $\lambda$
which satisfies $\epsilon^{\lambda}\cdot P=0$ and is normalized as
${\epsilon^{\lambda}}^{*}\\!\cdot\epsilon^{\lambda}=3$. This can again be
expressed by a loop integral:
$f_{V}m_{V}=\frac{\mathcal{Z}_{2}N_{c}}{3\sqrt{2}}\int^{\Lambda}\\!\\!\frac{d^{4}k}{(2\pi)^{4}}\,\operatorname{Tr_{D}}\left[\gamma_{\mu}\,\chi^{fg}_{V\mu}(k,P)\right]\
.$ (32)
In both expressions for the decay constants we define
$\mathcal{Z}_{2}(\mu,\Lambda)=\surd Z_{2}^{f}\surd Z_{2}^{g}$, as in Section
III.3, and $N_{c}=3$.
### III.2 Quark Gap Equation
Amongst the Green functions that enter the BSE, whether in Eq. (11) or Eq.
(12), are the flavor-dependent dressed quark propagators described by
Schwinger functions we obtain as solutions of the Dyson-Schwinger equation
(DSE),
$\displaystyle S^{-1}_{f}(p)=$ $\displaystyle\
Z_{2}^{f}\\!\left(i\,\gamma\cdot p+m^{\mathrm{bm}}_{f}\right)$
$\displaystyle+\ Z_{1}^{f}$ $\displaystyle
g^{2}\\!\\!\int^{\Lambda}\\!\frac{d^{4}k}{(2\pi)^{4}}\,D^{ab}_{\mu\nu}(q)\frac{\lambda^{a}}{2}\gamma_{\mu}S_{f}(k)\,\Gamma^{b}_{\nu,f}(k,p)\,,$
(33)
where $m^{\textrm{bm}}_{f}$ is the bare current-quark mass,
$Z_{1}^{f}(\mu,\Lambda)$ and $Z_{2}^{f}(\mu,\Lambda)$ are the vertex and wave-
function renormalization constants at the renormalization point $\mu$,
respectively. The integral in Eq. (33) represents the self-energy of the quark
and involves the dressed-quark propagator $S_{f}(k)$, the dressed-gluon
propagator $D_{\mu\nu}(q)$ with momentum $q=k-p$ and the quark-gluon vertex,
$\Gamma^{a}_{\mu}(k,p)=\frac{1}{2}\,\lambda^{a}\Gamma_{\mu}(k,p)$
Albino:2018ncl ; Albino:2021rvj ; El-Bennich:2022obe , where the SU(3) color
matrices $\lambda^{a}$ are in the fundamental representation. The Poincaré-
invariant regularization scale is $\Lambda\gg\mu$ and can be taken to
infinity. The solution of the DSE can be cast in the most general covariant
form as,
$\displaystyle S_{f}(p)$ $\displaystyle=\,-i\gamma\cdot p\,\sigma_{\rm
v}^{f}(p^{2})+\sigma_{\rm s}^{f}(p^{2})$
$\displaystyle=\,Z_{f}(p^{2})/\left[i\gamma\cdot p+M_{f}(p^{2})\right]\ .$
(34)
In this DSE, $Z_{f}(p^{2})$ defines the wave function and $M_{f}(p^{2})$ is
the running mass of the quark. The scalar functions $\sigma_{\rm
s}^{f}(p^{2})$ and $\sigma_{\rm v}^{f}(p^{2})$ thus depend on $Z_{f}(p^{2})$
and $M_{f}(p^{2})$. In a subtractive renormalization scheme the two
renormalization conditions,
$\displaystyle Z_{f}(\mu^{2})$ $\displaystyle=\,1\ ,$ (35) $\displaystyle
S^{-1}_{f}(\mu^{2})$ $\displaystyle=\,i\gamma\cdot p\ +m_{f}(\mu)\ ,$ (36)
are imposed, where $m_{f}(\mu)$ is the renormalized current-quark mass related
to the bare mass by,
$Z_{4}^{f}(\mu,\Lambda)\,m_{f}(\mu)=Z_{2}^{f}(\mu,\Lambda)\,m_{f}^{\rm
bm}(\Lambda)\ ,$ (37)
and $Z_{4}^{f}(\mu,\Lambda)$ is the renormalization constant that pertains to
the mass term in the QCD Lagrangian.
### III.3 Truncation Scheme
The rainbow-ladder (RL) truncation of the integral equation (33) and of the
BSE kernel has proven to be a robust and successful symmetry-preserving
approximation and allows for the description of light ground-state mesons in
the isospin-nonzero pseudoscalar and vector channels. The RL truncation is
realized by restricting the fully dressed quark gluon vertex to the
perturbative vertex: $\Gamma_{\nu,f}\to Z_{2}^{f}\gamma_{\nu}$. The DSE kernel
then reduces to Serna:2018dwk ,
$Z_{1}^{f}g^{2}D_{\mu\nu}(q)\Gamma_{\nu,f}(k,p)=\big{(}Z^{f}_{2}\big{)}^{\\!2}\mathcal{G}_{f}(q^{2})D_{\mu\nu}^{\mathrm{free}}(q)\frac{\lambda^{a}}{2}\gamma_{\nu}\,,$
(38)
in which an Abelianized Ward identity is enforced that leads to
$Z_{1}^{f}=Z_{2}^{f}$ Bashir:2012fs and implies the omission of the three-
gluon interaction in $\Gamma_{\mu}(k,p)$. An additional factor $Z_{2}^{f}$ in
Eq. (38) ensures multiplicative renormalizability of the DSE and therefore the
mass function $M_{f}(p^{2})$ is a renormalization-point invariant quantity
Bloch:2002eq .
We work in Landau gauge in which the free gluon propagator is transverse,
$D_{\mu\nu}^{\mathrm{free}}(q):=\delta^{ab}\left(\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\\!\frac{1}{q^{2}}\
,$ (39)
and introduce the flavor-dependent interaction,
$\frac{\mathcal{G}_{f}(q^{2})}{q^{2}}=\,\mathcal{G}_{f}^{\mathrm{IR}}(q^{2})+4\pi\tilde{\alpha}_{\mathrm{PT}}(q^{2})\
,$ (40)
where we deliberately absorb a factor $1/q^{2}$ from the gluon propagator
(39). The dressing function $\mathcal{G}_{f}(q^{2})$ consists of a term that
dominates in the infrared domain, $|k|<\Lambda_{\mathrm{QCD}}$, and is
suppressed at large momenta, and a second term that implements the regular
continuation of the perturbative QCD coupling and dominates large momenta. We
use the model of Ref. Qin:2011dd given by,
$\displaystyle\mathcal{G}_{f}^{\mathrm{IR}}(q^{2})$
$\displaystyle=\frac{8\pi^{2}}{\omega^{4}_{f}}D_{f}\,e^{-q^{2}/\omega^{2}_{f}}$
(41) $\displaystyle 4\pi\tilde{\alpha}_{\mathrm{PT}}(q^{2})$
$\displaystyle=\frac{8\pi^{2}\gamma_{m}\,\mathcal{E}(q^{2})}{\ln\left[\tau+\left(1+q^{2}/\Lambda^{2}_{\textrm{\tiny
QCD}}\right)^{\\!2}\right]}\ ,$ (42)
in which $\gamma_{m}=12/(33-2N_{f})$ is the anomalous mass dimension and
$N_{f}$ is the active flavor number, $\Lambda_{\textrm{\tiny QCD}}=0.234$ GeV,
$\tau=e^{2}-1$, $\mathcal{E}(q^{2})=[1-\exp(-q^{2}/4m^{2}_{t})]/q^{2}$ and
$m_{t}=0.5$ GeV.
The flavor dependence of the interaction
$\mathcal{G}_{f}^{\mathrm{IR}}(q^{2})$ was introduced in Refs. Serna:2017nlr ;
Serna:2020txe ; Chen:2019otg ; Serna:2022yfp to accommodate the strong
flavor-symmetry breaking effects that led to complications in the calculation
$D$\- and $B$-meson properties Rojas:2014aka ; Mojica:2017tvh ; see Refs.
Serna:2020txe ; Serna:2022yfp ; Qin:2019oar for the details of the
implementation of the BSE kernels (11) and (12) consistent with Eq. (38).
Suffice to say that herein we employ
$\mathcal{G}_{u}(q^{2})=\mathcal{G}_{d}(q^{2})=\mathcal{G}_{s}(q^{2})\neq\mathcal{G}_{c}(q^{2})\neq\mathcal{G}_{b}(q^{2})$.
Flavor | $m_{f}(19\,\mathrm{GeV})$ | $m_{f}(2\,\mathrm{GeV})$ | $\omega_{f}$ | $\kappa$ | $M^{E}_{f}$
---|---|---|---|---|---
$u,d$ | 0.0034 | 0.018 | 0.500 | 0.80 | 0.408
$s$ | 0.082 | 0.166 | 0.500 | 0.80 | 0.562
$c$ | 0.903 | 1.272 | 0.698 | 0.60 | 1.342
$b$ | 3.741 | 4.370 | 0.640 | 0.56 | 4.259
Table 1: Parameters of the interaction model in Eqs. (41) and (45):
$m_{f}(\mu)$, $\omega_{f}$ and $\kappa=(\omega D_{f})^{1/3}$ (in GeV).
$M^{E}_{f}$ is the Euclidean constituent quark mass:
$M^{E}_{f}=\\{p^{2}|p^{2}=M^{2}(p^{2})\\}$.
The form of the quark-antiquark ladder kernel we therefore employ differs
somewhat from the usual one in case of heavy-flavored mesons,
$K_{fg}(k,q,P)=-\mathcal{Z}_{2}^{2}\,\frac{\mathcal{G}_{fg}(l^{2})}{l^{2}}\,D_{\mu\nu}^{\mathrm{free}}(l)\,\frac{\lambda^{a}}{2}\gamma_{\mu}\frac{\lambda^{a}}{2}\gamma_{\nu}\
,$ (43)
in which the relative momentum is $l=k-q$. In other words, in Eq. (43) we
combine the wave-function renormalization constants of both quarks,
$\mathcal{Z}_{2}(\mu,\Lambda)=\surd{Z_{2}^{f}}\surd{Z_{2}^{g}}$, and use the
averaged interaction,
$\frac{\mathcal{G}_{fg}(l^{2})}{l^{2}}=\mathcal{G}_{fg}^{\mathrm{IR}}(l^{2})+4\pi\tilde{\alpha}_{\mathrm{PT}}(l^{2})\,,$
(44)
which leads to a different treatment of the light and heavy quarks. The
interaction in the low-momentum domain is given by the Gaussian form,
$\mathcal{G}_{fg}^{\mathrm{IR}}(l^{2})=\frac{8\pi^{2}}{(\omega_{f}\omega_{g})^{2}}\sqrt{D_{f}\,D_{g}}\,e^{-l^{2}/(\omega_{f}\omega_{g})}\,,$
(45)
while $4\pi\tilde{\alpha}_{\mathrm{PT}}(q^{2})$ is as in Eq. (42). The
parameters of this interaction model are listed in Table 1 Serna:2020txe .
| $m_{M}$ | $m^{\mathrm{exp}}_{M}$ | $\epsilon_{m_{M}}$ [%] | $f_{M}$ | $f^{\mathrm{exp/LQCD}}_{M}$ | $\epsilon_{f_{M}}$ [%]
---|---|---|---|---|---|---
$\pi(u\bar{d})$ | 0.140 | 0.138 | 1.45 | $0.094$ | 0.092(1) | 2.17
$K(u\bar{s})$ | 0.494 | 0.494 | 0.0 | $0.110$ | 0.110(2) | 0.0
$D(c\bar{d})$ | $1.867$ | 1.864 | 0.11 | $0.144$ | 0.150 (0.5) | 4.00
$\rho(u\bar{u})$ | 0.730 | 0.775 | 5.81 | 0.145 | 0.153(1) | 5.23
$\phi(s\bar{s})$ | 1.070 | 1.019 | 5.20 | 0.187 | 0.168(1) | 11.31
$K^{*}(u\bar{s})$ | 0.883 | 0.896 | 1.45 | 0.163 | 0.159(1) | 2.55
$D^{*}(c\bar{u})$ | 2.021 | 2.009 | 0.60 | 0.165 | 0.158(6) | 4.43
Table 2: Masses and weak decay constants [in GeV] of ground-state pseudoscalar
and vector mesons, $M=P,V$. The experimental mass values are taken from the
Particle Data Group [PDG] ParticleDataGroup:2022pth and the leptonic decay
constants for the $\rho$, $K^{*}$ and $\phi$ mesons are derived from their
experimental decay width via
$f_{V}^{2}=\frac{3m_{V}}{4\pi\alpha^{2}Q^{2}}\,\Gamma_{V\rightarrow
e^{+}e^{-}}$. The decay constant of the $D^{*}$ meson is a lattice-QCD
prediction by the ETM collaboration Lubicz:2017asp . The relative deviations
from experimental values are given by
$\epsilon_{v}=|v^{\textrm{exp.}}-v^{\textrm{th.}}|/v^{\textrm{exp.}}$.
The only missing ingredient now is the quark propagator for complex momenta,
$S_{f}(q_{\eta})=-i\gamma\cdot q_{\eta}\,\sigma_{\rm
v}^{f}(q_{\eta}^{2})+\sigma_{\rm s}^{f}(q_{\eta}^{2})\,,$ (46)
and likewise for $S_{f}(q_{\bar{\eta}})$, as in Euclidean space the arguments
$q_{\eta}^{2}$ and $q_{\bar{\eta}}^{2}$ define parabolas on the complex plane,
$\displaystyle q^{2}_{\eta}$ $\displaystyle=$ $\displaystyle
q^{2}-\eta^{2}m^{2}_{M}+2i\eta\,m_{M}|q|z_{q}\ ,$ $\displaystyle
q^{2}_{\bar{\eta}}$ $\displaystyle=$ $\displaystyle
q^{2}-{\bar{\eta}}^{2}m^{2}_{M}-2i\bar{\eta}\,m_{M}|q|z_{q}\,,$ (47)
where $z_{q}=q\cdot P/|q||P|$. We apply Cauchy’s integral theorem as
described, e.g., in Ref. Krassnigg:2008bob and obtain the solutions of the
DSE on the complex plane with the contour parametrization of the parabola
defined in Ref. Rojas:2014aka ; see Refs. El-Bennich:2016qmb ; Serna:2020txe
for graphic visualizations of $\sigma_{\rm s}^{u}(q_{\eta}^{2})$.
## IV Pseudoscalar and Vector Meson Properties
Using the quark propagators on the complex momentum plane (47) and the BSE
kernel (43) in Eqs. (11) and (12), we treat the BSE as an eigenvalue problem
El-Bennich:2015kja ; El-Bennich:2017brb . For instance, in case of the vector
mesons, the covariant decomposition in Eq. (15) along with the orthogonality
of the basis in Eqs. (20) to (27) allow to recast the homogeneous BSE (12)
with the kernel (43) in a set of eight coupled-integral equations,
$\displaystyle\mathcal{F}_{i}^{fg}$
$\displaystyle\big{(}k,P,z_{k}\big{)}=-\tfrac{4}{3}\,\mathcal{Z}_{2}^{2}\int^{\Lambda}_{q}\\!\mathcal{G}_{fg}\big{(}l^{2}\big{)}D^{\mathrm{free}}_{\mu\nu}(l)\mathcal{F}_{j}^{fg}(q,P,z_{q})$
$\displaystyle\times\,\mathrm{Tr}_{\mathrm{D}}\left[T^{i}_{\rho}(k,P)\gamma_{\mu}S_{f}(q_{\eta})T^{j}_{\rho}(q,P)S_{g}(q_{\bar{\eta}})\gamma_{\nu}\right],$
(48)
where the mnemonic shortcut for the integral represents the same integral with
Poincaré-invariant cut-off as before. In solving this equation system
numerically, we expand the scalar amplitudes in terms of Chebyshev moments,
$\mathcal{F}_{im}^{fg}(k,P)$,
$\mathcal{F}_{i}^{fg}(k,P,z_{k})=\sum_{m=0}^{\infty}\mathcal{F}_{im}^{fg}(k,P)\,U_{m}(z_{k})\
,$ (49)
which allows for a faster convergence. We consider $m=3$ Chebyshev
polynomials, $U_{m}(z_{p})$, of second kind. The eigenvalue problem for the
vector $\bm{\mathcal{F}}:=\\{\mathcal{F}_{i},\ldots,\mathcal{F}_{8}\\}$ is
then solved by means of Arnoldi factorization in the ARPACK library
Lehoucq1998 . Details of the practical implementation of ARPACK in a numerical
treatment of the BSE are reviewed, for instance, in Refs. Blank:2010bp ;
Rojas:2014aka .
Figure 2: Real and imaginary parts of the scalar amplitudes
$\mathcal{F}_{1}(k,P)$ and $\mathcal{F}_{2}(k,P)$ of the $D^{*}$ meson.
Figure 3: Real and imaginary parts of the leading BSA of the $D$ meson
analytically continued on the complex momentum plane with the representation
in Eq. (52). Note that the amplitude is plotted on the parabola spanned by
$k_{P}^{2}$ (7) whose vertex lies in the time-like region. On the real axis
the BSA normalization is $E_{D}(k_{p}=0,p_{2})=1$.
The masses, $m_{P}$ and $m_{V}$, of the ground state pseudoscalar and vector
mesons are the solutions of the eigenvalue trajectory
$\lambda(P^{2}=-m^{2}_{P,V})=1$. They are listed along with the leptonic decay
constants in Table 2 and are in very good agreement with experimental data,
when available, or lattice-QCD results otherwise. As a byproduct we obtain the
BSA of the mesons, which we illustrate with the real and imaginary parts of
the dominant scalar functions of the $D^{*}$ meson in Fig. 2.
## V Results
As we work in Euclidean space, the relative momenta, $k_{P}$ and
$k_{P}^{\prime}$ in the decay, Eqs. (7) and (8), of the final pseudoscalar
mesons and of the $D^{*}$ (6) are complex. This is because in the center of
mass of the initial vector meson its four-momentum is
$p_{1}=(\mathbf{0},im_{V})$ and thus $k_{V}=k+(w_{1}-w_{2})p_{1}/2$ is only
real if $w_{1}=w_{2}$. In case of the final-state mesons, the relative
momentum is inevitably complex. In principle, due to the Poincaré invariance
of the BSAs, our calculations are independent of the choice for the partition
parameters. Practically, though, we are limited by numerical constraints as
choosing $w_{2}=0.5$ in the case of $m_{V}=m_{D^{*}}$ implies probing the
light-quark propagator at large time-like momenta, much larger than the light
quark’s mass. In this region, the solutions of the quark propagator on the
complex momentum plane are characterized by branch cuts and/or complex-
conjugate poles El-Bennich:2016qmb ; Eichmann:2021vnj and a contour
deformation is not trivial.
Having in hand the numerical BSA for real momenta, we therefore parametrize it
with a Nakanishi type of representation which allows for an analytical
continuation of the $\mathcal{F}_{i}^{fg}(k,P)$ (49) in the complex plane. In
order to do so, we split the BSA in even and odd components,
$\mathcal{F}_{i}(k,P)=\mathcal{F}^{0}_{i}(k,P)+k\cdot
P\,\mathcal{F}^{1}_{i}(k,P)\,,$ (50)
in which $\mathcal{F}^{0,1}_{i}(k,P)$ are even under $k\cdot
P\rightarrow-k\cdot P$ and where we henceforth suppress the flavor indices
$fg$. As discussed, for instance, in Ref. Serna:2020txe ,
$\mathcal{F}^{1}_{i}(k,P)\equiv 0$ for flavorless pseudoscalar mesons, such as
the neutral pion, as they are eigenstates of the charge-conjugation operator
defined as,
$\Gamma_{M}(k,P)\,\stackrel{{\scriptstyle C}}{{\longrightarrow}}\
\bar{\Gamma}_{M}(k,P):=C\,\Gamma^{T}_{M}(-k,P)C^{T}\ .$ (51)
The constraint that the covariant basis (15) satisfies
$\bar{\Gamma}_{M}(k,P)=\lambda_{c}\Gamma_{M}(k,P)$ with $\lambda_{c}=+1$ for
pseudoscalar mesons and $\lambda_{c}=-1$ for vector mesons, respectively,
therefore imposes a definite parity of the scalar amplitudes
$\mathcal{F}_{i}(k,P)$ under $k\cdot P\rightarrow-k\cdot P$. For the neutral
vector mesons, $\rho$ and $\phi$, this implies that the $\mathcal{F}_{i}(k,P)$
are necessarily even and $\mathcal{F}^{1}_{i}(k,P)\equiv 0$ again. In case of
the $K$, $K^{*}$, $D$ and $D^{*}$ mesons, which are not eigenstates of $C$,
both amplitudes in Eq. (50) must be considered.
$g_{V\\!PP}$ | $(V,P)=(8,4)$ | $(V,P)=(5,4)$ | $(V,P)=(1,1)$ | Reference | $\epsilon_{g_{V\\!P\\!P}}\ [\%]$
---|---|---|---|---|---
$g_{\rho\pi\pi}$ | $5.13^{+0.24}_{-0.25}$ | 5.14 | 7.99 | $5.94\pm 0.44$ | 13.6
$g_{\phi KK}$ | $5.02^{+0.12}_{-0.22}$ | 5.03 | 10.12 | $5.53\pm 0.31$ | 9.2
$g_{K^{*}\\!K\pi}$ | $5.10^{+0.22}_{-0.26}$ | 5.25 | 9.08 | $5.47\pm 0.99$ | 6.8
$g_{D^{*}\\!D\pi}$ | $17.24^{+3.06}_{-2.30}$ | 16.41 | 37.22 | $17.9\pm 0.3\pm 1.90$ | 3.7
Table 3: Strong couplings $g_{V\\!PP}$ for the decay channels $\rho\to\pi\pi$,
$\phi\to\bar{K}K$, $K^{*}\to K\pi$ and $D^{*}\to D\pi$. The pair $(V,P)$
denotes the number of scalar amplitudes employed in the BSA of the vector and
pseudoscalar mesons, respectively. The theoretical errors stem from the fit to
Nakanishi representations of the BSA. The reference couplings are derived from
the experimental decay widths ParticleDataGroup:2022pth via
$\Gamma=g_{V\\!P_{1}\\!P_{2}}^{2}k^{3}/6\pi m_{V}^{2}$ with
$k^{2}=[m_{V}^{2}-(m_{P_{1}}+m_{P_{2}})^{2}][m_{V}^{2}-(m_{P_{1}}-m_{P_{2}})^{2}]/4m_{V}^{2}$.
The experimental $D^{*}\\!D\pi$ coupling was obtained by the CLEO
collaboration CLEO:2001sxb . The relative deviations from experimental values
are defined as
$\epsilon_{g_{V\\!P\\!P}}=|g_{V\\!PP}^{\textrm{exp.}}-g_{V\\!PP}^{\textrm{th.}}|/g_{V\\!PP}^{\textrm{exp.}}$.
When the relative momentum of the meson is complex, we choose the following
analytic representation of the scalar amplitudes for $l=0,1$,
$\mathcal{F}_{i}^{l}(k,P)=\sum_{j=1}^{N}\int_{-1}^{1}\\!d\alpha\,\rho_{j}(\alpha)\,\frac{U_{j}\Lambda^{2n_{j}}}{\Delta^{n_{j}}(k,\alpha,\Lambda)}\,,$
(52)
where $\Delta=k^{2}+\alpha k\cdot P+\Lambda^{2}$ and the spectral density
$\rho_{j}(\alpha)$ is given by,
$\rho_{j}(\alpha)=\tfrac{1}{2}\\!\left(C^{\nicefrac{{1}}{{2}}}_{0}(\alpha)+\sigma_{j}C^{\nicefrac{{1}}{{2}}}_{2}(\alpha)\right)\,.$
(53)
$C^{\nicefrac{{1}}{{2}}}_{0}(\alpha)$ and
$C^{\nicefrac{{1}}{{2}}}_{2}(\alpha)$ are Gegenbauer polynomials of order
$1/2$. The parameters $U_{j}$, $\Lambda$, $n_{j}$ and $\sigma_{j}$ are listed
in Tables 4, 5 and 7 of Appendix A for the pion, kaon, $D$ and $D^{*}$ mesons.
We use _real_ numerical BSA solutions for the remaining mesons considered in
this work, namely the $\rho$, $\phi$ and $K^{*}$. This approach, frequently
employed in calculations of distribution amplitudes Shi:2014uwa ;
Serna:2020txe , differs from the method in Ref. Jarecke:2002xd based on a 2nd
order Taylor expansion of $\mathcal{F}_{i}(k,P)$ about the closest point on
the positive real axis to the complex-valued relative momenta $k$. We checked
that the BSA parameterization of Eq. (52) produces the correct weak decay
constants of the pseudoscalar and vector mesons. An illustration of the
analytic continuation of the $D$-meson’s dominante amplitude with the
parametrization of Eq. (52) is given in Fig. 3.
With these technical considerations taken into account, we can take the trace
and calculate the loop integral in Eq. (2) for different initial and final
states in the strong two-body decay of a vector meson. The couplings,
$g_{V\\!PP}$, we obtain are listed in Table 3, where the errors are due to the
fit uncertainties of the Nakanishi representations. We remind that this error
would increase if we included a systematic error of the ladder truncation, as
explored in Ref. Serna:2020txe for example. More precisely, it is known that
some typical observables, such as the pion and kaon masses and weak decay
constants, are insensitive to a range $\omega_{i}\pm\Delta\omega_{i}$,
$i=u,d,s$, of the interaction parameter in Eq. (41). Varying $\omega_{i}$
alters the BSA of the mesons and this can add to the uncertainty in the strong
$V\to PP$ decay amplitude. However, modifying $\omega_{u,d}$ in the $D$ and
$D^{*}$ mesons is a numerically delicate matter, as no solution of the BSA is
found for the uncertainties $\pm\Delta\omega_{u,d}$. We therefore abstain from
including this source of error.
Figure 4: Comparison of theoretical values for $g_{D^{*}\\!D\pi}$ with the
experimental coupling extracted from the $D^{*}$ decay width by the CLEO
collaboration CLEO:2001sxb (shaded band). The couplings are taken from Ref.
El-Bennich:2010uqs (DSE-BSE), Refs. Becirevic:2009xp and Abada:2002xe
(LQCD), Ref. Bracco:2011pg (QCDSR) and Ref. Melikhov:2001zv (DQM).
We consider the cases of limiting the BSA to the dominant covariant in Eq.
(15), namely $\gamma_{5}$ and $\gamma_{\mu}$ for the pseudoscalar and vector
mesons, respectively. Using merely the dominant amplitude, $g_{\rho\pi\pi}$ is
56% larger than the value obtained with the complete BSA. We find a difference
of 102% for the $g_{\phi KK}$ coupling which increases to 116% for
$g_{D^{*}\\!D\pi}$. Clearly, this leading approximation is not adequate even
for couplings that involve only light quarks. On the other hand, including the
next four leading covariants of the vector meson’s BSA, the dominant physics
in the impulse approximation is captured and the couplings are within 1–5% of
the values obtained with $(V,P)=(8,4)$.
Our values for the strong couplings mostly agree with reference values, except
for the $\rho\to\pi\pi$ coupling which is found to be the 13.6% smaller than
the experimental coupling. As mentioned in Section II, this is expected given
our limitation to the impulse and ladder approximation which omits
intermediate $\pi\pi$, $\bar{K}K$ and $K\pi$ channels. Including explicit two-
pion exchange in the BSE kernel, the decay width of the $\rho$-meson can be
determined from the imaginary part of the resonance pole from which one
deduces a coupling constant $g_{\rho\pi\pi}=5.7$ Williams:2018adr . The widths
of the $\phi$, $K^{*}$ and $D^{*}$, on the other hand, are much smaller than
their masses and our approximation is more accurate, though we still notice a
deviation of 9.2% for $g_{\phi KK}$.
As our focus is on the $D^{*}\to D\pi$ decay, we also compare our result with
couplings obtained with lattice QCD (LQCD), QCD sum rules (QCDSR) and a
dispersion relation quark model (DQM) in Fig. 4. Our calculation is a
significant improvement on earlier work El-Bennich:2010uqs ; El-
Bennich:2011tme ; El-Bennich:2012hom ; El-Bennich:2016bno which also
considered the impulse approximation but employed model wave functions for the
mesons, based on the dominant covariant term of the BSA, and a simplified,
constant-mass propagator for the charm quark. This, as we noted in Table 3,
has detrimental effects on translational invariance and the couplings depend
on a _suitable_ choice of the partition parameters $w_{1}$ and $w_{2}$, see
the discussion in Ref. El-Bennich:2010uqs . Since our calculation is fully
Poincaré covariant, our decay amplitudes are independent of the momentum
distribution, as we verified with variations of $w_{1}$ and $w_{2}$ up to a
critical limit where we encounter singularities in the quark propagators on
the complex plane.
## VI Conclusion
We revisited the strong decays of vector mesons into two pseudoscalar mesons
within the framework of the DSE and BSE, having in mind the particular decay
$D^{*}\to D\pi$. As we argued, these decays are the simplest hadronic
observables beyond the meson’s masses, weak decay constants and
electromagnetic form factors, and thus provide additional information about
the dynamics of QCD in the nonperturbative regime. In particular, the vector-
meson decay to a pair of pseudoscalars proceeds via a $P$-wave interaction and
therefore involves the BSA differently than the meson’s weak decay constants.
The strong $D^{*}$ decay is then even more interesting, as it probes
nonperturbative QCD simultaneously at two distinct scales, namely the light-
and charm-quark masses.
We limited ourselves to the impulse approximation for the aforementioned
reason: our BSE kernel in ladder truncation is too simple to include $\pi\pi$,
$\bar{K}K$ and $K\pi$ channels in these decays and this is most likely the
largest source of error in our calculation of the $\rho\to\pi\pi$ coupling.
Nonetheless, this calculation represents an important theoretical and
numerical improvement over the simpler approaches in Ref El-Bennich:2010uqs ;
El-Bennich:2011tme ; El-Bennich:2012hom ; El-Bennich:2016bno , as the full
Poncaré invariant BSA structure of all mesons is included and the quark
propagators are calculated on the complex momentum plane for all flavors. This
present calculation can also serve as a guidance to reevaluate off-shell
space-like couplings between the $\rho$ and $D$\- and $D^{*}$ mesons in Ref.
El-Bennich:2016bno without resorting to model wave functions.
Our final value for $g_{D^{*}\\!D\pi}$ is 3.7% lower than that extracted from
the experimental decay width and well within the experimental errors. It
corresponds to a universal coupling in a chiral heavy meson Lagrangian which
at leading order in the heavy-mass expansion is given by:
$\hat{g}=\frac{g_{D^{*}\\!D\pi}}{2\sqrt{m_{D}^{*}m_{D}}}\,f_{\pi}=0.58^{+0.10}_{-0.08}\
.$ (54)
A consensus seems to be growing that the most recent theoretical couplings are
in good agreement with the CLEO value CLEO:2001sxb extracted from the $D^{*}$
decay width. Future improvements ought to consider strong $\pi\pi$
interactions, likely along the lines presented in Refs. Williams:2018adr ;
Miramontes:2019mco ; Miramontes:2021xgn , in the BSE kernels and to go beyond
the impulse approximation.
###### Acknowledgements.
B.E., F.E.S. and R.C.S. participate in the Brazilian network project _INCT-
Física Nuclear e Aplicações_ , no. 464898/2014-5. This work was supported by
the São Paulo Research Foundation (FAPESP), grant no. 2018/20218-4, and by the
National Council for Scientific and Technological Development (CNPq), grant
no. 428003/2018-4. F.E.S. is a CAPES-PNPD postdoctoral fellow financed by
grant no. 88882.314890/2013-01. We appreciated helpful communication with
Peter Tandy.
## Appendix A Parameters of Bethe-Salpeter Amplitude Representation
$\mathcal{F}_{i}^{l}$ | $\Lambda$ | $U_{1}$ | $U_{2}$ | $U_{3}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $n_{1}$ | $n_{2}$ | $n_{3}$
---|---|---|---|---|---|---|---|---|---|---
$E_{\pi}^{0}$ | 1.280 | 2.558 | $-1.559$ | 0.0 | 1.810 | 1.548 | 0.0 | 4 | 5 | 0
$F_{\pi}^{0}$ | 1.150 | 1.838 | $-0.948$ | $-0.381$ | $-2.679$ | $-2.547$ | $-5.107$ | 4 | 5 | 3
$G_{\pi}^{0}$ | 1.106 | 2.402 | $-1.950$ | 0.0 | $-0.4590$ | $-0.474$ | 0.0 | 6 | 7 | 0
$H_{\pi}^{0}$ | 1.056 | 1.253 | $-0.857$ | $-0.140$ | $-0.696$ | $-0.634$ | $-2.663$ | 5 | 6 | 3
Table 4: Parameters of the BSA representation in Eq. (52) for the pion. In the isospin limit, $m_{u}=m_{d}$, only $\mathcal{F}_{i}^{0}$ contributes which we fit to the sum of the 0th and 2nd Chebyshev moments. $\mathcal{F}_{i}^{l}$ | $\Lambda$ | $U_{1}$ | $U_{2}$ | $U_{3}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $n_{1}$ | $n_{2}$ | $n_{3}$
---|---|---|---|---|---|---|---|---|---|---
$E^{0}$ | 1.557 | 2.590 | $-1.590$ | 0.0 | 1.342 | 0.891 | 0.0 | 5 | 6 | 0
$E^{1}$ | 1.495 | $-0.810$ | 3.769 | $-2.251$ | $-1.039$ | $-0.680$ | $-0.680$ | 5 | 6 | 7
$F^{0}$ | 1.514 | 2.756 | $-3.558$ | 1.220 | $-0.527$ | $-0.173$ | 0.323 | 7 | 8 | 9
$F^{1}$ | 1.604 | 3.150 | $-5.480$ | 2.537 | $-1.074$ | $-0.881$ | $-0.718$ | 10 | 11 | 12
$G^{0}$ | 1.631 | $-0.613$ | 1.139 | $-0.522$ | 0.595 | 2.134 | 2.651 | 8 | 10 | 12
$G^{1}$ | 1.229 | 2.949 | $-4.542$ | 1.880 | $-0.558$ | $-0.747$ | $-0.920$ | 8 | 10 | 12
$H^{0}$ | 1.727 | $-0.276$ | 1.031 | $-0.594$ | 1.722 | 0.661 | 0.303 | 8 | 10 | 12
$H^{1}$ | 1.443 | $-1.564$ | 4.434 | $-2.663$ | $-0.8104$ | $-0.905$ | $-0.938$ | 8 | 9 | 10
Table 5: Parameters of the BSA representation in Eq. (52) for the kaon. Both, the $\mathcal{F}_{i}^{0}$ and $\mathcal{F}_{i}^{1}$ amplitudes contribute, as the kaon is not an eigenstate of charge conjugation. In the fit we include the sum of the 1st and 3rd Chebyshev moments in the odd component of the BSA. $\mathcal{F}_{i}^{l}$ | $\Lambda$ | $U_{1}$ | $U_{2}$ | $U_{3}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $n_{1}$ | $n_{2}$ | $n_{3}$
---|---|---|---|---|---|---|---|---|---|---
$E^{0}$ | 1.750 | 2.078 | $-1.077$ | 0.0 | $-1.274$ | $-1.126$ | 0.0 | 5 | 6 | 0
$E^{1}$ | 2.146 | $-0.207$ | 0.209 | 0.0 | $-1.115$ | $-1.115$ | 0.0 | 6 | 9 | 0
$F^{0}$ | 2.222 | 0.060 | 0.155 | 0.0 | $-0.934$ | $-2.211$ | 0.0 | 6 | 9 | 0
$F^{1}$ | 2.583 | 0.003 | $-0.043$ | 0.0 | $-1.372$ | $-1.757$ | 0.0 | 6 | 9 | 0
$G^{0}$ | 1.543 | $-1.596$ | 2.008 | $-0.427$ | $-1.439$ | $-1.596$ | $-1.795$ | 6 | 7 | 10
$G^{1}$ | 1.423 | 0.197 | 0.201 | $-0.233$ | $-1.289$ | $-2.657$ | $-2.097$ | 5 | 6 | 9
$H^{0}$ | 1.711 | $-0.249$ | 0.868 | $-0.530$ | $-1.535$ | $-1.618$ | $-1.621$ | 8 | 9 | 10
$H^{1}$ | 1.155 | $-0.481$ | 0.937 | $-0.470$ | $-1.917$ | $-2.011$ | $-2.086$ | 6 | 7 | 8
Table 6: Parameters of the BSA representation in Eq. (52) for the $D$ meson. Even and odd components of the BSA are in terms of Chebyshev moments as described in Tables 4 and 5. $\mathcal{F}_{i}^{l}$ | $\Lambda$ | $U_{1}$ | $U_{2}$ | $U_{3}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $n_{1}$ | $n_{2}$ | $n_{3}$
---|---|---|---|---|---|---|---|---|---|---
$\mathcal{F}_{1}^{0}$ | 1.942 | 3.510 | $-2.509$ | 0.0 | $-1.666$ | $-1.851$ | 0.0 | 7 | 8 | 0
$\mathcal{F}_{1}^{1}$ | 0.998 | 2.441 | $-1.533$ | $-0.680$ | $-1.856$ | $-2.027$ | $-1.167$ | 4 | 5 | 3
$\mathcal{F}_{2}^{0}$ | 1.107 | 0.440 | -0.216 | 0.0 | $-1.656$ | $-1.312$ | 0.0 | 4 | 6 | 0
$\mathcal{F}_{2}^{1}$ | 0.902 | 2.493 | $-3.754$ | 1.278 | $-2.500$ | $-2.522$ | $-2.550$ | 7 | 8 | 10
$\mathcal{F}_{3}^{0}$ | 1.445 | $-1.536$ | 1.372 | 0.0 | $-1.838$ | $-1.831$ | 0.0 | 7 | 8 | 0
$\mathcal{F}_{3}^{1}$ | 1.005 | 0.099 | $-0.032$ | 0.0 | $-1.030$ | $-4.365$ | 0.0 | 3 | 9 | 0
$\mathcal{F}_{4}^{0}$ | 1.411 | $-1.235$ | 0.837 | 0.0 | $-1.844$ | $-1.833$ | 0.0 | 5 | 6 | 0
$\mathcal{F}_{4}^{1}$ | 1.554 | 1.192 | $-2.367$ | 1.181 | $-1.949$ | $-2.040$ | $-2.10$ | 7 | 8 | 9
$\mathcal{F}_{5}^{0}$ | 1.895 | $-4.279$ | 2.958 | 0.0 | $-1.775$ | $-1.787$ | 0.0 | 7 | 8 | 0
$\mathcal{F}_{5}^{1}$ | 1.730 | 2.657 | $-4.797$ | 2.212 | $-1.831$ | $-1.932$ | $-2.00$ | 7 | 8 | 9
$\mathcal{F}_{6}^{0}$ | 1.384 | 0.815 | $-0.639$ | 0.0 | $-1.916$ | $-1.942$ | 0.0 | 6 | 8 | 0
$\mathcal{F}_{6}^{1}$ | 1.250 | $-3.728$ | 2.097 | 0.0 | $-2.229$ | $-3.453$ | 0.0 | 7 | 12 | 0
$\mathcal{F}_{7}^{0}$ | 1.183 | $-0.557$ | 0.971 | $-0.412$ | $-0.711$ | $-1.517$ | $-1.927$ | 4 | 5 | 6
$\mathcal{F}_{7}^{1}$ | 1.091 | $-0.190$ | 0.076 | 0.080 | $-1.453$ | $-2.005$ | $-0.520$ | 4 | 8 | 3
$\mathcal{F}_{8}^{0}$ | 1.316 | 0.601 | $-0.746$ | 0.147 | $-1.917$ | $-2.011$ | $-2.086$ | 4 | 5 | 8
$\mathcal{F}_{8}^{1}$ | 0.909 | 0.073 | $-0.647$ | 0.501 | $-2.127$ | $-2.312$ | $-2.331$ | 4 | 5 | 6
Table 7: Parameters of the BSA representation in Eq. (52) for the $D^{*}$
meson. Even and odd components of the BSA are in terms of Chebyshev moments as
described in Tables 4 and 5.
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|
# Machine learning potentials from transfer learning of periodic correlated
electronic structure methods: Application to liquid water with AFQMC, CCSD,
and CCSD(T)
Michael S. Chen Department of Chemistry, Stanford University, Stanford,
California, 94305, USA Joonho Lee Department of Chemistry, Columbia
University, New York, New York 10027, USA Hong-Zhou Ye Department of
Chemistry, Columbia University, New York, New York 10027, USA Timothy C.
Berkelbach<EMAIL_ADDRESS>Department of Chemistry, Columbia
University, New York, New York 10027, USA Center for Computational Quantum
Physics, Flatiron Institute, New York, New York 10010, USA David R. Reichman
<EMAIL_ADDRESS>Department of Chemistry, Columbia University, New York,
New York 10027, USA Thomas E. Markland<EMAIL_ADDRESS>Department of
Chemistry, Stanford University, Stanford, California, 94305, USA
###### Abstract
Obtaining the atomistic structure and dynamics of disordered condensed phase
systems from first principles remains one of the forefront challenges of
chemical theory. Here we exploit recent advances in periodic electronic
structure to show that, by leveraging transfer learning starting from lower
tier electronic structure methods, one can obtain machine learned potential
energy surfaces for liquid water from the higher tier AFQMC, CCSD, and CCSD(T)
approaches using $\leq$200 energies. By performing both classical and path
integral molecular dynamics simulations on these machine learned potential
energy surfaces we uncover the interplay of dynamical electron correlation and
nuclear quantum effects across the entire liquid range of water while
providing a general strategy for efficiently utilizing periodic correlated
electronic structure methods to explore disordered condensed phase systems.
## I Introduction
Ab initio molecular dynamics (AIMD) simulations, where the forces and energies
are generated at each time-step by performing an electronic structure
calculation, provide an appealing route to simulate reactive chemical
dynamics. However, for disordered condensed phase systems an accurate
description typically requires many 100’s of atoms to obtain a chemically
reasonable description of bulk systems (e.g., water) and this grows into the
1000’s for more heterogeneous systems (e.g., those with interfaces). Since
AIMD simulations require an electronic structure calculation to be performed
at each time-step, to statistically converge even simple thermodynamic
properties necessitates many tens of thousands of ab initio calculations (10’s
of picosecond timescale at a $\sim$1 fs time step) and for slower converging
properties many millions are needed (nanosecond timescale). The computational
expense of these simulations is further compounded if one wants to incorporate
nuclear quantum effects (NQE) via ab initio path integral molecular dynamics
simulations (PIMD). Due to its reasonable compromise between accuracy and
efficiency, density functional theory (DFT) is currently the most frequently
employed electronic structure method in condensed phase AIMD studies. However,
the results depend—sometimes sensitively—on the choice of the exchange-
correlation functional and the inclusion of dispersion corrections[1, 2]. This
issue motivates the use of beyond-DFT electronic structure theories, such as
those based on the many-electron wavefunction. For example, work performed
almost a decade ago demonstrated the AIMD simulations of liquid water using
second-order Möller-Plesset perturbation theory (MP2) [3]. However, the high
cost of more accurate methods precludes their direct use in AIMD.
Machine learned potentials (MLPs) have emerged as an extremely promising
approach to accurately model ab initio potential energy surfaces of condensed
phase systems while being orders of magnitude more computationally efficient
to evaluate. For liquid water, MLPs have been successfully developed at
various levels of electronic structure ranging from different levels of DFT[4,
5, 6, 7, 8, 9, 10] to more recently using the random phase approximation
(RPA)[11] and MP2[12, 13]. The modeling of liquid water and other molecular
systems with more accurate electronic structure methods, such as coupled-
cluster theory or quantum Monte Carlo, has been so far limited to training on
finite clusters of molecules[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
Recent advances have opened the door to efficiently calculating the properties
of periodic condensed phase systems using higher-level methods like coupled
cluster singles and doubles without and with perturbative triples (CCSD and
CCSD(T))[26, 27] and phaseless auxiliary-field quantum Monte Carlo
(AFQMC)[28]. However, while these advances allow the energies of many 100’s of
condensed phase configurations to be evaluated, this is still considerably
less than what would be typically required to train an accurate MLP.
Here, we demonstrate that by using an approach based on transfer learning
starting from a variety of lower tier electronic structure methods one can
generate a highly data-efficient approach to training MLPs using high level
electronic structure methods. With this approach, we show that only 200 high-
quality energies obtained from small periodic boxes containing 16 water
molecules provide sufficient data for training our MLPs. Specifically, we
train MLPs to periodic electronic structure calculations performed with AFQMC,
CCSD, and CCSD(T). The MLPs are then used to perform AIMD and PIMD simulations
of larger water boxes for the long times necessary to statistically converge
static and dynamic properties using both classical and quantum mechanical
treatment of nuclei. This allows us to achieve a careful comparison of the
quality of the underlying electronic structure theories in describing water
across its entire liquid temperature range and uncover the changes in the
properties of liquid water as the dynamical electron correlation captured is
increased. In addition, we provide a set of MLPs and a curated training set
that can form the basis of future studies of water and aqueous systems. Our
data-efficient approach provides a route to accurately obtain the properties
of other disordered condensed-phase systems by combining high-level electronic
structure theory and machine learning.
## II Methods
### II.1 Machine learning
To develop a highly data efficient strategy that only requires a minimal
number of energies from periodic high level electronic structure methods, such
as CCSD, CCSD(T), and AFQMC, and produce accurate MLPs, we exploited a
combination of approaches including an active learning procedure for curating
a small but comprehensive training set, the training of MLPs to energies for
small periodic boxes (16 waters) and showing that they reproduce properties
when used for larger simulations (64 waters) and a transfer learning approach
that leverages the transferability of physics from lower tier electronic
structure methods.
#### II.1.1 Developing an efficient training set by leveraging a committee of
machine learning potentials
We first applied an active learning procedure to curate a data efficient set
of configurations to train our MLPs. We invoked the commonly employed Query-
by-Committee (QbC)[29, 30, 8] approach to iteratively add configurations to
the training set. At every iteration, the current dataset of configuration
energies was used to train a committee of 8 MLPs, each a Behler-Parrinello
neural network potential[31, 4] with a different random initialization of
weights and random train-validation (90-10) splits of the dataset. To mitigate
overfitting, we applied early stopping to each of these MLPs in the committee,
monitoring the energy prediction error over a validation set. The mean
potential energy surface obtained for this committee MLP was used to run a
short MD simulation (SI Sec. LABEL:sec:si-md-details) that was terminated
either when the system becomes unphysical (i.e., reaching a temperature
greater than 400K) or when 50 ps of simulation trajectory was generated. We
then selected the 10 configurations where the committee MLP had the largest
standard deviation in its potential energy prediction to recalculate at the
target level of electronic structure theory and add to the training set for
the next iteration of this procedure. To prevent selected configurations from
being overly correlated with one another, no two selected configurations were
closer than 100 fs apart. The initial dataset of 50 configurations used to
initialize this procedure was also selected via an iterative QbC procedure, as
applied to a single 100 ps SCC-DFTB[32] (SI Sec. LABEL:sec:si-md-details)
trajectory, that started with 10 randomly selected configurations and added 10
additional configurations for each of 4 subsequent iterations.
We generated 5 different 200 configuration datasets by running 5 instances of
our QbC active learning scheme where the target level of electronic structure
theory was DFT using the revPBE0-D3[33, 34, 35] functional (SI Sec.
LABEL:sec:si-dft-hf-dftb) which was chosen due to being computationally
efficient compared to the high level electronic structure methods and since it
has previously been shown to produce the properties of liquid water accurately
when combined with path integral simulations[36]. To select the generated
dataset on which the higher tier electronic structure methods would be used we
then used each of the 5 datasets to train a committee MLP using revPBE0-D3. We
then selected the one that gave the lowest force prediction error (RMSE) when
evaluated on a test set of 1000 water configurations (64 water molecules)
drawn from previously published AIMD simulations[36]. To check the
transferability of the selected 200 configuration dataset, we recalculated the
energies and forces for the generated datasets and the test set using the
BLYP[37, 38] functional (SI Sec. LABEL:sec:si-dft-hf-dftb). After training and
evaluating a new set of BLYP trained MLPs, the same dataset that resulted in
the lowest error revPBE0-D3 trained MLP also gave the lowest error BLYP
trained MLP (SI Fig. LABEL:fig:trainingset_force_lc). Given the
transferability of the relative utility of this dataset, we used this same 200
configuration dataset to train our CCSD, CCSD(T), and AFQMC MLPs.
#### II.1.2 Training machine learned potentials on configurations of small
systems
Figure 1: Oxygen-oxygen RDFs for liquid water when running classical and PIMD
simulations using NNPs fitted to either CCSD, CCSD(T), and AFQMC energies at
both 300K and 370K. The PIMD oxygen-oxygen RDFs when using the CCSD(T) and
AFQMC correspond closely with the experimental results shaded in grey at both
295K and 366K.
Given the significant scaling of CCSD, CCSD(T), and AFQMC’s computational cost
with system size, we sought to reduce the system size of the configurations
used in our training set. We investigated the feasibility of training our
committee MLP to the energies of a set configurations for small periodic boxes
to accurately predict for properties associated with a larger simulation box.
For liquid water running periodic molecular dynamics simulations of small
water boxes (32 water molecules or fewer) leads to significant artifacts even
in simple properties such as radial distribution functions (RDFs) when
compared to larger system sizes[39]. However, as demonstrated in SI Sec.
LABEL:sec:si-revpbe0-refit-tests for the revPBE0-D3 functional, if one trains
a MLP on energies of small (16 molecule) periodic water configurations and
then uses the resulting model to perform dynamics of a larger system (64
molecule), then the results obtained are in excellent agreement with those
obtained from performing a AIMD simulation at the larger system size. Based on
this demonstration we therefore trained the MLPs for the higher level methods
on periodic configurations of 16 water molecules using the transfer learning
approach described in the following section and then report the properties
obtained in Sec. III by performing MD and PIMD simulations using 64 water
molecules.
#### II.1.3 Transfer learning approach to train the MLPs
With only hundreds of energies from the higher tier periodic electronic
structure methods available, to make efficient use of this data we employed a
transfer learning approach[40]. To achieve this we first trained a committee
MLP at a lower level of electronic structure theory, DFT or Hartree Fock (HF)
on 531 configurations using both energies and gradients to improve the
fitting. The parameters obtained for these MLPs were then used as the starting
point for the fits to the higher level methods. This strategy exploits the
idea that while the lower level methods may not reach the levels of chemical
accuracy required for some applications they do contain fundamental physical
information (e.g., about the fact that O and H when in close proximity form a
high frequency chemical bond) that can be used to structure the neural network
that underlies the MLP. Hence while the high level data is used to tune the
accuracy of the MLP, it is leveraging the physics learnt by the initial
training to the lower level method.
While using transfer learning to make efficient use of very small amounts of
high level electronic structure data has distinct advantages over starting
from a randomly initialized MLP, one must be mindful of the risk of hysteresis
in the final MLP, i.e., that by biasing the weights by taking them from a
model trained on a low level electronic structure method, that the final MLP
will incorrectly contain remnants of the failures of the low level method.
Hence to assess our transfer learning approach’s ability to produce a final
model that accurately reproduces the properties of the target high level
electronic structure method, we initialized the procedure starting from a
range of different low level methods that each give very different properties
of liquid water. By comparing the structural and dynamical properties of
liquid water obtained by performing MD and PIMD simulations on the final
models obtained from these different starting points, one can thus evaluate
which properties are obtained universally across the models initialized from
different low level methods and thus accurately reflect the high level
electronic structure approach.
In practice, following the convergence of the lower level fits to both energy
and gradient data (see SI Sec. LABEL:sec:si-ml-details), we retrained the
committee MLP (8 separate MLPs) to the energies of the higher level method
employing the extended Kalman filter optimizer as implemented in the n2p2[41]
package and using a 90-10 train-validation split in order to monitor the
prediction error over the validation set for early-stopping each individual
fit to prevent overfitting. Before applying this transfer learning approach to
the high level CCSD, CCSD(T) and AFQMC methods for which AIMD is not possible
for the timescales required, in SI Sec. LABEL:sec:si-revpbe0-refit-tests we
tested it by training an MLP with revPBE0-D3 as the higher level method and
BLYP and HF as the lower level methods from which the transfer learning was
performed. We chose DFT with the revPBE0-D3 exchange-correlation functional as
the higher level in this benchmark due to its accurate description of the
properties of liquid water and since we can compare the results of the
transfer learned MLPs to AIMD and AI-PIMD trajectories that we have previously
obtained [36]. We chose BLYP and HF as the lower level methods since the
former gives an incredibly overstructured and dynamically sluggish description
of water while the latter gives the opposite.
As shown in SI Sec. LABEL:sec:si-revpbe0-refit-tests, MD simulations using the
final transfer learned MLP models of revPBE0-D3 using only 200 energies
starting from either BLYP or HF correctly capture the target RDFs and VDOS for
liquid water at 300K obtained from AIMD simulations, which are both markedly
different from those given by low level methods themselves (BLYP water has a
more structured oxygen-oxygen RDF and higher wavenumber hydrogen VDOS bend and
stretch peak positions than revPBE0-D3 water, and vice-versa for HF water).
The agreement with the reference revPBE0-D3 results for both models is
particularly strong for the RDFs. For the VDOS, the BLYP-initialized model
outperforms the HF-initialized model in capturing the high frequency O-H
stretch peak. Both models also accurately reproduce the RDFs obtained from AI-
PIMD simulations of revPBE0-D3 for liquid water[36] at 300K (SI Fig.
LABEL:fig:revpbe0-D3_refits_rdfs_pimd_300K) with the only discrepancy again
being in the VDOS (SI Fig. LABEL:fig:revpbe0-D3_refits_vdos_pimd_300K_370K)
where the BLYP-initialized MLP captures the full spectrum whereas HF shows an
overstructured and blue shifted OH stretch region.
Additionally, we compared our transfer learning approach to two common
alternative machine learning approaches: directly training a committee MLP on
the high level energies starting from randomly initialized weights and
training a committee delta-learning model that corrects from the lower to
higher level method. To allow for a fair comparison, all three types of models
were trained to the same 200 configuration training set of energies
(revPBE0-D3) for liquid water using the same MLP architecture and optimization
settings. The randomly initialized model resulted in a potential that was
unstable for the purposes of running MD for a periodic simulation box of 64
water molecules, with the instantaneous temperature drifting severely after
the first simulation step. For the delta-learning model, we trained a
committee MLP to capture the energy difference between revPBE0-D3 and another
committee MLP trained to BLYP. The delta-learning model was similarly as
unstable as the randomly initialized model. Hence for liquid water with 200
energies at the higher level we found that the transfer learning approach we
have detailed above provides the most accurate results.
Given the demonstrated efficacy of our transfer learning procedure, in Sec.
III we applied this approach to train committee MLPs to CCSD, CCSD(T), and
AFQMC energies for liquid water using three different sets of MLP
initializations: HF, BLYP, and revPBE0-D3.
### II.2 Correlated electronic structure methods
We perform correlated electronic structure calculations of liquid water with
periodic boundary conditions at the $\Gamma$-point using AFQMC, CCSD, and
CCSD(T). Periodic CCSD and CCSD(T) calculations were performed using PySCF
[42, 43] where electron-repulsion integrals are handled using the range-
separated density fitting method [44, 45], and AFQMC calculations were
performed using QMCPACK[46, 47] and ipie[48]. These calculations all began
with a periodic spin-restricted HF calculation also perfomed using PySCF. We
provide brief details here, and further information can be found in SI Secs.
LABEL:sec:si-ccsd-ccsdt and LABEL:sec:si-afqmc.
AFQMC is a projector Monte Carlo method where the ground state of a given
Hamiltonian is obtained via imaginary time evolution. Without any
approximations, it scales exponentially as the system size grows due to the
fermionic sign problem. We use the phaseless approximation [28] to obtain an
algorithm that scales with the system size $N$ as $O(N^{3})-O(N^{4})$ for each
sample at the expense of introducing errors in the final ground state energy
estimate. The phaseless approximation sets a boundary condition in imaginary
time evolution using a priori chosen wavefunction called the trial
wavefunction. The bias from this approximation becomes smaller as one improves
the quality of trial wavefunctions. In this work we employ the simplest trial
wavefunction, the spin-restricted Hartree-Fock determinant. A recent benchmark
study examined the accuracy of AFQMC with Hartree-Fock trial wavefunctions
over 1004 data points [49], and based on these results at this level of
approximation we expect the accuracy of AFQMC to be between CCSD and CCSD(T)
for the problems considered here.
Coupled cluster (CC) parameterizes the electronic wavefunction using an
exponential ansatz,
$\ket{\Psi_{\textrm{CC}}}=\mathrm{e}^{\hat{T}}\ket{\Phi_{\textrm{HF}}}$, where
$\hat{T}$ creates all possible particle-hole excitations from the HF reference
and is determined by iteratively solving a set of coupled non-linear
equations[50]. CCSD approximates the full CC ansatz by truncating the
$T$-operator at single and double-excitation levels,
$\hat{T}=\hat{T}_{1}+\hat{T}_{2}$[26]. CCSD(T) improves upon CCSD by further
including the contribution from triple excitations in a perturbative (non-
iterative) manner[27] and is often referred to as the “gold standard” in the
quantum chemistry of simple molecules. The computational cost of CCSD and
CCSD(T) scales more steeply than that of AFQMC, as $O(N^{6})$ and $O(N^{7})$,
respectively. In this work, we avoid the high computational cost of full CCSD
and full CCSD(T) by systematically compressing the virtual space using the
frozen natural orbital (FNO) approximation [51, 52], which we confirmed to
introduce a negligible error (SI Sec. LABEL:sec:si-ccsd-ccsdt).
## III Results and Discussion
Figure 2: Tetrahedral order parameter $q$ distributions as sampled via
classical and PIMD simulations via NNPs fitted to either CCSD, CCSD(T), and
AFQMC energies at both 300K and 370K.
Having established the applicability of our data efficient approach to
training MLPs that accurately reproduce the target potential energy surfaces
on DFT, we now apply it to obtain the static and dynamic properties of liquid
water using three correlated methods: CCSD, CCSD(T), and AFQMC. We trained a
committee MLP, which as described in Sec. II.1.1 is formed of the mean of 8
independently trained MLPs, for each correlated electronic structure method.
By performing MD and PIMD simulations in the NVT ensemble at 300 K and 370 K
using the committee MLP for each electronic structure method, here we evaluate
how different treatments of dynamical electron correlation affect the selected
properties when the nuclei are treated classically or quantum mechanically.
To provide an assessment of the accuracy of our transfer learned committee
MLPs on different structural and dynamical properties as discussed in Sec.
II.1.3 we can compare the consistency of the results obtained from models that
have been initially trained to different lower level electronic structure
methods. Hence, for each of the correlated methods we trained three transfer
learned committee MLPs starting from HF, BLYP, and revPBE0-D3 as our lower
level methods which are known to understructure, overstructure, and accurately
reproduce the properties of water respectively. Performing classical MD at
300K from MLPs starting from these three different methods SI Fig.
LABEL:fig:afqmc-rdfs-vdos-classical shows that regardless of the lower level
that the transfer learning was initialized with, the properties obtained at
the high level, AFQMC in this case, coincide closely although the O-H stretch
for the HF-initialized model is blue-shifted slightly. This demonstrates that
our training set of 200 energies of periodic boxes consisting of 16 water
molecules is large and diverse enough to train an accurate model of liquid
water under these conditions using our transfer learning protocol. When
nuclear quantum effects are included by peforming PIMD simulations using the
committee MLPs, the oxygen-oxygen RDF and VDOS for the revPBE0-D3 and BLYP
initialized transfer learning models remain consistent with one another (SI
Fig. LABEL:fig:afqmc-rdfs-vdos-rpmd).
In this section we show the results from the committee MLP generated using
transfer learning from revPBE0-D3 for each of the correlated methods since, as
shown in SI Fig. LABEL:tab:si_electronic_structure_correlations, revPBE0-D3
shows the strongest correlation with the training set energies of CCSD,
CCSD(T), and AFQMC out of the three initialization methods we used.
For our AFQMC results, it is important to note that the AFQMC energies contain
stochastic error, with each of our N=200 training set AFQMC energies having a
corresponding estimate for the standard error that ranges from 1-2 mH
depending on the specific training set configuration. To evaluate how this
level of error might affect our reported properties, we employed a test where
we sample new training sets where the same N=200 configuration are used but a
random value is added to each AFQMC energy to reflect the uncertainty of our
AFQMC energies (see SI Sec. LABEL:sec:si-energy-noise-tests for details). In
total, 12 training sets were sampled and a transfer learned committee MLP was
fit to each. SI Figs. LABEL:fig:afqmc_noise_rdf and LABEL:fig:afqmc_noise_vdos
serve to quantify the variations in the oxygen-oxygen RDF and hydrogen VDOS
obtained from the 12 separate training sets due to the stochastic error in the
AFQMC energies, with the grey shading representing the standard deviations.
From this test we found that the stochastic error in our AFQMC calculations
does introduce noticeable variations in the OH stretch peak of the hydrogen
VDOS, particularly around the top of the peak, but the RDFs and VDOS are
otherwise consistent for the different training sets.
### III.1 Static properties of water from correlated electronic structure
methods
We first compare the static equilibrium properties for liquid water at 300 K
and 370 K obtained via classical MD and PIMD simulations using the committee
MLP for each correlated electronic structure method. For these properties PIMD
exactly includes the NQEs for distinguishable particles, which is a highly
reliable assumption for nuclei at this temperature. Figure 1 shows the oxygen-
oxygen RDFs for each of CCSD, CCSD(T), and AFQMC as compared to the
experimental results at 295 K[53] and 366 K[54]. At 300 K, classical MD
CCSD(T) and AFQMC both give a first peak that is slightly higher than observed
experimentally suggesting the liquid is overstructured. However, once NQEs are
accounted for in the PIMD simulations both RDFs become slightly less
structured and show better agreement with experiment, with that of CCSD(T)
coinciding quantitatively. CCSD gives good agreement with experiment when used
in classical MD simulations but is understructured when NQEs are included
which is consistent with this electronic structure approach starting from a HF
reference which gives a severely understructured liquid with the additional
dynamical correlation added through the tiers of CC theory progressively
structuring the liquid. At 370 K when used in PIMD simulations all methods
give good agreement with the experiment with AFQMC again exhibiting a first
peak that is higher than CCSD, CCSD(T) and experiment. SI Fig.
LABEL:fig:mbpol_comp_rdfs_classical_300K and Fig.
LABEL:fig:mbpol_comp_rdfs_pimd_300K show the hydrogen-hydrogen and oxygen-
hydrogen RDFs at 300 K sampled via classical MD and PIMD, respectively, where
all three electronic structure methods give similar results but AFQMC again
exhibits a more structured hydrogen bond network with the first intermolecular
OH peak at $\sim$1.85 $\AA$, which corresponds to hydrogen bonds, being
slightly higher than the other methods.
The tetrahedral order parameter provides a measure of higher order structural
correlations within water’s hydrogen bond network beyond the purely radial
information encoded in the RDFs. The tetrahedral order parameter $q$ is
defined for a given water molecule as[55],
$q=1-\frac{3}{8}\sum_{j=1}^{3}\sum_{k=j+1}^{4}\left(\cos{\theta_{jk}}+\frac{1}{3}\right),$
(1)
where $\theta_{jk}$ is the angle that a given water molecule’s oxygen atom
makes with two neighboring oxygen atoms $j$ and $k$. The tetrahedral order
parameter thus ranges from 0 to 1 with higher values indicating that the
hydrogen bond network possesses angles closer to that of a perfect tetrahedral
arrangement of the four nearest neighbour oxygen atoms around a central water
molecule. As shown in Fig. 2 at 300 K for both classical MD and PIMD this
property further highlights the understructured hydrogen bond network of CCSD
compared to the more accurate correlated methods, CCSD(T) and AFQMC, that are
in close agreement. At 370 K the distribution of the tetrahedral order
parameter for all three methods shifts to lower values.
The comparison of these static equilibrium properties suggests that the
inclusion of higher order electron correlation contributions in methods like
CCSD(T) and AFQMC, as compared to CCSD or HF, results in a greater degree of
structuring in liquid water at 300 K and 370 K. Given the directional nature
of this additional structuring, as seen from the greater probability density
at higher $q$ in Figure 2, this arises from slightly stronger hydrogen bonds
being formed when using the two higher level methods. Our comparisons of the
oxygen-oxygen RDFs obtained from classical and PIMD simulations at 300 K also
show that for these correlated methods the inclusion of NQEs works to slightly
destructure liquid water. The relatively subtle overall effect of NQEs on
liquid water around 300 K is known to arise from the close balance of
competing quantum effects in this system[56].
### III.2 Dynamical properties of water from correlated electronic structure
methods
| AFQMC | CCSD(T) | CCSD
---|---|---|---
Classical T=300K ($10^{-9}$ m2/s) | 2.09 (0.05) | 2.21 (0.06) | 2.60 (0.08)
TRPMD T=300K ($10^{-9}$ m2/s) | 2.16 (0.09) | 2.30 (0.08) | 2.80 (0.11)
TRPMD T=370K ($10^{-9}$ m2/s) | 7.09 (0.11) | 8.16 (0.14) | 8.29 (0.14)
Table 1: Diffusion coefficients for liquid water when running classical and
TRPMD simulations using NNPs fitted to either CCSD, CCSD(T), and AFQMC
energies at both 300K and 370K. Mean diffusion coefficients and standard
errors of the mean are reported and are calculated using 20 ps length
trajectories taken from 1 ns classical MD or 500 ps TRPMD trajectories. The
experimental diffusion coefficient for water at 300 K and 370 K are
2.41$\pm$0.05[57] and 8.26$\pm$0.02 ($10^{-9}$ m2/s)[58], respectively.
We now turn our attention to how the different correlated electronic structure
methods behave when used to compute dynamical properties of liquid water,
namely the self diffusion coefficient and VDOS. For these properties we
compare the results obtained from classical MD and TRPMD. For these
properties, since real time quantum dynamics is intractable for such a large
atomistic condensed phase system for the timescales required to compute these
properties, we use TRPMD to approximate the role of NQEs. TRPMD has previously
been shown to be an accurate way for treating NQEs in the dynamics of
condensed phase systems, however it is known to spuriously broaden high-
frequency vibrational modes[59, 36].
The diffusion coefficients obtained for the three correlated methods are shown
in Table 1 at 300 K and 370 K. Unlike the other properties we report,
diffusion coefficients exhibit a notable scaling with system size that must be
corrected for to make comparisons with experiment. Hence, as described in SI
Sec. LABEL:sec:si-diffusion-coefficients, the diffusion coefficients were
computed using simulations of 64 water molecules and then extrapolated to the
infinite system size limit using the previously derived system size scaling
relation[60] and the experimental viscosity of water[61]. At 300 K, where the
experimentally measured diffusion coefficient is 2.41$\pm$0.05$\times 10^{-9}$
m2/s [57], when classical MD is used AFQMC and CCSD(T) yield smaller diffusion
coefficients than observed experimentally, 2.09$\pm$0.05 and 2.21$\pm$0.06
$\times 10^{-9}$ m2/s respectively, while CCSD gives a larger value
2.60$\pm$0.08 $\times 10^{-9}$ m2/s. This behavior is consistent with the
trends observed for the electronic structure approaches in the structural
properties, where CCSD formed a slightly understructured hydrogen bond network
compared to the more accurate correlated methods, which here leads to faster
dynamics. Upon including NQEs using TRPMD, the diffusion coefficients for all
three electronic methods increase, consistent with the disruption of the
hydrogen bond network upon including zero-point energy, which brings the
CCSD(T) result (2.30$\pm$0.08 $\times 10^{-9}$ m2/s) to within the statistical
error bar of the experimentally observed value. Even with NQEs included the
AFQMC diffusion coefficient is lower (2.16$\pm$0.09 $\times 10^{-9}$ m2/s)
than experiment, consistent with it forming a more structured liquid. However,
it should be noted that the discrepancy in the diffusion coefficient is very
small; to change water’s diffusion coefficient from that observed at 300 K via
classical MD using our CCSD(T) model to the value obtained by performing TRPMD
dynamics would require less than a 2 K change in the temperature of the
liquid[58]. In addition, for dynamical properties TRPMD only approximately
includes NQEs and hence the better agreement of CCSD(T) with the experimental
value could be changed if a different approach was used to treat the quantum
dynamics of the nuclei. At 370 K, when TRPMD is used CCSD and CCSD(T) give
similar results (8.29$\pm$0.14 and 8.16$\pm$0.14 $\times 10^{-9}$ m2/s,
respectively), both of which are close to the experimental value of
8.26$\pm$0.02 $\times 10^{-9}$ m2/s[58]. The diffusion of AFQMC water is again
considerably slower, which is consistent with the relatively greater degree of
structure we saw in the latter’s respective oxygen-oxygen RDF and $q$
distribution.
Figure 3: The hydrogen VDOS for liquid water when running classical and TRPMD
simulations using NNPs fitted to either CCSD, CCSD(T), and AFQMC energies at
both 300K and 370K.
The VDOS in Figure 3 provides more information on the frequency dependence of
the dynamics of water for the three electronic structure methods, since the
diffusion coefficient is simply proportional to its zero frequency limit. All
three methods give qualitatively similar VDOS around the lower frequency
librational band ($\sim$500 cm-1) and peak associated with the H-O-H bending
mode ($\sim$1600 cm-1). At 300 K using classical MD, the main qualitative
difference lies in the OH stretch peak (3000-4000 cm-1), with CCSD(T) giving a
peak that is redshifted by $\sim$80 cm-1 with respect to the CCSD peak, while
the AFQMC peak is slightly broader than the others and is centered closer to
the CCSD result. At the low-frequency end of the O-H stretch peak, the VDOS
for CCSD(T) and AFQMC coincide with one another. Low frequency O-H stretches
are typically associated with stronger hydrogen bonds and the frequency of the
O-H stretch peak has previously been demonstrated to be inversely correlated
with the tetrahedrality parameter[5]. Hence, the consistency between CCSD(T)
and AFQMC at the low frequency part of their respective O-H stretch peaks is
consistent with the structural evidence in Sec. III.1 showing that these two
methods similarly structure water via slightly stronger hydrogen bonds as
compared to CCSD. The TRPMD simulations at 300 K, which include NQEs via TRPMD
simulations, lead to a $\sim$120 cm-1 red-shift and broadening of the stretch
peak and a $\sim$100 cm-1 shift in the bend for all three methods, which is
consistent with observations from previous studies[59, 36]. Although some of
the broadening likely arises from physical effects, TRPMD is known to
introduce spurious broadening of high-frequency vibrational modes[59, 36]. At
370 K the most significant difference is the shift in the zero frequency
intensity, which we expect since this is directly related to the self-
diffusion coefficient and otherwise the VDOS is qualitatively similar with
respect to the 300 K results. This is expected since for the high frequency
modes the zero point energy greatly exceeds the thermal energy in the the
mode, i.e. $k_{B}T<<\hbar\omega/2$ where $k_{B}$ is the Boltzmann constant,
$T$ is the temperature and $\omega$ is the frequency of the mode.
Overall, the trends we observe in the dynamical properties largely mirror the
evidence we presented for the structural properties showing that CCSD results
in an understructured description of liquid water at 300 K, as compared to
CCSD(T) and AFQMC, while at 300 K and 370 K AFQMC overstructures water. The
differences in the diffusion coefficients in Table 1 reflect this trend, with
CCSD overestimating the experimental diffusion coefficient at 300 K with its
understructured description of water and AFQMC underpredicting the diffusion
coefficient at 370 K. With the VDOS the main differences between the three
correlated methods manifest in the O-H stretch peak positions and breadth,
with peaks given by MD at 300 K for both CCSD(T) and AFQMC skewed more to
lower frequencies that are associated with stronger hydrogen bonds.
## IV Conclusion
In summary, we leveraged developments in high-level periodic electronic
structure theory and exploited methods to improve the data efficiency of
fitting MLPs to investigate the static and dynamical properties of liquid
water at the level of CCSD, CCSD(T), and AFQMC. We devised a data efficient
protocol for training MLPs that uses small periodic boxes of water (16
molecules) sampled judiciously via an iterative QbC active learning procedure.
To make the most out of the few configuration energies we can afford to
compute, we also employed a transfer learning approach that leverages the
transferability of physics between lower level electronic structure methods
(e.g. DFT with the BLYP functional, revPBE0-D3 functional, or HF) and our
target higher-level methods, using MLPs fit to the former to initialize a
fine-tuning transfer learning fit to the latter. Using this approach we showed
that we can train stable MLPs with as few as 50 configuration energies,
capture the RDFs with 100 (SI Fig.
LABEL:fig:revpbe0-D3_refits_rdfs_classical_300K), and with 200 configuration
energies obtain both accurate static and dynamical properties such as the
diffusion constant and VDOS (SI Fig.
LABEL:fig:revpbe0-D3_refits_vdos_classical_300K). In contrast, with these same
200 energies we were unable to train stable models using delta learning or
using random initialization of the model.
We used our MLPs trained to CCSD, CCSD(T), and AFQMC to examine how different
static and dynamical properties of liquid water are affected by the level of
dynamic electron correlation accounted for and the inclusion of NQEs. Our
results indicate that CCSD tends to understructure liquid water and
overpredict the diffusion coefficient, as compared to experiment. On the other
hand, both CCSD(T) and AFQMC give oxygen-oxygen RDFs and diffusion
coefficients that are more consistent with experimental values at 300 K,
suggesting that the more accurate treatment of dynamical correlation present
in these methods is sufficient for describing liquid water. The inclusion of
NQEs for our 300 K simulations bring the CCSD(T) and AFQMC results in even
closer agreement with experiment and seems to generally manifest as a slight
destructuring of liquid water for all three electronic structure descriptions.
This small destructuring upon including NQEs for these correlated methods is
in contrast to some DFT exchange correlation functionals where due to the
overprediction of the anharmonicity of the O-H coordinate the inclusion of
NQEs works to structure the liquid phase[36].
Ultimately, we envision that the configurations and energies that form the
training dataset, the resulting MLPs, and the protocols we employed here will
be useful in their own separate respects for future work in modeling potential
energy surfaces for condensed phase systems.
## Acknowledgments
T.E.M and M.S.C were supported by the National Science Foundation under Grant
No. CHE-2154291. This research also used resources of the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.
This work was also supported by the US Air Force Office of Scientific Research
under Grant No. FA9550-21-1-0400 (H.-Z.Y. and T.C.B.). We acknowledge the
computing resources provided by Columbia University’s Shared Research
Computing Facility project, which is supported by NIH Research Facility
Improvement Grant No. 1G20RR030893-01, and associated funds from the New York
State Empire State Development, Division of Science Technology and Innovation
(NYSTAR) Contract No. C090171, both awarded April 15, 2010. The Flatiron
Institute is a division of the Simons Foundation.
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# TetraFreeQ: tetrahedra-free quadrature on polyhedral elements
Alvise Sommariva<EMAIL_ADDRESS>Marco Vianello<EMAIL_ADDRESS>University of Padova, Italy Member of the INdAM Research group GNCS
###### Abstract
In this paper we provide a tetrahedra-free algorithm to compute low-
cardinality quadrature rules with a given degree of polynomial exactness,
positive weights and interior nodes on a polyhedral element with arbitrary
shape. The key tools are the notion of Tchakaloff discretization set and the
solution of moment-matching equations by Lawson-Hanson iterations for
NonNegative Least-Squares. Several numerical tests are presented. The method
is implemented in Matlab as open-source software.
###### keywords:
Polyhedral elements, tetrahedra-free algebraic quadrature, Positive Interior
rules, Tchakaloff theorem, Tchakaloff sets, Davis-Wilhelmsen theorem,
NonNegative Least Squares.
###### MSC:
[2020]65D32.
††journal: Applied Numerical Mathematics
## 1 Introduction
Let $\Omega\subset{\mathbb{R}}^{3}$ is a polyhedron (either convex or
nonconvex or even multiply-connected), and suppose that for a given continuous
function $f\in C(\Omega)$ one needs to approximate its integral over $\Omega$
by a quadrature rule with nodes $\\{Q_{j}\\}_{j=1,\ldots,\nu}\subset\Omega$
where $Q_{j}=(x_{j},y_{j},z_{j})$ and positive weights
$\\{w_{j}\\}_{j=1,\ldots,\nu}$,
$I_{\Omega}(f)=\int_{\Omega}f(x,y,z)\,\,\,dx\,dy\,dz\approx
I_{n}(f)=\sum_{j=1}^{\nu}w_{j}f(Q_{j})\;,$ (1)
that has Algebraic Degree of Exactness $ADE=n$, i.e. $I_{\Omega}(p)=I_{n}(p)$
for all $p\in{\mathbb{P}}^{3}_{n}$ (the space of trivariate polynomials of
total degree not exceeding $n$). Such a kind of rules are often called PI
(Positive Interior) in the literature and are optimally stable by the
positivity of the weights, being the quadrature conditioning
$\sum_{j}|w_{j}|/\left|\sum_{j}w_{j}\right|=1$. As a relevant motivation, we
may recall that the algebraic quadrature problem on polyhedra has been
extensively studied in the literature, especially during the last decade,
being at the core of many discretization methods for PDEs that use polyhedral
meshes, such as polyhedral FEM and discontinuous Galerkin, as well as the more
recent VEM. We may quote among others, without any pretence of exhaustivity,
the papers [1, 4, 15, 23, 24] with the numerous references therein.
Despite of such manifest interest in the framework of numerical PDEs, open-
source numerical codes to compute PI algebraic quadrature rules on polyhedra,
in particular Matlab codes, do not seem to be readily available. On the other
hand, differently from other approaches that do not provide an algebraic
quadrature formula but a numerical integration algorithm, the availability of
low-cardinality quadrature formulas allows to compute efficiently integrals
involving forcing terms, without the need of an explicit polynomial
approximation of such terms (cf. [1, Rem. 6]). Moreover, since the polyhedral
quadrature formulas can be computed once and for all for a fixed polyhedral
mesh, even more flexibility is gained when different forcing terms have to be
tested, or more generally when parametrized PDEs have to be solved, for
example by reduced basis methods or other model order reduction techniques
(cf., e.g., [10]).
Indeed, the main purpose of the present work is to provide an algorithm
implemented by an open-source Matlab package, named TetraFreeQ, for the
computation of tetrahedra-free PI algebraic quadrature rules on arbitrary
polyhedra (either convex or nonconvex or even multiply-connected), with
cardinality $\nu\leq N$ where
$N=N_{n}=dim(\mathbb{P}_{n}^{3})=(n+1)(n+2)(n+3)/6\;.$ (2)
Existence and computability of such rules are guaranteed by the well-known
Tchakaloff theorem [25] and the less known Wilhelmsen theorem on “Tchakaloff
sets” [28], the latter being at the core of our approach as described in the
nest section.
## 2 Tetrahedra-free quadrature
In this section we show how to determine a quadrature rule with $ADE=n$,
positive weights and interior nodes, on a general polyhedron $\Omega$, without
needing a decomposition of the domain by means of nonoverlapping tetrahedra
(tetrahedra-free quadrature, for short). It should be recalled that avoiding
“sub-tessellations” for integration on polyhedral elements is a common
requirement in the numerical PDEs framework, especially with high-order
methods.
In the sequel the polyhedron is assumed to be given via its polygonal faces,
say $\\{\mathcal{F}_{i}\\}$, each with counterclockwise oriented vertices
w.r.t. the outward normal. The procedure works essentially as follows:
Algorithm TetraFreeQ (tetrahedra-free quadrature on a general polyhedron
$\Omega$)
* $(i)$
compute the moment array
$\gamma=\\{\gamma_{k}\\}=\left\\{\int_{\Omega}{\phi_{k}(x,y,z)\,dx\,dy\,dz}\right\\}$
(3)
of a certain polynomial basis $\\{\phi_{k}\\}$ of ${\mathbb{P}}_{n}^{3}$,
$k=1,\ldots,N$, by the divergence theorem and bivariate quadrature rules with
ADE equal to $n+1$ on the polyhedron faces
$\\{{\mathcal{F}}_{i}\\}_{i=1,\ldots,M}$
* $(ii)$
using an in-polyhedron routine [11], in view of Wilhelmsen theorem [28],
determine iteratively a sufficiently dense pointset
$\\{P_{l}\\}_{l=1,\ldots,L}$ inside $\Omega$ (a so-called “Tchakaloff set”)
such that the underdetermined moment-matching system
$V^{t}u=\gamma\;,\;\;\mbox{where}\;\;V=[v_{lk}]=[\phi_{k}(P_{l})]\in\mathbb{R}^{L\times
N}$ (4)
has a nonnegative solution $u$ with at most $N\leq L$ positive components,
computed via Lawson-Hanson NNLS algorithm [13] applied to
$\min_{u\geq 0}{\|V^{t}u-\gamma\|_{2}}$ (5)
* $(iii)$
once $\\{P_{l}\\}_{l=1,\ldots,L}$ and the weight vector $u$ are determined,
the nonzeros of $u$ finally select the nodes of a PI-type quadrature rule with
cardinality at most $N$.
In spite of the simplicity of this approach, there are many theoretical and
practical aspects that deserve explanations. We stress that the key tools are
Wilhelmsen theorem on the existence of “Tchakaloff sets” within sufficiently
dense discrete subsets (a relevant but somehow overlooked result of
multivariate quadrature theory), and the capability of Lawson-Hanson active-
set iterative algorithm to compute a sparse solution to the NonNegative Least-
Squares problem (made more efficient by the recent Deviation Maximization
approach to QR factorization [6, 7]).
### 2.1 Moment computation
For an implementation of the algorithm sketched above, we intend to compute
first the moments on the polyhedron $\Omega$ of the product Chebyshev basis of
total degree $n$, relative to the smallest parallelepiped
${\mathcal{R}}=[a_{1},b_{1}]\times[a_{2},b_{2}]\times[a_{3},b_{3}]$ containing
$\Omega$.
In other words, defined the scaled Chebyshev polynomial in $[a,b]$
${\tilde{T}}^{(a,b)}_{m}(t):=T_{m}\left(\frac{2}{b-a}\cdot\left(t-\frac{a+b}{2}\right)\right)$
(6)
where $T_{m}$ is the Chebyshev polynomial of first kind, of degree $m$, we
compute the value of the moments
$\gamma_{\alpha}=\int_{\Omega}{\tilde{T}}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z){\mbox{
}}dx\,dy\,dz$
for all the triples $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ with
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{N}$ and
$0\leq\alpha_{1}+\alpha_{2}+\alpha_{3}\leq n$.
To this purpose, we recall that in view of the divergence theorem applied to a
continuously differentiable vector field ${\mathbf{F}}=(f,g,h)$ defined on a
neighborhood of $\Omega$, we have
$\int_{\Omega}\nabla\cdot{\mathbf{F}}{\mbox{
}}dx\,dy\,dz=\int_{\partial\Omega}{\mathbf{F}}\cdot{\mathbf{n}}{\mbox{ }}dS$
(7)
where ${\mathbf{n}}=(n_{1},n_{2},n_{3})$ is the outward pointing unit normal
at each point on the boundary ${\partial\Omega}$ and
$\nabla\cdot{\mathbf{F}}={\frac{\partial f}{\partial x}}+{\frac{\partial
g}{\partial y}}+{\frac{\partial h}{\partial z}}$ is the divergence of
$\mathbf{F}$. As a particular case of (7), we consider ${\mathbf{F}}=(f,0,0)$
obtaining
$\int_{\Omega}{\frac{\partial f(x,y,z)}{\partial
x}}dx\,dy\,dz=\int_{\partial\Omega}n_{1}(x,y,z)\,{{f(x,y,z)}}{\mbox{ }}dS\;,$
(8)
and we apply (8) to a certain $f$ such that
${\frac{\partial f(x,y,z)}{\partial
x}}={\tilde{T}}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z).$
In this regard, we exploit a fundamental property of Chebyshev polynomials of
first kind, namely that for $m\geq 2$
$\int_{0}^{x}T_{m}(t)dt=\frac{T_{m+1}(x)}{2(m+1)}-\frac{T_{m-1}(x)}{2(m-1)}.$
(9)
Consequently, setting $s=\frac{2}{b-a}\cdot(t-\frac{a+b}{2})$, by (6) and (9),
for $m\geq 2$,
$\displaystyle\int_{0}^{x}{\tilde{T}}^{(a,b)}_{m}(t)dt$ $\displaystyle=$
$\displaystyle\int_{0}^{x}T_{m}\left(\frac{2}{b-a}\cdot\left(t-\frac{a+b}{2}\right)\right)dt$
(10) $\displaystyle=$
$\displaystyle\int_{-\frac{a+b}{b-a}}^{\frac{2}{b-a}(x-\frac{a+b}{2})}T_{m}(s)\frac{2}{b-a}ds$
$\displaystyle=$
$\displaystyle\frac{2}{b-a}\left(\int_{-\frac{a+b}{b-a}}^{0}T_{m}(s)ds+\int_{0}^{\frac{2}{b-a}(x-\frac{a+b}{2})}T_{m}(s)ds\right)$
$\displaystyle=$
$\displaystyle\frac{2}{b-a}\left(\int_{-\frac{a+b}{b-a}}^{0}T_{m}(s)ds+\frac{T_{m+1}(\frac{2}{b-a}(x-\frac{a+b}{2}))}{2(m+1)}-\frac{T_{m-1}(\frac{2}{b-a}(x-\frac{a+b}{2}))}{2(m-1)}\right)$
$\displaystyle=$
$\displaystyle\frac{2}{b-a}\left(\int_{-\frac{a+b}{b-a}}^{0}T_{m}(s)ds+\frac{{\tilde{T}}^{(a,b)}_{m+1}(x)}{2(m+1)}-\frac{{\tilde{T}}^{(a,b)}_{m-1}(x)}{2(m-1)}\right).$
Thus, being the term $\frac{2}{b-a}\int_{-\frac{a+b}{b-a}}^{0}T_{m}(s)ds$
costant, setting
$\mathcal{Z}^{(a,b)}_{m}(x)=\frac{1}{b-a}\left(\frac{{\tilde{T}}^{(a,b)}_{m+1}(x)}{m+1}-\frac{{\tilde{T}}^{(a,b)}_{m-1}(x)}{m-1}\right)$
we get that for $m\geq 2$
$\displaystyle{\tilde{T}}^{(a,b)}_{m}(x)=\frac{\partial}{\partial
x}\mathcal{Z}^{(a,b)}_{m}(x).$ (11)
It can be easily checked that (11) holds also in the case $m=0$ and $m=1$, by
defining respectively
$\mathcal{Z}^{(a,b)}_{0}(x)=x-\frac{a+b}{2},\mbox{
}\mathcal{Z}^{(a,b)}_{1}(x)=\frac{1}{b-a}\left(x-\frac{a+b}{2}\right)^{2},$
and that $\mathcal{Z}^{(a,b)}_{m}\in{\mathbb{P}}_{m+1}$.
Now, denoting by $\\{{\mathcal{F}}_{i}\\}_{i=1,\ldots,M}$ the polygonal faces
that compose the boundary of the polyhedron $\Omega$, where
${\mathbf{n}}^{(i)}=(n^{(i)}_{1},n^{(i)}_{2},n^{(i)}_{3})$ is the outward
normal of ${\mathcal{F}}_{i}$, $i=1,\ldots,M$, by the form of the divergence
theorem (8) for
$f_{\alpha}(x,y,z)=\mathcal{Z}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z)$
from the application of (11) with $a=a_{1}$ and $b=b_{1}$, we have
$\displaystyle\gamma_{\alpha}$ $\displaystyle=$
$\displaystyle\int_{\Omega}{\tilde{T}}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z){\mbox{
}}dx\,dy\,dz$ $\displaystyle=$
$\displaystyle\int_{\Omega}\frac{\partial}{\partial
x}\mathcal{Z}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z){\mbox{
}}dx\,dy\,dz$ $\displaystyle=$
$\displaystyle\int_{\Omega}\frac{\partial}{\partial
x}\left(\mathcal{Z}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z)\right){\mbox{
}}dx\,dy\,dz$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}n_{1}(x,y,z)\mbox{
}\mathcal{Z}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z){\mbox{
}}dS$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\int_{{\mathcal{F}}_{i}}n^{(i)}_{1}\mbox{
}\mathcal{Z}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z){\mbox{
}}dS\;.$
Since the outward normal $n_{1}$ is constant on the face ${\mathcal{F}}_{i}$,
so is $n^{(i)}_{1}$, and thus the integrand
$n^{(i)}_{1}\mbox{
}\mathcal{Z}^{(a_{1},b_{1})}_{\alpha_{1}}(x){\tilde{T}}^{(a_{2},b_{2})}_{\alpha_{2}}(y){\tilde{T}}^{(a_{3},b_{3})}_{\alpha_{3}}(z)$
is a polynomial of degree not exceeding $n+1$ on the $i$-th face, with
$i=1,\ldots,M$, and consequently each moment $\gamma_{\alpha}$ can be
recovered by a quadrature rule with ADE equal to $n+1$ for all the polygons
${\mathcal{F}}_{1},\ldots,{\mathcal{F}}_{M}$.
Having this in mind, let ${\mathcal{F}}\subset\partial\Omega$ be a (polygonal)
face of the polyhedron $\Omega$, and suppose that the vertices are oriented
counterclockwise w.r.t. its outward normal ${\mathbf{n}}$. Since it is easier
to determine a quadrature rule on a planar polygon ${\mathcal{F}}^{*}$, we
apply a rototranslation $\Phi$, such that
${\mathcal{F}}^{*}=\Phi({\mathcal{F}})$ is a polygon belonging to the plane
$\pi^{*}=\\{(x,y,0),\,\,x,y\in\mathbb{R}\\}$, oriented counterclockwise w.r.t.
the vector $(0,0,1)$.
A quadrature formula on the planar polygon ${\mathcal{F}}^{*}$ with internal
nodes $\\{P_{i}\\}$ and positive weights $\\{w_{i}\\}$, having ADE equal to
$n+1$, can be determined as described in [3], by applying a rule with ADE
equal to $n+1$ on each element of a minimal triangulation of
${\mathcal{F}}^{*}$ or alternatively by a triangulation free routine as
discussed in [21]. Since the rototranslation $\Phi$ is an affine (bijective)
map with Jacobian modulus equal to 1, we conclude that
$\\{\Phi^{-1}(P_{i})\\}$ and $\\{w_{i}\\}$ are the nodes and the weights of a
quadrature rule over the polygonal face ${\mathcal{F}}\subset\partial\Omega$,
with ADE equal to $n+1$ and consequently we are able to compute the set of
moments $\\{\gamma_{\alpha}\\}$.
###### Remark 2.1.
The mapping $\phi$ can be easily obtained by applying a rotation $R$ to the
plane $\pi$ containing the vertices of the polygon $\mathcal{F}$, so that
$R\pi$ is parallel to the $xy$-plane
$\pi^{*}=\\{(x,y,0),\,\,x,y\in\mathbb{R}\\}$ and then by applying a vertical
translation $\tau$ so that
$\hat{\mathcal{F}}:=R{\mathcal{F}}+\tau\subset\pi^{*}$. If a counterclockwise
orientation of $\mathcal{F}$ is given by the user, then it is straightforward
to obtain the same for $\hat{\mathcal{F}}$. For the quadrature routines over
planar polygons it is a typical requirement that the orientation is
counterclockwise. If the planar polygon $\hat{\mathcal{F}}$ is oriented
clockwise w.r.t. the z-axis then, applying an additional rotation, we can take
as ${\mathcal{F}}^{*}$ the polygon symmetric to $\hat{\mathcal{F}}$ w.r.t. the
x-axis, otherwise it is sufficient to set
${\mathcal{F}}^{*}=\hat{\mathcal{F}}$. We recall that a necessary and
sufficient condition for a polygon with vertices $(x_{j},y_{j},0)$,
$j=1,\ldots,m$ to be oriented counterclockwise is that its signed area
$\sum_{j=1}^{m-1}(x_{j+1}-x_{j})(y_{j+1}+y_{j})$ be negative (for further
details see [27]).
In some instances, instead of the orientation of the vertices of
${\mathcal{F}}$, it is available its outward normal $\mathbf{n}$ (w.r.t.
$\Omega$). In this case, the procedure is the similar to that of the previous
remark. If in particular, after the application of the rotation $R$ (and the
translation $\tau$), the normal to ${\mathcal{\hat{F}}}$ correspond to
$\Phi({\mathbf{n}})=(0,0,-1)$ then as ${\mathcal{F}}^{*}$ we can consider the
polygon symmetric to $\hat{\mathcal{F}}$ w.r.t. the x-axis, otherwise we set
${\mathcal{F}}^{*}=\hat{\mathcal{F}}$.
### 2.2 Tchakaloff sets and computation of Tchakaloff-like rules
In the previous subsection we have implemented step $(i)$ of the Algorithm
TetraFreeQ sketched at the beginning of Section 2. With a little abuse of
notation, we shall consider an ordering of the multi-indices
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$,
$0\leq\alpha_{1}+\alpha_{2}+\alpha_{3}\leq n$, e.g. the graded lexicographical
ordering, and we shall term $\\{\phi_{k}(x,y,z)\\}$ the product Chebyshev
basis and $\\{\gamma_{k}\\}$ the corresponding moments, $1\leq k\leq N$.
Now, step $(ii)$ of Algorithm TetraFreeQ rests on the following relevant
(though often overlooked) result proved by Wilhelmsen in [28] where the key
notion of Tchakaloff set is introduced (extending a result of Davis [8] to
rather general functional spaces, including polynomial spaces).
###### Theorem 2.1.
(Wilhelmsen, 1976) Let $\Psi$ be the linear span of continuous, real-valued,
linearly independent functions $\\{\phi_{k}\\}_{k=1,\ldots,N}$ defined on a
compact set $\Omega\subset{\mathbb{R}}^{d}$. Assume that $\Psi$ satisfies the
Krein condition (i.e. there is at least one $f\in\Psi$ which does not vanish
on $\Omega$) and that $L$ is a positive linear functional, i.e. $Lf>0$ for
every $f\geq 0$ not vanishing everywhere in $\Omega$.
If $\\{P_{l}\\}_{i=1}^{\infty}$ is an everywhere dense subset of $\Omega$,
then for sufficiently large $L$, the set $X=\\{P_{l}\\}_{l=1,\ldots,L}$ is a
Tchakaloff set, i.e. there exist weights $w_{j}>0$, $j=1,\ldots,\nu$, and
nodes $\\{Q_{j}\\}_{j=1,\dots,\nu}\subset X\subset\Omega$, with
$\nu={\mbox{card}}(\\{Q_{j}\\})\leq N$, such that
$Lf=\sum_{j=1}^{\nu}w_{j}f(Q_{j})\;,{\mbox{ }}\forall f\in\Psi\;.$ (13)
In view of this existence result, we can the implement step $(ii)$ in our
framework, that is $Lf=I_{\Omega}(f)$ and $\Psi=\mathbb{P}_{n}^{3}$, by the
following iterative substeps:
* $(ii1)$
generate a set of uniform random or low-discrepancy points (e.g. Halton
points) with cardinality $K$ in the smallest box
$[a_{1},b_{1}]\times[a_{2},b_{2}]\times[a_{3},b_{3}]$ containing $\Omega$
* $(ii2)$
determine those points belonging to $\Omega$ by an in-polyhedron routine like
[11], say $X=\\{P_{l}\\}_{l=1,\ldots,L}$ with $L\leq K$, and compute the
Vandermonde-like matrix $V=[v_{lk}]=[\phi_{k}(P_{l})]\in\mathbb{R}^{L\times
N}$
* $(ii3)$
solve the NNLS problem $\min_{u\geq 0}{\|V^{t}u-\gamma\|_{2}}$ by Lawson-
Hanson active-set method
* $(ii4)$
if the residual $\|V^{t}u-\gamma\|_{2}<\varepsilon$ (where $\varepsilon$ is a
given tolerance)
then set $\\{w_{j}\\}=\\{u_{l}>0\\}$ and $\\{Q_{j}\\}=\\{P_{l}:\,u_{l}>0\\}$
else goto $(ii1)$ increasing $K$, e.g. by a fixed factor $\theta>1$
We stress that since the set of random points becomes denser along the
iterations, in view of Wilhelmsen theorem above termination of the iterative
cycle is (at least theoretically) guaranteed. Substep $(ii3)$ is based on the
fact that Lawson-Hanson algorithm computes a sparse solution of the NNLS
problem, and can be performed in Matlab by resorting to the built-in routine
lsqnonneg. On the other hand, a substantial acceleration can be obtained via
the recent Deviation-Maximization approach to $QR$ factorizations which are
the core of Lawson-Hanson method; cf. [5, 6, 7] for a detailed analysis of
this approach.
###### Remark 2.2.
(Tetrahedra-based quadrature)
As already stressed, the aim of the present work is to provide a completely
“tetrahedra-free” approach to quadrature on general polyhedral elements,
implemented in Matlab. On the other hand, a classical approach for polyhedral
quadrature, whenever a nonoverlapping “tetrahedralization” of the polyhedron
is available, consists in collecting and summing up the PI-formulas on the
single tetrahedra by integration additivity.
A first problem consists in the computation of the tetrahedralization. If
$\Omega$ is convex or star-shaped with a known center, this operation is
straightforward, but the task becomes not trivial for a general polyhedron
that may have a difficult geometry. We recall, incidentally, that differently
from the 2D case of polygons, there exist polyhedra that cannot be
triangulated using only their vertices (e.g. the well-known Schönardt
polyhedron, cf. [17]).
Moreover, though Matlab provides a minimal triangulation of a general
polygonal region via the environment polyshape, similar codes do not seem
available for polyhedra. As an alternative, sometimes useful, the built-in
alphashape command creates a bounding volume that envelops a given 3D point
cloud pointset. If the polyhedron of interest can be defined in such a way,
then alphaTriangulation returns a tetrahedralization that defines the domain
of the alpha shape (see [14] for additional details).
A second problem concerns the rule to be used in each tetrahedron. For degrees
of exactness up to $n=20$, there are in literature several PI-type rules on
the reference tetrahedron (the simplex) with vertices $(1,0,0)$, $(0,1,0)$,
$(0,0,0)$, $(0,0,1)$, which have near-minimal cardinality (see e.g. [12, 18]
and Table 1). For $n>20$, one can use the well-established Stroud rule [22,
p.28-32], again of PI-type, that in general is not minimal but still easy to
be implemented and with a moderate cardinality equal to
$\lceil{\frac{n+1}{2}}\rceil^{3}$. All these rules are typically written in
barycentric coordinates $(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$
and can be applied to a general tetrahedron with vertices $A,B,C,D$ by the
standard change of coordinates
$\lambda_{1}A+\lambda_{2}B+\lambda_{3}C+\lambda_{4}D$.
deg | card | deg | card | deg | card | deg | card
---|---|---|---|---|---|---|---
$1$ | $1$ | $6$ | $23$ | $11$ | $94$ | $16$ | $247$
$2$ | $4$ | $7$ | $31$ | $12$ | $117$ | $17$ | $288$
$3$ | $6$ | $8$ | $44$ | $13$ | $144$ | $18$ | $338$
$4$ | $11$ | $9$ | $57$ | $14$ | $175$ | $19$ | $390$
$5$ | $14$ | $10$ | $74$ | $15$ | $207$ | $20$ | $448$
Table 1: Cardinality of near-minimal rules on the reference tetrahedron.
Of course the overall cardinality of the resulting rule is proportional to the
number of tetrahedra and can be much larger than
$N=(n+1)(n+2)(n+3)/6=dim(\mathbb{P}_{n}^{3})$. In these cases, one can resort
to Tchakaloff-like compression of the corresponding discrete measure,
obtaining a rule with at most $N$ re-weighted nodes that are a subset of the
starting ones. Such a compression consists in computing a sparse positive
solution of a moment-matching system like (4), where the points and moments
are given by the high-cardinality rule itself. This approach inserts in the
more general setting of “Caratheodory-Tchakaloff discrete measure
compression”, and can be implemented by both Linear and Quadratic Programming
methods; cf. e.g. [9, 16, 20, 26] with the references therein.
In our Matlab package we have added, for the purpose of comparison, the
routines that provide a compressed Tchakaloff-like rule along the lines
sketched above, starting from a tetrahedralization of the polyhedral element.
We notice that even this approach does not seem to be yet readily available,
at least in Matlab, within existing open-source packages, and thus could be of
independent practical interest, especially because it resorts to the best
tetrahedral rules available in the literature.
## 3 Numerical tests
In this section we test Algorithm TetraFreeQ on several polyhedral domains.
All the routines have been implemented in Matlab and are freely available at
[19]. The numerical experiments were made using Matlab R2022a, on an Apple
MacBook Pro with M1 Chip and 16 GB of RAM.
We consider three polyhedra $\Omega_{j}$, $j=1,2,3$ with triangular facets,
where $\Omega_{1}$ is nonconvex with 30 facets, $\Omega_{2}$ is convex (an
approximation of a sphere with 760 facets) and $\Omega_{3}$ is multiply-
connected with 20 facets and a hole (see Figure 1). The domains, as well as
the facets and the tetrahedralization (used for comparison), are obtained by
Matlab built-in command alphashape, on suitable point clouds.
In particular we compute on the three polyhedra:
* 1.
a reference rule $Q_{T}$ corresponding to a tetrahedralization (cf. Remark
2.2)
* 2.
the corresponding compressed tetrahedral rule $Q_{TC}$
* 3.
the tetrahedra-free rule $Q_{TF}$ produced by Algorithm TetraFreeQ.
In our numerical tests, in $(ii1)-(ii4)$ of Algorithm TetraFreeQ we have
chosen the parameters $\varepsilon=10^{-14}$, $\theta=4$ and an initial value
$K=\lceil 8Nvol(B)/vol(\Omega)\rceil$ where $B$ is the smallest bounding box
for $\Omega$.
In order to test numerically polynomial exactness of $Q_{TF}$, in Figure 3 we
display for each domain the relative errors
$E(g_{k})=\frac{|Q_{T}(g_{k})-Q_{TF}(g_{k})|}{|Q_{T}(g_{k})|}$ (14)
corresponding to 100 trials of the polynomial
$g_{k}(x,y,z)=(a_{k}+b_{k}x+c_{k}y)^{n}$ (15)
where $n$ is the algebraic degree of precision of the rule and
$(a_{k},b_{k},c_{k})\in[0,1]$ are uniform random coefficients. In the same
figures, we have also represented with a circle the logarithmic averages
$\sum_{k=1}^{100}{\log(E(g_{k}))/100}$.
As additional investigation, we also compute the relative errors of these
rules in integrating on $\Omega_{j}$, $j=1,2,3$, the three test functions with
different degree of regularity
$f_{1}(x,y,z)=\exp(-\|P-P_{0}\|_{2}^{2})\;,\;\;f_{2}(x,y,z)=\|P-P_{0}\|_{2}^{5}\;,\;\;f_{3}(x,y,z)=\|P-P_{0}\|_{2}\;,$
(16)
where $P=(x,y,z)$ and $P_{0}(x_{0},y_{0},z_{0})$ was suitably chosen in the
domains, more precisely $P_{0}=(1.5,1.5,1.5)$ for $\Omega_{1}$ and
$\Omega_{3}$, $P_{0}=(1,1,1)$ for $\Omega_{2}$; cf. Figures 4-6. Moreover, in
Table 2 we report the cardinalities of the pointsets from which the
tetrahedra-free rules are extracted, varying the degree from 1 to 10.
In view of these numerical experiments, we can see that the rules $Q_{TC}$ and
$Q_{TF}$ give errors of comparable size, whereas the errors of $Q_{T}$ are
smaller. This is not surprising, since the basic tetrahedra-based rule uses a
much larger number of nodes than $N$, which turns out to be the number of
nodes of both $Q_{TC}$ and $Q_{TF}$, with a ratio increasing with the number
of facets (being for example more than 100 with the 760-facets sphere-like
polyhedron, cf. Figure 2). We stress once again, however, that $Q_{TF}$ avoids
completely any tetrahedralization of the polyhedron, a desiderable feature in
many applications (such as polyhedral FEM/VEM methods).
Concerning the cputime required by the moment computation in TetraFreeQ, it
grows with the algebraic degree of precision and of course with the number of
facets, ranging in our tests between $10^{-4}$ and $10^{-3}$ seconds. On the
other hand, there is numerical evidence that the bottleneck of TetraFreeQ is
due to the in-polyhedron routine and the NNLS algorithm, one dominating on the
other depending on the algebraic degree of exactness and on the number of
facets, with an overall cputime of the quadrature rule construction varying
from $10^{-2}$ seconds for $n=1,2,3$ up to $10^{0}$ seconds for $n=10$.
Figure 1: Examples of polyhedral domains. Left: $\Omega_{1}$ (nonconvex, 20
facets); Center: $\Omega_{2}$ (convex, 760 facets); Right: $\Omega_{3}$
(multiply connected, 20 facets).
Figure 2: Cardinality versus the exactness degree on the three polyhedra of
Figure 1: uncompressed tetrahedra-based rule (blue), compressed tetrahedra-
based and tetrahedra-free rule (brown).
Figure 3: Relative errors $E(g_{k})$ of the tetrahedra-free rule over 100
polynomial integrands of the form $g_{k}=(a_{k}x+b_{k}y+c_{k}z+d_{k})^{n}$ on
the three polyhedra of Figure 1, where $a_{k},b_{k},c_{k},d_{k}$ are uniform
random coefficients in $[0,1]$ and $n=1,2,\dots,10$; the circles correspond to
the average logarithmic error $\sum_{k=1}^{100}{\log(E(g_{k}))/100}$.
Figure 4: Relative errors versus the exactness degree in the integration of
the Gaussian $f_{1}$ in (16) on the three polyhedra of Figure 1: uncompressed
(blue) and compressed (brown) tetrahedra-based rules, tetrahedra-free
rule(yellow).
Figure 5: Relative errors versus the exactness degree in the integration of
the power function $f_{2}$ in (16) on the three polyhedra of Figure 1:
uncompressed (blue) and compressed (brown) tetrahedra-based rules, tetrahedra-
free rule(yellow).
Figure 6: Relative errors versus the exactness degree in the integration of
the distance function $f_{3}$ in (16) on the three polyhedra of Figure 1:
uncompressed (blue) and compressed (brown) tetrahedra-based rules, tetrahedra-
free rule(yellow).
deg | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$
---|---|---|---|---|---|---|---|---|---|---
$N$ | $4$ | $10$ | $20$ | $35$ | $56$ | $84$ | $120$ | $165$ | $220$ | $286$
$\Omega_{1}$ | $31$ | $77$ | $160$ | $286$ | $447$ | $676$ | $969$ | $1327$ | $1765$ | $2286$
$\Omega_{2}$ | $28$ | $78$ | $154$ | $273$ | $450$ | $669$ | $956$ | $1313$ | $1757$ | $2287$
$\Omega_{3}$ | $29$ | $77$ | $156$ | $276$ | $446$ | $662$ | $950$ | $1312$ | $1753$ | $2285$
Table 2: Cardinality of the Halton pointsets from which tetrahedra-free rules
with $N$ nodes are obtained.
## 4 Acknowledgements
Work partially supported by the DOR funds and the biennial project BIRD 192932
of the University of Padova, and by the INdAM-GNCS 2022 Project “Methods and
software for multivariate integral models”. This research has been
accomplished within the RITA “Research ITalian network on Approximation”, and
within the UMI Group TAA “Approximation Theory and Applications” (A.
Sommariva).
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|
# Vertex Classification of Planar C-Polygons 111
Illya Ivanov and Cameron Strachan
###### Abstract
Given a convex domain $C$, a $C$-polygon is an intersection of $n\geq 2$
homothets of $C$. If the homothets are translates of $C$ then we call the
intersection a translative $C$-polygon. This paper proves that if $C$ is a
strictly convex domain with $m$ singular boundary points, then the number of
singular boundary points a $C$-polygon has is between $n$ and $2(n-1)+m$. For
a translative $C$-polygon we show the number of singular boundary points is
between $n$ and $n+m$.
## 1 Introduction
Let $\mathbb{E}^{2}$ denote the 2-dimensional planar Euclidean space, and
$\mathbb{S}^{1}$ the unit sphere inside $\mathbb{E}^{2}$. A set
$X\subseteq\mathbb{E}^{2}$ is convex when the line segment connecting any two
points in $X$ is contained in $X$. We denote the interior, boundary and convex
hull of $X$, as $\text{int}(X)$, $\mathrm{bd}(X)$ and $\rm conv(X)$
respectively. A set $C\subseteq\mathbb{E}^{2}$ is convex domain when it is
compact, convex, and has non-empty interior. If the relative interior of the
line segment connecting any two boundary points is contained in the interior
of $C$, then we call $C$ a strictly convex domain.
Let $C$ be a convex domain, the Gauss mapping of $C$, which is denoted as
$\Gamma_{C}$, is the mapping between the boundary of $C$ and $\mathbb{S}^{1}$
where each boundary point, $x$, gets mapped to the set of normal unit vectors
of the supporting lines of $C$ at the boundary point $x$. The Gauss mapping of
any convex domain is always surjective and functional. When the Gauss mapping
of $C$ is well defined we call $C$ smooth, and when the Gauss mapping is
injective this is equivalent to $C$ being strictly convex. When a convex
domain is both strictly convex and smooth we call it a smooth strictly convex
domain, which is equivalent to having a bijective function as a Gauss mapping.
Let $X_{i}$ be a subset of $\mathbb{E}^{2}$ for each $i\in\\{1,2,\dots,n\\}$,
we call $\cap_{i=1}^{n}X_{i}$ a reduced intersection when
$\cap_{i=1}^{n}X_{i}\subsetneq\cap_{i=1,i\neq j}^{n}X_{i}$ for every
$j\in\\{1,2,\dots,n\\}$. We also denote $X_{1}+X_{2}=\\{x_{1}+x_{2}|x_{1}\in
X_{1},x_{2}\in X_{2}\\}$ and $\lambda X_{1}=\\{\lambda x_{1}|x_{1}\in
X_{1}\\}$ where $\lambda$ is a positive real.
###### Definition 1.
We call a finite intersection of sets, $\cap_{i=1}^{n}X_{i}$ a proper
intersection when $\cap_{i=1}^{n}X_{i}$ has non-empty interior and is a
reduced intersection. We call $\cap_{i=1}^{n}X_{i}$ an improper intersection
otherwise.
The main purpose of this paper is to provide a somewhat analogous result to
the famous upper bound theorem of McMullen [1], in the planar case. Given a
convex polytope in $d$-dimensional space, made by intersecting $n$ halfspaces,
the upper bound theorem provides sharp upper bounds for the number of
$i$-dimensional faces this polytope can have for $i\in\\{0,1,\dots,d-1\\}$. We
investigate the same question one could ask about $C$-polygons which are
defined as follows. It should be noted that the results and definitions in
this paper are strictly planar, although generalization to higher dimensions
can naturally be defined.
###### Definition 2.
Given a convex domain, $C\subseteq\mathbb{E}^{2}$, a set of $n$ points
$\\{x_{1},x_{2},\dots,x_{n}\\}\subseteq\mathbb{E}^{2}$, and a set of $n$
positive scalars
$\\{\lambda_{1},\lambda_{2},\dots,\lambda_{n}\\}\subseteq\mathbb{R}$ where
$n\geq 2$, we denote $H_{i}=x_{i}+\lambda_{i}C$ and $T_{i}=x_{i}+C$, and
define a $C$-polygon, $H$, and a translative $C$-polygon, $T$, as the
following intersections if these intersections are proper.
$H:=\cap_{i=1}^{n}H_{i}=\cap_{i=1}^{n}x_{i}+\lambda_{i}C~{}~{}\text{ and
}~{}~{}T:=\cap_{i=1}^{n}T_{i}=\cap_{i=1}^{n}x_{i}+C$
We call each $H_{i}$ or $T_{i}$ for $i\in\\{1,2,\dots n\\}$ a generating
homothet or generating translate respectively.
When $C$ is a ball, then the definitions of $C$-polygons and translative
$C$-polygons, define classical notions of generalized ball polygons and ball
polygons [3], respectively. We investigate a somewhat dual version of [2] and
study the complexity of the boundary structure of $C$-polygons and translative
$C$-polygons, and how varied the boundary structure can be if we fix the
number of generating translates or homothets. Specifically, we give upper and
lower bounds on how many singular boundary points these objects can have when
$C$ is a strictly convex domain. We call these singular boundary points the
vertices of the $C$-polygon and denote the set of them as $Vert(H)$. The
following is the main result established in this paper.
###### Theorem 1.
Let $T=\cap_{i=1}^{n}T_{i}$ and $H=\cap_{i=1}^{n}H_{i}$ be a translative
$C$-polygon and $C$-polygon respectively. If $C$ is a strictly convex domain
with $m$ singular boundary points then
$n\leq|Vert(T)|\leq n+m,\text{ and }n\leq|Vert(H)|\leq 2(n-1)+m.$
Our general approach in proving Theorem 1 is to handle the case where $m=0$,
or equivalently the case where $C$ is smooth, and then extend this to the case
where $m$ is positive. In section 2, we establish some basic properties of
$C$-polygons where $C$ is a smooth strictly convex domain. In section 3 we
prove Theorem 1 when $m=0$, and in section 4 we prove the case of Theorem 1
where $m$ is positive. Finally in section 5 we explore the sharpness of the
bounds established in Theorem 1.
## 2 Basic Properties for C-polygons where C is a smooth strictly convex
domain
We start with defining a natural face structure for $C$-polygons, where $C$ is
a smooth strictly convex domain. It suffices to define them for $C$-polygons
as translative $C$-polygons are a particular example of $C$-polygons. Let
$H=\cap_{i=1}^{n}H_{i}$ be a $C$-polygon, each generating homothet will be a
smooth strictly convex domain, which implies $H$ must also be a strictly
convex domain. Since $H$ is a strictly convex domain, its boundary can be
separated into singular points and smooth boundary arcs between singular
points, we define these to be vertices and edges respectively of our
$C$-polygon. However, in order to prove our result we come up with a more
useful equivalent characterization of vertices and edges. We can do this
because $H$ is not merely a strictly convex domain, it is a proper
intersection of $n$ homothets of $C$, which allows for the equivalence of the
following definitions with vertices and edges described above.
###### Definition 3.
Given a generating homothet $H_{j}$ of a $C$-polygon $H$, the edge family of
$H_{j}$ is denoted and defined as
$\mathcal{E}_{j}=(\mathrm{bd}(H)\cap\mathrm{bd}(H_{j}))\setminus A.$
The intersection $\mathrm{bd}(H)\cap\mathrm{bd}(H_{j})$ is a union of
maximally connected closed boundary arcs of $H_{j}$ and the set $A$ contains
all the arcs that are singleton points. We call each maximally connected
closed boundary arc of $\mathcal{E}_{j}$ an edge of $H$. We further denote
$E_{j}$ as the set whose elements are all the edges in $\mathcal{E}_{j}$.
###### Definition 4.
A point $v\in\mathrm{bd}(H)$ is a vertex if it lies in the boundary of at
least two generating homothets.
(a)
(b)
(c)
Figure 1: Three examples of $C$-polygons.
Diagram (c) in Figure 1 shows that there can be cases in which the non-
singleton maximally connected component of Definition 3 comes into play. If an
edge family has more than one edge in it we call it a multi-edge family, and
if it has only one edge we call it a singleton-edge family. In order to show
these definitions of vertices and edges align with singular points and smooth
boundary arcs of $H$ between singular boundary points, we need the following
lemma.
###### Lemma 2.
Suppose $H_{1}$ and $H_{2}$ are two homothets of a strictly convex domain,
$C$, whose intersection is proper, then
$|\mathrm{bd}(H_{1})\cap\mathrm{bd}(H_{2})|=2.$
###### Proof.
Let $H_{1}=x_{1}+\lambda_{1}C$ and $H_{2}=x_{2}+\lambda_{2}C$, first we
consider the case when $\lambda_{1}\neq\lambda_{2}$. In this case we can,
without loss of generality, choose an origin, $\mathbf{o}\in\mathbb{E}^{2}$,
such that $H_{2}=\lambda H_{1}$ for some $\lambda>1$. Since $H_{1}\cap H_{2}$
is a proper intersection, $\mathbf{o}$ must be outside $H_{1}$, and
$\mathrm{bd}(H_{1})\cap\mathrm{bd}(H_{2})$ must contain at least two points.
If the intersection of the boundary of the two homothets has cardinality
greater than two, then we have distinct
$q,r,s\in\mathrm{bd}(H_{1})\cap\mathrm{bd}(H_{2})$; notice that these points
can’t be colinear as they are boundary points of a strictly convex domain.
Naturally, $\lambda q,\lambda r,\lambda s\in\mathrm{bd}(\lambda
H_{1})=\mathrm{bd}(H_{2})$.
Since $\mathbf{o}\notin H_{1}$, one of the points $q,r,s$ must belong to the
positive hull of the other two. Without loss of generality, suppose
$s=\mu_{1}q+\mu_{2}r$ where $\mu_{1},\mu_{2}\geq 0$. In fact
$\mu_{1},\mu_{2}>0$, suppose for example $\mu_{2}=0$, then we would have the
points $q,\mu_{1}q,\lambda q,\lambda\mu_{1}q$ all lie on a straight line and
also in $\mathrm{bd}(H_{2})$, a contradiction since at least three out of
these four points must be distinct and $H_{1}\cap H_{2}$ is strictly convex.
Also note that $\mu_{1}+\mu_{2}\neq 1$, since $q,r,s$ aren’t colinear. Then
one can see that if $\mu_{1}+\mu_{2}<1$, then $\lambda s\in\text{int}(\rm
conv\left\\{q,r,s,\lambda q,\lambda r\right\\})$ and cannot lie in
$\mathrm{bd}(H_{2})$. If in turn $\mu_{1}+\mu_{2}>1$, then $s$ lies in the
interior of the convex hull of the other five points, and cannot be in
$\mathrm{bd}(H_{2})$.
For the other case suppose $\lambda_{1}=\lambda_{2}$, and let $H_{2}=H_{1}+x$.
Let $y$ be a normal vector to $x$. The inner product values $\langle
q,y\rangle,\langle r,y\rangle,\langle s,y\rangle$ are all distinct, otherwise
four of the points $q,r,s,q+x,r+x,s+x$ would lie on a same line and in the
boundary of $H_{1}$. Without loss of generality, suppose $\langle
q,y\rangle<\langle r,y\rangle<\langle s,y\rangle$. Then
$r=\mu_{1}q+\mu_{2}s+kx$ for some $k\neq 0$ and some $\mu_{1},\mu_{2}\in(0,1)$
such that $\mu_{1}+\mu_{2}=1$. If $k<0$, then $r+x\in\text{int}(\rm
conv\left\\{r,q,q+x,s,s+x\right\\})$, and if $k>0$, then $r\in\text{int}(\rm
conv\left\\{r+x,q,q+x,s,s+x\right\\})$. ∎
We can immediately see that the two points in
$\mathrm{bd}(H_{1})\cap\mathrm{bd}(H_{2})$, described in Lemma 2, are singular
points in the boundary of $H_{1}\cap H_{2}$. If they weren’t singular then the
Gauss image of one intersection point,
$x\in\mathrm{bd}(H_{1})\cap\mathrm{bd}(H_{2})$, with respect to the two
homothets, will be the same and a singleton unit vector. Since these homothets
are strictly convex their Gauss images will be injective, and hence the
inverse images of the point $x$ from the two homothets to $C$ will be the
same. This implies that the homothets are subsets of each other a
contradiction to the fact their intersection was assume to be proper.
Turning onto the implications Lemma 2 has to $C$-polygons, first we can
immediately see that any $C$-polygon will have a finite amount of vertices. As
a vertex is a boundary point that is in the boundary of two generating
homothets, and so the set of vertices is a subset of the set of all boundary
intersections between each pair of homothets. Which if we have $n$ generating
homothets we have at most $2\binom{n}{2}$ many boundary intersections between
all the pairs of generating homothets implying $|Vert(H)|\leq 2\binom{n}{2}$.
Since we are in the plane, this also implies the number of edges a $C$-polygon
has is finite.
We can also notice that if we have a boundary point $x\in\mathrm{bd}(H)$, then
it must be in the boundary of some generating homothet $H_{j}$. This means we
can classify boundary points of $H$ into two cases, the boundary points that
are in the boundary of only one generating homothet, and boundary points that
are in the boundaries of multiple generating homothets. We will soon see that
the latter will be vertices and the former will be elements in the relative
interior of edges. The next two lemmas establish that vertices are indeed
singular points and edges are smooth boundary arcs of $H$ between vertices.
###### Lemma 3.
Let $H$ be a $C$-polygon where $C$ is a smooth strictly convex domain, then a
point $v\in\mathrm{bd}(H)$ is a vertex if and only if it’s a singular point of
$H$.
###### Proof.
Suppose $v\in\mathrm{bd}(H)$ is a vertex, this means it is in the boundary
intersection of two generating homothets call them $H_{i}$ and $H_{j}$. By
construction we have that $H\subseteq H_{i}\cap H_{j}$ and that $v$ is a
boundary element of both sets. From this we obtain that $\Gamma_{H_{i}\cap
H_{j}}(v)\subseteq\Gamma_{H}(v)$, as supporting lines of $H_{i}\cap H_{j}$
will support subsets of $H_{i}\cap H_{j}$ that they intersect with. We
established $\Gamma_{H_{i}\cap H_{j}}(v)$ is not a singleton point and hence
$\Gamma_{H}(v)$ will also not be a singleton, showing that vertices are
singular points of $H$.
Conversely suppose $v\in\mathrm{bd}(H)$ is a singular point, then as it’s a
boundary point it must be either a boundary element of one generating homothet
or multiple. If it’s the latter we are done, so suppose this singular boundary
point was only in the boundary of one generating homothet $H_{j}$, and in the
interior of all the other generating homothets. Then the boundary points of
$H$ arbitrarily close to this point must also be only contained in the
boundary of $H_{j}$, implying $H_{j}$ is not smooth, a contradiction.
∎
###### Lemma 4.
Edges are smooth boundary arcs between vertices and vice versa.
###### Proof.
For the forward implication consider an edge in an edge family, this is a non-
singleton maximally connected boundary arc of some homothet that is contain in
$H$. Since this arc is in the boundary of one homothet it must be in the
boundary of $H$, this arc is also smooth as $C$ is smooth and it’s end point
must be vertices otherwise it would not be maximally connected. For the
reverse implication consider an arbitrary smooth boundary arc between two
vertices $v_{1},v_{2}\in Vert(H)$, and an arbitrary point, $x$, in this arc.
It is a smooth boundary point by assumption which implies $x$ must only be
contained in the boundary of one homothet, $H_{j}$. Notice that since the arc
is smooth, if one point in this smooth arc is in the boundary of only one
homothet, the entire smooth arc must be as well. If there was a transition to
another generating homothet along this arc, we must have a transition point.
This is because boundaries are closed, and so their intersection must have an
overlapping point, and that point would be a vertex and thus singular. So the
entire arc belongs to only one homothet, this arc is also contain in the
boundary of $H$ and is maximally connected and is not a singleton point which
means it’s an edge in the edge family $E_{j}$.
∎
## 3 Proof of Theorem 1, the case when m = 0
There are two statements to prove in Theorem 1, we will first prove the
$C$-polygon statement and then the translative $C$-polygon statement. Please
note throughout this entire section $C$ will be a smooth strictly convex
domain.
### 3.1 The C-polygon case.
Let $H=\cap_{i=1}^{n}H_{i}$ be a $C$-polygon, we need to show that $Vert(H)$
has cardinality between $n$ and $2(n-1)$. The lower bound is easily handled
with an upcoming observation, so we will focus on the upper bound. We will
prove this bound by induction on $n$, the number of generating homothets. We
start with the base case where $n=2$, we aim to show that $|Vert(H)|\leq
2=2(n-1)$. We have already seen in Lemma 2, that $|Vert(H)|=|Vert(H_{1}\cap
H_{2})|=|\mathrm{bd}(H_{1})\cap\mathrm{bd}(H_{2})|=2$ completing the base
case.
Our inductive hypothesis is that any $C$-polygon made up of $n$ generating
homothets will have at most $2(n-1)$ vertices. We aim to show that any
$C$-polygon made up of $n+1$ generating homothets will have at most $2n$
vertices. To that end let $H=\cap_{i=1}^{n+1}H_{i}$ be a $C$-polygon made up
of $n+1$ generating homothets.
The object we will be applying our inductive hypothesis on is denoted and
defined as $W_{j}=\bigcap_{i=1,i\neq j}^{n+1}H_{i}$. The notation $W_{j}$ is
meant to invoke that $H$ is without the $j$’th homothet in the intersection
creating it. This is in fact a $C$-polygon as $n\geq 2$, it has non-empty
interior as it’s a super-set of $H$ and is a reduced intersection by the
following lemma.
###### Lemma 5.
If $U_{i}\subseteq\mathbb{E}^{d}$ for all $i\in\\{1,2,\dots,n\\}$, and
$\cap_{i=1}^{n}U_{i}$ is reduced and $n\geq 2$, then for all
$j\in\\{1,2,\dots,n\\}$, $\cap_{i=1,i\neq j}^{n}U_{i}$ is reduced.
###### Proof.
Let $U=U_{1}\cap U_{2}\cap\dots\cap U_{n}$, and
$S\subseteq\left\\{1,\dots,n\right\\}$. We denote
$W_{S}=\bigcap_{i\in\left\\{1,\dots,n\right\\}\setminus S}U_{i},$
as defined before, $W_{\\{j\\}}=W_{j}$. Note that for any
$i\in\\{1,2,\dots,n\\}$ we have $U=W_{i}\cap U_{i}$.
By assumption, $U$ is reduced, which is equivalent to $U\neq W_{i}$ for any
$i\in\\{1,2,\dots,n\\}$. To show that $W_{i}$ is also reduced for any
$i\in\\{1,2,\dots,n\\}$, we need to demonstrate that for any distinct
$i,j\in\\{1,2,\dots,n\\}$, we have $W_{i}\neq W_{\left\\{i,j\right\\}}$.
Indeed if $W_{i}=W_{\left\\{i,j\right\\}}$, we would have $U=W_{i}\cap
U_{i}=W_{\left\\{i,j\right\\}}\cap U_{i}=W_{j}$. Thus, we contradict the
initial assumption that $U$ is reduced.
∎
An important property of $W_{j}$ is, if $H_{j}$ is a generating homothet of a
$C$-polygon $H$, then
$\mathrm{bd}(H_{j})\cap\mathrm{int}(W_{j})\neq\emptyset$ (1)
This can easily be proven by contradiction due to the fact $H$ is a proper
intersection. Another important observation is that the amount of boundary
arcs of $H_{j}$ that cross the interior of $W_{j}$ is the amount of edges in
the edge family $E_{j}$. With this observation we can immediately prove the
lower bound in Theorem 1 as each edge family has cardinality at least 1 by
equation (1). Notice that any new vertices found in $H$ that are not in
$W_{j}$ must be in the relative boundary of the edges in the edge family
$E_{j}$.
$W_{j}$$H_{j}$
$W_{j}$$H_{j}$
‘
Figure 2: Two examples of how $H_{j}$ can intersect $W_{j}$.
To complete the inductive step we will show that every $C$-polygon will
contain a singleton-edge family. This would complete the induction because if
we have this, lets call the singleton-edge family $E_{k}$, then the
$\mathrm{bd}(H_{k})$ intersects the interior of $W_{k}$ in one and only one
boundary arc. This means that the new vertices added on to $W_{k}$ by the
inclusion of $H_{k}$ in the intersection to form $H$ will be at most two, with
this we obtain:
$|Vert(H)|\leq|Vert(W_{j})|+|Vert(H)\setminus Vert(W_{j})|\leq 2(n-1)+2=2n$
So all we need to show is that every $C$-polygon contains a singleton-edge
family. We do this by utilizing a notion of gaps which are defined as follows.
###### Definition 5.
Let $H_{j}$ be a generating homothet of $H$, then consider
$\mathrm{bd}(H_{j})\setminus\mathcal{E}_{j}$ which is a union of $|E_{j}|$
many maximally connected open boundary arcs of $H_{j}$. Let us denote these
open boundary arcs of our generating homothet by
$g_{1},g_{2},\dots,g_{|E_{j}|}$. We will call $\rm conv({\rm cl}(g_{k}))$ a
gap of $H_{j}$ for $k\in\\{1,2,\dots,|E_{j}|\\}$ and we denote and call
$G_{j}=\cup_{i=1}^{|E_{j}|}\\{conv({\rm cl}(g_{i}))\\}$ the gap family of
$H_{j}$.
Figure 3: Three examples of gap families of a homothet.
###### Lemma 6.
For any $j\in\\{1,2,\dots n+1\\}$:
$(\mathrm{bd}(H)\setminus E_{j})\subseteq G_{j}$
###### Proof.
What this lemma says is that the boundary arcs of $H$ connecting the edges
generated by a particular homothet, must lie in the gaps of that homothet.
This can be proven by observing that the line segment connecting the two
vertices bounding a gap must be in the interior of $H$ because $H$ is strictly
convex, and the boundary of $H$ cannot go outside of the boundary of $H_{j}$
since $H\subseteq H_{j}$.
∎
###### Lemma 7.
If $H_{i}$ and $H_{j}$ are two distinct generating homothets of $H$, then the
intersection of $\mathrm{bd}(H_{i})$ with the gap family of $H_{j}$ is non-
empty and is contained in a single gap of $H_{j}$.
###### Proof.
$H_{i}$ must contain the edge family $E_{j}$, and so when the
$\mathrm{bd}(H_{i})$ intersects a gap, its boundary must enter and exit the
gap from the boundary of $H_{j}$. The homothets cannot be tangent to each
other as that would contradict Lemma 2, so when the $\mathrm{bd}(H_{i})$
intersects a gap it pierces the interior of it, and the $\mathrm{bd}(H_{i})$
must intersect at least one gap in order for the intersection to be reduced.
So every gap $\mathrm{bd}(H_{i})$ intersects generates two intersection points
in $\mathrm{bd}(H_{i})\cap\mathrm{bd}(H_{j})$, but of course Lemma 2 implies
this is at most two proving the lemma.
Figure 4: Example of what would happen if a homothet intersected multiple
gaps.
∎
As a result of the previous 2 lemmas we see the following.
###### Corollary 8.
No two edges in the same edge family may lie in different gaps of a gap
family.
It is with this final corollary that we are able to show that every
$C$-polygon will have a singleton-edge family. First as $n+1\geq 3$ we have at
least three edge families in $H$. Consider an arbitrary edge family $E_{k}$,
if this is a singleton-edge family we are done so suppose it’s a multi-edge
family, consider an arbitrary gap of the homothet $H_{k}$. This gap must have
at least one edge in it as the boundary of $H$ connecting the two vertices
bounding a gap, is a closed connected curve made out of edges. These edges
connecting the vertices must also be contained in the gap of $H_{k}$ according
to Lemma 6.
Take the edge adjacent to one of the vertices bounding the gap, that is also
contained in the gap of $H_{k}$ and let $E_{l}$ be the edge family of this
edge. If this is a singleton-edge family we are done, if it’s a multi-edge
family then the edges of this edge family must be contained in the gap of
$H_{k}$ by Corollary 8. So there is a gap of $H_{l}$ that is contained in the
gap of $H_{k}$ which we can repeat this process with. Since we have a finite
amount of edges this process must end and so we must obtain a singleton-edge
family eventually.
$H_{k}$$H_{l}$ Figure 5: Depiction of the process of how to find a singleton-
edge family.
The following theorem is the most general statement this section has proven
for the case where $m=0$. Notice we didn’t use the fact that the generating
homothets of our $C$-polygon are homothets of $C$ besides in deriving Lemma 2.
So if we assume our intersection satisfies Lemma 2, and remove the restriction
that the domains we are intersecting are homothets, our proof will remain
identical.
###### Theorem 9.
If $X=\cap_{i=1}^{n}X_{i}$ where $X_{i}$ is a smooth strictly convex domain
for each $i\in\\{1,2,\dots,n\\}$, where $X$ is a proper intersection, and for
every distinct $i,j\in\\{1,2,\dots,n\\}$,
$|\mathrm{bd}(X_{i})\cap\mathrm{bd}(X_{j})|=2$, then the cardinality of
singular points on $X$ is between $n$ and $2(n-1)$.
### 3.2 The translative C-polygon case:
First it should be noted that all the above results established in section 2
and 3.1 apply to translative $C$-polygons as well, this implies that a
translative $C$-polygon has at least as many edges as generating translates,
since every edge family is non-empty. This implies the desired lower bound for
translative $C$-polygons.
For the upper bound we prove it by induction in an analogous way to the
$C$-polygon case done previously, as a result we omit some details. Our base
case is also handled by Lemma 2, so let $T=\cap_{i=1}^{n+1}T_{i}$ be a
translative $C$-polygon where $C$ is a smooth strictly convex domain. Our
inductive hypothesis is that any translative $C$-polygon made up of $n$
generating translates will have at most $n$ vertices, we wish to show $T$ has
at most $n+1$ vertices. We already know there must be a singleton-edge family
by section 3.1, let $E_{j}$ be that singleton-edge family. In order to
complete the proof we only need to show that the inclusion of $T_{j}$ to
$W_{j}$, to form $T$, net total increases the vertex count by at most one. We
have seen that $T_{j}$ is an edge family that increases our count of vertices
by at most two in section 3.1. However, in our translative case we will show
we must exclude at least one of the vertices of $W_{j}$ and hence net total
can only increase our vertex count by at most one. We do this with the help of
the following lemma.
###### Lemma 10.
Let $T_{1}$ and $T_{2}$ be translates of $C$ that produce a proper
intersection, then the boundary of $T_{1}$ in the exterior of $T_{2}$ has a
Gauss image that contains a hemisphere.
###### Proof.
$\mathbb{S}^{1}$$\tau$$\rho$$H_{\rho}$$H_{-\rho}$ Figure 6: Intersection
between two translates of $C$.
First let $\tau\in\mathbb{S}^{1}$ be the direction of the translation to get
from $T_{1}$ to $T_{2}$, and let $\pm\rho\in\mathbb{S}^{1}$ be the two
perpendicular vectors of $\tau$. Since $C$ has a bijective function for its
Gauss mapping, $T_{1}$ and $T_{2}$ will each have two unique boundary points
with Gauss image $\pm\rho$. Notice by the choice of $\rho$ the support lines
will be equal, and both translates will be contained in the band between these
two support lines, which is depicted in Figure 6. Each open boundary arc of
$T_{1}$ and $T_{2}$ between the two points of contact with the support lines,
will have a Gauss image of an open hemisphere on $\mathbb{S}^{1}$ centered at
$\pm\tau$. Lemma 2 implies only two of these arcs have a boundary intersection
with the other translate proving the lemma.
∎
This lemma also implies that the relative interior of any edge has a Gauss
image that is contained in an open hemisphere. Since $T_{j}$ has one only edge
in its edge family, it has only one maximally connected boundary arc that
intersects the interior of $W_{j}$. This boundary arc will enter and exit the
boundary of $W_{j}$ in two places and the only way it can include all vertices
of $W_{j}$ is if the boundary of $T_{j}$ enters and leaves the boundary of
$W_{j}$ in the relative interior of same edge.
This would contradict Lemma 10 as the relative interior of edges have a Gauss
image that is contained in a hemisphere, but the pairwise intersection between
$T_{j}$ and the translate that generates the edge of $W_{j}$ that
$\textrm{bd}(T_{j})$ intersects, violates Lemma 10. This proves the induction
as we know $W_{j}$ has $n$ vertices by our inductive hypothesis and the
inclusion of $T_{j}$ to $W_{j}$ increases our vertex count by at most one
implying $|Vert(T)|=|Vert(W_{j}\cap T_{j})|\leq n+1$.
$W_{j}$$H_{j}$ Figure 7: Diagram depicting that in order to contain all
vertices of $W_{j}$ we need a homothetic copy of $C$ and not a translative
one.
## 4 Proof of Theorem 1: the positive m case
When we consider a $C$-polygon, $H=\cap_{i=1}^{n}H_{i}$, where $C$ has $m$
singular points, things behave similarly to the smooth case. Notice Lemma 2
holds for strictly convex domains so we know that the boundary intersection
between pairs of homothets will have a cardinality of two, and these two
points are singular in the intersection body. We define edge families
identically although now edges in edge families may not be smooth as boundary
arcs of $C$ may not be smooth. This observation lets us classify vertices of
$H$ into one of two kinds. Pairwise vertices are singular points of $H$ that
are in the boundary of two generating homothets. Inherited vertices are
singular points that are in the relative interior of an edge. Notice during
the proofs in section 3 our arguments did not use the fact that our edges were
smooth. This implies that we have at most $2(n-1)$ many pairwise vertices in
the $C$-polygon case, and $n$ many pairwise vertices in the translative
$C$-polygon case. Notice that each homothet has a non-empty edge family
implying our lower bound for both the $C$-polygon and translative $C$-polygon
cases.
Edges in an edge family will be non-singleton maximally connected components
of a generating homothet. We can view these edges as maximally connected
components of the boundary of $C$, by inverting the homothet map. The
inherited singular points on an edge will corresponds with a singular point of
$C$ since the inverted homothetic image of $C$ preserve the Gauss image. Since
$H$ is strictly convex it’s Gauss mapping is injective which implies when we
represent the edges of $H$ on $C$, they will not overlap. This implies $H$ has
at most $m$ inherited vertices completing the proof.
## 5 Sharpness of Results
In this section we discuss the sharpness of the bounds established in Theorem
1. Given a class of convex domains $\mathcal{C}$, and a bound on the number of
vertices a $C$-polygon has when $C\in\mathcal{C}$, we say the bound is
strongly sharp when for all $C\in\mathcal{C}$ and every natural $n\geq 2$
there is a $C$-polygon with $n$ generating homothets that realizes the bound.
If there exists $C\in\mathcal{C}$ such that for every $n\geq 2$ we can find a
$C$-polygon with $n$ generating homothets that can realize the bound, we call
the bound weakly sharp.
###### Claim 11.
The upper bound of $2(n-1)+m$ vertices for $C$-polygons where $C$ is a
strictly convex domain is strongly sharp.
###### Proof.
Given any strictly convex domain, $C$, with $m$ singular points, we must have
a smooth strictly convex boundary arc of $C$. Pick $n-1$ many points on the
relative interior of this boundary arc, by expanding $C$ with a homothet
centre at each of these boundary points we can create $n-1$ homothets to be
arbitrarily close to the supporting lines of the $n-1$ boundary points of $C$.
Then we move these $n-1$ homothets arbitrarily inward to pierce the interior
of $C$ and create $2$ pairwise vertices with each homothet giving us
$2(n-1)+m$ vertices. Figure 8 depicts the construction.
Figure 8: Sharpness of upper bound construction for $C$-polygons where $C$ is
a strictly convex domain.
∎
###### Claim 12.
The lower bound of $n$ vertices for $C$-polygons and translative $C$-polygons,
where $C$ is a strictly convex domain, is weakly sharp. However, the lower
bound of $n$ vertices for $C$-polygons and translative $C$-polygons is
strongly sharp when $C$ is a smooth strictly convex domain.
###### Proof.
The latter part of the claim is trivial as any translative $C$-polygon with
$n$ generating translates, where $C$ is a smooth strictly convex domain, will
have exactly $n$ vertices according to Theorem 1. For the former part of the
claim, the previous sentence shows that the bound is at least weakly sharp. So
we need only show this bound is not strongly sharp which means we need to show
there exists a strictly convex domain $C$ and a natural $n\geq 2$ where every
$C$-polygon with $n$ generating homothets has more than $n$ vertices. We need
only choose $n=2$ and $C$ as depicted in Figure 9. We construct $C$ by
intersecting three circles, this is a ball-polygon and we choose it such that
the antipodal point of the Gauss image of any boundary point of $C$ that is
smooth, is contained in the Gauss image of a vertex of $C$. By Lemma 2 we know
any intersection of two homothets of $C$ will have two pairwise vertices. So
we need to show every $C$-polygon generated by two homothets of $C$ will have
at least one inherited vertex. The following statement completes the proof and
is left to the reader. Given an intersection of two homothets, $H_{1}\cap
H_{2}$, of a strictly convex domain, there exists a point in the relative
interior of each of the two edges, call them $x_{1}\in\mathrm{bd}(H_{1})$ and
$x_{2}\in\mathrm{bd}(H_{2})$, such that $\Gamma_{H_{1}}(x_{1})$ contains a
point that is antipodal to a point in $\Gamma_{H_{2}}(x_{2})$.
Figure 9: Construction of $C$.
∎
###### Claim 13.
For every natural $n\geq 2$ there exist a convex domain, $C$, and a
$C$-polygon, $H,$ with $n$ generating homothets such that $H$ has zero
vertices.
###### Proof.
First notice that this condition on the bound of zero vertices is weaker than
weak sharpness. Given an arbitrary natural $n\geq 3$ we choose our convex
domain $C$ to be a polygon with $n$ vertices that have sufficiently small
rounded corners. To construct a $C$-polygon with $n$ generating homothets we
take $C$ and enlarge the other $n-1$ copies of $C$ to smoothly transition with
$n-1$ many smoothed corners of $C$ as depicted in Figure 10 for the case where
$n=3$ and $4$. For the case where $n=2$ two translates of the rounded square
can construct a translative $C$-polygon with zero vertices.
Figure 10: Depiction of the construction of a smooth $C$-polygon.
∎
We end this paper with an open problem.
###### Problem 14.
If $C$ is a convex domain with $m$ singular points, then the number of
singular points a $C$-polygon with $n$ generating domains will have is at most
$2(n-1)+m$.
## References
* [1] P. McMullen, On the upper-bound conjecture for convex polytopes, J. Combinat. Theorey, Ser.B. 10 (1971), 187–200.
* [2] P. K. Agarwal, J. Pach, and M. Sharir, State of the Union (of Geometric Objects), Technical report, American Mathematical Society (2008)
* [3] K. Bezdek, Z. Langi, M Naszodi, and P. Papez, Ball-Polyhedra, Discrete Comput Geom 38 (2007), 201–230.
Illya Ivanov
Department of Mathematics and Statistics, University of Calgary, Canada
E-mail<EMAIL_ADDRESS>
Cameron Strachan
Department of Mathematics and Statistics, University of Calgary, Canada
E-mail<EMAIL_ADDRESS>
|
# Values of binary partition function represented by a sum of three squares
Bartosz Sobolewski and Maciej Ulas
###### Abstract.
Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of
$n$ with parts being powers of 2, where each part can take $m$ colors. We show
that if $m=2^{k}-1$, then there exists the natural density of integers $n$
such that $b_{m}(n)$ can not be represented as a sum of three squares and it
is equal to $1/12$ for $k=1,2$ and $1/6$ for $k\geq 3$. In particular, for
$m=1$ the equation $b_{1}(n)=x^{2}+y^{2}+z^{2}$ has a solution in integers if
and only if $n$ is not of the form $2^{2k+2}(8s+2t_{s}+3)+i$ for $i=0,1$ and
$k,s$ are non-negative integers, and where $t_{n}$ is the $n$th term in the
Prouhet-Thue-Morse sequence. A similar characterization is obtained for the
solutions in $n$ of the equation $b_{2^{k}-1}(n)=x^{2}+y^{2}+z^{2}$.
###### Key words and phrases:
binary partitions, recurrence sequence, automatic sequence, sums of squares
The research of the authors is supported by the grant of the National Science
Centre (NCN), Poland, no. UMO-2019/34/E/ST1/00094
## 1\. Introduction
Let $\mathbb{N}$ be the set of non-negative integers and $\mathbb{N}_{+}$ the
set of positive integers. Moreover, for a given $n\in\mathbb{Z}$ we define the
2-adic valuation of $n$ as
$\nu_{2}(n)=\max\\{k\in\mathbb{N}:\;2^{k}|n\\},$
with the convention that $\nu_{2}(0)=+\infty$.
The problem of representation of integers by quadratic forms or, more
generally, by forms or polynomials in many variables, is a classical one. As
was proved by Lagrange in 1770, each non-negative integer can be represented
as a sum of four squares. On the other hand, there are infinitely many
positive integers which cannot be represented by three squares. More
precisely, in 1798 Legendre proved that a non-negative integer $N$ can be
represented as
$N=x^{2}+y^{2}+z^{2}$
for some $x,y,z\in\mathbb{Z}$ if and only if $N$ is not of the form
$4^{r}(8s+7)$ for $r,s\in\mathbb{N}$. In particular, the natural density of
the set of integers that cannot be represented by a sum of three squares is
equal to $1/6$. This raises an interesting question whether for a given
sequence of integers $(u_{n})_{n\in\mathbb{N}}$ there exist infinitely many
solutions of the Diophantine equation
(1) $u_{n}=x^{2}+y^{2}+z^{2}.$
To characterize the solutions of (1) it is necessary to have a good
understanding of the 2-adic behavior, or, to be more precise, the 2-adic
valuation of the terms of the sequence $(u_{n})_{n\in\mathbb{N}}$.
Especially interesting is the case where $u_{n}$ has a combinatorial meaning,
i.e., $u_{n}$ counts some discrete objects or structures. Equation (1) with
$u_{n}=\binom{2n}{n}$ was investigated by Granville and Zhu in [9]. They
characterized those $n\in\mathbb{N}$ such that (1) has no solutions in
$x,y,z$. In particular, the set of integers $n$, for which $\binom{2n}{n}$ can
be represented as a sum of three squares, has asymptotic density $7/8$ in the
set of all natural numbers (for an early study of this problem, see also the
paper of Robbins [18]). In the same paper they also obtained a
characterization of $n\in\mathbb{N}$ such that $n!$ is not a sum of three
squares, given in terms of existence of certain patterns in the binary
expansion of $n$. A different approach to this problem (via substitutions) was
presented by Deshouillers and Luca [8]. They showed that the natural density
of those $n$ such that $n!=x^{2}+y^{2}+z^{2}$ exists and is equal to $7/8$.
More precisely, they proved that
$\\#\\{n\leq x:\;n!\;\mbox{is a sum of three
squares}\\}=\frac{7}{8}x+O(r(x)),$
where $r(x)=x^{2/3}$. The error term was improved by Hajdu and Papp [15] to
$r(x)=x^{1/2}\log^{2}x$, and recently by Burns to $r(x)=x^{1/2}$ (see the
preprint [6]). On the other hand, Robbins obtained a precise characterzation
of the solutions in $n$ of (1) in the case where $u_{n}$ is the $n$-th term of
Fibonacci or Lucas sequence [17].
In this paper, we follow the same line of research and consider, in
particular, equation (1) with $u_{n}$ being the binary partition function
$b(n)$. More precisely, $b(n)$ counts the number of partitions of
$n\in\mathbb{N}$ into parts being powers of two. For example, $b(4)=4$ because
$4=2^{2}=2+2=1+1+2=1+1+1+1$
are all possible representations of 4 as a sum of powers of $2$. The sequence
$(b(n))_{n\in\mathbb{N}}$ was already introduced by Euler. However, it seems
that the first serious study of its properties was performed by Churchhouse
[7] in 1969. He computed the 2-adic valuation of $b(n)$. Further results,
motivated by Churchhouse’s computations, were obtained independently by Gupta
[11, 12, 13] and Rødseth [19] (see also recent studies by Rødseth and Sellers
[20]).
Besides being interesting in its own sake, the study of the equation
$b(n)=x^{2}+y^{2}+z^{2}$
connects two different areas: the Diophantine equations and partitions theory.
According to our best knowledge, there is no result in the literature
providing a characterization of the solutions of equation (1) with $u_{n}$
being a partition function of sub-exponential growth. Recall that $b(n)$ is
indeed of sub-exponential growth. More precisely, Mahler [16] proved that
$\log_{2}b(n)\sim\frac{1}{2}(\log_{2}n)^{2}$. Our study can be seen as a
continuation and extension of recent research concerning solvability of
Diophantine equations involving partitions, conducted by Tengely and Ulas
[21].
In the same context, we also study the $m$-colored binary partition function
$b_{m}(n)$, which counts binary partitions of $n$, where each part can have
one of $m\geq 1$ colors. In particular, we have $b(n)=b_{1}(n)$. The study of
arithmetic properties of this function was initiated in a paper of Gawron,
Miska and Ulas [10], where several useful results were obtained.
Let us describe the content of the paper in some detail. In Section 2 we
recall some basic properties and results concerning the function $b_{m}(n)$.
The main goal of the present investigation, pursued in Sections 3–5, is to
obtain an explicit characterization of the set
$S_{m}=\\{n\in\mathbb{N}:b_{m}(n)\neq x^{2}+y^{2}+z^{2}\text{ for any
}x,y,z\in\mathbb{Z}\\}.$
The reason for considering terms _not_ represented as a sum of three squares
is that the description turns out to be more concise (similarly as for non-
negative integers). We focus primarily on the case $m=2^{k}-1$ for
$k\in\mathbb{N}_{+}$ due to the fact that the $2$-adic valuation
$\nu_{2}(b_{2^{k}-1}(n))$ is known to be bounded [7, 10]. This allows us to
determine which terms $b_{2^{k}-1}(n)$ are of the form $4^{r}(8s+7)$ through
the reduction modulo a suitable power of $2$. The analysis is divided into
three cases: $k=1,k=2$, and $k\geq 3$, covered in Section 3, 4, and 5,
respectively. Based on these results, in Section 6 we give quite precise
bounds for the counting function of the set $S_{2^{k}-1}$, and determine its
natural density in the process. In the final section, we state some questions,
problems, and conjectures which may serve as a basis for further study. In
particular, we discuss the equation $b_{m}(n)=x^{2}+y^{2}+z^{2}$ when $m\neq
2^{k}-1$ as well as some interesting findings concerning the behavior of
$b_{m}(n)$ modulo powers of $2$. We also collect results of numerical
computations related to the equations $b(n)=x^{2}+y^{2}+z^{4}$ and
$b(n)=x^{2}+y^{2}$.
###### Remark 1.1.
In this paper, we focus on the problem of representation of $b_{2^{k}-1}(n)$
as a sum of three squares. However, it is possible to use our findings to
obtain similar results for other ternary quadratic forms $q$ such that the set
of $n\in\mathbb{N}$ represented by $q$ is given in terms of the binary
expansion of $n$. Such quadratic forms include, for example, the forms
$x^{2}+y^{2}+2z^{2},x^{2}+2y^{2}+2z^{2},x^{2}+2y^{2}+4z^{2}$.
## 2\. Preliminaries
In this section we collect known properties and results which will be used
throughout the paper. Recall that the ordinary generating function of the
sequence $(b(n))_{n\in\mathbb{N}}$ has the form
$B(x)=\prod_{n=0}^{\infty}\frac{1}{1-x^{2^{n}}}=\sum_{n=0}^{\infty}b(n)x^{n}.$
As a consequence, we see that $B(x)$ satisfies a Mahler-type functional
equation $(1-x)B(x)=B(x^{2})$. Comparing the coefficients on both sides, we
see that the sequence $(b(n))_{n\in\mathbb{N}}$ satisfies the recurrence:
$b(0)=b(1)=1$ and
$b(2n)=b(2n-1)+b(n),\quad b(2n+1)=b(2n).$
Churchouse [7] obtained a characterization of the $2$-adic valuation of the
terms $b(n)$.
###### Theorem 2.1.
For all $n\geq 2$ we have
$\nu_{2}(b(n))=\begin{cases}1&\text{if }\nu_{2}(n)\equiv 0\mkern
4.0mu({\operator@font mod}\mkern 6.0mu2),\\\ 2&\text{if }\nu_{2}(n)\equiv
1\mkern 4.0mu({\operator@font mod}\mkern 6.0mu2).\end{cases}$
Another useful result was obtained independently by Rødseth [19] and Gupta
[11], proving a conjecture of Churchhouse. More precisely, we have the
following theorem.
###### Theorem 2.2.
For all $s\in\mathbb{N}$ and odd $n\in\mathbb{N}$ the following congruence
holds:
$b(2^{s+2}n)\equiv b(2^{s}n)\mkern 4.0mu({\operator@font mod}\mkern
6.0mu2^{\mu(s)}),$
where $\mu(s)=\left\lfloor\frac{3s+4}{2}\right\rfloor$.
For $m\in\mathbb{N}_{+}$ we define the sequence $(b_{m}(n))_{n\in\mathbb{N}}$
as the convolution of $m$ copies of $(b(n))_{n\in\mathbb{N}}$. More precisely,
$b_{m}(n)=\sum_{i_{1}+\ldots+i_{m}=n}b(i_{1})\cdots b(i_{m}).$
It is clear that the generating function $B_{m}(x)$ of the sequence
$(b_{m}(n))_{n\in\mathbb{N}}$ is the $m$-th power of $B(x)$, and $b_{m}(n)$
also has a combinatorial interpretation. Indeed, $b_{m}(n)$ is the number of
binary partitions of $n$, where each part has one of $m$ possible colors. In a
recent paper by Gawron, Miska, and Ulas [10], it is proved that for
$m=2^{k}-1$ and $n\geq 2^{k}$ the 2-adic valuation of $b_{m}(n)$ belongs to
the set $\\{1,2\\}$ . More precisely, they gave the following characterization
of the $2$-adic valuation of the terms $b_{2^{k}-1}(n)$.
###### Theorem 2.3 (Theorem 4.6 in [10]).
Let $k\in\mathbb{N}_{+}$. For $n,i\in\mathbb{N}$ such that $i<2^{k+2}$ we have
$\nu_{2}(b_{2^{k}-1}(2^{k+2}n+i))=\begin{cases}\nu_{2}(b(8n))&\text{if }0\leq
i<2^{k},\\\ 1&\text{if }2^{k}\leq i<2^{k+1},\\\ 2&\text{if }2^{k+1}\leq
i<3\cdot 2^{k},\\\ 1&\text{if }3\cdot 2^{k+1}\leq i<2^{k+2}.\end{cases}$
In particular, $\nu_{2}(b_{2^{k}-1}(n))\in\\{0,1,2\\}$ and
$\nu_{2}(b_{2^{k}-1}(n))=0$ if and only if $n<2^{k}$.
The reciprocal of $B(x)$, denoted by
$T(x)=\frac{1}{B(x)}=\prod_{n=0}^{\infty}\left(1-x^{2^{n}}\right)=\sum_{n=0}^{\infty}t_{n}x^{n},$
is the ordinary generating function for the famous Prouhet-Thue-Morse sequence
$(t_{n})_{n\in\mathbb{N}}$ (PTM sequence for short). Recall that
$t_{n}=(-1)^{s_{2}(n)}$, where $s_{2}(n)$ is the number of $1$’s in the unique
expansion of $n$ in base 2. Equivalently, we have $t_{0}=1$ and
$t_{2n}=t_{n},\quad t_{2n+1}=-t_{n},\quad n\geq 0.$
The formula in Theorem 2.1 can then be written as
$\nu_{2}(b(n))=\frac{1}{2}|t_{n}-2t_{n-1}+t_{n-2}|$. We also consider a
variant of the PTM sequence, given by
$T_{n}=s_{2}(n)\mkern 4.0mu({\operator@font mod}\mkern 6.0mu2),$
i.e., the sequence $(T_{n})_{n\in\mathbb{N}}$ is related to the PTM sequence
by $t_{n}=1-2T_{n}$.
The PTM sequence is an example of an automatic sequence. More precisely, let
$k\geq 2$ be a fixed integer. A sequence $\mathbf{a}=(a_{n})_{n\in\mathbb{N}}$
is called $k$-automatic if its $k$-kernel, namely
$K_{k}(\mathbf{a})=\\{(a_{k^{j}n+i})_{n\in\mathbb{N}}:j\in\mathbb{N},0\leq
i<k^{j}\\},$
is a finite set. Equivalently, $\mathbf{a}$ is $k$-automatic if there exists a
deterministic finite automaton with output (DFAO) that reads the canonical
base-$k$ representation of $n$ and the outputs $a_{n}$. In the case of the PTM
sequence, it is clear that $K_{k}(\mathbf{t})=\\{\mathbf{t},-\mathbf{t}\\}$.
Equivalently, the sequence is generated by the DFAO in Figure 1. To generate
$t_{n}$, one moves between the states (symbolized by nodes) according to
subsequent digits in the binary representation of $n$. After the final digit
has been read, the DFAO returns the output corresponding to the current state.
For a detailed treatment of automatic sequences, we refer the reader to the
monograph by Allouche and Shallit [3].
$1$start$-1$0110 Figure 1. A DFAO generating the PTM sequence.
Finally, let us recall that the sequence ${\bf a}$ with values in
$\mathbb{Z}$-module $R$ is $k$-regular if there exist finitely many sequences
$\mathbf{a}_{i}=(a_{i}(n))_{n\in\mathbb{N}}$ with values in $R$ such that each
sequence in $K_{k}({\bf a})$ is a $\mathbb{Z}$-linear combination of the
$\bf{a}_{i}$. In other words, the $\mathbb{Z}$-submodule generated by the
$k$-kernel $K_{k}({\bf a})$. In particular, $k$-automatic sequences are
precisely $k$-regular sequences taking finitely many values. The class of
$k$-regular sequences with values in a ring $R$ has a ring structure itself. A
good introduction to the topic of regular sequences are the papers of Allouche
and Shallit [4, 5].
## 3\. The equation $b(n)=x^{2}+y^{2}+z^{2}$
We start with the characterization of the solutions (in variable $n$) of the
equation
$b(n)=x^{2}+y^{2}+z^{2}.$
Because the values $b(2n)$ and $b(2n+1)$ are equal, we restrict our attention
to even indices and consider the set
$S_{1}^{\prime}=\\{n\in\mathbb{N}:b(2n)\neq x^{2}+y^{2}+z^{2}\text{ for any
}x,y,z\in\mathbb{Z}\\}.$
The first few elements of $S_{1}^{\prime}$ are the following:
$10,18,34,40,58,66,72,90,106,114,130,136,154,160,170,178,202,210,226,\ldots.$
Using Theorem 2.1, we get the following characterization of $\nu_{2}(b(2n))$.
###### Proposition 3.1.
For all $n\in\mathbb{N}_{+}$ we have
$\nu_{2}(b(2n))=\begin{cases}1&\text{if }\nu_{2}(n)\equiv 0\pmod{2},\\\
2&\text{if }\nu_{2}(n)\equiv 1\pmod{2}.\end{cases}$
We can deduce that if $\nu_{2}(n)\equiv 0\pmod{2}$, then $b(2n)$ is a sum of
three squares. Hence, we only need to consider reduction modulo $32$ of
$b(4^{k}(8m+4))$, where $k,m\in\mathbb{N}$. More precisely, $b(4^{k}(8m+4))$
is a sum of three squares if and only if
(2) $b(4^{k}(8m+4))\equiv 28\pmod{32}.$
From Theorem 2.2 and the main result of Hirschhorn and Loxton [14] one can
extract suitable congruence relations, which reduce the general case to $k=0$
and describe the remaining terms $b(8m+4)$.
###### Proposition 3.2.
For all $m\in\mathbb{N}$ we have
$\displaystyle b(16m)$ $\displaystyle\equiv b(4m)\pmod{32},$ $\displaystyle
b(16m+4)$ $\displaystyle\equiv 4t_{m}\pmod{32},$ $\displaystyle b(16m+12)$
$\displaystyle\equiv 20t_{m}\pmod{32}.$
Using these relations, it is straightforward to describe the set consisting of
$n\in\mathbb{N}$ such that $b(2n)$ is (not) a sum of three squares.
###### Corollary 3.3.
The following conditions are equivalent:
1. (a)
The number $b(2n)$ is not a sum of three squares;
2. (b)
$n=2^{2k+1}(4s+1)$ for some $k,s\in\mathbb{N}$ such that $t_{s}=-1$;
3. (c)
$n=2^{2k+1}(8r+2t_{r}+3)$ for some $k,r\in\mathbb{N}$;
4. (d)
$\chi(n)=1$, where $\chi$ is defined by $\chi(0)=0$ and
$\chi(2n+1)=0,\quad\chi(4n)=\chi(n),\quad\chi(8n+2)=T_{n},\quad\chi(8n+6)=0.$
###### Proof.
As we have discussed earlier, $b(2n)$ is not a sum of three squares if and
only if $2n=4^{k}(8m+4)$ and (2) holds. By Proposition 3.2 this happens if and
only if $m$ is even and $t_{m}=t_{m/2}=-1$. Letting $m=2s$, we obtain the
equivalence of (a) and (b).
To prove that (b) is equivalent to (c), we use the following description from
[1]:
(3) $\displaystyle\\{n\in\mathbb{N}:\;T_{n}=0\\}$
$\displaystyle=\\{2m+T_{m}:\;m\in\mathbb{N}\\},$
$\displaystyle\\{n\in\mathbb{N}:\;T_{n}=1\\}$
$\displaystyle=\\{2m+1-T_{m}:\;m\in\mathbb{N}\\}.$
Hence, $t_{s}=-1$ if and only if $s=2r+1-T_{r}=2r+(t_{r}+1)/2$ for some
$r\in\mathbb{N}$, and our claim follows.
Finally, it is simple to check that the set on $n$ of the form given in (b) is
precisely $\\{n\in\mathbb{N}:\chi(n)=1\\}$. ∎
From the relation $b(2n+1)=b(2n)$ and part (c) of the corollary we get
$S_{1}=2S_{1}^{\prime}\cup(2S_{1}^{\prime}+1)=\\{2^{2k+2}(8r+2t_{r}+3)+i:k,r\in\mathbb{N},i\in\\{0,1\\}\\}.$
Furthermore, part (d) of the corollary directly shows that $S_{1}^{\prime}$
(and thus $S_{1}$) is a $2$-automatic set; i.e., its characteristic sequence
$(\chi(n))_{n\in\mathbb{N}}$ is $2$-automatic. A DFAO generating this sequence
is shown in Figure 2.
$0$start$0$$0$$0$$0$$1$$0$$0$$1$$0,1$$1$$1$$0$$0$$1$$1$$0$ Figure 2. A DFAO
generating $(\chi(n))_{n\in\mathbb{N}}$
We now turn to the problem of gaps between consecutive $n$ such that $b(2n)$
is a sum of three squares. More precisely, we define
$(f_{n})_{n\in\mathbb{N}}$ to be the increasing sequence consisting of the
elements of $S^{\prime}$. Let $(g_{n})_{n\in\mathbb{N}}$ be the sequence of
gaps, defined by
$g_{n}=f_{n+1}-f_{n}.$
In other words, $g_{n}$ is the distance between $n$th and $(n+1)$th $1$ in the
characteristic sequence $(\chi(n))_{n\in\mathbb{N}}$ (counting from $0$). The
following proposition shows that the gaps are bounded. Moreover, for each
possible gap length $g$ we provide in the proof an infinite set of
$n\in\mathbb{N}$ such that $g_{n}=g$.
###### Proposition 3.4.
For all $n\in\mathbb{N}$ we have
$g_{n}\in\\{6,8,10,16,18,24\\}$
and all possible values are attained infinitely often.
###### Proof.
Consider length $16$ subsequences $(\chi(16n+i))_{0\leq i\leq 15}$. A simple
case distinction together with the relations in Corollary 3.3(d) show that
there are only four possibilities, namely
$\begin{cases}1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0&\text{if }\chi(n)=1,\\\
0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0&\text{if }\chi(n)=0,T_{n}=1,\\\
0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0&\text{if }\chi(n)=0,T_{n}=0,2\mid n,\\\
0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0&\text{if }\chi(n)=0,T_{n}=0,2\nmid n.\\\
\end{cases}$
By inspecting the gaps within these subsequences and all their possible
concatenations, we can see that the gaps between subsequent $1$’s in
$(\chi(n))_{n\in\mathbb{N}}$ can only have lengths $6,8,10,16,18,24$.
It remains to show that each of these values is indeed attained infinitely
often. In Table 1 for each $g\in\\{6,8,10,16,18,24\\}$ we provide an infinite
set $I_{g}$ of indices $n$ such that $\chi(n)=\chi(n+g)=1$ and all the terms
inbetween are zero. We note that $I_{g}$ does not necessarily contain all such
indices $n$. The verification of each case is straightforward and left to the
reader. ∎
$g$ | $I_{g}$
---|---
$6$ | $\\{32m+2:T_{m}=1\\}$
$8$ | $\\{32m+10:T_{m}=0\\}$
$10$ | $\\{16m:\chi(m)=1\\}$
$16$ | $\\{64m+18:T_{m}=0\\}$
$18$ | $\\{32m+8:T_{m}=1\\}$
$24$ | $\\{256m+178:T_{m}=0\\}$
Table 1. Gaps between $1$’s in $(\chi(n))_{n\in\mathbb{N}}$
###### Remark 3.5.
Let us note that Hajdu and Papp proved that the gap sequence corresponding to
those values of $n$ such that $n!$ is a sum of three squares is bounded by 42
[15, Theorem 2.4].
It is also interesting to ask whether the sequence $(f_{n})_{n\in\mathbb{N}}$
itself is $2$-regular (equivalently, $(g_{n})_{n\in\mathbb{N}}$ is $2$
automatic), since its values form a $2$-automatic set. This question seems
hard, and we have not been able to give a definitive answer (see also Section
7). The problem comes from the fact that the description of elements of
$S_{1}^{\prime}$ in Corollary 3.3 does not give enough information about their
ordering. Instead, we consider a simpler version of this question, where we
restrict our attention to indices $n\in S_{1}^{\prime}$ with fixed $2$-adic
valuation. More precisely, we let $k=0$ in the description of Corollary 3.3,
so that $n$ is of the form $4m+2$. Then $b(2n)$ is not a sum of three squares
if and only if $b(8m+4)\equiv 28\pmod{32}$ (this is precisely (2) with $k=0$).
More generally, put
$\beta(m)=\frac{b(8m+4)}{4}\bmod{8},$
and for each $a\in\\{1,3,5,7\\}$ let ${\bf c}_{a}=(c_{a}(l))_{l\in\mathbb{N}}$
be the increasing sequence such that
$\\{m\in\mathbb{N}:\;\beta(m)=a\\}=\\{c_{a}(l):\;l\in\mathbb{N}\\}.$
It turns out that these sequences are described by surprisingly simple
formulas.
###### Theorem 3.6.
For each $a\in\\{1,3,5,7\\}$ the sequence ${\bf c}_{a}$ is 2-regular. More
precisely, for $m\in\mathbb{N}$ we have
$\displaystyle c_{1}(l)$ $\displaystyle=4l-t_{l}+1,$ $\displaystyle c_{3}(l)$
$\displaystyle=4l+t_{l}+2,$ $\displaystyle c_{5}(l)$
$\displaystyle=4l-t_{l}+2,$ $\displaystyle c_{7}(l)$
$\displaystyle=4l+t_{l}+1.$
###### Proof.
It is easy to see that each of the sequences from the statement is increasing.
To prove that $\beta(m)=a$ if and only if $m=c_{a}(l)$ for some
$l\in\mathbb{N}$, we restate the second and third relation of Proposition 3.2
in the following way:
$\beta(m)=\begin{cases}1&\text{if }2\mid m\text{ and }t_{m}=1,\\\ 3&\text{if
}2\nmid m\text{ and }t_{m}=1,\\\ 5&\text{if }2\nmid m\text{ and }t_{m}=-1,\\\
7&\text{if }2\mid m\text{ and }t_{m}=-1.\\\ \end{cases}$
We now use the relations (3). If $\beta(m)=1$, then $2\mid m$ and $m=2k+T_{k}$
for some $k\in\mathbb{N}$. This implies $T_{k}=0$, and thus $k=2l+T_{l}$ for
some $l\in\mathbb{N}$. As a result, we get $m=4l+2T_{l}=4l-t_{l}+1$.
Conversely, if $m$ is of this form, then also $\beta(m)=1$, and so we get the
claim for $a=1$.
The proof for $a=3,5,7$ is similar. ∎
To conclude this section, we point out that similar results can be obtained
for other quadratic forms given in Remark 1.1. More precisely, depending on
the chosen form, they can be derived from either Proposition 3.2 or the
following set of congruence relations (which again follows from Hirschhorn and
Loxton’s results).
###### Proposition 3.7.
For all $n\in\mathbb{N}$ we have
$\displaystyle b(16n+8)$ $\displaystyle\equiv b(4n+2)\pmod{16},$
$\displaystyle b(8n+2)$ $\displaystyle\equiv 2t_{n}\pmod{16},$ $\displaystyle
b(8n+6)$ $\displaystyle\equiv 6t_{n}\pmod{16}.$
## 4\. The equation $b_{3}(n)=x^{2}+y^{2}+z^{2}$
In this section we characterize the elements of the set $S_{3}$ containing
those $n$ such that $b_{3}(n)$ is not a sum of three squares. By virtue of
Theorem 2.3, to get the required characterization of $S_{3}$, we need to
understand of the behaviour of $b_{3}(16n+i)\mod{32}$ for
$i=0,1,2,3,8,9,10,11$. Let us recall that the sequence
$(b_{3}(n))_{n\in\mathbb{N}}$ satisfies the following recurrence relations:
$b_{3}(0)=1,b_{3}(1)=3,b_{3}(2)=9$ and
$\displaystyle b_{3}(2n)$
$\displaystyle=3b_{3}(2n-1)-3b_{3}(2n-2)+b_{3}(2n-3)+b_{3}(n),$ $\displaystyle
b_{3}(2n+1)$ $\displaystyle=3b_{3}(2n)-3b_{3}(2n-1)+b_{3}(2n-2).$
We start with the following lemma.
###### Lemma 4.1.
For all $n\in\mathbb{N}$ the following congruences hold:
$\displaystyle b_{3}(8n+i+4)$ $\displaystyle\equiv
2(2i+1+4(-1)^{n})t_{n}\mkern 4.0mu({\operator@font mod}\mkern 6.0mu32),$
$\displaystyle b_{3}(32n+i)$ $\displaystyle\equiv b_{3}(8n+i)\mkern
4.0mu({\operator@font mod}\mkern 6.0mu64),\;i=0,1,2,3,4$ $\displaystyle
b_{3}(8(2n+1)+i)$ $\displaystyle\equiv 4(3+3i-i^{2}-2(-1)^{n+i})t_{n}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu32)$
$\displaystyle\equiv\begin{cases}4(3-2(-1)^{n})t_{n}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu32)&\text{if }i=0,\\\
4(5+2(-1)^{n})t_{n}\mkern 4.0mu({\operator@font mod}\mkern 6.0mu32)&\text{if
}i=1,\\\ 4(5-2(-1)^{n})t_{n}\mkern 4.0mu({\operator@font mod}\mkern
6.0mu32)&\text{if }i=2,\\\ 4(3+2(-1)^{n})t_{n}\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu32)&\text{if }i=3.\end{cases}$
In particular, for each $k\in\mathbb{N}_{+}$ and $i\in\\{0,1,2,3\\}$, we have
$\displaystyle b_{3}(2^{2k}(2n+1)+i)$ $\displaystyle\equiv 2\mkern
4.0mu({\operator@font mod}\mkern 6.0mu4),$ $\displaystyle
b_{3}(2^{2k+1}(2n+1)+i)$ $\displaystyle\equiv b_{3}(8(2n+1)+i)\mkern
4.0mu({\operator@font mod}\mkern 6.0mu32),$
###### Proof.
The computation of the values of $b_{3}(8n+i+4)\mod{32}$ and
$b_{3}(8(2n+1)+i)\mod{32}$ for $i=0,1,2,3$ is based on a simple induction with
the help of recurrence relations satisfied by the PTM sequence
$(t_{n})_{n\in\mathbb{N}}$ and the sequence $(b_{3}(n))_{n\in\mathbb{N}}$.
Because of this, we omit the simple details. Essentially, the same approach
can be used in the case of the congruence $b_{3}(32n+i)\equiv
b_{3}(8n+i)\mkern 4.0mu({\operator@font mod}\mkern 6.0mu32)$.
However, a more conceptual proof is the following. Invoking [10, Lemma 4.7] we
know that for each $a\in\mathbb{N}_{+},b\in\\{0,1,\ldots,2^{a}-1\\}$ there is
a polynomial $P_{a,b}\in\mathbb{Z}[x]$ such that
$\sum_{n=0}^{\infty}b_{3}(2^{a}n+b)x^{n}=\frac{P_{a,b}(x)}{(1-x)^{3a}}B_{3}(x).$
In particular, in the case we are interested in, we have
$\sum_{n=0}^{\infty}(b_{3}(32n+i)-b_{3}(8n+i))x^{n}=\frac{P_{5,i}(x)-(1-x)^{6}P_{3,i}(x)}{(1-x)^{15}}B_{3}(x).$
A quick computation reveals that for each $i=0,1,2,3$, the polynomial
$P_{5,i}(x)-(1-x)^{6}P_{3,i}(x)$ is divisible by 64 in the ring
$\mathbb{Z}[x]$. Thus, as the function $(1-x)^{-15}B_{3}(x)$ has power series
expansion with integer coefficients, then each number
$b_{3}(32n+i)-b_{3}(8n+i)$ is divisible by 64 and we are done.
To obtain the first congruence from the “in particular” part, we apply
induction on $k$ and the congruence $b_{3}(8n+i+4)\equiv
2(2i+1+4(-1)^{n})t_{n}\mkern 4.0mu({\operator@font mod}\mkern 6.0mu32)$. In
particular, for $i=0,1,2,3$, the 2-adic valuation of $b_{3}(8n+i+4)$ is equal
to 1.
To obtain the congruence $b_{3}(2^{2k+1}(2n+1)+i)\equiv b_{3}(8(2n+1)+i)$ we
again use induction on $k$ and apply the congruence $b_{3}(32n+i)\equiv
b_{3}(8n+i)\mkern 4.0mu({\operator@font mod}\mkern 6.0mu64)$. ∎
We are ready to characterize the set $S_{3}$.
###### Theorem 4.2.
We have $n\in S_{3}$ if and only if
$n=2^{2k+1}\left(8p+2\left\lfloor\frac{i}{2}\right\rfloor+3+2(-1)^{i}t_{p}\right)+i$
for some $i\in\\{0,1,2,3\\}$ and $k\in\mathbb{N}_{+}$, $p\in\mathbb{N}$.
###### Proof.
From the characterization of the 2-adic valuation of $b_{3}(n)$ and Lemma 4.1
we know that if $n\in S_{3}$, then we necessarily have
$n\pmod{16}\in\\{0,1,2,3,8,9,10,11\\}$. We perform a case-by-case analysis.
Let $i\in\\{0,1,2,3\\}$. If $n\equiv i\mkern 4.0mu({\operator@font mod}\mkern
6.0mu16)$ and $n=2^{2k}(2s+1)+i$, then $\nu_{2}(b_{3}(n))=1$ and hence
$n\not\in S_{3}$. If $n=2^{2k+1}(2s+1)+i$, then we have
$b_{3}(n)\equiv b_{3}(8(2s+1)+i)\equiv 4(3+3i-i^{2}-2(-1)^{s+i})t_{s}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu32),$
and thus $n\in S_{3}$ if and only if
$c(i,s):=(3+3i-i^{2}-2(-1)^{s+i})t_{s}\equiv 7\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu8)$. A case-by-case analysis using the characterization (3),
reveals the following:
1. (1)
If $i=0$, then $c(i,s)\equiv 7\mkern 4.0mu({\operator@font mod}\mkern 6.0mu8)$
if and only if $s$ is even and $t_{s}=-1$. Thus, $s=4p+1+t_{p}$ for some
$p\in\mathbb{N}$.
2. (2)
If $i=1$, then $c(i,s)\equiv 7\mkern 4.0mu({\operator@font mod}\mkern 6.0mu8)$
if and only if $s$ is even and $t_{s}=1$. Thus $s=4p+1-t_{p}$ for some
$p\in\mathbb{N}$.
3. (3)
If $i=2$, then $c(i,s)\equiv 7\mkern 4.0mu({\operator@font mod}\mkern 6.0mu8)$
if and only if $s$ is odd, and $t_{s}=1.$ Thus $s=4p+2+t_{p}$ for some
$p\in\mathbb{N}$.
4. (4)
If $i=3$, then $c(i,s)\equiv 7\mkern 4.0mu({\operator@font mod}\mkern 6.0mu8)$
if and only if $s$ is odd and $t_{s}=-1.$ Thus $s=4p+2-t_{p}$ for some
$p\in\mathbb{N}$.
Gathering all the obtained characterizations, we get the statement of our
theorem. ∎
## 5\. The equation $b_{2^{k}-1}(n)=x^{2}+y^{2}+z^{2}$ with $k\geq 3$
In this section, we study for $k\geq 3$ representability of $b_{2^{k}-1}(n)$
as a sum of three squares. The main idea is to express
$(b_{2^{k}-1}(n))_{n\in\mathbb{N}}$ as the convolution of
$(b_{2^{k}}(n))_{n\in\mathbb{N}}$ and the PTM sequence, and apply the
following lemma [10, Lemma 4.4(1)] to $m=2^{k}$.
###### Lemma 5.1.
Let $m\in\mathbb{N}_{+}$. Then for all $n\in\mathbb{N}$ we have
$b_{m}(n)\equiv\binom{m}{n}+2^{\nu_{2}(m)+1}\binom{m-2}{n-2}\pmod{2^{\nu_{2}(m)+2}}.$
We split our reasoning into two parts: $n<2^{k}$ and $n\geq 2^{k}$. Starting
with the simpler case $n<2^{k}$, by Theorem 2.3, we have
$\nu_{2}(b_{2^{k}-1}(n))=0$. Therefore, it is sufficient for our purposes to
describe $b_{2^{k}-1}(n)$ modulo $8$.
###### Proposition 5.2.
Let $k\geq 3$ and $n<2^{k}$. Then
$b_{2^{k}-1}(n)\equiv t_{n}\cdot\begin{cases}1\pmod{8}&\text{if }0\leq
n<2^{k-2},\\\ 5\pmod{8}&\text{if }2^{k-2}\leq n<2^{k-1},\\\ 7\pmod{8}&\text{if
}2^{k-1}\leq n<3\cdot 2^{k-2},\\\ 3\pmod{8}&\text{if }3\cdot 2^{k-2}\leq
n<2^{k}.\end{cases}$
###### Proof.
By Lemma 5.1 we have
$b_{2^{k}-1}(n)\equiv\sum_{l=0}^{n}\binom{2^{k}}{l}t_{n-l}\pmod{8}$
Moreover, [10, Lemma 4.5] says that
$\binom{2^{k}}{l}\equiv\begin{cases}1\pmod{8}&\text{if }l=0,2^{k},\\\
4\pmod{8}&\text{if }l=2^{k-2},3\cdot 2^{k-2},\\\ 6\pmod{8}&\text{if
}l=2^{k-1},\\\ 0\pmod{8}&\text{otherwise}.\end{cases}$
From this description we immediately get the claim for the cases $0\leq
n<2^{k-2}$ and $2^{k-2}\leq n<2^{k-1}$. If $2^{k-1}\leq n<3\cdot 2^{k-2}$, we
get
$b_{2^{k}-1}(n)\equiv t_{n}+4t_{n-2^{k-2}}+6t_{n-2^{k-1}}\equiv
t_{n}+2t_{n-2^{k-1}}\pmod{8}.$
Since $n$ has $2^{k-1}$ in its binary expansion, we get
$t_{n-2^{k-1}}=-t_{n}$, and the required congruence follows. Finally, if
$3\cdot 2^{k-2}\leq n<2^{k}$, we again have $t_{n-2^{k-1}}=-t_{n}$ so
$b_{2^{k}-1}(n)\equiv t_{n}+4t_{n-2^{k-2}}+6t_{n-2^{k-1}}+4t_{n-3\cdot
2^{k-2}}\equiv-5t_{n}\pmod{8}.\qed$
As an immediate corollary, we can describe $n<2^{k}$ such that
$b_{2^{k}-1}(n)$ is (not) a sum of three squares.
###### Corollary 5.3.
Let $k\geq 3$ and $n<2^{k}$. Then $b_{2^{k}-1}(n)$ is not a sum of three
squares of integers if and only if one of the following cases holds:
1. (1)
$0\leq n<2^{k-2}$ and $t_{n}=-1$;
2. (2)
$2^{k-1}\leq n<3\cdot 2^{k-2}$ and $t_{n}=1$.
We move on to the case $n\geq 2^{k}$. This time we have
$\nu_{2}(b_{2^{k}-1}(n))\in\\{1,2\\}$ by Theorem 2.3, which means that it is
sufficient to consider $b_{2^{k}-1}(n)$ modulo $32$. To this end, we need a
standard lemma concerning the behavior of binomial coefficients modulo powers
of $2$ (we provide a proof for completeness).
###### Lemma 5.4.
The following statements hold:
1. (a)
For all $k,n\in\mathbb{N}$ such that $1\leq n\leq 2^{k}$, we have
$\nu_{2}\left(\binom{2^{k}}{n}\right)=k-\nu_{2}(n).$
2. (b)
For all $m,n\in\mathbb{N}$ we have
$\binom{2m}{2n}\equiv\binom{m}{n}\pmod{2^{\nu_{2}(m)+1}}.$
###### Proof.
By Legendre’s formula we get
$\nu_{2}\left(\binom{2^{k}}{n}\right)=2^{k}-1-(n-s_{2}(n)+2^{k}-n-s_{2}(2^{k}-n))=s_{2}(n)+s_{2}(2^{k}-n)-1.$
We can express $s_{2}(2^{k}-n)$ as
$s_{2}(2^{k}-n)=s_{2}((2^{k}-1)-(n-1))=k-s_{2}(n-1).$
Now, write $n=2^{\nu_{2}(n)}l$, which yields
$\displaystyle s_{2}(n-1)$
$\displaystyle=s_{2}(2^{\nu_{2}(n)}(l-1)+(2^{\nu_{2}(n)}-1))=s_{2}(l-1)+\nu_{2}(n)$
$\displaystyle=s_{2}(l)-1+\nu_{2}(n)=s_{2}(n)-1+\nu_{2}(n).$
Combining the above equalities, we get (a).
Moving on to (b), the claim clearly holds for $n=0$ so we can assume that
$n\geq 1$. We have the congruence
$\binom{2m}{2n}=\binom{m}{n}\frac{(2m-1)!!}{(2n-1)!!(2m-2n-1)!!}\equiv(-1)^{n}\binom{m}{n}\pmod{2^{\nu_{2}(m)+1}}.$
If $n$ is even, we immediately obtain (b). If $n$ is odd, we use the
inequality
$\nu_{2}\left(\binom{m}{n}\right)=\nu_{2}\left(\frac{m}{n}\binom{m-1}{n-1}\right)\geq\nu_{2}(m),$
which again leads to the desired result. ∎
We are now ready to describe $b_{2^{k}-1}(n)$ modulo $32$ for $n\geq 2^{k}$.
This time, the characterization involves two consecutive terms of the PTM
sequence.
###### Theorem 5.5.
Fix $k,i,j\in\mathbb{N}$ such that $k\geq 3$, $i<8$, and $j<2^{k-3}$. Then for
all $m\geq 1$ we have
$b_{2^{k}-1}(2^{k}m+2^{k-3}i+j)\equiv
t_{j}(c_{i}t_{m}+d_{i}t_{m-1})\pmod{32},$
where the coefficients $c_{i},d_{i}$ do not depend on $k$ and are given in
Table 2.
$i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$c_{i}$ | $1$ | $7$ | $3$ | $5$ | $9$ | $-1$ | $3$ | $5$
$d_{i}$ | $-5$ | $-3$ | $1$ | $-9$ | $-5$ | $-3$ | $-7$ | $-1$
Table 2. The coefficients $c_{i}$, $d_{i}$.
###### Proof.
Consider first the case $k\geq 4$. By Lemma 5.1 we have
$b_{2^{k}-1}(n)=\sum_{l=0}^{n}b_{2^{k}}(l)t_{n-l}\equiv\sum_{l=0}^{n}\binom{2^{k}}{l}t_{n-l}\pmod{32}.$
Now, by Lemma 5.4(a), the binomial coefficients with $v_{2}(l)<k-4$ vanish
modulo $32$. Therefore, assuming that $n\geq 2^{k}$, the above sum simplifies
to
$b_{2^{k}-1}(n)\equiv\sum_{l=0}^{16}\binom{2^{k}}{2^{k-4}l}t_{n-2^{k-4}l}\equiv\sum_{l=0}^{16}\binom{16}{l}t_{n-2^{k-4}l}\pmod{32},$
where the second congruence follows from Lemma 5.4(b). Furthermore, we can
eliminate the terms with $l$ odd, since there is an even number of them and
they are all congruent with $16$ modulo $32$. Therefore, we get the congruence
$b_{2^{k}-1}(n)\equiv\sum_{l=0}^{8}\binom{16}{2l}t_{n-2^{k-3}l}\pmod{32}.$
To simplify the right-hand side, consider $b_{2^{k}-1}$ at indices of the form
given in the statement, namely $n=2^{k}m+2^{k-3}i+j$, where $m\geq 1$, $0\leq
i<8$, and $0\leq j<2^{k-3}$. For the sake of clarity, we will now momentarily
use the notation $t(n)=t_{n}$. By the recurrences that define the Thue–Morse
sequence, we get
$t(2^{k}m+2^{k-3}i+j-2^{k-3}l)=t_{j}t_{8m+i-l}=t_{j}\cdot\begin{cases}t_{n}t_{i-l}&\text{if
}l\leq i,\\\ -t_{n-1}t_{l-i}&\text{if }l>i.\end{cases}$
Therefore, the claimed formula is valid with the coefficients
$c_{i}=\sum_{l=0}^{i}\binom{16}{2l}t_{i-l},\qquad
d_{i}=-\sum_{l=i+1}^{8}\binom{16}{2l}t_{l-i},$
and a direct calculation (modulo $32$) gives their values as in Table 2.
In the case $k=3$, the expression for $b_{2^{k}-1}(n)$ modulo $32$ obtained
from Lemma 5.1 also contains the sum
$16\sum_{l=0}^{n}\binom{6}{l-2}t_{n-l}.$
If $n\geq 8$, then the whole sum vanishes modulo $32$, so we again arrive at
the formula
$b_{7}(n)\equiv\sum_{l=0}^{8}\binom{8}{l}t_{8-l}\pmod{32}.$
After a similar calculation as before, we get the result. ∎
Using this result, we can determine the indices $n\geq 2^{k}$ such that
$b_{2^{k}-1}(n)$ is not a sum of three squares. The description turns out to
be surprisingly simple.
###### Corollary 5.6.
Let $k\geq 3$ and $n\geq 2^{k}$. The following conditions are equivalent:
1. (a)
$b_{2^{k}-1}(n)$ is not a sum of three squares;
2. (b)
$t_{n}=t_{n-2^{k}}=1$;
3. (c)
$n=2^{k}m+l,$ where $l,m\in\mathbb{N}$ are such that $l<2^{k}$, $t_{m}=t_{l}$,
and $\nu_{2}(m)\equiv 1\pmod{2}$.
###### Proof.
Write $n=2^{k}m+2^{k-3}i+j$ as in Theorem 5.5. Observe that
$c_{i}+d_{i}=-4t_{i}$, while $c_{i}-d_{i}$ is not divisible by $4$. Hence, the
term $b_{2^{k}-1}(2^{k}m+2^{k-3}i+j)$ is not a sum of three squares if and
only if
$t_{m}=t_{m-1}=t_{i}t_{j},$
which after multiplying both sides by $t_{i}t_{j}$ gives precisely (b).
The equivalence with (c) is obtained by writing $l=2^{k-3}i+j$ and observing
that $t_{m}=(-1)^{\nu_{2}(m)+1}t_{m-1}$. ∎
## 6\. Counting the solutions
The aim of this section is to provide estimates for the counting functions of
the sets $S_{2^{k}-1}$. For real $x\geq 0$ and $m\in\mathbb{N}_{+}$ let
$S_{m}(x)=S_{m}\cap[0,x]=\\#\\{n\leq x:b_{m}(n)\text{ is not a sum of three
squares}\\}.$
Using the descriptions of the sets $S_{2^{k}-1}$ obtained in the previous
sections for various $k$ it is straightforward to check that
$S_{2^{k}-1}(x)=\delta_{k}x+O(\log x),$
where $\delta_{1}=\delta_{2}=1/12$ and $\delta_{k}=1/6$ for $k\geq 3$. In the
following three results, we provide more precise bounds for
$S_{2^{k}-1}(x)-d_{k}x$ in the case $k=1,k=2$ and $k\geq 3$, respectively. In
particular, each lower and upper bound is of the form $C_{1}\log_{2}x+C_{2}$,
where the constant $C_{1}$ is optimal.
###### Theorem 6.1.
For every $x\geq 6$ we have
$-\frac{5}{3}<S_{1}(x)-\frac{x}{12}<\frac{1}{2}\log_{2}x-\frac{19}{12}.$
In particular, the density of the set $S_{1}$ in $\mathbb{N}$ exists and is
equal to
$\lim_{x\rightarrow+\infty}\frac{S_{1}(x)}{x}=\frac{1}{12}.$
Moreover, there exists an increasing sequence
$(m_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$ such that
$S_{1}(m_{l})-\frac{m_{l}}{12}\sim\frac{1}{2}\log_{2}m_{l}.$
###### Proof.
For real $x\geq 0$ define
$\displaystyle P(x)$ $\displaystyle=\\#\\{s\in\mathbb{N}:8s+2t_{s}+3\leq
x\\},$ $\displaystyle Q(x)$
$\displaystyle=\sum_{k=0}^{\infty}P\left(\frac{x}{4^{k}}\right).$
By Corollary 3.3(b) and the relation $b(2n+1)=b(2n)$, we get
$S_{1}(x)=Q\left(\frac{x}{4}\right)+Q\left(\frac{x-1}{4}\right).$
Hence, it is sufficient to focus on the function $Q$. For $m\in\mathbb{N}$ and
$i=0,1,2,3$ we have the recurrence relations
$Q(4m+i)=Q(m)+P(4m+i).$
Also, for $i<8$ we have
$P(8m+i)=m+\begin{cases}0&\text{if }i=0,\\\ T_{m}&\text{if }i=1,2,3,4,\\\
1&\text{if }i=5,6,7.\end{cases}$
Put
$R(x)=Q(x)-\frac{x}{6},$
so that
$S_{1}(x)-\frac{x}{12}=R\left(\left\lfloor\frac{x}{4}\right\rfloor\right)+R\left(\left\lfloor\frac{x-1}{4}\right\rfloor\right)+\frac{\left\lfloor\frac{x}{4}\right\rfloor+\left\lfloor\frac{x-1}{4}\right\rfloor}{6}-\frac{x}{12}.$
It is readily checked that
$-\frac{1}{3}<\frac{\left\lfloor\frac{x}{4}\right\rfloor+\left\lfloor\frac{x-1}{4}\right\rfloor}{6}-\frac{x}{12}\leq-\frac{1}{12}$
Therefore, to obtain the estimates for $S_{1}(x)-x/12$ (where $x\geq 9$), it
remains to prove that for each integer $m\geq 2$ there holds
$-\frac{2}{3}\leq R(m)\leq\frac{1}{4}\lfloor\log_{2}m\rfloor-\frac{1}{4},$
as then
$-\frac{4}{3}\leq
R\left(\left\lfloor\frac{x}{4}\right\rfloor\right)+R\left(\left\lfloor\frac{x-1}{4}\right\rfloor\right)<2\cdot\frac{1}{4}\left(\log_{2}\frac{x}{4}-1\right)=\frac{1}{2}\log_{2}x-\frac{3}{2}.$
This is done by induction on the length $L(m)$ of the binary expansion of $m$.
Direct computation shows that our claim holds when $2\leq L(m)\leq 6$. Hence,
let $L(m)\geq 7$. It is sufficient to prove that there exists an integer
$n\geq 2$ with $L(n)\leq L(m)-2$ such that
$0\leq R(m)-R(n)\leq\frac{1}{2}.$
This is indeed the case, as shown by the following set of identities (ordered
according to the residue class modulo $8$):
$\displaystyle R(8n)$ $\displaystyle=R(2n),$ $\displaystyle R(16n+1)$
$\displaystyle=R(4n+1),$ $\displaystyle R(16n+9)$
$\displaystyle=R(4n)+\frac{1}{2},$ $\displaystyle R(16n+2)$
$\displaystyle=R(4n+2),$ $\displaystyle R(16n+10)$
$\displaystyle=R(4n)+\frac{1}{3},$ $\displaystyle R(16n+3)$
$\displaystyle=R(4n+3),$ $\displaystyle R(16n+11)$
$\displaystyle=R(4n)+\frac{1}{6},$ $\displaystyle R(8n+4)$
$\displaystyle=R(2n+1)+T_{n}-\frac{1}{2},$ $\displaystyle R(64n+4)$
$\displaystyle=R(16n+4),$ $\displaystyle R(64n+20)$
$\displaystyle=R(16n+2)+1-T_{n},$ $\displaystyle R(64n+36)$
$\displaystyle=R(16n)+1-T_{n},$ $\displaystyle R(64n+52)$
$\displaystyle=R(16n+4),$ $\displaystyle R(16n+12)$ $\displaystyle=R(4n),$
$\displaystyle R(8n+5)$ $\displaystyle=R(2n+1)+\frac{1}{3},$ $\displaystyle
R(8n+6)$ $\displaystyle=R(2n+1)+\frac{1}{6},$ $\displaystyle R(8n+7)$
$\displaystyle=R(2n+1).$
We move on to the second part of the statement. Define $m_{0}=0$ and
$m_{l+1}=16m_{l}+36$ for $l\in\mathbb{N}$. First, we prove inductively that
$R(m_{l})=l$. This is clear for $l=0$. In general, we have
$R(m_{l+1})=R(16m_{l}+36)=R(4m_{l})+1-T_{m_{l}}=R(m_{l})+1,$
where we have used $4\mid m_{l}$, the recurrence relations above and
$T_{m_{l}}=0$ (easily shown by induction). We thus have
$S_{1}(m_{0})=S_{1}(m_{1})=0$ and for $l\geq 1$ the equality
$\displaystyle S_{1}(m_{l+1})-\frac{m_{l+1}}{12}$
$\displaystyle=R\left(\left\lfloor\frac{m_{l+1}}{4}\right\rfloor\right)+R\left(\left\lfloor\frac{m_{l+1}-1}{4}\right\rfloor\right)-\frac{1}{6}$
$\displaystyle=R(m_{l})+R(m_{l-1})+1=2l.$
The result follows. ∎
In Figure 3 we show the graph of the function $S_{1}(x)-x/12$ in the range
$[1,2^{10}]$ together with the bounds (as in the theorem).
Figure 3. The function $S_{1}(x)-x/12$.
From the presented graph, it appears it should be possible to obtain an even
better additive constant in the upper bound. To do this, one would need to
investigate closer the location of the “spikes” on the graph (some of which
correspond to $x=m_{l}$).
The following two results show that the function $S_{1}(x)$ is exceptional in
the sense that $S_{1}(x)-x/12$ is bounded from below by a constant.
###### Theorem 6.2.
For all $x\geq 1$ we have
$-\frac{1}{6}\log_{2}x-\frac{7}{12}<S_{3}(x)-\frac{x}{12}\leq\frac{1}{6}\log_{2}x-\frac{1}{6}.$
In particular, the density of the set $S_{3}$ in $\mathbb{N}$ exists and is
equal to
$\lim_{x\rightarrow+\infty}\frac{S_{3}(x)}{x}=\frac{1}{12}.$
Moreover, there exist increasing sequences
$(m_{l})_{l\in\mathbb{N}},(n_{l})_{l\in\mathbb{N}}\subset\mathbb{N}$ such that
$\displaystyle S_{3}(m_{l})-\frac{m_{l}}{12}$
$\displaystyle\sim\frac{1}{6}\log_{2}m_{l},$ $\displaystyle
S_{3}(n_{l})-\frac{n_{l}}{12}$ $\displaystyle\sim-\frac{1}{6}\log_{2}n_{l}.$
###### Proof.
For $i=0,1,2,3$ let
$P_{i}(x)=\\#\\{n\in\mathbb{N}:8n+2\left\lfloor\frac{i}{2}\right\rfloor+3+2(-1)^{i}t_{n}\leq
x\\},$
so that by Theorem 4.2 we have
$S_{3}(x)=\sum_{k=1}^{\infty}\sum_{i=0}^{3}P_{i}\left(\frac{x-i}{2\cdot
4^{k}}\right).$
This time, put
$\displaystyle P(x)$ $\displaystyle=\sum_{i=0}^{3}P_{i}(x),$ $\displaystyle
Q(x)$ $\displaystyle=\sum_{k=0}^{\infty}P\left(\frac{x}{4^{k}}\right).$
Then for any $x$ we have
$Q\left(\frac{x-3}{8}\right)\leq S_{3}(x)\leq Q\left(\frac{x}{8}\right).$
Therefore, we need to bound the function $R(x)=Q(x)-2x/3$. First, for
$n\in\mathbb{N}$ we have the easy to check equalities $P(n)=\lceil n/2\rceil$
and
$R(4n+i)=R(n)+\begin{cases}0&\text{for }i=0,3,\\\ \frac{1}{3}&\text{for
}i=1,\\\ -\frac{1}{3}&\text{for }i=2.\end{cases}$
In a similar fashion as in the previous proof, one can then prove that for
$m\in\mathbb{N}$ there holds
$-\frac{1}{6}\lfloor\log_{2}m\rfloor-\frac{1}{6}\leq
R(m)\leq\frac{1}{6}\lfloor\log_{2}m\rfloor+\frac{1}{3}$
The inequalities for $S_{3}(x)-x/12$ follow shortly by plugging in $m=\lfloor
x/8\rfloor$ and $m=\lfloor(x-3)/8\rfloor$.
If we define $m_{0}=0$ and $m_{l+1}=4m_{l}+8$, we can inductively compute
$R(m_{l}/8)=l/3$, and therefore
$S(m_{l})-\frac{1}{12}m_{l}\sim
Q\left(\frac{m_{l}}{8}\right)-\frac{1}{12}m_{l}=\frac{l}{3}\sim\frac{1}{6}\log_{2}m_{l}.$
Similarly, for $n_{0}=0$ and $n_{l+1}=4n_{l}+16$, we get $R(n_{l}/8)=-l/3$ so
$S(n_{l})-\frac{1}{12}n_{l}\sim-\frac{1}{6}\log_{2}n_{l},$
and the proof is finished. ∎
Figure 4. The function $S_{3}(x)-x/12$.
Figure 4 shows the graph of the function $S_{1}(x)-x/12$ in the range
$[1,2^{10}]$ together with the proved bounds (in red). Again, the bounds are
quite accurate, though the additive constants can probably be improved
further.
The final result of this section concerns the function $S_{2^{k}-1}(x)$. For
the sake of clarity, in the proof we make some rough estimates concerning the
additive constant (although the constant near $\log_{2}x$ remains optimal).
###### Theorem 6.3.
If $k\geq 3$, then for all $x\geq 2^{k}$ we have
$\left|S_{2^{k}-1}(x)-\frac{x}{6}\right|\leq\frac{2^{k-2}}{3}(\log_{2}x-k+26).$
In particular, the density of the set $S_{2^{k}-1}$ in $\mathbb{N}$ exists and
is equal to
$\lim_{x\rightarrow+\infty}\frac{S_{2^{k}-1}(x)}{x}=\frac{1}{6}.$
Moreover, there exist increasing sequences
$(m_{l})_{l\in\mathbb{N}},(n_{l})_{l\in\mathbb{N}}\subset\mathbb{N}$ such that
$\displaystyle S_{2^{k}-1}(m_{l})-\frac{m_{l}}{6}$
$\displaystyle\sim\frac{2^{k-2}}{3}\log_{2}m_{l},$ $\displaystyle
S_{2^{k}-1}(n_{l})-\frac{n_{l}}{6}$
$\displaystyle\sim-\frac{2^{k-2}}{3}\log_{2}n_{l}.$
###### Proof.
For $\varepsilon\in\\{1,-1\\}$ and non-negative $x\in\mathbb{R}$ we put
$\displaystyle P_{\varepsilon}(x)$ $\displaystyle=\\#\\{1\leq m\leq
x:t_{m}=\varepsilon\\},$ $\displaystyle Q_{\varepsilon}(x)$
$\displaystyle=\sum_{s=0}^{\infty}(-1)^{s}P_{\varepsilon}\left(\frac{x}{2^{s}}\right)=\\#\\{1\leq
m\leq x:t_{m}=\varepsilon\text{ and }\nu_{2}(m)\equiv 0\mkern
4.0mu({\operator@font mod}\mkern 6.0mu2)\\}.$
Then by Corollary 5.6 we get
$\displaystyle S_{2^{k}-1}(x)$
$\displaystyle=S_{2^{k}-1}(2^{k}-1)+\sum_{l=0}^{2^{k}-1}\\#\\{1\leq
m\leq\frac{x-l}{2^{k}}:t_{m}=t_{l}\text{ and }\nu_{2}(m)\equiv 1\mkern
4.0mu({\operator@font mod}\mkern 6.0mu2)\\}$
$\displaystyle=2^{k-2}+\sum_{l=0}^{2^{k}-1}Q_{t_{l}}\left(\frac{x-l}{2^{k+1}}\right),$
where $S_{2^{k}-1}(2^{k}-1)=2^{k-2}$ follows from Corollary 5.3. Furthermore,
we have the obvious inequality
$0\leq\sum_{l=0}^{2^{k}-1}Q_{t_{l}}\left(\frac{x}{2^{k+1}}\right)-\sum_{l=0}^{2^{k}-1}Q_{t_{l}}\left(\frac{x-l}{2^{k+1}}\right)\leq
2^{k}.$
Since for each $\varepsilon=\pm 1$ we have $t_{l}=\varepsilon$ for precisely
$2^{k-1}$ indices $l$, we obtain
(4)
$\left|S_{2^{k}-1}(x)-2^{k-1}\left(Q_{1}\left(\frac{x}{2^{k+1}}\right)+Q_{-1}\left(\frac{x}{2^{k+1}}\right)\right)\right|=5\cdot
2^{k-2}.$
Therefore, to bound $S_{2^{k}-1}(x)-x/6$ it remains to estimate for
$\varepsilon=\pm 1$ the functions
$R_{\varepsilon}(x)=Q_{\varepsilon}(x)-\frac{x}{3}.$
First, note that for $n\in\mathbb{N}$ we have
$P_{\varepsilon}(n)=\frac{n-\varepsilon}{2}+\begin{cases}\frac{\varepsilon}{2}t_{n}&\text{if
}2\mid n,\\\ 0&\text{if }2\nmid n.\end{cases}$
It follows that
$R_{\varepsilon}(4n+i)=R_{\varepsilon}(n)+\begin{cases}0&\text{if }i=0,3,\\\
\frac{1}{2}(1-\varepsilon t_{n})-\frac{1}{3}&\text{if }i=1,\\\
\frac{1}{2}(1-\varepsilon t_{n})-\frac{2}{3}&\text{if }i=2.\end{cases}$
This leads to the relations
$\displaystyle R_{\varepsilon}(4n)$ $\displaystyle=R_{\varepsilon}(n),$
$\displaystyle R_{\varepsilon}(8n+1)$ $\displaystyle=R_{\varepsilon}(2n+1),$
$\displaystyle R_{\varepsilon}(16n+5)$
$\displaystyle=R_{\varepsilon}(n)+\frac{1}{3},$ $\displaystyle
R_{\varepsilon}(16n+13)$ $\displaystyle=R_{\varepsilon}(4n+1),$ $\displaystyle
R_{\varepsilon}(16n+2)$ $\displaystyle=R_{\varepsilon}(4n+2),$ $\displaystyle
R_{\varepsilon}(8n+6)$ $\displaystyle=R_{\varepsilon}(2n),$ $\displaystyle
R_{\varepsilon}(16n+10)$ $\displaystyle=R_{\varepsilon}(n)-\frac{1}{3},$
$\displaystyle R_{\varepsilon}(4n+3)$ $\displaystyle=R_{\varepsilon}(n).$
By induction we obtain for $\varepsilon=\pm 1$ and all $n\in\mathbb{N}_{+}$
the inequality
$|R_{\varepsilon}(n)|\leq\frac{1}{12}\lfloor\log_{2}n\rfloor+\frac{2}{3},$
which implies
$\left|Q_{\varepsilon}(x)-\frac{x}{3}\right|\leq\frac{1}{12}\log_{2}x+1$
for all $x\geq 1$. The main part of the result follows shortly.
Finally, put $m_{0}=0$ and $m_{l+1}=16m_{l}+5\cdot 2^{k+1}$. Also, let
$\alpha=(k+1)\bmod{2}$. Using the fact that $2^{k+1}\mid m_{l}$, from the
recurrence relations for $R_{\varepsilon}$ we get
$\displaystyle R_{\varepsilon}(2^{\alpha}m_{l+1})$
$\displaystyle=R_{\varepsilon}(2^{4+\alpha}m_{l}+5\cdot
2^{k+1+\alpha})=R_{\varepsilon}(2^{3-k}m_{l}+5)$
$\displaystyle=R_{\varepsilon}(2^{-1-k}m_{l})+\frac{1}{3}=R_{\varepsilon}(2^{\alpha}m_{l})+\frac{1}{3}.$
It follows that
$\displaystyle
Q_{\varepsilon}\left(\frac{m_{l}}{2^{k+1}}\right)-\frac{m_{l}}{3\cdot
2^{k+1}}=R_{\varepsilon}(2^{\alpha}m_{l})=\frac{l}{3}\sim\frac{1}{12}\log_{2}m_{l},$
and it remains to use (4).
Similarly, we can take $n_{0}=0$ and $n_{l+1}=16n_{l}+10\cdot 2^{k+1}$. ∎
## 7\. Computational results, questions, problems and conjectures
In this section, we discuss possible directions for further research and
present some conjectures and computational results.
To begin, recall that in Section 3 we have defined $(f_{n})_{n\in\mathbb{N}}$
to be the increasing sequence such that
$S_{1}^{\prime}=\\{f_{n}:n\in\mathbb{N}\\}$, and asked whether it is regular.
We have performed some experimental computations in Mathematica 13 with the
help of the IntegerSequences package by Eric Rowland, available at
https://ericrowland.github.io/packages.html. More precisely, for each $m\leq
30$ we have used the FindRegularSequenceRecurrence function, which did not
find a finite set of (plausible) $\mathbb{Z}$-linear relations between the
elements of the $m$-kernel $K_{m}((f_{n})_{n\in\mathbb{N}})$. Hence, we expect
that following conjecture holds.
###### Conjecture 7.1.
The sequence $(f_{n})_{n\in\mathbb{N}_{+}}$ is not $m$-regular for any $m\geq
2$.
On the other hand, note that we if we consider the decomposition
$S_{1}^{\prime}=\bigcup_{k=0}^{\infty}U_{k}$
into pairwise disjoint sets
$U_{k}=\\{2^{2k+1}(8s+2t_{s}+3):\;s\in\mathbb{N}\\}$, then for each
$k\in\mathbb{N}$ the sequence $(2^{2k+1}(8s+2t_{s}+3))_{s\in\mathbb{N}}$ is
2-regular.
Next, it is natural to ask whether it is possible to obtain results on the
representation of $b_{m}(n)$ as a sum of three squares for any
$m\in\mathbb{N}_{+}$.
###### Problem 7.2.
Characterize the set $S_{m}$ for $m\in\mathbb{N}_{+}$.
If the valuations $\nu_{2}(b_{m}(n))$ are bounded, then the direct approach
used in this paper, namely reduction modulo a fixed power of $2$, is
sufficient to give a complete description of $S_{m}$. The following
proposition implies that in this case $S_{m}$ is a $2$-automatic set (its
characteristic sequence is $2$-automatic).
###### Proposition 7.3.
For each $m\in\mathbb{N}_{+}$ and $p\in\mathbb{N}$ the sequence
$(b_{m}(n)\bmod{2^{p}})_{n\geq 0}$ is $2$-automatic.
###### Proof.
Take any $k\geq p-1$ such that $2^{k}\geq m$. Note that $(b_{m}(n))_{n\geq 0}$
is the convolution of the sequence $(b_{2^{k}}(n))_{n\geq 0}$ with $2^{k}-m$
copies of the PTM sequence $(t_{n})_{n\geq 0}$. They are both $2$-regular when
treated as sequences over the ring $\mathbb{Z}/2^{p}\mathbb{Z}$ (for
$(b_{2^{k}}(n))_{n\geq 0}$ this follows from Lemma 5.1). Hence,
$(b_{m}(n)\bmod{2^{p}})_{n\geq 0}$ is $2$-regular as the convolution of
$2$-regular sequences. The result follows from the fact that a $2$-regular
sequence attaining finitely many values is necessarily $2$-automatic. ∎
Unfortunately, we do not know even for a single value $m\neq 2^{k}-1$, whether
or not the valuations $\nu_{2}(b_{m}(n))$ are bounded. It is conjectured that
they are unbounded for all $m\neq 2^{k}-1$ (see [10, Conjecture 5.3]).
Nevertheless, this does not rule out $2$-automaticity of the set $S_{m}$.
Surprisingly, numerical results for $m\leq 30$ (obtained with help of the
IntegerSequences package) suggest that the sets $S_{m}$ are $2$-automatic for
odd $m$, except for $m=17,21$.
It should be possible to get some partial results if we restrict our attention
to arithmetic progressions along which $\nu_{2}(b_{m}(n))$ is bounded. For
example, [10, Theorem 5.4] provides a collection of suitable arithmetic
progressions such that $\nu_{2}(b_{2}(2^{r}n+s))$ is constant. By Proposition
7.3, the set of $n$ such that $b_{2}(2^{r}n+s)$ is a sum of three squares, is
$2$-automatic.
A related interesting problem concerns the behavior of $b_{m}(n)$ modulo a
fixed power of $2$.
###### Problem 7.4.
For $m\in\mathbb{N}_{+}$ and $p\in\mathbb{N}$ characterize
$b_{m}(n)\bmod{2^{p}}$.
We already know that this sequence is $2$-automatic and may ask whether it can
be characterized in terms of simpler $2$-automatic sequences. The congruences
obtained in the previous sections for subsequences of the form
$b_{2^{k}-1}(2^{r}n+s)$ are all “admissible” in the sense of [14], that is,
only involve $t_{n}$ and $(-1)^{\nu_{2}(n)}$. In the case $k\geq 3$ Theorem
5.5 provides a congruence in terms of $t_{n},t_{n-1}$ that can be transformed
into an admissible one due to the relation $t_{n-1}=(-1)^{\nu_{2}(n)+1}t_{n}$.
It turns out that other interesting $2$-automatic sequences already appear if
we consider $b_{m}(n)$ modulo suitable powers of $2$. By inspecting modulo
$32$ the subsequences described in Proposition 3.7, we have found (without
proof) the following set of congruence relations:
$\displaystyle b(8n+2)$ $\displaystyle\equiv 2t_{n}+16\sigma_{n}\pmod{32},$
$\displaystyle b(8n+6)$ $\displaystyle\equiv
6t_{n}+16\sigma_{n}+16n\pmod{32},$ $\displaystyle b(16n+8)$
$\displaystyle\equiv(10+8n^{2})t_{n}+16\sigma_{n}\pmod{32},$
where $\sigma_{n}$ counts modulo $2$ the number of blocks of contiguous $1$’s
in the binary expansion of $n$. We have later learned that Alkauskas [2,
Theorem 2] obtained a set of relations that describe the same sequences and
involve the Rudin–Shapiro sequence instead of $(\sigma_{n})_{n\in\mathbb{N}}$.
It can be checked that both descriptions are equivalent.
Another sequence that arises in this way is the regular paperfolding sequence
$(p_{n})_{n\in\mathbb{N}_{+}}$ defined by $p_{2n}=p_{n}$ and
$p_{2n+1}=(-1)^{n}$ (see for example [3, Example 5.1.6]). If we let
$P(x)=\sum_{n\geq 1}p_{n}x^{n}$, then through manipulation of power series,
for $m$ even one can obtain the congruence relation
$B_{m}(x)\equiv(1-x)^{m}(1+2mP(x))\mkern 4.0mu({\operator@font mod}\mkern
6.0mu2^{\nu_{2}(m)+3}).$
This is essentially a generalization of Lemma 5.1.
We now consider some natural modifications of the original equation
$b_{m}(n)=x^{2}+y^{2}+y^{2}$. We have obtained precise characterization of
those $n\in\mathbb{N}$ such that $b(2n)$ is a sum of three squares. In
particular, the set of such numbers has natural density equal to $11/12$.
Analyzing, for a given $n$ not of the form $2^{2k+1}(8s+2t_{s}+3)$, the
solution set $(x,y,z)$ of the equation $b(2n)=x^{2}+y^{2}+z^{2}$, we found
that in many cases one of the values $x,y,z$ is a square, i.e., the
Diophantine equation
$b(2n)=x^{2}+y^{2}+z^{4}$
has a solution in non-negative integers. More precisely, for $n\leq 10^{3}$ we
know that there are exactly 916 values of $n$ such that $b(2n)$ is a sum of
three squares. Among them, there are exactly 831 values of $n$ such that
$b(2n)$ is a sum of two squares and a fourth power. This large number of
solutions suggest the following conjecture.
###### Conjecture 7.5.
Let $Q_{1}:=\\{n\in\mathbb{N}:\;b(2n)=x^{2}+y^{2}+z^{4}\;\mbox{for
some}\;x,y,z\in\mathbb{N}\\}$. The set $Q_{1}$ is infinite. Moreover, the set
$Q_{1}$ has positive natural density in $\mathbb{N}$.
On the other hand, there are exactly seven values of $n\leq 1000$ such that
$b(2n)$ is a sum of a square and two fourth powers. This may suggest that the
number of solutions of the equation $b(2n)=x^{2}+y^{4}+z^{4}$ is finite.
However, due to limited range of our computations we instead formulate the
following:
###### Question 7.6.
Is the set of $n$ such that $b(2n)=x^{2}+y^{4}+z^{4}$ has a solution in
integers $x,y,z$ infinite?
An even more interesting and difficult question is whether the set
$\mathcal{T}=\\{n\in\mathbb{N}:\;b(2n)=x^{2}+y^{2}\\}$
is infinite or not. Because we know the behaviour of $b(n)\mod{16}$ we can
easily prove that the complement of $\mathcal{T}$, i.e.,
$\mathbb{N}\setminus\mathcal{T}$ is infinite. Indeed, from Proposition 3.2 we
have $b(16n+4)\equiv 4t_{n}\mkern 4.0mu({\operator@font mod}\mkern 6.0mu16)$.
If $t_{n}=-1$, then $b(16n+4)\equiv 12\mkern 4.0mu({\operator@font mod}\mkern
6.0mu16)$ and thus $b(16n+4)$ is not a sum of two squares.
To get a clue what can be expected in the case of the set $\mathcal{T}$, we
computed the values of $b(2n)$ for $n\leq 2^{20}$ and check whether $b(2n)$ is
a sum of two squares. We put
$\mathcal{T}(x)=\\#\\{n\leq x:\;n\in\mathcal{T}\\}$
and in Table 3 we present the values of $\mathcal{T}(2^{n})$ for $n\leq 20$.
$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
$\mathcal{T}(2^{n})$ | 2 | 3 | 6 | 8 | 14 | 21 | 37 | 64 | 106 | 174
$n$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
$\mathcal{T}(2^{n})$ | 325 | 617 | 1089 | 2018 | 3699 | 6804 | 12551 | 23624 | 44606 | 84176
Table 3. The number $\mathcal{T}(2^{n})$ for $n\leq 20$.
We also define
$\mathcal{S}=\\{r_{2}(b(2n)):\;n\in\mathbb{N}\\},$
where $r_{2}(m)$ is the number of representations of $m$ as a sum of two
squares. Let us recall that
$r_{2}(m)=\sum_{d|m,d\equiv 1\mkern 4.0mu({\operator@font mod}\mkern
6.0mu2)}(-1)^{\frac{d-1}{2}}.$
In the considered range; i.e., $n\leq 2^{20}$ the set $\mathcal{S}$ contains
the number 0 and the 35 values $s_{1}\leq\ldots\leq s_{35}$. In Table 4 below,
we present the following values: $s_{i}$, $l_{i}$ – the number of times
$s_{i}$ is attained, and $n_{i}$ – the smallest value of $n$ such that
$r_{2}(b(2n))=s_{i}$.
$i$ | $s_{i}$ | $l_{i}$ | $n_{i}$ | $i$ | $s_{i}$ | $l_{i}$ | $n_{i}$
---|---|---|---|---|---|---|---
1 | 4 | 4 | 0 | 19 | 224 | 1 | 793875
2 | 8 | 13768 | 4 | 20 | 240 | 1 | 647317
3 | 12 | 2 | 21 | 21 | 256 | 1005 | 15113
4 | 16 | 26411 | 30 | 22 | 288 | 13 | 28561
5 | 24 | 760 | 431 | 23 | 320 | 19 | 113399
6 | 32 | 22889 | 115 | 24 | 384 | 149 | 24877
7 | 40 | 36 | 2522 | 25 | 512 | 202 | 11231
8 | 48 | 1400 | 117 | 26 | 576 | 5 | 420383
9 | 56 | 1 | 27502 | 27 | 640 | 2 | 210415
10 | 64 | 11710 | 482 | 28 | 768 | 23 | 88529
11 | 72 | 9 | 21880 | 29 | 1024 | 27 | 202049
12 | 80 | 46 | 36642 | 30 | 1152 | 1 | 938983
13 | 96 | 1094 | 309 | 31 | 1280 | 1 | 162157
14 | 112 | 2 | 84169 | 32 | 1536 | 5 | 379324
15 | 128 | 4130 | 1036 | 33 | 2048 | 2 | 324442
16 | 144 | 9 | 91925 | 34 | 2560 | 1 | 295411
17 | 160 | 24 | 10785 | 35 | 4096 | 1 | 105400
18 | 192 | 451 | 3085 | | | |
Table 4. Values of $s_{i},l_{i}$ and $n_{i}$ for $i\leq 35$.
Our numerical computations suggest the following.
###### Conjecture 7.7.
The set $\mathcal{T}$ is infinite.
The following heuristic reasoning provides further evidence towards our
conjecture. More precisely, recall that the counting function of the sums of
two squares up to $x$ is $O(x/\sqrt{\log x})$. Thus, one can say that the
probability that a random positive integer $n$ can be written as a sum of two
squares of integers is $c/\sqrt{\log n}$. Since,
$\log_{2}b(n)\approx\frac{1}{2}(\log_{2}n)^{2}$ one could conjecture that the
expectation that $b(n)$ is a sum of two squares is $c^{\prime}/\log n$ for
some positive constant $c^{\prime}$, provided that $b(n)$ behaves like a
random integer of its size. As a consequence, up to $x$, we would have at
least
$\sum_{n\leq x}\frac{1}{\log n}=\frac{x}{\log x}+O(x/\log^{2}x)$
values of $n$ such that $b(n)$ is a sum of two squares. We dare to formulate
the following statement.
###### Conjecture 7.8.
There exists a positive real number $c$ such that
$\mathcal{T}(x)=c\frac{x}{\log x}+O(x/\log^{2}x)$
as $x\rightarrow+\infty$.
Although limited, our computations confirm such an expectation. In Table 5 we
give the values $\mathcal{T}(2^{m})\frac{m}{2^{m}}$ for $m=10,\ldots,20$.
$m$ | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{T}(2^{m})\frac{m}{2^{m}}$ | 1.67 | 1.74 | 1.80 | 1.73 | 1.72 | 1.7 | 1.66 | 1.63 | 1.62 | 1.62 | 1.61
Table 5. Values of $\mathcal{T}(2^{m})\frac{m}{2^{m}}$ for $m=10,\ldots,20$.
###### Remark 7.9.
The expectation that $b(n)$ behaves like a random integer of its size is very
likely. Indeed, numerical computations suggest that for any odd integer $m$
the sequence $(b(n)\pmod{m})_{n\in\mathbb{N}}$ is uniformly distributed; i.e.,
for any $r\in\\{0,1,\ldots,m-1\\}$ we have
$\lim_{N\rightarrow+\infty}\frac{\\#\\{n\leq N:\;b(n)\equiv r\mkern
4.0mu({\operator@font mod}\mkern 6.0mum)\\}}{N}=\frac{1}{m}.$
However, according to the best knowledge of the authors, it is not even known
whether the set of prime numbers $p$ such that $p|b(n)$ for some $n$, is
infinite.
## References
* [1] J.-P. Allouche, B. Cloitre, V. Shevelev, Beyond odious and evil, Aequationes Math. 90(2) (2016), 341–353.
* [2] G. Alkauskas, Generalization of the Rødseth–Gupta theorem on binary partitions, Lithuanian Math. Jour. 43 (2) (2003/4), 103–110.
* [3] J.-P. Allouche, J. Shallit, Automatic sequences. Theory, applications, generalizations, Cambridge University Press, Cambridge, 2003.
* [4] J.-P. Allouche, J. Shallit, The ring of $k$-regular sequences, Theoret. Comput. Sci. 98 (1992), no. 2, 163–197.
* [5] J.-P. Allouche, J. Shallit, The ring of $k$-regular sequences II, Theoret. Comput. Sci. 307 (2003), no. 1, 3–29.
* [6] R. Burns, Factorials and Legendre’s three-square theorem: II: available at https://arxiv.org/pdf/2203.16469.pdf
* [7] R. F. Churchhouse, Congruence properties of the binary partition function, Proc. Cambridge Philos. Soc. 66 (1969), 371–376.
* [8] J.-M. Deshouillers, F. Luca, How often is $n!$ a sum of three squares? I: The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, pages 243–251. Springer New York, 2010.
* [9] A. Granville, Y. Zhu, Representing binomial coefficients as sums of squares, Amer. Math. Monthly, 97(6) (1990), 486–493.
* [10] M. Gawron, P. Miska, M. Ulas, Arithmetic properties of coefficients of power series expansion of $\prod_{n=0}^{\infty}\left(1-x^{2^{n}}\right)^{t}$ (with an Appendix by Andrzej Schinzel), Monatsh. Math. 185 (2018), 307–360.
* [11] H. Gupta, Proof of the Churchhouse conjecture concerning binary partitions, Proc. Cambridge Philos. Soc. 70 (1971), 53–56.
* [12] H. Gupta, A simple proof of the Churchhouse conjecture concerning binary partitions, Indian J. Pure Appl. Math. 5(3), (1972), 791–794.
* [13] H. Gupta, A direct proof of the Churchhouse conjecture concerning binary partitions, Indian J. Math., (1) N.18, (1976), 1–5.
* [14] M. D. Hirschhorn, J. H. Loxton, Congruence properties of the binary partition function, Math. Proc. Camb. Phil. Soc., 78(3) (1975), 437–442.
* [15] L. Hajdu, A. Papp, On asymptotic density properties of the sequence $n!$, Acta Arith., 184(4) (2018), 317–340.
* [16] K. Mahler, On a special functional equation, J. London Math. Soc. 1 (1940), 115–123.
* [17] N. Robbins, On Fibonacci and Lucas numbers which are sums of precisely four squares, Fib. Quarterly, 21 (1) (1983), 3–5.
* [18] N. Robbins, Representing $\binom{2n}{n}$ as a sum of squares, Fib. Quarterly, 25 (1987), 29–33.
* [19] Ø. Rødseth, Some arithmetical properties of $m$-ary partitions, Proc. Cambridge Philos. Soc. 68 (1970), 447–453.
* [20] Ø. Rødseth, J. A. Sellers, Binary partitions revisited, J. Comb. Theory, Series A 98 (2002), 33–-45.
* [21] Sz. Tengely, M. Ulas, Equal values of certain partition functions via Diophantine equations, Research in Number Theory 7 (4) (2021), Article number: 67.
Bartosz Sobolewski, Jagiellonian University, Faculty of Mathematics and
Computer Science, Institute of Mathematics, Łojasiewicza 6, 30 - 348 Kraków,
Poland
e-mail<EMAIL_ADDRESS>
Maciej Ulas, Jagiellonian University, Faculty of Mathematics and Computer
Science, Institute of Mathematics, Łojasiewicza 6, 30 - 348 Kraków, Poland
e-mail<EMAIL_ADDRESS>
|
# From spin liquid to magnetic ordering in the anisotropic kagome
Y-Kapellasite Y3Cu9(OH)19Cl8: a single crystal study
Dipranjan Chatterjee Université Paris-Saclay, CNRS, Laboratoire de Physique
des Solides, 91405, Orsay, France Pascal Puphal Max-Planck-Institute for
Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany Quentin
Barthélemy Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France Jannis Willwater Institut für Physik der
Kondensierten Materie, Technische Universität Braunschweig, Braunschweig,
Germany Stefan Süllow Institut für Physik der Kondensierten Materie,
Technische Universität Braunschweig, Braunschweig, Germany Christopher Baines
Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, 5232 Villigen,
Switzerland Sylvain Petit LLB, CEA, CNRS, Université Paris-Saclay, CEA
Saclay, Gif-sur-Yvette, France Eric Ressouche Univ. Grenoble Alpes, CEA,
IRIG, MEM, MDN, 38000 Grenoble, France Jacques Ollivier Institut Laue-
Langevin, 38042 Grenoble, France Katharina M. Zoch Physikalisches Institut,
Goethe-Universität Frankfurt, Frankfurt am Main, Germany Cornelius Krellner
Physikalisches Institut, Goethe-Universität Frankfurt, Frankfurt am Main,
Germany Michael Parzer Institute of Solid State Physics, TU Wien, 1040
Vienna, Austria Alexander Riss Institute of Solid State Physics, TU Wien,
1040 Vienna, Austria Fabian Garmroudi Institute of Solid State Physics, TU
Wien, 1040 Vienna, Austria Andrej Pustogow Institute of Solid State Physics,
TU Wien, 1040 Vienna, Austria Philippe Mendels Université Paris-Saclay,
CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France Edwin
Kermarrec Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France Fabrice Bert Université Paris-Saclay, CNRS, Laboratoire
de Physique des Solides, 91405, Orsay, France
###### Abstract
Y3Cu9(OH)19Cl8 realizes an original anisotropic kagome model hosting a rich
magnetic phase diagram [M. Hering _et al_ , npj Computational Materials 8, 1
(2022)]. We present an improved synthesis of large phase-pure single crystals
via an external gradient method. These crystals were investigated in details
by susceptibility, specific heat, thermal expansion, neutron scattering and
local $\mu$SR and NMR techniques. At variance with polycristalline samples,
the study of single crystals gives evidence for subtle structural
instabilities at 33 K and 13 K which preserve the global symmetry of the
system and thus the magnetic model. At 2.1 K the compound shows a magnetic
transition to a coplanar (1/3,1/3) long range order as predicted
theoretically. However our analysis of the spin wave excitations yields
magnetic interactions which locate the compound closer to the phase boundary
to a classical _jammed_ spin liquid phase. Enhanced quantum fluctuations at
this boundary may be responsible for the strongly reduced ordered moment of
the Cu2+, estimated to be $\sim$ 0.075 $\mu_{B}$ from $\mu$SR.
## I Introduction
Low-dimensional materials with strong magnetic frustration, such as compounds
with decoupled antiferromagnetic kagome layers, are prototypical systems to
search for an experimental realization of the quantum spin-liquid (QSL) state
[1, 2]. This long sought state features no static magnetic order, despite
sizeable magnetic interactions, but rather macroscopic entanglement and
fractional excitations. Experimental signatures of a QSL state in kagome
materials were found first in herbertsmithite, ZnCu3(OH)6Cl2 [3, 4] and in the
closely related Zn-doped barlowite [5, 6]. Thus Cu-based systems with a
perfect kagome layer have been strongly investigated.
Around the A2+Cu3(OH)6Cl2 family [7] a new similar series of compounds has
been found with A3+Cu3(OH)6Cl3 [7, 8, 9], which realizes perfect kagome layers
in a kapellasite-type structure. However, for Y3+ the compound is not stable
in water solutions, leading to a Cl-OH substitution YCu3(OH)6+xCl3-x with
$x=1/3$ (or Y3Cu9(OH)19Cl8) [10, 11]. This substitution induces a distortion
of the magnetic lattice as the Y atoms are moved out of the kagome plane [10].
The distorted kagome lattice of Cu2+ ions is scarcely studied with the
exception of the family around (Cs,Rb)2Cu3(Ti,Sn,Zr,Hf)F12 [12, 13, 14, 15].
However recent structural investigations of low temperature diffraction on
various kagome systems show that a structural instability induces distortions
in barlowite, claringbullite [16], volborthite [17] and vesigneite [18] making
it a typical structural motif at low temperatures.
In Y-kapellasite ($x=1/3$) that we investigate in this article, the distortion
yields two nonequivalent Cu sites and a unique magnetic model with three
different nearest neighbor interactions, while still retaining rotational
symmetry around the hexagons of the anisotropic kagome lattice (see inset Fig.
11 and Ref. [11]). The discovery of this compound has triggered a detailed
theoretical study [19] which disclosed the full classical phase diagram of the
model and its richness. Interestingly, besides two long range ordered phases
with propagation vectors $Q=(1/3,1/3)$ and $Q=(0,0)$, a large area in the
phase diagram, coined ”classical spin liquid” phase, which encompasses the
isotropic kagome model, opens up. It remains so far largely unexplored,
although it could realize an unprecedented ”jammed” spin liquid phase,
characterized by a discrete ground state degeneracy, in a disorder free model.
Besides, first principle calculations confirm the relevance of this
anisotropic nearest-neighbor model for Y-kapellasite and locates it in the
$(1/3,1/3)$ long range ordered phase [19]. Here, we report a comprehensive
study of the physical properties of large phase pure single crystals that can
be compared to the theoretical prediction and which show marked deviations
with respect to former studies of polycristalline samples [11] or other
synthesis route for crystals [20].
The paper is organized as follows. In sections II and III the synthesis of
phase-pure large single crystals is described and macroscopic measurements of
the susceptibility, heat capacity and thermal expansion are presented,
providing evidence for a series of structural and magnetic transitions below
33 K. In sections IV and V, the transitions are investigated in detail by
neutron diffraction and by Cl NMR, which shows in particular good agreement
with the theoretically predicted spin order in the ground state. In section
VI, muon spin spectroscopy is used to prove the bulk nature of the 2.1 K
magnetic transition. In section VII, the spin wave excitations are studied by
inelastic neutron scattering and give a novel insight into the determination
of the magnetic interactions beyond first principle calculations. Finally in
section VIII, we summarize and discuss our results in comparison with previous
ones on polycristalline samples and in light of theoretical studies.
## II Synthesis and susceptibility
The crystal growth of Y-Kapellasite was originally reported in Ref. [10],
where 0.59 g Y2O3, 0.82 g CuO and 0.89 g CuCl${}_{2}\cdot$2(H2O) in 10 ml H2O
were heated up to the dissolution point of Y2O3, followed by a slow cooling to
crystallize. However, these crystals of an average size of 1x1x1 mm3 suffered
small CuO inclusions since the growth takes place on the surface of the
polycrystalline CuO starting material, as the dissolution point of Y2O3 and
the crystallization point of the compound lie above the maximum in solubility
of CuO. Reference [20] describes a synthesis of inclusion free crystals using
LiOH, Y(NO3)${}_{3}\cdot 6$H2O and CuCl${}_{2}\cdot 2$H2O. However the
reported crystals are somewhat disordered as a partial occupation of the Y
site is observed, similarly to what was found in the related compound
YCu3(OH)6Cl3 [8]. This likely hints at a phase mixture in both cases due to
contact with water. Indeed, a low percentage of occupation of Y for
YCu3(OH)6Cl3 [8] lies out of the kagome plane, which represents structural
parts of Y-Kapellasite Y3Cu9(OH)19Cl8. For the reported inclusion free
Y3Cu9(OH)19Cl8 [20], the partial occupation of Y can be explained similarly,
since the specific heat shows a mixture of the signal observed for $x=0$ and
$x=1/3$ as shown in Fig. 2.
We thus improved the synthesis for inclusion-free large bulk single crystals
suitable for neutron studies via a horizontal external gradient growth method
in a thick-walled quartz ampule as described in detail for herbertsmithite in
Ref. [21]. The growth is realized by slowly dissolving CuO in a YCl3-H2O (or
YCl3-D2O for deuterated crystals) solution and transporting it to the cold
end. The growth is executed in a three zone furnace with a gradient of
25$\degree$C and a temperature of 240$\degree$C at the hot end (note that the
elevated pressure at this temperature requires especially thick quartz
ampules). The gradient was optimized as too low temperatures yielded a mixture
of Y-Kapellasite and Clinoatacamite. Afterwards, the inclusion-free hexagonal
single crystals have an average size of 3x3x1 mm3 up to 3x3x3 mm3, if grown
over several weeks. The pictures of single crystalline Y-kapellasite are
displayed in Fig. 1 a) and b), which show a transparent specimen without any
visible impurity inclusions.
The dc-susceptibility of a 15.7 mg single crystal of Y-Kapellasite measured in
a Quantum Design MPMS XL7 squid system with a 1 T field applied along the $c$
axis is shown in Fig. 1 c). It reproduces published results [10, 22], with an
antiferromagnetic Curie-Weiss constant $\theta_{\rm{cw}}=102(1)$ K and
$g_{\rm{c}}=2.40(1)$. Measurements in a lower 0.01 T magnetic field, shown in
the inset, reveal a bifurcation of the field cooled and zero field cooled
curves below $\sim 5$ K, and a maximum around 2.5 K. These low field
measurements point at a magnetic transition which is investigated in details
with neutron scattering and resonance techniques in the following sections. At
variance with reference [20], we observe no magnetic transition at 11 K. Such
a transition, reminiscent of the one observed at 12 K in the $x=0$
counterpart, supports a phase mixture scenario for the crystal of reference
[20].
Figure 1: a) Y3Cu9(OH)19Cl8 single crystal along the $c$ axis. b) Image in
transmission light of an as grown single crystal. c) Susceptibility (M/H)
versus temperature with a 1 T field applied along $c$. Bottom left inset:
field cooled (FC) and zero field cooled (ZFC) measurements at 0.01 T. Upper
right inset: anisotropic kagome lattice with 3 different nearest neighbor Cu-
Cu bonds and the related 3 different magnetic interactions.
## III Specific heat and thermal expansion
Figure 2: a) Specific heat divided by temperature in the temperature range
from 1.8 to 40 K for the $x=1/3$ optimized large crystals, compared to
literature results for $x=1/3$ and $x=0$ powder samples [11] and the reported
inclusion free $x=1/3$ crystals from Ref. [20]. b) Thermal expansion along the
$s$ direction of an optimized large Y-Kapellasite single crystal. c) and d)
Insets: enlarged views of the peaks detected in main panel b). Note the
different vertical scales in the two insets.
Specific heat was measured with the standard option of a Quantum Design
Physical Properties Measurement System (PPMS), using a 5.935 mg inclusion-free
single crystal formerly described. Its temperature evolution is shown in Fig.
2 a) and compared to the published results for $x=0$ and $x=1/3$
polycrystalline samples [11] and the $x=1/3$ single crystal reported in Ref.
[20]. Unexpectedly, our $x=1/3$ single crystal presents a clear double peak at
33 K, which is absent in powder samples of the same compound. The underlying
entropy amounts to 0.1537 J/mol K, below 3% of $R$ln2, calculated by using the
powder sample as a baseline. There is no associated sharp feature in the
susceptibility in this temperature range so that the double peak is likely
signaling a structural transition. Note that this 33 K anomaly was not
reported in the single crystals of Ref. [20] and that, on the contrary, the
large hump around 15 K seen in the latter crystals and in the $x=0$ powder
sample, associated to magnetic transitions, is absent from our phase-pure
$x=1/3$ single crystal. Only at 2.1 K do we find a sharp peak in specific
heat, which was already reported for this compound both in powder and crystals
[10, 11], and stems from a magnetic long range ordering transition as shown
later in this article. For the disordered powder sample $x=1/3$, the
underlying entropy of the transition is reduced and previous muon spin
relaxation studies had shown the absence of long range order [11], possibly
arising from the slightly altered structure due to the suppression of the 33 K
transition.
To inspect the 33 K anomaly in more detail, we performed thermal expansion
experiments along the crystallographic $c$-axis, i.e. perpendicular to the
kagome layers, using a capacitive dilatometer. Figure 2 b) displays the
thermal expansion coefficient $\alpha$ on the same temperature scale as the
specific heat data in panel a). We find a similar anomaly below 33 K as in
$C/T$, consisting of two slightly broadened, and thus overlapping peaks with a
ratio between the sizes of the higher-temperature and lower-temperature peaks
comparable to a). The relative length change incorporated in these two sharp
features amounts to $\Delta L/L=10^{-4}$. In addition, we discover another,
significantly smaller anomaly around 13 K, which can be seen in more details
in the inset c). Below 5 K an anomaly in $\alpha$ forms as approaching the
transition at 2.1 K, signalling the importance of magneto-elastic coupling as
antiferromagnetism sets in. Overall, the anomalies at 13 K and 33 K reveal
clear structural effects in this compound with a distorted kagome lattice,
without noticeable changes upon applying a magnetic field of 9 T along the
$c$-axis.
## IV Diffraction
T (K) | 0.065 | 8 | 20 | 40
---|---|---|---|---
| x/a | y/b | z/c | x/a | y/b | z/c | x/a | y/b | z/c | x/a | y/b | z/c
Y1 | 0 | 0 | 0.1318(7) | 0 | 0 | 0.1314(6) | 0 | 0 | 0.1324(7) | 0 | 0 | 0.1302(5)
Y2 | 0 | 0 | 0.5 | 0 | 0 | 0.5 | 0 | 0 | 0.5 | 0 | 0 | 0.5
Cu1 | 0.6661(10) | 0.8325(11) | 0.5025(4) | 0.6658(10) | 0.8340(11) | 0.5021(4) | 0.6675(11) | 0.8328(12) | 0.5026(4) | 0.6645(7) | 0.8317(7) | 0.5021(3)
Cu2 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5
Cl1 | 0.6745(9) | 0.0033(10) | 0.2845(2) | 0.6743(8) | 0.0037(10) | 0.28435(19) | 0.6748(9) | 0.0031(11) | 0.2845(2) | 0.6732(6) | 0.0027(6) | 0.28395(15)
Cl2 | 0 | 0 | 0.3368(5) | 0 | 0 | 0.3368(5) | 0 | 0 | 0.3365(6) | 0 | 0 | 0.3373(4)
O1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
D1 | 0.456(9) | 0.776(10) | 0.660(4) | 0.455(9) | 0.783(10) | 0.664(4) | 0.455(9) | 0.776(10) | 0.660(4) | 0.462(7) | 0.777(7) | 0.661(3)
O2 | 0.8119(14) | 0.8060(12) | 0.5435(7) | 0.8122(13) | 0.8046(12) | 0.5438(7) | 0.8129(14) | 0.8060(13) | 0.5438(8) | 0.8121(9) | 0.8035(9) | 0.5435(4)
D2 | 0.7769(19) | 0.7778(17) | 0.6011(9) | 0.7874(18) | 0.7858(17) | 0.5991(9) | 0.7723(17) | 0.7722(16) | 0.6030(8) | 0.7866(13) | 0.7827(13) | 0.6005(6)
O3 | 0.5293(12) | 0.6635(11) | 0.5540(6) | 0.5285(11) | 0.6638(10) | 0.5537(6) | 0.5283(12) | 0.6622(11) | 0.5542(6) | 0.5290(8) | 0.6634(8) | 0.5543(4)
D3 | 0.5494(17) | 0.6731(18) | 0.6065(8) | 0.5575(16) | 0.6709(17) | 0.6082(8) | 0.5452(15) | 0.6740(18) | 0.6051(8) | 0.5548(12) | 0.6692(13) | 0.6069(5)
O4 | 0.5142(12) | 0.8433(11) | 0.4631(5) | 0.5167(11) | 0.8430(10) | 0.4632(5) | 0.5154(12) | 0.8450(11) | 0.4634(6) | 0.5162(8) | 0.8441(8) | 0.4638(4)
D4 | 0.5008(16) | 0.8310(17) | 0.4096(8) | 0.5013(16) | 0.8328(16) | 0.4097(8) | 0.5018(17) | 0.8320(17) | 0.4099(8) | 0.5029(12) | 0.8337(12) | 0.4099(6)
Table 1: Rietveld refinement results of neutron diffraction data obtained at
65 mK, 8 K, 20 K and 40 K with R-3 (#148) $a=b=11.539$ Å, and $c=17.1355$ Å.
Figure 3: Temperature dependence of the neutron diffraction intensity around
the nuclear Bragg peaks a) (6 6 0) and b) (0 0 18). Structure viewed along the
$c$-axis obtained from neutron single crystal diffraction measurements at c)
40 K and d) 65 mK.
The structure of Y-Kapellasite Y3Cu9(OH)19Cl8 with R-3c was first reported in
Ref. [10], where hydrogen was placed by symmetry arguments. Later, from
neutron diffraction on deuterated powder samples, we found that the deuterium
atom next to the O1 refines to a distance above 1 Å, which is so far
unaccounted for an O-H bound. Thus in Ref. [11], we dismissed the D1, leading
to the revised chemical formula of Y3Cu9(OD)18OCl8. To tackle further this
issue we then measured the stoichiometry including the hydrogen content with a
combination of inductively coupled plasma mass spectroscopy (ICP-OES) and gas
extraction, the former with a SPECTRO CIROS CCD and the latter with an Eltra
ONH-2000 analyzer. For the determination of oxygen and hydrogen content, the
powder samples were placed in a Ni crucible and clipped and heated, where the
carrier gas takes the oxygen and hydrogen out of the sample. The oxygen reacts
with carbon, and CO2 is detected in an infrared cell, while hydrogen is
detected by a thermal conductivity cell. Each measurement is repeated three
times and compared to a standard. The quoted error bars provide the
statistical error. We thus measured three pieces of a hydrated single crystal
and a part of a powder sample of Ref. [11]. The results of:
Y3.00(3)Cu9.06(9)O19.0(2)H18.7(3)Cl8 for the crystals and
Y3.00(3)Cu9.07(9)O19.3(2)H18.8(3)Cl8 for the powder sample show a high enough
hydrogen content to assume the ideal stoichiometry of Y3Cu9(OH)19Cl8. Notably
we have also seen vibrational O-H modes that exist only for electrical field
parallel to the kagome plane solely matching to this hydrogen/deuterium
position further confirming the presence of the D1 atom. Therefore, from now
on, we use the ideal stoichiometry of Y3Cu9(OH)19Cl8.
We performed neutron diffraction on the 2 axis thermal neutron diffractometer
D23 instrument at the Institut Laue-Langevin (ILL) using an inclusion-free,
deuterated single crystal weighing around 30 mg. Measurements have been
performed in the temperature range of 40 K to 0.065 K using constant incident
neutron wavelength $\lambda$=1.27 Å. We find the R-3c structure that was
reported in Ref. [10] to be stable over the whole temperature range. The
previously mentioned D1 position can be reasonably refined in all cases to a
similar position and even a release of the occupation is possible. We find in
all cases an unusually large distance of d${}_{O-D}=1.3$ Å, which was
nevertheless considered as mentioned above. The transition around 33 K
observed in specific heat and thermal expansion is reflected here in an
increase of the Bragg peak intensity by approximately 30% as shown in Fig. 3
a) and b). Yet, it produces no change in the crystallographic cell, as the
crystallographic axes $a,b$ and $c$ remain unaltered in the investigated
temperature range. We find in Rietveld refinements a sudden jump around 33 K
for certain atomic positions, which in absolute numbers remain however
marginal. The Rietveld refinement results are summarized in table
LABEL:Rietveld and selected atomic positions are plotted normalized to the 40
K values versus temperature in Fig. 4 to visualize the sudden changes around
30 K (see also 111See Supplemental Material at [URL] for the Crystallographic
Information Files corresponding to the refined Y-Kapellasite Y3Cu9(OH)19Cl8
structure at 40 K, 20 K, 8 K and 0.065 K.). These changes lie below 1% of the
nominal value, except for deuterium and thus are hardly visible in Fig. 3 c)
and d) depicting the crystal structure along the $c$ axis at 40 K and 65 mK,
but will be relevant for the discussion of our NMR results.
Figure 4: Relative atomic position changes versus temperature of a) Chlorine
site 1 and 2, b) Deuterium site 1, 2, 3, 4 and c) Cu site 1.
At low temperature $T=65$ mK we have searched for magnetic ordering at
different (H,K,L) planes but we found no magnetic Bragg peaks probably due to
low magnetic moment of Cu2+ ions in the ground state. Hence extensive studies
of the low temperature regime has been done to investigate the magnetic
structure using NMR, $\mu$SR and inelastic neutron scattering, as presented in
the following sections.
## V Nuclear Magnetic Resonance
We further investigated the structural distortions and magnetic transition
using 35Cl NMR on a $\sim 30$ mg single crystal of Y-Kapellasite
Y3Cu9(OH)19Cl8 with an external magnetic field Bext applied parallel to the
$c$-axis, i.e. perpendicular to the large facet of the sample (see Fig. 1).
The spectra are measured from room temperature down to 5 K by sweeping the
frequency in a fixed field B${}_{ext}=7.553$ T which corresponds to the
reference frequency $\nu_{0}=31.510$ MHz. At lower temperatures, the broader
spectra were recorded by sweeping the field with a fixed irradiation frequency
$\nu_{0}$.
The spectrum measured at 100 K, far above the magnetic transition and the
structural distortions, is shown in Fig. 5. We observe two distinct central
lines (see inset of Fig. 5) corresponding to the two crystallographic Cl sites
with intensity ratio close to the expected 3:1. There are indeed 18 Cl1 (resp.
6 Cl2) sites per unit cell located in between the kagome layers, close to the
center of each triangle (resp. hexagon) of the kagome structure. For clarity
in this section we denote the Cl1 and Cl2 sites as the triangular ($tri$) and
hexagonal ($hex$) sites respectively.
Figure 5: Full NMR spectrum of 35Cl at 100 K with the configuration
$B_{ext}\parallel c$ which shows 2 central lines (C) of Cl1 (triangular site)
and Cl2 (hexagonal site) and the 2 corresponding pairs of satellites (Stri and
SHex). Inset: enhanced view of the central lines (C). The dashed vertical line
indicates the reference frequency $\nu_{0}=31.510$ MHz
Since 35Cl possesses a nuclear spin $I=3/2$ with a finite quadrupolar moment,
the three allowed transitions between the Zeeman-split adjacent nuclear levels
are modified by the quadrupolar interaction with the surrounding electric
charges. We therefore observe in the full spectrum three NMR lines per Cl
site, one central line ($-1/2\leftrightarrow 1/2$ transition) and two
satellite ones ($-3/2\leftrightarrow-1/2$ and $1/2\leftrightarrow 3/2$
transitions).
At first order in perturbation, the frequency difference between two satellite
lines of a given $\alpha=tri$ or $\alpha=hex$ Cl site arising from the
quadrupolar interaction is given by [24]
$\Delta\nu_{\alpha}^{(1)}=\nu^{\alpha}_{Q}(3\cos^{2}{\theta_{\alpha}}-1-\eta_{\alpha}\sin^{2}{\theta_{\alpha}}\cos
2\phi_{\alpha})$ (1)
where $\theta_{\alpha}$ and $\phi_{\alpha}$ are the polar and azimuthal angles
defining Bext in the local frame of the principal axes of the electric field
gradient tensor (EFG), $\nu_{Q}^{\alpha}$ is proportional to the quadrupolar
moment and the largest eigenvalue of the EFG, therefore reflecting the
strength of the quadrupolar interaction, and the asymmetry parameter
$\eta_{\alpha}$ reflects the departure of the EFG from cylindrical symmetry.
Because of the high symmetry of the hexagonal Cl site, the EFG at this site is
axially symmetric ($\eta_{hex}=0$) along the $c$ axis and, under the condition
of the experiment $B_{ext}\parallel c$, $\theta_{hex}=0$. For this site, the
two satellites are maximally separated by $2\nu_{Q}^{hex}$ and we measure
directly from Fig. 5 $\nu_{Q}^{hex}=3640(5)$ kHz. The triangular Cl site is
less symmetric. We used the structure determined at 40 K in section IV by
neutron diffraction to compute the EFG at this site with a point charge
approach. From the calculated parameters $\theta_{tri}=8.3^{\circ}$,
$\phi_{tri}=123.0^{\circ}$, $\eta_{tri}=0.33$ and the measured distance
between satellites, we obtain with Eq. 1 the value $\nu_{Q}^{tri}=2180(20)$
kHz. Note that if Bext is not strictly applied along the symmetry axis $c$,
the triangular sites become non-equivalent because of a different orientation
of their EFG with respect to Bext. The minute splitting of the triangular site
satellite lines in Fig. 5 reflects such a small misalignment $\pm 1.2^{\circ}$
of our crystal in the experiment.
The quadrupolar interaction shifts the position of the central line only at
second order in perturbation [25]:
$\nu^{(2)}_{\alpha}=-\frac{(\nu^{\alpha}_{Q})^{2}}{2\nu_{0}}f(\theta_{\alpha},\phi_{\alpha},\eta_{\alpha})$
(2)
where, for the simplest case of an axially symmetric EFG, $f$ reads
$f(\theta,\phi,0)=\frac{3}{8}(1-\cos^{2}{\theta})(9\cos^{2}{\theta}-1).$ (3)
In the present case at 100 K, the second order quadrupolar shift is either
$\nu^{(2)}_{hex}=0$ since $\theta_{hex}=0$ or negligible
$\nu^{(2)}_{tri}=-4(1)$ kHz. The positions of the two central lines
$\nu^{C}_{\alpha}=\nu_{0}(1+K_{\alpha}^{0}+K^{s}_{\alpha})+\nu^{(2)}_{\alpha}$
(4)
are then rather set by the first term of magnetic origin, where
$K_{\alpha}^{0}$ is the temperature-independent orbital shift and
$K^{s}_{\alpha}$ the spin shift arising from the polarization of the unpaired
electron of the neighbouring Cu2+ ions and therefore proportional to the spin
susceptibility. From the plots of $\nu^{C}_{\alpha}$ versus the susceptibility
$\chi_{c}$ measured along the $c$ direction in the range 100-300 K (not
shown), we extract for the configuration B${}_{ext}\parallel c$ the hyperfine
constants $A_{hex}=0.245(3)$ T/$\mu_{B}$ and $A_{tri}=-0.426(16)$ T/$\mu_{B}$
and the orbital terms $K_{hex}^{0}=120(10)$ ppm and $K_{tri}^{0}=52(10)$ ppm.
Figure 6: Evolution of the central lines of the 35Cl NMR spectra with
temperature. The spectra are shifted vertically for clarity. Note the change
of the width and the splitting of the line below 40 K which become prominent
below 20 K. The dashed vertical line indicates the reference frequency.
Figure 6 shows the evolution of the central lines upon cooling down to 5 K.
Marked changes of the spectral shape are observed below 40 K where the
structural distortions take place: the triangular site line broadens
significantly and the hexagonal one splits. Interestingly, although the
changes appear in the range 30-40 K, they really get prominent only below 20 K
where a second peak is observed in thermal expansion.
From the neutron diffraction study, the structural distortion at 33 K does not
affect the crystal symmetry and maintains only two distinct Cl sites. It
mainly involves small displacements of the protons, which naturally modify the
EFG and therefore the positions of the lines but cannot account for line
broadening or line splitting. Therefore, 35Cl NMR reveals more complex
distortions with local symmetry breaking but no spatial correlation so that
the average symmetry as detected by neutron diffraction is preserved. In
particular, the splitting of the hexagonal site line implies the occurrence of
at least two locally non-equivalent crystallographic sites and at least one
with a finite tilt $\theta_{hex}$ of its EFG so that $\nu^{(2)}_{hex}$ does
not vanish in equation 4. From the measured splitting and using Eq. 4, we
estimate $\theta_{hex}\sim 5^{\circ}$ at 20 K and $\theta_{hex}\sim
10^{\circ}$ at 10 K.
These changes of the local electrostatic environment of Cl affect even more
clearly, at first order, the satellite lines. As shown in the inset a) of Fig.
7, the hexagonal satellite acquires a complex structure below 33 K pointing at
even more than two nonequivalent sites. In order to determine the onset of the
structural transition, we have measured the width of this satellite versus
temperature as plotted in Fig. 7. Beside the abrupt change at the $33$ K
structural transition evidenced in heat capacity and neutron diffraction
experiments, a marked change of the slope is detected at 13(1) K suggesting a
secondary transition as detected in thermal expansion and also clearly in NMR
relaxation measurements as discussed further in this section.
Figure 7: Hexagonal satellite linewidth evolution with temperature which
indicates 2 structural changes (black arrows). Inset (a) depicts the line
shape changes occurring below 33 K for the hexagonal satellite. Inset (b)
shows the hexagonal NMR central lines for 37Cl and 35Cl at 10 K.
To confirm the structural nature of the 13 K transition we have compared the
35Cl and 37Cl NMR spectra at 10 K at the same reference frequency of 24.000
MHz. We have fitted the split hexagonal site central lines with gaussian
functions and compared the frequency differences $\Delta\nu^{i}$ between the
two peak maxima for the two isotopes $i$ (inset (b) Fig. 7). The ratio
$\Delta\nu^{35}/\Delta\nu^{37}=1.5(2)$ compares well to the square of the
ratio of the quadrupolar moments of the two isotopes 1.61 as expected from Eq.
4. At variance, in case of a magnetic origin with static internal fields, we
would have observed a scaling with the gyromagnetic ratios of the two
isotopes, i.e. a ratio $\sim 1.20$.
At lower temperatures ($T<4.2$ K), the NMR lines broaden significantly, likely
as a result of the building up of magnetic correlations, in line with the
FC/ZFC divergence observed in this $T$ range in the susceptibility (see Fig.
1). This magnetic broadening blurs the quadrupolar splitting and down to 2.1
K, we observe only two broad central lines, overlapping partially, for the two
different, triangular and hexagonal, chlorine sites (see Fig. 8). At lower
temperatures an abrupt broadening for the triangular site and more moderately
for the hexagonal one occurs, which represents the spectral NMR signature of
the magnetic transition detected in the bulk, $T_{1}$ or $\mu$SR measurements.
Actually, two triangular site peaks develop on both sides of the hexagonal
central line and move away from each other upon cooling down. To quantify
these evolutions, we compare in Fig. 9 the change of the linewidth $W(T)$ with
temperature for the two sites.
Figure 8: Central line spectra at 1.3 K, 2 K, 3.6 K and partial view of the
ground state spin texture proposed in Ref. [19] in the top right corner. The
spectra are shifted vertically for clarity. The blue line is a gaussian fit of
the hexagonal central line. This contribution is subtracted from the
experimental spectrum to expose the triangular site one, shown as a red line.
The triangle configurations a, b and c yield three different dipolar fields at
the triangular Cl site. Configurations a and c yield almost equal but opposite
dipolar fields whereas configuration b has almost zero dipolar field along the
$z$ axis, i.e. the applied field direction.
In the recent theoretical study Ref. [19], a $(1/3,1/3)$ long range magnetic
order has been proposed for this system with a coplanar spin configuration
reproduced partially in Fig. 8. Remarkably, for this predicted spin
configuration, the hyperfine field at the hexagonal sites is zero (hexagons
with different colors spins) or nearly zero (hexagons with same color spins).
At variance, for each triangle, two spins are nearly opposite leaving the
third one uncompensated and thus able to yield a finite field at the
triangular sites. There are actually three types of triangle configurations
labelled a, b and c in Fig. 8 corresponding to three different orientations of
this uncompensated spin. For an anisotropic hyperfine tensor, we thus expect
three different lines for the triangular Cl in the magnetically ordered phase.
Although we could not determine experimentally all the components of the
hyperfine tensor, we note that assuming a purely dipolar coupling, the shown
spin configuration yields one very small and two almost equal but opposite
$z$-components of the transferred fields at the triangular Cl site. Thus, the
low temperature spectrum supports at least qualitatively the theoretically
proposed order, both for the triangular and hexagonal Cl sites.
Figure 9: Evolution versus temperature of the linewidth $W(T)$ for the
triangular (red dots) and hexagonal (blue dots) sites.
Figure 10 shows the evolution with temperature of the spin lattice relaxation
rate ($T_{1}^{-1}$) measured using the saturation recovery sequence at both Cl
sites. Being quadrupolar active, Cl nuclei probe both the magnetic and
structural fluctuations. In the temperature evolution of $T_{1}^{-1}$ the
structural distortions at $\sim$33 K and $\sim$13 K discussed earlier are
clearly observed as a sharp and broad peak respectively, together with the
magnetic transition around $\sim$ 2.1 K.
To extract the $T_{1}$ values, the recovery curves of the nuclear
magnetization $M(t)$ at each temperature were fitted to [26]
$M(t)=M_{sat}\left(1-ae^{-\left(\frac{t}{T_{1}}\right)^{\beta}}-be^{-\left(\frac{6t}{T_{1}}\right)^{\beta}}\right)$
(5)
where $a+b=1$, $a=0.1$ and $0.23$ respectively for the hexagonal and
triangular sites, and $\beta\leq 1$ is a phenomenological stretching parameter
used to account for a possible distribution of $T_{1}$ [27]. The evolution of
$\beta(T)$ for both sites is shown as insets in Fig. 10. While a single value
of $T_{1}$ is measured at high temperature ($\beta=1$), an increasingly
broader distribution ($\beta<1$) is needed upon cooling down towards the
structural and magnetic transitions, likely pointing at the building up of
spatial correlations. During the transition regimes, we could fix the value of
$\beta$ with no noticeable reduction of the fit quality, to avoid artefacts in
the determination of the peaks of the correlated $T_{1}^{-1}$ quantity.
Figure 10: Plots of 35Cl NMR relaxation rate $T_{1}^{-1}$ versus temperature
measured at the hexagonal (top) and triangular (bottom) central line
positions. The two structural transition regimes (2) and (3) and the magnetic
one (1) are highlighted. The $T_{1}$ distribution parameter $\beta$ is plotted
in the inset. Below 5 K, when the NMR lines of the two sites overlap, $T_{1}$
was also measured at the isolated hexagonal satellite position (dark cyan
dots) and shows the same behaviour as the central line.
## VI Muon spin relaxation
To elucidate the nature of the ground state of Y- Kapellasite, we performed a
$\mu$SR experiment on the GPS and DOLLY instruments at the PSI muon facility
from the paramagnetic regime ($>$20 K) down to 0.28 K. The 100% spin polarized
muons are implanted at time zero in the sample, where they stop at the most
electronegative positions. The muon spins evolve in the local fields until the
muons decay into positrons, emitted preferentially in the last direction of
the muon spins after a mean life time of 2.2 $\mu$s. By collecting the
positrons emitted forward and backward with respect to the muon beam
direction, the muon decay asymmetry, proportional to muon spin polarization
can be recorded versus time and allows to characterize the internal fields and
their fluctuations [28, 29, 30].
For this $\mu$SR experiment, we used two coaligned phase pure single crystals
of sizes $\sim 3\times 3\times 1$ mm3 with their $c$ axis parallel to the
initial muon spin direction. The muon decay asymmetry in zero external field
(ZF) is depicted at some selected temperatures in Fig. 11. At low temperature,
the asymmetry evidently differs from previous measurements [11] on a powder
sample shown in the inset, with a much faster relaxation and a characteristic
damped oscillation signaling static internal fields.
Figure 11: Time evolution of the zero-field (ZF) muon decay asymmetry at some
selected temperatures in single crystal samples. Lines are fit to Eq. 9.
Inset: powder sample data from Ref. [11].
Above 2.5 K, the relaxation hardly depends on temperature nor on the nature of
the sample, powder or single crystal. The electronic spins are fast
fluctuating in their paramagnetic regime, and the relaxation is dominated by
the quasi static, weak and random, nuclear fields. Following the model used
for the powder sample [11], we fitted the muon decay asymmetry at high
temperature to
$a_{0}P_{para}(t)=a_{0}\left[fP_{OH}(t)+(1-f)KT_{\Delta_{Cl}}(t)\right]\\\ $
(6)
where
$P_{OH}(t)=e^{-\frac{\Delta^{2}_{OH}t^{2}}{2}}{\bigg{[}}\frac{1}{6}+\frac{1}{3}\cos{{\bigg{(}}\frac{\omega_{OH}t}{2}{\bigg{)}}}+\\\
\frac{1}{3}\cos{{\bigg{(}}\frac{3\hskip
2.84526pt\omega_{OH}t}{2}{\bigg{)}}}+\frac{1}{6}\cos{{\bigg{(}}{\omega_{OH}t}{\bigg{)}}}{\bigg{]}}.\
$ (7)
The latter $P_{OH}(t)$ stands for the formation of $\mu$OH complexes with a
pulsation $\omega_{OH}=\frac{\hbar\mu_{0}\gamma_{\mu}\gamma_{H}}{4\pi d^{3}}$
which depends on the $\mu$-H distance $d$, the gyromagnetic ratios
$\gamma_{H}=267.513$ Mrad/s/T and $\gamma_{\mu}=851.616$ Mrad/s/T respectively
for protons and muons, and a gaussian broadening $\Delta_{OH}$ due to the
other surrounding nuclear spins. The second term in Eq. 6 is a static Kubo-
Toyabe relaxation standing for the minority fraction $1-f$ of muon stopping
sites, likely close to the Cl${}^{\textbf{-}}$ ions, where they experience a
gaussian distribution of static nuclear fields with a width $\Delta_{Cl}$. The
parameter $a_{0}=0.255$ is the initial muon decay asymmetry in our
experimental conditions.
The nuclear relaxation parameters in Eq. 6 and Eq. 7 evaluated by fitting the
ZF muon polarisation at 20 K are presented in table 2.
Static parameters | Y3Cu9(OH)19Cl8
---|---
$f\%$ | 75.00$\pm$1.00
$\omega_{OH}$ (Mrad.s-1) | 0.55$\pm$0.02
d (Å) | 1.63$\pm$0.02
$\Delta_{Cl}~{}(\mu s^{-1})$ | 0.09$\pm$0.02
$\Delta_{OH}$ ($\mu$s-1) | 0.208$\pm$0.02
Table 2: Static nuclear parameters derived from high temperature fit of the ZF
asymmetry with Eq. 6.
Upon cooling, the relaxation from the electronic spins increases progressively
and below 2.1 K, at variance with the powder sample [11], a strongly damped
oscillation develops, conclusively indicating a magnetic transition in the
single crystal samples (Fig 11). Below 1.5 K, the asymmetry could be fitted
accordingly to
$a_{0}P_{f}(t)=a_{0}\left[\frac{2}{3}\cos(\omega_{f}t+\phi)e^{-\frac{\sigma^{2}t^{2}}{2}}+\frac{e^{-\lambda_{f}t}}{3}\right]$
(8)
which accounts for a magnetically frozen single phase with an average internal
field at the muon sites $B_{int}=\omega_{f}/\gamma_{\mu}$. The damping
parameter $\sigma$ encodes the width of the distribution of these internal
fields. The last exponential term accounts for residual fluctuations in the
frozen phase. At base temperature $T=0.28$ K, we obtain $B_{int}=8.6$ mT which
is almost twice lower than the lowest internal fields previously reported in
the closely related $x=0$ variant of YCu3(OH)6OxCl3-x [11, 31]. Although there
is an uncertainty about the exact position of the muons, from this dipolar
field value, we calculate a strongly reduced ordered magnetic moment for the
Cu2+ $\sim 0.075$ $\mu_{B}$ using the majority $\mu$OH complex forming sites.
Besides, the use in Eq. 8 of the coefficient $2/3$ and $1/3$ for the
oscillatory and non-oscillatory components in our single crystal, expected for
a spin-glass like state, suggests a rather disordered ground state. The strong
damping $\sigma/\gamma_{\mu}=3.2$ mT compared to $B_{int}$ also points to a
rather short range order and may simply reflect the absence of magnetic
correlations in the $c$ direction, as shown by inelastic neutron scattering
(see section VII).
In order to fit the asymmetry on the whole temperature range, we combined both
equations 6 and 8 in
$a_{0}P(t)=a_{0}\left[f_{f}P_{f}(t)+(1-f_{f})P_{para}(t)e^{-\lambda_{p}t}\right]$
(9)
with a switching parameter $f_{f}$ that tracks the frozen volume fraction from
fully ordered for $f_{f}=1$ to fully paramagnetic for $f_{f}=0$. All the
nuclear parameters are kept constant, so that the only varying parameters
shown in Fig. 12 are the frozen fraction, the oscillation frequency and its
damping rate, and the relaxation rate $\lambda_{p}$ or $\lambda_{f}$ for the
paramagnetic or the frozen phase. The latter show a clear peak at 2.1 K as
expected for the critical slowing down of the fluctuations at the magnetic
transition. Surprisingly, there is a singular point at 1.56K in the relaxation
rate plot but no significant changes in other variable parameters at the same
temperature. Although this feature demands further confirmation, it is
reminiscent of the spin dynamics re-entrance reported in clinoatacamite [32]
and echoes an anomaly reported in the low temperature magnetization
measurements [33].
Figure 12: Temperature evolution of the parameters used in Eq. 9 to fit the ZF
$\mu$SR data: a) frequency $\omega_{f}$ reflecting the internal field
magnitude, b) damping rate $\sigma$, c) fraction $f_{f}$ of the frozen phase
and d) relaxation rates in the paramagnetic ($\lambda_{p}$) and frozen
($\lambda_{f}$) phases.
To conclude, from our $\mu$SR measurements, Y-Kapellasite shows a completely
frozen ground state in the case of single crystals in contrast to the
fluctuating ground state observed in powder sample. The relaxation rate
variation with temperature indicates a phase transition at $\sim$2.1K where
the internal static fields set in. At 0.28 K, deep in the magnetic phase
state, we estimate a strongly reduced frozen moment for the Cu2+ of about
0.075 $\mu_{B}$.
## VII Time-of-flight neutron scattering
Figure 13: (a-c) Evolution in temperature of the dynamical structure factor
$S(q,E)$ along the $[H,H,0]$ direction (integrated over $L=[-6,6]$) at
$T=1.55$ K, $T=5$ K and $T=60$ K . (d) Brillouin zones boundaries (gray lines)
with $K$ and $\Gamma$ points and extended Brillouin zone (black dashed line)
in the reciprocal space. The red dotted line represents the $[H,H,0]$
direction. (e-g) Evolution in energy of $S(q,E)$ in the $(H,H,L)$ plane at
$T=1.55$ K. (h) INS measurement integrated in energy over the elastic
resolution showing a weak increase below 5 K at the $K^{\prime}_{1}$ and
$K^{\prime\prime}$ positions, characteristic of the $(1/3,1/3)$ magnetic
order.
With large deuterated single crystals of Y3Cu9(OD)19Cl8 at hand, we also
performed inelastic neutron scattering on this anisotropic kagome
antiferromagnet in order to probe the magnetic excitations in reciprocal space
and to identify the magnetic order [34]. Inelastic neutron scattering
measurements were performed on the IN5 disk chopper time-of-flight
spectrometer [35] at the Institut Laue-Langevin on a collection of single
crystals with a total mass of 0.3g. These crystals were co-aligned in the
$(H,H,L)$ horizontal plane and glued on a thin aluminium plate with the
hydrogen-free CYTOP solution (CTL-809M). Measurements runs were performed at a
constant temperature with an incident neutron wavelength $\lambda=4.8$ Å
(incident energy E${}_{i}=3.55$ meV, giving a FWHM energy resolution of 0.08
meV) or $\lambda=2.6$ Å (E${}_{i}=12.1$ meV), while the sample stage was
rotated around the direction perpendicular to the beam, to map out the
reciprocal space. The detectors efficiency was corrected by a preliminary
measurement on a vanadium can.
Magnetic scattering could be identified only below 3 meV, an unexpected
feature given the large antiferromagnetic Curie-Weiss temperature of
$\theta_{\rm cw}=-100$ K [10]. We thus focused on measurements using our low
incident energy of 3.55 meV and followed the temperature evolution of the
dynamical neutron structure factor $S(q,E)$. Figure 13 shows the intensity
contour color plots of $S(q,E)$ along the $[H,H,0]$ direction. At 60 K there
is a large phonon contribution that substantially reduces at 20 K and below,
where nearly vertical columns of intensity become apparent close to
$K^{\prime}_{1}=(2/3,2/3,0)$ and $K^{\prime\prime}=(4/3,4/3,0)$, with a region
of broad intensity located around 2 meV and close to $[1,1,0]$. At our base
temperature of 1.5 K, sharp dispersive features emerge from these two $Q$
positions which we identify as spin-wave excitations. They show a pronounced
two-dimensional character, as revealed by the presence of rods of scattering
intensity along the $[0,0,L]$ direction (Fig. 13 e-g), and thus justify the 2D
Heisenberg kagome antiferromagnet model applied below. From this time-of-
flight measurement, one can isolate the elastic contribution by integrating
over twice the energy resolution ($\sim 0.15$ meV) only. Figure 13 h) shows
such contribution with the apparition of tiny $q$-resolution limited magnetic
Bragg peaks below 5 K located at both $K^{\prime}_{1}$ and $K^{\prime\prime}$
positions. This elastic scattering points to a small ordered fraction of the
Cu2+ moment, that could be easily missed in a conventional energy-integrated
neutron diffraction experiment, and strongly supports a magnetic non-collinear
coplanar phase with a $Q=(1/3,1/3)$ ordering wave vector.
The inelastic scattering in the $(H,K,0)$ plane is displayed in Fig. 14. The
energy-integrated cuts show clear intensities located at the $K$ positions of
the Brillouin zone, mirroring the sharp spin wave dispersions already observed
below 1 meV in Fig. 13 a). Above 1 meV, the inelastic scattering broadens
significantly and extends toward the Brillouin zone center.
Figure 14: Dynamical structure factor $S(Q,E)$ for the $(H,K,0)$ plane
measured at $T=1.55$ K. The four color plot maps show its evolution in energy
with the corresponding integrated energy range. Gray lines indicate the
Brillouin zones boundaries. Bright spots of intensity are observed at the $K$
points, and corresponds to the dispersion of spin-waves.
To better isolate the inelastic magnetic contribution present at 1.5 K, we
subtracted a background constructed by the replication of the 5 K dataset,
integrated in the range $H=[1.6,2]$, along $[H,H,0]$, where the phonon and
magnetic contributions appear to be minimal. The result is shown in Fig. 15
a). The dispersive excitations rising from $K^{\prime}_{1}$ and
$K^{\prime\prime}$ are clearly singled out, and then merge into a broad arch
extending from 1 to 2.5 meV. Guided by the observation of an in-plane
distortion of the kagome lattice, and by the recent ab initio DFT results
[19], we performed linear spin-wave calculations using the SpinW package [36]
of the Heisenberg model with only nearest-neighbors interactions in order to
model the excitations on such an anisotropic kagome lattice:
$\mathcal{H}=\sum_{\left\langle
i,j\right\rangle}J_{ij}\bm{S_{i}}\cdot\bm{S_{j}}\\\ $ (10)
where $J_{ij}$ can take three different values $J_{\hexagon}$, $J$ and
$J^{\prime}$ depending respectively on the bonds $d_{1},d_{2}$ and $d_{3}$ as
defined in Fig. 1. From DFT calculation, Hering et al. [19] confirmed the
relevance of such spin models in the context of Y-Kapellasite using three
antiferromagnetic couplings $J\simeq J_{\hexagon}\simeq 140$ K and
$J^{\prime}\simeq 10$ K. The spin-wave simulations assumed an enlarged unit
cell of nine sites, because of the two different copper sublattices, and a
coplanar order with a $Q=(1/3,1/3)$ propagation vector, as described in
details in Ref. [19]. The $J=J_{\hexagon}=J^{\prime}$ case leads to the well-
known $\sqrt{3}\times\sqrt{3}$ phase of the classical kagome antiferromagnetic
model which fails to reproduce our data when $J>15$ K, due to the maximum of
the band observed around 2.5 meV. Given that $\theta_{\rm cw}=-100$ K imposes
a higher exchange value, we discard the isotropic case and instead study the
excitations for the parameters $J=J_{\hexagon}$ and $J^{\prime}\neq J$. From
the evolution of the spin-wave spectrum with $J$ or $J^{\prime}$, we find our
best model using $J\simeq J_{\hexagon}=140$ K and $J^{\prime}=63$ K. However,
we also find that a single set of interactions can not account for the spread
of intensity observed at $[1,1,0]$, from 1 to 2.5 meV. Instead, we consider a
$\sim 10$ % distribution of $J^{\prime}$, tentatively attributed to some local
disorder, and that could indeed satisfactorily fit our inelastic data, as
shown in Fig. 15 c), d) and e), leading to $J\simeq J_{\hexagon}=140(10)$ K
and $J^{\prime}=[56,70]$ K. Finally, we note that the intensity observed at
low $Q$, near $Q=(1/3,1/3,0)$, and below 0.5 meV (see Fig. 15 a) is much
broader than predicted by the simulation, and may hint at a more complex
ground state in Y-kapellasite with the presence of short-range order and
fluctuations. We note that this broad intensity extends mostly around the zone
boundaries of the first Brillouin zone (Fig. 14) and not close to the
boundaries of the _extended_ Brillouin zone, as predicted for the classical
spin liquid [19].
In summary, INS gives evidence for elastic scattering below 5 K, in agreement
with the static magnetism revealed by $\mu$SR and line broadening in NMR,
consistent with a $Q=(1/3,1/3)$ magnetic ground state. Furthermore, we found a
set of magnetic interactions in Y-kapellasite which is in very good agreement
with the predictions from DFT and variational Monte-Carlo studies [19]. The
main difference is our experimental value of $J^{\prime}\simeq 63(7)$ K as
compared to the theoretical estimate $J^{\prime}=10.3(7)$ K. This locates
Y-kapellasite in the $Q=(1/3,1/3)$ magnetic order of the phase diagram, but
very close to the classical spin liquid phase reported in Ref. [19].
Figure 15: Comparison between experimental data (a,b) and spin-waves
simulations (c,d) at $T=1.5$ K. The INS data in (a) measured at 1.5 K was
subtracted by a 5 K dataset (see text) to highlight the magnetic contribution.
(e) Energy-integrated cuts along $[H,H,0]$ of data shown in (a) and (c) for
different energies $E$ with an integration range of 0.15 meV. Symbols are for
data and lines for the simulation. They are both shifted vertically for
clarity. (f) Classical phase diagram reproduced from [19]. The black square
denotes the position found from DFT calculation [19] while the red star
locates Y-kapellasite using the exchange couplings found in our work.
## VIII Discussion and Conclusion
Our experimental investigation of Y3Cu9(OH)19Cl8 reveals an unusual
bidimensional coplanar $Q=(1/3,1/3)$ long range ordered ground state, overall
in very good agreement with the theoretical study in Ref. [19]. In particular,
the proposed spin arrangement in the ground state is found to be compatible
with the Cl NMR lineshape at low temperature. However, at variance with first
principle calculations where the third interaction of the model $J^{\prime}$
was found to be negligible, we determine experimentally a finite
$J^{\prime}\lesssim J\hexagon/2$. This finding of a sizeable antiferromagnetic
$J^{\prime}$ seems nonetheless consistent with the Cu-OH-Cu bond angles which
vary only by 4.6∘ among the three different bonds. The smallest angle
$113.1^{\circ}$ corresponding to $J^{\prime}$ remains significantly larger
than 105∘ which is known to yield small $\sim-15$ K ferromagnetic interaction
in Zn-kapellasite [37].
Besides, to reproduce the broad magnetic excitations in INS, a distribution
$\sim 10\%$ of the $J^{\prime}$ interaction was imposed. This suggests some
kind of disorder. Due to the large difference in the ionic radii of Y3+ and
Cu2+ we expect no significant intersite mixing at variance with Zn-kapellasite
or other Zn and Cu based kagome materials and indeed we found no evidence of
such disorder in the magnetic lattice above 33 K, neither in neutron
diffraction nor in NMR which on the contrary exposes well resolved narrow
lines. Therefore, the distribution of $J^{\prime}$ may arise only at low
temperature, as a result of the structural instabilities observed at 33 K and
13 K. The slightly different local environment detected below 33 K by Cl NMR
would then reflect in the moderate distribution of the interaction, while
preserving on average the high temperature magnetic model, since the symmetry
and lattice parameters remain unchanged on the whole temperature range, down
to 65 mK.
The experimentally determined interactions locate Y-kapellasite in the
classical phase diagram close to the boundary between the (1/3,1/3) long range
ordered and the ”classical spin liquid” phases (see Fig. 15 f). Although the
effect of quantum fluctuations has not been investigated so close to the phase
boundary, we may anticipate that they are responsible for the strongly reduced
value of the Cu2+ moments $\lesssim 0.1\mu_{B}$, which could not be detected
in standard neutron diffraction but yield static internal fields detected both
in NMR and $\mu$SR. The proximity to a phase boundary may also help
understanding the striking difference in the ground states of the present
large single crystal samples and formerly studied polycrystalline ones [11].
Indeed in the latter, the moments were found to remain fluctuating in the
ground state and also no structural transition was detected. We may assume
that some additional disorder in the polycrystalline samples or slightly
different interactions, due to the absence of structural transition,
destabilize the fragile ordered state observed in large single crystals. We
note that such a sensitivity to the sample crystallinity seems to be a common
feature of anisotropic kagome compounds like volborthite [38, 17] or
vesigneite [39, 40, 18]. The fragility of the ground state in all these
compounds seems to originate from a subtle interplay between structure and
magnetic frustration.
Beyond the proposed anisotropic nearest neighbor model, and despite its
apparent success to reproduce the physics in Y3Cu9(OH)19Cl8, one important
perspective to complete our understanding of this material is to quantify and
address the role of Dzyaloshinskii-Moriya interaction. Let us indeed recall
that this anisotropy of the interaction was pointed out as the main ingredient
driving long range order in the sister compound YCu3(OH)6Cl3 [41] with a
perfect kagome lattice. Quantum fluctuations and especially their role at the
phase boundaries and how they impact the ”jammed” classical spin liquid phase
is another key perspective of this work. Finally we note that a Br counterpart
[42] was recently synthesized, showing no sign of ordering, and likely
allowing to span further the phase diagram of the anisotropic kagome model.
## Acknowledgements
The authors acknowledge the support of the French Agence Nationale de la
Recherche, under Grant No. ANR-18-CE30-0022 ”LINK”, and funding from Deutsche
Forschungsgemeinschaft (DFG) through TRR 288-422213477 (project A03). We thank
Samir Hammoud for carrying out the ICP and gas extraction experiments, and A.
Razpopov, R. Valenti, H.O. Jeschke and J. Reuther for useful discussions and
for sharing INS simulation data.
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Email<EMAIL_ADDRESS>
# Electronic transport in copper-graphene composites
Kashi N. Subedi Department of Physics and Astronomy, Ohio University, Athens,
OH 45701, USA Kishor Nepal Department of Physics and Astronomy, Ohio
University, Athens, OH 45701, USA Chinonso Ugwumadu Department of Physics
and Astronomy, Ohio University, Athens, OH 45701, USA Keerti Kappagantula
Pacific Northwest National Laboratory, Richland, WA 99352, USA D. A. Drabold
[ Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA
###### Abstract
We investigate the electronic transport properties of copper-graphene
composites using a density-functional framework. Conduction in composites by
varying the interface distance of a copper/graphene/copper (Cu/G/Cu) interface
models was studied. The electronic density of states reveals increasing
contributions from both copper and carbon atoms near the Fermi level with
decreasing Cu-G interfacial distance. Electronic conductivity of the models
computed using the Kubo-Greenwood formula showed the conductivity increases
with decreasing Cu-G distance. We also find that the conductivity saturates
below a threshold Cu-G distance. By computing the space-projected conductivity
of the Cu/G/Cu models, we show that the graphene forms a bridge to the
electronic conduction at small copper-graphene distances, thereby enhancing
the conductivity.
Metal composites are key industrial materials that may be manufactured using
various novel techniques Heringhaus and Raabe (1996); Bouaziz, Kim, and Estrin
(2013); Bruschi _et al._ (2021), leading to metastable materials exhibiting
desirable mechanical, thermal, electronic, and transport properties without
the constraints typically seen in alloys. Copper is used extensively for
energy transport and electromagnetic applications such as motors, generators,
cable, busbars and transformers. In the last two decades, we have witnessed
copper composites with improved transport properties Singh and Gautam (2018),
devised as a means to reduce energy loss in applications.
One material that has demonstrated improved electrical conductivity and
current density is a copper-graphene (Cu-G) composite. Various Cu-G composites
ranging from a simple interface laminated to bulk Cu with interconnected
graphene flakes have been reported in the literature, with some showing
enhanced conductivity Weiping _et al._ (2015); Rho _et al._ (2017); An _et
al._ (2019); Yang _et al._ (2021), and more commonly enhancement in
mechanical performance Hwang _et al._ (2013); An _et al._ (2019).
Conventional wisdom suggests that the introduction of additives to metal would
increase scattering that would lead to increased electrical resistivity Ellis
_et al._ (1993). This is the reason why addition of silver to copper leads to
reduced electrical conductivity of Cu, even though Ag is 8% higher in
conductivity than Cu Felicia, Rochiem, and Laia (2018). The additives, on the
other hand, also arrest dislocation movement in the metals due to several
mechanisms such as precipitate hardening, solid solution strengthening and
dispersion strengthening, resulting in enhanced mechanical performance
Kuhlmann-Wilsdorf (1989).
At this juncture, it is known that the electrical conductivity of the Cu-G
composites are dependent on the quantity of graphene used in the composites,
the type of graphene and their defect density, and the arrangement of Cu
matrix and graphene in the composite microstructure, which eventually dictates
the carrier transport in the material during conduction Kappagantula _et al._
(2022); Pan _et al._ (2022). Processing routes adopted to manufacture the
composites are also known to influence electrical performance. Processes that
can control atomic scale deposition of Cu and graphene, such as vapor
deposition and molecular beam epitaxy, have shown promise in manufacturing
nano-to-micron scale samples demonstrating enhanced electrical performance
Zheng _et al._ (2018). At the bulk scale, solid phase processing techniques
have shown promise in enhancing electrical conductivity Kappagantula _et al._
(2022).
In all this, however, while there is an evolving understanding of processing
approaches and material chemistries that result in Cu-G composites with
improved electrical performance, little is known about the nature of charge-
carrier transfer between Cu and graphene. Several papers discuss carrier
mobility in graphene that is deposited on Cu foils Orofeo _et al._ (2012);
Banszerus _et al._ (2015). However, limited experimental and computational
research is published on transport phenomena across metal/graphene interfaces.
There is minimal understanding on how the nature of the interface (such as
atom arrangement, interfacial distance) affects the electron density at the
interface. In Cu-G composites demonstrating enhanced conductivity compared to
the corresponding Cu substrates, two causes were hypothesized for the
enhancement in conductivity. In one case, where the manufacturing conditions
allow for it, the Cu grains are surmised to be templated on the adjacent
graphene flakes leading to the formation of predominantly 111 textures in the
composite during the manufacturing process. Such crystallographic orientations
in Cu show higher conductivity compared to other textures such as 200 or 110.
If there is a large concentration of 111 grains in the Cu microstructure, it
is reasonable to expect an enhanced conductivity overall. In this hypothesis,
graphene is thought to only template the grains but not actually participate
in the conduction process. In the second case, graphene is assumed to
participate during conduction through the exchange of carriers with the
surrounding Cu matrix. One key consideration for this hypothesis is the nature
of interface between the Cu and graphene that may be suitable for carrier
transport across the interface. Some papers indicate that the 111 Cu grains
offer the least lattice mismatch with graphene, which may also help with
carrier transport. However, it is currently not well known how much
conductivity the introduction of graphene brings about for Cu per this
hypothesis.
In this Letter, we explore the electrical transport properties of the Cu-G
composites as a function of the interfacial distance between Cu and graphene,
a parameter that can be modulated during Cu-G manufacturing and a simple proxy
for a possible strained local configuration. We quantify the electronic
transport by computing the electronic conductivity using the Kubo–Greenwood
formula (KGF) Kubo (1957); Greenwood (1958), ideally suited for density-
functional simulations of materials with its single particle Kohn–Sham
orbitals and energies Martin (2008). The KGF method has been utilized in
tight-binding and DFT computations of the conductivity of liquids and solids
Allen and Broughton (1987); Galli _et al._ (1989). Besides quantifying the
conductivity, we determine the conduction-active sites in the Cu-G composites
by projecting the KGF conductivity onto real space using the space-projected
conductivity (SPC), described elsewhere Prasai _et al._ (2018). The method
has been implemented to study conduction processes in crystalline systems with
defects, amorphous and semi-conducting systems at atomistic level Subedi _et
al._ (2022); Thapa _et al._ (2022); Subedi _et al._ (2019); Subedi, Prasai,
and Drabold (2020).
We simulated the Cu-G composite by constructing an interface model with a
geometry of the form copper/graphene/copper (Cu/G/Cu) as shown in Figure 1a.
We adopted one of the low-indexed surfaces, namely 111, to construct the Cu
surface. We placed the graphene sheet with one sub-lattice site (A) above the
top of the first layer of Cu 111 and the other sub-lattice site (B) placed
above the third layer of Cu 111 as shown in Figure 1b. This arrangement is
also known as top-fcc, and has shown to represent the low-energy structure for
Cu-G surface models Yang _et al._ (2017). We simulated the external pressure
on Cu-G composites by varying the copper-graphene distance ($d_{Cu-G}$) of the
Cu/G/Cu interface model as shown in Figure 1a. Besides Cu/G/Cu models, we also
constructed an orthorhombic supercell of 216 atoms and simulated the external
pressure by reducing the vertical dimension of the supercell. A similar
approach has been used to simulate copper and aluminum under external pressure
in earlier work Lanzillo _et al._ (2014).
Figure 1: (a) Geometry of copper/graphene/copper (Cu/G/Cu) model with dCu-G
representing copper-graphene distance. b) Top-fcc configuration of graphene
above the Cu 111 surface with A and B representing sub-lattice sites above the
Cu atom at first and third layer respectively. The pink colored spheres
represent Cu atoms and brown colored spheres represent C atoms.
Density-functional calculations were carried out using the Vienna Ab Initio
Simulation Package Kresse and Hafner (1993) (VASP) code. For atomic structure
relaxations of Cu/G/Cu models, we used a plane-wave basis set with a kinetic
energy cutoff of 400 eV. For static calculations, we used a larger cutoff of
520 eV. Projected augmented wave (PAW) Blöchl (1994) potentials was used to
account for ion-election interactions, and the generalized gradient
approximation of Perdew-Burke-Ernzerhof (PBE) Perdew, Burke, and Ernzerhof
(1997) as the exchange-correlation functional. All calculations were performed
at 4 irreducible k-points, and periodic boundary conditions were implemented
throughout. To determine the partial occupancy of electrons near the Fermi
level, Fermi-Dirac distribution function with a smearing temperature of 1000 K
was considered. $\delta$ function in the expression of KGF by was approximated
by a Gaussian distribution function of width 0.05 eV. Factors like thermal
broadening, k-point sampling, and finite size effects discussed in references
Belmonte _et al._ (2015); Knyazev and levashov (2013); Calder’m, Karaseiv,
and Trickey (2017); Bulanchuk (2021) play a role in employing the KGF.
To obtain representative low-energy configurations, atomic structure
relaxation of the Cu/G/Cu models were performed. The initial and final Cu-G
distances after relaxation are tabulated in the first and the second column of
table 1 respectively. It is apparent that the extent of repulsion of Cu atoms
away from the graphene is higher for models with short Cu-G distances,
indicated by larger shift in Cu-G layers (third column of table 1). For long
Cu-G distances (up to $d_{Cu-G}$ = 2.61 Å ), most of the Cu-Cu bonds are
virtually unaffected and only few Cu-Cu bonds are formed up to $\approx$2.52
Å, whereas for short Cu-G distances, the vertical Cu-Cu bonds are also formed
at $\approx$2.49 and $\approx$2.46 Å.
Table 1: Variation of Cu-G distance for different Cu/G/Cu models after atomic structure relaxation and corresponding transverse component of Stress Tensor. $d_{Cu-G}$111$d_{Cu-G}$ is copper-graphene distance shown in Figure 1a) (Å) | $d_{Cu-G}$ (Å) | Shift (Å) | Pz222Transverse component of Stress Tensor [GPa]
---|---|---|---
(Initial) | (Relaxed) | |
2.88 | 2.94 | 0.06 | 2.7
2.71 | 2.78 | 0.07 | 3.2
2.54 | 2.61 | 0.07 | 3.1
2.37 | 2.46 | 0.09 | 4.2
2.20 | 2.37 | 0.17 | 6.5
2.02 | 2.29 | 0.27 | 10.3
1.86 | 2.23 | 0.37 | 14.9
1.69 | 2.19 | 0.50 | 20.2
KGF conductivity for the Cu/G/Cu models in the direction normal to the
graphene layer (i.e., along c-axis in Figure 1a) was computed. Figure 2a
displays conductivities for the models with different Cu-G distances
normalized by the KGF conductivity of copper crystal computed at 300 K
corresponding to $d_{Cu-Cu}$ = 2.56 Å. From Figure 2a, we see that the
conductivity increases with decreasing Cu-G distance in an almost exponential
manner up to $d_{Cu-G}$ = 2.37 Å. Beyond 2.37 Å, we find a sharp rise in
conductivity followed by a saturation below $\approx$2.23 Å. The increase in
conductivity may be attributed to increased constructive overlapping of copper
and graphene orbitals with decreasing Cu-G distance. For the short Cu-G
distances with $d_{Cu-G}$ = 2.23 Å and beyond, the conductivity is obtained to
be $\approx$10 % higher compared to that of copper computed at 300 K.
Figure 2: a) KGF conductivities for relaxed Cu/G/Cu models with different
Cu-G distance normalized by the KGF conductivity ($\sigma_{0}$) of a copper
crystal computed at 300K with $d_{Cu-Cu}$ = 2.56 Å. b) Relative conductivity
($\sigma/\sigma_{0}$) of orthorhombic Cu model computed as a function of
vertical Cu-Cu bond length. Figure 3: Electronic density of states for
different Cu/G/Cu models. Top subplot corresponds to the total EDOS and the
bottom subplot corresponds to the projected density of states (PDOS) onto
carbon atoms. The compression of models resulted in an enhanced EDOS near the
Fermi level, see inset for details. Figure 4: Space-projected conductivity
plotted as a RGB colormap along a plane normal to the graphene sheet for
different Cu/G/Cu models corresponding to $d_{Cu-G}$ = 3.25 Å, 2.94 Å, 2.29 Å
and 2.23 Å respectively. The colorbar on the left represents the magnitude of
SPC values that increases from red towards blue. The SPC values are scaled
from the minimum value for $d_{Cu-G}$ = 3.25 Å. Cu and C atoms are represented
by pink and brown-colored spheres respectively.
To understand such an increasing trend in the conductivity with decreasing
Cu-G distance, total electronic density of states (EDOS) and the projected
density of states (PDOS) for selected models were computed and are shown in
Figure 3. From Figure 3, we find relatively higher EDOS near the Fermi level
with decreasing Cu-G distance. Such increase in the EDOS is contributed by
both copper and carbon atoms of the Cu-G composites, which are more pronounced
for short Cu-G distances. According to Mott and Davis Mott and Davis (1979),
the DC conductivity, $\sigma_{dc}\propto(N(\epsilon_{f}))^{2}$, where $N$
represents the density of states. So, one expects increasing conductivity with
decreasing Cu-G distance for these composites as a combined effect of enhanced
EDOS near the Fermi level. We note that it was not at all obvious that the
graphene would enhance the DOS at the Fermi level, and we conjecture that this
is what underlies experiments showing improved conductivity of the composites.
Further exploration was made by investigating a spatial description of
conductivity as a function of Cu-G distance. We computed the SPC for selected
Cu/G/Cu models and are shown in Figure 4. Figure 4 displays the SPC projected
on 100 plane as a RGB colormap for four such models with different Cu-G
distance. The colorbar on the left depicts the magnitude of SPC values, where
red color represents low SPC values and blue color represents high SPC values.
From Figure 4, it is apparent that the contribution from both Cu and graphene
to the conduction increases with decreasing Cu-G distance, and is more
significant at short distances. The SPC plots at short Cu-G distances show
that the graphene directly participates in conduction and forms a bridge for
conduction between Cu atoms on opposite layers. In addition to the graphene,
we also see significant contributions from Cu atoms to conduction at short
Cu-G distances. This contribution from Cu atoms to conductivity can be
associated with different Cu bonding environments in the Cu-G composites.
KGF conductivity for the orthorhombic Cu model for different vertical Cu-Cu
distances ranging from 2.43 to 2.55 Åwere calculated. Figure 2b shows the
relative conductivity of Cu models for different Cu-Cu vertical distances. One
observes that the conductivity of Cu increases with reducing the Cu-Cu
distance from $\approx$2.55 to $\approx$2.46 Å after which it drops at
$\approx$2.44 Å. We observed that for short Cu-G distances ($d_{Cu-G}\leq$
2.23 Å), the Cu-Cu bond-lengths are $\approx$2.49 and $\approx$2.46 Å. So, the
contribution to the conductivity from the Cu atoms with nearest neighbors
close to 2.46 and 2.49 Å is higher compared with Cu atoms at other distances
and, in agreement with SPC plots in Figure 4.
Saturation in conductivity of Cu/G/Cu model for compression below $d_{Cu-G}$ =
2.23 Å (refer Figure 2a) were observed. Such a characteristic is attributed to
the contribution of the Cu atoms forming different nearest-neighbor distances.
At $d_{Cu-G}$ = 2.23 Å, the repulsion of Cu layers on both sides of graphene
leads to formation of Cu-Cu bonds at $\approx$2.49 and $\approx$2.46 Å. With
further reduction of Cu-G distance, here $d_{Cu-G}$ = 2.19 Å, the Cu-Cu bonds
are formed at $\approx$ 2.46 Å and also at $\approx$2.44 Å. So, the additional
disorder in Cu leads to slightly less contribution from the Cu atoms to the
conduction (refer Figure 2b) even as the graphene still contributes.
In conclusion, copper-graphene composites were modeled by considering the
Cu/G/Cu interface model with varying Cu-G distance. We showed that the DC
conductivity of the Cu/G/Cu interface models increases with decreasing copper-
graphene distance, and such a increase is a combined contributions from both
Cu atoms and the graphene sheet. We attribute this to an enhanced EDOS near
the Fermi-level. We also provided the spatial view of the electronic
conductivity by computing the SPC for varying copper-graphene distance. The
SPC calculations showed that graphene forms the bridge to electronic
conduction between Cu atoms that lie on the opposite layers. We also showed an
interesting characteristic of saturation of the conductivity at a short
copper-graphene distance (below $d_{Cu-G}$ = 2.23 Å), and such a
characteristic is mostly attributed to slightly reduced contribution to the
conduction from Cu atoms that form Cu-Cu bonds at $\approx$2.44 Å even when
the graphene showing enhanced contribution.
This paper provides one of the first demonstrations of graphene participating
in the conduction process along with the copper atoms in the composite,
especially at low copper-graphene distances. An implication of this finding is
that manufacturing process conditions may influence the electrical performance
metal-graphene composites by engineering the interfaces to encourage
graphene’s participation in the conduction processes, rather than leaving it
as a scattering site. It is important to note that in addition to the
interfacial distance, the arrangement of copper-carbon atoms (geometry) can
also influence electrical performance of the composite; however, those effects
were not explored in this work. Additional factors of importance, such as the
number of graphene layers, their dimensions and the defect density of the
graphene additives will be explored in future work.
Acknowledgements: We acknowledge Extreme Science and Engineering Discovery
Environment (XSEDE), supported by NSF Grant No. ACI-1548562 at the Pittsburgh
Supercomputing Center, for providing computational resources under allocation
DMR-190008P,Department of Energy Advanced Manufacturing Office, and Pacific
Northwest National Laboratory operated by the Battelle Memorial Institute for
the U.S. Department of Energy under Contract No. DE-AC06-76LO1830.
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|
# The structure of the density-potential mapping
Part I: Standard density-functional theory
Markus Penz Basic Research Community for Physics, Innsbruck, Austria Erik I.
Tellgren Hylleraas Centre for Quantum Molecular Sciences, Department of
Chemistry, University of Oslo, Norway Mihály A. Csirik Hylleraas Centre for
Quantum Molecular Sciences, Department of Chemistry, University of Oslo,
Norway Department of Computer Science, Oslo Metropolitan University, Norway
Michael Ruggenthaler Max Planck Institute for the Structure and Dynamics of
Matter, Hamburg, Germany Andre Laestadius<EMAIL_ADDRESS>Department of Computer Science, Oslo Metropolitan University, Norway
Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry,
University of Oslo, Norway
###### Abstract
The Hohenberg–Kohn theorem of density-functional theory (DFT) is broadly
considered the conceptual basis for a full characterization of an electronic
system in its ground state by just the one-body particle density. Part I of
this review aims at clarifying the status of the Hohenberg–Kohn theorem within
DFT and Part II at different extensions of the theory that include magnetic
fields. We collect evidence that the Hohenberg–Kohn theorem does not so much
form the basis of DFT, but is rather the consequence of a more comprehensive
mathematical framework. Such results are especially useful when it comes to
the construction of generalized DFTs.
###### Contents
1. I Introduction
2. II Preliminaries
3. III Representability of densities
4. IV The Hohenberg–Kohn theorem
5. V The unique-continuation property
6. VI Hierarchy of density functionals
7. VII Density-potential mappings from differentials
8. VIII Linking to a reference system: the Kohn–Sham scheme
9. IX Density-potential mixing and regularized DFT
10. X Abstract density-potential mapping
11. XI Summary
12. XII Outlook
## I Introduction
The theorem of Hohenberg and Kohn [1] (HK) is usually presented as the
theoretical justification of density-functional theory (DFT). It states that
the one-body particle density uniquely (up to an additive constant) determines
the scalar potential of a non-relativistic many-electron system in its ground
state. The Mathematical analysis of ground-state DFT was pioneered by Lieb
[2], using tools from convex analysis. In it, some important problems,
especially in relation with differentiability of the involved functionals that
map densities to energies, were left unanswered and remained as open
questions. Lammert [3] then demonstrated that the key functional of DFT is
indeed non-differentiable, but it remained unclear to what extent this
threatens the foundations of DFT and its algorithmic realization, the
Kohn–Sham scheme employed for practical calculations. Regularization as a
means to overcome non-differentiability has been applied to DFT [4] (Section
IX) and its extension, current DFT (CDFT) [5; 6]. The existence of functional
derivatives through regularization also avoids the problem of
$v$-representability that usually haunts DFT, i.e., that not every reasonable
density is the solution to a certain potential (Section III).
A central result in this work is a very convenient and novel formulation of
the HK theorem that restructures it into two sub-theorems, HK1 and HK2
(Section IV):
* •
(HK1) If two potentials share a common ground-state density then they also
share a common ground-state wave function or density matrix.
* •
(HK2) If two potentials share any common eigenstate and if that eigenstate is
non-zero almost everywhere (a property that is guaranteed if the the unique-
continuation property (UCP) holds; see Section V) then they are equal up to a
constant.
Combining HK1 and HK2, one obtains the classical HK theorem and with it a
well-defined density-potential mapping. The proof of HK1 will be shown to be
immediate from just the formulation of “ground-state energy”. Consequently, it
is also easily attainable in an abstract or extended formulation of DFT
(Section X). The situation for HK2, on the other hand, is more complicated
but, as will be demonstrated, it holds true with certain restrictions in the
standard DFT setting. It is known _not_ to hold in paramagnetic CDFT [7] and
has, to the best of our knowledge, an unknown status in total CDFT. In Part II
of this review, we will exemplify how different DFTs follow this structure
and, maybe more importantly, pinpoint why this route might fail.
After analyzing its basic structure, the status of the HK theorem within DFT
is scrutinized. If only the ground state of a system is the matter of
interest, a constrained-search approach seems to be sufficient for the
formulation of DFT, and the usual type of constrained-search functional [8; 9]
even _implicitly includes_ HK1 (Section VI). Besides being a mathematically
more transparent formulation than the HK theorem, the constrained-search
formalism is also a better starting point for deriving approximate density
functionals. Nonetheless, the full HK theorem remains important for going
beyond the bare minimum needed to set up a ground-state theory. For example,
the HK theorem implies that the ground-state density determines not only the
ground state but, by fixing the scalar potential, also all excited states.
This becomes relevant when thermostatistical properties are considered.
Furthermore, in order to be able to define the Kohn–Sham scheme (Section
VIII), one actually demands more than just the HK result, relying on
differentiability of the energy functional that in turn would imply the
_whole_ HK result (Section VII). Consequently, in a (Moreau–Yosida)
regularized setting, the Kohn–Sham scheme can be rigorously formulated and
even proven to converge in finitely many dimensions [10; 11; 12], and HK
becomes just a by-product.
Although we will do our best to orient the reader within the rich subject that
is DFT, the scope of this review is limited. We will mainly focus on, in our
opinion, matters closely related to the HK mapping and properties of the exact
functional(s). Many excellent reviews and textbooks are available on the
subject [13; 14; 15; 16; 17; 18]. For the interested reader, we also point out
the very recent article based on a round-table discussion [19].
## II Preliminaries
Density-functional theory is an approach to describe particles that obey the
laws of quantum mechanics, but that avoids their full description by a wave
function and instead switches to reduced quantities like the one-particle
density. In its basic form discussed here, the focus is solely on the ground-
state properties of the quantum system. For the configuration space of a
single particle we always choose $\mathbb{R}^{3}$ with the additional spin
degree-of-freedom for spin-$\frac{1}{2}$ particles. The Hamiltonian comprises
three parts,
$H[v]=T+W+V[v],$
relating to the kinetic energy, the Coulomb repulsion, and the external scalar
potential, respectively. The internal parts will be collected as $H_{0}=T+W$.
The kinetic-energy operator is $T=-\frac{1}{2}\sum_{i=1}^{N}\nabla_{i}^{2}$ in
standard DFT, where atomic units are employed. Notation-wise, we use small
letters for one-body objects. The external potential contribution $V[v]$ is
always defined from a one-body potential $v(\mathbf{r})$ and is of an additive
form,
$V[v](\underline{\mathbf{r}})=\sum_{i=1}^{N}v(\mathbf{r}_{i}),$
where $\underline{\mathbf{r}}=(\mathbf{r}_{1},\dots,\mathbf{r}_{N})$. For
later reference we also define
$\underline{\sigma}=(\sigma_{1},\dots,\sigma_{N})$ for the spin degrees-of-
freedom. The full quantum-mechanical description of a system in its ground
state is achieved by determining the eigenstate $\psi_{0}$ of $H[v]$ that has
the correct symmetry and the lowest eigenvalue $E_{0}$ (ground-state energy),
$H[v]\psi_{0}=E_{0}\psi_{0}.$ (1)
If such a lowest eigenstate is not unique, we speak of _degeneracy_ , a case
that will often appear in the discussion below and that leads to several
complicacies. Then a valid ground state can also be given as a statistical
mixture of the pure ground states $\psi_{k}$ in the form of a density matrix
$\Gamma=\sum_{k}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$ with
$\lambda_{k}\in[0,1]$ and $\sum_{k}\lambda_{k}=1$. It is natural to require
states of finite kinetic energy,
$\langle\psi|T|\psi\rangle=\frac{1}{2}\sum_{i=1}^{N}\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3N}}|\nabla_{i}\psi|^{2}\,\mathrm{d}\underline{\mathbf{r}}<+\infty,$
and we define the basic set for wave functions
$\mathcal{W}=\\{\psi\mid\,\psi\;\text{anti-
symmetric},\,\langle\psi|T|\psi\rangle<+\infty\\}.$
In cases where density matrices $\Gamma$ are considered, we require
$\psi_{k}\in\mathcal{W}$ for all their components.
The one-particle density of a given $\psi$ as the basic variable of standard
DFT is
$\rho_{\psi}(\mathbf{r}_{1})=N\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3(N-1)}}|\psi|^{2}\,\mathrm{d}\mathbf{r}_{\perp},$
(2)
where we used the shorthand notation
$\mathbf{r}_{\perp}=(\mathbf{r}_{2},\dots,\mathbf{r}_{N})$, and it is
$\rho_{\Gamma}(\mathbf{r})=\sum_{k}\lambda_{k}\rho_{\psi_{k}}(\mathbf{r})$ for
a given mixed state $\Gamma$. Since $\Gamma$ already includes the squared wave
function from Eq. (2), the mapping $\Gamma\mapsto\rho_{\Gamma}$ is linear.
Note that whenever we talk about a “density”, this will be assumed to be a map
$\rho:\mathbb{R}^{3}\to\mathbb{R}_{\geq 0}$ that is normalized to the particle
number $N$, $\int\rho(\mathbf{r})\,\mathrm{d}\mathbf{r}=N$, like it is
automatically the case for $\rho_{\psi}$ and $\Gamma_{\psi}$ if $\psi,\Gamma$
are normalized to 1.
The density alone suffices to give an expression for the potential energy
contribution. The resulting integral over the single-particle configuration
space will be written like an inner product $\langle\cdot,\cdot\rangle$, to
wit,
$\displaystyle\langle\psi|V[v]|\psi\rangle$
$\displaystyle=\sum_{i=1}^{N}\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3N}}v(\mathbf{r}_{i})|\psi|^{2}\,\mathrm{d}\underline{\mathbf{r}}$
(3)
$\displaystyle=N\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3}}v(\mathbf{r}_{1})\int_{\mathbb{R}^{3(N-1)}}|\psi|^{2}\,\mathrm{d}\mathbf{r}_{\perp}\,\mathrm{d}\mathbf{r}_{1}$
$\displaystyle=\int_{\mathbb{R}^{3}}v(\mathbf{r})\rho_{\psi}(\mathbf{r})\,\mathrm{d}\mathbf{r}=\langle
v,\rho_{\psi}\rangle.$
The notation $\langle v,\rho\rangle$ thus expresses a dual pairing between two
$L^{p}$ spaces or a combination of such, one for densities and the other one
for potentials. These density and potential spaces are the topic of the next
section. Without going into technicalities, the space $L^{p}(\mathbb{R}^{n})$,
$1\leq p\leq\infty$, can be thought of as all functions $f(\mathbf{r})$ that
have a finite $L^{p}$ norm
$\|f\|_{L^{p}}=\left(\int_{\mathbb{R}^{n}}|f(\mathbf{r})|^{p}\,\mathrm{d}\mathbf{r}\right)^{1/p}<\infty,$
where in the case $p=\infty$ a supremum norm is employed instead.
## III Representability of densities
The notion of “representability” is ubiquitous and conceptually important in
DFT. It generally refers to the situation that any density of a certain class
comes from a well-defined construction. Such a construction can simply be how
a density is calculated from an $N$-particle wave function of finite kinetic
energy following Eq. (2) and we then call the density “$N$-representable”. Or
one demands that the density should be that of an actual ground-state solution
of a Schrödinger equation with some given external potential $v$ and one calls
it “$v$-representable”. However, this definition of $v$-representability is a
bit naive since the set of permitted potentials to choose from was not even
specified [3]. One could argue that any potential that can be put into the
Schrödinger equation should be considered, but then the dual pairing $\langle
v,\rho\rangle$ appearing in Eq. (3) between the spaces of densities and
potentials might be “lost”, which has consequences for the density functionals
defined later in Section VI. So in order to talk about $v$-representability,
we will first have to choose a basic density space that includes the
$N$-representable densities.
The task of determining $N$-representable density classes was originally
tackled by Gilbert [20] and Harriman [21]. In the first work,
differentiability of the density was required, whereas in the second work no
further conditions on the density were assumed. Here, we rely on the version
by Lieb [2, Theorem 1.2] that gives the following class of $N$-representable
densities,
$N\text{-}\mathsf{rep}=\left\\{\rho\mid\rho(\mathbf{r})\geq
0,\smallint\rho\,\mathrm{d}\mathbf{r}=N,\nabla\sqrt{\rho}\in
L^{2}(\mathbb{R}^{3})\right\\}.$
The benefit of the additional constraint $\nabla\sqrt{\rho}\in L^{2}$ is that
one can always find a wave-function that not only gives the desired density
but also has finite kinetic energy and is thus in $\mathcal{W}$ (and in
addition is properly normalized). Lieb [2] further showed that
$N\text{-}\mathsf{rep}$ is convex and included in $X=L^{1}(\mathbb{R}^{3})\cap
L^{3}(\mathbb{R}^{3})$. This space $X$ is the basic density space in terms of
$L^{p}$ spaces, so by Eq. (3) this automatically yields a corresponding
potential space that is its dual,
$X^{*}=L^{3/2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$. Any element $v\in
X^{*}$ can thus be written as $v=v_{1}+v_{2}$ with $v_{1}\in
L^{3/2}(\mathbb{R}^{3})$ and $v_{2}\in L^{\infty}(\mathbb{R}^{3})$. Potentials
of Coulomb type, $v(\mathbf{r})=Cr^{-1}$, $r=|\mathbf{r}|$, are for example
elements of this $X^{*}$ (by virtue of
$\int_{0}^{R}|v(\mathbf{r})|^{3/2}r^{2}\,\mathrm{d}r<\infty$ for any finite
$R>0$ and $|v(\mathbf{r})|<\infty$ for $r>R$).
The issue of “$v$-representability” is much more profound. To date there is no
explicit description for the set of all $v$-representable densities
$v\text{-}\mathsf{rep}$. This issue is known as the “$v$-representability
problem”. We already noted that $v\text{-}\mathsf{rep}$ should contain all
densities that are a ground-state density for some potential $v\in X^{*}$. For
a glimpse of what densities have to be included in this set we refer to the
illustrative construction of Englisch and Englisch [22]. At this point one has
to differentiate between several levels of $v$-representability. We defined
$v\text{-}\mathsf{rep}$ as coming from a ground state of a Schrödinger
equation with some given external potential $v$. Within DFT we usually
consider two settings, the full system that contains a (Coulomb) interaction
$W$ and the Kohn–Sham system that does not. So whenever we talk about
$v$-representability, this can be amended by the attributes “interacting” or
“non-interacting” and it is not obvious at this point if the two classes are
equal, overlap, or are even disjoint. After all, the sets are not explicitly
known. Within each class we also have the possibility of ground-state
degeneracy. Then, instead of ground-state wave functions, the more general
concept of density matrices comes into play. The resulting notions are then
“pure-state $v$-representability” and “ensemble $v$-representability”. In the
second case such a density $\rho$ is then the convex combination of pure-state
$v$-representable densities $\rho_{k}$ that come from the degenerate ground-
states $\psi_{k}$ of $H[v]$, i.e., $\rho=\sum_{k}\lambda_{k}\rho_{k}$
($\lambda_{k}\in[0,1],\sum_{k}\lambda_{k}=1$). In the first case only
densities from pure states are allowed, but they might still individually come
from a set of degenerate ground-state wave functions. It was demonstrated by
Englisch and Englisch [22] by giving explicit examples that there are
$N$-representable densities that are not ensemble $v$-representable (an
obvious example is a density that vanishes on a set of positive measure,
however, for more elaborate examples we refer to Section 3.2 in Ref. 22). Levy
[23] and Lieb [2] gave arguments that an ensemble $v$-representable density
does not have to be pure-state $v$-representable. An explicit example for such
a density $\rho\in v\text{-}\mathsf{rep}_{\mathrm{ens}}\setminus
v\text{-}\mathsf{rep}_{\mathrm{pure}}$ was found within a finite-lattice
system of cuboctahedral symmetry [24]. So we can symbolically note that
$v\text{-}\mathsf{rep}_{\mathrm{pure}}\subsetneqq
v\text{-}\mathsf{rep}_{\mathrm{ens}}\subsetneqq
N\text{-}\mathsf{rep}\subsetneqq X.$ (4)
In Garrigue [25] it was demonstrated that the set
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$ is path-connected. There are further
topological relations between the sets appearing in Eq. (4) that are worth
mentioning. Since every $\rho\in v\text{-}\mathsf{rep}_{\mathrm{ens}}$ is a
convex combination $\rho=\sum_{k}\lambda_{k}\rho_{k}$ with $\rho_{k}\in
v\text{-}\mathsf{rep}_{\mathrm{pure}}$, it holds
$v\text{-}\mathsf{rep}_{\mathrm{ens}}\subseteqq\operatorname{conv}v\text{-}\mathsf{rep}_{\mathrm{pure}}\subsetneqq
N\text{-}\mathsf{rep}\subsetneqq X,$
where $\operatorname{conv}$ is the convex hull of a set. So while
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$ is definitely not convex because of
the mentioned counterexamples, $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ might
still be (to our understanding this is not known). Lastly,
$N\text{-}\mathsf{rep}$ is the closure of
$v\text{-}\mathsf{rep}_{\mathrm{ens}}$ within $L^{1}\cap L^{3}$, which means
that any $\rho\in N\text{-}\mathsf{rep}$ can be approximated arbitrarily well
by densities in $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ when distance is
measured in the $L^{1}\cap L^{3}$-norm [2, Theorem 3.14]. With the notion of
the “subdifferential” from Section VII, this result can be established as a
direct consequence of the Brøndsted–Rockafellar theorem [26, Corollary 2.44].
Still, potentials that lead to densities that are arbitrarily close could be
very far apart in the potential space $X^{*}$. On the other hand, it has been
suggested that $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ is not dense in
$N\text{-}\mathsf{rep}$ (see Conjecture 3.8 in Ref. 27).
## IV The Hohenberg–Kohn theorem
The classical HK theorem [1] states the existence of a well-defined density-
potential mapping for ground states. For a given potential $v$,
$\displaystyle E[v]$
$\displaystyle=\inf\left\\{\langle\psi|H_{0}+V[v]|\psi\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\right\\}$
(5) $\displaystyle=\inf\left\\{\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\right\\}$
is the _ground-state energy_ by the Rayleigh–Ritz variation principle. If a
minimizer exists then $\psi$ and $\rho_{\psi}$ are the corresponding _ground
state_ and _ground-state density_ that might not be unique in the case of
degeneracy. If a minimizer does not exist, there is still always a sequence
$\psi_{i}$ in $\mathcal{W}$ with $\|\psi_{i}\|=1$ such that
$\langle\psi_{i}|H_{0}+V[v]|\psi_{i}\rangle$ converges to $E[v]$. In Eq. (5),
$v$ should be selected from a class that makes $E[v]$ bounded below. See Reed
and Simon, Section X.2, for an extensive discussion on such potentials [28]. A
further demand on $v$ will later be that it guarantees a ground state that is
non-zero (almost everywhere), a property needed in the proof of the second
part of the HK theorem (HK2) below.
In Eq. (5) the problem of solving a partial-differential equation, the
stationary Schrödinger equation (1), has been transformed into a variational
problem: finding a minimizer for Eq. (5). The route backwards is also feasible
and any such minimizer is also a distributional solution to the Schrödinger
equation [29, Theorem 11.8].
We will now demonstrate that simply by virtue of the structure of $E[v]$,
where density and potential are combined in the term $\langle v,\rho\rangle$
that makes no explicit reference to the wave function while the remaining part
$\langle\psi|H_{0}|\psi\rangle$ (or $\operatorname{Tr}(H_{0}\Gamma)$, if
density matrices are used to describe the state) does not depend on $v$, we
can already define a mapping from ground-state densities to ground-state wave
functions or density matrices. This, then, is already half of a HK theorem,
that we will already give in a variant for ensemble $v$-representable
densities.
###### Theorem 1 (HK1).
Let $\Gamma_{1}$ be a ground state of $H[v_{1}]$ and $\Gamma_{2}$ a ground
state of $H[v_{2}]$. If $\Gamma_{1},\Gamma_{2}\mapsto\rho$, i.e., if these
states share the same density, then $\Gamma_{1}$ is also a ground state of
$H[v_{2}]$ and $\Gamma_{2}$ is also a ground state $H[v_{1}]$.
###### Proof 1.
Since we assumed the existence of ground states $\Gamma_{1},\Gamma_{2}$ for
the potentials $v_{1},v_{2}$, the infimum in Eq. (5), when varied over density
matrices, is actually a minimum. Further, the potential-energy contribution
$\langle v,\rho\rangle$ is fixed because $\rho$ is given and can be taken out
of the minimum,
$\displaystyle E[v_{1}]$
$\displaystyle=\min_{\Gamma^{\prime}_{1}\mapsto\rho}\operatorname{Tr}(H_{0}\Gamma^{\prime}_{1})+\langle
v_{1},\rho\rangle$ $\displaystyle=\operatorname{Tr}(H_{0}\Gamma_{1})+\langle
v_{1},\rho\rangle$ (6a) $\displaystyle E[v_{2}]$
$\displaystyle=\min_{\Gamma^{\prime}_{2}\mapsto\rho}\operatorname{Tr}(H_{0}\Gamma^{\prime}_{2})+\langle
v_{2},\rho\rangle$ $\displaystyle=\operatorname{Tr}(H_{0}\Gamma_{2})+\langle
v_{2},\rho\rangle$ (6b)
For completeness, we also give the same expression for a general $v$ in case
the state is pure.
$E[v]=\min_{\psi\mapsto\rho}\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho\rangle\\\ $ (7)
Here, the notation “$\Gamma\mapsto\rho$” and “$\psi\mapsto\rho$” means
variation over all states in $\mathcal{W}$ with density $\rho$. But the
remaining minima in Eq. (6) are then completely determined by the fixed
ground-state density and we can always choose $\Gamma_{1}=\Gamma_{2}$ [primes
removed] as a valid ground state. Thus the density alone already defines the
ground state, irrespective of the potential $v_{1}$ or $v_{2}$. ∎
As highlighted before, the above proof relies purely on the specific structure
of the energy function $E[v]$ that allows the potential part to be taken as a
separate, additive contribution that depends solely on the density. This idea
is due to Paul E. Lammert (during discussion at the workshop “Do Electron
Current Densities Determine All There Is to Know?” in Oslo, 2018). In contrast
to this, the usual proofs of this part of the HK theorem additionally depend
on the _linear_ structure of the density-potential pairing. Moreover, such
proofs are almost always performed indirectly (reductio ad absurdum), with a
few notable exceptions [30; 31]. For completeness, we will give an additional,
more traditional proof, yet one that is direct and does not work by raising a
contradiction.
###### Proof 2.
By the variational principle, we have
$\displaystyle E[v_{1}]$
$\displaystyle=\operatorname{Tr}(H[v_{1}]\Gamma_{1})\leq\operatorname{Tr}(H[v_{1}]\Gamma_{2}),$
$\displaystyle E[v_{2}]$
$\displaystyle=\operatorname{Tr}(H[v_{2}]\Gamma_{2})\leq\operatorname{Tr}(H[v_{2}]\Gamma_{1}).$
Exploiting the shared density $\rho$, this may be written as
$\begin{split}E[v_{1}]&=\operatorname{Tr}(H_{0}\Gamma_{1})+\langle
v_{1},\rho\rangle\\\ &\leq\operatorname{Tr}(H_{0}\Gamma_{2})+\langle
v_{1}+v_{2}-v_{2},\rho\rangle\\\ &=E[v_{2}]+\langle
v_{1}-v_{2},\rho\rangle\end{split}$
and analogously as
$E[v_{2}]\leq E[v_{1}]+\langle v_{2}-v_{1},\rho\rangle.$
Combining the inequalities gives
$E[v_{1}]-E[v_{2}]=\langle v_{1}-v_{2},\rho\rangle$
and from
$\operatorname{Tr}(H[v_{2}]\Gamma_{1})=\operatorname{Tr}(H[v_{1}]\Gamma_{1})-\langle
v_{1}-v_{2},\rho\rangle$
that $\operatorname{Tr}(H[v_{2}]\Gamma_{1})=E[v_{1}]$. So $\Gamma_{1}$ is also
a ground state of $H[v_{2}]$. Likewise,
$\operatorname{Tr}(H[v_{1}]\Gamma_{2})=E[v_{2}]$, so $\Gamma_{2}$ is also a
ground state of $H[v_{1}]$, as required. ∎
HK1 holds generally for mixed or pure ground states. The same proofs remain
valid when the theorem is specialized to a statement about pure states
$\Gamma_{i}=|\psi_{i}\rangle\langle\psi_{i}|$. An immediate but maybe
surprising consequence that is often referred to as the basis of DFT is that a
ground-state density ${\rho_{\mathrm{gs}}}$ alone already determines a ground
state. This result has been coined a _weak HK-like result_ before [32] and it
will be used to define the HK1 functionals on
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$ and
$v\text{-}\mathsf{rep}_{\mathrm{ens}}$ in Eq. (9) below. The ground state
(associated with ${\rho_{\mathrm{gs}}}$) is pure if ${\rho_{\mathrm{gs}}}\in
v\text{-}\mathsf{rep}_{\mathrm{pure}}$ but has to be an ensemble if
${\rho_{\mathrm{gs}}}\in v\text{-}\mathsf{rep}_{\mathrm{ens}}\setminus
v\text{-}\mathsf{rep}_{\mathrm{pure}}$. Any state that is a minimizer in Eq.
(6) is really a ground state for _all_ potentials that share the same ground-
state density. That all those potentials are in fact equal (up to a constant)
is then the statement of HK2, the second part of the HK theorem. It will be
formulated for eigenstates, in case of an ensemble we are free to just take
any of its components.
###### Theorem 2 (HK2).
If two potentials share any common eigenstate and if that eigenstate is non-
zero almost everywhere, then the potentials are equal up to a constant.
###### Proof.
If $v_{1},v_{2}$ share a common eigenstate $\psi$ it holds
$\displaystyle(H_{0}+V[v_{1}])\psi=E[v_{1}]\psi,$
$\displaystyle(H_{0}+V[v_{2}])\psi=E[v_{2}]\psi.$
Subtraction of the two equations and moving all potential parts that do not
depend on $\mathbf{r}_{1}$ to the right-hand side gives
$\displaystyle(v_{1}(\mathbf{r}_{1})-v_{2}(\mathbf{r}_{1}))\psi=$
$\displaystyle(E[v_{1}]-E[v_{2}])\psi$ (8)
$\displaystyle-\sum_{i=2}^{N}(v_{1}(\mathbf{r}_{i})-v_{2}(\mathbf{r}_{i}))\psi.$
Since we assumed $\psi$ non-zero almost everywhere, we can then divide by
$\psi$ and get $v_{1}(\mathbf{r}_{1})-v_{2}(\mathbf{r}_{1})=\mathrm{constant}$
(almost everywhere) because the right-hand side does not depend on
$\mathbf{r}_{1}$. ∎
Since HK2 states that sharing _any_ common eigenstate for two potentials means
that they are equal (up to a constant), this of course implies that the
potentials share _all_ eigenstates because they yield exactly the same
Hamiltonian (up to an additive constant that just shifts the spectrum). The
special requirement that the wave function is non-zero (almost everywhere) is
guaranteed for a large class of potentials by the _unique-continuation
property (UCP) from sets of positive measure_. This property will be further
discussed in Section V. That zeroes (nodes) in the wave function _are_ still
allowed on a set of measure zero is important here, since the fermionic many-
particle wave functions will exhibit nodal surfaces when particle positions
agree. Outside of the continuum setting, for example in finite-lattice
systems, such a UCP is _not_ at hand and there are actual counterexamples to
HK2, were two different potentials share a common eigenstate [24].
The complete HK result is then obtained by combining the two theorems above.
We will assume here that the potential is from the mentioned class that
guarantees a non-zero ground state. We should remember that such or similar
restrictions will always come into play if we want to show validity of a
density-potential mapping in other settings. The statement will be formulated
for densities in $v\text{-}\mathsf{rep}_{\mathrm{ens}}$, so it automatically
holds for $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ as well.
###### Corollary 3 (HK).
If two potentials share a common ensemble $v$-representable ground-state
density, then they are equal up to a constant.
###### Proof.
By HK1 there is a density matrix
$\Gamma=\sum_{k}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$ that is a ground
state for both potentials. Since at least one $\lambda_{k}\neq 0$ and since
the corresponding $\psi_{k}$ is a ground-state wave function for both
Hamiltonians, the proof can be completed by HK2. ∎
This structuring into two separate theorems was already used in Kohn, Savin,
and Ullrich [33], just in the reverse order, for a brief argument about DFT
with magnetization. Historically, the HK theorem was first given only for the
non-degenerate case and was only later extended to include degeneracy [22;
34]. The proof presented here does not suffer from any limitation to non-
degenerate ground states.
A final note is directed towards more general DFTs that will be briefly
discussed in Section X and especially in the forthcoming Part II of this
review. For other types of potentials, like vector potentials, the statement
in the HK theorem would not necessarily be that the potentials are equal “up
to a constant”, but for example “up to a gauge transformation”. The set of
gauge transformations that are possible without affecting the physical
properties of the system then have to be specified within the respective
theory.
## V The unique-continuation property
In this section, we summarize some important results on the unique-
continuation property (UCP) of solutions to the Schrödinger equation that is
heavily used in the context of (mathematical formulation of) HK-type theorems.
The current understanding is that the UCP cannot be avoided in a rigorous
proof of a HK-type theorem. The setting will be slightly more general than
before and allow for dimensionality $d$ of the spatial part of the single-
particle configuration space $\mathbb{R}^{d}$. The $N$-particle configuration
space is then $\mathbb{R}^{n}$ with $n=dN$.
Roughly speaking, the desired UCP result states that under certain conditions
on the potentials building up the operators $V$ and $W$ and if a solution
$\psi$ to the (distributional) equation $H[v]\psi=0$ vanishes on a set of
positive measure, then $\psi$ vanishes everywhere. That the right hand side is
zero comes as no restriction here, since the energy $E$ can always be absorbed
into the scalar potential $v$. The usual literature on the UCP shows _strong_
UCP, which means that $\psi$ is assumed to _vanish to infinite order_ at a
point $\underline{\mathbf{r}}_{0}\in\mathbb{R}^{n}$ and then the statement
follows. A function $f(\mathbf{r})$ is said to vanish to infinite order at
$\underline{\mathbf{r}}_{0}\in\mathbb{R}^{n}$ if for all $k\geq 1$ there is a
$c_{k}$ such that
$\int_{|\underline{\mathbf{r}}-\underline{\mathbf{r}}_{0}|<\epsilon}|f(\underline{\mathbf{r}})|^{2}\,\mathrm{d}\underline{\mathbf{r}}<c_{k}\epsilon^{k}$
for every $0<\epsilon<1$. Now a very convenient result by Regbaoui [35] shows
that the UCP on sets of positive measure actually follows from such a strong
UCP if the potentials are in $L^{n/2}_{\mathrm{loc}}$. This work apparently
built on de Figueiredo and Gossez [36] that again rests on an early estimate
for general Sobolev spaces by Ladyzenskaya and Ural’tzeva [37, Lemma 3.4]. The
result and its proof have been repeated in Lammert [38]. For us that means
that any strong UCP can also be used as a UCP on sets of positive measure
which is the one needed for the proof of HK2. Yet the traditional strong UCP
results, like most notably in Jerison and Kenig [39], also give dimension-
dependent constraints on the potentials like $L^{n/2}_{\mathrm{loc}}$, which
approaches $L^{\infty}$ for growing particle number and is thus too
restrictive for our use where singular potentials need to be considered. The
saving idea recently came from Garrigue [31] and was also extended to more
complex systems [40; 41]: To take the special $N$-body structure of the
potentials into account and thus avoid any dependence of the constraints on
the particle number $N$.
###### Theorem 4 (Garrigue’s UCP).
Suppose that the potentials are in $L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})$ with
$p>2$ for $d=3$ and $p=\max(2d/3,2)$ else. If a solution $\psi$ to the
Schrödinger equation vanishes on a set of positive measure or if it vanishes
to infinite order at any point, then $\psi=0$.
The most relevant case here is obviously $d=3$ which means that the potentials
need to be in $L_{\mathrm{loc}}^{p}(\mathbb{R}^{3})$ with $p>2$ but exactly
$p=2$ is not enough yet. This clearly does not fit our potential space
$X^{*}=L^{3/2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$, so while this UCP
result is the best one available, it cannot be used for a HK2 theorem that
covers the whole potential space of DFT in the formulation discussed here.
Lieb [2] also remarked on the UCP in the context of the HK theorem, which “is
believed to hold” for potentials in $X^{*}$, however in a weaker form that is
not sufficient for the current purpose. So whenever we state that the HK holds
in standard DFT, we actually mean under the given restrictions on the
potentials.
## VI Hierarchy of density functionals
The first part of the HK theorem, HK1, analogously holds in many different
varieties of DFT (that will be explored in Part II), simply because its
validity just depends on the form of the energy functional. HK1 then ensures
that we can map from pure-state $v$-representable ground-state densities
${\rho_{\mathrm{gs,pure}}}$ to ground-state wave functions
$\psi[{\rho_{\mathrm{gs,pure}}}]$ and from ensemble $v$-representable ground-
state densities ${\rho_{\mathrm{gs,ens}}}$ to ground-state density matrices
$\Gamma[{\rho_{\mathrm{gs,ens}}}]$. This makes it possible to define the HK1
functionals
$\displaystyle
F_{\mathrm{HK1,pure}}[{\rho_{\mathrm{gs}}}]=\langle\psi[{\rho_{\mathrm{gs}}}]|H_{0}|\psi[{\rho_{\mathrm{gs}}}]\rangle$
$\displaystyle\,\,\,\,\text{on}\;v\text{-}\mathsf{rep}_{\mathrm{pure}}$ (9a)
and $\displaystyle
F_{\mathrm{HK1,ens}}[{\rho_{\mathrm{gs}}}]=\operatorname{Tr}(H_{0}\Gamma[{\rho_{\mathrm{gs}}}])$
$\displaystyle\,\,\,\,\text{on}\;v\text{-}\mathsf{rep}_{\mathrm{ens}}$ (9b)
as the energy contribution only from the internal parts $H_{0}$ of the
Hamiltonian. The universal nature of such functionals, being independent of
any external $v$, justifies the usual attribution as universal functionals. It
is then possible to determine also the internal energy contributions for any
state with density ${\rho_{\mathrm{gs}}}$ just from ${\rho_{\mathrm{gs}}}$. To
get the total ground-state energy (5) with the help of the functional above,
it is enough to vary over $v$-representable densities alone, instead of the
much larger set of wave functions. We can write
$\displaystyle E[v]$
$\displaystyle=\inf\\{\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\\}$ (10)
$\displaystyle=\inf_{{\rho^{\prime}_{\mathrm{gs}}}}\\{\langle\psi[{\rho^{\prime}_{\mathrm{gs}}}],H_{0}\psi[{\rho^{\prime}_{\mathrm{gs}}}]\rangle+\langle
v,{\rho^{\prime}_{\mathrm{gs}}}\rangle\\}$
$\displaystyle=\inf_{\rho^{\prime}_{\mathrm{gs}}}\\{F_{\mathrm{HK1,pure}}[{\rho^{\prime}_{\mathrm{gs}}}]+\langle
v,{\rho^{\prime}_{\mathrm{gs}}}\rangle\\}\quad\text{on}\;X^{*},$
or equivalently with $F_{\mathrm{HK1,ens}}$. We see already that there is a
certain ambiguity in which density functional to use in the definition of
$E[v]$. The other density functionals presented here will all have the
property that they give the correct ground-state energy when applied in Eq.
(10) which makes them all _admissible_ functionals [42]. Yet, they will differ
with respect to their mathematical properties and we thus aim for the one with
the best features.
The first problem here is that the densities to be considered in the
variational problem are limited to those that are actual ground-state
densities ($v\text{-}\mathsf{rep}$), because else
$F_{\mathrm{HK1}}[{\rho_{\mathrm{gs}}}]$ is left undefined, and we already
learned in Section III that $v\text{-}\mathsf{rep}$ is not an explicitly
characterized set. Apart from that, HK1 just states the existence of a map
${\rho_{\mathrm{gs}}}\mapsto\psi$ or $\Gamma$ without giving any hints towards
a constructive scheme. A first step to overcome these problems is to inspect
Eq. (7). This suggests the definition of another pair of density functionals
that goes under the name of “constrained search”,
$\displaystyle
F_{\mathrm{CS,pure}}[\rho]=\inf_{\psi\mapsto\rho}\langle\psi|H_{0}|\psi\rangle$
$\displaystyle\quad\text{on}\;N\text{-}\mathsf{rep}\;\text{and}$ (11a)
$\displaystyle
F_{\mathrm{CS,ens}}[\rho]=\inf_{\Gamma\mapsto\rho}\operatorname{Tr}(H_{0}\Gamma)$
$\displaystyle\quad\text{on}\;N\text{-}\mathsf{rep}.$ (11b)
The domain is now the larger, convex, and explicitly defined
$N\text{-}\mathsf{rep}$ in both cases. Note that the literature mostly denotes
those functionals as $F_{\mathrm{CS,pure}}=F_{\mathrm{LL}}$ (“Levy–Lieb” [9;
2]) and $F_{\mathrm{CS,ens}}=F_{\mathrm{DM}}$ (from “density matrix” [2]). A
recent, comprehensive study of these functionals can be found in Lewin, Lieb,
and Seiringer [43]. Since the density is limited to the set
$N\text{-}\mathsf{rep}$ that guarantees finite kinetic energy, the infima in
Eq. (11) are always attained, though not necessarily by a possible ground
state (if $\rho$ is not $v$-representable), and can thus be replaced by minima
in both cases [2, Theorem 3.3]. The convex combination of pure-state
projections into density matrices translates to the functionals, so that
$F_{\mathrm{CS,ens}}$ is the convex envelope of $F_{\mathrm{CS,pure}}$ [24,
Proposition 18, the article treats DFT on a lattice but the statement and the
proof of proposition remains exactly the same in the continuum case.]. This
automatically ensures that $F_{\mathrm{CS,ens}}$ is convex, a fact that can
also be concluded from observing that $\Gamma\mapsto\rho$ linear [2, Section
4.B].
Since these density functionals appear in the optimization problem that
determines the ground-state energy and density, like in Eq. (10), convexity is
of great importance because only for a convex functional can we be sure that
identifying any _local_ minimum also means that a _global_ minimum has been
found. So while we now know that $F_{\mathrm{CS,ens}}$ is convex, the previous
functional $F_{\mathrm{HK1,pure}}$ does not even have a convex domain and
therefore cannot be convex. Levy [23] and Lieb [2] also gave arguments for the
non-convexity of $F_{\mathrm{CS,pure}}$. Since
$F_{\mathrm{CS,ens}}=\operatorname{conv}F_{\mathrm{CS,pure}}$, any density
where $F_{\mathrm{CS,ens}}[\rho]\neq F_{\mathrm{CS,pure}}[\rho]$ already shows
non-convexity of $F_{\mathrm{CS,pure}}$. But this is equivalent to saying that
$\rho$ is ensemble $v$-representable while it is _not_ pure-state
$v$-representable, so $\rho\in v\text{-}\mathsf{rep}_{\mathrm{ens}}\setminus
v\text{-}\mathsf{rep}_{\mathrm{pure}}$ [24, Proposition 21].
Note especially, that HK1 was necessary to define $F_{\mathrm{HK1}}$, but is
not needed any more for the constrained-search functional $F_{\mathrm{CS}}$.
Being able to define a universal constrained-search functional, one that is
independent of the potential like in Eq. (11), already fully facilitates the
proof of HK1 and thus implies this result. A potential-independent
constrained-search functional _already implicitly includes HK1_. This
implication was proven by Levy [9] along the lines of the usual HK proof and
is mentioned in textbooks like Parr and Yang [18, after their Eq. (3.4.4)] and
Tsuneda [44, after Eq. (4.5)]. Speaking generally though, a constrained search
is just as feasible if the constrained-search functional also depends on the
external potential $v$ (although it would not be universal), so indeed this
approach is more general than relying on HK1. Such a case turns up in CDFT
when the current variable is the total current that itself depends on the
vector potential (see Part II of this review for more on this).
By employing the constrained-search functional, the ground-state energy from
Eq. (5) can now be rewritten again as
$\displaystyle E[v]$
$\displaystyle=\inf\\{\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\\}$
$\displaystyle=\inf\\{F_{\mathrm{CS,pure}}[\rho_{\psi}]+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\\}$
$\displaystyle=\inf_{\rho}\\{F_{\mathrm{CS,pure}}[\rho]+\langle
v,\rho\rangle\\}\quad\text{on}\;X^{*},$
or equivalently with $F_{\mathrm{CS,ens}}$, where minimization is now
performed over $N\text{-}\mathsf{rep}$.
When looking at non-interacting systems, the definitions of
$F_{\mathrm{HK1}}$, Eq. (9), and $F_{\mathrm{CS}}$, Eq. (11), involve only the
kinetic-energy operator $T$ instead of $H_{0}$. We will then denote these
functionals with a zero superscript,
$F_{\mathrm{HK1}}^{0},F_{\mathrm{CS}}^{0}$, etc., that indicates that non-
interacting systems are considered. A further functional then comes into play
that is defined like $F_{\mathrm{CS,pure}}$, but where only Slater
determinants are considered as wave functions. We define on
$N\text{-}\mathsf{rep}$,
$F^{0}_{\mathrm{SD}}[\rho]=\inf_{\phi\mapsto\rho}\left\\{\langle\phi|T|\phi\rangle\mid\text{{\hbox{\phi}}
is a Slater determinant}\right\\}.$
The usual name in the literature is $F^{0}_{\mathrm{SD}}=T_{S}$. This
functional is of importance because it is the one used in Kohn–Sham theory
which will be discussed in Section VIII. In their original article, Kohn and
Sham [45] implicitly set $F^{0}_{\mathrm{SD}}=F_{\mathrm{HK1,pure}}^{0}$ for
all non-interacting pure-state $v$-representable densities, which has been
noted to be wrong because of possible degeneracy [2, Section 4.C]. On the
other hand, for non-degenerate ground states $\phi$, which by necessity are
always determinants in non-interacting systems, it holds that
$F^{0}_{\mathrm{SD}}[\rho_{\phi}]=F_{\mathrm{CS,pure}}^{0}[\rho_{\phi}]=F_{\mathrm{HK1,pure}}^{0}[\rho_{\phi}]$,
and else $F^{0}_{\mathrm{SD}}\geq F_{\mathrm{CS,pure}}^{0}$. Nevertheless, for
practical purposes, $F^{0}_{\mathrm{SD}}$ usually takes up the role of the
density functional when defining the energy functional in a non-interacting
setting.
The transformation from any density functional $F_{\bullet}$ for an
interacting system from above to the energy functional,
$E[v]=\inf_{\rho}\\{F_{\bullet}[\rho]+\langle
v,\rho\rangle\\}\quad\text{on}\;X^{*},$ (12)
is called the _convex conjugate_ or Legendre–Fenchel transformation [26,
Section 2.1.4]. There is also a way to reverse the transformation and we
define
$F[\rho]=\sup_{v}\\{E[v]-\langle v,\rho\rangle\\}\quad\text{on}\;X.$ (13)
This $F$ is the famous _Lieb functional_ [2], yet another density functional,
but this time the last one to be defined in standard DFT. It is the
_biconjugate_ of any $F_{\bullet}$ considered before. Defined this way, both
$E$ and $F$ are lower-semicontinuous and $E$ is concave while $F$ is convex
and has the property $F\leq F_{\bullet}$ [26, Proposition 2.19]. Actually, as
a biconjugate, $F$ is the largest convex and lower semicontinuous functional
that fulfills $F\leq F_{\bullet}$ which makes it the _convex envelope_ of
$F_{\bullet}$. The domain is now the whole $X=L^{1}(\mathbb{R}^{3})\cap
L^{3}(\mathbb{R}^{3})$, but automatically $F[\rho]=\infty$ for all densities
that are not in $N\text{-}\mathsf{rep}$ [2, Theorem 3.8], while at the same
time $F[\rho]<\infty$ if $\rho\in N\text{-}\mathsf{rep}$ [2, Theorem 3.9 and
the following Remark]. Let the _effective domain_ ‘$\operatorname{dom}$’ of a
convex functional be the elements from its domain where it is finite, then
this means that $\operatorname{dom}F=N\text{-}\mathsf{rep}$. Having reached
$F$, it does not matter any more which (admissible) functional has been used
in Eq. (12), which means the convex envelopes of all the functionals above
agree. Conversely, the Legendre–Fenchel transformation can also be utilized to
go back from $F$ to $E$ [26, Theorem 2.22],
$E[v]=\inf_{\rho}\\{F[\rho]+\langle v,\rho\rangle\\}.$ (14)
We already noted that $F$ is convex and lower-semicontinuous, which are both
important properties if we want to use the variational problem
$E[v]=\inf_{\rho}\\{F[\rho]+\langle v,\rho\rangle\\}$ to find a minimizing
density. The same properties come into play when defining the minimizers by
differentiation in Section VII. From the definition of $F$ it follows directly
that
$E[v]\leq F[\rho]+\langle v,\rho\rangle,$
a version of the Young inequality. Equality in the above estimate holds if the
density _is_ the ground-state density ${\rho_{\mathrm{gs}}}$ for the potential
$v$,
$E[v]=F[{\rho_{\mathrm{gs}}}]+\langle v,{\rho_{\mathrm{gs}}}\rangle.$
For $F_{\mathrm{CS,pure}}$ the converse holds too: If
$E[v]=F_{\mathrm{CS,pure}}[\rho]+\langle v,\rho\rangle$ then $\rho$ is a
ground-state density within $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ for the
potential $v$ and further
$F_{\mathrm{HK1,pure}}[\rho]=F_{\mathrm{HK1,ens}}[\rho]=F_{\mathrm{CS,pure}}[\rho]=F_{\mathrm{CS,ens}}[\rho]=F[\rho]$
[2, Theorem 3.10]. But what about using the more general functional $F$ for
the variational principle like in Eq. (14)? Can we find a real ground state
like this or will this variational principle yield additional artificial
solutions because it is too general? Because it is the convex envelope of the
other functionals, it cannot produce a functional value below the ground-state
energy, but it could produce a minimizing density where there are no
$v$-representable ground-state densities! The problem is solved if we allow
for ensembles of ground states: An “amusing fact” in Lieb [2, Eq. (4.5)] gives
$F=F_{\mathrm{CS,ens}}$ on $N\text{-}\mathsf{rep}$, which effectively means
$F=F_{\mathrm{CS,ens}}$ since we can just set
$F_{\mathrm{CS,ens}}[\rho]=\infty$ outside of its domain
$N\text{-}\mathsf{rep}$ to achieve equality globally on $X$. So any minimizer
of $F+\langle v,\cdot\rangle$ is also one of $F_{\mathrm{CS,ens}}+\langle
v,\cdot\rangle$ and it is further the convex combination of ground states for
the potential $v$. Consequently, when talking about ground states in the
context of the functional $F$, we will always actually mean _ensembles_ of
possibly degenerate ground states.
When comparing the functionals on $X$, we just set them to $\infty$ whenever
we are outside their domains. The following hierarchy can be set up and is
further laid out in Table 1.
$F=F_{\mathrm{CS,ens}}\leq\begin{array}[]{c}F_{\mathrm{CS,pure}}(\leq
F^{0}_{\mathrm{SD}})\\\ F_{\mathrm{HK1,ens}}\end{array}\leq
F_{\mathrm{HK1,pure}}.$
Here, $F^{0}_{\mathrm{SD}}$ appears in parentheses since it only comes into
play in the non-interacting setting where we can perform the same type of
transformations and have $F^{0}[\rho]$ and $E^{0}[v]$.
$F_{\bullet}$ | convex | domain | convex
---|---|---|---
$F_{\mathrm{HK1,pure}}$ | no | | $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ | no |
$F_{\mathrm{HK1,ens}}$ | ? | | $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ | ? | $\Downarrow$ cl
$F_{\mathrm{CS,pure}}$ | no | $\Downarrow$ conv | $N\text{-}\mathsf{rep}$ | yes
$F_{\mathrm{CS,ens}}$ | yes | $N\text{-}\mathsf{rep}$ | yes |
$F$ | yes | | $L^{1}\cap L^{3}$ | yes |
Table 1: The table shows the relations between the functionals discussed in
Section VI. From $F_{\mathrm{HK1,pure}}$ to $F_{\mathrm{HK1,ens}}$ the domain
gets extended to $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ while they agree on
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$. From $F_{\mathrm{HK1,ens}}$ to
$F_{\mathrm{CS,pure}}$ the domain gets closed (cl) within $L^{1}\cap L^{3}$
and from $F_{\mathrm{CS,pure}}$ to $F_{\mathrm{CS,ens}}$ the functional itself
gets convexified (conv) while the domain remains the same. Finally, $F$ is
just equal to $F_{\mathrm{CS,ens}}$ on $N\text{-}\mathsf{rep}$.
## VII Density-potential mappings from differentials
In the previous section it was stated that in order to get the ground-state
density of any system we have to find a solution to the variational problem
$E[v]=\inf_{\rho}\\{F[\rho]+\langle v,\rho\rangle\\},$ (15)
now relying on the density functional $F$ from Eq. (13). To find the global
minimum of a _convex and lower-semicontinuous_ functional we can perform
differentiation, i.e., demand that the differential of $F[\rho]+\langle
v,\rho\rangle$ with respect to $\rho$ must equal zero at the position of a
ground-state density ${\rho_{\mathrm{gs}}}$.
$\rho$$\rho_{0}$$\infty$ Figure 1: Example of a convex and lower-
semicontinuous function with a discontinuity at $\rho_{0}$ and some elements
from the subdifferential displayed as linear continuous tangent functionals at
$\rho_{0}$, represented by dashed lines.
The suitable notion of differentiation here is the subdifferential
$\underline{\partial}F$ that gives the _set_ of all linear continuous tangent
functionals to a convex functional $F$ at a given density $\rho$,
$\underline{\partial}F[\rho]=\\{v\in X^{*}\mid\forall\rho^{\prime}\in
X:F(\rho)\leq F(\rho^{\prime})+\langle v,\rho-\rho^{\prime}\rangle\\}.$
It is always well-defined, since the set $\underline{\partial}F[\rho]$ can
contain many elements, in case the functional $F$ has a kink (like the example
shown in Fig. 1), or can even be empty. Finally, if it contains exactly one
element, we found a _unique_ potential yielding that ground-state density. In
any case, the variational problem (15) has a minimizer ${\rho_{\mathrm{gs}}}$
if and only if the following condition is fulfilled [26, Proposition 2.33],
$\left.\underline{\partial}(F[\rho]+\langle
v,\rho\rangle)\right|_{\rho={\rho_{\mathrm{gs}}}}\ni
0\Longleftrightarrow\langle
v,\cdot\rangle\in-\underline{\partial}F[{\rho_{\mathrm{gs}}}].$ (16)
In what follows, we identify $v$ with the functional $\langle v,\cdot\rangle$
whenever the context implies a functional on density space instead of a
potential on configuration space, so Eq. (16) can be written
$v\in-\underline{\partial}F[{\rho_{\mathrm{gs}}}]$. The potential as the
subdifferential of the density functional means that potentials $v$ are from
the dual of the space of densities like already noted in Section III. This
general principle is not always respected in more complex versions of DFT, as
we will see in Section X and discuss further in Part II of this review.
If the set $\underline{\partial}F[{\rho_{\mathrm{gs}}}]$ is non-empty then
there is at least one potential $v\in X^{*}$ that yields the given ground-
state density. The set of all densities where
$\underline{\partial}F[{\rho_{\mathrm{gs}}}]\neq\emptyset$ is called the
domain of the subdifferential, so it follows that
$\operatorname{dom}\underline{\partial}F=v\text{-}\mathsf{rep}_{\mathrm{ens}}$.
Note that by a theorem of convex analysis [26, Corollary 2.44],
$\operatorname{dom}\underline{\partial}F$ is dense in $\operatorname{dom}F$,
so $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ is dense in $N\text{-}\mathsf{rep}$,
a fact already expressed with $N\text{-}\mathsf{rep}$ being the closure of
$v\text{-}\mathsf{rep}_{\mathrm{ens}}$ in Section III.
The meaning of a valid HK theorem for a class of densities is that they can
all be mapped as ground-state densities back to a unique potential (modulo a
constant) and consequently
$-\underline{\partial}F[{\rho_{\mathrm{gs}}}]=\\{v+c\mid c\in\mathbb{R}\\}$.
By eliminating the (physically unimportant) constant potentials from the
potential space, the subdifferential of a $v$-representable density is
precisely $-\underline{\partial}F[{\rho_{\mathrm{gs}}}]=\\{v\\}$ if the HK
theorem holds. If, on the other hand, $F$ is assumed differentiable, then the
directional derivative $-\frac{\delta}{\delta\rho}F[{\rho_{\mathrm{gs}}}]=v$
anyway always maps to a unique potential. One thus has a well-defined map from
densities in $v\text{-}\mathsf{rep}$ to the corresponding potentials, exactly
the content of the HK theorem! But where did it enter? The HK theorem is here
a _consequence_ from the assumption of differentiability of $F[\rho]$ at
$v$-representable densities. The situation will be summarized diagrammatically
in Section XI.
Because any potential $v$ that we determine by Eq. (16) will also be the
maximizer in the conjugate variational problem
$F[{\rho_{\mathrm{gs}}}]=\sup_{v}\\{E[v]-\langle
v,{\rho_{\mathrm{gs}}}\rangle\\},$
we can just as well say the same with the superdifferential of the concave
functional $E$,
$\left.\overline{\partial}(E[v^{\prime}]-\langle
v^{\prime},{\rho_{\mathrm{gs}}}\rangle)\right|_{v^{\prime}=v}=0\Longleftrightarrow{\rho_{\mathrm{gs}}}\in\overline{\partial}E[v].$
(17)
The right hand side, ${\rho_{\mathrm{gs}}}\in\overline{\partial}E[v]$, means
to find a density (or possibly many) that comes from a wave-function that
minimizes the total energy including $v$. It is thus a conceptual shortcut to
map from potentials to ground-state densities without any reference to an
underlying wave function or Schrödinger equation. The situation of a set
$\overline{\partial}E[v]$ with more than one element is known from
degeneracies of the Hamiltonian $H_{0}+V[v]$, where different linearly
independent ground states with eventually different densities all have the
same eigenvalue.
We showed in this section the important role of the generalized concepts of
sub/superdifferentials in the context of DFT, because indeed the functionals
from Section VI can _not_ be assumed differentiable as van Leeuwen [46] has
demonstrated for the $F_{\mathrm{HK1}}$ functionals and Lammert [3] for
$F_{\mathrm{CS}}$. The reason for non-differentiability even of
$F_{\mathrm{CS}}$ is that at any $\rho$ the functional $F[\rho+\delta\rho]$ is
infinite for various, arbitrarily small shifts $\delta\rho$ that lead out of
$N\text{-}\mathsf{rep}$, even if the normalization of the density is kept
constant. This happens by infinitely increasing the internal energy through
tiny oscillations of the density. A possible way to prevent that is to limit
the density space $X$ so that such shifts $\delta\rho$ are not possible any
more and Lammert [3] actually shows this for the Sobolev space
$H^{2}(\mathbb{R}^{3})$ when $\rho$ is also assumed $v$-representable. Another
way is to establish a coarse-grained model for DFT in which $F$ really becomes
differentiable and every density is ensemble $v$-representable [47]. In the
following section, in accordance with the vast majority of the literature, we
will assume functional differentiability of $F$ and consequently
$v$-representability. This strong assumption can be justified a posteriori, as
discussed later in Section IX, when a regularization procedure is applied.
## VIII Linking to a reference system: the Kohn–Sham scheme
In Section VI it was noted that a functional might be introduced for an
interacting or a non-interacting system. This means the respective Hamiltonian
has the internal part $T+\lambda W$ with $\lambda\in\\{0,1\\}$. We will now
write $F^{1}$ and $F^{0}$ to differentiate clearly between those two
situations. We then introduce the difference functional
$F_{\mathrm{Hxc}}=F^{1}-F^{0}$, which just corresponds to the internal-energy
difference between the interacting and the non-interacting system and that
will later be linked to the Hartree-exchange-correlation potential
$v_{\mathrm{Hxc}}$. This potential effectively compensates for the Hartree-
mean-field interaction as well as ‘exchange’ and ‘correlation’ effects. The
idea behind introducing this auxiliary non-interacting system is that the
energy difference between the (numerically tractable) non-interacting system
and the (numerically unfeasible) interacting system is small and can be
efficiently approximated. Since the reference system is non-interacting,
$F^{0}_{\mathrm{SD}}$ can be employed for $F^{0}$ if degeneracy for the ground
state does not have to be taken into account, like it was mentioned in Section
VI, and this switchover is performed in most practical situations. Then the
energy functional for the full system is
$\displaystyle E^{1}[v]$ $\displaystyle=\inf_{\rho}\\{F^{1}[\rho]+\langle
v,\rho\rangle\\}$
$\displaystyle=\inf_{\rho}\\{F^{0}[\rho]+F_{\mathrm{Hxc}}[\rho]+\langle
v,\rho\rangle\\}$
$\displaystyle=\inf_{\phi}\\{\langle\phi|T|\phi\rangle+F_{\mathrm{Hxc}}[\rho_{\phi}]+\langle
v,\rho_{\phi}\rangle\\}.$
In the last step the variation is changed from $N\text{-}\mathsf{rep}$
densities to single Slater determinants $\phi$, the minimizer – if it exists –
is then the Kohn–Sham Slater determinant. In order to link this to a partial
differential equation for the orbitals $\varphi_{i}$ constituting $\phi$, the
Kohn–Sham equation, variation of the energy expression above with respect to
$\varphi_{i}$ is performed under the constraint that all the $\varphi_{i}$
stay normalized. This means
$\rho_{\phi}(\mathbf{r})=\sum_{i=1}^{N}\sum_{\sigma}|\varphi_{i}(\mathbf{r}\sigma)|^{2}$
always stays in $N\text{-}\mathsf{rep}$, but generally the issue of non-
differentiability from Section VII remains. The resulting equation is a one-
particle Schrödinger equation with effective potential $v_{s}$ and eigenstates
$\varphi_{i}$,
$\left(-\tfrac{1}{2}\nabla^{2}+v_{s}(\mathbf{r})\right)\varphi_{i}(\mathbf{r}\sigma)=\varepsilon_{i}\varphi_{i}(\mathbf{r}\sigma).$
(18)
On the other hand this approach does not lead to the effective potential
$v_{s}$ for the Kohn–Sham equation right away, but requires the additional,
computationally challenging step of extracting the effective potential from
the variation of $F_{\mathrm{Hxc}}$ with respect to the orbitals (OEP integral
equation[48]).
To have a well defined $F_{\mathrm{Hxc}}[\rho]=F^{1}[\rho]-F^{0}[\rho]$, the
$\rho$ must be both, interacting and non-interacting $v$-representable. Both
systems then share the same ground-state density $\rho$ when the different
external potentials
$v\in-\underline{\partial}F^{1}[\rho]\quad\text{and}\quad
v_{s}\in-\underline{\partial}F^{0}[\rho]$ (19)
are assigned to them. That the density $\rho$ is simultaneously interacting
_and_ non-interacting $v$-representable is tacitly assumed here, else one of
the subdifferentials above is empty. This means that actually the
$v$-representability problem from Section III shows up at this point. A
purported solution [49; 50; 51] rests on an ill-founded notion of
differentiability where the functionals are extended to distributions, but
with an incorrect application of the calculus of distributions (see, e.g., Eq.
(0.24) in Gonis [49]).
The usual rationale of DFT is to assume that the potentials from Eq. (19)
exist and are unique (modulo a constant; after all the latter is the content
of the HK theorem). The difference $v_{\mathrm{Hxc}}=v_{s}-v$ is then known as
the Hartree-exchange-correlation potential: what needs to be added to the
fixed external potential $v$ in order to simulate all interactions in an non-
interacting system. Note that such missing effects from interactions do not
stem exclusively from the $W$-term in $F^{1}$, but also from the different
kinetic energy contributions between the interacting and non-interacting
system. Nevertheless, the usual understanding is that most of the kinetic
energy contributions can already be captured by a non-interacting system (with
an uncorrelated wave function) and that they thus practically cancel between
$F^{1}$ and $F^{0}$ when we calculate
$v_{\mathrm{Hxc}}[\rho]=v_{s}-v\in\underline{\partial}F^{1}[\rho]-\underline{\partial}F^{0}[\rho].$
(20)
At this point a problematic discrepancy is introduced, since the
subdifferential is not linear and thus $v_{\mathrm{Hxc}}$ and
$\underline{\partial}F_{\mathrm{Hxc}}$ need not match. If $v_{\mathrm{Hxc}}$
cannot be determined as $\underline{\partial}F_{\mathrm{Hxc}}$ we are left
with the necessity of individually solving the inverse problems $\rho\mapsto
v$ and $\rho\mapsto v_{s}$ in Eq. (20) for both systems, interacting and non-
interacting. In practice this means one cannot benefit from finding good
approximations to $F_{\mathrm{Hxc}}$ which are the most important elements of
applied DFT.
A possible circumvention lies in a conceptual shift from describing a system
in terms of energies to forces. The ground state is then characterized by a
certain force-balance equation that can be equally found in non-equilibrium
settings, just with an additional dynamical term [52; 53]. At a density that
is simultaneously interacting and non-interacting $v$-representable and where
the wave function has a sufficient regularity, the force-balance equation can
be employed to derive $v_{\mathrm{Hxc}}$ as the solution of a Poisson equation
instead of a functional derivative [54]. An alternative derivation for this
was already given earlier using line integrals describing the work it takes to
move an electron from infinity against the force field of the exchange-
correlation hole charge [55; 56].
Yet, we will proceed here for the sake of argument by _assuming_
differentiability for now. Since the functional derivative
$\frac{\delta}{\delta\rho}$ is linear and it holds
$\displaystyle v_{\mathrm{Hxc}}[\rho]$
$\displaystyle=v_{s}-v=-\frac{\delta}{\delta\rho}F^{0}[\rho]+\frac{\delta}{\delta\rho}F^{1}[\rho]$
(21)
$\displaystyle=\frac{\delta}{\delta\rho}(F^{1}[\rho]-F^{0}[\rho])=\frac{\delta}{\delta\rho}F_{\mathrm{Hxc}}[\rho].$
Also, several important properties that the Hxc potential needs to have are
automatically fulfilled when they are functional derivatives [57], which is
especially relevant for functional approximations to $v_{\mathrm{Hxc}}$.
The Kohn–Sham scheme is now introduced in order to find an unknown ground-
state density ${\rho_{\mathrm{gs}}}$ of an interacting system by starting from
an initial guess $\rho_{0}$ and by using $v_{\mathrm{Hxc}}$ (in practice a
suitable approximation to it) as the connection between the interacting system
and a non-interacting reference system. To this end, rewrite Eq. (19) with
assumed differentiability as
$\frac{\delta}{\delta\rho}F^{1}[{\rho_{\mathrm{gs}}}]+v=0$ and
$\frac{\delta}{\delta\rho}F^{0}[{\rho_{\mathrm{gs}}}]+v_{s}=0$ and set the two
equations equal,
$\frac{\delta}{\delta\rho}F^{1}[{\rho_{\mathrm{gs}}}]+v=\frac{\delta}{\delta\rho}F^{0}[{\rho_{\mathrm{gs}}}]+v_{s}.$
Now, apart from the fixed external potential $v$ of the interacting system,
all variables in this equation still remain generally unknown: the effective
potential of the non-interacting system $v_{s}$ and, especially, the density
$\rho$ of both systems that we would like to determine. The trick lies in
introducing sequences $\rho_{i}\to{\rho_{\mathrm{gs}}}$, $v_{i}\to v_{s}$ and
define an update rule,
$v_{i+1}=v+\frac{\delta}{\delta\rho}F^{1}[\rho_{i}]-\frac{\delta}{\delta\rho}F^{0}[\rho_{i}]=v+v_{\mathrm{Hxc}}[\rho_{i}].$
(22)
We see immediately that if $\rho_{i}$ has converged to the correct ground-
state density ${\rho_{\mathrm{gs}}}$ of the interacting system, then
$v+\frac{\delta}{\delta\rho}F^{1}[\rho_{i}]=0$ and the remaining equation
tells us that indeed $v_{i+1}$ is the potential that gives the same density
${\rho_{\mathrm{gs}}}$ in the non-interacting system. The next step after Eq.
(22) in the Kohn–Sham iteration lies in determining the density $\rho_{i+1}$
that comes from $v_{i+1}$ in the non-interacting system (which is comparably
easy achieved by solving the corresponding Kohn–Sham equation (18)) and then
iterate. Convergence problems are a big issue within this iteration scheme and
have been dealt with by either damping the iteration step from
$\rho_{i}\to\rho_{i+1}$ to $\rho_{i}\to\rho_{i}+\mu(\rho_{i+1}-\rho_{i})$,
$\mu\in(0,1)$, or mixing several of the previous steps $\\{\rho_{i}\\}$ into
the result $\rho_{i+1}$ [58; 59; 60]. Guaranteed convergence has been studied
and proven for the finite-lattice case [10; 11; 12] by combining an optimal
damping step and a regularization technique [4; 12], the latter truly making
$F$ differentiable and $E$ a strictly concave functional. This solves the
problem of defining $v_{\mathrm{Hxc}}$ in Eq. (21) and yields a curvature
bound on $F$ that is needed for guaranteed convergence. The regularization
method is briefly explained in Section IX below. For the Kohn–Sham iteration
in continuum DFT the convergence is still an open problem, a direct
generalization of the finite-lattice case has been found to be insufficient
[61]. In practical applications that suffer from convergence issues,
imaginary-time propagation in time-dependent DFT has recently been found as a
viable alternative to find a Kohn–Sham ground state [62].
## IX Density-potential mixing and regularized DFT
The full HK theorem guarantees a unique inversion from densities to
potentials, but the whole discussion, especially regarding the necessary
conditions for showing HK2, probably already made us a little bit sceptical
about its validity in different settings. We will thus introduce a method that
always guarantees a bijective mapping, not between densities and potentials,
but between _quasidensities_ (called _pseudo-densities_ in the original work
on regularization [4]) and potentials. The basic idea is simple: If for some
reason we cannot guarantee a unique (injective) mapping from potentials to
ground-state densities $v\mapsto\rho[v]$, meaning that different $v\neq
v^{\prime}$ map to the same $\rho[v]=\rho[v^{\prime}]$, then let us try it for
$v\mapsto\rho_{\varepsilon}[v]=\rho[v]-\varepsilon v$, where at least in the
previous example we would have
$\rho_{\varepsilon}[v]\neq\rho_{\varepsilon}[v^{\prime}]$ for sure. One could
argue that this could just as easily introduce new problems for injectivity,
like having $v\neq v^{\prime}$ such that
$\rho_{\varepsilon}[v]=\rho_{\varepsilon}[v^{\prime}]$, but we will show in
the following that this cannot be the case for the functionals considered
here. Remember that the mapping $v\mapsto\rho[v]$ can be defined by the
superdifferential of $E$, $\rho[v]=\overline{\partial}E[v]$, as explained in
Eq. (17). So what is the corresponding functional $E_{\varepsilon}$ such that
$\rho_{\varepsilon}[v]=\rho[v]-\varepsilon
v=\overline{\partial}E_{\varepsilon}[v]$? The superdifferential retains the
linear nature of a derivative if only concave functionals are added, so we can
look for a convex functional $\phi$ such that
$\overline{\partial}(-\phi)[v]=-\underline{\partial}\phi[v]=-v$. In a general
space, such a question proves hard [61], but it is easy to see that in the
usual space $L^{2}$ of square-integrable functions the norm square gives
exactly what we need, $\phi[v]=\tfrac{1}{2}\|v\|^{2}=\tfrac{1}{2}\langle
v,v\rangle$. In any case, we have established
$E_{\varepsilon}=E-\varepsilon\phi$ and
$\rho_{\varepsilon}[v]=\overline{\partial}E_{\varepsilon}[v]$ with such a
convex $\phi$. But in many cases, not only for the mentioned $L^{2}$ space,
the functional $\phi$ is not only convex, but _strictly convex_ , meaning that
any local minimizer is not only global but even unique. But this feature
transfers to $E_{\varepsilon}$ if $-\varepsilon\phi$, as a _strictly concave_
functional, is added to $E$. Consequently, $E_{\varepsilon}$ is also strictly
concave and any maximizing potential in
$F_{\varepsilon}[x]=\sup_{v}\\{E_{\varepsilon}[v]-\langle v,x\rangle\\}$ (23)
is necessarily unique (not just up to a constant). This means we can always
uniquely map $v\mapsto x=\rho[v]-\varepsilon v$ and back. We wrote $x$ now to
make clear that this is a quasidensity, a mixture between a density and its
associated potential. As such it is neither necessarily normalized nor
positive, just a general element of the density space, $x\in X$. By what we
learned in Section VII, the _quasidensity-potential mapping_ can also be
directly defined by $-\underline{\partial}F_{\varepsilon}[x]=\\{v\\}$ for all
$x$ without any “$v$-representability” restriction for $x$. Consequently, the
mapping is defined for all $x$ in the density space $X$ and thus bijective.
The whole maneuver of passing from $F$ to $F_{\varepsilon}$ corresponds to a
regularization strategy called Moreau–Yosida regularization [4; 12] by which
not only the concave $E$ transforms into a strictly concave $E_{\varepsilon}$,
but also the $F_{\varepsilon}$ defined by Eq. (23) is finally differentiable
if the spaces $X,X^{*}$ have some additional properties [5, Theorem 9]. The
only problem is that this requires the space $X$ to be reflexive, which it is
not in our current formulation as introduced in Section III, since it includes
the non-reflexive $L^{1}$ in its definition. So a different choice for the
basic spaces, like $X=L^{2}$ on a bounded domain [4] or $X=L^{3}$ as a larger
alternative to our space[5] has to be taken.
This section demonstrated how such a regularization that facilitates a unique
(quasi)density-potential mapping can be used to fully circumvent any reference
to the HK theorem. But to avoid confusion we will _not_ say that in a
regularized setting the HK theorem “holds” even though a unique and well-
defined (quasi)density-potential mapping exists. It is interesting to note
that the popular Zhao–Morrison–Parr method for density-potential inversion
already implicitly employs Moreau–Yosida regularization and a limit procedure
$\varepsilon\to 0$ [63].
## X Abstract density-potential mapping
The presented form of HK1 allows for an abstraction and thereby for
generalizations. Therein, the density is generalized to any system-inherent
quantity that seems suitable to describe other system parameters that we are
interested in. This could be the density together with the spin density, a
current-quantity etc. On the other side, we select a generalized form of the
potential that enters the Hamiltonian and that is able to steer the “density-
quantity” by coupling to it. Such a framework was developed in Laestadius _et
al._ [5], building on Banach spaces and their duals for density and potential
quantities. This enables us to employ the regularization technique from
Section IX to obtain a well-defined Kohn–Sham iteration scheme.
In order to be more concrete, let $\mathbf{x}$ be the density quantity
describing a state that will in general include many components, like
different densities, currents etc., and $\mathbf{v}$ the collection of
external potentials acting on them. At this point we do not even assume that
$\mathbf{x}$ and $\mathbf{v}$ have the same number or type of components like
a dual structure between densities and potentials would impose. Instead of a
linear pairing $\langle\mathbf{v},\mathbf{x}\rangle$ for the coupling to the
external potential we can introduce an arbitrary functional
$f[\mathbf{v},\mathbf{x}]$. Then the _only_ necessary condition left for an
abstract HK1 is that the ground-state energy expression has the form
$\displaystyle\tilde{F}[\mathbf{x}]=\inf_{\psi\mapsto\mathbf{x}}\left\\{\langle\psi|H_{0}|\psi\rangle\right\\},$
$\displaystyle
E[\mathbf{v}]=\inf_{\mathbf{x}}\\{\tilde{F}[\mathbf{x}]+f[\mathbf{v},\mathbf{x}]\\}.$
(24)
Since $\tilde{F}[\mathbf{x}]$ is independent of $\mathbf{v}$, the critical
argument in the first proof of HK1 still holds and thus two potentials that
share a common $\mathbf{x}$ in the ground state will also share a common
ground-state wave function or density matrix. Consequently, HK1 is secured in
any such formulation of DFT, while the situation for HK2 quite generally is
more problematic. Even if the coupling between $\mathbf{v}$ and $\mathbf{x}$
that enters the energy functional in Eq. (24) is linear like in
$f[\mathbf{v},\mathbf{x}]=\langle\mathbf{v},\mathbf{x}\rangle$, the critical
step (8) in the proof of HK2 will involve more degrees-of-freedom on the
potential side and the argument may fail.
In the literature, the presented situation with linear coupling corresponds to
what Schönhammer, Gunnarsson, and Noack [64] call $\\{a\\}$-functional theory.
Similarly, Higuchi and Higuchi [65; 66] allow for a more general choice of
basic variables in DFT next to the usual density. Xu _et al._ [67] derived
conditions that need to be fulfilled to also have a HK2 in such a general
setting. One can then try and extend DFT and the Kohn–Sham scheme
systematically to predict further system parameters, if good approximative
functionals can be found.
A first example would be the spin-resolved functional that has the usual one-
particle density $\rho=\rho_{\uparrow}+\rho_{\downarrow}$ and the spin-density
$\rho_{\uparrow}-\rho_{\downarrow}$ as basic variables,
$\mathbf{x}=(\rho_{\uparrow}+\rho_{\downarrow},\rho_{\uparrow}-\rho_{\downarrow})$.
An alternative possible choice would clearly be
$\mathbf{x}=(\rho_{\uparrow},\rho_{\downarrow})$ [18, Section 8.1]. The energy
functional is $E[v]=\inf_{\mathbf{x}}\\{\tilde{F}[\mathbf{x}]+\langle
v,\rho\rangle\\}$, with $v$ just the usual scalar potential that couples to
the one-particle density $\rho$. The involved spaces for densities and
potentials are not dual in this example, since they involve a different number
of components. But by choosing an $F_{\mathrm{Hxc}}[\mathbf{x}]$ that depends
on the spin-resolved density, the Hxc-potential as its derivative (and with it
the effective potential of the Kohn–Sham system) must be from the dual space
of $\mathbf{x}$ and thus include components that act on the different spin-
components individually.
A second example is CDFT and its variants that will be thoroughly discussed in
Part II of this review. The paramagnetic current density of a given state
$\psi\in\mathcal{W}$ is defined as
$\mathbf{j}^{\mathrm{p}}_{\psi}(\mathbf{r}_{1})=N\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3(N-1)}}\mathrm{Im}\left\\{\psi^{*}\nabla_{1}\psi\right\\}\,\mathrm{d}\mathbf{r}_{\perp}.$
Then the amended density quantity is
$\mathbf{x}=(\rho,\mathbf{j}^{\mathrm{p}})$ which couples linearly to
$\mathbf{v}=(v+\frac{1}{2}|\mathbf{A}|^{2},\mathbf{A})$ [68]. Since by this
the potential-energy contribution amounts exactly to the linear pairing
$f[\mathbf{v},\mathbf{x}]=\langle\mathbf{v},\mathbf{x}\rangle$ that allows to
define a potential-independent constrained-search functional, HK1 holds.
This means one can continue along the lines started in this work and try to
generalize many concepts and results from above to such extended DFTs. This
includes the definition of representable densities (Section III), different
functionals (Section VI), functional differentiability (Section VII), setting
up a Kohn–Sham scheme (Section VIII), as well as regularisation (Section IX),
since also there the existence of a full HK theorem was hardly ever assumed.
$\begin{array}[]{ccccccc}&\lx@intercol\hfil\text{$F$
differentiable}\hfil\lx@intercol&&&\text{$F_{\varepsilon}$ from}\\\
&\lx@intercol\hfil\text{at $\rho\in
v\text{-}\mathsf{rep}$}\hfil\lx@intercol&&&\text{regularization}\\\
&&\Downarrow&&&&\Downarrow\\\
-\underline{\partial}F[\rho]=\\{v+c\\}&\hbox{\multirowsetup$\Longleftrightarrow$}&\hbox{\multirowsetup
HK}&\hbox{\multirowsetup$\Longleftrightarrow$}&\text{unique
density-}&\hbox{\multirowsetup$\Longrightarrow$}&\text{unique
\emph{quasi}density-}\\\ \text{at $\rho\in
v\text{-}\mathsf{rep}$}&&&&\text{potential mapping}&&\text{potential
mapping}\\\ &&\Updownarrow\\\
&\lx@intercol\hfil$\downbracefill$\hfil\lx@intercol\\\
&\text{HK1}&&\text{HK2}\\\ &\Uparrow&&\Uparrow\\\
\lx@intercol\hfil$\text{potential-
independent}$\hfil\lx@intercol&\lx@intercol$\text{UCP
result}$\hfil\lx@intercol\\\ \lx@intercol\hfil$\text{constrained-search
fctl.}$\hfil\lx@intercol\end{array}$ Figure 2: Logical implications between
the different statements relating to a “unique density-potential mapping” and
the HK theorem in standard DFT.
## XI Summary
We will give a brief summary of the structure of the density-potential mapping
and its relation to the HK theorem. Following the last section on abstract
DFTs, at least the HK1 result does not only hold for standard DFT (that maps
one-particle densities to scalar potentials), but it holds for all variants of
DFTs that offer the required structures. This will be especially useful with
foresight towards CDFT, the topic of the second part of this review.
In standard DFT, with a setting that yields the unique-continuation property
that in turn prevents the ground-state density from being zero on a set of
non-zero measure (Section V) and due to the simple relation (8) in the proof
of HK2, a full HK result can be established. In any higher DFT this proof
strategy potentially fails. The status of HK1, on the other hand, is much less
critical, since this result holds automatically whenever a _potential-
independent_ (“universal”) constrained-search functional can be set up. But
also in cases where the constrained-search functional depends on the external
potential, a valid statement like in the HK theorem, that two potentials that
share a common ground-state density are equal up to gauge changes, is still
possible in general. The more general way how to think and talk about a HK
result is by calling it a “unique density-potential mapping” and we explained
how such a mapping can be established as the subdifferential of the density
functional $F$ at $v$-representable densities. If the potentials in the
resulting subdifferential are equal up to a gauge transformation, then this is
just the HK result again. Assuming full differentiability of $F$ implies a
one-element subdifferential, so there would not even be any room for gauge
changes, and a unique density-potential mapping would be the result once more.
This property of differentiability of the density functional $F$ is desirable
also in the context of Kohn–Sham theory in order to be able to link the
functional $F_{\mathrm{Hxc}}$ to the Hxc potential like in Eq. (21).
But since differentiability is _not_ a property of the usual DFTs, a
regularization strategy was devised and briefly explained in Section IX. This
yields a unique _quasi_ density-potential mapping, where quasidensities are
actually mixtures between ground-state densities and their potentials. The
mixing parameter $\varepsilon$ could be set to zero to retrieve the
unregularized theory together with the problem of non-differentiability. The
whole structure is laid out diagrammatically in Figure 2.
## XII Outlook
In this outlook, we first want to collect the problems that still remain open
within the foundations of standard DFT and that will surely be the topic in
upcoming works. Considering Lieb’s mathematical formulation of DFT, summarized
above in Section III and VI, there are two main issues. Firstly, HK2 is
guaranteed only for eigenstates that are non-zero almost everywhere, a
property that is secured by the UCP explained in Section V. But the potential
space required for this does not cover all potentials from the Lieb setting
and a sufficiently general UCP result is not available to date. Secondly, the
issue of $v$-representability, explained in Section III, still remains open.
While regularization as described in Section IX formally allows us to
circumvent this problem, it has not yet been put to practical use. Since the
overlap between interacting and non-interacting $v$-representability is poorly
understood, this has direct implications for Kohn–Sham theory. But even with
$v$-representability assumed, convergence of the Kohn–Sham self-consistent
field iterations in the standard setting is still an open problem. Both
issues, availability of UCP and $v$-representability, relate to the function
spaces for densities and potentials. Possibly, with a more refined choice of
these spaces, full $v$-representability or even differentiability of $F$ might
be achievable. However, it also cannot be ruled out that non-differentiability
is fundamental to DFT.
This non-differentiability of $F$, that has been repeatedly stressed in this
work, implies that the exchange-correlation potential cannot be found as a
functional derivative with respect to the density, as it is usually assumed in
standard DFT. Orbital-dependent functionals [69] can be formally viewed as
relying on the HK1 map $\rho\mapsto\phi$ to obtain the Kohn–Sham wave function
from a density. Non-differentiability of $F[\rho]$ might then be represented
in the noninteracting wave function $\phi[\rho]$, which may benefit the
functional approximations if they rely directly on the Kohn–Sham orbitals. The
lack of differentiability also favors approaches based on forces instead of
energies, as mentioned in Section VIII. However, practical functionals that
are derived from this approach remain unexplored and there is still a
dependence on $v$-representability.
It is interesting to note which useful structures of DFT carry over to
“higher” density-functional theories, and in Part II we will discuss density-
functional theory for systems involving magnetic fields. While one of its
flavours, paramagnetic CDFT, already briefly discussed in Section X, still
allows for a constrained-search functional (HK1), the realization of a full
density-potential mapping is highly problematic. For this reason, in the
classical formulation of paramagnetic CDFT [70] the HK2 result that different
potentials lead to different ground states was just tacitly assumed with the
words: “Let $\psi$ and $\psi^{\prime}$ be the two different ground states
corresponding to the two sets of fields [$(v,\mathbf{A})$ and
$(v^{\prime},\mathbf{A}^{\prime})$].” Later, Capelle and Vignale [7] even
found counterexamples to HK2 which shows that a density-potential mapping
cannot be constructed in paramagnetic CDFT. But this clearly does not mean
that in different versions of CDFT the density-potential mapping is impossible
to achieve in general. A formulation utilizing the total current will be
studied as well, but here the constrained-search functional would depend on
$\mathbf{A}$ and thus HK1 is not available in the fashion as it was presented
here. So while for paramagnetic CDFT the HK2 fails, for total (physical) CDFT
already HK1 does not hold. Overall, the existence of a well-defined density-
potential mapping in CDFT is still an open issue that will be considered in
the second part of this review.
## Acknowledgement
EIT, MAC and AL thank the Research Council of Norway (RCN) under CoE
(Hylleraas Centre) Grant No. 262695, for AL and MAC also CCerror Grant No.
287906 and for EIT also “Magnetic Chemistry” Grant No. 287950, and MR
acknowledges the Cluster of Excellence “CUI: Advanced Imaging of Matter” of
the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, project ID 390715994. AL
and MAC was also supported by the ERC through StG REGAL under agreement No.
101041487. The authors thank Centre for Advanced Studies (CAS) in Oslo, since
this work includes insights gathered at the YoungCAS workshop “Do Electron
Current Densities Determine All There Is to Know?”, held July 9-13, 2018, in
Oslo, Norway.
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|
longtable
# Understanding Complex Patterns in Social, Geographic, and Economic
Inequities in COVID-19 Mortality at the County Level in the US Using
Generalized Additive Models
Christian Testa1,*
(November 29, 2022)
###### Abstract
I present three types of applications of generalized additive models (GAMs) to
COVID-19 mortality rates in the US for the purpose of advancing methods to
document inequities with respect to which communities suffered
disproportionate COVID-19 mortality rates at specific times during the first
three years of the pandemic. First, GAMs can be used to describe the changing
relationship between COVID-19 mortality and county-level covariates
(sociodemographic, economic, and political metrics) over time. Second, GAMs
can be used to perform spatiotemporal smoothing that pools information over
time and space to address statistical instability due to small population
counts or stochasticity resulting in a smooth, dynamic latent risk surface
summarizing the mortality risk associated with geographic locations over time.
Third, estimation of COVID-19 mortality associations with county-level
covariates conditional on a smooth spatiotemporal risk surface allows for more
rigorous consideration of how socio-environmental contexts and policies may
have impacted COVID-19 mortality. Each of these approaches provides a valuable
perspective to documenting inequities in COVID-19 mortality by addressing the
question of which populations have suffered the worst burden of COVID-19
mortality taking into account the nonlinear spatial, temporal, and social
patterning of disease.
Abbreviations used: United States (US), Coronavirus Disease 2019 (COVID-19),
Generalized Additive Model (GAM), Centers for Disease Control and Prevention
(CDC), Index of Concentration at the Extremes (ICE), Confidence Interval (CI)
1 Department of Social and Behavioral Sciences, Harvard T.H. Chan School of
Public Health
* Correspondence: Christian Testa<EMAIL_ADDRESS>
## Introduction
As we enter the third winter with the novel coronavirus disease COVID-19 in
the United States, evidence documenting the intense disparities in COVID-19
mortality rates comparing socially advantaged and disadvantaged populations
continues to mount. Eliminating inequities in health outcomes has been stated
as a major policy goal of the Biden administration (The White House, 2021,
2022), representing a revitalized commitment to health equity and underlining
the importance of adequate data reporting systems that report timely estimates
of prevalent health inequities. To this end, I use generalized additive models
(GAMs) as a flexible regression framework to illustrate the evolving roles and
relationships sociodemographic, geographic, and economic conditions have with
respect to trends in COVID-19 mortality. The code, data, and documentation
necessary to reproduce the analyses contained in this paper are online and
free to access at https://github.com/ctesta01/covid.gradient.estimation.
#### Background
Having passed over 1 million COVID-19 deaths in the United States in 2022
(Donovan, 2022), and facing uncertain prospects for the third COVID-19 winter
looming even as new iterations on the COVID-19 vaccines become available, it
remains critical that inequities in COVID-19 outcomes are documented and
analyzed to reckon with the unjust and unfair burden of preventable illness.
Even though the first vaccines were granted emergency use authorization by the
U.S. Food and Drug Administration in 2020 (Mayo Clinic, 2022), with the first
shots going in arms in December 2020, COVID-19 is still continuing to cause
hundreds of deaths a day in the US in the fall of 2022 (“United States COVID -
Coronavirus Statistics - Worldometer,” 2022). The new bivalent vaccines
released at the end of August 2022 contain mRNA sequences from both the
original strain as well as the recently emergent BA.4 and BA.5 lineages in an
effort to make the nation’s immunity more up-to-date and robust against the
myriad of phylogenetic directions the COVID-19 virus is evolving to explore
(Office of the Commissioner, 2022). Despite the updated bivalent boosters
representing a significant step forward in prevention strategy, less than 4%
of eligible Americans had taken the booster in the first month after it became
available (Bendix, 2022; Lambert, 2022). As such, and with an enduring history
of inequities in health care access in the US (Bailey et al., 2021; Blendon et
al., 2002; Carpenter, 2021; Chrisler et al., 2016; Feldman et al., 2021;
Okonkwo et al., 2021; Ortega & Roby, 2021; Rapp et al., 2022; Whitehead et
al., 2016), it is clear that without further intervention not all communities
will be equally able to benefit from the updated vaccines and inequities in
COVID-19 illness and mortality may persist despite the technological
innovations in vaccine technology.
#### The Role of Geography in COVID-19
Figure 1: Estimates of monthly COVID-19 mortality rates per 100,000 person-
years by county organized by Census Division. For each division, the median
trendline and quantile ranges are shown weighted by county population size.
Prior literature has demonstrated that the spread and impact of COVID-19 has
varied geography over time reflecting dynamics in the timing of introduction
and transmission events. Methods employed to highlight the geographic
patterning in COVID-19 outcomes have included quantile regression (Sigler et
al., 2021), Besag-York-Molli'e mixed models (Whittle & Diaz-Artiles, 2020),
spatial cluster analysis (Sugg et al., 2021), geographically weighted
regression (Mollalo et al., 2020; Park et al., 2021), and others.
Figure 1 shows the monthly COVID-19 mortality rates for the counties grouped
within each of the nine U.S. Census Divisions (U.S. Department of Commerce
Economics and Statistics Administration & U.S. Census Bureau, 2000). The
figure summarizes each division’s median mortality rates weighted by county
population size. Notably, the mortality associated with the early surge of
cases starting in New York City and spreading through New York, New Jersey,
and Massachusetts is visible in the Middle Atlantic and New England division
figures. The figure also illustrates how the first peak in the mortality time-
series for states in the Midwest (West North Central, East North Central)
occurred later, in late 2020 and going into early 2021.
In the US context, one of the key aspects to the geographic story of
COVID-19’s spread and diffusion was the early surge of cases and epicenter in
New York City during March 2020 (Thompson, 2020) followed by subsequent waves
of cases in the South and Midwest (Glenza, 2020; Scott, 2020; Shumaker & Wu,
2020). As Park et al. stated summarized the trends in the US from March 2020
to May 2021, “hot spots have shifted from densely populated cities and the
states with a high percentage of socially vulnerable individuals to the states
with relatively relaxed social distancing requirements, and then to the states
with low vaccination rates” (2021).
When considering the drivers of the COVID-19 pandemic, it’s necessary to note
that geography and social conditions are inextricably linked. In July 2021,
the CDC reported that “the COVID-19 cumulative death rate in non-metropolitan
areas has exceeded that of metropolitan areas since December 2020,” noting
that of the approximately 1/5th of Americans who live in rural areas, many
“are considered highly vulnerable according to CDC’s Social Vulnerability
Index (SVI), which includes factors such as housing, transportation,
socioeconomic status, race, and ethnicity” (CDC, 2021). Moreover, rural
communities often have lower health insurance rates, higher disability rates,
older populations, and limited access to health care. One of CDC’s Morbidity
and Mortality Weekly Reports found that vaccination against COVID-19 was lower
in rural communities than in urban communities between December 2020 and April
2021 (Murthy, 2021).
#### The Social Determinants of COVID-19 Mortality
Even since the beginning of the COVID-19 outbreak in the US, data reflected
sharp inequities in mortality rates. During January 22nd to May 5th 2020,
county COVID-19 mortality rates were 4.94 (95% CI 4.78, 5.09) times higher in
counties in the highest quintile of percent People of Color (61%-100%)
compared to counties in the lowest quintile of percent People of Color
(0%-17.2%) (Chen & Krieger, 2021). This was not wholly unanticipated: as
COVID-19 was beginning to take off in the US, some were already calling
attention to the fact that COVID-19 threatened to exacerbate existing
disparities (Kim et al., 2020). Kim, Marrast, and Conigliaro noted at least
three structural barriers in COVID-19 prevention and care: 1) originally
requiring residents to have a doctor’s prescription for a COVID-19 test
reduced the opportunity for healthcare for People of Color as they are less
likely to have a primary care provider; 2) drive-through testing made testing
disadvantaged those who rely on public transportation; and 3) quarantining at
home while waiting the 7-10 days originally required for COVID-19 test results
to come back posed an economic and social challenge that many in already
financially difficult situations may not have been able to take on (2020).
Others noted yet more reasons why COVID-19 threatened to worsen an already
inequitable healthcare landscape in the US: in particular, those who reside in
prisons and jails, immigrants and undocumented people, people with
disabilities, and people experiencing homeless all face additional challenges
in seeking and getting the healthcare they deserve (Okonkwo et al., 2021).
Even though stay-at-home orders designed to mitigate spread that were
prevalent in many states (Moreland et al., 2020), workers functioning in
capacities essential to the functioning of critical infrastructure operations
(later termed “essential workers”) were exposed to heightened risk of COVID-19
transmission (Hanage et al., 2020; National Bureau of Economic Research, 2021;
National Conference of State Legislatures, 2021; The Lancet, 2020; Wei et al.,
2022). It’s clear that those with more structurally enfranchised privileges
have been more able to mitigate their risk of negative health outcomes
associated with COVID-19 — during February 1 to April 1, 2020, New York City
residents who lived in more affluent neighborhoods were more likely to have
left the city, while New York City residents from more Black and Hispanic
neighborhoods were more likely to continue working (Coven & Gupta, 2020).
> Contrary to the oft used phrase that the ‘virus does not discriminate’, the
> data presented here suggest that this virus, as many other infectious
> diseases, has the greatest implications for the most vulnerable people. The
> intersections between health and human rights are clear—the health of a
> society and vulnerability to a pandemic are directly related to its human
> rights track record for those who are marginalised. (Okonkwo et al., 2021)
When vaccines became available, vaccine appointments were often only available
to be scheduled through online web-portals contributing to the inequities
between those who had internet access and technological literacy and those who
didn’t (Press et al., 2021). Vaccination sites have not been equally
distributed and areas determined to be vaccine deserts have been found to have
disproportionately more Black and Hispanic residents (Rader et al., 2021). In
fact, healthcare facilities in counties with higher Black composition had 32%
(95% CI 14%-47%) lower odds of serving as vaccine sites (Hernandez et al.,
2022).
What vaccination has been administered hasn’t suddenly erased the unequal
burden of COVID-19 either; in August 2022 the New York Times was reporting
“Black death rates at this winter’s peak were greater than those of white
people by 34 percent in rural areas, 40 percent in small or medium cities and
57 percent in big cities and their suburbs” (Goldstein, 2022). As COVID-19
case and mortality rates have waxed and waned, the inequities have widen and
shrunken, often with racial/ethnic inequities growing during times when
COVID-19 rates have surged (Hill & Artiga, 2022). During 2022, the age-
standardized COVID-19 mortality rates for white people have, at times, been
slightly higher than those of Black and Hispanic people, predominantly because
the mortality rate among white people has increased (Johnson & Keating, 2022).
It’s important to note that white COVID-19 mortality rates overtaking the
Black mortality rates does not imply that equity has been established: neither
does this undo the cumulative impact of mortality (which has been twice as
high for Black people compared to white people (Hill & Artiga, 2022)), nor
does it imply the underlying systemic barriers to equity have been overturned
(Del Rios et al., 2022). As Del Rios, Chomilo, and Lewis note, instead,
COVID-19 leaves in its wake more years of life expectancy lost, wages lost,
and degradation of mental health in Communities of Color.
## Methods
### Data Sources
The following variables were retrieved at the county level:
* •
Counts of COVID-19 deaths (The New York Times, 2021).
* •
Population size estimates for 2020 from the U.S. Census (US Census Bureau,
2021).
* •
Median age, median household income, racial/ethnic composition, population
density, percent below the federal poverty line, and number of households with
high ($100k+)/low (<$25k) household income by racial/ethnic group from the
2014-2019 5-year American Community Survey (US Census Bureau, 2020) through
the tidyverse R package (Walker & Herman, 2022).
* •
Votes cast in the 2020 presidential election (MIT Election Data and Science
Lab, 2022)
### Generalized Additive Models
Generalized additive models (GAMs) improve upon generalized linear models by
allowing for the fitting of smooth functions that transform right-hand-side
variables. This is a convenient means to account for nonlinear relationships
between the outcome and predictor variables. Whereas a generalized linear
model may look like
$g(\mu_{i})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\beta_{3}x_{i3}...$
a generalized additive model could look like
$g(\mu_{i})=\mathbf{A}_{i}\mathbf{\theta}+f_{1}(x_{i1})+f_{2}(x_{i2})+f_{3}(x_{i3},x_{i4})+\dots$
where the expected value of the outcome is given as
$\mu_{i}\equiv\mathbb{E}(Y_{i})$, $\mathbf{A}_{i}$ is a row of the model
matrix for any strictly non-parametric model components, $\mathbf{\theta}$ is
the corresponding parameter vector, and the $f_{j}$ are smooth functions of
the covariates $x_{k}$ (S. N. Wood, 2017).
Figure 2: A demonstration of fitting regression models using B-spline basis
functions. Part A. shows the simulated dataset. Part B. shows the B-spline
basis functions. Part C. shows the weighting of the B-spline basis functions
and their weighted sum with an intercept added to create the regression model
fit to the data.
The smooth functions estimated as part of fitting a GAM are constructed using
spline basis functions. These spline basis functions allow for the smooth
interpolation of trends in the data allowing for the incorporation of
nonlinearity. To avoid overfitting the data, a penalty term is introduced that
controls the degree of “wiggliness” or smoothness and this penalty term is fit
using generalized cross validation. B-splines are one kind of spline basis
function that is commonly used and are especially popular because of their
property that they are non-negative on only a finite interval.
While a variety of spline-based approaches exist (cubic splines, B- and
P-splines, thin-plate splines, etc.), our particular application setting
warrants the use of tensor-product smooths because we require anisotropic
penalization. That is to say, when we seek to create spatiotemporally smoothed
model estimates, it’s inappropriate to assume that the amount of smoothing
across space should be the same as the amount of smoothing across time because
they’re in fundamentally different units (space being measured in Cartesian
coordinates and time being measured in years, months, days, etc.). More
details about tensor-product smooths are available in Simon Wood’s
_Generalized Additive Models - An Introduction with R_ (2017).
### Overdispersion and Event Modeling
Above and beyond using the GAM framework to allow for flexible, nonlinear
relationships between our observed county-level variables and COVID-19
mortality, we must have a model specification that agrees with the data
generating process. In our case, the data generated are counts of deaths per
population, and the class of models most suited to represent counts of events
are Poisson, quasi-Poisson, and negative binomial models. Here we’ve chosen to
use the negative binomial model as it accounts for overdispersion and has the
intuitive interpretation of a Poisson model with gamma distributed underlying
rate parameter (Gelman et al., 2013).
A negative binomial model in the context of modeling count data can be written
$y_{i}\sim\text{NegativeBinomial}(u_{i}\theta_{i},\phi),$
where $y_{i}$ are the counts observed, $u_{i}$ is the “exposure”, $\theta_{i}$
are the rates, and $\phi$ determines the amount of overdispersion. The rates
are modeled as $\theta_{i}=e^{X_{i}\beta}$ where $X_{i}$ are the observed
covariates and $\beta$ are the coefficients on the covariates corresponding to
log rate ratios. The logarithm of the exposure, $\log(u_{i})$ is often called
(and later herein referenced as) the offset. In epidemiological contexts, the
offset is often representative of the amount of person-time during which
observations were recorded. Whereas the Poisson model holds that
$\text{var}(y)=\mu$ where $\mu$ is the average rate, the negative binomial
model instead assumes that $\text{var}(y)=\mu+\mu^{2}/\phi$. The Poisson model
is a special case of the negative binomial model when $\phi\to\infty$.
An alternative and equivalent formulation of the negative binomial that is
commonly used makes the connection to the Poisson model even more clear:
instead of $\theta_{i}$ and $\phi$, using $\alpha$ and $\beta$,
$y\sim\text{NegativeBinomial}(\alpha,\beta),\text{ and}$
$\text{NegativeBinomial}(y|\alpha,\beta)=\int\text{Poisson}(y|\theta)\text{Gamma}(\theta|\alpha,\beta)d\theta.$
Introductions to and additional exposition on the negative binomial model,
especially in the context of modeling counts of events outcome data, are
available in _Bayesian Data Analysis_ and _Regression and Other Stories_
(Gelman et al., 2020, 2013).
### Variables of Interest
The following variables have been included as covariates of interest:
* •
Median Age
* •
Population Density per Square Mile
* •
Median Household Income
* •
Proportion in Poverty
* •
the Index of Concentration at the Extremes for Racialized Economic Segregation
* •
Political Lean in the 2020 Election (1 = 100% Republican votes, -1 = 100%
Democratic Votes)
##### Index of Concentration at the Extremes for Racialized Economic
Segregation
The Index of Concentration at the Extremes (ICE) is a measure which describes
how concentrated a given area’s population is in terms of the extreme ends of
privilege and marginalization (Krieger et al., 2016). In general, the ICE
measure is formulated as
$\text{ICE}=\frac{\text{Number of People in Most Privileged
Category}-\text{Number of People in Least Privileged Category}}{\text{Total
Population}}$
Applying the ICE approach to a specific context involves defining the axes of
privilege of interest. In this case, data on racialized economic segregation
are used from the US Census American Community Survey on white households
earning more than $100,000 a year (the most privileged group) or households of
People of Color earning less than $25,000 a year (the least privileged group).
This variable is referred to throughout as the ICE for racialized economic
segregation, or ICEraceinc in the code.
Compared with the Gini coefficient which is one of the most popular methods
for summarizing area-level rates of inequities, the ICE has the advantage that
it is suitable for describing inequities at smaller area levels (Krieger et
al., 2016). While the Gini coefficient measures within-area dissimilarity (as
in, for example, how unequal wealth is distributed within a county), the ICE
measure establishes where on a spectrum a given county’s population falls
allowing for comparison across counties. The Gini coefficient suffers from the
fact that areas which are made up of relatively homogeneous populations will
appear as having low within-area inequality (and therefore low Gini
coefficient). Instead, the ICE measures how much of the population is either
privileged or not. The Gini coefficient remains useful for reporting on the
degree of inequity in larger areal units (like countries, states, and
regions), but at smaller areal units (like counties, ZIP codes, census tracts)
can be more difficult to interpret and compare. Maps of ICE measure can
elucidate what spatial social segregation and polarization exist, and the ICE
for racialized economic segregation has been repeatedly and significantly
associated with COVID-19 outcomes (Brown et al., 2021; Chen & Krieger, 2021;
Eichenbaum & Tate, 2022; Hanage et al., 2020; Krieger et al., 2022; Saha &
Feldman, 2020; Sonderlund et al., 2022).
##### Political Lean
Political lean has been associated with COVID-19 mortality in numerous
studies, with plausible mechanisms explaining the association including
differences in non-pharmaceutical intervention uptake (mask usage, social
distancing, quarantining), differences in rates of vaccination, differences in
political leadership’s messaging, resource allocation, and the adoption of
policy interventions (Gonzalez et al., 2021; Grossman et al., 2020; Kaashoek
et al., 2021; Krieger et al., 2022; Leonhardt, 2021).
## Results
### Application 1: Non-Spatial Covariate Effects Over Time
The GAM models shown in Figure 3 are fit with the following structure using
the gam function from the mgcv package (S. Wood, 2022):
gam(
formula = deaths ~ s(median_age) + te(covariate, date, d=c(1,1)), # regression
formula
offset = log(popsize/1e5/12), # our offset represents the person-time
data = df, # our data-set of county-level observations
family = nb(link='log') # indicates negative binomial family and a log-link
function
)
The formula used puts a one-dimensional smoothing spline on median age to
represent a nonlinear age-effect and a two-dimensional tensor-product smooth
on the interaction between the given covariate and the date. The d=c(1,1)
argument provides the instruction necessary to consider covariate and date as
being on separate scales and therefore to fit the tensor-product smooth with
anisotropy — that is, to allow for independent amounts of scaling in the
dimensions of the covariate and time. The offset used structures the
regression to estimate rates in units of person-time per 100,000 person-years.
Since the death counts are aggregated to the monthly level, the person-time in
units of 100,000 person-years are calculated by taking each county’s
population size, dividing by 100,000, and dividing by 12 (for the 12 months in
a year).
The above structure is used to estimate models for our different covariate
variables of interest: median income, percent in poverty, the ICE for
racialized economic segregation, political lean. The model presenting median
age treats median age as the main covariate of interest including it as the
te(covariate, date, d=c(1,1)) and dropping the s(median_age) term which
otherwise becomes redundant. Results of these models are summarized in Figure
3 where the additional COVID-19 mortality associated with each covariate is
visualized.
Figure 3: Additional COVID-19 mortality associated with covariates over time.
A) age and time, B) log (base 10) population density and time, C) median
income and time, D) proportion in poverty and time, E) ICEraceinc and time, F)
political lean and time. In panels B-F the effect of age is marginalized out
using the median age in the US, 38.8 (US Census Bureau, 2022).
Likelihood ratio tests confirmed that models with covariates interacted with
time had significantly lower residual deviance ($p\leq 2.2e^{-16}$ for all
models) compared to models only including a spline term on median age and the
given covariate not interacted with time. The model interacting median age and
time was compared to a model with a spline for median age alone. Akaike
Information Criteria values were also lower for all models compared to models
that did not interact the covariates with time.
A more complex non-spatial application of GAMs to describe the distribution of
COVID-19 mortality in the US over time is to consider the associations of
mortality with three-way interactions of time and two covariates taken
together. In the following example, the formula used is deaths ~ te(date,
ICEraceinc, median_age, d=c(1,1,1)). Again, the d argument specifying the
marginal basis dimensions is used indicate that each of the date, ICEraceinc,
and median_age measures are in different units and should not be smoothed
assuming that a one unit difference in one variable is comparable to a one
unit difference in another variable. This approach is represented in Figure 4.
Figure 4: The changing interacted effect of the ICE for racialized economic
segregation and age over time. A) May 2020, B) November 2020, C) May 2021, D)
October 2021, E) February 2022, F) August 2022. An animated version is
available online at
https://github.com/ctesta01/covid.gradient.estimation/blob/main/analysis/05_two_variables_at_a_time/animation_ICEraceinc_age/readme.md
### Application 2: Spatiotemporal Smoothing
By fitting GAMs with a tensor-product smoothing term on latitude, longitude,
and time we can estimate a spatiotemporally smoothed trend in COVID-19
mortality. To do this, the GAM is constructed similarly as in Application 1
but with the smoothing term specified as te(latitude, longitude, time,
d=c(2,1)) where d=c(2,1) indicates that latitude and longitude share the same
dimensions (i.e., both are spatial and in units of degrees) while the time
data are in separate units. Note that results presented in Application 2 and 3
are based on county-level data from the contiguous US excluding Alaska and
Hawaii.
Using GAMs to present spatiotemporally smoothed estimates of COVID-19
mortality allows for the synthesis of trends over time. Whereas the raw rates
of COVID-19 mortality are noisy due to 72% of the land area in the US being
classified as rural and low population density (Health Resources & Services
Administration, 2022), the spatiotemporally smoothed GAM estimates use
population weighting via the offset specified to estimate a latent surface
that represents localized mortality rate averages in spatial coordinates and
in time.
Figure 5 shows the difference between crude mortality rates and spatially
smoothed mortality rates in January 2022 to illustrate the level of noise
present in raw rates and how spatially smoothing synthesizes local geographic
patterns into trends that can be meaningfully interpreted as local area
mortality risk levels with information information pooled across nearby county
rates.
Figure 6 shows the results from a spatiotemporally smoothed model in select
months to highlight how spatiotemporal smoothing can yield results that
synthesize trends in space and time.
Figure 5: Comparison of crude vs. smoothed mortality rates. A) Crude mortality
rates per 100,000 person-years in January 2022. B) Smoothed mortality rates
from a GAM applied to mortality rates from January 2022. Figure 6:
Spatiotemporally smoothed COVID-19 mortality estimates from a GAM fit to data
from March 2020-August 2022. Panels show months selected to highlight the
changing spatial patterns of COVID-19 mortality risk over time. An animated
version is available online at
https://github.com/ctesta01/covid.gradient.estimation/blob/main/analysis/09_spatiotemporal_models/animation/readme.md
### Application 3: Estimating Covariate Effects Adjusted for Spatiotemporal
Autocorrelation
The final application of GAMs presented here is to estimate the effects of
covariates after adjusting for a spatiotemporally smoothed latent risk
surface. This has the interpretation of asking what are the changes to
COVID-19 mortality rates associated with each covariate after taking into
account regional geographic patterns in COVID-19 mortality over time. In
particular, this is relevant for understanding what county-level measures are
associated with elevated COVID-19 mortality even after adjusting for where
COVID-19 rates were locally elevated or depressed due to variation in the
timing of local introduction, transmission, and diffusion.
Figure 7: The spatial autocorrelation of COVID-19 rates in January 2022. Panel
A shows the graph of US counties connected according to which are neighbors.
Panel B shows a scatterplot depicting each US counties’ COVID-19 mortality
rate (x-axis) compared to the average of its neighbors’ COVID-19 mortality
rates (y-axis). A regression line shows the association between counties’
COVID-19 rates and their neighboring counties’ COVID-19 mortality rates.
Dotted lines indicated the average for US county-level crude COVID-19
mortality rates (vertical) and for the average of each counties’ neighboring
counties’ COVID-19 mortality rates (horizontal). Figure 8: The median (black)
and quantiles (shaded bands) of temporal autocorrelation of counties’ COVID-19
mortality rates over time during April 2020 — August 2022.
To motivate the need for and application of spatiotemporal smoothing in
assessing the associated changes in COVID-19 mortality with county-level
covariates, Figure 7 presents the Moran’s I diagnostic plot for January 2022
which summarizes the amount of autocorrelation between counties’ COVID-19
mortality rates and the average of each counties’ neighboring counties’
COVID-19 mortality rates. The observed autocorrelation indicates that there is
an association between counties’ COVID-19 rates and their neighboring
counties’ COVID-19 mortality rates, implying that without taking this
patterning into account, regression models that do not explicitly model the
effect of spatial autocorrelation may be biased due to inappropriately
assuming that county-level data are independent of one another. Second, while
the Moran’s I plot demonstrates the spatial-correlation present in the data at
a single time-point, the following Figure 8 shows the tendency of county
COVID-19 observations to be autocorrelated between subsequent months.
Figure 9: Estimates of increases in COVID-19 mortality associated with county-
level covariates without (first) and with (second) adjustment for
spatiotemporal autocorrelation. Panels A and B show COVID-19 mortality
increases associated with varying levels of median age over time; C and D show
associations with log population density; E and F median income; G and H for
proportion in poverty; I and J the ICE for racialized economic segregation; K
and L political lean.
Figure 9 shows the additional COVID-19 mortality associated with each
covariate over time without and with adjustment for spatiotemporal trends. The
models for population density and median income show particularly marked
changes with the spike at high population density and high median incomes
disappearing after adjusting for a spatiotemporally smoothed term. This likely
reflects the early emergence and surge of COVID-19 in the greater New York
City area as being predominantly a feature of the combined effect of human
geography and infectious disease dynamics where the earliest introductions of
COVID-19 were made to dense social networks in the US. Other effects remain
qualitatively changed little, suggesting that the trends in COVID-19 mortality
associated with these county-level measures remains durable when taking into
account spatiotemporal autocorrelation.
## Discussion
This paper presents three applications of GAMs to county-level COVID-19
mortality data from the US during March 2020 to August 2022. This paper
addresses an important need, which is to assess what measures of the
socioeconomic and geopolitical landscape are associated with increases in
COVID-19 mortality in a way that takes into account how infectious disease
dynamics are locally correlated in space and time. However, the approaches
outlined in this paper are not without several limitations which include (at
least) the following: the potential for cross-level confounding, lack of
individual level data, an inability to include all possible metrics, further
improvements to the methods used may be warranted, and an inability to make
claims about the causal relationships between the variables investigated.
In analyses that use area-level aggregated data, the ecologic fallacy refers
to the possibility that associations between area-level measures and outcomes
may not reflect the associations that would be observed if analyzing
individual level data on the same measures and outcomes as a result of cross-
level confounding (Greenland, 2001). It’s important to note that some of the
variables in this analysis are included to contextualize the places in which
people live, reside, and work and area-based social metrics should be included
even in individual-level analyses so as to accurately capture the effects of
both individual-level risk factors and risk factors that stem from the
contexts in which people exist (Testa et al., 2022).
One particular improvement that could be made in future work is to address age
adjustment by using direct or indirect age standardization (Anderson et al.,
1998) or to create age-group stratified models. In the applications presented
here, county-level median age is adjusted for in order to control for age
effects as age-specific data were not publicly available without substantial
levels of suppression.
Additional variables like vaccination rates, mobility data, further variables
that relate to risk factors for COVID-19, and other COVID-19 related outcomes
like case trends and hospitalizations may be worthwhile to investigate under a
similar framework. The emphasis on trends in COVID-19 mortality in the
applications in this paper was done to prioritize presenting data that are
less subject to the reporting inconsistencies in COVID-19 cases and
hospitalizations (Galaitsi et al., 2021).
Further improvements to the methods employed here could include use of soap
film smoothing splines which are more appropriate in settings where non-convex
geographic boundaries are common (as in with peninsulas and bays) (S. N. Wood
et al., 2008). Taking a critical eye to the assumptions employed reveals yet
another area of potential improvement, which is the nature of how
smoothing splines are penalized in the GAM framework. It is conceivable that
geographic changes in COVID-19 mortality rates are not equally smooth across
all regions of the US, potentially warranting rapid changes in some areas and
more smooth changes in other areas in a risk surface model. The Bayesian
wombling literature provides the means to identify areas of rapid change in
latent surface models and may pose a fruitful direction of exploration for
future efforts to understand trends in COVID-19 mortality rates in the US
(Gelfand & Banerjee, 2015).
## Conclusions
This work introduces generalized additive models (GAMs) applied to COVID-19
mortality data to document the disparities and inequities in which US counties
suffered the worst mortality outcomes and at what times. The effort to
document inequities and explain the driving mechanisms behind them is crucial
in building up the evidence necessary to enact policy changes that can
mitigate unjust and preventable harm in the future. To this end, it is
critical that efforts to model infectious disease dynamics take into account
how disease transmission and outcomes are spatiotemporally patterned,
violating any misplaced assumptions that area-level disease rates can be
treated as independent observations. By presenting novel applications of GAMs
to trends in COVID-19 mortality, this paper illustrates new ways of
understanding and visualizing the disproportionate burden some communities
have suffered by incorporating complex patterns of interaction across
socioeconomic and geopolitical variables and spatiotemporal trends.
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p
|
# Hierarchical Transformer for Survival Prediction Using Multimodality Whole
Slide Images and Genomics
Chunyuan Li, Xinliang Zhu, Jiawen Yao, Junzhou Huang Department of Computer
Science and Engineering
University of Texas at Arlington
Arlington, Texas 76013 Email: {chunyuan.li, xinliang.zhu, jiawen.yao,
<EMAIL_ADDRESS>
###### Abstract
Learning good representation of giga-pixel level whole slide pathology images
(WSI) for downstream tasks is critical. Previous studies employ multiple
instance learning (MIL) to represent WSIs as bags of sampled patches because,
for most occasions, only slide-level labels are available, and only a tiny
region of the WSI is disease-positive area. However, WSI representation
learning still remains an open problem due to: (1) patch sampling on a higher
resolution may be incapable of depicting microenvironment information such as
the relative position between the tumor cells and surrounding tissues, while
patches at lower resolution lose the fine-grained detail; (2) extracting
patches from giant WSI results in large bag size, which tremendously increases
the computational cost. To solve the problems, this paper proposes a
hierarchical-based multimodal transformer framework that learns a hierarchical
mapping between pathology images and corresponding genes. Precisely, we
randomly extract instant-level patch features from WSIs with different
magnification. Then a co-attention mapping between imaging and genomics is
learned to uncover the pairwise interaction and reduce the space complexity of
imaging features. Such early fusion makes it computationally feasible to use
MIL Transformer for the survival prediction task. Our architecture requires
fewer GPU resources compared with benchmark methods while maintaining better
WSI representation ability. We evaluate our approach on five cancer types from
the Cancer Genome Atlas database and achieved an average c-index of $0.673$,
outperforming the state-of-the-art multimodality methods.
## I Introduction
Survival analysis is an essential task in cancer study and diagnosis. Instead
of observing the biopsies under the microscope, whole slide pathology images
(WSI) are widely used to present tumor growth and morphology information.
Automatically analyzing histology provides valuable help for histologists on
precision medicine and reduces diagnosis bias. However, WSIs are typically in
multi-gigapixel level with high morphological variance, which brings a
tremendous challenge to the feature representation of WSIs.
Many solutions have been proposed to solve the representation problem of WSIs
[1, 2]. A widely used strategy is patch-based processing, where thousands of
patches are extracted from a WSI using either a non-overlap sliding window or
a random sample method [3, 4, 5]. In the clinics, the tumor area usually only
takes a tiny portion of the WSI, which means most patches sampled from the
gigapixel WSI are unrelated to survival prediction. Unfortunately, the patch-
wise annotation for WSIs is laborious and even infeasible in most situations.
Many researchers consider weakly supervised learning approaches to solve the
problem with a lack of annotations. Zhu et al. [3] proposed a two-stage
framework for survival prediction on WSIs without annotations. They applied
K-means clustering to group the patches according to visual appearance and
then aggregated patches from the selected clusters for survival prediction.
Recent approaches [6, 7, 8, 9] use multiple instance learning (MIL) [10] to
formulate the WSI representation, where each patient is considered as a bag
containing a set of instances of patches. A bag is annotated as disease-
positive if there is any disease-positive patch in the bag. Patch features are
integrated with fully connected (FC) layers for specific tasks. On the basis
of attention-based MIL, Chen et al., [8] first proposed a Multimodal Co-
Attention Transformer (MCAT) architecture that investigates early fusion
mechanisms for identifying informative patches. MCAT learns the long-range
relationship between image and gene using attention mechanism and visually
shows the interpretability of multimodal interaction. However, MCAT uses non-
overlapping patches sampling and maintains a bag size of over ten thousand,
which essentially increases the computation requirement during training and is
time-consuming.
Two major limitations exist in the above approaches for WSI representation.
Firstly, current studies use a fixed scale for patch extraction. The widely
used patch sizes, either $[256\times 256]$ or $[512\times 512]$, have limited
ability to capture coarser level tissue morphology characteristics, such as
tumor shape, size, and circularity that are essential to determine grades of
glioma [11, 12, 13, 14]. As shown in Figure 1, WSIs are scanned at $20\times$
with a resolution of $0.5\mu p$ per pixel. The $256\times 256$ patches sampled
from WSI at $20\times$ magnification can capture a small set of cell
characteristics and interactions, while patches extracted from downsampled
WSIs at $10\times$ or $5\times$ present rich microenvironmental information
between different sets of cells or tissue. The multi-scale patches depict
significant inter- and intra-instance differences in shape, size, and pattern.
Thus they better represent the overall heterogeneity of the tumor
microenvironment [15]. Secondly, existing approaches use VGG and Resnet models
pre-trained on ImageNet to obtain patch features. Compared with vision
Transformer (ViT), CNN models do not incorporate spatial location information
and maintain less global information. The different structures result in
quantitative variability in out-of-distribution generalization ability [16,
17].
Figure 1: Hierarchy structure and example patches. Left: WSIs at $5x$, $10x$,
and $20x$ magnification. Right: Patches extracted from WSIs at different
magnifications. The blue box indicated the corresponding region in upper
magnification.
To address these challenges, we present a weakly supervised, Transformer-based
hierarchical architecture named Hierarchical MIL Transformer (HiMT) that
learns a mapping between genetics and WSIs at different scales for survival
prediction. Specifically, HiMT randomly samples a few patches from different
scales, then constructs bag representation using processed images and gene
categories. HiMT learns a co-attention imaging-genomic mapping to reduce the
space complexity of patches feature. The gene-guided visual concept can be
used on various tasks with proper FC layers. Here we demonstrate the power
with the survival prediction task. During the patch sampling stage, our
hierarchy strategy ensures the patient bag has a reasonable size of
approximately 3000, while our benchmark architecture [8] collected an average
bag size of 15,000, which requires a plenty of GPU resources during training.
Our main contributions can be summarized as follows.
* •
We introduce a hierarchical mechanism to identify the informative instance
across different scales using genetic features as queries.
* •
We use a pre-trained ViT for feature extraction and showed in ablation studies
that even with a single WSI scale, features extracted from $80\%$ fewer
patches than MCAT could achieve competitive performance.
* •
Instead of using a non-overlapping patch sampling procedure, we randomly
sampled fewer patches and essentially reduced the computation stress while
maintaining outperforming results.
We evaluate HiMT on five popular TCGA cancer datasets. Our results outperform
the state-of-the-art MIL models on imaging-genetic data mapping tasks, which
show that our architecture can capture representative features with vastly
reduced feature space. Experiments further show that although the number of
patches dramatically reduced compared to previous work [8] which collects on
average over ten thousand patches from each WSI, our architecture performs
better in survival prediction. In addition, we conduct an ablation study on
the five TCGA datasets and demonstrate that HiMT improves with patches from
different magnification. Our code is publicly available at
https://github.com/chunyuan1/HiMT.
## II Related Work
### II-A Weakly Supervised Learning for WSI Analysis
Recent works have developed many weakly supervised models for medical image
diagnosis to overcome the unavailability of manual annotation in clinical data
[18, 7, 19]. Edwards and Storkey and Zaheer et al. first proposed network
architecture on set-based data [20, 21]. Ilse et al. extend the set-based
concept by employing an attention mechanism and applying it on WSIs [6]. Yao
et al. incorporated attention-based MIL on clustered phenotype and achieved
promising results [7]. Instead of only focusing on the instance-level feature,
Chen et al. proposed co-attention MIL Transformer that learns an interpretable
mapping and visually demonstrates the relationship between imaging and
genetics [8]. However, the approaches at a single scale cannot maintain high-
level morphological features. We believe that fewer patches with the multi-
scale patches feature can achieve promising results compared to tens of
thousands of patches.
### II-B Self-Attention and Vision Transformer
Solving computer vision tasks using non-convolutional neural networks has been
an active research area [22, 23, 24]. The original ViT takes $256\times 256$
images as input and processes it as a sequence of $[16\times 16]$ tokens with
positional information. Since ViT was first proposed by Dosovitskiy et al.,
plenty of recent work is developed to analyze aspects of ViT, such as
robustness [25], effects of self-attention [26], designing improved ViT models
[27, 28] and comparison with CNN [16, 17]. ViT has been proved for its out-of-
distribution generalization and outstanding feature representation due to the
accessibility to global information at an early stage [16, 17]. In the
research of medical images, pre-trained Resnet is widely used for feature
extraction, while ViT is rarely explored. The reason may be that training a
transformer model from scratch requires an immense amount of data and
intensive computational resources. We highlight that the out-of-distribution
generalization ability of ViT makes the ImageNet-pretrained ViT model been
able to extract high-quality representative features from the medical image.
Based on the fact that ViT model relationships between the tokens, we design a
hierarchy model that allows ViT to learn interaction at different scales of
the WSIs.
## III Method
Figure 2: Framework overview of HiMT contains three stages. Bag
representation: 1) Patches sampled from WSIs with different magnification are
sent into a pre-trained ViT model to extract instance-level patch feature; 2)
DNA microarray are categorized according to functional expression. Early
feature fusion: This stage learns a gene-guided co-attention mapping that
directly models the pairwise relationship between pathological images and
genomics. The GCA is used to reduce the sequence length of WSI bags, leading
to more flexibility in choosing aggregation strategies in the following stage.
MIL transformer and survival prediction: applies set-based MIL transformer on
gene-guided imaging feature and genomics for survival prediction.
Figure 2 depicts an overview of HiMT architecture. Given a set of $\Omega$
patients ${X_{i}},i=1,...,\Omega$, each patient has its feature vector
$(Surv_{i},\delta_{i})$. The $Surv_{i}$ indicates the time until the event of
interest occurs, and vital status $\delta_{i}\in\\{0,1\\}$ indicates whether
the event is censored or uncensored (death occurs). The objective is to
predict a corresponding target variable $o_{i}$ for patient $X_{i}$ from the
imaging data.
### III-A Patient Bag Construction
This stage aims to formulate the problem by constructing bags using pathology
images and genomics. Multiple Instance Learning (MIL) uses bags as the data
samples, each containing a set of unordered instances. In our work, patient
$X_{i}$ is a bag of instance, which contains feature vectors
$\\{x_{1},\ldots,x_{M}\\}\in\mathbb{R}^{M\times d}$ from all WSIs of $X_{i}$
and functional gene sets $\\{G\\}_{i}$, where the number of instances $M$
various for each image bag.
Hierarchy Feature Embedding using ViT. Due to the considerable size, it is
infeasible to use WSIs as input. We extract image features by sampling patches
at different magnifications from all WSIs belonging to the same patients.
Instead of sampling non-overlapping patches, we randomly generate patches from
multi-scale WSIs. A thousand patches are extracted randomly from WSIs at
$5\times$, $10\times$, and $20\times$ magnification, respectively, with a
fixed size of $256\times 256\times 3$. The patches that contain background are
excluded. The above procedure results in three thousand patches for each WSI.
The number of patches is about eight times less than the non-overlapping
sampling strategy, while it contains more morphology information at different
resolutions. Then a ViT-L16 model [23] (pre-trained on ImageNet) is used to
extracts $d_{k}$-dim feature embeddings $h\in\mathbb{R}^{d_{k}\times 1}$. The
$M$ extracted patch embeddings from all WSIs of patient $X_{i}$ are packed
into a bag $H_{i}$, where $H_{bag}\in\mathbb{R}^{M\times d_{k}}$ and $M$ is
approximately $3000$.
Genomic Feature Embedding. The heterogeneity within a gene set limited the
utility of the gene database. A Gene alone from the gene set cannot coherently
describe the biological impact. To use the semantic information within the
gene set, genes that contribute to the same biological function are
categorized together and end up with $N$ gene sets. Let $D_{i}=\\{d_{j}\\}$ be
the gene expression array of patient $X_{i}$ and $\\{S_{n}\\}^{N}_{n=1}$
represents the functional categories according to [29, 30]. For each gene
expression in $\\{d_{j}\\}$, if its attribute $attr_{j}\in S_{n}$, $d_{j}$ is
assigned to gene set $g_{n}$. In the resulting gene sets
$\\{g_{n}\\}_{n=1}^{N}$, each set $g_{n}$ has various size. We then apply a FC
layer over all the gene sets to obtain the final genomic embeddings
$\\{G_{n}\in\mathbb{R}^{d_{k}\times 1}\\}^{N}_{n=1}$ and pack it into a bag
$G_{bag}\in\mathbb{R}^{N\times d_{k}}$. In our experiments, $N$ is set to $6$,
indicating the following functional categories: Tumor Suppression,
Oncogenesis, Protein Kinases, Cellular Differentiation, Transcription, and
Cytokines and Growth.
### III-B Multimodality Feature Aggregation
To capture interpretable interactions between genes and pathological images,
we use an early fusion mechanism, Genomic-Guided Co-Attention (GCA) [8], to
discover the genotype-phenotype interactions from the tumor microenvironment.
Inspired by Transformer attention [22], GCA directly model the pairwise
relationship by learning a co-attention matrix $A_{coa}$ between patch
embeddings in $H_{bag}\in\mathbb{R}^{M\times d_{k}}$ and gene embeddings
$G_{bag}\in\mathbb{R}^{N\times d_{k}}$. Then $A_{coa}$ is used to map the
$H_{bag}$ to a set of gene-guided visual concept $\widehat{H}_{bag}$. The
mapping can be expressed as
$\displaystyle CoA(G,H)=softmax(\frac{QK^{\top}}{\sqrt{d_{k}}})$ (1)
$\displaystyle=softmax(\frac{W_{q}GH^{\top}W_{k}^{\top}}{\sqrt{d_{k}}})W_{v}H$
$\displaystyle\to A_{coa}W_{v}H\to\widehat{H}$
where weight matrices $W_{q},W_{k},W_{v}\in\mathbb{R}^{d_{k}\times d_{k}}$ are
trainable parameters to map the query $G_{bag}$ and key-value pairs
$(H_{bag},H_{bag})$ to an output. In the experiment, bag size of $Q$ is much
smaller than that of $K$ and $V$. Consequently, GCA can largely reduce the
complexity of WSI bags.
### III-C MIL Transformers with Survival Prediction
With image embeddings $\widehat{H}_{coa}\in\mathbb{R}^{N\times d_{v}}$ and
genomic embeddings $G_{bag}\in\mathbb{R}^{N\times d_{k}}$ as inputs, we use
two set-based MIL Transformers $\mathcal{T}_{H}$ and $\mathcal{T}_{G}$ to
aggregate image and genomic features, respectively [21, 22, 31, 8]. This step
can be written as
$\begin{split}&\mathcal{E}^{(l)}(H^{(l)})=\zeta^{(l)}(\psi^{(l)}(\\{\phi^{(l)}(x_{i}):h_{i}^{(l)}\in
H^{(l)}\\}))\\\
&\mathcal{F}^{(L)}(H^{(L)})=\zeta^{(L)}(\rho^{(L)}(\\{\phi^{(L)}(x_{i}):h_{i}^{(L)}\in
H^{(L)}\\}))\\\
&\mathcal{T}(X)=\mathcal{F}^{(L)}(\mathcal{E}^{(L-1)}(...\mathcal{E}^{(1)}\\{(x_{i}):x_{i}\in
X\\}))\end{split}$ (2)
Among the above equations: $h_{i}^{(l)}$ is an arbitrary instance in the set
$H^{(l)}$ at hidden layer $l$.
$\phi:\mathbb{R}^{d_{in}}\to\mathbb{R}^{d_{out}}$ is an instance-level
functions. $\psi^{(l)}$ is the self-attention layer. $\rho:\mathbb{R}^{m\times
d_{out}}\to\mathbb{R}^{d_{out}}$ is a permutation-invariant function that
aggregate instances to bag-level feature.
$\zeta:\mathbb{R}^{d_{out}}\to\mathbb{R}^{\text{\\# class}}$ is a bag-level
classifier that undertake target-specific task. In our work, $\zeta$ is a
position-wise FC layer and is used to estimate risk score for survival
analysis. $\mathcal{E}^{(l)}$ is an encoder block. $\mathcal{F}^{(L)}$ is the
MIL network that perform global pooling at the last layer $L$. Precisely,
$\psi^{(l)}$ can be written as the set function which is permutation-
equivariant:
$\psi^{(l)}(\\{h_{i}^{(l)}\\}^{M}_{i=1})=\\{\sum^{M}_{i=1}\frac{\exp{(h_{i}^{(l)}h_{j}^{(l)\top})}}{d_{k}\sum_{j}\exp{(h_{i}^{(l)}h_{j}^{(l)\top})}}\cdot
h_{i}^{(l)}\to h_{i}^{(l+1)}\\}$ (3)
The expression indicates that Transformer is a generalization of shallow set-
based data structure. $\rho_{H}$ and $\rho_{G}$ are implemented following [6]:
$\begin{split}&\phi^{(L)}(h_{i}^{(L)})=W_{\phi}h_{i}^{(l)}\\\
&\rho^{(L)}(\\{h_{i}^{(L)}\\}^{M}_{i=1})=\sum^{M}_{i=1}a_{i}\phi^{(L)}(h_{i}^{(L)})\to
h^{(l)}\text{ where}\\\
&a_{i}=\frac{\exp\\{W_{\rho}(\tanh(V_{\rho}h_{i}^{(L)\top})\odot\text{sigm}(U_{\rho}h_{i}^{(L)\top}))\\}}{\sum^{M}_{j=1}\exp\\{W_{\rho}(\tanh(V_{\rho}h_{j}^{(L)\top})\odot\text{sigm}(U_{\rho}h_{j}^{(L)\top}))\\}}\\\
&\zeta^{(L)}(h^{(L)})=W_{\zeta}h^{(L)}\end{split}$ (4)
in which $W_{\phi}$, $W_{\rho}$, $V_{\rho}$, $U_{\rho}$,
$W_{\zeta}\in\mathbb{R}^{d_{v}\times d_{v}}$ are trainable parameters,
$\phi^{(L)}$ is a FC layer processing instance-level feature. $\rho^{(L)}$ is
a bag-level attention pooling layer and $a_{i}$ controls weight of embedding
$h_{i}^{(L)}$ contributing to bag-level $h^{(L)}$.
Finally, HiMT integrates output of $\mathcal{T}_{H}$ and $\mathcal{T}_{G}$ to
bag-level features by concatenating, denoted as
$[\zeta_{h}^{(L)}(h^{(L)}),\zeta_{g}^{(L)}(g^{(L)})]$. Several FC layers are
added in the end to obtain the final risk score $o_{risk}$.
### III-D Loss Function
For $i$-th patient, the output hazard risk is denoted as $o_{risk,i}$. Let
continuous random variable $T$ represent overall survival time. Following the
discovery from chen et al. [8] that loss function is mini-batch dependent, we
model the survival prediction problem using discrete time intervals. Given
right-censored survival data, we partition the time scale into non-overlapping
intervals: $[t_{0},t_{1}),[t_{1},t_{2}),[t_{2},t_{3}),[t_{3},t_{4})$ depending
on the quantiles of survival time. For each patient, the discrete event time
with the continuous event time $T_{i,cont}$ can be defined as
$T_{i}=r\text{ if }T_{i,cont}\in[t_{r},t_{r+1})\text{ for }r\in\\{0,1,2,3\\}$
(5)
Denote the discrete time ground truth of $i^{th}$ patient as $Y_{i}$. For a
patient with bag-level risk $o_{risk,i}$, the hazard function that measures
the probability of patient die at time $r$ can be expressed as
$f_{hazard}(r|o_{risk,i})=P(T_{i}=r|T_{i}\geq r,o_{risk,i})$ (6)
while the survival function that estimate the probability of a patient live
longer than time point $R$ can then be defined as
$\begin{split}f_{surv}(r|o_{risk,i})=P(T_{i}>r|o_{risk,i})\\\
=\prod^{r}_{u=1}(1-f_{hazard}(u|o_{risk,i}))\end{split}$ (7)
When updating the parameters of discrete survival model, we use log likelihood
[32] that consider the vital status of patients:
$\begin{split}L=-c_{i}\cdot\log(f_{surv}(Y_{i}|o_{risk,i}))\\\
-(1-c_{i})\cdot\log(f_{surv}(Y_{i}-1|o_{risk,i}))\\\
-(1-c_{i})\cdot\log(f_{surv}(Y_{i}|o_{risk,i}))\\\ \end{split}$ (8)
Weighted sum is applied on $L$ and $L_{uncensored}$ that up-weight the
contribution of uncensored patients.
$L_{surv}=(1-\beta)\cdot L+\beta\cdot L_{uncensored}$ (9)
where the uncensored loss is computed as
$\begin{split}L_{uncensored}=-(1-c_{i})\cdot\log(f_{surv}(Y_{i}-1|o_{risk,i}))\\\
-(1-c_{i})\cdot\log(f_{hazard}(Y_{i}|o_{risk,i}))\end{split}$ (10)
## IV Experiments
### IV-A Dataset Description
To show the performance of our architecture, we conduct experiments on five
largest cancer datasets from The Cancer Genome Atlas (TCGA): Bladder
Urothelial Carcinoma (BLCA), Breast Invasive Carcinoma (BRCA), Glioblastoma
Multiforme (GBM) $\&$ Brain Lower Grade Glioma (LGG), Lung Adenocarcinoma
(LUAD), Uterine Corpus Endometrial Carcinoma (UCEC). The numbers of WSIs and
patients in each dataset are shown in Table I. For each WSI, $1000$ patches
are randomly sampled form each hierarchy layer. 5-fold cross-validation is
performed on each dataset.
TABLE I: The number of WSIs and patients in each TCGA dataset. Dataset | BLCA | BRCA | GBMLGG | LUAD | UCEC
---|---|---|---|---|---
# patients | 373 | 957 | 569 | 453 | 480
# WSIs | 437 | 1022 | 1011 | 515 | 538
TABLE II: Performance comparison with different methods using C-index values. Methods | BLCA | BRCA | GBMLGG | LUAD | UCEC | Overall
---|---|---|---|---|---|---
DeepMISL (WSI Only) [6] | 0.536$\pm$0.038 | 0.564$\pm$0.050 | 0.787$\pm$0.028 | 0.559$\pm$0..060 | 0.625$\pm$0.057 | 0.614
DeepMISL (Concat) | 0.605$\pm$0.045 | 0.551$\pm$0.077 | 0.816$\pm$0.011 | 0.563$\pm$0.050 | 0.614$\pm$0.052 | 0.630
DeepMISL (Bilinear Pooling) | 0.567$\pm$0.034 | 0.536$\pm$0.074 | 0.812$\pm$0.005 | 0.578$\pm$0.036 | 0.562$\pm$0.058 | 0.611
DeepAttnMISL (WSI Only) [7] | 0.504$\pm$0.042 | 0.524$\pm$0.043 | 0.734$\pm$0.029 | 0.548$\pm$0.050 | 0.597$\pm$0.059 | 0.581
DeepAttnMISL (Concat) | 0.611$\pm$0.049 | 0.545$\pm$0.071 | 0.805$\pm$0.014 | 0.595$\pm$0.061 | 0.615$\pm$0.020 | 0.634
DeepAttnMISL (Bilinear Pooling) | 0.575$\pm$0.032 | 0.577$\pm$0.063 | 0.813$\pm$0.022 | 0.551$\pm$0.038 | 0.586$\pm$0.036 | 0.621
MCAT [8] | 0.624$\pm$0.034 | 0.580$\pm$0.069 | 0.817$\pm$0.021 | 0.620$\pm$0.032 | 0.622$\pm$0.019 | 0.653
ours | 0.660$\pm$0.021 | 0.606$\pm$0.028 | 0.823$\pm$0.019 | 0.616$\pm$0.016 | 0.658$\pm$0.047 | 0.673
### IV-B Implementation Details
HiMT is trained with one Geforce GTX 1080 Ti GPU. The functional signatures to
category the gene embeddings are obtained from [29]. After collecting the six
gene categories, several FC layers are applied to convert the gene categories
into similar lengths, followed by one encoder layer. Patch features are
extracted by ViT-Large model pre-trained on ImageNet-21k with $16\times 16$
token embedding strategy [23], followed by a FC layer that obtains a feature
vector of length $1024$ for each patch. The third phase of HiMT uses two
Vision Transformer from [23] with an attention pooling layer to deal with WSI
and genomic features, respectively. The processed logits are then concatenated
together and go through several FC layer with sigmoid function to obtain the
final risk score $o_{risk}$. During training, we follow the experimental
settings in [8] that use Adam optimizer [33] with a learning rate of $2\times
10^{-4}$ and weight decay of $1\times 10^{-5}$. Batch size is set to $1$ due
to the various bag size.
### IV-C Evaluation metrics
Concordance index (C-index) is a popular metric in evaluating survival
prediction models [4, 3, 7]. We also use it as evaluation metric in our
experiments. The C-index quantifies the correlation of risk score and survival
time, calculated as follows:
$c=\frac{1}{n}\sum_{i\in\\{1...N|\sigma_{i}=1\\}}\sum_{s_{j}>s_{i}}I[X_{i}\hat{\beta}>X_{j}\hat{\beta}]$
(11)
where $n$ is the number of comparable pairs, $I[.]$ is the indicator function
and $s.$ is the actual observation. The value of C-index ranges from 0 to 1.
The larger CI value means the better prediction performance of the model and
vice versa.
### IV-D Experimental Results
We compare our architecture with several state-of-the-art methods for survival
prediction using the same 5-fold cross-validation splits, training hyper-
parameters, and loss function.
* •
DeepMISL [6]: Deep multiple survival learning is a set-based network that
first applies global attention into MIL algorithm.
* •
DeepAttnMISL [7]: DeepAttnMISL first applies K-Mean clustering to group
instant-level feature into phenotypes. Then an attention-based global pooling
is used to aggregate the risk score of each patient.
* •
MCAT [8]: A state-of-the-art multimodal survival architecture that incorporate
self-attention mechanism to learn an interpretable mapping between imaging and
genomics in early fusion manner.
Table II presents the results using the above methods on five cancer datasets.
Compared with WSI-based MIL methods (DeepMISL, DeepAttnMISL), HiMT increases
the overall c-index by $9.6\%$ and $15.8\%$. Against the similar methods with
multimodal approaches, HiMT achieves $6.8\%$, $6.2\%$, and $3.1\%$ advancement
over DeepMISL, DeepAttnMISL, and MCAT, respectively. DeepMISL and DeepAttnMISL
both attain more reasonable results than baseline models that only use WSIs.
However, DeepMISL shows a considerable variance among all experiments with
$p>0.1$, indicating the result is unreliable. DeepMISL exhibits a marginal
performance boost over the baseline method with statistically significant for
the BLCA ($p<0.01$). Unlike the late fusion strategy used in DeepMISL and
DeepAttnMISL, MCAT proposed an early fusion mechanism called co-attention and
performs notable progress over previous work with $p<0.01$ for BLCA, GBMLGG,
and UCEC. Our method achieves the best average c-index among all the
approaches with $p<0.01$ for all five cancer datasets. Specifically, HiMT
achieves the highest c-index in 4 out of 5 cancer datasets in comparison
concerning each cancer. This indicates that the hierarchy architecture in HiMT
can capture representative features from medical images while also requiring
less computational resources.
Figure 3: Boxplots of C-index values of five cancer datasets compared with
MCAT. The red dot indicates the mean C-index of the 5-fold cross-validation.
The boxplot of C-index in Figure 3 depicts the variation of C-index values
among the 5-fold and further demonstrate that our algorithm not only performs
better in terms of mean C-index but also is more robust than previous work.
TABLE III: Ablation study assessing C-index performance with respect to different magnification settings. 1000 $256\times 256$ patches are sampled from each magnification for experiments shown below. Methods | BLCA | BRCA | GBMLGG | LUAD | UCEC | Overall (C-index)
---|---|---|---|---|---|---
Patches at $5\times$ | 0.636$\pm$0.022 | 0.612$\pm$0.050 | 0.806$\pm$0.006 | 0.593$\pm$0.061 | 0.556$\pm$0.045 | 0.641
Patches at $10\times$ | 0.634$\pm$0.010 | 0.586$\pm$0.047 | 0.804$\pm$0.009 | 0.598$\pm$0.045 | 0.635$\pm$0.071 | 0.651
Patches at $20\times$ | 0.638$\pm$0.028 | 0.609$\pm$0.046 | 0.821$\pm$0.027 | 0.609$\pm$0.030 | 0.638$\pm$0.031 | 0.663
Patches at $20\times$ and $10\times$ | 0.626$\pm$0.018 | 0.592$\pm$0.029 | 0.805$\pm$0.008 | 0.617$\pm$0.048 | 0.645$\pm$0.078 | 0.657
Patches at $20\times$, $10\times$ and $5\times$ | 0.660$\pm$0.021 | 0.606$\pm$0.028 | 0.823$\pm$0.019 | 0.616$\pm$0.016 | 0.658$\pm$0.047 | 0.673
TABLE IV: Ablation study assessing AUC values with respect to different magnification settings. 1000 $256\times 256$ patches are sampled from each magnification for experiments shown below. Methods | BLCA | BRCA | GBMLGG | LUAD | UCEC | Overall (AUC)
---|---|---|---|---|---|---
Patches at $5\times$ | 0.664$\pm$0.051 | 0.648$\pm$0.055 | 0.843$\pm$0.016 | 0.612$\pm$0.073 | 0.583$\pm$0.047 | 0.670
Patches at $10\times$ | 0.659$\pm$0.031 | 0.611$\pm$0.045 | 0.844$\pm$0.016 | 0.616$\pm$0.050 | 0.651$\pm$0.082 | 0.676
Patches at $20\times$ | 0.663$\pm$0.055 | 0.626$\pm$0.040 | 0.862$\pm$0.028 | 0.624$\pm$0.047 | 0.655$\pm$0.028 | 0.686
Patches at $20\times$ and $10\times$ | 0.654$\pm$0.048 | 0.615$\pm$0.009 | 0.841$\pm$0.007 | 0.633$\pm$0.067 | 0.662$\pm$0.075 | 0.681
Patches at $20\times$, $10\times$ and $5\times$ | 0.697$\pm$0.040 | 0.632$\pm$0.035 | 0.856$\pm$0.037 | 0.631$\pm$0.022 | 0.678$\pm$0.067 | 0.699
### IV-E Ablation Studies
To evaluate the impact of hierarchical architecture in solving MIL tasks, we
conduct an ablation study to evaluate patches sampled on fixed magnification
with pretrained ViT for feature extraction. Table III and table IV shows the
C-index and AUC results, respectively, for HiMT models using: 1) patches
sampled from single magnification, 2) patches sampled from multi-scale WSIs.
The results illustrate that both feature extraction using pre-trained ViT
model and hierarchical architecture contribute to the overall c-index.
Specifically, from Table III and Table IV, the models use patches from all
three resolutions have the best overall performance, with an average C-index
of $0.673$ and average AUC of $0.699$. In terms of each cancer dataset,
although it beat almost all the benchmark methods, the performance of LUAD and
BRCA do not show significant improvement with different experimental settings.
LUAD stands for lung cancer, which is well-known for the hybrid entities, high
genetic instability, high malignancy, and mortality [34]. Even in the clinic,
the diversity causes problems for experts to diagnose. Surprisingly, both the
mean C-index and mean AUC of BRCA dataset with $5\times$ magnification achieve
the highest score among multi-scale settings. BRCA is a heterogeneous complex
of diseases with many subtypes that has distinct biological features [35]. The
result might suggest that tissue-level information is more important to HiMT
on survival prediction.
Noticeably, in the third row of Table III and IV where the model only takes
$1000$ $256\times 256$ patches sampled from each $20\times$ WSI as input, HiMT
outperforms the MCAT[8], which uses over $15,000$ patches per WSI with a
similar patch size for feature representation. The result further suggests
that pre-trained ViT has out-of-distribution generalization ability and could
be used for image feature extraction in a wide variety of research.
## V Conclusion
This paper presents a Transformer-based hierarchical architecture for weakly
supervised survival prediction on multimodality data. Compared with the state-
of-the-art methods, our work requires remarkably less computational resources
by reducing the number of patches from WSIs without losing performance.
Obtaining features from different magnification capture representative
information from the microenvironment. Our results have quantitatively shown
significant improvement over the state-of-the-art methods on survival
prediction.
Future work. The relative position of different types of cells is vital to
identifying the cancer stage, which is a critical measurement during
diagnosis. Future research could include integrating spatial relations to
capture relative position between different cell types, potentially improving
further. Besides, the ViT models used for feature extraction are pre-trained
on ImageNet-1k, while the pathology image is distinct from the nature image. A
ViT model pretrained on an ensemble of cancers might be able to extract more
representative patch features.
## Acknowledgment
This work was partially supported by the NSF CAREER grant IIS-1553687 and
Cancer Prevention and Research Institute of Texas (CPRIT) award (RP190107).
The results in this paper are in whole based upon data generated by the TCGA
Research Network: https://www.cancer.gov/tcga.
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|
# On the dynamics of mortality and the ephemeral nature of mammalian megafauna
Taran Rallings1, Christopher P Kempes2, Justin D. Yeakel1,∗ 1School of Natural
Sciences, University of California Merced
2Santa Fe Institute
∗Corresponding author<EMAIL_ADDRESS>
###### Abstract
Energy flow through consumer-resource interactions is largely determined by
body size. Allometric relationships govern the dynamics of populations by
impacting rates of reproduction, as well as alternative sources of mortality,
which have differential impacts on smaller to larger organisms. Here we derive
and investigate the timescales associated with four alternative sources of
mortality for terrestrial mammals: mortality from starvation, mortality
associated with aging, mortality from consumption by predators, and mortality
introduced by anthropogenic subsidized harvest. The incorporation of these
allometric relationships into a minimal consumer-resource model illuminates
central constraints that may contribute to the structure of mammalian
communities. Our framework reveals that while starvation largely impacts
smaller-bodied species, the allometry of senescence is expected to be more
difficult to observe. In contrast, external predation and subsidized harvest
have greater impacts on the populations of larger-bodied species. Moreover,
the inclusion of predation mortality reveals mass thresholds for mammalian
herbivores, where dynamic instabilities may limit the feasibility of
megafaunal populations. We show how these thresholds vary with alternative
predator-prey mass relationships, which are not well understood within
terrestrial systems. Finally, we use our framework to predict the harvest
pressure required to induce mass-specific extinctions, which closely align
with previous estimates of anthropogenic megafaunal exploitation in both
paleontological and historical contexts. Together our results underscore the
tenuous nature of megafaunal populations, and how different sources of
mortality may contribute to their ephemeral nature over evolutionary time.
## Introduction
Consumer-resource interactions are the fundamental unit from which complex
food webs arise [deangelis1980energy]. In such dynamics, the rates governing
transitions of biomass and energy from one species to another are largely
determined by body size [Yodzis:1992hg]. Specifically, the allometric
relationships between consumer body mass and metabolic rate constrain
energetic assimilation [hou2008], storage [Lindstedt:2002td], and growth
[West:2001bv], all of which govern the dynamics of populations
[hennemann1983relationship, West:2002it, Kempes:2012hy, yeakel2018dynamics].
Because allometrically-constrained models of population dynamics apply
generally across large taxonomic clades, they are useful for examining dynamic
constraints that may contribute to community structure across
macroevolutionary timescales [DeLong:2012fjb, DeLong2012carnivores, Pawar2012,
yeakel2018dynamics, bhat2020scaling]. Furthermore, examination of community
dynamics at these scales enables the investigation of extinct communities
where body size distributions were different than those in contemporary
ecosystems [alroy2001multispecies, brook2008synergies, bradshaw2021relative].
The dynamics of populations represent an energetic balance between
reproduction and mortality [murdoch:2003]. Across Mammalia, the average rate
of reproduction can be predicted from allometric scaling relationships
[CalderIII:1983jd], though individual clades demonstrate a large variety of
reproductive strategies – from changing reproductive cycles, litter sizes, and
dynamic responses to changing resource conditions to name a few
[roff1993evolution]. As these strategies are typically evolved responses to
particular conditions and clade-specific, they are not universally
experienced. On the other hand, mortality has a variety of forms that nearly
all species must deal with to a greater or lesser extent, and do not all scale
similarly with body size [weitz2006size]. Mortality originates from both
internal and external drivers, where the former depends on an organism’s
internal state to initiate death. For example, senescence and starvation
involve physiological states that change with respect to clock time, metabolic
rate, and resource depletion [yeakel2018dynamics, robert2015actuarial]. In
contrast, external drivers of mortality consist of an outside force that
induces death more independently of an organism’s internal state, such as
mortality due to natural predation or subsidized anthropogenic harvest. Often,
mortality occurs through correlations between internal and external drivers,
where for example, the starvation state of prey may alter the success rates of
predators [Alonzo:2002ub]. While virtually all primary consumer populations
must deal with the effects of resource limitation, aging, and predation, the
effects of anthropogenic harvesting (the subsidized extraction of prey) are
uniquely limited to those species serving as resources for human populations
[Dunne:2016jp].
How do different sources of mortality impact the dynamics of mammalian
populations? Here we construct a general consumer-resource framework to
examine mammalian herbivore populations as a function of consumer body size
$M_{C}$, as well as size-dependent vulnerability to different internal and
external pressures. Our approach integrates relationships governing specific
timescales of physiology and assimilation from a process-based energetic
perspective [West:2001bv]. Our model is low-dimensional and compact [cf.
yeakel2018dynamics], but due to its close connection to fundamental energetic
mechanism, it is also capable of reproducing observed large-scale empirical
patterns of mammalian communities. We begin by describing our approach,
reproducing key macroecological relationships such as Damuth’s law
[Damuth1987], and examine how changes to energetic parameters impact these
predictions. We then derive timescales associated with four sources of
mortality experienced by mammalian consumers: _i_) natural mortality, _ii_)
starvation mortality, _iii_) natural predation, and _iv_) subsidized
anthropogenic harvest. By examining each source of mortality in turn, our
framework illuminates central constraints governing mass-specific behaviors,
strategies, and risks experienced by mammalian consumers.
Our results reveal four key insights into the constraints structuring
mammalian communities. First, our allometric consumer-resource system
accurately captures both the central tendency and variability of Damuth’s law,
suggesting that the included vital rates capture mass-specific dynamics.
Second, our results demonstrate that natural and starvation mortality
differentially impact small mammals, confirming expectations, and point to why
the allometric effects of senescence are difficult to observe in nature.
Third, we detail the differences in how mortality under different levels of
predation intensity induce dynamic instabilities for large-bodied
megaherbivores. We also show that the body size at which these instabilities
occur is dependent on the prevailing predator-prey mass relationship (PPMR).
Finally, we evaluate the harvest pressure required to induce mass-specific
extinction, and show that our predictions are comparable to estimates of both
paleontological and historical exploitation of mammalian megafauna.
## Methods
We model a consumer-resource interaction, where the resource $R$ (g/m2) grows
logistically with intrinsic growth rate $\alpha$ to a carrying capacity $k$,
and declines due to consumption by an herbivore consumer population $C$ (g/m2)
(Eq. 3). Consumed resources govern both consumer somatic maintenance and
reproduction. The rate of consumption to fuel somatic maintenance is given by
$\rho$, and is independent of resource density, as these are invariant
requirements of the consumer population [yeakel2018dynamics]. In contrast, the
rate of consumption to fuel reproduction is proportional to resource density
and is given by $\lambda_{C}(R)/Y_{C}$, where $\lambda_{\rm C}(R)$ is the
consumer growth rate and $Y_{\rm C}$ is the consumer yield coefficient, or the
grams of consumer produced per gram of resource consumed. As in DeLong &
Vasseur [DeLong:2012fjb], the consumer’s growth rate $\lambda_{C}(R)$ follows
a Type II (saturating) functional response given the resource density $R$,
where the maximum growth is $\lambda^{\rm max}_{\rm C}$ and the resource half-
saturation density is $\hat{k}=k/2$, such that
$\lambda_{\rm C}(R)=\lambda^{\rm max}_{\rm
C}\left(\frac{R}{\hat{k}+R}\right).$ (1)
While the consumer population density grows at rate $\lambda_{\rm C}(R)$, we
assume for now that consumer mortality is a function of both natural mortality
$\mu$ and starvation $\sigma(R)$, where the rate of starvation,
$\sigma(R)=\sigma^{\rm max}\left(1-\frac{R}{k}\right),$ (2)
increases as resources become scarce. In this context, $\sigma^{\rm max}$ is
the maximal rate of starvation that occurs when the environment is devoid of
resources. The full system describing resource and consumer dynamics is given
by
$\displaystyle\frac{{\rm d}}{{\rm dt}}C$ $\displaystyle=\lambda_{\rm
C}(R)C-\left(\mu+\sigma(R)+...\right)C,$ $\displaystyle\frac{{\rm d}}{{\rm
dt}}R$ $\displaystyle=\alpha
R\left(1-\frac{R}{k}\right)-\left(\frac{\lambda_{\rm C}(R)}{Y_{\rm
C}}+\rho\right)C,$ (3)
where the ‘$...$’ denotes where additional mortality terms, described later,
will be included. The dynamic outcomes of this system of equations include two
trivial steady states at $(R^{*}=0,C^{*}=0)$ and $(R^{*}=k,C^{*}=0)$, and one
internal steady state where both the consumer and resource population coexist.
See tab. 1 for a description of parameters.
The rate laws describing resource consumption as well as consumer growth and
mortality all vary as a function of consumer body mass $M_{C}$, where the
consumer is assumed to be a mammalian herbivore, and the resource is an
unspecified primary producer with characteristic growth rate, carrying
capacity, and energy density $E_{d}$. We approach the derivation of vital
rates with respect to consumer mass by solving for multiple timescales
associated with ontogenetic growth, maintenance, and expenditure. The growth
of an individual consumer from birth mass $m=m_{0}$ to its reproductive size
$m=0.95M_{C}$ is given by the solution to the general balance condition
$B_{0}m^{\eta}=E_{m}\frac{\rm d}{\rm dt}m+B_{m}m$, where $E_{m}$ is the energy
needed to synthesize a unit of biomass, $B_{m}$ is the metabolic rate to
support an existing unit of biomass (tab. 1), and the metabolic exponent
$\eta=3/4$ [Pirt1965, West:2001bv, hou2008]. From this balance condition, the
time required for an organism starting from mass $m_{1}$ to reach mass $m_{2}$
follows
$\tau(m_{1},m_{2})=\ln\left(\frac{1-(m_{1}/M_{C})^{1-\eta}}{1-(m_{2}/M_{C})^{1-\eta}}\right)\frac{M_{C}^{1-\eta}}{a(1-\eta)}$
(4)
where $a=B_{0}/E_{m}$ [West:2001bv]. We use this general timescale equation to
calculate maximal rates of growth and starvation as a function of organismal
body size $M_{C}$, which are then modified by resource density $R$ to provide
realized timescales (eqs. 1,2). We note that a more complex framework could
include the effects of changing resource densities on timescales directly,
where individual growth is itself variable, effectively introducing dynamic
population structure [deRoos2020]. From this general equation, we calculate
the timescale of reproduction for an herbivore consumer of mass $M_{C}$ as
$t_{\lambda_{C}}=\tau(m_{0},0.95M_{C})$, such that the maximal reproductive
rate is $\lambda^{\rm max}_{C}=\ln(\nu)/t_{\lambda_{C}}$, where $\nu=2$ is the
set number of offspring per reproductive cycle [Savage2004,
yeakel2018dynamics]. The consumer yield coefficient is given by
$Y_{C}=M_{C}E_{d}/B_{\lambda_{C}}$ (g consumer per g resource), where
$B_{\lambda_{C}}$ is the lifetime energy use required by the herbivore to
reach maturity $B_{\lambda_{C}}=\int_{0}^{t_{\lambda_{C}}}B_{0}m(t)^{\eta}{\rm
d}t$, and the maintenance rate is given by $\rho=B_{0}M_{C}^{\eta}/M_{C}E_{d}$
[yeakel2018dynamics].
To determine the rate of mortality from starvation, we calculate the time
required for an organism to metabolize its endogenous energetic stores,
estimated from its cumulative fat and muscle mass, where the remaining mass is
given by $M_{C}^{\rm starve}=M_{C}-(M_{C}^{\rm fat}+M_{C}^{\rm musc})$ (see
app. C). During starvation, we assume that an organism burns its existing
endogenous stores as its sole energy source, such that the balance condition
is altered to $\frac{\rm d}{\rm dt}mE^{\prime}_{m}=-B_{m}m$, where
$E_{m}^{\prime}$ is the amount of energy stored in a unit of biomass
(differing from the amount of energy used to synthesize a unit of biomass
$E_{m}$; tab. 1) [moses2008rmo, hou2008]. The starvation timescale is then
given by
$t_{\sigma}=-\frac{M_{C}^{1-\eta}}{a^{\prime}}\ln(M_{C}^{\rm starve}/M_{C}),$
(5)
where $a^{\prime}=B_{0}/E^{\prime}_{m}$, such that the starvation rate is the
$\sigma^{\rm max}=1/t_{\sigma}$. Importantly, the starvation mortality
expressed here is specifically that experienced by adult organisms (as the
timescale of metabolizing fat stores is conditioned on adult mass), and does
not capture potential starvation mortality of juveniles.
To determine the rate of mortality from aging, we note that population cohorts
experience two primary sources of natural mortality: the initial cohort
mortality rate $q_{0}$ and the annual rate of increase in mortality as the
cohort ages, or the actuarial aging rate, $q_{a}$ over lifetime $t_{\ell}$. We
begin by assuming that the number of survivors over time follows a Gompertz
relationship [CalderIII:1983jd] from which we derive the average rate of
natural mortality
$\mu=\frac{q_{0}}{q_{a}t_{\ell}}\big{(}{\rm exp}(q_{a}t_{\ell})-1\big{)}.$ (6)
The three parameters $(q_{0},q_{a},t_{\ell})$ each have well-documented
allometric relationships for terrestrial mammals, such that natural mortality
can be written as a function of consumer mass $\mu(M_{C})$ (see app. A).
Because both cohort and actuarial mortality are not subdivided into specific
categories, we may assume that this rate is capturing the combined effects of
all sources of mortality, particularly during early development when
starvation and predation risks are highest.
As the sizes of physiological biomass compartments are obtained from empirical
observations, the rates determining biomass flux are derived from process-
based energetic relationships (eq. 4). Together, the allometric rate laws and
the dynamic system presented in Eq. 3 allow us to assess the dynamics of
consumer-resource systems for mammalian herbivores spanning the observed range
of terrestrial body sizes, from the smallest (the Etruscan shrew at roughly
$1$ g) to the largest (the Oligocene paraceratheres and Miocene deinotheres at
ca. $1.5-1.74\times 10^{7}$ g) [Smith:2010p3442]. We next examine how this
minimal framework is well-suited to provide general insight into several key
allometric constraints that contribute to the functioning and limitations of
terrestrial mammalian communities.
## Results & Discussion
### Recovering Damuth’s mass-density relationship
Our consumer-resource system is related to the nutritional state model (NSM)
proposed in yeakel2018dynamics, where an explicit starvation dynamic was
incorporated by separating the consumer population density into ‘full’ and
‘hungry’ states. Here we eliminate the transition between these states, and
because the timescales of transitioning between full and hungry states are
short relative to those of reproduction, have sacrificed only a modest degree
of physiological realism to enable analytical expression of steady states with
additional sources of mortality. If we ignore the negligible effects of $\rho$
(see app. B), analytical expression of the consumer steady state as a function
of mass – or the mass-density relationship – follows
$C^{*}(M_{C})\approx\alpha kY_{C}\frac{\sigma^{\rm max}-2\lambda_{C}^{\rm
max}+A}{4{\sigma^{\rm max}}^{2}},$ (7)
where $A=\sqrt{8{\sigma^{\rm max}}^{2}+(\sigma^{\rm max}-2\lambda_{C}^{\rm
max})}$, where $\lambda_{C}^{\rm max}$, $\sigma^{\rm max}$ and $Y_{C}$ are
functions of mass $M_{C}$.
Figure 1: Model predictions of mammalian steady states ($\rm inds\cdot
m^{-2}$) as a function of herbivore consumer body mass $M_{C}$ (thick blue
line) compared to observational data from Damuth [Damuth1987] (black points).
The black line denotes the best-fit linear regression on observed densities
based on ordinary least squares. Variation in steady state densities is
captured by allowing the plant resource growth rate to vary as
$\alpha=2.81\times 10^{-10}:2.19\times 10^{-8}\leavevmode\nobreak\ {\rm
s}^{-1}$ (dark blue shaded region), and both $\alpha$ and the plant resource
carrying capacity to vary as $k=2.3:34\leavevmode\nobreak\ {\rm kg/m}{}^{2}$
(light blue shaded region).
The scaling of mammalian population densities was originally observed by
Damuth1981, Damuth1981b as the reciprocal of energy use requirements with an
exponent of ca. -3/4. Consumer-resource models parameterized using allometric
relationships can effectively predict this mass-density relationship
[Yodzis:1992hg, DeLong:2012fjb, yeakel2018dynamics], while the addition of
predator-prey size ratios and consumer capture relationships enable similar
predictions at higher trophic levels [weitz2006size, DeLong2012carnivores]. By
integrating dimensional scaling into search and consumption rates, Pawar2012
captured the mass-density relationship while highlighting potential
instabilities arising in 3-Dimensional (aquatic) environments. Our approach
differs from most prior efforts by deriving timescales associated with
reproduction and mortality directly from the energetic trade-offs associated
with somatic growth and maintenance. After substituting allometric
relationships into the rate laws in Eq. 3, we observe that the internal steady
state of consumer densities in our framework is very close, though slightly
elevated, to observed mammalian densities, similarly approximating Damuth’s
Law (blue line, fig. 1).
Our predicted mass-density relationship is premised on the assumption of
resource growth rates and carrying capacities characteristic of grasses (tab.
1), contributing to the slightly elevated mass-density relationship compared
to the observed best-fit (black line, fig. 1). As the resource growth rate and
carrying capacity are in the numerator of eq. 7, they determine the intercept
of the relationship such that lower values will more closely match observed
densities. Along these lines, incorporating observed ranges of $\alpha$ and
$k$ reveal strong alignment between model predictions and the variability of
empirical mammalian densities (fig. 1; see app. B for details). Compared to
the NSM [yeakel2018dynamics], and similar to DeLong2012carnivores, our
prediction reveals exaggerated densities for smaller-bodied consumers, though
within the observed range of variation, resulting in a predicted mass-density
relationship with a steeper slope than expected. An elevated mass-density
slope is not observed when explicit starvation and recovery are included
[yeakel2018dynamics], suggesting these dynamics play an important role in
depressing the populations of smaller-bodied species, in particular.
While eq. 7 cannot be readily expressed when allometric relationships are
included, for larger body sizes the maximal starvation rate $\sigma^{\rm
max}\propto M_{C}^{-0.3}$, the yield coefficient $Y_{C}\propto M_{C}^{-1/4}$,
the maximal consumer growth rate $\lambda_{C}^{\rm max}\propto M_{C}^{-1/4}$,
and the quantity $A\propto M_{C}^{-0.37}$. For larger body masses, this
results in a predicted mass-density relationship $\propto M_{C}^{-0.82}$
inds/m2, only slightly steeper than Damuth’s mass-density relationship
$\propto M_{C}^{-0.77}$ inds/m2. At unrealistically large body sizes, the
consumer steady state encounters a vertical asymptote [also noted in
yeakel2018dynamics]. In this region, the superlinear body fat allometry (tab.
1) predicts the organism to be 100% fat, such that the starvation timescale is
infinite. While this is mathematically entertaining, we restrict our
interpretations to realistic body size ranges, thereby avoiding this
particular physiological singularity. We examine additional effects of altered
vital rates on the slope and intercept of the mass-density relationship in
app. B.
### Senescence and starvation have a larger impact on smaller consumers
We first consider two internal sources of mortality: that due to the effects
of aging, where mortality changes with an organism’s temporal state, and that
due to starvation, where mortality scales with an organism’s energetic state.
To understand the effect of changes to $\mu(M_{C})$ on consumer steady states,
we examine variations in the principle components of $\mu$: initial cohort
mortality $q_{0}$ and actuarial mortality $q_{a}$. The initial cohort
mortality represents the mortality experienced by a cohort prior to accruing
effects of age. We observe that the mortality rate changes proportionally with
$q_{0}$ independent of consumer mass, where the ratio $\mu/\lambda_{C}^{\rm
max}<1$ even with respect to large increases in $q_{0}$, unless $q_{a}$ is
similarly magnified (fig. 2A,B). For survivorship mortality to approach the
rate of reproduction ($\mu/\lambda_{C}^{\rm max}=1$), where perceptible
declines in population densities result, the initial cohort mortality must
increase by roughly an order of magnitude (shaded region in fig. 2C). Due to
the steepness of the scaling of $\mu$ relative to $\lambda_{C}^{\rm max}$,
this effect is felt exclusively by small-bodied organisms.
Figure 2: Changes in natural mortality as a function of initial cohort
mortality $q_{0}$, and actuarial aging, $q_{a}$ for two different consumer
body masses, $M_{C}$. A,B. The ratio reproduction $\lambda_{C}^{\rm max}$ to
natural mortality $\mu$ for a mammalian herbivore of A. $M_{C}=10^{2}$ g and
B. $M_{C}=10^{6}$ g, across proportional changes to the initial cohort
mortality rate $q_{0}$ and the actuarial aging rate $q_{a}$. The black contour
denotes $\mu/\lambda=1$. C,D. Natural mortality $\mu$ (green) relative to
reproduction $\lambda_{C}^{\rm max}$ (orange) as a function of consumer body
mass $M_{C}$. The range of variation (light green shaded region) shows
proportional changes to the C. cohort mortality rate $q_{0}$ and the D.
actuarial aging rate $q_{a}$ from -0.99 to 10.
Actuarial mortality represents the cumulative effects of aging, or senescence,
across the organism’s expected lifetime. We observe that as $q_{a}$ increases,
the magnitude of mortality increases disproportionately (fig. 2A,B), while the
slope of $\mu(M_{C})$ becomes more shallow (fig. 2D), primarily due to the
cumulative nature of senescence magnifying its effects across the longer
lifetimes of larger mammals. As such, an increase in $q_{a}$ overwhelms
reproduction such that $\mu/\lambda_{C}^{\rm max}>1$, resulting in population
instability (fig. 2A,B). The extinction risk imposed by senescence has been
explored across mammalian taxa, and while some life-history characteristics
such as the inter-birth interval appear to correlate strongly with these
risks, the role of body size is notably ambiguous [robert2015actuarial].
Though our model – which considers averaged effects across terrestrial mammals
– predicts that the risks of increased actuarial mortality are
disproportionately felt by smaller size-classes, we also show that
$\mu(M_{C})$ increasingly resembles $\lambda_{C}^{\rm max}(M_{C})$ with
increasing $q_{a}$ (the top border of the shaded region in fig. 2D). This
increased similarity implies that relatively small variations in other
demographic processes or interactions may have potentially large and
destabilizing effects on population size that cannot be predicted from body
mass, a potential source for the noted ambiguity between size and actuarial
extinction risk [robert2015actuarial].
Figure 3: The relative change in consumer steady state $\Delta C^{*}_{s}$ as
a function of consumer body mass $M_{C}$ given an altered rate of starvation
$\sigma(R)\cdot(1+\chi_{s})$ across the proportional change
$\chi_{s}\in(-0.99,1)$.
While the temporal state of an organism is unidirectional and linear, other
internal states, such as an organism’s energetic state, fluctuate nonlinearly
over time. In this case, the rate of starvation is low when resources become
plentiful ($R\rightarrow k$) and increases to $\sigma^{\rm max}$ as resources
become scarce ($R\rightarrow 0$). Because organisms metabolize their fat and
muscle tissue during starvation, and die from starvation when these energetic
stores are metabolized, the timescale of starvation varies with the amount of
endogenous energetic stores an organism carries. Larger organisms carry a
larger proportion of body mass as fat [Lindstedt:2002td], such that they are
more protected from the effects of short-term resource scarcity
[Millar:1990p923]. We observe this effect by modifying the starvation rate and
examining how the steady state population size is altered. We introduce
variation to the rate of starvation as $\sigma(R)\cdot(1+\chi_{s})$, from
which the altered steady state $C^{*}_{s}$ is calculated. The relative change
in steady states introduced by the altered starvation rate is then given by
$\Delta C^{*}_{s}=(C^{*}_{s}-C^{*})/C^{*}$, where positive values indicate a
relative gain in steady state densities from the proportional change
$\chi_{s}$, and negative values indicate a relative loss (fig. 3). We observe
that, while all mammals benefit from reduced starvation rates ($\chi_{s}<0$),
smaller-bodied mammals benefit to a much greater extent, and this effect
tapers off with increasing body mass. Because fat biomass scales super-
linearly with body mass, the populations of larger consumers are more
resilient to the effects of starvation, whereas those of smaller consumers are
more prone.
An organism’s rate of starvation emerges from two governing forces – the
amount of energy storage and the rate of its use – and as such can be be
manipulated both physiologically and behaviorally. For instance, behaviorally
supplementing endogenous fat stores with exogenous caches magnifies an
individual’s energetic stores [Lucas1991, yeakel2020caching], whereas
physiologically-mediated responses to starvation risk such as torpor can
introduce significant temporal delays to the effects of resource scarcity
[schubert2010daily]. In both cases the time required to pass from a
replenished to a starved state is effectively increased, lowering the rate of
starvation. The predicted benefits of such adaptations to mammalian steady
state densities will be realized primarily by smaller mammals (fig. 3, app.
B), and it is the smaller body size range where traits such as caching and
torpor are most commonly observed [geiser1998evolution, smith1984evolution,
yeakel2020caching].
### Predation mortality and the feasibility of megatrophic interactions
Predators introduce an external source of mortality on prey populations,
fueling their own population growth in whole (trophic specialists) or in part
(trophic generalists), by the rate at which prey are consumed. We account for
the effects of an implicit predator density $P$ with body size $M_{P}$ on the
herbivore consumer density $C$ with body size $M_{C}$, where we assume the
predator population to exist at a fixed density $P\equiv P^{*}$. The mortality
rate of the herbivore consumer from an external predator is given by
$\beta(C,P)=w\frac{\lambda_{P}(C)P}{CY_{P}},$ (8)
where $\lambda_{P}(C)$ is the growth rate of the predator and $Y_{P}$ is the
predator yield coefficient, describing the grams of predator produced per gram
of prey consumed, and $w$ is the predation intensity. Mirroring the
calculation of the consumer yield coefficient,
$Y_{P}=M_{C}E_{C}/B_{\lambda_{P}}$, where $E_{C}$ is the energy density of
consumable biomass carried by herbivore prey, and the $B_{\lambda_{P}}$ is the
lifetime energy requirement of the predator (app. C).
Assuming a linear functional response for predation mortality,
$\lambda_{P}(C)$ is maximized when the consumer reaches its theoretical
maximum population density, which we calculate by converting the resource
carrying capacity directly to grams of consumer produced, or $C^{\rm
max}=Y_{C}k$. While this is an ultimately unattainable theoretical bound, it
allows for a direct calculation of the predator growth rate as a function of
$C$, written as
$\lambda_{P}(C)=\lambda_{P}^{\rm max}\frac{C}{C^{\rm max}}=\lambda_{P}^{\rm
max}\frac{C}{Y_{C}k},$ (9)
where $\lambda_{P}^{\rm max}=\ln(\nu)/t_{\lambda_{P}}$ is the maximum predator
growth rate, given $\nu=2$, and $t_{\lambda_{P}}$ is the time required for the
predator to reach maturity (following eq. 4). The theoretical boundary density
for herbivore consumers $C^{\rm max}$ can similarly be used to calculate the
boundary density for predators, $P^{\rm max}=Y_{P}C^{\rm max}$, both of which
accurately capture the upper-bounds of herbivore and carnivore mass-density
observations (dashed lines in fig. 4A). Because the effects of the predator
are implicit, we assume that the predator population remains at empirically
measured steady state densities for mammalian carnivores, where
$P^{*}=p_{0}M_{P}^{p_{1}}$ given $p_{0}=8.62\times 10^{-4}\leavevmode\nobreak\
{\rm inds}^{1-p_{1}}/{\rm m^{2}}$ and $p_{1}=-0.88$ [carbone2002common]. As we
are employing this framework to evaluate longer-term evolutionary
consequences, this condition assumes that predator densities do not have long-
term feasibility if they stray far from $P^{*}$.
The predation mortality rate depends on both the body size of the herbivore
consumer and its respective predator. Trophic interactions are constrained by
body size [Sinclair2003, Brose2005, Hatton:2015fk], though the nature of the
predator-prey mass relationship (PPMR) varies across communities
[barnes2010global] and size classes [Carbone:1999ju, Brose2005, Rohr2010,
riede2011stepping, pires2015pleistocene, nakazawa2017individual].
Compellingly, PPMRs for many clades can be predicted from the scaling of
handling time [delong2020], suggesting that the signatures of body size
evolution has cascading effects on community structure and function. While
prior work has largely focused on the expected prey mass for a given predator
mass, because our framework is prey-centric we require a prediction of the
expected predator mass $M_{P}$ given an herbivore of body size $M_{C}$. For
larger predators and prey ($>10^{5}$ g), the expected predator mass given a
particular herbivore mass follows roughly ${\rm
E}\\{M_{P}\\}=v_{0}M_{C}^{v_{1}}$, where $v_{0}=9.76\times 10^{3}$
g${}^{1-v_{1}}$ and $v_{1}=0.21$ [fig. 4B; see app. C; hayward2005lion,
Hayward2006hyena, hayward2006leopard, hayward2006lycaon, hayward2006cheetah,
Hayward2008]. [Here and throughout the prefix ‘mega’ is used to signify size
classes $>5\times 10^{5}$ g; hayward2005lion]. Accordingly, larger terrestrial
herbivores tend to suffer mortality from proportionately smaller predators, an
asymmetry that becomes more pronounced with increasing size [cf.
Sinclair2003]. We note that smaller terrestrial predator/prey size classes
tend to be much larger than prey [e.g. rodent- or insect-specialist
mesocarnivores; cruz2022geography, Cruz2022], also captured by the ${\rm
E}\\{M_{P}\\}$ scaling.
Figure 4: The feasibility of predation. A. Empirical mammalian herbivore
(blue points) and predator (red points) mass-densities shown alongside the
theoretical maximum herbivore $C^{\rm max}$ and predator $P^{\rm max}$
densities across body size (dashed lines). The solid blue curves denote the
predicted herbivore consumer steady state $C^{*}(M_{C})$ with predation
mortality given high ($w=1$) and low ($w=0.37$) predation intensities. B.
Predator body mass as a function of prey body mass observed among contemporary
mammalian fauna provides the allometric predator-prey mass relationship
(PPMR). The green line denotes the best-fit, where $M_{P}=9.76\times
10^{3}M_{C}^{0.21}$ g. C. Threshold herbivore mass $M_{C}^{\dagger}$ given
changes to the PPMR intercept $\chi_{\rm int}$ and slope $\chi_{\rm slope}$
(see Eq. A25 and legend in B.). White shaded region highlights the mass range
enabling feasible megatrophic interactions, shown in D. for herbivore mass
thresholds $M_{C}^{\dagger}$ with the expected mass of associated carnivores
${\rm E}\\{M_{P}\\}$, assuming alternative predation intensities.
Integrating the large-bodied PPMR into the predation mortality rate reveals
the emergence of a dynamic instability at megaherbivore size classes (fig.
4A,B), the product of a transcritical bifurcation at consumer mass
$M_{C}^{\dagger}$ (app. C), and similar to the more general instability
documented by weitz2006size. An implicit predator population with body size
${\rm E}\\{M_{P}\\}$ is thus able to withdraw sufficient biomass from an
herbivore population – without crashing the herbivore population – below a
threshold herbivore size of $M_{C}^{\dagger}=2.58\times 10^{6}$ g (fig. 4A).
Above this critical size threshold, the herbivore population has such low
densities that it is unable to sustain a specialist predator species large
enough to consume it, introducing a strong upper-bound to mammalian carnivore
body size driven by a trophic cascade. This boundary matches the herbivore
maximum size limit observed in contemporary terrestrial systems
[Sinclair2003], at roughly the size of an elephant (fig. 4B; app. C), though
the exact placement of $M_{C}^{\dagger}$ varies with the resource growth rate
and carrying capacity. While we have assumed values representative of grass
resources, decreasing $\alpha$ and/or $k$ lowers the steady state mass-density
intercept (eq. 7), setting the mass threshold at a lower body size (app. C).
This means that lower-productivity environments, or environments subject to
large and long-term oscillations in productivity, may be expected to have more
severe limitations on feasible megaherbivore sizes. And while we do not
consider stochastic or transient effects directly, we may also assume
actualized extinction risk to emerge at smaller-than-predicted masses where
transient or stochastic effects may push populations below the point of
recovery.
$M_{C}^{\dagger}$ marks the threshold herbivore mass above which predation is
unsustainable, though Sinclair2003 revealed contemporary herbivores to escape
predation at ca. $4.22\times 10^{5}$ g. This change-point reflects the
limitations of contemporary carnivores, which reach a maximum body size of
$1.15$ to $2.60\times 10^{5}$ g [Sinclair2003], and have preferences for prey
up to $5.50\times 10^{5}$ g [hayward2005lion]. Importantly, the sole predators
of contemporary giants are not megaherbivore specialists, instead
opportunistically subsidizing their preferred prey with larger taxa. While we
have so far assumed a predator-prey interaction where the entirety of predator
growth is fueled by the focal herbivore, the largest predators in natural
systems tend be dietary generalists [Sinclair2003, gross2009generalized]. We
observe that reducing the predation intensity (such that $w<1$) increases
$M_{C}^{\dagger}$ to a larger threshold mass (app. C). For example, $w=0.37$
increases the herbivore body mass boundary to $M_{C}^{\dagger}=1.75\times
10^{7}$ g (fig. 4B; app. C), roughly the body mass attained by the largest
terrestrial herbivores, the Oligocene paraceratheres and Miocene deinotheres
[Smith:2010p3442].
That the threshold herbivore mass decreases with increasing predation
intensity suggests that larger predators are dynamically constrained to be
dietary generalists [Sinclair2003], while also pointing to an amplifying
feedback mechanism [brook2008synergies] that may operate in diverse
communities undergoing megafaunal extinctions. As megaherbivore species are
lost, the largest predators must respond by increasing the intensity of
predation on those remaining. Our results suggest that this energetic
redirection reduces the threshold herbivore mass $M_{C}^{\dagger}$ to lower
size classes, increasing the likelihood of additional extinctions and
attendant increases in predation intensity on survivors. Together, this
demonstrates a dynamic mechanism for the previously proposed influence of top-
down dietary ratcheting hypothesized for the Pleistocene extinctions, in
particular [kay2002false, ripple2010linking].
Adaptations to hypercarnivorous strategies are difficult to reverse on
macroevolutionary time scales, resulting in the so-called ‘hypercarnivore
ratchet’ [VanValkenburgh1991, Holliday2004]. Moreover, there is a strong
correlation between hypercarnivorous adaptations and body size among
terrestrial carnivores, the combination of which may promote vulnerability to
extinction [VanValkenburgh:2004p2451]. While deinotheres and paraceratheres
top the megaherbivore scale, the Eocene artiodactyl _Andrewsarchus_ may have
been the largest terrestrial mammalian predator at ca. $1\times 10^{6}$ g
[burness2001dinosaurs], while the Miocene Hyaenodontid _Megistotherium
osteothlastes_ ranged between $5$ to $8\times 10^{5}$ g and the early Eocene
Oxyaenodont _Sarkastodon mongoliensis_ weighed ca. $8\times 10^{5}$ g
[sorkin2008biomechanical]. A theoretical maximum mammalian carnivore size of
$1.1\times 10^{6}$ g has been proposed based on the intersection of daily
energetic uptake requirements against metabolic expenditures [Carbone:2007dz],
closely aligning with the largest known megapredators. While our consumer-
resource framework provides a range of predicted megaherbivore body mass
thresholds depending on the fraction of predator growth it fuels, we next ask
under what conditions megatrophic relationships between megaherbivores and
megapredators are dynamically feasible.
A central relationship in our framework is the allometric PPMR observed for
the largest contemporary herbivores and carnivores, however empirical
observations of PPMRs reveal tremendous variability both across and within
clades [delong2020], and are completely unknown for megatrophic interactions
in terrestrial systems (fig. 4B; app. C). While it is unknown whether these
super-sized carnivores were specialists on deinothere size-classes, our
framework allows us to investigate whether and to what extent changes to the
contemporary PPMR enable megatrophic interactions (fig. 4C,D). To examine
this, we allow the expected predator mass given a particular prey mass to vary
as
${\rm E}\\{M_{P}\\}=v_{0}(1+\chi_{\rm int})M_{C}^{v_{1}(1+\chi_{\rm slope})},$
(10)
where the proportional changes in the PPMR intercept and slope are given by
$\chi_{\rm int}$ and $\chi_{\rm slope}\in(-0.99,2)$ (see the legend in fig.
4B). We note that while $\chi_{\rm slope}$ explores changes to the inferred
steepness of the PPMR, $\chi_{\rm int}$ explores variation around the scaling
relationship for a particular prey mass.
We observe that only a small range of values for PPMR intercepts and slopes
permit the existence of dynamically feasible megatrophic interactions, where
megaherbivores serve as prey for specialized megapredators (white band in fig.
4C,D; app. C). Such interactions could be realized if the PPMR in fig. 4B had
an increased intercept or alternatively both a lower intercept and higher
slope for mega size-classes. That alternative PPMRs could characterize
different size-classes across foraging guilds has been previously examined
[vezina1985empirical, Carbone:1999ju], and the clear disconnect between that
shown for contemporary large-bodied mammals and those in the megatrophic range
(fig. 4B) supports this notion. When predation intensity is high ($w=1$), the
PPMR enabling feasible megatrophic interactions lowers the herbivore mass
threshold $M_{C}^{\dagger}$, resulting in megapredators consuming relatively
smaller prey. However, if predation intensity is lowered ($w=0.37$), we
observe feasible megatrophic interactions for size-classes capturing
megaherbivores and megapredators at their largest documented sizes in the
fossil record (fig. 4D; app. C).
That decreased predation intensity enables feasibility of the largest
mammalian size-classes agrees with previous conjectures that the largest
mammalian terrestrial predators were likely dietary generalists
[farlow1993rareness]. Our framework thus highlights dynamic constraints
existing between predators and prey that may serve to structure mammalian
communities over evolutionary time, in particular revealing the susceptibility
of megaherbivores to perturbations. As carnivorous clades evolved body sizes
enabling megaherbivore predation, their super-sized appetites may have
suppressed megaherbivores to unsustainable densities where the risk of
extinction became overwhelming – an evolutionary trap marking the final tooth
in the hypercarnivore ratchet [cf. VanValkenburgh:2004p2451].
### Harvesting to extinction
We last consider the effects of anthropogenic harvest-induced mortality on
herbivore populations. While the predation rate is naturally limited by the
energetic needs of the predator, we consider harvest to be a comparatively
unconstrained source of mortality. This may be the case if the human
population(s) engaged in harvesting are subsidized by alternative resources
[Brook2005]. Harvest pressure has potentially varying relationships with
consumer (prey) body mass, a complex product of environment, climate, culture,
and technology [churchill1993weapon]. For example, hunting traditions
specializing in mass-collecting, by way of trapping or netting
[churchill1993weapon, ugan2005does] are expected to exhibit harvest
allometries biased towards smaller species, whereas a purely opportunistic
strategy may be expected to have very little allometric dependence. While
smaller mammals do not appear to offer a significant return on investment, the
mass-collecting of invertebrates, such as grasshoppers, and fish can offer
significant returns [ugan2005does]. In contrast, the innovation of advanced
projectiles is thought to have enabled harvest of terrestrial megafauna
[churchill1993weapon, prates2022changes], and archeological evidence points to
many Pleistocene human populations as potential megafaunal specialists
[Smith:2018gm].
Figure 5: The effects of harvest mortality on herbivore consumers. A.
Proportion mortality due to an extinction-inducing harvest rate $h^{\dagger}$
without predation ($w=0$; blue line), and with low ($w=0.37$; orange line) or
high predation intensity ($w=1$; red line), as a function of consumer body
mass $M_{C}$. B. Harvest pressure $\psi^{\dagger}$ resulting from extinction-
inducing harvest (inds/year/$A_{\rm CA}$) without predation ($w=0$; blue
line), and with low ($w=0.37$; orange line) or high predation intensity
($w=1$; red line), as a function of consumer body mass $M_{C}$. Black point
and line: median and range of estimated harvest rates for woolly mammoths
[_Mammuthus primigenius_ ; fordham2022process]; Green point: estimated harvest
pressure for the Australian _Diprotodon_ [bradshaw2021relative]; lower and
higher yellow point: estimated harvest rates for contemporary _Loxodonta_
during the early 1800s and just prior to 1987, respectively
[milner1993exploitation].
Because harvest scaling may be difficult to measure and idiosyncratic, we
instead calculate the harvest rate required to induce extinction,
$h^{\dagger}$, as a function of body size $M_{C}$, and find a scaling
relationship proportional to the rate of reproduction where
$h^{\dagger}\propto M_{C}^{-1/4}$. This is a natural result, as the effort
required to suppress a population is expected to be proportional to its
reproductive rate, reflecting the increased susceptibility of large-bodied
organisms to extinction [enquist2020megabiota]. As a proportion of the other
sources of consumer mortality that we have considered (excluding predation;
$w=0$), extinction-level harvesting is lower for smaller consumers, saturating
at close to unity at large size classes, reflecting the elevated role of
starvation mortality among smaller-sized organisms (Fig 5A). With predation
mortality included at both low ($w=0.37$) and high ($w=1$) intensities,
extinction-level harvesting accounts for an increasingly smaller proportion of
mortality for larger organisms (orange and red lines, fig. 5A). This
highlights the delicate nature of the megafaunal niche, where smaller changes
in mortality rates can induce population collapse [enquist2020megabiota].
To examine how our estimate of extinction-level harvesting rates $h^{\dagger}$
compare to those estimated for human hunting of paleontological and historical
mammalian populations, we converted $h^{\dagger}$ to harvest pressure
$\psi^{\dagger}$, or the number of individuals harvested per year to reduce
the population to a fraction of its steady state $\epsilon C^{*}$ where we set
$\epsilon=0.01$. We calculate $\psi^{\dagger}$ for an arbitrary area (see app.
D), which we standardize to the area of California ($A_{\rm CA}=4.24\times
10^{5}\leavevmode\nobreak\ {\rm km}^{2}$), such that
$\psi^{\dagger}\propto-h^{\dagger}\frac{C^{*}(1-\epsilon)}{M_{C}\log(\epsilon)}.$
(11)
Though the annual harvesting pressure is unrealistically high for smaller
organisms, we observe that it is ca. $4.3\times 10^{3}\leavevmode\nobreak\
{\rm inds/yr/}A_{\rm CA}$ for elephant-sized mammals (ca. $2.5\times 10^{6}$
g) in the absence of predation mortality ($w=0$). With increasing predation
intensity, the harvest pressure required to induce extinction is much less for
these larger consumers (orange and red lines in fig. 5B). We note that this
calculation of harvest pressure should be viewed as a minimum estimate given
that we do not account for demographic rebound. As such, this measure is
appropriate only if the timescale of harvest is less than the generational
timescale, which is the case for the megafauna considered here.
Our predictions of extinction-inducing harvest pressure compare well with
paleontological and historical estimates of harvest pressure on mammalian
megafauna (fig. 5B; see app. D). For example, using a formulation similar to
that of alroy2001multispecies, fordham2022process estimate the harvest
pressure required to collapse mammoth (_Mammuthus primigenius_) populations,
revealing a range of values consistent with our expectation for similar size-
classes (est. $\psi^{\dagger}=$ ca. $1.24\times 10^{4}\leavevmode\nobreak\
{\rm inds/yr/}A_{\rm CA}$), as did estimates of extinction-inducing harvest of
the Australian _Diprotodon_ [est. $\psi^{\dagger}=$ ca. 763 ${\rm
inds/yr/}A_{\rm CA}$; bradshaw2021relative]. Within the historical record,
elephant (_Loxodonta_) populations experienced comparatively lower harvest
pressure through 1850 [ca. $466\leavevmode\nobreak\ {\rm inds/yr/}A_{\rm CA}$,
derived from the volume of ivory exports; milner1993exploitation]. While
fluctuating over the next century, harvest pressure increased to a maximum of
ca. $13.3\times 10^{5}\leavevmode\nobreak\ {\rm inds/yr/}A_{\rm CA}$ just
prior to 1987 (fig. 5B). This level of harvest was not sustained, as ivory
export volume plummeted following the implementation of trade restrictions in
1989 [milner1993exploitation]. Both the fordham2022process estimate for
Pleistocene mammoths and the short-lived harvest maximum for African elephants
in 1987 [milner1993exploitation] achieved pressures greater than
$\psi^{\dagger}$ under the conservative assumption of no natural predation
(fig. 5B). While estimates for Diprotodon harvest are considerably lower
[bradshaw2021relative], it is important to note that our framework is
parameterized for eutherian rather than marsupial mammals. Nevertheless, the
estimated _Diprotodon_ $\psi^{\dagger}$ is well within range of extinction-
inducing harvest rates if natural predation pressures are also included, and
there is evidence to suggest that _Diprotodon_ likely served as prey for
marsupial lions [horton1981cuts, wroe1999estimating], and both giant
crocodylians (_Pallimnarchus_ spp.) and varanid lizards [_Megalania_ spp.;
webb2009late].
## Conclusion
We have shown that the inclusion of mass-specific energetic transfer between
resources and consumers, combined with the unique timescales governing
consumer mortality, both predict Damuth’s Law [Damuth1987] and provide insight
into dynamic thresholds constraining populations. While natural and starvation
mortality primarily impact small-bodied species, trophic mortality primarily
impacts large-bodied species with longer generational timescales. Moreover,
while mass-specific predation gives rise to dynamic thresholds for herbivore
populations, these effects are sensitive to both predation intesnity as well
as the associated predator-prey mass relationship, which isn’t well understood
in terrestrial ecosystems [nakazawa2017individual]. While assessment of
particular communities and/or species requires more detailed approaches –
integrating, for example, life history dynamics as in bradshaw2021relative –
we suggest that a lower-dimensional framework is useful for extracting
general, first-order energetic constraints that both shape and potentially
limit the nature of mammalian communities.
That extinction risk appears to increase with body size [Cardillo:2005et] is
integral to our understanding of the Pleistocene extinctions
[alroy2001multispecies, johnson2002determinants, brook2008synergies,
Smith:2018gm, bradshaw2021relative] and anthropogenic effects throughout the
Holocene [Estes:2011eo]. Because megafaunal loss may have disproportionately
large impacts on ecosystem functioning [enquist2020megabiota], understanding
the mechanistic drivers that may lead these species to the brink is of
paramount importance. Assessing which energetic walls close in and why as body
size increases, is a fundamental aspect of reconciling the nature of
extinction [brook2008synergies], particularly when there is size-selectivity
[Smith:2018gm]. That we observe dynamically-feasible megatrophic interactions
to occupy a narrow band of predator-prey mass relationships points to a
broader range of interaction structures than are realized in contemporary
communities. As the threshold consumer mass decreases with increased predation
intensity, how megafaunal trophic structure changes during extinction cascades
may be central for understanding the dynamics of community disassembly
[Yeakel2014]. And while these dynamics may arise naturally from the energetic
limitations of mammalian interactions, it may be that the added pressure of
subsidized harvest, particularly on megafauna, inevitably leads to collapse.
Table 1: Model parameters and values/units Definition | Parameter | Value/Units
---|---|---
Resource | |
density | $R$ | ${\rm g/m^{2}}$
reproduction rate | $\alpha$ | $9.49\times 10^{-9}\leavevmode\nobreak\ ({\rm 1/s})$
carrying capacity | $k$ | $23\times 10^{3}\leavevmode\nobreak\ ({\rm g/m^{2}})$
energy density | $E_{d}$ | $1.82\times 10^{4}$ (J/g)
Consumer | |
density | $C$ | ${\rm g/m^{2}}$
body mass | $M_{C}$ | g
timescale of growth from $m_{1}$ to $m_{2}$ | $\tau(m_{1},m_{2})$ | s
reproduction rate | $\lambda_{C}^{\rm max}$ | ${\rm 1/s}$
yield coefficient | $Y_{C}$ | $({\rm g/m^{2}\leavevmode\nobreak\ }C)/({\rm g/m^{2}\leavevmode\nobreak\ }R)$
maintenance rate | $\rho$ | ${\rm 1/s}$
natural mortality rate | $\mu$ | ${\rm 1/s}$
starvation rate | $\sigma^{\rm max}$ | ${\rm 1/s}$
harvest rate | $h$ | ${\rm 1/s}$
Predator | |
steady state density1 | $P^{*}$ | $P_{0}M_{P}^{-0.88}\leavevmode\nobreak\ {\rm inds/m^{2}}$
body mass | $M_{P}$ | g
growth rate | $\lambda_{P}^{\rm max}$ | ${\rm 1/s}$
yield coefficient | $Y_{P}$ | $({\rm g/m^{2}\leavevmode\nobreak\ }P)/({\rm g/m^{2}\leavevmode\nobreak\ }C)$
predation intensity | $w$ | (0,1)
Metabolic normalization constant | $B_{0}$ | 0.047 (W g-3/4)
Energy to synthesize a unit of mass | $E_{m}$ | 5774 (J g-1)
Energy stored in a unit of mass | $E_{m}^{\prime}$ | 7000 (J g-1)
Prop. change PPMR intercept1 | $\chi_{\rm int}$ | (-0.99,2)
Prop. change PPMR slope1 | $\chi_{\rm slope}$ | (-0.99,2)
Extinction-inducing harvest rate | $h^{\dagger}$ | ${\rm 1/s}$
Extinction-inducing harvest pressure | $\psi^{\dagger}$ | inds/yr/$A_{\rm CA}$
1PPMR: Predator-Prey Mass Relationship |
## Acknowledgments
We would like to thank Irina Birskis-Barros, Uttam Bhat, Jessica Blois,
Nathaniel Fox, Jacquelyn Gill, Paulo Guimarães Jr., Emily Lindsey, Mathias
Pires, Megha Suswaram, and Ritwika VPS for insightful comments and discussions
that greatly improved the ideas and concepts that contributed to this
manuscript. These ideas benefited greatly from travel funds provided to JDY
from the Santa Fe Institute. This project was supported by National Science
Foundation grant EAR-1623852 to JDY.
## Statement of Authorship
JDY, CPK, and TR conceived of the model. JDY and TR developed the code and
oversaw model analysis. All authors reviewed and edited the writing at all
stages of composition.
## Data and Code Availability
Code and data archived on Zenodo: https://doi.org/10.5281/zenodo.8213158
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## Appendix A: Natural mortality
The natural mortality rate is obtained by first assuming that the number of
surviving individuals in a cohort $N$ follows a Gompertz relationship
[CalderIII:1983jd], where
$N=N_{0}{\rm exp}\left(\frac{q_{0}}{q_{a}}\Big{(}1-{\rm
exp}({-q_{a}t})\Big{)}\right),$ (A12)
given that $q_{0}$ is the initial cohort mortality rate, and $q_{a}$ is the
annual rate of increase in mortality, or the actuarial mortality rate. The
change in the cohort’s population over time then follows
$\frac{\rm d}{\rm dt}N=-dN,$ (A13)
such that
$d=-\frac{1}{N}\frac{\rm d}{\rm dt}N.$ (A14)
If $t_{\rm\ell}$ is the expected lifetime of the organism, then the average
rate of mortality over a lifetime $t_{\ell}$ is
$\displaystyle\mu$
$\displaystyle=\frac{1}{t_{\rm\ell}}\int_{0}^{t_{\rm\ell}}q_{0}{\rm
exp}(q_{a}t_{\ell})$ $\displaystyle=\frac{q_{0}}{q_{a}t_{\ell}}\Big{(}{\rm
exp}(q_{a}t_{\ell})-1\Big{)}.$ (A15)
The cohort mortality rate $q_{0}$, the actuarial mortality rate $q_{a}$ and
the expected lifetime $t_{\ell}$ of a mammal with mass $M_{C}$ all follow
allometric relationships, where $q_{0}=1.88\times 10^{-8}M_{C}^{-0.56}$ (1/s)
and $q_{a}=1.45\times 10^{-7}M_{C}^{-0.27}$ (1/s) where $M_{C}$ is in grams.
Together, we obtain the allometric relationship
$\mu(M_{C})=\frac{3.21\times 10^{-8}\left({\rm
exp}({0.586M_{C}^{0.03}})-1\right)}{M_{C}^{0.59}}.$ (A16)
## Appendix B: Variations in model parameters and allometric rates
While our framework dictates that plant growth rates and carrying capacities
are directly proportional to consumer steady states, we can gain insight into
what drives the very large range of observed consumer densities by exploring
the observed ranges of $\alpha$ and $k$ in terrestrial systems. We assume an
intrinsic growth rate roughly that of grass where $\alpha=9.45\times 10^{-9}$
(s-1), whereas observations among terrestrial plants reveal a range in growth
rates from $2.81\times 10^{-10}$ to $2.19\times 10^{-8}$
[michaletz2014convergence], according with a change in $\alpha$ of roughly 97%
lower and 130% higher than the set value. By incorporating this range into the
estimated resource growth rate, we observe that we can account for a large
portion of consumer steady state densities around the mean density (inner
shaded region, Fig. 1, main text). If we additionally adjust the carrying
capacity $k$ of the resource to 90% less-than and 150% more-than the assumed
value of $23\times 10^{3}$ g/m2, our framework accounts for nearly the full
range of mammalian steady state densities (outer shaded region, Fig. 1, main
text). In this context, the upper-boundary of $k$ observed to capture most
higher herbivore densities is ca. 34 kg/m2, which is on the higher end of
estimated live above-ground biomass densities in terrestrial forests such as
in Isle Royal and the Allegheny National Forest [de2017simulating].
Our model’s ability to capture the bounds of mammalian densities at low and
high productivity invites some speculation into the actual steepness of the
mass-density relationship. While the best-fit slope to Damuth’s Law is -0.77
we also observe that the steeper relationship given by our framework better
captures the boundaries of mass-density data, whereas varying the intercept of
the statistical best-fit would not capture the lower-density outer-boundary of
larger species. While within-clade mass-density relationships often reveal a
shallower slope than if measured across clades [Pedersen:2017he], it is
possible that the absence of data for larger mammals may bias estimates of the
slope towards smaller (shallower) values. Mammalian communities have undergone
significant anthropogenic restructuring throughout the Holocene such that many
larger species are excluded from the mass-density relationship by way of
extinction [Koch:2006vt], and the greater prevalence of smaller species may
introduce size-dependent biases. For example, if species $<100$ g are
excluded, the empirical mass-density slope steepens from $-0.77$ to $-0.85$.
Considering how variations to the underlying energetic parameters driving
consumer-resource dynamics alters the expected mass-density relationship may
shed light on key constraints shaping mammalian communities. We next explore
how variations in the vital rates included in the consumer-resource model
modify the expected intercept and slope of the mammalian mass-density
relationship. Different vital rates impact the mass-density relationship in
three distinct ways, by either _i_) influencing only the mass-density slope,
_ii_) influencing only the mass-density intercept, or _iii_) influencing both.
Aside from the resource growth rate and carrying capacity, our framework also
includes the intrinsic consumer reproductive rate $\lambda^{\rm max}_{C}$, the
consumer yield coefficient $Y_{C}$, and the maximum rate of starvation
$\sigma^{\rm max}$. We introduce changes to these rates as, for example,
$\lambda^{\rm max\prime}_{C}=\lambda^{\rm max}_{C}(1+\chi)$, where
$\chi\in(-1,2)$ represents the proportion increase or decrease of the altered
parameter denoted by ′. We note that the recovery rate $\rho$ is sufficiently
small that alterations do not have an influence on either the consumer mass-
density intercept or slope (fig. B1).
Figure B1: The predicted mass-density relationship with $\rho$ set to its
allometric quantity versus $\rho=0$. The two relationships cannot be
distinguished, allowing us to extract insight from the consumer steady state
(eq. 7, main text) without its inclusion.
Ignoring the effects of $\rho$, we can more easily intuit analytical
expressions of the steady state conditions for both the consumer $C$ and
resoure $R$, where
$\displaystyle C^{*}$ $\displaystyle=\alpha kY_{C}\frac{\sigma^{\rm
max}-2\lambda_{C}^{\rm max}+\sqrt{8{\sigma^{\rm max}}^{2}+(\sigma^{\rm
max}-2\lambda_{C}^{\rm max})^{2}}}{4{\sigma^{\rm max}}^{2}}$ $\displaystyle
R^{*}$ $\displaystyle=k\frac{\sigma^{\rm max}-2\lambda_{C}^{\rm
max}+\sqrt{8{\sigma^{\rm max}}^{2}+(\sigma^{\rm max}-2\lambda_{C}^{\rm
max})^{2}}}{4\sigma^{\rm max}}.$ (A17)
We thus observe that the consumer steady state can also be expressed as
$C^{*}=\frac{\alpha}{\sigma^{\rm max}}Y_{C}R^{*}.$ (A18)
We discuss how the specific mass-scalings of the relationships impacting the
steady states provide more intuition into Damuth’s mass-density relationship
in the main text.
We can gain additional insight into the role of each vital rate by exploring
their quantitative effects on the mass-density relationship directly. Changes
to the starvation rate have a large effect on both the consumer-density
intercept and slope (Figs. B2,B3). We observe that decreasing $\sigma^{\rm
max}$ from the expected value ($\chi<0$) serves to increase the steady state
intercept, while decreasing the mass-density slope. By comparison, increasing
$\sigma$ from the expected value ($\chi>0$) has less effect on the mass-
density relationship. In the consumer-resource model described in Eq. 2.3
(main text), starvation is the primary source of consumer mortality, and
therefore plays an out-sized role in determining consumer steady states. As
this mortality is reduced, consumer densities increase, raising the intercept.
However, as consumer starvation rates decline we observe a steeper mass-
density slope. Reduced starvation rates therefore principally benefit the
steady state densities of smaller species, with reduced effects observed for
larger-bodied mammals. Because fat biomass scales super-linearly with body
mass (see Table 1, main text), the populations of larger consumers are more
resilient to the effects of starvation, whereas those of smaller consumers are
more prone.
Figure B2: The effects of changes to metabolic parameters on the prediction
of the mass-density relationship. Figure B3: The effects of changes to
metabolic parameters on the prediction of the mass-density relationship.
The consumer’s maximal rate of reproduction $\lambda^{\rm max}_{C}$ influences
only the mass-density slope except for the case $\chi\rightarrow-1$, where
growth becomes zero. Above this trivial limit, we observe the consumer growth
rate to have a negative effect on the mass-density slope, such that as the
growth rate increases, the mass-density relationship becomes steeper (Figs.
B2,B3). As the intercept does not change, this means that the steady states of
larger bodied consumers decline with increasing $\lambda_{C}^{\rm max}$, while
those of smaller-bodied consumers remain unaltered, though the effect is
slight. Of more interest is the effect of the yield coefficient $Y_{C}$ and
starvation rate $\sigma^{\rm max}$ (Figs. B2,B3). The yield coefficient
represents the conversion of resources to consumer biomass, where an increase
in $\chi$ correlates to large increases in consumer steady state without
altering the mass-density slope. Here we observe that increased efficiency in
converting resource to consumer biomass will have an effect similar to
increasing resource productivity, as the effective abundance of the resource
is greater when relatively fewer resources fuel a given unit of consumer
biomass. Because $Y_{C}\propto E_{d}$, where $E_{d}$ is the energy density of
the resource (see methods), resource quality is therefore expected to
translate directly to higher consumer steady state densities.
## Appendix C: Mortality from predation
Per-capita mortality rate from predation The per-capita mortality rate from
predation of the herbivore consumer with mass $M_{C}$ and population density
$C$ by a mammalian predator with body mass $M_{P}$ and population density $P$
is given by
$\beta(C,P)=w\frac{\lambda_{P}(C)P}{CY_{P}},$ (A19)
where $\lambda_{P}(C)$ is the growth rate of the predator, $Y_{P}$ is the
predator yield coefficient, describing the grams of predator produced per gram
of prey consumed, and $w$ is the degree of predation intensity ($w=1$ denotes
high predation intensity, whereas $w<1$ denotes lower predation intensity).
Assuming a linear functional response for predation mortality,
$\lambda_{P}(C)$ is maximized to $\lambda_{P}^{\rm max}$ when the consumer
reaches its theoretical maximum population density, which we calculate by
converting the resource carrying capacity directly to grams of consumer
produced, or $C^{\rm max}=Y_{C}k$. The growth rate of the predator is then
given by
$\lambda_{P}(C)=\lambda_{P}^{\rm max}\frac{C}{C^{\rm max}}=\lambda_{P}^{\rm
max}\frac{C}{Y_{C}k}.$ (A20)
Together, we observe the per-capita mortality rate to be (as expected)
independent of the consumer density $C$, and is simplified to
$\beta(P)=w\frac{\lambda_{P}^{\rm max}P}{Y_{P}Y_{C}k},$ (A21)
where we assume that the predator population remains at empirically measured
steady state densities for mammalian carnivores, where $P\equiv
P^{*}=P_{0}M_{P}^{-0.88}$ [carbone2002common]. This assumption is required
because the effects of predation are implicit rather than explicit, and
effectively assumes that predator populations operating far below this
relationship are not viable. While there is bound to be a range of viable
densities for a predator of a given body size, that mass-density relationships
exist at all indicates that population densities are highly constrained over
evolutionary time and therefore represent a predator energetic demand as a
function of body size. Accordingly, if the predator mass-density relationship
$P^{*}$ represents an expected energetic requirement for a functioning
predator population, our assumption of predation as a constant, rather than
dynamic, influence on herbivore mortality reveals the dynamic consequence of
such energetic relationships. We suggest that it is these energetic mismatches
that may constrain longer-timescale macroevolutionary forces, even if the
shorter-timescale ecological dynamics may be more idiosyncratic and complex
than our minimal model captures.
Herbivore and predator yields As described in the main text, consumer yield is
calculated
$Y_{C}=\frac{M_{C}E_{d}}{\int_{0}^{t_{\lambda_{C}}}B_{0}m(t)^{\eta}{\rm dt}},$
(A22)
where $E_{d}$ is the energy density of the plant resource $R$ (Joules/g) and
the denominator is the lifetime energy use required by the herbivore consumer
to reach maturity (Joules). The parameters $t_{\lambda_{C}}$ and $B_{0}$ are
the timescale associated with reaching reproductive maturity and the metabolic
coefficient for herbivorous mammals, respectively, and $\eta=-3/4$ is the
metabolic exponent (see Table 1, main text). The predator yield is calculated
similarly, where
$Y_{P}=\frac{M_{C}E_{C}}{\int_{0}^{t_{\lambda_{P}}}{B_{0}}_{P}m(t)^{\eta}{\rm
dt}},$ (A23)
where $E_{C}$ is the energy density of the herbivore being consumed, and the
denominator is the lifetime energy use required by the predator to reach
maturity. The parameters $t_{\lambda_{P}}$ and ${B_{0}}_{P}$ are the timescale
associated with reaching reproductive maturity and the metabolic coefficient
for predatory mammals, respectively, and $\eta=-3/4$ is the metabolic
exponent. We note that the metabolic coefficient for predators is different
than that for mammals [MunozGarcia2005].
The energy density of herbivore consumers changes with body mass $M_{C}$. For
example, small mammals have very low percent body fat, whereas very large
mammals have high percent body fat. We assume that predators consume all non-
skeletal mass of prey. Because the amount of consumable tissues with different
energy densities within an herbivore varies allometrically, so too should the
energy density $E_{C}$. We consider four primary tissue groups: a consumable
set composed of muscle, fat, and _other_ tissues, and an non-consumable set
composed only of skeletal tissues. If the scalings associated with fat,
muscle, and skeletal tissues are $M_{C}^{\rm fat}=f_{0}M_{C}^{1.19}$,
$M_{C}^{\rm musc}=g_{0}M_{C}^{1.00}$, and $M_{C}^{\rm skel}=h_{0}M_{C}^{1.09}$
[with normalization constants $f_{0}=0.02$, $g_{0}=0.38$, and $h_{0}=0.0335$;
prange1979scaling], the scaling of the _other_ tissue (gut tissue, organ
tissue, etc) is given by $M_{C}^{\rm other}=M_{C}-(M_{C}^{\rm fat}+M_{C}^{\rm
musc}+M_{C}^{\rm skel})$. The energy density of fat is $E_{\rm fat}=37700$
J/g, whereas the energy density of muscle is $E_{\rm musc}=17900$ J/g
[merrill1973part]. If we assume that gut and organ tissues have roughly the
same energy density as muscle, the attainable energy density for an herbivore
of size $M_{C}$ is given by
$E_{C}(M_{C})=E_{\rm fat}\frac{M_{C}^{\rm fat}}{M_{C}}+E_{\rm
musc}\left(\frac{M_{C}^{\rm musc}}{M_{C}}+\frac{M_{C}^{\rm
other}}{M_{C}}\right).$ (A24)
Large-bodied Predator-Prey Mass Relationship (PPMR) The predator growth rate
$\lambda_{P}^{\rm max}$, the time required for the predator to reach
reproductive maturity $t_{\lambda_{P}}$, and the predator’s steady state
population density $P^{*}$ are allometric relationships that depend on
predator body mass $M_{P}$. Accordingly, for an herbivore of a given mass
$M_{C}$, we must anticipate the size of its likely predator $M_{P}$. This is
very different than the predator-centric perspective of anticipating the
average prey size for a given predator [Carbone:1999ju]. For example, the most
preferred prey mass for an African lion is ca. $350$ kg [hayward2005lion],
where the inclusion of megaherbivores to diet is comparatively low. However
from a megaherbivore’s perspective, lions may represent the only potential
predator. In other words, because the range of prey body mass increases for
predators of larger body mass [Sinclair2003], it is the upper limit of the
range that impacts the populations of larger herbivores.
To obtain an herbivore-centric measure of the expected predator mass given a
particular herbivore mass ${\rm E}\\{M_{P}|M_{C}\\}$, we first compiled the
known diets of large-bodied predators, including tigers, lions, hyenas,
leopards, dhole, wild dogs, and cheetahs [hayward2005lion, Hayward2006hyena,
hayward2006leopard, hayward2006lycaon, hayward2006cheetah, Hayward2008].
Because smaller mammalian predators and prey have very different PPMRs than
larger-bodied mammalian predators and prey, we here focus exclusively on the
predators of large-bodied herbivore prey $>10^{5}$ g. From the mean
proportional reliance of predators on large-bodied prey [hayward2005lion,
Hayward2006hyena, hayward2006leopard, hayward2006lycaon, hayward2006cheetah,
Hayward2008], we repeatedly sampled predator dietary distributions to reflect
each predator’s reliance as a function of prey mass. We introduced variability
in predator and prey masses by assuming that body sizes were normally
distributed about the expected value with a standard deviation of $\pm 25\%$,
allowing us to obtain a distribution of expected predator diets as a function
of prey mass. From this relationship, we then evaluated the expected predator
mass for a given prey mass range to obtain ${\rm E}\\{M_{P}\\}$ (Fig. 4b, main
text), demonstrating the allometric relationship of ${\rm
E}\\{M_{P}\\}=9.76\times 10^{3}M_{C}^{0.21}$, where we used the output of 100
independent replicates to robustly estimate the best fit. We emphasize that
this relationship only pertains to large-bodied predators and prey $>10^{5}$
g. Alterations to and variations from this relationship are explored in the
main text.
Including the empirically-measured PPMR (or a variant of the empirically
measured PPMR – see below) results in the appearance of a transcritical
bifurcation at consumer mass $M_{C}^{\dagger}$. We observe that this critical
mass threshold results in the extinction of the consumer population
characterized by body sizes $M_{C}\geq M_{C}^{\dagger}$ (fig. C1A). At this
body mass, the Determinant of the Jacobian matrix characterizing the system
presented in eq. 3 (main text) with predation mortality included (eq. 8, main
text) is zero (fig. C1B), aligning with the real component of a single
eigenvalue crossing zero and becoming positive (fig. C1C). While we do not
derive a normal form for this bifurcation, these features strongly suggest the
observed bifurcation is transcritical in nature [Kuznetsov1995].
Figure C1: A. Consumer population density as a function of mass $M_{C}$. B.
The determinant of the Jacobian matrix for the system presented in eq. 3 (main
text) with predation mortality included (eq. XX, main text) as a function of
mass $M_{C}$. C. The Real component of the two eigenvalues of the Jacobian
assessed in panel B. as a function of mass $M_{C}$. The dashed line in each
panel is the measured consumer mass threshold $M_{C}^{\dagger}=2.5*10^{6}$ g.
The Gray horizontal line in panels B,C denotes zero on the y-axis.
As explored in the main text, the empirically-measured PPMR for large-bodied
mammals results in a threshold body size for herbivore consumers
$M_{C}^{\dagger}$. This size marks the point where the predator population,
with a body mass derived from the PPMR, cannot sustain its own growth from the
predated herbivore population, thereby driving the herbivore population to
extinction. The size at which $M_{C}^{\dagger}$ occurs is both dependent on
the nature of the PPMR, as well as predation intensity $w$. As predation
intensity $w$ decreases, $M_{C}^{\dagger}$ increases (Fig. C2).
Figure C2: The effect of changing the predation intensity $w$ on the single
herbivore consumer population. If $w=1$, predation intensity is maximized, is
lower if $0<w<1$, and is above the maximal level required to support a
predator population if $w>1$. The blue region denotes herbivore threshold mass
range characterizing $w=1\pm 0.1$. The yellow line denotes the mass range of
contemporary elephantids. Vertical dashed line denotes the size of the largest
terrestrial mammal (_Deinotherium_ at ca. $1.74\times 10^{7}$, corresponding
to $w=0.37$, such that predation intensity is moderate.
By allowing the PPMR to vary as
${\rm E}\\{M_{P}\\}=v_{0}(1+\chi_{\rm int})M_{C}^{v_{1}(1+\chi_{\rm slope})},$
(A25)
where the proportional changes in the PPMR intercept and slope are given by
$\chi_{\rm int}$ and $\chi_{\rm slope}\in(-0.99,2)$, so does the threshold
herbivore body mass $M_{C}^{\dagger}$ and, by extension, the related threshold
predator body mass $M_{P}^{\dagger}$. From Fig. 4D (main text), we observe
that changing the intercept and slope of the PPMR has a large influence on
$M_{C}^{\dagger}$ and $M_{P}^{\dagger}$. Across this range of potential PPMRs,
we highlight those values for the intercept and slope of the PPMR that permit
megatrophic interactions, where both megapredators subsist on megaherbivores
at the threshold body mass (highlighted region in Fig. 4D, main text). Fig. C3
shows the relationship between megapredator and megaherbivore body masses
highlighted within this region. Allowing both the PPMR to vary and assuming
lower predation intensity ($w=0.37$) enables much larger body sizes for
megaherbivores and their associated megapredators (Fig. C4).
Figure C3: Mass ranges corresponding to feasible megatrophic interactions
(where herbivore and predator threshold masses are $>5\times 10^{5}$ g) across
variations to the assumed PPMR, demarcated by the white bands in Fig. 4C,D
(main text), and assuming high predation intensity ($w=1$). Figure C4: The
effects of lower predation intensity ($w=0.37$) on A. threshold herbivore mass
$M_{C}^{\dagger}$ and B. threshold predator mass $M_{P}^{\dagger}$ across
variable PPMRs, where ${\rm E}\\{M_{P}\\}=v_{0}(1+\chi_{\rm
int})M_{C}^{v_{1}(1+\chi_{\rm slope})}$ and both $\chi_{\rm int}$ and
$\chi_{\rm slope}\in(-0.99,2)$, and $v_{0}=9.76\times 10^{3}$ and $v_{1}=0.21$
are set as in the main text. White bands denote regions of $\chi_{\rm int}$
and $\chi_{\rm slope}$ where megatrophic interactions are feasible (i.e. both
predator and herbivore threshold masses are $>5\times 10^{5}$ g). C. Mass
ranges corresponding to feasible megatrophic interactions in the white bands
in A. and B. Figure C5: The effect of an altered carrying capacity $k$ on
the consumer mass threshold $M_{C}^{\dagger}$. Carrying capacity values that
vary from -20% to +20% give rise to an $M_{C}^{\dagger}$ roughly 65% and 140%
the original estimated mass of $2.55\times 10^{6}$ g. Equivalent variations in
$\alpha$ will result in the same changes to $M_{C}^{\dagger}$ given its role
in eq. 7 (main text). The shaded region highlights the mass range captured by
the altered carrying capacity.
Finally, we note that changes to both the resource growth rate $\alpha$ as
well as to the resource carrying capacity $k$ can impact the consumer size at
which populations become infeasible $M_{C}^{\dagger}$. Because the consumer
steady state is directly proportional to both of these parameters (see eq. 7,
main text), a lower growth rate and/or carrying capacity lowers the intercept
of the steady state mass-density relationship $C^{*}(M_{C})$. Analysis of the
effect of changes to $k$ (and this will be the same for $\alpha$) reveals that
while it has influence on $M_{C}^{\dagger}$, it is not incredibly large (Fig.
C5. However this relationship carries with it an important message: in
environments with lower carrying capacities and/or plant growth rates, we
would expect a lower mass threshold bounding feasible megaherbivore
populations.
## Appendix D: Derivation of harvesting mortality
We first determined the harvest rate $h=h^{\dagger}$ required to drive an
herbivore population to extinction, thereby satisfying the condition
$C^{*}(M_{C}|h)=0$ as a function of herbivore body mass $M_{C}$. This
extinction-inducing harvest rate, itself now a function of consumer body mass
$h^{\dagger}(M_{C})$, defines the rate at which the population must be
harvested to drive the steady state to zero. To compare this rate against
measures of harvest both in nature and predicted from other mathematical or
computational treatments of harvest-induced extinction, we calculated the
harvest pressure $\psi^{\dagger}$, which we defined as the number of herbivore
individuals per area harvested at this rate to reduce the population to some
proportion $\epsilon$ of its steady state. This harvest pressure is thus
defined by some number of individuals harvested per year over a certain number
of years to reduce the population from $C^{*}$ to its post-harvest density
$\epsilon C^{*}$.
To calculate harvest pressure, we first assume that at the steady state,
harvest is occurring on a shorter-than-generational timescale. For
megaherbivores such as elephants, a generation is roughly 25 years
[wittemyer2013comparative], and for harvest pressures that must be applied
beyond this period of time, we would expect population growth to counter the
negative effects of harvest. Assuming harvest-only change, we simplify the
dynamics to
$\frac{\rm d}{\rm dt}C=-h^{\dagger}(M_{C})C,$ (A26)
where the time to reduce $C^{*}$ to $\epsilon C^{*}$ is
$\displaystyle C(t)$ $\displaystyle=C_{0}{\rm e}^{-h^{\dagger}(M_{C})t},$
$\displaystyle\epsilon C^{*}$ $\displaystyle=C^{*}{\rm
e}^{-h^{\dagger}(M_{C})t}$ $\displaystyle t_{\epsilon}$
$\displaystyle=-\frac{\log(\epsilon)}{h^{\dagger}(M_{C})}.$ (A27)
We note that for elephant-sized herbivores and larger, $t_{\epsilon}\leq 23$
years. While the time required to harvest the population to $\epsilon C^{*}$
is only just approaching generational timescales, it should be treated as a
minimum $t_{\epsilon}$ given the effects of population growth will prolong the
imposed harvest effort. Harvest pressure is then calculated as
$\psi^{\dagger}=\frac{C^{*}(1-\epsilon)}{M_{C}t_{\epsilon}}c_{0}=-h^{\dagger}(M_{C})\frac{C^{*}(1-\epsilon)}{M_{C}\log(\epsilon)}c_{0}$
(A28)
where the constant $c_{0}$ denotes the conversion from inds/m2/second to
inds/$A_{\rm CA}$/year, where $A_{\rm CA}=4.24\times
10^{11}\leavevmode\nobreak\ {\rm m}^{2}$ is the arbitrarily-chosen area of
California. This conversion is particularly important for evaluating other
harvest measures from the historical record and estimates from independent
models and simulations for extinct species. As described in the main text, the
extinction-inducing harvest pressure is calculated to be $4.3\times 10^{3}$
inds/yr/$A_{\rm CA}$ for an elephant-sized organism of $M_{C}=2.5\times
10^{6}$ g (see Fig 5, main text).
Harvest pressure on Pleistocene mammoths We compare our measure of harvest
pressure to that calculated for mammoths (_Mammuthus primigenius_) by
fordham2022process. Because fordham2022process employ a much more specific and
detailed assessment of the effects of harvest specifically for mammoths over a
spatially explicit landscape, we must make a few simplifications in order to
derive a comparable estimate. First, the harvest interaction between mammoth
populations and humans is modeled as a Type II functional response, where,
again isolating population-level effects to that of harvest we obtain
$\frac{\rm d}{\rm dt}C=-\frac{sNFC}{G+\frac{C}{C_{\rm max}M_{C}}},$ (A29)
where $N$ is the normalized human population density maximized at unity, the
constant $s=7.884\times 10^{-8}$ generations/second (where a generation is 25
years), $F$ represents the effectiveness of human hunting, ranging from
$(0.01,0.34)$, $C_{\rm max}=1.875\times 10^{-6}$ g/${\rm m^{2}}$ is the
maximum mammoth population density (converted from the average degree-by-
degree grid cells in Siberia), $G=0.4$ is the half-saturation constant, and
$M_{C}=2.5\times 10^{6}$ grams.
Solving for the time required to reduce the population to $\epsilon C^{*}$, we
obtain
$t_{\epsilon}^{\rm mammoth}=\frac{C^{*}-C_{\rm
max}GM_{C}\log\left[C^{*}\exp(\frac{C^{*}\epsilon}{C_{\rm
max}GM_{C}})\epsilon\right]}{sC_{\rm max}FM_{C}N}.$ (A30)
We then calculate the harvest pressure as
$\psi^{\rm mammoth}=c_{0}\frac{sC_{\rm
max}FN(C^{*}-C^{*}\epsilon)}{C^{*}-C_{\rm
max}GM_{C}\log\left[C^{*}\exp(\frac{C^{*}\epsilon}{C_{\rm
max}GM_{C}})\epsilon\right]},$ (A31)
where the constant $c_{0}$ again denotes the conversion from inds/m2/second to
inds/$A_{\rm CA}$/year, where $A_{\rm CA}$ is the arbitrarily-chosen area of
California. Given a range in $F\in(0.01,0.35)$ and $N\in(0.01,1)$, we obtain a
distribution of values for mammoth harvest pressure with a median value of
$1.24\times 10^{4}$ inds/yr/$A_{\rm CA}$ over the course of 9.8 years. The
bounds of the estimated range from $5\times 10^{4}$ inds/yr/$A_{\rm CA}$ over
the course of 2 years to $5\times 10^{2}$ inds/yr/$A_{\rm CA}$ over the course
of ca. 200 years (the range is plotted as the vertical black line in Fig. 5,
main text).
Harvest pressure on Pleistocene _Diprotodon_ The harvest rate needed to
collapse _Diprotodon_ populations was calculated by Bradshaw et al.
[bradshaw2021relative], where a harvest pressure of between 400-500
inds/year/area of Australia was sufficient to collapse the population.
Translating this to the area of California, we obtain between 678 to 848
inds/yr/$A_{\rm CA}$, with a mean of $763.2$ inds/yr/$A_{\rm CA}$.
Harvest pressure on historical elephants _Loxodonta africana_ Elephant harvest
rates are estimated from historical documentation of the ivory trade detailed
in milner1993exploitation. While the trade volume oscillates with changes in
technology, access to habitats within Africa, and the feedbacks of trade on
elephant population size, we compare our results against estimates taken at
two points in time: early in the ivory trade (1810), and late in the ivory
trade (1987). From milner1993exploitation we assume that each elephant killed
contributes 1.88 tusks, and that tusk mass begins at 15 kg per tusk early in
trade to 5 kg per tusk in later years. While the area from which elephants
were harvested is largely unknown, we assume the area harvested is that
assessed to be suitable elephant habitat in sub-Saharan Africa, estimated at
$3.22\times 10^{12}\leavevmode\nobreak\ {\rm m}^{2}$ [thouless2016african].
From rates of ca. $1\times 10^{5}$ kg/yr of ivory harvested in 1810 to ca.
$9.7\times 10^{5}$ kg/yr of ivory harvested in 1987, normalized to habitat
area and converted to the area of California, we obtain estimates of ca. $467$
inds/yr/$A_{\rm CA}$ in 1810 to ca. $1.33\times 10^{4}$ inds/yr/$A_{\rm CA}$
in 1987 (see Fig. 5, main text).
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|
# Unitary Howe dualities for fermionic and bosonic algebras and related Dirac
operators
Guner Muarem Campus Middelheim
Middelheimlaan 1
M.G.221
2020 Antwerpen
België<EMAIL_ADDRESS>
###### Abstract.
In this paper we use the canonical complex structure $\mathbb{J}$ on
$\mathbb{R}^{2n}$ to introduce a twist of the symplectic Dirac operator. As a
matter of fact, these operators can be interpreted as the bosonic analogues of
the Dirac operators on a Hermitian manifold. Moreover, we prove that the
algebra of these symplectic Dirac operators is isomorphic to the Lie algebra
$\mathfrak{su}(1,2)$ which leads to the Howe dual pair
$(\mathsf{U}(n),\mathfrak{su}(1,2))$.
## 1\. Introduction
The CCR (canonical commuting relation) and CAR algebras (canonical
anticommuting relation) are fundamental algebras in theoretical physics used
for the study of bosons and fermions. From a mathematical viewpoint, these
algebras are named the Weyl algebra (or symplectic Clifford algebra) and
Clifford algebra. These algebras can be constructed in a very analogous way.
The Clifford algebra is constructed on a vector space $V$ equipped with a
symmetric bilinear form $B$, whereas the Weyl algebra requires an even
dimensional vector space equipped with a skew-symmetric bilinear form (or
symplectic form) $\omega$. In both cases, one then constructs the tensor
algebra $T(V)$ where an ideal $I(V)$ is divided out. In the orthogonal
setting, this is the ideal $I_{B}(V)$ with elements subject to the relation
$\\{u,v\\}=2B(u,v)$. In the symplectic setting, this is the ideal
$I_{\omega}(V)$ generated by $[u,v]=-\omega(u,v)$.
There is, however, a fundamental difference: the Clifford algebra is finite-
dimensional, whereas the Weyl algebra is infinite-dimensional. For the spinors
(orthogonal versus symplectic) the same infinite-dimensional principle holds
as for the Clifford algebras. As a matter of fact, the symplectic spinors are
the smooth vectors in the metaplectic representation [2]. Using the generators
of the Clifford (resp. Weyl) algebra, one can associate a natural first order
spin (resp. metaplectic) invariant differential operator by contracting the
Clifford algebra elements using the bilinear form $B$ (resp. the symplectic
form $\omega)$ with derivatives. This gives rise to the Dirac operator
$\partial_{x}=\sum_{k=1}^{n}e_{k}\partial_{x_{k}}$ where
$\\{e_{j},e_{k}\\}=-2\delta_{ij}$ and the symplectic Dirac operator
$\sum_{k=1}^{n}\left(iq_{k}\partial_{y_{k}}-\partial_{q_{k}}\partial_{x_{k}}\right)$
where $[\partial_{q_{j}},iq_{k}]=i\delta_{jk}$ are the Heisenberg relations.
The theory which studies the solutions of the Dirac operator is known as
Clifford analysis and can be seen as a hypercomplex function theory. Moreover,
quite some generalisations have occurred in the last two decades. This
involves e.g. Clifford analysis on superspace and Clifford analysis on
(hyper)Kähler spaces (see for instance [3]). It is in the latter framework in
which this paper is situated, but then from a symplectic point of view. More
precisely, we provide the foundations of what we will call a hermitian variant
of symplectic Clifford analysis, where we incorporate the additional datum of
a compatible complex structure $\mathbb{J}$ on the flat symplectic space
$\mathbb{R}^{2n}$. This leads to the study of the symplectic Dirac operator on
a Kähler manifold as was already initiated in [2, 4]. However, the underlying
invariance symmetry and the algebra generated by these type of operators (and
their duals) was never investigated.
## 2\. Rudiments of symplectic Clifford analysis
Let us consider the canonical symplectic space $\mathbb{R}^{2n}$ with
coordinates $({x},{y})$ and the usual symplectic form
$\omega_{0}=\sum_{j=1}^{n}dx_{j}\wedge dy_{j}$ which has the matrix
representation
$\Omega_{0}=\begin{pmatrix}0&I_{n}\\\ -I_{n}&0\end{pmatrix}.$
Recall that the symplectic group $\mathsf{Sp}(2n,\mathbb{R})$ is the group
given by invertible linear transformations preserving the non-degenerate skew-
symmetric bilinear form from above and is given (in terms of matrices) by
$\mathsf{Sp}(2n,\mathbb{R})=\\{M\in\mathsf{GL}(2n,\mathbb{R})\mid
M^{T}\Omega_{0}M=\Omega_{0}\\}.$
The group is non-compact and has dimension $2n^{2}+n$. Moreover, the
corresponding Lie algebra is denoted by $\mathfrak{sp}(2n,\mathbb{R})$. The
main difference with the orthogonal case, lies in the fact that the
metaplectic group (the double cover of the symplectic group) does not admit a
finite dimensional representation (it is not a matrix group). This is a strong
contrast with the spin representation in the orthogonal case. Moreover, the
orthogonal spinors $\mathbb{S}$ are realised as a idempotent left ideal in the
Clifford algebra, which is not the case for the symplectic spinors. As
mentioned, the symplectic equivalent of the spin representations are infinite
dimensional, which means that one needs to work with the theory of unitary
representations.
### 2.1. The Schwartz space and metaplectic representation
For further convenience, we fix notation and define the Schwartz space, which
plays a crucial role in the construction of the metaplectic representation. On
the space $\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{C})$ we define (using
the multi-index notation) the norm
$||f||_{\alpha,\beta}:=\sup_{{q}\in\mathbb{R}^{n}}|{q}^{\alpha}(D^{\beta}f)({q})|$
for all $\alpha,\beta\in\mathbb{N}^{n}$. The Schwartz space
$\mathcal{S}(\mathbb{R}^{n})$ is the subspace of $L^{p}(\mathbb{R}^{n})$ (for
$1\leq p\leq\infty$) consisting of rapidly decreasing functions and is given
by
$\mathcal{S}(\mathbb{R}^{n},\mathbb{C}):=\\{f\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{C}):||f||_{\alpha,\beta}<\infty\text{
for all }\alpha,\beta\in\mathbb{N}^{n}\\}.$
We now describe (following [2]) the infinite-dimensional Segal-Shale-Weil
representation (also oscillator or metaplectic representation) of the
metaplectic group. The smooth vectors of the unitary representation
$\mathfrak{m}:\mathsf{Mp}(2n)\to\mathsf{U}(L^{2}(\mathbb{R}^{n}))$ coincide
with the Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ and are a model for the
symplectic spinors $\mathbb{S}^{\infty}$. Due to Stone-Von Neumann theorem the
representation is unique (up to unitary equivalence).
### 2.2. The symplectic Clifford algebra and the related Dirac operator
Let $(V,\omega)$ be a symplectic vector space. The symplectic Clifford algebra
$\mathsf{Cl}_{s}(V,\omega)$ is defined as the quotient algebra of the tensor
algebra $T(V)$ of $V$, by the two-sided ideal
$\mathcal{I}_{\omega}:=\\{v\otimes u-u\otimes v+\omega(v,u):u,v\in V\\}.$
In other words $\mathsf{Cl}_{s}(V,\omega):=T(V)/\mathcal{I}_{\omega}$ is the
algebra generated by $V$ in terms of the relation $[v,u]=-\omega(v,u)$, where
we have omitted the tensor product symbols. We refer to the symplectic
Clifford algebra on $\mathbb{R}^{2n}$ as the $n$th Weyl algebra
$\mathcal{W}_{n}$ with generators
$iq_{1},\dots,iq_{n},\partial_{q_{1}},\dots,\partial_{q_{n}}$ satisfying the
commutation relations $[q_{j},q_{k}]=0$ and
$[\partial_{q_{j}},q_{k}]=\delta_{jk}$.
Denote by $\mathcal{F}$ a suitable function space (e.g. the space of
polynomials, or smooth funtions).
###### Definition 2.1.
The symplectic Dirac operator on $(\mathbb{R}^{2n},\omega_{0})$ is the first-
order (in the base variables ${x}$ and ${y}$) differential operator acting on
a symplectic spinor-valued functions space
$\mathcal{F}\otimes\mathbb{S}^{\infty}$ given by
$D_{s}=\sum_{j=1}^{n}(iq_{j}\partial_{y_{j}}-\partial_{q_{j}}\partial_{x_{j}}).$
With respect to the symplectic Fischer inner product (see [5]), we obtain the
dual operator $X_{s}=\sum_{j=1}^{n}(iq_{j}{x_{j}}+\partial_{q_{j}}{y_{j}}).$
These operators satisfy the relations:
$\displaystyle[\mathbb{E}+n,X_{s}]=X_{s},$
$\displaystyle[\mathbb{E}+n,D_{s}]=-D_{s},$
$\displaystyle[D_{s},X_{s}]=-i(\mathbb{E}+n)$
where $\mathbb{E}=\sum_{j=1}^{n}(x_{j}\partial_{x_{j}}+y_{j}\partial_{y_{j}})$
is the Euler operator. In other words, the three operators give rise to a copy
of the Lie algebra $\mathfrak{sl}(2)$.
## 3\. The interaction with a complex structure
### 3.1. Definition of the twisted symplectic Dirac operators
We will now introduce a complex structure $\mathbb{J}$ on the symplectic
manifold $(\mathbb{R}^{2n},\omega_{0})$ which is compatible with the
symplectic form $\omega_{0}$. This means that $\omega_{0}(x,\mathbb{J}y)$
defines a Riemannian metric $g$. Otherwise said, we will be working with the
canonical Kähler manifold $(\mathbb{R}^{2n},\omega_{0},g,\mathbb{J})$. By
Darboux’s theorem, we obtain, with respect to the canonical symplectic basis
$\\{e_{j}\\}_{j=1}^{2n}$ the following complex structure
$\mathbb{J}=\left(\begin{smallmatrix}0&-I_{n}\\\
I_{n}&0\end{smallmatrix}\right).$ The action of the complex structure
$\mathbb{J}$ on $\mathbb{R}^{2n}$ is given by
$\displaystyle(x_{1},\dots,x_{n},y_{1},\dots,y_{n})\mapsto(y_{1},\dots,y_{n},-x_{1},\dots,-x_{n}).$
###### Definition 3.1.
The new differential operators acting on symplectic spinor-valued functions
$\displaystyle\tilde{D}_{s}$
$\displaystyle=\sum_{j=1}^{n}iq_{j}\partial_{x_{j}}+\partial_{y_{j}}\partial_{q_{j}}$
$\displaystyle\tilde{X}_{s}$
$\displaystyle=\sum_{j=1}^{n}x_{j}\partial_{q_{j}}-iy_{j}q_{j}$
$\displaystyle\mathbb{E}$
$\displaystyle=\sum_{j=1}^{n}x_{j}\partial_{x_{j}}+y_{j}\partial_{y_{j}}$
also give rise to a copy of the Lie algebra $\mathfrak{sl}(2)$. We call these
first two operators the twists of $D_{s}$ and $X_{s}$.
Both sets of operators, i.e. $(D_{s},X_{s})$ and
$(\tilde{D}_{s},\tilde{X}_{s})$, are symplectic invariant, albeit under the
following two different realisations of the symplectic Lie algebra given by:
$\displaystyle\begin{cases}X_{jk}=x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}}-(q_{k}\partial_{q_{j}}+\frac{1}{2}\delta_{jk})&1\leq
j\leq k\leq n\\\
Y_{jk}=x_{j}\partial_{y_{k}}+x_{k}\partial_{y_{j}}+i\partial_{q_{j}}\partial_{q_{k}}&j<k=1,\dots,n\\\
Z_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}+iq_{j}q_{k}&j<k=1,\dots,n\\\
Y_{jj}=x_{j}\partial_{y_{j}}+\frac{i}{2}\partial_{q_{j}}^{2}&j=1,\dots,n\\\
Z_{jj}=y_{j}\partial_{x_{j}}+\frac{i}{2}q_{j}^{2}&j=1,\dots,n\end{cases}$
and
$\displaystyle\begin{cases}\tilde{X}_{jk}=x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}}+q_{k}\partial_{q_{j}}+\frac{1}{2}\delta_{jk}&1\leq
j\leq k\leq n\\\
\tilde{Y}_{jk}=x_{j}\partial_{y_{k}}+x_{k}\partial_{y_{j}}-iq_{j}q_{k}&j<k=1,\dots,n\\\
\tilde{Z}_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}-i\partial_{q_{j}}\partial_{q_{k}}&j<k=1,\dots,n\\\
\tilde{Y}_{jj}=x_{j}\partial_{y_{j}}-\frac{i}{2}q_{j}^{2}&j=1,\dots,n\\\
\tilde{Z}_{jj}=y_{j}\partial_{x_{j}}-\frac{i}{2}\partial_{q_{j}}^{2}&j=1,\dots,n\end{cases}$
Of course, it is not very useful that $D_{s}$ and $\widetilde{D}_{s}$ are
invariant under different (yet isomorphic)
$\mathfrak{sp}(2n,\mathbb{R})$-realisations. Therefore, we will perform a
symmetry reduction so that both operators become invariant under one and the
same Lie algebra. To that end, we need to find the symplectic matrices which
commute with the complex structure.
###### Lemma 3.2.
We have that
$\mathsf{Sp}_{\mathbb{J}}(2n,\mathbb{R}):=\\{M\in\mathsf{Sp}(2n,\mathbb{R})\mid
M\mathbb{J}=\mathbb{J}M\\}$
defines a realisation for the unitary Lie group.
###### Proof.
In order to see this, assume that $M$ is of the block-form:
$\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right),$ where $A,B,C$
and $D$ are $(n\times n)-$matrices. The condition that $M$ is symplectic is
equivalent to the one of the following conditions: the matrices $A^{T}C$ and
$B^{T}D$ are symmetric and $A^{T}D-C^{T}B=I$. So, in order to determine
$\operatorname{Sp}_{\mathbb{J}}(2n,\mathbb{R})$ we need to determine the
symplectic matrices $M$ which commute with the complex structure $\mathbb{J}$.
The latter conditions means that
$\displaystyle M\mathbb{J}=\mathbb{J}M$
$\displaystyle\iff\mathbb{J}^{-1}M\mathbb{J}=M$
$\displaystyle\iff\begin{pmatrix}0&I\\\ -I&0\end{pmatrix}\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}$ $\displaystyle\iff\begin{pmatrix}D&-C\\\
-B&A\end{pmatrix}=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$
This implies $A=D$ and $B=-C$. In other words the matrix $M$ is of the form:
$M=\left(\begin{smallmatrix}A&B\\\ -B&A\end{smallmatrix}\right).$ Next, we
still have the condition that $M$ is symplectic, i.e.
$\displaystyle M^{T}\Omega M=\Omega$
$\displaystyle\iff\begin{pmatrix}A^{T}&C^{T}\\\
B^{T}&D^{T}\end{pmatrix}\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}\begin{pmatrix}A&B\\\ C&D\end{pmatrix}=\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}$ $\displaystyle\iff\begin{pmatrix}-C^{T}&A^{T}\\\
-D^{T}&B^{T}\end{pmatrix}\begin{pmatrix}A&B\\\
C&D\end{pmatrix}=\begin{pmatrix}-C^{T}A+A^{T}C&-C^{T}B+A^{T}D\\\
-D^{T}A+B^{T}C&-D^{T}B+B^{T}D\end{pmatrix}=\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}$ $\displaystyle\iff\begin{cases}A^{T}C=C^{T}A\\\
A^{T}D-C^{T}B=I\\\ B^{T}C-D^{T}A=-I\\\ B^{T}D=D^{T}B\end{cases}$
This means that $A^{T}C$ and $B^{T}D$ should be symmetric matrices and
$A^{T}D-C^{T}B=I$. But now, due to the first condition this reduces to
$B^{T}A=A^{T}B$ and $A^{T}A+B^{T}B=I$. In other words, the matrices we are
looking for must be of the form $M=\left(\begin{smallmatrix}A&B\\\
-B&A\end{smallmatrix}\right)$ with $B^{T}A=A^{T}B$ and $A^{T}A+B^{T}B=I$, i.e.
$\displaystyle\begin{pmatrix}A^{T}&-B^{T}\\\
B^{T}&A^{T}\end{pmatrix}\begin{pmatrix}A&B\\\
-B&A\end{pmatrix}=\begin{pmatrix}A^{T}A+B^{T}B&A^{T}B-B^{T}A\\\
B^{T}A-A^{T}B&B^{T}B+A^{T}A\end{pmatrix}=\begin{pmatrix}I&0\\\
0&I\end{pmatrix}$
Which is exactly the condition for a unitary matrix. The map
$\Phi:\mathsf{Sp}_{\mathbb{J}}(2n,\mathbb{R})\to\mathsf{U}(n):M\mapsto A+iB$
gives the wanted isomorphism. ∎
### 3.2. Unitary invariant symplectic Dirac operators
One can now check that the symplectic Dirac operator and its twist, are
unitary invariant differential operators. This can be done by verifying that
the operators commute with the following realisation of unitary Lie algebra
$\mathfrak{u}(n)$:
###### Lemma 3.3.
We have the following operator realisation of the Lie algebra
$\mathfrak{u}(n)$:
$\displaystyle\begin{cases}A_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}-x_{j}\partial_{y_{k}}-x_{k}\partial_{y_{j}}+i(q_{j}q_{k}-\partial_{q_{j}}\partial_{q_{k}})&\quad
1\leq j<k\leq n\\\
B_{jj}=y_{j}\partial_{x_{j}}-x_{j}\partial_{y_{j}}+\frac{i}{2}\left(q_{j}^{2}-\partial_{q_{j}}^{2}\right)&\quad
1\leq j\leq n\\\
C_{jk}=x_{j}\partial_{x_{k}}-x_{k}\partial_{x_{j}}+y_{j}\partial_{y_{k}}-y_{k}\partial_{y_{j}}+q_{j}\partial_{q_{k}}-q_{k}\partial_{q_{j}}&\quad
1\leq j<k\leq n\end{cases}$
This means that we can refine the $\mathfrak{sp}(2n)$-invariant PDE $D_{s}f=0$
into two $\mathfrak{u}(n)$-invariant PDEs given by $D_{s}f=0$ and
$\widetilde{D}_{s}f=0$, for a symplectic spinor valued polynomial
$f\in\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathcal{S}(\mathbb{R}^{n})$.
In analogy with the orthogonal case, we call the solutions hermitian
symplectic monogenics (or $h$-symplectic monogenics in short).
### 3.3. Symplectic Dolbeault operators
Moreover, there is a second way of introducing the twist of the symplectic
Dirac operator. Let us define the following operators which are known in the
literature as the symplectic Dolbeault operators [4] defined by means of
$D_{z}=\frac{D_{s}+i\widetilde{D}_{s}}{2}$
and
$D_{z}^{\dagger}:=\frac{D_{s}-i\widetilde{D}_{s}}{2}.$
One easily verifies that
$\displaystyle\frac{1}{2}(D_{s}+i\widetilde{D}_{s})$
$\displaystyle=-\sum_{j=1}^{n}\mathfrak{F}_{j}\partial_{z_{j}}\quad$
$\displaystyle\ \frac{1}{2}(D_{s}-i\widetilde{D}_{s})$
$\displaystyle=\sum_{j=1}^{n}\mathfrak{F}_{j}^{\dagger}\partial_{\overline{z}_{j}}.$
where we have introduced the symbols
$\mathfrak{F}_{j}=(q_{j}+\partial_{q_{j}})$ and
$\mathfrak{F}_{j}^{\dagger}=(q_{j}-\partial_{q_{j}})$. The structure of the
operators $D_{z}$ and $D_{z}^{\dagger}$ is similar to the orthogonal case.
However, the raising/lowering operators $\mathfrak{F}_{j}$ and
$\mathfrak{F}_{j}^{\dagger}$ are used instead of isotropic Witt vectors
$\mathfrak{f}_{j}$ and $\mathfrak{f}_{j}^{\dagger}$ (see for instance [6] and
the references therein).
### 3.4. Class of simultaneous solutions of $D_{s}$ and $\widetilde{D}_{s}$
We will now describe a wide class of examples of $h$-symplectic monogenics, by
making the link with holomorphic functions in several variables. Let
$f:\Omega\subset\mathbb{C}^{n}\to\mathbb{C}$ be a complex-valued function in
several complex variables which is of the class $\mathcal{C}^{1}(\Omega)$
(i.e. continuously differentiable). We say that $f$ is holomorphic (in several
variables) if $\partial_{\overline{z}_{j}}f(z_{1},\dots,z_{n})=0$ for all
$1\leq j\leq n$, where
$\partial_{\overline{z}_{j}}:=\frac{1}{2}(\partial_{x_{j}}+\partial_{y_{j}})$
is the Cauchy-Riemann operator in the relevant variable. Moreover, we denote
the set of holomorphic functions in $\Omega$ by $\operatorname{Hol}(\Omega)$.
In order not to overload notations, we use the summation convention. Suppose
that we have a function of the form
$F({x},{y},{q})=e^{-\frac{1}{2}|{q}|^{2}}H({x},{y})$. Letting the symplectic
Dirac operator act on $F$ gives:
$\displaystyle D_{s}\left(e^{-\frac{1}{2}|{q}|^{2}}H({x},{y})\right)$
$\displaystyle=(iq_{k}\partial_{y_{k}}-\partial_{x_{k}}\partial_{q_{k}})\left(e^{-\frac{1}{2}|{q}|^{2}}H({x},{y})\right)$
$\displaystyle=iq_{k}e^{-\frac{1}{2}|{q}|^{2}}\partial_{y_{k}}H({x},{y})+e^{-\frac{1}{2}|{q}|^{2}}q_{k}\partial_{x_{k}}H({x},{y})$
$\displaystyle=e^{-\frac{1}{2}|{q}|^{2}}q_{k}(\partial_{x_{k}}+i\partial_{y_{k}})H({x},{y})$
We note that this equals zero if
$(\partial_{x_{k}}+i\partial_{y_{k}})H({x},{y})=0$ for all $k=1,\dots,n$, i.e.
if $H({x},{y})$ is a holomorphic function in several variables. Completely
similar,
$\displaystyle\widetilde{D_{s}}\left(e^{-\frac{1}{2}|{q}|^{2}}H({x},{y})\right)$
$\displaystyle=(iq_{k}\partial_{x_{k}}+\partial_{y_{k}}\partial_{q_{k}})\left(e^{-\frac{1}{2}|{q}|^{2}}H({x},{y})\right)$
$\displaystyle=e^{-\frac{1}{2}|{q}|^{2}}q_{k}(i\partial_{x_{k}}-\partial_{y_{k}})H({x},{y})$
$\displaystyle=ie^{-\frac{1}{2}|{q}|^{2}}q_{k}(\partial_{x_{k}}+i\partial_{y_{k}})H({x},{y}),$
which is zero for holomorphic $H({x},{y})$.
This means that every function of the form
$e^{-\frac{1}{2}(q_{1}^{2}+\dots+q_{n}^{2})}H({x},{y})$ with $H$ an
holomorphic function is several variables is a solution of both $D_{s}$ and
$\widetilde{D}_{s}$. This observation generalises the class of solutions
obtained by Habermann in [2] for $n=1$. As a matter of fact, it turned out
that there are much more solutions than the ones of this form. In order to
describe these systematically, we will need the notion of Howe dualities and
corresponding Fischer decompositions. This will be done in full detail in our
upcoming paper [7]. In the following section we will reveal the algebraic
structures required for this approach.
## 4\. A unitary Howe duality associated with $D_{s}$ and $\widetilde{D}_{s}$
Recall that the Lie algebra $\mathfrak{su}(1,2)$ is a quasi-split real form of
the complex Lie algebra $\mathfrak{sl}(3)$ and is defined in terms of matrices
as
$\displaystyle\mathfrak{su}(1,2)=\left\\{\begin{pmatrix}\alpha&\beta&ic\\\
\gamma&\overline{\alpha}-\alpha&-\overline{\beta}\\\
id&-\overline{\gamma}&-\overline{\alpha}.\end{pmatrix}\mid c,d\in\mathbb{R}\
\&\ \alpha,\beta,\gamma\in\mathbb{C}\right\\}$
Some calculations now lead to observation that the Lie algebra generated by
the symplectic Dirac operators $D_{s},\widetilde{D}_{s}$ and their duals
$X_{s},\widetilde{X_{s}}$ gives rise to a copy of the Lie algebra
$\mathfrak{su}(1,2)$. In order to close the algebra, we introduce the
following differential operators:
$\displaystyle\mathcal{O}:=\sum_{j=1}^{n}i(x_{j}\partial_{y_{j}}-y_{j}\partial_{x_{j}})+\partial_{q_{j}}^{2}-q_{j}^{2},$
$\displaystyle\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}+\partial_{y_{j}}^{2}$
and $\displaystyle r^{2}:=\sum_{j=1}^{n}{x_{j}}^{2}+{y_{j}}^{2}.$
Together with symplectic Dirac operators $D_{s},\widetilde{D}_{s}$ and their
duals $X_{s},\widetilde{X}_{s}$ these operators satisfy the commutator
relations of $\mathfrak{su}(1,2)$. Moreover, their commutators are given by:
$[\cdot,\cdot]$ | $D_{s}$ | $\widetilde{D}_{s}$ | $\Delta$ | $X_{s}$ | $\mathbb{E}$ | $\mathcal{O}$ | $\widetilde{X}_{s}$ | $r^{2}$
---|---|---|---|---|---|---|---|---
$D_{s}$ | 0 | $\Delta$ | 0 | -$\mathbb{E}$ | $D_{s}$ | $-3\widetilde{D}_{s}$ | $-\mathcal{O}$ | $-2\widetilde{X}_{s}$
$\widetilde{D}_{s}$ | $-\Delta$ | 0 | 0 | $\mathcal{O}$ | $\widetilde{D}_{s}$ | $3D_{s}$ | $-\mathbb{E}$ | $2X_{s}$
$\Delta$ | 0 | 0 | 0 | $\widetilde{D}_{s}$ | $\Delta$ | 0 | $-2{D}_{s}$ | $\mathbb{E}$
$X_{s}$ | $\mathbb{E}$ | $-\mathcal{O}$ | $-\widetilde{D}_{s}$ | 0 | $X_{s}$ | $-3\widetilde{X}_{s}$ | $-r^{2}$ | 0
$\mathbb{E}$ | $-D_{s}$ | $-\widetilde{D}_{s}$ | $-\Delta$ | $-X_{s}$ | 0 | 0 | $-\widetilde{X}_{s}$ | 0
$\mathcal{O}$ | $3\widetilde{D}_{s}$ | $-3D_{s}$ | 0 | $3\widetilde{X}_{s}$ | 0 | 0 | $-3X_{s}$ | 0
$\widetilde{X}_{s}$ | $\mathcal{O}$ | $\mathbb{E}$ | $2D_{s}$ | $r^{2}$ | $\widetilde{X}_{s}$ | $-3X_{s}$ | 0 | 0
$r^{2}$ | $2\widetilde{X}_{s}$ | $-2X_{s}$ | $-\mathbb{E}$ | 0 | $-2r^{2}$ | 0 | 0 | 0
1. (1)
We have two copies of the Heisenberg algebra:
$\operatorname{Alg}\\{D_{s},\tilde{D}_{s},\Delta\\}\cong\operatorname{Alg}\\{X_{s},\widetilde{X}_{s},r^{2}\\}\cong\mathfrak{h}_{3}.$
2. (2)
We have two copies of the Lie algebra $\mathfrak{sl}(2)$
$\operatorname{Alg}\\{D,D^{\dagger},\mathbb{E}\\}\cong\operatorname{Alg}\\{\tilde{D},\tilde{D}^{\dagger},\mathbb{E}\\}\cong\operatorname{Alg}\\{\Delta,N,\mathbb{E}\\}\cong\mathfrak{sl}(2).$
This means that there is a canonical $\mathfrak{su}(1,2)$-action on the space
of spinor valued polynomials
$\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathcal{S}(\mathbb{R}^{n})$
where restricting to the subalgebra $\operatorname{Alg}(D_{s},X_{s})$
corresponds to $\mathfrak{sl}(2)$-copy obtained in the Howe duality for
symplectic Clifford analysis (see [5] for more details). Now, taking into
account the symplectic Dirac operators and its twists, we obtain the dual pair
$\mathsf{U}(n)\times\mathfrak{su}(1,2)$ (i.e. the underlying group of
invariance, together with the algebra generated by the operators and their
duals).
We now focus on the reduction of the symplectic spinor space. In the
orthogonal case, the spinor space $\mathbb{S}$ decomposes under the action of
the unitary group $\mathsf{U}(n)$ as
$\mathbb{S}=\bigoplus_{r}\mathbb{S}_{(r)},$ with $\mathbb{S}_{(r)}$
inequivalent irreducible pieces, which are eigenspaces of the fermionic
quantum oscillator (also called spin-Euler operator, see for instance [6]). In
the symplectic case, the relevant operator for decomposing the infinite
dimensional spinor space is the bosonic quantum oscillator. The hamiltonian of
the quantum oscillator, the so-called Hermite operator, is given by
$\displaystyle\mathcal{H}:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n}),\quad
f({q})\mapsto\frac{1}{2}\sum_{j=1}^{n}(\partial_{q_{j}}^{2}-q_{j}^{2})f({q}).$
Note that we can write
$\mathcal{O}=\sum_{j=1}^{n}i(x_{j}\partial_{y_{j}}-y_{j}\partial_{x_{j}})+2\mathcal{H},$
so that the Hermite operator is in fact the spinor-valued part of the operator
$\mathcal{O}$, i.e. the differential operator in $\partial_{q_{j}}$ and the
variables $q_{j}$. Moreover, the eigenspaces can be identified with the
irreducible decomposition of $\mathbb{S}^{\infty}$ into
$\mathfrak{u}(n)$-irreducible representations. This means that the symplectic
spinor space $\mathbb{S}^{\infty}$ decomposes into
$\mathfrak{u}(n)$-irreducible representations ${\widetilde{S}}^{\infty}_{(k)}$
of dimension ${n+k-1\choose k}$ which can be thought of as $k$-homogeneous
polynomials or the eigenspaces of the Hermite operator $\mathcal{H}$.
Moreover, the solutions of the corresponding Dirac operators, called
monogenics, can be introduced from a purely representation theoretical
viewpoint. In general, this boils down to determining the decomposition (this
is called a Fischer decomposition)
$\mathcal{P}_{k}(\mathbb{R}^{m},\mathbb{C})\otimes\mathbf{S}$ where
$\mathbf{S}$ is the spinor space, which is $\mathbb{S}$ in the orthogonal case
and $\mathbb{S}^{\infty}$ in the symplectic case, where we take $m=2n$ in
particular. Moreover, the space of $k$-homogeneous polynomials
$\mathcal{P}_{k}$ coincides with the $k$-symmetric power of the fundamental
representation of resp. the orthogonal or symplectic algebra. We denote by
$\mathcal{M}_{k}$ the $k$-homogeneous solutions of the Dirac operator
$\partial_{x}$, these are called monogenics. They can be defined as follows:
$\displaystyle\mathcal{M}_{k}\leftrightarrow(k,0,\dots,0)\boxtimes\mathbb{S}=(k)\boxtimes\left(\frac{1}{2},\dots,\frac{1}{2}\right)\cong\left(k+\frac{1}{2},\dots,\frac{1}{2}\right),$
where $\boxtimes$ denotes the Cartan product of the
$\mathfrak{so}(m)$-representations. In the symplectic case, we analoguously
obtain:
$\displaystyle\mathcal{M}_{k}^{s}\leftrightarrow(k,0,\dots,0)_{s}\boxtimes\mathbb{S}^{\infty}$
$\displaystyle=(k)_{s}\boxtimes\left(\left(-\frac{1}{2},\dots,-\frac{1}{2}\right)\oplus\left(-\frac{1}{2},\dots,-\frac{3}{2}\right)\right)$
$\displaystyle\cong\left(k-\frac{1}{2},\dots,-\frac{1}{2}\right)\oplus\left(k-\frac{1}{2},\dots,-\frac{3}{2}\right).$
In order to obtain an algebraic characterisation of the space of
$h$-symplectic monogenics, one proceeds as follows. First of all, we note that
we need to consider the symplectic spinors $\mathbb{S}^{\infty}$ from an
unitary viewpoint. We saw that $\mathbb{S}^{\infty}$ decomposes as an infinite
direct sum of finite dimensional $\mathfrak{u}(n)$-modules
${\widetilde{S}}^{\infty}_{(k)}$ which are in fact eigenspaces of the Hermite
operator. We denote the branched spinor space (which is in fact a direct sum
of $\mathfrak{u}(n)$-irreps) by $\widetilde{\mathbb{S}^{\infty}}$. However,
the space of $k$-homogeneous polynomials is not irreducible as a
$\mathfrak{u}(n)$-module and we denote the branched module by
$\widetilde{(k)}$. This means that we are left with the following Cartan
product
$\mathcal{M}_{k}^{hs}\leftrightarrow\widetilde{(k)}\boxtimes\widetilde{\mathbb{S}^{\infty}}$
as a representation theoretical definition of the $h$-symplectic monogenics.
Recall that this are the symplectic spinor-valued polynomial functions
$f\in\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathcal{S}(\mathbb{R}^{n})$
that satisfy the system of unitary unitary-invariant partial differential
equations
$\begin{cases}D_{s}f&=0\\\ \widetilde{D}_{s}f&=0\end{cases}$
The explicit calculation of the Cartan product (and more generally the tensor
product) will be done in [7]. Moreover, as an application we will prove a
Fischer decomposition for the Howe dual pair we obtained in this paper.
## 5\. Conclusion
In this paper we investigated a new Howe dual pair occurring in symplectic
Clifford analysis by allowing a compatible complex structure. This Howe
duality is of the form $(G,\mathfrak{g}^{\prime})$ where $G$ is the underlying
invariance group for which the relevant Dirac operators are invariant and
$\mathfrak{g}^{\prime}$ is the algebra generated by the Dirac operators and
their duals. Depending on the orthogonal or symplectic framework, we have the
following ‘types’ of Clifford analysis and refinements thereof:
1. (1)
Orthogonal geometry (giving rise to a Clifford algebra)
1. (a)
Clifford analysis: $\mathsf{SO}(m)\times\mathfrak{osp}(1|2)$
2. (b)
Hermitian Clifford analysis $\mathsf{U}(m)\times\mathfrak{osp}(2|2)$
3. (c)
Quaternionic Clifford analysis $\mathsf{USp}(m)\times\mathfrak{osp}(4|2)$
2. (2)
Symplectic geometry (giving rise to a Weyl algebra)
1. (a)
Symplectic Clifford analysis: $\mathsf{Sp}(2m)\times\mathfrak{sl}(2)$
2. (b)
Hermitian symplectic Clifford analysis:
$\mathsf{U}(m)\times\mathfrak{su}(1,2)$
3. (c)
Quaternionic symplectic Clifford analysis: $\mathsf{USp}(m)\times$?
Thus far, we extended the framework of hermitian Clifford analysis in the
presence of a symplectic structure in the case of the (flat) Kähler manifold
$\mathbb{R}^{2n}$. It is an interesting question to further reduce the
symmetry to the compact symplectic group $\mathsf{USp(n)}$ so that we have the
chain $\mathsf{Sp}(2n)\supset\mathsf{U}(n)\supset\mathsf{USp}(n)$. In our
furture work [7], we will describe the Fischer decomposition accompanying this
new Howe dual pair.
### Funding information
The author is supported by the FWO-EoS project ‘Symplectic Techniques in
Differential Geometry’ G0H4518N.
## References
* [1] D. Eelbode, G. Muarem, The orthogonal branching problem for symplectic monogenics, Advances in Applied Clifford Algebras 32, (2022)
* [2] K. Habermann, L. Habermann, Introduction to symplectic Dirac operators, In Lecture Notes in Mathematics. Springer Berlin Heidelberg (2006)
* [3] F. Brackx, H. De Schepper, D. Eelbode, R. Lávička, V. Soucek, $\mathfrak{osp}(4|2)$–monogenicity in Clifford analysis, In 15th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2015) (Vol. 1, pp. 240-243) (2015)
* [4] E. Korman, Symplectic Dolbeault operators on Kähler manifolds, Annals of Global Analysis and Geometry 44(3), 339-358 (2013)
* [5] H. De Bie, P. Somberg, V. Soucek, The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator, Journal of Geometry and Physics 75, 120-128 (2014)
* [6] D. Eelbode, Stirling numbers and spin-Euler polynomials, Experimental Mathematics, 16(1), 55-66 (2007)
* [7] Eelbode, D., Muarem G. (2022). A unitary refinement of the symplectic Fischer decomposition. In preparation.
|
# Security Issuance, Institutional Investors and Quid Pro Quo: Insights from
SPACs††thanks: We would like to thank Christian Opp, Joanna Wu, Michael
Gofman, and Bill Wilhelm for their helpful comments and suggestions. We thank
the 3Cavaliers at the University of Virginia for their generous research
support and Lam Bui and Hengxiang Cao for their excellent research assistance.
Gaurab Aryal Zhaohui Chen Yuchi Yao Chris Yung Department of Economics,
Washington University in St. Louis<EMAIL_ADDRESS>School of
Commerce, University of Virginia<EMAIL_ADDRESS>School of Business,
University of Rochester<EMAIL_ADDRESS>School of Commerce,
University of Virginia<EMAIL_ADDRESS>
###### Abstract
Security issuance is subject to informational and agency-related frictions.
However, it is difficult to separate their effects. We consider SPACs and
assess those effects separately. To this end, we identify premium investors
who produce value-relevant information: their participation is associated with
higher SPAC success and announcement-day returns. Non-premium investors,
however, engage only in quid pro quo arrangements. Their high returns today
from issuers (quid) mean they are more likely to participate in low-returns
future deals initiated by those issuers (quo). Thus quid pro quo is not pure
agency cost but partly a transfer that enables more firms to go public.
Keywords: Security Issuance, Institutional Investors, SPAC, Underpricing JEL
classification: G14, G32, G34.
## 1 Introduction
Firms issuing securities through intermediaries face two types of friction.
One, there may be an informational gap between them (issuers) and the
investors, which may prevent securities from being sold at full value. Second,
there is an agency problem if intermediaries do not maximize sale proceeds but
instead engage in quid pro quo arrangements with investors. Specifically,
intermediaries may favor certain buyers (investors) at the expense of the
issuers, expecting their favoritism to be returned through profitable future
transactions.
Ritter and Welch (2002) point out that both frictions coexist and emphasize
the need and difficulty in separating those frictions. In a survey paper,
Lowry et al. (2017) similarly pose this separation as an unresolved question,
using IPO underpricing as an example:
> Does [IPO underpricing] reflect a reward to institutions for sharing value-
> relevant information with underwriters, or does it reflect a quid pro quo
> arrangement where the bank is rewarding its best clients? [..] This is an
> immensely important issue and one that we hope future research will be able
> to address.
We revisit this problem in the context of special purpose acquisition
companies (SPACs) and assess and separate the effects of these frictions on
security issuance.111We focus on broader issues in security issuance rather
than questions specific to SPACs and their regulatory oversight. Thus we
sidestep questions about the issuer’s choice between traditional IPO and SPAC,
whether SPACs have misled public investors, and how SPACs should be regulated.
See Gahng, Ritter and Zhang (Forthcoming) and Klausner, Ohlrogge and Ruan
(2022) for comprehensive reviews. To this end, we identify two types of
institutional investors: “premium” and “non-premium” (defined shortly below).
We find evidence consistent with premium investors acting in the role of
information production and non-premium investors engaging in only quid pro quo
arrangements. However, unlike the extant literature that identifies any quid
pro quo as purely value-destroying, our estimates suggest that in SPACs, quid
pro quo can create value by enabling marginal businesses to access public
markets.
We can separate the effects of information and quid pro quo by relying on the
institutional features of SPACs. A SPAC is a publicly traded shell (i.e.,
blank check) corporation formed by a sponsor and public investors with the
sole purpose of effecting a merger with a private firm to enable the latter to
go public. As such, SPACs offer four research design advantages over a
traditional IPO.
First, unlike traditional IPOs, a SPAC divides the process of going public
into two distinct phases that are affected by either informational or agency-
related friction, but not both. In the initial stage, the _sponsor_ of the
SPAC raises capital from investors. The SPAC starts as a shell company without
an operating business, which limits the amount of information asymmetry
compared to a traditional IPO.222Consistent with this, in our sample, we find
a small (1.3%) underpricing at SPAC IPO. The next phase consists of a
“business combination” in which the SPAC acquires or merges with a private
business.
Second, during the business combination phase, SPACs report the allocations to
each institutional investor or the _private investment in public equity_
(acronym, “PIPE”) investor. These data enable us to measure the investment
return history for every sponsor-PIPE pair and create in-sample measures of
past “relationships” based on their track records. In traditional IPOs,
allocations to institutional investors are not reported, so one cannot
credibly measure past relationships between underwriters and investors.333In a
traditional IPO, although one can read quarterly 13F filings after the IPO, it
may be insufficient to quantify favoritism if preferred clients are allowed to
“flip” shares in secondary market trading. Furthermore, 13F filings do not
distinguish between shares allocated at the IPO and purchased in the secondary
market.
Third, SPACs are publicly traded firms. Consequently, we observe the effect of
major events (e.g., securing new external capital or identifying a target) on
share prices. Fourth, the quid pro quo channel is easier to observe in SPACs
than in traditional IPOs. Unlike the investment banks in traditional IPOs,
SPAC sponsors typically do not operate in other lines of business (e.g.,
trading and wealth management). So, to the extent that sponsors earn quid pro
quo from investors in future SPACs deals, we can measure those gains.
Once the SPAC finds a target and negotiates a “business combination” deal, the
existing SPAC investors have to approve the deal. Those who do not like the
deal can redeem their investments. Such redemptions reduce the money available
for the post-merger firm. As a safeguard, once the target is identified but
before the target firm begins public trading, the SPAC sponsor brings in PIPE
investors.
These investors that participate in the target announcements are the analogs
to the institutional buyers in the traditional IPO bookbuilding models.
Specifically, the sponsor finds a target and approaches PIPE (at) investors to
solicit their interest. Then the sponsor builds a book of PIPE demand, sets
prices, and allocates shares. Thus, when it comes to the business combination
phase, SPAC and traditional IPO are similar in that they both are subject to
information and agency frictions–the subject of this paper.
Much of our analysis focuses on the role of such PIPE investors. We classify
them into two types of investors$\\--$premium and non-premium$\\--$based on
their participation frequency and investments in SPACs, using k-means
clustering, which is an objective approach to classification. The underlying
assumption is that the PIPE investors who participate frequently and invest
large amounts are likely more reputable than others (Benveniste and Spindt,
1989), which behooves them to conduct due diligence and evaluate the proposed
merger deal.
Using data on all 1,072 SPACs from January 2010 to December 2021, we find
evidence consistent with premium investors offering value-relevant information
and playing certification roles. In particular, the participation of premium
investors is associated with significantly less share redemption by public
SPAC investors and a large positive announcement effect on the SPAC stock
price. The estimates suggest that one standard deviation (32.85 pps) increase
in premium PIPE investors’ participation is associated with a seven pps
decrease in the redemption rate (which is 12.4% of the sample mean) and a four
pps increase in the SPAC’s announcement-day return (which is 108.8% of the
sample mean).
Creating a SPAC and finding a suitable target falls on the sponsors, who act
as intermediaries between the target (who issues shares) and PIPE investors
(who buy shares). Sponsors profit an average of $126.9 million when the merger
is successful: otherwise, the SPAC is liquidated, their securities expire
worthlessly, and they lose their initial investment, which, on average, is
$8.2 million. Therefore, the sponsors benefit from participation by PIPE
investors which improves the chances of a successful merger.
So, the sponsors may have incentives to offer discounted shares to PIPE
investors in strong deals in exchange for a tacit promise to help out in
(future) weaker deals. Under such a quid pro quo arrangement, high-quality
firms (inadvertently) transfer to (in part) low-quality targets by allowing
the latter to go public.444This transfer between issuers is reminiscent of the
mechanism described in Benveniste, Busaba and Wilhelm (2002), who argue that
underwriters cross-subsidize issuers depending on their position in a wave.
There, the leader is subsidized, whereas, in SPACs, the follow-on SPACs (by
the same sponsor) are subsidized.
Such a quid pro quo arrangement suggests the following testable predictions.
First, for each sponsor-PIPE pair, money left on the table (acronym, “MLOT”)
should be mean-reverting, i.e., high profits in a current deal predict low
profits in a future deal, and vice versa. As is standard in the literature
(e.g., Lowry and Schwert, 2004), we measure MLOT using the first-day closing
price, taken as a proxy for the true value. Second, when the deal is weak, the
sponsor allocates shares disproportionately to those who earned strong returns
from that sponsor in the past. Third, when a sponsor with strong relationships
with investors, accumulated through past deals, leads a SPAC, that SPAC takes
fewer days to find a target and is less likely to be liquidated.
These predictions have no analog in the information production paradigm:
investors who did well in the past receive new signals of the current deal’s
quality and are under no obligation to support future weak deals. If anything,
the MLOTs of the premium investors should exhibit positive dependence if
investor skill is persistent.
We find evidence consistent with all three quid pro quo predictions, but
_only_ when considering non-premium investors. Specifically, non-premium
investors profit less in the current deal when they have high returns on past
deals. In particular, one dollar _above_ average MLOT in the last deal leads
to a $2.72 _below_ average MLOT from the current deal. We interpret this
_mean-reversion_ as the sponsor building a relationship and drawing it down,
likely for weaker deals.
We refer to weaker deals as “cold” deals and use the difference between the
offer share price of the SPAC, i.e., the share price at SPAC IPO, and the
share price on the closing day of the merger to measure it. Indeed, we find
that investors with a stronger relationship are more inclined to participate
in weaker deals. In particular, non-premium investors with a strong
relationship with a sponsor (with a total MLOT from this sponsor that is one
standard deviation above the sample mean) are 3.54 pps more likely to
participate in deals with the cold measure that are in the 90th percentile of
sample than deals with the cold measure that are at 10th percentile.
Conversely, non-premium investors without a prior relationship with the
sponsor are less likely to participate in cold(er) deals. Moreover, we observe
that when a non-premium investor with a strong relationship does participate,
the sponsor allocates a larger portion of securities to them in weaker deals
than in stronger deals.
We contribute to the literature by establishing a new institutional theory in
security issuance based on a sponsor’s relationship with institutional
investors. We provide evidence that the quid pro quo arrangement enables
cross-deal subsidies that enable weaker firms to go public by reducing the
SPACs’ liquidation risk. In contrast, the existing literature treats quid pro
quo as pure agency costs. More precisely, we find that if sponsors allocate
one standard deviation more MLOT above average ($300 million) to non-premium
investors, the liquidation probability of the current deal falls by 48.93 pps.
As a reference, the average liquidation rate in our sample is 17.82%, so this
decrease is economically meaningful. However, the MLOT allocated to the
premium investors is uncorrelated with the liquidation risk.555In Appendix A,
we verify that our main findings are robust to the presence of outlier (large
and small) investors and placebo assignments.
We also contribute to the information production mechanism by measuring the
information content of institutional investors. The traditional IPO literature
argues that price-limited bids are likely more informative than market-order
bids (Cornelli and Goldreich, 2001; Jenkinson, Jones and Suntheim, 2018).
SPACs are publicly traded and thus afford us the advantage of quantifying
price reactions to information conveyed by PIPE investments.
Overall, premium and non-premium investors play different but complementary
roles in SPACs and are paid accordingly. In our sample, each premium investor,
on average, is paid $6.96 million, and a non-premium investor $3.17 million.
Moreover, their observable differences can only explain $0.04 million of the
$4.79 million gap between their pay. We interpret this result as evidence of
their different “business models.” That is, a premium investor is paid $6.96
million to produce information for the current target, and a non-premium
investor is paid $3.17 million to reduce liquidation risk for the future SPAC.
Related Literature. Our research contributes to a long line of research that
uses underpricing to infer the importance of agency and informational
problems. For example, Benveniste and Spindt (1989) predict positive average
underpricing because investors are rewarded for value-relevant information.
The implication is that in a traditional IPO, when prices rise from the
midpoint of the filing range until the offer price (i.e., when demand is
high), there should be further price increases when the stock starts trading.
In other words, underwriters partially adjust to good information revealed by
investors but fully adjust to bad information.666See Hanley (1993) for
evidence of partial adjustment. Bradley and Jordan (2002) and İnce (2014)
argue that prices respond to public information, which is inconsistent with
bookbuilding models, whereas Lowry and Schwert (2004) argue that the IPO price
is “almost efficient” with respect to public information. In contrast, Beatty
and Ritter (1986); Loughran and Ritter (2002) and Loughran and Ritter (2004)
argue that underpricing is pure unproductive rent.
We complement this literature by looking at a broader range of outcomes:
redemption, announcement price reactions, liquidation, and allocations, and
show that both information and quid pro quo arrangements matter.
We also contribute to a nascent literature that seeks to understand the role
and functioning of SPACs. For instance, Gahng, Ritter and Zhang (Forthcoming)
study the cost and benefit of using SPACs to go public; Gofman and Yao (2022)
study the conflict of interest and associated harm from directors serving on
the boards of multiple SPACs; and Klausner, Ohlrogge and Ruan (2022) analyze
the structure of SPACs and the associated costs.
Our findings about the differential effects of premium and non-premium
investors complement the findings in Wang and Yung (2011) that partial
adjustment is present only in the subset of highly reputable underwriters. In
particular, they show that low-reputation underwriters cluster heavily on
exactly zero-dollar price revision. This behavior is consistent with no
information being produced or revealed. We also complement Chemmanur, Hu and
Huang (2010), who study the role of institutional investors in IPOs.
Our finding that information production coexists with quid pro quo is
reminiscent of the findings in Jenkinson, Jones and Suntheim (2018). Using
data on the initial allocations of investment banks participating in
traditional IPOs, they find evidence that investors are rewarded more when
they submit price-limited bids and participate more actively in the meetings
before or during bookbuilding. Both behaviors are consistent with an
informational role. However, they also find that allocations are affected by
how much revenue the investor generates for the bank, which is consistent with
the quid pro quo argument.
As previously mentioned, SPACs offer an opportunity to examine this issue in a
context where we can directly observe the effect of PIPE investor
participation on redemption decisions, announcement period returns, and the
likelihood of liquidation, none of which are observable in a traditional IPO.
The notion of quid pro quo employed in our papers is slightly different.
Jenkinson, Jones and Suntheim (2018) document that investors build a
relationship with a bank to get underpriced shares. We instead document that
_sponsors_ build relationships with non-premium PIPE investors to use them to
complete future deals.
## 2 Institutional Details
This section introduces important institutional details about SPACs. In
particular, these features of SPACs are important for understanding how they
differ from traditional IPOs and the measurement of underpricing in the data
that follow in the next section.
### 2.1 A Primer on SPACs
A SPAC is formed when a sponsor begins the SEC-mandated initial public
offering process. A SPAC is, first and foremost, an IPO and must comply with
all SEC IPO regulations. In addition, SPACs are subject to additional
requirements as “blank check company”, including how they use their escrow
fund (described below). The SPAC’s purpose is to identify and acquire a
private company, thereby bringing it public. During its IPO process, the SPAC
usually announces its intended target industry. This industry is meant to
reflect the expertise of its management team and founders and to inform
potential investors of its acquisition strategy. More often than not, these
industries are broad, e.g., financial services, healthcare, and technology, so
as not to constrain the search. A SPAC, however, is not obligated to pursue a
target in the announced industry, although most do.777For example, _Hunter
Maritime Acquisition Corp._ announced it would target the international
maritime shipping industry but merged with a fintech company named _NCF Wealth
Holdings_.
During their IPOs, SPACs offer securities to public investors called _units_ ,
typically priced at $10 per unit.888One exception is _Periphas Capital
Partnering Corporation_ , which priced its unit at $25. A unit consists of one
share of Class A common stock and a proportion (e.g., 1, 1/2, 1/3) of one
redeemable warrant. Initially, these units trade as a whole. However, usually
around the 52nd calendar day following the IPO, owners have the option to
either continue to hold units as a whole or trade shares and warrants
separately.999After the separation, SPACs do not issue fractional warrants and
only whole warrants trade. In other words, the number of warrants issued after
separation equals the fraction rounded down to the nearest whole number. For
example, if a unit contains one share and 1/3 of a warrant, then an investor
with 1 unit receives no warrant, but an investor with 30 units receives ten
warrants. Once the merger deal between the SPAC and a target is closed, a
warrant gives an investor the right to purchase additional shares of common
stock. Typically, the strike price is set well above the original SPAC price;
a common choice is $11.50 per share. Any shares created in this manner
automatically convert into ordinary shares after a merger.
Sponsors purchase shares before SPACs go public. These shares are called the
“founder shares” or the sponsor’s “promote”. These founder shares are
identical to the Class A common shares included in the units owned by public
investors, except that only the founder shares (i) contain the voting rights
to elect directors, (ii) are subject to transfer restrictions (“lock-up”), and
(iii) are irredeemable and become worthless if the SPAC liquidates.
A SPAC generally has 24 months from its IPO to identify a target company,
negotiate a deal, and complete the merger. Failing to complete the merger on
time leads to its liquidation, and the SPAC has to redeem all public shares
and pay its investors with the fund held in the trust account, including any
interest earned.
Once the SPAC locates a target, it uses proceeds from the trust fund for the
merger. Until then, the trust fund cannot be touched except to pay public
shareholders who redeem. To meet operating expenses, the sponsor creates a
“risk capital” by selling private placement warrants. These warrants are
irredeemable when the SPAC liquidates and are subject to transfer
restrictions.
Next, SPAC shareholders vote on the deal, and the transaction occurs only if a
majority votes in favor. Shareholders also can redeem their shares, typically
having at least 20 business days to redeem their shares before the merger is
complete. The merger agreement with the target company requires that the SPAC
have a minimum net worth or a certain amount of cash (as per the negotiation)
as a closing condition. Thus, if too many public shareholders redeem, the SPAC
may not meet the closing conditions of the merger. These financing shortfalls
create a need for _private investments in public equities_ (PIPE) investors.
### 2.2 PIPE Investment
Sometimes the funds raised in the SPAC IPO are sufficient to purchase the
target located by the sponsor. More typically, the sponsor needs (or chooses)
to raise additional funds. Because the SPAC is a publicly traded firm, these
funds are referred to as PIPEs.
PIPE investors are accredited institutional investors, e.g., hedge funds and
mutual funds, that agree to purchase securities issued by a company. Such an
agreement requires the company to file a resale registration statement (in the
future) to allow investors to resell their securities. Compared to raising
capital in the public market, PIPE investments tend to have lower transaction
costs, and companies can disclose transaction details _after_ receiving
commitments from PIPE investors.
We categorize PIPE transactions based on the timing of their commitment:
before, at, and after the business combination. In particular, the sponsor can
invite PIPE investors before the SPAC IPO to commit to making certain
investments in connection with the business combination. The PIPE transactions
formed at the time of the IPO but before a target is identified are called
forward purchase agreements; henceforth, PIPE (pre). The second type of PIPE
transaction is the one announced along with the news about the business
combination; henceforth, PIPE (at). The third type of PIPE transaction occurs
after the target is announced but before the merger is completed; henceforth,
PIPE (post).
Table 1: Example of Key Events and Timeline for a SPAC
Key Events | Dates
---|---
S-1 Filed and PIPE (pre) Announced | 08-Jan-2021
IPO | 29-Jan-2021
Deal Announced, PIPE (at) Announced | 06-Jul-2021
Proxy Filed | 12-Aug-2021
PIPE (post) Announced | 18-Jan-2021
Shareholder Vote | 24-Jan-2022
Deal Closed | 25-Jan-2022
Liquidation Deadline | 02-Feb-2023
Note: Key events and the dates for the SPAC _CF Acquisition Corp V_.
In principle and practice, a SPAC can have any combination of these three
types of PIPEs. For example, 1 shows an example SPAC, CF Capital Acquisition
Corp V, which had all three types of PIPEs. The PIPE (at) investors are
similar to the institutional investors in traditional IPO. After finding a
target, sponsors engage in an “roadshow” to raise PIPE investments. During
these roadshows, sponsors share material and proprietary information about the
target, creating a scope for information production.
### 2.3 SPAC vs. Traditional IPO
SPAC IPOs are different from traditional IPO in that SPAC is a blank-check
company without a business model. However, the merger between a SPAC and a
target, or de-SPAC transaction, is analogous to the traditional IPO because,
during the de-SPAC process, a private company transitions to public trading
and receives new financial capital in exchange for newly created
securities.101010For more about differences and similarities see the SEC
report “ _Special Purpose Acquisition Companies, Shell Companies, and
Projections_ ” available at
https://www.sec.gov/rules/proposed/2022/33-11048.pdf. In Figure 1, we compare
the timeline of a SPAC and a traditional IPO.
Figure 1: Events timing for Traditional IPO and SPAC
Note: Illustration of the timelines of a traditional IPO (top) and a SPAC
(bottom). Here, the acronym “BC" stands for a business combination.
A private operating company’s path to becoming a publicly listed company
through a SPAC begins when the SPAC considers it a candidate for the business
combination. The sponsors perform due diligence and negotiate a letter of
intent with the target company. Once the letter is signed, the second type of
PIPE fundraising process begins. In particular, sponsors and the target
company management consult with potential PIPE investors, pitch the company’s
business plan, and take subscriptions for shares from PIPE investors. Thus,
PIPE investors are similar to institutional investors in traditional IPOs.
After PIPE investments are secured, the deal is announced to the public
investors, and the de-SPAC process starts. Private companies become publicly
listed on the secondary market when the business combination is closed.
Finally, the _Private Securities Litigation Reform Act_ provides a safe harbor
for forward-looking statements made by SPACs in their de-SPAC transactions.
There is no such safe harbor for the traditional IPO. Thus, the private
company’s performance projections can be reported to investors during the de-
SPAC announcement.
## 3 Testable Hypotheses
We develop testable hypotheses based on two competing paradigms – the
information production and quid pro quo arrangements – that we assess in our
data.
Information Production. Under the first paradigm, investors have or obtain
private information about the issuer and earn rents according to the value of
their information. We refer to this paradigm as “information production”
because it views the central goal of the “going public” process as producing
or revealing value-relevant information about the issuer.
The process of going public via SPACs has three instances that involve
investors (see Figure 1). So, information production will have different
implications for those steps.
In particular, information production predicts zero underpricing during the
SPAC IPO because there is yet to be a target identified and, thus, no due
diligence to perform. There is virtually no scope for investors to have
private information; they exchange cash for a proportion of a trust fund of a
fixed value.
In the business combination or de-SPAC phase, investors that participate in
the target announcement (PIPE (at)) are the analogs to the buyers in the IPO
bookbuilding models. Specifically, the sponsor finds a target and approaches
PIPE (at) investors to solicit their interest. Then the sponsor builds a book
of PIPE demand, sets prices, and allocates shares.
However, these investors may differ in their abilities to produce information.
For instance, Benveniste and Spindt (1989) suggest that premium investors who
frequently access the market are likely more informed than others. In our
context, such investors have private information about the quality of the
issuing (target) firm and overall conditions affecting the market valuation.
Conversely, non-premium investors are less informed and consequently have less
impact on prices. Under these assumptions, the securities of the target firm
may be underpriced to compensate premium investors for truthfully disclosing
their private information about the target.
Furthermore, a premium PIPE (at) investor is incentivized to purchase more
shares when its private information indicates the de-SPAC deal is profitable.
Additionally, sponsors prioritize allocating to premium investors to encourage
them to disclose their information. Consequently, a larger allocation to
premium investors signals the good quality of the deal, which reduces
redemption by public investors and leads to a positive market reaction.
However, as non-premium PIPEs have no certification role, their participation
should neither affect redemption nor lead to higher announcement-day returns.
Conversely, we posit that PIPE (pre) and PIPE (post) investors do not play
certification roles. In the case of PIPE (pre), there has yet to be any
information about the target, so there is nothing to certify. PIPE (post)
investors are used less often than PIPE (at) investors and can be used to fill
the financing gap caused by redemption. If the sponsor intended a PIPE
investor to be used for certification purposes (as opposed to “gap filling”),
the sponsor would have announced the participation of the PIPE
contemporaneously when the deal was announced rather than waiting until after
public investors reacted.
Finally, suppose a premium PIPE (at) investor acts purely as an information
producer and is under no obligation to the sponsor to maintain a relationship.
In this case, the PIPE will participate in deals it likes and be absent from
deals otherwise. Such an arrangement should not generate any mean reversion
when the unit of observation is sponsor-PIPE.
###### Hypothesis 1.
Under the information production paradigm, the following should hold:
1. 1.
Allocations to premium PIPE (at) investors are associated with lower
redemption and positive announcement-day returns.
2. 2.
For a sponsor-premium PIPE (at) pair, MLOT is not mean-reverting.
Quid Pro Quo. We now discuss the implications of the quid pro quo paradigm.
Under this paradigm, the seller of securities acts through a self-interested
intermediary. Rather than selling securities at full price, say, to the
premium investors, the intermediary gives discounts to some buyers, with an
implicit understanding that this favor is rewarded through profits on future
SPAC transactions.
The SPAC IPO presents the underwriter with such an opportunity. Underwriters
likely deal with institutional investors more frequently (and in more varied
capacities) than sponsors. As an intermediary, the underwriter may choose to
favor institution investors (the buyers) over sponsors (the sellers), just as
in a traditional IPO. If so, the SPAC offering will be underpriced.
The business combination phase presents another opportunity for us to observe
favoritism, except the actors are swapped. The issuer is the private firm
rather than the SPAC because it is the private firm, not the SPAC, that issues
new securities. The buyers are the PIPE investors who infuse new capital in
exchange for securities. The intermediary is the sponsor that negotiates the
terms under which PIPE investors buy these securities and profits only if the
deal is completed.
Suppose that the underpricing at the business combination phase is not because
of information production but because the sponsor allocates rents to PIPE (at)
investors. Then, the sponsor should receive returns from these investors in
the future. If so, then including PIPE investors in a SPAC would not benefit
the SPAC. Therefore, higher allocations to any PIPE (at) investors should
_not_ be associated with either lower redemption rates or higher announcement-
day returns.
In any given deal, sponsors can favor particular PIPE investors and insist on
favorable terms from the target firm. Favoritism may manifest differently at
this stage than in traditional IPOs. In a traditional IPO, underwriters
operate in many lines of business, including trading and wealth management. An
institutional investor receiving underpriced shares in a traditional IPO may
repay the underwriter by providing business through other channels. In
contrast, the sponsors operate only in the SPAC industry, and the primary
objective of a SPAC sponsor is to close the existing fund and start future
SPACs
We hypothesize that quid pro quo arrangements are employed only with non-
premium PIPE investors. A sponsor showing favoritism to a particular non-
premium PIPE investor expects reciprocity in future deals. Specifically, non-
premium PIPE investors participate in an underpriced deal (acting as a tax on
the current private firm) only in exchange for implied participation in a
future weak deal. If a PIPE is involved only in “strong” deals, then the
relationship is one-sided and involves “quid” with no “quo.”
While a long-term relationship (e.g., Levin, 2003) between a non-premium PIPE
investor and a sponsor can provide necessary incentives to the investors to
participate in weaker future deals, the observed interactions in any data are
likely finite. Then the question remains if non-premium investors have
incentives to participate in weaker future deals. In our context, it is
appropriate to view non-premium investors are long-lived players who interact
with sponsors who are short-lived players. In such a setting, results from
Fudenberg and Levine (1994) indicate that an investor does not need to have
repeated interactions with a particular sponsor for the sponsor to have the
incentive to honor their commitment to participate in weak future deals. That
is because an investor’s participation is publicly observed. Suppose they
refrain from participating in a weak deal led by a sponsor from whom they have
received favorable returns. In that case, they risk losing their market-wide
reputation and opportunities to participate in future SPACs.
Furthermore, under quid pro quo arrangements, the sponsor benefits in creating
goodwill among investors, hoping to “draw down” on this goodwill in the future
when the deals are weak. So, we hypothesize that the liquidation risk of a
SPAC is negatively correlated with the amount of MLOT investors have received
from this SPAC’s sponsor in the past.
###### Hypothesis 2.
Under the quid pro quo paradigm, the following should hold:
1. 1.
Allocations to non-premium PIPE (at) investors are uncorrelated with
redemption and announcement-day returns.
2. 2.
For a given sponsor -non-premium PIPE (at) pair, MLOT is mean-reverting.
3. 3.
When a deal is weak, the sponsor is likelier to allocate shares to non-premium
PIPE investors who earned large past MLOT.
4. 4.
SPACs are less likely to liquidate when the sponsor has earned goodwill
through high past MLOT. This connection occurs through only non-premium and
not premium PIPEs.
## 4 Data
We consider all 1,072 SPACs traded in major exchanges between January 2010 and
December 2021. See Figure A4 for the monthly scatter plot in Figure A4, which
includes the so-called boom and bust cycle for SPACs. We construct our sample
by merging five datasets: (i) the _SPAC Research_
,111111https://www.spacresearch.com/. (ii) the _PrivateRaise_ –SPACs
Platform,121212https://www.privateraise.com/about/platform_spac.php. (iii) the
_PrivateRaise_ –PIPEs Platform (iv) _Bloomberg_ data on SPAC prices, and (v)
Super 8-K filings from SEC.131313https://www.sec.gov/edgar/search-and-
access.,141414Thus, we exclude SPACs traded in the over-the-counter markets.
After a SPAC’s merger with a target is complete, the SPAC is delisted. The new
(merged) company may be assigned a new ticker symbol. We use Bloomberg-
assigned unique identifiers for matching across samples.
The SPAC Research database and the PrivateRaise’s SPACs platform database
contain information on SPACs, including dates for important events, and the
PrivateRaise’s PIPEs platform database contains comprehensive information on
PIPE transactions. PrivateRaise’s PIPEs platform covers more than 20,000
placement profiles of PIPEs transactions. It also includes contract terms,
investment amounts, types of transactions, prices, identities, and allocations
of the PIPE investors.
For these 1,072 SPACs, we manually merge the first two databases. The issuer’s
name recorded in the PIPE transactions can be either that of SPAC or that of
the post-merger company. In those cases, we use both names to match the SPAC
sample with the PrivateRaise database. To ensure match accuracy, we also use
the total PIPE amount from PrivateRaise to cross-check the match. If there are
inconsistencies, we use the SPAC’s SEC filings documents to record the correct
amount. We then merge our sample with Bloomberg’s pricing data; warrant prices
are unavailable in CRSP. Our matching process yields detailed information on
transactions for all three types of PIPE investments for each SPAC in our
sample.
In summary, we observe each SPAC’s life-cycle status, IPO structure, IPO
outcomes, sponsor and management team, redemption, merger details, merger
outcome, PIPE transactions, and price movement across events. Out of these
1,072 SPACs, as of February 17, 2023, 473 have successfully merged, 128 have
announced that they have identified a target, 280 are in the process of
identifying targets, and 191 have failed and liquidated.
The number of SPAC shares changes over time on account of i) investors
exercising warrants, ii) redemption of shares, iii) securities purchased by
PIPES, and iv) the fact that founders sometimes voluntarily forfeit
securities. Consequently, the number of shares listed in the IPO prospectus is
no longer current as of the business combination date. We use Super 8-K (Super
20-F for foreign issuer) filings to collect information on the number and
types of outstanding securities on the first listing day of the post-merger
company. Information about securities at the time of the SPAC IPO may not be
comprehensive either because the sponsors have forfeited their founder shares
and private placement securities or the units that contain rights would have
become common shares or both. To accurately measure the number of outstanding
securities post-merger, we manually collect information from the Super 8-K (or
Super 20-F) filings. SEC mandates that a SPAC file Super 8-K within four
business days since the merger is complete, and it records the shares and
warrants held by sponsors and public investors.
Table 2 provides summary data. On average, SPACs raise $272.42 million
proceeds during their IPO, with a median of $240 million and 90th percentile
at $440 million. In terms of securities, typically, a SPAC unit contains 0.44
warrants and 0.01 rights. During the IPO process, SPACs’ underwriters choose,
on average, to exercise the 11.59% over-allotment option to meet the
additional demand.
SPACs take, on average, 361 days to find a target and an additional 168 days
to complete the merger process. After targets’ announcements, 56.54% of SPAC’s
public shareholders redeem their shares on average. Table 2 also displays
information about the participation of PIPE investors for 473 SPACs that
completed the merger process during our sample period. At the time of their
IPOs, 10% of SPACs have pre-committed investment from PIPE investors, i.e.,
PIPE (pre). When SPACs announce their targets, 66% of them also announce
investments from PIPE (at), and 12% of SPACs obtain additional investments
from PIPE (post) after the target announcement.
Returns. We measure SPAC returns at its IPO and when it announces a target.
Following the literature on IPO underpricing, we measure SPAC’s IPO return as
$\text{Listing-day return}=\frac{\text{First-Trading Day Closing
Price}}{\text{Offer Price}}-1,$ and announcement-day return as
$\text{Announcement-day return}=\frac{\text{Announcement Day Closing
Price}}{\text{One-day-before-announcement Day Closing Price}}-1.$ As seen in
Table 2, the average return for SPAC IPO investors on the first trading day is
1.3%, and when the target is announced, SPAC investors receive an average
return of 3.74%.
Money Left on the Table. To measure underpricing, i.e., the return earned by
investors, we use _money left on the table_ (“MLOT”), which is the difference
between the value of securities on the first day of listing the post-merger
company and their initial investment. We begin with the total investments made
by public investors, which we denote using superscript $\mathfrak{pu}$. Their
MLOT is the sum of the value of their shares and the value of their warrants
minus their leftover investments during IPO, i.e.,
$\text{MLOT}^{\mathfrak{pu}}=\text{Shares}^{\mathfrak{pu}}\times\text{P}_{share}+\text{Warrants}^{\mathfrak{pu}}\times\text{P}_{warrant}-\text{IPO
Proceeds}\times(1-\text{Redemption Rate}).$
Table 2: Summary Statistics
| Mean | Std.Dev. | P10 | P50 | P90 | Obs.
---|---|---|---|---|---|---
SPAC Measures | | | | | |
IPO proceeds (mm USD) | 272.42 | 218.22 | 100.00 | 230.00 | 440.00 | 1,072
No. of Warrant | 0.44 | 0.24 | 0.20 | 0.50 | 0.75 | 1,072
No. of Right | 0.01 | 0.03 | 0.00 | 0.00 | 0.00 | 1,072
Overallotment (%) | 11.59 | 5.37 | 0.00 | 15.00 | 15.00 | 1,072
Listing-day return (%) | 1.30 | 3.02 | -0.70 | 0.40 | 4.00 | 857
Days searching | 361.71 | 215.79 | 114.00 | 329.50 | 677.00 | 614
Days remaining | 340.98 | 189.87 | 120.00 | 303.00 | 611.00 | 614
Announcement-day return (%) | 3.74 | 12.56 | -0.68 | 0.34 | 9.95 | 586
Redeemption(%) | 56.54 | 37.49 | 0.00 | 70.00 | 97.00 | 445
Days deSPAC | 167.99 | 67.06 | 100.00 | 160.00 | 241.00 | 474
PIPE Participation | | | | | |
$\mathbbm{1}${PIPE (pre)} | 0.10 | 0.30 | 0.00 | 0.00 | 0.00 | 473
$\mathbbm{1}${PIPE (at)} | 0.66 | 0.47 | 0.00 | 1.00 | 1.00 | 473
$\mathbbm{1}${PIPE (post)} | 0.12 | 0.33 | 0.00 | 0.00 | 1.00 | 473
$\mathbbm{1}${Premium PIPE (at)} | 0.50 | 0.50 | 0.00 | 1.00 | 1.00 | 473
$\mathbbm{1}${Non-Premium PIPE (at)} | 0.65 | 0.48 | 0.00 | 1.00 | 1.00 | 473
No. of PIPE (at) | 10.76 | 11.83 | 0.00 | 8.00 | 26.00 | 472
No. of Premium PIPE (at) | 3.02 | 4.36 | 0.00 | 1.00 | 10.00 | 472
No. of Non-Premium PIPE (at) | 7.75 | 8.64 | 0.00 | 6.00 | 18.00 | 472
PIPE Investment/IPO Proceeds | | | | | |
PIPE (pre) (%) | 2.93 | 11.22 | 0.00 | 0.00 | 0.00 | 473
Premium PIPE (at) (%) | 17.84 | 32.85 | 0.00 | 0.00 | 52.39 | 473
Non-Premium PIPE (at) (%) | 41.94 | 58.64 | 0.00 | 28.57 | 101.16 | 473
PIPE (post) (%) | 7.45 | 30.39 | 0.00 | 0.00 | 22.68 | 473
MLOT (mm USD) | | | | | |
Public Investors | 113.22 | 433.26 | -3.76 | 11.24 | 226.41 | 433
Sponsors | 126.93 | 324.28 | 15.66 | 64.32 | 195.48 | 425
PIPE | 201.22 | 1,200.54 | -25.60 | 3.75 | 248.17 | 358
PIPE (pre) | 29.37 | 120.76 | -10.52 | 3.17 | 35.70 | 45
PIPE (at) | 193.07 | 1,135.58 | -25.36 | 4.77 | 248.17 | 303
PIPE (post)∗ | 8.07 | 42.62 | -15.74 | 0.25 | 31.76 | 56
Premium PIPE (at) | 76.63 | 374.15 | -10.22 | 1.68 | 129.46 | 231
Non-Premium PIPE (at) | 134.65 | 838.67 | -17.73 | 3.40 | 138.48 | 303
MLOT/Investment (%) | | | | | |
Public Investors | 182.10 | 898.42 | -7.89 | 24.87 | 179.08 | 433
Sponsors | 1,400.46 | 2,570.29 | 370.04 | 861.15 | 2,220.70 | 324
PIPE | 91.26 | 472.71 | -16.50 | 4.47 | 85.80 | 358
PIPE (pre) | 70.33 | 289.37 | -8.05 | 4.36 | 91.80 | 45
PIPE (at) | 63.74 | 307.47 | -16.00 | 4.20 | 80.90 | 303
PIPE (post)∗ | 70.74 | 285.16 | -36.02 | 0.80 | 110.60 | 56
Premium PIPE (at) | 64.05 | 291.75 | -13.00 | 5.30 | 94.90 | 225
Non-Premium PIPE (at) | 64.10 | 308.46 | -16.00 | 4.20 | 80.90 | 301
Note: The sample includes 1,072 SPACs that went through the IPO process
between January 2010 and December 2021. All variables are defined in Table B1.
$(^{*})$ We exclude a PIPE (post) investor named _Quinpario Acquisition
Corporation 2_ , with unusually large returns.
Here Sharespu is the total number of outstanding shares held by public
investors on the first-listing day of the post-merger company. Thus, Sharespu
includes non-redeemed public shares, and shares are automatically converted
from rights included in the SPAC unit. Warrantspu is the total number of
warrants held by public investors on the first-listing day of the post-merger
company. Similarly, $\text{P}_{share}$ and $\text{P}_{warrant}$ denote the
price for shares and warrants of the post-merger company at the end of the
first trading day.
For the sponsor’s MLOT, we can replace the IPO proceeds that were not redeemed
in the above definition of $\text{MLOT}^{\mathfrak{p}}$ with risk capital,
i.e.,
$\text{MLOT}^{\mathfrak{s}}=\text{Shares}^{\mathfrak{s}}\times\text{P}_{share}+\text{Warrants}^{\mathfrak{s}}\times\text{P}_{warrant}-\text{Risk
Capital},$ where Sharess measures the total number of shares held by the
sponsor on the first-listing day of the post-merger company. Thus Sharess
includes the founder shares and private placement shares purchased using the
sponsor’s “risk capital” and excludes shares forfeited by the sponsor during
the de-SPAC process. Warrantss measures the private placement warrants
purchased by the sponsor and excludes warrants forfeited by the sponsor.
For a PIPE investor’s MLOT, a similar definition applies. However, instead of
risk capital, we subtract the issuance amount, i.e.,
$\text{MLOT}^{\mathfrak{pi}}=\text{Shares}^{\mathfrak{pi}}\times\text{P}_{share}+\text{Warrants}^{\mathfrak{pi}}\times\text{P}_{warrant}-\text{Issuance
Amount}^{\mathfrak{pi}},$ where Sharespi, Warrantspi are the number of shares
and warrants purchased by PIPE investors, and Issuance Amountpi is the dollar
amount of the issuance. We calculate MLOTpi for all three types of (pre, at,
and post) PIPE transactions.
Table 2 highlights the MLOT for various investor types. Investing in a SPAC
generates an average of $113.22 million for public investors, $126.93 million
for sponsors, and $201.22 million for PIPE investors. Out of the 45 SPAC deals
in which they participate, PIPE (pre) investors make $29.37 million on
average. From the 303 SPAC deals in which they take part, PIPE (at) investors
earn $193.07 million per deal. PIPE (post) investors participate in 56 deals
and earn $8.07 million per deal on average.
On average, a SPAC deal generates 182.1% return for public investors and
1,400% for the sponsors. PIPE investors get an average return of 91.26%, where
PIPE (pre) investors earn 70.33%, PIPE (at) investors earn 63.74%, and PIPE
(post) investors earn 70.74%.
Premium PIPE Investors. In total, there are 2,005 PIPE investors in our
sample. Out of them, 76 invested in PIPE (pre) transaction, 249 in PIPE
(post), and 1,826 invested in at least one PIPE (at) transaction, but only a
few large institutional investors participated frequently and invested
significantly more than others. To capture this distinction, we use k-means
clustering on the total investments in all SPACs and the number of SPACs to
classify PIPE investors into two groups. We refer to a subset of PIPE
investors who frequently participate as “premium” PIPE investors and the rest
as non-premium PIPE investors. There are 34 premium PIPE investors in our
sample (Table 3).151515As a way to corroborate this classification, we counted
the number of articles written about each of these 34 premium investors in
_The New York Times_ and found that these premium investors have the highest
counts. For example, excluding research universities, e.g., Columbia
University, and non-financial companies, e.g., General Motors and Microsoft
Inc., top ten investors include Morgan Stanley, Fidelity, USB, Wells Fargo,
Blackstone, Citadel, Bain Capital, BlackRock, T. Rowe Price, and SoftBank.
Table 3: Premium PIPE Investors
Investor | #SPAC | Volume (million USD)
---|---|---
Fidelity Management & Research Company | 69 | 4,342
BlackRock, Inc. | 57 | 2,317
Capital Research and Management Company | 29 | 2,231
Alyeska Investment Group L.P. | 106 | 1,704
Morgan Stanley | 14 | 1,238
Millennium Management, LLC | 77 | 1,059
Luxor Capital Group | 52 | 972
Blackstone Inc. (f/k/a The Blackstone Group Inc.) | 34 | 965
Koch Industries Inc. | 21 | 926
Third Point, LLC | 12 | 910
Hedosophia Services Limited | 23 | 910
T. Rowe Price Group, Inc. | 27 | 899
SoftBank Group Corp. | 16 | 795
Franklin Resources, Inc. (d/b/a Franklin Templeton Investments) | 24 | 741
Senator Investment Group LP | 36 | 713
Moore Capital Management, Inc. | 66 | 629
Neuberger Berman Group LLC | 20 | 626
Baron Capital Group, Inc. | 25 | 621
Citadel LLC | 51 | 604
Hudson Bay Capital Management LP | 56 | 577
Suvretta Capital Management, LLC | 31 | 576
UBS AG | 57 | 494
Wellington Management Company, LLP | 26 | 437
Heights Capital Management, Inc. | 65 | 431
Magnetar Capital LLC | 39 | 424
D.E. Shaw & Co., L.P. | 38 | 411
Monashee Investment Management LLC | 61 | 408
Ghisallo Capital Management LLC | 62 | 389
Jane Street Group, LLC | 49 | 368
Park West Asset Management, LLC | 27 | 342
Kepos Capital LP | 51 | 286
Linden Advisors LP | 43 | 282
Schonfeld Strategic Advisors LLC | 32 | 224
BlueCrest Capital Management Ltd. | 31 | 181
Note: This table lists the name of premium PIPE investors identified using
k-means clustering based on the number of SPAC deals and total investment
volumes using data on SPACs that went through an IPO from January 2010 to
December 2021.
Table 2 provides summary statistics for premium and non-premium PIPE
investors. A typical SPAC deal includes 10 PIPE (at) investors, with around
three premium and seven non-premium investors. In the 231 SPAC deals in which
they participate, premium investors earn an average of $76.63 million (MLOT)
with an average return of 64.05%. Non-premium investors participate in 303
SPACs and make an average of $134.65 million (MLOT) and earn an average return
of 64.10%. As the standard deviations suggest, the high average returns for
the non-premium PIPEs are driven by five outlier SPACs with extreme returns.
We revisit this difference in premium and non-premium investors’ MLOTs in
Section 6.
## 5 Empirical Results
In this section, we investigate whether the data are consistent with the
hypotheses stated in Section 3. We begin measuring underpricing at the SPAC
IPO level and compare it with the traditional IPO underpricing. Then, we show
that the premium PIPE investors who participate in business combinations
generate value-relevant information, whereas non-premium PIPE investors engage
in a quid pro quo relationship. We end by quantifying the benefit of this quid
pro quo relationship in terms of lowering the liquidation risk of future
SPACs.
### 5.1 Underpricing
#### 5.1.1 Underpricing during SPAC IPO
We first examine underpricing during the SPAC IPO. Table 2 reports the average
equal-weighted first-day return of SPACs is 1.3%. Although our data start from
2010, most SPAC IPOs occurred between 2020 and 2021. In 2020, 248 SPACs went
public, with an average equal-weighted first-day return of 1.3%; in 2021, 613
SPACs went public, and their average equal-weighted first-day return was
1.51%. Although these measures are statistically different from zero (p-values
$\approx 0$ in both cases), they are small.161616For comparison, the average
equal-weighted first-day return for traditional IPOs was 41.6% in 2020 and 32%
in 2021. For more on the IPO underpricing, see
https://site.warrington.ufl.edu/ritter/files/IPOs-Underpricing.pdf, accessed
09/22/2022. In addition, SPACs almost always trade at $10 per share until the
target announcement date. This uniformity is inconsistent with any information
about sponsor quality having been revealed during the SPAC IPO.
#### 5.1.2 Underpricing during de-SPAC
This section focuses on the underpricing associated with the de-SPAC stage. We
say that a private company’s shares are underpriced if the first-day trading
value of the post-merger company’s securities held by all SPAC investors
(including sponsors, PIPEs, and original SPAC investors who did not redeem)
exceeds the proceeds invested in the private firm.
There are at least two sources of asymmetric information at this stage. First,
the target company may be better informed about its business prospect than
anyone else (Smith Jr, 1986; Booth and Smith II, 1986). Second, the PIPE
investors may also have better information about factors affecting the target
firm than the public investors. In response to PIPE investors’ information,
sponsors may underprice the securities issued to them as an incentive payment
for their value-relevant information. We focus on the second type of
information.
From Table 2, we see that, on average, a private company that uses the SPAC
route to access the public market leaves $126.93 million MLOT to the sponsors
and $201.22 million to the PIPE investors. Furthermore, we also see that PIPE
investors who participate in different stages of SPAC are compensated
differently. During the SPAC IPO, PIPE (pre) investors earn an average of
$29.37 million through forward-purchase agreements, but PIPE (at) investors
get, on average, a total of $193.07 million per deal. After the target is
disclosed, PIPE (post) investors who participate receive an additional $8.07
million MLOT. Even among PIPE (at) investors, approximately one-third of the
MLOT accrues to 34 premium investors. This difference suggests that the two
types of investors have different business models, which we return to later in
Section 6.
Overall, the SPAC shares offered to PIPE (at) investors, especially the
premium PIPE investors, during the de-SPAC phase are underpriced. These
results are consistent with Hypothesis 1 that PIPE (at) investors
participating in the de-SPAC stage are privately informed about the quality of
the target company.
### 5.2 Information Production
#### 5.2.1 Redemption Rates
To test the first part of Hypothesis 1-1, we use the following regression:
$\displaystyle\text{Redemption rate}_{i}$ $\displaystyle=$
$\displaystyle\alpha_{0}+\alpha_{1}\times\%\text{Premium PIPE
(at)}_{i}+\alpha_{2}\times\%\text{Non-Premium PIPE (at)}_{i}$ (1)
$\displaystyle+\alpha_{3}\times\%\text{PIPE
(pre)}_{i}+\alpha_{4}\times\%\text{PIPE (post)}_{i}+{\bf
X}^{\top}\alpha_{5}+\varepsilon_{i},$
where $\text{Non-Premium PIPE (at)}_{i}$ denotes the investment allocated to
premium PIPE (at) investors as a fraction of the total IPO proceeds in SPAC
$i$, similarly for other allocations, and ${\bf X}$ is a vector of covariates
including total IPO proceeds and fixed effects that may vary across
specifications. If premium investors generate value-relevant information, the
higher the share allocated to premium investors lower the redemption rate,
i.e., $\alpha_{1}<0$.
Table 4 provides evidence consistent with the hypothesis. In column (1), we
include only the allocations to four types of PIPE investors. In column (2),
we control for the size of the SPAC IPO and the liquidation risk of the SPAC
when the target is announced using Days Remaining until the expiration. In
column (3), we control for the quarter-fixed effects. In column (4), which is
also our preferred specification, we additionally control for the target
company’s sector fixed effects to alleviate the problem that investors may
have varying investment preferences across industries. In all these
specifications, we cluster the robust standard errors at the sponsor level to
capture correlations within a sponsor but across investors and SPACs.
We find that the allocations to premium PIPE (at) investors negatively
correlate with the redemption rates, and this result is robust across all
these specifications. The estimates from Table 4 column (4) suggest that one
standard deviation (32.85 pps) increase in premium PIPE investors’
participation, is associated with a 7.03 pps decrease in the redemption rate,
which equals 12.4% of the sample mean.
Furthermore, we find no association between non-premium PIPE (at) or PIPE
(pre) allocations and redemption rates, as in Hypothesis and Hypothesis 2-1.
These results are consistent with the hypothesis that premium PIPE (at)
investors are better informed about the deal quality than either the non-
premium PIPE (at) investors or PIPE (pre).
Table 4: Redemption Rates
| (1) | (2) | (3) | (4)
---|---|---|---|---
PIPE (pre) (%) | 0.125 | 0.192 | 0.148 | 0.156*
| (0.123) | (0.119) | (0.116) | (0.095)
Premium PIPE (at) (%) | -0.419*** | -0.380*** | -0.366*** | -0.214***
| (0.069) | (0.064) | (0.059) | (0.050)
Non-Premium PIPE (at) (%) | -0.013 | -0.009 | -0.002 | -0.037
| (0.034) | (0.035) | (0.034) | (0.030)
PIPE (post) (%) | 0.120*** | 0.103*** | 0.114*** | 0.148***
| (0.035) | (0.031) | (0.033) | (0.035)
Log IPO proceeds | ✗ | ✓ | ✓ | ✓
Days Remaining | ✗ | ✓ | ✓ | ✓
IPO Month FEs | ✗ | ✗ | ✓ | ✓
Target Sector FEs | ✗ | ✗ | ✗ | ✓
Adj. $R^{2}$ | 0.161 | 0.182 | 0.183 | 0.400
Obs. | 444 | 444 | 444 | 430
Note: This table shows the results from estimating (1) using SPACs that went
through an IPO from January 2010 to December 2021. The independent variable is
the investment amount as a percentage of the SPAC’s IPO proceeds for all PIPE
(pre) investors, premium PIPE (at) investors, non-premium PIPE (at) investors,
and all PIPE (post) investors. Premium PIPE (at) investors are listed in Table
3. All other variables are defined in Table B1. All unbounded continuous
variables are winsorized at their $1^{st}$ and $99^{th}$ percentile values.
Robust standard errors are clustered at the sponsor level and are reported in
parentheses. ${}^{*},^{**}$, and ∗∗∗ denote p-values less than 0.1, 0.05, and
0.01, respectively.
The negative association between premium PIPE (at) allocation and redemption
rates is consistent with premium investors producing information. Suppose not,
and instead, suppose that sponsors underprice and distribute them to the
premium investors in exchange for a promise to help them in the future on an
unrelated SPAC. In that case, because the timing of the investors should be
immaterial, we should also expect the allocations to PIPE (post) investors to
correlate negatively with the redemption rate. However, as we see from Table
4, the two are positively correlated, further strengthening our finding.
Lastly, we note that almost all (231 out of 25) PIPE (post) investors are non-
premium investors. In fact, among the top 20 PIPE (post) investors by
investment size, only Luxor Capital Group is a premium investor. Thus, it is
reasonable to assume that sponsors include premium investors at the time of
the announcement as certification and use non-premium investors later as
needed to backfill the trust fund.
#### 5.2.2 Announcement-day Return
To test the second part of Hypothesis 1-1 about the announcement-day return,
we use the following regression:
$\displaystyle\text{Announcement-day return}_{i}$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}\times\%\text{Premium PIPE (at)}_{i}$ (2)
$\displaystyle+\beta_{2}\times\%\text{Non-Premium PIPE (at)}_{i}$
$\displaystyle+\beta_{3}\times\%\text{PIPE (pre)}_{i}+\varpi_{i}.\qquad$
The estimation results are shown in Table 5. There is a strong positive
association between announcement-day returns (%) and allocations to premium
PIPE (at) investors. In particular, the estimates in Table 5 column (4)
suggest that one standard deviation (32.85 pps) increase in premium PIPE (at)
investors’ participation is associated with a 4.07 pps increase in the SPAC’s
announcement-day return, which is 108.8% of the sample mean of the
announcement-day returns. Thus the effect of premium investors on
announcement-day return is economically substantial and statistically
significant.
Furthermore, the announcement-day returns are uncorrelated with the
allocations to non-premium PIPE (at) investors. Again, this finding is
consistent with the hypothesis that larger investment by premium PIPE (at)
investors acts as a certification of the quality of the deal. In contrast,
investment by non-premium PIPE (at) is non-informative.
We also observe a negative correlation between announcement-day returns and
allocations to PIPE (pre) investors. Specifically, the estimates in column (4)
suggest that one standard deviation (11.22 pps) increase in PIPE (pre)
investors’ participation is associated with 0.81 pps lower in announcement-day
return, which equals 21.7% of the sample mean.
Table 5: Announcement-day Return
| (1) | (2) | (3) | (4)
---|---|---|---|---
PIPE (pre) (%) | -0.118*** | -0.119*** | -0.099*** | -0.072
| (0.042) | (0.042) | (0.037) | (0.044)
Premium PIPE (at) (%) | 0.148*** | 0.140*** | 0.136*** | 0.124***
| (0.041) | (0.041) | (0.038) | (0.040)
Non-Premium PIPE (at) (%) | -0.003 | -0.007 | -0.010 | -0.002
| (0.014) | (0.014) | (0.013) | (0.015)
Log IPO proceeds | ✗ | ✓ | ✓ | ✓
Days Remaining | ✗ | ✓ | ✓ | ✓
IPO Month FEs | ✗ | ✗ | ✓ | ✓
Target Sector FEs | ✗ | ✗ | ✗ | ✓
Adj. $R^{2}$ | 0.099 | 0.104 | 0.098 | 0.095
Obs. | 457 | 457 | 431 | 417
Note: This table shows the results from estimating (2) using SPACs that went
through an IPO from January 2010 to December 2021. The key independent
variable is the investment amount as a percentage of the SPAC’s IPO proceeds
for all PIPE (pre) investors, premium PIPE (at) investors, non-premium PIPE
(at) investors, and all PIPE (post) investors. Premium PIPE (at) investors are
listed in Table 3. All other variables are defined in Table B1. All unbounded
continuous variables are winsorized at their $1^{st}$ and $99^{th}$ percentile
values. Robust standard errors are clustered at the sponsor level and are
reported in parentheses. ${}^{*},^{**}$, and ∗∗∗ denote p-values less than
0.1, 0.05, and 0.01, respectively.
Likewise, the hypothesis predicts a non-positive market reaction to the
announcement of PIPE (post) investments. To assess this claim, we use an event
study approach. In particular, we use different asset-pricing factor models to
estimate the cumulative abnormal returns (CAR) (Campbell, Lo and MacKinlay,
1997) associated with announcements about PIPE investors in the process of
announcing the target. Specifically, we estimate the following linear
regression model:
$\displaystyle r_{i,t}=\kappa_{i}+\bm{\rho}_{i}\texttt{W}_{t}+\upsilon_{i,t},$
(3)
where $r_{i,t}$ is the share return of SPAC $i$, at date $t$, X includes
portfolio returns of different factor models. For each SPAC, we estimate (3)
using an event window of $[t-\kappa,t+\kappa]$, which is $\kappa$ days around
each PIPE (post) transaction’s announcement date. The estimation period is
chosen to start from the SPAC’s unit separation date and ends before the event
window. Then we obtain the predicted return $\hat{r}_{i,t}$, which gives us
the “abnormal” return as $\hat{\upsilon}_{i,t}\equiv r_{s,t}-\hat{r}_{i,t}$.
The CAR associated with target announcement by $i$ is given by accumulating
$\hat{\upsilon}_{i,t}$ during the event window, i.e.,
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\texttt{CAR}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\texttt{CAR}}{\tmpbox}_{i,t}=\sum_{t=t_{i}-\kappa}^{t_{i}+\kappa}\hat{\upsilon}_{i,t}.$
Table 6 reports the average of the
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\texttt{CAR}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\texttt{CAR}}{\tmpbox}$
and their respective standard deviation across all SPACs using a three-day,
five-day, and 10-day window around the event date. For each event window, we
estimate CAPM, the Fama-French three-factor model, and Fama-French five-factor
model. In all cases, the average
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\texttt{CAR}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\texttt{CAR}}{\tmpbox}$
estimates are not statistically different from zero. This result suggests that
the investors do not respond to the announcement of PIPE (post), likely
because almost all of them are non-premium and unlikely to add information.
Table 6: Cumulative (average) Abnormal Returns
Event Window | CAPM | FF3 | FF5
---|---|---|---
[-3, 3] | 0.71 | 0.74 | 0.75
| (0.80) | (0.79) | (0.78)
[-5, 5] | -1.04 | -1.04 | -1.06
| (1.14) | (1.13) | (1.12)
[-10, 10] | -2.37 | -2.30 | -2.28
| (1.92) | (1.92) | (1.91)
Note: This table reports the SPAC share’s cumulative average abnormal return
in percentage for three different event windows around announcements of 81
PIPE (post) transactions. Here, CAPM stands for capital asset pricing model,
and FF3 and FF5 stand for Fama-French three- and five-factor models,
respectively. The cross-sectional standard deviations of the cumulative
abnormal returns are reported in parentheses. ${}^{*},^{**}$, and ∗∗∗ denote
p-values less than 0.1, 0.05, and 0.01, respectively.
### 5.3 Quid Pro Quo
Next, we test Hypothesis 2 that there is a quid pro quo arrangement between
sponsors and non-premium PIPE (at) investors and that it helps a weaker firm
go public. Recall that under favoritism, the current MLOT should be negatively
correlated with the past MLOT. In particular, if the sponsor allocates
securities to non-premium investors as a favor, then if the MLOT in the past
is high, the sponsor is likely to ask this investor to finance a weak deal
today. We verify that the probability that a sponsor allocates securities to a
“relationship” investor increases if the investor has higher MLOT from the
past deals _and_ the SPAC deal is cold. Lastly, we show that quid pro quo
lowers the liquidation risks.
#### 5.3.1 Mean-reverting MLOT
As a first step, we assess if the MLOT exhibits mean reversion for the non-
premium PIPE (at) investors, i.e., MLOT is stable around a mean. In contrast,
under the information production hypothesis, the correlation between the
current MLOT and past MLOT should be zero if the information production across
deals is independent and, as (Benveniste et al., 2003) show, they can be
positively correlated if there is information spillover.
A necessary condition to test favoritism is the presence of repeated deals
between a sponsor and a PIPE investor. In Table 7, we list the top 20 sponsors
in our sample in terms of the total number of initialized and completed SPAC
deals. In our sample, 139 sponsors have initialized more than one SPAC,
totaling 439 SPACs. Furthermore, non-premium PIPE investors are more likely to
participate in repeat deals (with the same sponsor) than premium investors.
To gauge the difference, consider the following. On average, the probability
that a non-premium investor from the previous deal participates in the current
deal is 17%. As there are around eight non-premium investors in a SPAC, one
out of these eight non-premium investors in the current deal is from the
previous deal. If these investors were randomly (uniformly) participating in a
SPAC deal, the probability would be much smaller because there are 1,792 non-
premium investors in our sample. If non-premium investors participate randomly
in a SPAC, then the probability that at least one non-premium investor
participated in the previous deal is negligible.
Table 7: Top 20 Sponsors
Initialized | Completed
---|---
Sponsor | #SPAC | Sponsor | #SPAC
The Gores Group | 13 | The Gores Group | 9
Cohen & Company | 11 | Chardan | 7
Social Capital | 10 | Social Capital | 6
Hennessy Capital Group | 8 | Cohen & Company | 6
Cantor Fitzgerald | 8 | TPG | 5
TPG | 8 | Cantor Fitzgerald | 5
Michael Klein | 8 | Riverstone Investment Group | 5
Jonathan J. Ledecky | 7 | Hennessy Capital Group | 5
Chardan | 7 | Harry L. You | 5
Hedosophia | 6 | Eagle Equity Partners | 5
Harry L. You | 6 | Michael Klein | 4
Riverstone Investment Group | 6 | Jonathan S. Huberman | 4
Fortress Investment Group | 6 | Hedosophia | 4
GigCapital Global | 6 | GigCapital Global | 4
Apollo Global Management | 6 | Chinh Chu | 4
Bill Foley | 6 | Niccolo de Masi | 4
Chinh Chu | 5 | Bill Foley | 4
Jaws Estates Capital | 5 | Apollo Global Management | 4
Craig-Hallum Capital Group | 5 | Union Acquisition Group | 3
Niccolo de Masi | 5 | Casdin Capital | 3
Note: This table presents the top 20 sponsors in terms of the total number of
SPACs initialized and completed.
The probability gap reverses for the premium investors. The probability that a
premium investor from the previous deal participates in the current deal is
36.60%. There are, on average, three premium investors in a SPAC, so a sponsor
includes one premium investor from the previous deal. There are 34 premium
investors in our sample, so if they participate randomly in a SPAC, then the
probability that at least one premium investor in the current deal has
participated in the previous deal by the same sponsor is 90.31%.
Let $\mu$ be the average MLOT for all PIPE (at) investors in our sample. Table
8 is the summary statistics. As we see, on average, 28% of SPACs have at least
one premium PIPE investor. Then we estimate the following linear model,
$\displaystyle\text{MLOT}_{p,i,s}$ $\displaystyle=$
$\displaystyle\theta_{0}+\theta_{1}\times(\text{lag-MLOT}_{p,i,s}-\mu)+{\bf
X}^{\top}\theta_{2}+\eta_{p,i,s},$ (4)
where MLOTp,i,s is the MLOT that PIPE (at) investor $p$ gets in the current
SPAC deal $i$ initialized by the sponsor $s$, and $\text{lag-MLOT}_{p,i,s}$ is
the MLOT from the most recent deal in the past. As before, ${\bf X}$ includes
several covariates, including fixed effects (see Table 9).
Table 8: Summary Statistics of PIPE Investors
| Mean | Std.Dev. | P10 | P50 | P90 | Obs.
---|---|---|---|---|---|---
At Least 1 Premium | 0.28 | 0.45 | 0.00 | 0.00 | 1.00 | 5,812
All PIPE Investors | | | | | |
Investment Amount ($mm) | 16.67 | 37.62 | 1.00 | 7.00 | 35.00 | 5,535
MLOT ($mm) | 9.76 | 75.33 | -0.98 | 0.28 | 13.05 | 4,816
No. of Past SPACs | 0.13 | 0.47 | 0.00 | 0.00 | 0.00 | 5,812
MLOT (past) | 2.48 | 23.66 | 0.00 | 0.00 | 0.00 | 5,812
Premium PIPE Investors | | | | | |
Investment Amount ($mm) | 20.50 | 38.79 | 3.00 | 10.00 | 45.00 | 1,609
MLOT ($mm) | 11.01 | 45.60 | -1.17 | 0.58 | 23.97 | 1,462
No. of Past SPACs | 0.20 | 0.53 | 0.00 | 0.00 | 1.00 | 1,647
MLOT (past) | 3.98 | 19.82 | 0.00 | 0.00 | 2.55 | 1,647
Non-premium PIPE Investors | | | | | |
Investment Amount ($mm) | 15.10 | 37.01 | 0.75 | 5.00 | 30.00 | 3,926
MLOT ($mm) | 9.22 | 85.10 | -0.96 | 0.17 | 10.37 | 3,354
No. of Past SPACs | 0.10 | 0.43 | 0.00 | 0.00 | 0.00 | 4,165
MLOT(past) | 1.88 | 25.00 | 0.00 | 0.00 | 0.00 | 4,165
Note: This table contains summary statistics for SPACs that went through an
IPO from January 2010 to December 2021. Premium PIPE investors are identified
using k-means clustering and listed in Table 3. All variables are defined in
Table B1.
Under information production, the deviation of lagged MLOT from its mean is
either uncorrelated or positively correlated with the current MLOT; hence
$\theta_{1}\geq 0$ in (4). Under quid pro quo arrangements, however, the MLOT
should increase if the past deal was lower than the average and vice versa;
hence $\theta_{1}<0$. The results are in column (1) of Table 9. As expected
under favoritism, MLOT exhibits mean reversion.
Table 9: Mean-Reverting MLOT
| MLOT
---|---
| (1) | (2) | (3)
All PIPE Investors | -0.099* | |
| (0.052) | |
Premium PIPE Investors | | -0.025 | 0.235**
| | (0.062) | (0.109)
Non-premium PIPE Investors | | -0.218*** | -2.729***
| | (0.076) | (0.756)
Instrumental Variables | ✗ | ✗ | ✓
F-stat (first-stage) | | | 92.82
Adjusted $R^{2}$ | 0.418 | 0.419 |
Observations | 3,790 | 3,790 | 3,790
| | Mean | Std.Dev.
$\text{Non-Premium}\times$(lag-MLOT-$\mu$) | | -8.12 | 5.41
(lag-MLOT-$\mu$)$\times$(lag2-MLOT-$\mu$) | | 121.96 | 85.54
(lag2-MLOT-$\mu$)2 | | 135.78 | 5.77
(lag-MLOT-$\mu$)2 | | 166.93 | 297.54
Note: This table shows results from estimating (5) using SPACs that went
through IPO between January 2010 to December 2021. Each observation is at
(SPAC, sponsor, PIPEat investor) level. The dependent variable, MLOT, measures
the money left on the table the PIPEat investor makes from the SPAC deal. The
key independent variables are the MLOT earned with the same sponsor in the
last deal for all PIPEat investors, premium PIPEat investors, and non-premium
PIPEat investors. Additional controls include the log of IPO proceeds, the
number of PIPE investors, the number of premium PIPE investors, days
remaining, and separate fixed effects for the month, sponsor, and PIPE. All
continuous variables are winsorized at the $1^{st}$ and $99^{th}$ percentile
values. All other variables are defined in Table B1. Robust standard errors
are reported in parentheses. ${}^{*},^{**}$, and ∗∗∗ denote p-values less than
0.1, 0.05, and 0.01, respectively.
Next, we differentiate between premium and non-premium investors and estimate
$\displaystyle\text{MLOT}_{p,i,s}$ $\displaystyle=$
$\displaystyle\delta_{0}+\delta_{1}\times(\text{lag-MLOT}_{p,i,s}-\mu)$ (5)
$\displaystyle+\delta_{2}\times\text{Non-Premium}_{p}\times(\text{lag-
MLOT}_{p,i,s}-\mu)+{\bf X}^{\top}\delta_{3}+\tilde{\eta}_{p,i,s},$
where $\text{Non-Premium}_{p}$ is a binary variable equal to one if the PIPE
(at) investor $p$ is a non-premium investor. The results from the estimation
are in column (2) of Table 9. Comparing the second and third rows, we find
that MLOT exhibits mean reversion for non-premium PIPE investors but not for
premium investors. This result is consistent with the previous results that
only the premium PIPE investors generate information, and Hypothesis 1-2.
A potential concern with the estimate of $\delta_{2}$ in column (2) of Table 9
is that sponsors’ decisions to include non-premium investors may be correlated
with unobserved deal quality. Indeed, favoritism implies that the sponsor is
more likely to involve non-premium investor(s) for a weak deal. If so, the
sample used to identify the parameter is selected and not representative of
the population. Because the sponsor is likely to allocate higher MLOT to non-
premium investors in weak deals, the above estimate can be biased upwards.
We use the instrumental variable (IV) approach to address this problem. The
idea behind our choice of IV is based on the following observation: suppose
$v=v_{1}\times v_{2}$ is endogenous because $v_{1}$ is endogenous, then
$v_{2}$ can be an IV for $v$ under appropriate additional assumptions. For a
recent application of this intuition, see Fabra and Reguant (2014) and Aryal,
Ciliberto and Leyden (2022). In our setting, $v$ is $\text{Non-
Premium}\times(\text{lag-MLOT}-\mu)$ and $v_{1}$ is the Non-Premium dummy, and
given the specification in (5) $(\text{lag-MLOT}-\mu)$ is uncorrelated with
the quality of the current deal.
However, $(\text{lag-MLOT}-\mu)$ may not satisfy the exclusion restriction
because it directly affects the outcome variable. Based on the intuition
above, we propose to use mean deviation from the second-to-last deal, i.e.,
$(\text{lag${}^{2}$-MLOT}-\mu)$ as an excluded IV. Using lagged variables as
IVs is widely used in the dynamic panel data under the Arellano-Bond
estimator. Furthermore, to capture the nonlinearity of the interaction terms,
we include $\\{(\text{lag-
MLOT}-\mu)\times(\text{lag${}^{2}$-MLOT}-\mu),(\text{lag${}^{2}$-MLOT}-\mu)^{2},(\text{lag-
MLOT}-\mu)^{2}\\}$ as additional excluded variables.
The results from this estimation are in column (3) of Table 9. The first-stage
Cragg-Donald Wald F-statistic of 92.82 suggests that our excluded variables
are not weak instruments. Comparing these estimates with the ones in column
(2), we see that, as expected, the fixed-effect estimate for the non-premium
investor is biased upwards. More importantly, and consistent with the quid pro
quo hypothesis, we find that one dollar higher mean deviation of MLOT in the
last deal leads to a $2.729 reduction in the mean deviation from the current
deal and vice versa presumably because now the non-premium investor is “asked”
to participate in tepid deals that have smaller underpricing. In contrast, for
the premium PIPE investors, the effect is the opposite: if the investor made
one more dollar in MLOT above the mean in the last deal, on average, the MLOT
in the next deal with the same sponsor increases by 23 cents consistent with
the hypothesis that premium PIPE investors generate value-relevant
information.171717Some deals have more than one sponsor. In such cases, we
treated each sponsor separately. Although we do not show this here, we verify
that the results are qualitatively the same if we exclude any SPAC with more
than one sponsor or consider only the sponsor with the maximum number of past
SPAC deals.
#### 5.3.2 Allocations and Past Relationship
Next, we test the Hypothesis 2-3. A sponsor builds a relationship with a non-
premium PIPE investor by providing MLOT, so when the deal is tepid, it expects
the _relationship investors_ with high past MLOT to return the favor by
participating in the deal. To this end, we evaluate if a sponsor is more
likely to use non-premium PIPE (at) investors if the investor has made larger
returns from the sponsor in the past and the current target is cold.
Although we do not observe the quality of the target, we propose to proxy it
by the difference between the offer share price of the SPAC, i.e., the share
price at SPAC IPO, and the share price on the closing day of the merger. We
denote this gap as “Cold.”
Then, we estimate the probability that a sponsor allocates securities to a
non-premium investor as a function of the past MLOT and a measure of cold,
among other variables. For a sponsor $s$, let ${\mathcal{R}}_{s}$ denote the
set of relationship investors as the set of all non-premium investors that
invested in at least one SPAC with sponsor $s$. Let
$\text{Participation}_{p,i,s}\in\\{0,1\\}$ be a binary variable equal to one
if investor $p\in{\mathcal{R}}_{s}$ is allocated securities from SPAC $i$
started by sponsor $s$, and zero otherwise. Then we estimate the participation
probability under the assumption that the probability takes a logit form,
i.e.,
$\displaystyle\Pr(\text{Participation}_{p,i,s}=1\mid\mathbf{Z}_{p,i,s}^{\top}\bm{\delta})$
$\displaystyle=$
$\displaystyle\frac{\exp(\mathbf{Z}_{p,i,s}^{\top}\bm{\delta})}{1+\exp(\mathbf{Z}_{p,i,s}^{\top}\bm{\delta})},$
(6) $\displaystyle\mathbf{Z}_{p,i,s}^{\top}\bm{\delta}$ $\displaystyle=$
$\displaystyle\mathbf{X}^{\top}\bm{\delta_{0}}+\delta_{1}\times\log(1+\text{MLOT(past)}_{p,s})+\delta_{2}\times\text{Cold}_{i}$
$\displaystyle+\delta_{3}\times\log(1+\text{MLOT(past)}_{p,s})\times\text{Cold}_{i},$
where ${\bf X}$ is the vector of controls shown in Table 9;
$\log(1+\text{MLOT(past)}_{p,s})$ is the log of the total MLOT $p$ made in
past deals with $s$ and measures the past relationship between $s$ and $p$;
and, and as we defined above, $\text{Cold}_{i}$ is the measure of the quality
of the deal $i$.
We estimate (6) using the method of maximum likelihood and present the results
in column (1) of Table 10. The estimated coefficient of the interaction term
(third row) in column (1) is positive (p-value $<0.01$), supporting the quid
pro quo hypothesis that the sponsor first builds a relationship and draws it
down as and when needed.
Table 10: Sponsor and Non-Premium PIPE Investors
| Participation | Allocation
---|---|---
| (1) | (2) | (3) | (4) | (5)
$\log(\text{1+MLOT(past)})$ | 0.085 | -0.004 | -0.088 | -0.335∗∗∗ | 0.019
| (0.136) | (0.123) | (0.098) | (0.081) | (0.010)
| | | | | [0.013]
Cold | 0.006∗∗∗ | 0.004∗∗∗ | -0.003 | 0.003∗∗ | -0.00003
| (0.001) | (0.001) | (0.002) | (0.002) | (.00006)
| | | | | [0.0006]
$\log(\text{1+MLOT(past)})\times\text{Cold}$ | 0.008∗∗∗ | 0.011∗∗∗ | 0.011∗∗∗ | 0.008∗∗∗ | 0.0003∗∗∗
| (0.002) | (0.003) | (0.003) | (0.002) | (0.0001)
| | | | | [0.00012]
Outside Option | | -0.017∗∗∗ | -0.007∗∗∗ | -0.002 |
| | (0.003) | (0.002) | (0.002) |
$\widehat{\mathbb{E}(\text{Participation}\mid{\mathbf{Z}})}$ | | | | | 0.239∗∗∗
| | | | | (0.018)
| | | | | [0.024]
Log IPO proceeds | ✗ | ✓ | ✓ | ✓ | ✓
Days Remaining | ✗ | ✓ | ✓ | ✓ | ✓
No. Premium Investors | ✗ | ✓ | ✓ | ✓ | ✗
Sponsor Dummies | ✗ | ✗ | ✓ | ✓ | ✗
Month Dummies | ✗ | ✗ | ✗ | ✓ | ✓
Pseudo/Adj. $R^{2}$ | 0.0098 | 0.0644 | 0.2763 | 0.4061 | 0.8114
Obs. | 10,104 | 10,044 | 7,917 | 7,913 | 7,838
Note: Columns (1) to (4) of this table show the MLE of participation (6), and
column (5) shows the estimates of the allocation (5.3.2). Each observation is
at (SPAC, sponsor, non-premium PIPE investor) level. For each SPAC, non-
premium investors include all relationship investors that participate in at
least one SPAC deal initialized by the sponsor during the sample period. For
the first four columns, the dependent variable, $Participation$, equals one if
the non-premium investor participates in the focal SPAC deal. The dependent
variable for the last column, $Allocation$, is the investor’s allocation as a
fraction of all relationship investors’ allocations. The key independent
variables include the non-premium investor’s MLOT earned with this sponsor in
the past in log form, $\log(1+\text{MLOT ({past})})$, and a measure of the
(reciprocal) of the deal quality, Cold. For the allocation regression, we also
include the estimated probability of participation using estimates from column
(4) as an additional variable. All continuous variables at winsorized at
$99^{th}$ percentile. Robust standard errors are clustered at the sponsor-PIPE
pair level and reported in parentheses. For column (5), bootstrapped standard
errors (500 replications) are reported in the square brackets. ${}^{*},^{**}$,
and ∗∗∗ denote p-values less than 0.1, 0.05, and 0.01, respectively.
One possible concern with the estimation is that it ignores the outside option
of the sponsor, which may be correlated with the unobserved error. A better
outside option lowers the probability of issuing securities to a non-premium
investor and vice versa. Ignoring outside options could introduce attenuation
bias if cold deals have profitable outside options. To alleviate this concern,
we use the highest MLOT among all premium PIPE investors _not included_ in the
current deal as an additional control variable to capture the strength of the
outside option. We also include the log of IPO proceeds, days remaining, and
the number of premium investors as additional covariates.
The estimates from this new model are in column (2) of Table 10. A better
outside option lowers the probability of selecting the non-premium investor,
and our previous finding becomes slightly stronger. The results are robust
when we include dummy variables for the sponsor and the month when the deal is
announced. In all cases, the coefficient for the interaction term is positive
and estimated precisely (p-value $<0.01$).
To help interpret these estimates, we evaluate the probability in (6) at a
different relationship and tepidness while holding other variables at their
average value using the estimates from column (4) of Table 10. We find that
investors with a stronger relationship have higher participation rates in
deals with lower demand. In SPAC deals with Cold at its $90^{th}$-percentile,
non-premium investors who have a strong relationship with the sponsor (one
standard deviation above the mean) are 3.54 pps more likely to participate
than in SPAC deals with Cold at its $10^{th}$-percentile. Conversely, non-
premium investors without a prior relationship with the sponsor are less
likely to participate in cold deals.
Next, we examine whether, conditional on participation, the sponsor allocates
a higher portion of securities to non-premium investors with a stronger
relationship when a deal is cold and vice versa. To this end, we determine the
allocation of SPAC $i$’s securities from sponsor $s$ to PIPE investor $p$
among all relationship investors ${\mathcal{R}}_{s}$,
$\text{Allocation}_{p,i,s}\in[0,1]$, as the ratio of $p$’s shares to the total
shares allocated to ${\mathcal{R}}_{s}$. We set the share to zero if a non-
premium investor does not participate in a deal. We find that the mean
allocation is $2.93$, and the standard deviation of the allocation is $8.77$.
Furthermore, the allocation is skewed towards zero, with the median of $0$ and
the $99^{th}$-percentile at $42.37$. Given that there are 1,792 non-premium
investors, and so the allocation in our sample is skewed to the left with a
mass at zero, we estimate the following Tobit model:
$\displaystyle\text{Allocation}_{p,i,s}^{*}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{p,i,s}^{\top}\bm{\gamma^{*}}+u_{s},$
$\displaystyle\text{Allocation}_{p,i,s}$ $\displaystyle=$
$\displaystyle\min\\{\max\\{0,\mathbf{W}_{p,i,s}^{\top}\bm{\gamma}+\nu_{1}\\},1\\}$
(7) $\displaystyle\mathbf{W}_{p,i,s}^{\top}\bm{\gamma}$ $\displaystyle=$
$\displaystyle\mathbf{X}^{\top}\bm{\gamma_{0}}+\gamma_{1}\times\log(1+\text{MLOT(past)}_{p,s})+\gamma_{2}\times\text{Cold}_{i}$
$\displaystyle+\gamma_{3}\times\log(\text{MLOT(past)}_{p,s}+1)\times\text{Cold}_{i}+\gamma_{4}\times\widehat{\mathbb{E}(\text{Participation}|{\mathbf{Z}})}_{p,i,s},$
where $(\mathbf{W},\text{Allocation}_{p,i,s})$ is always observed but the
latent variable $\text{Allocation}_{p,i,s}^{*}$ is observed only when
$\text{Allocation}_{p,i,s}\in[0,1]$. We further assume that $\mathbf{W}$ is
independent of $(u_{1},]\nu_{1})$ and that $\nu_{1}$ is a mean-zero Normal
random variable with unknown variabce and
$\mathbb{E}(u_{1}\mid\nu_{1})=\rho\nu_{1}$. Furthermore,
$\widehat{\mathbb{E}(\text{Participation}\mid{\mathbf{Z}})}_{p,i,s}$ is the
estimated participation probability of allocation given ${\bf Z}$ and is
determined by the estimates in column (4) of Table 10. Thus, the outside
option serves as our excluded variable, with the underlying exclusion
restriction being that the outside option should not directly affect a non-
premium investor’s allocation.
In Column (5) of Table 10, the estimates indicate a positive association
between a higher probability of participation and an increased allocation of
shares. Moreover, non-premium investors who have previously profited from
deals with the sponsor receive a larger share of allocation than their
counterparts when a deal is considered cold.
In economic terms, a one standard deviation increase in the interaction term
leads to a 5.8% increase in allocation relative to its standard deviation,
amounting to 14.80% of the sample mean. In our sample, an average non-premium
investor is allocated 2.68% of the total non-premium investors’ allocations.
Furthermore, as the non-premium investor’s MLOT (past) times the deal’s Cold
measure increase together by one standard deviation, the allocation fraction
rises by 0.40 pps to 3.08%. In terms of dollar amount, the allocation
increases from $0.40 million to $0.46 million. These findings support the
hypothesis that non-premium investors and sponsors engage in quid pro quo
agreements.
#### 5.3.3 Relationship Capital and Liquidation Risk
We have shown that the relationship investors support the sponsor by
participating in a tepid deal and purchasing a larger fraction of shares.
These non-premium investors received larger MLOTs from the same sponsor during
the previous deals. Therefore, it stands to reason that a sponsor builds these
relationships with an eye toward potential tepid deals in the future. If so,
some of the underpricing driven by quid pro quo could be advantageous to
private companies which want to go public.
Next, we verify Hypothesis 2-4 by estimating the liquidation risk and
liquidation probability of SPACs as a function of the total MLOT paid by the
sponsor to non-premium investors. Here, we use the number of days from the
SPAC’s IPO until a target is announced or liquidation is announced to denote
liquidation risk because a SPAC has two years to find a target, or else it
will have to liquidate. That is, the sooner a SPAC can find a deal, the lower
the liquidation risk.
A sponsor with more relationship capital can afford to take more risk with a
potential tepid target because it can draw down the relationship capital to
close the deal and move on to a new SPAC If the risk and the probability
decrease with the MLOT, then we consider that as evidence of a benefit of quid
pro quo.
Table 11: Liquidation and Relationship
| Liquidation Risk | Liquidation
---|---|---
| (1) | (2)
Premium | 0.092 | 0.363
| (0.061) | (0.222)
Non-premium | -0.138∗∗ | -0.406∗
| (0.066) | (0.246)
Log IPO proceeds | ✓ | ✓
No. Past Successful Deals | ✓ | ✓
Obs. | 679 | 680
Note: This table demonstrates that SPACs initiated by sponsors with a history
of stronger relationships with non-premium investors are less likely to
liquidate and face reduced liquidation risk when a target is announced or
liquidated. Column (1) uses a Poisson regression on the dependent variable
_Liquidation Risk_ , which represents the number of days from the SPAC’s IPO
until a target or liquidation is announced. Column (2) uses a Logit regression
on the dependent variable _Liquidation_ , which is equal to one if the SPAC
liquidates and zero if the SPAC successfully completes its merger. The primary
independent variable is the total MLOT earned by premium and non-premium
investors in previous SPACs initiated by the same sponsor. All continuous
variables are winsorized at the $1^{st}$ and $99^{th}$ percentiles. Robust
standard errors, clustered at the sponsor level, are reported in parentheses.
${}^{*},^{**}$, and ∗∗∗ denote p-values less than 0.1, 0.05, and 0.01,
respectively.
We begin with a count (Poisson) regression of the number of days from the
SPAC’s IPO until a target is announced or liquidation is announced on the MLOT
earned by premium and non-premium investors in previous SPACs initiated by the
same sponsor. We also include the log of the IPO proceeds and the number of
past successful SPACs by the same sponsor as additional controls. The results
from this estimation exercise are in column (1) of Table 11, which suggests
that the days it takes for a SPAC to find a target decreases with the MLOT
earned by non-premium investors in the past. In particular, a one standard
deviation increase in the log of MLOT decreases the number of days to find a
target by 64 days, which is 15.17% of the average 422 days it takes to either
find a target or liquidate in the sample, holding all other variables constant
at their respective average values. However, consistent with the information
creation hypothesis, we find that the MLOT earned by premium investors in the
past is uncorrelated with the liquidation risk.
Next, we consider the probability of liquidation. In our sample, we observe
191 liquidations and 473 successful mergers. We verify the extent to which the
difference between these two groups is attributable to the relationship
payments the SPAC sponsor made to non-premium investors. Let
$\text{Liquidation}_{i}\in\\{0,1\\}$ be a binary variable equal to one if the
SPAC $i$ has liquidated and zero otherwise. Under the assumption that we can
use the Logit model to specify the conditional distribution of liquidation, we
estimate the probability of liquidation and present it in column (2) of Table
11. The estimates suggest that SPACs initialized by sponsors who have built a
stronger relationship with non-premium investors are less likely to liquidate.
However, the coefficient is imprecisely estimated (p-value $<0.1$). In
contrast, but consistent with the information production hypothesis, the MLOT
earned by premium PIPE investors from the same sponsor’s deals in the past is
uncorrelated with the success of the current deal.
In terms of economic magnitude, we find that if sponsors had allocated one
standard deviation more MLOT ($300.53 million) to non-premium investors above
average, the liquidation probability of the current deal would fall by 48.93
pps. As a reference, the average liquidation rate in our sample is 17.82%. In
contrast, MLOT allocated to the premium investors is uncorrelated with the
liquidation risk.
In summary, we find evidence consistent with these conditions: (i) premium
PIPE investors generate value-relevant information, (ii) non-premium PIPE
investors are paid agency fees, and (iii) the non-premium investors help close
weaker deals in the future. In Appendix A, we show that these results are
robust to removing outlier investors, placebo treatments, and the definition
of premium PIPE investors.
## 6 Observed and Unobserved Heterogeneity
In this section, we determine how much of the difference in average earnings
between premium and non-premium investors can be explained by differences in
their observable characteristics. If they have different “business models”
(information production vs. agency service), we expect observed heterogeneity
to explain a smaller portion of the difference.
To minimize the effect of outliers in this exercise, we exclude seven out of
473 successful SPACs with closing share prices higher than $100.181818These
SPACs are: Queen’s Gambit Growth Capital, Insurance Acquisition Corp., Matlin
& Partners Acquisition Corporation, Northern Genesis Acquisition Corp. II, VPC
Impact Acquisition Holdings III, Inc., Reinvent Technology Partners Z (f/k/a
Reinvent Technology Partners B), and Ajax I. In Figure 3 (left panel), we
display the histogram of total MLOT paid to all PIPE (at) investors across
SPACs that have successfully merged with a private target firm. The average
MLOT per deal is $62.38 million, with a large standard deviation of $199.07
million. These earnings are divided among, on average, 10.76 PIPE investors
(see Table 2), out of which 3.02 are premium investors. In Figure 3 (right
panel), we show the shares of premium PIPE investors (left vertical line) and
the total MLOT (right vertical line). Averaging these shares, we find that
premium PIPE investors get (around) 26% of the MLOT. In other words, 26% of
the underpricing at the time of the business closing announcement is for
information production, and 74% is for quid pro quo.
Figure 2: Money Left on the Table
Note: The left panel is the histogram of total MLOT for all
$\texttt{PIPE}_{\texttt{at}}$ investors. The right panel shows the premium
PIPE investors’ share of MLOT (vertical axis on the left) and the total MLOT
in that deal (vertical axis on the right).
Considering each investor separately, the MLOT of a premium investor, on
average, is $6.96 million, and that of a non-premium investor is $3.17
million. Next, we determine what part of this $3.79 million difference in
average MLOT can be explained by differences in the observable characteristics
of these two types of investors.
First, we introduce new variables capturing some of the observed differences
between these two groups. In particular, for each PIPE investor and each SPAC,
we determine the number of SPACs they have participated in before the current
one; how much they have invested; the number of past SPACs that involved a
sponsor different from the current one; total MLOT from other sponsors; and
the number of SPACs with sponsors different from the current one, and whether
the target is in the same industry as the current target. These variables
capture market-wide experience and successes.
We have 4,152 deals at the SPAC and PIPE (at) investor level, of which 1,245
are with premium investors and the rest with non-premium investors. Table 12
presents the descriptive statistics and the results from the tests of mean
differences between premium and non-premium investors.
Table 12: Difference in Characteristics of Premium and Non-premium Investors
| Premium PIPE Sample | Non-Premium PIPE Sample | Difference
---|---|---|---
MLOT ($mm) | 6.95 | 3.17 | 3.78∗∗∗
SPAC Characteristics | | |
IPO Proceeds ($mm) | 397.25 | 363.87 | 33.38∗∗∗
Days Remaining | 492.98 | 467.59 | 25.38∗∗∗
PIPE Investor Characteristics | | |
No. of past SPACs | 23.99 | 5.51 | 18.48∗∗∗
Past Investment Amount ($mm) | 460.09 | 50.77 | 409.32∗∗∗
No. of SPACs with other sponsors | 24.40 | 5.44 | 18.95∗∗∗
Past MLOT with other sponsors ($mm) | 292.85 | 24.61 | 268.25∗∗∗
No. of SPACs with other sponsors, same industry | 3.59 | 0.80 | 2.79∗∗∗
Past MLOT with other sponsors, same industry ($mm) | 45.83 | 3.65 | 42.18∗∗∗
Note: This table reports means and differences in our sample’s average
characteristics for premium and non-premium investors. ${}^{*},^{**}$, and ∗∗∗
denote p-values less than 0.1, 0.05, and 0.01, respectively.
The two groups display distinct MLOTs and characteristics. Specifically,
premium investors are involved in larger SPACs that take fewer days to
identify a target. Moreover, these investors possess greater experience in the
SPAC market, as evidenced by a higher number of previous SPAC participations
(23.99 vs. 5.51) and a larger investment amount ($460.09 million vs. $50.77
million). They also work with more SPACs (24.40 vs. 5.44) and obtain a higher
MLOT ($292.85 million vs. $24.61 million) from other sponsors. In addition,
premium investors operate more in the same industry with different sponsors
(3.59 vs. 0.80).
Figure 3: Kitagawa-Blinder-Oaxaca Decomposition
Note: The figure displays the explained components of a twofold Kitagawa-
Blinder-Oaxaca decomposition of the premium vs. non-premium MLOT. The x-axis
is the mean difference in MLOT (in millions), and the y-axis lists observable
characteristics.
Next, we use the Kitagawa-Blinder-Oaxaca decomposition to document the extent
and drivers of the MLOT differences between premium and non-premium investors.
In particular, we begin with the following:
$\text{MLOT}_{i,g}={\mathbf{W}}_{g}^{\top}\omega_{g}+\iota_{i,g},\quad
g\in\\{\text{premium, non-premium}\\},$ where ${\bf W}$ is the vector of
observed characteristics in Table 12, and $\iota$ is a mean zero random
variable. Let $\Delta\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{MLOT}=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}_{premium}^{\top}\hat{\omega}_{\text{premium}}-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}_{non-premium}^{\top}\hat{\omega}_{\text{non-premium}}$, where
$\overline{W}_{g}$ denotes the sample average of the group $W_{g}$. We can
then re-write $\Delta\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{MLOT}$ as
$\displaystyle\Delta\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{MLOT}=\underbrace{(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\bf
W}_{premium}-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\bf
W}_{non-premium})^{\top}\hat{\omega}_{non-premium}}_{\text{explained
portion}}+\underbrace{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\bf
W}^{\top}_{premium}(\hat{\omega}_{premium}-\hat{\omega}_{non-
premium})}_{\text{unexplained portion}}.$ (8)
The explained portion is the portion of the MLOT difference explained by
differences in observed characteristics between premium and non-premium
investors. In other words, if we change $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\bf
W}_{non-premium}$ to $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\bf
W}_{premium}$, it measures how much additional MLOT a non-premium investor
gets. The unexplained portion is the part driven by differences in the
coefficients.
We find that the mean increase in non-premium investors’ MLOT is $0.04 million
(s.e. $1.38 million) if they had the same characteristics as premium
investors. So the differences in observed characteristics do not account for
the MLOT gap. Figure 3 displays how much each of the eight components in ${\bf
W}$ explains, along with their confidence intervals. Insofar as ${\bf W}$
contains all characteristics that affect MLOT, we find that most of the
difference in average MLOT cannot be explained by differences in average ${\bf
W}$, suggesting that the difference in the business models of these two groups
is important.
## 7 Conclusion
Understanding reasons for underpricing hinges on the relative importance of
information production and quid pro quo. In this paper, we determine the
relative importance within the context of SPACs. Distinguishing between
premium and non-premium PIPE (institutional) investors, we show that the
former produce information and, on average, are paid 26% of the total
underpricing. The latter extracts the remainder 74% as quid pro quo payments.
While we have relied on some institutional features specific to SPACs,
understanding how these features arise to alleviate information and agency
frictions have implications for security issuance under asymmetric information
in general and IPOs in particular. After all, the SPAC business combination
process is also a case of security issuance through sponsors as an
intermediary with PIPE investors as buyers and the target company as the
seller.
There are two main takeaways for IPOs. First, information production and quid
pro quo can play important roles in IPO underpricing. In other words, they are
not necessarily mutually exclusive. Second, non-premium PIPE investors help
sponsors with weak deals, suggesting that, on average, they are value-
enhancing for targets and issuers. This role is unlike the traditional IPO,
where some investment bank clients get favorable allocations of “hot” IPO
shares in exchange for paying high trading fees to the bank.
The benefit of quid pro quo payment is a function of its positive effect on
successful business combinations. However, to determine if quid pro quo
payments enhance welfare, we have to show that the total welfare is higher
when the sponsor is allowed to build and deplete relationships than if not.
These are interesting questions for future research.
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APPENDIX
## Appendix A Robustness
In Section 5, we found evidence consistent with the hypotheses that premium
investors produce value-relevant information and non-premium investors engage
in quid pro quo arrangements. We also found evidence that the quid pro quo
relationship between a sponsor and non-premium investors enables weaker
private firms to go public. In this section, we perform a series of robustness
exercises.
### A.1 Possible Outliers.
Figure A4: Scatter Plots of SPACs
Note: The figure shows the scatter plot of SPACs by month and year. The top
figure shows the number of SPAC mergers, and the bottom figure shows the deal
sizes (total IPO proceeds) in billions of dollars. The size of each dot is
proportional to the number of deals in a month (top) and the total IPO
proceeds (bottom) in that month.
As we can see from Figure A4, although our sample starts in 2010, most of the
variation comes from SPAC from 2020 to 2022. Although our sample includes the
so-called boom and bust cycle for SPAC, we may worry about the effects of
outliers in our analysis. One such concern is that not all 34 premium PIPE
investors (in Table 3) produce value-relevant information and that our results
are driven by either the large premium PIPE investors or that the small
premium PIPE investors are, in fact, non-premium investors.
To address this concern, we exclude six (large) premium investors with at
least $1 billion invested in SPACs and re-estimate all our models. Then, we
repeat the same exercise, but this time we exclude seven (small) premium
investors with at most $400 million invested in SPACs and re-estimate all
models.
Table A1: Robustness: Excluding Outliers
| Redemption | Announcement-day | Mean Reversion | Liquidation Risk | Liquidation
---|---|---|---|---|---
| Rate (%) | Return (%) | MLOT | | | |
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
All PIPE Investors (pre) | 0.180* | 0.159* | -0.085* | -0.074* | | | | | |
| (0.099) | (0.095) | (0.045) | (0.044) | | | | | |
Premium PIPE Investors (at) | -0.241*** | -0.214*** | 0.143** | 0.126*** | 0.153 | 0.149 | 0.114** | 0.080 | 0.423* | 0.356
| (0.082) | (0.051) | (0.060) | (0.041) | (0.104) | (0.099) | (0.058) | (0.061) | (0.226) | (0.222)
Non-premium PIPE Investors (at) | -0.042 | -0.038 | 0.001 | -0.001 | -1.633*** | -2.075*** | -0.155** | -0.126* | -0.448* | -0.391
| (0.030) | (0.030) | (0.014) | (0.015) | (0.531) | (0.615) | (0.062) | (0.065) | (0.235) | (0.241)
All PIPE Investors (post) | 0.152*** | 0.148*** | | | | | | | |
| (0.036) | (0.036) | | | | | | | |
Exclude Large Premium Investors | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗
Exclude Small Premium Investors | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓
Pseudo/Adj. $R^{2}$ | 0.389 | 0.399 | 0.074 | 0.094 | | | 0.0248 | 0.0211 | 0.0153 | 0.0133
F-stat (first-stage) | | | | | 175.54 | 140.73 | | | |
Observations | 430 | 430 | 417 | 417 | 3,487 | 3,578 | 680 | 680 | 681 | 680
Note: This table presents estimates after excluding outliers in premium
investors. Large premium investors are PIPE investors that have invested a
total of more than $1 billion in more than 50 SPACs. They include Fidelity
Management & Research Company, BlackRock, Inc., Capital Research and
Management Company, Alyeska Investment Group L.P., Morgan Stanley, and
Millennium Management, LLC. Small premium investors are those that invested
less than $0.4 billion and participated in at most 50 SPAC deals. They include
Ghisallo Capital Management LLC, Jane Street Group, LLC, Park West Asset
Management LLC, Kepos Capital LP, Linden Advisors LP, Schonfeld Strategic
Advisors LLC, and BlueCrest Capital Management Ltd. Columns (1) and (2) have
the same specification as Table 4, column (4). Columns (3) and (4) have the
same specification as Table 5, column (4). Columns (5) and (6) have the same
specification as Table 9, column (3). Columns (7) and (8) have the same
specification as Table 11, column (1). Columns (9) and (10) have the same
specification as Table 11, column (2). ${}^{*},^{**}$, and ∗∗∗ denote p-values
less than 0.1, 0.05, and 0.01, respectively.
The results are presented in Table A1. In particular, the estimates from
excluding large premium investors are in columns (1), (3), (5), and (7) of
Table A1, and the estimates from excluding small premium investors are in the
remaining columns. Comparing these estimates with those in columns (4), (4),
(3), and (2) of Tables 4, 5, 9, and 11, respectively, we see that our main
findings are neither due only to the large premium investors nor the small
premium investors. The only exception is the liquidation result when we
exclude the large investors (in column (9)), which is less stable.
### A.2 Placebo Exercises.
Next, we verify that our results are not purely noise and that they capture
the effects of premium and non-premium investors. To this end, we explore the
effect of choosing an arbitrary set of PIPE investors and designating them as
premium PIPE investors on the estimates. If our results were only noise, then
a random group of premium PIPE investors should also give similar estimates.
To implement this “placebo” exercise, we randomly picked 34 out of all 1,826
PIPE investors and labeled them premium investors. Then we estimate all the
models using this new set of premium investors. We repeat this exercise 500
times and present only the coefficients for premium and non-premium investors
and the 95% confidence intervals in Figure A5, in the first and the second
column, respectively. The five rows are redemption rate, announcement day
return, mean-reversion, liquidation risk, and probability of
liquidation.191919For the mean-reversion exercise, we present the first 500
cases with the first-stage F-stat $\geq 50$.
Now, let us consider each row separately. For the redemption rates, comparing
the first row in Figure A5 with the estimates in Table 4 column (4), we see
that the randomly chosen premium PIPE investors do not affect redemption
rates. However, because the non-premium PIPE investors include some (actual)
premium PIPE investors, the redemption rate is low but negatively correlated
with allocations for non-premium PIPE investors. Similarly, comparing the
second row in Figure A5 with the estimates in Table 5 column (4), we find no
effect of premium PIPE investors on the announcement-day return but a small
and positive effect of non-premium PIPE investors on first-day return because
the latter set now contains the premium investors identified in Table 3. These
effects are consistent with only the premium investors identified in Table 3
producing information.
For the mean-reversion (in the third row), some estimates are negative, which
is consistent with the new premium PIPE investor containing non-premium PIPE
investors and vice versa. Finally, for liquidation risk and probability of
liquidation, comparing rows four and five in Figure A5 with Table 11 columns
(1) and (2), we also find that the new estimates are mixed, which is again
consistent with the new premium investors containing non-premium investors,
although the estimates have wider confidence intervals.
Figure A5: “Placebo” Exercises
Note: This figure shows estimation results using randomly generated premium
PIPE identities. We randomly pick 34 PIPE investors out of 1,826, label them
as premium PIPE investors, and re-estimate our models. We repeat this exercise
so that we have 500 sets of randomly chosen premium PIPE samples. The first
column presents the coefficients corresponding to (the new) premium PIPE
investors and the second column corresponds to the (new) non-premium PIPE
investors. The five rows correspond to the (i) redemption rates, (ii)
announcement-day return, (iii) mean reversion, (iv) liquidation risk, and (v)
probability of liquidation. In all these five cases, the specifications are
the same as in Tables 4 and 5, column (4), Table 9 column (3), and Table 11
columns (1) and (2), respectively.
Table B1: Variable Definition
Variable | Unit | Definition
---|---|---
Panel A: SPAC-Level Characteristics
SPAC Measures |
IPO proceeds | MM USD | Gross proceeds raised in the IPO, including any full or partial exercise of the Greenshoe.
No. of Warrant | | Number of warrants included in the unit issued at the SPAC’s IPO.
No. of Right | | Number of rights included in the unit issued at the SPAC’s IPO.
Overallotment | % | Amount of the overallotment option exercised by the underwriter(s) as a fraction of total IPO proceeds.
Listing-day return | % | SPAC IPO investors’ first-day return.
Days searching | Day | The number of days between the SPAC’s IPO date and the target announcement date.
Days remaining | Day | The number of days between the SPAC’s target announcement date and the liquidation deadline.
Announcement-day return | % | One-day return on the SPAC’s shares on the target’s announcement date.
Redemption | % | Redeemed SPAC common shares as a percentage of total shares issued at IPO.
Days deSPAC | Day | The number of days between the SPAC’s target announcement date and the merger closing date.
Cold | pps | The difference between the SPAC’s offer price and the SPAC’s share price on the closing day of the merger.
Liquidation Risk | Day | The number of days between the SPAC’s IPO date, the target announcement date, or the liquidation date.
Liquidation | binary | One if the SPAC liquidated during our sample period, zero otherwise.
PIPE Participation |
$\mathbbm{1}${PIPE (pre)} | Binary | One if PIPE (pre) investors participate in the deal.
$\mathbbm{1}${PIPE (at)} | Binary | One if PIPE (at) investors participate in the deal.
$\mathbbm{1}${PIPE (post)} | Binary | One if PIPE (post) investors participate in the deal.
$\mathbbm{1}${Premium PIPE (at)} | Binary | One if premium PIPE (at) investors participate in the deal.
$\mathbbm{1}${Non-Premium PIPE (at)} | Binary | One if non-premium PIPE (at) investors participate in the deal.
No. of PIPE (at) | | Number of PIPE (at) investors.
No. Premium PIPE (at) | | Number of premium PIPE (at) investors.
No. of Non-Premium PIPE (at) | | Number of non-premium PIPE (at) investors.
PIPE Investment/IPO Proceeds |
%PIPE (pre) | % | PIPE (pre) investors’ investment as a percentage of total IPO proceeds.
%Premium PIPE (at) | % | Premium PIPE (at) investors’ investment as a percentage of total IPO proceeds.
%Non-Premium PIPE (at) | % | Non-premium PIPE (at) investors’ investment as a percentage of total IPO proceeds.
%PIPE (post) | % | PIPE (post) investors’ investment as a percentage of total IPO proceeds.
Panel B: PIPE-Investor-Level Characteristics
Non-Premium | Binary | One if either the PIPE (at) investor’s number of participated PIPE deals or the total commitment amount is not in the top 5th percentile.
Investment Amount | MM USD | Dollar amount of the PIPE investor’s investment in the SPAC deal.
MLOT | MM USD | The difference between the value of securities held by the PIPE investor on the first day of listing of the post-merger company and the PIPE investor’s initial investment.
lag-MLOT | MM USD | The PIPE investor’s MLOT from the previous SPAC deal with the same sponsor.
lag2-MLOT | MM USD | The PIPE investor’s MLOT from the one before the previous SPAC deal with the same sponsor.
MLOT(past) | MM USD | Total MLOT a PIPE investor made in past deals with the sponsor.
No. of past SPACs | | Number of SPAC deals participated by the PIPE investor before the focal SPAC deal.
Past Investment Amount | MM USD | The PIPE investor’s total amount of investments in SPAC deals before the focal SPAC deal.
No. of SPACs with other sponsors | | Number of other sponsors’ SPAC deals participated in by the PIPE investor before the focal SPAC deal.
Past MLOT with other sponsors | MM USD | The PIPE investor’s total MLOT made in other sponsors’ SPACs before the focal SPAC deal.
No. of SPACs with other sponsors, same industry | | Number of other sponsors’ SPAC deals that found targets in the same industry as the focal SPAC and participated by the PIPE investor before the focal SPAC deal.
Past MLOT with other sponsors, same industry | MM USD | The PIPE investor’s total MLOT made in other sponsors’ SPACs that found targets in the same industry as the focal SPAC.
Outside Option | MM USD | Highest MLOT among all premium PIPE investors not included in the current deal.
Participation | Binary | One if the sponsor allocates securities to the non-premium PIPE investor, and zero otherwise.
Allocation | | Ratio of the non-premium investor’s shares to the total shares allocated to the sponsor’s relationship investors.
|
# Asymmetric Dependence Measurement and Testing
H. D. Vinod address: H. D. Vinod, Professor of Economics, Fordham University,
Bronx, New York, USA 10458. E-mail<EMAIL_ADDRESS>JEL codes C30, C51.
Keywords: Kernel regression, Standardized beta coefficients, Partial
Correlation.
###### Abstract
Measuring the (causal) direction and strength of dependence between two
variables (events), $X_{i}$ and $X_{j}$, is fundamental for all science. Our
survey of decades-long literature on statistical dependence reveals that most
assume symmetry in the sense that the strength of dependence of $X_{i}$ on
$X_{j}$ exactly equals the strength of dependence of $X_{j}$ on $X_{i}$.
However, we show that such symmetry is often untrue in many real-world
examples, being neither necessary nor sufficient. Vinod’s (2014) asymmetric
matrix $R^{*}\in[-1,1]$ of generalized correlation coefficients provides
intuitively appealing, readily interpretable, and superior measures of
dependence. This paper proposes statistical inference for $R^{*}$ using
Taraldsen’s (2021) exact sampling distribution of correlation coefficients and
the bootstrap. When the direction is known, proposed asymmetric (one-tail)
tests have greater power.
## 1 Introduction
A great deal of science focuses on understanding the dependence between
variables. Its quantification has a long history starting with Galton-Pearson
correlation coefficient $r_{ij}$ from the 1890s and its cousins, including
Spearman’s $\rho$, Kendall’s $\tau$, and Hoeffding’s $D$. Let
$dep(X_{i}|X_{j})$ measure the strength of dependence of $X_{i}$ on $X_{j}$
given a measurement $X_{j}$. Many measures of dependence try to satisfy the
symmetry postulate by Renyi (1959), which posits that the two strengths based
on opposite conditioning are identical:
$dep(X_{i}|X_{j})\equiv dep(X_{j}|X_{i}).$ (1)
We regard the symmetry postulate akin to an avoidable dogma. The following
subsection explains why attempting to satisfy this symmetry equation (1)
provides misleading measures of dependence in practice.
### 1.1 Four Examples of Asymmetric Dependence
A correct notion of dependence in nature or data is rarely (if ever)
symmetric.
* •
A newborn baby boy depends on his mother for his survival, but it is ludicrous
to expect that his mother must exactly equally depend on the boy for her
survival, as implied by (1).
* •
Meteorologists know that the average daily high of December temperatures in
New York city is 44 degrees Fahrenheit and that this number depends on New
York’s latitude (40.7). The latitude is a geographical given and does not
depend on anything like city temperatures. Symmetric dependence by (1) between
temperature and latitude implies the ludicrous claim that latitude depends on
temperature with equal strength.
* •
For a third example, imagine a business person B with several shops. B’s 30%
earnings depend on the hours worked by a key employee in one shop. Now the
symmetry by (1) means that hours worked by the key employee always depend on
B’s earnings, exactly 30%.
* •
Our fourth example assumes $Y$ as complete data, but a subset of $Y$ is
unavailable. The available subset $X$ is a proxy that depends on $Y$, but the
complete set $Y$ does not equally depend on its subset $X$.
These four examples are enough to convince the reader that the symmetry
postulate is neither necessary nor sufficient for real-world dependence.
However, it is interesting that the unrealistic property (1) is an old
established sacrosanct postulate from the 1950s, Renyi (1959). Even in 2022,
Geenens and de Micheaux (2022)(“GM22”), still adhere to the symmetry postulate
(dogma) by proposing an ingenious new definition of dependence to fit the
model in (1). Actually, measure of dependence satisfying (1) can be ludicrous
some contexts analogous to the four examples above.
### 1.2 Sources of the Symmetry Dogma
What is the origin of symmetry dogma?
(i) The definitional and numerical equality of covariances,
$Cov(X_{i},X_{j})=Cov(X_{j},X_{i})$ may have been the initial reason for the
symmetry result.
(ii) In a bivariate linear regression $X_{1}=a+bX_{2}+\epsilon$, the strength
of dependence of $X_{1}$ on $X_{2}$ is clearly measured by the coefficient of
determination $R^{2}_{1|2}$. If we consider a flipped linear regression,
$X_{2}=a^{\prime}+b^{\prime}X_{1}+\epsilon^{\prime}$, the strength of
dependence is $R^{2}_{2|1}$. The assumption of linearity makes the two
strengths equal to each other $R^{2}_{2|1}=R^{2}_{1|2}$. The equality of two
$R^{2}$ strengths supports the symmetry dogma. When we consider the signed
square roots of the two $R^{2}$ values, we have a symmetric matrix of
correlation coefficients, $r_{ij}=r_{ji}$. These signed measures of dependence
further support the dogma.
The symmetry dogma depends on the harmless-looking linearity assumption. Back
in 1784, the German philosopher Kant said: “Out of the crooked timber of
humanity, no straight thing was ever made.” Since social sciences and medicine
deal with human subjects, evidence supporting linearity and the implied
symmetry dogma is missing.
(iii) Since all distances satisfy symmetry, it may have been another reason
behind Renyi’s postulate.
(iv) The concept of statistical independence in probability theory is
symmetric. It can be formulated in terms of the absence of any divergence
between a joint density and a product of two marginal densities,
$f(X_{i}X_{j})=f(X_{i})\,f(X_{j}).$ (2)
Since dependence is the opposite of independence, it is tempting (but
unhelpful) to impose symmetry on dependence as well.
### 1.3 Statistical Independence in Contingency Tables
Two-way contingency tables refer to tabulated data on a grand total of $T$
observations distributed over an $r\times c$ matrix. There are two categorical
variables represented by $r$ manifestations of row characteristics $R_{i}$
along ($i=1,2,\ldots r$) rows, and $c$ column characteristics $C_{j}$ along
($j=1,2,\dots c$) columns. The body of the ($r\times c$) contingency table has
observed values $O_{ij}$ in a matrix cell located at row number $i$ and column
number $j$. The joint probability $P(R_{i},C_{j})$ is simply $O_{ij}/GT$,
where GT denotes the grand total of the tabulated numbers. The row margins of
the contingency table have row totals, $R_{i}=\Sigma_{j}O_{ij}$. The column
margin has column totals, $C_{j}=\Sigma_{i}O_{ij}$. The marginal probabilities
are $P(R_{i})=R_{i}/GT$ and $P(C_{j})=C_{j}/GT$, which are also called
unconditional probabilities.
A conditional probability restricts the sample space to the part of the Table
which satisfies the specified condition, referring to a particular row $R_{i}$
or column $C_{j}$. The direct computation of conditional probability has the
respective row or column sums in its denominator instead of the grand total
$GT$. An equivalent calculation of conditional probability defines
$P(R_{i}|C_{j})=P(R_{i},C_{j})/P(C_{j})$, as a ratio of the joint probability
to the marginal probability of the conditioning characteristic. Analogous
conditional probability conditioning on row characteristic is a ratio of the
same joint probability to the marginal probability of the conditioning row,
$P(C_{j}|R_{i})=P(R_{i},C_{j})/P(R_{i})$.
In probability theory based on contingency tables, the notion of statistical
independence is studied by considering the following three criteria.
(a) $P(R_{i},C_{j})=P(R_{i})P(C_{j})$, joint probability equals the product of
marginals.
(b) $P(R_{i}|C_{j})=P(R_{i})$, conditional probability equals unconditional or
marginal probability.
(c) $P(C_{j}|R_{i})=P(C_{j})$, the other conditional probability equals
unconditional or marginal probability.
Note that criterion (a) is both necessary and sufficient for independence. It
is symmetric in that the joint probability is the same even if we interchange
the order and write it as $P(C_{j},R_{i})$. However, data can satisfy (b)
without satisfying (c), and vice versa. Hence tests of independence typically
rely on the symmetric criterion (a). However, dependence is the opposite of
independence and is generally asymmetric. We find that using (b) and (c) helps
avoid the misleading symmetry postulate in the context of dependence.
It is customary to imagine a population of thousands of contingency tables;
the observed table is one realization from that population. The null
hypothesis ($H_{0}$) is that row and column characteristics are statistically
independent. The sample table may not exactly satisfy independence in the
sense of (a) to (c) above. The testing problem is whether the observed table
of $O_{ij}$ values could have arisen from a population where conditions (a) to
(c) are satisfied. That is, $O_{ij}$ are numerically close enough to the
expected values $E_{ij}=R_{i}C_{j}$ obtained from the cross-product of
relevant marginal totals.
Pearson’s Chi-square test statistic for ($H_{0}$) or independence of row
effect and column effect in a contingency table is
$\chi^{2}=(O_{ij}-E_{ij})^{2}/E_{ij},\quad df=(r-1)(c-1),$ (3)
where $df$ denotes the degrees of freedom. Note that $\chi^{2}\in[0,\infty)]$
of (3) cannot be computed unless we have contingency tables. Statisticians
have long recognized that the magnitude of $\chi^{2}$ cannot reliably measure
the direction and strength of dependence. This paper assumes that a
practitioner would want to know both the general direction and strength of
dependence.
## 2 Symmetric Measures of Dependence
Granger et al. (2004) (“Gr04”) is an important paper on formal testing for
statistical independence, especially for time series data. They cite a survey
by Tjostheim (1996) on the topic. The novelty in Gr04 is in using
nonparametric nonlinear kernel densities in testing the equality (2) in their
test of independence. Unfortunately, Gr04 authors adhere to the symmetry dogma
by insisting that, similar to independence, a measure of dependence should be
a symmetric distance-type ‘metric.’
### 2.1 Dependence Measures and Entropy
Shannon defined information content in 1948 as the amount of surprise in a
piece of information. His information is inversely proportional to the
probability of occurrence and applies to both discrete and continuous random
variables with probabilities defined by a probability distribution $f(y)$. In
the context of entropy, let us use the fourth example of Section 1.1, where
$Y$ is the complete data and $X$ is a subset with some missing observations.
How does $X$ depend on $Y$? We develop a measure of dependence using
information theory, especially entropy.
Intuitively, entropy is our ignorance or the extent of disorder in a system.
The entropy $H(Y)$ is defined by the mathematical expectation of the Shannon
information or $E(-logf(y))$. And the conditional entropy of Y given X
averaged over $X$is
$H(Y|X)=-E[E[log(f_{Y|X}(Y|X))|X]].$ (4)
The reduction in our ignorance $H(Y)$ by knowing the proxy $X$ is
$H(Y)-H(Y|X)$. Mutual information $I_{mu}(X,Y)$ is defined as
$H(x)+H(Y)-H(X,Y).$ It is symmetric since $I_{mu}(X,Y)=I_{mu}(Y,X)$. The
entropy-based measure of dependence is
$D(X;Y)=\frac{H(Y)-H(Y|X)}{H(Y)},$ (5)
or proportional reduction in entropy of $Y$ by knowing $X$. Reimherr and
Nicolae (2013) complain that (5) is not symmetric. By contrast, we view
asymmetry as a desirable property.
Neyman-Pearson showed that a way to distinguish between two distributions
$f(X)$ and $f(Y)$ for parameter $\theta$ is the difference between logs of
their likelihood functions. Shannon’s relative entropy, also known as
Kullback–Leibler (KL) divergence, is the expected value of that difference,
$KLD=E(logf(\theta|X)-logf(\theta|Y)).$ (6)
It is easy to verify that KLD or relative entropy is not symmetric.
Gr04 authors state on page 650 that “Shannon’s relative entropy and almost all
other entropies fail to be ‘metric’, as they violate either symmetry, or the
triangularity rule, or both.” We argue that asymmetry is an asset, not a
liability, in light of four examples in Section 1.1. Hence, we regard (5) or
(6) as superior measures compared to the symmetric measure by Gr04.
D(X;Y) of (5) and KLD of (6) cannot be used directly on data vectors. They
need frequency distribution counts as input based on the grouping of data into
bins (histogram class intervals). The choice of the number of bins is
arbitrary, and D(X;Y) and KLD are sensitive to that choice. Hence, we do not
recommend D(X;Y) or KLD as a general-purpose measure of dependence.
### 2.2 Dependence Measures and Fisher Information
Fisher information measures the expected amount of information given by a
random variable $Y$ about a parameter $\theta$ of interest. Under Gaussian
assumptions, the Fisher information is inversely proportional to the variance.
Reimherr and Nicolae (2013) use the Fisher information to define a measure of
dependence. Consider the estimation of a model parameter $\theta$ using $X$ as
a proxy for unavailable $Y$. That is, $X$ is a subset of $Y$ with missing
observations, as in the fourth example of Section 1.1. If the Fisher
information for $\theta$ based on proxy $X$ is denoted by
$\mathcal{I}_{X}(\theta)$, they define a measure of dependence as:
$D(X;Y)=\frac{\mathcal{I}_{X}(\theta)}{\mathcal{I}_{Y}(\theta)},$ (7)
where $\mathcal{I}_{X}(\theta)\leq\mathcal{I}_{Y}(\theta)$. Consider the
special case where a proportion $p$ of the $Y$ data are missing in $X$ at
completely random locations. Then, the measure of dependence (7) equals $p$.
This measure of dependence is almost acceptable because it is asymmetric,
where subset $X$ being a proxy for $Y$ cannot be interchanged with $Y$, except
that $D(X;Y)$ of (7) cannot be negative. Later, we recommend in Section 3 a
more generally applicable and intuitive measure of dependence.
### 2.3 Regression Dependence from Copulas
Consider a two-dimensional joint (cumulative) distribution function $F(X,Y)$
and marginal densities $U=F_{1}(X)$ and $V=F_{2}(Y)$ obtained by probability
integral transformations. Sklar proved in 1959 that a copula function
$C(F_{1},F_{2})=F$ is unique if the components are continuous. The copula
function $C:[0,1]^{2}\to[0,1]$ is subject to certain conditions forcing it to
be a bivariate uniform distribution function. It is extended to the
multivariate case to describe the dependence structure of the joint density.
We have noted in section 1.3 that a contingency table represents the joint
dependence structure of row and column characteristics. Copulas represent
similar joint dependence when row and column characteristics are continuous
variables rather than simple categories.
Dette et al. (2013) (“DSS13”) define joint density as $F_{X,Y}$, and
conditional density of $Y$ given $X$ as $F_{Y|X=x}$. They use uniform random
variables $U$ and $V$ to construct copula $C$ as a joint distribution
function. The copula serves as their measure of dependence based on the
quality of regression-based prediction of $Y$ from $X$. The flipped prediction
of $X$ from $Y$ ignored by DSS13 is considered in Section 3 in the sequel.
DSS13 assume Lipschitz continuity, which implies that a copula is absolutely
continuous in each argument, so that it can be recovered from any of its
partial derivatives by integration. The conditional distribution $F_{V|U=u}$
is related to the corresponding copula $C(X,Y)$ by
$F_{V|U=u}(v)=\partial_{1}C_{X,Y}(u,v)$.
A symmetric measure of dependence proposed by DSS13 is denoted here as
$r_{D}(X,Y)=6\int_{0}^{1}\int_{0}^{1}F_{V|U=u}(v)^{2}dvdu,$ (8)
where $r_{D}=0$ represents independence, and $r_{D}=1$ represents almost sure
functional dependence. DSS13 focus on $r_{D}$ filling the intermediate range
of the closed interval $[0,1]$ while ignoring the negative range $[-1,0)$.
Section 3 covers $[-1,1]$, including the negative range. DSS13 rely on
parametric copulas, making them subject to identification problems, as
explained by Allen (2022).
The numerical computation of (8) is involved since it requires the estimation
of the copula’s partial derivative. DSS13 authors propose a kernel-based
estimation method without providing any ready-to-use computational tools for
$r_{D}$.
Remark 3.7 in Beare (2010) states that symmetric copulas imply time
reversibility, which is unrealistic for economic and financial data. Bouri et
al. (2020) reject the symmetry dogma and note that their parametric copula can
capture tail dependence, which is important in a study of financial markets.
Allen (2022) uses nonparametric copula construction and asymmetric $R^{*}$
from Vinod (2014). Allen’s application to financial data shows that
cryptocurrencies do not help portfolio diversification.
### 2.4 Hellinger Correlation $\eta$ as a Dependence Measure
Now we turn to the recent GM22 paper mentioned earlier, which proposes
Hellinger correlation $\eta$ as a new symmetric measure of the strength of
dependence. They need to normalize to ensure that $\eta\in[0,1]$. GM22 denote
the normalized version as $\hat{\eta}$. GM22 authors explain why dependence
axioms by Renyi (1959) need updating, while claiming that their $\eta$
satisfies all updated axioms. Unfortunately, GM22 retain the symmetry axiom
criticized in Section 1.1 above. An advantage of $\eta$ over Pearson’s
$r_{ij}$ is that it incorporates some nonlinearities.
Let $F_{1}$ and $F_{2}$ denote the known marginal distributions of random
variables $X_{1}$ and $X_{2}$, and let $F_{12}$ denote their joint
distribution. Now, GM22 authors ask readers to imagine reconstructing the
joint distribution from the two marginals. The un-intuitive (convoluted?)
definition of the strength of dependence by GM22 is the size of the “missing
link” in reconstructing the joint from marginals. This definition allows GM22
to claim that symmetry is “unquestionable.”
GM22 authors define squared Hellinger distance $\mathcal{H}^{2}(X_{1},X_{2})$
as the missing link between $F_{12}$ and $F_{1}F_{2}$. They approximate a
copula formulation of $\mathcal{H}^{2}$ using the Bhattacharyya (1943)
affinity coefficient $\mathcal{B}$. Let $C_{12}$ denote the copula of
$(X_{1},X_{2})$, and $c_{12}$ denote its density. The computation of
$\hat{\eta}$ in the R package HellCor uses numerical integrals
$\mathcal{B}=\int\int\surd c_{12}$. Hellinger correlation $\eta$ is
$\eta=\frac{2}{\mathcal{B}^{2}}\\{\mathcal{B}^{4}+(4-3\mathcal{B}^{4})^{1/2}-2\\}^{1/2}.$
(9)
The Hellinger correlation is symmetric, $\eta(X_{1},X_{2})=\eta(X_{2},X_{1})$.
GM22 provide an R package HellCor to compute $\hat{\eta}$ from data as a
measure of dependence, and test the null hypothesis of independence of two
variables.
A direct and intuitive measure of dependence in a regression framework is the
multiple correlation coefficient (of determination) $R_{1|2}^{2}$. It is
symmetric because even if we flip $X_{1}$ and $X_{2}$, the $R_{2|1}^{2}$ from
linear regressions is exactly the same. The reason for the equality of two
flipped $R^{2}$ values, $R^{2}_{(2|1)}=R^{2}_{(1|2)}$, is the assumption of
linearity of the two regressions. When we relax linearity, the two $R^{2}$
values generally differ, $R_{2|1}^{2}\neq R^{2}_{1|2}$. We argue that
quantitative researchers should reject the unrealistic linearity assumption in
the presence of ready-to-use kernel regression (np package) software.
Kernel-based $R^{2}$ values of flipped regressions are rarely equal. GM22 cite
Janzing et al. (2013) only to reject such asymmetric dependence suggested by
nonparametric regressions.
## 3 Recommended Measures of Dependence
We have noted earlier that covariances satisfy symmetry
$Cov(X_{i},X_{j})=Cov(X_{j},X_{i})$. However, the sign of symmetric
covariances suggests the overall direction of the dependence between the two
variables. For example, $Cov(X_{i},X_{j})<0$ means when $X_{i}$ goes up
$X_{j}$ goes down, by and large. Most of the symmetric measures of dependence
discussed above fail to provide this type of useful directional information
except Pearson’s correlation coefficients $r_{ij}$. Hence, $r_{ij}$ has
retained its popularity as a valuable measure of dependence for over a
century, despite assuming unrealistic linearity.
Zheng et al. (2012) reject the dogma by introducing nonsymmetric generalized
measures of correlation ($GMC\in[0,1]$), proving that
$GMC(Y|X)\neq GMC(X|Y).$ (10)
Since GMCs fail to provide directional information in covariances needed by
practitioners, Vinod (2014) and Vinod (2017) extend Zheng et al. (2012) to
develop a non-symmetric correlation matrix $R^{*}=\\{r^{*}_{ij}\\}$, where
$r^{*}_{ij}\neq r^{*}_{ji}$, while providing an R package. The R package
generalCorr uses kernel regressions to overcome the linearity of $r_{ij}$ from
the np package by Hayfield and Racine, which can handle kernel regressions
among both continuous and discrete variables.
Sometimes the research interest is focused on the strength of dependence,
while the direction is ignored, perhaps because it is already established. In
that case, one can use the R package generalCorr and the R function
depMeas(,). It is defined as appropriately signed larger of the two
generalized correlations, or
$depMeas(X_{i},X_{j})=sgn*max(|r^{*}(i|j)|,|r^{*}(j|i)|),$ (11)
where $sgn$ is the sign of the covariance between the two variables.
In general, both the strength and general direction of quantitative dependence
matter. Hence, we recommend two asymmetric measures $r^{*}(X_{i}|X_{j})$ and
$r^{*}(X_{j}|X_{i})$. The generalCorr package functions for computing $R^{*}$
elements are rstar(x,y) and gmcmtx0(mtx). The latter converts a data matrix
argument (mtx) with $p$ columns, into a $p\times p$ asymmetric matrix $R^{*}$
of generalized correlation coefficients. Regarding the direction of
dependence, the convention is that the variable named in the column is the
“cause” or the right-hand regressor, and the variable named along the row is
the response. Thus the recommended measures from $R^{*}$ are easy to compute.
See an application to forecasting the stock market index of fear (VIX) and
causal path determination in Allen and Hooper (2018).
### 3.1 Statistical Inference for Recommended Measures
We recommend the signed generalized correlation coefficients $-1\leq
r^{*}_{ij}\neq r^{*}_{ji}\leq 1$ from the $R^{*}$ matrix as the best
dependence measure. This is because they do not adhere to the potentially
misleading symmetry dogma while measuring arbitrary nonlinear dependence
dictated by the data. An additional reason is its potential for more powerful
(one-tail) inference, discussed in this section.
The sign of each element of the $R^{*}$ matrix is based on the sign of the
covariance $Cov_{ij}=Cov(X_{i},X_{j})$. A two-tail test of significance is
appropriate only when $Cov_{ij}\approx 0$. Otherwise, a one-tail test is
appropriate. Any one-tailed test provides greater power to detect an effect in
one direction by not testing the effect in the other direction, Kendall and
Stuart (1977), sections 22.24 and 22.28.
Since sample correlation coefficient $r_{ij}$ from a bivariate normal parent
has a non-normal distribution, Fisher developed his famous z-transformation in
the 1920s. He proved that the following transformed statistic $r^{T}_{ij}$ is
approximately normal with a stable variance,
$r^{T}_{ij}=(1/2)\quad log\frac{(1+r_{ij})}{(1-r_{ij})}\sim N(0,1/n),$ (12)
provided $r_{ij}\neq 1$. Recent work has developed the exact distribution of a
correlation coefficient. It is now possible to directly compute a confidence
interval for any hypothesized value $\rho$ of the population correlation
coefficient.
Let $r$ be the empirical correlation of a random sample of size $n$ from a
bivariate normal parent. Theorem 1 of Taraldsen (2021) generalized Fisher’s
famous z-transformation extended by C. R. Rao. The exact density with
$v=(n-1)>1$ is
$\displaystyle f(\rho|r,v)$
$\displaystyle=\frac{v(v-1)\Gamma(v-1))}{\surd(2\pi)\Gamma(v+0.5)}(1-r^{2})^{\frac{v-1}{2}}\,(1-\rho^{2})^{\frac{v-2}{2}}(1-r\rho)^{\frac{1-2v}{2}}$
$\displaystyle\quad F(\frac{3}{2};-0.5;v+0.5;\frac{1+r\rho}{2}),$
where F(.;.;.;.) denotes the Gaussian hypergeometric function, available in
the R package hypergeom by R.K.S Hankin. The following R code readily computes
(3.1) over a grid of 2001 $r$ values.
library(hypergeo); r=seq(-1,1,by=0.001)
Tarald=function(r,v,rho,cum){ #find quantile r given cum
Trm1=(v*(v-1)*gamma(v-1))/((sqrt(2*pi)*gamma(v+0.5)))
Trm2=(1-r^2)^((v-1)/2)
Trm2b=((1-rho^2)^((v-2)/2))*((1-rho*r)^((1-2*v)/2))
Trm3b=hypergeo(3/2,-1/2,(v+0.5),(1+r*rho)/2)
y0=Re(Trm1*Trm2*Trm2b*Trm3b)
p=y0/sum(y0)
cup=cumsum(p)
loc=max(which(cup<cum))+1
return(r[loc])}
Tarald(r=seq(-1,1,by=0.001),v=11,rho=0,cum=0.05) #example
Assuming that the data come from a bivariate normal parent, the sampling
distribution of any correlation coefficient is (3.1). Hence, the sampling
distribution of unequal off-diagonal elements of the matrix of generalized
correlations $R^{*}$ also follows (3.1). When we test the null hypothesis
$H_{0}:\rho=0$, the relevant sampling distribution is obtained by plugging
$\rho=0$ in (3.1) depicted in Figure 1 for two selected sample sizes. Both
distributions are centered at the null value $\rho=0$.
A two-tail (95%, say) confidence interval is obtained by using the 2.5% and
97.5% quantiles of the density. If the observed correlation coefficient $r$ is
inside the confidence interval, we say that the observed $r$ is statistically
insignificant, as it could have arisen from a population where the null value
$\rho=0$ holds.
Figure 1: Taraldsen’s exact sampling density of a correlation coefficient
under the null of $\rho=0$, solid line n=50, dashed line n=15
Similarly, one can test the nonzero null hypothesis $H_{0}:\rho=0.5$ using the
equation obtained by plugging $\rho=0.5$ in (3.1) depicted in Figure 2 .
Figure 2: Taraldsen’s exact sampling density of correlation coefficient under
the null of $\rho=0.5$, solid line n=50, dashed line n=15
Figures 1 and 2 show that the formula (3.1) and our numerical implementation
are ready for practical use. These exact densities depend on the sample size
and on the value of the population correlation coefficient, $-1\leq\rho\leq
1$. Given any hypothesized $\rho$ and sample size, a computer algorithm
readily computes the exact density, similar to Figures 1 and 2. Suppose we
want to help typical practitioners who want the tail areas useful for testing
the null hypothesis $\rho=0$. Then, we need to create a table of a set of
typical quantiles evaluated at certain cumulative probabilities and a
corresponding selected set of common sample sizes with a fixed $\rho=0$.
Because of the complicated form of the density (3.1), it is not surprising
that its (cumulative) distribution function $\int_{-1}^{r}f(\rho|r,v)$ by
analytical methods is not available in the literature. Hence, let us compute
cumulative probabilities by numerical integration defined as the rescaled area
under the curve $f(r,v)$ for $\rho=0$. See Figure 1 for two choices of
$v(=n-1)$ for sample sizes (n=50, 15). The cumulative probability becomes a
sum of rescaled areas of small-width rectangles whose heights are determined
by the curve tracing $f(r,v)$. The accuracy of numerical approximation to the
area is obviously better, the larger the number of rectangles.
We use a sequence of $r\in[-1,1]$ created by the R command r=seq(-1,1, by
=0.001), yielding 2001 rectangles. Denote the height of $f(r,v)$ by
$H_{f}=H_{f(r,v)}$. Now, the area between any two $r\in[-1,1]$ limits, say
$r_{Lo}$ and $r_{Up}$ is a summation of areas (height times width=0.001) of
all rectangles. Now, the cumulative probabilities in the range are
$\Sigma_{r_{Lo}}^{r_{Up}}H_{f}/\Sigma_{-1}^{1}H_{f},$ (14)
where the common width cancels, and where the denominator
$\Sigma_{-1}^{1}H_{f}$ converts the rectangle areas into probabilities. More
generally, we can use $f(\rho,r,v)$ for any $\rho\in[-1,1]$.
Thus we have a numerical approximation to the exact (cumulative) distribution
function under the bivariate normality of the parent,
$F(\rho,r,v)=\int_{-1}^{r}f(\rho|r,v).$
The transform from $f(.)$ to $F(.)$ is called the probability integral
transform, and its inverse $F^{-1}(c|\rho,v)$ gives relevant correlation
coefficients $r$ as quantiles for specified cumulative probability $c$ as the
argument. A computer algorithm can readily find such quantiles.
The exact $F^{-1}(c|\rho,v)$ allows the construction of confidence intervals
based on quantiles for each $\rho$ and sample size. For example, a 95% two-
tail confidence interval uses the 2.5% quantile $F^{-1}(c=0.025)$ as the lower
limit, and 97.5% quantile $F^{-1}(c=0.975)$ as the upper limit. These limits
depend on hypothesized $\rho$ and sample size. Since $\rho=0$ is a common null
hypothesis for correlation coefficients, let us provide a table of $F^{-1}(c)$
quantiles for eleven sample sizes (listed in row names) and eight cumulative
probabilities listed in column titles of Table 1.
The p-values in statistical inference are defined as the probability of
observing the random variable (correlation coefficient) as extreme or more
extreme than the observed value of the correlation coefficient $r$ for a given
null value $\rho=0$. Any one-tail p-values based on $f(\rho|r,v)$ of (3.1) for
arbitrary nonzero “null” values of $\rho$ can be similarly computed by
numerical integration defined as the area under the curve. Some code for R
functions Tarald(.), and pTarald(.) is included in Sections 3.1 and 4,
respectively.
Table 1: Correlation coefficients as quantiles evaluated at specified cumulative probabilities (c=.) using Taraldsen’s exact sampling distribution for various sample sizes assuming $\rho=0$ | | c= | | | | | c= |
---|---|---|---|---|---|---|---|---
| c=0.01 | 0.025 | c=0.05 | c=0.1 | c=0.9 | c=0.95 | 0.975 | c=0.99
n=5 | -0.83 | -0.75 | -0.67 | -0.55 | 0.55 | 0.67 | 0.75 | 0.83
n=10 | -0.66 | -0.58 | -0.50 | -0.40 | 0.40 | 0.50 | 0.58 | 0.66
n=15 | -0.56 | -0.48 | -0.41 | -0.33 | 0.33 | 0.41 | 0.48 | 0.56
n=20 | -0.49 | -0.42 | -0.36 | -0.28 | 0.28 | 0.36 | 0.42 | 0.49
n=25 | -0.44 | -0.38 | -0.32 | -0.26 | 0.26 | 0.32 | 0.38 | 0.44
n=30 | -0.41 | -0.35 | -0.30 | -0.23 | 0.23 | 0.30 | 0.35 | 0.41
n=40 | -0.36 | -0.30 | -0.26 | -0.20 | 0.20 | 0.26 | 0.30 | 0.36
n=70 | -0.27 | -0.23 | -0.20 | -0.15 | 0.15 | 0.20 | 0.23 | 0.27
n=90 | -0.24 | -0.20 | -0.17 | -0.14 | 0.14 | 0.17 | 0.20 | 0.24
n=100 | -0.23 | -0.20 | -0.16 | -0.13 | 0.13 | 0.16 | 0.20 | 0.23
n=150 | -0.19 | -0.16 | -0.13 | -0.10 | 0.10 | 0.13 | 0.16 | 0.19
For the convenience of practitioners, we explain how to use the cumulative
probabilities in Table 1 in the context of testing the null hypothesis
$\rho=0$. The Table confirms that the distribution is symmetric around
$\rho=0$ as in Figure 1. Let us consider some examples. If n=100, the critical
value from Table 1 for a one-tail 95% test is 0.16 (line n=100, column
c=0.95). Let the observed positive r be 0.3. Since $r$ exceeds the critical
value, $(r>0.16$), we reject $\rho=0$. If n=25, the critical value for a 5%
left tail in Table 1 is $-0.32$. If the observed $r=-0.44$, is less than the
critical value $-0.32$ it falls in the left tail, and we reject $\rho=0$ to
conclude that it is significantly negative.
Table 1 can be used for constructing a few two-tail 95% confidence intervals
as follows. If the sample size is 30, we see along the row n=30, and column
c=0.025 gives $-0.35$ as the lower limit, and column c=0.975 gives $0.35$ as
the upper limit. In other words, for n=30, any correlation coefficient smaller
than 0.35 in absolute value is statistically insignificant.
If the standard bivariate normality assumption is not believed, one can use
the maximum entropy bootstrap (R package meboot) designed for dependent data.
A bootstrap application creates a large number $J=999$, say, versions of data
($X_{i\ell},X_{j\ell}$) for $\ell=1,\ldots J$. Each version yields
$r^{*}(i|j;\ell),r^{*}(j|i;\ell)$ values. A large set of $J$ replicates of
these correlations give a numerical approximation to the sampling distribution
of these correlations. Note that such a bootstrap sampling distribution is
data-driven. It does not assume bivariate normality needed for the
construction of Table 1 based on (3.1).
Sorting the replicated $r^{*}(i|j;\ell),r^{*}(j|i;\ell)$ values from the
smallest to the largest, one gets their “order statistics” denoted upon
inserting parentheses by replacing $\ell$ by $(\ell)$. Now a left-tail 95%
confidence interval for $r^{*}(i|j)$ leaves a 5% probability mass in the left
tail. The interval is approximated by the order statistics as
$[r^{*}(i|j;(50)),1]$. If the hypothesized $\rho=0$ is inside the one-tail
interval, one fails to reject (accepts) the null hypothesis $H_{0}:\rho=0$.
We conclude this section by noting that recommended measures of dependence
based on the $R^{*}$ matrix and their formal inference are easy to implement.
The tabulation of Taraldsen’s exact sampling distribution of correlation
coefficients in Table 1 is new and should be of broader applicability. It is
an improvement over standard significance tests of correlation coefficients
based on Fisher’s z-transform. The next section illustrates with examples the
use of Table 1, newer dependence measures, and other inference tools.
## 4 Dependence Measure Examples & Tests
This section considers some examples of dependence measures. Our first example
deals with fuel economy in automobile design. R software comes with ‘mtcars’
data on ten aspects of automobile design and performance for 32 automobiles.
We consider two design features for illustration, miles per gallon $mpg$ and
horsepower $hp$. Vinod (2014) reports the Pearson correlation coefficient
$r(mpg,hp)=-0.78$ in his Figure 2. The negative sign correctly shows that one
gets reduced $mpg$ when a car has larger horsepower $hp.$ Table 2 in Vinod
(2014) reports two generalized correlation coefficients obtained by using
kernel regressions as $r^{*}(mpg|hp)=-0.938$ and $r^{*}(hp|mpg)=-0.853$.
One can interpret these $r^{*}(X_{i}|X_{j})$ values as signed strengths of
dependence of $X_{i}$ on the conditioning variable $X_{j}$. The strengths are
asymmetric, $|r^{*}(mpg|hp)|>|r^{*}(hp|mpg)|$, the absolute value of a
dependence strength using both generalized correlation coefficients is larger
than the dependence strength suggested under linearity. Thus Pearson’s
correlation coefficient can underestimate dependence by assuming linearity.
For the ‘mtcars’ data depMeas based on (11) is $-0.938$. Now consider Table 1
for n=30 row and c=0.05 for a one-tail critical value $-0.30$. The observed
correlation $-0.938$ is obviously in the left tail (rejection region) of the
exact sampling distribution of the correlation coefficient. Thus the negative
dependence of fuel economy (mpg) on the car’s horsepower is statistically
significant. We re-confirm the significance by computing the one-tail p-value
(= 1e-16) using an R function pTarald(.). Our R code for p-values from
Taraldsen’s exact density of correlation coefficients is given next.
pTarald=function(r,n,rho,obsr){
v=n-1
if(v<=164) Trm1=(v*(v-1)*gamma(v-1))/((sqrt(2*pi)*gamma(v+0.5)))
if(v>164) Trm1=(164*(163)*gamma(163))/((sqrt(2*pi)*gamma(163.5)))
Trm2=(1-r^2)^((v-1)/2)
if(rho!=0) Trm2b=((1-rho^2)^((v-2)/2))*((1-rho*r)^((1-2*v)/2))
if(rho==0) Trm2b=1
Trm3b=Re(hypergeo(3/2,-1/2,(v+0.5),(1+r*rho)/2))
y0=Re(Trm1*Trm2*Trm2b*Trm3b)
p=y0/sum(y0)
cup=cumsum(p)
loc=max(which(r<obsr))+1
if(obsr<0) ans=cup[loc]
if(obsr>=0) ans=1-cup[loc]
return(ans)}
pTarald(r=seq(-1,1,by=0.001),n=32,rho=0,obsr=-0.938)
The first term (Trm1) in the R function computing the p-values involves a
ratio of two gamma (factorial) functions appearing in (3.1). For $n>164$ each
gamma becomes infinitely large, and Trm1 becomes ‘NaN’ or not a number. Our
code winsorizes large $n$ values.
Since mtcars data has n=32, and the observed generalized correlation is
$r^{*}=0.938$, we use the command on the last line of the code to get the
p-value of 1e-16, or extremely small, suggesting statistical significance. If
we use the same automobile data using GM22’s R package called HellCor we find
that $\eta=0.845>0$, giving no hint that mpg and hp are negatively related. If
we compare numerical magnitudes, we have $\eta>|r(mpg,hp)|=0.78$. Since $\eta$
exceeds Pearson’s correlation in absolute value, $\eta$ is seen to incorporate
nonlinear dependence. However, $\eta=0.78$ may be an underestimation of the
absolute value of depMeas=$-0.938$. We fear that either $\eta$ may be failing
to incorporate some nonlinear dependence or is paying an unknown penalty for
adhering to the symmetry dogma.
### 4.1 Further Real-Data Applications in GM22
GM22 illustrate the Hellinger correlation measure of dependence using two sets
of data where the Pearson correlation is statistically insignificant, yet
their Hellinger correlation is significant. Their first data set refers to the
population of seabirds and coral reef fish residing around n = 12 islands in
the British Indian Ocean Territory of Chagos Archipelago. Ecologists and other
scientists cited by GM22 have determined that fish and seabirds have an
ecologically symbiotic relationship. The seabirds create an extra nutrient
supply to help algae. Since fish primarily feed on those algae, the two
variables should have a significantly positive dependence.
GM22 begin with the low Pearson correlation $r(fish,seabirds)=0.374$ and a 95%
confidence interval $[-0.2548,0.7803]$ that contains a zero, suggesting no
significant dependence. The p-value using pTarald(..,obsr=0.374) is 0.0935,
which exceeds the benchmark of 0.05, confirming statistical insignificance.
The wide confidence interval, which includes zero, is partly due to the small
sample size (n=12).
Our Table 1 with the exact distribution of correlations suggests that when
$n=10$, more conservative than the correct n=12, the exact two-tail 95%
confidence interval (leaving 2.5% probability mass in both tails) also has a
wide range $[-0.58,0.58]$, which includes zero. Assuming the direction is
known, a one-tail interval with 5% in the right tail (n=10) value is 0.50.
That is, only when the observed correlation is larger than 0.50 it is
significantly positive (assuming a bivariate normal parent density).
Figure 3: Marginal densities of fish and seabirds data are skewed, not Normal
GM22 find that their Hellinger correlation $\eta$ needs to be normalized to
ensure that $\eta\in[0,1]$, because their estimate of $\mathcal{B}$ can exceed
unity. They denote the normalized version as $\hat{\eta}$ and claim an easier
interpretation of $\hat{\eta}$ on the “familiar Pearson scale,” though
Pearson’s $r_{ij}\in[-1,1]$ scale admits negative values. GM22 employ
considerable ingenuity to achieve the positive range [0, 1] described in their
Section 5.3. They state on page 650 that their range normalization “comes at
the price of a lower power when it comes to test for independence.”
Using the population of seabirds and coral reef fish residing around n = 12
islands, GM22 report the estimate $\hat{\eta}$(fish, seabirds)=0.744. If one
assumes bivariate normal parent distribution and uses Taraldsen’s exact
density from Table 1, $\hat{\eta}$(fish, seabirds)$=0.744>0.50$ suggests
statistical significance. The p-value using pTarald(..,obsr=0.744) is 0.0027,
which is smaller than the benchmark 0.05, confirming significance.
In light of Figure 3, it is unrealistic to assume that the data come from a
bivariate normal parent distribution. Hence the evidence showing a
significantly positive correlation between fish and seabirds based on
Taraldsen’s exact density is suspect. Accordingly, GM22 report a bootstrap
p-value of $0.045<0.05$ as their evidence. Since this p-value is too close to
0.05, we check for unintended p-hacking. When one runs their HellCor(.)
function with set.seed(99) and default settings, the bootstrap p-value becomes
$0.0513>0.05$, which exceeds the benchmark suggesting insignificant
$\hat{\eta}(fish,seabirds).$ Then, GM22’s positive Hellinger correlation
estimate of $\hat{\eta}$= 0.744 is not statistically significant at the usual
95% level. Thus, the Hellinger correlation fails to be strongly superior to
Pearson’s correlation $r$ because $\hat{\eta}$ is also insignificantly
positive.
Now, let us compare $\hat{\eta}$ with off-diagonal elements of the generalized
correlation matrix $R^{*}$ recommended here. Our gmcmtx0(cbind(fish,seabirds))
suggests the “causal” direction (seabirds$\to$ fish), to be also positive,
$r^{*}(fish|\break seabirds)=0.6687$. The p-value using
pTarald(..,obsr=0.6687) is 0.0086, which is smaller than the benchmark 0.05,
confirming significance. There is no suspicion of p-hacking here. A 95%
bootstrap two-tail confidence interval using the meboot R package is [0.3898,
0.9373]. A one-tail interval is [0.4394, 1], which includes the observed
0.6687 with a p-value of zero. See Figure 4, where almost the entire density
has positive support. Note that the interval does not include a zero,
suggesting significant positive dependence consistent with what the ecologists
expect. The lower limit of our meboot confidence interval is not close to
zero. More importantly, our $R^{*}$ generalized correlation coefficients do
not impose symmetric dependence, revealing sign information borrowed from the
covariance, absent in the Hellinger correlation.
Figure 4: Bootstrap density of generalized correlation coefficient
r*(seabirds, fish).
The second example in GM22 has the number of births ($X_{1}$) and deaths
($X_{2}$) per year per 1000 individuals in n=229 countries in 2020. A data
scatterplot in their Figure 7 displays a C-shaped nonlinear relation.
Pearson’s correlation $r_{12}=-0.13$ is negative and insignificant at level
$\alpha=0.05$. This is based on a two-tail traditional Fisher approximation to
the sampling distribution of a correlation coefficient. It is reversed by our
more powerful one-tail p-value using Taraldsen’s exact sampling distribution.
Our pTarald(..,n=229, obsr=-0.13) is $(0.0246<0.05)$, implying a statistically
significant negative correlation. On the other hand, GM22 estimate
$\hat{\eta}=0.69$ with a two-tail 95% bootstrap confidence interval [0.474,
0.746], hiding important information about the negative direction of
dependence. Since zero is outside the confidence interval, GM22 claim that
they have correctly overcome an apparently incorrect inference based on
traditional methods. We have shown that traditional inference was incorrect
because more accurate Taraldsen distribution was not used.
Our gmcmtx0(cbind(birth,death)) estimates that $r^{*}(death|birth)$ is
$=-0.6083$. A one-tail 95% confidence interval using the maximum entropy
bootstrap (R package meboot) is [-1, -0.5693]. A somewhat less powerful two-
tail interval [$-0.6251,-0.5641$] is also entirely negative. The null
hypothesis states that the true unknown $r^{*}$ is zero. Since our random
interval excludes zero, the dependence is significantly negative. The p-value
is zero in Figure 5, since almost the entire density has negative support. A
larger birth rate significantly leads to a lower death rate in 229 countries
in 2020.
Figure 5: Bootstrap density of generalized correlation coefficient r*(birth,
death).
In summary, the two examples used by GM22 to sell their Hellinger correlation
have a discernible advantage over Pearson’s $r_{ij}$, but not over our
generalized correlation $R^{*}$. The examples confirm four shortcomings of
Hellinger’s correlation $\hat{\eta}$ over $R^{*}$. (a) It imposes an
unrealistic symmetry assumption. (b) It provides no information about the
direction of dependence. (c) It forces the use of less powerful two-tail
confidence intervals. (d) It is currently not implemented for discrete
variables.
## 5 Final Remarks
Many scientists are interested in measuring the directions and strengths of
dependence between variables. This paper surveys quantitative measures of
dependence between two variables. We use four real-world examples in Section
1.1 to show that any symmetric measure of dependence alleging equal strength
in both directions is unacceptable. Yet, the majority of statistical
dependence extant in the literature adheres to the symmetry dogma. A 2022
paper (GM22) develops an intrinsically flawed symmetric measure of dependence
while proposing Hellinger correlations.
We show that off-diagonal elements of the asymmetric $R^{*}$ matrix of
generalized correlation coefficients provide an intuitively sensible measure
of dependence after incorporating nonlinear and nonparametric relations among
the variables involved. The R package generalCorr makes it easy to implement
our proposal. Its six vignettes provide ample illustrations of the theory and
applications of $R^{*}$.
We discuss statistical inference for elements of the $R^{*}$ matrix, providing
a new Table 1 of quantiles of Taraldsen (2021) exact density of a correlation
coefficient for eleven typical sample sizes and eight cumulative
probabilities. We illustrate with two data sets used by GM22 to support their
Hellinger correlation. Directional information is uniquely provided by our
asymmetric measure of dependence in the form of generalized correlation
coefficients $\\{r^{*}(i|j)\\}$. It allows the researcher to achieve somewhat
better qualitative results and more powerful one-tail tests compared to
symmetric measures of dependence in the literature.
We claim that one-tail p-values of the Taraldsen’s density can overcome the
inaccuracy of the traditional Pearson correlation inference based on Fisher’s
z-transform. We illustrate the claim using GM22’s second example where Pearson
correlation $r(birth,death)$ is shown to be significantly negative using
Taraldsen’s density. Hence, the complicated Hellinger correlation inference is
not really needed to achieve correct significance. Interestingly, both hand-
picked examples designed to show the superiority of GM22’s $\hat{\eta}$ over
$r_{ij}$ show the merit of our proposal based on $R^{*}$ over $\hat{\eta}$.
Almost every issue of every quantitative journal refers to correlation
coefficients at least once, underlining its importance in measuring
dependence. We hope that $R^{*}$ and our implementation of Taraldsen’s exact
sampling distribution of correlation coefficients receive further attention
and development.
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|
# Ultrafast learning of 4-node hybridization cycles in phylogenetic networks
using algebraic invariants
Zhaoxing Wu
Wisconsin Institute for Discovery
Department of Statistics
University of Wisconsin-Madison
Madison, WI 53706
&Claudia Solís-Lemus
Wisconsin Institute for Discovery
Department of Plant Pathology
University of Wisconsin-Madison
Madison, WI 53706
Corresponding author<EMAIL_ADDRESS>
###### Abstract
Motivation: The abundance of gene flow in the Tree of Life challenges the
notion that evolution can be represented with a fully bifurcating process, as
this process cannot capture important biological realities like hybridization,
introgression, or horizontal gene transfer. Coalescent-based network methods
are increasingly popular, yet not scalable for big data, because they need to
perform a heuristic search in the space of networks as well as numerical
optimization that can be NP-hard.
Results: Here, we introduce a novel method to reconstruct phylogenetic
networks based on algebraic invariants. While there is a long tradition of
using algebraic invariants in phylogenetics, our work is the first to define
phylogenetic invariants on concordance factors (frequencies of 4-taxon splits
in the input gene trees) to identify level-1 phylogenetic networks under the
multispecies coalescent model. Our novel inference methodology is
optimization-free as it only requires the evaluation of polynomial equations,
and as such, it bypasses the traversal of network space, yielding a
computational speed at least 10 times faster than the fastest-to-date network
methods. We illustrate the accuracy and speed of our new method on a variety
of simulated scenarios as well as in the estimation of a phylogenetic network
for the genus Canis.
Availability and Implementation: We implement our novel theory on an open-
source publicly available Julia package PhyloDiamond.jl available at
https://github.com/solislemuslab/PhyloDiamond.jl with broad applicability
within the evolutionary biology community.
Contact<EMAIL_ADDRESS>
_K_ eywords Hybridization $\cdot$ Polynomials $\cdot$ Reticulate evolution
$\cdot$ Coalescent model
## 1 Introduction
The Tree of Life is the graphical structure that represents the evolutionary
process from single-cell organisms at the origin of life to present day
biodiversity. Mathematically, a phylogenetic tree is a fully bifurcating graph
in which internal nodes represent ancestral species that over time
differentiate into two separate species giving rise to its two children nodes.
Recent evidence [28, 11, 1, 33, 35, 44] has challenged the notion that
evolution across the Tree of Life can be represented with a fully bifurcating
process, as this process cannot capture important biological realities like
hybridization, introgression, or horizontal gene transfer, that require two
fully separated branches to join again. These processes of reticulate
evolution are more prevalent in certain groups like plants [9], fungi [14, 43]
and prokaryotes [13]. To accurately include these groups in the Tree of Life,
recent years have seen an explosion of methods to reconstruct phylogenetic
networks, which naturally account for reticulate evolution [12, 15, 7, 26].
Existing methods, however, are still not scalable enough to tackle the
complexities of the present day’s biological big data. The most scalable
alternatives infer split (or implicit) networks [8, 24, 21] which are not
biologically interpretable as internal nodes no longer represent biological
events (speciation or hybridization), and the resulting network is completely
unrooted, even with the inclusion of an outgroup.
Among methods to infer explicit networks (those whose internal nodes do
represent speciation or hybridization events), likelihood-based approaches are
among the most popular. PhyloNet [47, 48] has been the pioneer of likelihood
inference of phylogenetic networks, later expanding to Bayesian [46] and
pseudolikelihood alternatives [49]. However, even in its most scalable option
(pseudolikelihood), the method is still not suitable beyond dozens of taxa and
hundreds of genes. SNaQ [38] within the PhyloNetworks Julia package [39] has
proven a scalable alternative to infer large phylogenetic networks from
multilocus alignments. The method is based on a pseudolikelihood model under
the coalescent, and running time does not increase as a function of number of
genes because the input data for SNaQ are split frequencies on subsets of $4$
taxa (concordance factors). That is, after the estimation of the concordance
factors, the running time is the same for data with 10 genes or 10,000 genes.
Despite its popularity, SNaQ still lacks scalability as the number of taxa
increases and remains a suitable alternative only for $50$ or fewer taxa.
Last, BEAST2 [51] provides the only co-estimation method that infers
simultaneously the phylogenetic networks and the gene trees, and it allows
users to infer a variety of relevant biological parameters (such as divergence
times), yet the complexity of this method renders it unsuitable for anything
beyond a handful of taxa.
Because of the complexity in inferring a phylogenetic network, the
evolutionary biology community has embraced the use of hybrid detection
methods such as ABBA-BABA test [32], MSCquartets [30, 36], and HyDe [27] to
identify hybridization events on a fixed phylogeny. These methods take subsets
of taxa and test whether the current subset is well-explained with a tree-like
pattern or if there is a hybridization event in this subset. While fast, these
methods can be inaccurate in the presence of multiple hybridization events
affecting the same taxa [6] or by ghost lineages [31, 45, 6]. Furthermore, the
process of first reconstructing a phylogenetic tree and then adding
hybridization events can be flawed given the bias that gene flow causes on the
inference of the backbone phylogenetic tree [41].
Here, we introduce a novel method to infer hybridization events using
phylogenetic invariants. Because our method only requires evaluation of
polynomial equations, it bypasses optimization in the space of networks and it
is at least $10$ times faster than current inference network methodologies.
Our method uses gene trees as input and it exploits the signal in the
frequencies of splits in the data, also well-known as concordance factors.
Indeed, under the coalescent model, splits of taxa display certain
probabilities of appearing in the sample of gene trees. Under the true
network, these frequencies need to satisfy certain polynomial equations,
denoted phylogenetic invariants. By plugging in the observed frequencies on
the phylogenetic invariants of every possible placement of the hybridization
cycle, we can identify the hybridization cycle that better agrees with the
data as the one whose evaluated invariants are closest to zero.
Algebraic invariants have been widely used in phylogenetics to identify trees
under a variety of models of evolution (Jukes-Cantor [16, 42], GTR [2]), and
more recently, to identify phylogenetic networks [19, 20, 3, 10, 4]. Our work,
however, is the first to define phylogenetic invariants on concordance factors
under the multispecies coalescent model on networks as existing work on
phylogenetic invariants on networks has been restricted to models of
evolution.
This work first introduces the phylogenetic invariants that concordance
factors need to satisfy under level-1 phylogenetic networks with a 4-node
hybridization cycle under the multispecies coalescent model. Next, we describe
an inference methodology to use the phylogenetic invariants to infer the
correct 4-node hybridization cycle among $n$ taxa. Moreover, we demonstrate
the performance of our method with simulated data, and we revisit the
inference of the Canis phylogenetic network that had already been published in
[17]. We prove that our method is faster than all existing network inference
methods, and it accurately identifies the correct hybridization cycle in the
Canis genus. Additionally, in all simulating scenarios, our method identifies
the correct placement of hybridization events with high accuracy. Even in the
few cases when our method identifies a wrong network (usually cases when there
are not sufficient sampling of taxa that descends from the hybrid node), the
correct network is still within the top 5 ranked networks given the invariant
score (evaluated phylogenetic invariants that need to vanish to zero). This
means that, regardless of sampling, our method is always able to reduce the
space of candidate networks that can later be tested with a likelihood-based
approach.
We highlight that our method has two main limitations: 1) it is only suitable
to identify level-1 hybridization cycles with 4 nodes, and 2) it sometimes
fails to identify the correct network as the one with the top 1 ranked
invariant score when there is only one taxon sampled that is descendant of the
hybrid node. Despite these limitations, we think that our method is a valuable
innovation in the landscape of phylogenetic network methods, if anything, to
reduce the space of candidate networks to be used in likelihood-based
methodologies. Indeed, we do not see our method as a replacement for other
network inference approaches. On the contrary, we believe that our ultrafast
learning methodology can serve to identify the true phylogenetic network when
it involves a simple hybridization event, or it can serve to provide several
candidate networks to be later tested with a likelihood-based approach,
effectively bypassing heuristic optimization in the space of networks. We
further highlight that our method is open source, publicly available as the
new PhyloDiamond.jl Julia package in
https://github.com/solislemuslab/PhyloDiamond.jl.
The structure of the paper is as follows. In Section 2, we introduce the
phylogenetic invariants under the multispecies coalescent model on networks,
and we describe an inference methodology that uses the phylogenetic invariants
to identify the 4-node hybridization cycle that generates the data. In this
Section, we also describe the simulation study and the re-analysis of the
Canis data from [17]. In Section 3, we describe the results on the simulated
data and the Canis phylogeny. Last, in Section 4, we describe the main
limitations of our method and potential future work.
## 2 Materials and Methods
### 2.1 Phylogenetic invariants for 4-node hybridization cycles in level-1
phylogenetic networks
Under the coalescent model, the distribution of gene trees estimated from
multilocus sequence alignments provides information on the true network that
generated the data [47]. In [38], it was shown that the split frequencies on
subsets of 4 taxa, namely the concordance factors, also provide information on
the true network. Namely, a concordance factor (CF) of a given quartet (or
split) is the proportion of genes whose true tree displays that quartet (or
split) [5]. For example, for a taxon set $s=\\{a,b,c,d\\}$, there are only
three possible quartets, represented by the splits $q_{1}=ab|cd$,
$q_{2}=ac|bd$ and $q_{3}=ad|bc$. The concordance factor for the split $ab|cd$
is the proportion of gene trees that display this split.
As in [38], our method uses the CFs as input data, and we focus on the case of
4-node hybridization cycles on level-1 phylogenetic networks as shown on
Figure 1 (left). A level-1 phylogenetic networks is a network with no vertex
belonging to more than one hybridization cycle. The semi-directed network in
Figure 1 (left) is unrooted, yet the direction of the hybrid edges (in blue)
is known, so the root placement of the network is constrained. That is, the
root cannot be anywhere below the hybrid node (in blue). The hybridization
cycle partitions the taxa into 4 subsets with $n_{0},n_{1},n_{2}$ and $n_{3}$
as the number of taxa in each subset. For example, the clade below the hybrid
node in blue in Figure 1 (left), also known as the hybrid clade, has $n_{0}$
taxa, and the two sister clades to the hybrid clade have $n_{1}$ and $n_{2}$
taxa. Even when $n_{i}$ represents the number of taxa in a specific clade, we
use this terminology to refer to the clade. That is, we refer to the hybrid
clade as the $n_{0}$ clade. Last, we highlight that each clade could be a
subtree or a subnetwork as long as the overall network is level-1 [23].
Figure 1: Left: 4-node hybridization cycle on a level-1 phylogenetic network
where $t_{i}$ represents the branch length in coalescent units. The hybrid
node and hybrid edges are in blue. The minor hybrid edge is labelled with
$\gamma$ that represents the inheritance probability (or proportion of genes
transferred through the minor edge). Top right: Two examples of level-1
networks, each with one 4-node hybridization cycle, and its corresponding
notation (N2222 for the 8-taxon network and N2112 for the 6-taxon network).
Bottom right: Examples of quartets drawn from each network in top right along
with their vector notations. For example, the quartet $AB|CD$ drawn from the
N2222 network corresponds to the 4-taxon subset $(0,2,0,2)$ where each entry
in this vector corresponds to the number of taxa drawn from each of the four
clades: $n_{0},n_{1},n_{2},n_{3}$.
In this work, we introduce a method to infer 4-node hybridization cycles in
level-1 semi-directed networks. Since we focus on level-1 4-node hybridization
cycles (as in Figure 1 (left)), we utilize a simplifying notation to represent
each network. Namely, let $N$represent an $n-$taxon semi-directed level-1
phylogenetic network. We represent this network by the number of taxa in its
four clades. For example, the network in the top right of Figure 1 is denoted
N2222 as there are two taxa in each of the four clades, while the network
N2112 contains two taxa in clades $n_{0}$ and $n_{3}$, and one taxon in clades
$n_{1}$ and $n_{2}$.
Another relevant notation is the vector representation of every 4-taxon
subset. For example, in the bottom right of Figure 1, the 4-taxon subset
$(0,2,0,2)$ drawn from network N2222 corresponds to the case when two taxa are
taken from $n_{1}$ and $n_{3}$ and no taxon is taken from $n_{0}$ and $n_{2}$.
For this specific network (N2222), this 4-taxon subset corresponds to the
quartet $AB|CD$. Another example is the 4-taxon subset $(0,1,1,2)$ drawn from
network N2112. Note that since this network only has one taxon in clades
$n_{1}$ and $n_{2}$, any 4-taxon subset drawn from this network must have at
most one taxon in the second and third value in the 4-taxon vector notation.
In general, a 4-taxon subset is represented by a 4-dimensional vector so that
each element in this vector corresponds to the number of taxa drawn from each
of the four clades $n_{0},n_{1},n_{2},n_{3}$.
Next, we describe the formulas for the expected CFs under the coalescent model
for every 4-taxon subset on a level-1 semi-directed network with branch
lengths in coalescent units ($t_{i}$) and inheritance probability ($\gamma$).
These formulas have already been derived in [38]. Let $N$ be the network on
Figure 1 (left), and let $(0,0,2,2)$ be the 4-taxon subset for which we want
to compute the CF formulas under the coalescent model. Let taxa
$k_{1},k_{2}\in n_{2}$ and taxa $l_{1},l_{2}\in n_{3}$. The probability of
taxa $k_{1}$ and $k_{2}$ coalescing in a branch with length
$t_{2}+t_{23}+t_{3}$ is given by $1-\dfrac{2}{3}z_{2}z_{23}z_{3}$ where
$z_{i}=\exp(-t_{i})$. Then, the formulas for the three CF under the coalescent
model are defined as:
$\displaystyle P(k_{1},k_{2}|l_{1},l_{2})$
$\displaystyle=1-\dfrac{2}{3}z_{2}z_{23}z_{3}=a_{1}$ $\displaystyle
P(k_{1},l_{1}|k_{2},l_{2})$ $\displaystyle=\dfrac{1}{3}z_{2}z_{23}z_{3}=a_{2}$
$\displaystyle P(k_{1},l_{2}|k_{2},l_{1})$
$\displaystyle=\dfrac{1}{3}z_{2}z_{23}z_{3}=a_{3}$
where $a_{1},a_{2}$ and $a_{3}$ denote the values for the true concordance
factors corresponding to the splits $k_{1},k_{2}|l_{1},l_{2}$,
$k_{1},l_{1}|k_{2},l_{2}$ and $k_{1},l_{2}|k_{2},l_{1}$ respectively. We thus
note that for the 4-taxon subset $(0,0,2,2)$, there are three polynomial
equations that represent the probabilities for each quartet under the
coalescent model. These three formulas are equal to the true CF values
($a_{1},a_{2},a_{3}$) which will later be replaced by the observed CF
estimated from the sample of gene trees.
In the Appendix, we list of all the CF formulas for all 4-taxon subsets on a
level-1 semi-directed network with one 4-node hybridization cycle (Figure 1
left). Because of the theoretical work in [38, 40], we know that we only need
two taxa per clade to represent all CF formulas involving the hybridization
cycle, and thus, we can restrict to the case of $n=8$ taxa for the definition
of all CF formulas (yet our method is not restricted to 8 taxa, see Section
2.2). Furthermore, not all ${8\choose 4}=70$ 4-taxon subsets are required.
Indeed, we ignore 4-taxon subsets of the form $(3,1,0,0)$ since three or more
taxa from one of the clades (in this case, $3$ taxa from clade $n_{0}$) do not
provide information about the hybridization cycle. Recall that only the length
of internal branches appears in the CF formulas but not the length of external
branches. Therefore, if there are three or more taxa on one clade (e.g. $3$
taxa in clade $n_{0}$), then branches involved in the hybridization cycle will
correspond to an external branch and thus, will not be included in any of the
CF formulas. The same is true if all four taxa in the 4-taxon subset all come
from the same clade. Thus, only 4-taxon subsets where at most $2$ taxa come
from a given clade contain information about the hybridization cycle. There
are $19$ of such 4-taxon subsets in which $0\leq i,j,k,l\leq 2$ for
$(i,j,k,l)$, and thus, there are $19\times 3=57$ concordance factor polynomial
equations along with $57$ true CF values ($a_{i}$).
As just described, the semi-directed network (denoted $N$) in Figure 1 (left)
can be represented by a set of $57$ polynomial equations, denoted $CF(N)$ in
$66$ variables: $9$ variables corresponding to branch lengths and inheritance
probability ($z_{0},z_{1},z_{2},z_{3},z_{01},z_{02},z_{13},z_{23},\gamma$ for
$z_{i}=\exp(-t_{i})$) and $57$ variables corresponding to the true CF values
($a_{i}$ for $i=1,\dots,57$). Here, we identify the relationships that the
$a_{i}$ values need to satisfy for the polynomial system $CF(N)$ to be
consistent. These relationships are denoted phylogenetic invariants. For
example, because the three concordance factors corresponding to a given
4-taxon subset (say $(0,0,2,2)$ as described above) need to sum up to one, we
have one phylogenetic invariant for these three numbers:
$a_{1}+a_{2}+a_{3}=1$, or similarly
$i_{1}(a_{1},a_{2},a_{3})=a_{1}+a_{2}+a_{3}-1$ with
$i_{1}(a_{1},a_{2},a_{3})=0$. In addition, since the two minor concordance
factors are equal, we have a second phylogenetic invariant defined for these
numbers: $a_{2}=a_{3}$, or similarly $i_{2}(a_{2},a_{3})=a_{2}-a_{3}$ with
$i_{2}(a_{2},a_{3})=0$. It turns out that the set of all concordance factor
values for all 4-taxon subsets (the 57 polynomial equations denoted $CF(N)$)
define a set of phylogenetic invariants (equations only in the $a_{i}$
variables) that need to vanish to zero whenever the concordance factors come
from the true network.
Note that not all networks will have the same number of CF polynomial
equations. For example, for the 8-taxon network $N=2222$, all 19 4-taxon
subsets can be extracted, and thus, all 57 CF equations are defined in
$CF(N)$. For the 5-taxon network $N=1112$, the 4-taxon subset $(2,1,1,0)$
cannot be considered since this subset requires two taxa from the clade
$n_{0}$ and the 5-taxon network only has one taxon in this clade. Therefore,
the 5-taxon network $N=1112$ defines a set of fewer polynomial equations (12
to be exact) in $CF(N)$ involving only the CF values:
$a_{7},a_{8},a_{9},a_{22},a_{23},a_{24},a_{28},a_{29},a_{30},a_{31},a_{32}$
and $a_{33}$.
To obtain the phylogenetic invariants defined by $N$, denoted
$\mathcal{I}(N)$, we get the Gröbner basis of $CF(N)$ on the $a_{i}$ variables
using any elimination method in Macaulay2 [18]. All Macaulay2 scripts (and
output files) are publicly available in
https://github.com/solislemuslab/PhyloDiamond.jl.
For example, for the network $N=1112$, there are 10 phylogenetic invariants in
the Gröbner basis of $CF(N)$ on the $a_{i}$ variables:
1. 1.
$a_{32}-a_{33}$
2. 2.
$a_{31}+2a_{33}-1$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{23}-a_{24}$
5. 5.
$a_{22}+2a_{24}-1$
6. 6.
$a_{8}-a_{9}$
7. 7.
$a_{7}+2a_{9}-1$
8. 8.
$3a_{9}*a_{30}+a_{9}-a_{24}-a_{33}$
9. 9.
$a_{24}*a_{29}+2a_{24}*a_{30}+a_{29}*a_{33}-a_{30}*a_{33}-a_{33}$
10. 10.
$3a_{9}*a_{29}-2a_{9}+2a_{24}-a_{33}$
We note that the first 7 invariants correspond to the "trivial" invariants
related to the sum-to-one property and the equality of the minor concordance
factors. The last three invariants, however, identify relationships that the
true concordance factors need to satisfy if they originated from the $N=1112$
network.
We present the phylogenetic invariants corresponding to each $n-$taxon network
($5\leq n\leq 8$) in the Appendix.
### 2.2 Inference of n-taxon phylogenetic networks with one 4-node
hybridization cycle using phylogenetic invariants
The procedure to infer a phylogenetic network with one 4-node hybridization
cycle using phylogenetic invariants starts with a table of estimated
concordance factors. Each value in this table is mapped to the vector of
$\begin{bmatrix}a_{1}&\dots&a_{57}\\\ \end{bmatrix}$ and this vector is
plugged in the invariants for each candidate network. For example, as
mentioned, the candidate network $N=1112$ has 10 phylogenetic invariants, and
thus, after evaluation on the observed CFs, we have a $10$-dimensional vector.
The invariant score is defined as the $L_{2}$ norm of the vector of evaluated
invariants and the candidate network with the smallest invariant score is
identified as the network that agrees best with the observed CFs. See Figure 2
for a graphical abstract of the procedure and Algorithm 1 for a more detailed
description of the inference steps.
Figure 2: Graphical abstract of the inference methodology using phylogenetic
invariants to identify the best phylogenetic network to fit the table of
observed concordance factors. In this case, there are 8 taxa shown as colored
circles. The invariant score is defined as the $L_{2}$ norm of the vector of
evaluated invariants.
Input : Table of estimated concordance factors; optional: the number of
optimal networks ($m$) to return (default $m=5$)
Output : Top $m$ optimal networks with smallest invariant score
scores $\leftarrow$ an empty array;
nets $\leftarrow$ an empty array;
for _$N_{i}$ , a possible partition of taxa_ do
append(nets,$N_{i}$);
map observed CF values to $a_{i}$ values;
score $\leftarrow$ L2norm($\mathcal{I}(N_{i})$);
append(scores, score);
rank(scores, nets);
return scores[1:m], nets[1:m];
Algorithm 1 Inference of n-taxon phylogenetic network ($5\leq n\leq 8$) with
phylogenetic invariants
Given that we only have phylogenetic invariants for the cases of $5\leq n\leq
8$ taxa, if the true network has $5\leq n\leq 8$ taxa, the procedure is just
as described (Figure 2 and Algorithm 1). If the true network has more than 8
taxa, then the procedure is slightly different. Namely, our method iterates
through all possible subsets of 8 taxa, and fit the phylogenetic invariants on
each subset to identify the top subnetwork with $n=8$ taxa. Let $N^{*}_{8}$ be
the best 8-taxon network to fit the observed CFs. The missing taxa are added
to $N^{*}_{8}$ according their placement in the second best network, or if the
missing taxa are not in the second best network, the algorithm adds them
according to their placement in the third best or the fourth best and so on.
For example, assume that we have $n=9$ taxa (A,B,C,D,E,F,G,H,I) and the best
network with $n=8$ taxa ($N^{*}_{8}$) contains A,B,C,D,E,F,G,H in the
following partition: A,B in $n_{0}$; C,D in $n_{1}$; E,F in $n_{2}$, and G,H
in $n_{3}$. We then find the best network containing taxon I. Assume that this
network puts taxon I in the clade $n_{2}$, then the resulting best network
would have E,F, and I in $n_{2}$. Note that our algorithm is unable to resolve
the bifurcation structure within a given clade (e.g. the $n_{2}$ clade here
with E,F, and I), but it is able to 1) correctly identify the partition of
taxa among the four clades defined by the 4-node hybridization cycle and 2)
correctly identify who are in the hybrid clade ($n_{0}$), who are in the
sister clades to the hybrid clade ($n_{1}$ and $n_{2}$), and who are in the
remaining fourth clade ($n_{3}$). See Algorithm 2 for the steps in the
procedure for $n>8$ taxa.
Input : Table of estimated concordance factors; optional: the number of
optimal networks ($m$) to return (default $m=5$
Output : Top $m$ optimal networks with smallest invariant score
scores $\leftarrow$ an empty array;
subnets $\leftarrow$ an empty array;
for _$P_{i}$ , a subset of 8 taxa_ do
$\text{scores}_{i}$,$\text{subnets}_{i}$ $\leftarrow$ Algorithm 1(CF, $P_{i}$,
m=2520); /* Note total number of subnets = 2520 */
append(scores,$\text{scores}_{i}$);
append(subnets, $\text{subnets}_{i}$);
subnets_sorted $\leftarrow$ sort subnets in descending order according to its
score;
result $\leftarrow$ an empty array;
for _i in 1:length(subnets_sorted)_ do
miss_taxa $\leftarrow$ array of missing taxa in subnets_sorted[i];
for _t in miss_taxa_ do
for _j in (i+1):length(subnets_sorted)_ do
if _t in $n_{0}$ of subnets_sorted[j]_ then
add t to $n_{0}$ of subnets_sorted[i];
break;
else if _t in $n_{3}$ of subnets_sorted[j]_ then
add t to $n_{3}$ of subnets_sorted[i];
break;
else if _t in $n_{1}$ of subnets_sorted[j]_ then
if _( $n_{2}$ of subnets_sorted[j] == $n_{1}$ of subnets_sorted[i]) &
($n_{1}$ of subnets_sorted[j] without t is in $n_{2}$ of subnets_sorted[i])_
then
add t to $n_{2}$ of subnets_sorted[i];
break;
else
add t to $n_{1}$ of subnets_sorted[i];
break;
else if _t in $n_{2}$ of subnets_sorted[j]_ then
if _( $n_{1}$ of subnets_sorted[j] == $n_{2}$ of subnets_sorted[i]) &
($n_{2}$ of subnets_sorted[j] without t is in $n_{1}$ of subnets_sorted[i])_
then
add t to $n_{1}$ of subnets_sorted[i];
break;
else
add t to $n_{2}$ of subnets_sorted[i];
break;
if _subnets_sorted[i] not in result_ then
append(result, subnets_sorted[i])
if _length(result)==m_ then
break;
return result;
Algorithm 2 Inference of n-taxon phylogenetic network ($n>8$) with
phylogenetic invariants
### 2.3 Simulation study
#### 2.3.1 Networks with eight or fewer taxa
In this section, we focus on the case of six taxa
($N=2211,2121,2112,1122,1212,1221$), seven taxa ($N=2221,2212,2122,1222$) and
eight taxa ($N=2222$). Later, Section 2.3.2 describes the simulations with
more than eight taxa.
For a given network $N$, we generate data under four settings: 1) true
concordance factors (proof of concept); 2) true concordance factors perturbed
by Gaussian error; 3) simulated gene trees, and 4) estimated gene trees. To
generate the true concordance factors on a given network in the first
scenario, we use the Julia package PhyloNetworks.jl [39], and this scenario is
designed to prove that the invariants method works under ideal conditions. For
the second setting, we add Gaussian noise to the true concordance factors with
zero mean and standard deviation of $\sigma=0.0005,0.00005,0.000005$. We
choose Gaussian noise as is standard practice. In the third setting, we use
ms-converter [25] that runs ms [22] to simulate $k=100,1000,10000$ gene trees
under the coalescent model on a given network $N$. All internal branches in
the network are set to 1.0 coalescent unit so that the amount of incomplete
lineage sorting is controlled, but not trivial, and we set an inheritance
probability parameter of $\gamma=0.3$. For the fourth setting, we simulate
$L=500,2000$ sequences on each simulated gene tree with seq-gen [34] under the
HKY model, scale the branch lengths by $0.036$, and set nucleotide frequencies
as $0.300414$, $0.191363$, $0.196748$, and $0.311475$. Then, we estimate gene
trees with IQ-Tree [29] with ModelFinder Plus for model selection [37]. We
repeat each simulation setting $30$ times. All simulation scripts are in the
GitHub repository https://github.com/solislemuslab/PhyloDiamond.jl.
#### 2.3.2 Networks with more than eight taxa
We also analyze cases of nine taxa ($N=2223,2232,2322,3222$) and ten taxa
($N=3322,3232,3223,2233,2323,2332$). Given that the algorithm to infer a
network with more than 8 taxa relies entirely on the algorithm on 8 taxa, we
focus on extensive simulations for the algorithm on 8 taxa in the previous
section, and simply show proof-of-concept simulations for the case of more
than 8 taxa in this section. In particular, we focus on testing the following
scenarios: 1) true concordance factors, and 2) true concordance factors
perturbed by Gaussian error as described above ($\sigma=0.0005$).
### 2.4 Reticulate evolution in the Genus Canis
We further test our method on the Canis dataset from [17]. The original
genomic dataset contained $12$ gray wolves, $14$ dogs, five coyotes, one
Ethiopian wolf, three golden jackals, six African golden wolves, two dholes,
four African hunting dogs, and one Andean fox. Given the widespread gene flow
reported in the original study [17], we need to subsample the taxa so that
there is one 4-node hybridization event among the selected taxa: one African
hunting dog, one coyote, one dhole, one dog, one golden jackal, and one grey
wolf. We use the estimated gene trees from [50] available in
https://github.com/chaoszhang/Weighted-ASTRAL_data to infer the concordance
factors with the Julia package PhyloNetworks [39].
To better evaluate our method performance, we also run other methods on this
dataset for comparison. In addition to our method, we also run SNaQ [38] on
$h=1$ hybridization event, 10 independent runs, and using one randomly
selected gene tree as the starting tree. We use all 449,450 gene trees to
infer the table of CFs and then map the allele names to species names. We also
run PhyloNet on two options: 1) maximum likelihood (ML) and 2) maximum
pseudolikelihood (MPL). We choose $h=1$ hybridization event on both cases,
with 10 independent runs for each. We choose one representative individual per
species in the gene trees and removed gene trees that contained fewer than 5
taxa. In total, we use 448,758 gene trees in the PhyloNet analyses. We root
the gene trees on the known outgroup (African hunting dog) or on dhole if
African hunting dog is not in the gene tree.
## 3 Results
### 3.1 Simulation study on eight or fewer taxa
#### 3.1.1 Identifying the correct network as the top 1 ranked network with
the smallest invariant score
Figure 3 shows the proportion of times (out of the 30 replicates) that our
invariants method identifies the true network as the top 1 ranked network with
the smallest invariant score for the cases of true concordance factors and the
Gaussian-perturbed CFs with increasing standard deviation (from left to
right). We also quantify the proportion of times that our invariants method
identifies the symmetric network (when clades $n_{1}$ and $n_{2}$ in Figure 1
are switched) because in the 4-node hybridization cycle it is difficult for
the method to distinguish clades $n_{1}$ and $n_{2}$ given that they both
contribute to the hybridization node. Even for the case of true CFs, our
method identifies the symmetric network as the top 1 network, instead of the
true network, in two cases ($N=1221$ and $N=2121$), yet the difference in
invariance score ($L_{2}$-norm of evaluated polynomials) of the true and
symmetric networks in these cases is of the order of $10^{-16}$. For the
noisiest setting ($\sigma=0.0005$, far right), it is evident that some
networks are difficult to be detected by our method, but it is worth
highlighting that whenever there are two taxa sampled on each of the four
clades ($n_{0},n_{1},n_{2},n_{3}$) as in $N=2222$, our method always
identifies the true (or symmetric) network as the top 1.
Figure 3: Proportion of times that the true network or the symmetric network
(with clades $n_{1}$ and $n_{2}$ inverted) are the top 1 ranked networks
identified by the phylogenetic invariants on the cases of true and Gaussian-
perturbed CFs. Y-axis is between 0 and 100. Each panel corresponds to a type
of simulation: using true concordance factors (left) and using concordance
factors with added Gaussian noise (with increasing standard deviation for
noise from left to right). Whenever there are at least two taxa in each of the
four clades (network $2222$), our method is very accurate in detecting the
true (or its symmetric) network. Our method is also accurate to identify the
true network in the top 5 ranked networks (see Figure 6) which will greatly
reduce the space of candidate networks to be compared with a likelihood
approach.
Figure 4 shows the proportion of times (out of the 30 replicates) that our
invariants method identifies the true network as the top 1 ranked network with
smallest invariant score for the case of true simulated gene trees with
increasing number of gene trees (from left to right), where “g.t." is
abbreviation for “gene trees". Unlike the case of true or Gaussian-perturbed
CFs, our method only identifies the true (or symmetric) network when there are
2 taxa sampled per clade ($N=2222$) or at least 2 taxa sampled on the hybrid
clade and sister clades ($N=2221$). With 1,000 gene trees or more, the
networks $N=2121$ and $N=2211$ are also accurately identified by our method.
Figure 4: Proportion of times that the true network or the symmetric network
(with clades $n_{1}$ and $n_{2}$ inverted) are the top 1 ranked networks
identified by the phylogenetic invariants for the case of true simulated gene
trees (“g.t."). Y-axis is between 0 and 100. Whenever there are at least two
taxa in each of the four clades (network $2222$), our method is very accurate
in detecting the true (or its symmetric) network. Our method is also accurate
to identify the true network in the top 5 ranked networks (see Figure 7) which
will greatly reduce the space of candidate networks to be compared with a
likelihood approach.
Figure 5 shows the proportion of times (out of the 30 replicates) that our
invariants method identifies the true network as the top 1 ranked network with
smallest invariant score for the case of estimated gene trees with increasing
number of gene trees and sequence length (from left to right). Again, our
method only identifies the true (or symmetric) network when there are 2 taxa
sampled per clade ($N=2222$) or at least 2 taxa sampled on the hybrid clade
and sister clades ($N=2221$). As before, with 1,000 gene trees, the networks
$N=2121$ and $N=2211$ are also accurately identified by our method, and with
10,000 gene trees, the networks $N=1221$ and $N=1222$ are also accurately
identified.
Summarizing all figures related to our method’s ability to identify the true
(or symmetric) network as top 1 ranked network by its invariant score (Figures
3, 4, 5), we can highlight that networks with only one sampled taxon in some
of the clades are harder to be identified. As long as two taxa are sampled
from each of the clades, our method is able to identify the network with high
accuracy on all simulation settings. The number of genes is more important
than the sequence length, and with at least 100 genes, our method is able to
correctly identify networks with fewer than two taxa sampled from some clades
(like $2221$). We highlight that we are equally interested in the true and
symmetric networks since a follow-up optimization of branch lengths and
inheritance probability ($\gamma$) on the fixed network will estimate the
correct $\gamma$ and identify which of the sister clades is the major
($\gamma>0.5$) and which is the minor ($\gamma<0.5$).
Figure 5: Proportion of times that the true network or the symmetric network
(with clades $n_{1}$ and $n_{2}$ inverted) are the top 1 ranked networks
identified by the phylogenetic invariants for the case of estimated gene
trees. Y-axis is between 0 and 100. Each panel corresponds to a number of gene
trees (g.t. from 1000 to 10,000) and sequence length ($L$ from 500 to 2000).
Whenever there are at least two taxa in each of the four clades (network
$2222$) or only clade $n_{3}$ having one taxon ($N=2221$), our method is very
accurate in detecting the true (or its symmetric) network. Our method is also
accurate to identify the true network in the top 5 ranked networks (see Figure
8) which will greatly reduce the space of candidate networks to be compared
with a likelihood approach.
#### 3.1.2 Reducing the space of candidate networks using the invariant score
Figure 6 shows the proportion of times (out of the 30 replicates) that the
true (or symmetric) network are within the top 5 ranked networks based on
smallest invariant score for the cases of true concordance factors and the
Gaussian-perturbed CFs with increasing standard deviation (from left to
right). In all cases, our method includes the true (and symmetric) networks
within the top 5 which means that our method accurately and in a fast manner
reduces the space of candidate networks to just 5 candidates that can later be
tested with another accurate methodology like likelihood.
Figure 6: Proportion of times that the true network or the symmetric network
(with clades $n_{1}$ and $n_{2}$ inverted) are in the top 5 ranked networks
identified by the phylogenetic invariants on the cases of true and Gaussian-
perturbed CFs. Y-axis is between 0 and 100. Each panel corresponds to a type
of simulation: using true concordance factors (left) and using concordance
factors with added Gaussian noise (with increasing standard deviation for
noise from left to right). All networks are accurately identified in the top 5
ranked based on invariant score which provides evidence that our method fast
and accurately reduce the space of candidate networks to just 5 alternatives.
Figure 7 shows the proportion of times (out of the 30 replicates) that the
true (or symmetric) network are within the top 5 ranked networks based on
smallest invariant score for the case of true simulated gene trees. Only the
networks with one sampled taxon on the hybrid clade ($n_{0}$) cannot be
recovered with 100 or 1000 gene trees. Whenever there are 10,000 gene trees in
the sample, the networks $N=1221$ and $N=1222$ can now be recovered as part of
the top 5, yet the networks $N=1122$ or $N=1212$ (when there is one sampled
taxon in the hybrid clade and one of the sister clades) still refuse
detection. The same conclusions are true for the case of estimated gene trees
(Figure 8) with number of gene trees having more weight on the accuracy than
sequence length.
Figure 7: Proportion of times that the true network or the symmetric network
(with clades $n_{1}$ and $n_{2}$ inverted) are in the top 5 ranked networks
identified by the phylogenetic invariants for the case of true simulated gene
trees (“g.t."). Y-axis is between 0 and 100. Only the networks that have one
taxon below the hybrid node (1122, 1212, 1221, 1222) do not allow accurate
reconstruction with fewer than 10,000 gene trees which brings into attention
the importance of taxon sampling for this method.
Summarizing all figures related to our method’s ability to identify the true
(or symmetric) network within the top 5 ranked networks by its invariant score
(Figures 6, 7, 8), we can highlight that as long as there are at least 2 taxa
sampled from the hybrid clade, then our method is able to place the true (and
symmetric) network within the top 5 networks. This effectively reduces the
space of candidate networks to compare with a likelihood approach. As
mentioned, we do not view our method as a substitute to existing network
inference methods. On the contrary, we believe that our invariants method will
serve as a complement to identify a small subset of network possibilities that
will help bypass the optimization on network space.
Figure 8: Proportion of times that the true network or the symmetric network
(with clades $n_{1}$ and $n_{2}$ inverted) are in the top 5 ranked networks
identified by the phylogenetic invariants for the case of estimated gene
trees. Y-axis is between 0 and 100. Each panel corresponds to a number of gene
trees (g.t. from 1000 to 10,000) and sequence length ($L$ from 500 to 2000).
Only the networks that have one taxon below the hybrid node (1122, 1212, 1221,
1222) do not allow accurate reconstruction with fewer than 10,000 gene trees
which brings into attention the importance of taxon sampling for this method.
Figure 9 shows the rank in the invariant scores of the true and symmetric
networks on the cases of true and Gaussian-perturbed CFs. Ideally, the true
network (or at least its symmetric version) should be ranked as 1 by our
invariants method. While the true and symmetric networks are not always ranked
in number 1, they are ranked within the top 5 for most of the cases. This
further confirms the advantage of our current method. It is a fast way to
reduce the space of candidate networks (to 5) that can later be tested using a
likelihood approach.
Figure 9: Rank (y-axis) in the invariant score for each network (x-axis) for
the true network (red) and its symmetric network (inverted clades $n_{1}$ and
$n_{2}$, in blue) on the cases of true and Gaussian-perturbed CFs. Dashed line
corresponds to rank 5. Each panel corresponds to a type of simulation: using
true concordance factors (left) and using concordance factors with added
Gaussian noise (with increasing standard deviation for noise from left to
right). Both the true and symmetric networks are within the top 5 ranked
networks by the method in all cases, and are thus, easy to distinguish from
wrong networks.
Figure 10 shows the rank in the invariant scores of the true and symmetric
networks for the case of true simulated gene trees (increased number of gene
trees from left to right). The true (and symmetric) networks are not within
the top 5 networks only for the cases of one sampled taxon from the hybrid
clade ($N=1122,1212,1221,1222$). This same behavior is evident in Figure 11
for estimated gene trees. Again, these figures show our method’s ability to
reduce the possible networks that fit the data.
Figure 10: Rank (y-axis) in the invariant score for each network (x-axis) for
the true network (red) and its symmetric network (inverted clades $n_{1}$ and
$n_{2}$, in blue) in the case of true simulated gene trees (“g.t"). Each panel
corresponds to a number of simulated gene trees from 100 (left) to 10,000
(right). Both the true and symmetric networks are within the top 5 ranked
networks by the method as the number of genes increases, and are thus, easy to
distinguish from wrong networks. Only the networks that have one taxon below
the hybrid node (1122, 1212, 1221, 1222) do not allow accurate reconstruction
which brings into attention the importance of taxon sampling for this method.
Figure 11: Rank (y-axis) in the invariant score for each network (x-axis) for
the true network (red) and its symmetric network (inverted clades $n_{1}$ and
$n_{2}$, in blue) in the case of estimated gene trees. Each panel corresponds
to a number of gene trees (g.t. from 1000 to 10,000) and sequence length ($L$
from 500 to 2000). Only the networks that have one taxon below the hybrid node
(1122, 1212, 1221, 1222) do not allow accurate reconstruction which brings
into attention the importance of taxon sampling for this method.
In the Appendix, we also show plots with the values of the invariant scores
for the true and symmetric networks.
Last, we present a comparison of the running times of four network methods in
Table 1. While all methods are able to identify the correct network, our
phylogenetic invariants method (top row) only takes 7.17 seconds to correctly
infer the network $N=2222$ which is over twice as fast as the second fastest
(PhyloNet MPL [49]). The difference in running times (and even accuracy) is
more evident in the results on the Canis dataset (Section 3.3).
Method | Time (seconds)
---|---
Phylogenetic invariants (our method) | 7.17
SNaQ | 80.23
PhyloNet ML | 84.34
PhyloNet MPL | 17.99
Table 1: Running times (in seconds) on the inference of network $N=2222$ from
100 true simulated gene trees by four network methods: 1) our phylogenetic
invariants (top row); 2) SNaQ [38]; 3) PhyloNet ML [48], and 4) PhyloNet MPL
[49].
### 3.2 Simulation study on more than eight taxa
Table 2 shows the proof-of-concept results on simulated data with more than 8
taxa. Since the algorithm for more than 8 taxa (Algorithm 2) relies entirely
on the algorithm for 8 or fewer taxa (Algorithm 1), we simply show here that
our steps to append taxon to the optimal subset of 8 taxa identified by the
method indeed yields the desired network with more than 8 taxa.
Network | True CF | Noisy CF
---|---|---
2223 | 1 | 1
2232 | 1 | 1 (symmetric)
2322 | 1 | 2
3222 | 1 | 1
2233 | 1 | 1 (symmetric)
2323 | 1 | 2 (symmetric)
3223 | 1 | 1
2332 | 1 | 2 (symmetric)
3232 | 1 | 2 (symmetric)
3322 | 1 | 1 (symmetric)
Table 2: Rank of the true (or symmetric) network under the two types of
simulations: true CFs (left column) and Gaussian-perturbed CFs (right column)
for a level of noise of $\sigma=0.0005$. In all cases, either the true network
or the symmetric networks are within the top 2 of networks identified by the
method which proves that our algorithm for more than 8 taxa that builds on
Algorithm 1 works appropriately.
### 3.3 Phylogenetic network for the Canis genus
The original analysis of the Canis genus identified nine gene flow events
using the D statistics [32] (Figure 3a in [17]). Here, we are able to
replicate the hybridization event involving the ancestor of dog and grey wolf
into golden jackal (Figure 12 top). The same network is correctly inferred by
SNaQ and by PhyloNet ML, but both methods take much longer to run (Table 3).
PhyloNet MPL identifies a different hybridization event that has not been
reported in previous studies, so it is impossible to validate it at this
point. This hybridization involves an unsampled or extinct taxon (Figure 12
bottom).
Figure 12: Left top: Semi-directed phylogenetic network estimated by SNaQ
[38], PhyloNet ML [48], and our method based on phylogenetic invariants. Left
bottom: Semi-directed phylogenetic network estimated by PhyloNet MPL [49]. The
star marks the place where the root should be. Right: Rooted version of the
semi-directed networks. The network estimated with PhyloNet MPL (bottom) is
different from the network inferred by the other methods, and it involves an
unsampled (or extinct) taxon in the hybridization. SNaQ, PhyloNet ML and our
phylogenetic invariants method (top) identify the same hybridization event as
the original publication [17] between the ancestral species to dog and grey
wolf and the ancestral species of golden jackal, yet PhyloNet ML takes 389
times longer and SNaQ takes 20 times longer than our invariants method (see
running times in Table 3).
In terms of running time, our invariants method is able to identify the
correct hybridization in under 7 seconds while SNaQ (which also identifies the
correct hybridization) takes over 140 seconds. PhyloNet MPL takes 40 times
longer than our invariants method (with running time of over 281 seconds) and
identifies a different hybridization event to those originally published (so,
not possible to validate at this point). PhyloNet ML is the slowest method
with over 45 minutes of running time on this small network of only 6 taxa. The
inferred network by PhyloNet ML also agrees with the one originally published
in [17].
Method | Time (seconds)
---|---
Phylogenetic invariants (our method) | 6.78
SNaQ | 140.58
PhyloNet ML | 2723.99
PhyloNet MPL | 281.25
Table 3: Running times (in seconds) on the Canis dataset. Our method based on
evaluation of phylogenetic invariants is 20 times faster than the second
fastest method (SNaQ).
## 4 Conclusion
Here, we introduce a novel method to infer 4-node hybridization cycles on
level-1 phylogenetic networks of over five taxa based on ultrafast evaluations
of phylogenetic invariants. Our method bypasses optimization on network space
and is able to accurately detect the hybridization cycles based on simulated
and real data, especially when there are at least 2 taxa sampled from each of
the four clades defined by the hybridization cycle (Figure 1 left). While our
method is at least $\sim 10$ times faster than other existing network methods,
it is not meant to replace existing methods. On the contrary, we believe that
our ultrafast algorithm can work in conjunction with existing methodologies by
reducing the space of candidate networks that can later be evaluated based on
likelihood or Bayesian approaches.
Nevertheless, there are limitations in our current method. Specifically, we
are only able to detect hybridization cycles with 4 nodes on level-1 networks.
For the case of three nodes in the hybridization cycle, it has been
investigated already in [38] that the set of phylogenetic invariants is empty.
However, there is still room to extend the current methodology to
hybridization cycles of five nodes or more. In addition, even when our method
can infer networks with any number of taxa, it is unable to resolve the
topology inside any of the four clades defined by the hybridization cycle
(Figure 1 left). That is, our method is only able to identify the placement of
the 4-node hybridization cycle. For this reason, we view our method as in line
with other hybrid detection methods such as ABBA-BABA test [32], MSCquartets
[30, 36], and HyDe [27] that are also only able to detect specific
hybridization patterns in subsets of taxa.
Future work will involve the generation of the phylogenetic invariants related
to hybridization cycles of five nodes or more, and the development of a
merging algorithm that can produced a fully resolved $n$-taxon phylogenetic
network from the 8-taxon estimated networks inferred by our invariants method.
In addition, our proposed method is unable to infer biological parameters like
inheritance probability or branch length, but future work could exploit the CF
formulas (which depend on branch lengths and inheritance probabilities) to
provide estimated values to these parameters from the observed CFs.
## 5 Data and code availability
The Canis dataset was made publicly available by the original publication [17]
and can be accessed through the GitHub repository of [50] here
https://github.com/chaoszhang/Weighted-ASTRAL_data. All the scripts for our
work are publicly available in the GitHub repository
https://github.com/solislemuslab/PhyloDiamond.jl. The new Julia package
PhyloDiamond.jl is open-source, publicly available in
https://github.com/solislemuslab/PhyloDiamond.jl.
## 6 Acknowledgements
This work was supported by the National Science Foundation [DEB-2144367 to
CSL]. This was also funded by the UW-Madison Fall Competition [to CSL]. The
authors thank Marianne Bjørner and Nathan Kolbow for help with phylogenetics
code.
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## Appendix A Representation of a level-1 semi-directed phylogenetic network
with a set of polynomial equations corresponding to the expected concordance
factors under the coalescent model
Notation:
* •
$n$ corresponds to the number of individuals from each of the 4 clades:
$(n_{0},n_{1},n_{2},n_{3})$. For example, $n=(0,0,2,2)$ means that you have
the quartet with 2 individuals in $n_{2}$ and 2 individuals in $n_{3}$
* •
"Individuals" list the individuals taken from each of the four clades. For
example, $i_{1},i_{2}\in n_{0}$ means that we took two species from clade
$n_{0}$ denoted $i_{1}$ and $i_{2}$. We use these species names to define the
splits for the formulas: e.g. $P(i_{1},i_{2}|j_{1},j_{2})$ represents the
probability that $i_{1}$ and $i_{2}$ are together in one side of the split
(and $j_{1},j_{2}$ together)
* •
"Type" corresponds to the type of quartet which matches the types in [38]
* •
"CF Formula" corresponds to the formula of the expected CF for that given
quartet under the multispecies coalescent model
* •
"CF value" is the variable we give to the observed CF we will read from the
data table
By [40], we know that we only need to select at most two individuals per
subgraph $(n_{0},n_{1},n_{2},n_{3})$ to define all the CF formulas that
involve the hybridization cycle. We list all the CF formulas in the table
below assuming that we do have 2 individuals per clade
$(n_{0},n_{1},n_{2},n_{3})$ (that is, we have at least eight species).
Table 4: Concordance factor equations for all 4-taxon subsets that involve the hybridization cycle. $n$ | Type | CF Formula | CF value
---|---|---|---
$(0,0,2,2)$ | $5$ | $1-\dfrac{2}{3}z_{2}z_{2,3}z_{3}$ | $a1$
| | $\dfrac{1}{3}z_{2}z_{2,3}z_{3}$ | $a2$
| | $\dfrac{1}{3}z_{2}z_{2,3}z_{3}$ | $a3$
$(0,1,2,1)$ | $5$ | $1-\dfrac{2}{3}z_{2,3}z_{2}$ | $a4$
| | $\dfrac{1}{3}z_{2,3}z_{2}$ | $a5$
| | $\dfrac{1}{3}z_{2,3}z_{2}$ | $a6$
$(0,1,1,2)$ | $5$ | $1-\dfrac{2}{3}z_{3}$ | $a7$
| | $\dfrac{1}{3}z_{3}$ | $a8$
| | $\dfrac{1}{3}z_{3}$ | $a9$
$(0,2,2,0)$ | $5$ | $1-\dfrac{2}{3}z_{2}z_{2,3}z_{1,3}z_{1}$ | $a10$
| | $\dfrac{1}{3}z_{2}z_{2,3}z_{1,3}z_{1}$ | $a11$
| | $\dfrac{1}{3}z_{2}z_{2,3}z_{1,3}z_{1}$ | $a12$
$(0,2,1,1)$ | $5$ | $1-\dfrac{2}{3}z_{1,3}z_{1}$ | $a13$
| | $\dfrac{1}{3}z_{1,3}z_{1}$ | $a14$
| | $\dfrac{1}{3}z_{1,3}z_{1}$ | $a15$
$(0,2,0,2)$ | $5$ | $1-\dfrac{2}{3}z_{3}z_{1,3}z_{1}$ | $a16$
| | $\dfrac{1}{3}z_{3}z_{1,3}z_{1}$ | $a17$
| | $\dfrac{1}{3}z_{3}z_{1,3}z_{1}$ | $a18$
$(1,0,2,1)$ | $2$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{2,3}z_{2}\right)+\gamma\left(1-\dfrac{2}{3}z_{2}\right)$ | $a19$
| | $(1-\gamma)\dfrac{1}{3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}$ | $a20$
| | $(1-\gamma)\dfrac{1}{3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}$ | $a21$
$(1,0,1,2)$ | $2$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{3}\right)+\gamma\left(1-\dfrac{2}{3}z_{2,3}z_{3}\right)$ | $a22$
| | $(1-\gamma)\dfrac{1}{3}z_{3}+\gamma\dfrac{1}{3}z_{2,3}z_{3}$ | $a23$
| | $(1-\gamma)\dfrac{1}{3}z_{3}+\gamma\dfrac{1}{3}z_{2,3}z_{3}$ | $a24$
$(1,1,2,0)$ | $2$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{1,3}z_{2,3}z_{2}\right)+\gamma\left(1-\dfrac{2}{3}z_{2}\right)$ | $a25$
| | $(1-\gamma)\dfrac{1}{3}z_{1,3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}$ | $a26$
| | $(1-\gamma)\dfrac{1}{3}z_{1,3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}$ | $a27$
$(1,1,1,1)$ | $3$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{1,3}\right)+\gamma\dfrac{1}{3}z_{2,3}$ | $a28$
| | $(1-\gamma)\dfrac{1}{3}z_{1,3}+\gamma\left(1-\dfrac{2}{3}z_{2,3}\right)$ | $a29$
| | $(1-\gamma)\dfrac{1}{3}z_{1,3}+\gamma\dfrac{1}{3}z_{2,3}$ | $a30$
$(1,1,0,2)$ | $2$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{1,3}z_{3}\right)+\gamma\left(1-\dfrac{2}{3}z_{3}\right)$ | $a31$
| | $(1-\gamma)\dfrac{1}{3}z_{1,3}z_{3}+\gamma\dfrac{1}{3}z_{3}$ | $a32$
| | $(1-\gamma)\dfrac{1}{3}z_{1,3}z_{3}+\gamma\dfrac{1}{3}z_{3}$ | $a33$
$(1,2,1,0)$ | $2$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{1}\right)+\gamma\left(1-\dfrac{2}{3}z_{2,3}z_{1,3}z_{1}\right)$ | $a34$
| | $(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{2,3}z_{1,3}z_{1}$ | $a35$
| | $(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{2,3}z_{1,3}z_{1}$ | $a36$
$(1,2,0,1)$ | $2$ | $(1-\gamma)\left(1-\dfrac{2}{3}z_{1}\right)+\gamma\left(1-\dfrac{2}{3}z_{1,3}z_{1}\right)$ | $a37$
| | $(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{1,3}z_{1}$ | $a38$
| | $(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{1,3}z_{1}$ | $a39$
$(2,0,2,0)$ | $4$ | $(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{2}z_{0}z_{0,1}z_{1,3}z_{2,3}\right)+2\gamma(1-\gamma)\left(1-\dfrac{2}{3}z_{2}z_{0}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{2}z_{0}z_{0,2}\right)$ | $a40$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,1}z_{1,3}z_{2,3}+2\gamma(1-\gamma)\dfrac{1}{3}z_{2}z_{0}+\gamma^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,2}$ | $a41$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,1}z_{1,3}z_{2,3}+2\gamma(1-\gamma)\dfrac{1}{3}z_{2}z_{0}+\gamma^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,2}$ | $a42$
$(2,0,1,1)$ | $1$ | $(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{0}z_{1,3}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-z_{0}+\dfrac{1}{3}z_{0}z_{2,3}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,2}\right)$ | $a43$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{1,3}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}$ | $a44$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{1,3}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}$ | $a45$
$(2,0,0,2)$ | $4$ | $(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{3}z_{0}z_{1,3}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-\dfrac{2}{3}z_{3}z_{0}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{3}z_{0}z_{2,3}z_{0,2}\right)$ | $a46$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{3}z_{0}z_{1,3}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{3}z_{0}+\gamma^{2}\dfrac{1}{3}z_{3}z_{0}z_{2,3}z_{0,2}$ | $a47$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{3}z_{0}z_{1,3}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{3}z_{0}+\gamma^{2}\dfrac{1}{3}z_{3}z_{0}z_{2,3}z_{0,2}$ | $a48$
$(2,1,1,0)$ | $1$ | $(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-z_{0}+\dfrac{1}{3}z_{0}z_{2,3}z_{1,3}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,2}\right)$ | $a49$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}$ | $a50$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}$ | $a51$
$(2,1,0,1)$ | $1$ | $(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-z_{0}+\dfrac{1}{3}z_{0}z_{1,3}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,2}z_{2,3}\right)$ | $a52$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}z_{2,3}$ | $a53$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}z_{2,3}$ | $a54$
$(2,2,0,0)$ | $4$ | $(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{1}z_{0}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-\dfrac{2}{3}z_{1}z_{0}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{1}z_{0}z_{0,2}z_{2,3}z_{1,3}\right)$ | $a55$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{1}z_{0}+\gamma^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,2}z_{2,3}z_{1,3}$ | $a56$
| | $(1-\gamma)^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{1}z_{0}+\gamma^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,2}z_{2,3}z_{1,3}$ | $a57$
For the mapping of observed CFs to $a_{i}$ values, we need to know to which of
the three splits each of the $a_{i}$’s corresponds to. So, next, we elaborate
on which specific split each of the $a_{i}$ corresponds to:
#### $n=(0,0,2,2)$
* •
Individuals $k_{1},k_{2}\in n_{2}$ and $l_{1},l_{2}\in n_{3}$
* •
$P(k_{1},k_{2}|l_{1},l_{2})=1-\dfrac{2}{3}z_{2}z_{2,3}z_{3}=a_{1}$
* •
$P(k_{1},l_{1}|k_{2},l_{2})=\dfrac{1}{3}z_{2}z_{2,3}z_{3}=a_{2}$
* •
$P(k_{1},l_{2}|k_{2},l_{1})=\dfrac{1}{3}z_{2}z_{2,3}z_{3}=a_{3}$
#### $n=(0,1,2,1)$
* •
Individuals $j_{1}\in n_{1}$; $k_{1},k_{2}\in n_{2}$ and $l_{1}\in n_{3}$
* •
$P(k_{1},k_{2}|j_{1},l_{1})=1-\dfrac{2}{3}z_{2}z_{2,3}=a_{4}$
* •
$P(k_{1},j_{1}|k_{2},l_{1})=\dfrac{1}{3}z_{2}z_{2,3}=a_{5}$
* •
$P(k_{1},l_{1}|k_{2},j_{1})=\dfrac{1}{3}z_{2}z_{2,3}=a_{6}$
#### $n=(0,1,1,2)$
* •
Individuals $j_{1}\in n_{1}$; $k_{1}\in n_{2}$ and $l_{1},l_{2}\in n_{3}$
* •
$P(l_{1},l_{2}|j_{1},k_{1})=1-\dfrac{2}{3}z_{3}=a_{7}$
* •
$P(l_{1},j_{1}|l_{2},k_{1})=\dfrac{1}{3}z_{3}=a_{8}$
* •
$P(l_{1},k_{1}|l_{2},j_{1})=\dfrac{1}{3}z_{3}=a_{9}$
#### $n=(0,2,2,0)$
* •
Individuals $j_{1},j_{2}\in n_{1}$; $k_{1},k_{2}\in n_{2}$
* •
$P(j_{1},j_{2}|k_{1},k_{2})=1-\dfrac{2}{3}z_{2}z_{2,3}z_{1,3}z_{1}=a_{10}$
* •
$P(j_{1},k_{1}|j_{2},k_{2})=\dfrac{1}{3}z_{2}z_{2,3}z_{1,3}z_{1}=a_{11}$
* •
$P(j_{1},k_{2}|j_{2},k_{1})=\dfrac{1}{3}z_{2}z_{2,3}z_{1,3}z_{1}=a_{12}$
#### $n=(0,2,1,1)$
* •
Individuals $j_{1},j_{2}\in n_{1}$; $k_{1}\in n_{2}$; $l_{1}\in n_{3}$
* •
$P(j_{1},j_{2}|k_{1},l_{1})=1-\dfrac{2}{3}z_{1,3}z_{1}=a_{13}$
* •
$P(j_{1},k_{1}|j_{2},l_{1})=\dfrac{1}{3}z_{1,3}z_{1}=a_{14}$
* •
$P(j_{1},l_{1}|j_{2},k_{1})=\dfrac{1}{3}z_{1,3}z_{1}=a_{15}$
#### $n=(0,2,0,2)$
* •
Individuals $j_{1},j_{2}\in n_{1}$; $l_{1},l_{2}\in n_{3}$
* •
$P(j_{1},j_{2}|l_{1},l_{2})=1-\dfrac{2}{3}z_{3}z_{1,3}z_{1}=a_{16}$
* •
$P(j_{1},l_{1}|j_{2},l_{2})=\dfrac{1}{3}z_{3}z_{1,3}z_{1}=a_{17}$
* •
$P(j_{1},l_{2}|j_{2},l_{1})=\dfrac{1}{3}z_{3}z_{1,3}z_{1}=a_{18}$
#### $n=(1,0,2,1)$
* •
Individuals $i_{1}\in n_{0}$; $k_{1},k_{2}\in n_{2}$; $l_{1}\in n_{3}$
* •
$P(k_{1},k_{2}|i_{1},l_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{2,3}z_{2}\right)+\gamma\left(1-\dfrac{2}{3}z_{2}\right)=a_{19}$
* •
$P(k_{1},i_{1}|k_{2},l_{1})=(1-\gamma)\dfrac{1}{3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}=a_{20}$
* •
$P(k_{1},l_{1}|k_{2},i_{1})=(1-\gamma)\dfrac{1}{3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}=a_{21}$
#### $n=(1,0,1,2)$
* •
Individuals $i_{1}\in n_{0}$; $k_{1}\in n_{2}$; $l_{1},l_{2}\in n_{3}$
* •
$P(l_{1},l_{2}|i_{1},k_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{3}\right)+\gamma\left(1-\dfrac{2}{3}z_{2,3}z_{3}\right)=a_{22}$
* •
$P(l_{1},i_{1}|l_{2},k_{1})=(1-\gamma)\dfrac{1}{3}z_{3}+\gamma\dfrac{1}{3}z_{2,3}z_{3}=a_{23}$
* •
$P(l_{1},k_{1}|l_{2},i_{1})=(1-\gamma)\dfrac{1}{3}z_{3}+\gamma\dfrac{1}{3}z_{2,3}z_{3}=a_{24}$
#### $n=(1,1,2,0)$
* •
Individuals $i_{1}\in n_{0}$; $j_{1}\in n_{1}$; $k_{1},k_{2}\in n_{2}$
* •
$P(k_{1},k_{2}|i_{1},j_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{1,3}z_{2,3}z_{2}\right)+\gamma\left(1-\dfrac{2}{3}z_{2}\right)=a_{25}$
* •
$P(k_{1},i_{1}|k_{2},j_{1})=(1-\gamma)\dfrac{1}{3}z_{1,3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}=a_{26}$
* •
$P(k_{1},j_{1}|k_{2},i_{1})=(1-\gamma)\dfrac{1}{3}z_{1,3}z_{2,3}z_{2}+\gamma\dfrac{1}{3}z_{2}=a_{27}$
#### $n=(1,1,1,1)$
* •
Individuals $i_{1}\in n_{0}$; $j_{1}\in n_{1}$; $k_{1}\in n_{2}$; $l_{1}\in
n_{3}$
* •
$P(i_{1},j_{1}|k_{1},l_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{1,3}\right)+\gamma\dfrac{2}{3}z_{2,3}=a_{28}$
* •
$P(i_{1},k_{1}|j_{1},l_{1})=(1-\gamma)\dfrac{1}{3}z_{1,3}+\gamma\left(1-\dfrac{2}{3}z_{2,3}\right)=a_{29}$
* •
$P(i_{1},l_{1}|j_{1},k_{1})=(1-\gamma)\dfrac{1}{3}z_{1,3}+\gamma\dfrac{1}{3}z_{2,3}=a_{30}$
#### $n=(1,1,0,2)$
* •
Individuals $i_{1}\in n_{0}$; $j_{1}\in n_{1}$; $l_{1},l_{2}\in n_{3}$
* •
$P(l_{1},l_{2}|i_{1},j_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{1,3}z_{3}\right)+\gamma\left(1-\dfrac{2}{3}z_{3}\right)=a_{31}$
* •
$P(l_{1},i_{1}|l_{2},j_{1})=(1-\gamma)\dfrac{1}{3}z_{1,3}z_{3}+\gamma\dfrac{1}{3}z_{3}=a_{32}$
* •
$P(l_{1},j_{1}|l_{2},i_{1})=(1-\gamma)\dfrac{1}{3}z_{1,3}z_{3}+\gamma\dfrac{1}{3}z_{3}=a_{33}$
#### $n=(1,2,1,0)$
* •
Individuals $i_{1}\in n_{0}$; $j_{1},j_{2}\in n_{1}$; $k_{1}\in n_{2}$
* •
$P(j_{1},j_{2}|i_{1},k_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{1}\right)+\gamma\left(1-\dfrac{2}{3}z_{2,3}z_{1,3}z_{1}\right)=a_{34}$
* •
$P(j_{1},i_{1}|j_{2},k_{1})=(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{2,3}z_{1,3}z_{1}=a_{35}$
* •
$P(j_{1},k_{1}|j_{2},i_{1})=(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{2,3}z_{1,3}z_{1}=a_{36}$
#### $n=(1,2,0,1)$
* •
Individuals $i_{1}\in n_{0}$; $j_{1},j_{2}\in n_{1}$; $l_{1}\in n_{3}$
* •
$P(j_{1},j_{2}|i_{1},l_{1})=(1-\gamma)\left(1-\dfrac{2}{3}z_{1}\right)+\gamma\left(1-\dfrac{2}{3}z_{1,3}z_{1}\right)=a_{37}$
* •
$P(j_{1},l_{1}|j_{2},i_{1})=(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{1,3}z_{1}=a_{38}$
* •
$P(j_{1},i_{1}|j_{2},l_{1})=(1-\gamma)\dfrac{1}{3}z_{1}+\gamma\dfrac{1}{3}z_{1,3}z_{1}=a_{39}$
#### $n=(2,0,2,0)$
* •
Individuals $i_{1},i_{2}\in n_{0}$; $k_{1},k_{2}\in n_{2}$
* •
$P(i_{1},i_{2}|k_{1},k_{2})=(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{2}z_{0}z_{0,1}z_{1,3}z_{2,3}\right)+2\gamma(1-\gamma)\left(1-\dfrac{2}{3}z_{2}z_{0}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{2}z_{0}z_{0,2}\right)=a_{40}$
* •
$P(i_{1},k_{1}|i_{2},k_{2})=(1-\gamma)^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,1}z_{1,3}z_{2,3}+2\gamma(1-\gamma)\dfrac{1}{3}z_{2}z_{0}+\gamma^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,2}=a_{41}$
* •
$P(i_{1},k_{2}|k_{1},i_{2})=(1-\gamma)^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,1}z_{1,3}z_{2,3}+2\gamma(1-\gamma)\dfrac{1}{3}z_{2}z_{0}+\gamma^{2}\dfrac{1}{3}z_{2}z_{0}z_{0,2}=a_{42}$
#### $n=(2,0,1,1)$
* •
Individuals $i_{1},i_{2}\in n_{0}$; $k_{1}\in n_{2}$; $l_{1}\in n_{3}$
* •
$P(i_{1},i_{2}|k_{1},l_{1})=(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{0}z_{1,3}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-z_{0}+\dfrac{1}{3}z_{0}z_{2,3}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,2}\right)=a_{43}$
* •
$P(i_{1},k_{1}|l_{1},i_{2})=(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{1,3}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}=a_{44}$
* •
$P(i_{1},l_{1}|i_{2},k_{1})=(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{1,3}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}=a_{45}$
#### $n=(2,0,0,2)$
* •
Individuals $i_{1},i_{2}\in n_{0}$; $l_{1},l_{2}\in n_{3}$
* •
$P(i_{1},i_{2}|l_{1},l_{2})=(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{3}z_{0}z_{1,3}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-\dfrac{2}{3}z_{3}z_{0}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{3}z_{0}z_{2,3}z_{0,2}\right)=a_{46}$
* •
$P(i_{1},l_{1}|i_{2},l_{2})=(1-\gamma)^{2}\dfrac{1}{3}z_{3}z_{0}z_{1,3}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{3}z_{0}+\gamma^{2}\dfrac{1}{3}z_{3}z_{0}z_{2,3}z_{0,2}=a_{47}$
* •
$P(i_{1},l_{2}|i_{2},l_{1})=(1-\gamma)^{2}\dfrac{1}{3}z_{3}z_{0}z_{1,3}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{3}z_{0}+\gamma^{2}\dfrac{1}{3}z_{3}z_{0}z_{2,3}z_{0,2}=a_{48}$
#### $n=(2,1,1,0)$
* •
Individuals $i_{1},i_{2}\in n_{0}$; $j_{1}\in n_{1}$; $k_{1}\in n_{2}$
* •
$P(i_{1},i_{2}|j_{1},k_{1})=(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-z_{0}+\dfrac{1}{3}z_{0}z_{2,3}z_{1,3}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,2}\right)=a_{49}$
* •
$P(i_{1},j_{1}|i_{2},k_{1})=1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}=a_{50}$
* •
$P(i_{1},k_{1}|i_{2},j_{1})=1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{2,3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}=a_{51}$
#### $n=(2,1,0,1)$
* •
Individuals $i_{1},i_{2}\in n_{0}$; $j_{1}\in n_{1}$; $l_{1}\in n_{3}$
* •
$P(i_{1},i_{2}|j_{1},l_{1})=(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-z_{0}+\dfrac{1}{3}z_{0}z_{1,3}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{0}z_{0,2}z_{2,3}\right)=a_{52}$
* •
$P(i_{1},j_{1}|i_{2},l_{1})=(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}z_{2,3}=a_{53}$
* •
$P(i_{1},l_{1}|i_{2},j_{1})=(1-\gamma)^{2}\dfrac{1}{3}z_{0}z_{0,1}+\gamma(1-\gamma)z_{0}\left(1-\dfrac{1}{3}z_{1,3}\right)+\gamma^{2}\dfrac{1}{3}z_{0}z_{0,2}z_{2,3}=a_{54}$
#### $n=(2,2,0,0)$
* •
Individuals $i_{1},i_{2}\in n_{0}$; $j_{1},j_{2}\in n_{1}$;
* •
$P(i_{1},i_{2}|j_{1},j_{2})=(1-\gamma)^{2}\left(1-\dfrac{2}{3}z_{1}z_{0}z_{0,1}\right)+2\gamma(1-\gamma)\left(1-\dfrac{2}{3}z_{1}z_{0}\right)+\gamma^{2}\left(1-\dfrac{2}{3}z_{1}z_{0}z_{0,2}z_{2,3}z_{1,3}\right)=a_{55}$
* •
$P(i_{1},j_{1}|i_{2},j_{2})=(1-\gamma)^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{1}z_{0}+\gamma^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,2}z_{2,3}z_{1,3}=a_{56}$
* •
$P(i_{1},j_{2}|i_{2},j_{1})=(1-\gamma)^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,1}+2\gamma(1-\gamma)\dfrac{1}{3}z_{1}z_{0}+\gamma^{2}\dfrac{1}{3}z_{1}z_{0}z_{0,2}z_{2,3}z_{1,3}=a_{57}$
## Appendix B Phylogenetic invariants for $n$-taxon phylogenetic networks
with one 4-node hybridization cycle
Below, we present the invariants for different networks $N$ all with one
4-cycle, but with different number of species on the clades
$n_{0},n_{1},n_{2},n_{3}$. For example, the network $N=1112$ corresponds to a
network with 5 species: one in $n_{0}$, one in $n_{1}$, one in $n_{2}$ and two
in $n_{3}$. The number of species defines the number of CF formulas. For
example, for 6 species, there are ${6\choose 4}=15$ 4-taxon subsets, each with
3 CF formulas. So, for 6 species, we have 45 CF formulas and thus, 45 CF
values. However, we only want to focus on the 4-taxon subsets that involve the
hybridization cycle. For the case of $N=1112$, they are only 4:
$(0,1,1,2),(1,0,1,2),(1,1,1,1),(1,1,0,2)$. Note that our table of CF formulas
has 57 different CF formulas (and therefore, values). This discrepancy is due
to the fact that the table is listing all possible CF formulas and we will
have fewer formulas if we have less than 8 species.
For some examples, for computational restrictions, we had to include just a
subset of the original CF equations to obtain the Gröbner basis in $a_{i}$. We
denote these cases with "subset". All Macaulay2 scripts (and output) can be
found in the GitHub repository: https://github.com/solislemuslab/phylo-
diamond.jl.
* •
$N=1112$
1. 1.
$a_{32}-a_{33}$
2. 2.
$a_{31}+2a_{33}-1$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{23}-a_{24}$
5. 5.
$a_{22}+2a_{24}-1$
6. 6.
$a_{8}-a_{9}$
7. 7.
$a_{7}+2a_{9}-1$
8. 8.
$3a_{9}*a_{30}+a_{9}-a_{24}-a_{33}$
9. 9.
$a_{24}*a_{29}+2a_{24}*a_{30}+a_{29}*a_{33}-a_{30}*a_{33}-a_{33}$
10. 10.
$3a_{9}*a_{29}-2a_{9}+2a_{24}-a_{33}$
* •
$N=1121$
1. 1.
$a_{28}+a_{29}+a_{30}-1$
2. 2.
$a_{26}-a_{27}$
3. 3.
$a_{25}+2a_{27}-1$
4. 4.
$a_{20}-a_{21}$
5. 5.
$a_{19}+2a_{21}-1$
6. 6.
$a_{5}-a_{6}$
7. 7.
$a_{4}+2a_{6}-1$
8. 8.
$a_{6}*a_{29}+2a_{6}*a_{30}-a_{6}+a_{21}-a_{27}$
* •
$N=1122$
1. 1.
$a_{32}-a_{33}$
2. 2.
$a_{31}+2a_{33}-1$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{26}-a_{27}$
5. 5.
$a_{25}+2a_{27}-1$
6. 6.
$a_{23}-a_{24}$
7. 7.
$a_{22}+2a_{24}-1$
8. 8.
$a_{20}-a_{21}$
9. 9.
$a_{19}+2a_{21}-1$
10. 10.
$a_{8}-a_{9}$
11. 11.
$a_{7}+2a_{9}-1$
12. 12.
$a_{5}-a_{6}$
13. 13.
$a_{4}+2a_{6}-1$
14. 14.
$a_{2}-a_{3}$
15. 15.
$a_{1}+2a_{3}-1$
16. 16.
$3a_{9}*a_{30}+a_{9}-a_{24}-a_{33}$
17. 17.
$a_{24}*a_{29}+2a_{24}*a_{30}+a_{29}*a_{33}-a_{30}*a_{33}-a_{33}$
18. 18.
$3a_{9}*a_{29}-2a_{9}+2a_{24}-a_{33}$
19. 19.
$a_{6}*a_{29}+2a_{6}*a_{30}-a_{6}+a_{21}-a_{27}$
20. 20.
$a_{3}*a_{29}+2a_{3}*a_{30}-3a_{6}*a_{33}$
21. 21.
$3a_{6}*a_{24}-3a_{3}*a_{30}+3a_{6}*a_{33}-a_{3}$
22. 22.
$3a_{9}*a_{21}-3a_{9}*a_{27}+3a_{6}*a_{33}-a_{3}$
23. 23.
$3a_{6}*a_{9}-a_{3}$
* •
$N=1211$
1. 1.
$a_{38}-a_{39}$
2. 2.
$a_{37}+2a_{39}-1$
3. 3.
$a_{35}-a_{36}$
4. 4.
$a_{34}+2a_{36}-1$
5. 5.
$a_{28}+a_{29}+a_{30}-1$
6. 6.
$a_{14}-a_{15}$
7. 7.
$a_{13}+2a_{15}-1$
8. 8.
$a_{15}*a_{29}-a_{15}*a_{30}+a_{36}-a_{39}$
* •
$N=1212$
1. 1.
$a_{38}-a_{39}$
2. 2.
$a_{37}+2a_{39}-1$
3. 3.
$a_{35}-a_{36}$
4. 4.
$a_{34}+2a_{36}-1$
5. 5.
$a_{32}-a_{33}$
6. 6.
$a_{31}+2a_{33}-1$
7. 7.
$a_{28}+a_{29}+a_{30}-1$
8. 8.
$a_{23}-a_{24}$
9. 9.
$a_{22}+2a_{24}-1$
10. 10.
$a_{17}-a_{18}$
11. 11.
$a_{16}+2a_{18}-1$
12. 12.
$a_{14}-a_{15}$
13. 13.
$a_{13}+2a_{15}-1$
14. 14.
$a_{8}-a_{9}$
15. 15.
$a_{7}+2a_{9}-1$
16. 16.
$a_{18}*a_{30}-a_{15}*a_{33}-a_{9}*a_{36}+a_{9}*a_{39}$
17. 17.
$3a_{9}*a_{30}+a_{9}-a_{24}-a_{33}$
18. 18.
$a_{24}*a_{29}+2a_{24}*a_{30}+a_{29}*a_{33}-a_{30}*a_{33}-a_{33}$
19. 19.
$a_{18}*a_{29}-a_{15}*a_{33}+2a_{9}*a_{36}-2a_{9}*a_{39}$
20. 20.
$a_{15}*a_{29}-a_{15}*a_{30}+a_{36}-a_{39}$
21. 21.
$3a_{9}*a_{29}-2a_{9}+2a_{24}-a_{33}$
22. 22.
$3a_{15}*a_{24}-3a_{9}*a_{36}+3a_{9}*a_{39}-a_{18}$
23. 23.
$3a_{9}*a_{15}-a_{18}$
* •
$N=1221$
1. 1.
$a_{38}-a_{39}$
2. 2.
$a_{37}+2a_{39}-1$
3. 3.
$a_{35}-a_{36}$
4. 4.
$a_{34}+2a_{36}-1$
5. 5.
$a_{28}+a_{29}+a_{30}-1$
6. 6.
$a_{26}-a_{27}$
7. 7.
$a_{25}+2a_{27}-1$
8. 8.
$a_{20}-a_{21}$
9. 9.
$a_{19}+2a_{21}-1$
10. 10.
$a_{14}-a_{15}$
11. 11.
$a_{13}+2a_{15}-1$
12. 12.
$a_{11}-a_{12}$
13. 13.
$a_{10}+2a_{12}-1$
14. 14.
$a_{5}-a_{6}$
15. 15.
$a_{4}+2a_{6}-1$
16. 16.
$a_{15}*a_{29}-a_{15}*a_{30}+a_{36}-a_{39}$
17. 17.
$a_{12}*a_{29}-a_{12}*a_{30}+3a_{6}*a_{36}-3a_{6}*a_{39}$
18. 18.
$a_{6}*a_{29}+2a_{6}*a_{30}-a_{6}+a_{21}-a_{27}$
19. 19.
$3a_{15}*a_{21}-3a_{15}*a_{27}+3a_{12}*a_{30}-3a_{6}*a_{36}+3a_{6}*a_{39}-a_{12}$
20. 20.
$3a_{6}*a_{15}-a_{12}$
21. 21.
$a_{21}*a_{29}*a_{36}+2a_{21}*a_{30}*a_{36}-a_{27}*a_{29}*a_{39}+a_{27}*a_{30}*a_{39}-a_{15}*a_{27}+a_{12}*a_{30}-a_{6}*a_{36}-a_{21}*a_{36}-a_{27}*a_{36}+2a_{21}*a_{39}$
* •
$N=1222$
1. 1.
$a_{38}-a_{39}$
2. 2.
$a_{37}+2a_{39}-1$
3. 3.
$a_{35}-a_{36}$
4. 4.
$a_{34}+2a_{36}-1$
5. 5.
$a_{32}-a_{33}$
6. 6.
$a_{31}+2a_{33}-1$
7. 7.
$a_{28}+a_{29}+a_{30}-1$
8. 8.
$a_{26}-a_{27}$
9. 9.
$a_{25}+2a_{27}-1$
10. 10.
$a_{23}-a_{24}$
11. 11.
$a_{22}+2a_{24}-1$
12. 12.
$a_{20}-a_{21}$
13. 13.
$a_{19}+2a_{21}-1$
14. 14.
$a_{17}-a_{18}$
15. 15.
$a_{16}+2a_{18}-1$
16. 16.
$a_{14}-a_{15}$
17. 17.
$a_{13}+2a_{15}-1$
18. 18.
$a_{11}-a_{12}$
19. 19.
$a_{10}+2a_{12}-1$
20. 20.
$a_{8}-a_{9}$
21. 21.
$a_{7}+2a_{9}-1$
22. 22.
$a_{5}-a_{6}$
23. 23.
$a_{4}+2a_{6}-1$
24. 24.
$a_{2}-a_{3}$
25. 25.
$a_{1}+2a_{3}-1$
26. 26.
$a_{18}*a_{30}-a_{15}*a_{33}-a_{9}*a_{36}+a_{9}*a_{39}$
27. 27.
$3a_{9}*a_{30}+a_{9}-a_{24}-a_{33}$
28. 28.
$a_{24}*a_{29}+2a_{24}*a_{30}+a_{29}*a_{33}-a_{30}*a_{33}-a_{33}$
29. 29.
$a_{18}*a_{29}-a_{15}*a_{33}+2a_{9}*a_{36}-2a_{9}*a_{39}$
30. 30.
$a_{15}*a_{29}-a_{15}*a_{30}+a_{36}-a_{39}$
31. 31.
$a_{12}*a_{29}-a_{12}*a_{30}+3a_{6}*a_{36}-3a_{6}*a_{39}$
32. 32.
$3a_{9}*a_{29}-2a_{9}+2a_{24}-a_{33}$
33. 33.
$a_{6}*a_{29}+2a_{6}*a_{30}-a_{6}+a_{21}-a_{27}$
34. 34.
$a_{3}*a_{29}+2a_{3}*a_{30}-3a_{6}*a_{33}$
35. 35.
$3a_{15}*a_{24}-3a_{9}*a_{36}+3a_{9}*a_{39}-a_{18}$
36. 36.
$3a_{6}*a_{24}-3a_{3}*a_{30}+3a_{6}*a_{33}-a_{3}$
37. 37.
$a_{18}*a_{21}-a_{12}*a_{24}-a_{18}*a_{27}+a_{12}*a_{33}+a_{3}*a_{36}-a_{3}*a_{39}$
38. 38.
$3a_{15}*a_{21}-3a_{15}*a_{27}+3a_{12}*a_{30}-3a_{6}*a_{36}+3a_{6}*a_{39}-a_{12}$
39. 39.
$3a_{9}*a_{21}-3a_{9}*a_{27}+3a_{6}*a_{33}-a_{3}$
40. 40.
$a_{6}*a_{18}-a_{12}*a_{24}+a_{3}*a_{36}-a_{3}*a_{39}$
41. 41.
$3a_{9}*a_{15}-a_{18}$
42. 42.
$3a_{6}*a_{15}-a_{12}$
43. 43.
$a_{3}*a_{15}-a_{12}*a_{24}+a_{3}*a_{36}-a_{3}*a_{39}$
44. 44.
$a_{9}*a_{12}-a_{12}*a_{24}+a_{3}*a_{36}-a_{3}*a_{39}$
45. 45.
$3a_{6}*a_{9}-a_{3}$
46. 46.
$a_{21}*a_{29}*a_{36}+2a_{21}*a_{30}*a_{36}-a_{27}*a_{29}*a_{39}+a_{27}*a_{30}*a_{39}-a_{15}*a_{27}+a_{12}*a_{30}-a_{6}*a_{36}-a_{21}*a_{36}-a_{27}*a_{36}+2a_{21}*a_{39}$
47. 47.
$6a_{9}*a_{27}*a_{36}-3a_{6}*a_{33}*a_{36}-3a_{21}*a_{33}*a_{36}-3a_{9}*a_{27}*a_{39}-3a_{24}*a_{27}*a_{39}+6a_{6}*a_{33}*a_{39}+a_{18}*a_{27}-a_{12}*a_{33}+a_{3}*a_{36}-a_{3}*a_{39}$
* •
$N=2111$
1. 1.
$a_{53}-a_{54}$
2. 2.
$a_{52}+2a_{54}-1$
3. 3.
$a_{50}-a_{51}$
4. 4.
$a_{49}+2a_{51}-1$
5. 5.
$a_{44}-a_{45}$
6. 6.
$a_{43}+2a_{45}-1$
7. 7.
$a_{28}+a_{29}+a_{30}-1$
* •
$N=2112$
1. 1.
$a_{53}-a_{54}$
2. 2.
$a_{52}+2a_{54}-1$
3. 3.
$a_{50}-a_{51}$
4. 4.
$a_{49}+2a_{51}-1$
5. 5.
$a_{47}-a_{48}$
6. 6.
$a_{46}+2a_{48}-1$
7. 7.
$a_{44}-a_{45}$
8. 8.
$a_{43}+2a_{45}-1$
9. 9.
$a_{32}-a_{33}$
10. 10.
$a_{31}+2a_{33}-1$
11. 11.
$a_{28}+a_{29}+a_{30}-1$
12. 12.
$a_{23}-a_{24}$
13. 13.
$a_{22}+2a_{24}-1$
14. 14.
$a_{8}-a_{9}$
15. 15.
$a_{7}+2a_{9}-1$
16. 16.
$3a_{9}*a_{30}+a_{9}-a_{24}-a_{33}$
17. 17.
$a_{24}*a_{29}+2a_{24}*a_{30}+a_{29}*a_{33}-a_{30}*a_{33}-a_{33}$
18. 18.
$3a_{9}*a_{29}-2a_{9}+2a_{24}-a_{33}$
* •
$N=2121$
1. 1.
$a_{4}+2a_{6}-1$
2. 2.
$a_{5}-a_{6}$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{19}+2a_{21}-1$
5. 5.
$a_{20}-a_{21}$
6. 6.
$a_{25}+2a_{27}-1$
7. 7.
$a_{26}-a_{27}$
8. 8.
$a_{40}+2a_{42}-1$
9. 9.
$a_{41}-a_{42}$
10. 10.
$a_{43}+2a_{45}-1$
11. 11.
$a_{44}-a_{45}$
12. 12.
$a_{49}+2a_{51}-1$
13. 13.
$a_{50}-a_{51}$
14. 14.
$a_{52}+2a_{54}-1$
15. 15.
$a_{53}-a_{54}$
16. 16.
$a_{5}*a_{29}+2a_{5}*a_{30}-a_{5}+a_{20}-a_{26}$
17. 17.
$2a_{20}*a_{29}^{3}a_{41}+a_{26}*a_{29}^{3}a_{41}+3a_{20}*a_{29}^{2}a_{30}*a_{41}-3a_{20}*a_{29}*a_{30}^{2}a_{41}-3a_{26}*a_{29}*a_{30}^{2}a_{41}-2a_{20}*a_{30}^{3}a_{41}+2a_{26}*a_{30}^{3}a_{41}-3a_{20}*a_{26}*a_{29}^{2}a_{44}-3a_{26}^{2}a_{29}^{2}a_{44}-3a_{20}*a_{26}*a_{29}*a_{30}*a_{44}+6a_{26}^{2}a_{29}*a_{30}*a_{44}+6a_{20}*a_{26}*a_{30}^{2}a_{44}-3a_{26}^{2}a_{30}^{2}a_{44}-6a_{20}^{2}a_{29}^{2}a_{50}+3a_{20}*a_{26}*a_{29}^{2}a_{50}-15a_{20}^{2}a_{29}*a_{30}*a_{50}-6a_{20}*a_{26}*a_{29}*a_{30}*a_{50}-6a_{20}^{2}a_{30}^{2}a_{50}+3a_{20}*a_{26}*a_{30}^{2}a_{50}+3a_{20}*a_{26}*a_{29}^{2}a_{53}+3a_{20}*a_{26}*a_{29}*a_{30}*a_{53}-6a_{20}*a_{26}*a_{30}^{2}a_{53}-4a_{20}*a_{29}^{2}a_{41}-2a_{26}*a_{29}^{2}a_{41}-a_{20}*a_{29}*a_{30}*a_{41}+a_{26}*a_{29}*a_{30}*a_{41}+5a_{20}*a_{30}^{2}a_{41}+a_{26}*a_{30}^{2}a_{41}+3a_{20}^{2}a_{29}*a_{44}+6a_{20}*a_{26}*a_{29}*a_{44}+3a_{26}^{2}a_{29}*a_{44}+6a_{20}^{2}a_{30}*a_{44}+18a_{5}*a_{26}*a_{30}*a_{44}-6a_{20}*a_{26}*a_{30}*a_{44}-3a_{26}^{2}a_{30}*a_{44}+6a_{20}*a_{26}*a_{29}*a_{50}-9a_{5}*a_{20}*a_{30}*a_{50}+9a_{20}^{2}a_{30}*a_{50}-9a_{5}*a_{26}*a_{30}*a_{50}+12a_{20}*a_{26}*a_{30}*a_{50}-3a_{20}^{2}a_{29}*a_{53}-3a_{26}^{2}a_{29}*a_{53}-6a_{20}^{2}a_{30}*a_{53}+3a_{26}^{2}a_{30}*a_{53}+3a_{26}*a_{29}*a_{41}-3a_{26}*a_{30}*a_{41}-3a_{5}^{2}a_{44}+3a_{5}*a_{20}*a_{44}-3a_{5}*a_{26}*a_{44}-6a_{26}^{2}a_{44}+3a_{5}^{2}a_{50}-6a_{5}*a_{20}*a_{50}+3a_{20}^{2}a_{50}+6a_{5}*a_{26}*a_{50}-6a_{20}*a_{26}*a_{50}+3a_{5}*a_{20}*a_{53}-3a_{20}^{2}a_{53}-3a_{5}*a_{26}*a_{53}+6a_{20}*a_{26}*a_{53}$
* •
$N=2211$
1. 1.
$a_{13}+2a_{15}-1$
2. 2.
$a_{14}-a_{15}$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{34}+2a_{36}-1$
5. 5.
$a_{35}-a_{36}$
6. 6.
$a_{37}+2a_{39}-1$
7. 7.
$a_{38}-a_{39}$
8. 8.
$a_{43}+2a_{45}-1$
9. 9.
$a_{44}-a_{45}$
10. 10.
$a_{49}+2a_{51}-1$
11. 11.
$a_{50}-a_{51}$
12. 12.
$a_{52}+2a_{54}-1$
13. 13.
$a_{53}-a_{54}$
14. 14.
$a_{55}+2a_{57}-1$
15. 15.
$a_{56}-a_{57}$
16. 16.
$a_{14}*a_{29}-a_{14}*a_{30}+a_{35}-a_{38}$
17. 17.
$3a_{29}^{2}a_{35}*a_{38}*a_{44}+3a_{29}*a_{30}*a_{35}*a_{38}*a_{44}-6a_{30}^{2}a_{35}*a_{38}*a_{44}+3a_{29}^{2}a_{35}*a_{38}*a_{50}+12a_{29}*a_{30}*a_{35}*a_{38}*a_{50}+12a_{30}^{2}a_{35}*a_{38}*a_{50}-6a_{29}^{2}a_{38}^{2}a_{50}+3a_{29}*a_{30}*a_{38}^{2}a_{50}+3a_{30}^{2}a_{38}^{2}a_{50}-3a_{29}^{2}a_{35}^{2}a_{53}-12a_{29}*a_{30}*a_{35}^{2}a_{53}-12a_{30}^{2}a_{35}^{2}a_{53}-3a_{29}^{2}a_{35}*a_{38}*a_{53}-3a_{29}*a_{30}*a_{35}*a_{38}*a_{53}+6a_{30}^{2}a_{35}*a_{38}*a_{53}-a_{29}^{3}a_{35}*a_{56}-3a_{29}^{2}a_{30}*a_{35}*a_{56}+4a_{30}^{3}a_{35}*a_{56}-2a_{29}^{3}a_{38}*a_{56}-3a_{29}^{2}a_{30}*a_{38}*a_{56}+3a_{29}*a_{30}^{2}a_{38}*a_{56}+2a_{30}^{3}a_{38}*a_{56}+3a_{29}*a_{35}^{2}a_{44}+6a_{30}*a_{35}^{2}a_{44}-6a_{29}*a_{35}*a_{38}*a_{44}-3a_{30}*a_{35}*a_{38}*a_{44}+3a_{29}*a_{38}^{2}a_{44}-3a_{30}*a_{38}^{2}a_{44}-9a_{14}*a_{30}*a_{35}*a_{50}-9a_{14}*a_{30}*a_{38}*a_{50}-12a_{29}*a_{35}*a_{38}*a_{50}-6a_{30}*a_{35}*a_{38}*a_{50}+12a_{29}*a_{38}^{2}a_{50}+6a_{30}*a_{38}^{2}a_{50}+18a_{14}*a_{30}*a_{35}*a_{53}+3a_{29}*a_{35}^{2}a_{53}+6a_{30}*a_{35}^{2}a_{53}-9a_{30}*a_{35}*a_{38}*a_{53}-3a_{29}*a_{38}^{2}a_{53}+3a_{30}*a_{38}^{2}a_{53}+a_{29}^{2}a_{35}*a_{56}+a_{29}*a_{30}*a_{35}*a_{56}-2a_{30}^{2}a_{35}*a_{56}+2a_{29}^{2}a_{38}*a_{56}-a_{29}*a_{30}*a_{38}*a_{56}-a_{30}^{2}a_{38}*a_{56}-3a_{14}*a_{35}*a_{44}-3a_{35}^{2}a_{44}+3a_{14}*a_{38}*a_{44}+9a_{35}*a_{38}*a_{44}-6a_{38}^{2}a_{44}+3a_{14}^{2}a_{50}+6a_{14}*a_{35}*a_{50}-6a_{14}*a_{38}*a_{50}+3a_{35}*a_{38}*a_{50}-3a_{38}^{2}a_{50}-3a_{14}^{2}a_{53}-3a_{14}*a_{35}*a_{53}-6a_{35}^{2}a_{53}+3a_{14}*a_{38}*a_{53}+3a_{35}*a_{38}*a_{53}+3a_{38}^{2}a_{53}-2a_{29}*a_{35}*a_{56}-4a_{30}*a_{35}*a_{56}+2a_{29}*a_{38}*a_{56}+4a_{30}*a_{38}*a_{56}+2a_{35}*a_{56}-2a_{38}*a_{56}$
* •
$N=2212$
1. 1.
$a_{16}+2a_{18}-1$
2. 2.
$a_{17}-a_{18}$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{13}+2a_{15}-1$
5. 5.
$a_{14}-a_{15}$
6. 6.
$a_{31}+2a_{33}-1$
7. 7.
$a_{32}-a_{33}$
8. 8.
$a_{7}+2a_{9}-1$
9. 9.
$a_{8}-a_{9}$
10. 10.
$a_{34}+2a_{36}-1$
11. 11.
$a_{35}-a_{36}$
12. 12.
$a_{37}+2a_{39}-1$
13. 13.
$a_{38}-a_{39}$
14. 14.
$a_{22}+2a_{24}-1$
15. 15.
$a_{23}-a_{24}$
16. 16.
$a_{43}+2a_{45}-1$
17. 17.
$a_{44}-a_{45}$
18. 18.
$a_{46}+2a_{48}-1$
19. 19.
$a_{47}-a_{48}$
20. 20.
$a_{49}+2a_{51}-1$
21. 21.
$a_{50}-a_{51}$
22. 22.
$a_{52}+2a_{54}-1$
23. 23.
$a_{53}-a_{54}$
24. 24.
$a_{55}+2a_{57}-1$
25. 25.
$a_{56}-a_{57}$
26. 26.
$a_{17}*a_{30}-a_{14}*a_{32}-a_{8}*a_{35}+a_{8}*a_{38}$
27. 27.
$3a_{8}*a_{30}+a_{8}-a_{23}-a_{32}$
28. 28.
$a_{23}*a_{29}+2a_{23}*a_{30}+a_{29}*a_{32}-a_{30}*a_{32}-a_{32}$
29. 29.
$a_{17}*a_{29}-a_{14}*a_{32}+2a_{8}*a_{35}-2a_{8}*a_{38}$
30. 30.
$a_{14}*a_{29}-a_{14}*a_{30}+a_{35}-a_{38}$
31. 31.
$3a_{8}*a_{29}-2a_{8}+2a_{23}-a_{32}$
32. 32.
$3a_{14}*a_{23}-3a_{8}*a_{35}+3a_{8}*a_{38}-a_{17}$
33. 33.
$3a_{8}*a_{14}-a_{17}$
34. 34.
$3a_{32}*a_{38}*a_{44}-2a_{29}*a_{38}*a_{47}-a_{30}*a_{38}*a_{47}-3a_{32}*a_{38}*a_{50}+3a_{32}*a_{35}*a_{53}+3a_{8}*a_{38}*a_{53}-3a_{23}*a_{38}*a_{53}+a_{29}*a_{32}*a_{56}-a_{30}*a_{32}*a_{56}-a_{17}*a_{44}+a_{14}*a_{47}-a_{35}*a_{47}+a_{38}*a_{47}+a_{17}*a_{50}-a_{17}*a_{53}-a_{8}*a_{56}+a_{23}*a_{56}$
35. 35.
$3a_{8}*a_{35}*a_{44}-3a_{32}*a_{35}*a_{44}-3a_{8}*a_{38}*a_{44}+a_{29}*a_{35}*a_{47}+2a_{30}*a_{35}*a_{47}+3a_{8}*a_{35}*a_{50}-3a_{23}*a_{38}*a_{50}+a_{29}*a_{32}*a_{56}-a_{30}*a_{32}*a_{56}+a_{17}*a_{44}-a_{14}*a_{47}-a_{35}*a_{47}+a_{38}*a_{47}-a_{8}*a_{56}+a_{23}*a_{56}$
36. 36.
$3a_{17}*a_{32}*a_{44}^{2}-6a_{14}*a_{32}*a_{44}*a_{47}+3a_{32}*a_{35}*a_{44}*a_{47}+3a_{14}*a_{30}*a_{47}^{2}-a_{29}*a_{35}*a_{47}^{2}-2a_{30}*a_{35}*a_{47}^{2}-3a_{8}*a_{17}*a_{44}*a_{50}-3a_{17}*a_{32}*a_{44}*a_{50}+3a_{14}*a_{32}*a_{47}*a_{50}-3a_{8}*a_{35}*a_{47}*a_{50}+3a_{23}*a_{38}*a_{47}*a_{50}+3a_{8}*a_{17}*a_{50}^{2}-3a_{17}*a_{23}*a_{44}*a_{53}+3a_{8}*a_{35}*a_{47}*a_{53}-3a_{8}*a_{38}*a_{47}*a_{53}-3a_{17}*a_{23}*a_{50}*a_{53}+6a_{8}*a_{32}*a_{44}*a_{56}-a_{29}*a_{32}*a_{47}*a_{56}+a_{30}*a_{32}*a_{47}*a_{56}-3a_{8}^{2}*a_{50}*a_{56}+3a_{8}*a_{23}*a_{50}*a_{56}-3a_{8}*a_{32}*a_{50}*a_{56}+3a_{8}^{2}*a_{53}*a_{56}-3a_{8}*a_{23}*a_{53}*a_{56}+3a_{8}*a_{32}*a_{53}*a_{56}-a_{35}*a_{47}^{2}+a_{38}*a_{47}^{2}+a_{17}*a_{47}*a_{50}+a_{17}*a_{47}*a_{53}-2a_{32}*a_{47}*a_{56}$
37. 37.
$9a_{8}*a_{23}*a_{38}*a_{44}*a_{50}-9a_{8}*a_{23}*a_{38}*a_{50}^{2}+9a_{23}*a_{32}*a_{35}*a_{44}*a_{53}+9a_{8}*a_{23}*a_{38}*a_{44}*a_{53}+3a_{29}*a_{32}*a_{35}*a_{47}*a_{53}-3a_{30}*a_{32}*a_{35}*a_{47}*a_{53}+9a_{23}^{2}*a_{38}*a_{50}*a_{53}+9a_{23}*a_{30}*a_{32}*a_{53}*a_{56}+3a_{29}*a_{32}^{2}*a_{53}*a_{56}-3a_{30}*a_{32}^{2}*a_{53}*a_{56}-3a_{17}*a_{23}*a_{44}^{2}-3a_{23}*a_{35}*a_{44}*a_{47}+6a_{32}*a_{35}*a_{44}*a_{47}-3a_{29}*a_{35}*a_{47}^{2}-3a_{30}*a_{35}*a_{47}^{2}+3a_{17}*a_{23}*a_{44}*a_{50}-9a_{8}*a_{35}*a_{47}*a_{50}+3a_{8}*a_{38}*a_{47}*a_{50}+3a_{23}*a_{38}*a_{47}*a_{50}-3a_{17}*a_{23}*a_{44}*a_{53}+3a_{8}*a_{35}*a_{47}*a_{53}-3a_{32}*a_{35}*a_{47}*a_{53}-3a_{8}*a_{38}*a_{47}*a_{53}-3a_{23}*a_{38}*a_{47}*a_{53}-6a_{8}*a_{23}*a_{44}*a_{56}-3a_{23}*a_{30}*a_{47}*a_{56}-3a_{29}*a_{32}*a_{47}*a_{56}+3a_{30}*a_{32}*a_{47}*a_{56}+6a_{8}*a_{23}*a_{50}*a_{56}-3a_{23}^{2}*a_{50}*a_{56}-3a_{8}*a_{23}*a_{53}*a_{56}-3a_{32}^{2}*a_{53}*a_{56}+a_{14}*a_{47}^{2}+2a_{35}*a_{47}^{2}-a_{38}*a_{47}^{2}-a_{17}*a_{47}*a_{50}+a_{17}*a_{47}*a_{53}+2a_{8}*a_{47}*a_{56}+a_{32}*a_{47}*a_{56}$
38. 38.
$9a_{23}*a_{32}*a_{35}*a_{44}^{2}+9a_{23}*a_{30}*a_{35}*a_{44}*a_{47}+a_{29}^{2}*a_{35}*a_{47}^{2}+4a_{29}*a_{30}*a_{35}*a_{47}^{2}-5a_{30}^{2}*a_{35}*a_{47}^{2}-18a_{23}*a_{32}*a_{35}*a_{44}*a_{50}-6a_{29}*a_{32}*a_{35}*a_{47}*a_{50}+6a_{30}*a_{32}*a_{35}*a_{47}*a_{50}+27a_{23}*a_{30}*a_{38}*a_{47}*a_{50}+9a_{29}*a_{32}*a_{38}*a_{47}*a_{50}-9a_{30}*a_{32}*a_{38}*a_{47}*a_{50}+18a_{8}*a_{23}*a_{35}*a_{50}^{2}-9a_{8}*a_{23}*a_{38}*a_{50}^{2}-9a_{23}^{2}*a_{38}*a_{50}^{2}-9a_{23}^{2}*a_{35}*a_{44}*a_{53}+9a_{23}*a_{32}*a_{35}*a_{44}*a_{53}+9a_{8}*a_{23}*a_{38}*a_{44}*a_{53}+9a_{23}*a_{30}*a_{35}*a_{47}*a_{53}+6a_{29}*a_{32}*a_{35}*a_{47}*a_{53}-6a_{30}*a_{32}*a_{35}*a_{47}*a_{53}-9a_{23}^{2}*a_{35}*a_{50}*a_{53}+9a_{23}^{2}*a_{38}*a_{50}*a_{53}+9a_{23}*a_{30}*a_{32}*a_{44}*a_{56}+3a_{29}*a_{32}^{2}*a_{44}*a_{56}-3a_{30}*a_{32}^{2}*a_{44}*a_{56}+9a_{23}*a_{30}^{2}*a_{47}*a_{56}+a_{29}^{2}*a_{32}*a_{47}*a_{56}+a_{29}*a_{30}*a_{32}*a_{47}*a_{56}-2a_{30}^{2}*a_{32}*a_{47}*a_{56}+9a_{23}^{2}*a_{30}*a_{50}*a_{56}-27a_{23}*a_{30}*a_{32}*a_{50}*a_{56}-9a_{29}*a_{32}^{2}*a_{50}*a_{56}+9a_{30}*a_{32}^{2}*a_{50}*a_{56}-9a_{23}^{2}*a_{30}*a_{53}*a_{56}+18a_{23}*a_{30}*a_{32}*a_{53}*a_{56}+6a_{29}*a_{32}^{2}*a_{53}*a_{56}-6a_{30}*a_{32}^{2}*a_{53}*a_{56}-3a_{17}*a_{23}*a_{44}^{2}-9a_{23}*a_{35}*a_{44}*a_{47}-3a_{32}*a_{35}*a_{44}*a_{47}+9a_{23}*a_{38}*a_{44}*a_{47}-3a_{29}*a_{35}*a_{47}^{2}+3a_{29}*a_{38}*a_{47}^{2}-3a_{30}*a_{38}*a_{47}^{2}+6a_{17}*a_{23}*a_{44}*a_{50}-3a_{8}*a_{35}*a_{47}*a_{50}-9a_{23}*a_{35}*a_{47}*a_{50}+6a_{32}*a_{35}*a_{47}*a_{50}+6a_{8}*a_{38}*a_{47}*a_{50}+6a_{23}*a_{38}*a_{47}*a_{50}-9a_{32}*a_{38}*a_{47}*a_{50}-3a_{17}*a_{23}*a_{44}*a_{53}+3a_{8}*a_{35}*a_{47}*a_{53}+3a_{23}*a_{35}*a_{47}*a_{53}-6a_{32}*a_{35}*a_{47}*a_{53}-3a_{8}*a_{38}*a_{47}*a_{53}-3a_{23}*a_{38}*a_{47}*a_{53}-3a_{8}*a_{23}*a_{44}*a_{56}+3a_{23}^{2}*a_{44}*a_{56}-3a_{32}^{2}*a_{44}*a_{56}-12a_{23}*a_{30}*a_{47}*a_{56}-4a_{29}*a_{32}*a_{47}*a_{56}+a_{30}*a_{32}*a_{47}*a_{56}-3a_{23}*a_{32}*a_{50}*a_{56}+9a_{32}^{2}*a_{50}*a_{56}-3a_{8}*a_{23}*a_{53}*a_{56}+3a_{23}^{2}*a_{53}*a_{56}+3a_{23}*a_{32}*a_{53}*a_{56}-6a_{32}^{2}*a_{53}*a_{56}+a_{14}*a_{47}^{2}+4a_{35}*a_{47}^{2}-4a_{38}*a_{47}^{2}-2a_{17}*a_{47}*a_{50}+a_{17}*a_{47}*a_{53}-a_{8}*a_{47}*a_{56}+a_{23}*a_{47}*a_{56}+4a_{32}*a_{47}*a_{56}$
39. 39.
$3a_{14}*a_{17}^{2}*a_{44}^{2}-6a_{14}^{2}*a_{17}*a_{44}*a_{47}+3a_{14}*a_{17}*a_{35}*a_{44}*a_{47}+3a_{14}^{3}*a_{47}^{2}-3a_{14}^{2}*a_{35}*a_{47}^{2}-3a_{14}*a_{17}^{2}*a_{44}*a_{50}-3a_{17}^{2}*a_{38}*a_{44}*a_{50}+3a_{14}^{2}*a_{17}*a_{47}*a_{50}+3a_{14}*a_{17}*a_{38}*a_{47}*a_{50}+3a_{17}^{2}*a_{38}*a_{50}^{2}-3a_{17}^{2}*a_{35}*a_{44}*a_{53}+3a_{14}*a_{17}*a_{35}*a_{47}*a_{53}-3a_{17}^{2}*a_{35}*a_{50}*a_{53}+3a_{8}*a_{17}*a_{35}*a_{50}*a_{56}-3a_{8}*a_{17}*a_{38}*a_{50}*a_{56}-3a_{8}*a_{17}*a_{35}*a_{53}*a_{56}+3a_{8}*a_{17}*a_{38}*a_{53}*a_{56}+2a_{17}^{2}*a_{44}*a_{56}-2a_{14}*a_{17}*a_{47}*a_{56}+a_{17}*a_{35}*a_{47}*a_{56}-a_{17}*a_{38}*a_{47}*a_{56}-a_{17}^{2}*a_{50}*a_{56}+a_{17}^{2}*a_{53}*a_{56}$
40. 40.
$3a_{29}^{2}*a_{35}*a_{38}*a_{44}+3a_{29}*a_{30}*a_{35}*a_{38}*a_{44}-6a_{30}^{2}*a_{35}*a_{38}*a_{44}+3a_{29}^{2}*a_{35}*a_{38}*a_{50}+12a_{29}*a_{30}*a_{35}*a_{38}*a_{50}+12a_{30}^{2}*a_{35}*a_{38}*a_{50}-6a_{29}^{2}*a_{38}^{2}*a_{50}+3a_{29}*a_{30}*a_{38}^{2}*a_{50}+3a_{30}^{2}*a_{38}^{2}*a_{50}-3a_{29}^{2}*a_{35}^{2}*a_{53}-12a_{29}*a_{30}*a_{35}^{2}*a_{53}-12a_{30}^{2}*a_{35}^{2}*a_{53}-3a_{29}^{2}*a_{35}*a_{38}*a_{53}-3a_{29}*a_{30}*a_{35}*a_{38}*a_{53}+6a_{30}^{2}*a_{35}*a_{38}*a_{53}-a_{29}^{3}*a_{35}*a_{56}-3a_{29}^{2}*a_{30}*a_{35}*a_{56}+4a_{30}^{3}*a_{35}*a_{56}-2a_{29}^{3}*a_{38}*a_{56}-3a_{29}^{2}*a_{30}*a_{38}*a_{56}+3a_{29}*a_{30}^{2}*a_{38}*a_{56}+2a_{30}^{3}*a_{38}*a_{56}+3a_{29}*a_{35}^{2}*a_{44}+6a_{30}*a_{35}^{2}*a_{44}-6a_{29}*a_{35}*a_{38}*a_{44}-3a_{30}*a_{35}*a_{38}*a_{44}+3a_{29}*a_{38}^{2}*a_{44}-3a_{30}*a_{38}^{2}*a_{44}-9a_{14}*a_{30}*a_{35}*a_{50}-9a_{14}*a_{30}*a_{38}*a_{50}-12a_{29}*a_{35}*a_{38}*a_{50}-6a_{30}*a_{35}*a_{38}*a_{50}+12a_{29}*a_{38}^{2}*a_{50}+6a_{30}*a_{38}^{2}*a_{50}+18a_{14}*a_{30}*a_{35}*a_{53}+3a_{29}*a_{35}^{2}*a_{53}+6a_{30}*a_{35}^{2}*a_{53}-9a_{30}*a_{35}*a_{38}*a_{53}-3a_{29}*a_{38}^{2}*a_{53}+3a_{30}*a_{38}^{2}*a_{53}+a_{29}^{2}*a_{35}*a_{56}+a_{29}*a_{30}*a_{35}*a_{56}-2a_{30}^{2}*a_{35}*a_{56}+2a_{29}^{2}*a_{38}*a_{56}-a_{29}*a_{30}*a_{38}*a_{56}-a_{30}^{2}*a_{38}*a_{56}-3a_{14}*a_{35}*a_{44}-3a_{35}^{2}*a_{44}+3a_{14}*a_{38}*a_{44}+9a_{35}*a_{38}*a_{44}-6a_{38}^{2}*a_{44}+3a_{14}^{2}*a_{50}+6a_{14}*a_{35}*a_{50}-6a_{14}*a_{38}*a_{50}+3a_{35}*a_{38}*a_{50}-3a_{38}^{2}*a_{50}-3a_{14}^{2}*a_{53}-3a_{14}*a_{35}*a_{53}-6a_{35}^{2}*a_{53}+3a_{14}*a_{38}*a_{53}+3a_{35}*a_{38}*a_{53}+3a_{38}^{2}*a_{53}-2a_{29}*a_{35}*a_{56}-4a_{30}*a_{35}*a_{56}+2a_{29}*a_{38}*a_{56}+4a_{30}*a_{38}*a_{56}+2a_{35}*a_{56}-2a_{38}*a_{56}$
* •
$N=2122$ (subset)
1. 1.
$a_{7}+2a_{9}-1$
2. 2.
$a_{8}-a_{9}$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{22}+2a_{24}-1$
5. 5.
$a_{23}-a_{24}$
6. 6.
$a_{31}+2a_{33}-1$
7. 7.
$a_{32}-a_{33}$
8. 8.
$a_{4}+2a_{6}-1$
9. 9.
$a_{5}-a_{6}$
10. 10.
$a_{19}+2a_{21}-1$
11. 11.
$a_{20}-a_{21}$
12. 12.
$a_{25}+2a_{27}-1$
13. 13.
$a_{26}-a_{27}$
14. 14.
$a_{1}+2a_{3}-1$
15. 15.
$a_{2}-a_{3}$
16. 16.
$3a_{8}*a_{30}+a_{8}-a_{23}-a_{32}$
17. 17.
$a_{23}*a_{29}+2a_{23}*a_{30}+a_{29}*a_{32}-a_{30}*a_{32}-a_{32}$
18. 18.
$3a_{8}*a_{29}-2a_{8}+2a_{23}-a_{32}$
19. 19.
$a_{5}*a_{29}+2a_{5}*a_{30}-a_{5}+a_{20}-a_{26}$
20. 20.
$a_{2}*a_{29}+2a_{2}*a_{30}-3a_{5}*a_{32}$
21. 21.
$3a_{5}*a_{23}-3a_{2}*a_{30}+3a_{5}*a_{32}-a_{2}$
22. 22.
$3a_{8}*a_{20}-3a_{8}*a_{26}+3a_{5}*a_{32}-a_{2}$
23. 23.
$3a_{5}*a_{8}-a_{2}$
* •
$N=2221$ (subset)
1. 1.
$a_{13}+2a_{15}-1$
2. 2.
$a_{14}-a_{15}$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{34}+2a_{36}-1$
5. 5.
$a_{35}-a_{36}$
6. 6.
$a_{37}+2a_{39}-1$
7. 7.
$a_{38}-a_{39}$
8. 8.
$a_{10}+2a_{12}-1$
9. 9.
$a_{11}-a_{12}$
10. 10.
$a_{4}+2a_{6}-1$
11. 11.
$a_{5}-a_{6}$
12. 12.
$a_{19}+2a_{21}-1$
13. 13.
$a_{20}-a_{21}$
14. 14.
$a_{25}+2a_{27}-1$
15. 15.
$a_{26}-a_{27}$
16. 16.
$a_{14}*a_{29}-a_{14}*a_{30}+a_{35}-a_{38}$
17. 17.
$a_{11}*a_{29}-a_{11}*a_{30}+3a_{5}*a_{35}-3a_{5}*a_{38}$
18. 18.
$a_{5}*a_{29}+2a_{5}*a_{30}-a_{5}+a_{20}-a_{26}$
19. 19.
$3a_{14}*a_{20}-3a_{14}*a_{26}+3a_{11}*a_{30}-3a_{5}*a_{35}+3a_{5}*a_{38}-a_{11}$
20. 20.
$3a_{5}*a_{14}-a_{11}$
21. 21.
$a_{20}*a_{29}*a_{35}+2a_{20}*a_{30}*a_{35}-a_{26}*a_{29}*a_{38}+a_{26}*a_{30}*a_{38}-a_{14}*a_{26}+a_{11}*a_{30}-a_{5}*a_{35}-a_{20}*a_{35}-a_{26}*a_{35}+2a_{20}*a_{38}$
* •
$N=2222$ (subset)
1. 1.
$a_{16}+2a_{18}-1$
2. 2.
$a_{17}-a_{18}$
3. 3.
$a_{28}+a_{29}+a_{30}-1$
4. 4.
$a_{13}+2a_{15}-1$
5. 5.
$a_{14}-a_{15}$
6. 6.
$a_{31}+2a_{33}-1$
7. 7.
$a_{32}-a_{33}$
8. 8.
$a_{7}+2a_{9}-1$
9. 9.
$a_{8}-a_{9}$
10. 10.
$a_{34}+2a_{36}-1$
11. 11.
$a_{35}-a_{36}$
12. 12.
$a_{37}+2a_{39}-1$
13. 13.
$a_{38}-a_{39}$
14. 14.
$a_{22}+2a_{24}-1$
15. 15.
$a_{23}-a_{24}$
16. 16.
$a_{10}+2a_{12}-1$
17. 17.
$a_{11}-a_{12}$
18. 18.
$a_{4}+2a_{6}-1$
19. 19.
$a_{5}-a_{6}$
20. 20.
$a_{19}+2a_{21}-1$
21. 21.
$a_{20}-a_{21}$
22. 22.
$a_{25}+2a_{27}-1$
23. 23.
$a_{26}-a_{27}$
24. 24.
$a_{1}+2a_{3}-1$
25. 25.
$a_{2}-a_{3}$
26. 26.
$a_{17}*a_{30}-a_{14}*a_{32}-a_{8}*a_{35}+a_{8}*a_{38}$
27. 27.
$3a_{8}*a_{30}+a_{8}-a_{23}-a_{32}$
28. 28.
$a_{23}*a_{29}+2a_{23}*a_{30}+a_{29}*a_{32}-a_{30}*a_{32}-a_{32}$
29. 29.
$a_{17}*a_{29}-a_{14}*a_{32}+2a_{8}*a_{35}-2a_{8}*a_{38}$
30. 30.
$a_{14}*a_{29}-a_{14}*a_{30}+a_{35}-a_{38}$
31. 31.
$a_{11}*a_{29}-a_{11}*a_{30}+3a_{5}*a_{35}-3a_{5}*a_{38}$
32. 32.
$3a_{8}*a_{29}-2a_{8}+2a_{23}-a_{32}$
33. 33.
$a_{5}*a_{29}+2a_{5}*a_{30}-a_{5}+a_{20}-a_{26}$
34. 34.
$a_{2}*a_{29}+2a_{2}*a_{30}-3a_{5}*a_{32}$
35. 35.
$3a_{14}*a_{23}-3a_{8}*a_{35}+3a_{8}*a_{38}-a_{17}$
36. 36.
$3a_{5}*a_{23}-3a_{2}*a_{30}+3a_{5}*a_{32}-a_{2}$
37. 37.
$a_{17}*a_{20}-a_{11}*a_{23}-a_{17}*a_{26}+a_{11}*a_{32}+a_{2}*a_{35}-a_{2}*a_{38}$
38. 38.
$3a_{14}*a_{20}-3a_{14}*a_{26}+3a_{11}*a_{30}-3a_{5}*a_{35}+3a_{5}*a_{38}-a_{11}$
39. 39.
$3a_{8}*a_{20}-3a_{8}*a_{26}+3a_{5}*a_{32}-a_{2}$
40. 40.
$a_{5}*a_{17}-a_{11}*a_{23}+a_{2}*a_{35}-a_{2}*a_{38}$
41. 41.
$3a_{8}*a_{14}-a_{17}$
42. 42.
$3a_{5}*a_{14}-a_{11}$
43. 43.
$a_{2}*a_{14}-a_{11}*a_{23}+a_{2}*a_{35}-a_{2}*a_{38}$
44. 44.
$a_{8}*a_{11}-a_{11}*a_{23}+a_{2}*a_{35}-a_{2}*a_{38}$
45. 45.
$3a_{5}*a_{8}-a_{2}$
46. 46.
$a_{20}*a_{29}*a_{35}+2a_{20}*a_{30}*a_{35}-a_{26}*a_{29}*a_{38}+a_{26}*a_{30}*a_{38}-a_{14}*a_{26}+a_{11}*a_{30}-a_{5}*a_{35}-a_{20}*a_{35}-a_{26}*a_{35}+2a_{20}*a_{38}$
47. 47.
$6a_{8}*a_{26}*a_{35}-3a_{5}*a_{32}*a_{35}-3a_{20}*a_{32}*a_{35}-3a_{8}*a_{26}*a_{38}-3a_{23}*a_{26}*a_{38}+6a_{5}*a_{32}*a_{38}+a_{17}*a_{26}-a_{11}*a_{32}+a_{2}*a_{35}-a_{2}*a_{38}$
## Appendix C Simulation study
Figure 13: Invariant score (y-axis) measured as $L_{2}$-norm of the
phylogenetic invariants for each network (x-axis) for the true network (red)
and its symmetric network (inverted clades $n_{1}$ and $n_{2}$, in blue) on
the cases of true and Gaussian-perturbed CFs. Each panel corresponds to a type
of simulation: using true concordance factors (left) and using concordance
factors with added Gaussian noise (with increasing standard deviation for
noise from left to right). Both the true and symmetric networks have invariant
score close to zero, and are thus, easy to distinguish from wrong networks
(whose invariant score is far from zero). As the noise increases, the
invariant scores moves away from zero, but still within $10^{-4}$. Figure 14:
Invariant score (y-axis) measured as $L_{2}$-norm of the phylogenetic
invariants for each network (x-axis) for the true network (red) and its
symmetric network (inverted clades $n_{1}$ and $n_{2}$, in blue) for the case
of true simulated gene trees (“g.t."). Each panel corresponds to a number of
simulated gene trees from 100 (left) to 10,000 (right). With true and
perturbed concordance factors (top left and top right, respectively). As the
number of gene trees increases, the invariant scores of the true and symmetric
networks converge to zero, and they are thus, easy to distinguish from wrong
networks (whose invariant score is far from zero). Figure 15: Invariant score
(y-axis) measured as $L_{2}$-norm of the phylogenetic invariants for each
network (x-axis) for the true network (red) and its symmetric network
(inverted clades $n_{1}$ and $n_{2}$, in blue) for the case of estimated gene
trees. Each panel corresponds to a number of gene trees (g.t. from 1000 to
10,000) and sequence length ($L$ from 500 to 2000). As the number of gene
trees increases, the invariant scores of the true and symmetric networks
converge to zero, and they are thus, easy to distinguish from wrong networks
(whose invariant score is far from zero).
|
# CLIP-Nav: Using CLIP for Zero-Shot Vision-and-Language Navigation
Vishnu Sashank Dorbala
University of Maryland, College Park
&Gunnar Sigurdsson
Amazon Alexa AI
&Robinson Piramuthu
Amazon Alexa AI
&Jesse Thomason
Amazon Alexa AI
&Gaurav S. Sukhatme
Amazon Alexa AI
###### Abstract
Household environments are visually diverse. Embodied agents performing
Vision-and-Language Navigation (VLN) in the wild must be able to handle this
diversity, while also following arbitrary language instructions. Recently,
Vision-Language models like CLIP have shown great performance on the task of
zero-shot object recognition. In this work, we ask if these models are also
capable of zero-shot language grounding. In particular, we utilize CLIP to
tackle the novel problem of zero-shot VLN using natural language referring
expressions that describe target objects, in contrast to past work that used
simple language templates describing object classes. We examine CLIP’s
capability in making sequential navigational decisions without any dataset-
specific finetuning, and study how it influences the path that an agent takes.
Our results on the coarse-grained instruction following task of REVERIE
demonstrate the navigational capability of CLIP, surpassing the supervised
baseline in terms of both success rate (SR) and success weighted by path
length (SPL). More importantly, we quantitatively show that our CLIP-based
zero-shot approach generalizes better to show consistent performance across
environments when compared to SOTA, fully supervised learning approaches when
evaluated via Relative Change in Success (RCS).
## 1 Introduction
Vision-and-Language Navigation (VLN) requires agents to follow human
instructions in unseen environments. This is a challenging problem since
instructions heavily rely on the contents of the scene, possibly unknown to a
general agent. A common approach to VLN is to use supervised learning; we
argue that this is not practical, given the drastic shift in semantics from
scene to scene that impacts the performance of a trained model. We observe
this phenomenon in SOTA supervised learning approaches for VLN, which have
large performance drops on “unseen” environments that are absent from the
training dataset.
In this work, we seek to address this issue of generalizing to new
environments, and propose to solve VLN in a fully zero-shot manner. The agent
is assumed to have no prior knowledge about the instructions or the
environment. This setting is does not rely on a particular VLN dataset and is
free from any environmental bias that datasets may inherently have.
A substantial body of prior work tackles VLN using supervised learning Gu et
al. (2022); Guhur et al. (2021); Wang et al. (2019b); Hong et al. (2021).
Models are first trained on a set of “seen” environments and instructions,
before being evaluated on both “seen” and “unseen” test data. This paradigm
often involves encoding instructions and learning topological relations during
training Chen et al. (2021); Wang et al. (2021); Chen et al. (2022a), and
imitating navigational decisions during inference. These approaches usually
see a significant drop in performance on unseen data. Figure 1 compares
results of popular VLN approaches on the coarse-grained instruction following
task of REVERIE Qi et al. (2020). Observe the significant drop in performance
on the “Unseen Val” and “Unseen Test” sets, when compared with “Seen Val”
which validates on previously seen environments. This drop holds across all
approaches. Performance in comparison with humans is also significantly worse,
not just for REVERIE, but also for other popular instruction following tasks
such as R2R Anderson et al. (2018) and SOON Zhu et al. (2021).
Figure 1: Comparing Model Performance on REVERIE Seen and Unseen Splits:
Observe the significant drop in performance of the Unseen Val and Unseen Test
sets (blue) when compared with the Val Seen set (green), across all approaches
as measured by both Success Rate (SR) (Left) and Success Weighted by Path
Length (SPL) (Right). Also observe the poor Unseen Test set performance when
compared to the human baseline.
Current VLN models are also dataset specific; a trained model from one will
not generalize to another. For instance, training on REVERIE and testing on
SOON may not give comparable results, despite both involving the coarse-
grained instruction following task. A critical factor affecting this
generalization is the lack of sufficient training data, and a few articles
address this issue by proposing data augmentation techniques Chen et al.
(2022b); Li et al. (2022). However, data augmentation has only shown minor
improvements in performance.
Figure 2: Household images generated from text using a latent diffusion model
Rombach et al. (2022). Observe the variance in layout, positioning and
lighting. Each home is visually unique, and we hypothesize that this causes
people to use unique, environment-specific language in giving out instructions
(Orange-striped bathroom for instance). This hypothesis is substantiated by
our inspection of REVERIE, and motivates us to treat VLN as a fully zero-shot
problem.
Household environments are visually diverse, each with unique layouts and
objects. As such, instructions given by humans to describe target locations
usually contain environment-specific cues. For example, the same target item
“bottle” could be described as “next to the mirror” in one instruction and as
“above the sink” in another. There could also be unique environment-specific
identifiers such as “orange-striped bathroom” or “painting depicting crudely-
drawn people”.
We hypothesize that the visual distinctiveness of environments causes people
to use unique language involving those scenes. In our inspection of the
REVERIE dataset we noticed a low pairwise average cosine similarity of $0.32$
between all seen and unseen instructions, substantiating this claim. As all
environments are potentially distinct from one another, training datasets fail
to capture all the visuo-linguistic diversity needed to generalize (Figure 2).
Our work makes the following contributions:
* •
We present a novel approach to solve coarse-grained instruction following
tasks: breaking down guidance language into keyphrases, visually grounding
them, and using the grounding scores to drive CLIP-Nav, a novel “zero-shot”
navigation scheme.
* •
Further, we incorporate backtracking into our scheme, and present Seq CLIP-Nav
which drastically improves CLIP-Nav’s results, showcasing the importance of
backtracking in solving such tasks.
* •
Our results establish a zero-shot baseline on the task of REVERIE Qi et al.
(2020), surpassing the unseen supervised baseline without any form of dataset-
specific finetuning in terms of SR and SPL. Our SPL results on this dataset
also improves on the SOTA.
* •
Finally, we establish a new metric to measure generalizability in VLN tasks —
Relative Change in Success (RCS). This metric quantitatively showcases the
improved performance of our CLIP-based approaches over other supervised
methods.
## 2 Related Work
Vision-and-Language Navigation (VLN): Coarse-grained Qi et al. (2020); Zhu et
al. (2021) and fine-grained Anderson et al. (2018); Ku et al. (2020);
Vasudevan et al. (2021) instruction following tasks have recently been of
great interest. A majority of approaches attempting to solve these tasks use
some form of supervision, employing behavior cloning Huang et al. (2019),
reinforcement learning Wang et al. (2018), or even imitation learning Wang et
al. (2019a). These approaches are limited to solving VLN either on a single or
a specific set of datasets, and do little to analyze cross-dataset
performance. Since VLN by definition is intended to be solved in arbitrary
environments, we argue that methods should give consistent performance across
all scenes, previously seen or not.
Zero-Shot VLN: A recent paradigm shift in machine learning has led to the
emergence of large deep learning models that are pre-trained on vast amounts
of unlabeled data, and finetuned on a variety of downstream tasks. Examples of
this include BERT Devlin et al. (2018) and GPT-3 Brown et al. (2020), which
have shown SOTA performance on various natural language tasks. For zero-shot
VLN, we are particularly interested utilizing CLIP Radford et al. (2021),
which has shown improved zero-shot performance on several downstream vision-
language tasks Shen et al. (2021).
Gadre et al. (2022) in CLIP on Wheels (CoW) attempt to utilize CLIP to perform
zero-shot object navigation. Object navigation involves exploring the
environment to find a target object without any guidance instructions. CoW
modularizes this task into exploration and object localization, and uses CLIP
for the latter.
While underlying task of navigating to a target location is the same, VLN is
very different from object navigation. VLN uses natural human instruction
language as opposed to template language. For example, a coarse-grained VLN
instruction would be “Go to the kitchen on your right and water the plant
there.”, while object navigation uses only the target word “plant”. CoW
augments these words with templates (a photo of a $<$object$>$ in a video
game) and then grounds them with CLIP. We instead use CLIP not just to ground
keyphrases extracted from the VLN instruction, but also to drive our
exploration policy.
To the best of our knowledge, there are no works that attempt to solve coarse-
grained instruction following in a generalized, zero-shot setting, which forms
the basis of our research. We look at solving VLN without any form of dataset-
specific finetuning, and seek to transfer CLIP’s powerful language grounding
capabilities into a sequential decision making pipeline for navigation.
## 3 Approach
Performing zero-shot VLN in a household environment requires the agent to have
a sense of structural priors to make sequential decisions about how to get to
an unseen target location. Consider Figure 3 for instance. The command is for
the robot to “Go to the kitchen”, but the panoramic view does not contain
visual elements associated with a kitchen. In such a case, we require the
agent to pick the best view (and consequently the best direction) that would
potentially lead to a kitchen. In this work, we use CLIP to make sequential
navigational decisions, asserting its capability in capturing structural
priors of indoor hosuehold environments.
Figure 3: We look at CLIP’s ability to make sequential navigational decisions.
Here, the instruction “Go to the kitchen” suggests that the agent needs to
leave the room. However, in order for it to make this decision, it needs to
ground this instruction within the panorama, to choose a view with the door
leading outwards. Notice that there are no clear visual entities (i.e. spoons
or sinks) to suggest the chosen image (in red) is a “kitchen”. The decision is
based on pretrained CLIP’s structural prior of the household in picking a view
that might lead to the kitchen.
Our approach consists of three steps:
1. 1.
Instruction Breakdown \- Decompose coarse-grained instructions into
keyphrases.
2. 2.
Vision-Language Grounding \- Ground keyphrases in the environment using CLIP.
3. 3.
Zero-Shot Navigation \- Utilize the CLIP scores to make navigational
decisions.
Given that our approach is zero-shot and requires no finetuning, we could
choose any fixed set of datapoints for evaluation. We opt to use the Seen and
Unseen validation splits of the REVERIE Qi et al. (2020), a popular dataset to
conduct our experiments. Cross-comparing results between these splits helps us
infer the generalizability of our approach, while also enabling similar
comparisons on other supervised learning approaches on this dataset.
REVERIE contains several human-annotated instructions for paths taken from the
Matterport3D Chang et al. (2017) environment. The language used in the
instructions uniquely capture various aspects of the environment. Each path is
described as a discrete set of adjacent photorealistic panoramic images. They
contain around 8 adjacent pano images (or hops) on average, which have been
annotated with coarse-grained language guidance to reach target locations.
### 3.1 Instruction Breakdown
We first look at breaking down coarse-grained guidance into keyphrases. Our
objective is for the keyphrases to contain intermediate goals for the agent to
use for sequential decision-making.
Consider the instruction:
On the second level go the bathroom inside the second bedroom on the right
and
replace the towels on the towel rack with the clean towels from the linen
closet.
Observe that the conjunction and separates the Navigational Component (NC) of
the instruction from the Activity Component (AC). As such, a coarse-grained
instruction can be broken down as,
Coarse Instruction (I) = Navigation Component (NC) + Activity Component (AC)
We empirically observe that this sentence form holds true with most
instructions in REVERIE. The NC tells the agent about how to get to the target
location, while the AC tells it what to do once it has reached there. In our
case, since our objective is solely navigation, we focus on breaking up the NC
into keyphrases and using the AC for target object grounding.
One way to obtain keyphrases is to use prepositions in the instruction to
break it up. Using this form of retrieval on the example above gives us:
While this approach is simplistic, and breaks down instructions into
rudimentary keyphrases for grounding, it often fails to capture the temporal
nature of the instruction. Notice that “go to the bathroom” comes before
“inside the second bedroom”, when the agent needs to enter the second bedroom
before going to the bathroom.
Alternatively, we look at using a Large Language Model to break it down for
us. Using GPT-3 Brown et al. (2020) with the phrase “Break this down into
steps.” gives:
The instruction breakdown while being far more articulate, and is able to
capture the sequential timeline i.e., “Enter the second bedroom” is before “Go
to the bathroom”.
### 3.2 Language Grounding using CLIP
After breaking down instructions into keyphrases, we use them to navigate
inside a Matterport3D environment. Each node in Matterport3D is a panoramic
image covering a $360$ degree space around the agent. In order to select a
direction for navigation within this panorama, we split it into 4 separate
images. Each of these images covers approximately a $90$ degree space around
the agent, at a uniform horizontal elevation.
We use CLIP to ground keyphrases from both the Navigational (NC) and Activity
components (AC) obtained in the instruction breakdown stage, and obtain two
scores - a Keyphrase Grounding Score (KGS) and an AC grounding score. KGS
helps the agent make sequential navigational decisions, while AC grounding
score helps identify if the agent has reached the target location.
At each timestep, the image with the highest KGS is selected as the CLIP-
chosen image, and is used to drive our navigational scheme. The AC grounding
scores are for target object grounding, which gives us threshold or ‘Stop
Condition’ for when the agent needs to stop navigating.
Figure 4: CLIP Grounding \- We ground the Navigational Component (NC) on all
the split images to obtain Keyphrase Grounding Scores (KGS). The “CLIP-chosen
image” (highlighted in red) represents the one with the highest KGS, which
drives our navigation algorithms. We also simultaneously ground the AC, and
use the grounding score to determine if the agent has reached the target
location—our “Stop Condition”.
The red image in Figure 4 represents the CLIP-chosen image for the given
instruction. We observe that changing the instruction by adding more
information to it also changes the CLIP-chosen image. For example, if the
instruction here were “Go into to the kitchen that is next to the balcony on
the second level” instead, the leftmost image is no longer the one with the
highest grounding score. CLIP is sensitive to the language given to it for
grounding, and this can hinder navigation performance. As such, we limit the
words in the extracted keyphrases before computing CLIP scores.
### 3.3 Zero-Shot Navigation
The “CLIP-chosen image” obtained drives both our zero-shot navigation
strategies.
#### 3.3.1 CLIP-Nav
Figure 5 shows the overview of CLIP-Nav. At each time step, we split the
panorama into 4 images, and obtain the CLIP-chosen image as explained in
section 3.2. We obtain adjacent navigable nodes visible from this image using
the Matterport Simulator, and choose the closest node. This is done
iteratively till the ‘Stop Condition’ described in the previous section is
reached.
Beyond extracting CLIP-chosen images, the NC grounding score also determines
when to select the next keyphrase. For instance, for “Go to the kitchen”, if
the grounding score is above a certain threshold, we assume that the agent has
successfully navigated to the kitchen, and needs to execute the next keyphrase
- “next to the balcony”. In this way, we utilize CLIP not just for choosing
navigational directions, but also for determining when the agent has reached
intermediate goal locations.
Figure 5: CLIP-Nav \- We present a novel approach for zero-shot VLN, that
utilizes CLIP to make sequential navigational decisions. At each timestep, a
CLIP-Chosen Image is determined by grounding the current NC to each of the
panoramic splits. The chosen image represents the direction our model has
chosen for zero-shot navigation. In this case, it refers to the bedroom
potentially being somewhere in the chosen direction. The AC grounding score
gives us a stopping threshold for when to our agent believes it has reached
the target. CLIP-Nav runs iteratively until this threshold is reached.
#### 3.3.2 Seq CLIP-Nav
In order to improve CLIP-Nav, we incorporate a simple backtracking mechanism.
Figure 6 presents an overview of this method.
We aggregate Keyphrase Grounding Scores across a sequence of $n$ nodes, and
average them out to obtain a Sequence Grounding Score (SGS) as follows,
$\text{SGS}\>=\frac{\sum_{n}\text{KGS}_{i}}{n},$
where $\text{KGS}_{i}$ is the KGS value of the CLIP-chosen image at at a
particular timestep, and $n$ is the number of nodes. This score acts as a
backtracking threshold to determine if the agent is heading in the right
direction or needs to go back.
Figure 6: Seq CLIP-Nav \- In order to improve the performance of CLIP-Nav, we
incorporate a backtracking mechanism. CLIP scores across a sequence of
timesteps are averaged to obtain a Sequence Grounding Score (SGS). This score
is then used to determine if the agent needs to go back a few nodes
(backtrack) or not.
A low SGS value suggests that the agent might have gone down a path where it
is not able to find the intermediate goal defined by the current NC.
Alternatively, a high SGS shows that the agent is more confident of its
ability in navigating the environment in a zero-shot manner.
## 4 Results
Our results establish a baseline for zero-shot VLN, on the task of REVERIE. We
use the following metrics to evaluate agent performance.
1. 1.
Success Rate (SR) \- This is the fraction of episodes where the agent
successfully reaches and stops at the target location.
2. 2.
Success Weighted by Inverse Path Length (SPL) \- We use the definition
provided by Batra et.al. in Batra et al. (2020). This is a standard metric
used in several VLN tasks, and tells us about the optimality of the agent’s
success route, when compared with the ground-truth oracle path.
3. 3.
Oracle Succss Rate (OSR) \- This is the fraction of episodes where the agent
passes through the target location in its path, but does not stop there. This
accounts for paths that may have overshot the target, due to inaccurate
grounding.
4. 4.
Relative Change in Success (RCS) \- We also compute the percentage relative
change in performance between Seen and Unseen data, which gives us insight
into the generalizability of each approach. This is defined as -
$RCS=\frac{\lvert Seen-Unseen\rvert}{\max\\{Seen,Unseen\\}}\times 100$
A lower score indicates that the agent is performing similarly across the
splits, while a higher score indicates overfitting on the seen training
environments.
SR($\%$) $\uparrow$ OSR (%) $\uparrow$ SPL $\uparrow$ Approach Seen Unseen RCS
$\downarrow$ Seen Unseen RCS $\downarrow$ Seen Unseen RCS $\downarrow$ ZS
Random Walk $3.99$ $5.19$ $23.12$ $8.92$ $11.93$ $25.23$ $0.006$ $0.043$
$86.04$ CLIP-Nav $4.56$ $5.79$ $21.89$ $17.53$ $27.63$ $36.55$ $0.152$ $0.248$
38.70 Seq CLIP-Nav $12.34$ $14.97$ 17.56 $19.47$ $24.46$ 20.40 $0.212$ 0.450
$52.88$ SV FAST-MATTN $50.53$ $14.40$ $71.50$ $55.17$ $28.20$ $50.69$ $0.455$
$0.072$ $84.17$ AirBERT $47.01$ $27.89$ $40.67$ $48.98$ $34.51$ $29.54$
$0.423$ $0.218$ $46.46$ DUET 71.75 46.98 $34.52$ 73.86 51.07 $30.85$ 0.639
$0.337$ $47.26$
Table 1: Zero-Shot (ZS) and Supervised (SV) results on the Seen and Unseen Val
splits of REVERIE. The lowest RCS scores across all metrics indicate that our
methods generalize significantly better in new environments over supervised
approaches. On the unseen split, Seq. CLIP-Nav gives us the SOTA SPL result,
and also improves upon the REVERIE baseline in terms of SR.
Table 1 compares our Zero-Shot (ZS) results on the Seen and Unseen Val splits
of REVERIE with SOTA fully supervised methods. We also compare with a Random
Walk approach that chooses a random neighboring node for $8$ steps. This is
the upper limit on the average path length of the dataset which is between
$5$-$8$ steps. Observe the improvement in performance of Seq CLIP-Nav over the
REVERIE Baseline approach (FAST-MATTN), in terms of SR and SPL on the Unseen
split. The Unseen SPL even outperforms the SOTA supervised learning approaches
- Airbert and DUET, showing that when our agent does take the right path in an
unseen environment, it tends to do so in a more optimal manner. Also notice
the significant improvement in the SR performance of Seq CLIP-Nav over CLIP-
Nav, even while the OSRs are similar. This indicates the strong influence of
backtracking, in preventing the agent from overshooting once it reaches a
target location.
On the Seen split, the supervised approaches outperform our methods in terms
of SR and SPL, and this is quite evident since they have been trained on these
environments. The higher RCS scores on these methods however indicate that
they perform much worse on unseen environments in comparison to seen ones.
This shows poor generalizability, and that they might have overfit on the
training dataset. In contrast, the lower RCS scores across SR, OSR and SPL on
our CLIP-based approaches indicates a far more consistent performance. We can
thus quantitatively infer that our approaches generalize better in new
environments over supervised approaches, satisfying one of our primary
objectives for zero-shot navigation. This is a promising result, as it shows
that our CLIP-based models without any finetuning are able to make consistent
embodied navigational decisions irrespective of the type of environment that
the agent is placed in.
Additionally, we also obtain “Best OSR” results of $47.51\%$ on CLIP-Nav, and
$48.41\%$ on Seq CLIP-Nav on the unseen split. These are the single best
values across all the house scans. The higher values when compared to the
overall OSR shows that our approach is able to perform particularly well in
certain types of environments, and analyzing the influencing visuo-lingual
factors to transfer knowledge is part of a future study.
## 5 Conclusion and Future Scope
We tackle the problem of Vision-and-Language Navigation (VLN) in a fully zero-
shot manner to address the issue of generalizability in unseen environments.
Our approaches CLIP-Nav, and Seq CLIP-Nav give a far more consistent
performance over fully supervised SOTA approaches on the REVERIE dataset when
assessed by our generalizability metric — Relative Change in Success (RCS).
Seq CLIP-Nav in particular gives improved results over the REVERIE baseline on
the unseen validation set in terms of SR, and also improves upon SOTA
supervised learning approaches in terms of SPL. This showcases the capability
of CLIP without any dataset-specific finetuning in being able to make accurate
sequential navigational decisions necessary for zero-shot VLN.
In the future, we intend to study the cross-dataset performance of CLIP-Nav on
other fine and coarse-grained instruction sets in indoor and outdoor settings
as well as when there is a cooperative dialog for instruction following
Padmakumar et al. (2022); Gao et al. (2022). Our current backtracking score
while simplistic, still drastically influences performance; improving this via
meta-learning is a future study. Yet another exciting direction is a Virtual
Reality (VR) experiment to learn human patterns in zero-shot instruction
following. Analyzing people’s zero-shot paths in VR environments could give us
insight into their intuition for sequential decision making, and can be
modeled to improve CLIP-Nav.
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# The H$\alpha$ and [O III] $\lambda 5007$ Luminosity Functions of $1.2<z<1.9$
Emission-Line Galaxies from HST Grism Spectroscopy
Gautam Nagaraj Department of Astronomy & Astrophysics, The Pennsylvania State
University, University Park, PA 16802, USA Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Robin Ciardullo Department of Astronomy & Astrophysics, The Pennsylvania State
University, University Park, PA 16802, USA Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
William P. Bowman Astronomy Department, Yale University, New Haven, CT 06511,
USA Department of Astronomy & Astrophysics, The Pennsylvania State
University, University Park, PA 16802, USA Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Alex Lawson Department of Astronomy & Astrophysics, The Pennsylvania State
University, University Park, PA 16802, USA Caryl Gronwall Department of
Astronomy & Astrophysics, The Pennsylvania State University, University Park,
PA 16802, USA Institute for Gravitation and the Cosmos, The Pennsylvania
State University, University Park, PA 16802, USA
###### Abstract
Euclid and the Roman Space Telescope (Roman) will soon use grism spectroscopy
to detect millions of galaxies via their H$\alpha$ and [O III] $\lambda 5007$
emission. To better constrain the expected galaxy counts from these
instruments, we use a vetted sample of 4,239 emission-line galaxies from the
3D-HST survey to measure the H$\alpha$ and [O III] $\lambda 5007$ luminosity
functions between $1.16<z<1.90$; this sample is $\sim 4$ times larger than
previous studies at this redshift. We find very good agreement with previous
measurements for H$\alpha$, but for [O III], we predict a higher number of
intermediate-luminosity galaxies than previous works. We find that for both
lines, the characteristic luminosity, $\mathcal{L}_{*}$, increases
monotonically with redshift, and use the H$\alpha$ luminosity function to
calculate the epoch’s cosmic star formation rate density. We find that
H$\alpha$-visible galaxies account for $\sim 81\%$ of the epoch’s total star
formation rate, and this value changes very little over the $1.16<z<1.56$
redshift range. Finally, we derive the surface density of galaxies as a
function of limiting flux and find that previous predictions for galaxy counts
for the Euclid Wide Survey are unchanged, but there may be more [O III]
galaxies in the Roman High Latitude Survey than previously estimated.
Galaxy evolution (594), Luminosity function (942), Spectral energy
distribution (2129), High-redshift galaxies (734)
††facilities: HST (WFC3), Spitzer (MIPS), Herschel (PACS, SPIRE), GALEX,
Swift(UVOT)††software: AstroPy (Astropy Collaboration et al., 2013, 2018),
SciPy (Jones et al., 2001–; Virtanen et al., 2020), CLOUDY (Ferland et al.,
1998, 2013), FSPS (Conroy et al., 2009; Conroy & Gunn, 2010), MCSED(Bowman et
al., 2020)
## 1 Introduction
Since its observational discovery by Riess et al. (1998) and Perlmutter et al.
(1999), dark energy has been at the forefront of astronomical research. While
the $\Lambda$CDM paradigm has been extremely successful in predicting the
properties of the cosmological microwave background and reproducing
observables such as large scale structure and cosmic abundances, there are
many open questions remaining, especially about the nature of dark matter and
the evolution of dark energy (e.g., Bull et al., 2016; Amendola et al., 2018,
and references therein). In order to better constrain cosmological models, we
must continue accruing more accurate and precise observational probes.
Such efforts include the use of large-scale spectroscopic surveys to measure
baryonic acoustic oscillations and redshift space distortions throughout
cosmic time. The most efficient mechanisms for generating these data are
multi-fiber spectroscopic surveys, such as WiggleZ (Drinkwater et al., 2010;
Blake et al., 2011), the Baryon Oscillation Spectroscopic Survey (BOSS; Dawson
et al., 2013), the extended Baryon Oscillation Spectroscopic Survey (eBOSS;
Dawson et al., 2016) and the Dark Energy Spectroscopic Instrument (DESI)
survey (DESI Collaboration et al., 2016; Abareshi et al., 2022); integrated
field unit (IFU) spectroscopic surveys, such as the Hobby-Eberly Telescope
Dark Energy Experiment (HETDEX; Gebhardt et al., 2021; Hill et al., 2021);
wide-field imaging using large, comprehensive sets of narrow-band filters
(e.g., J-PAS; Cepa et al., 2016; Salzano et al., 2021), and slitless (grism)
spectroscopic surveys, including Euclid (Laureijs et al., 2011, 2012) and the
Nancy Grace Roman Space Telescope (Roman; Green et al., 2012; Dressler et al.,
2012; Spergel et al., 2015).
Slitless spectroscopy in particular is an extremely efficient method of
obtaining spectra over an entire field with no need for the pre-selection of
targets. Narrow-band surveys, such as the High-$z$ Emission Line Survey
(HiZELS; Geach et al., 2008; Sobral et al., 2009) and the NOAO Extremely Wide
Field Infrared Imager (NEWFIRM) H$\alpha$ Survey (Ly et al., 2011), achieve
the same efficiency but are usually restricted to a very limited slice of
redshift space, and thus require many different filters to survey large
volumes.
Normal galaxies have no strong emission lines between Ly$\alpha$ at 1216 Å and
[O II] $\lambda 3727$, so beyond $z\sim 1$, redshift surveys are most
efficiently performed at near-infrared wavelengths, with lines such [O II]
$\lambda 3727$, H$\beta$, [O III] $\lambda 5007$, and H$\alpha$. While it is
possible to photometrically-select $z\gtrsim 1$ objects and then refine their
redshifts with follow-up spectroscopy (e.g., Davis et al., 2003; Steidel et
al., 2004; Lilly et al., 2007; DESI Collaboration et al., 2016), emission line
galaxy (ELG) surveys to probe large swaths of cosmic time are most easily
performed from space. GRAPES (Pirzkal et al., 2004), which used the ACS G800L
grism of the Hubble Space Telescope, as well as WISP (Atek et al., 2010) and
3D-HST (Brammer et al., 2012; Momcheva et al., 2016), which used the HST/WFC3
G102 and G141 grisms, represent some of the first efforts to create space-
based ELG samples. In this study, we use the 3D-HST sample described in
Nagaraj et al. (2021b) and Nagaraj et al. (2021a), hereafter referred to as
Paper I and Paper II, to further explore the emission-line properties of
$1.2\lesssim z\lesssim 1.9$ galaxies.
Two near-future missions, Euclid (Laureijs et al., 2011, 2012) and Roman
(Green et al., 2012; Dressler et al., 2012; Spergel et al., 2015), will
identify millions of galaxies at redshifts $0.7\lesssim z\lesssim 2.7$ using
their H$\alpha$ and [O III] $\lambda 5007$ emission. Given the similarity in
the spectroscopic survey designs of 3D-HST, the Euclid Deep Survey, and the
Roman High Latitude Survey (see Figure 5 in Paper I for a visual depiction of
the survey limits and observed 3D-HST fluxes), we can use 3D-HST as a
pathfinder for these missions. In particular, by evaluating the H$\alpha$ and
[O III] $\lambda 5007$ luminosity functions and biases with respect to other
galaxy samples and dark matter distributions, we can estimate how many
galaxies these programs will find and how well they will be able to measure
cosmological parameters.
Several efforts have been made to calculate the H$\alpha$ and [O III] $\lambda
5007$ luminosity functions in both the local and distant universe, and these
have led to deeper understanding of galaxy evolution (e.g., Gallego et al.,
1995; Tresse & Maddox, 1998; Sullivan et al., 2000; Jones & Bland-Hawthorn,
2001; Fujita et al., 2003; Hippelein et al., 2003; Glazebrook et al., 2004;
Treyer et al., 2005; Wyder et al., 2005; Ly et al., 2007; Geach et al., 2008;
Shim et al., 2009; Sobral et al., 2009, 2011; Ly et al., 2011; Tadaki et al.,
2011; Sobral et al., 2012; Colbert et al., 2013; Pirzkal et al., 2013; Sobral
et al., 2013; Khostovan et al., 2015; Mehta et al., 2015; Sobral et al., 2015;
Comparat et al., 2016; Hayashi et al., 2020; Khostovan et al., 2020). At
$z\gtrsim 1$, most of these studies are narrow-band surveys targeted at
specific redshifts; while these programs involve large numbers of sources and
extremely deep exposures (e.g., Khostovan et al., 2020), their ability to
investigate cosmic evolution is limited due to the small volumes covered.
Other investigations have been limited by sample size (e.g., Shim et al.,
2009) or spectral resolution (e.g., Pirzkal et al., 2013, where [O III] and
H$\beta$ are a blended feature). Particularly notable is the work of Colbert
et al. (2013) and the follow-up study by Mehta et al. (2015), which used data
from the HST/WFC3 Infrared Spectroscopic Parallel (WISP) survey (Atek et al.,
2010) to create a sample of approximately 1,000 ELGs between $0.3<z<2.3$.
Recently, Nagaraj et al. (2021b) carefully vetted a sample of 4350 $1.2<z<1.9$
emission-line galaxies which were originally identified on the 3D-HST grism
frames by Momcheva et al. (2016). Here we use a subsample of 3,187 sources to
determine the luminosity function and equivalent width distribution of [O III]
$\lambda 5007$ and H$\alpha$ in this redshift range. Since the depth and
resolution of the 3D-HST data are similar to the grism surveys planned for
Euclid and Roman, we can use our measurements to refine the predictions for
these studies, and improve our measurement of the amount of intermediate-
redshift star formation that is occurring in emission-line galaxies. These
data will also allow us to examine the relationship between [O III] $\lambda
5007$ emission and star-formation rate at an epoch intermediate between the
local universe, where emission from oxygen is mostly from [O II] $\lambda
3727$, and the redshifts studied by Bowman et al. (2021), where [O III]
$\lambda 5007$ dominates.
This paper is the third in series that analyses 3D-HST sources at redshifts
$1.2<z<1.9$. Paper I showed empirical relations between stellar mass and
various observational and physical properties, such as absolute magnitude in a
rest-frame optical filter. Paper II focused on the relationships among stars,
gas, and dust in galaxies with available mid- and far-IR data.
Throughout this paper, we assume a $\Lambda$CDM cosmology with
$\Omega_{\Lambda}=0.69$, $\Omega_{M}=0.31$ and $H_{0}=68$ km s-1 Mpc-1
(Bennett et al., 2013). All magnitudes given in the paper are in the AB
magnitude system (Oke, 1974).
## 2 Data and Selection Effects
In this section we describe the fluxes used for the H$\alpha$ and [O III]
$\lambda 5007$ luminosity function calculations, including the critical issue
of incompleteness for low-luminosity objects.
### 2.1 Data
As the data used to compute the H$\alpha$ and [O III] $\lambda 5007$
luminosity functions have been described in Papers I and II, we give only a
brief overview here. From an initial list of 9341 $1.2\lesssim z\lesssim 1.9$
candidates identified on WFC3/G141 grism frames by the 3D-HST survey
(GO-11600, 12177, 12328; Brammer et al., 2012; Momcheva et al., 2016), we
carefully identified a clean sample of 4350 ELGs brighter than the catalog’s
F125W (J) + F140W (JH) + F160W (H) magnitude limit of $m_{J+JH+H}$ $=26$.
Since the 625 arcmin2 covered by the 3D-HST survey coincides with regions of
the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS;
Grogin et al., 2011; Koekemoer et al., 2011), the fields have a wealth of
photometric observations, and these data helped fuel the results of Paper I
and Paper II.
Paper I describes the process we used to identify active galactic nuclei (AGN)
masquerading as normal galaxies in our sample. First, we used X-ray matching
from the deep surveys of the CANDELS fields (especially GOODS-S), to identify
72 AGN in our ELG sample. Specifically, we used a cross-correlation search
radius of 1″ to identify possible X-ray counterparts to our ELGs. Any galaxy
with an X-ray luminosity greater than $10^{42}$ $\text{erg s}^{-1}$ in the
$2-10$ keV band was considered an AGN and eliminated from the analysis.
Stacking of the remaining ELGs then found X-rays levels consistent with those
expected from simple star formation, suggesting that the vast majority of
objects remaining in our sample are normal galaxies with no obvious AGN
activity.
While X-rays are very effective for AGN identification, they fail when the gas
column densities are too high ($N_{H}\gtrsim 5-50\times
10^{23}~{}\textrm{cm}^{-2}$), which tends to correspond to highly dusty,
Compton-thick systems (e.g., Brandt & Alexander, 2015, and references
therein). To find such objects, we used the IRAC AGN selection criteria
described by Donley et al. (2012); the method identified 50 AGN candidates
(with 11 being previously excluded via their X-ray luminosity). This whittled
down our ELG sample to 4239 objects.
Bowman et al. (2019) ran nearly the same procedure on 3D-HST galaxies at
$1.9\leq z\leq 2.35$. After removing AGN, they were left with a sample of 1964
ELGs with trustworthy redshifts and clean spectra. These data are used in
§4.3, where we combine the datasets to examine the evolution of the [O III]
$\lambda 5007$ luminosity function over the full redshift range from $1.16\leq
z\leq 2.35$.
### 2.2 Completeness and Selection Effects
In this work, we use the H$\alpha$ and [O III] fluxes derived from the 3D-HST
grism spectra by Momcheva et al. (2016). Due to the relatively low resolution
($R\sim 130$) of the G141 grism and the morphological broadening that occurs
in extended sources, the two components of [O III], [O III] $\lambda 5007$ and
[O III] $\lambda 4959$, appear as a single blended feature. However, since the
ratio of [O III] $\lambda 5007$ to [O III] $\lambda 4959$ is fixed at 2.98:1
(Storey & Zeippen, 2000), we can simply rescale the Momcheva et al. (2016)
measurements to give the de-blended fluxes for primary [O III] $\lambda 5007$
emission line.
A larger concern is our inability to separate H$\alpha$ from the bracketing
forbidden lines of [N II] $\lambda\lambda 6548,6584$. While corrections for [N
II] exist in the literature (e.g., the procedure and references outlined in
Price et al., 2014), gas-phase metallicities defined through strong line
indicators are subject to degeneracies, inaccuracies, and other issues (e.g.,
Kewley & Ellison, 2008). Furthermore, the ionization balance between the first
and second excited states of oxygen changes dramatically, from predominantly
O+ at low redshift to O++ at $z\sim 2$ (e.g., Kewley et al., 2019, and
references therein). Since N+ has an ionization potential only slightly less
than O+, we can expect a similar trend in our data. This shift in ionization
balance means that the [N II]/H$\alpha$ ratio must also evolve with redshift.
Another concern associated with our understanding of the data involves the
issue of dust attenuation. The wavelength dependence of attenuation has a
variety of shapes in different galaxies (see Shivaei et al., 2020; Salim &
Narayanan, 2020, and references therein). An accurate modeling of this
behavior is difficult, especially given its degeneracy with the other
parameters of the complex stellar populations that make up a galaxy’s SED.
Even at the wavelength of H$\alpha$ (6563 Å), dust attenuation represents an
uncertainty. Given the issues associated with [N II] and dust, we do not
attempt to disentangle the flux of H$\alpha$ from that of [N II]; instead, we
present a luminosity function for the combined lines. We discuss this further
in §4.2 where we calculate the star formation rate density between
$1.2\lesssim z\lesssim 1.9$.
An extremely important issue for any luminosity function analysis is
completeness. Following Bowman et al. (2021), we parameterize the completeness
of the [O III] $\lambda 5007$ and H$\alpha$ detections as a function of flux,
$f$, using a modified Fleming et al. (1995) function,
$\displaystyle F_{F}(f)$
$\displaystyle=\frac{1}{2}\Bigg{[}1+\frac{\alpha_{F}\log(f/f_{{\rm{50}}})}{\sqrt{1+(\alpha_{F}\log(f/f_{{\rm{50}}}))^{2}}}\Bigg{]}$
(1) $\displaystyle\tau(f)$ $\displaystyle=1-e^{-f/f_{\rm 10}}$ (2)
$\displaystyle F_{c}(f)$ $\displaystyle=[F_{F}(f)]^{1/\tau(f)}$ (3)
In the equations, $F_{c}(f)$ is the completeness, $\alpha_{F}$ describes how
quickly the completeness drops off as a function of flux, $f_{\rm 50}$ is the
flux where the recovery fraction of objects is 50%, and $f_{\rm 10}$ is the
flux at which the sample is 10% complete. As the Fleming function is
completely described by $f_{\rm 50}$ and $\alpha_{F}$, $f_{\rm 10}$ is not an
independent parameter but a quantity directly derived from Equation 1. An
important point to note is that for G141 grism data, the behavior of the
completeness curve is virtually independent of wavelength (e.g., Zeimann et
al., 2014; Bowman et al., 2021).
There are two ways to estimate the parameters $\alpha_{F}$ and $f_{\rm 50}$.
The first method, which was used by Bowman et al. (2021) in their analysis of
the [O III] $\lambda 5007$ luminosity function of $1.90<z<2.35$ grism-selected
galaxies, is to use the data themselves and simultaneously fit the galaxy
luminosity function (§3) and each field’s completeness parameters to the
observed distribution of emission-line luminosities. Such a procedure is
complex, since, even if $\alpha_{F}$ is assumed to be the same across all five
CANDELS fields, the calculation still involves at least 9 separate variables
(a minimum of three for the luminosity function, one value of $f_{\rm 50}$ for
each field, and $\alpha_{F}$).
Alternatively, it is possible to fit the completeness curves separately from
the luminosity function by assuming the intrinsic flux distribution of faint
emission lines is a power law. This approach is reasonable, given the expected
nature of the faint galaxy number counts, and was the method employed by
Bowman et al. (2019) in their study of the physical properties of $z\sim 2$
grism-selected galaxies.
Our experiments show that both methods yield similar results for the galaxy
luminosity function. Therefore, to avoid any degeneracies associated with
high-dimensional fits and to simplify the analysis, we chose to decouple the
question of completeness from the luminosity function calculation and adopt
the values of $f_{\rm 50}$ and $\alpha_{F}$ found by Bowman et al. (2021).
These parameters, which were derived for $1.90\leq z\leq 2.35$ [O III]
$\lambda 5007$ emission on the same 3D-HST frames used here, were found using
the simultaneous fitting technique described above. Since completeness should
only be a function of line flux, and not depend on the specific line being
observed, the Bowman et al. (2021) values should be equally applicable to our
survey.
There is one potential caveat to this last assertion. As detailed by Momcheva
et al. (2016), the $1\sigma$ sensitivity limit of the 3D-HST survey depends
not only on the strength of an emission line, but also the angular size of the
emitting source. Since the present study focuses on galaxies at lower redshift
than those measured by Bowman et al. (2021), this difference has the potential
to cause a shift in the survey’s 50% completeness limit. However, based on
values from the 3D-HST catalog (Momcheva et al., 2016), the mean size of
$1.16\leq z\leq 1.90$ emission-line galaxies is only 1.15 times that of the
Bowman et al. (2021) systems. In comparison, the size spread amongst the
galaxies in the sample is a factor of $\sim 4$. For this reason, we do not
account for this size difference in our analysis. We list the completeness
parameters in Table 1 and show the completeness curves in Figure 1.
For the purposes of the luminosity functions presented in this paper, we only
consider flux measurements above the 50% levels given in Table 1. The
inclusion of objects much below this limit induces strong effects on the
luminosity function by amplifying the uncertainties in the completeness
curves. Conversely, if the flux limit is too strict, the sample of galaxies
becomes too small for any reliable measurement of the function’s low-
luminosity end. Our 50% cutoff results in 2947 [O III] and 1892 H$\alpha$
measurements being used for our luminosity function calculations. For the
analysis of evolution in the [O III] $\lambda 5007$ luminosity function, this
number can be incremented using the $1.9\leq z\leq 2.35$ ELGs found by Bowman
et al. (2021), which were identified in the same manner as the galaxies used
here. This results in a sample of 4519 [O III] $\lambda 5007$ ELGs above the
50% completeness limit.
Table 1: Completeness Parameters
Field | $f_{\rm 50}~{}(10^{-17}$ $\text{erg cm}^{-2}\text{ s}^{-1}$)
---|---
AEGIS | 2.35
COSMOS | 3.12
GOODS-N | 2.20
GOODS-S | 2.86
UDS | 2.85
Note. — For all fields, $\alpha_{F}=4.56$.
Figure 1: Completeness curve for H$\alpha$ and [O III] $\lambda 5007$ in each
of the five 3D-HST fields. The parameter values used to generate these curves
are listed in Table 1.
## 3 Methodology
### 3.1 Deriving the Luminosity Function
The maximum likelihood estimator (MLE) is commonly used in astronomy to
measure and fit the parameterized variables of a luminosity function.
Simplifications to the MLE integral have also led to computationally efficient
procedures for creating discrete luminosity functions derived from techniques
such as the $1/V_{\rm max}$ (Schmidt, 1968, 1970; Huchra & Sargent, 1973; Avni
& Bahcall, 1980) and the $C^{-}$ (Lynden-Bell, 1971) methods. Here we do both,
and derive both a digital representation of the number of galaxies versus
emission-line (log) flux, and a convenient analytical representation of the
luminosity function.
We assume Poisson statistics hold, and define the likelihood of a luminosity
function (scripted as $\mathcal{P}$ to avoid confusion with luminosity)
following the derivation by Ciardullo et al. (2013). For computational
convenience, we use $\mathcal{L}\equiv\log L$ rather than the linear
luminosity; in both cases, the math is identical, although the units are
different. With the equations in this section, we are able to both measure the
galaxies’ discrete luminosity function and fit the data to a parameterization
of our choosing (i.e., the Schechter, 1976, function).
From Ciardullo et al. (2013), the likelihood of observing any luminosity
function, $\phi^{\prime}$ is given by
$\ln\mathcal{P}=\sum_{i}^{N}\ln\phi^{\prime}(\mathcal{L}_{i},z_{i})-\int_{z_{1}}^{z_{2}}\int_{\mathcal{L}_{\rm
min}(z)}^{\infty}\phi^{\prime}(\mathcal{L},z)\frac{dV}{dz}d\mathcal{L}\,dz$
(4)
where $N$ is the number of galaxy luminosities included in the sample. Note
that there is a distinction between the true luminosity function
$\phi(\mathcal{L},z)$, which we model as a Schechter (1976) function with
parameters $\alpha$, $\mathcal{L_{*}}$, and $\log\phi_{*}$, i.e.,
$\phi(\mathcal{L},z)=\ln(10)10^{\log\phi_{*}(z)}10^{\left[\mathcal{L}-\mathcal{L}_{*}(z)\right]\left[\alpha+1\right]}\exp\left(-10^{\mathcal{L}-\mathcal{L}_{*}(z)}\right)$
(5)
and the observed luminosity function, $\phi^{\prime}$, which is affected by
incompleteness and measurement error.
Following Marshall et al. (1983) and Marshall (1985), we define a function
$\Omega(\mathcal{L},z)$ that takes into account both the flux completeness and
the limits of the survey area. If $f$ is the flux, which can be thought of as
a function of luminosity and redshift, and $p(f,\hat{n})$ is the completeness
function in a given direction $\hat{n}$, then we have Equation 6 below, where
$d^{2}\hat{n}$ is the integral over the unit sphere.
In our case, we consider $p(f,\hat{n})$ constant over any given field used in
the analysis. If $\Omega_{i}$ is the effective survey area of field $i$, then
$\Omega(\mathcal{L},z)$ can be approximated as
$\Omega(\mathcal{L},z)=\int\frac{d^{2}\hat{n}}{4\pi}\,p(f,\hat{n})\approx\sum_{i=0}^{N_{\rm
fields}}\frac{\Omega_{i}}{4\pi}p_{i}(f)$ (6)
Assuming that the true luminosity is isotropic, we can simply connect the
observed and true luminosity functions through
$\phi^{\prime}(\mathcal{L},z)=\Omega(\mathcal{L},z)\phi(\mathcal{L},z)$ (7)
Going back to Equation 4, $z_{1}$ and $z_{2}$ represent the survey redshift
limits. The minimum luminosity, $L_{\rm min}(z)$, is somewhat arbitrary and is
simply taken to be a luminosity lower than what is observable. In practice,
the upper limit of the integral is also fixed at a value at, or above, that
which no galaxies are expected to exist. Finally, $dV/dz$ is the differential
volume element. Assuming a spatially flat universe, this is simply
$\frac{dV}{dz}=\frac{4\pi d_{A}^{2}(z)}{H(z)}$ (8)
where $d_{A}(z)$ represents the comoving angular diameter distance and $H(z)$
is the Hubble parameter.
Given this definition, the true luminosity function represents the number of
galaxies per dex (log luminosity units) per comoving volume element. Moreover,
given the nature of the 3D-HST survey, we assume that the flux completeness
function is constant within each CANDELS field, so in a given field,
$p(f,\hat{n})\equiv p(f)$. As described in §2, we use the modified Fleming et
al. (1995) completeness curve $F_{c}(f)$ for $p(f)$.
The likelihood analysis presented here has not included measurement errors in
the luminosity (i.e., heteroscedasticity). We follow the convention adopted in
Mehta et al. (2015), in which the luminosity errors are assumed to be normal
with mean $\mathcal{L}_{i}$ and standard deviation $\sigma_{i}$. In that case,
we can replace Equation 4 with
$\ln\mathcal{P}=\sum_{i}^{N}\ln\int_{\mathcal{L}_{\rm low}}^{\mathcal{L}_{\rm
high}}\phi^{\prime}(\mathcal{L}_{i},z_{i})N(\mathcal{L}|{\mathcal{L}_{i},\sigma_{i}})\frac{dV}{dz}d\mathcal{L}-\int_{z_{1}}^{z_{2}}\int_{\mathcal{L}_{\rm
min}(z)}^{\infty}\phi^{\prime}(\mathcal{L},z)\frac{dV}{dz}d\mathcal{L}\,dz$
(9)
The drawback of this approach is that it is computationally expensive, as the
normal distribution has a possibly different mean and standard deviation for
every flux measurement. The results given in §4 are therefore based on
Equation 4. We find that including luminosity errors has little to no effect
at the bright end of the [O III] luminosity function fit but does slightly
lessen the low-luminosity slope $\alpha$. In other words, the observed faint-
end slope is affected by Eddington (1913) bias. Bowman et al. (2021) provides
a more detailed analysis of the effects of photometric uncertainties on the
shape of the luminosity function.
If we assume that the luminosity function remains unchanged over the redshift
interval of interest, the MLE for discrete points becomes much simpler to
compute. Let us parameterize the luminosity function as a sum of discrete
Dirac-delta functions at $M$ different luminosities, i.e.,
$\phi(L)=\sum_{j=1}^{M}\phi_{j}\delta^{D}(L-L_{j})$ (10)
Then, using Equations 4 and 10, the likelihood can be expressed as
$\ln\mathcal{P}=\sum_{i=1}^{N}\ln\phi_{i}-\sum_{j=1}^{M}\int
dz\frac{dV}{dz}\Omega(L_{j},z)\phi_{j}$ (11)
Setting the derivative of the likelihood to zero, the MLE solution for the
luminosity function $\phi_{i}$ at $L_{i}$ becomes
$\phi_{i}^{-1}=\int dz\frac{dV}{dz}\Omega(L_{i},z)\equiv V_{\rm eff}(L_{i})$
(12)
This is a slight generalization of the historic $1/V_{\rm max}$ method
(Schmidt, 1968, 1970; Huchra & Sargent, 1973; Avni & Bahcall, 1980) for any
given completeness curve $p(f,\hat{n})$. We use this formalism, which we will
call the $V_{\rm eff}$ method, as a benchmark for our more sophisticated MCMC
approaches.
### 3.2 Computing the Luminosity Function
Our simplest method of computing the luminosity function is the $V_{\rm eff}$
approach introduced in §3.1. We calculate $\phi_{i}$ (Equation 12) at every
source luminosity in our sample and collect the results into luminosity (or
log luminosity) bins. Wide bins allow for larger numbers of sources per bin
and are thus more reliable, but do not convey as much information given their
coarseness. We find that dividing the measurements into $\sim 50$ bins of
equal size in log luminosity space works quite well for balancing the number
of sources per bin against the complexity of the results.
The errors on our fitted parameters are generated via a bootstrap analysis. We
take the true $V_{\rm eff}$-method result using the original set of
$N$($L_{i},\phi_{i}$) values, and then generate $B$ bootstrap samples, in
which the $N$ values of $L_{i}$ and $\phi_{i}$ are generated randomly with
replacement. The sample variance is taken to be the error on the luminosity
function measurement,
$\hat{\sigma}^{2}=\frac{1}{B-1}\sum_{i=1}^{B}\left[\theta_{i}-\left(\frac{1}{B}\sum_{i=1}^{B}\theta_{i}\right)\right]^{2}$
(13)
where $\theta$ is the binned luminosity function measured at a particular
interval, and $B$ is typically set to $B=100$. We include all fluxes down to
the 50% completeness limit (§2) in the $V_{\rm eff}$ method.
For the computation of the Schechter (1976) function parameters, our MCMC code
uses uniform priors on $\alpha$, $\log\phi_{*}$, and $\mathcal{L}_{*}$ with
bounds $[-3,1]$, $[-8,5]$, and $[40,45]$, respectively, while the completeness
parameters are fixed at the values given in Table 1. We define the relative
likelihood of a solution either through Equation 4 (no observational errors)
or Equation 9 (with observational errors) and employ the emcee package
(Foreman-Mackey et al., 2013) to explore the parameter space. We compute both
a static (non-evolving) luminosity function and one that evolves over time.
To explore time evolution in the luminosity function, we use a method based on
the technique described in Leja et al. (2020). We let both $\log\phi_{*}$ and
$\mathcal{L}_{*}$ be quadratic functions of redshift. However, rather than
fitting the coefficients of the quadratic, whose priors are difficult to
physically motivate (Leja et al., 2020), we use the values of $\log\phi_{*}$
and $\mathcal{L}_{*}$ at three specific redshifts ($z_{1}=1.20$, $z_{2}=1.76$,
and $z_{3}=2.32$ for [O III] $\lambda 5007$ and $z_{1}=1.18$, $z_{2}=1.36$,
and $z_{3}=1.54$ for H$\alpha$) to define the quadratic formulation of
$\mathcal{L}_{*}(z)$ and $\log\phi_{*}(z)$. As in our static luminosity-
function calculation, the prior on $\mathcal{L}_{i*}$ is uniform over the
range $[40,45]$ and the prior on $\log\phi_{i*}$ uniform on $[-8,5]$.
For the analysis, we fix the completeness parameters and also constrain
$\alpha$ to be constant over time. Doing so avoids exacerbating degeneracies
between $\alpha$ and the other two Schechter parameters. (In fact, in the case
of [O III], we find that leaving $\alpha$ as a free parameter over the large
redshift range $1.16\leq z\leq 2.35$ leads to failures in the MCMC fits. For
the [O III] $\lambda 5007$ line, we therefore we fix $\alpha=-1.5$.)
For the reader’s convenience, in Table 2 we have listed all the parameters
used in the non-evolving and redshift-varying luminosity function
calculations, as well as the priors applied in the MCMC code.
Table 2: H$\alpha$ and [O III] $\lambda 5007$ Luminosity Function Fitting Parameters Parameter(s) | Priors
---|---
$\alpha$ | Uniform on $[-3,1]$; Fixedaa$\alpha$ is fixed at $-1.5$ only for the redshift-varying [O III] $\lambda 5007$ luminosity function.
$\mathcal{L}_{*}$, $\mathcal{L}_{1*}$, $\mathcal{L}_{2*}$, $\mathcal{L}_{3*}$ | Uniform on $[40,45]$
$\log\phi_{*}$, $\log\phi_{1*}$, $\log\phi_{2*}$, $\log\phi_{3*}$ | Uniform on $[-8,5]$
$\alpha_{F}$ | Fixed
$f_{\rm 50}$ (5 Fields) | Fixed
In all cases, we employ 100 walkers and 1000 steps, resulting in 100,000 MCMC
realizations. In other words, 100 points in the parameter space are randomly
selected as initial states, and the MCMC algorithm takes 1000 steps from each
initial state to find regions of higher likelihood. There is no “best-fit”
solution, but we find that given these generous numbers for walkers and steps,
the solutions generally do converge, suggesting that the true best-fit is
closely approached.
### 3.3 Cosmic Variance
One source of uncertainty in the normalization of our emission-line luminosity
functions is cosmic variance. To estimate the expected amplitude of this
effect, we use the cosmic variance calculator111
https://www.ph.unimelb.edu.au/$\sim$mtrenti/cvc/ of Trenti & Stiavelli (2008),
which employs both the extended Press-Schechter formalism (Press & Schechter,
1974) and numerical simulations to compute the expected variance in any
pencil-beam region of the sky. Following Colbert et al. (2013) and Bowman et
al. (2021), we compute the cosmic variance by taking the result for each of
the five disconnected CANDELS fields and then dividing the average of these
estimates by $\sqrt{5}$.
The results of this calculation show that for a non-evolving luminosity
function over the redshift range $1.16\leq z\leq 1.56$, the cosmic variance
expected for our H$\alpha$-emission luminosity function is $\sim 6.7\%$, while
that for [O III] $\lambda 5007$ galaxies between $1.16\leq z\leq 1.90$, this
number is $\sim 5.3\%$. For the redshift-varying case, the process of
measuring the cosmic variance is not so straightforward, as we have modeled
cosmic evolution using a quadratic equation, represented using the values of
$\phi_{*}$ and $\mathcal{L}_{*}$ at three redshifts. However, if we divide the
surveyed redshift range into three bins, we can estimate the effect of cosmic
variance on each bin. We find that for both H$\alpha$ and [O III] $\lambda
5007$, the variance should slightly increase with redshift, with the
uncertainties being roughly 12% and 9%, respectively.
We include the effects of cosmic variance in our calculations for the expected
galaxy counts (Tables 3 and 4) as well as §4.4) and the star formation rate
density (§4.2). Given the sizes of our galaxy samples and the volumes of space
being surveyed ($\gtrsim 0.7$ and $1.4\times 10^{6}$ Mpc3 for the H$\alpha$
and [O III] $\lambda 5007$ studies, respectively), the effects of cosmic
variance should be small in the case of the static luminosity function, but
non-negligible for our analysis of cosmic evolution.
## 4 Results
In this section, we present the H$\alpha$ \+ [N II] and [O III] $\lambda 5007$
luminosity functions of 3D-HST ELGs, along with the H$\alpha$-based cosmic
star formation rate density contained in the emission-line galaxies. Tables 3,
4, and 5 present the overall results for our sample.
In Tables 3 and 4, we list our best-fit static luminosity functions along with
those of Shim et al. (2009), Colbert et al. (2013), Sobral et al. (2013),
Khostovan et al. (2015), and Sobral et al. (2015). As a cautionary statement,
the luminosity functions being given in the tables are not directly
comparable, as detailed in the columns labeled “Notes”. Moreover, because of
the well-known degeneracies between the three Schechter (1976) parameters, our
values of $\alpha$, $\mathcal{L}$, and $\phi_{i}$ are not necessarily in
agreement with those of the previous studies. Nevertheless, we find that the
overall form of our H$\alpha$ luminosity function is compatible with the
luminosity functions derived by Colbert et al. (2013) and Sobral et al.
(2013), but somewhat distinct from the literature measurements for [O III]
$\lambda 5007$.
In Table 5, we give the parameters of our best-fit H$\alpha$ and [O III]
$\lambda 5007$ redshift-evolving luminosity functions. For the latter, we also
extend the redshift range to $z=2.35$ using the measurements of Bowman et al.
(2021), since their galaxy sample was defined in exactly the same manner as
our dataset.
We note that in a grism survey, the true survey area is difficult to calculate
as contamination from overlapping spectra and edge effects reduce the number
of objects included in analyses. Bowman et al. (2021) studied this censoring
by masking out those regions of the 3D-HST survey where emission-line
detections are compromised, and fitting a luminosity function using only those
galaxies in the unmasked areas. They found that the effective survey area of
3D-HST is $\sim 85\%$ of the total survey area; this is consistent with the
correction applied by Colbert et al. (2013) and the estimate made by Ciardullo
et al. (2014). In this work, we reduce the quoted area of the 3D-HST survey by
15% to approximate the effects of overlapping spectra and edge-losses.
Table 3: H$\alpha$ Luminosity Function Schechter Parameters Reference | $z$ | Sample Size | $\log\phi_{*}$ | $\mathcal{L}_{*}$ | $\alpha$ | $\log\,\int_{0.03L^{*}}^{\infty}\phi(L)\,dL$ | Notes
---|---|---|---|---|---|---|---
Shim et al. (2009) | 0.7 - 1.9 | 80 | $-2.48\pm 0.07$ | $42.54\pm 0.06$ | $-1.39$ (fixed) | $-1.67\pm 0.07$ | HST-NICMOS, H$\alpha$
Colbert et al. (2013) | 0.9 - 1.5 | 517 | $-2.70\pm 0.12$ | $42.18\pm 0.10$ | $-1.43\pm 0.17$ | $-1.8\pm 0.2$ | WISPS, H$\alpha$
Sobral et al. (2013) | 1.47 | 515 | $-2.61^{+0.08}_{-0.09}$ | $42.56^{+0.06}_{-0.05}$ | $-1.62^{+0.25}_{-0.29}$ | $-1.6\pm 0.3$ | HiZELS, H$\alpha$, dust-corrected
This work | 1.16 - 1.56 | 1892 | $-2.87^{+0.08}_{-0.09}$ | $42.39^{+0.05}_{-0.05}$ | $-1.60^{+0.07}_{-0.07}$ | $-1.84\pm 0.04$aaIncludes cosmic variance (§3.3) in error budget. | 3D-HST, H$\alpha$ +[N II]
This work | 1.16 - 1.56 | 1892 | $-2.86^{+0.03}_{-0.03}$ | $42.39^{+0.02}_{-0.02}$ | $-1.60$ (fixed) | $-1.84\pm 0.04$aaIncludes cosmic variance (§3.3) in error budget. | 3D-HST, H$\alpha$ +[N II]
Table 4: [O III] $\lambda 5007$ Luminosity Function Schechter Parameters Reference | $z$ | Sample Size | $\log\phi_{*}$ | $\mathcal{L}_{*}$ | $\alpha$ | $\log\,\int_{0.03L^{*}}^{\infty}\phi(L)\,dL$ | Notes
---|---|---|---|---|---|---|---
Colbert et al. (2013) | 0.7 - 1.5 | 192 | $-3.28\pm 0.09$ | $42.39\pm 0.08$ | $-1.5$ (fixed) | $-2.36\pm 0.09$ | WISPS, [O III] $\lambda\lambda 4959,5007$
Colbert et al. (2013) | 1.5 - 2.3 | 58 | $-3.60\pm 0.14$ | $42.83\pm 0.11$ | $-1.5$ (fixed) | $-2.68\pm 0.16$ | WISPS, [O III] $\lambda\lambda 4959,5007$
Khostovan et al. (2015) | 1.42 | 371 | $-2.61^{+0.10}_{-0.09}$ | $42.06^{+0.06}_{-0.05}$ | $-1.60$ (fixed) | $-1.58\pm 0.09$ | HiZELS, H$\beta$ \+ [O III]
Sobral et al. (2015) | 1.37 | 1343 | $-2.71^{+0.08}_{-0.09}$ | $42.10^{+0.05}_{-0.04}$ | $-1.60$ (fixed) | $-1.68\pm 0.09$ | CF-HiZELS, H$\beta$ \+ [O III]
This work | 1.16 - 1.9 | 2947 | $-2.67^{+0.06}_{-0.06}$ | $42.23^{+0.04}_{-0.04}$ | $-1.50^{+0.07}_{-0.07}$ | $-1.75\pm 0.03$aaIncludes cosmic variance (§3.3) in error budget. | 3D-HST, [O III] $\lambda 5007$
This work | 1.16 - 1.9 | 2947 | $-2.68^{+0.02}_{-0.02}$ | $42.24^{+0.02}_{-0.02}$ | $-1.50$ (fixed) | $-1.76\pm 0.03$aaIncludes cosmic variance (§3.3) in error budget. | 3D-HST, [O III] $\lambda 5007$
Table 5: Redshift-Evolving Schechter Function Parameters Line | $z$ | Size | $\mathcal{L}_{{z_{1}}*}$ | $\mathcal{L}_{{z_{2}}*}$ | $\mathcal{L}_{{z_{3}}*}$ | $\log\phi_{{z_{1}}*}$ | $\log\phi_{{z_{2}}*}$ | $\log\phi_{{z_{3}}*}$ | $\alpha$ | $z_{1}$ | $z_{2}$ | $z_{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---
H$\alpha$ | 1.16 - 1.56 | 1892 | $42.31^{+0.09}_{-0.08}$ | $42.37^{+0.06}_{-0.06}$ | $42.52^{+0.09}_{-0.08}$ | $-2.84^{+0.11}_{-0.12}$ | $-2.81^{+0.09}_{-0.10}$ | $-3.02^{+0.10}_{-0.12}$ | $-1.60^{+0.08}_{-0.09}$ | 1.18 | 1.36 | 1.54
[O III] | 1.16 - 1.90 | 2947 | $42.34^{+0.10}_{-0.09}$ | $42.30^{+0.06}_{-0.05}$ | $42.56^{+0.08}_{-0.07}$ | $-3.20^{+0.15}_{-0.18}$ | $-2.87^{+0.10}_{-0.11}$ | $-3.00^{+0.13}_{-0.15}$ | $-1.94^{+0.09}_{-0.09}$ | 1.20 | 1.53 | 1.86
[O III] | 1.16 - 2.35 | 4519 | $42.09^{+0.04}_{-0.03}$ | $42.19^{+0.02}_{-0.02}$ | $42.88^{+0.04}_{-0.03}$ | $-2.70^{+0.04}_{-0.04}$ | $-2.47^{+0.03}_{-0.02}$ | $-3.01^{+0.04}_{-0.04}$ | $-1.50$ (fixed) | 1.20 | 1.76 | 2.32
### 4.1 H$\alpha$ \+ [N II] Luminosity Function
As mentioned in §1, the low resolution of the G141 grism and the morphological
broadening associated with grism observations prevent us from separating [N
II] from H$\alpha$. While prescriptions for correcting the H$\alpha$
luminosity function for [N II] do exist (e.g., Pettini & Pagel, 2004), their
applicability at high-redshift is uncertain. Not only are high-$z$
metallicities generally measured via strong line indicators, which are prone
to degeneracies and inconsistencies (e.g., Kewley & Ellison, 2008), but at
$z\gtrsim 1$, the ionization parameter of emission-line regions is typically
larger than that seen at $z\sim 0$ (see Kewley et al., 2019, and references
therein). As the ionization parameter rises, the dominant form of nitrogen
should shift to N++, thereby weakening then strength of the [N II] lines and
decreasing their contribution to the H$\alpha$ \+ [N II] complex. In this
paper, we present H$\alpha$ luminosity functions that are uncorrected for the
(presumably minor) contribution of [N II].
In Figure 2, the top plot shows the H$\alpha$ \+ [N II] luminosity function
with the three fitted parameters from Equation 5: $\alpha$, $\mathcal{L}_{*}$,
and $\log\phi_{*}$. Fits from two hundred MCMC iterations are shown in red,
and the median fit in displayed gray. The small spread in the solutions
suggests a stable and well-characterized result.
The plots in the lower triangle show the marginal posterior distributions for
the three parameters, along with 2-D contour plots of the MCMC chains. The
strong correlations in the contour plots confirm that the three parameters are
not independent. This is expected, as the Schechter parameters are not
orthogonal variables. However, the ubiquity of the function in the literature
makes continued efforts in such a parameterization worthy.
Figure 2: Result of our Bayesian MCMC computation for the H$\alpha$ \+ [N II]
luminosity function when assuming invariance over the $1.16\leq z\leq 1.95$
redshift interval. The plots in the left/lower part of the figure show the
marginal posterior distributions for each parameter and the 2-D cross-sections
of the MCMC realizations. As expected, all three parameters are tightly
correlated; this is a common feature of Schechter fits when the data extend
less then $\sim 1$ dex below $\mathcal{L}_{*}$. Nevertheless, as illustrated
by the plot on the top right, the shape of the luminosity function is very
well-defined, with the 200 randomly chosen MCMC runs (shown in red) displaying
very little scatter about the most-likely solution (plotted in gray).
While a non-evolving luminosity function for H$\alpha$ \+ [N II] $\lambda
6584$ is valuable, the non-negligible redshift range of the data,
$1.16<z<1.56$ enables us to study the luminosity function’s evolution. As
mentioned in §3.2, we did this by employing an approach based on Leja et al.
(2020), in which both $\mathcal{L}_{*}$ and $\log\phi_{*}$ are assigned to be
quadratic functions of redshift. Because of the dearth of galaxies at very low
luminosities, we forced $\alpha$ to be the same across all redshifts; this
avoids the issue of degeneracies between $\alpha$, $\mathcal{L}_{*}$, and
$\log\phi_{*}$ seen in Figure 2.
We find that the redshift-evolving luminosity function solution is quite
similar to the static case, with a statistically indistinguishable low-
luminosity slope $\alpha$ and similar $\mathcal{L}_{*}$ and $\log\phi_{*}$
values. All relevant quantities are presented in Table 5. Figure 3 shows the
best-fit luminosity functions with redshift indicated by the color. The use of
quadratic functions for $\mathcal{L}_{*}$ and $\log\phi_{*}$ allows for
smooth, differentiable evolution of the luminosity function. The red points on
top of the color curve show the $V_{\rm eff}$ result for the entire redshift
range. The points are in greater agreement with the redshift-constant
luminosity function but are still within the bounds of the redshift-varying
function showed here.
We also show how our results compare to the literature. We include comparisons
to Shim et al. (2009), who found 80 H$\alpha$ emitters at redshifts
$0.7<z<1.9$ through a Hubble-NICMOS grism survey; Colbert et al. (2013), who
analyzed a sample of 517 H$\alpha$ emitters at $0.9<z<1.5$ found through the
Hubble WISP program; and Sobral et al. (2013), who obtained a dataset of 515
H$\alpha$ emitters at $z=1.47$ through the HiZELS narrow-band imaging. To
ensure an apples-to-apples comparison, the literature luminosity functions
have been modified to reflect the inclusion of [N II] in our data. Shim et al.
(2009) and Colbert et al. (2013) assume $F_{{\rm H}\alpha}=0.71F_{{\rm
H}\alpha+{\rm[NII]}}$, while Sobral et al. (2013) uses a formula for
correcting [N II] based on equivalent width, which gives an average correction
of 25% over all their data. We undo these corrections, along with the 1 mag
internal extinction correction applied by Sobral et al. (2013).
Figure 3 shows the results. Our sample of H$\alpha$-emitting galaxies is $\sim
4$ times larger than that of any previous study. But from the figure, it is
clear that the H$\alpha$ luminosity functions of Colbert et al. (2013) and
Sobral et al. (2013) are in good agreement with our work. The only serious
discrepancy is with the curve produced by Shim et al. (2009), but since that
measurement was based on only 80 objects, this difference is not a concern.
From both Table 5 and Figure 3, we see that $\mathcal{L}_{*}$ increases with
redshift. Given that we are highly complete at all redshifts above
$\mathcal{L}>41.8$, this finding must reflect a physical difference as we go
back in cosmic time: there are more high-luminosity ELGs at earlier epochs.
From Figure 3, we also find fewer low-luminosity objects at higher redshifts,
but this result is more subject to completeness issues, and our result may not
be robust.
We explore the effects of incompleteness in Figure 4. For our main results, we
have fixed the minimum completeness fraction at 50%, i.e., we have excluded
from the analysis all objects with monochromatic fluxes fainter than the 50%
limit shown in Figure 1. But in Figure 4, we perform an experiment in which we
vary the minimum threshold from 1% to 80% and observe the effects on the best-
fit luminosity function. As shown in the left panels of the figure, the best-
fit value for $\alpha$ decreases (gets steeper), $\mathcal{L}_{*}$ increases,
and $\phi_{*}$ decreases when the minimum completeness fraction increases.
Nevertheless, from the right panel, we see that the overall luminosity
function does not vary significantly in the luminosity range we are able to
observe. This is because the three Schechter parameters are correlated:
various sets of values can lead to the same overall luminosity function. We
see from this experiment that the decision of which flux measurements to
include changes the parameterization of the luminosity function, but not the
overall shape curve.
Figure 3: Redshift evolution of the H$\alpha$ \+ [N II] luminosity function.
The curves show the MCMC solutions for the redshift-varying Schechter (1976)
function, using each redshift’s median parameter values (closely related to
the highest-likelihood solution). Also shown are the best-fit curves from Shim
et al. (2009), Colbert et al. (2013), and Sobral et al. (2013). Our results
are in general agreement with the literature, especially the results of
Colbert et al. (2013). The MCMC results are also compatible with our $V_{\rm
eff}$ data point (red triangles). In regard to evolution, we find that the
knee of the luminosity function, $\mathcal{L}_{*}$, increases with redshift
while the normalization factor $\log\phi_{*}$ decreases, although this may not
be due to true redshift evolution (see text for more details). In any case,
the evolution is not particularly strong in our $[1.16,1.56]$ redshift range.
Figure 4: Effects of the minimum completeness fraction considered for
H$\alpha$ fluxes on the fitted parameters. The left panels show that $\alpha$
decreases (gets steeper), $\mathcal{L}_{*}$ increases, and $\phi_{*}$
decreases as the completeness limit for the analysis increases. This
experiment demonstrates the importance of completeness in determining the
Schechter (1976) function parameters. The right panel shows the luminosity
functions for the Schechter parameters in the left panels. The lines are
colored according to the minimum completeness fraction. We can see that the
effects of completeness on the shape of the luminosity function are much less
pronounced than that variables used in the parameterization.
### 4.2 Cosmic Star Formation Rate Density
The evolution of the star formation rate density (SFRD) of the universe is an
important indicator of galaxy growth and evolution. As reviewed by Madau &
Dickinson (2014), we have the general picture that star formation peaked
around $z\sim 2$, an era dubbed as “cosmic noon,” and has been declining at
$\sim 0.1$ dex per Gyr ever since. However, while this outline is known,
further measurements of the SFRD, especially for certain populations of
galaxies such as ELGs, are useful for improving our knowledge of the evolution
of star formation and quantifying how selection effects propagate into this
understanding. In this section, we use the H$\alpha$ luminosity function to
calculate the SFRD between $1.16\leq z\leq 1.56$.
#### 4.2.1 Conversion between H$\alpha$ luminosity and Star Formation Rate
H$\alpha$ luminosity is often used as a direct proxy for very recent star
formation (under 10 Myr; e.g., Kennicutt & Evans, 2012). As such, applying a
conversion formula from H$\alpha$ luminosity to star formation rate (SFR) is a
common process. Typically, the conversion from Hao et al. (2011) and Murphy et
al. (2011) as compiled by Kennicutt & Evans (2012) is used.
Nevertheless, the aforementioned relation is calibrated in the local universe,
therefore raising the concern of its application to higher redshifts. For
example, at lower metallicities, main sequence stars tend to be bluer and
hotter, changing the amount of ionizing radiation emitted by massive stars,
and thus the calibration.
Furthermore, the H$\alpha$-SFR calibration summarized by Kennicutt & Evans
(2012) applies to dust-corrected H$\alpha$ brightness, and at higher
redshifts, the details of dust attenuation constitute a major source of
uncertainty (e.g., Bouwens et al., 2012; Nagaraj et al., 2021a). This error
propagates directly into the SFR conversion, and is then compounded by the
fact that our H$\alpha$ data is contaminated by [N II] $\lambda\lambda
6548,6584$, while the SFR calibration of Kennicutt & Evans (2012) is for
H$\alpha$ only. Thus, the applicability of the local conversion of H$\alpha$
luminosity to SFR is unclear.
We can address this issue directly by calculating our own relation between SFR
and observed H$\alpha$ \+ [N II] luminosity. In Paper I, we discussed the SED
fitting procedure of the entire sample of $1.16\leq z\leq 1.90$ 3D-HST
emission-line galaxies. We used the Bayesian MCMC SED code MCSED (Bowman et
al., 2020) to estimate the physical properties of our ELG sample, assuming a
Kroupa (2001) initial mass function (IMF), a single-valued (but free) stellar
metallicity, a binned star formation history, with the SFR in each bin a free
parameter, and a dust attenuation parameterization from Noll et al. (2009) and
Kriek & Conroy (2013).
MCSED uses simple stellar population (SSP) SEDs from the Flexible Stellar
Population Synthesis (FSPS) library (Conroy et al., 2009; Conroy & Gunn, 2010)
with Padova isochrones (Bertelli et al., 1994; Girardi et al., 2000; Marigo et
al., 2008). Nebular emission is treated via interpolation in tables of CLOUDY
models (Ferland et al., 1998, 2013) computed by Byler et al. (2017). Gas-phase
metallicity ($Z_{\rm gas}$) is set equal to the stellar metallicity in MCSED,
and we fixed the ionization parameter at $\log U=-2.5$ based on the high [O
III]/H$\beta$ ratios observed for our sources.
Our SED fits are based primarily on the 3D-HST photometry collected by Skelton
et al. (2014), which consists of 147 filter bands distributed over the five
CANDELS fields and covering the wavelength range from 3,000 Å to 80,000 Å
(observed frame). In Paper I, we merged these data with photometry from Swift
and GALEX, which extended the wavelength coverage for over 400 sources to
$\sim 2,000$ Å (observed-frame). Also, mid- and far-IR measurements from
Spitzer and Herschel were added for over 600 sources in Paper II, but to
maintain consistency within the sample, we do not include these dust-sensitive
wavelengths here. In addition, MCSED is able to employ emission line fluxes in
its SED-solution, making it an ideal analysis tool for grism-based surveys. We
included H$\alpha$, H$\beta$, and [O III] $\lambda 5007$ fluxes for galaxies
whenever available.
Our MCSED-based SFR estimates are fairly robust. We find that the mean
uncertainty on our SFRs is $0.24$ dex, with a standard deviation of $0.08$
dex. Moreover, in Paper I, we examined how the basic assumptions underlying
our SED fits affected the derived properties of the 3D-HST galaxies. For
example, we found that changing the ionization parameter, modifying the
weights assigned to the emission line fluxes, or fixing galaxy metallicity all
led to statistically indistinguishable distributions for the galaxies’ SFRs.
In other words, ionization parameter, metallicity, and the inclusion of
emission line fluxes do not noticeably affect the systematics of an SFR
measurement. In addition, Bowman et al. (2020) showed that for a large sample
of 3D-HST galaxies, the choice of dust attenuation curve does not strongly
affect the SFR estimate.
On the other hand, we do find that the details of a galaxy’s assumed star
formation history (SFH) do affect our SFR estimates. Our fits are based on a
“non-parametric” SFH, i.e., one in which the SFR of each epoch in a galaxy’s
history is fit independently of the other epochs. Such fits have been proven
to reduce a bias in SFR measurements that is introduced by the use of
parameterized SFHs (e.g., Conroy, 2013; Leja et al., 2017, 2019; Bowman et
al., 2020), though the magnitude of this bias reduction depends on the prior
used in each age bin (Leja et al., 2019). Our quoted SFRs represent the star
formation rate during the most recent age bin, i.e., over the last 100 Myr of
cosmic time.
Given that we are trying to calibrate the SFR-H$\alpha$ relation, the lack of
influence of the H$\alpha$ flux on the SFR measurement suggests that
correlations found between SFR and H$\alpha$ are not artificially induced by
the method of measuring SFR. Furthermore, the benefit of using MCSED SFRs is
that we have a way of connecting the observed H$\alpha$ \+ [N II] fluxes with
SFRs that takes into account dust attenuation and contamination by [N II].
This reduces the bias and uncertainty introduced by applying single values for
the dust and [N II] corrections.
To calculate the mean SFR-H$\alpha$ relation, we use a procedure very similar
to that employed by Bowman et al. (2021) in their analysis of the [O III]
$\lambda 5007$ luminosity function. We first divide the H$\alpha$ \+ [N II]
luminosities into 25 bins with equal numbers of objects each interval. In each
bin, we adopt the mean linear SFR and linear H$\alpha$ luminosity as
representative values. We then take the logarithm of those values and fit a
line. We chose this process because if we average the measurements in
logarithmic space, we would underestimate the total SFR of the population. As
suggested by Feigelson & Babu (1992), we use the orthogonal distance
regression technique to fit the line, since there is non-negligible
heteroscedasticity associated with the measurement errors for both luminosity
and SFR.
To estimate errors in our solution, we bootstrap the H$\alpha$ \+ [N II]
luminosity measurements and repeat the procedure above 1000 times. This error
estimation process is the same as that described in §3.2 for the $V_{\rm eff}$
method.
We present the results for our sample in Figure 5. The data are plotted as
blue dots, while the linear mean binned values are shown as amber diamonds.
Finally, the best-fit line is shown in red, with the $1\sigma$ uncertainty
displayed via the shaded region (nearly too small to be visible). The
correlation between apparent H$\alpha$ \+ [N II] luminosity and SFR is quite
clear.
The best-fit relation is given by Equation 14. The line has a slope consistent
with $1$, suggesting that the relation between SFR and uncorrected H$\alpha$
\+ [N II] luminosity is very close to linear. To be consistent with the SFRD
compilation of Madau & Dickinson (2014), we have divided the (linear) SFR by
$0.67$ as done in their work; this converts an SFR based on the Kroupa (2001)
IMF to one based on the Salpeter (1955) IMF with limits of 0.1 and
$100\,M_{\odot}$. In the equation $L_{{\rm H}\alpha+{\rm[NII]}}$ is in units
of erg s-1 and SFR is in units of $M_{\odot}$/year.
$\begin{split}\log{\rm SFR}&=(1.02\pm 0.02)(\log L_{\rm uncorr}-42)\\\
&+(1.056\pm 0.006)-\log(0.67)\end{split}$ (14)
Figure 5: Calibration between SFR and observed (not corrected for dust)
H$\alpha$ \+ [N II] luminosity for the sample used in this paper. We show the
data as blue dots, the logarithm of the mean linear SFR and luminosity in 25
bins as amber diamonds, and the best-fit line to the binned mean values in
red, which has a slope nearly identical to one (a linear relationship in
linear space). There is a clear correlation in the data.
To be complete, we also calculate the SFRD using the Kennicutt & Evans (2012)
relation (shown below in Equation 15), after applying corrections for uniform
dust attenuation and [N II] contamination.
$\log{\rm SFR}=\log L_{{\rm H}\alpha}-41.27-\log(0.67)$ (15)
Both Shim et al. (2009) and Colbert et al. (2013) use a 29% correction for [N
II], whereas Sobral et al. (2013) use a formula based on equivalent width and
find a median correction of 25%. The 29% correction, which we now adopt, is
equivalent to a $-0.15$ dex shift in log luminosity.
To test the appropriateness of this 29% correction ($\log$ [N II]/H$\alpha$
$=-0.54$), we determined how [N II]/H$\alpha$ should vary as a function of
population age and metallicity using the CLOUDY (Ferland et al., 1998, 2013)
nebular emission tables created by Byler et al. (2017), under the assumption
of $\log U=-2.5$ (the same value used in our MCSED fits). We show the result
in Figure 6. According to the CLOUDY lookup tables, the correction is valid
only for most galaxies with $\log(Z/Z_{\odot})\gtrsim-0.2$. For galaxies with
$\log(Z/Z_{\odot})\lesssim-0.2$, the true relative strength of [N II] is lower
than the correction.
Figure 6: CLOUDY prediction for $\log$ [N II]/H$\alpha$ as a function of
metallicity and age when $\log U=-2.5$. While $\log$ [N II]/H$\alpha$ is
nearly independent of age, it is highly dependent on metallicity. For galaxies
with $\log(Z/Z_{\odot})\lesssim-0.7$, [N II] emission is less than 1% of the
H$\alpha$ emission.
Stellar metallicity is difficult to measure using SED fits to mostly broadband
photometric data (e.g., Conroy, 2013; Lower et al., 2020). Nevertheless, from
our analysis (see Paper I) we find that 82% of the galaxies in our sample have
$\log(Z/Z_{\odot})<-0.2$; this is consistent with the relatively small stellar
masses of the galaxies (median $\log M_{*}/M_{\odot}\sim 9.5$). In other
words, for the majority of H$\alpha$-emitting galaxies, our 29% correction
overestimates the contamination by [N II], and thus underestimates the
population’s SFRD. Our MCSED-calibration bypasses this issue by removing the
need to correct for [N II].
Topping et al. (2021) do a stacking analysis on $z\sim 1.5$ galaxies found in
the Multi-Object Spectrometer for Infra-Red Exploration Deep Evolution Field
(MOSDEF) survey. They find that $\log$ [N II]/H$\alpha$ is a strong function
of stellar mass, which is correlated with metallicity. From their Figure 2, we
notice that for galaxies with $\log(M_{*}/\hbox{$M_{\odot}$})\lesssim 10.5$,
the 29% correction for [N II] is an overestimate. We find that 81% of our
galaxies have masses $\log(M_{*}/\hbox{$M_{\odot}$})<10.5$, which is in
perfect agreement with the aforementioned finding using CLOUDY lookup tables.
As for dust, based on results from Paper I, we find that the average
differential extinction, $E(B-V)$, for our galaxy sample is 0.16 mag. To
calculate $A({\rm H}\alpha)$ for our sample, we use the conversions given by
Reddy et al. (2020),
$\displaystyle E(B-V)_{\rm nebular}$ $\displaystyle=2.07E(B-V)_{\rm stellar}$
(16) $\displaystyle A({\rm H}\alpha)$ $\displaystyle=2.66E(B-V)_{\rm nebular}$
(17)
and apply an average value of $A(H\alpha)=0.88$ to the entire sample. The
combination of the [N II] and dust corrections results in a multiplicative
factor of $1.60$.
#### 4.2.2 Star Formation Rate Density Results
Given the relation between SFR and uncorrected H$\alpha$ \+ [N II] luminosity
(Equation 14), we can calculate the total SFRD using the luminosity function
results in this paper. If $\phi(L)$ is the true luminosity function,
$\textrm{SFR}(L)$ is the relation given by Equation 14, and $L$ is the
uncorrected H$\alpha$ \+ [N II] luminosity, then the total SFRD contained in
3D-HST ELGs brighter than some luminosity $L_{\rm min}$ is
${\rm SFRD}(z)=\int_{L_{\rm min}}^{\infty}{\rm SFR}(L)\,\phi(L,z)\,dL$ (18)
Alternatively, we can use the local calibration between SFR and dust-corrected
H$\alpha$ luminosity (Equation 15) and apply the average dust and [N II]
corrections as described in §4.2.1. The net effect of these two factors is
included in the constant $k=1.6$, and
${\rm SFRD}(z)={\rm SFR}_{\rm corr}\left(k\,\int_{L_{\rm
min}}^{\infty}L\,\phi(L,z)\,dL\right)$ (19)
which simplifies to
${\rm SFRD}(z)={\rm SFR}_{\rm corr}\left(k\,\Gamma\left(\alpha+2,\frac{L_{\rm
min}}{L_{*}}\right)\,\phi_{*}(z)\,L_{*}(z)\right)$ (20)
where $\Gamma$ represents the incomplete gamma function.
Figure 7 shows the evolution of the cosmic SFRD based on results compiled by
Madau & Dickinson (2014) from both UV-based and IR-based measurements, along
with the best-fit curve from their review paper. We also include SFRD
measurements from Gruppioni et al. (2020) using sub-mm data. Our SFRD results
for both the non-evolving and redshift-varying luminosity functions, based on
our SED-based SFR calibration, are shown as a green star and red line,
respectively. Our SFRD measurement derived from the non-evolving luminosity
function with the Kennicutt & Evans (2012) relation is shown as an amber
diamond. In order to make a fair comparison to the literature, we use $L_{\rm
min}=0.03\mathcal{L}_{*}$, as adopted by Madau & Dickinson (2014).
Our measurements derived via the SFR calibration shown in Figure 5 yield an
SFRD that is larger by $0.12$ dex than that found using the local
H$\alpha$-SFR calibration given by Kennicutt & Evans (2012). This discrepancy
is consistent with the conclusion of Section 4.2.1, i.e., that the application
of a 29% correction for the contribution of [N II] to the H$\alpha$
measurement leads to an underestimate in star-formation rate and the epoch’s
SFRD. We therefore believe that our SED-based SFR calibration is more
accurate. However, we do note that the uncertainty on the local-calibration-
based SFRD is underestimated since the uncertainties on the dust and [N II]
corrections are not propagated into the analysis. Thus, the two numbers may
still be consistent.
We also observe that the cosmic SFRD calculated for our sample of ELGs is at
least $\sim 0.09$ dex below the value expected from best-fit curve of Madau &
Dickinson (2014). This implies that H$\alpha$-selected galaxies contain
$\lesssim 81\%$ of the star formation in the $z\sim 1.4$ universe.
The difference between the cosmic SFRD at $z\sim 1.4$, as determined by Madau
& Dickinson (2014), and our H$\alpha$-based measurement suggests that not all
of the epoch’s star-formation is detectable via surveys for rest-frame optical
emission lines. Such a result is easily explained if some star-forming
galaxies are heavily obscured by dust; since emission-line gas is generally
attenuated more than star-light, this is a reasonable hypothesis (e.g.,
Charlot & Fall, 2000; Calzetti et al., 2000; Reddy et al., 2020). These dusty
star forming galaxies have been shown to contribute significantly to the star
formation rate around cosmic noon, i.e., $1<z<3$ (e.g., Casey et al., 2013).
At higher redshifts ($z\gtrsim 3$), far-IR and sub-mm studies have shown that
UV-based studies miss a significant portion of the star formation due to dust
attenuation in extremely high-SFR galaxies (e.g., Gruppioni et al., 2015;
Rowan-Robinson et al., 2016; Wang et al., 2019; Williams et al., 2019;
Gruppioni et al., 2020; Loiacono et al., 2021; Khusanova et al., 2021).
Nevertheless, the discrepancy between the UV-NIR and FIR-sub-mm methods of
calculating star formation (e.g., see the extensive discussion by Katsianis et
al., 2021) is much less pronounced at the redshifts of our sample. For
example, in Figure 7, we include sub-mm results from Gruppioni et al. (2020)
and find that for $z=1$ and $z=2$, the results are still consistent with the
Madau & Dickinson (2014) best-fit curve.
Coming back to our results, we cannot discount the possibility that our SFRD
is consistent with the Madau & Dickinson (2014) curve, as the discrepancy is
still within the variance of the literature values. Thus, there is no
conclusive evidence that the emission line surveys are missing a significant
fraction of sources found in UV-based and/or IR/sub-mm-based surveys between
redshifts $1.16<z<1.56$.
While the redshift-evolving SFRD is not a flat curve, given the non-negligible
uncertainties due to cosmic variance, we cannot dismiss the possibility of no
evolution. In other words, our H$\alpha$ emission-line measurements find no
strong evidence for SFRD evolution over our $1.16<z<1.56$ redshift range.
Figure 7: Our calculated star formation rate density (SFRD) compared to
literature values (UV-based results are in blue dots, IR results in red, and
sub-mm-based in brown) and the best-fit curve (shown in black) from Madau &
Dickinson (2014). The green star and red line show the SFRD when using our
MCSED-based H$\alpha$ \+ [N II] SFR calibration with the static and redshift-
varying luminosity functions, respectively. By directly connecting the
observed H$\alpha$ +[N II] fluxes to SFR, we bypass the issues associated with
dust attenuation and [N II] contamination, while propagating in their
uncertainties. The amber diamond shows the SFRD calculated using the non-
evolving luminosity function with the Kennicutt & Evans (2012) H$\alpha$-SFR
calibration; here the uncertainties from dust and [N II] are not included in
the error bar. This latter calibration produces an SFRD that is smaller by
$0.12$ dex than SED-based value; this is consistent with the overestimate of
[N II] in low-mass (low-metallicity) galaxies. At $z\sim 1.4$, the SFRD from
ELGs is $\gtrsim 0.09$ dex lower than the Madau & Dickinson (2014) curve, but
this difference is still within the variance seen in the literature. We are
not able to find conclusive evidence of SFRD evolution between $1.16\leq z\leq
1.56$.
### 4.3 [O III] $\lambda 5007$ Luminosity Function
Unlike H$\alpha$, [O III] $\lambda 5007$ does not suffer from uncertainties
due to blending. At the redshifts under consideration, the G141 grism has
enough resolution and the sources are sufficiently small so that [O III] and
H$\beta$ are easily distinguishable. [O III] $\lambda 5007$ is blended with [O
III] $\lambda 4959$, but since the ratio of the two lines is fixed by basic
physics (2.98:1; Storey & Zeippen, 2000), converting the observed [O III]
feature into [O III] $\lambda 5007$ is trivial. We quote only the luminosity
of [O III] $\lambda 5007$ in the subsequent analysis.
In Figure 8, we show our fitted non-evolving [O III] $\lambda 5007$ luminosity
function (over the range $1.16<z<1.90$) and place it in the context of the
literature. This includes fits to 192 $0.7<z<1.5$ and 58 $1.5<z<2.3$ [O III]
emitters by Colbert et al. (2013), 371 $z=1.42$ [O III] + H$\beta$ emitters
from Khostovan et al. (2015), and 1343 $z\sim 1.4$ [O III] + H$\beta$ emitters
from Sobral et al. (2015). For our comparison, the [O III] $\lambda\lambda
4959,5007$ values of these works have been converted to [O III] $\lambda 5007$
to match our measurements.
The studies by Khostovan et al. (2015) and Sobral et al. (2015) identify
H$\beta$ emitters as well as [O III] galaxies, since their narrow-band
photometry is unable to distinguish the two object classes without
spectroscopic follow-up. Thus, it is unclear how their samples compare to
ours. Sobral et al. (2015) find that in their set of sources with
spectroscopic confirmation, H$\beta$ emitters constitute around 16% of their
$z\sim 1.4$ sample. Interestingly, they notice that these H$\beta$ emitters
tend to have lower luminosities than the [O III]-identified emitters, and this
may be reflected in the lower $\mathcal{L}_{*}$ values implied by the plot.
Still 5/6’s of their (spectroscopically confirmed) sample is made up of [O
III] emitters, so the effect of these contaminants should not be large.
Khostovan et al. (2015) find a similar phenomenon in their sample.
As shown in Figure 8, at $41.5\lesssim\log L\lesssim 42.3$, our luminosity
function predicts more [O III] galaxies than any of the other studies, though
at higher luminosities our results are well within the bounds of the
literature. Moreover, our MCMC result is in good agreement with the $V_{\rm
eff}$ result (blue triangles).
There are a few factors that may lead to our distinct result for [O III]. The
first is that our redshift range is unique. No other study focuses
specifically on $1.16<z<1.90$ galaxies. As the luminosity function is an
evolving quantity, the particular redshifts involved play a role in
determining the measured parameters. Another important consideration is that
our sample size is the largest to date, and the larger the sample size, the
more accurate the luminosity function. In fact, as shown in Table 4, the value
for $\log\,\int_{0.03L^{*}}^{\infty}\phi(L)\,dL$ derived by Sobral et al.
(2015) is closest to our study, and it is also the measurement that is most
consistent with our value.
Alternatively, we note that our analysis encounters problems at both the faint
and bright ends of the luminosity function. At the faint end, we are subject
to rapid loss of completeness while at the bright end, our measurements of the
luminosity function suffer from cosmic variance as we are dealing with small
numbers of galaxies. Coupled with degeneracies between Schechter parameters,
these effects can lead to divergences between different studies.
Extrapolations to low-luminosity galaxies, especially when $\alpha$ is fixed,
are often uncertain and should be considered as such. Nevertheless, given our
large sample size and the number of galaxy measurements at luminosities
significantly less than $L_{*}$, we believe that the differences between our
results and those of previous studies are real and not an artifact of our
analysis.
Figure 8: [O III] $\lambda 5007$ luminosity function from this work (MCMC as
the red curves and $V_{\rm eff}$ as blue triangles), as well as fits from
Colbert et al. (2013), Khostovan et al. (2015), and Sobral et al. (2015). We
predict higher counts of intermediate-luminosity galaxies than the other
studies, but at the high-luminosity end, our data are well within the range of
values previously derived. There is good agreement between the MCMC and
$V_{\rm eff}$ methods.
In Figure 9, we combine our [O III] $\lambda 5007$ measurements with those of
Bowman et al. (2021) to show the redshift evolution of the [O III] $\lambda
5007$ luminosity function between $1.16\leq z\leq 2.35$. For this analysis, we
fix $\alpha=-1.5$, as it very difficult to fit $\alpha$ as a free parameter
when the limiting luminosity depends so much on redshift. Like in Figure 2, we
show the 1D and 2D cross sections of the MCMC chains in the lower left panels.
Once again, we find that $\mathcal{L}_{*}$ and $\log\phi_{*}$ are correlated,
but only at the same redshift. In other words, $\mathcal{L}_{1*}$ and
$\log\phi_{1*}$ are highly correlated but $\mathcal{L}_{1*}$ and
$\log\phi_{2*}$ are not.
The data of Figure 9 show that the characteristic luminosity $\mathcal{L}_{*}$
increases with redshift. Moreover, as redshift increases, the overall
luminosity function increases at all but the lowest luminosities. In other
words, there are many more [O III] $\lambda 5007$-visible galaxies, and
especially more [O III] $\lambda 5007$-bright systems, at earlier epochs of
cosmic history. This is consistent with the findings of Zeimann et al. (2014),
Khostovan et al. (2015), and Bowman et al. (2019), among others, that show the
prevalence of [O III] compared to [O II] $\lambda 3727$ at high redshift.
Figure 9: MCMC result for the redshift varying [O III] $\lambda 5007$
luminosity function. In the lower left, we show the 1D and 2D cross sections
of the parameter chains. Note that at each redshift, $\mathcal{L}_{*}$ and
$\log\phi_{*}$ are correlated, but the correlation is not strong across
redshifts. The upper-right panel shows the evolution of the luminosity
function. Our data show that $\mathcal{L}_{*}$ increases with redshift, and,
at most luminosities, the luminosity function is larger at higher redshift. In
other words, [O III] $\lambda 5007$-bright galaxies were much more common at
$z\sim 2$ than at $z\sim 1$.
### 4.4 Number Counts
The precision to which one can measure cosmological parameters through galaxy
surveys depends on the number of galaxies and the square of the bias of the
observed galaxy population relative to dark matter. Surveys planned for Euclid
and Roman will observe millions of H$\alpha$-visible galaxies at $0.9\lesssim
z\lesssim 1.8$ and [O III]-visible galaxies at $1.5\lesssim z\lesssim 2.7$,
and thus measure quantities such as the angular diameter distance and Hubble
parameter at these distant epochs. Specifically, the Euclid Wide Survey (WS)
will observe $\sim 15000~{}{\rm deg}^{2}$ of the sky down to a flux limit of
$\sim 2\times 10^{-16}$ $\text{erg cm}^{-2}\text{ s}^{-1}$ (Euclid
Collaboration et al., 2022), while the Roman High Latitude Survey (HLS) will
observe $\sim 2200~{}{\rm deg}^{2}$ down to a flux limit of $\sim 6\times
10^{-17}$ $\text{erg cm}^{-2}\text{ s}^{-1}$ (Spergel et al., 2015). The
Euclid Deep Survey will observe $50~{}{\rm deg}^{2}$ in three fields to a
similar flux limit (Vavrek et al., 2016).
In this section, we use our measurements of the H$\alpha$ and [O III] $\lambda
5007$ luminosity functions to calculate the number of galaxies these surveys
are likely to measure, corrected for completeness. One factor to note is that
since we have removed AGN from the sample, the number counts we derive will be
a slight underestimate.
We begin with the total number of galaxies with luminosities greater than
$L_{\rm min}(z)$ from redshifts $z_{0}$ to $z_{1}$ over area $\Omega$,
$N_{\rm tot}=\int
d\Omega\int_{z_{0}}^{z_{1}}dz\frac{dV}{dz\,d\Omega}\int_{L_{\rm
min}(z)}^{\infty}dL\,\Omega(L,z)\phi(L,z)$ (21)
Given the Schechter function with parameters $\alpha$, $L_{*}$, and $\phi_{*}$
(that are assumed to be invariant over the effective survey area
$\Omega_{0}$), we can simplify Equation 21 to
$N_{\rm
tot}=\Omega_{0}\int_{z_{0}}^{z_{1}}dz\frac{dV}{dzd\Omega}\,\Gamma\left(\alpha+1,\frac{L_{\rm
min}(z)}{L_{*}}\right)\phi_{*}(z)L_{*}(z)$ (22)
where $\Gamma$ once again represents the incomplete gamma function.
In Figure 10, we show the total number of galaxies per square degree that are
expected to have H$\alpha$ at $1.2\leq z\leq 1.6$ (left) and [O III] $\lambda
5007$ at $1.5\leq z\leq 1.9$ (right) above a given threshold. We calculate
these values using Equation 22 with $\Omega_{0}=1$ deg2, while translating
$L_{\rm min}$ to emission line flux using the luminosity distance $D_{L}$ via
$F_{\rm min}=\frac{L_{\rm min}(z)}{4\pi D_{L}(z)^{2}}$ (23)
For the figure, we use the non-evolving luminosity functions (though the
redshift varying luminosity functions yield similar results) and perform the
same calculations for the luminosity functions given in the literature. In all
cases, the values represent galaxy number counts, assuming 100% completeness.
In addition, we include the direct measurements of counts obtained by Bagley
et al. (2020) using samples of ELGs from the WFC3 Infrared Parallel
Spectroscopic Survey (WISPS), 3D-HST, and A Grism H-Alpha SpecTroscopic survey
(AGHAST). This work also made corrections for completeness.
In the left panel, we observe that our H$\alpha$ counts are in excellent
agreement with those of Colbert et al. (2013) at all values of the limiting
flux. The counts also agree with those of Bagley et al. (2020) around the
limiting flux of the Euclid WS, but less so at brighter fluxes. Given the
consistency of our results with those of Colbert et al. (2013) and Bagley et
al. (2020), there is no need to update the prediction of $\sim 3300$ deg-2 for
the Euclid WS at 100% completeness (Bagley et al., 2020; Euclid Collaboration
et al., 2022) and 16.4 million $7\sigma$ H$\alpha$ galaxy detections at
$1.06<z<1.88$ for the Roman HLS assuming 70% completeness (Spergel et al.,
2015).
On the other hand, our result for [O III] (right side of Figure 10) is
distinct from the literature. For limiting fluxes of 1 to $3\times 10^{-16}$
$\text{erg cm}^{-2}\text{ s}^{-1}$, our predicted galaxy counts agree with the
results of Colbert et al. (2013) and Bagley et al. (2020). As these limits are
similar to those of the Euclid WS ($2\times 10^{-16}$ $\text{erg
cm}^{-2}\text{ s}^{-1}$) and the $\geq 7\sigma$ limit of $10^{-16}$ $\text{erg
cm}^{-2}\text{ s}^{-1}$ used by Spergel et al. (2015) for predicting Roman HLS
counts, our results do not change the predictions for these surveys; at 70%
completeness, the Roman HLS should detect 1.4 million [O III] $1.88<z<2.77$
galaxies with $>7\sigma$ confidence (Spergel et al., 2015). However, as we
push to lower flux limits, our luminosity function predicts higher counts. For
example, for the Roman HLS nominal flux limit of $6.0\times 10^{-17}$
$\text{erg cm}^{-2}\text{ s}^{-1}$, we predict twice as many galaxies as
Colbert et al. (2013) in the redshift range $1.5<z<1.9$.
Figure 10: Predictions for total number of galaxies per square degree
corrected for completeness, as a function of the limiting flux for H$\alpha$
at $1.2\leq z\leq 1.6$ (left) and [O III] $\lambda 5007$ at $1.5\leq z\leq
1.9$ (right). We use our non-evolving luminosity functions for these
calculations and include cosmic variance in the error budget. Our H$\alpha$
counts agree remarkably well with those from Colbert et al. (2013) at all
limiting fluxes as well as with the direct counts from Bagley et al. (2020)
around $F_{\rm lim}\sim 2\times 10^{-16}$ $\text{erg cm}^{-2}\text{ s}^{-1}$.
This agreement suggests no changes to the H$\alpha$ predictions for Euclid and
Roman are necessary. While our [O III] $\lambda 5007$ count predictions are
distinct from the literature, they still do not greatly change the expected
counts from Euclid. However, for the HLS flux limit of $6.0\times 10^{-17}$
$\text{erg cm}^{-2}\text{ s}^{-1}$, we predict twice as many galaxies as
Colbert et al. (2013).
## 5 Conclusion
In Papers I and II, we used the G141 grism data from the Hubble 3D-HST
Treasury program (GO-11600, 12177, 12328; Brammer et al., 2012; Momcheva et
al., 2016) to identify a clean sample of 4350 normal (i.e., non-AGN) star-
forming emission lines galaxies in the redshift range $1.16\leq z\leq 1.90$.
In this work, we use the line fluxes of these galaxies, along with a similar
set of fluxes measured by Bowman et al. (2021), to measure the galaxies’
emission-line luminosity functions. These data include 1892 H$\alpha$ emitting
galaxies between $1.16\leq z\leq 1.56$ and 4519 [O III] emitters with
$1.16\leq z\leq 2.35$, all with line fluxes above the 3D-HST 50% completeness
limit. While there have been several previous efforts to calculate H$\alpha$
and [O III] luminosity functions in these redshift ranges (see §1 for a
detailed list), none used the large sample sizes used here.
We employ a generalization of the classical $1/V_{\rm max}$ method to derive
the emission-line luminosity functions for our entire sample of galaxies, and
samples of galaxies broken down by redshift. We then use Markov Chain Monte
Carlo (MCMC) Bayesian techniques to fit these data to the Schechter (1976)
luminosity function with $\alpha$ held constant across redshift. We find very
good agreement between our H$\alpha$ results and those from the literature,
and our [O III] luminosity function is also a good match to prior measurements
for line luminosities brighter than $\log L=42.3$ (ergs s-1). However, at
fainter [O III] luminosities (where completeness corrections might be an
issue), we infer an excess of objects. These results are shown in Figures 2,
3, 8, and 9 and summarized in Tables 3 \- 5.
We also compute the star formation rate density (SFRD) of the $1.16\leq z\leq
1.56$ epoch using the H$\alpha$ luminosity function. We find that our SFRD is
$\sim 19\%$ smaller than the best-fit $z\sim 1.4$ value found by Madau &
Dickinson (2014) (Figure 7), though this discrepancy is within the variance
found in the literature. If the difference is real, then one possible
explanation is that not all $z\sim 1.4$ star formation takes place in galaxies
with observable H$\alpha$ emission lines. In particular, surveys such as
3D-HST will miss heavily obscured galaxies where the emission lines are too
extinguished to make it into the sample. We find no evidence for or against
cosmic evolution of the SFRD between $1.16<z<1.56$
Finally, we predict total galaxy counts per square degree as a function of the
limiting flux (Figure 10). For H$\alpha$ \+ [N II] $\lambda 6584$, our results
are consistent with those from Colbert et al. (2013) at all limiting fluxes
and Bagley et al. (2020) down to a limiting flux of $\sim 2\times 10^{-16}$
$\text{erg cm}^{-2}\text{ s}^{-1}$, suggesting the previous predictions for
the Euclid (Euclid Collaboration et al., 2022) and Roman (Spergel et al.,
2015) surveys are accurate. For [O III] $\lambda 5007$, our numbers agree with
previous estimates for the Euclid Wide Survey, but depending on where exactly
we define the flux limit for the Roman High Latitude Survey, our data may
imply a significantly larger number of detectable galaxies. For example, at
$10^{-16}$ $\text{erg cm}^{-2}\text{ s}^{-1}$, our results are consistent with
the analysis of Colbert et al. (2013), but at $6\times 10^{-17}$ $\text{erg
cm}^{-2}\text{ s}^{-1}$, our number counts are higher by a factor of two.
Roman may find more faint [O III] emitters than previously anticipated.
The $\Lambda$CDM paradigm has garnered many resounding successes in explaining
observations of our universe at a variety of scales. However, there are still
inconsistencies and unknowns leaving the cosmological model incomplete. To
better constrain the model as well as alternate or additional theories, we
need to continue honing our observations. Large galaxy surveys represent an
important avenue to constrain cosmological parameters through the measurement
of baryonic acoustic oscillations and redshift space distortions.
Galaxy surveys with precise redshifts will be especially useful for generating
the necessary constraints, and IFU and slitless spectroscopy are the most
efficient ways of performing these surveys. In the near future, Euclid and
Roman will greatly enhance samples of emission line galaxies. The similarities
between 3D-HST and these planned surveys make it a perfect pathfinder mission.
Our measurements of the H$\alpha$ and [O III] $\lambda 5007$ luminosity
functions with galaxy samples that are several times larger than any previous
study of the $z\sim 1.5$ redshift range help cement the predictions for the
expected yield of the ongoing and future surveys.
We thank the anonymous referee for their insightful advice that helped make
the paper more thorough. This work has made use of the Rainbow Cosmological
Surveys Database, which is operated by the Centro de Astrobiología (CAB/INTA),
partnered with the University of California Observatories at Santa Cruz
(UCO/Lick,UCSC). This work is based on observations taken by the CANDELS
Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. This research has made use of NASA’s Astrophysics Data
System. This research has made use of the SVO Filter Profile Service
(http://svo2.cab.inta-csic.es/theory/fps/) supported from the Spanish MINECO
through grant AYA2017-84089. Computations for this research were performed on
the Pennsylvania State University’s Institute for Computational and Data
Sciences’ Roar supercomputer. The Institute for Gravitation and the Cosmos is
supported by the Eberly College of Science and the Office of the Senior Vice
President for Research at the Pennsylvania State University. This material is
based upon work supported by the National Science Foundation Graduate Research
Fellowship under Grant No. DGE1255832. Any opinion, findings, and conclusions
or recommendations expressed in this material are those of the authors(s) and
do not necessarily reflect the views of the National Science Foundation.
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|
# Supercurrent-induced Anomalous Thermal Hall Effect as a New Probe to
Superconducting Gap Anisotropy
Xiaodong Hu Department of Physics, Boston College, Chestnut Hill,
Massachusetts 02467, USA Jung Hoon Han Department of Physics, Sungkyunkwan
University, Suwon 16419, Korea Ying Ran Department of Physics, Boston
College, Chestnut Hill, Massachusetts 02467, USA
###### Abstract
Two-dimensional superconductors have been realized in various atomically thin
films such as the twisted bilayer graphene, some of which are anticipated to
involve unconventional pairing mechanism. Due to their low dimensionality,
experimental probes of the exact nature of superconductivity in these systems
have been limited. We propose, by applying a _vertical_ supercurrent to a
bilayer superconductor where the mirror symmetry is naturally broken by the
twisting, there will be anomalous thermal Hall effect induced by the
supercurrent that can serve as a sharp probe for the _in-plane_ anisotropy of
the superconducting gap function. This effect occurs in the _absence_ of an
external magnetic field and spontaneous breaking of the time-reversal symmetry
in the ground state. We derive explicit formulas for the induced thermal Hall
conductivity and show them to be significant in the examples of twisted
cuprates and twisted FeSe where monolayer superconductivity have already been
observed. Though technical challenges still exist, we propose this to be a
generic probe of the gap anisotropy in a twisted bilayer superconductor.
Introduction.— The discovery of the correlated insulating phases and
superconductivity in twisted bilayer graphene (TBG) Cao _et al._ (2018a, b)
has spurred intense interest in twisted two-dimensional (2D) heterostructures,
leading to the notion of _twistronics_ as a new form of electronic device. To
understand the (possibly unconventional) superconductivity realized in these
systems, intense experimental investigations have been on-going Oh _et al._
(2021); Kim _et al._ (2022); Lake _et al._ (2022). While the majority of
transport studies on twisted bilayers focuses on electrical transport at the
moment, thermal transport has long been established as a powerful and
complementary tool for investigating the nature of elementary excitations,
particularly in superconductors where ordinary electric transport measurement
is ineffective Krishana _et al._ (1997); Chiao _et al._ (2000); Sutherland
_et al._ (2003); Durst _et al._ (2003); Zhang _et al._ (2001); Cvetkovic and
Vafek (2015). More recently, quantized thermal Hall conductivity at low
temperature became a signature of the topologically ordered ground states in
correlated materials Banerjee _et al._ (2018); Yokoi _et al._ (2021). In
this work, we show that the thermal Hall response can be a sensitive probe of
the gap anisotropy in twisted bilayer superconductors, or TBS for short.
The gap function $\varDelta_{\bm{k}}$ of a superconductor has immense
implications for the underlying pairing mechanism and the quasiparticle
transport. It may be deduced in the angle-resolved photoemission spectroscopy
(ARPES) Shen _et al._ (1993); Ding _et al._ (1996); Zhang _et al._ (2016),
or through the quasiparticle interference (QPI) imaging in scanning tunneling
spectroscopy Hanaguri _et al._ (2007); Sprau _et al._ (2017). Resolving the
gap anisotropy in ARPES becomes challenging though for low temperature
superconductors where $\varDelta_{\bm{k}}$ is smaller than the experimental
resolution. For twisted heterostructures, resolving the momentum space
structure within a small moiré Brillouin zone (BZ) necessitates the QPI
imaging over a formidably large area in the real space. As the demand to
resolve the gap structure in TBS grows, limitations of existing experimental
probes seem to loom larger. A natural question to ask, at this stage, is
whether it is possible to invent a new probe of the gap anisotropy for very
small $\varDelta_{\bm{k}}$. Here we propose a supercurrent-induced anomalous
thermal Hall effect (SATHE) as one possible way to directly probe the gap
anisotropy in TBS.
Fig 1.(a) shows the schematic setup for SATHE. The TBS may be a vertical
Josephson junction (JJ) formed by stacking two atomically thin superconducting
films with a certain twist angle, or an intrinsic twisted bilayer
superconductor as in TBG. SATHE is a nonlinear response of heat, created by
simultaneously applying a _vertical_ supercurrent $\bm{J}_{S}$ and an in-plane
temperature gradient, with the resulting transverse in-plane flow of heat. In
contrast to the conventional thermal Hall effect (THE) which occurs when the
ground state breaks the time-reversal symmetry (TRS), SATHE can occur for
ground states that preserve the TRS. No external magnetic field is required to
observe SATHE.
Figure 1: (a) SATHE for bilayer-SC. The vertical supercurrent $\bm{J}_{S}$, or
equivalently, a pairing phase twist $\varphi$ between the two layers of,
either a vertical JJ or an intrinsic bilayer superconductor, would induce an
in-plane thermal Hall effect in a bilayer superconductor. In the former case,
an insulating buffer layer may be present (not shown). (b) an illustration for
the dimensionless function $\xi(x)$ in Eq.(2).
Our proposal thus differs for most proposals of THE including, for instance,
Ref. Can _et al._ (2021) where the JJ spontaneously breaks TRS in the absence
of applied supercurrent. It bear resemblance to the Sodemann-Fu proposal for
nonlinear electrical Hall effect Sodemann and Fu (2015); Ma _et al._ (2019),
in that in both proposals the _unperturbed_ ground state preserves TRS. One
can think of the nonlinear electrical Hall effect as arising from the first
electric field driving an imbalance of the fermion distribution, and the
second electric field is used to probe the Hall response. In our proposal, the
supercurrent is employed to drive Berry curvature out of its equilibrium form,
then a temperature gradient is applied to induce the thermal Hall response. In
this sense, SATHE can be taken as a thermal analogue of the nonlinear Hall
effect.
When the supercurrent is not too large, one can linearize the current-phase
relation $\bm{J}_{S}\propto\varphi$ and SAHTE becomes the perturbative change
of the thermal Hall conductivity $\delta\kappa_{xy}$ proportional to the phase
twist, $\delta\kappa_{xy}\simeq\varphi\cdot\chi_{\varphi xy}$, with
$\chi_{\varphi xy}$ capturing the nonlinear response of the system. Similar to
nonlinear Hall effect, the breaking of inversion symmetry is necessary to
elicit the desired responses. The observation of SATHE additionally requires
the breaking of in-plane and out-of-plane mirror symmetries, which are not
conditions normally present in the family of nonlinear Hall effects, but are
naturally satisfied in the TBS. The conditions for observing SATHE are
therefore not any more stringent than those of other proposed nonlinear Hall
effect, at least from the perspective of symmetry requirement.
Main results.— For TBS with a multiband electronic structure, the pairing
Hamiltonian is generally
$\sum_{a,b}\varDelta_{\bm{k}}^{a,b}c^{\dagger}_{a,\bm{k}}c^{\dagger}_{b,-\bm{k}}$,
with $a,b$ ranging over the bands. We firstly study the simple situation where
only the _intraband pairing_ is present: $\varDelta_{\bm{k}}^{a,b}=0$ for
$a\neq b$. Assuming that $|\varDelta_{\bm{k}}|$ is much smaller than the
energy difference between bands in the normal state, we find that a small
pairing phase difference $\varphi$ between the layers in the TBS induces a
change of the Berry curvature and leads to the SATHE formula
$\displaystyle\frac{\delta\kappa_{xy}^{\text{intra.}}}{T}=\varphi\cdot\frac{\chi_{\varphi
xy}^{\text{intra.}}}{T}=\varphi\cdot\frac{k_{B}^{2}}{16\pi^{2}\hbar}\sum_{\text{FS}}\mathop{\mathrm{sign}}(v_{F})$
$\displaystyle\qquad\times\oint_{\text{FS}}\mathop{}\\!\mathrm{d}k_{\parallel}\,\xi\bigg{(}\frac{\varDelta_{\bm{k}}(T)}{k_{B}T}\bigg{)}\cdot\partial_{k_{\parallel}}[\langle
u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle].$ (1)
Here $\chi_{\varphi xy}^{\text{intra.}}$ is the intraband transport
coefficient for SATHE, $\xi(x)$ is the dimensionless function (see Fig.1 (b))
$\displaystyle\xi(x)\equiv\int_{|x|}^{\infty}\mathrm{d}x^{\prime}\sqrt{x^{\prime
2}-x^{2}}\frac{x^{\prime}}{1+\cosh(x^{\prime})},$ (2)
and $|u_{\bm{k}}\rangle$ is the Bloch state at the Fermi level in the _normal_
state. The dimensionless Hermitian operator $\hat{L}$ is the generator of
Doppler shift due to $\varphi$, and equals
$\hat{L}=\mathop{\mathrm{diag}}\\{\mathbf{1},\mathbf{-1}\\}$ in the layer
space for a vertical JJ but gets more complicated for an intrinsic bilayer
superconductor (see below and Supplemental Material (SM) SM ). The loop
integral is performed over each Fermi surface (FS) and $k_{\parallel}$ is the
counterclockwise tangential momentum at the Fermi surface. The
$\mathop{\mathrm{sign}}(v_{F})=\pm 1$ characterizes whether the FS is
electron-like or hole-like (a single band may host multiple FS’s). Clearly if
$\varDelta_{\bm{k}}$ is $\bm{k}$-independent, the loop integral over the FS
reduces to a total derivative and vanishes identically.
The $\hat{L}$ operator and Doppler shift.— The normal state tight-binding
Hamiltonian for the bilayer can be written in a block form:
$\displaystyle
H_{0}(\bm{k})=\begin{pmatrix}H_{0}^{t}(\bm{k})&T_{\perp}(\bm{k})\\\
T^{\dagger}_{\perp}(\bm{k})&H_{0}^{b}(\bm{k})\end{pmatrix},$ (3)
where $t/b$ labels the top/bottom layer, and $T_{\perp}$ represents the
interlayer hopping. In the superconducting state, pairing terms are
introduced. Introducing the pairing phase difference $\varphi$ due to the
applied supercurrent is equivalent to performing a gauge transformation
$H_{0}\rightarrow U(\varphi)H_{0}U^{\dagger}(\varphi)$ (Doppler shift) while
keeping the pairing terms unchanged, and $\hat{L}$ is its generator:
$U(\varphi)\equiv e^{-i\frac{\varphi}{4}\hat{L}}$. For a vertical JJ,
$\hat{L}=\mathop{\mathrm{diag}}\\{\mathbf{1},\mathbf{-1}\\}$. For an intrinsic
superconductor, $\hat{L}$ depends on the atomic orbital positions normal to
the bilayer plane (see SM SM ).
For a JJ, since $|u_{\bm{k}}\rangle$ is the eigenstate of $H_{0}(\bm{k})$,
$\langle u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle\in[1,-1]$ serve as the indicator
of the layer-component of the state. If $T_{\perp}$ is absent, top/bottom
layers decouple and $\langle u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle=\pm 1$,
leading to vanishing SATHE according to Eq.(1), consistent with physical
intuitions.
Derivation of Eq. (1).— Here we show how to derive the SATHE formula of Eq.
(1) while leaving the more technical parts to SM . The thermal Hall
conductance in a 2D superconductor is related to the superconducting Berry
curvature Qin _et al._ (2011); Sumiyoshi and Fujimoto (2013) as
$\displaystyle\frac{\kappa_{xy}}{T}=-\frac{k_{B}^{2}}{2\hbar}\int_{-\infty}^{\infty}dE\frac{E^{2}}{(k_{B}T)^{2}}\bm{\sigma}(E)f^{\prime}(E),$
(4)
where $f(E)$ is the Fermi-Dirac function, and
$\displaystyle\bm{\sigma}(E)\equiv-\sum_{\mathbf{a}}\int_{E_{\bm{k}}<E}\frac{d^{2}k}{(2\pi)^{2}}\bm{\Omega}^{\mathbf{a}}_{\bm{k}}.$
(5)
(Bold fonts are used for quantities related to the superconducting BdG state,
to be distinguished from the normal state ones.) Here
$\bm{\Omega}^{\mathbf{a}}_{\bm{k}}=-2\mathop{\mathrm{Im}}\langle\partial_{k_{x}}\mathbf{u}^{\mathbf{a}}_{\bm{k}}|\partial_{k_{y}}\mathbf{u}^{\mathbf{a}}_{\bm{k}}\rangle$
is the superconducting Berry curvature while $\mathbf{a}$ labels bands of the
Bogoliubov–de Gennes (BdG) Hamiltonian
$\displaystyle\mathbf{H}(\bm{k})=\left(\begin{array}[]{cc}H_{0}(\bm{k})&\bm{\Delta}_{\bm{k}}\\\
\bm{\Delta}_{\bm{k}}&-H_{0}(\bm{k})\end{array}\right).$ (8)
The BdG Hamiltonian (8) is written in the Nambu basis
$\\{\psi_{\bm{k}},\Lambda^{\dagger}\psi_{-\bm{k}}^{\dagger}\\}$ where
$\psi_{\bm{k}}$ is a collection of fermion operators in the band basis, and
$\bm{\Delta}_{\bm{k}}^{\dagger}=\bm{\Delta}_{\bm{k}}$. The time reversal
transformation works on $\psi_{\bm{k}}$ as $\psi_{\bm{k}}\rightarrow\Lambda
K\psi_{\bm{k}}$, where $K$ is complex conjugation and $\Lambda$ is some
unitary operation.
The energy eigenvalues of the BdG Hamiltonian are ordered into pairs
$\pm\mathbf{E}_{1,\bm{k}},\pm\mathbf{E}_{2,\bm{k}},\cdots$ with
$\mathbf{E}_{1,\bm{k}}<\mathbf{E}_{2,\bm{k}}<\cdots$ and $\mathbf{a}=\pm 1,\pm
2,\cdots$ are used to label the bands such that
$\mathbf{E}_{-\mathbf{a},\bm{k}}=-\mathbf{E}_{\mathbf{a},\bm{k}}$. Assuming
only the intraband pairing is present, we have
$\mathbf{E}_{|\mathbf{a}|,\bm{k}}=(\epsilon_{|\mathbf{a}|,\bm{k}}^{2}+\varDelta_{|\mathbf{a}|,\bm{k}}^{2})^{1/2}$,
where $\epsilon_{a,\bm{k}}$ is the normal state band energy ($a=1,2,...$). We
assume that only the $a=1$ band crosses the Fermi level, and all other bands
lie strictly above or below it. This allows us to define the interband energy
scale as $t\equiv\min_{b}\min_{\bm{k}}|\epsilon_{b,\bm{k}}|$ ($b\neq 1$). The
gap function of the first band is defined as
$\varDelta_{\bm{k}}\equiv\varDelta_{1,\bm{k}}$. We will consider the weak-
pairing limit $|\varDelta_{\bm{k}}|\ll t$, so only the lowest-energy states
with $\mathbf{a}=\pm 1$ need to be included to the leading-order
$\varDelta_{\bm{k}}/t$ expansion of the Berry curvature
$\displaystyle\bm{\Omega}^{\mathbf{a}}_{\bm{k}}\doteq-\mathop{\mathrm{Im}}\left\\{\frac{\mathop{\mathrm{Tr}}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{(\mathbf{E}_{\mathbf{a}}-\mathbf{E}_{-\mathbf{a}})^{2}}-(x\leftrightarrow
y)\right\\},$ (9)
where
$\mathbf{P}_{\mathbf{a}}\equiv|\mathbf{u}^{\mathbf{a}}_{\bm{k}}\rangle\langle{\mathbf{u}}^{\mathbf{a}}_{\bm{k}}|$
is the projector in the Nambu space.
After a small phase twist $\varphi$ is turned on,
$\mathbf{H},\mathbf{E}_{\pm\mathbf{a}},\mathbf{P}_{\pm\mathbf{a}}$ appearing
in Eq. (9) all receive some corrections, which propagate through the THE
formulas in Eqs. (4) and (5). However, as shown in SM , only the change in
$\mathbf{P}_{\pm\mathbf{a}}$ is important when the unperturbed ground state
obeys TRS. In the end, the intraband pairing contribution gives
$\displaystyle\delta\bm{\Omega}^{\mathbf{a},\text{intra.}}_{\bm{k}}$
$\displaystyle\doteq\varphi\sum_{\mathbf{b}\neq\pm\mathbf{a}}4\mbox{Im\,Tr}[(\partial_{k_{x}}\mathbf{P}_{\mathbf{a}})(\partial_{k_{y}}\mathbf{P}_{-\mathbf{a}})\mathbf{P}_{\mathbf{b}}(\partial_{\varphi}\mathbf{P}_{\mathbf{a}})\mathbf{P}_{\mathbf{a}}]$
$\displaystyle\qquad-(x\leftrightarrow y),$ (10)
To the leading order of $\varDelta_{\bm{k}}/t$ expansion and focusing on the
$\mathbf{a}=\mathbf{1}$ band, we find
$\delta\bm{\Omega}^{\mathbf{1},\text{intra.}}_{\bm{k}}=-\varphi\cdot\frac{\varDelta_{\bm{k}}^{2}}{4\mathbf{E}_{\mathbf{1}}^{3}}\big{(}v_{x}\partial_{k_{y}}-v_{y}\partial_{k_{x}}\big{)}\langle
u^{1}_{\bm{k}}|\hat{L}|u^{1}_{\bm{k}}\rangle,$ (11)
with $v_{x,y}\equiv\partial_{k_{x,y}}\epsilon_{1,\bm{k}}$ the normal state
Fermi velocity.
The change in Berry curvature is now fully described with normal state wave
functions, and exhibits high concentration near the Fermi surface due to
$\mathbf{E}_{\mathbf{1}}^{3}$ in the denominator. Identifying
$\hat{L}|u_{\bm{k}}^{1}\rangle=4i\partial_{\varphi}|u_{\bm{k}}^{1}\rangle$,
the derivative $\partial_{k_{y}}\langle
u_{\bm{k}}^{1}|\hat{L}|u_{\bm{k}}^{1}\rangle=4\Omega_{y,\varphi}$ becomes the
_$\varphi$ -twist Berry curvature_, with one component along the momentum
direction and the other along the phase twist $\varphi$. Equation (11) thus
demonstrates how the change of Berry curvature in the superconducting state is
intricately composed of the gap function, and a mixed Berry curvature in the
momentum-phase space. Plugging Eq. (11) into Eq. (4) and after some efforts,
the main result of our paper Eq. (1) is established.
Interband pairing and nodal superconductivity.— When interband pairing is
present, the calculation becomes more sophisticated but the final result turns
out to be simple. Introducing the normal state projector
$P_{c}\equiv|u^{c}_{\bm{k}}\rangle\langle u^{c}_{\bm{k}}|$, the interband
pairing gives $\bm{\Delta}_{\bm{k}}^{\text{inter.}}=\sum_{b\neq
c}P_{b}\bm{\Delta}_{\bm{k}}P_{c}$, which can be eliminated from the BdG
Hamiltonian in Eq. (8) by a small unitary rotation
$e^{i\mathcal{S}\otimes\bm{\tau}_{2}}\mathbf{H}(\bm{k})e^{-i\mathcal{S}\otimes\bm{\tau}_{2}}\doteq
H_{0}(\bm{k})\otimes\bm{\tau}_{3}+\bm{\Delta}_{\bm{k}}^{\text{intra.}}\otimes\bm{\tau}_{1}.$
(12)
Here $\bm{\tau}$ are Pauli matrices in the Nambu space, and
$\mathcal{S}\equiv\sum_{a\neq
b}P_{a}\bm{\Delta}_{\bm{k}}P_{b}/(\epsilon_{a,\bm{k}}-\epsilon_{b,\bm{k}})$
can be viewed small simply because
$\bm{\Delta}_{\bm{k}}/(\epsilon_{a,\bm{k}}-\epsilon_{b,\bm{k}})\sim\varDelta_{\bm{k}}/t$
is small in the weak-pairing limit. Based on this useful property, all
previous perturbative analysis for the intraband pairing can be extended to
interband pairing as well, with only one modification replacing the projector
$\mathbf{P}_{\mathbf{a}}$ by
$\widetilde{\mathbf{P}}_{\mathbf{a}}=e^{-i\mathcal{S}\otimes\bm{\tau}_{2}}\mathbf{P}_{\mathbf{a}}e^{i\mathcal{S}\otimes\bm{\tau}_{2}}$.
Collecting all $\varphi$-linear terms contributing to the change of the Berry
curvature in Eq. (9) seems to be more complicated, but to the leading order of
$\varDelta_{\bm{k}}/t$ the interband contribution can be neatly arranged as SM
$\displaystyle\delta\bm{\Omega}^{\mathbf{1},\text{inter.}}_{\bm{k}}=\varphi\cdot\frac{-{\mathbf{d}}_{\bm{k}}\cdot(\partial_{k_{x}}{\mathbf{d}}_{\bm{k}}\times\partial_{k_{y}}{\mathbf{d}}_{\bm{k}})}{2\mathbf{E}^{3}_{\mathbf{1}}},$
(13)
$\displaystyle{\mathbf{d}}_{\bm{k}}\equiv(\varDelta_{\bm{k}},G_{\bm{k}},\epsilon_{1,\bm{k}}),\quad
G_{\bm{k}}\equiv\frac{-1}{2}\mbox{Re}[\langle
u^{1}_{\bm{k}}|\bm{\Delta}^{\text{inter.}}_{\bm{k}}\hat{L}|u^{1}_{\bm{k}}\rangle].$
Equation (13) is clearly reminiscent of the Berry curvature of the effective
two-band model
$\mathbf{H}_{\text{eff}}=\varphi\cdot\partial_{\varphi}\epsilon_{1,\bm{k}}\bm{\tau}_{0}+\epsilon_{1,\bm{k}}\bm{\tau}_{3}+\varDelta_{\bm{k}}\bm{\tau}_{1}+\varphi\cdot
G_{\bm{k}}\bm{\tau}_{2}.$ (14)
within the perturbative regime. Indeed, we show that $\mathbf{H}_{\text{eff}}$
is exactly the low-energy effective Hamiltonian of the TBS itself with
insertion of phase twist SM
$\mathbf{H}_{\text{eff}}^{\mathbf{a}}=\widetilde{\mathbf{P}}_{\mathbf{a}}\mathbf{H}[\varphi]\widetilde{\mathbf{P}}_{\mathbf{a}}\doteq\widetilde{\mathbf{P}}_{\mathbf{a}}\mathbf{H}\widetilde{\mathbf{P}}_{\mathbf{a}}+\varphi\cdot\widetilde{\mathbf{P}}_{\mathbf{a}}(\partial_{\varphi}\mathbf{H})\widetilde{\mathbf{P}}_{\mathbf{a}}$,
so could serve as a faithful model to compute
$\delta\mathbf{\Omega}_{\bm{k}}^{\mathbf{1},\text{inter.}}$.
In calculating $\delta\kappa_{xy}$ in Eq. (4) one should add the two Berry
curvature contributions
$\delta\bm{\Omega}^{\mathbf{1}}_{\bm{k}}=\delta\bm{\Omega}^{\mathbf{1},\text{intra.}}_{\bm{k}}+\delta\bm{\Omega}^{\mathbf{1},\text{inter.}}_{\bm{k}}$
in Eq. (5). Among the two, the intraband Berry curvature part can be converted
to the Fermi surface integral form in Eq. (1) but not the interband part. For
calculating the latter contribution we can directly use Eqs. (4)-(5) with
$\delta\bm{\Omega}^{\mathbf{1},\text{inter.}}_{\bm{k}}$ given in Eq. (13).
Although in principle the interband contribution for SATHE cannot be
neglected, there are many physical situations in which the interband pairing
and thus the interband contribution to THE does become small. For instance, if
each monolayer of TBS itself is superconducting as in a vertical JJ,
$\bm{\Delta}_{\bm{k}}$ in Eq. (8) is diagonal in the monolayer band basis. For
small twist angle $\theta\ll 1$, a unitary transformation to the band basis
also induces interband pairing components in $\bm{\Delta}_{\bm{k}}$, which are
proportional to $\theta$ and can be ignored. If the system is an intrinsic TBS
like TBG, the intra- and inter-band pairings should be determined self-
consistently. For example, for boson-mediated superconductivity, the multiband
Eliashberg formulation may be applied, which is characterized by the electron-
boson coupling matrix $[\alpha^{2}(\omega)F(\omega)]_{ab}$, where $a,b$ label
bands. Either in the regime that intraband coupling is dominant
$[\alpha^{2}(\omega)F(\omega)]_{a=b}\gg[\alpha^{2}(\omega)F(\omega)]_{a\neq
b}$, or in the regime that the intraband coupling is dominant
$[\alpha^{2}(\omega)F(\omega)]_{a=b}\ll[\alpha^{2}(\omega)F(\omega)]_{a\neq
b}$, it is easy to show that only the intraband pairing is significant Dolgov
_et al._ (2009).
However, there is one particular scenario in which the interband contributions
may be dominant, and that is when the superconductivity in the TBS is nodal.
In this case, the node would develop a mass gap $m_{\bm{k}}=\varphi\cdot
G_{\bm{k}}$ right at the nodal point upon the application of a phase twist
$\varphi$, and a Chern number transfer $\Delta C=\pm\frac{1}{2}$ between the
low-energy BdG bands occurs per node. When the net transferred Chern numbers
$n$ from all the nodes is nonzero, the system becomes a chiral topological
superconductor with quantized THE
$\delta\kappa_{xy}^{\text{inter.}}\equiv\varphi\cdot\chi_{\varphi
xy}^{\text{inter.}}=ng_{0}$ in the low-temperature limit $k_{B}T\ll|m|$, where
$\chi_{\varphi xy}^{\text{inter.}}$ is the interband transport coefficient for
SATHE, and $g_{0}\equiv\pi k_{B}^{2}T/(6\hbar)$ is the quantum of thermal
conductance. In our perturbative treatment, $\chi_{\varphi
xy}^{\text{inter.}}$ diverges in the low-$T$ regime. This supercurrent-driven
topological superconductivity has been discussed in the context of twisted
cuprate bilayers Volkov _et al._ (2023a, b); Can _et al._ (2021); Song _et
al._ (2022) and twisted NbSe2 heterostructures Hu and Ran (2022).
Applications to FeSe and cuprates.— To demonstrate the possibility of
observing SATHE in real materials we consider two examples: vertical JJ’s
formed by twisted nodeless FeSe, and the nodal cuprate superconducting films.
The $\varDelta_{\bm{k}}$ in both of these 2D superconductors has been well
characterized by ARPES due to the large energy scale of $\varDelta_{\bm{k}}$
and high transition temperatures. Note that due to the dimensionless nature of
the SATHE response, the supercurrent-induced $\kappa_{xy}$ only depends on the
ratio $\varDelta_{\bm{k}}/k_{B}T$ as in Eq. (1) and the SATHE response of
low-$T_{c}$ superconductors after appropriate rescaling must be similar to the
examples we consider now.
Monolayer FeSe is reported to host a significant nodeless gap anisotropy of
the form $\varDelta_{\bm{k}}=\varDelta_{0}+\varDelta_{2}\cos
2\theta_{\bm{k}}+\varDelta_{4}\cos 4\theta_{\bm{k}}$, where
$\varDelta_{0}=9.9$meV, $\varDelta_{2}=-1.4$meV, and $\varDelta_{4}=1.2$meV
Zhang _et al._ (2016). The $C_{4}$-rotation-related elliptic $M$-pockets
positioned at $(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$ within the two-iron BZ
Brouet _et al._ (2012); Borisenko _et al._ (2016); Yi _et al._ (2015) can
be described by the $\bm{k}\cdot\bm{p}$ expansion within $\mathrm{Fe}$’s
$\\{3d_{xz},3d_{yz}\\}$ orbitals Agterberg _et al._ (2017); Ran _et al._
(2009) as
$H_{M}=\left(\frac{1}{2m}(k_{x}^{2}+k_{y}^{2})-\mu\right)+ak_{x}k_{y}\tau_{z}$,
where $\tau_{z}$ is a Pauli matrix in the orbital space, $\mu=0.08$eV,
$1/2m=1.4$eV$\cdot\text{\AA}^{2}$ and $a=0.6$eV$\cdot\text{\AA}^{2}$. Since
all Fermi pockets are near the zone boundary, moiré zone folding effect comes
into play, and Bistritzer-MacDonald model Bistritzer and MacDonald (2011) is
used to construct the normal state Hamiltonian. For example, to the lowest
truncation of the moiré BZ, two hopping proccesses corresponding to
$\delta\bm{q}^{t,b}=\pm 2K_{M}\sin\frac{\theta}{2}\hat{k}_{y}$ are included
for all four of the $M$-pockets (see Fig.2 in SM SM ). We set
$T_{\perp}^{d_{xz}^{t}\text{-}d_{xz}^{b}}(\delta\bm{q}^{t,b})=T_{\perp}^{d_{yz}^{t}\text{-}d_{yz}^{b}}(\delta\bm{q}^{t,b})\simeq
T_{\perp}=15$meV and ignore all interorbital hoppings. After turning on such
simple $T_{\perp}$ with a twist angle $\theta=11.5^{\circ}$, the Fermi
surfaces reconstruct within the moiré BZ (see Fig. 2 (a1)).
We secondly consider a twisted bilayer of cuprates 111Note that the currently
available cuprate van der Waals materials is Bi2212, which is a bilayer of
Cu-O planes. Consequently, twisted _double_ bilayer is more experimental
relevant at present Zhu _et al._ (2021); Zhao _et al._ (2021). Our twisted
bilayer model can viewed as a minimal illustration on SATHE for twisted nodal
superconductors. with a $d$-wave gap anisotropy
$\varDelta_{\bm{k}}=\varDelta_{N}\cos 2\theta_{\bm{k}}$ Kanigel _et al._
(2007); Kondo _et al._ (2015) and the reported relation
$8.5k_{B}T_{c}=2\varDelta_{N}$ Kendziora _et al._ (1996); Anzai _et al._
(2013). The normal state bilayer Hamiltonian is constructed by first taking
the tight-binding model in Ref. Eschrig and Norman (2003) as the monolayer
Hamiltonian, and then obtaining the interlayer tunneling
$T_{\perp}(\bm{k})=t_{z}\big{(}\frac{1}{4}(\cos k_{x}^{t}-\cos k_{y}^{t})(\cos
k_{y}^{b}-\cos k_{y}^{b})+a_{0}\big{)}$ from the detailed orbital analysis in
Refs. Song _et al._ (2022); Markiewicz _et al._ (2005). The constant
$a_{0}=0.4$ is determined from DFT simulations Markiewicz _et al._ (2005). We
set (i) $t_{z}=0.025t_{0}$ ($t_{0}$=leading hopping strength within the
$\mathrm{CuO}_{2}$ plane) with $\theta=0.6^{\circ}$ (chiral topological
superconductor) and (ii) $t_{z}=0.01t_{0}$ with $\theta=17.2^{\circ}$
(topologically trivial superconductor) and plot the corresponding Fermi
surfaces in Fig. 2 (b1) and (c1), respectively Volkov _et al._ (2023a, b).
Here we neglect the moiré zone folding effect for several reasons. First,
based on the two-center approximation Bistritzer and MacDonald (2011), far
away from the zone boundary the moiré zone folding effect may be neglected.
Second, the moiré zone folding effect for the Fermi surfaces near the zone
boundary does not modify the Fermi surface topology, leaving the results
qualitatively unchanged.
The interband and intraband contributions to SATHE for twisted bilayers of
FeSe and cuprates are plotted in Fig. 2 (a2)-(c2). In our linearized scheme,
$\kappa_{xy}$ and $\chi_{\varphi xy}$ are related simply by
$\kappa_{xy}=\varphi\cdot\chi_{\varphi xy}$. We plot $\kappa_{xy}$ rather than
$\chi_{\varphi xy}$ because the latter apparently diverges as
$\varphi\rightarrow 0$ while $\kappa_{xy}^{\text{inter.}}$ remains finite in a
supercurrent-driven topological superconductor as shown in Fig. 2 (b2). For
illustrative purposes, phase twist $\varphi=1$ rad is chosen for computation
222Josephson phase twist $\varphi=1$ rad is taken, primarily to exhibit the
strength of the transport coefficient $\chi_{\varphi xy}^{\text{intra.}}$ from
the intraband contribution along with the quantized behavior of $\kappa_{xy}$
arising from the interband one. The former component contributes to the
thermal Hall conductivity simply as
$\kappa_{xy}^{\text{intra.}}=\varphi\cdot\chi_{\varphi xy}^{\text{intra.}}$,
as along as we remain within the _linear regime_ of the sine current-phase
relation, which is satisfied simply because $\sin 1.0\doteq 0.84\approx 1.0$.
Therefore, the choice $\varphi=1$ rad can still be considered small in our
perturbative treatment, and with such choice the plot of $\kappa_{xy}$ also
directly reflects the strength of the SATHE transport coefficient
$\chi_{\varphi xy}$ that we are concerned with.. The interband contribution is
evaluted by using the integral formula of Eq. (1). For the interband
contribution we rely on Eqs. (4)-(5) with the Berry curvature obtained in Eq.
(13).
As shown in Fig. 2, the intraband contribution plays a dominant role for SATHE
in both twisted FeSe (a2) and the topological trivial case of twisted cuprates
(c2), while interband contribution dominates in topological nontrivial case of
twisted cuprates (b2), especially in the low-temperature regime. Note that the
$\kappa_{xy}$ in the proposed SATHE can reach $\sim 10^{-1}$ of the thermal
conductance quantum when the gap anisotropy is significant, e.g., in the FeSe
example. We conclude that SATHE can be a sizable effect, detectable in the
foreseeable future.
Figure 2: (a): JJ formed by twisted bilayer FeSe. (b) and (c): the topological
non-trivial and trivial phases induced by the supercurrent in JJ formed by
twisted bilayer cuprates. The twisted Fermi surfaces are shown on the left
panels. For FeSe the moiré zone folding effect plays an important role on the
topology of reconstructed Fermi surfaces. The color scheme on each Fermi
surface represents the strength of $\langle
u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle\in[-1,1]$, serving as the indicator of
the layer-component of states. For all three cases, $\kappa_{xy}$ is computed
in unit of the thermal conductance quantum $g_{0}$ as a function of
temperature, and a simple mean-field temperature-dependence of
$\varDelta_{\bm{k}}(T)=\varDelta_{\bm{k}}(T=0)\sqrt{1-T/T_{c}}$ is
implemented. For the topologically nontrivial case (b), gaps $\sim 0.2$meV are
generated around superconducting nodes, leading to a Chern number $C=8$
topological superconductivity, while for the topological trivial case (c) the
net Chern number is zero.
Conclusion and Discussion.— We propose the supercurrent-induced anomalous
thermal Hall effect, SATHE, as a new probe to the in-plane gap anisotropy of
bilayer superconductors. Different from the conventional thermal Hall effect,
the ground state preserve the time reversal symmetry. A pair of probes — a
vertically applied supercurrent and a horizontal temperature gradient — is
applied to induce the in-plane nonlinear thermal Hall response. Being a
thermal response, it works on 2D superconductors where usual electrical probes
fail. Since no external magnetic field needs to be applied, SATHE avoids the
complications of vortices in the mixed state and probes purely the
quasiparticle dynamics in superconductors.
Within the BdG framework we showed that SATHE is sensitive to the in-plane gap
anisotropy in the twisted bilayer superconductor, and could be large enough to
serve as a new experimental probe of gap structure for atomically thin
superconducting 2D crystals including twisted bilayer graphene systems, for
which experimental probes have been limited due to the low dimensionality.
Motivated by the discovery of novel quantum states in van der Waals materials,
the experimental community has recently made significant progress towards
measuring thermal transports in low dimensional systems. For instance,
graphene-based Johnson noise thermometry has been developed Fong and Schwab
(2012); Talanov _et al._ (2021) and used to measure thermal transports in
graphene, carbon nanotubes and $\alpha$-RuCl3 Fong _et al._ (2013); Waissman
_et al._ (2022). We believe SATHE can eventually find its place as an
effective probe of 2D twisted materials, in particular for low $T_{c}$
superconductors where SATHE can serve as a sensitive measure of the gap
anisotropy.
Our results can be straightforwardly generalized to multi-layer
superconductors, by replacing the operator $\hat{L}$ with the generator of the
multi-layer Doppler shift.
###### Acknowledgements.
Y.R. and X.H. acknowledge the support from National Science Foundation under
Grant No. DMR-1712128. J.H.H. was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.
2023R1A2C1002644).
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Supplemental Material for “Supercurrent-induced Anomalous Thermal Hall Effect
as a New Probe to Superconducting Gap Anisotropy”
Xiaodong Hu, Jung Hun Han, and Ying Ran
## I The Doppler shift and the gauge transformation due to the supercurrent
The Doppler shift effect due to supercurrent has been discussed in textbooks
and carefully in a recent paper by Crowley and Fu Crowley and Fu (2022). Below
we follow the notation in the Appendix B of Ref. Crowley and Fu (2022).
To model a superconducting state in the presence of a finite supercurrent with
supercurrent velocity $\mathbf{u}$, it is convenient to consider a moving
reference frame $S^{\prime}$ which is moving at a velocity $\mathbf{u}$
relative to the lab frame $S$. In the moving frame $S^{\prime}$, there is no
supercurrent and the superconducting order parameter is spatially uniform.
Microscopically, the Galilean transformation between the frame-$S$ and
frame-$S^{\prime}$ is implemented by the unitary:
$\displaystyle U_{\mathbf{u}}$
$\displaystyle=e^{\frac{i}{\hbar}\mathbf{u}\cdot\mathbf{g}},$
$\displaystyle\mathbf{g}$ $\displaystyle=M\mathbf{R}-\mathbf{K}t,$ (S1)
where $\mathbf{K}$ is the total momentum and $\mathbf{R}$ is the center of
mass:
$\displaystyle\mathbf{K}$ $\displaystyle=\int
d\mathbf{r}\,c_{\mathbf{r},\sigma}^{\dagger}(-i\hbar\nabla)c_{\mathbf{r},\sigma},$
$\displaystyle\mathbf{R}$ $\displaystyle=\frac{1}{M}\int
d\mathbf{r}\,c_{\mathbf{r},\sigma}^{\dagger}c_{\mathbf{r},\sigma}m\mathbf{r}.$
(S2)
Here $M$ is the total mass, and $\sigma$ labels electron’s spin. Under
Galilean transformation, in the real-space basis:
$\displaystyle
U^{\dagger}_{\mathbf{u}}c^{\dagger}_{\mathbf{r},\sigma}U_{\mathbf{u}}=c^{\dagger}_{\mathbf{r}-\mathbf{u}t,\sigma}e^{\frac{-i}{\hbar}m\mathbf{r}\cdot\mathbf{u}+\frac{i}{\hbar}mu^{2}t/2}.$
(S3)
Namely, the electron state $c^{\dagger}_{\mathbf{r},\sigma}$ in frame-$S$ is
transformed into the electron state
$c^{\dagger}_{\mathbf{r}-\mathbf{u}t,\sigma}$ in frame-$S^{\prime}$. The
_spatially uniform_ pairing field in frame-$S^{\prime}$ then can be
represented in the real-space basis:
$\displaystyle\hat{\Delta}_{S^{\prime}}=\int
d\mathbf{r}_{1}d\mathbf{r}_{2}\Delta(\mathbf{r}_{1}-\mathbf{r}_{2})c^{\dagger}_{\mathbf{r}_{1}-\mathbf{u}t,\sigma_{1}}\epsilon_{\sigma_{1},\sigma_{2}}c^{\dagger}_{\mathbf{r}_{2}-\mathbf{u}t,\sigma_{2}},$
(S4)
where we assumed spin-singlet pairing for simplicity, and
$\epsilon_{\sigma_{1},\sigma_{2}}$ is the Levi-Civita symbol. Performing the
inverse Galilean transformation, one finds the pairing field in frame-$S$ is:
$\displaystyle\hat{\Delta}_{S}=U_{\mathbf{u}}\hat{\Delta}_{S^{\prime}}U^{\dagger}_{\mathbf{u}}=\int
d\mathbf{r}_{1}d\mathbf{r}_{2}e^{\frac{i}{\hbar}m(\mathbf{r}_{1}+\mathbf{r}_{2})\cdot\mathbf{u}}\Delta(\mathbf{r}_{1}-\mathbf{r}_{2})c^{\dagger}_{\mathbf{r}_{1},\sigma_{1}}\epsilon_{\sigma_{1},\sigma_{2}}c^{\dagger}_{\mathbf{r}_{2},\sigma_{2}},$
(S5)
One concludes that in the lab frame-$S$, the Cooper pair carries a nonzero
center-of-mass momentum $2m\mathbf{u}$ due to the supercurrent. The mean-field
Hamiltonian in the frame-$S$ is given by:
$\displaystyle H^{MF}_{S}=H_{0}+\hat{\Delta}_{S},$ (S6)
where $H_{0}$ is the normal state Hamiltonian. One may now perform a space-
dependent and time-independent gauge transformation $\mathbf{U}$ (which is
different from the Galilean transformation) to eliminate the spatial
dependence in the pairing field $\hat{\Delta}_{S}$:
$\displaystyle\mathbf{U}$ $\displaystyle=e^{\frac{-i}{\hbar}\int
d\mathbf{r}\,c^{\dagger}_{\mathbf{r},\sigma}c_{\mathbf{r},\sigma}m\mathbf{r}\cdot\mathbf{u}},$
$\displaystyle\mathbf{U}c^{\dagger}_{\mathbf{r},\sigma}\mathbf{U}^{\dagger}$
$\displaystyle=e^{\frac{-i}{\hbar}m\mathbf{r}\cdot\mathbf{u}}c^{\dagger}_{\mathbf{r},\sigma}$
(S7)
$H^{MF}_{S}$ after this unitary becomes:
$\displaystyle\tilde{H}^{MF}_{S}\equiv\mathbf{U}H^{MF}_{S}\mathbf{U}^{\dagger}=\mathbf{U}H_{0}\mathbf{U}^{\dagger}+\int
d\mathbf{r}_{1}d\mathbf{r}_{2}\Delta(\mathbf{r}_{1}-\mathbf{r}_{2})c^{\dagger}_{\mathbf{r}_{1},\sigma_{1}}\epsilon_{\sigma_{1},\sigma_{2}}c^{\dagger}_{\mathbf{r}_{2},\sigma_{2}},$
(S8)
In $\tilde{H}^{MF}_{S}$, the pairing field restores the form as if
supercurrent is absent, and the normal state Hamiltonian becomes Doppler-
shifted due to the gauge transformation. If we denote the phase factor
$e^{\frac{-i}{\hbar}m\mathbf{r}\cdot\mathbf{u}}\equiv
e^{-i\varphi(\mathbf{r})/2}$, the supercurrent density is:
$\displaystyle\mathbf{J}_{s}=-en_{s}\mathbf{u}=-e\frac{\hbar
n_{s}}{2m}\nabla\varphi(\mathbf{r}),$ (S9)
where we introduced the electron’s superfluid density $n_{s}$, and
$\varphi(\mathbf{r})$ can be identified as the phase of the pairing order
parameter, which recovers a well-known result. Generally speaking, the
superfluid density may be spatially dependent $n_{s}(\mathbf{r})$. For a
uniform supercurrent density, due to current conservation, this implies that
the superfluid velocity $\mathbf{u}(\mathbf{r})$ would be spatially dependent.
Namely, $\mathbf{u}(\mathbf{r})$, or $\nabla\varphi$, would be larger when
$n_{s}(\mathbf{r})$ is smaller. In this situation, the unitary $\mathbf{U}$
should be modified as:
$\displaystyle\mathbf{U}$ $\displaystyle=e^{\frac{-i}{\hbar}\int
d\mathbf{r}\,c^{\dagger}_{\mathbf{r},\sigma}c_{\mathbf{r},\sigma}m\int_{\mathbf{0}}^{\mathbf{r}}d\mathbf{r}^{\prime}\cdot\mathbf{u}(\mathbf{r}^{\prime})},$
$\displaystyle\mathbf{U}c^{\dagger}_{\mathbf{r},\sigma}\mathbf{U}^{\dagger}$
$\displaystyle=e^{\frac{-i}{\hbar}m\int_{\mathbf{0}}^{\mathbf{r}}d\mathbf{r}^{\prime}\cdot\mathbf{u}(\mathbf{r}^{\prime})}c^{\dagger}_{\mathbf{r},\sigma}\equiv
e^{-i\varphi(\mathbf{r})/2}c^{\dagger}_{\mathbf{r},\sigma},$ (S10)
and one still has
$\mathbf{u}(\mathbf{r})=\frac{\hbar}{2m}\nabla\varphi(\mathbf{r})$.
The bottom line is that no matter the system is a bilayer Josephson junction
or an instrinsic bilayer superconductor, a finite vertical supercurrent state
is always modeled by the gauge transformation
$c^{\dagger}_{\mathbf{r},\sigma}\rightarrow
e^{-i\varphi(\mathbf{r})/2}c^{\dagger}_{\mathbf{r},\sigma}$ that transforms
the normal state electronic structure
$H_{0}\rightarrow\mathbf{U}H_{0}\mathbf{U}^{\dagger}$ while leaving the
pairing field spatially uniform (as if the supercurrent is absent), as we have
done in Eq.(4) in the main text. Here $\varphi(\mathbf{r})$ satisfies Eq.(S9),
and is only depending on the vertical coordinate $z$:
$\varphi(\mathbf{r})=\varphi(z)$.
The difference between the two cases lies in the details of the profile of
$\varphi(z)$. In the case of a bilayer Josephson junction, the superfluid
density is concentrated in each monolayers, while between the two layers
$n_{s}$ is small. Consequently, the gradient of the phase $\varphi(z)$ is
essentially located between the two layers. In the case of an intrinsic
bilayer superconductor, $\varphi(z)$ would be a more smooth function of $z$.
We schematically plot the phase $\varphi(z)$ of the gauge transformation for
the two cases in Fig.S1.
Figure S1: The schematic plot of the phase $\varphi$ involved in the gauge
transformation to describe a finite supercurrent state. The two curves
represent the case of a Josephson junction (JJ) and an intrinsic bilayer
superconductor(Intrinisic) respectively.
We now consider the situation that $\varphi(z)$ interpolates the top layer
$\delta\varphi/2$ to the bottom layer $-\delta\varphi/2$. The Doppler shift
unitary is generated via $\hat{L}$:
$\mathbf{U}=e^{-i\frac{\delta\varphi}{4}\hat{L}}$.
* •
In the case of a vertical JJ, $\hat{L}$ can written in the tight-binding
basis, where the $\pm\mathbf{1}$ blocks correspond to the top/bottom layers:
$\displaystyle\hat{L}_{\text{JJ}}=\hat{c}^{\dagger}\begin{pmatrix}\mathbf{1}&0\\\
0&\mathbf{-1}\end{pmatrix}c.$ (S11)
* •
In the case of an intrinsic bilayer superconductor, assuming the superfluid
density is uniform, $\hat{L}$ is a diagonal matrix whose diagonal matrix
elements are given by $\frac{z_{\alpha}}{z_{0}}$, where $z_{\alpha}$ is the
atomic coordinate of the orbital-$\alpha$ and $\pm z_{0}$ is the position of
the top/botom layers.
$\displaystyle\hat{L}_{\text{Intrinsic}}=\sum_{\alpha}c_{\alpha}^{\dagger}\frac{z_{\alpha}}{z_{0}}c_{\alpha}$
(S12)
## II Details on the Derivation of the Main Result
### II.1 Supercurrent-indcued Berry Curvature for Intraband Pairing
Here we provide details of the derivation of the main result Eq.(1) in the
main text. Because only $\mathbf{a}=\pm 1$ bands contribute to the low-energy
Berry curvature, we will start with Eq.(8) in the main text.
Denoting the $c$-th eigenstate of $H_{0}(\bm{k})$ as $|u^{c}_{\bm{k}}\rangle$:
$H_{0}(\bm{k})|u^{c}_{\bm{k}}\rangle=\epsilon_{c,\bm{k}}|u^{c}_{\bm{k}}\rangle$,
and introducing the normal state projector
$P_{c}\equiv|u^{c}_{\bm{k}}\rangle\langle u^{c}_{\bm{k}}|$, we can define the
BdG projector into the low energy Hilbert space:
$\displaystyle\mathbf{P}\equiv P_{1}\otimes\bm{\tau}_{0},$ (S13)
where $\bm{\tau}_{0}$ is the identity Pauli matrix in the particle-hole space.
The low-energy BdG Hamiltonian is simply the two-by-two matrix, which
perfectly decouples from the high energy Hilbert space if only intraband
pairing is present (which we call _intraband pairing assumption_ below):
$\displaystyle\mathbf{P}\mathbf{H}\mathbf{P}=P_{1}\otimes(\epsilon_{1,\bm{k}}\bm{\tau}_{3}+\varDelta_{\bm{k}}\bm{\tau}_{1}).$
(S14)
Here $\varDelta_{\bm{k}}$ is chosen to be real due to the time-reversal
symmetry. In addition, Projector $\mathbf{P}$ satisfies an important identity
with any operator of the following form:
$\displaystyle\mathbf{A}^{\vee}\equiv
A\otimes\bm{\tau}_{0}\Rightarrow\mathbf{P}\mathbf{A}\mathbf{P}=\langle
u^{1}_{\bm{k}}|A|u^{1}_{\bm{k}}\rangle\mathbf{P},$ (S15)
which in turn indicates
$\mathbf{P}_{-\mathbf{a}}\mathbf{A^{\vee}}\mathbf{P}_{\mathbf{a}}=0$.
After the pairing phase twist $\delta\varphi$ are turned on,
$\mathbf{H},\mathbf{E}_{\pm\mathbf{a}},\mathbf{P}_{\pm\mathbf{a}}$ all receive
perturbations in Eq.(8) in the main text. $\delta\mathbf{H}$ and
$\delta\mathbf{E}_{\pm\mathbf{a}}$ do not contribute because
$\partial_{\varphi}\mathbf{H}$ and
$\partial_{k_{x}}\partial_{\varphi}\mathbf{H}$ both have the form of
$\mathbf{A}^{\vee}$ _under the intraband assumption_. For instance:
$\displaystyle\partial_{\varphi}\mathbf{H}=\partial_{\varphi}H_{0}(\bm{k})\otimes\bm{\tau}_{0}.$
(S16)
Thus we are left with the $\delta\mathbf{P}_{\pm\mathbf{a}}$ contribution
only:
$\displaystyle\frac{\delta\bm{\Omega}^{\mathbf{a}}}{\delta\varphi}\doteq\frac{-\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}\partial_{\varphi}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}^{2}}-(x\leftrightarrow
y).$ (S17)
Here we used the time-reversal symmetry combined with the particle-hole
symmetry, which dictates that the $\delta\mathbf{P}_{-\mathbf{a}}$
contribution and $\delta\mathbf{P}_{\mathbf{a}}$ contribution are identical.
After inserting $\mathbf{1}=\sum_{\mathbf{c}}\mathbf{P}_{\mathbf{c}}$ in front
of $\partial_{\varphi}\mathbf{P}_{\mathbf{a}}$, and noting that only terms
with $\mathbf{c}\neq\pm\mathbf{a}$ contributes _under the intraband pairing
assumption_ , the above expression can be rewritten as:
$\displaystyle\frac{\delta\bm{\Omega}^{\mathbf{a}}}{\delta\varphi}\doteq$
$\displaystyle\sum_{\mathbf{b}\neq\pm\mathbf{a}}\frac{\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}\mathbf{P}_{\mathbf{b}}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}^{2}(\mathbf{E}_{\mathbf{b}}-\mathbf{E}_{\mathbf{a}})}-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{b}\neq\pm\mathbf{a}}\frac{\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{P}_{\mathbf{b}}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}^{2}(\mathbf{E}_{\mathbf{b}}-\mathbf{E}_{\mathbf{a}})}(\mathbf{E}_{\mathbf{b}}+\mathbf{E}_{\mathbf{a}})-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{b}\neq\pm\mathbf{a}}\frac{2\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{P}_{\mathbf{b}}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}(\mathbf{E}_{\mathbf{b}}-\mathbf{E}_{\mathbf{a}})}-(x\leftrightarrow
y)$
$\displaystyle\qquad+\sum_{\mathbf{b}\neq\pm\mathbf{a}}\frac{\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{P}_{\mathbf{b}}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}^{2}}-(x\leftrightarrow
y).$ (S18)
The term in the last line vanishes _under the intraband pairing assumption_ :
$\mathbf{P}=\mathbf{P}_{1}+\mathbf{P}_{-1}$,
$\sum_{\mathbf{b}\neq\pm\mathbf{a}}\partial_{k_{y}}\mathbf{P}_{\mathbf{b}}=-\partial_{k_{y}}\mathbf{P}$,
and the fact that $\partial_{k_{y}}\mathbf{P}\partial_{\varphi}\mathbf{H}$
takes the form of $\mathbf{A}^{\vee}$. We therefore arrive at:
$\displaystyle\frac{\delta\bm{\Omega}^{\mathbf{a},\text{intra.}}}{\delta\varphi}\doteq$
$\displaystyle\sum_{\mathbf{b}\neq\pm\mathbf{a}}4\mbox{Im\,Tr}[\partial_{k_{x}}\mathbf{P}_{\mathbf{a}}\partial_{k_{y}}\mathbf{P}_{-\mathbf{a}}\mathbf{P}_{\mathbf{b}}\partial_{\varphi}\mathbf{P}_{\mathbf{a}}\mathbf{P}_{\mathbf{a}}]-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{b}\neq\pm\mathbf{a}}\frac{2\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}\mathbf{P}_{\mathbf{b}}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}(\mathbf{E}_{\mathbf{b}}-\mathbf{E}_{\mathbf{a}})(\mathbf{E}_{\mathbf{b}}-\mathbf{E}_{-\mathbf{a}})}-(x\leftrightarrow
y)$ $\displaystyle\doteq$ $\displaystyle\sum_{b\neq
1}\frac{2\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}(\mathbf{P}_{b}+\mathbf{P}_{-b})\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}(\epsilon_{b}-\epsilon_{1})^{2}}-(x\leftrightarrow
y)$ $\displaystyle=$ $\displaystyle\sum_{b\neq
1}\frac{2\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}P_{b}\otimes\bm{\tau}_{0}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}(\epsilon_{b}-\epsilon_{1})^{2}}-(x\leftrightarrow
y)$ (S19)
Now we are ready to connect with the properties of the normal state.
Introducing two-dimensional eigenvectors $|\bm{\alpha}_{\mathbf{c}}\rangle$ in
the particle-hole subspace:
$\displaystyle|\mathbf{u}^{\mathbf{c}}_{\bm{k}}\rangle=|u^{|\mathbf{c}|}_{\bm{k}}\rangle\otimes|\bm{\alpha}_{\mathbf{c}}\rangle,$
(S20)
and noting that
$\displaystyle\partial_{k}\mathbf{H}=\partial_{\bm{k}}H_{0}(\bm{k})\otimes\bm{\tau}_{3}+O(\partial_{\bm{k}}\bm{\Delta}_{\bm{k}}),$
(S21)
together with Eq.(S16), one has:
$\displaystyle\frac{\delta\bm{\Omega}^{\mathbf{a},\text{intra.}}}{\delta\varphi}\doteq$
$\displaystyle\sum_{b\neq
1}\frac{2|\langle\bm{\alpha}_{\mathbf{a}}|\tau_{3}|\bm{\alpha}_{-\mathbf{a}}\rangle|^{2}\mbox{Im\,Tr}[P_{1}\partial_{k_{x}}H_{0}P_{1}\partial_{k_{y}}H_{0}P_{b}\partial_{\varphi}H_{0}P_{1}]}{\mathbf{E}_{\mathbf{a}}(\epsilon_{b}-\epsilon_{1})^{2}}-(x\leftrightarrow
y)$ $\displaystyle=$ $\displaystyle\sum_{b\neq
1}\frac{2|\langle\bm{\alpha}_{\mathbf{a}}|\tau_{3}|\bm{\alpha}_{-\mathbf{a}}\rangle|^{2}v_{x}\mbox{Im\,Tr}[P_{1}\partial_{k_{y}}H_{0}P_{b}\partial_{\varphi}H_{0}P_{1}]}{\mathbf{E}_{\mathbf{a}}(\epsilon_{b}-\epsilon_{1})^{2}}-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle-\frac{v_{x}|\langle\bm{\alpha}_{\mathbf{a}}|\tau_{3}|\bm{\alpha}_{-\mathbf{a}}\rangle|^{2}}{\mathbf{E}_{\mathbf{a}}}\Omega_{k_{y},\varphi}-(x\leftrightarrow
y)=-\frac{\varDelta_{\bm{k}}^{2}}{\mathbf{E}_{\mathbf{a}}^{3}}v_{x}\Omega_{k_{y},\varphi}-(x\leftrightarrow
y),$ (S22)
where $v_{x}\equiv\partial_{k_{x}}\epsilon_{1,\bm{k}}$, and
$\Omega_{k_{y},\varphi}\equiv-2\mbox{Im}[\langle\partial_{k_{y}}u^{1}_{\bm{k}}|\partial_{\varphi}u^{1}_{\bm{k}}\rangle]$
is the _normal-state_ Berry curvature w.r.t. $k_{y}$ and $\varphi$. Using the
fact that
$|\partial_{\varphi}u^{1}_{\bm{k}}\rangle=\frac{-i}{4}\hat{L}|u^{1}_{\bm{k}}\rangle$,
the normal-state Berry curvature $\Omega_{k,\varphi}$ can also be expressed as
$\displaystyle\Omega_{k,\varphi}=\frac{1}{4}\partial_{\bm{k}}\langle
u^{1}_{\bm{k}}|\hat{L}|u^{1}_{\bm{k}}\rangle,$ (S23)
so that Eq.(10) in the main text is obtained.
### II.2 Fermi Surface Integral
The SATHE can be computed using Eq.(5) and Eq.(6) in the main text. Define
$\displaystyle\bm{\Omega}(E)\equiv\sum_{\mathbf{a}}\int\frac{d^{2}k}{(2\pi)^{2}}\bm{\Omega}^{\mathbf{a}}_{\bm{k}}\delta(E-\mathbf{E}_{\mathbf{a},\bm{k}}),$
(S24)
we have
$\bm{\sigma}(E)=\bm{\sigma}(0)-\int_{0}^{E}dE^{\prime}\bm{\Omega}(E^{\prime})$.
Here $\bm{\sigma}(0)=0$ due to the time-reversal symmetry and the nodeless
assumption.
Because the change of the Berry curvature is concentrated near the Fermi
surface due to the $\mathbf{E}_{\mathbf{a}}$ denominator, and proportional to
$1/\varDelta_{\bm{k}}$ right at the Fermi surface, it is convenient to
represent $\bm{\Omega}(E)$ as a Fermi surface integral as following:
$\displaystyle\frac{\delta\bm{\Omega}(E)}{\delta\varphi}\doteq$
$\displaystyle\sum_{\text{FS}}\frac{1}{(2\pi)^{2}}\oint_{\text{FS},\epsilon_{\bm{k}}\equiv\pm\sqrt{E^{2}-\varDelta_{\bm{k}}^{2}}}dk_{\parallel}\frac{|E|}{|\epsilon_{\bm{k}}|}\frac{1}{\hbar|v_{F}|}\mathbf{\Omega}^{\mathop{\mathrm{sign}}(E)}_{\bm{k}}$
$\displaystyle\doteq$
$\displaystyle\sum_{\text{FS}}\frac{-2}{(2\pi)^{2}}\oint_{\text{FS}}dk_{\parallel}\frac{|E|}{\sqrt{E^{2}-\varDelta_{\bm{k}}^{2}}}\frac{1}{\hbar|v_{F}|}\frac{\varDelta_{\bm{k}}^{2}}{E^{3}}(v_{x}\Omega_{k_{y},\varphi}-v_{y}\Omega_{k_{x},\varphi})$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{sign}}(E)\sum_{\text{FS}}\frac{-1}{2\pi^{2}}\oint_{\text{FS}}dk_{\parallel}\frac{\varDelta_{\bm{k}}^{2}}{E^{2}\sqrt{E^{2}-\varDelta_{\bm{k}}^{2}}}\frac{1}{\hbar|v_{F}|}(v_{x}\Omega_{k_{y},\varphi}-v_{y}\Omega_{k_{x},\varphi})$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{sign}}(E)\sum_{\text{FS}}\frac{-\mathop{\mathrm{sign}}(v_{F})}{2\pi^{2}\hbar}\oint_{\text{FS}}d\vec{k}_{\parallel}\cdot(\Omega_{k_{x},\varphi},\Omega_{k_{y},\varphi})\frac{\varDelta_{\bm{k}}^{2}}{E^{2}\sqrt{E^{2}-\varDelta_{\bm{k}}^{2}}}$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{sign}}(E)\sum_{\text{FS}}\frac{-\mathop{\mathrm{sign}}(v_{F})}{8\pi^{2}\hbar}\oint_{\text{FS}}dk_{\parallel}\partial_{k_{\parallel}}\langle
u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle\frac{\varDelta_{\bm{k}}^{2}}{E^{2}\sqrt{E^{2}-\varDelta_{\bm{k}}^{2}}}.$
(S25)
Integrating out $E$ in advance, we get
$\displaystyle\bm{\sigma}(E)=\sum_{\text{FS}}\frac{\mathop{\mathrm{sign}}(v_{F})}{8\pi^{2}\hbar}\oint_{\text{FS}}dk_{\parallel}\partial_{k_{\parallel}}\langle
u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle\frac{\sqrt{E^{2}-\varDelta_{\bm{k}}^{2}}}{|E|}\Theta(|E|-|\varDelta_{\bm{k}}|).$
(S26)
We finally arrive at the result using Eq.(5) in the main text:
$\displaystyle\frac{\delta\kappa_{xy}}{\delta\varphi
T}=\sum_{\text{FS}}\frac{\mathop{\mathrm{sign}}(v_{F})k_{B}^{2}}{16\pi^{2}\hbar}\oint_{\text{FS}}dk_{\parallel}\partial_{k_{\parallel}}\langle
u_{\bm{k}}|\hat{L}|u_{\bm{k}}\rangle\xi\bigg{(}\frac{\varDelta_{\bm{k}}}{T}\bigg{)}.$
(S27)
## III Supercurrent-indcued Berry Curvature for Interband Pairing
When interband pairing is present, additional terms need to be considered.
However, one can always perform a small unitary rotation
$e^{iS\otimes\bm{\tau}_{2}}$ to eliminate the interband pairing:
$\displaystyle S\equiv\sum_{a\neq
b}\frac{P_{a}\bm{\Delta}^{\text{inter.}}_{\bm{k}}P_{b}}{\epsilon_{a,\bm{k}}+\epsilon_{b,\bm{k}}},\quad
S^{\dagger}=S.$ (S28)
Since
$\mathbf{H}=H_{0}\otimes\bm{\tau}_{3}+(\bm{\Delta}^{\text{intra.}}+\bm{\Delta}^{\text{inter.}})\otimes\bm{\tau}_{1}$,
we have:
$\displaystyle
e^{iS\otimes\bm{\tau}_{2}}\mathbf{H}e^{-iS\otimes\bm{\tau}_{2}}\doteq$
$\displaystyle
H_{0}\otimes\bm{\tau}_{3}+(\bm{\Delta}^{\text{intra.}}+\bm{\Delta}^{\text{inter.}})\otimes\bm{\tau}_{1}+(i)[S\otimes\bm{\tau}_{2},H_{0}\otimes\bm{\tau}_{3}]$
$\displaystyle=$ $\displaystyle
H_{0}\otimes\bm{\tau}_{3}+(\bm{\Delta}^{\text{intra.}}+\bm{\Delta}^{\text{inter.}})\otimes\bm{\tau}_{1}+(i)(SH_{0}+H_{0}S)\otimes(i)\bm{\tau}_{1}$
$\displaystyle=$ $\displaystyle
H_{0}\otimes\bm{\tau}_{3}+(\bm{\Delta}^{\text{intra.}}+\bm{\Delta}^{\text{inter.}})\otimes\bm{\tau}_{1}+(-1)\bm{\Delta}^{\text{inter.}}_{\bm{k}}\otimes\bm{\tau}_{1}=H_{0}\otimes\bm{\tau}_{3}+\bm{\Delta}^{\text{intra.}}\otimes\bm{\tau}_{1}$
(S29)
Denoting the projectors in the presence of interband pairing as
$\widetilde{\mathbf{P}}_{\mathbf{c}}$, we have
$\widetilde{\mathbf{P}}_{\mathbf{c}}\doteq
e^{-iS\otimes\bm{\tau}_{2}}\mathbf{P}_{\mathbf{c}}e^{iS\otimes\bm{\tau}_{2}}$.
Basically, in all the previous derivations for the intraband case, we only
need to replace $\mathbf{P}_{\mathbf{c}}$ by
$\widetilde{\mathbf{P}}_{\mathbf{c}}$, which corresponds to performing the
small unitary transformation for the operators like
$\partial_{\varphi}\mathbf{H}\rightarrow
e^{iS\otimes\bm{\tau}_{2}}\partial_{\varphi}\mathbf{H}e^{-iS\otimes\bm{\tau}_{2}}$.
In this way $\partial_{\varphi}\mathbf{H}$ has a correction
$[iS,\partial_{\varphi}H_{0}]\otimes\bm{\tau}_{2}$ and is no longer
$\propto\bm{\tau}_{0}$. We can leave all the energies
$\mathbf{E}_{\mathbf{c}}$ unchanged since they receive second order
contributions from $\bm{\Delta}^{\text{inter.}}$.
After inspection, to the leading order of $\varDelta/t$, one can identify
three sources of interband contributions. First, we do need to consider
$\delta_{\varphi}\partial_{\bm{k}}\mathbf{H}$ in Eq.(8) in the main text.
Second, in the first line of Eq.(S18), $\mathbf{b}=-\mathbf{a}$ needs to be
included. Third, the term in the last line of Eq.(S18) is no longer vanishing.
We term them as _Part-A,B,C_ and compute them one by one below.
_Part-A:_ For the $\delta_{\varphi}\partial_{\bm{k}}\mathbf{H}$ contributions,
$\displaystyle\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{A}=$
$\displaystyle\frac{-\mbox{Im\,Tr}\big{[}\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}[iS\otimes\bm{\tau}_{2},\partial_{k_{y}}\partial_{\varphi}\mathbf{H}]\mathbf{P}_{\mathbf{a}}\big{]}}{2\mathbf{E}^{2}_{\mathbf{a}}}-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{-\mbox{Im\,Tr}\big{[}\mathbf{P}_{\mathbf{a}}(\partial_{k_{x}}H_{0}\otimes\bm{\tau}_{3}+\partial_{k_{x}}\bm{\Delta}\otimes\bm{\tau}_{1})\mathbf{P}_{-\mathbf{a}}[iS\otimes\bm{\tau}_{2},\partial_{k_{y}}\partial_{\varphi}H_{0}\otimes\bm{\tau}_{0}]\mathbf{P}_{\mathbf{a}}\big{]}}{2\mathbf{E}^{2}_{\mathbf{a}}}-(x\leftrightarrow
y)$ $\displaystyle=$ $\displaystyle-\langle
u^{1}_{\bm{k}}|[iS,\partial_{k_{y}}\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle$
$\displaystyle\qquad\times\frac{v_{x}\mbox{Im}\big{[}\langle\bm{\alpha}_{\mathbf{a}}|\bm{\tau}_{3}|\bm{\alpha}_{-\mathbf{a}}\rangle\langle\bm{\alpha}_{-\mathbf{a}}|\bm{\tau}_{2}|\bm{\alpha}_{\mathbf{a}}\rangle\big{]}+\langle
u^{1}_{\bm{k}}|\partial_{k_{x}}\bm{\Delta}|u^{1}_{\bm{k}}\rangle\mbox{Im}\big{[}\langle\bm{\alpha}_{\mathbf{a}}|\bm{\tau}_{1}|\bm{\alpha}_{-\mathbf{a}}\rangle\langle\bm{\alpha}_{-\mathbf{a}}|\bm{\tau}_{2}|\bm{\alpha}_{\mathbf{a}}\rangle\big{]}}{2\mathbf{E}^{2}_{\mathbf{a}}}-(x\leftrightarrow
y).$ (S30)
Noting a few identities:
$\displaystyle\bm{\tau}_{2}|\bm{\alpha}_{-\mathbf{a}}\rangle\langle\bm{\alpha}_{-\mathbf{a}}|\bm{\tau}_{2}=|\bm{\alpha}_{\mathbf{a}}\rangle\langle\bm{\alpha}_{\mathbf{a}}|$
$\displaystyle\partial_{k_{x}}\mathbf{E}_{\mathbf{a}}=\mbox{Tr}[\mathbf{P}_{\mathbf{a}}(\partial_{k_{x}}H_{0}\otimes\bm{\tau}_{3}+\partial_{k_{x}}\bm{\Delta}\otimes\bm{\tau}_{1})\mathbf{P}_{\mathbf{a}}]=v_{x}\frac{\epsilon_{1,\bm{k}}}{\mathbf{E}_{a}}+\langle
u^{1}_{\bm{k}}|\partial_{k_{x}}\bm{\Delta}|u^{1}_{\bm{k}}\rangle\frac{\varDelta_{\bm{k}}}{\mathbf{E}_{\mathbf{a}}}$
$\displaystyle\partial_{k_{x}}\mathbf{E}_{\mathbf{a}}=\frac{\varDelta_{\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}+\epsilon_{1,\bm{k}}v_{x}}{\mathbf{E}_{\mathbf{a}}}$
$\displaystyle\Rightarrow\langle
u^{1}_{\bm{k}}|\partial_{k_{x}}\bm{\Delta}|u^{1}_{\bm{k}}\rangle=\partial_{k_{x}}\varDelta_{\bm{k}}$
$\displaystyle\langle\bm{\alpha}_{\mathbf{a}}|\bm{\tau}_{1}|\bm{\alpha}_{\mathbf{a}}\rangle=\frac{\varDelta_{\bm{k}}}{\mathbf{E}_{\mathbf{a}}},\;\;\langle\bm{\alpha}_{\mathbf{a}}|\bm{\tau}_{3}|\bm{\alpha}_{\mathbf{a}}\rangle=\frac{\epsilon_{1,\bm{k}}}{\mathbf{E}_{\mathbf{a}}}$
(S31)
we have:
$\displaystyle\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{A}=$
$\displaystyle-\frac{\langle
u^{1}_{\bm{k}}|[iS,\partial_{k_{y}}\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle}{2\mathbf{E}^{3}_{\mathbf{a}}}\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}-(x\leftrightarrow
y)$
_Part-B:_ This term is:
$\frac{\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}^{2}(\mathbf{E}_{-\mathbf{a}}-\mathbf{E}_{\mathbf{a}})}-(x\leftrightarrow
y)$ (S33)
Note that
$\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}=\partial_{k_{y}}\mathbf{E}_{\mathbf{-}a}\mathbf{P}_{-\mathbf{a}}$,
$\displaystyle\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{B}=$
$\displaystyle\frac{\partial_{k_{y}}\mathbf{E}_{\mathbf{a}}}{2\mathbf{E}_{\mathbf{a}}^{3}}\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}(\partial_{k_{x}}H_{0}\otimes\bm{\tau}_{3}+\partial_{k_{x}}\bm{\Delta}\otimes\bm{\tau}_{1})\mathbf{P}_{-\mathbf{a}}[iS,\partial_{\varphi}H_{0}]\otimes\bm{\tau}_{2}\mathbf{P}_{\mathbf{a}}]-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{\partial_{k_{y}}\mathbf{E}_{\mathbf{a}}}{2\mathbf{E}_{\mathbf{a}}^{3}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}(\partial_{k_{x}}H_{0}\otimes\bm{\tau}_{3}+\partial_{k_{x}}\bm{\Delta}\otimes\bm{\tau}_{1})\mathbf{P}_{-\mathbf{a}}\bm{\tau}_{2}\mathbf{P}_{\mathbf{a}}]-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{\partial_{k_{y}}\mathbf{E}_{\mathbf{a}}}{2\mathbf{E}_{\mathbf{a}}^{3}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}(\partial_{k_{x}}H_{0}\otimes\bm{\tau}_{3}+\partial_{k_{x}}\bm{\Delta}\otimes\bm{\tau}_{1})\bm{\tau}_{2}\mathbf{P}_{\mathbf{a}}]-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{\partial_{k_{y}}\mathbf{E}_{\mathbf{a}}}{2\mathbf{E}_{\mathbf{a}}^{3}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle(-v_{x}\langle\bm{\alpha}_{\mathbf{a}}|\bm{\tau}_{1}|\bm{\alpha}_{\mathbf{a}}\rangle+\langle
u^{1}_{\bm{k}}|\partial_{k_{x}}\bm{\Delta}|u^{1}_{\bm{k}}\rangle\langle\bm{\alpha}_{\mathbf{a}}|\bm{\tau}_{3}|\bm{\alpha}_{\mathbf{a}}\rangle)-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{\partial_{k_{y}}\mathbf{E}_{\mathbf{a}}}{2\mathbf{E}_{\mathbf{a}}^{4}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{(v_{y}\epsilon_{1,\bm{k}}+\varDelta_{\bm{k}}\partial_{k_{y}}\varDelta_{\bm{k}})}{2\mathbf{E}_{\mathbf{a}}^{5}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}-(x\leftrightarrow
y)$ (S34)
After $(x\leftrightarrow y)$ antisymmetrization, we get
$\displaystyle\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{B}=$
$\displaystyle\frac{-1}{2\mathbf{E}_{\mathbf{a}}^{3}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle
v_{x}\partial_{k_{y}}\varDelta_{\bm{k}}-(x\leftrightarrow y)$ (S35)
_Part-C:_ Note that
$\displaystyle\partial_{k_{y}}\widetilde{\mathbf{P}}_{\mathbf{b}}\doteq\partial_{k_{y}}\big{[}\mathbf{P}_{\mathbf{b}}+[-iS\otimes\bm{\tau}_{2},\mathbf{P}_{\mathbf{b}}]\big{]}\doteq
e^{-iS\otimes\bm{\tau}_{2}}\partial_{k_{y}}\mathbf{P}_{\mathbf{b}}e^{iS\otimes\bm{\tau}_{2}}+[-i\partial_{k_{y}}S\otimes\bm{\tau}_{2},\mathbf{P}_{\mathbf{b}}]$
(S36)
The contribution from the last line of Eq.(S18) becomes:
$\displaystyle\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{C}\doteq$
$\displaystyle\frac{-\mbox{Im\,Tr}[\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}\partial_{k_{y}}\mathbf{P}[iS\otimes\bm{\tau}_{2},\partial_{\varphi}\mathbf{H}]\mathbf{P}_{\mathbf{a}}+\mathbf{P}_{\mathbf{a}}\partial_{k_{x}}\mathbf{H}\mathbf{P}_{-\mathbf{a}}[-i\partial_{k_{y}}S\otimes\bm{\tau}_{2},\mathbf{P}]\partial_{\varphi}\mathbf{H}\mathbf{P}_{\mathbf{a}}]}{\mathbf{E}_{\mathbf{a}}^{2}}-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{-1}{\mathbf{E}_{\mathbf{a}}^{3}}\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}\mbox{Re\,Tr}[\partial_{k_{y}}P_{1}[iS,\partial_{\varphi}H_{0}]P_{1}+P_{1}i\partial_{k_{y}}S\partial_{\varphi}H_{0}P_{1}]-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{-1}{2\mathbf{E}_{\mathbf{a}}^{3}}\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}\mbox{Re\,Tr}[2\partial_{k_{y}}P_{1}[iS,\partial_{\varphi}H_{0}]P_{1}+P_{1}[i\partial_{k_{y}}S,\partial_{\varphi}H_{0}]P_{1}]-(x\leftrightarrow
y)$ (S37)
Let’s add part-A and part-C together.
$\displaystyle\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{A+C}=$
$\displaystyle\frac{-1}{2\mathbf{E}_{\mathbf{a}}^{3}}\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}\partial_{k_{y}}\Big{[}\mbox{Re\,Tr}[P_{1}[iS,\partial_{\varphi}H_{0}]P_{1}]\Big{]}-(x\leftrightarrow
y)$ $\displaystyle=$
$\displaystyle\frac{-1}{2\mathbf{E}_{\mathbf{a}}^{3}}\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}\partial_{k_{y}}\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle-(x\leftrightarrow
y)$ (S38)
The quantity $\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle$ here and in
part-B can be easily computed, which we denote as $G_{\bm{k}}$:
$\displaystyle G_{\bm{k}}\equiv$ $\displaystyle\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle=-2\mbox{Im}[\langle
u^{1}_{\bm{k}}|S\partial_{\varphi}H_{0}|u^{1}_{\bm{k}}\rangle]=-2\sum_{b\neq
1}\mbox{Im}[\langle u^{1}_{\bm{k}}|S|u^{b}_{\bm{k}}\rangle\langle
u^{b}_{\bm{k}}|\partial_{\varphi}H_{0}|u^{1}_{\bm{k}}\rangle]$
$\displaystyle=-2\sum_{b\neq 1}\mbox{Im}\Big{[}\frac{\langle
u^{1}_{\bm{k}}|\bm{\Delta}^{\text{inter.}}|u^{b}_{\bm{k}}\rangle\langle
u^{b}_{\bm{k}}|\partial_{\varphi}H_{0}|u^{1}_{\bm{k}}\rangle}{\epsilon_{1,\bm{k}}+\epsilon_{b,\bm{k}}}\Big{]}\doteq-2\sum_{b\neq
1}\mbox{Im}\Big{[}\frac{\langle
u^{1}_{\bm{k}}|\bm{\Delta}^{\text{inter.}}|u^{b}_{\bm{k}}\rangle\langle
u^{b}_{\bm{k}}|\partial_{\varphi}H_{0}|u^{1}_{\bm{k}}\rangle}{-\epsilon_{1,\bm{k}}+\epsilon_{b,\bm{k}}}\Big{]}$
$\displaystyle=2\sum_{b\neq 1}\mbox{Im}[\langle
u^{1}_{\bm{k}}|\bm{\Delta}^{\text{inter.}}|u^{b}_{\bm{k}}\rangle\langle
u^{b}_{\bm{k}}|\partial_{\varphi}u^{1}_{\bm{k}}\rangle]=2\mbox{Im}[\langle
u^{1}_{\bm{k}}|\bm{\Delta}^{\text{inter.}}|\partial_{\varphi}u^{1}_{\bm{k}}\rangle]=\frac{-1}{2}\mbox{Re}[\langle
u^{1}_{\bm{k}}|\bm{\Delta}^{\text{inter.}}\hat{L}|u^{1}_{\bm{k}}\rangle]$
(S39)
Putting together, we finally get
$\displaystyle\frac{\delta\bm{\Omega}^{\mathbf{a},\text{inter.}}_{\bm{k}}}{\delta\varphi}=\left.\frac{\delta\bm{\Omega}^{\mathbf{a}}_{\bm{k}}}{\delta\varphi}\right|_{A+B+C}=$
$\displaystyle\frac{-1}{2\mathbf{E}_{\mathbf{a}}^{3}}\big{(}-\varDelta_{\bm{k}}v_{x}+\epsilon_{1,\bm{k}}\partial_{k_{x}}\varDelta_{\bm{k}}\big{)}\partial_{k_{y}}G_{\bm{k}}+\frac{-1}{2\mathbf{E}_{\mathbf{a}}^{3}}G_{\bm{k}}v_{x}\partial_{k_{y}}\varDelta_{\bm{k}}-(x\leftrightarrow
y)$ (S40)
It is also instructive to study the behavior of $\partial_{\varphi}\mathbf{H}$
in the low energy subspace:
$\displaystyle\widetilde{\mathbf{P}}\partial_{\varphi}\mathbf{H}\widetilde{\mathbf{P}}=\mathbf{P}\partial_{\varphi}\mathbf{H}\mathbf{P}+\mathbf{P}[iS\otimes\bm{\tau}_{2},\partial_{\varphi}\mathbf{H}]\mathbf{P}=\langle
u^{1}_{\bm{k}}|\partial_{\varphi}H_{0}|u^{1}_{\bm{k}}\rangle\mathbf{P}+\langle
u^{1}_{\bm{k}}|[iS,\partial_{\varphi}H_{0}]|u^{1}_{\bm{k}}\rangle\mathbf{P}\bm{\tau}_{2}\mathbf{P}$
(S41)
The low energy effective Hamiltonian in the presence of $\delta\varphi$
becomes:
$\displaystyle\mathbf{H}_{\text{eff}}=\delta\varphi\partial_{\varphi}\epsilon_{1,\bm{k}}\bm{\tau}_{0}+\epsilon_{1,\bm{k}}\bm{\tau}_{3}+\varDelta_{\bm{k}}\bm{\tau}_{1}+\delta\varphi
G_{\bm{k}}\bm{\tau}_{2}=\delta\varphi\partial_{\varphi}\epsilon_{1,\bm{k}}\bm{\tau}_{0}+\mathbf{d}_{\bm{k}}\cdot\vec{\bm{\tau}},$
(S42)
where we introduced vector
$\mathbf{d}_{\bm{k}}\equiv(\varDelta_{\bm{k}},\delta\varphi
G_{\bm{k}},\epsilon_{1,\bm{k}})$. We merely showed that the interband
contribution can be faithfully computed using this effective 2-by-2
Hamiltonian:
$\displaystyle\delta\bm{\Omega}^{\mathbf{a},\text{inter.}}_{\bm{k}}=-2\mbox{Im\,Tr}[p_{\mathbf{a}}\partial_{k_{x}}p_{\mathbf{a}}\partial_{k_{y}}p_{\mathbf{a}}]=\frac{-d_{\bm{k}}\cdot(\partial_{k_{x}}\mathbf{d}_{\bm{k}}\times\partial_{k_{y}}\mathbf{d}_{\bm{k}})}{2\mathbf{a}|\mathbf{d}_{\bm{k}}|^{3}}\doteq\frac{-\mathbf{d}_{\bm{k}}\cdot(\partial_{k_{x}}\mathbf{d}_{\bm{k}}\times\partial_{k_{y}}\mathbf{d}_{\bm{k}})}{2\mathbf{E}_{\mathbf{a}}^{3}}\propto\delta\varphi$
(S43)
$p_{\mathbf{a}}=\frac{1}{2}(1+\mathbf{a}\frac{\mathbf{d}_{\bm{k}}\cdot\vec{\bm{\tau}}}{|\mathbf{d}_{\bm{k}}|})$
is the projector in this effective model.
When the superconductivity is nodal, near a node, the effective theory becomes
a Dirac equation:
$\displaystyle\mathbf{H}^{\text{node}}_{\text{eff}}=\delta\varphi\partial_{\varphi}\epsilon_{1,\bm{k}}\bm{\tau}_{0}+\hbar
v_{F}k_{\perp}\bm{\tau}_{3}+\hbar
v_{\Delta}k_{\parallel}\bm{\tau}_{1}+m\bm{\tau}_{2},$ (S44)
where $\hbar v_{\Delta}=\frac{\partial\varDelta_{\bm{k}}}{\partial
k_{\parallel}}$, and $m=\delta\varphi G_{\bm{k}_{\text{node}}}$ is the mass
gap generated by the supercurrent. It is easy to show that the nodal
contribution to $\bm{\sigma}(E)$ is
$\displaystyle\bm{\sigma}^{\text{node}}(E)=\begin{cases}\frac{C}{2\pi},&\text{if
}-|m|<E<|m|,\\\ \frac{C|m|}{2\pi|E|},&\text{otherwise}.\end{cases}$ (S45)
Here $C=\pm\frac{1}{2}$ is the Chern number transferred due to $m$. Performing
the energy integral in Eq.(5) in the main text, in the low temperature limit
$k_{B}T\ll|m|$ one recovers the quantized
$\kappa^{\text{node}}_{xy}/T=\frac{k_{B}^{2}}{\hbar}\frac{C\pi}{12}$. However,
in the high-temperature limit $m\ll k_{B}T$, we have:
$\displaystyle\frac{\kappa_{xy}^{\text{node}}}{T}\doteq-\frac{k_{B}^{2}}{\hbar}\frac{C|m|}{2\pi
k_{B}T}\int^{\infty}_{0}xf^{\prime}(x)dx=\frac{k_{B}^{2}}{\hbar}\frac{C\ln
2}{2\pi}\frac{|m|}{k_{B}T}.$ (S46)
Namely there is a $1/T$ tail in $\frac{\kappa_{xy}}{T}$.
As a final remark, in the absence of $\delta\varphi$, it is easy to show that
the Berry curvature is nonsingular near the Fermi surface. Consequently, one
does not need to consider the contribution to SATHE from
$\partial_{\varphi}\mathbf{E}_{\mathbf{a}}$ in the leading order of
$\varDelta/t$ expansion.
## IV Fermi Surface Reconstruction for Twisted Bilayer FeSe
The effective $\bm{k\cdot p}$ model in the main text is written in the oribtal
space of $\\{3d_{xz},3d_{yz}\\}$. Because the dominant hopping processes occur
within _the same_ orbitals, we can treat each orbital separately, or
equivalently go back to the single-iron Brillouin zone to work with one
ellipse on each direction. The right horizontal elliptic $M$-pocket of
monolayer FeSe is simply described with
$h=-\widetilde{\mu}+a(k_{x}-k_{M})^{2}+bk_{y}^{2}$ (S47)
with $\mu=-0.08$eV, $a=1.08\text{eV}\cdot\text{\AA}^{2}$ and
$b=1.6\text{eV}\cdot\text{\AA}^{2}$. Denoting the rotated monolayer
Hamiltonian as $h_{\pm\theta/2}(\bm{k})$, the rotated bilayer Hamiltonian
_without_ interlayer tunnelings is then simply a two-by-two diagonal matrix
$H_{\text{w/o
}T_{\perp}}=\mathop{\mathrm{diag}}\\{h_{\theta/2}(\bm{k}),h_{-\theta/2}(\bm{k})\\}$.
Under small twisting angles, two-center approximation applies Bistritzer and
MacDonald (2011) and the general tunneling strength between the top/bottom
Bloch states $|\psi^{t}_{\alpha,\bm{k}^{t}}\rangle$ and
$|\psi^{b}_{\beta,\bm{k}^{b}}\rangle$ takes the form of
$t_{\bm{k}^{t},\bm{k}^{b}}^{\alpha\beta}\equiv\langle\psi_{\alpha}^{t}|H|\psi_{\beta}\rangle=\dfrac{1}{V}\sum_{\bm{G}^{t},\bm{G}^{b}}\delta_{\bm{k}^{t}+\bm{G}^{t},\bm{k}^{b}+\bm{G}^{b}}\cdot
e^{-i\bm{G}^{b}\cdot\bm{\tau}^{b}_{\alpha}}\cdot
t_{\alpha\beta}(\bm{k}^{t}+\bm{G}^{t})\cdot
e^{i\bm{G}^{t}\cdot\bm{\tau}^{t}_{\beta}},$ (S48)
where $\bm{G}^{t,b}$ are the reciprocal vector of the top/bottom layer and
$\bm{\tau}_{\alpha}^{t}$ and $\bm{\tau}_{\beta}^{b}$ are the sublattice
position of the wannier centers of top/bottom states. Now that only one state
is left in the single-iron Brillouin zone, we can simply take
$t_{\alpha\beta}(\bm{k}^{t}+\bm{G}^{t})=\delta_{\alpha\beta}t(\bm{k}^{t}+\bm{G}^{t})$.
The low-energy form of the interlayer tunnelings can be obtained with the
expansion $\bm{k}^{t,b}\equiv\bm{q}^{t,b}+\bm{K}_{M}^{t,b}$,
$|\bm{q}^{t,b}|\ll 1$. Assuming that the tunneling function
$t(\bm{k}^{t}+\bm{G}^{t})$ only depends on the norm of its arguments, we can
take the approximation that
$t(\bm{k}^{t}+\bm{G}^{t})\equiv t(\bm{q}^{t}+\bm{K}_{M}^{t}+\bm{G}^{t})\doteq
t(|\bm{K}_{M}|).$
Since $|\bm{K}_{M}|\gg|\bm{q}^{t,b}|$, only terms with momentum differences
$\bm{k}^{t}+\bm{G}^{t}-\bm{k}^{b}-\bm{G}^{b}\propto\theta$ will be left due to
the delta function in Eq.(S48). Taking the right $M$-pocket as an example, up
to the lowest-order truncation on the grids of the scattering vectors, only
two terms of $\delta\bm{q}^{t}$ and $\delta\bm{q}^{b}$ need to be included, as
is seen in Fig.S2.
Figure S2: (a) Rotated top layer (green) and bottom layer (orange). The
interference of all different hopping processes capture the spatial variation
of interlayer tunnelings, which determines the moiré Brillouin zone (gray
square). Up to the lowest truncation on such moiré k-shell, there are just two
vertical hopping vectors $\delta\bm{q}^{t}$ and $\delta\bm{q}^{b}$ involved in
the reconstruction of the left/right $M$-pockets (ditto for the top/bottom
ones under a $C_{4}$-rotation). (b) Illustration for the moire-pattern-
involved Fermi surface reconstruction. Taking the top layer elliptic Fermi
surface (green dashed lines) as the reference, both the rotated bottom layer
(down orange dashed lines) and the copied $M$-pockets due to $G_{x}$ (up
orange dashed lines) involve in the reconstruction of the Fermi surfaces (red
solid lines).
Denoting $t(|\bm{K}_{M}|)/V\equiv w$, such truncation gives rise to the
minimal moiré Hamiltonian
$H^{\text{moir\'{e}}}(\bm{k})=\left(\begin{array}[]{ccc}h_{\theta/2}(\bm{k})&w&w\\\
w&h_{-\theta/2}(\bm{k}+\delta\bm{q}^{t})&0\\\
w&0&h_{-\theta/2}(\bm{k}+\delta\bm{q}^{b})\end{array}\right).$ (S49)
The moiré pattern-induced Fermi surface reconstruction is then obtained by
diagonalization Eq.(S49), which recovers the two vertical _open_ curves in
(a1) of Fig.2 in the main text. The other two horizontal curves comes from the
interference of the hopping processes between the up/down $M$-pockets.
|
# Canards in a bottleneck
Annalisa Iuorio<EMAIL_ADDRESS>Gaspard Jankowiak
<EMAIL_ADDRESS>Peter Szmolyan<EMAIL_ADDRESS>Marie-
Therese Wolfram<EMAIL_ADDRESS>
###### Abstract
In this paper we investigate the stationary profiles of a nonlinear Fokker-
Planck equation with small diffusion and nonlinear in- and outflow boundary
conditions. We consider corridors with a bottleneck whose width has a global
nondegenerate minimum in the interior. In the small diffusion limit the
profiles are obtained constructively by using methods from geometric singular
perturbation theory (GSPT). We identify three main types of profiles
corresponding to: (i) high density in the domain and a boundary layer at the
entrance, (ii) low density in the domain and a boundary layer at the exit, and
(iii) transitions from high density to low density inside the bottleneck with
boundary layers at the entrance and exit. Interestingly, solutions of the last
type involve canard solutions generated at the narrowest point of the
bottleneck. We obtain a detailed bifurcation diagram of these solutions in
terms of the in- and outflow rates. The analytic results based on GSPT are
further corroborated by computational experiments investigating corridors with
bottlenecks of variable width.
[label1]organization=University of Vienna, Faculty of Mathematics,
addressline=Oskar-Morgenstern-Platz 1, city=Vienna, postcode=1090,
country=Austria [label2]organization=Universität Konstanz, Fachbereich
Mathematik und Statistik, addressline=Fach D 197, city=Konstanz,
postcode=78457, country=Germany [label3]organization=Technische Universität
Wien, Institute for Analysis and Scientific Computing, addressline=Wiedner
Hauptstr. 8-10, city=Vienna, postcode=1040, country=Austria
[label4]organization=University of Warwick, Mathematics Institute,
city=Coventry, postcode=CV47AL, country=UK lox
## 1 Introduction
In this paper we investigate the stationary profiles of a nonlinear Fokker-
Planck equation with inflow and outflow boundary conditions, describing the
unidirectional cross-sectional average flow of pedestrians in corridors with a
single entrance and exit. Changes in the cross section lead to an in- or
decrease of the possible flow inside the corridor; different in- and outflow
conditions to the formation of boundary layers at the entrance and exit. In
[7] the authors derived the investigated 1D area averaged model from a
nonlinear convection diffusion equation that was originally proposed by Burger
and Pietschmann in [3]. They studied the formation of boundary layers in the
case of strictly monotone cross sectional profiles using geometric singular
perturbation theory (GSPT). In this paper we extend our analysis to corridors
with a unique point of minimal width which we denote as bottlenecks in the
following.
There has been an increased interest in the analysis of PDE models for
pedestrian flows within the applied mathematics community in the last years.
These models usually describe the dynamics of a single group of pedestrians
having a common goal; for example unidirectional flows in corridor; or several
groups with different objectives; as in bidirectional flows see [2, 1]. The
resulting PDEs or systems of PDEs are usually highly nonlinear and coupled. In
addition to nonlinear boundary conditions, convection dominated terms as well
as nonlinear interaction terms require the use of non-standard analytical and
computational techniques to show existence of solutions, analyse their long
time behavior and perform computational experiments. Stationary profiles of
these PDE models provide useful insights into the complex dynamics and allow
to predict seggregation dynamics (in the case of multi-species flows) or the
formation of boundary layers or high density regions (in the case of low or
high inflow and outflow rates or at bottlenecks), see for example [1, 3]. For
a general overview on mathematical modeling, analysis and simulation we refer
to [4, 13].
We reiterate that the investigated PDE model for area averaged flows comprises
a nonlinear convection and linear diffusion term, as well as nonlinear in- and
outflow at the entrance and exit. The interplay of small diffusion, the
geometry of the domain as well as the in- and outflow rates lead to the
formation of boundary layers, which we analyse using Geometric Singular
Perturbation Theory (GSPT).
GSPT is a dynamical systems approach to singularly perturbed ordinary
differential equations started by the pioneering work of Fenichel [6]. The
most common form of GSPT considers slow-fast systems of the form
$\displaystyle\dot{u}$ $\displaystyle=f(u,v),$ (1)
$\displaystyle\varepsilon\dot{v}$ $\displaystyle=g(u,v),$
where $u$ and $v$ are functions of $t$ and $0<\varepsilon\ll 1$. Often $t$ has
the interpretation of time but it may represent equally well a spatial
variable. For $f=O(1)$ and $g=O(1)$ the variable $u$ varies on the slow time-
scale $t$ and the variable $v$ on the fast time-scale
$\tau:=\frac{t}{\varepsilon}$, which explains the name slow-fast system.
Written on the fast time-scale the equation has the form
$\displaystyle u^{\prime}$ $\displaystyle=\varepsilon f(u,v),$ (2)
$\displaystyle v^{\prime}$ $\displaystyle=g(u,v).$
Under suitable assumptions, solutions of System (1) for small values of
$\varepsilon$ can be constructed as perturbation of concatenations of
solutions of the two limiting problems obtained by setting $\varepsilon=0$ in
systems (1) and (2), which are referred to as the reduced problem and the
layer problem, respectively. In GSPT, these constructions are carried out in
the framework of dynamical systems theory; with the theory of invariant
manifolds playing a particularly important role. In the specific problem
analysed in this paper, well established results and methods from GSPT are
used and adapted for the analysis of a boundary value problem.
Therefore, we do not give a more detailed summary of GSPT, but refer to [10,
11] for more background on GSPT and its many applications. To name a few
recent applications we mention the analysis of multi-scale structures in
Micro-Electro-Mechanical Systems [8] and in vegetation patterns [9]. In the
context of pedestrian dynamics, GSPT has been successfully applied to study
closing channels in [7].
The necessary concepts and results from GSPT are explained in Section 2 as
needed in the context of the specific problem at hand.
### 1.1 The mathematical model
In the following we briefly discuss the underlying modeling assumption of the
area averaged PDE under investigation. A more detailed derivation can be found
in [7].
We consider a undirectional flow of a large pedestrian crowd, whose density if
given by $\rho=\rho(x,y,t)$, in a 2D domain with a single entrance (at $x=0$)
and a single exit (at $x=L$). Furthermore we assume that the pedestrian
density is constant across the cross section, that is for fixed $x$. This
assumption is satisfied if
* •
the domain is symmetric with respect to the $x$-axis, and
* •
the initial pedestrian distribution is symmetric with respect to the $x$-axis.
We assume that the dynamics are driven by convective transport and diffusion,
in particular the total normalised pedestrian flow is given by
$\displaystyle\mathbf{j}=-\varepsilon\nabla\rho+\rho(1-\rho)\mathbf{u},$ (3)
where $\mathbf{u}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a normalised vector
field in the desired direction (in our case pointing in the general direction
of the exit) and $\varepsilon>0$ is the diffusion coefficient. We see that the
average velocity corresponds to $1-\rho$, hence individuals move at maximum
speed $1$ at density $\rho\equiv 0$ and vanishes if the density reaches its
maximum value $\rho\equiv 1$. Note that the relation of the average density to
the average velocity is commonly referred to as the fundamental diagram, and
that similar relations have been investigated in traffic flow; consider for
example the well known Lighthill-Whitham-Richard model [12, 14].
In [7] the authors derived a 1D area averaged PDE model, which is based on the
above assumptions and a suitable rescaling in space. It reads as
$\displaystyle\partial_{t}\rho(x,t)=\partial_{x}\left(k(x)(-\varepsilon\partial_{x}\rho(x,t)+\rho(x,t)\,(1-\rho(x,t)))\right)=0$
(4a) The equation is supplemented with in- and outflow conditions
$\displaystyle j(0,t)$ $\displaystyle=\alpha(1-\rho(0,t))$ (4b) $\displaystyle
j(L,t)$ $\displaystyle=\beta\rho(0,t)$ (4c)
where $j(x,t)=-\varepsilon\partial_{x}\rho(x,t)+\rho(x,t)\,(1-\rho(x,t))$ is
the 1D equivalent of (3). In the derivation of (4), the function $k$ is the
product of the width with the cross-sectional average of first component of
$\mathbf{u}$. For simplicity we refer to $k$ as the width of the bottleneck,
which amounts to assuming that the cross-sectional average is 1. The
parameters $\alpha>0$ and $\beta>0$ are the inflow and outflow rate,
respectively.
The boundary condition (4b) describes the inflow at the entrance; the inflow
is maximal if the entrance is empty ($\rho\equiv 0$), but decreases to zero
when approaching the maximum density $\rho\equiv 1$. At the exit (4c) we do
not assume that the outflow is limited by the maximum capacity. Hence, the
outflow rate is proportional to the density of individuals at the exit.
In [7] the existence of a unique stationary solution of (4) has been
established in great generality by PDE methods, thus, it remains to understand
its structure and dependence on parameters.
### 1.2 Content and organisation of the paper
In this paper we continue and extend the analysis of stationary profiles for
system (4) in [7], where a detailed analysis of stationary profiles and their
dependence on the in- and outflow rates $\alpha$ and $\beta$ was given for
corridors with monotonically decreasing (or increasing) functions $k$. Recall,
that smaller values of the function $k$ account for reduced mobility in
narrower regions. It was shown in [7] that the nonlinear in- and outflow
conditions lead to the formation of boundary layers at the entrance or exit.
Building on the approach in [7] we now consider the important case of
corridors, whose width has a unique minimum. This setting corresponds to
functions $k$ which have a unique global minimum at $x=x_{0}\in(0,L)$. In the
following we refer to domains of this type as corridors with bottlenecks (or
sometimes only bottlenecks). Throughout this paper we will without loss of
generality, assume that $L=1$. Therefore, we investigate the stationary states
of system (4) described by
$\partial_{x}\mathrm{J}=\partial_{x}(k(x)j(x))=0\,,$ (5a) where
$j(x)=-\varepsilon\partial_{x}\rho(x)+\rho(x)\,(1-\rho(x))$ coupled with the
following boundary conditions $\displaystyle j$
$\displaystyle=\alpha\left(1-\rho\right)$ $\displaystyle\text{ at }x$
$\displaystyle=0\,,$ (5b) $\displaystyle j$ $\displaystyle=\beta\rho$
$\displaystyle\text{ at }x$ $\displaystyle=1\,.$
We characterise all profiles for different inflow and outflow rates $\alpha$
and $\beta$ in the singular limit $\varepsilon=0$. We identify 8 regions in
parameter space corresponding to profiles with different structures. Two of
these regions correspond to high-density profiles, two other regions
correspond to low density profiles. These profiles are quite similar to
profiles considered in [7] and are only weakly affected by the presence of the
bottleneck. Due to the bottleneck a new interesting class of profiles exists,
which corresponds to solutions starting at high density and making a
transition to low density in the region where the function $k$ attains its
minimum. Since this type of solutions allows four possible configurations of
boundary layers this leads to four types of transitional density profiles. We
refer to these four types of profiles as transitional profiles.
Our GSPT analysis shows that these transitional profiles are caused by the
existence of canard solutions passing through a folded saddle [15]. Canard
solutions are solutions of singularly perturbed ODEs which follow repelling
slow manifolds for a considerable time. The essence of the canard phenomenon
is that these solutions lie exponentially close to the repelling slow manifold
and are therefore able to follow it for some time before they are ultimately
repelled from it. Clearly, special mechanisms are needed to bring solutions of
interest exponentially close to the repelling slow manifold. The occurrence of
canard solutions in boundary value problems is conceptually less surprising
than their occurrence in initial value problems, nevertheless we are not aware
of similar works or results in the context of nonlinear boundary value
problems.
The three profile types have a similar structure as the low density, high
density and maximum current phases observed in Totally Asymmetric Simple
Exclusion Process (TASEP). Note that the proposed model (4) was derived from a
2D TASEP, see [3]. In the TASEP, $\alpha$ and $\beta$ are the entry and exit
rates, respectively, see for example [5, 17].
The rest of the paper is organised as follows. The GSPT analysis leading to
the main result on the structure of solutions is carried out in Section 2. In
Section 3 the analytical results are illustrated and confirmed by
computational experiments for different channels. We conclude with an
interpretation of the main features of the constructed solutions in the
various regimes in a manner which could be useful in further studies of
pedestrian dynamics.
## 2 GSPT analysis
In this section, the stationary states associated to (5b) are investigated in
a bottleneck scenario. The problem is rewritten as an equivalent boundary
value problem for an autonomous three-dimensional system of first order
differential equations in slow-fast form. As explained in the introduction, we
will identify 8 regions in the $(\alpha,\beta)$ parameter space, in which the
stationary profiles have the same structure in the singular limit
$\varepsilon=0$. We construct singular solutions of the boundary value problem
as concatenations of solutions of the corresponding layer- and reduced
problem. These singular solutions are then shown to persist for $\varepsilon$
small. The profiles which exist in four of these regions involve a canard
solution generated at a point corresponding to the minimum of $k$.
For the rest of this paper we make the following assumption which is crucial
for our approach and results.
Main Assumption: The function $k\in C^{2}([0,1])$ is positive and has a unique
global nondegenerate minimum at $x=x^{\ast}\in(0,1)$ satisfying
$k^{\prime}(x^{\ast})=0,\qquad k^{\prime\prime}(x^{\ast})>0.$ (6)
###### Remark 1.
Here we denote the derivative of the coefficient function $k$ as $k^{\prime}$.
Below we will also consider the function $g:=k^{\prime}/k$, its derivative
will also be denoted as $g^{\prime}$. We would like to point out that starting
with Equation (10) the symbol ′ will be mainly used to denote derivatives of
the sought solution with respect to a rescaled fast variable. The above slight
use of notations should not lead to any confusion.
By introducing the function
$g:=\frac{k^{\prime}}{k}$
we can rewrite Equation (5b) as the system
$\displaystyle\frac{dj}{dx}$ $\displaystyle=-g(x)j,$ (7)
$\displaystyle\varepsilon\frac{d\rho}{dx}$ $\displaystyle=\rho(1-\rho)-j.$
Analogously to [7], this system can be transformed into an autonomous system
by introducing the variable $\xi=x$ as a new dynamic variable and including
the trivial equation $\frac{d\xi}{dx}=1$. From now on, we use the notation
$\dot{~{}}=\frac{d}{dx}$. Thus, we obtain the following autonomous
reformulation of Equation (7)
$\displaystyle\dot{j}$ $\displaystyle=-g(\xi)j,$ (8) $\displaystyle\dot{\xi}$
$\displaystyle=1,$ $\displaystyle\varepsilon\dot{\rho}$
$\displaystyle=\rho(1-\rho)-j,$
where the above assumptions on $k$ identically apply with
$\xi^{\ast}=x^{\ast}$, with boundary conditions
$\displaystyle j$ $\displaystyle=\alpha\left(1-\rho\right)$
$\displaystyle\text{ at }\xi$ $\displaystyle=0\,,$ (9) $\displaystyle j$
$\displaystyle=\beta\rho$ $\displaystyle\text{ at }\xi$ $\displaystyle=1\,.$
System (8) is a slow-fast system, where the dynamics of $\rho$ occur on the
fast scale, while the dynamics of $j$ and $\xi$ take place on the slow scale.
By transforming to the fast variable $\chi=\frac{x}{\varepsilon}$, and using
the notation ${}^{\prime}=\frac{d}{d\chi}$, we can rewrite System (8) as
$\displaystyle j^{\prime}$ $\displaystyle=-\varepsilon g(\xi)j,$ (10)
$\displaystyle\xi^{\prime}$ $\displaystyle=\varepsilon,$
$\displaystyle\rho^{\prime}$ $\displaystyle=\rho(1-\rho)-j.$
As explained in the introduction, letting $\varepsilon\to 0$ in Equations (8)
and (10) leads to two limiting subproblems – i.e. the _reduced_ problem and
the _layer_ problem, respectively – which are simpler to analyse. The layer
problem ($\varepsilon=0$ in (10)) is given by
$\displaystyle j^{\prime}$ $\displaystyle=0,$ (11) $\displaystyle\xi^{\prime}$
$\displaystyle=0,$ $\displaystyle\rho^{\prime}$
$\displaystyle=\rho(1-\rho)-j,$
and describes the dynamics of the fast variable $\rho$ for fixed $j$ and $\xi$
values. The manifold of its equilibria is known as the _critical manifold_
$\mathcal{C}_{0}:=\left\\{(j,\xi,\rho)~{}:~{}j=\rho(1-\rho)\right\\},$ (12)
which is a folded surface in $(j,\xi,\rho)$ space. The critical manifold
$\mathcal{C}_{0}$ is the union of two submanifolds $\mathcal{C}_{0}^{a}$
($\rho>\frac{1}{2}$) and $\mathcal{C}_{0}^{r}$ ($\rho<\frac{1}{2}$) – which
are attracting and repelling, respectively – and a line of fold points
$F:=\left\\{(j,\xi,\rho)~{}:~{}j=\frac{1}{4},~{}\rho=\frac{1}{2}\right\\},$
(13)
as shown in Figure 1. Fenichel Theory [6] implies that away from the fold line
$F$ the submanifolds $\mathcal{C}_{0}^{a}$ and $\mathcal{C}_{0}^{r}$ perturb
to (non-unique) attracting and repelling slow manifolds
$\mathcal{C}_{\varepsilon}^{a}$ and $\mathcal{C}_{\varepsilon}^{r}$ for
$\varepsilon$ small.
If the reduced flow reaches the fold line $F$ transversally at a point $p\in
F$, the point $p$ is a jump point where a transition to fast motion close to
solutions of the layer problem occurs, see [16]. At exceptional points $p\in
F$ where this transversality condition is violated solutions of the reduced
flow may cross through $p$ from $\mathcal{C}_{0}^{a}$ to
$\mathcal{C}_{0}^{r}$, or vice versa. Such solutions are called (singular)
canards, the corresponding $p\in F$ is a canard point. The least degenerate
canard points have been classified and analysed by the blow-up method as
folded saddles and folded nodes in [15]. There it is shown that these
(singular) canards persist as canard solutions, i.e. solutions corresponding
to intersections of the slow manifolds $\mathcal{C}_{\varepsilon}^{a}$ and
$\mathcal{C}_{\varepsilon}^{r}$ near $p$ for $\varepsilon$ small. Thus, the
existence of canard solutions provides a mechanism that solutions lying in (or
exponentially close to) the attracting slow manifold
$\mathcal{C}_{\varepsilon}^{a}$ can be continued in (or exponentially close
to) the repelling slow manifold $\mathcal{C}_{\varepsilon}^{r}$. The less
counter-intuitive situation that solutions lying in the repelling slow
manifold can be continued in (or close to) the attracting slow manifold is
also possible. Canard solutions of this second type are often referred to as
faux canards.
\begin{overpic}[scale={.5}]{laypb}
\put(40.0,20.0){\footnotesize$\mathcal{C}_{0}^{r}$}
\put(40.0,70.0){\footnotesize$\mathcal{C}_{0}^{a}$}
\put(58.0,45.0){\footnotesize$F$} \end{overpic} Figure 1: Fast dynamics in
$(j,\rho)$-space for a fixed value of $\xi$. The blue curve represents
$\mathcal{C}_{0}$ consisting of the two branches $\mathcal{C}_{0}^{a}$
(attracting), $\mathcal{C}_{0}^{r}$ (repelling), and the fold line $F$. The
green lines indicate orbits of the layer problem (11), while the blue dot
represents the line of fold points $F$.
In the following we analyse the reduced flow on $F$. We will show that a
canard point of folded saddle type occurs at the point
$p^{\ast}=\left(\frac{1}{4},\xi^{\ast},\frac{1}{2}\right)$ (14)
where $\xi^{\ast}$ is the location of the global minimum of the function $k$.
The reduced problem is very simple
$\displaystyle\dot{j}$ $\displaystyle=-g(\xi)j,$ (15a)
$\displaystyle\dot{\xi}$ $\displaystyle=1.$ (15b)
The phase space for the reduced problem is $[0,1/4]\times[0,1]$, where $j=1/4$
corresponds to the fold line. It follows from Equation (5a) that $k(\xi)\,j$
is a conserved quantity, hence the level lines of this function give the
orbits of the reduced problem (15).
However, as always for folded critical manifolds, the classification of the
reduced flow – in particular at the fold line and at canard points – is more
conveniently carried out in the variables $(\xi,\rho)$ by using the constraint
$j=\rho(1-\rho)$ which defines $\mathcal{C}_{0}$. Differentiating the
constraint with respect to $x$ gives $\dot{j}=(1-2\rho)\dot{\rho}$, which
allows to rewrite the reduced problem as
$\displaystyle\dot{\xi}$ $\displaystyle=1,$ (16)
$\displaystyle(1-2\rho)\,\dot{\rho}$ $\displaystyle=-g(\xi)\,\rho(1-\rho).$
with $\xi\in[0,1]$ and $\rho\in[0,1]$. System (16) is singular at the fold
line $F$, i.e. for $\rho=1/2$. This system can be desingularised by
multiplying the right hand-side by $1-2\rho$ and dividing out this factor in
the $\rho$ equation. This gives the desingularised reduced system
$\displaystyle\dot{\xi}$ $\displaystyle=1-2\rho,$ (17)
$\displaystyle\dot{\rho}$ $\displaystyle=-g(\xi)\,\rho(1-\rho).$
This multiplication of the right hand side by $(1-2\rho)$ corresponds to a
position dependent rescaling of the independent variable $x$, which does not
change orbits of the system away from the fold line. However, for $\rho>1/2$
the flow direction is reversed, which needs to be taken into account.
We now collect the properties of the reduced problem, which are needed in the
analysis of the boundary value problem (8)-(9). These properties depend on
properties of the function $g=k^{\prime}/k$. Our main Assumption implies that
the global nondegenerate minimum of $k$ at $\xi^{\ast}$ is a simple zero of
$g$ corresponding to a saddle point $(\xi^{\ast},1/2)$ of the desingularised
reduced problem. Other zeros of $g$ lead to additional equilibria, which are
discussed only briefly, since we show later that these play no role in the
analysis of the boundary value problem.
###### Lemma 1.
The reduced problem (16) has the following properties:
1. 1.
The phase portrait is symmetric with respect to the line $\rho=1/2$, which
corresponds to the fold line $F$.
2. 2.
The variable $\xi$ is increasing along all orbits, i.e. the flow is from left
to right.
3. 3.
The lines $\rho=0$ and $\rho=1$ are invariant.
4. 4.
In regions with $g(\xi)>0$ the variable $\rho$ is decreasing along orbits for
$1/2<\rho<1$ and is increasing for $0<\rho<1/2$. In regions with $g(\xi)<0$
this monotonicity is reversed. The variable $\rho$ is constant in regions with
$g(\xi)=0$, corresponding to regions where the width of the corridor is
constant.
5. 5.
The line $\rho=1/2$ is a line of singularities. Points $(\xi,1/2)$ with
$g(\xi)>0$ are reached in finite time by the forward flow and the derivative
$\dot{\rho}$ blows up there. Similarly, points $(\xi,1/2)$ with $g(\xi)<0$ are
reached in finite time by the backward flow.
6. 6.
The point $p^{\ast}=\left(\xi^{\ast},\frac{1}{2}\right)$ is a canard point of
folded saddle type.
7. 7.
There exist two (symmetric with respect to the line $\rho=1/2$) singular
canard solutions with orbits $S_{c}$ and $\tilde{S}_{c}$ passing smoothly
through the singularity located at $p^{\ast}$. The canard $S_{c}$ crosses from
the attracting part of the critical manifold to the repelling one, the (faux)
canard $\tilde{S}_{c}$ crosses from the repelling part of the critical
manifold to the attracting one.
8. 8.
The (faux) canard orbit $\tilde{S}_{c}$ starts at $\xi=0$,
$\rho=\rho_{c}^{0}\in(0,1/2)$ and reaches $\xi=1$ at
$\rho=\rho_{c}^{1}\in(0,1/2)$. The canard orbit $S_{c}$ starts at $\xi=0$,
$\rho=1-\rho_{c}^{0}\in(1/2,1)$ and reaches $\xi=1$ at
$\rho=1-\rho_{c}^{1}\in(1/2,1)$.
9. 9.
Solutions starting at $\xi=0$ with $\rho\in[0,\rho_{c}^{0})$ reach $\xi=1$
with $\rho\in[0,1-\rho_{c}^{1})$. Solutions starting at $\xi=0$ with
$\rho\in(1-\rho_{c}^{0},1]$ reach $\xi=1$ with $\rho\in(\rho_{c}^{1},1])$.
10. 10.
Solutions starting at $\xi=0$ with $\rho\in(\rho_{c}^{0},1-\rho_{c}^{0})$ do
not cross the line $\xi=\xi^{\ast}$, in particular they do not reach the line
$\xi=1$. Solutions reaching $\xi=1$ with
$\rho\in(1-\rho_{c}^{1},\rho_{c}^{0})$ do not cross the line $\xi=\xi^{\ast}$
in backwards time, in particular they do not reach the line $\xi=0$.
11. 11.
An isolated zero of $g$ at say $\xi_{0}\neq\xi^{\ast}$ corresponds to another
folded singularity at $(\xi_{0},1/2)$, which is a folded saddle for
$g^{\prime}(\xi_{0})>0$ and a folded center for $g^{\prime}(\xi_{0})<0$. A
more degenerate zero of $g$ corresponds to a more degenerate folded
singularity. If $g$ is zero on an interval $[\xi_{1},\xi_{2}]$, the density
$\rho$ is constant there. In this situation $[\xi_{1},\xi_{2}]\times\\{1/2\\}$
is a line of equilibria, the endpoints of this line are again degenerate
folded singularities.
The properties of the reduced problem described in the Lemma are illustrated
in Figure 2 for a function $k$ which satisfies $k^{\prime}<0$ in
$(1,\xi^{\ast})$ and $k^{\prime}>0$ in $(\xi^{\ast},1)$.
###### Remark 2.
1. (a)
The notation $S_{c}$ and $\tilde{S}_{c}$ for the canard orbits is chosen to be
consistent with the notation we introduce below for other orbits of the
reduced problem in the construction of singular solutions of the boundary
value problem.
2. (b)
The property 11. associated with additional zeros of $g$ (which may occur
under our rather general main Assumption on the function $k$) are mainly
included for completeness. In Remark 3 below, we show that they play no role
in the construction of solutions of the boundary value problem, due to
property 10. of the Lemma.
###### Proof.
Properties 1.-5. follow directly from the equations. The point $p^{\ast}$ is
an equilibrium for the desingularised system (17). The matrix associated with
the linearisation of (17) at $p^{\ast}$ is
$A:=\left(\begin{array}[]{cc}0&-2\\\\[2.84526pt]
-\frac{g^{\prime}(\xi^{\ast})}{4}&0\end{array}\right).$ (18)
Since $g^{\prime}(\xi^{\ast})=k^{\prime\prime}(\xi^{\ast})/k(\xi^{\ast})$, the
assumption $k^{\prime\prime}(\xi^{\ast})>0$ translates into
$g^{\prime}(\xi^{\ast})>0$. This gives $\det A=-\frac{g(\xi^{\ast})}{2}<0$,
hence $p^{\ast}$ is a saddle point for (17) with associated smooth stable and
unstable manifolds. For the reduced problem (16) – with the flow direction
reversed for $\rho>1/2$ – the point $p^{\ast}$ is a folded saddle [15]. Due to
a cancellation of a simple zero on both sides of the $\rho$-equation in (16),
the stable manifold of the saddle is now the (faux) canard $\tilde{S}_{c}$,
corresponding to a smooth solution passing through through the point
$p^{\ast}$. Similarly, the unstable manifold of the saddle becomes the canard
$S_{c}$. This proves properties 6. and 7.
The conserved quantity $k(\xi)j$ of equation (15a) translates into the
conserved quantity
$H(\xi,\rho)=k(\xi)\rho(1-\rho)>0$ (19)
of the desingularised system (17), i.e. the level lines of $H$ give the phase
portrait. The canard orbits $S_{c}$ and $\tilde{S}_{c}$ are the level lines
$H(\xi,\rho)=\frac{k(\xi^{\ast})}{4}$. Since $k$ has its global minimum at
$\xi^{\ast}$, the canard orbits cannot intersect the (fold) line $\rho=1/2$.
Since in addition, the canard orbits cannot intersect the lines $\rho=0$,
$\rho=1$ where $H=0$, the canard orbits extend to $\xi=0$ and $\xi=1$. Thus
assertion 8. follows, with $\rho_{c}^{0}\in(0,1/2)$ and
$\rho_{c}^{1}\in(1/2,1)$ defined as the solutions of the equations
$H(0,\rho_{c}^{0})=\frac{k(\xi^{\ast})}{4},\qquad
H(1,\rho_{c}^{1})=\frac{k(\xi^{\ast})}{4}.$
The solutions described in Assertion 9. lie on level lines with
$H(\xi,\rho)>H(\xi^{\ast},1/2)$, the solutions described in Assertion 10. lie
on level lines with $H(\xi,\rho)<H(\xi^{\ast},1/2)$. Together with 8. this
implies 9. and 10. ∎
The canard $S_{c}$ and the (faux) canard $\tilde{S}_{c}$ on $\mathcal{C}_{0}$
can be described as graphs by means of the following functions
$\displaystyle\rho_{c}^{+}(\xi)$
$\displaystyle:=\frac{1}{2}\left(1+\sqrt{1-\frac{k\left(\xi^{\ast}\right)}{k(\xi)}}\right),$
(20a) $\displaystyle\rho_{c}^{-}(\xi)$
$\displaystyle:=\frac{1}{2}\left(1-\sqrt{1-\frac{k(\xi^{\ast})}{k(\xi)}}\right),$
(20b)
as follows
$\displaystyle S_{c}$
$\displaystyle:=\left\\{(\xi,\rho)\,:\,0\leq\xi\leq\xi^{\ast},\,\rho=\rho_{c}^{+}(\xi)\right\\}\cup\left\\{(\xi,\rho)\,:\,\xi^{\ast}\leq\xi\leq
1,\,\rho=\rho_{c}^{-}(\xi)\right\\},$ (21) $\displaystyle\tilde{S}_{c}$
$\displaystyle:=\left\\{(\xi,\rho)\,:\,0\leq\xi\leq\xi^{\ast},\,\rho=\rho_{c}^{-}(\xi)\right\\}\cup\left\\{(\xi,\rho)\,:\,\xi^{\ast}\leq\xi\leq
1,\,\rho=\rho_{c}^{+}(\xi)\right\\}.$
The values $\rho_{c}^{0}$ and $\rho_{c}^{1}$ introduced in Lemma 1
(corresponding to the $\rho$-values of the (faux) canard) at $\xi=0$ and
$\xi=1$, respectively, are then given by
$\rho_{c}^{0}=\rho_{c}^{-}(0),\qquad\rho_{c}^{1}=\rho_{c}^{+}(1).$ (22)
The points of the canard $S_{c}$ corresponding to $\xi=0$ and $\xi=1$ in
$(j,\xi,\rho)$-space which play an important role in the following analysis
are
$\displaystyle p_{c}^{0}$
$\displaystyle:=\left(\rho_{c}^{0}(1-\rho_{c}^{0}),0,1-\rho_{c}^{0}\right),$
(23) $\displaystyle p_{c}^{1}$
$\displaystyle:=\left(\rho_{c}^{1}(1-\rho_{c}^{1}),1,1-\rho_{c}^{1}\right).$
###### Remark 3.
The function $g$ may have zeros $\xi\neq\xi^{\ast}$. All these points
$(\xi,1/2)$ are equilibria of the desingularised system (17) but these
equilibria and possible canard solutions associated with them are confined to
the open region $\mathcal{N}$ bounded by $\tilde{S}_{c}$ from below and by
$S_{c}$ from above for $\xi<\xi^{\ast}$, and by $S_{c}$ from below and by
$\tilde{S}_{c}$ from above for $\xi>\xi^{\ast}$ (see Fig. 2). Since no
transitions from $\xi=0$ to $\xi=1$ are possible through the region
$\mathcal{N}$, it plays no role in the construction of solutions of the
boundary value problem. Since other folded singularities associated with local
minima or maxima of $k$ and their associated canard solutions are confined to
$\mathcal{N}$ these also play no role for boundary value problem.
\begin{overpic}[scale={0.7}]{red_flow_new_shade.pdf}
\put(1.0,2.0){\small{$0$}} \put(61.0,2.0){$\xi^{\ast}$}
\put(-2.0,17.0){$\rho_{c}^{0}$} \put(-8.0,70.0){$1-\rho_{c}^{0}$}
\put(83.0,27.0){$1-\rho_{c}^{1}$} \put(83.0,61.0){$\rho_{c}^{1}$}
\put(55.0,53.0){$S_{c}$} \put(55.0,33.0){$\tilde{S}_{c}$} \end{overpic} Figure
2: Illustration of the reduced flow associated to Equations (8)-(9) described
in Lemma 1 for $k(\xi)=1+a\,\mathrm{cos}\left(\frac{2\pi\xi}{b}\right)$ and
$a=0.3$, $b=1.5$. The solid gray line indicates the line of fold points $F$
(see (13)), whereas the blue dot corresponds to the canard point of folded
saddle type $p^{\ast}$. The cyan curves correspond to the canard $S_{c}$
(solid line) and the (faux) canard $\tilde{S}_{c}$ (dashed line). The shaded
red area represents the region $\mathcal{N}$ defined in Remark 3, which plays
no role in the construction of solutions of the boundary value problem
(8)-(9).
We now begin the construction of solutions of the boundary value problem
(8)-(9) by combining solutions of the reduced problem with solutions of the
layer problem in such a way that the boundary conditions are satisfied. Here
it is important to keep in mind that solutions can jump from points on the
repelling branch $\mathcal{C}_{0}^{r}$ of the critical manifold to the
attracting branch $\mathcal{C}_{0}^{a}$, but not vice versa. In [7] we have
constructed singular solutions in the case of a closing channel using a
shooting strategy: we evolved the manifold of boundary conditions at $\xi=0$
forward and checked whether it intersected the manifold of boundary conditions
at $\xi=1$. This constructive procedure allowed to identify the initial and
final values of $\rho$ (namely $\rho_{0}$ and $\rho_{1}$) for $\varepsilon=0$.
In the bottleneck scenario, however, the presence of a canard point lying in
the interior of the spatial domain $[0,1]$ implies that singular orbits
containing segments of the canards $S_{c}$ or $\tilde{S}_{c}$ can make slow
transitions between the branches of the critical manifold. Most importantly,
this allows transitions from the attracting branch back to the repelling
branch. We will show that this leads to the new type of transitional profiles,
described in the introduction. Due to the special role of the canard point
$p^{\ast}$ we modify the shooting strategy by evolving also the manifold of
boundary conditions at $\xi=1$ (backwards) and checking the intersection with
the forward evolution of the manifold of left boundary conditions at
$\xi=\xi^{\ast}$, where the canard point $p^{\ast}$ lies.
In the dynamical systems framework, boundary conditions (9) correspond to two
lines in the $(j,\xi,\rho)$-space, satisfying $j=\alpha(1-\rho)$ at $\xi=0$
and $j=\beta\rho$ at $\xi=1$, respectively. However, due to the fast-slow
structure, the set of admissible boundary conditions is restricted to (see
Figures 3-4)
$\displaystyle\mathcal{L}$
$\displaystyle:=\left\\{\left(\alpha(1-s),\,0,\,s\right)\ :\ \rho_{\alpha}\leq
s\leq 1\right\\},$ (24a) $\displaystyle\mathcal{R}$
$\displaystyle:=\left\\{\left(\beta t,\,1,\,t\right)\ :\ 0\leq
t\leq\rho_{\beta}\right\\}.$ (24b)
Here
$\rho_{\alpha}=\begin{cases}\alpha&\quad\text{if }\alpha\leq\frac{1}{2},\\\
1-\frac{1}{4\alpha}&\quad\text{if }\alpha\geq\frac{1}{2},\end{cases}$ (25)
and
$\rho_{\beta}=\begin{cases}1-\beta&\quad\text{if }\beta\leq\frac{1}{2},\\\
\frac{1}{4\beta}&\quad\text{if }\beta\geq\frac{1}{2}.\end{cases}$ (26)
The lower and upper bounds $\rho_{\alpha}$ and $\rho_{\beta}$ for the density
$\rho$ are caused by the fast-slow structure of the flow: if we would consider
a starting point $(\alpha(1-\rho),\,0,\,\rho)$ with $0\leq\rho<\rho_{\alpha}$,
the orbit would be immediately repelled to infinity from $\mathcal{C}_{0}$,
hence connecting to the boundary conditions at $\xi=1$ is impossible.
Analogously, points satisfying $(\beta\rho,\,1,\,\rho)$ with
$\rho_{\beta}<\rho\leq 1$ cannot be endpoints of the singular orbits, since
they are repelling for the layer problem.
Thus, the initial and final points of the singular orbits – $p_{0}$ and
$p_{1}$, respectively – must satisfy
$p_{0}\in\mathcal{L},\textrm{ and }p_{1}\in\mathcal{R}.$ (27)
The manifold $\mathcal{L}$ intersects with $\mathcal{C}_{0}$ at $(0,0,1)$ and
$l=(\alpha(1-\alpha),0,\alpha),$ (28)
while $\mathcal{R}$ intersects with $\mathcal{C}_{0}$ at $(0,1,0)$ and
$r=(\beta(1-\beta),1,1-\beta).$ (29)
For $\varepsilon=0$, the variable $\xi$ evolves only on $\mathcal{C}_{0}$
according to the reduced flow (16). Therefore, in order for the singular
solution to evolve from $\xi=0$ to $\xi=1$, we must connect $\mathcal{L}$ and
$\mathcal{R}$ to $\mathcal{C}_{0}$. The points $l$ and $r$ already belong to
$\mathcal{C}_{0}$. Other points on $\mathcal{L}$ and $\mathcal{R}$ can reach
$\mathcal{C}_{0}$ using the layer problem (11). Tracking the evolution of
$\mathcal{L}$ by means of the layer problem at $\xi=0$ until it reaches
$\mathcal{C}_{0}$, and analogously the evolution of $\mathcal{R}$ backwards
until the layer problem at $\xi=1$ intersects $\mathcal{C}_{0}$, yields two
sets (shown in Figures 3-4):
$\displaystyle\mathcal{L}^{+}$
$\displaystyle:=\left\\{\left(\alpha(1-s),\,0,\,\rho^{+}(0,s)\right)\ :\
\rho_{\alpha}\leq s\leq 1\right\\},$ (30a) $\displaystyle\mathcal{R}^{-}$
$\displaystyle:=\left\\{\left(\beta t,\,1,\,\rho^{-}(1,t)\right)\ :\ 0\leq
t\leq\rho_{\beta}\right\\}.$ (30b)
In the following, we use the symbol $\rho^{+}(0,s)$ to indicate the
$\rho$-value (greater than or equal to $\frac{1}{2}$) reached by the point
$(\alpha(1-s),\,0,\,s)$ after its transition from $\mathcal{L}$ to
$\mathcal{C}_{0}$ by means of the layer problem. If the solution $(\xi,\rho)$
of the reduced flow 16 starting at $(0,\rho^{+}(0,s))$ reaches
$\xi=\xi^{\ast}$, we denote its value of $\rho$ at $\xi=\xi^{\ast}$ by
$\rho^{+}\left(\xi^{\ast},s\right)$. In an analogous manner, we introduce the
symbol $\rho^{-}(1,t)$ to indicate the $\rho$-value (less than or equal to
$\frac{1}{2}$) reached by the point $(\beta t,\,1,\,t)$ after its transition
from $\mathcal{R}$ to $\mathcal{C}_{0}$ by means of the layer problem. If the
solution $(\xi,\rho)$ of the reduced flow starting at $(1,\rho^{-}(1,t))$ and
flowing backwards reaches $\xi=\xi^{\ast}$, we denote its value of $\rho$ at
$\xi=\xi^{\ast}$ by $\rho^{-}\left(\xi^{\ast},t\right)$.
When $\alpha<\frac{1}{2}$, the reduced flow can either start on
$\mathcal{L}^{+}$ or at $l$, while for $\alpha>\frac{1}{2}$ it must start on
$\mathcal{L}^{+}$. Analogously, when $\beta<\frac{1}{2}$, the reduced flow can
either end on $\mathcal{R}^{-}$ or at $r$, while for $\beta>\frac{1}{2}$ it
must end on $\mathcal{R}^{-}$.
(a)
(b)
Figure 3: Schematic representation of $\mathcal{L}$ (orange line) and
$\mathcal{L}^{+}$ (orange curve) for (a) $0<\alpha<\frac{1}{2}$ and (b)
$\frac{1}{2}<\alpha<1$. The orange dot corresponds to $l$, the blue curve
represents $\mathcal{C}_{0}$, and the green lines correspond to the orbits of
the layer problem.
(a)
(b)
Figure 4: Schematic representation of $\mathcal{R}$ (purple line) and
$\mathcal{R}^{-}$ (purple curve) for (a) $0<\beta<\frac{1}{2}$ and (b)
$\frac{1}{2}<\beta<1$. The purple dot corresponds to $r$, the blue curve
represents $\mathcal{C}_{0}$, and the green lines correspond to the orbits of
the layer problem.
Based on this geometric interpretation of the boundary conditions, we proceed
with the construction of the singular orbits by connecting
$\mathcal{L}^{+}\cup l$ and $\mathcal{R}^{-}\cup r$ by means of the reduced
flow (16) on $\mathcal{C}_{0}$. In doing so, we first let $\mathcal{L}^{+}$ in
(30a) flow forward and $\mathcal{R}^{-}$ in (30b) flow backwards by means of
the reduced flow until $\xi=\xi^{\ast}$: we call the corresponding sets
$\mathcal{L}_{\xi^{\ast}}^{+}$ and $\mathcal{R}_{\xi^{\ast}}^{-}$,
respectively.
$\displaystyle\mathcal{L}_{\xi^{\ast}}^{+}$
$\displaystyle:=\left\\{\left(\rho^{+}\left(\xi^{\ast},s\right)\left(1-\rho^{+}\left(\xi^{\ast},s\right)\right),\,\xi^{\ast},\,\rho^{+}\left(\xi^{\ast},s\right)\right)\
:\ \rho_{\alpha}\leq s\leq 1\right\\},$ (31a)
$\displaystyle\mathcal{R}_{\xi^{\ast}}^{-}$
$\displaystyle:=\left\\{\left(\rho^{-}\left(\xi^{\ast},t\right)\left(1-\rho^{-}\left(\xi^{\ast},t\right)\right),\,\xi^{\ast},\,\rho^{-}\left(\xi^{\ast},t\right)\right)\
:\ 0\leq t\leq\rho_{\beta}\right\\}.$ (31b)
If $\alpha\geq\frac{1}{2}$ then $l\in\mathcal{L}^{+}$, and the evolution of
$l$ by means of the reduced flow is already included in
$\mathcal{L}_{\xi^{\ast}}^{+}$. If $\alpha<\frac{1}{2}$ then
$l\notin\mathcal{L}^{+}$, and therefore the corresponding point at
$\xi=\xi^{\ast}$ must be defined separately as
$l_{\xi^{\ast}}:=\left\\{\left(\rho^{+}\left(\xi^{\ast},\alpha\right)\left(1-\rho^{+}\left(\xi^{\ast},\alpha\right)\right),\,\xi^{\ast},\,1-\rho^{+}\left(\xi^{\ast},\alpha\right)\right)\right\\}.$
(32)
Analogously, if $\beta\geq\frac{1}{2}$ then $r\in\mathcal{R}^{-}$, and the
backwards evolution of $r$ by means of the reduced flow is already included in
$\mathcal{R}_{\xi^{\ast}}^{-}$. If $\beta<\frac{1}{2}$, however,
$r\notin\mathcal{R}^{-}$, and therefore the corresponding point at
$\xi=\frac{1}{2}$ must be defined separately as
$r_{\xi^{\ast}}:=\left\\{\left(\rho^{-}\left(\xi^{\ast},1-\beta\right)\left(1-\rho^{-}\left(\xi^{\ast},1-\beta\right)\right),\,\xi^{\ast},\,1-\rho^{-}\left(\xi^{\ast},1-\beta\right)\right)\right\\}.$
(33)
We note that the point $l_{\xi^{\ast}}$ exists if and only if
$\alpha\leq\rho_{c}^{0}$; analogously, the point $r_{\xi^{\ast}}$ exists if
and only if $\beta\leq 1-\rho_{c}^{1}$ (see Remark 3).
A singular orbit is then given by matching the slow and fast pieces obtained
by investigating the reduced and layer problems, respectively. More
specifically, a singular orbit exists if and only if the intersection between
the sets $\mathcal{L}_{\xi^{\ast}}^{+}\cup l_{\xi^{\ast}}$ and
$\mathcal{R}_{\xi^{\ast}}^{-}\cup r_{\xi^{\ast}}$ is non-empty, and it is
unique if this intersection consists of one point.
In addition to the canards $S_{c}$ and $\tilde{S}_{c}$ introduced above, our
analysis of the existence and structure of singular orbits is based on four
special orbits $S_{\alpha}$, $S_{\beta}$, $\tilde{S}_{\alpha}$,
$\tilde{S}_{\beta}$ of the reduced flow (see Figure 5):
* •
The orbit $S_{\alpha}$, defined for each $\alpha\in(0,1)$, is the one starting
at $\rho=\alpha$ at $\xi=0$. For $\alpha<\rho_{c}^{0}$ or
$\alpha>1-\rho_{c}^{0}$, the corresponding final value of $\rho$ at $\xi=1$ is
denoted by $\rho^{\ast}(\alpha)$. For $\rho_{c}^{0}<\alpha<1-\rho_{c}^{0}$,
$S_{\alpha}$ ends on the fold line $F$ and hence
$S_{\alpha}\subset\mathcal{N}$. For $\alpha=\rho_{c}^{0}$ or
$\alpha=1-\rho_{c}^{0}$, $S_{\alpha}$ ends on the canard point $p^{\ast}$ at
$\xi=\xi^{\ast}$, and its continuation for $\xi\in[\xi^{\ast},1]$ is therefore
not uniquely defined.
* •
The orbit $S_{\beta}$, defined for each $\beta\in(0,1)$, is the one ending at
$\rho=1-\beta$ at $\xi=1$. For $\beta<1-\rho_{c}^{1}$ or $\beta>\rho_{c}^{1}$,
the corresponding initial value of $\rho$ at $\xi=0$ is denoted by
$\rho_{\ast}(\beta)$. For $1-\rho_{c}^{1}<\beta<\rho_{c}^{1}$, $S_{\beta}$
ends on the fold line $F$ and hence $S_{\beta}\subset\mathcal{N}$. For
$\beta=1-\rho_{c}^{1}$ or $\beta=\rho_{c}^{1}$, $S_{\beta}$ ends on the canard
point $p^{\ast}$ at $\xi=\xi^{\ast}$ backward in $\xi$, and its continuation
for $\xi\in[0,\xi^{\ast}]$ is therefore not uniquely defined.
* •
For $i=\alpha,\beta$, we define $\tilde{S}_{i}$ as the reflection of the orbit
$S_{i}$ with respect to $\rho=\frac{1}{2}$.
Depending on the values of $\alpha$ and $\beta$, one of the orbits $S_{i}$,
$\tilde{S}_{i}$, $i=c,\alpha,\beta$, corresponds to the slow part of the
singular orbits we will construct.
Changing $\alpha$ and $\beta$ influences the orbits $S_{\alpha}$,
$\tilde{S}_{\alpha}$ and $S_{\beta}$, $\tilde{S}_{\beta}$. We will show in the
following that the $\alpha$, $\beta$ dependent mutual position of these orbits
determines the type of singular solution of the boundary value problem.
\begin{overpic}[scale={0.7}]{spec_sol_2.pdf} \put(0.0,0.0){$0$}
\put(68.0,0.0){$\xi^{\ast}$} \put(90.0,0.0){$1$} \put(100.0,0.0){\Large$\xi$}
\put(-11.0,8.0){\footnotesize$1-\rho_{\ast}(\beta)$}
\put(0.0,14.0){\footnotesize$\alpha$}
\put(0.0,19.0){\footnotesize$\rho_{c}^{0}$} \put(0.0,47.0){$\frac{1}{2}$}
\put(-6.0,76.0){\footnotesize$1-\rho_{c}^{0}$}
\put(-5.0,80.0){\footnotesize$1-\alpha$}
\put(-5.0,85.0){\footnotesize$\rho_{\ast}(\beta)$} \put(0.0,90.0){$1$}
\put(0.0,100.0){\Large$\rho$} \put(92.0,10.0){\footnotesize$\beta$}
\put(92.0,22.0){\footnotesize$\rho^{\ast}(\alpha)$}
\put(92.0,29.0){\footnotesize$1-\rho_{c}^{1}$}
\put(92.0,65.0){\footnotesize$\rho_{c}^{1}$}
\put(92.0,73.0){\footnotesize$1-\rho^{\ast}(\alpha)$}
\put(92.0,83.0){\footnotesize$1-\beta$} \put(60.0,35.0){$\tilde{S}_{c}$}
\put(60.0,58.0){$S_{c}$} \put(55.0,20.0){$S_{\alpha}$}
\put(55.0,73.0){$\tilde{S}_{\alpha}$} \put(72.0,85.0){$S_{\beta}$}
\put(72.0,8.0){$\tilde{S}_{\beta}$} \end{overpic} Figure 5: Schematic
illustration of the special orbits $S_{c}$, $S_{\alpha}$, $S_{\beta}$,
$\tilde{S}_{c}$, $\tilde{S}_{\alpha}$, $\tilde{S}_{\beta}$ in
$(\xi,\rho)$-space in the case $0<\alpha<\rho_{c}^{0}$ and
$\rho_{c}^{1}<\beta<1$. Here
$k(\xi)=1+a\,\mathrm{cos}\left(\frac{2\pi\xi}{b}\right)$ with $a=0.3$,
$b=1.5$. The orbit $S_{c}$ (solid cyan curve) connects $(0,1-\rho_{c}^{0})$
and $(1,1-\rho_{c}^{1})$, the orbit $S_{\alpha}$ (solid magenta curve)
connects $(0,\alpha)$ and $(1,\rho^{\ast}(\alpha))$, and the orbit $S_{\beta}$
(solid brown curve) connects $(0,\rho_{\ast}(\beta))$ and $(1,1-\beta)$ with
$\alpha,\beta<\frac{1}{2}$ as in (34). The dashed curves represent the orbits
$\tilde{S}_{c}$ (cyan), $\tilde{S}_{\alpha}$ (magenta), $\tilde{S}_{\beta}$
(brown), which are symmetric to the corresponding solid ones $S_{c}$,
$S_{\alpha}$, $S_{\beta}$ with respect to $\rho=\frac{1}{2}$. We note that the
orbits $S_{\alpha}$, $\tilde{S}_{\alpha}$ and $S_{\beta}$, $\tilde{S}_{\beta}$
switch position in the above diagram as $\alpha$, $\beta>\frac{1}{2}$.
By using the conserved quantity (19) the respective values of
$\rho^{\ast}(\alpha)$ and $\rho_{\ast}(\beta)$ can be computed explicitly:
$\displaystyle\rho^{\ast}(\alpha)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\frac{1}{2}\left(1-\sqrt{1-4\alpha(1-\alpha)\frac{k(0)}{k(1)}}\right)\text{
if }\alpha<\rho_{c}^{0},\\\
\frac{1}{2}\left(1+\sqrt{1-4\alpha(1-\alpha)\frac{k(0)}{k(1)}}\right)\text{ if
}\alpha>1-\rho_{c}^{0},\end{array}\right.$ (34c)
$\displaystyle\rho_{\ast}(\beta)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\frac{1}{2}\left(1+\sqrt{1-4\beta(1-\beta)\frac{k(1)}{k(0)}}\right)\text{
if }\beta<1-\rho_{c}^{1},\\\
\frac{1}{2}\left(1-\sqrt{1-4\beta(1-\beta)\frac{k(1)}{k(0)}}\right)\text{ if
}\beta>\rho_{c}^{1}.\\\ \end{array}\right.$ (34f)
Note that $\alpha=1-\rho_{\ast}(\beta)$ is equivalent to
$\beta=\rho^{\ast}(\alpha)$.
Based on this, we divide the $(\alpha,\beta)$-parameter space into eight
regions $\mathcal{G}_{i}$, $i=1,\dots,8$ defined via the following curves
$\gamma_{ij}$ (here the indices refer to the adjacent regions):
$\displaystyle\gamma_{12}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;0<\alpha\leq\rho_{c}^{0},\,\beta=1-\rho^{\ast}(\alpha)\right\\},$ (35a)
$\displaystyle\gamma_{13}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\alpha=\rho_{c}^{0},\,1-\rho_{c}^{1}\leq\beta\leq\rho_{c}^{1}\right\\},$
(35b) $\displaystyle\gamma_{17}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\alpha=1-\rho_{\ast}(\beta),\,0<\beta\leq 1-\rho_{c}^{1}\right\\},$ (35c)
$\displaystyle\gamma_{24}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\alpha=\rho_{c}^{0},\,\rho_{c}^{1}\leq\beta<1\right\\},$ (35d)
$\displaystyle\gamma_{34}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\rho_{c}^{0}\leq\alpha\leq 1-\rho_{c}^{0},\,\beta=\rho_{c}^{1}\right\\},$
(35e) $\displaystyle\gamma_{35}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\alpha=1-\rho_{c}^{0},\,1-\rho_{c}^{1}\leq\beta\leq\rho_{c}^{1}\right\\},$
(35f) $\displaystyle\gamma_{37}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\rho_{c}^{0}\leq\alpha\leq 1-\rho_{c}^{0},\,\beta=1-\rho_{c}^{1}\right\\},$
(35g) $\displaystyle\gamma_{46}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\alpha=1-\rho_{c}^{0},\,\rho_{c}^{1}\leq\beta<1\right\\},$ (35h)
$\displaystyle\gamma_{56}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;1-\rho_{c}^{0}\leq\alpha<1,\,\beta=\rho_{c}^{1}\right\\},$ (35i)
$\displaystyle\gamma_{58}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;1-\rho_{c}^{0}\leq\alpha<1,\,\beta=1-\rho_{c}^{1}\right\\},$ (35j)
$\displaystyle\gamma_{78}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\alpha=\rho_{\ast}(\beta),\,0<\beta\leq 1-\rho_{c}^{1}\right\\}.$ (35k)
The above curves correspond to situations where some of the orbits $S_{i}$,
$\tilde{S}_{i}$, $i=1,2,3$ defined above coincide. In particular:
* •
for $(\alpha,\beta)\in\gamma_{12}$, we have $S_{\alpha}=S_{\beta}$ (lying in
$\mathcal{C}_{0}^{r}$);
* •
for $(\alpha,\beta)\in\gamma_{13}\cup\gamma_{24}$, we have
$\tilde{S}_{c}=S_{\alpha}$ for $\xi\in[0,\xi^{\ast}]$ (i.e. up to the canard
point $p^{\ast}$);
* •
for $(\alpha,\beta)\in\gamma_{17}$, we have $S_{\alpha}=\tilde{S}_{\beta}$;
* •
for $(\alpha,\beta)\in\gamma_{34}\cup\gamma_{56}$, we have $S_{c}=S_{\beta}$
for $\xi\in[\xi^{\ast},1]$ (i.e. up to the canard point $p^{\ast}$);
* •
for $(\alpha,\beta)\in\gamma_{35}\cup\gamma_{46}$, we have $S_{c}=S_{\alpha}$
for $\xi\in[0,\xi^{\ast}]$ (i.e. up to the canard point $p^{\ast}$);
* •
for $(\alpha,\beta)\in\gamma_{37}\cup\gamma_{58}$, we have
$\tilde{S}_{c}=S_{\beta}$ for $\xi\in[\xi^{\ast},1]$ (i.e. up to the canard
point $p^{\ast}$);
* •
for $(\alpha,\beta)\in\gamma_{78}$, we have $S_{\alpha}=S_{\beta}$ (lying in
$\mathcal{C}_{0}^{a}$).
###### Remark 4.
Whenever two orbits coincide, their symmetric reflections with respect to
$\rho=\frac{1}{2}$ coincide as well.
The eleven curves in (35) split $(0,1)^{2}$ into $8$ regions
$\mathcal{G}_{i}$, $i=1,\dots,8$ (shown in Figure 6):
$\displaystyle\mathcal{G}_{1}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;0<\alpha<\rho_{c}^{0},\,\rho^{\ast}(\alpha)<\beta<1-\rho^{\ast}(\alpha)\right\\}$
(36a) $\displaystyle\mathcal{G}_{2}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;0<\alpha<\rho_{c}^{0},\,1-\rho^{\ast}(\alpha)<\beta<1\right\\},$ (36b)
$\displaystyle\mathcal{G}_{3}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\rho_{c}^{0}<\alpha<1-\rho_{c}^{0},\,1-\rho_{c}^{1}<\beta<\rho_{c}^{1}\right\\},$
(36c) $\displaystyle\mathcal{G}_{4}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\rho_{c}^{0}<\alpha<1-\rho_{c}^{0},\,\rho_{c}^{1}<\beta<1\right\\},$ (36d)
$\displaystyle\mathcal{G}_{5}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;1-\rho_{c}^{0}<\alpha<1,\,1-\rho_{c}^{1}<\beta<\rho_{c}^{1}\right\\},$
(36e) $\displaystyle\mathcal{G}_{6}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;1-\rho_{c}^{0}<\alpha<1,\,\rho_{c}^{1}<\beta<1\right\\},$ (36f)
$\displaystyle\mathcal{G}_{7}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;1-\rho_{\ast}(\beta)<\alpha<\rho_{\ast}(\beta),\,0<\beta<1-\rho_{c}^{1}\right\\},$
(36g) $\displaystyle\mathcal{G}_{8}$ $\displaystyle:=\left\\{(\alpha,\beta)\
:\;\rho_{\ast}(\beta)<\alpha<1,\,0<\beta<1-\rho_{c}^{1}\right\\}.$ (36h)
In short terms, moving from one region to the other in the $(\alpha,\beta)$
-parameter space leads to a corresponding change in the structure of the
singular solutions.
Figure 6: Representation of the $(\alpha,\beta)$ bifurcation diagram for
$\varepsilon=0$. Here $k(\xi)=1+a\,\mathrm{cos}\left(\frac{2\pi\xi}{b}\right)$
with $a=0.3$, $b=1.5$. Red regions correspond to high density, blue regions to
low density, and green regions to transitions from high to low density
regimes. In the insets, the density $\rho$ is shown as a function of $\xi$.
The blue parts correspond to solutions of the reduced problem (16), whereas
the green parts indicate boundary layers. The gray line represents
$\rho=\frac{1}{2}$
We will show (in Proposition 1) that within each of those region the structure
of the singular solutions is the same. Note that our construction of singular
solutions works also on all the boundary curves defined in (35) except for
$\gamma_{17}$, $\gamma_{13}\cup\gamma_{24}$, and $\gamma_{37}\cup\gamma_{58}$,
where singular solutions are not unique (see Remark 8).
To this aim, we introduce the following eight types of singular solutions (see
Figure 8-9):
Type 1. Singular solutions which start on $\mathcal{C}_{0}^{r}$ at $\xi=0$,
follow the reduced flow on $\mathcal{C}_{0}^{r}$ (where $\rho$ increases), and
have a layer at $\xi=1$ in which $\rho$ increases.
Type 2. Singular solutions which start on $\mathcal{C}_{0}^{r}$ at $\xi=0$,
follow the reduced flow on $\mathcal{C}_{0}^{r}$ (where $\rho$ increases), and
have a layer at $\xi=1$ in which $\rho$ decreases.
Type 3. Singular solutions which have a layer at $\xi=0$ in which $\rho$
increases, follow the reduced flow on $\mathcal{C}_{0}$ (where $\rho$
decreases) passing through the point $p^{\ast}$, and have another layer at
$\xi=1$ in which $\rho$ increases.
Type 4. Singular solutions which have a layer at $\xi=0$ in which $\rho$
increases, follow the reduced flow on $\mathcal{C}_{0}$ (where $\rho$
decreases) passing through the point $p^{\ast}$, and have another layer at
$\xi=1$ in which $\rho$ decreases.
Type 5. Singular solutions which have a layer at $\xi=0$ in which $\rho$
decreases, follow the reduced flow on $\mathcal{C}_{0}$ (where $\rho$
decreases) passing through the point $p^{\ast}$, and have another layer at
$\xi=1$ in which $\rho$ increases.
Type 6. Singular solutions which have a layer at $\xi=0$ in which $\rho$
decreases, follow the reduced flow on $\mathcal{C}_{0}$ (where $\rho$
decreases) passing through the point $p^{\ast}$, and have another layer at
$\xi=1$ in which $\rho$ decreases.
Type 7. Singular solutions which have a layer at $\xi=0$ in which $\rho$
increases, and follow the reduced flow on $\mathcal{C}_{0}^{a}$ (where $\rho$
decreases).
Type 8. Singular solutions which have a layer at $\xi=0$ in which $\rho$
decreases and follow the reduced flow on $\mathcal{C}_{0}^{a}$ (where $\rho$
decreases).
More details about the construction and structure of these singular orbits are
given in the proof of the following proposition.
###### Proposition 1.
Let $k\in C^{2}([0,1])$ be a positive function satisfying Assumption (6). Then
for each $(\alpha,\beta)\in\mathcal{G}_{i}$, $i=1,\ldots,8$ there exists a
unique singular solution $\Gamma^{i}$ of type $i$ to (5b) composed of segments
of orbits of the layer problem (11) and the reduced problem (16) satisfying
the boundary conditions.
###### Proof.
The proof is based on the shooting technique outlined above. Technically
speaking, we show that the intersection of the sets
$\mathcal{L}_{\xi^{\ast}}^{+}\cup l_{\xi^{\ast}}$ in (31a)-(32) and
$\mathcal{R}_{\xi^{\ast}}^{-}\cup r_{\xi^{\ast}}$ in (31b)-(33) is non-empty,
and in particular consists of one point. This gives us the unique values of
$\rho_{0},\,\rho_{1}$ for which a singular orbit exists depending on $\alpha$
and $\beta$, which in turn allows us to identify the eight types of singular
solutions corresponding to the eight regions defined in (36). While we claim
the existence of singular solutions only in the open regions $\Gamma^{i}$,
$i=1,\ldots,8$ we also comment on the singular configurations where
$(\alpha,\beta)$ lies on the curves $\gamma_{ij}$ from (35).
In principle there are four possible ways for the intersection between
$\mathcal{L}_{\xi^{\ast}}^{+}\cup l_{\xi^{\ast}}$ and
$\mathcal{R}_{\xi^{\ast}}^{-}\cup r_{\xi^{\ast}}$ to occur; one of these
defines four possible profiles corresponding to four regions in
$(\alpha,\beta)$-parameter space, two of these lead to two possible profiles
corresponding to two regions in $(\alpha,\beta)$-parameter space, while the
fourth case ($l_{1}\cap r$) leads to an empty intersection, since
$l_{\xi^{\ast}}$ and $r_{\xi^{\ast}}$ are separated from
$\mathcal{L}_{\xi^{\ast}}^{+}$ and $\mathcal{R}_{\xi^{\ast}}^{-}$,
respectively, only for $\alpha\leq\rho_{c}^{0}$ and $\beta\leq
1-\rho_{c}^{1}$, and in this case they can never coincide. Thus, we are left
with:
Case 1: $l_{\xi^{\ast}}\cap\mathcal{R}_{\xi^{\ast}}^{-}\neq\emptyset$.
From the investigation of this case we obtain orbits of type 1, 2.
Case 2:
$\mathcal{L}_{\xi^{\ast}}^{+}\cap\mathcal{R}_{\xi^{\ast}}^{-}\neq\emptyset$.
From the investigation of this case we obtain orbits of type 3, 4, 5, 6.
Case 3: $\mathcal{L}_{\xi^{\ast}}^{+}\cap r_{\xi^{\ast}}\neq\emptyset$.
From the investigation of this case we obtain orbits of type 7, 8.
In the following, we examine Cases 1-3 in more detail.
_Case 1: $l_{\xi^{\ast}}\,\cap\,\mathcal{R}_{\xi^{\ast}}^{-}\neq\emptyset$._
By definition of $l_{\xi^{\ast}}$, this occurs only when
$\alpha\leq\rho_{c}^{0}$. In this case, we have
$l_{\xi^{\ast}}\in\mathcal{R}_{\xi^{\ast}}^{-}$, which implies that $p_{0}=l$
and, consequently, $\rho_{0}=\alpha$. This implies that in this regime no
boundary layers exist at $\xi=0$. Moreover, since
$\rho(1,s)=\rho^{\ast}(\alpha)$, following the flow of the layer problem until
it hits $\mathcal{R}$ we obtain
$\rho_{1}=\frac{\alpha(1-\alpha)k(0)}{\beta k(1)}.$ (37)
In this case, the singular orbit consists in a slow motion along
$\mathcal{C}_{0}^{r}$ followed by a layer at $\xi=1$. The nature of this layer
– in particular its orientation – depends on $\alpha$ and $\beta$ as follows:
* •
When $\alpha<\rho_{c}^{0}$ and
$\rho^{\ast}(\alpha)<\beta<1-\rho^{\ast}(\alpha)$, i.e. for
$(\alpha,\beta)\in\mathcal{G}_{1}$, $\rho$ increases along the boundary layer
at $\xi=1$. The corresponding singular solution is therefore of type $1$ (see
Figure 8(a)).
* •
When $\alpha<\rho_{c}^{0}$ and $\beta>1-\rho^{\ast}(\alpha)$, i.e. for
$(\alpha,\beta)\in\mathcal{G}_{2}$, $\rho$ decreases along the boundary layer
at $\xi=1$. Therefore, the corresponding singular solution is of type $2$ (see
Figure 8(b)).
We note that when $\alpha<\rho_{c}^{0}$ and $\beta=1-\rho^{\ast}(\alpha)$
(i.e. on $\gamma_{12}$) there is no layer at $\xi=1$.
_Case 2:
$\mathcal{L}_{\xi^{\ast}}^{+}\cap\mathcal{R}_{\xi^{\ast}}^{-}\neq\emptyset$._
We observe that by definition
$\mathcal{L}_{\xi^{\ast}}^{+}\subset\mathcal{C}_{0}^{a}$ and
$\mathcal{R}_{\xi^{\ast}}^{-}\subset\mathcal{C}_{0}^{r}$. Thus, this case
corresponds to having $\alpha\geq\rho_{c}^{0}$ and $\beta\geq 1-\rho_{c}^{1}$
and their non-empty intersection is realised at the canard point $p^{\ast}$
(see (14)). This implies that the slow segment of these singular orbits is the
canard orbit $S_{c}$.
In particular, since
$\rho^{+}(\xi^{\ast},s)=\frac{1}{2}=\rho^{-}(\xi^{\ast},t)$, it follows that
$\rho^{+}(0,s)=1-\rho_{c}^{0}$ and $\rho^{-}(1,t)=1-\rho_{c}^{1}$.
Consequently, the start/end point of the reduced flow are fixed by the canard
and correspond to $p_{c}^{0}$ and $p_{c}^{1}$ respectively, whereas boundary
layers at $\xi=0$, $1$ may arise depending on $\alpha$ and $\beta$. It is then
possible to determine the starting and ending points of the orbit by following
the flow of the layer problem (backwards at $\xi=0$ and forward at $\xi=1$);
this leads to
$\rho_{0}=1-\frac{k(\xi^{\ast})}{4\alpha
k(0)},\qquad\rho_{1}=\frac{k(\xi^{\ast})}{4\beta k(1)}.$ (38)
In particular, we have:
* •
When $\rho_{c}^{0}<\alpha<1-\rho_{c}^{0}$, $\rho$ increases along the boundary
layer at $\xi=0$. Additionally:
* –
If $1-\rho_{c}^{1}<\beta<\rho_{c}^{1}$, i.e. for
$(\alpha,\beta)\in\mathcal{G}_{3}$, $\rho$ increases along the boundary layer
at $\xi=1$. This implies that the singular orbit is of type $3$ (see Figure
8(c)).
* –
If $\rho_{c}^{1}<\beta<1$, i.e. for $(\alpha,\beta)\in\mathcal{G}_{4}$, $\rho$
decreases along the boundary layer at $\xi=1$. This implies that the singular
orbit is of type $4$ (see Figure 8(d)).
* •
When $1-\rho_{c}^{0}<\alpha<1$, $\rho$ decreases along the boundary layer at
$\xi=0$. Additionally:
* –
If $1-\rho_{c}^{1}<\beta<\rho_{c}^{1}$, i.e. for
$(\alpha,\beta)\in\mathcal{G}_{5}$, $\rho$ increases along the boundary layer
at $\xi=1$. This implies that the singular orbit is of type $5$ (see Figure
9(a)).
* –
If $\rho_{c}^{1}<\beta<1$, i.e. for $(\alpha,\beta)\in\mathcal{G}_{6}$, $\rho$
decreases along the boundary layer at $\xi=1$. This implies that the singular
orbit is of type $6$ (see Figure 9(b)).
We note that when $\alpha=1-\rho_{c}^{0}$ and $\beta\geq 1-\rho_{c}^{1}$ (i.e.
on $\gamma_{35}\cup\gamma_{46}$) we have no boundary layer at $\xi=0$.
Moreover, when $\beta=\rho_{c}^{1}$ and $\alpha\geq\rho_{c}^{0}$ (i.e. on
$\gamma_{34}\cup\gamma_{56}$) we have no boundary layer at $\xi=1$.
_Case 3: $\mathcal{L}_{\xi^{\ast}}^{+}\cap r_{\xi^{\ast}}\neq\emptyset$._ By
definition of $r_{\xi^{\ast}}$, this occurs only when $\beta<1-\rho_{c}^{1}$.
In this case, we have $r_{\xi^{\ast}}\in\mathcal{L}_{\xi^{\ast}}^{+}$, which
implies that $p_{1}=r$ and, consequently, $\rho_{1}=1-\beta$ (i.e., no
boundary layers emerge at $\xi=1$). Moreover, since
$\rho(0,s)=\rho_{\ast}(\beta)$, following the layer problem backwards until it
hits $\mathcal{L}$, we obtain
$\rho_{0}=1-\frac{\beta(1-\beta)k(1)}{\alpha k(0)}.$ (39)
Consequently, the slow motion is here entirely contained in
$\mathcal{C}_{0}^{a}$ and there is a boundary layer at $\xi=0$, whose nature
depends on $\alpha$ as follows:
* •
If $1-\rho_{\ast}(\beta)<\alpha<\rho_{\ast}(\beta)$ and
$0<\beta<1-\rho_{c}^{1}$, i.e. if $(\alpha,\beta)\in\mathcal{G}_{7}$, $\rho$
is increasing and the singular solution is of type $7$ (see Figure 9(c)).
* •
If $\rho_{\ast}(\beta)<\alpha<1$ and $0<\beta<1-\rho_{c}^{1}$, i.e. if
$(\alpha,\beta)\in\mathcal{G}_{8}$, $\rho$ is decreasing, and we have a
singular solution of type $8$ (see Figure 9(d)).
We note that when $\alpha=\rho_{\ast}(\beta)$ and $\beta<1-\rho_{c}^{1}$ (i.e.
on $\gamma_{78}$), there are no boundary layers. ∎
###### Remark 5.
The construction in Case 3 is essentially the same as the one in Case 1 upon
reversal of the flow direction in (8).
###### Remark 6.
Singular solutions of type 1, 2, 7, and 8 can be obtained also applying the
same strategy used in [7, Proposition 2], as their slow portion is entirely
contained in one of the two halves of the critical manifold
($\mathcal{C}_{0}^{r}$ in the case of type 1, 2, $\mathcal{C}_{0}^{r}$ in the
case of type 7, 8). Therefore, it would be possible to only focus on the flow
of the manifold $\mathcal{L}$ of left boundary conditions up to $\xi=1$ and
check its intersection with the projection of the manifold $\mathcal{R}$ of
right boundary conditions on $\mathcal{C}_{0}$.
###### Remark 7.
Different values of $k(0)$, $k(1)$, and $k(\xi^{\ast})$ influence the
structure of the bifurcation diagram sketched in 6 only quantitatively. In
particular, the smaller $k(\xi^{\ast})$ is, the larger regions
$\mathcal{G}_{i}$, $i=3,4,5,6$ are, consequently reducing the sizes of regions
$\mathcal{G}_{i}$, $i=1,2,7,8$. Recall that smaller values of $k(\xi^{\ast})$
correspond to a narrower bottleneck.
Figure 7: Schematic representation of a singular solution of (8)-(9) with
$a=0.3$, $b=1.5$, $\alpha=0.3$, and $\beta=0.6$ (i.e.
$(\alpha,\beta)\in\mathcal{G}_{3}$). The solution has boundary layers at
$\xi=0$ and $\xi=1$, while the slow portion of the orbit coincides with
$S_{c}$.
(a) Region $\mathcal{G}_{1}$: $\alpha=0.1$, $\beta=0.4$
(b) Region $\mathcal{G}_{2}$: $\alpha=0.1$, $\beta=0.9$
(c) Region $\mathcal{G}_{3}$: $\alpha=0.3$, $\beta=0.6$
(d) Region $\mathcal{G}_{4}$: $\alpha=0.3$, $\beta=0.8$
Figure 8: Schematic representation of singular solutions of type 1-4 (rows
1-4, respectively). _First column_ : Boundary conditions at $\xi=0$ in
$(j,\rho)$-space: the orange line is $\mathcal{L}$, while the orange curve is
$\mathcal{L}^{+}$. The red dot represents $p_{0}$ and the green line
illustrates the layer where $\rho$ increases (type 3, 4). _Second column_ :
Slow evolution on $\mathcal{C}_{0}$ (blue curve). The orange lines are the
projection of $\mathcal{L}$ and $\mathcal{L}^{+}$ on $\mathcal{C}_{0}$, while
the purple one represents the projection of $\mathcal{R}^{-}$ on
$\mathcal{C}_{0}$. The orange dot corresponds to $l$, while the purple dot
corresponds to $r$. For orbits of type 3 and 4 the slow flow involves the
passage through the canard point $p^{\ast}$. _Third column_ : Boundary
conditions at $\xi=1$ in $(j,\rho)$-space. The red dot corresponds to $p_{1}$,
while the purple line and curve represent the manifolds $\mathcal{R}$ and
$\mathcal{R}^{-}$, respectively. The green line corresponds to the layer of
the singular orbit where $\rho$ increases (type 1-3)/decreases (type 2-4).
_Fourth column_ : Singular solution in $(\xi,\rho)$-space.
(a) Region $\mathcal{G}_{5}$: $\alpha=0.9$, $\beta=0.6$
(b) Region $\mathcal{G}_{6}$: $\alpha=0.9$, $\beta=0.8$
(c) Region $\mathcal{G}_{7}$: $\alpha=0.7$, $\beta=0.2$
(d) Region $\mathcal{G}_{8}$: $\alpha=0.9$, $\beta=0.2$
Figure 9: Schematic representation of singular solutions of type 5-8 (rows
1-4, respectively). _First column_ : Boundary conditions at $\xi=0$ in
$(j,\rho)$-space: the orange line is $\mathcal{L}$, while the orange curve is
$\mathcal{L}^{+}$. The red dot represents $p_{0}$ and the green line
illustrates the layer where $\rho$ increases (type 7)/decreases (type 5, 6,
8). _Second column_ : Slow evolution on $\mathcal{C}_{0}$ (blue curve). The
orange lines are the projection of $\mathcal{L}$ and $\mathcal{L}^{+}$ on
$\mathcal{C}_{0}$, while the purple one represents the projection of
$\mathcal{R}^{-}$ on $\mathcal{C}_{0}$. The orange dot corresponds to $l$,
while the purple dot corresponds to $r$. For orbits of type 5 and 6 the slow
flow involves the passage through the canard point $p^{\ast}$. _Third column_
: Boundary conditions at $\xi=1$ in $(j,\rho)$-space. The red dot corresponds
to $p_{1}$, while the purple line and curve represent the manifolds
$\mathcal{R}$ and $\mathcal{R}^{-}$, respectively. The green line corresponds
to the layer of the singular orbit where $\rho$ increases (type 5)/decreases
(type 6). _Fourth column_ : Singular solution in $(\xi,\rho)$-space.
###### Remark 8 (Degenerate cases including continua of singular solutions).
When $\alpha\leq\rho_{c}^{0}$ and $\beta=\rho^{\ast}(\alpha)$ \- i.e. when
$(\alpha,\beta)\in\gamma_{17}$ \- we have that both
$\mathcal{L}_{\xi^{\ast}}^{+}\cap r_{\xi^{\ast}}$ and
$l_{\xi^{\ast}}\cap\mathcal{R}_{\xi^{\ast}}^{-}$ are non-empty. Consequently,
there are two possible reduced solutions, satisfying (see Figure 10(a))
$\text{(a)}\,\begin{cases}\rho(0,s)=\alpha,\\\
\rho(1,s)=\beta,\end{cases}\text{ or }\quad\text{ (b)
}\begin{cases}\rho(0,s)=1-\alpha,\\\ \rho(1,s)=\rho_{1}=1-\beta.\end{cases}$
(40)
In this case, we have a continuum of singular solutions, since at any
$\xi\in[0,1]$ it is possible to jump from the slow trajectory of the reduced
flow in (a) to the one in (b) via the flow of the layer problem. Analogously,
we obtain a continuum of singular solutions when $\alpha=\rho_{c}^{0}$,
$\beta\geq 1-\rho_{c}^{1}$, i.e. when
$(\alpha,\beta)\in\gamma_{13}\cup\gamma_{24}$. In this case, in fact, we have
that both $\mathcal{L}_{\xi^{\ast}}^{+}\cap\mathcal{R}_{\xi^{\ast}}^{-}$ and
$l_{\xi^{\ast}}\cap\mathcal{R}_{\xi^{\ast}}^{-}$ are non-empty, and therefore
there are two possible reduced solutions (with jumps possible at any
$\xi\in[0,\xi^{\ast}]$ via the flow of the layer problem) satisfying (see
Figure 10(b))
$\text{(c)}\,\begin{cases}\rho(0,s)=\rho_{c},\\\
\rho(1,s)=1-\rho_{c}^{1},\end{cases}\text{ or }\quad\text{ (d)
}\begin{cases}\rho(0,s)=1-\rho_{c},\\\ \rho(1,s)=1-\rho_{c}^{1}.\end{cases}$
(41)
A last example of such a situation is given by $\alpha\geq\rho_{c}^{0}$,
$\beta=1-\rho_{c}^{1}$, i.e. when
$(\alpha,\beta)\in\gamma_{37}\cup\gamma_{58}$. Here, both
$\mathcal{L}_{\xi^{\ast}}^{+}\cap\mathcal{R}_{\xi^{\ast}}^{-}$ and
$\mathcal{L}_{\xi^{\ast}}^{+}\cap r_{\xi^{\ast}}$ are non-empty, leading again
to two possible reduced solutions (with jumps possible at any
$\xi\in[\xi^{\ast},1]$ via the flow of the layer problem) satisfying (see
Figure 10(c))
$\text{(c)}\,\begin{cases}\rho(0,s)=1-\frac{k(\xi^{\ast})}{4\alpha k(0)},\\\
\rho(1,s)=\rho_{c}^{1},\end{cases}\text{ or }\quad\text{ (d)
}\begin{cases}\rho(0,s)=1-\frac{k(\xi^{\ast})}{4\alpha k(0)},\\\
\rho(1,s)=1-\rho_{c}^{1}.\end{cases}$ (42)
Since in these degenerate cases singular solutions are not unique, our method
based on transversality arguments to infer persistence of singular solutions
to (5a)-(5b) for $0<\varepsilon\ll 1$ do not apply. Moreover, at the point
$\alpha=\rho_{c}^{0}$, $\beta=1-\rho_{c}^{1}$ – i.e. at the intersection of
$\gamma_{17}$, $\gamma_{13}$, and $\gamma_{37}$ – the situation is even more
degenerate as the three previous scenarios collide. We leave the analysis of
these more delicate situations for future work.
(a)
(b)
(c)
Figure 10: Schematic representation in $(\xi,\rho)$-space of the slow portions
(blue curves) of the possible singular orbits for (a)
$(\alpha,\beta)\in\gamma_{17}$, (b)
$(\alpha,\beta)\in\gamma_{13}\cup\gamma_{24}$, and (c)
$(\alpha,\beta)\in\gamma_{37}\cup\gamma_{58}$. The orange and purple curves
correspond to the projection of $\mathcal{L}^{+}$ and $\mathcal{R}^{-}$,
respectively, on the $(\xi,\rho)$-space. Fast jumps from the slow solution in
$\mathcal{C}_{0}^{r}$ to the slow solution in $\mathcal{C}_{0}^{a}$ are
possible (a) at each $\xi\in[0,1]$, (b) for $\xi\in[0,\xi^{\ast}]$, and (c)
for $\xi\in[\xi^{\ast},1]$.
We now prove that the singular solutions from Proposition 1 perturb to
solutions of (5a)-(5b) for $\varepsilon$ sufficiently small.
###### Theorem 1.
Let $k\in C^{2}([0,1])$ be a positive function satisfying Assumption (6). For
each $(\alpha,\beta)\in\mathcal{G}_{i}$, $i=1,\dots,8$, the boundary value
problem (5b) has a unique solution $\rho(x,\alpha,\beta,\varepsilon)$ for
$\varepsilon$ sufficiently small. In the phase-space formulation (10), this
solution corresponds to an orbit $\Gamma^{i}_{\varepsilon}$ which is
$\mathcal{O}(\varepsilon^{\mu})$-close to $\Gamma^{i}$ in terms of Hausdorff
distance, with $\mu=1$ for $i=1,2,7,8$ and $\mu=1/2$ for $i=3,4,5,6$.
###### Proof.
The solutions for $\varepsilon$ small are obtained by perturbing from the
singular solutions $\Gamma^{i}$, $i=1,\dots,8$. More precisely, we show that
the manifold obtained by flowing the line $\mathcal{L}$ of points
corresponding to the boundary conditions at $\xi=0$ to $\xi=\xi^{\ast}$ for
$\varepsilon$ small intersects the manifold obtained by flowing the line
$\mathcal{R}$ of points corresponding to the boundary conditions at $\xi=1$ to
$\xi=\xi^{\ast}$ in a point which is close to the corresponding point of the
singular solution. Analogously to Proposition 1, this is done by considering
three cases.
_Case 1: $(\alpha,\beta)\in\mathcal{G}_{i}$, $i=1,2$_. In this case, the proof
is completely analogous to Case 1 in [7, Theorem 2]. In particular, it is
possible to show that for $0<\varepsilon\ll 1$ the (forward) flow defined by
(8) takes a suitable small segment of $\mathcal{L}$ to a smooth, two-
dimensional manifold $\mathcal{M}_{0,\varepsilon}$, which reduces to a curve
$\mathcal{L}_{1,\varepsilon}$ when projected in the plane $\xi=1$. Such curve
intersects $\mathcal{R}$ in a point $p_{1,\varepsilon}$ which corresponds to
the right end-point of the solution of the boundary value problem. The full
solution for $\xi\in[0,1]$ is then obtained by following the flow backward
from $p_{1,\varepsilon}$ to $\xi=0$. In this case, the perturbed orbits are
$\mathcal{O}(\varepsilon)$ close to the corresponding singular ones as all
perturbations are $C^{1}$ in $\varepsilon$.
_Case 2: $(\alpha,\beta)\in\mathcal{G}_{i}$, $i=3,4,5,6$_. In this case, the
singular solution starts with a layer connecting the point
$p_{0}\in\mathcal{L}$ to the point $p_{c}^{0}$ on $S_{c}$, then follows the
canard through the canard point $p^{\ast}$ up to $\xi=1$, and finally ends
with another layer connecting $p_{c}^{1}$ with the point
$p_{1}\in\mathcal{R}$.
To prove the persistence of this singular orbit, we flow the line
$\mathcal{L}$ of boundary conditions at $\xi=0$ forward, the line
$\mathcal{R}$ of boundary conditions at $\xi=1$ backward, and show that they
intersect transversally at $\xi=\xi^{\ast}$ for $\varepsilon$ small. Since the
singular solution involves the point $p^{\ast}$ on the non-hyperbolic fold
line $F$ and the emergence of a canard, results on extending GSPT to such
problems [15] are needed here.
Fenichel theory [6] implies that away from the fold line $F$ (compact subsets
of) $\mathcal{C}_{0}^{a}$ and $\mathcal{C}_{0}^{r}$ perturb smoothly to the
slow manifolds $\mathcal{C}_{\varepsilon}^{a}$ and
$\mathcal{C}_{\varepsilon}^{r}$, respectively. The results in [15, Theorem
4.1] imply that in a neighbourhood of the canard point $p^{\ast}$ the
manifolds $\mathcal{C}_{\varepsilon}^{a}$ and $\mathcal{C}_{\varepsilon}^{r}$
intersect transversally in a maximal canard $S_{c}^{\varepsilon}$ (close to
$S_{c}$) for $\varepsilon$ sufficiently small. As in case 1, consider a small
segment of $\mathcal{L}$ containing $p_{0}$ and denote its extension by the
forward flow of (10) by $\mathcal{M}_{0,\varepsilon}$ for $\varepsilon$ small.
Analogously, consider a small segment of $\mathcal{R}$ containing $p_{1}$ and
denote its extension by the backward flow of (10) by
$\mathcal{M}_{1,\varepsilon}$ for $\varepsilon$ small (again a smooth, two-
dimensional manifold). By Fenichel theory, the manifolds
$\mathcal{M}_{0,\varepsilon}$ and $\mathcal{M}_{1,\varepsilon}$ are
exponentially close to $\mathcal{C}_{\varepsilon}^{a}$ and
$\mathcal{C}_{\varepsilon}^{r}$, respectively. Therefore,
$\mathcal{M}_{0,\varepsilon}$ and $\mathcal{M}_{1,\varepsilon}$ also intersect
transversally in a unique orbit, which is the unique solution to the boundary
value problem (see Figure 11). Here, the $\mathcal{O}(\varepsilon^{1/2})$
distance between the perturbed and the corresponding singular solutions
follows from the blow-up analysis in [15], since the effect of the
perturbation in the scaling chart of the blow-up transformation is of the
order $\varepsilon^{1/2}$.
\begin{overpic}[scale={.7}]{constr_eps_2.pdf}
\put(30.0,80.0){$\mathcal{M}_{0,\varepsilon}$}
\put(60.0,70.0){$\mathcal{M}_{1,\varepsilon}$} \end{overpic} Figure 11:
Schematic representation of a solution (continuous red curve) to the full
problem (5a)-(5b) obtained for $(\alpha,\beta)\in\mathcal{G}_{4}$ with the
strategy discussed in Case 2 of Theorem 1. The orange and purple manifolds
represent $\mathcal{M}_{0,\varepsilon}$ in $[0,\xi^{\ast}+\eta]$ and
$\mathcal{M}_{1,\varepsilon}$ in $[\xi^{\ast}-\eta,1]$ with $\eta=1/18$,
respectively, whereas the red dots at $\xi=0$ and $\xi=1$ correspond to the
initial and final point of the orbit. These manifolds intersect transversally
at $\xi=\xi^{\ast}$, providing the uniqueness of the obtained solution. The
passage close to the canard point $p^{\ast}$ leads to the
$\mathcal{O}(\varepsilon^{1/2})$ distance from the corresponding singular
orbit.
_Case 3: $(\alpha,\beta)\in\mathcal{G}_{i}$, $i=7,8$_. This case can be proved
following the same approach as in [7, Theorem 2], and in particular is
completely analogous to Case 1 upon reversal of the flow direction. ∎
## 3 Numerical experiments
In this section we present some numerical results for the steady-state problem
(5b) which support the analysis of Section 2. More details about the numerical
method employed here can be found in [7]. All results are obtained for
$\varepsilon=10^{-3}$.
We first set $k(\xi)=1+a\cos\left(\frac{2\pi\xi}{b}\right)$, a choice that was
already considered in Figure 6 in the singular case. For $\varepsilon\neq 0$,
Figure 12 illustrates some typical profiles, one per region defined by the
GSPT analysis. The values chosen for $\alpha$ and $\beta$ are the same as in
Figure 6, the solutions are qualitatively very close.
Figure 12: $(\alpha,\beta)$ bifurcation diagram for $\varepsilon=10^{-3}$.
Here $k(\xi)=1+a\,\mathrm{cos}\left(\frac{2\pi\xi}{b}\right)$ with $a=0.3$,
$b=1.5$. The insets show the density $\rho$ as a function of $\xi$.
Next investigate more realistic choices for $k$, which should mimic a corridor
with a bottleneck. We consider two regions of constant width that are
connected by a narrower section in the middle. In particular, we consider a
“supergaussian” profile for $k$:
$k(\xi)=w_{e}-(w_{e}-w_{m})e^{-\left|\frac{\xi-\xi_{0}}{d}\right|^{6}}\,,$
where $w_{e}$, $w_{m}$, $d$ and $\xi_{0}$ are positive parameters,
corresponding to the width of the wider regions at the left and right, the
width of the narrow middle section, the neck length and the neck position,
respectively. We pick $w_{e}=1$ and consider both $w_{m}=0.9$, corresponding
to wider neck and $w_{m}=0.5$ which gives a more pronounced neck. The other
parameters are taken as $w_{e}=1$, $d=0.2$, $\xi_{0}=0.6$, which gives a
satisfactory, asymmetric width profile as shown in Figure 13.
$0$$0.2$$0.4$$0.6$$0.8$$1$$-1/2$$-1/4$$0$$1/4$$1/2$$x$$y$$w_{m}=0.9$$w_{m}=0.5$
Figure 13: Representation of the 2D domain associated with a supergaussian
$k$. The thick and bold lines correspond to $w_{m}=0.5$ and $w_{m}=0.9$,
respectively.
Some characteristic profiles are shown in Figure 14, along with the 8 regions
defined by the GSPT analysis above. The selected values of the parameter pair
$\alpha$ and $\beta$ are picked with exactly one pair value per region, and
the same for both $w_{m}=0.9$ and $w_{m}=0.5$. The parameters $\alpha$ and
$\beta$ are also chosen away from the $0$ and $1$, since those values lead to
almost constant solutions for regions $\mathcal{G}_{1}$, $\mathcal{G}_{2}$,
$\mathcal{G}_{7}$ and $\mathcal{G}_{8}$, which correspond to the blue and red
shaded areas. All chosen values are stated in Table 1.
| $\mathcal{G}_{1}$ | $\mathcal{G}_{2}$ | $\mathcal{G}_{3}$ | $\mathcal{G}_{4}$ | $\mathcal{G}_{5}$ | $\mathcal{G}_{6}$ | $\mathcal{G}_{7}$ | $\mathcal{G}_{8}$
---|---|---|---|---|---|---|---|---
$\alpha$ | $0.119$ | $0.128$ | $0.5$ | $0.5$ | $0.881$ | $0.881$ | $0.5$ | $0.941$
$\beta$ | $0.5$ | $0.941$ | $0.5$ | $0.881$ | $0.5$ | $0.881$ | $0.119$ | $0.128$
Table 1: Parameters for Figure 14
Generally speaking, we have three parameters ranges of interest, within which
the stationary solutions share the same qualitative behaviour:
* •
Small $\alpha$ (which corresponds to low inflow as in the blue regions
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$), which leads to _low_ density
stationary states with $\rho<\frac{1}{2}$ and a boundary layer on the right
boundary.
* •
Small $\beta$ (which corresponds to low outflow as in the red regions
$\mathcal{G}_{7}$ and $\mathcal{G}_{8}$), which leads to _high_ density
stationary states with $\rho>\frac{1}{2}$ and a boundary layer on the left
boundary.
* •
Large values of $\alpha$ and $\beta$ (corresponding to high inflow and outflow
regimes as in the green regions $\mathcal{G}_{3}$ to $\mathcal{G}_{6}$),
leading to density profiles going from high density on the left (before the
bottleneck) to low density on the right (after). In this case, boundary layers
a present on both boundaries.
Inside these three areas, solutions seem to depend only weakly on $\alpha$ and
$\beta$, which affect the height of the boundary layers only. It is only
across the boundary between these areas (white lines) that pronounced
qualitative changes occur.
#### Impact of the width of the bottleneck
We now turn our attention to the influence of $w_{m}$ on the solutions. The
first obvious difference in Figure 14 is the larger square in the center for
small $w_{m}$, which corresponds to the region $\mathcal{G}_{3}$ in the
singular analysis. This is explained by the very simple dependency of both
$\rho_{c}^{0}$ and $\rho_{c}^{1}$ (which bound $\mathcal{G}_{3}$) on $k$, see
(34).
Figure 14: Phase diagrams in the $(\alpha,\beta)$ parameter space from the
GSPT analysis for the supergaussian $k$ along with some typical _non-singular_
solutions for $w_{m}=0.9$ (top) and $w_{m}=0.5$ (bottom).
In the regions of low (resp. high) density, in blue (resp. red) in Figure 14,
the density $\rho$ is roughly constant on large parts of the domain, with
variations at the boundaries as well as at the front and back of the narrow
section. Outside of it, $\rho$ takes similar values for both $w_{m}=0.9$ and
$w_{m}=0.5$. Inside however, $\rho$ takes values much closer to $1/2$ for
$w_{m}=0.5$. Indeed, where $\rho$ is almost constant, the flux can be
approximated as $\mathrm{J}=k\rho(1-\rho)$ ; since $\mathrm{J}$ is independent
of $x$, lower values of $k$ correspond to $\rho$ closer to $1/2$. This also
seems to indicate that in both the low and high density phases, the flux
$\mathrm{J}$ for given $(\alpha,\beta)$ only depends weakly on $w_{m}$.
This numerical observation confirms the analytical results of Proposition 1
for the singular case ($\varepsilon=0$). We have in fact that in the regions
of low (resp. high) density, studied in Case $1$, corresponding to regions
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ (resp. $3$, corresponding to
$\mathcal{G}_{7}$ and $\mathcal{G}_{8}$), the density at the entrance (resp.
exit) is given by $\alpha$ (resp. $\beta$) and hence is not affected by the
features of the bottleneck. It follows that $\mathrm{J}=\alpha(1-\alpha)$
(resp. $\mathrm{J}=\beta(1-\beta)$).
In the green region (which could be argued to correspond to the so-called
maximum flux phase for constant $k$), the situation is different. Although the
profiles are qualitatively similar with a transition between a high density to
a low density plateau, the densities for $w_{m}=0.5$ (wider bottleneck, top)
are much closer to $1/2$ for the values of $\alpha$ and $\beta$ which are
considered. This relates to a higher flux for the wider bottleneck.
The computational results in these regions correspond to the analysis of Case
$2$ (regions $\mathcal{G}_{3}$ to $\mathcal{G}_{6}$), a situation in which the
density changes significantly (i.e. a boundary layer) in proximity of both the
entrance and the exit, immediately preceded by a region where it is
approximately constant. The density value in these areas is defined by
$1-\rho_{c}^{0}$ and $1-\rho_{c}^{1}$, respectively. With the choice of
parameters in Table 1, the approximation $k\simeq 1$ holds, at least in the
first and last $10\%$ of the domain; we then get from its definition that
$\rho_{c}^{0}$ (resp. $\rho_{c}^{1}$) is increasing (resp. decreasing) w.r.t
$w_{m}$. In fact we have
$1-\rho_{c}^{0}\simeq\rho_{c}^{1}\simeq\frac{1}{2}\left(1+\sqrt{1-w_{m}}\right)\quad\text{and}\quad\mathrm{J}\simeq\frac{1}{4}w_{m}\,,$
so that $\mathrm{J}$ grows linearly with the width of the neck and eventually
reaches $1/4$ for $w_{m}=1$, the maximum value for a straight channel. This is
in agreement with the numerical observations described above.
To summarize, the influence of the width of the neck in this case is two-fold.
First, in terms of $(\alpha,\beta)$, the green region grows larger and
eventually completely fills the parameter space as the neck-width goes to
zero. Second, it is in this region that $w_{m}$ has a noticeable effect on the
flux $\mathcal{J}$, which depends linearly on $w_{m}$, as one would expect
intuitively.
###### Remark 9.
The observations above are independent of the choice of $k$, provided that
$k(0)=k(1)=1$ and that the minimum of $k$ is non degenerate. In the singular
case, one obtains an explicit expression for $\mathrm{J}$, which we write as
$\mathrm{J}_{k}(\alpha,\beta)$ to emphasize the dependency on $k$, $\alpha$
and $\beta$:
$\mathrm{J}_{k}(\alpha,\beta)=\begin{cases}\alpha(1-\alpha)&\alpha\leq\rho_{c}^{0}\wedge\alpha\leq\beta\,,\\\
\beta(1-\beta)&\beta\leq 1-\rho_{c}^{1}\wedge\beta\leq\alpha\,,\\\
\frac{1}{4}\min_{\xi}k(\xi)&\text{otherwise}\,.\end{cases}$
In particular, we have that
$\mathrm{J}_{k}(\alpha,\beta)=\min\left\\{\mathrm{J}_{1}(\alpha,\beta),\min_{\xi}k(\xi)\right\\}$.
This is illustrated in Figure 15. This means that as $\min_{\xi}k(\xi)$
decreases, the flow $\mathrm{J}_{k}(\alpha,\beta)$ will saturate, _i.e._ reach
its maximum, faster as $\alpha$ and $\beta$ increase. The maximum of
$\mathrm{J}_{k}$ will also decrease linearly with $\min_{\xi}k(\xi)$.
Numerical experiments with two narrow sections of varying width suggest that
this applies also for functions $k$ with several (nondegenerate) critical
points.
Figure 15: Illustration of the flow $\mathrm{J}_{1}$ (solid colors on the
left, wireframe on the right) and $\mathrm{J}_{k}$ (solid colors on the right)
as a function of $\alpha$ and $\beta$. Recall that
$\rho_{c}^{0,1}\rightarrow\frac{1}{2}$ as $k(\xi^{*})\rightarrow 1$, so that
from the green rectangles, only the darker one (top right) remains in the
limit.
## Conclusion
In this work, we investigate the steady-states of a 1D area averaged model
describing pedestrian dynamics for unidirectional flows in domains that have a
bottleneck. In the proposed model, information about the geometry enters as a
nonhomogeneous factor acting both on the diffusive and convective terms. We
investigate the case in which this factor admits an isolated minimum, which
corresponds to the bottleneck. The stationary profiles exhibit a multi-scale
nature, which we analyse using GSPT. This allows us to thoroughly understand
the influence of inflow and outflow rates ($\alpha$ and $\beta$, respectively)
on the structure of the solutions and, in particular, on the formation of
boundary layers. In this framework, the isolated minimum inside the bottleneck
corresponds to a canard point where an unusual passage through a repelling
branch of the critical manifold occurs. The more complex geometry therefore
induces the emergence of two additional regions in the singular bifurcation
diagram which have not been observed and investigated before. In general,
orbits which include such passage exist for a wide area in the
$(\alpha,\beta)$-parameter space, whose size decreases as the neck becomes
wider.
In order to test the ability of our 1D reduction to capture the essential
dynamics of the original two-dimensional model, we plan to suitably calibrate
and validate our model as a next step. As observed in [7], the quality of the
proposed 1D area averaged approximation depends on the parameter regime
considered; we will therefore investigate further averaging assumptions to
overcome these issues in the next steps of our research.
## Declaration of competing interests
The authors declare no conflict of interest.
## Acknowledgements
AI acknowledges support from an FWF Hertha Firnberg Research Fellowship (T
1199-N).
## References
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* [9] A. Iuorio and F. Veerman. The influence of autotoxicity on the dynamics of vegetation spots. Physica D: Nonlinear Phenomena, 427:133015, 2021.
* [10] C. K. R. T. Jones. Geometric singular perturbation theory. In Dynamical Systems, pages 44–118. Springer Berlin Heidelberg, 1995.
* [11] C. Kuehn. Multiple Time Scale Dynamics. Springer International Publishing, 2015.
* [12] M. J. Lighthill and G. B. Whitham. On kinematic waves II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229(1178):317–345, 1955.
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|
# High order asymptotic preserving finite difference WENO schemes with
constrained transport for MHD equations in all sonic Mach numbers
###### Abstract.
In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and
divergence-free finite difference weighted essentially nonoscillatory (WENO)
scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the
sonic Mach number $\varepsilon$ ranging from $0$ to $\mathcal{O}(1)$. High-
order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-
RK) time discretization. High-order accuracy in space is achieved by finite
difference WENO schemes with characteristic-wise reconstructions. A
constrained transport method is applied to maintain a discrete divergence-free
condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA)
in the incompressible MHD limit is obtained if the implicit part of the SI
IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to
validate the AP, AA, and divergence-free properties of our proposed approach.
Besides, the scheme can well capture discontinuities such as shocks in an
essentially non-oscillatory fashion in the compressible regime, while it is
also a good incompressible solver with uniform large-time step conditions in
the low sonic Mach limit.
###### Key words and phrases:
all sonic Mach number; MHD equations; divergence-free; asymptotic preserving;
SI IMEX-RK; finite difference WENO
Wei Chen
School of Mathematical Sciences, Xiamen University
Xiamen, Fujian, 361005, P.R. China
Email<EMAIL_ADDRESS>
Kailiang Wu***This work is supported in part by NSFC grant 12171227
Department of Mathematics & SUSTech International Center for Mathematics
Southern University of Science and Technology
National Center for Applied Mathematics Shenzhen (NCAMS)
Shenzhen, Guangdong 518055, China
Email<EMAIL_ADDRESS>
Tao Xiong†††Corresponding author. The work of this author was partially
supported by NSFC grant No. 11971025, and the Strategic Priority Research
Program of the Chinese Academy of Sciences Grant No. XDA25010401.
School of Mathematical Sciences, Xiamen University
Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance
Scientific Computing
Xiamen, Fujian, 361005, P.R. China
Email<EMAIL_ADDRESS>
## 1\. Introduction
Ideal magnetohydrodynamic (MHD) equations are widely used in the modeling of
weather prediction, astrophysics, as well as laboratory plasma applications
such as flows in tokamaks and stellarators. Many shock capturing schemes with
explicit time discretizations have been developed for solving compressible
ideal MHD equations, including high order discontinuous Galerkin [43, 44, 60,
59], finite difference [49, 15, 18, 16, 17], and finite volume schemes [42,
63, 2, 54, 62], etc.
For MHD equations, divergence free of the magnetic field is a very important
property. If the initial magnetic field is divergence-free, this condition
will be maintained for all later times. Numerically, if this divergence-free
condition is violated seriously, an unphysical force will be created to
parallel the magnetic field [16], which can produce some numerical
instabilities or nonphysical features in the computed solution, cause the loss
of pressure positivity [58, 61], and/or lead to the failure of the simulation.
The constrained transport (CT) methodology, which was originally introduced by
Evans and Hawley[26], is an effective way to maintain the (discrete)
divergence-free condition up to machine precision. Many improvements and
extensions [4, 1, 20, 21, 15, 16, 17, 18, 29] are followed.
Recently, the MHD system in the low sonic Mach limit has attracted a lot of
interests [45, 42, 25, 33, 34, 19, 48]. Jiang, Ju, and Li [33] have shown that
a weak solution of compressible MHD equations will converge to a strong
solution of their corresponding incompressible MHD equations in this low sonic
Mach limit. They have also investigated the low Mach limit of full MHD
equations with a heat conductivity [34]. Cui, Ou, and Ren [19] have
established a uniform convergence from full compressible MHD equations to
isentropic incompressible MHD equations with well-prepared initial conditions
in the three-dimensional case. Numerically it is also very attractive to
develop schemes for MHD flows at any speed. For the above mentioned explicit
shock capturing schemes, when applied to the MHD system in the low sonic Mach
regime, the time step is subject to a very strict CFL condition which is
proportional to the sonic Mach number, making it very undesirable for all
speed flows. An implicit time discretization can release the small time step
restrictions from an explicit one, but usually it results in a highly
nonlinear system, which is not easy to be solved efficiently with an iterative
method and it is hard to guarantee a fast convergence. Furthermore, numerical
viscosities for such schemes are inversely proportional to the sonic Mach
number, which introduce excessive numerical dissipations. Instead, in recent
years some researchers are in pursuit of semi-implicit (SI) schemes with low
dissipations. In [25], Dumbser et. al. developed a pressure-based SI finite
volume all Mach number flow solver for compressible MHD equations. Minoshima
and Miyoshi [48] proposed a multistate low-dissipation advection upstream
splitting method for MHD with both high and low Mach numbers, by using a
Harten-Lax-van Leer discontinuities (HLLD) approximate Riemann solver. Leidi
et al. [42] presented an SI finite volume solver with a 5-wave HLLD and a
well-balanced method to efficiently simulate MHD flows at low Mach numbers
with a gravitational source.
For the MHD system with all sonic Mach numbers, it is important to design
schemes with asymptotic stability and consistency as the sonic Mach number
goes to $0$ in the incompressible limit, namely, asymptotic preserving (AP).
AP schemes have been widely used in a variety of areas, such as hydrodynamic
or diffusive limits of kinetic models, relaxation limits of hyperbolic models,
and low Mach number limits of compressible fluid models. We refer readers to
the papers [35, 37, 36, 38, 27, 24, 23] for more information. To achieve high
order AP schemes in time, SI implicit-explicit Runge-Kutta (IMEX-RK) methods
are widely used, which apply an explicit time discretization for nonstiff
terms, while an implicit time discretization for stiff terms [5, 7, 6, 9, 10,
8, 40, 39, 50, 13, 47, 11]. With a suitable choice of explicit and implicit
discretizations, both high efficiency and uniform stability independent of the
stiff parameter can be obtained. As related, in [57] Tavelli and Dumbser
developed an SI space-time discontinuous Galerkin method for all Mach-number
flow, which involves staggered meshes and a Picard iteration for solving a
large nonlinear system. In [10, 8], Boscarino, Qiu, Russo, and Xiong presented
high-order SI IMEX weighted essentially nonoscillatory (WENO) schemes for all-
Mach isentropic Euler equations and all-Mach full Euler system, respectively.
In [11], Boscheri and Pareschi developed high order pressure-based semi-
implicit IMEX schemes for the three dimensional Navier-Stokes equations at all
Mach numbers. In [30], Huang, Xing, and Xiong designed a high-order well-
balanced SI AP scheme for shallow water equations in all Fraude numbers. We
refer to many other related works from the references therein.
In this paper, we would like to develop a high-order AP finite difference WENO
scheme with CT for MHD equations in all sonic Mach numbers. To our best
knowledge, there are very few AP schemes for MHD systems in literature. The
new contributions and innovations of this work are outlined as follows:
* •
In the low sonic Mach number regime, SI IMEX-RK methods are adopted for time
discretization to obtain a uniform time step independent of the sonic Mach
number. Besides, to avoid a nonlinearity from the equation of state (EOS), an
SI approach is used, leading to a linearized elliptic equation for the
pressure. With carefully designed explicit and implicit discretizations, only
a linear system needs to be solved. The resulting scheme is more efficient
than explicit shock capturing schemes, especially in the low sonic Mach
regime.
* •
Numerical viscosities are carefully designed to avoid excessive numerical
dissipations, and a CT method is applied to maintain a discrete divergence-
free property. The scheme can well capture discontinuities such as shocks in
an essentially non-oscillatory fashion in the compressible regime. Meanwhile,
the scheme is also a good incompressible solver as $\varepsilon\rightarrow 0$.
Due to less dissipation, our scheme can be shown to perform better than
explicit schemes for low sonic Mach problems.
* •
Formal AP and asymptotically accurate (AA) properties in the stiff limit as
the sonic Mach number $\varepsilon\rightarrow 0$ are proved, by assuming the
implicit part of an SI IMEX-RK scheme is stiffly accurate (SA).
The rest of the paper is as follows. In Section 2, we will briefly review the
ideal MHD system, corresponding to its low sonic Mach limit based on
asymptotic expansions, and a CT methodology for the divergence-free condition.
A first-order SI scheme in time will first be introduced in Section 3, and
then a high-order SI scheme with IMEX-RK in time and finite difference WENO in
space is followed. Formal proofs of AP and AA properties are given in Section
4. Numerical experiments are performed in Section 5. A brief conclusion will
be drawn in the last section.
## 2\. Equations of compressible ideal MHD
The ideal MHD equations in a conservative form can be written as
(2.1) $\frac{\partial}{\partial t}\begin{bmatrix}\rho\\\ \rho\mathbf{u}\\\
\mathbf{B}\\\ E\end{bmatrix}+\nabla\cdot\begin{bmatrix}\rho\mathbf{u}\\\
\rho\mathbf{u}\otimes\mathbf{u}+(p+\frac{1}{2}{\|\mathbf{B}\|}^{2})\mathbf{I}-\mathbf{B}\otimes\mathbf{B}\\\
\mathbf{u}\otimes\mathbf{B}-\mathbf{B}\otimes\mathbf{u}\\\
(E+p+\frac{1}{2}{\|\mathbf{B}\|}^{2})\mathbf{u}-\mathbf{B}(\mathbf{u}\cdot\mathbf{B})\end{bmatrix}=\mathbf{0},$
where $\rho$ is the fluid mass density, $\mathbf{u}=(u,v,w)$ is the velocity,
$\mathbf{B}=(B_{x},B_{y},B_{z})$ is the magnetic field, $E$ is the total
energy density, $p$ is the gas pressure, and $\|\cdot\|$ is the Euclidean
vector norm. The total energy is given by
(2.2)
$E=\frac{p}{\gamma-1}+\frac{1}{2}(\rho{\|\mathbf{u}\|}^{2}+{\|\mathbf{B}\|}^{2}),$
where $\gamma$ is the specific heat ratio. The system is subject to a
divergence-free condition of the magnetic field
(2.3) $\nabla\cdot\mathbf{B}=0.$
If initially, the divergence-free condition holds, from (2.1), it also holds
for all later times [18]. For this reason, (2.3) is usually not regarded as a
constraint (like the $\nabla\cdot{\bf u}=0$ constraint for the incompressible
Navier-Stokes equations), but rather an involution [29, 18, 16].
### 2.1. Eigenstructure of the MHD system
For (2.1), the eigenvalues of the Jacobian for the flux function along a
normal direction ${\bf n}$ are given as:
(2.4) $\lambda_{1,8}={\bf u}\cdot{\bf n}\mp c_{f},\quad\lambda_{2,7}={\bf
u}\cdot{\bf n}\mp c_{a},\quad\lambda_{3,6}={\bf u}\cdot{\bf n}\mp
c_{s},\quad\lambda_{4,5}={\bf u}\cdot{\bf n},\quad$
where $\lambda_{1,8}$, $\lambda_{2,7}$, $\lambda_{3,6}$, $\lambda_{4}$, and
$\lambda_{5}$ are the left/right fast magnetosonic waves, left/right Alfvén
waves, left/right slow magnetosonic waves, entropy wave, and divergence wave,
correspondingly. The sound speed $a$, Alfvén speed $c_{a}$, fast magnetosonic
speed $c_{f}$, and slow magnetosonic speed $c_{s}$ are defined as
(2.5) $\left\\{\begin{aligned} a&:=\sqrt{\frac{\gamma p}{\rho}},\\\
c_{a}&:=\sqrt{\frac{{({\bf B}\cdot{\bf n})}^{2}}{\rho}},\\\
c_{f,s}&:={\left\\{\frac{1}{2}\left[a^{2}+\frac{{\|{\bf
B}\|}^{2}}{\rho}\pm\sqrt{{\left(a^{2}+\frac{{\|{\bf
B}\|}^{2}}{\rho}\right)}^{2}-4a^{2}{c^{2}_{a}}}\right]\right\\}}^{\frac{1}{2}}.\end{aligned}\right.$
A fixed ordering for the MHD eigenvalues is
$\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\lambda_{4}\leq\lambda_{5}\leq\lambda_{6}\leq\lambda_{7}\leq\lambda_{8}.$
Spectral decomposition for this system has already been described in [32, 12,
14, 51, 52, 55], and we will follow the one in [32] for all simulations in the
current paper.
### 2.2. Non-dimensionalized MHD
To derive a low sonic Mach limit for the ideal MHD system, we start with
rewriting (2.1), (2.2), and (2.3) into a dimensionless form. We choose some
reference or characteristic values, such as a length $x_{0}$, a time $t_{0}$,
a fluid mass density $\rho_{0}$, a velocity $u_{0}$, a magnetic field $B_{0}$,
and a gas pressure $p_{0}$, with $u_{0}=x_{0}/t_{0}$. For the sake of
simplicity, we consider a regime where the characteristic Alfvén Mach number
$u_{0}\sqrt{\rho_{0}}/B_{0}=1$. Then we define the following dimensionless
variables:
$\hat{{{\bf x}}}=\frac{{{\bf
x}}}{x_{0}},\quad\hat{t}=\frac{t}{t_{0}},\quad\hat{\rho}=\frac{\rho}{\rho_{0}},\quad\hat{{\bf
u}}=\frac{{\bf u}}{u_{0}},\quad\hat{{\bf B}}=\frac{{\bf
B}}{B_{0}},\quad\hat{p}=\frac{p}{p_{0}}.$
Inserting these new variables into (2.1), we obtain the following non-
dimensionalised MHD equations[42]:
(2.6) $\frac{\partial}{\partial t}\begin{bmatrix}\rho\\\ \rho\mathbf{u}\\\
\mathbf{B}\\\ E\\\ \end{bmatrix}+\nabla\cdot\begin{bmatrix}\rho\mathbf{u}\\\
\rho\mathbf{u}\otimes\mathbf{u}+(\frac{p}{\varepsilon^{2}}+\frac{1}{2}{\|\mathbf{B}\|}^{2})\mathbf{I}-\mathbf{B}\otimes\mathbf{B}\\\
\mathbf{u}\otimes\mathbf{B}-\mathbf{B}\otimes\mathbf{u}\\\
(E+p+\frac{\varepsilon^{2}}{2}{\|\mathbf{B}\|}^{2})\mathbf{u}-\varepsilon^{2}\mathbf{B}(\mathbf{u}\cdot\mathbf{B})\end{bmatrix}=\mathbf{0},$
where we have dropped the hats of the dimensionless variables for ease of
presentation. The EOS, divergence-free condition and the characteristic sonic
Mach number $\varepsilon$ are now given by
(2.7) $E:=\frac{p}{\gamma-1}+\frac{\varepsilon^{2}}{2}(\rho{\|{\bf
u}\|}^{2}+{\|{\bf B}\|}^{2}),\quad\nabla\cdot{\bf
B}=0,\quad\varepsilon:=u_{0}\sqrt{\frac{\rho_{0}}{p_{0}}}.$
In this dimensionless MHD system (2.6) with (2.7), the expressions of those
eigenvalues (2.4) remain unchanged, except the sound speed $a$, Alfvén speed
$c_{a}$, fast magnetosonic speed $c_{f}$, and slow magnetosonic speed $c_{s}$
from (2.5) now become
$\left\\{\begin{aligned} a&:=\frac{1}{\varepsilon}\sqrt{\frac{\gamma
p}{\rho}},\\\ c_{a}&:=\sqrt{\frac{{({\bf B}\cdot{\bf n})}^{2}}{\rho}},\\\
c_{f,s}&:={\left\\{\frac{1}{2}\left[a^{2}+\frac{{\|{\bf
B}\|}^{2}}{\rho}\pm\sqrt{{\left(a^{2}+\frac{{\|{\bf
B}\|}^{2}}{\rho}\right)}^{2}-4a^{2}{c^{2}_{a}}}\right]\right\\}}^{\frac{1}{2}}.\end{aligned}\right.$
Following the eigenstructure of the dimensionless MHD system discussed above,
it is natural to find that if the background flow velocity $u_{0}$ is slow,
then the sonic Mach number $\varepsilon=u_{0}\sqrt{\rho_{0}}/\sqrt{p_{0}}\ll
1$, and the fast magnetosonic speed $c_{f}$ becomes very large. We can see
that large magnetosonic speed $c_{f}$ will make the fast magnetosonic waves
$\lambda_{1,8}$ to be much faster as compared to other waves. In this
situation, for explicit shock-capturing schemes, the time step subjects to
$\Delta t={\rm CFL}\frac{\Delta x}{\mathop{\max}\limits_{1\leq i\leq
8}|\lambda_{i}|}=\mathcal{O}(\varepsilon\Delta x),$
where $\Delta t$ is the time step size, $\Delta x$ is the mesh size, and ${\rm
CFL}$ is the time stability CFL number. On one hand, highly inaccurate
solutions will appear due to the excessive numerical viscosity (scales as
$\varepsilon^{-1}$) in standard upwind schemes. On the other hand, this
restriction results in an increasingly large computational cost for low sonic
Mach fluid flows. A straight approach is to apply an implicit time
discretization to avoid such severe time step constraints. However, a fully
implicit scheme will result in a complicated nonlinear system which is usually
very hard to be solved efficiently. Besides, it may not be able to guarantee a
correct asymptotic limit. Thus, it is important to design efficient schemes
with both asymptotic stability and consistency in the low sonic Mach limit,
namely, the AP property.
### 2.3. Low sonic Mach limit
Next, we recall a formal derivation of the incompressible MHD system from the
non-dimensionalized compressible MHD equations (2.6) with (2.7). We consider
an asymptotic expansion ansatz for the following variables [46]:
(2.8) $\left\\{\begin{aligned}
&p(\mathbf{x},t)=p_{0}(\mathbf{x},t)+\varepsilon
p_{1}(\mathbf{x},t)+\varepsilon^{2}p_{2}(\mathbf{x},t)+\cdots,\\\
&\rho(\mathbf{x},t)=\rho_{0}(\mathbf{x},t)+\varepsilon\rho_{1}(\mathbf{x},t)+\varepsilon^{2}\rho_{2}(\mathbf{x},t)+\cdots,\\\
&\mathbf{u}(\mathbf{x},t)=\mathbf{u}_{0}(\mathbf{x},t)+\varepsilon\mathbf{u}_{1}(\mathbf{x},t)+\varepsilon^{2}\mathbf{u}_{2}(\mathbf{x},t)+\cdots,\\\
&\mathbf{B}(\mathbf{x},t)=\mathbf{B}_{0}(\mathbf{x},t)+\varepsilon\mathbf{B}_{1}(\mathbf{x},t)+\varepsilon^{2}\mathbf{B}_{2}(\mathbf{x},t)+\cdots.\end{aligned}\right.$
Inserting (2.8) into (2.6) and (2.7), equating to zero for different orders of
$\varepsilon$, corresponding to the first three leading orders, we have
* •
$\mathcal{O}(\varepsilon^{-2})$
$\nabla p_{0}=0,$
* •
$\mathcal{O}(\varepsilon^{-1})$
$\nabla p_{1}=0,$
* •
$\mathcal{O}(\varepsilon^{0})$
(2.9) $\frac{\partial}{\partial t}\begin{bmatrix}\rho_{0}\\\
\rho_{0}\mathbf{u}_{0}\\\ \mathbf{B}_{0}\\\ E_{0}\\\
\end{bmatrix}+\nabla\cdot\begin{bmatrix}\rho_{0}\mathbf{u}_{0}\\\
\rho_{0}\mathbf{u}_{0}\otimes\mathbf{u}_{0}+(p_{2}+\frac{1}{2}{\|\mathbf{B}_{0}\|}^{2})\mathbf{I}-\mathbf{B}_{0}\otimes\mathbf{B}_{0}\\\
\mathbf{u}_{0}\otimes\mathbf{B}_{0}-\mathbf{B}_{0}\otimes\mathbf{u}_{0}\\\
(E_{0}+p_{0})\mathbf{u}_{0}\end{bmatrix}=\mathbf{0},$
where we have assumed a similar expansion of $E$ as $p$ in (2.8), and take
$E_{0}=p_{0}/(\gamma-1)$. Denoting the material derivative ${\rm
d/d}t=\partial/\partial t+{\bf u}_{0}\cdot\nabla$, from the energy equation in
(2.9), $E_{0}=p_{0}/(\gamma-1)$, with $\nabla p_{0}=0$, we have
(2.10) $\nabla\cdot{\bf u}_{0}=-\frac{1}{p_{0}\gamma}\frac{{\rm d}p_{0}}{{\rm
d}t}.$
Taking into account periodic or no-slip ${\bf u}\cdot{\bf n}=0$ boundary
conditions on the spatial domain $\Omega$, and integrating (2.10) over
$\Omega$, we get $\int_{\Omega}\nabla\cdot{\bf u}_{0}d{\bf
x}=\int_{\partial\Omega}{\bf u}_{0}\cdot{\bf n}ds=0$, where ${\bf n}$ is the
unit outward normal vector along $\partial\Omega$. This implies $p_{0}$ is
constant not only in space but also in time, that is $p_{0}={\rm Const}$. A
direct result $\nabla\cdot{\bf u}_{0}=0$ can be obtained through (2.10) with
$p_{0}={\rm Const}$. Inserting $\nabla\cdot{\bf u}_{0}=0$ into the mass-
continuity equation in (2.9), we get ${\rm d}\rho_{0}/{\rm d}t=0$ which means
that the density is constant along the characteristic trajectories.
For the order $\mathcal{O}(\varepsilon)$ in (2.6), we have:
$\partial E_{1}/\partial t+\nabla\cdot[(E_{0}+p_{0}){\bf
u}_{1}+(E_{1}+p_{1}){\bf u}_{0}]=0.$
Due to $E_{1}=p_{1}/(\gamma-1)$ from the EOS in (2.7), similarly as above, we
get $p_{1}={\rm Const}$ and $\nabla\cdot{\bf u}_{1}=0$. In this case, we
include $\varepsilon\mathbf{u}_{1}$ into $\mathbf{u}_{0}$, and $\varepsilon
p_{1}$ into $p_{0}$, the asymptotic expansions of ${\bf u}$ and $p$ become
(2.11) $\mathbf{u}=\mathbf{u}_{0}+\varepsilon^{2}\mathbf{u}_{2}+\cdots,$
and
(2.12) $p=p_{0}+\varepsilon^{2}p_{2}+\cdots.$
Using such expansions for $\mathbf{u}$ and $p$, as $\varepsilon\rightarrow 0$,
the leading order of the system (2.6) yields
(2.13) $\left\\{\begin{aligned} &\frac{\partial}{\partial
t}\begin{bmatrix}\rho_{0}\mathbf{u}_{0}\\\ \mathbf{B}_{0}\\\
\end{bmatrix}+\nabla\cdot\begin{bmatrix}\rho_{0}\mathbf{u}_{0}\otimes\mathbf{u}_{0}+(p_{2}+\frac{1}{2}{\|\mathbf{B}_{0}\|}^{2})\mathbf{I}-\mathbf{B}_{0}\otimes\mathbf{B}_{0}\\\
\mathbf{u}_{0}\otimes\mathbf{B}_{0}-\mathbf{B}_{0}\otimes\mathbf{u}_{0}\end{bmatrix}=\mathbf{0},\\\
&\frac{{\rm d}\rho_{0}}{{\rm d}t}=0,\quad p_{0}={\rm
Const},\quad\nabla\cdot{\bf u}_{0}=0,\quad\nabla\cdot{\bf B}=0.\\\
\end{aligned}\right.$
Numerically, in order to have a correct asymptotic limit from (2.6) to (2.13),
it is important to impose well-prepared initial conditions, which are
consistent with the expansions given in (2.8), with ${\bf u}$ and $p$ from
(2.11) and (2.12). According to [22, 41], the well-prepared initial conditions
for (2.6) are given as
(2.14) $\left\\{\begin{aligned} &\rho(t=0,{\bf x})=\rho_{0}({\bf
x})+\mathcal{O}(\varepsilon)>0,\quad&&p(t=0,{\bf
x})=p_{0}+\varepsilon^{2}p_{2}({\bf x})+\cdots,\\\ &{\bf u}(t=0,{\bf x})={\bf
u}_{0}({\bf x})+\mathcal{O}(\varepsilon^{2}),\quad&&{\bf B}(t=0,{\bf x})={\bf
B}_{0}({\bf x})+\mathcal{O}(\varepsilon),\end{aligned}\right.$
where $\rho_{0}({\bf x})$ is a strictly positive function, $p_{0}={\rm
Const}$, $\nabla\cdot{\bf u}_{0}=0$, and $\nabla\cdot{\bf B}_{0}=0$.
### 2.4. CT for $\nabla\cdot{\bf B}=0$
Many CT methods have been proposed to satisfy the discrete divergence-free
condition $\nabla\cdot{\bf B}=0$ for the ideal MHD equations [18, 17, 16, 15].
Here we consider the one described in [18], in which the magnetic field is
written as the curl of a magnetic vector potential ${\bf A}$:
(2.15) ${\bf B}=\nabla\times{\bf A}.$
Due to $\nabla\cdot({\bf u}\otimes{\bf B}-{\bf B}\otimes{\bf
u})=\nabla\times({\bf B}\times{\bf u})$, the equation of the magnetic field
${\bf B}$ in the MHD system can be rewritten as
(2.16) $\frac{\partial{\bf B}}{\partial t}+\nabla\times({\bf B}\times{\bf
u})=0.$
Inserting (2.15) into (2.16), we further get
$\nabla\times\left(\frac{\partial{\bf A}}{\partial t}+(\nabla\times{\bf
A})\times{\bf u}\right)=0,$
which implies the existence of a scalar function $\psi$ such that
$\frac{\partial{\bf A}}{\partial t}+(\nabla\times{\bf A})\times{\bf
u}=-\nabla\psi.$
Stable solutions can be obtained by introducing a Weyl gauge, i.e. $\psi\equiv
0$ [29], and the evolution equation for the vector potential becomes
$\frac{\partial{\bf A}}{\partial t}+(\nabla\times{\bf A})\times{\bf u}=0.$
It is easy to check that $\nabla\cdot{\bf B}=\nabla\cdot(\nabla\times{\bf
A})=0$, i.e. the divergence-free condition will be satisfied for all times. In
summary, we get the following MHD system:
(2.17) $\left\\{\begin{aligned} &\frac{\partial\rho}{\partial
t}+\nabla\cdot\left(\rho{\bf u}\right)=0,\\\ &\frac{\partial\rho{\bf
u}}{\partial
t}+\nabla\cdot\left(\rho\mathbf{u}\otimes\mathbf{u}+\left(\frac{p}{\varepsilon^{2}}+\frac{1}{2}{\|\mathbf{B}\|}^{2}\right)\mathbf{I}-\mathbf{B}\otimes\mathbf{B}\right)=\mathbf{0},\\\
&\frac{\partial{\bf A}}{\partial t}+(\nabla\times{\bf A})\times{\bf
u}=\mathbf{0},\\\ &\frac{\partial E}{\partial
t}+\nabla\cdot\left(\left(E+p+\frac{\varepsilon^{2}}{2}{\|\mathbf{B}\|}^{2}\right)\mathbf{u}-\varepsilon^{2}\mathbf{B}\left(\mathbf{u}\cdot\mathbf{B}\right)\right)=0,\\\
&{\bf B}-\nabla\times{\bf A}=\mathbf{0}.\\\ \end{aligned}\right.$
Corresponding to (2.17), with well-prepared initial conditions (2.14), the low
sonic Mach limit incompressible MHD system becomes
(2.18) $\left\\{\begin{aligned} &\frac{d\rho_{0}}{dt}=0,\quad p_{0}={\rm
Const},\quad\nabla\cdot{\bf u}_{0}=0,\quad\nabla\cdot\mathbf{B}_{0}=0,\\\
&\frac{\partial\rho_{0}\mathbf{u}_{0}}{\partial
t}+\nabla\cdot\left(\rho_{0}\mathbf{u}_{0}\otimes\mathbf{u}_{0}+\left(p_{2}+\frac{1}{2}{\|\mathbf{B}_{0}\|}^{2}\right)\mathbf{I}-\mathbf{B}_{0}\otimes\mathbf{B}_{0}\right)=\mathbf{0},\\\
&\frac{\partial{\bf A}_{0}}{\partial t}+(\nabla\times{\bf A}_{0})\times{\bf
u}_{0}=\mathbf{0},\\\ &{\bf B}_{0}-\nabla\times{\bf
A}_{0}=\mathbf{0}.\end{aligned}\right.$
## 3\. Numerical schemes
In this section, we will construct and analyze a high-order finite difference
scheme with AP and divergence-free properties for the non-dimensionalized MHD
system (2.17). The features of our scheme include the following: we design an
SI IMEX-RK time discretization so that the scheme is stable with a time step
constraint independent of the sonic Mach number $\varepsilon$; the AP property
is preserved in the zero sonic Mach number limit; the scheme can be
implemented efficiently without solving a nonlinear system. We will adopt the
high order finite difference WENO scheme in [8, 10, 30], and the CT method in
[18]. The final scheme can well capture discontinuities such as shocks in the
compressible regime, while keeping a discrete divergence-free condition in the
low sonic Mach regime, with a uniform time step condition independent of
$\varepsilon$. We start with a first-order SI scheme. The high-order IMEX-RK
scheme in time and a finite difference WENO reconstruction in space will be
followed. After that, an overall scheme will be summarized.
### 3.1. First order SI IMEX scheme
In the following, we start with a first-order SI IMEX time discretization for
(2.17), while keeping space continuous:
(3.1a) $\frac{\rho^{n+1}-\rho^{n}}{\Delta t}+\nabla\cdot\mathbf{q}^{n}=0,$
(3.1b) $\frac{\mathbf{q}^{n+1}-\mathbf{q}^{n}}{\Delta
t}+\nabla\cdot\left(\frac{\mathbf{q}^{n}\otimes\mathbf{q}^{n}}{\rho^{n}}-\mathbf{B}^{n}\otimes\mathbf{B}^{n}+\left(\frac{1}{2}{\|\mathbf{B}^{n}\|}^{2}+p^{n}\right)\mathbf{I}\right)+\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\nabla
p^{n+1}=\mathbf{0},$ (3.1c) $\frac{\mathbf{A}^{n+1}-\mathbf{A}^{n}}{\Delta
t}+\left(\nabla\times{\bf
A}^{n}\right)\times\frac{\mathbf{q}^{n}}{\rho^{n}}=\mathbf{0},$ (3.1d)
$\frac{E^{n+1}-E^{n}}{\Delta
t}+\nabla\cdot\left(\frac{E^{n}+p^{n}}{\rho^{n+1}}\mathbf{q}^{n+1}+\varepsilon^{2}\left(\frac{1}{2}{\|\mathbf{B}^{n}\|}^{2}\frac{\mathbf{q}^{n}}{\rho^{n}}-\left(\frac{\mathbf{q}^{n}}{\rho^{n}}\cdot\mathbf{B}^{n}\right)\mathbf{B}^{n}\right)\right)=0,$
(3.1e) ${\bf B}^{n+1}-\nabla\times{\bf A}^{n+1}=\mathbf{0},$
where $\mathbf{q}=\rho{\bf u}$. The updating flow chart based on the SI IMEX
scheme (3.1) is as follows:
* •
Step 1. Update $\rho^{n+1}$ and ${\bf A}^{n+1}$ from (3.1a) and (3.1c)
respectively, then update ${\bf B}^{n+1}$ from (3.1e).
* •
Step 2. We rewrite (3.1b) as
(3.2) $\mathbf{q}^{n+1}=\mathbf{q}^{*}-\Delta
t\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\nabla p^{n+1},$
with
$\mathbf{q}^{*}=\mathbf{q}^{n}-\Delta
t\nabla\cdot\left(\frac{\mathbf{q}^{n}\otimes\mathbf{q}^{n}}{\rho^{n}}-\mathbf{B}^{n}\otimes\mathbf{B}^{n}+\left(\frac{1}{2}{\|\mathbf{B}^{n}\|}^{2}+p^{n}\right)\mathbf{I}\right).$
Substituting $\mathbf{q}^{n+1}$ into (3.1d), we can further get
(3.3) $E^{n+1}=E^{*}+\Delta
t^{2}\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\nabla\cdot(H^{n}\nabla
p^{n+1}),$
where $H^{n}=(E^{n}+p^{n})/\rho^{n+1}$ and
$E^{*}=E^{n}-\Delta t\nabla\cdot(H^{n}\mathbf{q}^{*})-\Delta
t\varepsilon^{2}\nabla\cdot\left(\frac{1}{2}{\|\mathbf{B}^{n}\|}^{2}\frac{\mathbf{q}^{n}}{\rho^{n}}-\left(\frac{\mathbf{q}^{n}}{\rho^{n}}\cdot\mathbf{B}^{n}\right)\mathbf{B}^{n}\right).$
Corresponding to the pressure in the incompressible limit, we introduce a
pressure perturbation $p^{n+1}_{2}$, defined as
(3.4) $p^{n+1}_{2}=(p^{n+1}-\bar{p}^{n})/\varepsilon^{2},$
where $\bar{p}^{n}=\int_{\Omega}p^{n}d{\bf x}/|\Omega|$, and we replace
$E^{n+1}$ by
$p^{n+1}/(\gamma-1)+\varepsilon^{2}({\|\mathbf{q}^{n}\|}^{2}/(2\rho^{n})+{\|\mathbf{B}^{n}\|}^{2}/2)$
from (3.3). We rewrite (3.3) as
(3.5) $\frac{\varepsilon^{2}}{\gamma-1}p^{n+1}_{2}-\Delta
t^{2}(1-\varepsilon^{2})\nabla\cdot(H^{n}\nabla p^{n+1}_{2})=E^{**},$
where
$E^{**}=E^{*}-\bar{p}^{n}/(\gamma-1)-\varepsilon^{2}({\|\mathbf{q}^{n}\|}^{2}/(2\rho^{n})+{\|\mathbf{B}^{n}\|}^{2}/2)$.
We can now update $p_{2}^{n+1}$ from the elliptic equation (3.5).
* •
Step 3. We update $\mathbf{q}^{n+1}$ from (3.2) by using (3.4), and then
$E^{n+1}$ from (3.1d).
As described in [8], we may set $\nabla\cdot(\alpha p^{n}{\bf I})$ and
$(1-\alpha\varepsilon^{2})\nabla p^{n+1}/\varepsilon^{2}$ to replace
$\nabla\cdot(p^{n}{\bf I})$ and $(1-\varepsilon^{2})\nabla
p^{n+1}/\varepsilon^{2}$ in (3.1b) with $\alpha=1$ for $\varepsilon<1$ and
$\alpha=1/\varepsilon^{2}$ for $\varepsilon\geq 1$. Note that if
$\varepsilon\geq 1$ the implicit pressure contribution in (3.1b) vanishes, so
the momentum $\mathbf{q}^{n+1}$ is evaluated explicitly. With updated
$\rho^{n+1}$, ${\bf A}^{n+1}$, ${\bf B}^{n+1}$, and $\mathbf{q}^{n+1}$,
$E^{n+1}$ can also be updated in an explicit way from (3.1d). In one
dimension, the divergence-free condition is satisfied automatically, so there
is no need to apply a CT method. We can replace (3.1c) and (3.1e) with
(3.6) $\frac{\mathbf{B}^{n+1}-\mathbf{B}^{n}}{\Delta
t}+\nabla\cdot\left(\frac{\mathbf{q}^{n}\otimes\mathbf{B}^{n}}{\rho^{n}}-\frac{\mathbf{B}^{n}\otimes\mathbf{q}^{n}}{\rho^{n}}\right)=0.$
The first step in the flow chart becomes: update $\rho^{n+1}$ and ${\bf
B}^{n+1}$ from (3.1a) and (3.6) respectively, and other steps remain
unchanged.
### 3.2. High order SI IMEX-RK scheme
In this subsection, we will follow a similar procedure as in [8, 10] to extend
the first-order SI IMEX scheme (3.1) to high order. For simplicity, we only
consider (2.6) with CT methods, namely, (2.17).
First, (2.17) can be written as an autonomous system:
$\left\\{\begin{aligned} &U_{t}=\mathcal{H}(U,U,{\bf B}),\\\ &{\bf
B}=\nabla\times{\bf A},\end{aligned}\right.$
where $U={(\rho,\mathbf{q},{\bf A},E)}^{T}$ and
$\mathcal{H}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
is a sufficiently regular mapping. We use subscript “E” to express an explicit
treatment for the first argument $U$ and the third one ${\bf B}$, and
subscript “I” to express an implicit treatment for the second $U$. For the
system (2.17), the function $\mathcal{H}(U_{E},U_{I},{\bf B}_{E})$ is defined
as
$\displaystyle\mathcal{H}(U_{E},U_{I},{\bf B}_{E})=$
$\displaystyle-\nabla\cdot\begin{pmatrix}\mathbf{q}_{E}\\\
\frac{\mathbf{q}_{E}\otimes\mathbf{q}_{E}}{\rho_{E}}-\mathbf{B}_{E}\otimes\mathbf{B}_{E}+(\frac{1}{2}{\|\mathbf{B}_{E}\|}^{2}+p_{E})\mathbf{I}\\\
\mathbf{0}\\\ 0\\\ \end{pmatrix}-\begin{pmatrix}0\\\ \mathbf{0}\\\
(\nabla\times{\bf A}_{E})\times\frac{\mathbf{q}_{E}}{\rho_{E}}\\\
0\end{pmatrix}$ $\displaystyle-\nabla\cdot\begin{pmatrix}0\\\ \mathbf{0}\\\
\mathbf{0}\\\
\varepsilon^{2}(\frac{1}{2}{\|\mathbf{B}_{E}\|}^{2}\frac{\mathbf{q}_{E}}{\rho_{E}}-(\frac{\mathbf{q}_{E}}{\rho_{E}}\cdot\mathbf{B}_{E})\mathbf{B}_{E})\end{pmatrix}-\nabla\cdot\begin{pmatrix}0\\\
(1-\varepsilon^{2})p_{I,2}\mathbf{I}\\\ \mathbf{0}\\\
H_{I}\mathbf{q}_{I}\end{pmatrix}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\mathcal{F}_{E}^{1}-\mathcal{G}_{E}-\nabla\cdot\mathcal{F}_{E}^{2}-\nabla\cdot\mathcal{F}_{SI},$
with $H_{I}=(E_{E}+p_{E})/\rho_{I}$ and
$p_{I,2}=(p_{I}-\bar{p}_{E})/\varepsilon^{2}$.
To obtain high-order accuracy in time, we can apply a multi-stage IMEX-RK time
discretization with a double Butcher $tableau$,
$\begin{array}[]{c|c}\tilde{c}&\tilde{A}\\\ \hline\cr\vspace{-0.25cm}\hfil\\\
&\tilde{b}^{T}\end{array}\ \ \ \ \ \qquad\begin{array}[]{c|c}{c}&{A}\\\
\hline\cr\vspace{-0.25cm}\hfil\\\ &{b^{T}}\end{array},$
where $\tilde{A}=(\tilde{a}_{ij})$ is an $s\times s$ matrix for an explicit
scheme, with $\tilde{a}_{ij}=0$ for $j\geq i$, and $A=({a}_{ij})$ is an
$s\times s$ matrix for an implicit scheme. For the implicit part of the
methods, we use a diagonally implicit scheme, i.e. $a_{ij}=0$, for $j>i$, to
guarantee simplicity and efficiency in solving the algebraic equations
corresponding to the implicit part of the discretization. The vectors
$\tilde{c}=(\tilde{c}_{1},...,\tilde{c}_{s})^{T}$,
$\tilde{b}=(\tilde{b}_{1},...,\tilde{b}_{s})^{T}$, $c=(c_{1},...,c_{s})^{T}$,
and $b=(b_{1},...,b_{s})^{T}$ complete the characterization of the scheme. The
coefficients $\tilde{c}$ and $c$ are given by the usual relation
$\displaystyle\tilde{c}_{i}=\sum_{j=1}^{i-1}\tilde{a}_{ij},\ \ \
c_{i}=\sum_{j=1}^{i}a_{ij}.$
Later, for the consideration of AP property, we consider implicit schemes with
SA property, i.e., the implicit part of the Butcher table satisfies the
condition $b^{T}=e_{s}^{T}A$, with $e_{s}=(0,\cdots 0,1)^{T}$. We will see
that the SA property guarantees that the numerical solution is identical to
the last internal stage value of the scheme. Now, we may update the solutions
as follows. Starting from $U_{E}^{(0)}=U_{I}^{(0)}=U^{n}$ and ${\bf
B}_{E}^{(0)}={\bf B}^{n}$, for inner stages $i=1\text{ to }s$:
* •
First update the solution $U_{E}^{(i)}$ for the explicit part
(3.7) $U_{E}^{(i)}=U^{n}+\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\mathcal{H}(U_{E}^{(j)},U_{I}^{(j)},{\bf
B}_{E}^{(j)}),\quad{\bf B}_{E}^{(i)}=\nabla\times{\bf A}_{E}^{(i)}.$
* •
Update the known values for the implicit part $U_{*}^{(i)}$, where
(3.8) $U_{*}^{(i)}=U^{n}+\Delta
t\sum^{i-1}_{j=1}a_{ij}\mathcal{H}(U_{E}^{(j)},U_{I}^{(j)},{\bf
B}_{E}^{(j)}),$
and then solve
(3.9) $U_{I}^{(i)}=U_{*}^{(i)}+\Delta
ta_{ii}\mathcal{H}(U_{E}^{(i)},U_{I}^{(i)},{\bf B}_{E}^{(i)}).$
* •
Finally, compute the solutions $U^{n+1}$ and ${\bf B}^{n+1}$ at time level
$t^{n+1}$ from
(3.10) $U^{n+1}=U^{n}+\Delta
t\sum_{i=1}^{s}b_{i}\mathcal{H}(U_{E}^{(i)},U_{I}^{(i)},{\bf
B}_{E}^{(i)}),\quad{\bf B}^{n+1}=\nabla\times{\bf A}^{n+1}.$
The first order scheme (3.1) is the same as applying the following Butcher
table to (2.17):
$\begin{array}[]{c|c}0&0\\\
\hline\cr&1\end{array}\qquad\qquad\begin{array}[]{c|c}1&1\\\
\hline\cr&1\end{array},$
namely $U_{E}=U^{n}$ and $U_{I}=U^{n+1}$.
### 3.3. High order finite difference WENO scheme
Below we will describe our spatial discretization strategies which incorporate
a WENO mechanism in order to capture shocks in the compressible regime and
produce a high order incompressible solver for the flow in the zero Mach
limit. In particular, we propose to apply a fifth order characteristic-wise
WENO procedure to $\mathcal{F}_{E}^{1}$ of the ideal MHD system so that in the
compressible regime numerical oscillations could be well controlled, and to
apply a component-wise WENO procedure for the flux functions of
$\mathcal{F}_{E}^{2}$ and $\mathcal{F}_{SI}$. As for $\mathcal{G}_{E}$, which
is a Hamiltonian, a WENO procedure described in [31] will be applied.
* •
Component-wise WENO $\nabla_{W}$ for $\nabla\cdot\mathcal{F}_{E}^{2}$ and
$\nabla\cdot\mathcal{F}_{SI}$:
Let ${\bf v}=(\rho,\mathbf{q},{\bf B},E)^{T}$ and $\mathbf{f}=\mathbf{f}({\bf
v})$ be the unknown variables and flux of the ideal MHD system (2.1). For
simplicity, we consider the 1D case as an example, and we take a uniform mesh
with $N+1$ grid points: $a=x_{0}<x_{1}<\cdots<x_{N}=b$. A conservative finite
difference WENO scheme for system (2.1) can be written in the following flux-
difference form:
$(\mathbf{v}_{i})_{t}=\frac{1}{\Delta
x}(\hat{\mathbf{f}}_{i+\frac{1}{2}}-\hat{\mathbf{f}}_{i-\frac{1}{2}}),$
where $\hat{\mathbf{f}}_{i\pm\frac{1}{2}}$ are the numerical fluxes, which can
be reconstructed by the following procedures:
1. (1)
Compute the physical flux at each grid point: $\mathbf{f}_{i}=\mathbf{f}({\bf
v}_{i})$.
2. (2)
Perform a Lax–Friedrichs flux vector splitting for each component of the
physical variables to get
(3.11)
$\mathbf{f}^{\pm}_{i}=\frac{1}{2}(\mathbf{f}_{i}\pm\alpha\mathbf{v}_{i}),$
with $\alpha=\mathop{\max}\limits_{0\leq k\leq N}(|{\bf
u}_{k}|+\hat{c}_{f}^{(k)})$, where
(3.12) $\left\\{\begin{aligned} &\hat{a}^{(k)}:={\rm
min}\left(\frac{1}{\varepsilon},1\right)\sqrt{\frac{\gamma
p_{k}}{\rho_{k}}},\quad\hat{c}_{a}^{(k)}:=\sqrt{\frac{{({\bf B}_{k}\cdot{\bf
n})}^{2}}{\rho_{k}}},\\\
&\hat{c}_{f,s}^{(k)}:={\left\\{\frac{1}{2}\left[\left(\hat{a}^{(k)}\right)^{2}+\frac{{\|{\bf
B}_{k}\|}^{2}}{\rho_{k}}\pm\sqrt{{\left(\left(\hat{a}^{(k)}\right)^{2}+\frac{{\|{\bf
B}_{k}\|}^{2}}{\rho_{k}}\right)}^{2}-\left(2{\hat{a}^{(k)}}{\hat{c}^{(k)}_{a}}\right)^{2}}\right]\right\\}}^{\frac{1}{2}}.\end{aligned}\right.$
3. (3)
Perform a finite difference WENO reconstruction to obtain upwind and downwind
numerical fluxes:
$\hat{\mathbf{f}}^{+}_{i+\frac{1}{2}}=\Phi_{\rm
WENO5}(\mathbf{f}^{+}_{i-2},\mathbf{f}^{+}_{i-1},\mathbf{f}^{+}_{i},\mathbf{f}^{+}_{i+1},\mathbf{f}^{+}_{i+2}),\quad\hat{\mathbf{f}}^{-}_{i+\frac{1}{2}}=\Phi_{\rm
WENO5}(\mathbf{f}^{-}_{i+3},\mathbf{f}^{-}_{i+2},\mathbf{\mathbf{f}}^{-}_{i+1},\mathbf{f}^{-}_{i},\mathbf{f}^{-}_{i-1}).$
Here $\Phi_{\rm WENO5}$ denotes a 5th-order WENO reconstruction [31], while
other orders can also be used. Then we set
$\hat{\mathbf{f}}_{i+\frac{1}{2}}=\hat{\mathbf{f}}^{+}_{i+\frac{1}{2}}+\hat{\mathbf{f}}^{-}_{i+\frac{1}{2}}.$
* •
Characteristic-wise WENO $\nabla_{CW}$ for $\nabla\cdot\mathcal{F}_{E}^{1}$:
The Jacobian matrix for the flux function
$\partial\mathbf{f}/\partial\mathbf{v}$, has a spectral decomposition of the
form $\partial\mathbf{f}/\partial\mathbf{v}=RDL$ [18], where $D$ is a diagonal
matrix of real eigenvalues, $R$ is the matrix of right eigenvectors, and
$L=R^{-1}$ is the matrix of left eigenvectors. Then, for a characteristic-wise
WENO reconstruction $\nabla_{CW}$, we have the following procedures:
1. (1)
Compute the physical flux at each grid point: $\mathbf{f}_{i}=\mathbf{f}({\bf
v}_{i})$.
2. (2)
Compute the average state ${\bf v}_{i+\frac{1}{2}}$ by using an arithmetic
mean:
${\bf v}_{i+\frac{1}{2}}=\frac{1}{2}({\bf v}_{i}+{\bf v}_{i+1}).$
3. (3)
Compute the right and left eigenvectors of the flux Jacobian matrix
$\partial\mathbf{f}/\partial\mathbf{v}$ by taking ${\bf v}={\bf
v}_{i+\frac{1}{2}}$ at $x=x_{i+\frac{1}{2}}$:
$R_{i+\frac{1}{2}}=R({\bf v}_{i+\frac{1}{2}}),\quad L_{i+\frac{1}{2}}=L({\bf
v}_{i+\frac{1}{2}}).$
4. (4)
Project the solution and physical flux into the characteristic space:
$\mathbf{w}_{j}=L_{i+\frac{1}{2}}{\bf
v}_{j},\quad\mathbf{g}_{j}=L_{i+\frac{1}{2}}\mathbf{f}_{j},$
for all $j$ in the numerical stencil associated with $x=x_{i+\frac{1}{2}}$. In
the case of a 5th-order finite difference WENO scheme:
$j=i-2,i-1,i,i+1,i+2,i+3$.
5. (5)
Perform a Lax–Friedrichs flux vector splitting for each component of the
characteristic variables to compute
$\mathbf{g}^{\pm}_{j}=\frac{1}{2}(\mathbf{g}_{j}\pm\alpha\mathbf{w}_{j}),$
where $\alpha$ has the same expression in the component-wise WENO (3.11).
6. (6)
Perform a WENO reconstruction $\Phi_{\rm WENO5}$ on each component of
$\mathbf{g}^{\pm}_{j}$ to obtain their corresponding numerical fluxes, and
then set
$\hat{\mathbf{g}}_{i+\frac{1}{2}}=\hat{\mathbf{g}}^{+}_{i+\frac{1}{2}}+\hat{\mathbf{g}}^{-}_{i+\frac{1}{2}}.$
7. (7)
Project the numerical flux $\hat{\mathbf{g}}_{i+\frac{1}{2}}$ back to the
physical space:
$\hat{\mathbf{f}}_{i+\frac{1}{2}}=R_{i+\frac{1}{2}}\hat{\mathbf{g}}_{i+\frac{1}{2}}.$
However, when the sonic Mach number $\varepsilon$ is small, the fast
eigenvalue $c_{f}$ will make such a spectral characteristic decomposition to
be unstable. Since a characteristic decomposition is mainly required in the
compressible regime where $\varepsilon=\mathcal{O}(1)$, while it does not make
any significant differences for small Mach numbers; in this case we directly
use the eigenvectors in the expression of (2.1) by taking $\varepsilon=1$,
which works well from our numerical results.
* •
A Hamiltonian WENO reconstruction WENO-HJ $(\cdot)_{HJ}$ for
$\mathcal{G}_{E}$:
We take a 1D Hamilton–Jacobi equation as an example
(3.13) $v_{t}+\mathcal{HT}(t,x,v,v_{x})=0,$
where $\mathcal{HT}$ is the Hamiltonian. As described in [31], a semi-discrete
form for (3.13) is
$v_{t,i}=-\widehat{\mathcal{HT}}(t,x_{i},v_{i},v^{-}_{x,i},v^{+}_{x,i}),$
where $\widehat{\mathcal{HT}}$ is a numerical Hamiltonian which approximates
(3.13), and $v^{-}_{x,i},v^{+}_{x,i}$ are upwind and downwind approximations
of $v_{x}$ at $x=x_{i}$. A Lax-Friedrichs Hamiltonian from [31] is given as
follows
(3.14)
$\widehat{\mathcal{HT}}\left(t,x,v,u^{-},u^{+}\right)=\mathcal{HT}\left(t,x,v,\frac{u^{-}+u^{+}}{2}\right)-\beta\left(\frac{u^{+}-u^{-}}{2}\right),$
where
$\beta=\mathop{\max}\limits_{u\in I(u^{-},u^{+})}|\mathcal{HT}_{u}(t,x,v,u)|,$
and $I(u^{-},u^{+})$ is the interval between $u^{-}$ and $u^{+}$. The values
of $v^{+}_{x,i}$ and $v^{-}_{x,i}$ are obtained from
$\displaystyle v^{+}_{x,i}$ $\displaystyle=\Phi_{\rm
WENO5}\left(\frac{\Delta^{+}v_{i+2}}{\Delta x},\frac{\Delta^{+}v_{i+1}}{\Delta
x},\frac{\Delta^{+}v_{i}}{\Delta x},\frac{\Delta^{+}v_{i-1}}{\Delta
x},\frac{\Delta^{+}v_{i-2}}{\Delta x}\right),$ $\displaystyle v^{-}_{x,i}$
$\displaystyle=\Phi_{\rm WENO5}\left(\frac{\Delta^{+}v_{i-3}}{\Delta
x},\frac{\Delta^{+}v_{i-2}}{\Delta x},\frac{\Delta^{+}v_{i-1}}{\Delta
x},\frac{\Delta^{+}v_{i}}{\Delta x},\frac{\Delta^{+}v_{i+1}}{\Delta
x}\right),$
with $\Delta^{+}v_{j}=v_{j+1}-v_{j}$. Such a Hamiltonian WENO reconstruction
helps us to control unphysical oscillations both in $v_{x}$ and $v$. For
$\mathcal{G}_{E}$ in 2D, the Hamiltonian is
$\mathcal{HT}\left(t,x,y,\frac{\partial A_{z}}{\partial x},\frac{\partial
A_{z}}{\partial y}\right)=u(t,x,y)\frac{\partial A_{z}}{\partial
x}+v(t,x,y)\frac{\partial A_{z}}{\partial y},$
and the numerical Hamiltonian (3.14) can be applied along each direction for
$u(t,x,y)\frac{\partial A_{z}}{\partial x}$ and $v(t,x,y)\frac{\partial
A_{3}}{\partial y}$, respectively. Here $A_{z}$ is the third component of
${\bf A}$.
### 3.4. Flowchart of a high order SI IMEX-RK finite difference WENO scheme
With the above space and time discretizations, we now summarize our high order
scheme as follows:
* •
Step 1. Starting from $U^{n}$ at time $t^{n}$, for an intermediate stage $i$
($1\leq i\leq s$), we first compute $U^{(i)}_{E}$ from (3.7)
$\rho^{(i)}_{E}=\rho^{n}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\nabla_{CW}\cdot\mathbf{q}^{(j)}_{E},$
$\mathbf{q}^{(i)}_{E}=\mathbf{q}^{n}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\left(\nabla_{CW}\cdot\left(\frac{\mathbf{q}^{(j)}_{E}\otimes\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}-\mathbf{B}^{(j)}_{E}\otimes\mathbf{B}^{(j)}_{E}+\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{E}\|}^{2}+p^{(j)}_{E}\right)\mathbf{I}\right)+(1-\varepsilon^{2})\nabla_{W}p^{(j)}_{I,2}\right),$
$\mathbf{A}^{(i)}_{E}=\mathbf{A}^{n}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\left(\left(\nabla\times{\bf
A}^{(j)}_{E}\right)\times\frac{\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}\right)_{HJ},$
$E^{(i)}_{E}=E^{n}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\left(\nabla_{W}\cdot\left(\varepsilon^{2}\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{E}\|}^{2}\frac{\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}-\left(\frac{\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}\cdot\mathbf{B}^{(j)}_{E}\right)\mathbf{B}^{(j)}_{E}\right)\right)+\nabla_{W}\cdot\left(\bar{H}^{(j)}\mathbf{q}^{(j)}_{I}\right)\right),$
with $\bar{H}^{(j)}=(E_{E}^{(j)}+p_{E}^{(j)})/\rho_{I}^{(j)}$. Then we obtain
${\bf B}_{E}^{(i)}=\nabla\times{\bf A}_{E}^{(i)},$
with a 4th-order central difference discretization.
* •
Step 2. We compute $U^{(i)}_{*}$ in (3.8)
$\rho^{(i)}_{*}=\rho^{n}-\Delta
t\sum^{i-1}_{j=1}a_{ij}\nabla_{CW}\cdot\mathbf{q}^{(j)}_{E},$
$\mathbf{q}^{(i)}_{*}=\mathbf{q}^{n}-\Delta
t\sum^{i-1}_{j=1}a_{ij}\left(\nabla_{CW}\cdot\left(\frac{\mathbf{q}^{(j)}_{E}\otimes\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}-\mathbf{B}^{(j)}_{E}\otimes\mathbf{B}^{(j)}_{E}+\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{E}\|}^{2}+p^{(j)}_{E}\right)\mathbf{I}\right)+(1-\varepsilon^{2})\nabla_{W}p^{(j)}_{I,2}\right),$
$E^{(i)}_{*}=E^{n}-\Delta
t\sum^{i-1}_{j=1}a_{ij}\left(\nabla_{W}\cdot\left(\varepsilon^{2}\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{E}\|}^{2}\frac{\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}-\left(\frac{\mathbf{q}^{(j)}_{E}}{\rho^{(j)}_{E}}\cdot\mathbf{B}^{(j)}_{E}\right)\mathbf{B}^{(j)}_{E}\right)\right)+\nabla_{W}\cdot\left(\bar{H}^{(j)}\mathbf{q}^{(j)}_{I}\right)\right).$
* •
Step 3. Solve $U_{I}^{(i)}$ from (3.9).
1. (1)
In components, $U_{I}^{(i)}$ satisfies
(3.15a) $\rho^{(i)}_{I}=\rho^{(i)}_{*}-\Delta
ta_{ii}\nabla_{CW}\cdot\mathbf{q}^{(i)}_{E},$ (3.15b)
$\mathbf{q}^{(i)}_{I}=\mathbf{q}^{(i)}_{**}-\Delta
ta_{ii}(\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\nabla p^{(i)}_{I}),$ (3.15c)
$E^{(i)}_{I}=E^{(i)}_{**}-\Delta
ta_{ii}\nabla\cdot(\bar{H}^{(i)}\mathbf{q}^{(i)}_{I}),$
where
$\mathbf{q}^{(i)}_{**}=\mathbf{q}^{(i)}_{*}-\Delta
ta_{ii}\nabla_{CW}\cdot\left(\frac{\mathbf{q}^{(i)}_{E}\otimes\mathbf{q}^{(i)}_{E}}{\rho^{(i)}_{E}}-\mathbf{B}^{(i)}_{E}\otimes\mathbf{B}^{(i)}_{E}+\left(\frac{1}{2}{\|\mathbf{B}^{(i)}_{E}\|}^{2}+p^{(i)}_{E}\right)\mathbf{I}\right),$
$E^{(i)}_{**}=E^{(i)}_{*}-\Delta
ta_{ii}\nabla_{W}\cdot\left(\varepsilon^{2}\left(\frac{1}{2}{\|\mathbf{B}^{(i)}_{E}\|}^{2}\frac{\mathbf{q}^{(i)}_{E}}{\rho^{(i)}_{E}}-\left(\frac{\mathbf{q}^{(i)}_{E}}{\rho^{(i)}_{E}}\cdot\mathbf{B}^{(i)}_{E}\right)\mathbf{B}^{(i)}_{E}\right)\right).$
2. (2)
We can obtain $\rho_{I}^{(i)}$ from (3.15a) directly. To solve (3.15b) and
(3.15c), we substitute $\mathbf{q}^{(i)}_{I}$ of (3.15b) into (3.15c), and
obtain
(3.16) $E^{(i)}_{I}=E^{\circ}_{I}+\left(1-\varepsilon^{2}\right)\Delta
t^{2}a^{2}_{ii}\nabla\cdot\left(\bar{H}^{(i)}\left(\frac{\nabla
p^{(i)}_{I}}{\varepsilon^{2}}\right)\right),$
with $E_{I}^{(i)}$ following the EOS
$E_{I}^{(i)}=p_{I}^{(i)}/(\gamma-1)+\varepsilon^{2}({\|\mathbf{q}_{E}^{(i)}\|}^{2}/(2\rho_{E}^{(i)})+{\|\mathbf{B}_{E}^{(i)}\|}^{2}/2)$
and $E^{\circ}_{I}=E^{(i)}_{**}-\Delta
ta_{ii}\nabla_{W}\cdot(\bar{H}^{(i)}\mathbf{q}^{(i)}_{**})$. By substituting
(3.17) $p^{(i)}_{I}=\bar{p}^{(i)}_{E}+\varepsilon^{2}p^{(i)}_{I,2}$
into (3.16), where $\bar{p}^{(i)}_{E}=\int_{\Omega}p_{E}^{(i)}d{\bf
x}/|\Omega|$, we solve
(3.18)
$\frac{\varepsilon^{2}}{\gamma-1}p^{(i)}_{I,2}-\left(1-\varepsilon^{2}\right)\Delta
t^{2}a^{2}_{ii}\nabla\cdot\left(\bar{H}^{(i)}\left(\nabla
p^{(i)}_{I,2}\right)\right)=E^{\circ\circ}_{I},$
with
$E^{\circ\circ}_{I}=E^{\circ}_{I}-\bar{p}^{(i)}_{E}/(\gamma-1)-\varepsilon^{2}({\|\mathbf{q}^{(i)}_{E}\|}^{2}/(2\rho^{(i)}_{E})+{\|\mathbf{B}^{(i)}_{E}\|}^{2}/2)$.
Notice that in the process of substitution to obtain (3.16) or (3.18), the
gradient and the divergence operators are kept to be continuous, obtaining the
second order operator $\nabla\cdot(\bar{H}^{(i)}\nabla p^{(i)}_{I,2})$. The
second-order spatial derivative is then discretized by a compact
discretization as proposed in [10].
3. (3)
With $p^{(i)}_{I,2}$ solved from (3.18), we can update $\mathbf{q}^{(i)}_{I}$
from
$\mathbf{q}^{(i)}_{I}=\mathbf{q}^{(i)}_{**}-\Delta
ta_{ii}\left(\left(1-\varepsilon^{2}\right)\nabla_{W}p^{(i)}_{I,2}\right),$
and successively update $E^{(i)}_{I}$ from
$E^{(i)}_{I}=E^{(i)}_{**}-\Delta
ta_{ii}\nabla_{W}\cdot\left(\bar{H}^{(i)}\mathbf{q}^{(i)}_{I}\right).$
* •
Step 4. Finally, we update the numerical solution $U^{n+1}=U^{(s)}_{I}$, and
${\bf B}^{n+1}=\nabla\times{\bf A}^{n+1}$.
## 4\. Asymptotic preserving (AP) and Asymptotically Accurate (AA) properties
### 4.1. AP property
In this section, we will prove the AP property for the first order SI IMEX
scheme (3.1), and the AA property for the high order SI IMEX-RK scheme
(3.7)-(3.10). We focus on time discretizations while keeping the space
continuous when we discuss the AP or AA property. First, we have the following
theorem:
###### Theorem 4.1.
The first order SI IMEX scheme (3.1) with space continuous is AP, in the sense
that, with well-prepared initial conditions (2.14), at the leading order
asymptotic expansions, the scheme (3.1) is a consistent approximation of the
incompressible MHD equations (LABEL:S2_E26).
###### Proof.
We assume the following expansions of the solutions at time $t^{n}$:
(4.1) $\left\\{\begin{aligned}
&p^{n}(\mathbf{x})=p^{n}_{0}+\varepsilon^{2}p^{n}_{2}(\mathbf{x}),\quad\rho^{n}(\mathbf{x})=\rho^{n}_{0}(\mathbf{x})+\mathcal{O}(\varepsilon),\\\
&\mathbf{u}^{n}(\mathbf{x})=\mathbf{u}^{n}_{0}(\mathbf{x})+\mathcal{O}(\varepsilon^{2}),\quad\mathbf{A}^{n}(\mathbf{x})=\mathbf{A}^{n}_{0}(\mathbf{x})+\mathcal{O}(\varepsilon),\\\
&\mathbf{B}^{n}(\mathbf{x})=\mathbf{B}^{n}_{0}(\mathbf{x})+\mathcal{O}(\varepsilon),&\end{aligned}\right.$
with periodic or no-slip boundary conditions, and well-prepared conditions
$p_{0}^{n}=(\gamma-1)E^{n}_{0}={\rm Const}$,
$\nabla\cdot\mathbf{u}^{n}_{0}(\mathbf{x})=0$ and $\nabla\cdot{\bf B}^{n}({\bf
x})=0$. No matter what value ${\bf A}^{n+1}$ takes, we can obtain
$\nabla\cdot{\bf B}^{n+1}=\nabla\cdot(\nabla\times{\bf A}^{n+1})=0$ from
(3.1e). We plug (4.1) into the semi-discrete scheme (3.1) and EOS (3.3). For
the leading order $\mathcal{O}(\varepsilon^{-2})$, we obtain
$\nabla p_{0}^{n+1}=0,$
i.e. $p_{0}^{n+1}$ is constant in space. Equating to zero the
$\mathcal{O}(\varepsilon^{0})$ terms, we have the following equations:
(4.2a) $\frac{\rho_{0}^{n+1}-\rho_{0}^{n}}{\Delta
t}+\nabla\cdot(\rho_{0}^{n}{\bf u}_{0}^{n})=0,$ (4.2b)
$\frac{\rho_{0}^{n+1}\mathbf{u}_{0}^{n+1}-\rho_{0}^{n}\mathbf{u}_{0}^{n}}{\Delta
t}+\nabla\cdot\left(\rho_{0}^{n}{\bf u}_{0}^{n}\otimes{\bf
u}_{0}^{n}-\mathbf{B}_{0}^{n}\otimes\mathbf{B}_{0}^{n}+\frac{1}{2}{\|\mathbf{B}_{0}^{n}\|}^{2}\mathbf{I}\right)+\nabla
p^{n+1}_{2}=\mathbf{0},$ (4.2c)
$\frac{\mathbf{A}_{0}^{n+1}-\mathbf{A}_{0}^{n}}{\Delta t}+(\nabla\times{\bf
A}_{0}^{n})\times{\bf u}_{0}^{n}=\mathbf{0},$ (4.2d)
$\frac{E_{0}^{n+1}-E_{0}^{n}}{\Delta
t}+\nabla\cdot\left(\bar{H}_{0}^{n}\left(\rho_{0}^{n+1}{\bf
u}_{0}^{n+1}\right)\right)=0,$ (4.2e) ${\bf B}_{0}^{n+1}-\nabla\times{\bf
A}_{0}^{n+1}=\mathbf{0},$ (4.2f) $E_{0}^{n+1}=\frac{p_{0}^{n+1}}{\gamma-1},$
with
(4.3)
$\bar{H}_{0}^{n}=\frac{p_{0}^{n}+E_{0}^{n}}{\rho_{0}^{n+1}}=\frac{\gamma}{\gamma-1}\frac{p_{0}^{n}}{\rho_{0}^{n+1}}.$
Due to (4.2f) and (4.3), it is easy to check that (4.2d) is equivalent to
(4.4) $\frac{p_{0}^{n+1}-p_{0}^{n}}{\Delta t}+\gamma p_{0}^{n}\nabla\cdot{\bf
u}_{0}^{n+1}=0.$
Taking into account a spatial domain $\Omega$ with periodic or no-slip
boundary conditions and integrating (4.4) over $\Omega$, we further get
$\int_{\Omega}\nabla\cdot{\bf u}_{0}^{n+1}d{\bf x}=\int_{\partial\Omega}{\bf
u}_{0}^{n+1}\cdot{\bf n}ds=0$, where ${\bf n}$ is the unit outward normal
vector along $\partial\Omega$. This implies $p_{0}^{n+1}=p^{n}_{0}={\rm
Const}$. A direct result $\nabla\cdot{\bf u}_{0}^{n+1}=0$ can be obtained
through combing (4.4) with $p_{0}^{n+1}=p^{n}_{0}$. The equations
(4.2a)-(4.2c) and (4.2e) correspond to consistent discretizations for other
equations in (LABEL:S2_E26), so that the AP property is obtained. ∎
### 4.2. AA property
Similarly to the AP property, here we focus on the AA analysis on time
discretizations, while keeping space continuous. Then we have the following
theorem:
###### Theorem 4.2.
We consider a high order SI IMEX-RK scheme (3.7)-(3.10) of order r applied to
system (2.17) on a bounded domain $\Omega\subset\mathbb{R}$ with periodic or
compact support boundary conditions, and assume that the implicit part of the
SI IMEX-RK scheme is SA and the initial conditions $(\rho^{0}({\bf
x}),\rho^{0}({\bf x}){\bf u}^{0}({\bf x}),p^{0},$ ${\bf A}^{0}({\bf x}),{\bf
B}^{0}({\bf x}))^{T}$ are well-prepared, namely in the form of (4.1). If we
denote $(\rho^{1}({\bf x};\varepsilon),\rho^{1}({\bf x};\varepsilon){\bf
u}^{1}({\bf x};\varepsilon),$ $p^{1}({\bf x};\varepsilon),{\bf A}^{1}({\bf
x};\varepsilon),{\bf B}^{1}({\bf x};\varepsilon))^{T}$ as the numerical
solution after one-time step, then, with $p_{*}$ being a constant, we have:
(4.5) $\lim_{\varepsilon\to 0}p^{1}({\bf
x};\varepsilon)=p_{*},\quad\lim_{\varepsilon\to 0}\nabla\cdot{\bf u}^{1}({\bf
x};\varepsilon)=0,\quad\nabla\cdot{\bf B}^{1}({\bf x};\varepsilon)=0.$
Furthermore, let ${\bf V}_{inc}({\bf x};t)=(\rho_{inc}({\bf
x};t),\rho_{inc}({\bf x};t){\bf u}_{inc}({\bf x};t),p_{inc}({\bf x};t),{\bf
A}_{inc}({\bf x};t),{\bf B}_{inc}({\bf x};t))^{T}$ be the exact solution of
the incompressible MHD equations (LABEL:S2_E26) with the same initial data,
one has the following one-step error estimate
(4.6) $\lim_{\varepsilon\to 0}{\bf V}^{1}({\bf x};\varepsilon)={\bf
V}_{inc}({\bf x};\Delta t)+\mathcal{O}(\Delta t^{r+1}),$
i.e. the scheme is AA.
###### Proof.
We consider the first step from $t^{0}=0$ to $t^{1}=\Delta t$ for the SI IMEX-
RK scheme (3.7)-(3.10) of order $r$ applied to system (2.17) with well-
prepared initial data (4.1):
$\left\\{\begin{aligned}
&p^{0}(\mathbf{x})=p_{*}+\varepsilon^{2}p^{0}_{2}(\mathbf{x}),\quad\rho^{0}(\mathbf{x})=\rho^{0}_{inc}+\mathcal{O}(\varepsilon),\\\
&\mathbf{u}^{0}(\mathbf{x})=\mathbf{u}^{0}_{inc}+\mathcal{O}(\varepsilon^{2}),\quad\mathbf{A}^{0}(\mathbf{x})=\mathbf{A}^{0}_{inc}+\mathcal{O}(\varepsilon),\quad\mathbf{B}^{0}(\mathbf{x})=\mathbf{B}^{0}_{inc}+\mathcal{O}(\varepsilon),\end{aligned}\right.$
where
$\displaystyle\rho^{0}_{inc}:=\rho_{inc}({\bf
x},0),\quad\mathbf{u}^{0}_{inc}:=\mathbf{u}_{inc}({\bf
x},0),\quad\mathbf{A}^{0}_{inc}:=\mathbf{A}_{inc}({\bf
x},0),\quad\mathbf{B}^{0}_{inc}:=\mathbf{B}_{inc}({\bf x},0).$
By well-prepared assumptions we have: $p_{inc}({\bf x},0):=p_{*}$ is a
constant independent of time and space, $\nabla\cdot{\bf u}_{inc}^{0}=0$ and
$\nabla\cdot{\bf B}^{0}({\bf x})=0$.
Now we consider a formal $\varepsilon$-expansion of the quantities:
$U_{I}^{(i)}=(\rho_{I}^{(i)},\mathbf{q}_{I}^{(i)},{\bf
A}_{I}^{(i)},E_{I}^{(i)}),\quad
U_{E}^{(i)}=(\rho_{E}^{(i)},\mathbf{q}_{E}^{(i)},{\bf
A}_{E}^{(i)},E_{E}^{(i)}),$
and ${\bf B}_{E}^{(i)}$, with $\mathbf{q}_{I}^{(i)}=\rho_{I}^{(i)}{\bf
u}_{I}^{(i)}$ and $\mathbf{q}_{E}^{(i)}=\rho_{E}^{(i)}{\bf u}_{E}^{(i)}$. For
example, the expansions of the density and pressure are
(4.7) $\left\\{\begin{aligned}
&\rho_{I}^{(i)}=\rho_{0,I}^{(i)}+\varepsilon\rho_{1,I}^{(i)}+\cdots,\quad&\rho_{E}^{(i)}&=\rho_{0,E}^{(i)}+\varepsilon\rho_{1,E}^{(i)}+\cdots,\\\
&p_{I}^{(i)}=p_{0,I}^{(i)}+\varepsilon^{2}p_{2,I}^{(i)}+\cdots,\quad&p_{E}^{(i)}&=p_{0,E}^{(i)}+\varepsilon^{2}p_{2,E}^{(i)}+\cdots.\end{aligned}\right.$
To prove the theorem, we use mathematical induction.
* •
AA property for internal stages $i=1,\cdots,s.$ When $i=1$, it leads to the
same AP analysis for the scheme (3.1) with $\Delta t$ replaced by
$a_{11}\Delta t$. To prove the result for $i>1$, we make use of the induction
hypothesis, assuming the property holds for $j\leq i-1$, and prove that it
still holds for $j=i$. For $j=1,\cdots,i-1$, we have
(4.8) $p_{0,E}^{(j)}=p_{0,I}^{(j)}=p_{*},\quad
E_{0,E}^{(j)}=E_{0,I}^{(j)}=\frac{p_{*}}{\gamma-1},\quad\nabla\cdot{\bf
u}_{0,I}^{(j)}=0,\quad\nabla\cdot{\bf B}_{E}^{(j)}=0.$
Now we insert the expansions (4.7) into the explicit step in (3.7), and from
the energy equation we get
(4.9) $E_{0,E}^{(i)}=E_{inc}^{0}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\nabla\cdot\left(\bar{H}_{0}^{(j)}\mathbf{q}_{0,I}^{(j)}\right),$
with $E_{inc}^{0}=p_{*}/(\gamma-1)$ and for $j=1,\cdots,i-1$,
(4.10)
$\bar{H}_{0}^{(j)}=\frac{E_{0,E}^{(j)}+p_{0,E}^{(j)}}{\rho_{0,I}^{(j)}}=\frac{\gamma}{\gamma-1}\frac{p_{*}}{\rho_{0,I}^{(j)}}.$
Now substituting (4.8), (4.10), and $E_{0,E}^{(i)}=p_{0,E}^{(i)}/(\gamma-1)$
into (4.9), we obtain
$p_{0,E}^{(i)}=p_{*}-\Delta t\gamma
p_{*}\sum^{i-1}_{j=1}\tilde{a}_{ij}\nabla\cdot{\bf u}_{0,I}^{(j)}=p_{*},$
and $E_{0,E}^{(i)}=p_{*}/(\gamma-1)$ is also a constant for the stage $i$.
From (3.7), we have
(4.11) $\left\\{\begin{aligned} &\mathbf{A}^{(i)}_{E}=\mathbf{A}^{n}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\left(\left(\nabla\times{\bf
A}^{(j)}_{E}\right)\times\mathbf{u}^{(j)}_{E}\right),\\\ &{\bf
B}_{E}^{(i)}=\nabla\times{\bf A}_{E}^{(i)}.\end{aligned}\right.$
From (LABEL:416), we get $\nabla\cdot{\bf
B}_{E}^{(i)}=\nabla\cdot\left(\nabla\times{\bf A}_{E}^{(i)}\right)=0$. From
(3.7), in the order of $\mathcal{O}(1)$, the density and momentum equations
yield
(4.12) $\left\\{\begin{aligned} &\rho_{0,E}^{(i)}=\rho_{inc}^{0}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\nabla\cdot\mathbf{q}_{0,E}^{(j)},\\\
&\mathbf{q}^{(i)}_{0,E}=\mathbf{q}^{0}_{inc}-\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\left(\nabla\cdot\left(\rho_{0,E}^{(j)}\mathbf{u}_{0,E}^{(j)}\otimes\mathbf{u}_{0,E}^{(j)}-\mathbf{B}^{(j)}_{0,E}\otimes\mathbf{B}^{(j)}_{0,E}+\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{0,E}\|}^{2}\right)\mathbf{I}\right)+\nabla
p^{(j)}_{2,I}\right),\end{aligned}\right.$
with $\mathbf{q}^{0}_{inc}=(\rho{\bf u})^{0}_{inc}$ and $\nabla
p_{0,E}^{(j)}=0$ for $j=1,\cdots,i-1$.
Similarly, inserting expansions (4.7) into (3.8), in the order of
$\mathcal{O}(1)$, we get
(4.13) $\left\\{\begin{aligned} &\rho_{0,*}^{(i)}=\rho_{inc}^{0}-\Delta
t\sum^{i-1}_{j=1}{a}_{ij}\nabla\cdot\mathbf{q}_{0,E}^{(j)},\\\
&\mathbf{q}^{(i)}_{0,*}=\mathbf{q}^{0}_{inc}-\Delta
t\sum^{i-1}_{j=1}{a}_{ij}\left(\nabla\cdot\left(\rho_{0,E}^{(j)}\mathbf{u}_{0,E}^{(j)}\otimes\mathbf{u}_{0,E}^{(j)}-\mathbf{B}^{(j)}_{0,E}\otimes\mathbf{B}^{(j)}_{0,E}+\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{0,E}\|}^{2}\right)\mathbf{I}\right)+\nabla
p^{(j)}_{2,I}\right),\end{aligned}\right.$
where from (3.17) and (4.8), it follows $\nabla\bar{p}^{(j)}_{E}=0$ for
$j=1,\cdots,i-1$. Using (4.10), we get
(4.14) $E_{0,*}^{(i)}=E_{inc}^{0}-\Delta
t\sum^{i-1}_{j=1}{a}_{ij}\nabla\cdot\left(\bar{H}_{0}^{(j)}\mathbf{q}_{0,I}^{(j)}\right)=\frac{p_{*}}{\gamma-1}-\frac{\gamma
p_{*}}{\gamma-1}\Delta t\sum^{i-1}_{j=1}{a}_{ij}\nabla\cdot{\bf
u}_{0,I}^{(j)}.$
Therefore, from (4.8) and (4.14), we get
$E_{0,*}^{(i)}=\frac{p_{*}}{\gamma-1}.$
Now from (4.14) and (3.9), it follows for the energy equation
$E^{(i)}_{0,I}=E^{(i)}_{0,*}-\Delta
ta_{ii}\nabla\cdot(\bar{H}^{(i)}_{0}\mathbf{q}^{(i)}_{0,I})=\frac{p_{*}}{\gamma-1}-\frac{\gamma
p_{*}}{\gamma-1}\Delta t{a}_{ii}\nabla\cdot{\bf u}_{0,I}^{(i)}$
Considering the EOS
$E_{I}^{(i)}=p_{I}^{(i)}/(\gamma-1)+\varepsilon^{2}({\|\mathbf{q}_{E}^{(i)}\|}^{2}/(2\rho_{E}^{(i)})+{\|\mathbf{B}_{E}^{(i)}\|}^{2}/2)$,
to zeroth order in $\varepsilon$, we get
$E_{0,I}^{(i)}=p_{0,I}^{(i)}/(\gamma-1)$, and we obtain for the pressure
$p^{(i)}_{0,I}=p_{*}-\gamma p_{*}\Delta t{a}_{ii}\nabla\cdot{\bf
u}_{0,I}^{(i)}.$
Integrating it over the spatial bounded domain $\Omega$, and assuming periodic
or compact support boundary conditions, we first obtain $p^{(i)}_{0,I}=p_{*}$,
and then we get $\nabla\cdot{\bf u}_{0,I}^{(i)}=0$ at the stage $i$.
Finally, from (3.9), considering (4.13), we get for the density and momentum:
(4.15) $\left\\{\begin{aligned} &\rho_{0,I}^{(i)}=\rho_{inc}^{0}-\Delta
t\sum^{i}_{j=1}{a}_{ij}\nabla\cdot\mathbf{q}_{0,E}^{(j)},\\\
&\mathbf{q}^{(i)}_{0,I}=\mathbf{q}^{0}_{inc}-\Delta
t\sum^{i}_{j=1}{a}_{ij}\left(\nabla\cdot\left(\rho_{0,E}^{(j)}\mathbf{u}_{0,E}^{(j)}\otimes\mathbf{u}_{0,E}^{(j)}-\mathbf{B}^{(j)}_{0,E}\otimes\mathbf{B}^{(j)}_{0,E}+\left(\frac{1}{2}{\|\mathbf{B}^{(j)}_{0,E}\|}^{2}\right)\mathbf{I}\right)+\nabla
p^{(j)}_{2,I}\right).\end{aligned}\right.$
Then equations (LABEL:416), (4.12), and (4.15), with $\nabla\cdot{\bf
B}_{E}^{(i)}=0$, $p_{0,E}^{(i)}=p_{0,I}^{(i)}=p_{*}$, and the divergence-free
leading order velocity, i.e. $\nabla\cdot{\bf u}_{0,I}^{(i)}=0$, provide a
multi-stage SI IMEX-RK discretization of the limiting system (LABEL:S2_E26)
for the internal stage $i$.
* •
AA property for the updated numerical solution. Assuming that SI IMEX-RK
scheme in Section 3.2 is SA, then the numerical solution coincides with the
last internal stage $s$, and by setting $i=s$, we get
(4.16) $p_{0}^{1}=p_{0,I}^{(s)}=p_{*},\quad\nabla\cdot{\bf
u}_{0}^{1}=\nabla\cdot{\bf u}_{0,I}^{(s)}=0,$
and for the magnetic field
(4.17) $\nabla\cdot{\bf B}_{E}^{1}=\nabla\cdot(\nabla\times{\bf
A}_{E}^{(1)})=0,$
namely we have (4.5). Now if we denote
${\bf V}_{inc}({\bf x};t)=(\rho_{inc}({\bf x};t),\rho_{inc}({\bf x};t){\bf
u}_{inc}({\bf x};t),p_{inc}({\bf x};t),{\bf A}_{inc}({\bf x};t),{\bf
B}_{inc}({\bf x};t))^{T}$
as the exact solution of the incompressible MHD equations (LABEL:S2_E26), with
initial data ${\bf V}_{inc}({\bf x};0)=(\rho^{0}({\bf x}),\rho^{0}({\bf
x}){\bf u}^{0}({\bf x}),p^{0},{\bf A}^{0}({\bf x}),{\bf B}^{0}({\bf x}))^{T}$,
from equations (LABEL:416), (4.12), (4.15), (4.16), and (4.17), one gets in
the low sonic Mach limit where $\varepsilon=0$, a high order SI IMEX-RK scheme
of order $r$ for the numerical solutions of (LABEL:S2_E26), that is, the
scheme (3.7)-(3.10) is AA, and (4.6) holds.
∎
## 5\. Numerical tests
In this section, we perform some numerical experiments to validate the high
order accuracy, AP, AA, divergence-free, and good performances of our proposed
scheme for the MHD system with all sonic Mach numbers. The third order SI
IMEX-RK with an SA property from [8, 30] is used for time discretizations. The
fifth order finite difference WENO reconstruction[8, 30, 53, 31] for the first
order spatial derivatives, and a fourth order compact central difference
discretization [8, 10] for the second order derivatives will be applied for
spatial discretizations. Our scheme is referred as the “IMEX” scheme. For some
of the following examples, we will compare our results to reference solutions,
which are produced by a fifth-order finite difference WENO scheme with a
third-order strong stability preserving RK method. It is referred as an “ERK”
scheme if without a CT method [32], while as an “ERKC” scheme if with a CT
method [18]. To avoid excessive numerical dissipations for low sonic Mach
numbers, for the explicit schemes “ERK” and “ERKC”, here we also take a Lax-
Friedrichs flux the same as in (3.11) and (3.12), and numerically we find that
it leads to much better results as we can see in the following. Unless
otherwise specified, the ${\rm CFL}$ number is taken as $0.25$, and the gas
constant is $\gamma=5/3$. For 1D problems, the time step is
$\Delta t=\frac{{\rm CFL}\Delta x}{\mathop{\max}\limits_{0\leq k\leq
N}\left(|u_{k}|+\hat{c}_{f}^{(k)}\right)},$
with $N+1$ computational grid points, and the definition of
$\hat{c}_{f}^{(k)}$ has been given in (3.12). For 2D problems, the time step
is
$\Delta t={\rm CFL}/\left(\frac{\mathop{\max}\limits_{0\leq k\leq N_{x},0\leq
l\leq N_{y}}\left(|u_{k,l}|+\hat{c}_{f,x}^{(k,l)}\right)}{\Delta
x}+\frac{\mathop{\max}\limits_{0\leq k\leq N_{x},0\leq l\leq
N_{y}}\left(|v_{k,l}|+\hat{c}_{f,y}^{(k,l)}\right)}{\Delta y}\right),$
with $(N_{x}+1)\times(N_{y}+1)$ computational grid points. Here, the fast
speeds in the $x$-direciton ($\hat{c}_{f,x}^{(k,l)}$) and in the $y$-direciton
($\hat{c}_{f,y}^{(k,l)}$) are defined as
$\left\\{\begin{aligned}
\hat{c}_{f,x}^{(k,l)}={\left\\{\frac{1}{2}\left[\left(\hat{a}^{(k,l)}\right)^{2}+\frac{{\|{\bf
B}_{k,l}\|}^{2}}{\rho_{k,l}}\pm\sqrt{{\left(\left(\hat{a}^{({k,l})}\right)^{2}+\frac{{\|{\bf
B}_{k,l}\|}^{2}}{\rho_{k,l}}\right)}^{2}-\left(2{\hat{a}^{({k,l})}}\sqrt{\frac{B_{x;{k,l}}^{2}}{\rho_{k,l}}}\right)^{2}}\right]\right\\}}^{\frac{1}{2}},\\\
\hat{c}_{f,y}^{(k,l)}={\left\\{\frac{1}{2}\left[\left(\hat{a}^{(k,l)}\right)^{2}+\frac{{\|{\bf
B}_{k,l}\|}^{2}}{\rho_{k,l}}\pm\sqrt{{\left(\left(\hat{a}^{({k,l})}\right)^{2}+\frac{{\|{\bf
B}_{k,l}\|}^{2}}{\rho_{k,l}}\right)}^{2}-\left(2{\hat{a}^{({k,l})}}\sqrt{\frac{B_{y;{k,l}}^{2}}{\rho_{k,l}}}\right)^{2}}\right]\right\\}}^{\frac{1}{2}},\end{aligned}\right.$
where
$\hat{a}^{({k,l})}=\min\left(\frac{1}{\varepsilon},1\right)\sqrt{\frac{\gamma
p_{k,l}}{\rho_{k,l}}}.$
### 5.1. One dimensional case
Since the divergence-free condition is satisfied automatically in 1D,
therefore, the CT method will not need to be applied in the following 1D
numerical simulations.
###### Example 5.1.
(Accuracy test) To assess the convergence of our proposed scheme, we simulate
the nonlinear circularly polarized Alfvén wave problem in 1D. The smooth
initial conditions are given as [16]:
(5.1) $\displaystyle(\rho,u,v,w,B_{x},B_{y},B_{z},p)(0,x)$
$\displaystyle\quad=(1,0,0.1\sin(2\pi x),0.1\cos(2\pi x),1,0.1\sin(2\pi
x),0.1\cos(2\pi x),0.1)\quad x\in[0,1],$
with a periodic boundary condition. The exact solution to (5.1) propagates
with a unit Alfvén speed (that is $U(t,x)=U(0,x+t)$.). We set the time step to
be
$\Delta t=\frac{{\rm CFL}\Delta x^{\frac{5}{3}}}{\mathop{\max}\limits_{0\leq
k\leq N}\left(|u_{k}|+\hat{c}_{f}^{(k)}\right)}.$
The sonic Mach number is $\varepsilon=1$ and we compute up to a final time
$T=1$. The numerical $L_{1}$, $L_{2}$, and $L_{\infty}$ errors and orders are
shown in Table 5.1. We can observe that a fifth-order spatial accuracy is
obtained.
Table 5.1. Example 5.1. The $L_{1}$, $L_{2}$, and $L_{\infty}$ errors and orders for $\rho v$ with $\varepsilon=1$ at $T=1$. $N$ | 10 | 20 | 40 | 80 | 160
---|---|---|---|---|---
$L_{1}$ error | 6.24E-04 | 2.03E-05 | 6.41E-07 | 2.01E-08 | 6.27E-10
order | – | 4.94 | 4.99 | 5.00 | 5.00
$L_{2}$ error | 6.83E-04 | 2.25E-05 | 7.11E-07 | 2.23E-08 | 6.97E-10
order | – | 4.93 | 4.98 | 5.00 | 5.00
$L_{\infty}$ error | 9.65E-04 | 3.15E-05 | 1.00E-06 | 3.15E-08 | 9.85E-10
order | – | 4.94 | 4.97 | 4.99 | 5.00
###### Example 5.2.
(Accuracy test for a range of $\varepsilon$) We take a well-prepared initial
data which is $\varepsilon-$dependent:
$\left\\{\begin{aligned} \rho(x,0)&=1+\varepsilon^{2}\sin^{2}(2\pi
x),\quad&p(x,0)&=(1+\varepsilon^{2}\sin^{2}(2\pi x))^{\gamma},\\\
u(x,0)&=\varepsilon^{2}\sin(2\pi x),\quad&B_{x}(x,0)&=0.5,\\\
v(x,0)&=\sin(2\pi x)+\varepsilon^{2}\cos(2\pi
x),\quad&B_{y}(x,0)&=(1+\varepsilon^{2})\sin(2\pi x),\\\
w(x,0)&=0,\quad&B_{z}(x,0)&=(1+\varepsilon^{2})\cos(2\pi x),\\\
\end{aligned}\right.$
on the domain $\Omega=[0,1]$ with $\gamma=1.4$ and periodic boundary
conditions. Reference solutions are computed with $N=320$. Four different
sonic Mach numbers $\varepsilon=1,10^{-2},10^{-6}$, and $0$ are taken, with a
final time $T=0.05$. Numerical errors and orders are shown in Table 5.2. For
the intermediate value of $\varepsilon=10^{-2}$, order reduction is observed,
which is a typical behavior of high-order IMEX schemes for these multi-scale
problems [8]. Around 5th-order accuracy for $\varepsilon=1,10^{-6}$, and $0$
is observed.
Table 5.2. Example 5.2. The $L_{1}$ errors and orders for $\rho v$ with $\varepsilon=1,10^{-2},10^{-6}$, and $0$. $N$ $\varepsilon$ | $\varepsilon=1$ | $\varepsilon=10^{-2}$ | $\varepsilon=10^{-6}$ | $\varepsilon=0$
---|---|---|---|---
$L_{1}$ error | order | $L_{1}$ error | order | $L_{1}$ error | order | $L_{1}$ error | order
10 | 3.08E-02 | – | 4.93E-04 | – | 4.82E-04 | – | 4.82E-04 | –
20 | 6.47E-03 | 2.25 | 7.18E-05 | 2.78 | 1.61E-05 | 4.90 | 1.61E-05 | 4.90
40 | 8.32E-04 | 2.96 | 3.33E-05 | 1.11 | 5.22E-07 | 4.95 | 5.22E-07 | 4.95
80 | 3.74E-05 | 4.48 | 1.01E-05 | 1.72 | 1.76E-08 | 4.89 | 1.76E-08 | 4.89
160 | 1.22E-06 | 4.94 | 3.34E-06 | 1.60 | 6.65E-10 | 4.72 | 6.65E-10 | 4.72
###### Example 5.3.
(MHD shock tube) In this example, we consider a 1D MHD shock tube in the
compressible regime where the sonic Mach number is of $\mathcal{O}(1)$. The
initial conditions are as follows
$\displaystyle(\rho,u,v,w,B_{x},B_{y},B_{z},p)(0,x)$
$\displaystyle\quad=\left\\{\begin{aligned}
&(1,0,0,0,0.75,1,0,1)\quad&x&\in[0,0.5),\\\
&(0.125,0,0,0,0.75,-1,0,0.1)\quad&x&\in[0.5,1],\end{aligned}\right.$
with a final time $T=0.1$. The specific heat ratio is set to be $\gamma=2$
with $\varepsilon=1$ on the domain $\Omega=[0,1]$. Reflective boundary
conditions are considered, and we take $N=200$. The results are shown in Fig.
5.1, and the solutions obtained by the ERK scheme with $N=2400$ are displayed
as references. From these results, it can be seen that our IMEX scheme can
capture the MHD waves very well.
Figure 5.1. MHD shock tube solutions for Example 5.3. Left: density $\rho$;
Right: magnetic field $B_{y}$. Every two points are plotted for the IMEX
scheme.
### 5.2. Two dimensional case
A CT method described in [18] will be used to maintain the divergence-free
condition. In 2D, ${\bf B}=\nabla\times{\bf A}$ can be written as
$B_{x}=\partial A_{z}/\partial y$ and $B_{y}=-\partial A_{z}/\partial x$.
Therefore, we only need to update the third component $A_{z}$ of ${\bf A}$,
and ignore $A_{x}$ and $A_{y}$. All solutions of the following tests maintain
$\nabla\cdot{\bf B}=0$ up to machine round off errors[16].
###### Example 5.4.
(Accuracy test) The 2D version of the smooth Alfvén wave problem is obtained
by rotating the direction of propagation by an angle of $\theta$, so that the
wave now propagates along the direction ${\bf
n}=(-\cos{\theta},-\sin{\theta})$ on the domain
$\Omega=[0,1/\cos{\theta}]\times[0,1/\sin{\theta}]$ [16]. In this numerical
test, periodic boundary conditions are applied on all both directions with
$\varepsilon=1,\theta=\pi/4$, and $T=1$. Similarly, if we take
$\Delta t={\rm CFL}/\left(\frac{\mathop{\max}\limits_{0\leq k\leq N_{x},0\leq
l\leq N_{y}}\left(|u_{k,l}|+\hat{c}_{f,x}^{(k,l)}\right)}{\Delta
x^{\frac{5}{3}}}+\frac{\mathop{\max}\limits_{0\leq k\leq N_{x},0\leq l\leq
N_{y}}\left(|v_{k,l}|+\hat{c}_{f,y}^{(k,l)}\right)}{\Delta
y^{\frac{5}{3}}}\right),$
from Table 5.3 we can find around 5th-order accuracy for the momentum $\rho
u$, and 4th-order accuracy for the magnetic field component $B_{x}$ where a
fourth-order central difference discretization is used for ${\bf
B}=\nabla\times{\bf A}$.
Table 5.3. Example 5.4. The $L_{1}$, $L_{2}$, and $L_{\infty}$ errors and orders for $\rho u$ and $B_{x}$ with $\varepsilon=1$ and $T=1$. | $N_{x}\times N_{y}$ | $8\times 8$ | $16\times 16$ | $32\times 32$ | $64\times 64$ | $128\times 128$
---|---|---|---|---|---|---
$\rho u$ | $L_{1}$ error | 1.54E-03 | 5.73E-05 | 1.86E-06 | 5.88E-08 | 1.84E-09
order | – | 4.75 | 4.94 | 4.99 | 5.00
$L_{2}$ error | 1.80E-03 | 6.47E-05 | 2.08E-06 | 6.55E-08 | 2.07E-09
order | – | 4.80 | 4.96 | 4.99 | 4.99
$L_{\infty}$ error | 2.55E-03 | 9.26E-05 | 2.98E-06 | 9.44E-08 | 3.00E-09
order | – | 4.79 | 4.96 | 4.98 | 4.98
$B_{x}$ | $L_{1}$ error | 9.29E-04 | 6.36E-05 | 3.17E-06 | 1.69E-07 | 9.64E-09
order | – | 3.87 | 4.33 | 4.23 | 4.13
$L_{2}$ error | 1.08E-03 | 7.02E-05 | 3.51E-06 | 1.88E-07 | 1.07E-08
order | – | 3.94 | 4.32 | 4.23 | 4.13
$L_{\infty}$ error | 1.29E-03 | 9.69E-05 | 5.06E-06 | 2.70E-07 | 1.52E-08
order | – | 3.73 | 4.26 | 4.23 | 4.15
###### Example 5.5.
(Accuracy test for a range of $\varepsilon$) In this 2D case, we set a well-
prepared initial condition as follows
$\left\\{\begin{aligned}
\rho(x,y,0)&=1+\varepsilon^{2}\sin^{2}(2\pi(x+y)),\quad&p(x,y,0)&=(1+\varepsilon^{2}\sin^{2}(2\pi(x+y)))^{\gamma},\\\
u(x,y,0)&=\sin(2\pi(x-y))+\varepsilon^{2}\sin(2\pi(x+y)),\quad&B_{x}(x,y,0)&=-\frac{1}{\sqrt{2}}\sin(2\pi(x+y)),\\\
v(x,y,0)&=\sin(2\pi(x-y))+\varepsilon^{2}\cos(2\pi(x+y)),\quad&B_{y}(x,y,0)&=\frac{1}{\sqrt{2}}\sin(2\pi(x+y)),\\\
w(x,y,0)&=0,\quad&B_{z}(x,y,0)&=\cos(2\pi(x+y)),\\\ \end{aligned}\right.$
with the initial magnetic potential
$A_{z}(x,y,0)=\cos(2\pi(x+y))/(2\sqrt{2}\pi)$. As in Example 5.2, four
different sonic Mach numbers are considered with periodic boundary conditions
on the domain $\Omega=[0,1]\times[0,1]$. We compute the solution up to a final
time $T=0.01$ on mesh grid points of $N^{2}$. The $L_{1}$ errors and orders of
the accuracy are shown in Table 5.4. From this Table, we can see high-order
accuracy can be obtained for $\varepsilon=1,10^{-6}$, and $0$. However, under
the current mesh sizes, order degeneracy can also be found for
$\varepsilon=10^{-2}$, which is similar to the 1D case.
Table 5.4. Example 5.5. The $L_{1}$ errors and orders for $\rho u$ with $\varepsilon=1,10^{-2},10^{-6}$, and $0$. $N$ $\varepsilon$ | $\varepsilon=1$ | $\varepsilon=10^{-2}$ | $\varepsilon=10^{-6}$ | $\varepsilon=0$
---|---|---|---|---
$L_{1}$ error | order | $L_{1}$ error | order | $L_{1}$ error | order | $L_{1}$ error | order
8 | 8.98E-02 | – | 5.18E-03 | – | 7.65E-03 | – | 7.65E-03 | –
16 | 1.39E-02 | 2.69 | 2.09E-03 | 1.31 | 8.74E-04 | 3.13 | 8.74E-04 | 3.13
32 | 6.76E-04 | 4.36 | 2.09E-03 | – | 1.68E-05 | 5.70 | 1.68E-05 | 5.70
64 | 2.28E-05 | 4.89 | 2.41E-03 | – | 7.42E-07 | 4.50 | 7.43E-07 | 4.50
128 | 7.15E-07 | 5.00 | 6.41E-04 | 1.91 | 1.42E-08 | 5.70 | 1.40E-08 | 5.73
###### Example 5.6.
(Orszag-Tang vortex) Next we consider the Orszag–Tang vortex problem, which is
widely considered as a standard test for MHD [18, 16, 56, 3]. The problem has
smooth initial conditions
$\displaystyle(\rho,u,v,w,B_{x},B_{y},B_{z},p)(x,y,0)$ $\displaystyle=$
$\displaystyle(\gamma^{2},-\sin{(y)},\sin{(x)},0,-\sin{(y)},\sin{(2x)},0,\gamma),$
with the initial magnetic potential: $A_{z}=0.5\cos{(2x)}+\cos{(y)}$ on the
domain $\Omega=[0,2\pi]\times[0,2\pi]$. Periodic boundary conditions are
imposed on all boundaries. As time evolves, the solution forms several shock
waves and a vortex structure in the middle of the computational domain. We set
$\varepsilon=1$, $N_{x}=N_{y}=192$ and present the density at $T=0.5,T=2,T=3$,
and $T=4$ in Fig. 5.2. A slice of the pressure at $y=0.625\pi$ and $T=3$ is
shown on the right panel of Fig. 5.3. We find our IMEX scheme can successfully
capture the shocks. We do not observe significant oscillations in any of the
conserved quantities, and our results are in good agreement with those given
in [18, 16].
Figure 5.2. Density for Orszag-Tang problem of Example 5.6 on the mesh grid
$192^{2}$.
Figure 5.3. Example 5.6. Thermal Pressure for Orszag-Tang problem on the mesh
grid $192^{2}$. Every two points are plotted for the IMEX scheme for the
cutting plot.
###### Example 5.7.
(MHD blast wave) MHD blast wave has been commonly used to test numerical
methods for the MHD system [15, 28]. Here, we set periodic boundary conditions
on the domain $\Omega=[-0.5,0.5]\times[-0.5,0.5]$. The initial conditions are
$\displaystyle(\rho,u,v,w,B_{x},B_{y},B_{z},p)(x,y,0)$
$\displaystyle\quad=\left\\{\begin{aligned}
&(1,0,0,0,10\sin{\theta},10\cos{\theta},0,100)\quad&r&\leq 0.125,\\\
&(1,0,0,0,10\sin{\theta},10\cos{\theta},0,10)\quad&r&>0.125,\end{aligned}\right.$
with $\theta=\pi/4$ and $r=\sqrt{x^{2}+y^{2}}$. The initial magnetic potential
is given by $A_{z}(x,y,0)=5\sqrt{2}(-x+y)$ with a first order extrapolation if
outside the boundary. In Fig. 5.4, we show the density and magnetic pressure
obtained by the IMEX and ERKC schemes on a $200\times 200$ mesh with
$\varepsilon=0.9$ at $T=0.02$. The slices of the density and the magnetic
pressure along $y=-x+0.5$ are also shown in Fig. 5.5. We see that the solution
of the IMEX scheme is close to the reference ERKC solutions and no obvious
oscillations are observed. In our test, the maximum error of discrete
divergence of the magnetic field is $2.87\times 10^{-11}$ for the IMEX scheme
and $2.77\times 10^{-11}$ for the ERKC scheme, both are close to machine
precision.
Figure 5.4. Example 5.7. MHD blast wave problem on the mesh grid $200^{2}$.
Top: IMEX; Bottom: ERKC. 35 equally spaced contours are used.
Figure 5.5. Example 5.7 MHD blast wave problem. 1D cutting plots along
$y=-x+0.5$. Every two points are plotted for the IMEX scheme for the cutting
plots.
###### Example 5.8.
(Field loop advection) Now, we show the 2D field loop advection test [45, 49]
to assess the capability of capturing a tangential discontinuity for a
multidimensional flow. The computational domain is
$\Omega=[-1,1]\times[-0.5,0.5]$ with periodic boundary conditions. The domain
is divided with $256\times 128$ uniform cells. The initial conditions are set
as
$\displaystyle(\rho,u,v,w,B_{x},B_{y},B_{z},p)(r,0)$
$\displaystyle\quad=(1,v_{0}\cos{(\theta_{0})},v_{0}\sin{(\theta_{0})},0,\partial
A_{z}/\partial y,-\partial A_{z}/\partial x,0,1)$
with
$v_{0}=1,\cos{(\theta_{0})}=-2/\sqrt{5},\sin{(\theta_{0})}=1/\sqrt{5},r=\sqrt{x^{2}+y^{2}}$,
and
$\displaystyle A_{z}(r)=\left\\{\begin{aligned} &10^{-3}(0.3-r)\quad&r\leq
0.3,&\\\ &0\quad&otherwise.&\end{aligned}\right.$
In this test, a loop of the magnetic field is advected. In Fig. 5.6, we show
both the deviation of the density $\rho$ away from 1 and the magnetic pressure
$\|{\bf B}\|^{2}/2$ for the case of $\varepsilon=0.1$ at $T=\sqrt{5}$. For the
magnetic pressure, both schemes preserve the initial symmetric loop structure
very well, and the IMEX scheme has almost the same results as the ERKC scheme.
The one-dimensional cutting plots for the magnetic pressure along $x=0.2$ show
that both schemes slightly distort the profile at $y=0.15$. For the deviation
of the density, we can find the IMEX scheme is better than the ERKC scheme for
preserving a symmetric structure, as compared with the results of the ERKC
scheme on a uniform mesh of $384\times 192$ cells. For this problem in the
intermediate regime with $\varepsilon=0.1$, we also compare the CPU cost of
two schemes in Table 5.5. We can observe that the ERKC scheme costs less when
$\varepsilon$ is of $\mathcal{O}(1)$. However, the IMEX scheme is much more
efficient in the intermediate and low sonic Mach regimes. The computational
cost for the ERKC scheme increases very rapidly, as the sonic Mach number
$\varepsilon$ becomes smaller. This example demonstrates the main advantage of
an AP scheme for such multiscale problems.
Figure 5.6. Example 5.8. Field loop advection on the mesh grid $256\times 128$. Top left: magnetic pressure (IMEX); Middle left: magnetic pressure (ERKC); Bottom left: 1D cutting plot for the magnetic pressure; Top right: deviation of density (IMEX); Middle right: deviation of density (ERKC); Bottom right: 1D cutting plot for the deviation of density. Table 5.5. Example 5.8. The CPU cost (seconds) for the IMEX and ERKC schemes on the mesh grid $256\times 128$ at $T=\sqrt{5}$. $\varepsilon$ | 0.5 | 0.1 | 0.05
---|---|---|---
IMEX | 7634.880 | 5687.467 | 8015.959
ERKC | 4736.466 | 19719.964 | 38666.192
###### Example 5.9.
(Magnetized Kelvin–Helmholtz instability) Finally, we consider a simulation of
the magnetized Kelvin–Helmholtz instability problem[8, 42]. The initial
conditions are
$\displaystyle(\rho,u,v,w,B_{x},B_{y},B_{z},p)(x,y,0)$
$\displaystyle\quad=(\gamma,1-2\eta(x),0.1\sin{(2\pi x)},0,0.1,0,0,1),$
with a magnetic potential $A_{z}(x,y,0)=0.1y$ on the domain
$\Omega=[0,2]\times[-0.5,0.5]$, and
$\displaystyle\eta(x)=\left\\{\begin{aligned}
&\frac{1}{2}\left(1+\sin\left(16\pi\left(y+\frac{1}{4}\right)\right)\right)\quad&y\in[-\frac{9}{32},-\frac{7}{32}],&\\\
&1\quad&y\in[-\frac{7}{32},\enspace\;\frac{7}{32}],&\\\
&\frac{1}{2}\left(1-\sin\left(16\pi\left(y-\frac{1}{4}\right)\right)\right)\quad&y\in[\enspace\;\frac{7}{32},\enspace\;\frac{9}{32}],&\\\
&0\quad&otherwise.&\end{aligned}\right.$
Periodic boundary conditions are used for the conserved quantities, while a
first-order extrapolation is utilized when the magnetic potential is outside
of the computational domain. We run the solution up to $T=0.8$ with
$\gamma=1.4$ and $\varepsilon=10^{-6}$ on a mesh grid $N_{x}\times
N_{y}=256\times 128$. We define the local sonic Mach number as
$M_{a}=\sqrt{u^{2}+v^{2}}/\sqrt{\gamma p/\rho}$, and the ratio of the local
sonic Mach number $M_{ratio}$ to the maximum value of $M_{a}$ as
$M_{ratio}=M_{a}/\max(M_{a})$, where the maximum is taken over all
computational grid points. The vorticity $\omega=v_{x}-u_{y}$, where $v_{x}$
and $u_{y}$ are discretized by the 4th order central difference, and
$M_{ratio}$ are shown in Fig. 5.7. We observe that they are comparable to the
results in [42, 8], namely, our IMEX scheme can capture the incompressible MHD
system in the low sonic Mach number regime very well, which verifies the AP
property of our scheme.
Figure 5.7. Magnetized Kelvin–Helmholtz instability problem for Example 5.9
on the mesh grid $256\times 128$. Left: Vorticity $\omega=v_{x}-u_{y}$; Right:
The ratio of the local sonic Mach number $M_{ratio}$.
## 6\. Conclusion
In this paper, a high-order SI AP scheme with a discrete divergence-free
property for the MHD equations has been developed. High-order accuracy in time
has been obtained by SI temporal integrator based on an IMEX-RK framework.
High-order accuracy in space has been achieved by finite difference WENO
reconstructions. We have formally proved that the scheme is AP. We have also
proved the AA property in the stiff limit as the sonic Mach number
$\varepsilon\rightarrow 0$, if the SI IMEX-RK scheme is SA. Numerical
experiments in 1D and 2D have demonstrated the divergence-free for all ranges
of sonic Mach numbers, AP and AA properties in the low sonic Mach limit.
Compared to the explicit schemes as reference solutions, such as the ERK [32]
and ERKC [18] schemes in 1D and 2D respectively, our IMEX scheme has almost
the same performances as the ERK or ERKC scheme, but is much more efficient in
the intermediate and low sonic Mach regimes.
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# Some Comments on Unitary Qubit Lattice Algorithms for Classical Problems
Paul Anderson 1 Lillian Finegold-Sachs 1 George Vahala 1 Linda Vahala 2 Abhay
K. Ram 3 Min Soe 4 Efstratios Koukoutsis 5 Kyriakos Hizandis 51
1 Department of Physics William & Mary Williamsburg VA23185
2 Department of Electrical & Computer Engineering Old Dominion University
Norfolk VA
23529
3 Plasma Science and Fusion Center MIT Cambridge MA 02139
4 Department of Mathematics and Physical Sciences Rogers State University
Claremore
OK 74017
5 School of Electrical and Computer Engineering National Technical University
of Athens
Zographou 15780 Greece
###### Abstract
A qubit lattice algorithm (QLA), which consists of a set of interleaved
unitary collision-streaming operators, is developed for electromagnetic wave
propagation in tensor dielectric media. External potential operators are
required to handle gradients in the refractive indices, and these operators
are typically non-unitary. A similar problem arises in the QLA for the
Korteweg-de Vries equation, as the potential operator that models the KdV
nonlinear term is also non-unitary. Several QLAs are presented here that avoid
the need of this non-unitary potential operator by perturbing the collision
operator. These QLAs are fully unitary.
## 1 Introduction
We have been investigating qubit lattice algorithms (QLA) for some time
[1-21]. The aim of QLA is to develop a unitary interleaved sequence of
collision-streaming operators which in the continuum limit reduces
perturbatively to the desired differential equations describing the system of
interest. The first step is to associate a basis set of qubits for the
lattice, which on taking appropriate moments will recover the classical fields
of interest. Thus the QLA would be immediately encodable onto a quantum
computer. However, from our earlier nonlinear studies of 2D and 3D quantum
turbulence [8,9, 11], the QLA is ideally parallelized on classical
supercomputers with no degradation in parallel performance as the number of
cores are ramped up [e.g., to over 750 000 cores on the $IBM$ BlueGene $Mira$
supercomputer at Argonne]. Some care is needed in the choice of the qubit
basis. For example, it will be shown in Sec. 2 that the simple basis choice of
$(\mathbf{E,H})$ will never lead to a unitary representation of the Maxwell
equations of electrodynamics. [$\mathbf{E}$ is the electric field, and
$\mathbf{H}$ is the magnetic field].
At the heart of an efficient algorithm on a quantum computer is quantum
entanglement of the qubits. For example, consider a 2-qubit representation
[e.g., for a basis for a differential equation like the scalar nonlinear
Schrodinger (NLS) equation or the Korteweg de Vries (KdV) equation]. A basis
is the $2^{2}$ elements $(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$. Now
consider a unitary $2\times 2$ collision operator
$C=\left[\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right]$ (1)
acting on the qubit subspace of $(|01\rangle,|10\rangle)$. One of the post-
collision qubit elements is
$\cos\theta\,|01\rangle+\sin\theta\,|10\rangle.$ (2)
However, this post-collision state cannot be represented by a tensor product
of the $2^{2}$ \- basis, since the most general tensor product state is
$a_{0}b_{0}|00\rangle+a_{0}b_{1}|01\rangle+a_{1}b_{0}|10\rangle+a_{1}b_{1}|11\rangle$
(3)
where one qubit state is $a_{0}|0\rangle+a_{1}|1\rangle$ , and the other state
is $b_{0}|0\rangle+b_{1}|1\rangle$ , for some coefficients
$a_{0}\quad\ldots\quad b_{1}$. To recover the state Eq. (2) from the tenor
product state Eq. (3) one would have to eliminate the $|00\rangle$-term.
However one must then set either $a_{0}=0$ or $b_{0}=0$. But this would
eliminate either the state $|01\rangle$ or the state $|10\rangle$ \- both of
which are needed to recover Eq. (2). States which cannot be represented in a
tensor product basis of qubits are called entangled states. A maximally
entangled state is achieved on taking $\theta=\pi/4$, and is known as a Bell
state [22]
$B_{1}=\frac{|01\rangle+|10\rangle}{\sqrt{2}}$
Note that the quantum entanglement is achieved here by the unitary collision
operator. The streaming operator in QLA will then propagate this entanglement
throughtout the lattice.
In Sec. 2 we will develop a QLA for the solution of 2D Maxwell equations in a
tensor Hermitian dielectric medium. All our previous Maxwell QLA [18, 21] were
restricted to scalar dielectrics. We will present a simplified discussion of
the Dyson map [23] that will permit us to transform from a non-unitary to
unitary basis for the representation of the two curl equations of Maxwell.
From these qubit equations we will generate a QLA for tensor dielectric media
that is second order accurate. The QLA that we discuss here is not fully
unitary. While the collide-stream operator sequence is fully unitary, the
external potential operators required to recover the Maxwell equations are
not. However these non-unitary matrices are very sparse and could be amenable
to some unitary approximate representation. The role of the perturbation
parameter $\epsilon$ introduced in the QLA is quite subtle.
As to an understanding of the subtlety of $\epsilon$ in QLA we return to the
KdV equation in Sec. 3. Our original QLA for KdV [1] consisted of maximally
entanglement Bell collision operators together with a required external
potential operator to model the KdV nonlinearity. This external potential was
not unitary. Here we present a modified collision operator that is fully
unitary and which leads to a QLA-KdV that does not require any external
potential to be introduced.
Finally, in Sec. 4 we make some concluding remarks about future QLA
simulations that are needed to elucidate the perturbation parameter
$\epsilon$. This parameter is required in order to move the discrete QLA into
a continuum representation.
## 2 QLA for Maxwell Equations
### 2.1 Qubit-Electromagnetic field representation
Consider a simple dielectric non-magnetic medium with the constitutive
equations
$\mathbf{D}=\epsilon\mathbf{E},\quad\mathbf{B}=\mu_{0}\mathbf{H}.$ (4)
Treating $\mathbf{u}=(\mathbf{E},\mathbf{H})^{\mathbf{T}}$ as the fundamental
fields, and $\mathbf{d}=(\mathbf{D},\mathbf{B})^{\mathbf{T}}$ the derived
fields , Eq. (2) can be written in matrix form
$\mathbf{d}=\mathbf{Wu}$ (5)
where $\mathbf{W}$ is a Hermitian $6\times 6$ matrix
$\mathbf{W}=\left[\begin{array}[]{cc}\epsilon_{3\times 3}&0\\\
0&\mu_{0}\mathbf{I}_{3\times 3}\end{array}\right]$ (6)
with $\mathbf{I}_{3\times 3}$ the $3\times 3$ identity matrix. and
$\mathbf{T}$ is the transpose operator. The curl-curl (source-free) Maxwell
equations $\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$, and
$\nabla\times\mathbf{H}=\partial\mathbf{D}/\partial t$ in matrix form are just
$i\frac{\partial\mathbf{d}}{\partial t}=\mathbf{Mu}$ (7)
where, under standard boundary conditions, the curl-matrix operator
$\mathbf{M}$ is Hermitian
$\mathbf{M}=\left[\begin{array}[]{cc}0_{3\times 3}&i\nabla\times\\\
-i\nabla\times&0_{3\times 3}\end{array}\right].$ (8)
Since $\mathbf{W}$ is invertible, Eq. (5) can be written in terms of the basic
electromagnetic fields $\mathbf{u}=(\mathbf{E},\mathbf{H})$
$i\frac{\partial\mathbf{u}}{\partial
t}=\mathbf{W}^{-\mathbf{1}}\mathbf{M}\mathbf{u}$ (9)
In continuum applications, one typically treats the two Maxwell divergence
equations $\nabla\cdot\mathbf{B}=0$ and $\nabla\cdot\mathbf{D}=0$ as initial
conditions. From the curl-curl equations we see that they will then be
satisfied for all time.
#### 2.1.1 homogeneous dielectric medium
If one is dealing with a homogeneous dielectric medium (e.g., a vacuum), then
the constitutive matrix $\mathbf{W}$ is a constant and trivially commutes with
the curl-operator $\mathbf{M}$. As a result, the product of the two Hermitian
matrices, $\mathbf{W}^{-\mathbf{1}}\mathbf{M}$ is itself Hermitian, and Eq.
(7) gives a unitary evolution of the electromagnetic fields
$\mathbf{u}=(\mathbf{E},\mathbf{H})^{\mathbf{T}}$. Thus $\mathbf{u}$ is an
appropriate basis for the qubit fields and for quantum computation.
#### 2.1.2 inhomogeneous dielectric media
However, when the matrix $\mathbf{W}$ is spatiaily dependent, then
$\mathbf{W}^{-\mathbf{1}}\mathbf{M}\neq\mathbf{M}\mathbf{W}^{-\mathbf{1}}$ and
$\mathbf{W}^{-\mathbf{1}}\mathbf{M}$ is not Hermitian. Under these conditions,
the qubit representation of the electromagnetic fields
$\mathbf{u}=(\mathbf{E},\mathbf{H})^{\mathbf{T}}$ will not yield a unitary
evolution of these qubits. However Koukoutsis et. al. [23] have shown how to
determine the so-called Dyson map from the fields $\mathbf{u}$ to a new field
representation $\mathbf{U}$ such that the resultant representation in terms of
the new field $\mathbf{U}$ will result in a unitary evolution of these fields.
Indeed, it can be shown [23], that the Dyson map
$\mathrm{U}=\mathrm{W}^{1/2}\mathrm{u}$ (10)
will yield a unitary evolution equation for $\mathbf{U}$ with
$i\frac{\partial\mathbf{U}}{\partial
t}=\mathbf{W}^{-1/2}\mathbf{M}\mathbf{W}^{-1/2}\mathbf{U}$ (11)
as the matrix operator
$\mathbf{W}^{-\mathbf{1}/\mathbf{2}}\mathbf{M}\mathbf{W}^{-\mathbf{1}/\mathbf{2}}$
is Hermitian.
Thus one could start to build a QLA based on the electromagnetic fields
$\mathbf{U}=\left(\epsilon^{1/2}\mathbf{E},\mu_{0}^{1/2}\mathbf{H}\right)^{T}$
(12)
or under the rotation matrix
$\mathbf{L}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}I_{3\times
3}&iI_{3\times 3}\\\ I_{3\times 3}&-iI_{3\times 3}\end{array}\right]$ (13)
one could base a QLA on the field representation
$\mathbf{U}_{\mathbf{RSW}}=\mathbf{LU}$ where
$\mathbf{U}_{\mathbf{RSW}}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}\epsilon^{1/2}\mathbf{E}+i\mu_{0}^{1/2}\mathbf{H}\\\
\epsilon^{1/2}\mathbf{E}-i\mu_{0}^{1/2}\mathbf{H}\end{array}\right].$ (14)
This is nothing but the unitary evolution of the Riemann-Silberstein-Weber
(RSW) vector - a representation used to represent Maxwell equations from the
early 1920’s [24-26].
Moreover, the theory can be readily extended to diagonal tensor dielectric
media, with (assuming non-magnetic materials) the 6-qubit representation
$\mathbf{Q}$ of the field
$\mathbf{U}=\left(n_{x}E_{x},n_{y}E_{y},n_{z}E_{z},\mu_{0}^{1/2}\mathbf{H}\right)^{T}=\mathbf{Q}$
(15)
$(n_{x},n_{y},n_{z})$ is the vector (diagonal) refractive index, with
$\epsilon_{x}=n_{x}^{2}$ … . We work in Cartesian coordinates.
### 2.2 $\mathrm{2D}$ QLA for $\mathrm{x-y}$ dependent propagation of Maxwell
Equations
From Eqs. (9) and (13), Maxwell equations for 2D x-y spatially dependent
fields written in terms of the 6-$\mathbf{Q}$ vector components
$\displaystyle\frac{\partial q_{0}}{\partial t}=\frac{1}{n_{x}}\frac{\partial
q_{5}}{\partial y},\qquad\frac{\partial q_{1}}{\partial
t}=-\frac{1}{n_{y}}\frac{\partial q_{5}}{\partial x},\qquad\frac{\partial
q_{2}}{\partial t}=\frac{1}{n_{z}}\left[\frac{\partial q_{4}}{\partial
x}-\frac{\partial q_{3}}{\partial y}\right]$ (16) $\displaystyle\frac{\partial
q_{3}}{\partial t}=-\frac{\partial(q_{2}/n_{z})}{\partial
y},\qquad\frac{\partial q_{4}}{\partial
t}=\frac{\partial(q_{2}/n_{z})}{\partial x},\qquad\frac{\partial
q_{5}}{\partial t}=-\frac{\partial(q_{1}/n_{y})}{\partial
x}+\frac{\partial(q_{0}/n_{x})}{\partial y}$
This representation is unitary.
Our QLA representation focusses on recovering Eq. (14) perturbatively. One can
thus consider developing the representation dimension by dimension. In
particular we introduce the following unitary collision operator with
collision angles $\theta_{1}$ and $\theta_{2}$ (to be specified later):
$C_{X}=\left[\begin{array}[]{cccccc}1&0&0&0&0&0\\\
0&cos\,\theta_{1}&0&0&0&-sin\,\theta_{1}\\\
0&0&cos\,\theta_{2}&0&-sin\,\theta_{2}&0\\\ 0&0&0&1&0&0\\\
0&0&sin\,\theta_{2}&0&cos\,\theta_{2}&0\\\
0&sin\,\theta_{1}&0&0&0&cos\,\theta_{1}\end{array}\right]$ (17)
and the unitary collision operator
$C_{Y}=\left[\begin{array}[]{cccccc}cos\,\theta_{0}&0&0&0&0&sin\,\theta_{0}\\\
0&1&0&0&0&0\\\ 0&0&cos\,\theta_{2}&0&sin\,\theta_{2}&0\\\
0&0&-sin\,\theta_{2}&cos\,\theta_{2}&0&0\\\ 0&0&0&0&1&0\\\
-sin\,\theta_{0}&0&0&0&0&cos\,\theta_{0}\end{array}\right]$ (18)
with collision angles $\theta_{0}$ and $\theta_{2}$. The unitary streaming
operator $S^{+x}_{14}$ shifts qubits $q_{1}$ and $q_{4}$ one lattice unit in
the $+x$ direction, while leaving the remaining 4 qubits alone. We finally
need to introduce the external potential operators
$V_{X}=\left[\begin{array}[]{cccccc}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&-sin\,\beta_{2}&0&cos\,\beta_{2}&0\\\
0&sin\,\beta_{0}&0&0&0&cos\,\beta_{0}\end{array}\right]$ (19)
and
$V_{Y}=\left[\begin{array}[]{cccccc}1&0&0&0&0&o\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&\cos\,\beta_{3}&\sin\,\beta_{3}&0&0\\\ 0&0&0&0&1&0\\\
-sin\,\beta_{1}&0&0&0&0&cos\,\beta_{1}\end{array}\right]$ (20)
for particular angles $\beta_{0}$, $\beta_{1}$ and $\beta_{2}$. These
potential operators are not unitary, but very sparse.
We now consider the following unitary sequence of interleaved collision-
streaming operators:
$\mathbf{U_{X}}=S^{+x}_{25}.C_{X}^{\dagger}.S^{-x}_{25}.C_{X}.S^{-x}_{14}.C_{X}^{\dagger}.S^{+x}_{14}.C_{X}.S^{-x}_{25}.C_{X}.S^{+x}_{25}.C_{X}^{\dagger}.S^{+x}_{14}.C_{X}.S^{-x}_{14}.C_{X}^{\dagger}$
(21)
and
$\mathbf{U_{Y}}=S^{+y}_{25}.C_{Y}^{\dagger}.S^{-y}_{25}.C_{Y}.S^{-y}_{03}.C_{Y}^{\dagger}.S^{+y}_{03}.C_{Y}.S^{-y}_{25}.C_{Y}.S^{+y}_{25}.C_{Y}^{\dagger}.S^{+y}_{03}.C_{Y}.S^{-y}_{03}.C_{Y}^{\dagger}$
(22)
with the discrete time advancement of the 6-qubit $\mathbf{Q}$ given by
$\mathbf{Q}(t+\delta
t)=V_{Y}.V_{X}.\mathbf{U_{Y}}.\mathbf{U_{X}}.\mathbf{Q}(t)$ (23)
To recover the desired Maxwell equations (14) perturbatively, one employ a
small parameter $\epsilon$ as the spatial lattice shift unit (assuming a
square $x-y$ lattice), and the unitary collision angles
$\theta_{0}=\frac{\epsilon}{4n_{x}}\quad,\qquad\theta_{1}=\frac{\epsilon}{4n_{y}}\quad,\qquad\theta_{2}=\frac{\epsilon}{4n_{z}}.$
(24)
so as to recover the coefficients of the $\partial\mathbf{Q}/\partial(x,y)$
terms. Finally, the nonunitary external potential angles need to be defined as
$\beta_{0}=\epsilon^{2}\frac{\partial n_{y}/\partial
x}{n^{2}_{y}}\quad,\quad\beta_{1}=\epsilon^{2}\frac{\partial n_{x}/\partial
y}{n^{2}_{x}}\quad,\quad\beta_{2}=\epsilon^{2}\frac{\partial n_{z}/\partial
x}{n^{2}_{z}}\quad,\quad\beta_{3}=\epsilon^{2}\frac{\partial n_{z}/\partial
y}{n^{2}_{z}}$ (25)
Indeed, using Mathematica to evaluate Eq. (21), one obtains in the continuum
spatial limit the desired Maxwell equations to errors of $\epsilon^{4}$
$\displaystyle\frac{\partial q_{0}}{\partial t}=\epsilon^{2}\delta
t\frac{1}{n_{x}}\frac{\partial q_{5}}{\partial y},\qquad\frac{\partial
q_{1}}{\partial t}=-\epsilon^{2}\delta t\frac{1}{n_{y}}\frac{\partial
q_{5}}{\partial x},\qquad\frac{\partial q_{2}}{\partial t}=\epsilon^{2}\delta
t\frac{1}{n_{z}}\left[\frac{\partial q_{4}}{\partial x}-\frac{\partial
q_{3}}{\partial y}\right]$ (26) $\displaystyle\frac{\partial q_{3}}{\partial
t}=-\epsilon^{2}\delta t\frac{\partial(q_{2}/n_{z})}{\partial
y},\qquad\frac{\partial q_{4}}{\partial t}=\epsilon^{2}\delta
t\frac{\partial(q_{2}/n_{z})}{\partial x},\qquad\frac{\partial q_{5}}{\partial
t}=-\epsilon^{2}\delta t\left(\frac{\partial(q_{1}/n_{y})}{\partial
x}+\frac{\partial(q_{0}/n_{x})}{\partial y}\right)$
i.e., under diffusion ordering, $\epsilon^{2}\delta t\approx O(1)$, one
recovers the continuum Maxwell equations to errors $O(\epsilon^{2})$. We will
explore QLA simulations of 2D Maxwell equations in a subsequent paper.
## 3 QLA for KdV without external non-unitary potential operators
The KdV equation is an important nonlinear equation and was developed to
explore the evolution of shallow water waves. Interestingly, it [27] has also
been associated with the Fermi-Pasta-Ulam-Tsingou simulations of the 1950’s.
Fermi wanted to examine the equipartition of energy among the modes of a many
body problem of weakly coupled nonlinear oscillators. Statistical mechanics
indicates that the time- asymptotic state will be one in which there is
equipartition of energy among all the oscillator modes. Instead, in the
parameter regime they considered, Fermi et. al. found recurrence of initial
conditions, but a recurrence that was not a usually extremely long Poincare
recurrence time of Hamiltonian systems. Interestingly, this would turn out to
be a precursor to soliton theory.
The general $\mathrm{KdV}$ equation for arbitrary positive constants $a$ and
$b$
$\frac{\partial\psi}{\partial t}+a\psi\frac{\partial\psi}{\partial
x}+b\frac{\partial^{3}\psi}{\partial x^{3}}=0$ (27)
is exactly integrable. One of its solutions is the right traveling soliton
with speed $c$ \- a free parameter
$\psi(x,t)=\frac{3c}{a}\operatorname{sech}^{2}\left(\frac{1}{2}\sqrt{\frac{c}{b}}[x-ct]\right)$
(28)
Notice that for the KdV soliton, the amplitude and its speed are correlated
(unlike the NLS soliton).
Since the $\mathrm{KdV}$ equation is a scalar equation for the real function
$\psi(x,t)$ one need only to employ 2 qubits / lattice site. First we shall
reconsider the QLA for KdV with the use of an external potential to model the
nonlinear term in KdV [1]. The collision operator is nothing but Eq. (1). We
denote the operator $S_{0}^{+}$ to be the streaming operator that translates
the qubit $q_{0}$ one lattice unit in the $+x$-direction. To eliminate the 2nd
order spatial derivative one must choose the interleaved sequence of
collision-stream unitary operators carefully. In particular the following
sequence will generate a second order QLA for the KdV equation
$Q(t+\Delta t)=V_{pot}\cdot S_{0}^{+}C\cdot S_{1}^{-}C^{T}\cdot
S_{0}^{-}C\cdot S_{1}^{+}C^{T}\cdot S_{0}^{-}C^{T}\cdot S_{1}^{+}C\cdot
S_{0}^{+}C^{T}\cdot S_{1}^{-}C\cdot Q(t)$ (29)
where the unitary collision operator $C$ is nothing but the maximally
entangling operator. Eq, (1), with $\theta=\pi/4$.
$Q=\left(q_{0}\,q_{1}\right)^{T}$. The external potential $V_{\text{pot }}$ is
the Hermitian matrix
$V_{\text{pot }}=\left[\begin{array}[]{cc}\cos\alpha&-\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right]\quad\text{ with
}\quad\alpha=\epsilon^{3}m[x].$ (30)
In the continuum limit, one recovers
$\frac{\partial\psi}{\partial
t}+\epsilon^{3}\left(m[x]\cdot\psi(x,t)+\frac{1}{2}\frac{\partial^{3}\psi}{\partial
x^{3}}\right)=0+O\left(\epsilon^{5}\right)$ (31)
on defining $\psi=q_{0}+q_{1}$. With the choice of $m[x]=\partial\psi/\partial
x$ we have a second order accurate QLA for KdV. Note that the QLA of Eq. (26)
is not fully unitary because of the non-unitary property of the external
potential operator $V_{\text{pot }}$.
### 3.1 Fully unitary QLAs for KdV
There is a large class of unitary QLAs all of which recover the KdV equation
to second order accuracy. Here, we will present two QLAs, both having the same
unitary collision operator, but with different streaming sequences on the two
qubits. Indeed, using Mathematica, it can be shown that the following QLA
$Q(t+\Delta t)=S_{0}^{-}C_{1}\cdot S_{0}^{+}C_{1}\cdot S_{1}^{+}C_{1}\cdot
S_{1}^{-}C_{1}^{T}\cdot S_{0}^{-}C_{1}^{T}\cdot S_{0}^{+}C_{1}^{T}\cdot
S_{1}^{+}C_{1}^{T}\cdot S_{1}^{-}C_{1}\cdot Q(t)$ (32)
with unitary collision operator $C_{1}$
$C_{1}=\left[\begin{array}[]{cc}\cos\alpha_{1}&\sin\alpha_{1}\\\
-\sin\alpha_{1}&\cos\alpha_{1}\end{array}\right]\quad\text{ with
}\quad\alpha_{1}=\frac{\pi}{4}+\epsilon^{2}m_{1}[x].$ (33)
leads in the continuum limit to
$\frac{\partial\psi_{1}}{\partial
t}+\epsilon^{3}\left(4m_{1}[x]\cdot\frac{\partial\psi_{1}}{\partial
x}+\frac{1}{2}\frac{\partial^{3}\psi_{1}}{\partial
x^{3}}\right)=0+O\left(\epsilon^{5}\right)$ (34)
so that the choice of $m[x]=\psi_{1}$ will recover $\mathrm{KdV}$.
Another fully unitary QLA that recovers KdV has the following interleaved
sequence of unitary collision-streaming operators:
$Q(t+\Delta t)=C_{1}S_{0}^{-}\cdot C_{1}S_{1}^{+}\cdot C_{1}S_{0}^{-}\cdot
C_{1}S_{1}^{+}\cdot C_{1}^{T}S_{0}^{+}\cdot C_{1}^{T}S_{1}^{-}\cdot
C_{1}^{T}S_{0}^{+}\cdot C_{1}^{T}S_{1}^{-}\cdot Q(t)$ (35)
$C_{1}$ is the same collision operator, Eq. (33). In the continuum limit, we
find
$\frac{\partial\psi_{1}}{\partial
t}+\epsilon^{3}\left(-4m_{1}[x]\cdot\frac{\partial\psi_{1}}{\partial
x}+\frac{1}{2}\frac{\partial^{3}\psi_{1}}{\partial
x^{3}}\right)=0+O\left(\epsilon^{5}\right)$ (36)
so that the choice of $m_{1}[x]=-\psi_{1}$ will recover the KdV equation.
The implementation of these fully unitary algorithms may not necessarily be
straightforward as the perturbation parameter $\epsilon$ introduced into the
Mathematica algorithm requires a perturbation in the collision angle of
$O\left(\epsilon^{2}\right)$, Eq. (33), while the continuum limit has scaling
proceeds as $O\left(\epsilon^{3}\right)$. In previous QLA for nonlinear
physics, the order of the function $\psi$ controlled the $\epsilon$-factor.
## 4 SUMMARY
The development of a fully unitary QLA for plasma physics [28-31] in
particular, is of considerable interest to us as these algorithms are
immediately encodable on quantum computers. Of course, since they are time
evolution algorithms, they will have to wait till there are error-correcting
quantum computers available. In developing QLAs for plasma physics we have
taken the tack of first concentrating on the Maxwell equations in a given
scalar dielectric media. Then one would eventually generalize to a tensor
unitary dielectric description of a cold magnetized plasma.
Here, we have shown how to generalize our QLA-scalar dielectric Maxwell
equations to handle tensor Hermitian dielectric media. This was facilitated by
the use of the Dyson map [23]. Indeed, the explicit determination of the Dyson
map proves that there exists a unitary quantum algorithm to describe such a
Maxwell system. The problem, of course, is to explicitly construct such an
algorithm. We have concentrated on the QLA approach, which employs a non-
trivial sequence of interleaved collide-stream unitary operators. To proceed
explicitly with QLA, one must resort to perturbation theory and the
introduction of a small parameter $\epsilon$. The uinitary collide-stream
operator sequence does not in all our Maxwell equation considerations recover
the required full set of evolution equations. This has resulted in the need to
introduce so-called potential operators in order to recover the equation of
interest. At least one of these potential operators turns out to be non-
unitary. As these QLAs have parallelized outstandingly on classical
supercomputer architectures, outperforming standard computational fluid
dynamic codes for the study of quantum turbulence, we have proceeded with the
numerical implementation of such QLAs. This seems prudent as an error-
correcting quantum computer with long qubit coherence times is still on the
somewhat distant horizon. Nevertheless we are also pursuing a fully unitary
QLA. In particular, we are revisiting our original non-unitary QLA-KdV [1] to
determine a fully unitary QLA. We have found a large class of such unitary
QLA-KdV, based on changing the particular collide-stream sequences. The
implementation of these fully unitary algorithms may not necessarily be
straightforward as the perturbation parameter $\epsilon$ introduced into the
Mathematica symbolic manipulations requires a perturbation in the collision
angle of $O\left(\epsilon^{2}\right)$, Eq. (33), while the continuum limit has
scaling proceeds as $O\left(\epsilon^{3}\right)$, Eq. (34) or (36). In
previous QLA for nonlinear physics, the order of the function $\psi$ was used
as the $\epsilon$-factor. We believe that understanding the role of $\epsilon$
in the QLA-KdV simulations will be pivotal in handling the role of $\epsilon$
in QLA-Maxwell in both scalar and tensor dielectric media. These simulations
will be reported in a future publication.
## 5 Acknowledgments
This research was partially supported by Department of Energy grants DE-
SC0021647, DE-FG0291ER-54109, DE-SC0021651, DE-SC0021857, and DE-SC0021653.
This work has been carried out partially within the framework of the
EUROfusion Consortium. E.K has received funding from the Euratom research and
training program WPEDU under grant agreement no. 101052200 as well as from the
National Program for Controlled Thermonuclear Fusion, Hellenic Republic. K.H
is supported by the National Program for Controlled Thermonuclear Fusion,
Hellenic Republic. The views and opinions expressed herein do not necessarily
reflect those of the European Commission.
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|
# A Blockchain-based Semantic Exchange Framework for Web 3.0 toward
Participatory Economy
Yijing Lin, Zhipeng Gao*, Yaofeng Tu, Hongyang Du, Dusit Niyato, Jiawen Kang,
Hui Yang
###### Abstract
Web 3.0 is the next-generation Internet that enables participants to read,
write, and own contents in a decentralized manner. It is mainly driven by
blockchain, semantic communication, edge computing, and artificial
intelligence, which can construct value networks to achieve participatory
economics based on participatory decision making. Web 3.0 can capture the
characteristics of blockchain, semantic extraction, and communication to
achieve decentralized semantic sharing and transfer information precisely.
However, current Web 3.0 solutions focus on the blockchain while overlooking
other new technologies’ roles in Web 3.0. To further unleash the advantages of
semantic extraction and communication in Web 3.0, in this paper, we propose a
blockchain-based semantic exchange framework to realize fair and efficient
interactions. In this framework, we first attempt to tokenize semantic
information into Non-Fungible Token (NFT) for semantic exchange. Then we
utilize a Stackelberg game to maximize buying and pricing strategies for
semantic trading. We also leverage Zero-Knowledge Proof to share authentic
semantic information without publishing it before receiving payments, which
can achieve a fair and privacy-preserving trading compared with current NFT
marketplaces. A case study about urban planning is given to show clearly the
proposed mechanisms. Finally, several challenges and opportunities are
identified.
###### Index Terms:
Web 3.0, NFT, Optimization, Zero-Knowledge Proof
## I Introduction
Web 3.0 is now attracting the interest of researchers from academics and
industries. It is considered to be the next generation of the Internet where
participants can control their data, identities, and capabilities to make
decentralized decisions and construct participatory economy. There are two
mainstream developments of Web 3.0 as follows, including Blockchain Web 3.0
and Semantic Web 3.0. When the concept of Web 3.0 first emerged, it was
identified as a semantic web, which can make Internet data machine-readable
[1] and reduce energy consumption for processing and transmission. The current
definition of Web 3.0 [2] was prevailing with the help of blockchain
technologies, which can make participants get rid of tech giants and control
user-generated content. Besides, since artificial intelligence and edge
computing play important roles to connect Web 3.0 participants in a smart and
lightweight way, they are also considered vital components of Web 3.0.
The diversified underlying technologies of Web 3.0 are similar to Web 2.0
mainly driven by cloud computing, social networks, and mobile connectivity.
Instead of being controlled by tech giants, Web 3.0 allows all participants to
make decisions to participate in constructing value networks, which should
permit participants without the need of having the expertise of underlying
technologies, like DeFi Money Legos [3]. Thus, Web 3.0 can be supported by
multiple technologies like blockchain, artificial intelligence, edge
computing, and semantic communication, which can easily coordinate to
construct value networks to achieve participatory economy based on
decentralized decision making among participants.
Although several new technologies are emerging as main components to construct
value networks in Web 3.0, blockchain is considered the main technology for
the participatory economy in many cases. Few researchers focus on how to
integrate the components of new technologies into blockchain-based Web 3.0
toward the participatory economy. DeCAST [4] utilizes the integration of
blockchain and federated learning to construct parallel intelligent
transportation systems, which are effective and practical for smart mobility.
A unified blockchain-semantic framework [5] is proposed to capture the
benefits of blockchain, semantic extraction, and communication to improve
interaction efficiency for Web 3.0. Besides, given the steps of the
integration of new technologies in Web 3.0, it is necessary to consider how to
make value circulated in the participatory economy to attract participants to
join in Web 3.0 decision-making.
Figure 1: Difference Between Web 1.0, Web 2.0 and Web 3.0 Figure 2:
Applications of Blockchain and Semantic Communication-enabled Web 3.0
Blockchain, semantic extraction and communication can be utilized to reduce
information overloaded, reduce energy consumption, and construct efficient
interactions toward the participatory economy by extracting semantic
information from raw data and implementing sharing in a decentralized manner.
However, even capitalizing on the benefits of the above technologies, the
design toward the participatory economy still faces many difficulties. On the
one hand, it is difficult to quantify and transact semantic information
circulated in Web 3.0. The reason is that producers and consumers are
difficult to maximize their revenue and utility [6]. On the other hand, Web
3.0 is supported by a public blockchain platform, which makes pending semantic
information exposed to both parties. Consumers can copy semantic information
without payments to maximize their benefits [7] in current marketplaces.
In this paper, we attempt to complement the understanding of blockchain,
semantic extraction and communication playing an important role in Web 3.0. We
also develop a fair and efficient mechanism based on blockchain and semantic
communication to tokenize semantic information and enable semantic exchange by
utilizing the Non-Fungible Token (NFT), Stackelberg Game, and Zero Knowledge
Proof (ZKP) toward participatory economy. Our contributions are summarized as
follows.
* •
We propose a new semantic exchange framework, that tokenize semantic
information to facilitate semantic exchange by using blockchain.
* •
In this framework, we formulate a Stackelberg game-based semantic trading
mechanism to design optimal pricing strategies.
* •
To facilitate fair and privacy-preserving transactions between producers and
consumers, we utilize zero-knowledge proof to transact transformed semantic
information without publishing authentic one before payments.
* •
To show clearly the applications and mechanisms, we utilize urban planning to
display how the above contributions work.
## II The Future of Blockchain, Semantic Extraction and Communication-enabled
the Internet
### II-A An Accelerating Growth of The Internet
The accelerating growth of the iterations of the Internet is witnessed from
Web 1.0 to Web 3.0, as shown in Fig. 1. According to Fig. 1, we illustrate the
difference between Web 1.0, Web 2.0, and Web 3.0, the connection between Web
3.0, blockchain, edge computing, and semantic communication, and the
applications of blockchain and semantic communication-enabled Web 3.0.
#### II-A1 What Is The Difference Between Web 1.0, Web 2.0, And Web 3.0
The development of the Internet experiences three iterations of Web 1.0, Web
2.0, and Web 3.0 eras. Web 1.0 is a read-only web, where users can only read
information produced by centralized content producers from centralized
servers. Web 2.0 is a read-write web driven by mobile connectivity, social
networks, and cloud computing, where users can read and write content on
platforms provided by giant tech companies. Platforms can utilize content
produced by users to make a profit like advertising without paying any to
producers. Therefore, the need for Web 3.0 is that users expect to control
their data and protect their privacy in a decentralized manner. Web 3.0 is a
read-write-own web driven by blockchain, edge computing, artificial
intelligence, and semantic communication, where users can read, write, and own
content on decentralized value networks.
#### II-A2 What Is The Connection Between Web 3.0, Blockchain, Edge
Computing, And Semantic Communication
Whereas Web 2.0 was mainly driven by mobile connectivity, social networks, and
cloud computing to allow users to participate in constructing platform
networks, Web 3.0 is enabled by blockchain, edge computing, artificial
intelligence, and semantic communication to design an open, trustless,
seamless, ubiquitous, and efficient next-generation Internet. Blockchain is
often considered the same as Web 3.0, while it is actually an infrastructure
to support decentralized data sharing in an open and trustless manner. Edge
computing is also vital to construct Web 3.0, which can leverage the computing
and storage capabilities of ubiquitous edge devices and servers to make
Internet services available and near to users. Artificial intelligence can
leverage tons of data produced by edge computing and blockchain to train
models, which can improve services and provide seamless experiences. Instead
of transmitting information accurately, semantic communication [8] based on
edge computing and artificial intelligence pursues how precisely the
transmitted information can convey the desired meanings to reduce information
overloaded and energy consumed to provide Web 3.0 with efficiency.
#### II-A3 What Are Applications of Blockchain And Semantic Communication-
enabled Web 3.0
Whereas Web 2.0 mainly focuses on applications on platform networks for
controllers to make a profit from user data, blockchain and semantic
communication-enabled Web 3.0 concentrate on value networks to facilitate
transparent use of data, and decentralized data exchange and sharing.
Blockchain and semantic communication-enabled Web 3.0 toward participatory
economy can be applied for embracing the digital transformation of healthcare
services, the automotive industry, and urban planning, as shown in Fig. 2.
### II-B Research Gaps of Blockchain and Semantic Communication-enabled Web
3.0
Instead of a mansion in the air, Web 3.0 is the next-generation infrastructure
consisting of multiple building blocks like blockchain and semantic
communication, which is also similar to blockchain made up of cryptography and
distributed algorithms. Web 3.0 is expected to construct an entirely new
Internet for participants to transact and interact in the participatory
economy.
Previous works only focus on the blockchain or semantic-enabled Web 3.0, while
there are few works focusing on the integration between Web 3.0, blockchain,
semantic extraction and communication. DeSci [9] utilized blockchain and
decentralized autonomous organizations to construct a reference model for
scientific systems. A unified blockchain-semantic framework enabled Web 3.0
was proposed to maintain service security and improve interaction efficiency
in Web 3.0 [5]. HyperService [10] was designed to construct an interoperable
cross-chain platform for decentralized applications to interact data across
blockchains.
Besides, blockchain and semantic-enabled Web 3.0 still suffers from the
following challenges: 1) how to release the value of semantic information when
combining semantic communication and blockchain with Web 3.0, 2) how to
maximize the revenue and utility of semantic information producers and
consumers in the participatory economy, 3) how to trade semantic information
among producers and consumers in a fair manner. To solve the above challenges,
we propose a fair and efficient blockchain-based semantic exchange framework
for Web 3.0 by NFT-based semantic exchange, Stackelberg game-based semantic
trading, and zero-knowledge proof-based semantic sharing mechanisms to form
the participatory economy.
## III Fair and Efficient Mechanism Based on Semantic Blockchain
In this section, we describe the NFT-based semantic exchange scheme to
facilitate semantic circulation. We also propose a Stackelberg game-based
dynamic semantic trading mechanism to maximize the utility and revenue of
semantic information. Moreover, we construct a zero-knowledge proof-based
semantic sharing mechanism to enable fair authentic semantic sharing.
### III-A NFT-based Semantic Exchange Scheme
Semantic information extracted from producers and exchanged in semantic
communications has been considered how to determine the importance of
information [8]. Thus, for semantic information of high importance, it is
necessary to decide digital ownerships to facilitate the circulation of
valuable semantic information. Therefore, we propose an NFT-based semantic
exchange scheme to tokenize semantic information and unalterably prove the
digital ownership of producers.
Figure 3: Case Study in Urban Planning
The NFT-based semantic exchange scheme is composed of three functional parts,
including blockchain, off-chain storage, and semantic communication systems.
The blockchain is constructed by edge servers to enable consensus and smart
contracts, which can support NFT-based semantic exchange transactions between
edge devices. Since blockchain has limited storage spaces, off-chain storage
is utilized to supplement spaces for blockchain to interact with texts,
images, and videos in an efficient way. Edge devices can contribute spare
storage spaces to extend the off-chain storage to expand the storage ability
of blockchain. Once semantic information is uploaded into off-chain storage,
the off-chain storage will return a hash value to uniquely map the semantic
information to prevent tampering. The semantic communication systems include
semantic information producers and consumers to extract, exchange, and consume
semantic information to reduce information overloaded. Moreover, since
semantic communication systems need to interact with the blockchain and off-
chain storage, edge devices (producers and consumers) should equip with key
generation, blockchain client, and off-chain storage client modules to
implement semantic information circulating. The workflow of the NFT-based
semantic exchange scheme is described as follows.
* •
Step 1: Write and Deploy NFT Smart Contracts. NFTs are on-chain credentials
for semantic information and proofs of the ownership of producers, which are
implemented by smart contracts. The function of NFT smart contracts mainly
include Mint, Transfer, and Burn.
Mint enables producers to utilize smart contracts to release NFT tokens
mapping semantic information which is stored in off-chain storage. Transfer
can shift the ownership of NFT tokens from producers to consumers, which means
the exchange of semantic information. Burn destroys NFT tokens on the
blockchain, which excludes semantic information from circulation between
producers and consumers.
* •
Step 2: Publish Semantic information. Due to the limited storage of
blockchain, semantic information is first published in off-chain storage by
producers via IPFS clients. Producers obtain metadata to access the semantic
information via Distributed Hash Table. Metadata can be considered as the hash
commitment of semantic information to prevent it from tampering.
* •
Step 3: Mint Semantic information into NFT. Producers can bind metadata of
semantic information, mint them into NFT tokens, and generate token URIs by
smart contracts and blockchain clients. Besides metadata, NFT tokens should
contain attributes to introduce types of semantic tasks, producers, and time,
which can be transmitted in JSON format.
* •
Step 4: Transfer NFT-mapped Semantic Information. Producers can send a
transferFrom transaction to NFT smart contract to assign NFT tokens to
consumers, which can exchange semantic information stored in off-chain
storage. When consumers own NFT tokens mapped semantic information, they can
transfer it to other consumers to make a profit. NFT tokens can be observed to
track how the ownership of semantic information changes over time.
Besides, producers can send approved transactions to authorize trading
marketplaces as operators, and operators can substitute for owners to transfer
semantic information from producers to consumers.
### III-B Efficient Stackelberg Game-based Semantic Trading Mechanism
Although the proposed NFT-based semantic exchange scheme can circulate
semantic information between producers and consumers, it is difficult for both
parties to maximize the revenue and utility of semantic information. The fixed
pricing strategies may be unfair for producers when prices are lower than
cost, while the unlimited pricing strategies may be disadvantaged for
consumers when prices are higher than utilities. Thus, we propose a
Stackelberg game-based semantic trading mechanism to provide dynamic
marketplaces to trade semantic information efficiently. The Stackelberg game
approach considers the realistic utility and revenue functions of producers
and consumers to obtain the optimal buying and pricing strategies.
The revenue function of producers can be defined as income by providing
semantic information to consumers minus their costs for computing and
extracting semantic information. The income can be obtained from the required
amount multiplied by the unit pricing, while the costs should consider the
product of the number of CPU cycles, the CPU’s frequency, the energy
consumption and the network cost.
The utility function of consumers can be given that the precision of received
semantic information minuses the computing latency, and minuses the required
amount multiplied by the unit pricing. The first-order derivative of the
utility function should be greater than zero to incentive consumers to buy
semantic information from producers.
The above functions are modeled as a Stackelberg game and analyzed the
existence and uniqueness of Nash equilibrium. In the first stage, Karush-Kuhn-
Tucker (KKT) conditions are utilized to help consumers determine their
required amount of semantic information. The optimal solution is reported to
producers for consideration. In the second stage, producers maximize their
revenue given buying strategies.
### III-C Fair Zero-Knowledge Proof-based Semantic Sharing Mechanism
Although tokenizing semantic information can help producers to transact with
consumers, it is difficult to preserve the privacy of semantic information.
The reason is that blockchain is a public platform where everyone can submit
and retrieve semantic information mapped in NFTs. Moreover, since semantic
information can be easily duplicated, it is unfair for producers to release it
before getting payments, while it is also disadvantageous for consumers to pay
before verifying it. Therefore, we propose a fair and privacy-preserving zero-
knowledge proof-based semantic sharing mechanism to enable fair authentic
semantic sharing. Instead of directly transmitting semantic information once
extracted from encoders, producers should transform or process it to protect
fairness between both parties. Combining with the above mechanisms, the
workflow of the zero-knowledge proof-based semantic sharing mechanism is
described as follows. There are three stages corresponding to the three
mechanisms.
(a) Communication Overhead (b) Revenue of Semantic Trading
Figure 4: An Illustration of Efficiency of The Proposed Framework
Stage I: Transform semantic information with zero-knowledge proof (ZKP).
Producers take a security parameter and a transform ZKP circuit as input, and
output a common reference string for setup. The transform ZKP circuit carries
on the logic of transformation in a private way to generate proofs of
authentic information. Then they take the common reference string, and
semantic information, and transformed semantic information as inputs to
generate proofs.
Stage II: Publishe and mint NFTs for trading given transformed semantic
information. Producers bind metadata and proofs of transformed semantic
information into NFT. The transformed semantic information is not only mapped
by metadata but also represents the source semantic information by proofs.
Stage III: Implement transactions between producers and consumers. Producers
and consumers configure buying and pricing strategies according to Stackelberg
game-based semantic trading. After consumers pay for NFT mapped by transformed
semantic information, the NFT is transferred from producers to consumers.
Consumers can input proofs, transformed semantic information, and the common
string to verify the authenticity of transformed semantic information without
revealing the semantic information.
## IV Case Study for Urban Planning in Web 3.0
Urban planning activities are aided significantly by leveraging the proposed
framework, which can incorporate the benefits of blockchain and semantic
communication to construct Web 3.0 infrastructures. Semantic communication can
reduce information overloaded to support real-time semantic interactions and
minimize energy consumption for sustainable blockchain implementations.
Blockchain can provide traceable semantic information status to enable
semantic sharing in untrusted environments. Urban planning requires a
tremendous amount of data from other counterparts to support interactions from
different locations, as shown in Fig. 3. Producers can extract the contour
from images of a certain category of construction, and the corresponding
snapshots (semantic information) can be transmitted to respective consumers
[11]. Instead of transmitting original images, the extracted contour can
reflect key objects and the appearance of certain urban areas to reduce
information overloaded. Once urban designers obtain the contour of certain
areas from other designers, they can design a panoramic for urban planning.
Table I: An Overview of Open Challenges in Blockchain and Semantic Communication-Enabled Web 3.0 Type | Layers | Open Problem
---|---|---
Blockchain-Enabled Web 3.0 | On-Chain | Storage Structure Design, Consensus Protocol Design, Regulation Solutions, Quantum-Resistance Protocol Design
Off-Chain | Cross-Chain Solutions, Oracle Protocol Design, Rollup Protocol Design, Zero-Knowledge Proof Layer 2 Solutions
Collaboration | Decentralized Identity, Decentralized Finance, Metaverse, Decentralized Autonomous Organization
Semantic Communication -Enabled Web 3.0 | Theory | Explainability, Impact of Transmission Rate, Capacity of Semantic Channel [12], Quantity of Semantic Information [13]
Metric | Importance of Semantic Information, Tradeoff Between Accuracy and Communication Overhead [8], Inconsistent Background Knowledge Between Source and Destination [13]
Application | Implementation of Semantic Communication, Multi-User Collaboration, Video Transmission, Personalization
Blockchain and Semantic -Enabled Web 3.0 | Architecture | Unified, Secure, Efficient, Decentralized, Fault-Tolerant, Verifiable, Interoperable, Regulated
Collaboration | Verification and Price Bubbles of Semantic Information, Semantic Interoperability, Task-Oriented Layer 2 Solutions
Application | Metaverse, Digital Twin, Health Services, Digital Asset Management [5], Automotive Industry
However, information circulated in the urban planning activities can be
duplicated easily, which makes that other urban planning designers are not
willing to exchange it without any benefit and profit. For the same reason, it
is unfair for producers to release semantic information first, which makes
other urban planning designers can pay nothing to receive the core contour of
certain areas. Moreover, since semantic information is valuable, it is
necessary to construct a trading mechanism to guide semantic exchange in urban
planning activities.
The proposed framework is based on the above two infrastructures to build fair
and efficient semantic exchange mechanisms, as shown in Fig. 3. The designed
zero-knowledge proof-based semantic sharing mechanism can implement fair and
efficient transactions among urban designers by downsizing semantic
information to exchange the contour of semantic information without releasing
core designs first. It utilizes the NFT-based semantic exchange scheme to
facilitate downsized semantic information produced from spatial data to
circulation in urban planning activities. The proposed Stackelberg game-based
semantic trading mechanism can facilitate urban planning activities
constructing efficient trading marketplaces to facilitate the circulation of
semantic information.
To evaluate the proposed framework and mechanisms, we implement the simulation
case to illustrate the efficiency, as exhibited in Fig. 4. Fig. 4(a) shows the
communication overhead among tradition communication (Tradition), blockchain-
based semantic communication (Unified) [5], and our proposed mechanism (Ours).
The blockchain module is supported by a practical byzantine fault tolerance
algorithm with four nodes. The semantic module is implemented by Pytorch. The
size of the transmitted messages among those mechanisms is 1 MB, 100 KB, and 1
KB, referring to Fig. 3. The amount of semantic information represents
exchanged number of messages. The communication overhead is the time that
semantic information is recorded in the blockchain module. As shown in Fig.
4(a), the proposed mechanism is more efficient than the compared mechanisms.
Fig. 4(b) illustrates how the revenue of producers is affected by different
parameters of the semantic trading mechanism as the number of iterations
increases. With the increase of parameters, the costs take up more weight in
the revenue function, which causes a decrease in the income of producers.
## V Open Challenges and Future Directions
We overview recent open challenges in terms of blockchain, semantic
communication, and Web 3.0, as shown in Table I. Then we also elaborate on
several of them in the following.
Verification of Semantic Information: In scenarios that require high privacy
and security, e.g., automotive industries, verification of semantic
information is critical to perform operations correctly. It is impossible to
publish semantic information on public Web 3.0 platforms, and leverage
semantic information without any verification. Thus, zero-knowledge proof
combined with pragmatic functions of semantic communication is expected to
share private and authentic semantic information to protect privacy and
security in Web 3.0. Besides, since proofs generated by ZKP can be verified
given mathematical relationships, reputation mechanisms can be utilized to
classify and exclude malicious Web 3.0 participants.
Price Bubbles of NFT-mapped Semantic Information: Although semantic
information can be considered as NFT-based Web 3.0 assets circulating in the
marketplaces, it is difficult to avoid price bubbles in current market
mechanisms, like OpenSea [14]. The English Dutch auction mechanism [15] is
promising to eliminate price bubbles in the circulation of semantic
information.
Semantic Interoperability: Since Web 3.0 is integrated with blockchain and
semantic communication, semantic interoperability not only includes semantic
information circulated among multiple blockchains but also interacted with
multiple semantic tasks. The former can be implemented by cross-chain
technologies combined with verification, while the latter should consider how
to extract the same characteristics between different types of semantic tasks.
Task-Oriented Layer 2 Solutions: Participants have to continuously exchange
semantic information after perceiving environments to improve communication
quality, which is time and resource-consuming. Current Layer 2 solutions may
be a practical method to improve the performance of blockchain and semantic
communication-based Web 3.0, which can provide off-chain interactions in a
decentralized and traceable way.
## VI Conclusion
In this paper, we present our understanding and research gaps in Web 3.0,
Blockchain, and Semantic Communication. To this end, we propose a fair and
efficient blockchain-based semantic exchange framework for the participatory
economy. In this framework, we utilize NFT to circulate semantic information
in the blockchain to construct value networks. After that, we consider optimal
buying and pricing strategies for counterparts transacting with semantic
information by a Stackelberg game approach. We also take fairness into account
and leverage the zero-knowledge proof to enable authentic semantic sharing. A
case study of urban planning and simulation results are implemented to
illustrate the efficiency of the proposed framework. Finally, key challenges,
opportunities, and future research related to blockchain and semantic
communication for Web 3.0 are discussed.
## References
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* [2] G. Wood, “Đapps: What web 3.0 looks like,” Apr 2014. [Online]. Available: http://gavwood.com/dappsweb3.html
* [3] S. Cousaert, J. Xu, and T. Matsui, “Sok: Yield aggregators in defi,” in _2022 IEEE International Conference on Blockchain and Cryptocurrency (ICBC)_. IEEE, 2022, pp. 1–14.
* [4] C. Zhao, Y. Lv, J. Jin, Y. Tian, J. Wang, and F.-Y. Wang, “Decast in transverse for parallel intelligent transportation systems and smart cities: Three decades and beyonds,” _IEEE Intelligent Transportation Systems Magazine_ , 2022.
* [5] Y. Lin, Z. Gao, H. Du, D. Niyato, J. Kang, R. Deng, and X. S. Shen, “A unified blockchain-semantic framework for wireless edge intelligence enabled web 3.0,” _arXiv preprint arXiv:2210.15130_ , 2022.
* [6] Z. Li, W. Wang, Q. Wu, and X. Wang, “Multi-operator dynamic spectrum sharing for wireless communications: A consortium blockchain enabled framework,” _IEEE Transactions on Cognitive Communications and Networking_ , 2022.
* [7] R. Song, S. Gao, Y. Song, and B. Xiao, “Zkdet: A traceable and privacy-preserving data exchange scheme based on non-fungible token and zero-knowledge,” in _2022 IEEE 42nd International Conference on Distributed Computing Systems (ICDCS)_. IEEE, 2022, pp. 224–234.
* [8] W. Yang, H. Du, Z. Liew, W. Y. B. Lim, Z. Xiong, D. Niyato, X. Chi, X. S. Shen, and C. Miao, “Semantic communications for 6g future internet: Fundamentals, applications, and challenges,” _arXiv preprint arXiv:2207.00427_ , 2022.
* [9] W. Ding, J. Hou, J. Li, C. Guo, J. Qin, R. Kozma, and F.-Y. Wang, “Desci based on web3 and dao: A comprehensive overview and reference model,” _IEEE Transactions on Computational Social Systems_ , vol. 9, no. 5, pp. 1563–1573, 2022\.
* [10] Z. Liu, Y. Xiang, J. Shi, P. Gao, H. Wang, X. Xiao, B. Wen, Q. Li, and Y.-C. Hu, “Make web3. 0 connected,” _IEEE Transactions on Dependable and Secure Computing_ , 2021.
* [11] W. C. Ng, H. Du, W. Y. B. Lim, Z. Xiong, D. Niyato, and C. Miao, “Stochastic resource allocation for semantic communication-aided virtual transportation networks in the metaverse,” _arXiv preprint arXiv:2208.14661_ , 2022.
* [12] G. Shi, Y. Xiao, Y. Li, and X. Xie, “From semantic communication to semantic-aware networking: Model, architecture, and open problems,” _IEEE Communications Magazine_ , vol. 59, no. 8, pp. 44–50, 2021.
* [13] X. Luo, H.-H. Chen, and Q. Guo, “Semantic communications: Overview, open issues, and future research directions,” _IEEE Wireless Communications_ , 2022.
* [14] B. White, A. Mahanti, and K. Passi, “Characterizing the opensea nft marketplace,” in _Companion Proceedings of the Web Conference 2022_ , 2022, pp. 488–496.
* [15] C. Deck, M. Servátka, and S. Tucker, “Designing call auction institutions to eliminate price bubbles: Is english dutch the best?” _American Economic Review: Insights_ , vol. 2, no. 2, pp. 225–36, 2020.
BIOGRAPHIES
Yijing Lin is currently pursuing the Ph.D degree at the State Key Laboratory
of Networking and Switching Technology, Beijing University of Posts and
Telecommunications, China (e-mail: yjlin@bupt.edu.cn).
Zhipeng Gao is the corresponding author of this paper. He is a professor with
the State Key Laboratory of Networking and Switching Technology, Beijing
University of Posts and Telecommunications, China (e-mail:
gaozhipeng@bupt.edu.cn).
Yaofeng Tu is the Director of Center Institute of ZTE, China (e-mail:
tu.yaofeng@zte.com.cn).
Hongyang Du is currently pursuing the Ph.D degree at the School of Computer
Science and Engineering, Nanyang Technological University, Singapore (e-mail:
hongyang001@e.ntu.edu.sg).
Dusit Niyato is a professor with the School of Computer Science and
Engineering, Nanyang Technological University, Singapore (e-mail:
dniyato@ntu.edu.sg).
Jiawen Kang is a professor with the School of Automation, Guangdong University
of Technology, China (e-mail: kjwx886@163.com).
Hui Yang is a professor with the State Key Laboratory of information Photonics
and Optical Communications, Beijing University of Posts and
Telecommunications, China (e-mail: yanghui@bupt.edu.cn).
|
# Geoclidean: Few-Shot Generalization
in Euclidean Geometry
Joy Hsu
Computer Science
Stanford University
<EMAIL_ADDRESS>
&Jiajun Wu
Computer Science
Stanford University
<EMAIL_ADDRESS>
&Noah D. Goodman
Psychology and Computer Science
Stanford University
<EMAIL_ADDRESS>
###### Abstract
Euclidean geometry is among the earliest forms of mathematical thinking. While
the geometric primitives underlying its constructions, such as perfect lines
and circles, do not often occur in the natural world, humans rarely struggle
to perceive and reason with them. Will computer vision models trained on
natural images show the same sensitivity to Euclidean geometry? Here we
explore these questions by studying few-shot generalization in the universe of
Euclidean geometry constructions. We introduce _Geoclidean_ , a domain-
specific language for Euclidean geometry, and use it to generate two datasets
of geometric concept learning tasks for benchmarking generalization judgements
of humans and machines. We find that humans are indeed sensitive to Euclidean
geometry and generalize strongly from a few visual examples of a geometric
concept. In contrast, low-level and high-level visual features from standard
computer vision models pretrained on natural images do not support correct
generalization. Thus Geoclidean represents a novel few-shot generalization
benchmark for geometric concept learning, where the performance of humans and
of AI models diverge. The Geoclidean framework and dataset are publicly
available for download.***The Geoclidean framework can be found at
https://github.com/joyhsu0504/geoclidean_framework. †††Datasets can be found
at https://downloads.cs.stanford.edu/viscam/Geoclidean/geoclidean.zip.
## 1 Introduction
The built world we inhabit is constructed from geometric principles. Yet
geometric primitives such as perfect lines and circles, which are the
foundations of human-made creations, are uncommon in the natural world.
Whether for efficiency or for visual aesthetics, whether innate or learned,
humans are sensitive to geometric forms and relations. This natural
understanding of geometry enables a plethora of applied skills such as design,
construction, and visual reasoning; it also scaffolds the development of
rigorous mathematical thinking, historically and in modern education. Thus,
understanding the visually-grounded geometric universe is an important
desideratum for machine vision systems.
Ancient Greek philosophers were amongst the earliest to formalize geometric
notions, culminating in Euclid’s geometry in the 4th century BC. With a
compass and straight edge, Euclid’s axioms can construct a geometric world
that reflects an idealized, or Platonic, physical reality. We hypothesize that
Euclidean constructions are intrinsic to human visual reasoning. We thus build
a library to define and render such concepts, allowing systematic exploration
of geometric generalization. In this paper, we present Geoclidean, a domain-
specific language (DSL) for describing Euclidean primitives and construction
rules. Sets of construction rules define concepts that can then be realized
into infinitely many rendered images capturing the same abstract geometric
model.
Figure 1: Rendered realizations of Euclidean geometry concepts from the
Geoclidean datasets.
Based on Geoclidean, we introduce two datasets to study few-shot
generalization to novel rendered realizations of Euclidean geometry concepts
(See Figure 1). To succeed in solving these tasks, one must understand the
underlying geometric concept of a set of rendered images. The first dataset,
Geoclidean-Elements, covers mathematical definitions from the first book of
Euclid’s Elements (Simson et al., 1838). The second dataset, Geoclidean-
Constraints, simplifies and more systematically explores possible
relationships between primitives. We publicly release both datasets, as well
as the dataset generation library based on Geoclidean.
We report findings on the Geoclidean few-shot generalization tasks from human
experiments, as well as from evaluation on low-level and high-level visual
features from ImageNet-pretrained VGG16 (Deng et al., 2009; Simonyan and
Zisserman, 2014), ResNet50 (He et al., 2016), InceptionV3 (Szegedy et al.,
2016), and Vision Transformer (Dosovitskiy et al., 2020). We show that humans
significantly outperform pretrained vision models, generalizing in a way that
is highly consistent with abstract target concepts. ImageNet-pretrained vision
models are not as sensitive to intrinsic geometry and do not as effectively
encode these geometric abstractions. Our benchmarking process illustrates this
gap between humans and models, establishing Geoclidean as an interesting and
challenging generalization task for visual representation and geometric
concept learning.
## 2 Foundations of Geoclidean
In this section, we give an overview of the Geoclidean DSL, which builds on
foundations of Euclid’s axioms for constructing with a compass and
straightedge. We present a Python library that renders Euclidean geometry
concepts described from our DSL into images. We first describe the Euclidean
geometry universe in Section 2.1, and then introduce the Geoclidean language
and construction rules in Section 2.2. Finally, we show geometric concept
realizations into rendered images in Section 2.3.
### 2.1 Euclidean Geometry Universe
There exist numerous systems of geometry, each with its own logical system.
Each logic can be described in a formal language, each formal language
describes construction rules, and each set of construction rules describe
concepts in the geometric universe. Geometric concepts can be realized and
rendered into images. Euclidean geometry was among the earliest formalized,
first described by Euclid in Elements (Simson et al., 1838). The first book of
Elements details plane geometry, laying the foundation for basic properties
and propositions of geometric objects. Importantly, Euclid’s geometry is
constructive—objects only exist if they can be created with a compass and
straightedge. Euclidean constructions are abstract models of geometric
objects, specified without the use of coordinates and without concrete
realizations into images. Hence, changes to the image rendering, such as
alterations to size and rotation, do not change an object’s intrinsic
Euclidean geometry.
Euclid’s axioms build the foundations of this geometric universe via 1) draw a
straight line from any point to any point, 2) describe a circle with any
centre and distance. The Euclidean geometry universe is determined by these
base geometric primitives, and the constraints that parameterize object
relationships. There are other universes that contain different logic systems,
from Descartes’ analytic geometry, to Euler’s affine geometry, to Einstein’s
special relativity. Because of its simplicity and abstraction we are
particularly interested in the universe of Euclidean geometry – do humans and
machines find these constructions natural? We later explore whether they
spontaneously generalize image classes according to Euclidean rules.
### 2.2 Domain-Specific Language: Geoclidean
We create a domain-specific language (DSL), Geoclidean, defining Euclidean
constructions that arise from the mathematics of Euclidean geometry.
Geoclidean allows us to define construction rules for objects and their
relations to each other, encompassing concepts in the Euclidean geometry
universe. It includes three simple primitives. The first is a point,
parameterized by constraints, if any, to an object or a set of objects
previously defined. The point is defined without specific coordinates, and
only when realized, would be assigned coordinate values $x$ and $y$. The
second is a line, which, following the first axiom, is parameterized by two
points, representing the beginning and end. The third is a circle which,
following the second axiom, is defined by a center point and an edge point.
These primitives represent the compass and straightedge constructions that
Euclid introduced. Lines and circles are defined by points, while points can
be constrained to previously built objects. In this way primitives are
sequentially defined to form a concept.
Table 1: The Geoclidean DSL for building Euclidean geometry concepts. We assume a pool of variable names for points and objects (point_name, object_name). As a shorthand for point creation followed by reference, we later use e.g. Line(p1(),p2()) to represent p1 = Point(); … Line(p1,p2). The marker * indicates that the object will not be visible in the final rendering. Concept | $\rightarrow$ | Statement; Concept
---|---|---
Statement | $\rightarrow$ | object_name${}^{\text{Visibility}}$ = Object(point_name, point_name) |
| | point_name = point(Constraints)
Visibility | $\rightarrow$ | [] | *
Object | $\rightarrow$ | line | circle
Constraints | $\rightarrow$ | [] | [object_name] | [object_name, object_name]
Geoclidean’s syntax is defined in Table 1. We can initialize a new point,
line, or circle via the constructors Point, Line, Circle, assigning them to a
named variable. For points we generally use a shorthand to define and use the
variable inline. For example, p1() is a free point that is unconstrained to
any objects, and can be realized anywhere in the image, p2(circle1) is a
partially constrained point that lies on the object circle1, and p3(line1,
line2) is a fully constrained point that lives in the intersection of line1
and line2. A point can be reused by referring to its name. The line Line(p1,
p2) is parameterized by two end points, and the circle Circle(p1, p2) is
parameterized by points at the center and edge. Not indicated in Table 1 is
the semantic constraint that variable names must be defined before being used
within constructors. Visibility of rendering is denoted for each object, with
* indicating that the object is not visible in the final rendering, as some
objects in Euclidean geometry are used solely as helper constructions for
other core objects in the concept. Each construction rule is a geometric
object, and a sequence of construction rules define a Euclidean geometry
concept.
### 2.3 Concept Realization
Geoclidean implements this Euclidean language and renders realizations of
concepts from the language into images. For each line or circle object, we
realize its rendering based on the point parameters required. During
rendering, each point parameter is given randomly sampled real-valued
coordinates bound by its constraints. If the point exists already, Geoclidean
reuses the past component; if it is a free point without constraints,
Geoclidean randomly samples values for $x$ and $y$; if constrained, Geoclidean
randomly samples a point that lies constrained on the object or the
intersection of a set of objects. It is possible for intersection sets to be
empty, and we reject sample realizations until finding a satisfying
realization for all points. Geoclidean creates the objects sequentially and
renders them into an image, if visible.
Algorithm 1 Construction rules for the equilateral triangle concept, with
colored steps corresponding to the rendered realization in Figure 2. The two
circles c1 and c2 are not rendered in the final image as denoted by *.
1:
2:l1 = Line(p1(), p2()) $\triangleright$ Red line
3:c1*= Circle(p1(), p2()) $\triangleright$ Light orange circle
4:c2*= Circle(p2(), p1()) $\triangleright$ Light yellow circle
5:l2 = Line(p1(), p3(c1, c2)) $\triangleright$ Green line
6:l3 = Line(p2(), p3(c1, c2)) $\triangleright$ Blue line
Figure 2: Rendered realization of the equilateral triangle concept.
We see in Figure 2 an example of the Geoclidean language describing the
equilateral triangle concept realized into an image. The construction rules in
Algorithm 1 are created step by step; we color each object for clarity. The
first construction rule creates a red line between two unconstrained free
points, p1() and p2(); when realized with sampled real-valued points, this
line can be anywhere in the image with any length and rotation. The second
rule creates an invisible orange circle with the ends of the first line as its
center and edge point. The third rule creates another invisible yellow circle
with the center and edge point flipped, forming intersecting helper circles
with the same radius. Then, the fourth rule creates a green line between p1
and a new point p3(c1, c2), which is a point constrained to the intersection
of the previously created orange and yellow circles (the realization randomly
chooses one of the two intersection points). The last rule creates the final
blue line between p2 and p3, completing the constructed triangle and enforcing
all sides to be of equal length. This concept is that of the equilateral
triangle, and we see that the invisible helper circle objects serve as
essential constraints to the final rendering.
Importantly, the construction rules do not specify any coordinates, and our
Geoclidean framework creates the coordinates upon realization of the concept
into an image. Hence, the rendered equilateral triangle can be of any size and
orientation, while always respecting the underlying geometric concept. In
Geoclidean, every random realization of this concept creates equilateral
triangles, as represented in Euclid’s geometric universe. The Geoclidean
framework allows us to create rendered image datasets that follow the
specified concept language.
## 3 Geoclidean Task and Datasets
Do the realizations of Euclidean constructions form natural categories for
humans? For computer vision models? To study these questions we introduce a
few-shot generalization task based on Geoclidean and two image datasets that
realize $37$ Euclidean geometry concepts.
#### Task.
We explore few-shot generalization from positive examples of a target concept;
to the extent that participants generalize to other realizations of the target
concept, and not realizations of more general concepts, we conclude the
concept is an intuitive kind.
For each concept, the task includes five reference examples of the target
concept and a test set of 15 images. Among the 15 images, there are five
positive examples and two sets of five negative examples. Here, positive
examples in the test set are realizations of the target concept, as are the
reference examples, and negative examples are realizations of related but
different concepts (which are not realizations of the target concept). The
goal is to correctly categorize positive examples as positive and negative
examples as negative in the test set. The ten negative examples are divided
into five Close and five Far examples, where the negative examples in Close
are from a closely related concept with a fewer number of constraint
differences from the target concept, and negative examples in Far are from a
further, less related concept with a larger number of constraint differences.
These constraint differences consist of altering a point to have fewer
constraints compared to that point in the target concept (yielding a more
general and less specified geometric concept). See Figure 3 for examples, with
the top representing reference examples, and the bottom representing the test
set. Note that, because we are interested in intrinsic sensitivity of visual
representations to geometric concepts, we are not introducing a meta-learning
task: there are no few-shot generalization sets intended for model training.
We now introduce the two datasets we created based on Geoclidean for the
generalization task. The first dataset, Geoclidean-Elements, includes the
tests of $17$ concepts derived from the first book of Euclid’s Elements; the
second dataset, Geoclidean-Constraints, includes the tests of $20$ concepts
based on constraints defining relationships between Geoclidean primitives. See
Figure 3 for examples from both splits.
Figure 3: Examples of Geoclidean-Elements and Geoclidean-Constraints tasks.
Each task consists of few-shot reference examples as well as test examples.
Few-shot examples and positive test examples derive from the same concept
(contained in red boxes), while negative test examples derive from a related
but different concept (contained in blue boxes); Close test examples differ by
fewer point differences in the construction rules than Far, seen bolded in the
Geoclidean language.
#### Geoclidean-Elements.
The Geoclidean-Elements dataset is derived from definitions in the first book
of Euclid’s Elements, which focuses on plane geometry. Geoclidean-Elements
includes $17$ target concepts, which, along with Geoclidean primitives, covers
definitions in the first book of Elements. These concepts require complex
construction rules with helper objects that are not visible in the final
renderings. Realizations from Geoclidean-Elements test sensitivity to the
exactness of Euclidean constructions without explicit visual constraint
differences.
The concepts in Geoclidean-Elements include angle (Book I definition IX),
perpendicular bisector (def X), angle bisector (def X), sixty degree angle
(def XI and XII), radii (def XV), diameter (def XVII and XVIII), segment (def
XIX), rectilinear (def XX and XXIII), triangle (def XXI), quadrilateral (def
XXII and XXXIV), equilateral triangle (def XXIV, XXV, XXVI in Close and Far),
right angled triangle (def XXVII, XXVIII, XXIX in Close and Far), square (def
XXX), rhombus (def XXXI), oblong (def XXXII), rhomboid (def XXXIII), and
parallel lines (def XXXV). The rest of the definitions are descriptions of
Geoclidean primitives (e.g. points, lines).
#### Geoclidean-Constraints.
The Geoclidean-Constraints dataset consists of $20$ concepts, created from
permutations of line and circle construction rules with various constraints
describing the relationship between objects. This dataset focuses on explicit
constraints between geometric objects. We denote the objects as the
following—lines as L, circles as C, and triangles (constructed from three
lines) as T. Tasks include three, four, and five object variants, each with
specific ordering; the different ordering of objects is significant, as
constraints may only depend on previously defined objects. Each concept is
defined by object ordering as well as constraints describing the relationships
between them; the full set of construction rules for each concept is released
with the dataset.
The three object concepts are [LLL, CLL, LLC, CCL, LCC, CCC], the four object
concepts are [LLLL, LLLC, CLLL, CLCL, LLCC, CCCL, CLCC, CCCC], and the five
object concepts are [TLL, LLT, TCL, CLT, TCC, CTT]. These $20$ concepts test
the few-shot generalization capability in constrained Euclidean geometry
concepts.
## 4 Findings
We present our findings on the Geoclidean dataset in benchmarking human
performance (Section 4.1) and pretrained vision models’ capabilities (Section
4.2). We show that humans are indeed sensitive to Euclidean geometry concepts,
generalizing strongly from five examples across the $37$ concepts. This
establishes Geoclidean as an interesting task for evaluating the human-like
visual competencies of machine vision. Indeed, we find that state-of-the-art
pretrained visual representations perform poorly on this few-shot
generalization task.
### 4.1 Human Performance
We collected human judgements for the Geoclidean few-shot concept learning
task. We recruited $30$ participants for each concept using the Prolific
crowd-sourcing platform (Palan and Schitter, 2018). As mentioned above,
participants are given five example realizations of the target concept and
$15$ test images including five positive examples, five Close negative
examples, and five Far negative examples. Each of the test questions states:
“Each of the five images shown is a ‘wug’. Is the image below a ‘wug’ or not a
‘wug’?”, where ‘wug’ is a random made-up word for each task. The order of
tasks and realizations is randomized for each participant to remove order
effects. The experiment interface was implemented on Qualtrics, with details
described in the Appendix.
Figure 4: Histogram of human accuracies on the Geoclidean-Elements and the
Geoclidean-Constraints datasets, as well as accuracies with Close and Far
negative examples. The y-axis indicates the percentage of tasks with the
specified accuracy on the x-axis.
We report task accuracy in Table 2, scored as the percentage of participants
correctly categorizing test images, averaged across all examples. We split the
$15$ test images into two tasks, with the Close task consisting of $5$
positive examples and $5$ negative examples from Close, and the Far task
consisting of the same $5$ positive images with $5$ negative examples from
Far. Each task contains $10$ examples in total. We see that human performance
is strong across all tasks, with on-average higher scores in Far compared to
Close, showing that the number of differences in construction rules affects
the semantic distance between rendered realizations. Only tasks LLL, CLCL, and
rhomboid yielded slightly better performance in Close than Far. Out of all
tasks, LLC, CCC, LLLC with Close negatives are more difficult for humans (with
equally poor performance across all test images), which we hypothesize is due
to more subtle constraint intersections. In general, humans perform well on
this generalization task.
We show accuracy histograms in Figure 4, with the left two plots depicting
results from concepts in Geoclidean-Elements and Geoclidean-Constraints, and
the right two plots depicting results when calculated with Close negative
examples and Far negative examples. Humans are more sensitive to concepts in
Geoclidean-Elements, which are complex constructions that test the exactness
of shapes (e.g., squares and equilateral triangles), and slightly less
sensitive to concepts in Geoclidean-Constraints, which test the precise
relationships between objects (e.g., constrained contact point between the end
of a line and the center of a circle).
Participants could generate infinitely many rules consistent with the positive
images seen in the few-shot examples (e.g., a “wug” can have its own prototype
for each example, as there are no negative examples), and there is potential
ambiguity as to which are the correct construction rules of the concept.
Despite this wide range of possible generalization patterns, the
generalization rule chosen by humans corresponds well to the Euclidean
construction universe.
Table 2: Human accuracy across all $74$ tasks in Geoclidean. concept | Close | Far | concept | Close | Far
---|---|---|---|---|---
angle | 0.9767 | 0.9833 | lll | 0.9700 | 0.9667
perp bisector | 0.9367 | 0.9833 | cll | 0.9467 | 0.9667
ang bisector | 0.9433 | 0.9533 | llc | 0.6767 | 0.9233
sixty ang | 0.8233 | 0.9533 | ccl | 0.8700 | 0.8833
radii | 0.9233 | 0.9600 | lcc | 0.8867 | 0.9633
diameter | 0.9567 | 1.0000 | ccc | 0.6667 | 0.8767
segment | 0.9300 | 0.9833 | llll | 0.8833 | 0.9767
rectilinear | 0.9000 | 0.9033 | lllc | 0.6667 | 0.8867
triangle | 0.9633 | 0.9767 | clll | 0.8367 | 0.9033
quadrilateral | 0.9167 | 0.9267 | clcl | 0.8700 | 0.8567
eq t | 0.9533 | 0.9800 | llcc | 0.8867 | 0.9333
right ang t | 0.7200 | 0.8133 | cccl | 0.9233 | 0.9333
square | 0.8933 | 0.9867 | clcc | 0.8633 | 0.9000
rhombus | 0.9367 | 0.9667 | cccc | 0.8167 | 0.8800
oblong | 0.9666 | 0.9900 | tll | 0.9467 | 0.9800
rhomboid | 0.9700 | 0.9300 | llt | 0.9267 | 0.9400
parallel l | 0.9500 | 0.9567 | tcl | 0.9533 | 0.9633
| | | clt | 0.9533 | 0.9633
| | | tcc | 0.9533 | 0.9633
| | | cct | 0.9533 | 0.9633
### 4.2 Model Benchmarks
Figure 5: Histogram of human and maximum low-level and high-level feature
accuracies of various vision models.
We benchmark pretrained vision models’ performance on the Geoclidean task, to
evaluate few-shot generalization (with no meta-learning or fine-tuning) in the
Euclidean geometry universe. We measure performance of features from ImageNet-
pretrained VGG16 (Simonyan and Zisserman, 2014), ResNet50 (He et al., 2016),
InceptionV3 (Szegedy et al., 2016), and Vision Transformer (Dosovitskiy et
al., 2020), and evaluate both low-level features and high-level features for
each of the models. Low-level features are outputs of earlier layers in neural
networks, which tend to capture low-level information such as edges and
primitive shapes, while high-level features are outputs of later layers that
tend to capture more high-level semantic information (Zeiler and Fergus,
2014). We detail how we define the layers for each baseline in the Appendix.
To evaluate these features, we extract features $\phi$ from the few-shot $t$
reference images of the target concept, to create a prototype target feature,
$T=\frac{1}{n}\sum_{i=1}^{n}\phi(t_{i})$. We classify a test image $r$ as in-
concept if it is closer than a threshold to the prototype:
$f|T-\phi(r)|<\theta$, where $f$ is the normalizing function between $0\sim
1$. The threshold is fit by selecting the best-performing normalized distance
threshold across all 74 tasks, for given features. (By fitting this free
threshold, we bias the reported accuracy in favor of models.)
Table 3: Human accuracy compared to low and high-level feature accuracy of vision models across $37$ concepts in Geoclidean, each concept containing averaged accuracies between Close and Far tasks. | Human | VGG16 | | RN50 | | InV3 | | ViT |
---|---|---|---|---|---|---|---|---|---
auc | | low | high | low | high | low | high | low | high
angle | 0.98 | 0.50 | 0.45 | 0.45 | 0.50 | 0.40 | 0.45 | 0.50 | 0.40
perp bisector | 0.96 | 0.70 | 0.85 | 0.70 | 0.50 | 0.65 | 0.90 | 0.70 | 0.85
ang bisector | 0.95 | 0.50 | 0.60 | 0.50 | 0.65 | 0.60 | 0.60 | 0.50 | 0.75
sixty ang | 0.89 | 0.25 | 0.65 | 0.35 | 0.45 | 0.55 | 0.45 | 0.35 | 0.70
radii | 0.94 | 0.75 | 0.70 | 0.75 | 0.60 | 0.65 | 0.80 | 0.75 | 0.75
diameter | 0.98 | 0.30 | 0.55 | 0.40 | 0.40 | 0.50 | 0.75 | 0.45 | 0.85
segment | 0.96 | 0.45 | 0.60 | 0.60 | 0.40 | 0.30 | 0.65 | 0.55 | 0.65
rectilinear | 0.90 | 0.65 | 0.35 | 0.60 | 0.45 | 0.55 | 0.60 | 0.65 | 0.45
triangle | 0.97 | 0.65 | 0.45 | 0.50 | 0.40 | 0.40 | 0.55 | 0.35 | 0.60
quadrilateral | 0.92 | 0.50 | 0.70 | 0.60 | 0.50 | 0.55 | 0.75 | 0.60 | 0.60
eq t | 0.97 | 0.40 | 0.85 | 0.50 | 0.70 | 0.65 | 0.65 | 0.50 | 0.55
right ang t | 0.77 | 0.65 | 0.75 | 0.60 | 0.70 | 0.60 | 0.55 | 0.70 | 0.55
square | 0.94 | 0.70 | 0.80 | 0.65 | 0.45 | 0.85 | 0.70 | 0.60 | 0.75
rhombus | 0.95 | 0.70 | 0.55 | 0.60 | 0.55 | 0.60 | 0.55 | 0.70 | 0.60
oblong | 0.98 | 0.45 | 0.55 | 0.45 | 0.45 | 0.50 | 0.70 | 0.45 | 0.70
rhomboid | 0.95 | 0.70 | 0.60 | 0.65 | 0.55 | 0.50 | 0.50 | 0.65 | 0.75
parallel l | 0.95 | 0.55 | 0.35 | 0.50 | 0.40 | 0.45 | 0.55 | 0.45 | 0.85
lll | 0.97 | 0.50 | 0.70 | 0.55 | 0.40 | 0.55 | 0.35 | 0.55 | 0.80
cll | 0.96 | 0.25 | 0.60 | 0.40 | 0.30 | 0.30 | 0.60 | 0.30 | 0.60
llc | 0.80 | 0.65 | 0.55 | 0.60 | 0.55 | 0.55 | 0.65 | 0.60 | 0.60
ccl | 0.88 | 0.65 | 0.50 | 0.55 | 0.65 | 0.30 | 0.60 | 0.55 | 0.75
lcc | 0.93 | 0.45 | 0.70 | 0.45 | 0.70 | 0.65 | 0.70 | 0.45 | 0.85
ccc | 0.77 | 0.60 | 0.55 | 0.60 | 0.70 | 0.70 | 0.75 | 0.60 | 0.90
llll | 0.93 | 0.60 | 0.30 | 0.50 | 0.30 | 0.30 | 0.50 | 0.50 | 0.50
lllc | 0.78 | 0.30 | 0.65 | 0.40 | 0.55 | 0.50 | 0.55 | 0.30 | 0.80
clll | 0.87 | 0.55 | 0.65 | 0.60 | 0.65 | 0.65 | 0.50 | 0.70 | 0.65
clcl | 0.86 | 0.60 | 0.65 | 0.65 | 0.45 | 0.40 | 0.35 | 0.60 | 0.85
llcc | 0.91 | 0.75 | 0.65 | 0.85 | 0.55 | 0.80 | 0.70 | 0.80 | 0.70
cccl | 0.93 | 0.35 | 0.60 | 0.50 | 0.65 | 0.35 | 0.65 | 0.45 | 0.75
clcc | 0.88 | 0.60 | 0.55 | 0.55 | 0.60 | 0.60 | 0.75 | 0.55 | 0.80
cccc | 0.85 | 0.80 | 0.65 | 0.70 | 0.60 | 0.75 | 0.50 | 0.75 | 0.95
tll | 0.96 | 0.70 | 0.65 | 0.65 | 0.60 | 0.60 | 0.55 | 0.75 | 0.85
llt | 0.93 | 0.55 | 0.40 | 0.55 | 0.65 | 0.55 | 0.65 | 0.50 | 0.85
tcl | 0.96 | 0.65 | 0.70 | 0.70 | 0.55 | 0.60 | 0.60 | 0.65 | 0.45
clt | 0.88 | 0.50 | 0.60 | 0.45 | 0.50 | 0.55 | 0.65 | 0.45 | 0.70
tcc | 0.84 | 0.30 | 0.45 | 0.35 | 0.35 | 0.55 | 0.45 | 0.30 | 0.45
cct | 0.79 | 0.55 | 0.70 | 0.45 | 0.60 | 0.60 | 0.55 | 0.50 | 0.85
average | 0.91 | 0.55 | 0.60 | 0.55 | 0.53 | 0.54 | 0.60 | 0.55 | 0.70
In Table 3, we present accuracy of ImageNet-pretrained low-level and high-
level features across different models. Humans substantially outperform
features from vision models, showcasing the gap between human and model
capabilities in Euclidean geometry concept learning.
We see that, on average across tasks, pretrained vision models perform poorly
compared to humans. In general, high-level features perform slightly better
than low-level features, though there are cases where this differs.
Interestingly, the high-level features from the Vision Transformer (ViT)
outperform its convolutional network counterparts, and achieve human-level
performance in some tasks. There are a few tasks where the visual feature
accuracy outperforms humans, notably tasks CCC, LLLC, CCCC, CCT, where high-
level features from ViT perform strongly. We hypothesize that this is because
humans are not as sensitive to details of intersections between circles.
In Figure 5, we present histograms comparing maximum low and high-level
feature accuracy to human accuracy, illustrating the gap in performance. In
Table 4, we report the Pearson correlation coefficient between the answers of
humans and models. We see that humans are generally more aligned with high-
level features than low-level ones. Additionally, though ViT achieves high
accuracy, it is not correlated to human performance, indicating failure to
generalize in the most human-like way.
Table 4: Pearson’s correlation to human accuracy across all $74$ tasks in Geoclidean. VGG16 low | high | RN50 low | high | InV3 low | high | ViT low | high
---|---|---|---|---|---|---|---
-0.08947 | 0.0943 | 0.0341 | -0.1522 | -0.0820 | 0.2575 | -0.0056 | -0.0259
We report additional comparisons of low-level and high-level visual features
from ImageNet-pretrained VGG16, ResNet50, InceptionV3, and Vision Transformer
in the Appendix, and show that similar trends follow across data splits and
performance metrics. We also include low-level and high-level feature
visualizations in the Appendix, comparing Geoclidean tasks that require
reasoning, to perception tasks involving simple geometric primitives and
perturbations. These comparisons highlight Geoclidean as a unique and
interesting test for vision models.
## 5 Related Work
#### Geometric reasoning datasets.
Prior geometric datasets generally fall into two main categories—with
geometric objects for computer-aided design (CAD) and for plane geometry. In
the first category, the SketchGraphs dataset models relational geometry in CAD
design (Seff et al., 2020), and the ABC-Dataset includes parametric
representations of 3D CAD models (Koch et al., 2019). In the latter category,
CSGNet presented a generated dataset of constructive solid geometry based on
2D and 3D synthetic programs with squares, circles, and triangles (Sharma et
al., 2018), while Ellis et al. (2018) connected high–level Latex graphics
programs with 2D geometric drawings. Works such as Zhang et al. (2022); Lu et
al. (2021) proposed using datasets with annotated geometric primitives and
relationships such as containment from geometry diagrams in textbooks. Others
introduced reasoning benchmarks with geometric shapes, including Raven’s
progressive matrices (Matzen et al., 2010; Wang and Su, 2015; Barrett et al.,
2018; Zhang et al., 2019), Bongard problems (Depeweg et al., 2018; Nie et al.,
2020), odd-one-out tasks (Mańdziuk and Żychowski, 2019), and a variety of
reasoning challenges (Hill et al., 2019; Zhao et al., 2021; El Korchi and
Ghanou, 2020; Zhang et al., 2020). Our work is more related to the latter of
geometric shapes, and Geoclidean differs by targeting Euclidean geometry
concept learning whose construction language 1) does not require specific
coordinates, and 2) focuses on the construction steps that form semantically-
complex geometric concepts and the constraints between geometric primitives
that humans are intrinsically sensitive to.
#### Few-shot concept learning.
Few-shot learning tasks range in complexity on both the input and task
description axis. In the natural language processing domain, tasks such as
FewRel (Han et al., 2018) and Few-NERD (Ding et al., 2021) have been proposed
for few-shot relation classification and entity recognition. Goodman et al.
(2008) introduced concept learning tasks with sequences generated from
specified logical rules. In the vision domain, which we are interested in,
commonly used tasks include those from Lake et al. (2015), which introduced
Omniglot as a collection of simple visual concepts collected from 50 writing
systems, and from Vinyals et al. (2016), which proposed miniImageNet (Deng et
al., 2009), both for the task of one-shot classification. Works in other
vision domains include Massiceti et al. (2021), which explores the few-shot
video recognition challenge, and Xiao et al. (2020); Gehler et al. (2008),
which examines the few-shot color constancy problem. Triantafillou et al.
(2019) created the meta-dataset as a diverse dataset for few-shot learning,
with multiple tasks for meta-training such as Maji et al. (2013); Wah et al.
(2011); Cimpoi et al. (2014); Nilsback and Zisserman (2008); Houben et al.
(2013); Lin et al. (2014). In comparison, we propose Geoclidean as a zero-shot
meta-trained, few-shot generalization task that consists of labeled image
renderings from a single target concept.
## 6 Discussion
An important contribution of our task is that it allows for better testing of
vision models that aim to incorporate reasoning and high-level semantics.
Additionally, Geoclidean’s zero-shot meta-trained evaluation is especially
significant, as many downstream tasks that may leverage pretrained models
would greatly benefit from geometric reasoning, such as construction (LegoTron
(Walsman et al., 2022), Physical Construction Tasks Bapst et al. (2019)),
physical reasoning (CLEVRER (Yi et al., 2019), ThreeDWorld (Gan et al.,
2020)), and shape understanding tasks (PartNet (Mo et al., 2019), ShapeNet
(Chang et al., 2015)). Furthermore, numerous additional evaluation tasks can
be built with the Geoclidean DSL and rendering library, such as those
involving natural language or generated large-scale datasets. We include
further analyses and discussion in the Appendix.
## 7 Conclusion
We have introduced Geoclidean, a domain-specific language for the realization
of the Euclidean geometry universe, and presented two datasets of few-shot
concept learning to test generalization capability in the geometry domain.
Humans considerably outperform vision models on Geoclidean tasks, and we
believe that this gap illustrates the potential for improvement in learning
visual features that align with human sensitivities to geometry. Geoclidean is
thus an important generalization task that vision models are not yet sensitive
to, and an effective benchmark for geometric concept learning. Furthermore,
such explorations of geometric generalization may help us to understand how
human vision made the leap from natural forms to the Platonic forms so
prevalent in modern design and engineering.
We expect minimal negative societal impact from the release of Geoclidean. We
hope future work can build on the foundations of Geoclidean for augmenting
vision models in areas such as geometric reasoning and construction, as well
as in applications such as education, where geometry is both an essential
academic subject and an introduction to proof-based mathematics.
#### Acknowledgements
We thank Gabriel Poesia and Stephen Tian for providing valuable feedback on
the paper. This work is in part supported by the Stanford Institute for Human-
Centered Artificial Intelligence (HAI), Center for Integrated Facility
Engineering (CIFE), Analog, Autodesk, IBM, JPMC, Salesforce, and Samsung. JH
is supported by the Knight Hennessy fellowship and the NSF Graduate Research
Fellowship.
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# 3D fictitious wave domain CSEM inversion by adjoint source estimation
Pengliang Yang1
1 School of Mathematics, Harbin Institute of Technology, 150001, Harbin, China
E-mail<EMAIL_ADDRESS>
###### Abstract
Marine controlled-source electromagnetic (CSEM) method has proved its
potential in detecting highly resistive hydrocarbon bearing formations. A
novel frequency domain CSEM inversion approach using fictitious wave domain
time stepping modelling is presented. Using Lagrangian-based adjoint state
method, the inversion gradient with respect to resistivity can be computed by
the product between the forward and adjoint fields. Simulation of the adjoint
field using the same modelling engine is challenging as it requires time
domain adjoint source time functions while only a few discrete frequencies of
the data residual are available for the inversion. A regularized linear
inverse problem is formulated in order to estimate a long time series from
very few frequency samples. It can then be solved using linear optimization
technique, yielding a matrix-free implementation. Instead of computing adjoint
source time function one by one at each receiver location, a basis function
implementation has been developed such that the inverse problem can be solved
only once and reused every time to construct all time-domain adjoint sources.
The method allows computing all frequencies of the EM fields in one go without
heavy memory and computational overhead, making efficient 3D CSEM inversion
feasible. Numerical examples are employed to demonstrate the application of
our method.
## 1 Introduction
Controlled-source electromagnetic (CSEM) method is a well established
technology to do geophysical exploration (Chave and Cox, 1982; Constable et
al., 1986). It can also be configured for air-borne (Chang-Chun et al., 2015)
and cross-well (Alumbaugh and Newman, 1997) geometries. Land CSEM has been a
commonplace to find mineral deposit (Ward and Hohmann, 1988; Zhdanov and
Keller, 1994; Grayver et al., 2014). As a complement to seismic measurement,
marine CSEM have been successfully applied to detect hydrocarbon bearing
formations in oil and gas industry (Eidesmo et al., 2002; Ellingsrud et al.,
2002; Constable and Srnka, 2007; MacGregor and Tomlinson, 2014), using vessel-
towed dipole source over an array of receivers deployed on the seabed (hence
the name seabed logging). The key to these applications is the good
sensitivity of the electromagnetic signals to distinguish high contrast in
resistivity between saline-filled rocks and hydrocarbons (Abubakar et al.,
2008). This property makes CSEM an ideal tool for de-risking (MacGregor et
al., 2007) in reservoir exploration prior to drilling, as well as 4D
monitoring during the production (Shantsev et al., 2020). A topical review on
marine CSEM inversion has been given by Constable (2010).
CSEM inversion, also known as resistivity tomography, has been a standard
technique to translate CSEM data into electrical properties of the subsurface.
To determine the resistivity model, a nonlinear inverse problem is then
formulated to iteratively minimize the difference between the observed EM data
and the synthetic data derived from numerical modelling, using different
optimization schemes, such as nonlinear conjugate gradient method (Gribenko
and Zhdanov, 2007; Commer and Newman, 2008), quasi-Newton l-BFGS (Plessix and
Mulder, 2008; Schwarzbach and Haber, 2013), and Gauss-Newton method (Constable
et al., 1987; Abubakar et al., 2008; Zaslavsky et al., 2013).
The most computation intensive part of CSEM inversion is the numerical
simulation of 3D electromagnetic field. One can consider the time-domain
finite-difference method (Oristaglio and Hohmann, 1984; Wang and Hohmann,
1993; Taflove and Hagness, 2005), the frequency-domain finite-difference
method (Newman and Alumbaugh, 1995; Smith, 1996a; Mulder, 2006; Streich,
2009), and the frequency-domain finite-element method (Li and Key, 2007; da
Silva et al., 2012; Key, 2016; Rochlitz et al., 2019).
A plethora of CSEM modelling tools directly solve the linear system based on
the frequency domain solution of the discretized Maxwell equation. This avoids
the high computational cost by direct discretization of time-domain diffusive
Maxwell equation involving extremely large number of time steps dictated by
the restrictive stability condition. Both the frequency domain finite
difference method and finite element method formulate Maxwell equation as a
matrix-based linear system, which may be solved using direct (Streich, 2009)
or iterative (Smith, 1996b; Mulder, 2006; Puzyrev et al., 2013) solvers.
Modelling by direct solver is very attractive for multi-source CSEM problems
(Streich, 2009), as the matrix system can be factorized only once and reused
for all sources by forward-backward substitution. Since the inversion of the
large sparse matrix for 3D problems involves huge amount of memory resources,
direct methods may easily go beyond the memory capacity of a desktop computer.
Iterative solvers require much less memory storage but may be time-consuming
and difficult to converge due to ill-conditioning of the discretized Helmholtz
matrix when the conductivity/resistivity model is highly heterogeneous or the
modelling grid is severely stretched (Mulder, 2006). While the frequency
domain finite element method is flexible to address complex model geometries
thanks to the pre-computed meshes, the meshing in 3D geometries using
tetrahedral and hexahedral mesh itself is a challenging and time consuming
task.
In the time-domain, the fields are updated at each time step, allowing the
modelling over the same memory units. The frequency domain fields can be
integrated on the fly during time-stepping, such that multiple frequencies can
be extracted from the same simulation. This avoids repeating several times of
the modelling for different frequencies in frequency domain methods. It
motivates Maaø (2007) to propose a modified wave domain approach to
significantly speed up the computation of diffusive electromagnetic modelling.
The method has been adapted in Støren et al. (2008) to efficiently compute the
inversion gradient for 3D industrial scale applications. The modified wave
domain approach by Maaø (2007) has an attenuation/diffusive term. Inspired by
the fictitious wave domain approach initially proposed in Lee et al. (1989),
Mittet (2010) transformed the diffusion domain Maxwell equation into a pure
wave domain.
A straightforward time-domain discretization of the diffusive Maxwell equation
leads to the temporal sampling proportional to the square of grid spacing, due
to the requirement of the stability condition ($\Delta t\propto\Delta x^{2}$).
This means taking half the spatial sampling will lead to a quadratic increase
of the number of time steps for a simulation of the same duration.
Transforming the diffusive domain into wave domain using fictitious wave
domain approach allows a linear increase of time step with the use of finer
grid spacing ($\Delta t\propto\Delta x$). This yields a highly efficient
scheme to compute the same frequency domain EM field, as it significantly
reduces the required number of time steps. These advantages inspire us to
adopt the fictitious wave domain method for CSEM inversion.
The major contribution of this paper is to develop an efficient frequency
domain CSEM inversion scheme based on the fictitious wave domain modelling.
The gradient of the misfit functional for frequency domain CSEM inversion
requires both a forward and an adjoint field. Efficient computation of the
frequency domain adjoint field by time domain modelling manifests itself as a
major challenge in this development as it requires the time domain adjoint
source time functions at all receiver locations which are unavailable due to
the formulation of CSEM inversion in frequency domain. The key novelty of our
approach is to estimate the time-domain adjoint source based on only a few
discrete frequency samples of the data residual. To do so, we formulate a
regularized linear inverse problem, and then present an efficient, matrix-free
implementation using very few basis functions. The linear inverse problem can
be solved only once and the resulting solution can be reused to derive the
adjoint source time functions at all receiver locations. This novel
development delivers an efficient 3D CSEM inversion methodology in fictitious
wave domain. We finally apply our method to two examples for numerical
demonstration.
## 2 CSEM modelling in fictitious wave domain
Within a three dimensional space $X$, the diffusive Maxwell equations are
governed in the frequency domain
$\begin{cases}\nabla\times
E(\mathbf{x},\omega;\mathbf{x}_{s})-\mathrm{i}\omega\mu(\mathbf{x})H(\mathbf{x},\omega;\mathbf{x}_{s})=M(\mathbf{x}_{s},\omega),\quad\mathbf{x}\in
X,\omega\in\Omega\\\ \nabla\times
H(\mathbf{x},\omega;\mathbf{x}_{s})-\sigma(\mathbf{x})E(\mathbf{x},\omega;\mathbf{x}_{s})=J(\mathbf{x}_{s},\omega),\quad\mathbf{x}\in
X,\omega\in\Omega\end{cases},$ (1)
where the variables $\mathbf{x}$, $t$ and $\omega$ denote space, time and
frequency, respectively. The convention of Fourier transform
$\partial_{t}\leftrightarrow-\mathrm{i}\omega$ has been adopted above. The
electrical and magnetic fields ($E$ and $H$) are vectors consisting of 3
components in x, y and z directions; $J$ and $M$ stand for electrical and
magnetic sources at the location $\mathbf{x}_{s}$, respectively. The magnetic
permeability is $\mu$. The conductivity $\sigma$ is a symmetric $3\times 3$
tensor, i.e., $\sigma_{ij}=\sigma_{ji}$, $i,j\in\\{x,y,z\\}$. In the isotropic
medium, only the diagonal elements of the conductivity tensor are non-zeros
and the same in all directions: $\sigma_{xx}=\sigma_{yy}=\sigma_{zz}$;
$\sigma_{ij}=0,i\neq j$. In the vertical transverse isotropic (VTI) medium,
the diagonal elements are different in horizontal and vertical directions:
$\sigma_{h}:=\sigma_{xx}=\sigma_{yy},\quad\sigma_{v}=\sigma_{zz}$, where
$\sigma_{h}$ and $\sigma_{v}$ stand for horizontal conductivity and vertical
conductivity, respectively. The resistivity is defined as the inverse of the
conductivity, i.e., $\rho_{ij}=1/\sigma_{ij}$.
The above diffusive Maxwell equation can be converted into wave domain
(Mittet, 2010), by defining a fictitious di-electrical permittivity in
equation (1) as $\sigma=2\omega_{0}\varepsilon$, while multiplying the 2nd
equation in (1) with $\sqrt{-\mathrm{i}\omega/2\omega_{0}}$:
$\begin{cases}\nabla\times
E^{\prime}(\mathbf{x},\omega;\mathbf{x}_{s})-\mathrm{i}\omega^{\prime}\mu
H^{\prime}(\mathbf{x},\omega;\mathbf{x}_{s})=M^{\prime}(\mathbf{x}_{s},\omega),\quad\mathbf{x}\in
X,\omega\in\Omega\\\ \nabla\times
H^{\prime}(\mathbf{x},\omega;\mathbf{x}_{s})+\mathrm{i}\omega^{\prime}\varepsilon
E^{\prime}(\mathbf{x},\omega;\mathbf{x}_{s})=J^{\prime}(\mathbf{x}_{s},\omega),\quad\mathbf{x}\in
X,\omega\in\Omega\end{cases}$ (2)
based on the following correspondence relation
$E^{\prime}=E,\;H^{\prime}=\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}H,\;M^{\prime}=M,\;J^{\prime}=\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}J,\;\omega^{\prime}=(1+\mathrm{i})\sqrt{\omega\omega_{0}}.$
(3)
The time domain counterpart of equation (2) reads
$\begin{cases}\nabla\times
E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})+\mu(\mathbf{x})\partial_{t^{\prime}}H^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})=M^{\prime}(\mathbf{x}_{s},t^{\prime}),\quad\mathbf{x}\in
X,t^{\prime}\in T\\\ \nabla\times
H^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})-\varepsilon(\mathbf{x})\partial_{t^{\prime}}E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})=J^{\prime}(\mathbf{x}_{s},t^{\prime}),\quad\mathbf{x}\in
X,t^{\prime}\in T,\end{cases}$ (4)
which allows us to do efficient modelling using leap-frog finite-difference
time-domain (FDTD) method over the staggered grid. From the electromagnetic
fields in the time domain, the frequency domain fields can be integrated
during modelling using the transformation
$u(\mathbf{x},\omega;\mathbf{x}_{s})=\int_{0}^{T_{\max}}u(\mathbf{x},t^{\prime};\mathbf{x}_{s})e^{\mathrm{i}\omega^{\prime}t^{\prime}}\mathrm{d}t^{\prime}=\int_{0}^{T_{\max}}u(\mathbf{x},t^{\prime};\mathbf{x}_{s})e^{-\sqrt{\omega\omega_{0}}t^{\prime}}e^{\mathrm{i}\sqrt{\omega\omega_{0}}t^{\prime}}\mathrm{d}t^{\prime},$
(5)
where
$u(\mathbf{x},t^{\prime};\mathbf{x}_{s})\in\\{E^{\prime}_{x},E^{\prime}_{y},E^{\prime}_{z},H^{\prime}_{x},H^{\prime}_{y},H^{\prime}_{z}\\}$
denotes the electric/magnetic field at the spatial location $\mathbf{x}\in X$
and the fictitious time $t^{\prime}\in[0,T_{\max}]$ excited by a source from
the spatial location $\mathbf{x}_{s}$; $T_{\max}$ is the final time that the
frequency domain field $u(\mathbf{x},\omega;\mathbf{x}_{s})$ reaches its
steady state. Equation (5) shows that the complex-valued frequency
$\omega^{\prime}=(1+\mathrm{i})\sqrt{\omega\omega_{0}}$ leads to an
exponentially decay factor in the time integration, which is the key to
realize the attenuation effect of diffusive EM field during time evolution.
To model the CSEM response as precise as possible, we resort to our newly
developed high-order finite-difference time-domain method over non-uniform
grid (Yang and Mittet, 2023; Yang, 2023). To satisfy homogeneous boundary
condition approximately, perfectly matched layers (PML) (Komatitsch and
Martin, 2007) are padded surrounding the domain of interest to absorb the
reflections in the truncated domain. The implementation of (5) can be carried
out on the fly during timestepping modelling thanks to the discrete time
Fourier transform (DTFT):
$u(\mathbf{x},\omega_{k};\mathbf{x}_{s})=\sum_{n=0}^{N_{t}-1}u(\mathbf{x},t^{\prime}_{n};\mathbf{x}_{s})\exp(-\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n})\exp(\mathrm{i}\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n}),$
(6)
where the discretized fictitious time is $t^{\prime}_{n}=n\Delta t$ with time
step $\Delta t$ and time index $n=0,\cdots,N_{t}$, while the interested
frequencies are $\omega_{k}$, $k=1,\cdots,N_{\omega}$. Note that the total
number of time steps $N_{t}$ is much large than the total number of discrete
frequencies $N_{\omega}$, i.e., $N_{t}\gg N_{\omega}$. The modelling
terminates when the electric field
$E^{\prime}(\mathbf{x},\omega;\mathbf{x}_{s})$ and the magnetic field
$H^{\prime}(\mathbf{x},\omega;\mathbf{x}_{s})$ evolves to the stead state.
This implies that we may check the convergence of the field regularly during
the modelling to avoid additional timesteppings which have negligible
contributions to the time integral. It is noteworthy that multiple frequencies
can be integrated on the fly during the same timestepping procedure. These
advantages make the method attractive for forward modelling.
## 3 Fictitious wave domain CSEM inversion
The CSEM inversion is an iterative nonlinear optimization procedure. At each
iteration, the gradient of the data misfit with respect to conductivity can be
computed via the product between forward and adjoint fields. To use fictitious
wave domain modelling engine for adjoint simulation, we present a novel
approach to estimating the adjoint source time functions by formulating a
regularized linear inverse problem. An in-depth analysis shows that this
linear inverse problem needs to be solved only once to result in a limited
number of basis functions, which can then be used to construct the time-domain
adjoint source at different receiver locations. This key development enables
efficient and practical 3D CSEM inversion by fictitous wave domain modelling
engine.
### 3.1 CSEM inverse problem
The Maxwell equations forms a linear system as
$\underbrace{\begin{bmatrix}-\sigma&\nabla\times\\\
\nabla\times&-\mathrm{i}\omega\mu\end{bmatrix}}_{\mathbf{A}(m)}\underbrace{\begin{bmatrix}E\\\
H\end{bmatrix}}_{\mathbf{u}}=\underbrace{\begin{bmatrix}J\\\
M\end{bmatrix}}_{\mathbf{f}},$ (7)
where the electromagnetic propagator $\mathbf{A}(m)$ is a linear operator
applied to the vector field $\mathbf{u}$, which gathers electrical and
magnetic fields (each component has been denoted by
$u:=u(\mathbf{x},\omega;\mathbf{x}_{s})$ in the previous section). The source
vector with nonzero excitation at the source location $\mathbf{x}_{s}$ is
prescribed by $\mathbf{f}:=\mathbf{f}(\mathbf{x},\omega;\mathbf{x}_{s})$. Note
that all field variables are functions of both frequency and space
($u(\mathbf{x},\omega;\mathbf{x}_{s}),\mathbf{x}\in X,\omega\in\Omega$), while
the medium property is only a function of space ($m(\mathbf{x}),\mathbf{x}\in
X$), which can be the conductivity $\sigma_{ij}$ or the permeability $\mu$.
To find the resistivity of the subsurface, the data misfit is defined in
least-squares sense
$\phi_{d}(m)=\frac{1}{2}\Big{\|}\underbrace{\begin{bmatrix}\mathbf{W}_{1}&0\\\
0&\mathbf{W}_{2}\end{bmatrix}}_{\mathbf{W}}\Big{(}\underbrace{\begin{bmatrix}E^{obs}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})\\\
H^{obs}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})\end{bmatrix}}_{\mathbf{d}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})}-\underbrace{\begin{bmatrix}E(\mathbf{x}_{r},\omega;\mathbf{x}_{s})\\\
H(\mathbf{x}_{r},\omega;\mathbf{x}_{s})\end{bmatrix}}_{\mathbf{d}_{syn}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})[m]}\Big{)}\Big{\|}^{2}=\frac{1}{2}\|\mathbf{W}(\mathbf{d}-\mathbf{R}\mathbf{u})\|^{2},$
(8)
where $\mathbf{d}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})$ denotes the observed
EM data at the receiver location $\mathbf{x}_{r}$ due to the source at the
location $\mathbf{x}_{s}$, while the synthetic data
$\mathbf{d}_{syn}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})[m]:=\mathbf{u}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})=\mathbf{R}\mathbf{u}(\mathbf{x},\omega;\mathbf{x}_{s})$
(simulated with the model parameter $m$) are extracted by the restriction
operator $\mathbf{R}$ from the modelled wavefield at the receiver location
$u(\mathbf{x}_{r},\omega;\mathbf{x}_{s})=\int_{X}u(\mathbf{x},\omega;\mathbf{x}_{s})\delta(\mathbf{x}-\mathbf{x}_{r})\mathrm{d}\mathbf{x},\quad\mathbf{x}\in
X,\omega\in\Omega$ (9)
where $u(\mathbf{x}_{r},\omega;\mathbf{x}_{s})$ is one component of the vector
field $\mathbf{u}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})$. In the remainder of
the paper, we shall drop the dependence of
$(\mathbf{x}_{r},\omega;\mathbf{x}_{s})$ and the summation over sources and
receivers without loss of clarity unless clearly stated if necessary. Two
weighting matrices, $\mathbf{W}_{1}$ and $\mathbf{W}_{2}$ together forming the
diagonal weighting matrix $\mathbf{W}$, are employed to weight the electric
and magnetic fields respectively. These weighting matrices may be specified
according to the uncertainty model in terms of the real acquisition and the
equipment (Mittet and Morten, 2012). If the magnetic data are not considered
in the inversion, we simply set $\mathbf{W}_{2}=0$.
Taking the first derivative of equation (7) with respect to model parameter
$m$ gives
$\frac{\partial\mathbf{u}}{\partial
m}=-\mathbf{A}^{-1}(m)\frac{\partial\mathbf{A}(m)}{\partial m}\mathbf{u}.$
(10)
The gradient of the data misfit with respect to the model parameter $m$ is
$\begin{split}\frac{\partial\phi_{d}(m)}{\partial
m}=&\Re\langle\mathbf{W}\mathbf{R}\frac{\partial\mathbf{u}}{\partial
m},\mathbf{W}(\mathbf{R}\mathbf{u}-\mathbf{d})\rangle\\\
=&-\Re\langle\mathbf{W}\mathbf{R}\mathbf{A}^{-1}(m)\frac{\partial\mathbf{A}(m)}{\partial
m}\mathbf{u},\mathbf{W}(\mathbf{R}\mathbf{u}-\mathbf{d})\rangle\\\
=&\Re\langle\frac{\partial\mathbf{A}(m)}{\partial
m}\mathbf{u},\underbrace{(\mathbf{A}^{\dagger})^{-1}(m)\mathbf{R}^{\dagger}\mathbf{W}^{\dagger}\mathbf{W}(\mathbf{d}-\mathbf{R}\mathbf{u}}_{\mathbf{v}})\rangle\\\
=&\Re\langle\frac{\partial\mathbf{A}(m)}{\partial
m}\mathbf{u},\mathbf{v}\rangle\\\
=&\Re\sum_{s}\sum_{\omega}\bar{\mathbf{v}}^{\mathrm{T}}\frac{\partial\mathbf{A}(m)}{\partial
m}\mathbf{u},\end{split}$ (11)
where $\Re$ takes the real part of a complex number, $\dagger$ is the complex
conjugate transpose. It is noteworthy that the newly introduced variable
$\mathbf{v}$, coined adjoint variable, or co-state variable in optimal control
theory, must satisfy
$\mathbf{A}^{\dagger}(m)\mathbf{v}=\mathbf{R}^{\mathrm{T}}\mathbf{W}^{\mathrm{T}}\mathbf{W}(\mathbf{d}-\mathbf{R}\mathbf{u}).$
(12)
This shows that $\mathbf{v}$ satisfies another Maxwell equation based on
adjoint Maxwell operator $\mathbf{A}^{\dagger}(m)$. The right hand side of the
adjoint equation acts as the virtual source to emanate the adjoint field.
Equation (11) is a generic expression applicable in fully anisotropic medium
for the model parameters $m\in\\{\sigma_{ij},\mu\\}$. Let us point out that
the above gradient expression may be scaled with the local cell volume $\Delta
V(\mathbf{x})$ due to the discretization of the spatial integral
$\int_{X}\mathrm{d}\mathbf{x}$. In the CSEM settings, the magnetic
permeability $\mu$ is considered as the same constant as in the vacuum. We are
interested in retrieving the conductivity $\sigma$ (or the resistivity
$\rho=1/\sigma$) which may exhibit anisotropy. Equation (7) implies
$\frac{\partial\mathbf{A}(m)}{\partial\sigma}=\frac{\partial}{\partial\sigma}\begin{bmatrix}-\sigma&\nabla\times\\\
\nabla\times&-\mathrm{i}\omega\mu\end{bmatrix}=\begin{bmatrix}-1&0\\\
0&0\end{bmatrix}.$ (13)
Denote the adjoint field
$\mathbf{v}=(\underline{E},\underline{H})^{\mathrm{T}}$, where the underline
is utilized to distinguish it from the forward field. The gradient in equation
(11) becomes
$\frac{\partial\phi_{d}}{\partial\sigma_{ij}}=-\Re\sum_{s}\sum_{\omega}\underline{\bar{E}}_{j}\cdot
E_{i},\qquad i,j\in\\{x,y,z\\}.$ (14)
For commonly used VTI medium, the gradients of the misfit with respect to
horizontal conductivity $\sigma_{h}$ and vertical conductivity $\sigma_{v}$
are
$\frac{\partial\phi_{d}}{\partial\sigma_{h}}=-\Re\sum_{s}\sum_{\omega}(\underline{\bar{E}}_{x}\cdot
E_{x}+\underline{\bar{E}}_{y}\cdot
E_{y}),\quad\frac{\partial\phi_{d}}{\partial\sigma_{v}}=-\Re\sum_{s}\sum_{\omega}\underline{\bar{E}}_{z}\cdot
E_{z}.$ (15)
In isotropic medium ($\sigma_{ii}=\sigma$, $i=x,y,z$), we have the simple
gradient expression
$\frac{\partial\phi_{d}}{\partial\sigma}=-\Re\sum_{s}\sum_{\omega}\underline{E}^{\dagger}\cdot
E=-\Re\sum_{s}\sum_{\omega}(\underline{\bar{E}}_{x}\cdot
E_{x}+\underline{\bar{E}}_{y}\cdot E_{y}+\underline{\bar{E}}_{z}\cdot E_{z}).$
(16)
### 3.2 Adjoint modelling in fictitious wave domain
To facilitate the computation of the adjoint field and the inversion gradient,
one may take the conjugate of equation (12):
$\mathbf{A}^{\mathrm{T}}(m)\bar{\mathbf{v}}=\overline{\mathbf{R}^{\mathrm{T}}\mathbf{W}^{\mathrm{T}}\mathbf{W}(\mathbf{d}-\mathbf{R}\mathbf{u})}.$
(17)
Since $\mathbf{A}^{\mathrm{T}}(m)=\mathbf{A}(m)$, the above operation allows
us to use the same modelling engine to compute the conjugate of the adjoint
field, as long as proper adjoint source can be provided for adjoint
simulation. Unfortunately, for CSEM inversion in the frequency domain, only
few discrete frequencies are available, leading to the adjoint source (i.e.
the data residual) specified only at few discrete frequencies. A time domain
adjoint source is therefore not directly accessible.
To still use fictitious wave domain timestepping modelling for adjoint
simulation, let us now split the weighted data residual into electrical and
magnetic components
$\overline{\mathbf{R}^{\mathrm{T}}\mathbf{W}^{\mathrm{T}}\mathbf{W}(\mathbf{d}-\mathbf{R}\mathbf{u})}=(\overline{\delta
d_{E}},\overline{\delta d_{H}})^{\mathrm{T}}$ and repeat the substitution and
multiplication operations as in Section 2:
$\begin{cases}\nabla\times\underbrace{\overline{\underline{E}}}_{\underline{\overline{E}}^{\prime}}+\mu\underbrace{\sqrt{-\mathrm{i}2\omega\omega_{0}}}_{-\mathrm{i}\omega^{\prime}}\underbrace{\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\underline{\overline{H}}}_{\underline{\overline{H}}^{\prime}}=\overline{\delta
d_{E}},\\\
\nabla\times\underbrace{\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\underline{\overline{H}}}_{\underline{\overline{H}}^{\prime}}-\underbrace{\sqrt{-\mathrm{i}2\omega\omega_{0}}}_{-\mathrm{i}\omega^{\prime}}\varepsilon\underline{\overline{E}}=\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\overline{\delta
d_{H}}.\end{cases}$ (18)
We need to consider the time domain counterpart of the above system for
efficient adjoint modelling by fictitious timestepping modelling. Switching
from diffusive frequency domain back to fictitious time domain with wave
should be understood as a linear transformation rather than the true inverse
Fourier transform. This is because the mapping of fictitious transform is not
bijective, as the number of time steps $N_{t}$ for numerical simulation is
significantly larger than the number of discrete frequencies $N_{\omega}$ used
for CSEM investigation ($N_{\omega}\ll N_{t}$). There exists an infinite
number of time series which may match the few discrete frequencies. Searching
for the inverse is therefore an under-determined problem.
Let us denote the right hand side of equation (18) ($\overline{\delta d_{E}}$
or $\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\overline{\delta d_{H}}$) at a
specific receiver location as $s(\omega)$. To retrieve a long time series for
adjoint modelling, a linear inverse problem is formulated to convert the
adjoint source $s(\omega)$ from frequency to time domain based on DTFT in
equation (6) by minimizing the following misfit functional
$\begin{split}\Psi&=\sum_{k=1}^{N_{\omega}}|s(\omega_{k})-\sum_{n=0}^{N_{t}-1}e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n}}e^{\mathrm{i}\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n}}s(t^{\prime}_{n})|^{2}+\gamma\sum_{n=0}^{N_{t}-1}|s(t^{\prime}_{n})|^{2}\\\
&=\sum_{k=1}^{N_{\omega}}\Big{(}|\Re\\{s(\omega_{k})\\}-\sum_{n=0}^{N_{t}-1}e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n}}\cos(\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n})s(t^{\prime}_{n})|^{2}\\\
&\qquad+|\Im\\{s(\omega_{k})\\}-\sum_{n=0}^{N_{t}-1}e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n}}\sin(\sqrt{\omega_{k}\omega_{0}}t^{\prime}_{n})s(t^{\prime}_{n})|^{2}\Big{)}+\gamma\sum_{n=0}^{N_{t}-1}|s(t^{\prime}_{n})|^{2}\end{split}$
where $\Im$ takes the imaginary part of the complex variable. If we discretize
the time as $t_{n}=n\Delta t$ and define
$\mathbf{s}(\omega):=\begin{bmatrix}\Re\\{s(\omega_{1})\\}\\\ \vdots\\\
\Re\\{s(\omega_{N_{\omega}})\\}\\\ \Im\\{s(\omega_{1})\\}\\\ \vdots\\\
\Im\\{s(\omega_{N_{\omega}})\\}\end{bmatrix},\mathbf{s}(t^{\prime}):=\begin{bmatrix}s(t^{\prime}_{0})\\\
\vdots\\\ s(t^{\prime}_{N_{t}-1})\end{bmatrix}$ (19)
and
$\mathbf{B}:=\begin{bmatrix}e^{-\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{0}}\cos(\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{0})&\cdots&e^{-\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{N_{t}-1}}\cos(\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{N_{t}-1})\\\
\vdots&\ddots&\vdots\\\
e^{-\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{0}}\cos(\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{0})&\cdots&e^{-\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{N_{t}-1}}\cos(\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{N_{t}-1})\\\
e^{-\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{0}}\sin(\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{0})&\cdots&e^{-\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{N_{t}-1}}\sin(\sqrt{\omega_{1}\omega_{0}}t^{\prime}_{N_{t}-1})\\\
\vdots&\ddots&\vdots\\\
e^{-\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{0}}\sin(\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{0})&\cdots&e^{-\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{N_{t}-1}}\sin(\sqrt{\omega_{N_{\omega}}\omega_{0}}t^{\prime}_{N_{t}-1})\\\
\end{bmatrix},$ (20)
the objective of the above linear inverse problem can be compactly written in
matrix form
$\Psi=\|\mathbf{s}(\omega)-\mathbf{B}\mathbf{s}(t^{\prime})\|^{2}+\gamma\|\mathbf{s}(t^{\prime})\|^{2},$
(21)
where a regularization term taking into account the minimum energy of the
solution has been penalized by a parameter $\gamma$ to stabilize the linear
inversion and to determine a unique solution. Note that the complex factors
have been split into real and imaginary part. To solve this ill-posed problem,
regularization has been introduced to recover the well-posedness during the
inversion of the matrix $\mathbf{B}\in\mathbb{R}^{2N_{\omega}\times N_{t}}$.
The solution $\mathbf{s}(t)$ can easily be found using linear optimization
algorithms, e.g. LSQR (Paige and Saunders, 1982) and the conjugate gradient
method for the resulting normal equation (CGNR) (Saad, 2003). Based on the
frequency and time index, the each element of $\mathbf{B}$ can be formed on
the fly when computing matrix vector product, implying a matrix free
implementation.
It should be noted that fictitious wave domain modelling is simply a
mathematical tool to efficiently compute frequency domain EM fields.
Disguising a diffusive phenomenon as a wave event will not change the
diffusive nature of the underlying physics. Indeed, Mittet (2010) proposed
another method to calculate the adjoint source time function by superposition
of delayed causal wavelets. However, it yields a nonlinear inverse problem
which is very difficult to solve. Clearly, embedding a linear inverse problem
in a nonlinear inverse problem is a much better option than embedding double
nonlinear inverse problems. The truncated singular value decomposition (SVD)
have been applied (Støren et al., 2008) to invert for an adjoint source,
following the modified wave domain formulation of Maaø (2007). Mathematically,
our approach using damped least-squares should give the same solution as the
one from truncated SVD, since SVD is equivalent to finding the Moore-Penrose
pseudo-inverse for $\mathbf{B}$ (Björck, 1996, theorem 1.2.10 in p. 15, and
section 2.7.2, p. 101). However, our empirical experience shows that the
approximate inverse given by SVD suffers from numerical instability while
consuming too much memory. The above linear inversion method gets rid of these
issues, incorporating all frequencies in one run with better memory and
computational efficiency.
### 3.3 A matrix-free basis function implementation
Let us denote
$\mathbf{B}^{+}=(\mathbf{B}^{\mathrm{T}}\mathbf{B}+\gamma\mathbf{I})^{-1}\mathbf{B}^{\mathrm{T}}\in\mathbb{R}^{N_{t}\times
2N_{\omega}}$ the Moore-Penrose pseudo-inverse of
$\mathbf{B}\in\mathbb{R}^{2N_{\omega}\times N_{t}}$ such that
$\mathbf{s}(t)=\mathbf{B}^{+}\mathbf{s}(\omega).$ (22)
The matrix $\mathbf{B}^{+}$ may be written down using the $2N_{\omega}$
columns:
$\mathbf{B}^{+}=\begin{bmatrix}\mathbf{b}_{1},\cdots,\mathbf{b}_{N_{\omega}},\mathbf{b}_{N_{\omega}+1},\cdots,\mathbf{b}_{2N_{\omega}}\end{bmatrix}$
(23)
in which $\mathbf{b}_{k}\in\mathbb{R}^{N_{t}}$ is the $i$-th column of
$\mathbf{B}^{+}$. As a result, equation (22) translates into
$\mathbf{s}(t)=\sum_{k=1}^{N_{\omega}}\Re\\{s(\omega_{k})\\}\mathbf{b}_{k}+\Im\\{s(\omega_{k})\\}\mathbf{b}_{k+N_{\omega}}.$
(24)
It then becomes clear that the columns of $\mathbf{B}^{+}$ are the very few
number of basis functions to construct the adjoint source time function. Given
$\Delta t$ and $N_{t}$, these basis functions are uniquely determined, hence
independent of the frequency spectrum $s(\omega)$ (which are simply the
coefficients for the linear combination of these basis functions). This means
the iterative solution procedure needs to be performed only once to find all
$\mathbf{b}_{k}$, which can then be used for each receiver locations to infer
the corresponding adjoint source time function. Equation (24) also gives a
recipe to efficiently compute these functions:
* •
If we set $\Re\\{s(\omega_{k})\\}=1$, $\Im\\{s(\omega_{k})\\}=0$ and
$\Re\\{s(\omega_{j})\\}=\Im\\{s(\omega_{j})\\}=0$ ($j\neq k$), we obtain
$\mathbf{s}(t)=\mathbf{b}_{k}$.
* •
If we set $\Im\\{s(\omega_{k})\\}=1$, $\Re\\{s(\omega_{k})\\}=0$ and
$\Re\\{s(\omega_{j})\\}=\Im\\{s(\omega_{j})\\}=0$ ($j\neq k$), we obtain
$\mathbf{s}(t)=\mathbf{b}_{k+N_{\omega}}$.
By repeatedly feeding the iterative optimization algorithm $2N_{\omega}$ times
using different Dirac delta input
$\mathbf{s}(\omega)=(0,\cdots,0,1,0,\cdots,0)^{\mathrm{T}}$, we obtain the
$2N_{\omega}$ columns of $\mathbf{B}^{+}$.
Figure 1 gives an example of these basis functions using the parameters
$\Delta t=0.002$, $N_{t}=1000$ and $f_{0}=1$ and $N_{\omega}=3$. To solve this
linear optimization problem, Claerbout’s conjugate gradient algorithm
(Claerbout and Fomel, 2008, chapter 2.3.6) (which is essentially a CGNR
algorithm according to Saad (2003, chapter 8.3, pp. 266-268)) has been
applied. As can be seen from Figure 2, within 15 CG iterations, the proposed
method leads to highly accurate estimation for these basis functions, with the
error less than $10^{-8}$. In this test, the regularization parameter $\gamma$
was chosen to be $10^{-3}$. It is found the shape of the basis functions do
not change much using different choices of $\gamma$, indicating that the
method is robust.
For different receiver locations, the data residuals in the frequency domain
may be dramatically different, but these basis functions are the same. We
therefore only solve the above regularized linear inverse problem once to
construct all adjoint source time functions through their linear combination.
The frequency domain data residuals are the weighting coefficients to modulate
these basis.
Figure 1: The basis functions using the parameters $\Delta t=0.004$ s,
$N_{t}=1000$ and $N_{\omega}=3$. The amplitude of these basis functions decays
to zero with the increase of the time, which is quite suitable for frequency
domain EM fields converging to steady state after simulation using sufficient
number of time steps. Figure 2: The convergence of CG iterations for the
solution of basis functions
One thing to remark is that the adjoint modelling in the time domain should
have reverse time order compared with the forward modelling. Taking the
conjugate reverse the time again. This is the reason why a decaying trend of
the estimated basis function is observed in Figure 1.
### 3.4 Connections and distinctions with existing methods
Note the incident field $E_{i}$ is the product between the Green’s function
with the source current
$E_{i}(\mathbf{x},\omega;\mathbf{x}_{s})=\sum_{k}G_{ik}^{EE}(\mathbf{x},\omega;\mathbf{x}_{s})J_{k}(\mathbf{x}_{s},\omega),$
(25)
while the conjugate of adjoint field can be specified as
$\overline{\underline{E}}_{j}(\mathbf{x},\omega;\mathbf{x}_{s})=\int_{X}(\sum_{k}G_{jk}^{EE}(\mathbf{x},\omega;\mathbf{x}_{r})\overline{\delta
d^{E}_{k}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})}+\sum_{i}G_{ji}^{EH}(\mathbf{x},\omega;\mathbf{x}_{r})\overline{\delta
d^{H}_{i}(\mathbf{x}_{r},\omega;\mathbf{x}_{s})})\mathrm{d}\mathbf{x}_{r},$
(26)
where $i,j,k\in\\{x,y,z\\}$ indicate different components of the EM field,
while $G_{ij}^{EE}(\mathbf{x},\omega;\mathbf{x}_{s})$ and
$G_{ik}^{EH}(\mathbf{x},\omega;\mathbf{x}_{s})$ stand for the $i$th component
of the electrical Green’s function for angular frequency $\omega$ at the
spatial location $\mathbf{x}$ due to the electrical and magnetic sources
directed in $j$\- and $k$\- th directions. This implies that we can still use
fictitious wave modelling for adjoint simulation: one can, for each frequency
and each receiver, first compute the Green’s function (which is independent of
the source time function) in frequency domain, and then form the adjoint field
by linear combination of them using the data residual. This method, ensured by
the superposition principle thanks to the linearity of the Maxwell equation,
enables the approach still working out. Such a computing scheme repeats many
times of simulation depending on the number of receivers, which is very
inefficient when the number of receivers becomes extremely large.
Equation (17) shows that the adjoint field can be computed using only one
modelling, provided that a time-domain adjoint source time function is
available and then injected at once to do time-domain simulation. The spirit
of our method is to perform frequency domain inversion using time domain
modelling engine, similar to Sirgue et al. (2008, 2010) for seismic full
waveform inversion (FWI). However, there are a number of differences.
Switching between time and frequency domain is straightforward in seismic FWI
using the definition of discrete inverse Fourier transform. One can convert
frequency domain data residual using the definition of inverse discrete
Fourier transform
$s(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}s(\omega)e^{-i\omega
t}\mathrm{d}\omega\approx\sum_{k=1}^{N_{\omega}}s(\omega_{k})e^{-i\omega_{k}t}.$
(27)
Of course, the same issue presented in Sirgue et al. (2010) approach: only
limited number of frequencies are available to reconstruct a long time series.
The last equality in (27) assumes all the absent frequencies are zeros. Since
Fourier basis is orthonormal, the computation of adjoint source time function
under this assumption leads to a minimum energy solution equivalent to solving
the following least-squares minimization problem
$\min_{s(t)}\sum_{k=1}^{N_{\omega}}\|s(\omega_{k})-\sum_{j=1}^{N_{t}}s(t_{j})e^{\mathrm{i}\omega_{k}t_{j}}\|^{2}.$
(28)
For fictitious wave domain approach, plugging the complex-valued frequency
$\omega^{\prime}=(1+\mathrm{i})\sqrt{\omega\omega_{0}}$ into exponential
factor yields an exponentially decay factor
$e^{\mathrm{i}\omega^{\prime}t^{\prime}}=e^{\mathrm{i}(1+\mathrm{i})\sqrt{\omega\omega_{0}}t^{\prime}}=e^{-\sqrt{\omega\omega_{0}}t^{\prime}}e^{\mathrm{i}\sqrt{\omega\omega_{0}}t^{\prime}},$
(29)
leaving the converted EM field possessing strong attenuation behavior, while
making the basis no more orthogonal. Precisely speaking, the transformation in
(5) is a Laplace transform rather than normal Fourier transform. As a result,
the inverse fictitious transformation is not well defined, that is,
$u(\mathbf{x},t^{\prime};\mathbf{x}_{s})\neq\frac{1}{2\pi}\int_{\Omega}\mathrm{d}\omega
u(\mathbf{x},\omega;\mathbf{x}_{s})e^{-\mathrm{i}\omega^{\prime}t^{\prime}}.$
(30)
Since direct use of inverse Fourier transform as done by Sirgue et al. (2010)
does not apply, a matrix-free iterative solution of adjoint source time
function in this work is evidently desirable.
### 3.5 The final algorithm
The total misfit functional for CSEM inversion consists of both data fitting
and model regularization:
$\phi(m)=\phi_{d}(m)+\beta\phi_{m}(m),$ (31)
where $\phi_{m}(m)$, penalized by the parameter $\beta$, is the model misfit
term to enforce smoothness on the inverted model. A popular choice is to use
Tikhonov regularization to minimize the roughness of the model compared with a
reference model $m_{ref}$, i.e.,
$\phi_{m}(m)=\frac{1}{2}\|\nabla^{\alpha}(m-m_{ref})\|^{2}$, where the
anisotropic first order difference operator
$\nabla^{\alpha}=(\alpha_{x}\partial_{x},\alpha_{y}\partial_{y},\alpha_{z}\partial_{z})^{\mathrm{T}}$
with coefficients $\alpha_{x}$, $\alpha_{y}$ and $\alpha_{z}$ can be tuned by
the user. This leads to the total gradient of the misfit functional prescribed
by $\frac{\partial\phi}{\partial m}=\frac{\partial\phi_{d}}{\partial
m}+\frac{\partial\phi_{m}}{\partial m}$. Based on the gradient information
computed above, a descent direction $\delta m^{k}$ can be constructed to
update the model parameters iteratively
$m^{k+1}=m^{k}+\alpha\delta m^{k},$ (32)
where $\alpha$ is the step length estimated by line search method. In this
paper, the descent direction $\delta m^{k}$ at the $k$-th iteration is
computed using l-BFGS algorithm (Nocedal and Wright, 2006) by storing the
gradients in the previous iterations.
It is worth noting that the inversion gradient computed using the proposed
approach is approximate rather than exact. Consequently, the descent direction
estimated using l-BFGS after several iterations may have difficulty to succeed
in the line search procedure. We therefore propose to restart l-BFGS using the
steepest descent direction after the first failure of the line search. The
algorithm will eventually be terminated if the maximum number of iterations is
reached.
Using conductivity as model parameter creates some kinds of ill-conditioning
due to its value varying in a large range (the seawater is around 0.3
$\Omega\cdot$m while a thin layer of hydrocarbon bearing sediment is more than
100 $\Omega\cdot$m). To capture the high contrast of the parameter variations,
the inversion is re-parameterized using the logarithmic transformation
$m:=\ln\rho$, which is a dimensionless parameter with comparable magnitude
varying in a much smaller dynamic range. Indeed, there exists other types of
parameter scaling, see another example in Abubakar et al. (2008). This re-
parametrization gives a relation $\rho=e^{m}$. Switching the parametrization
is trivial thanks to the chain rule:
$\frac{\partial\phi}{\partial m}=\frac{\partial\rho}{\partial
m}\cdot\frac{\partial\sigma}{\partial\rho}\cdot\frac{\partial\phi}{\partial\sigma}=-\frac{1}{\rho}\frac{\partial\phi}{\partial\sigma}.$
(33)
During the iterative inversion, the model parameter may be bounded within an
upper bound $m_{\max}$ and a lower bound $m_{\min}$ according to a priori
knowledge. These bounds help to stabilize the inversion in a more physically
sensible manner.
## 4 Application examples
In this section we present two numerical examples to demonstrate the
application of our method. The two examples are designed to validate the
gradient expressions for the isotropic (the first example) and the VTI
anisotropic (the second example) cases. In both examples, I use in-line and
azimuth data with $E_{x}$ and $E_{y}$ components. This motivates us to utilize
the uncertainty model proposed in Morten et al. (2009) to compute the diagonal
weight matrix $\mathbf{W}$ in order to capture the varying sensitivity of
broadside EM data. The observed data are poluted by 3% of Gaussian white noise
before inversion. The regularization parameters are configured with
$\beta=0.01$ using a cooling factor (between 0.7 and 1) through iterations. We
set $\alpha_{x}=\alpha_{y}=1$ and $\alpha_{z}=0.1$ in both tests. We choose
the initial model as the reference model for both inverse exercises. To
mitigate the weak sensitivity of the EM field to the deeper part of the model,
a depth preconditioning has been applied following the work of Plessix and
Mulder (2008). The code is parallelized over the sources to achieve the best
scaling performance since the sources are independent of each other.
### 4.1 A two-block model
Our first example is a two-block land model similar to the one presented in
Grayver et al. (2013). The model has a background resistivity of 5
$\Omega\cdot$m, including two anomalies: a low resistivity inclusion of 1
$\Omega\cdot$m and a high resistivity inclusion of 100 $\Omega\cdot$m, as
shown in Figure 3a. The physical dimension of the model spans over a 3D domain
$X=X_{1}\times X_{2}\times X_{3}$, where $X_{1}=[-2000,2000]$ m,
$X_{2}=[-2000,2000]$ m and $X_{3}=[0,1500]$ m. There are 16 receivers and 256
transmitters with equal distance on the surface of this model. Figure 4 gives
the survey layout sheet of the acquisition geometry. The model has been
discretized with the regular grid using $\Delta x=\Delta y=60$ m and $\Delta
z=25$ m. The observed data are generated at two frequencies: 0.25 Hz and 1 Hz.
Figure 3: (a) The true land model (5 $\Omega\cdot$m background with two
anomalous inclusions of 1 $\Omega\cdot$m (left) and 100 $\Omega\cdot$m (right)
); (b) The homogeneous initial model of 5 $\Omega\cdot$m. The models are
clipped at 25 $\Omega\cdot$m for display purpose. Figure 4: Survey layout
sheet for the two-block model. In total, 16 transmitters (marked by dots) and
256 receivers (labeled with triangles) are deployed.
This 3D CSEM inversion runs for 30 iterations. In order to check how the
synthetic data fits the observations, we plot both the amplitude and the phase
of the synthetic data at all receivers together with the observed data using
the initial model and the retrieved resistivity model after inversion. In
Figure 5, the synthetic data created from the homogeneous initial model cannot
match the observed data very well. In Figure 6, the synthetic data generated
from inverted resistivity volume indeed are better aligned with the observed
data. Note that the horizontal axis in Figures 5 and 6 are receiver indices
rather than offset. This is due to the broadside survey configuration. To
better visualize the data matching connected to acquisition geometry, we
design a scatter plot with hot colors indicating the data error at each
receiver location. Figures 7 shows that at the beginning of the inversion, the
significant misfit (defined as
$\|\mathbf{W}(\mathbf{d}_{obs}-\mathbf{d}_{syn})\|$ at each receiver location)
is very large. Since we mute the data within 600 m offset, the significant
misfit surrounding transmitter Tx-10 are zeros. The plot in Figure 7 also
highlights the importance of azimuthal data has much larger significant misfit
based than inline directions. After 30 inversion iterations, the significant
misfit becomes much lower, see Figure 8.
Figure 5: Comparison between observed data and synthetic data from initial
model according to (a) amplitude and (b) phase for transmitter Tx-10. The
horizontal axis is the index of the receivers rather than offset, since we
consider broadside configuration. It is clear that both the amplitude and
phase does not match very well. Note also that the level of the data matching
has a strong correlation with the receiver indices. Figure 6: Comparison
between observed data and synthetic data from inverted model according to (a)
amplitude and (b) phase for transmitter Tx-10. The synthetic data are now
better aligned with the observed data after overlapping display. Figure 7:
Scatter plot of the significant misfit for transmitter Tx-10 at iteration 1.
Figure 8: Scatter plot of the significant misfit for transmitter Tx-10 at
iteration 30. The significant misfit becomes much lower after inversion.
The above data comparison gives us a confidence of our inversion scheme
working properly, according to a specific source (Tx-10). In Figure 9, we plot
the normalized data misfit which is a global measure of the inversion. We see
that after 30 iterations, it arrives at a relatively low normalized misfit.
According to Figure 10a, the frequency of large significant misfit at receiver
locations are distributed in a large range from 0 to 15. After the inversion,
it has been compressed a lot (most of them are clustered within 3), as shown
in Figure 10b. The final inverted model is capable to retrieve both the low
resistivity inclusion (Figure 11a) and the high resistivity inclusion (Figure
11b).
Figure 9: The convergence history in terms of the normalized misfit for the
inversion of the two-block model. Figure 10: The histogram of the significant
misfit at (a) iteration 1 and (b) iteration 30. Figure 11: Inversion result
for (a) low resistivity anomaly and (b) high resistivity anomaly.
### 4.2 A marine CSEM example
In practical CSEM survey for reservoir exploration, a number of towlines using
very low-frequency (0.1-10 Hz), high-energy electrical source will be deployed
with hundreds of receivers. The source-receiver offset extends up to 10 km is
very standard. Here, a 3D CSEM inversion is carried out for 30 iterations
based on a model of size $X=X_{1}\times X_{2}\times X_{3}$, where
$X_{1}=X_{2}=[-9000,9000]$ m and $X_{3}=[0,3500]$ m. Figure 12 illustrates the
survey layout sheet for the test: 10 towlines (5 in x direction and 5 in y
direction) are deployed in order to cover the region of interest. Each towline
has 81 source locations with a separation of 200 m. The receivers (25 in
total) are deployed at the crossings of these towlines. We consider the
reciprocity to switch the source and receiver, to achieve efficiency in
inversion.
As shown in Figure 13a, this synthetic model includes the seawater of 0.3
$\Omega\cdot$m. The resistivity of the background sediment is varying from 1.5
to 2.5 $\Omega\cdot$m along the depth. The most striking feature is a resistor
of 10 $\Omega\cdot$m at shallow part and a disk-shaped resistor of 100
$\Omega\cdot$m sitting at the depth between 2200 m and 2350 m, which mimics
the canonical reservoir. The starting model for the inversion in Figure 13b
takes a homogeneous value 1.5 $\Omega\cdot$m to mimic the situation that no
accurate a priori information is known for inversion. A seafloor bathymetry
with varying depth has been embedded in the model. The model is densely
gridded around bathymetry and above, with the grid spacing $\Delta z=25$ m,
see Figure 14. The grid has been stretched with a constant growing factor at a
certain distance below the seabed.
Three frequencies (0.25 Hz, 0.75 Hz and 2.25 Hz) have been used to perform
this inversion. Figure 15 shows that this CSEM inversion was converging well.
The comparison between Figure 16 and Figure 17 shows that the significant
misfit has been largely reduced for the receiver at the location (0, 0) m.
Note that in Figure 17c, the data misfit for the third frequency at the near
offset is still large, indicating the presence of shallow resistor, which has
not been well recovered in the inverted model (cf. Figure 19). To recover also
the shallow resistor, we should use more near offset data rather than drop off
them since 1000 m. A frequency dependent weighting strategy can be applied to
boost the importance of high frequencies, as they have shallow penetrating
depth according to skin depth. Since the scatter plot of the signficant misfit
does not reflect the global misfit, the significant misfit at each receiver
location has been shown in Figure 18: at the beginning of the inversion, they
are far from 1; after the inversion, the significant misfit at all receiver
locations are close to 1, which is the target value after adding noise into
the observed data.
After 30 l-BFGS iterations, the inversion was successful to recover the
resevoir in both vertical resistivity ($\rho_{v}=1/\sigma_{h}$) and the
horizontal resistivity ($\rho_{h}=1/\sigma_{v}$), but the shallow resistor is
difficult to obtain. The imprint of the shallow resistor can be found just
above the resevoir in our reconstructed model: the horizontal location is
correct but the depth is obviously incorrect. It can be seen that the
recovered anomaly in $\rho_{v}$ is of 100-150 m shallower than the true
location. This highlights the low resolution of CSEM inversion and the
ambuiguity of the depth. The inverted vertical resistivity in Figure 19a is
much better resolved than the horizontal resistivity in Figure 19b. The value
of the retrieved $\rho_{v}$ is higher than the retrieved $\rho_{h}$, while
both are smaller than 100 $\Omega\cdot$m and thicker than the truth. This is
due to the effect of anomalous transverse resistance (ATR) (Mittet and Morten,
2012, equation 11): because the CSEM response is proportional to the product
of resistivity difference times the thickness, lower resistivity with larger
thickness can create similar amplitude response compared with higher
resistivity with thin depth distribution. There is also weak increase of the
resistivity in $\rho_{h}$ in Figure 19b, suggesting that the initial model of
homogeneous background lower than the true model, as can be seen by comparing
with Figure 19c.
Figure 12: Survey layout sheet for marine CSEM inversion. Figure 13: (a) The
true resistivity model (note that the resistivity of the sediment below the
seabed is varying from 1.5 to 2.5 $\Omega\cdot$m); (b) The initial resistivity
model (the resistivity of the sediment takes a homogeneous value 1.5, to mimic
the situation that no accurate a priori information is known). The models are
clipped at 5 $\Omega\cdot$m for display purpose. Figure 14: The vertical
section of the nonuniform grid to mesh the 3D marine CSEM resistivity Figure
15: The convergence history of the normalized misfit for marine CSEM
inversion. Figure 16: Scatter plot of the significant misfit for receiver at
(0, 0) m in iteration 1. Figure 17: Scatter plot of the significant misfit for
receiver at (0, 0) m in iteration 30. The significant misfit becomes much
lower after inversion. Figure 18: The significant misfit at each receiver
location before and after inversion Figure 19: The result of 3D marine CSEM
inversion: (a) vertical resistivity $\rho_{v}$, (b) horizontal resistivity
$\rho_{h}$ and (c) true resistivity model. The display is in logorithmic
scale. Note that in the inverted vertical resistivity, there is a small uphill
variation just above the deep resistor which has a good correspondence to the
shallow resistor in horizontal direction, where the highest resistivity has
been retrieved at the center.
## 5 Discussion
In our method, the modelling jobs for each source can be parallelized
independently with perfect scalability. Since it is based on time stepping,
the EM fields at the next time step will overwrite the memory unit at the
previous step. This makes the memory usage of the method very economically.
Our computation was performed on a Intel(R) Xeon(R) Gold 6258R CPU @ 2.70GHz,
possessing 56 CPU cores (each core has 2 threads) and 128 GB memory. For the
two-block model of 3D grid size 139*139*99 (after padding with PML boundaries
on each side), combination of forward and adjoint modelling for building the
gradient associated with one source only takes the memory of approximately 402
MB to compute all frequencies. (The memory requirement using frequency domain
finite difference direct solver for such a large model size could not even fit
128 GB memory.) This implies that we can simulateneously launch more than 100
modelling jobs (corresponding to 100 independent sources) to fully explore the
small memory footprint of fictitous wave domain modelling in performing large
scale 3D inversion. For 16 sources, we can use 4 threads to parallelize the
computation by OpenMP using shared memory. The modelling takes less time at
the beginning of the inversion, as the model is conductive and has less
inhomogeneties. At the later iterations, each gradient building takes more
computing time than before, because the resistive anomalies gradually come
into the model, the time step $\Delta t$ becomes smaller and the number of
timesteps becomes larger due to stability condition. The inversion with 30
iterations can be completed within 10 hours. On average, building one gradient
(equivalent to one forward plus one adjoint solve) takes roughly 20 minutes.
It should be noted that the time series generated by our method only allows
the matching of the field at given frequencies. The above showcases simply
demonstrate that the proposed method works well, making fictitious wave domain
an attractive modelling engine for 3D practical inversion. However, one has to
be aware that the choice of regularization parameters $\beta$, $\alpha_{x}$,
$\alpha_{y}$, $\alpha_{z}$ are of paramount importance to the final inversion
result. Lelièvre and Farquharson (2013) have made a thorough investigation and
concluded that regularization of noisy data usually requires a lot of apriori
knowledge about the geologic structure, such that the related parameters can
be specified properly. Our choices may be far from perfect, and better choices
should be possible to significantly improve the inversion result, while
avoiding the false positives in imaging. These are, however, irrelevant to the
core contribution of this work.
The derivation in the appendix A develops an alternative adjoint formulation
to avoid solving the regularized linear inverse problem. Equation (40) gives
an explicit expression to build CSEM inversion gradient directly by cross-
correlation of fictitious time domain forward and adjoint fields. However, 3D
implementation of such a strategy is not trivial as it is mandatory to
simultaneously access fictitious time domain forward and adjoint field at each
time step. This is challenging due to the opposite directions in time between
forward and adjoint fields. In doing so, one may think of storing all 3D
wavefield snapshots during thunsands of times and accessing them one by one in
reverse order, which is extremely inefficient due to slow memory acess speed
and heavy IO traffic. A feasible approach is to use the wavefield
reconstruction method by storing the boundary values (Yang et al., 2016b), or
resort to optimal checkpointing strategy (Symes, 2007; Yang et al., 2016a).
Wavefield reconstruction by boundary storage implies each gradient computation
involves three times of simulation, two for forward fields and one for adjoint
fields. Optimal checkpointing requires much higher recomputation ratio and
thus even more computationally expensive. In summary, these strategies demand
either significant memory consumption for storing boundaries of forward fields
at every time step, or sophisticated operations to achieve optimal
combinatorial recomputation of the forward fields, thus left as the future
work. The scheme used in this paper is a hybrid combination of time domain
modelling and frequency domain inversion. To build the gradient, only two
times of modelling are needed, one for forward fields and one for adjoint
fields. Consequently, the gradient computed by product between forward and
adjoint fields in frequency domain becomes the most efficient option among all
equivalent formulations.
## 6 Conclusion
A nonlinear CSEM inversion based on fictitious wave domain modelling using
timestepping simulation has been presented. To construct the gradient using
forward and adjoint fields, the adjoint source time functions are estimated by
solving a regularized linear inverse problem. A matrix-free implementation
using the basis functions is proposed such that the linear inverse problem is
iteratively solved only once and then the resulting basis functions can be
reused to build up all adjoint source time functions at every receiver
location. The method enables efficient 3D CSEM inversion which are
demonstrated by numerical examples. The method will become more attractive for
land CSEM which normally requires much more number of frequencies than marine
CSEM. The proposed method can equally be applied to magnetotelluric (MT)
inversion to decipher the subsurface structure in even larger scale using much
lower frequencies of the EM recordings.
## Computer Code Availability
Name of the code: `libEMMI`
Functionality: Library for 3D controlled-source electromagnetic modelling and
inversion
Contact: Pengliang Yang, Harbin Institute of Technology, China
E-mail<EMAIL_ADDRESS>
System requirements: Linux OS
Programming language: C, Fortran
Result visualization: Python3, Madagascar (Scons) and Gnuplot
Compilation requirement: gcc compiler, mpicc compiler and make
Software required: fftw3 (http://fftw.org/)
The code has been released publically via github repository
https://github.com/yangpl/libEMMI.
## Acknowledgements
The author acknowledges the support from Chinese Fundamental Research Funds
for the Central Universities (AUGA5710010121) and National Natural Science
Fundation of China (42274156). The author is indebted to Rune Mittet for the
inspiring discussions and the encouragement of this work. The alternative
adjoint formulation in the appendix A was due to René-Édourt Plessix.
## Appendix A An alternative adjoint formulation
The above approach uses diffusive Maxwell equation in frequency domain as the
state equation, which holds no matter what kind of modelling engine is used as
the solver. The misfit in (8) can also be expanded as
$\phi_{d}(m)=\frac{1}{2}\sum_{s}\sum_{k}|\mathbf{W}_{1}(E^{obs}(\mathbf{x}_{r},\omega_{k};\mathbf{x}_{s})-RE(\mathbf{x},\omega_{k};\mathbf{x}_{s}))|^{2}+|\mathbf{W}_{2}(H^{obs}(\mathbf{x}_{r},\omega_{k};\mathbf{x}_{s})-RH(\mathbf{x},\omega_{k};\mathbf{x}_{s}))|^{2}.$
Based on equation (4) and the correspondence between wave and diffusive EM
fields, the Lagrangian functional translates a constrained minimization into
an unconstrained optimization
$\begin{split}\mathcal{L}=&\frac{1}{2}\sum_{s}\sum_{k}|\mathbf{W}_{1}(E^{obs}(\mathbf{x}_{r},\omega_{k};\mathbf{x}_{s})-RE(\mathbf{x},\omega_{k};\mathbf{x}_{s}))|^{2}+|\mathbf{W}_{2}(H^{obs}(\mathbf{x}_{r},\omega_{k};\mathbf{x}_{s})-RH(\mathbf{x},\omega_{k};\mathbf{x}_{s}))|^{2}\\\
+&\Re\sum_{s}\sum_{k}\int_{X}\mathrm{d}\mathbf{x}\overline{\lambda_{1}(\mathbf{x},\omega_{k};\mathbf{x}_{s})}\bigg{(}E(\mathbf{x},\omega_{k};\mathbf{x}_{s})-\int_{0}^{T_{\max}}dt^{\prime}E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}}e^{\mathrm{i}\sqrt{\omega_{k}\omega_{0}}t^{\prime}}\bigg{)}\\\
+&\Re\sum_{s}\sum_{k}\int_{X}\mathrm{d}\mathbf{x}\overline{\lambda_{2}(\mathbf{x},\omega_{k};\mathbf{x}_{s})}\bigg{(}H(\mathbf{x},\omega_{k};\mathbf{x}_{s})-\sqrt{-\frac{2\omega_{0}}{\mathrm{i}\omega_{k}}}\int_{0}^{T_{\max}}dt^{\prime}H^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}}e^{\mathrm{i}\sqrt{\omega_{k}\omega_{0}}t^{\prime}}\bigg{)}\\\
+&\sum_{s}\int_{0}^{T_{\max}}dt^{\prime}\int_{X}\mathrm{d}\mathbf{x}\underline{H}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})\cdot\bigg{(}\nabla\times
E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})+\mu^{\prime}\partial_{t}H^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})-M^{\prime}(\mathbf{x}_{s},t^{\prime})\bigg{)}\\\
+&\sum_{s}\int_{0}^{T_{\max}}dt^{\prime}\int_{X}\mathrm{d}\mathbf{x}\underline{E}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})\cdot\bigg{(}\nabla\times
H^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})-\epsilon\partial_{t}E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})-J^{\prime}(\mathbf{x}_{s},t^{\prime})\bigg{)},\end{split}$
(34)
where $\lambda_{1}(\mathbf{x},\omega;\mathbf{x}_{s})$ and
$\lambda_{2}(\mathbf{x},\omega;\mathbf{x}_{s})$ are Lagrangian multipliers in
the frequency domain, while
$\underline{E}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})$ and
$\underline{H}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})$ are Lagrangian
multipliers in the fictitious time domain.
Setting the Lagrangian with respect to state variables to zero
($\partial\mathcal{L}/\partial\lambda_{1}=\partial\mathcal{L}/\partial\lambda_{2}=\partial\mathcal{L}/\partial\underline{E}^{\prime}=\partial\mathcal{L}/\underline{H}^{\prime}=0$)
gives exactly the state equations. The adjoint equations are obtained by
setting the Lagrangian with respect to state variables, i.e.,
$\displaystyle\frac{\partial\mathcal{L}}{\partial
E(\mathbf{x},\omega_{k};\mathbf{x}_{s})}=0$
$\displaystyle\Leftrightarrow\lambda_{1}(\mathbf{x},\omega_{k};\mathbf{x}_{s})=R^{\dagger}\mathbf{W}_{1}^{\dagger}\mathbf{W}_{1}(E^{obs}(\mathbf{x}_{r},\omega_{k})-RE(\mathbf{x},\omega_{k};\mathbf{x}_{s})),$
(35a) $\displaystyle\frac{\partial\mathcal{L}}{\partial
H(\mathbf{x},\omega_{k};\mathbf{x}_{s})}=0$
$\displaystyle\Leftrightarrow\lambda_{2}(\mathbf{x},\omega_{k};\mathbf{x}_{s})=R^{\dagger}\mathbf{W}_{2}^{\dagger}\mathbf{W}_{2}(H^{obs}(\mathbf{x}_{r},\omega_{k};\mathbf{x}_{s})-RH(\mathbf{x},\omega_{k};\mathbf{x}_{s})),$
(35b) $\displaystyle\frac{\partial\mathcal{L}}{\partial
E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})}=0$
$\displaystyle\Leftrightarrow\nabla\times\underline{H}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})+\epsilon\partial_{t}\underline{E}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})=\Re\sum_{k}\overline{\lambda_{1}(\mathbf{x},\omega_{k};\mathbf{x}_{s})}e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}}e^{\mathrm{i}\sqrt{\omega_{k}\omega_{0}}t^{\prime}},$
(35c) $\displaystyle\frac{\partial\mathcal{L}}{\partial
H^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})}=0$
$\displaystyle\Leftrightarrow-\mu\partial_{t}\underline{H}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})+\nabla\times\underline{E}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})=\Re\sum_{k}\sqrt{-\frac{2\omega_{0}}{\mathrm{i}\omega_{k}}}\overline{\lambda_{2}(\mathbf{x},\omega_{k};\mathbf{x}_{s})}e^{-\sqrt{\omega_{k}\omega_{0}}t^{\prime}}e^{\mathrm{i}\sqrt{\omega_{k}\omega_{0}}t^{\prime}}.$
(35d)
Equations (35a) and (35b) reveals that $\lambda_{1}$ and $\lambda_{2}$ are
nothing more than the electric and magnetic components of the weighted data
residual, while equations (35c) and (35d) form a new set of Maxwell system. It
is important to remark that in deriving the above equations, we have tacitly
applied integration by parts in time and space, assuming zero initial
condition of the forward EM fields
$E^{\prime}(\mathbf{x},t^{\prime}=0)=H^{\prime}(\mathbf{x},t^{\prime}=0)=0,\quad\mathbf{x}\in
X,$ (36)
zero final conditions of the adjoint fields
$\underline{E}^{\prime}(\mathbf{x},t^{\prime}=T_{\max})=\underline{H}^{\prime}(\mathbf{x},t^{\prime}=T_{\max})=0,\quad\mathbf{x}\in
X,$ (37)
and homogeneous boundary condition in space for both forward and adjoint
fields
$E^{\prime}(\mathbf{x},\cdot)=H^{\prime}(\mathbf{x},\cdot)=\underline{E}^{\prime}(\mathbf{x},\cdot)=\underline{H}^{\prime}(\mathbf{x},\cdot)=0,\quad\mathbf{x}\in\partial
X.$ (38)
At the saddle point, we again obtain
$\frac{\partial\phi_{d}}{\partial\epsilon}=\frac{\partial\mathcal{L}}{\partial\epsilon}=-\sum_{s}\int_{0}^{T_{\max}}\mathrm{d}t^{\prime}\underline{E}^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s})\cdot\partial_{t}E^{\prime}(\mathbf{x},t^{\prime};\mathbf{x}_{s}).$
(39)
Since $\sigma_{ij}=2\omega_{0}\epsilon_{ij}$, the application of the chain
rule yields
$\frac{\partial\phi_{d}}{\partial\sigma_{ij}}=\frac{\partial\phi_{d}}{\partial\epsilon_{ij}}\frac{\partial\epsilon_{ij}}{\partial\sigma_{ij}}=-\sum_{s}\frac{1}{2\omega_{0}}\int_{0}^{T_{\max}}\mathrm{d}t^{\prime}\underline{E}^{\prime}_{j}(\mathbf{x},t^{\prime};\mathbf{x}_{s})\cdot\partial_{t}E^{\prime}_{i}(\mathbf{x},t^{\prime};\mathbf{x}_{s}).$
(40)
With the gradient at hand, one can then construct the descent direction based
on nonlinear minimization methods to solve the inverse problem.
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|
# FedGPO: Heterogeneity-Aware Global Parameter Optimization for Efficient
Federated Learning
Young Geun Kim Carole-Jean Wu
Korea University ASU / Meta
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Federated learning (FL) has emerged as a solution to deal with the risk of
privacy leaks in machine learning training. This approach allows a variety of
mobile devices to collaboratively train a machine learning model without
sharing the raw on-device training data with the cloud. However, efficient
edge deployment of FL is challenging because of the system/data heterogeneity
and runtime variance. This paper optimizes the energy-efficiency of FL use
cases while guaranteeing model convergence, by accounting for the
aforementioned challenges. We propose FedGPO based on a reinforcement
learning, which learns how to identify optimal global parameters (B, E, K) for
each FL aggregation round adapting to the system/data heterogeneity and
stochastic runtime variance. In our experiments, FedGPO improves the model
convergence time by 2.4 times, and achieves 3.6 times higher energy efficiency
over the baseline settings, respectively.
## 1 Introduction
Federated learning (FL) has recently emerged as a practical framework for
training machine learning (ML) models on a large variety of mobile devices
privately [4, 36, 51, 24]. A shared ML model is trained over E epochs with a
minibatch size of B on K selected devices, where K is a subset of N client
devices participating in the FL task. The K devices then upload the respective
model gradients (or trained model parameters) to the cloud, in order to update
the global model while keeping all the raw data on the device. Thus, FL deals
with the risk of privacy leaks for deep neural network (DNN) training.
While many existing approaches have been proposed to deploy FL efficiently
[51, 11, 6, 42, 35, 36, 26, 69], a fundamental challenge remains — setting
global parameters (B, E, K) round-by-round to ensure efficient edge execution.
These global parameters significantly affect the model quality and convergence
time, as they directly determine the amount of data reflected on the model
gradients [29]. Moreover, they also affect the energy efficiency of
participant devices, because the amount of training computation on each client
device depends on the parameter settings [35]. Therefore, to achieve efficient
FL execution at the edge, finding optimal global parameter settings is
crucial.
Hyperparameter optimization (HPO) has been extensively studied for the
centralized training. Common approaches include grid- and genetic-based
searches [1]. ML-based methods, such as Bayesian Optimization [66, 25], are
also applicable. Typically, ML-based HPO methods tune the hyperparameters
using the accuracy results obtained from iterative DNN training on the entire
dataset. Since it is infeasible to train a model on the entire dataset for
each set of global parameters int the resource-constrained edge execution
environment, tuning the global parameters round-by-round has been considered
as a common practice for FL [49, 29]. However, round-by-round global parameter
tuning is still challenging due to the following unique aspects of FL:
* •
System Heterogeneity and Runtime Variance. There exist a variety types of
system-on-chips (SoCs) with distinct computing performance at the edge [70],
which results in large performance gaps across participating devices.
Furthermore, stochastic runtime variance, including on-device interference
[63] and network stability [30], can even exacerbate the performance
variability [15, 16] across the devices. This results in the straggler
problem, where the training time per aggregation round is determined by the
slowest device, making it difficult to find the optimal global parameters for
each round.
* •
Data Heterogeneity. For model convergence, ensuring training data are
independently and identically distributed (IID) for each and every
participating device is crucial. However, in edge execution environment,
client training data are not guaranteed to be non-IID, as training samples of
an individual user are often not representative of the entire population [5,
50]. The inclusion of non-IID data in training can defer model convergence
[44, 51]. Since global parameters influence the degree of non-IID data
reflected in the model gradients; it is also crucial to adjust the parameters
considering data heterogeneity.
The high degree of heterogeneity and runtime variance makes it challenging to
optimize global parameters using offline, server-based simulations. Static
simulation-based optimization studies cannot adapt to dynamic system/data
heterogeneity or runtime variance. To identify efficient global parameters for
each FL training round under system/data heterogeneity, several approaches
have been proposed recently [49, 29]. However, these prior approaches do not
consider the stochastic nature of edge computing, including performance
interference and network variability. In addition, prior work did not consider
optimizing the energy efficiency of FL, which can lead to increasing energy
footprint at scale [71]. To the best of our knowledge, this is the first work
to tackle energy-efficient global parameter optimization for FL.
This paper proposes an FL global parameter optimization framework based on
reinforcement learning — FedGPO — that dynamically adjusts the global
parameters (B, E, K) to maximize the FL energy efficiency guaranteeing model
quality. The optimization is performed round-by-round over the entire training
process, considering system and data heterogeneity, as well as runtime
variance. Since the optimal (B, E, K) varies with the computation
characteristics of neural networks (NN), performance profiles of participant
device systems, local training sample distributions, and runtime variance, the
enormous design space makes it difficult to enumerate exhaustively. Hence, we
propose a technique based on a reinforcement learning to addres this
optimization formulation. FedGPO identifies the characteristics of NNs and
profiles of devices such as the intensity on-device interference, network
instability, and the distributions of data samples every aggregation round. It
then determines the global parameters for the round, which maximizes energy
efficiency while not deteriorating the training accuracy. Based on the result
of the decision, FedGPO continuously learns and predicts the efficient global
parameters. The key contributions of this work are as follows:
* •
We present performance and energy efficiency characterization for the FL
global parameter design space. The characterization results show that optimal
settings vary across the FL training rounds due to the varying level of the
system/data heterogeneity and the stochastic runtime variance (Section 2).
* •
We propose a global parameter optimization framework, FedGPO, that identifies
the near-optimal global parameter setting for each round, enabling energy-
efficient federated learning (Section 3).
* •
We implement and evaluate FedGPO for FL use cases with 200 mobile devices
encompassing three performance categories: high, medium, and low (Section 5).
Real system-based experiments demonstrate that FedGPO improves the energy
efficiency of the participant devices by 3.6x, while satisfying the accuracy
requirements.
## 2 Background and Motivation
### 2.1 Global Impact of FL Parameters
Algorithm 1 FedAvg
Variable: B, E, K
B is the local minibatch size
E is the number of local epochs
K is the number of participant devices
Constants: N, $\eta$
N is the number of entire devices
$\eta$ is the learning rate
Server executes:
initialize $(B,E,K)$
initialize $w_{0}$
for each round t = 1,2,… do
$S_{t}$ $\leftarrow$ (random set of K clients among N clients)
for each client k $\in$ $S_{t}$ in parallel do
$w^{k}_{t+1}$ $\leftarrow$ ClientUpdate($k$, $w_{t}$)
$w_{t+1}$ $\leftarrow$ $\sum_{k=1}^{K}\frac{n_{k}}{n}w_{t+1}^{k}$
ClientUpdate(k, w): // Run on client k
$B$ $\leftarrow$ (split $P_{k}$ into batches of size B
for each local epoch i from 1 to E do
for batch $b\in B$ do
$w$ $\leftarrow$ $w-\eta\nabla\ell(w;b)$
return w to server
To prevent privacy leaks in ML training, FL is proposed, where edge devices
train a shared global model collaboratively without sharing the on-device data
samples with the cloud [4, 36, 51]. FedAvg is the de-facto FL algorithm [36,
51] (Algorithm 1). For N devices, the server initializes a global model along
with the number of local training epochs E, the local training minibatch size
B, and the number of participant devices K. It also initializes the model
parameters $w_{0}$. In every round t, the server randomly selects K devices
among the N devices and transmits the global model to them. Each selected
device trains the model locally using the on-device data with a batch size of
B over E epochs. After the local execution of the training is finished, each
device transmits the model parameters back to the server. The server then
updates the global model with the average of local parameters.
Global parameters (B, E, K) significantly impact the FL model convergence and
energy efficiency. It is therefore crucial to carefully select the parameters
for better model quality and the energy efficiency of participating devices.
Figure 1 shows the (a) convergence time and (b) global performance per watt
(PPW) of a CNN model with the MNIST dataset (CNN-MNIST) [38, 67] for varying
FL settings of (B, E, K).
B determines the number of local data samples used for one iteration of
training on each device. Typically, B is associated with the generalization
problem — using larger batch sizes usually yield poor generalizability [64,
21]. For this reason, B largely affects the model convergence and global
energy efficiency, as shown in Figure 1.
E represents the number of training iterations for each device with the same
data samples. Since E is related to the over- versus under-fitting to specific
data samples [51], it also has a global impact on the model convergence, as
shown in Figure 1.
The number of participant devices K can be considered as the global batch size
in FL, as it is related to the amount of global data used per round. Although
smaller values of K enable efficient FL deployment by reducing the impact of
communication overhead [51], a careful selection is still required — K is also
associated with the generalization problem [51], affecting the model
convergence and global energy efficiency as shown in Figure 1.
Figure 1: Depending on the global parameters, the FL convergence performance
and global energy efficiency vary significantly. The convergence round and
global PPW are normalized to (1, 10, 20). Figure 2: The most energy-efficient
global parameter combination shifts in accordance with the NN characteristics.
The convergence round and global PPW are normalized to (1, 10, 20).
It is also important to consider the NN characteristics when selecting the
global parameters, as the two are interrelated. Figure 2 shows the global PPW
of two NNs under the different FL settings of (B, E, K). In case of CNN-MNIST
[38, 67], (B, E, K) of (8, 10, 20) shows the best energy efficiency among the
selected global parameter combinations. In contrast, when we use the LSTM
model with the Shakespeare dataset (LSTM-Shakespeare) [51, 36], the best
energy efficient global parameter combination shifts to (4, 20, 20) as the
learning characteristics of LSTM-Shakespeare differ from those of CNN-MNIST.
Furthermore, LSTM-Shakespeare comprises more memory-intensive RC layers,
whereas CNN-MNIST mainly consists of computation-intensive convolutional and
fully-connected layers. Owing to the memory pressure, LSTM-Shakespeare
exhibits higher energy efficiency with smaller input batch sizes and more
iterations.
### 2.2 FL Global Parameter Optimization
Figure 3: Training time per round of devices significantly varies with (a) B
and (b) E introducing large performance gaps across the devices. The training
time per round is normalized to that of H with B of 1 and that of H with E of
10 for (a) and (b), respectively.
In traditional centralized training, hyperparameter optimization (HPO) is the
process of finding a set of hyperparameters that minimizes loss or maximizes
accuracy for a DNN [66]. Generally, HPO first randomly selects a set of
hyperparameters, and trains the DNN using the entire dataset. The training
results (i.e., loss and accuracy) obtained with the selected hyperparameters
are measured, and used to select the next set of hyperparameters. Since the
hyperparameters can be any integer or floating point values, their search
space is usually large. Hence, to efficiently explore the search space,
machine learning-based optimization techniques, such as Bayesian Optimization
(BO) or Tree Parzan Estimator (TPE), are widely employed] for the HPO process.
Despite a variety of previous works on centralized training, global parameter
optimization in FL has unique challenges:
Straggler Problem: For each device, the training time of each device
substantially varies with the global parameters, introducing the straggler
problem — the overall training time per round is determined by the slowest
device. Figure 3 illustrates the training time per round for different device
categories (i.e., H, M, and L for high-end, mid-end, and low-end devices,
respectively111A detailed specification of each device category is presented
in Section 4.1.), depending on different B and E values. As B determines the
amount of on-device data to be processed in each iteration, training time on
each device category significantly depends on its computation- and memory-
capabilities. In addition, as E determines the number of iterations, it has a
linear impact on the training time per round.
Figure 4: Runtime variance significantly affects computation and communication
time exacerbating the straggler problem. The training time per round is
normalized to that of H in the absence of runtime variance.
Stochastic runtime variance exacerbates the straggler problem. In a real use
case, there can be several applications co-running with the FL execution,
since modern mobile device support multi-tasking features [62]. This causes
on-device shared resource interference [31, 34, 40] degrading computation
performance of FL. In addition, signal strength variations in wireless network
can affect the performance and energy efficiency of global aggregations in FL
— the data transmission latency and energy increase exponentially at weak
signal strength [33, 12].
Figure 4 shows the training time per round on different device categories, (a)
when there is no runtime variance, (b) when there is on-device interference,
and (c) when the network is not stable. As shown in Figure 4(a) and (b), the
on-device interference deteriorates the computation time for each device.
Since the impact of interference depends on the capabilities of each device in
terms of computation and memory, it exacerbates the inter-device performance
gaps. Further, as shown in Figure 4(a) and (c), the network instability
deteriorates the communication time of each device, thus affecting the
percentage of performance gaps across the devices.
Figure 5: The adaptive adjustment of global parameters resolves the straggler
problem saving energy consumption of each device category. The energy
consumption is normalized to H with fixed parameters. Figure 6: Adaptive
parameters can improve global PPW by resolving the straggler problem while
guaranteeing model convergence.
The adaptive round-by-round adjustment of the global parameters for different
devices can resolve the straggler problem, improving the FL energy efficiency.
Figure 5 shows the energy consumption of each device (a) when using the same
fixed parameters and (b) when adaptively adjusting the parameters for
different devices round-by-round. In the former case, faster devices (e.g., H
and M) need to wait for the slower devices (e.g., L) consuming energy, as
shown in Figure 5(a). Using smaller B or E for the slower devices can reduce
the performance gaps across the devices, saving the energy as shown in Figure
5(b). This significantly improves the average training time per round (2.3x)
and global PPW (3.6x), as shown in Figure 6(b) and (c) respectively — a
careful adjustment of the parameters can still guarantee the model convergence
as shown in Figure 6(a).
Figure 7: The optimal global parameters shift depending on the presence of
data heterogeneity.
Data Heterogeneity: It is also crucial for parameter optimization strategies
to consider the impact of data heterogeneity. Figure 7(a) shows the global PPW
over different global parameter (B, E, K) setting, in the absence of data
heterogeneity. In this case, the most energy-efficient (B, E, K) is (8, 10,
20). In the presence of data heterogeneity, however the global energy
efficiency of all (B, E, K) is degraded as shown in Figure 7(b), since the
data heterogeneity significantly affects model convergence [35]. In this case,
the most energy-efficient (B, E, K) shifts to (8, 5, 10), as decreasing E or K
reduces the amount of non-IID data reflected to the model parameters — E
affects the number of iterations for parameter updates with the given data and
K affects the number of non-IID devices participating for gradient updates.
Since the degree of data heterogeneity can vary round-by-round depending on
the participant compositions, it is also important to carefully tune the
global parameters round-by-round taking into account the data heterogeneity.
## 3 FedGPO
Figure 8: FedGPO Design Overview.
Under the heterogeneity and runtime variance, it is infeasible to enumerate
the large search space associated with FL global parameter optimization. To
efficiently explore the optimization space and accurately predict the optimal
global parameters, we propose an approach called FedGPO, based on
reinforcement learning (RL). Due to its low complexity yet high sample
efficiency [28, 73], RL has been widely used for the system optimization in
the edge domain [58]. In the following sections, we first provide an overview
of the FedGPO design, and then elaborate on its RL design and algorithm.
### 3.1 Overview
Figure 8 presents an overview of the FedGPO design. Based on RL, FedGPO
attempts to learn an optimal action decision (i.e., global parameter
selection) from prior information based on the current state (i.e.,
heterogeneity and runtime variance) and the given reward (i.e., the amount of
global/local energy efficiency and model accuracy improvement of the selected
action).
In each aggregation round, FedGPO identifies the global execution states of FL
(\small{1}⃝), such as characteristics of neural network model architectures
and the composition of randomly-selected K’ participant devices. Note K’ is K
determined in the previous aggregation round. FedGPO then identifies the local
execution states of the selected devices (\small{1}⃝), such as the usage of
resources, network instability, and the number of data classes each device
has. With the identified state information, FedGPO selects the action
(\small{2}⃝), i.e., sets per-device global parameters expected to improve the
FL energy efficiency without degrading the model convergence and accuracy. It
selects the actions using lookup tables (i.e., Q-tables shared across the
devices in the same performance category) [14, 37, 58], which store the
accumulated rewards of previously selected parameter combinations. Using the
selected global parameters, FedGPO executes the training on each selected
device (\small{3}⃝). After the aggregation round ends, FedGPO measures its
result (i.e., training time, energy consumption, and test accuracy) for
calculating the reward (\small{4}⃝). Finally, FedGPO updates the Q-tables with
the calculated reward (\small{5}⃝). By repeating the aforementioned process
(\small{1}⃝-\small{5}⃝), FedGPO learns how to select the optimal global
parameters.
### 3.2 FedGPO RL Design
Table 1: Discrete values for FedGPO states. State | Discrete Values
---|---
$S_{CONV}$ | Small ($<$10), medium ($<$20), large ($<$30),
larger ($>=$40)
$S_{FC}$ | Small ($<$10), large ($>=$10)
$S_{RC}$ | Small ($<$5), medium ($<$10), large ($>=$10)
$S_{Co\\_CPU}$ | None (0%), small ($<$25%), medium ($<$75%),
large ($<=$100%)
$S_{Co\\_MEM}$ | None (0%), small ($<$25%), medium ($<$75%),
large ($<=$100%)
$S_{Network}$ | Regular ($>$40Mbps), bad ($<=$40Mbps)
$S_{Data}$ | Small ($<$25%), medium ($<$100%), large (=100%)
To produce accurate predictions of RL, it is crucial to model the core
components in a realistic execution environment: 1) state, 2) action, and 3)
reward. This section defines the core components for system energy efficiency
optimization of FL.
State: We define the FL execution state based on our observations in Section
2.
As demonstrated in Section 2.1, the optimal global parameters depend on the NN
characteristics. To model the impact of these characteristics, we identify
$S_{CONV}$, $S_{FC}$, and $S_{RC}$ which represent the numbers of
convolutional, fully-connected, and recurrent layers, respectively — the three
layers are typically highly interrelated with the training/inference
efficiency [35].
In addition, as shown in Section 2.2, the optimal global parameters are also
affected by on-device interference. To model this impact, we identify per-
device states of $S_{Co\\_CPU}$ and $S_{Co\\_MEM}$ which represent the CPU
utilization and memory usage of co-running applications, respectively. Because
the optimal global parameters also depend on network stability, we identify
the per-device network stability of $S_{Network}$ with the wireless network
(e.g., Wi-Fi and 5G) bandwidth. We also model the data heterogeneity impact by
identifying $S_{Data}$ which represents the number of data classes of each
device for the aggregation round.
It is difficult to encode continuous values into the RL lookup table. Thus, we
convert continuous values of each state into discrete values, by applying a
clustering algorithm [9, 35]. Table 1 summarizes the discretized values. Note
FedGPO can support larger search space by further reducing the search space
size with different clustering algorithms.
Action: Actions in RL model the customizable control knobs of the system. In
the context of FL global parameter optimization, we define the actions as the
selection of global parameters for each aggregation round. For the local mini-
batch size of B, we define the discrete batch size numbers as a feasible range
for resource-constrained edge devices [51, 35]. We also define the discrete
numbers of local epochs E and participant devices K as ranges based on the de-
facto FL algorithm [51] to ensure a balanced computation-communication ratio.
For example, larger E with smaller K typically increases the computation-
communication ratio. Table 2 summarizes the discrete values of all global
parameters as actions.
Table 2: Discrete global parameter values for FedGPO actions. Parameter | Discrete Values
---|---
B | {1, 2, 4, 8, 16, 32}
E | {1, 5, 10, 15, 20}
K | {1, 5, 10, 15, 20}
Reward: The reward in RL models the optimization objective. To ensure that
FedGPO selects global parameters that maximize energy efficiency without
degrading model convergence and accuracy, we define the reward R as in (1).
Here, $R_{energy\\_local}$ represents the energy consumption of each
participant device, whereas $R_{energy\\_global}$ denotes that of all devices.
$R_{accuracy}$ is the test accuracy of the NN model, while
$R_{accuracy\\_prev}$ is that of the prior round. Note, since time-to-
convergence is not measurable before the convergence, we substitute it with
the improvement in accuracy — a similar practice has been used for the
hyperparameter optimization of ML training [65].
$\displaystyle if\;\;\;R_{accuracy}\;-\;R_{accuracy\\_prev}\;\;<=\;\;0,$ (1)
$\displaystyle\qquad R=R_{accuracy}-100$ $\displaystyle else$
$\displaystyle\qquad R=-R_{energy\\_global}-R_{energy\\_local}$
$\displaystyle\qquad\;\;\;\;\;\;\;\;+\alpha
R_{accuracy}+\beta(R_{accuracy}-R_{accuracy\\_prev})$
Among the encoded reward values, we estimate $R_{energy\\_local}$ using a
commonly-used energy formulation, such as [27, 32, 30].
Algorithm 2 Training the Q-learning model
Variable: S, A
S is the variable for the state
A is the variable for the action
Constants: $\gamma$, $\mu$, $\epsilon$
$\gamma$ is the learning rate
$\mu$ is the discount factor
$\epsilon$ is the exploration probability
Initialize $Q(S,A)$ as random values
Repeat (whenever an aggregation round begins):
Observe state and store in S
if rand() $<\epsilon$ then
Choose action A randomly
else
Choose action A which maximizes Q(S,A)
Run training with global parameters defined by A
(when local training and aggregation terminate)
Obtain $R_{energy\\_global}$, $R_{energy\\_local}$, and $R_{accuracy}$
Calculate reward R
Observe new state S’
Choose action A’ which maximizes Q(S’,A’)
Q(S,A) $\leftarrow$ Q(S,A) \+ $\gamma$[R \+ $\mu$Q(S’,A’) \- Q(S,A)]
S $\leftarrow$ S’
For each device, we calculate computation energy, $E_{comp}$, using
utilization-based CPU and GPU power models [27, 32], as in (2). Here,
$E^{i}_{CPU\\_core}$ and $E_{GPU}$ represent the energy consumed by the ith
CPU core and GPU, respectively, $E_{PU\\_core}$ is the energy consumed by
either of the CPU cores or GPU, $t^{f}_{busy}$ is the time spend in the busy
state at frequency f and $t_{idle}$ is that in the idle state, and
$P^{f}_{busy}$ is the power consumed during $t^{f}_{busy}$ at f and $P_{idle}$
is that during $t_{idle}$. Note $P^{f}_{busy}$ and $P_{idle}$ for CPU/GPU are
obtained by measuring the corresponding processing unit’s power at each
frequency.
$\displaystyle E_{comp}$ $\displaystyle=\sum_{i}{E_{CPU\\_core}^{i}}+E_{GPU},$
(2) $\displaystyle E_{PU\\_core}$ $\displaystyle=\sum_{f}(P_{busy}^{f}\times
t_{busy}^{f})+P_{idle}\times t_{idle}$
We also calculate communication energy [30], $E_{comm}$, as in (3), where
$t_{TX}$ is the measured transmission latency of the gradient (or parameter)
updates, and $P^{S}_{TX}$ is the power consumed by the device during $t_{TX}$
at signal strength S. Note $P^{S}_{TX}$ is obtained based on the measured
transmission power consumption of devices at each signal strength.
$\displaystyle E_{comm}=$ $\displaystyle P_{TX}^{S}\times t_{TX}$ (3)
We then calculate the idle energy, $E_{idle}$, for the rest of devices (i.e.,
devices not participating in the round), as in (4), where $t_{round}$ denotes
the training time of the round.
$\displaystyle E_{idle}=$ $\displaystyle P_{idle}\times t_{round}$ (4)
Based on $E_{comp}$, $E_{comm}$, and $E_{idle}$, $R_{energy\\_local}$ is
calculated for each device, as in (5).
$\displaystyle if\;\;device\;\;\subset\;\;S_{t}$ (5) $\displaystyle\qquad
R_{energy\\_local}=E_{comp}+E_{comm}$ $\displaystyle else$
$\displaystyle\qquad R_{energy\\_local}=E_{idle}$
Based on $R_{energy\\_local}$, $R_{energy\\_global}$ is also calculated for
entire N devices (6).
$\displaystyle R_{energy\\_global}=$
$\displaystyle\sum_{i}^{N}R_{energy\\_local}$ (6)
### 3.3 RL Algorithm
To enable real-time decision making for each FL round, we employ Q-learning
[9], as it provides the advantage of low latency overhead by employing lookup
tables to determine the best action. For RL, it is also crucial to consider
the balance between exploitation and exploration to avoid the local optima
[14, 37, 35]. To overcome this issue, we use the epsilon-greedy algorithm [57,
58] along with the Q-learning, which chooses an action with the highest
reward, or a uniformly-random action based on a pre-specified exploration
probability ($\epsilon$).
In Q-learning, the value function $Q(S,A)$ accepts the state S and the action
A as parameters in the form of a lookup table (Q-table). To permit a large
number of participants and address the scalability requirement of FL, FedGPO
exploits shared Q-tables222Shared Q-tables can be a potential source of system
usage leakage. To overcome this, FedGPO can exploit per-device Q-tables
instead, which imporves the prediction accuracy by 2.7% degrading 12.2% of
convergence overhead. for devices within the same performance category.
Sharing the learned results across the devices can also expedite the design
space exploration process [35], as each client device experiences different
level of heterogeneity and runtime variance.
Algorithm 2 presents the algorithm for training the shared Q-tables. FedGPO
first randomly initializes the Q-tables. During each aggregation round, it
observes the execution state S as identified in Section 3.1. It then generates
a random value and compares it with $\epsilon$ 333We use 0.1 for $\epsilon$.
To determine $\epsilon$, we tested the accuracy and convergence overhead of
FedGPO with 0.1, 0.5, and 0.9 of $\epsilon$.. If the value is lower than
$\epsilon$, FedGPO selects the per-device global parameters randomly for
exploration. Otherwise, it selects A with the largest $Q(S,A)$ using the
Q-tables.
After the local training execution in an FL aggregation round is finished
using the selected A, FedGPO calculates the reward R as described in Section
3.1. FedGPO then identifies the new execution state S’, and selects the
corresponding A’ using $Q(S^{\prime},A^{\prime})$ for each device. It then
updates the $Q(S,A)$ based on the equation in Algorithm 2, where $\gamma$ and
$\mu$ are hyperparameters of the learning rate and discount factor,
respectively — $\gamma$ and $\mu$ determined based on the sensitivity
analysis.
When the learning phase is completed, i.e., the largest $Q(S,A)$ value is
converged for each S, FedGPO uses the shared Q-tables to select A (i.e.,
global parameters) for each device to maximize $Q(S,A)$ for S.
## 4 Experimental Methodology
### 4.1 System Infrastructure
Table 3: System profiles of Amazon EC2 instances. Category | Instance | Performance | RAM
---|---|---|---
(GFLOPS) | (GB)
H | m4.large | 153.6 | 8
M | t3a.medium | 80.0 | 4
L | t2.small | 52.8 | 2
We emulate FL with 200 mobile devices referring to prior studies pertaining to
FL [42, 51, 35, 36]. Since it is difficult to run experiments with 200 real
smartphones, we emulate the FL performance by using Amazon EC2 instances [2]
(Table 3), which feature equivalent theoretical GFLOP performance and memory
capacity to those of the three smartphone performance categories: high-end
(H), mid-end (M), and low-end (L) devices. By referring to in-the-field system
performance distribution [70], we composed 200 instances with 30 H, 70 M, and
100 L devices. For the model aggregation server, we use a c5d.24xlarge Amazon
EC2 instance, whose theoretical performance and RAM are 448 GFLOPS and 32GB,
respectively.
For measuring the power, we use three representative smartphones for each
performance category [34]: Mi8Pro [23], Galaxy S10e [61], and Moto X Force
[55] (Table 4). We use an external Monsoon Power Meter [54] to measure three
smartphones’ power consumption while running on-device training (implemented
with DL4j [13]) — a similar approach was used in prior studies [59, 63, 35].
Based on the measured performance and power consumption, we evaluate the
energy efficiency of participant devices in FL.
To evaluate the effectiveness of FedGPO, we implement it on top of the FedAvg
algorithm [36, 51] with PyTorch [60]. We compare FedGPO with three baselines:
* •
Fixed (Best), which uses the most energy-efficient parameter combination
identified by grid search, yet fixed during the entire FL rounds.
* •
Adaptive (BO) which adjusts the global parameters each round using a Bayesian
Optimization algorithm where many state-of-the-art approaches are based [66].
* •
Adaptive (GA) which adjusts the global parameters every round using a genetic
algorithm [1].
We also compare FedGPO with two previous approaches: FedEX [29] and ABS [49].
FedEX adjusts the parameters with an exponentiated gradient updates, whereas
ABS only adjusts the local minibatch size with a Deep RL.
Table 4: Specifications of mobile devices. Device | CPU | GPU
---|---|---
Mi8Pro (H) | Cortex-A75 (2.8GHz) | Adreno 630 (0.7GHz)
23 V/F steps | 7 V/F steps
5.5 W | 2.8 W
Galaxy S10e (M) | Mongoose (2.7GHz) | Mali-G76 (0.7GHz)
21 V/F steps | 9 V/F steps
5.6 W | 2.4 W
Moto X Force (L) | Cortex-A57 (1.9GHz) | Adreno 430 (0.6GHz)
15 V/F steps | 6 V/F steps
3.6 W | 2.0 W
We determine two hyperparameters (i.e., learning rate and discount factor) of
FedGPO by evaluating the three values of 0.1, 0.5, and 0.9 for each one [9,
35]. We find that a higher learning rate improves prediction accuracy, as
FedGPO works better when a higher amount of the reward is reflected in the
Q-tables — FedGPO needs to adapt to the heterogeneity and stochastic variance
within the limited rounds. In contrast, we observe that a lower discount
factor improves prediction accuracy, as FedGPO exhibits higher performance
when a lower amount of the reward or the following state is reflected in that
of the current state — sequential states have a weak mutual relationship
because of their stochastic nature. Accordingly, in our evaluation, we use 0.9
and 0.1 for the respective hyperparameter.
### 4.2 Workloads and Execution Scenarios
Workloads: We evaluate FedGPO with two workloads [42, 51, 36]: (1) training a
CNN model with the MNIST dataset (CNN-MNIST) for image classification [38, 67]
and (2) training an LSTM model with the Shakespeare dataset (LSTM-Shakespeare)
for the next character prediction [51, 36]. We also use a state-of-the-art NN
workload: (3) training the MobileNet with the ImageNet dataset (MobileNet-
ImageNet) for image classification [10, 22]. Note our study focuses on mobile-
centric neural networks by referring to recent FL deployment and use cases
[18, 17, 3, 52, 19, 75, 24], as larger networks have been infeasible for
resource-constrained mobile devices [20].
Runtime variance: To emulate realistic on-device interference, we run a
synthetic co-running application on a random subset of devices. The synthetic
application exhibits the same CPU and memory usage as the real-world mobile
application of web browsing [63, 59]. In addition, as the variability of real-
world network follows a Gaussian distribution [12, 30], we generate a random
network bandwidth following such a distribution using a Wi-Fi AP.
Data distribution: We evaluate different degrees of data heterogeneity with
two different data sample distributions [5, 44]: Ideal IID and Non-IID. For
Ideal IID, all the data classes are evenly distributed to the devices. In
contrast, for non-IID, each data class is randomly distributed following a
Dirichlet distribution with a 0.1 concentration parameter [35, 5, 41, 44, 47,
39, 56, 72].
## 5 Evaluation Results and Analysis
### 5.1 Result Overview
Figure 9 compares the PPW energy efficient, the convergence performance, and
the training accuracy among the three FL applications. The PPW and the
convergence time speedup are normalized to the Fixed (Best) case. Compared to
the Fixed (Best), Adaptive (BO), and Adaptive (GA), FedGPO achieves 3.6x,
3.1x, and 1.7x of the average FL energy efficiency improvement, respectively.
It also maintains the training accuracy with the improved convergence time.
For all FL use cases, FedGPO alleviates the performance gaps across the
participant devices, by identifying more efficient per-device global
parameters compared to the baseline settings. This improves the average
training time per round by 2.3x over the Fixed (Best). Additionally, by
reducing the performance gaps across the devices, the redundant energy
consumption of the devices is saved by 57.5% over the Fixed (Best). As a
result, significantly better energy efficiency is achieved compared to the
baseline settings.
The global parameters selected by FedGPO also guarantees the convergence
(i.e., the training loss settles to a certain value [53] while the training
accuracy gets to an error range of the value achieved by the baseline in an
ideal environment). FedGPO maintains the similar number of convergence rounds
with Fixed (Best), improving the convergence time while maintaining the model
quality; the convergence round difference of Fixed (Best) and FedGPO is only
0.2%.
Compared to Adaptive (BO), FedGPO shows 2.4x better convergence time. This is
mainly because the Adaptive (BO) has lower sample efficiency than FedGPO, and
thus fails to adapt to the heterogeneity and runtime variance round-by-round.
GA has a relatively higher sample efficiency than BO [28], but it still
requires a number of mutation/crossover generations for the convergence. Due
to the quick adaptability to the hetergeneity and runtime variance every
round, FedGPO exhibits 1.6x better convergence time, compared to Adaptive
(GA).
Figure 9: FedGPO improves the PPW by 4.1x, 3.2x, and 3.5x compared to the
baseline Fixed (Best) for CNN-MNIST, LSTM-Shakespeare, and MobileNet-ImageNet,
respectively. It also maintains the training accuracy with the improved
convergence time.
### 5.2 Adaptability and Accuracy Analysis
Adaptability to stochastic variance: Figure 10 compares the PPW, convergence
performance, and model accuracy of CNN-MNIST, (a) in the absence of runtime
variance, (b) in the presence of on-device interference from co-running
applications, and (c) in the presence of network variance. When there exists
runtime variance, FedGPO achieves 5.0x, 4.2x, and 3.0x of the average energy
efficiency improvement, compared to Fixed (Best), Adaptive (BO), and Adaptive
(GA), respectively. Furthermore, it also improves the convergence time while
maintaining the training accuracy. Note other NNs show similar result trends.
Under the runtime variance, the performance gap across participating devices
significantly varies round-by-round due to the varying on-device computation
and communication time. Nevertheless, FedGPO selects better per-device global
parameters every round compared to the baseline settings, by quickly adapting
varying on-device computation or communication time with high sample
efficiency. As a result, the convergence time is improved by 3.2x, 2.9x, and
2.5x, compared with that of Fixed (Best), Adaptive (BO), and Adaptive (GA),
respectively. Additionally, by eliminating the redundant energy consumption of
the participating devices, FedGPO also significantly improves the energy
efficiency compared to the baseline settings. Note, in this case, the training
accuracy of the baseline setting is significantly degraded due to the
exacerbated straggler problems — previous works just drop the gradient updates
from the stragglers [41, 42].
Adaptability to data heterogeneity: Figure 11 illustrates the energy
efficiency, convergence time, and training accuracy of CNN-MNIST, (a) in the
absence of data heterogeneity (i.e., Ideal IID) and (b) in the presence of
data heterogeneity (Non-IID). Even under the latter scenario, FedGPO still
improves the PPW by 6.2x, 1.9x, and 1.3x, compared with Fixed (Best), Adaptive
(BO), and Adaptive (GA), respectively. It also improves convergence time and
training accuracy against the baseline settings.
Figure 10: Under the runtime variance, FedGPO significantly improves the FL
energy efficiency by 5.0x, 4.2x, and 3.0x, on average, compared to Fixed
(Best), Adaptive (BO), and Adaptive (GA), respectively. It also improves the
convergence time maintaining the training accuracy.
In the presence of non-IID participants, neither E nor K is adjusted by the
baseline settings depending on the degree of data heterogeneity, maintaining
the amount of non-IID data reflected on the model gradients. On the other
hand, FedGPO learns how data heterogeneity affects the energy efficiency and
convergence performance, and adjusts gradient updates with E and K along with
B. Therefore, it significantly improves the PPW, convergence performance, and
model accuracy even under the data heterogeneity. The prediction accuracy of
the baseline settings is also significantly degraded in this case, as they
accept the gradient (or parameter) updates from non-IID participants as
equally as those from IID-participants [68].
Prediction accuracy: FedGPO accurately selects the near-optimal global
parameters round-by-round. Table 5 lists the mean absolute percentage accuracy
of FedGPO over the optimal global parameters for each round — these parameters
are identified in terms of minimizing the performance gap across the devices,
rather than global convergence. FedGPO achieves an average prediction accuracy
of 94.7%.
FedGPO also adapts to the stochastic features of edge execution environments.
As shown in Table 5, in the presence of runtime variance (i.e., on-device
interference and network variability), FedGPO successfully selects the near-
optimal global parameters, achieving a 94.4% average prediction accuracy. In
the presence of data heterogeneity, FedGPO exhibits relatively lower
prediction accuracy (88.9% on average), as the minimization of the performance
gaps across devices does not guarantee the model convergence in this case,
whereas FedGPO selects the parameters that guarantee the model convergence
while improving the energy efficiency, as shown in Figure 11.
### 5.3 Comparison with Prior Work
Figure 11: Even in the presence of data heterogeneity, FedGPO achieves 6.2x,
1.9x, and 1.3x higher energy efficiency than Fixed (Best), Adaptive (BO), and
Adaptive (GA), respectively, by adaptively adjusting E and K along with B
round-by-round.
We evaluate FedGPO compared to two prior works for CNN-MNIST: FedEX [29] and
ABS [49]. On average, FedGPO achieves 1.5x and 2.1x improvements in energy
efficiency compared to FedEX and ABS, respectively (Figure 12). Note other NNs
show similar result trends.
Under the runtime variance, FedEX and ABS improves the convergence performance
and PPW over the baseline, as they reduce the device performance gap, by
explicitly considering the straggler problem. However, FedEX does not adapt to
the runtime variance as quickly as FedGPO does, due to the lower sample
efficiency of exponentiated gradient updates. Furthermore, ABS does not adjust
E and K, which helps to deal with the straggler problem and data
heterogeneity. As a result, FedGPO further improves PPW by 1.5x and 1.7x over
FedEX and ABS, respectively (Figure 12).
Compared to the baseline, FedEX is robust to data heterogeneity by adjusting
the amount of non-IID data reflected on the model gradients with E and K.
However, it still fails to quickly adapt to the data heterogeneity due to its
low sample efficiency. In contrast, ABS is not robust to data heterogeneity,
as B is not closely related to the data heterogeneity as explained in Section
2. Consequently, FedGPO achieves 1.4x and 3.6x higher PPW over FedEX and ABS,
respectively (Figure 12).
### 5.4 Convergence and Overhead Analysis
Table 5: Accuracy for Global Parameter Selection. Runtime | Data | Prediction
---|---|---
Variance | Heterogeneity | Accuracy
No | No | 94.7%
Yes (On-device Interference) | No | 94.2%
Yes (Unstable Network) | No | 94.5%
No | Yes | 87.7%
Yes | Yes | 90.1%
When training the shared Q-tables of FedGPO, the reward converges after 30-40
aggregation rounds. Prior to convergence, FedGPO shows 24.2% lower energy
efficiency than Fixed (Best), on average. After the convergence, however,
FedGPO selects more efficient global parameters than the baselines, as we
observed in Section 5.2. Consequently, the global energy efficiency is
eventually improved compared with that of the baselines.
The average runtime cost associated with training the shared Q-tables is 499.6
$\mu$s except for the FL execution time, corresponding to 0.7% of the training
time per round. The overhead includes 1) identifying the per-device states
(496.8 $\mu$s), 2) choosing global parameters (0.2 $\mu$s), 3) calculating the
reward (2.1 $\mu$s), and 4) updating the Q-tables (0.5 $\mu$s). The total
memory requirement of FedGPO is also efficient — in our experiments with three
device categories, 0.4MB, 0.0125% of the 32GB DRAM capacity, was only
required.
## 6 Related Work
Optimization for FL: FL enables collaborative training of a shared ML model in
a private manner [4, 36, 51, 46]. To deploy FL on the edge efficiently, FedAvg
has been employed as the standard algorithm [36, 51]. This algorithm
alleviates the communication overheads by employing fewer participants with
higher number of per-device training iterations [51, 11, 6]. Subsequently,
various methods have been proposed for model accuracy improvement [43] and
security robustness [45, 48].
Although FedAvg demonstrates the potential of FL deployment, it is still
faceing the important challenges for optimization — namely the straggler
problem and data heterogeneity. As the straggler problem leads to the excess
energy consumption of participant devices, prior studies attempted to mitigate
it via asynchronous gradient updates [8] or intelligent participant selection
[26, 51, 35]. In addition, to deal with the adverse impact of data
heterogeneity, other approaches attempted to reduce the amount of non-IID data
reflected on model gradients by allowing asynchronous aggregations [8, 7], the
use of globally shared data [74], or partial updates [42]. As the main target
of the techniques does not encompass the global parameter adjustment, they can
be applied with FedGPO.
Global parameter optimization for FL: Along with the general optimizations,
global parameter optimization is also crucial in FL, as the global parameters
can significantly affect the FL execution efficiency.
In traditional centralized training, many hyperparameter optimization (HPO)
techniques have been proposed to expedite the space exploration based on ML.
Such techniques include Bayesian Optimization (BO) [66, 25] and the Genetic
Algorithm (GA) [1]. However, as these HPO techniques require training with the
entire dataset for each set of global parameters, they are not feasible for
the resource-constrained edge execution environment. Furthermore, they also do
not directly address the unique challenges of FL including the straggler
problem and data heterogeneity.
Considering the system and data heterogeneity in FL, several approaches have
been proposed to adjust the global parameters round-by-round based on
exponentiated gradient updates [29] or deep RL [49]. However, these techniques
do not consider the stochastic nature of the edge-cloud execution environment
including performance interference and network variability. In addition, they
do not account for the energy efficiency of the participant devices. Different
from the prior approaches, FedGPO explores the energy efficient global
parameter optimization for FL, when there exists system/data heterogeneity and
runtime variance. Based on a customized reinforcement learning, FedGPO can
identify a near-optimal global parameters for each round by adapting to the
heterogeneity and runtime variance.
Figure 12: FedGPO outperforms both FedEX [29] and ABS [49], in terms of
convergence time and energy efficiency, regardless of the presence of runtime
variance or data heterogeneity.
## 7 Conclusion
To enable energy-efficient FL on the edge, we propose a global parameter
optimization framework called FedGPO. The FL performance and energy efficiency
characterization in the edge execution environment demonstrates that optimal
global parameters depend on various features: NN characteristics, system/data
heterogeneity, and stochastic runtime variance. FedGPO successfully determines
near-optimal global parameters in consecutive FL aggregation rounds, by
considering these features. We implement representative FL use cases on an
emulated edge-cloud execution environment using off-the-shelf systems. FedGPO
improves the average FL energy efficiency by 3.6x, compared with the baseline
settings. Compared to FedEX and ABS, FedGPO improves the energy efficiency by
1.5x and 2.1x, on average, respectively, by considering system/data
heterogeneity and runtime variance. We demonstrate the viability of FedGPO as
a solution to global parameter optimization for energy-efficient FL in
realistic edge-cloud execution environments.
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# Regret Pruning for Learning Equilibria in Simulation-Based Games
Bhaskar Mishra, Cyrus Cousins, Amy Greenwald
###### Abstract
In recent years, empirical game-theoretic analysis (EGTA) has emerged as a
powerful tool for analyzing games in which an exact specification of the
utilities is unavailable. Instead, EGTA assumes access to an oracle, i.e., a
simulator, which can generate unbiased noisy samples of players’ unknown
utilities, given a strategy profile. Utilities can thus be empirically
estimated by repeatedly querying the simulator. Recently, various progressive
sampling (PS) algorithms have been proposed, which aim to produce PAC-style
learning guarantees (e.g., approximate Nash equilibria with high probability)
using as few simulator queries as possible. A recent work by Areyan Viqueira,
Cousins, and Greenwald introduces a pruning technique called _regret-pruning_
which further minimizes the number of simulator queries placed in PS
algorithms which aim to learn _pure_ Nash equilibria. In this paper, we
address a serious limitation of this original regret pruning approach – it is
only able to guarantee that _true_ pure Nash equilibria of the empirical game
are approximate equilibria of the true game, and is unable to provide any
strong guarantees regarding the efficacy of approximate pure Nash equilibria.
This is a significant limitation since in many games, pure Nash equilibria are
computationally intractable to find, or even non-existent. We introduce three
novel regret pruning variations. The first two variations generalize the
original regret pruning approach to yield guarantees for approximate pure Nash
equilibria of the empirical game. The third variation goes further to even
yield strong guarantees for all approximate mixed Nash equilibria of the
empirical game. We use these regret pruning variations to design two novel
progressive sampling algorithms, PS-REG+ and PS-REG-M, which experimentally
outperform the previous state-of-the-art algorithms for learning pure and
mixed equilibria, respectively, of simulation-based games.
## Introduction
Game theory is the standard conceptual framework used to analyze strategic
interactions among rational agents in multi-agent systems. A game comprises a
collection of players, each with a set of strategies and a utility function,
mapping strategy profiles (i.e., combinations of strategies) to values.
Traditionally, game-theoretic analysis presumes complete access to a game’s
structure, including the utility functions.
In recent years, empirical game-theoretic analysis (EGTA) has emerged as a
powerful tool for analyzing games in which such an exact specification of the
utilities is unavailable. Instead, EGTA assumes access to an oracle, i.e., a
(stochastic) simulator, which produces unbiased noisy samples of players’
unknown utilities given a strategy profile (Wellman 2006; Tuyls et al. 2020;
Areyan Viqueira, Cousins, and Greenwald 2020). Such games are called
_simulation-based games_ (Vorobeychik and Wellman 2008), or _black-box_ games
(Picheny, Binois, and Habbal 2016), and their empirical counterparts, which
are derived from simulation data, are called _empirical games_. Simulation-
based games have been studied in many practical settings including trading
agent analyses in supply chains (Vorobeychik, Kiekintveld, and Wellman 2006;
Jordan, Kiekintveld, and Wellman 2007), ad auctions (Jordan and Wellman 2010;
Areyan Viqueira et al. 2019), and energy markets (Ketter, Peters, and Collins
2013); designing network routing protocols (Wellman, Kim, and Duong 2013);
strategy selection in real-time games (Tavares et al. 2016); and the dynamics
of RL algorithms, like AlphaGo (Tuyls et al. 2018).
A typical EGTA goal is to produce PAC-style learning guarantees (e.g.,
approximate Nash equilibria with high probability) with minimal _query
complexity_ , i.e., the number of simulation queries placed (Tuyls et al.
2020; Areyan Viqueira, Cousins, and Greenwald 2020; Cousins et al. 2022). This
goal has led to the development of _progressive sampling_ algorithms (Areyan
Viqueira, Cousins, and Greenwald 2020; Cousins et al. 2022), which place
simulation queries in progressive batches, until the desired guarantee is
reached. On top of progressive sampling, two papers introduce various pruning
techniques, to further minimize query complexity. One of these techniques,
_well-estimated pruning_ , prunes strategy profiles whose utilities are very
likely to already be sufficiently close to the true utilities (Cousins et al.
2022). This technique is useful for learning a variety of game properties,
including regret, pure or mixed Nash equilibria, welfare-maximizing outcomes,
and more. A second technique, called _regret pruning_ , is intended for use
only when learning pure Nash equilibria, as it prunes strategy profiles that
are highly unlikely to be best responses, and hence unlikely to be necessary
for finding pure Nash equilibria (Areyan Viqueira, Cousins, and Greenwald
2020).
In this paper, we focus predominantly on regret pruning. Areyan Viqueira,
Cousins, and Greenwald (2020) claim that their regret pruning criterion can be
used to learn an empirical game which, with high probability, satisfies a
certain dual pure Nash containment guarantee – all pure Nash equilibria of the
simulation-based game are approximate equilibria of the empirical game, and
all approximate pure Nash equilibria of the empirical game are approximate
equilibria of the simulation-based game. We show via a direct counterexample,
however, that their progressive sampling algorithm using regret pruning fails
to satisfy this second inclusion. Rather, their algorithm only yields the
guarantee that all _true_ pure Nash equilibria of the empirical game are
approximate equilibria of the simulation-based game, and is unable to directly
yield any non-trivial guarantee regarding _approximate_ pure Nash equilibria
of the empirical game. This difference is crucial, since in many games,
computing a pure Nash equilibrium is computationally intractable (e.g., it is
NP-complete in graphical games (Gottlob, Greco, and Scarcello 2005)), and
sometimes such an equilibrium does not even exist. In such cases, an
approximate pure Nash equilibrium is the best that can be hoped for, but
Areyan Viqueira, Cousins, and Greenwald’s progressive sampling algorithm using
regret pruning yields no guarantees for such equilibria.
In response to this limitation of the original regret pruning technique, we
design three new variations of regret pruning (and new corresponding
progressive sampling algorithms). At the cost of a slightly tighter regret
pruning criterion, our first regret pruning variation yields the guarantee
regarding approximate equilibria of the empirical game which Areyan Viqueira,
Cousins, and Greenwald’s regret pruning technique was originally intended to
yield. Our second variation incorporates the non-uniform utility deviation
bounds used by Cousins et al. (2022) for well-estimated pruning in order to
design a looser regret pruning criterion which nonetheless yields the same
guarantees as the first variation. Finally, our third variation also takes
advantage of non-uniform bounds, but uses different proof techniques than
those used in the first two variations to yield both pure _and_ mixed Nash
containment guarantees.
The third regret pruning variation is a particularly significant contribution,
as it is one of the first pruning techniques beyond simple well-estimated
pruning for learning mixed equilibria. The only available alternative is
rationalizability pruning introduced by Areyan Viqueira, Cousins, and
Greenwald (2020), which requires the use of a computationally expensive
iterative dominance algorithm, has a very tight pruning criterion which often
prunes few to no strategy profiles in practice, and most importantly, like the
original regret pruning technique, only yields guarantees regarding _true_
mixed Nash equilibria of the empirical game. In contrast, our novel third
regret pruning variation utilizes a pruning criterion which is very cheap to
compute, can prune a very significant number of simulation queries (which we
confirm experimentally), and yields Nash containment guarantees for
approximate mixed equilibria of the empirical game.
In addition to presenting the guarantees progressive sampling algorithms using
these novel variations of regret pruning can satisfy, we also derive sample
complexity bounds for these new algorithms. In particular, we present PAC-
style upper bounds on the number of samples our progressive sampling
algorithms will take to prune each respective strategy profile. Finally, we
conclude by demonstrating experimentally that our novel progressive sampling
algorithms which incorporate both well-estimated pruning and novel regret
pruning variations significantly outperform Cousins et al.’s progressive
sampling algorithm which used well-estimated pruning alone, in some cases
requiring up to 50% fewer simulation queries to learn equilibria of similar
quality.
### Related Works
The EGTA literature, while relatively young, is growing rapidly, with
researchers actively contributing methods for myriad game models. Some of
these methods are designed for normal-form games (Cousins et al. 2022; Areyan
Viqueira, Cousins, and Greenwald 2020; Areyan Viqueira et al. 2019; Tavares et
al. 2016; Fearnley et al. 2015; Vorobeychik and Wellman 2008), and others, for
extensive-form games (Marchesi, Trovò, and Gatti 2020; Gatti and Restelli
2011; Zhang and Sandholm 2021). Most methods apply to games with finite
strategy spaces, but some apply to games with infinite strategy spaces
(Marchesi, Trovò, and Gatti 2020; Vorobeychik, Wellman, and Singh 2007;
Wiedenbeck, Yang, and Wellman 2018). A related line of work aims to
empirically design mechanisms via EGTA methodologies (Vorobeychik,
Kiekintveld, and Wellman 2006; Areyan Viqueira et al. 2019).
The progressive sampling algorithms and pruning techniques which we design in
this paper extend work done by Cousins et al. (2022); Areyan Viqueira,
Cousins, and Greenwald (2020); Areyan Viqueira et al. (2019) in designing
algorithms for learning equilibria in normal-form simulation-based games with
finite strategy spaces.
## Learning Framework
We begin with the standard definition of standard normal-form games, and some
related properties . We then introduce our formal model of simulation-based
games and empirical games. Finally, we state the concentration inequalities we
use to guide pruning in our progressive sampling algorithms.
### Basic Game Theory
###### Definition 1 (Normal-Form Game).
A _normal-form game_ $\Gamma\doteq\langle P,\\{S_{p}\\}_{p\in
P},\bm{u}\rangle$ consists of a set of players $P$, each with a corresponding
_pure strategy set_ $S_{p}$. We define $\bm{S}\doteq S_{1}\times\dots\times
S_{|P|}$ to be the _pure strategy profile space_ , and then
$\bm{u}:\bm{S}\to\mathbb{R}^{|P|}$ is a vector-valued _utility function_
(equivalently, a vector of $|P|$ scalar utility functions $\bm{u}_{p}$).
Given an NFG $\Gamma$, we denote by $S_{p}^{\diamond}$ the set of
distributions over $S_{p}$; this set is called player $p$’s _mixed strategy
set_. We define
$\bm{{S}^{\diamond}}={S}^{\diamond}_{1}\times\dots\times{S}^{\diamond}_{|P|}$
to be the _mixed strategy profile space_ , and, overloading notation, we write
$\bm{u}(\bm{s})$ to denote the expected utility of a mixed strategy profile
$\bm{s}\in\bm{{S}^{\diamond}}$. Each pure strategy profile $\bm{s}\in\bm{S}$
is contained in the mixed strategy profile space $\bm{{S}^{\diamond}}$,
represented by the profile with each mixed strategy concentrated entirely at
the respective pure strategy.
Given player $p$ and strategy profile $\bm{s}\in\bm{{S}^{\diamond}}$, the set
$\operatorname{Adj}_{p,\bm{s}}\doteq\\{(\bm{s}_{1},\dots,\bm{s}_{p-1},t,\bm{s}_{p+1},\dots,\bm{s}_{\left\lvert{}P\right\rvert{}})\mid
t\in S_{p}\\}$ contains all adjacent strategy profiles, meaning those in which
the strategies of all players $q\neq p$ are fixed at $\bm{s}_{q}$, while
player $p$’s strategy may vary across their pure strategy set.
###### Definition 2 (Regret).
A player $p$’s regret at strategy profile $\bm{s}\in\bm{{S}^{\diamond}}$ is
defined as
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\bm{u})\doteq\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\bm{u}_{p}(\bm{s}^{\prime})-\bm{u}_{p}(\bm{s})$.
We further define $\operatorname{Reg}(\bm{s};\bm{u})\doteq\max_{p\in
P}\operatorname{Reg}_{p}(\bm{s};\bm{u})$.
A strategy profile $\bm{s}\in\bm{S}$ is player $p$’s _best response_ if
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$ (i.e., the player does not regret
choosing this strategy profile as opposed to an adjacent one). We say it is an
$\varepsilon$-_best response_ if
$\operatorname{Reg}_{p}(\bm{s};\bm{u})\leq\varepsilon$.
A strategy profile $\bm{s}\in\bm{{S}^{\diamond}}$ is an $\varepsilon$-_Nash
equilibrium_ if it is an $\varepsilon$-best response for each player $p\in P$
(i.e., if $\operatorname{Reg}(\bm{s};\bm{u})\leq\varepsilon$). If $\bm{s}$
corresponds to a pure strategy profile, then we call it an $\varepsilon$-_pure
Nash equilibrium_ ($\varepsilon$-PNE); otherwise, we call it an
$\varepsilon$-_mixed Nash equilibrium_ ($\varepsilon$-MNE). A $0$-PNE is
simply called a PNE, and a $0$-MNE is called an MNE. The set of
$\varepsilon$-pure (resp. mixed) Nash equilibria is denoted
$\textup{E}_{\varepsilon}(\bm{u})$ (resp.
$\textup{E}^{\diamond}_{\varepsilon}(\bm{u})$), and the set of pure (resp.
mixed) Nash equilibria is denoted $\textup{E}(\bm{u})$ (resp.
$\textup{E}^{\diamond}(\bm{u})$).
### Formal Model of Simulation Based Games
In simulation-based games, we assume access to a _simulator_
$\mathscr{S}(\cdot)$, which can be queried to produce unbiased noisy samples
of the players’ utilities when $\bm{s}\in\bm{S}$ is played. We denote such a
sample by $\dot{\bm{u}}(\bm{s})\sim\mathscr{S}(\bm{s})$, where
$\dot{\bm{u}}(\bm{s})$ is a $\left\lvert{}P\right\rvert{}$-vector comprising
utilities for each player.
###### Definition 3 (Simulation-Based Game).
A simulation-based game $\Gamma_{\mathscr{S}}\doteq\langle
P,\bm{S},\mathscr{S}\rangle$ consists of a set of players $P$, a pure strategy
profile space $\bm{S}\doteq S_{1}\times\dots\times
S_{\left\lvert{}P\right\rvert{}}$, and a simulator $\mathscr{S}(\cdot)$ that
produces noisy samples $\dot{\bm{u}}(\bm{s})\sim\mathscr{S}(\bm{s})$ upon
simulation of a strategy profile $\bm{s}\in\bm{S}$.
Corresponding to each simulation-based game $\Gamma_{\mathscr{S}}$ is an
expected normal-form game.
###### Definition 4 (Expected Normal-Form Game).
Given a simulation-based game $\Gamma_{\mathscr{S}}\doteq\langle
P,\bm{S},\mathscr{S}\rangle$, we define the “underlying” utility function
$\bm{u}:\bm{S}\to\mathbb{R}^{\left\lvert{}P\right\rvert{}}$ by
$\bm{u}(s)\doteq\operatorname*{\mathbb{E}}_{\dot{\bm{u}}(\bm{s})\sim\mathscr{S}(\bm{s})}\left[\dot{\bm{u}}(\bm{s})\right]$.
The expected game corresponding to $\Gamma_{\mathscr{S}}$ is then the normal-
form game $\langle P,\bm{S},\bm{u}\rangle$. Overloading notation, we also let
$\Gamma_{\mathscr{S}}$ denote this (unknown) expected normal-form game.
Since we do not have direct access to the expected normal-form game, its
utilities must be learned by repeatedly querying the simulator. The resulting
empirical estimate of the expected game is called an empirical game.
###### Definition 5 (Empirical Normal-Form Game).
Given a simulation-based game $\Gamma_{\mathscr{S}}\doteq\langle
P,\bm{S},\mathscr{S}\rangle$, let $\dot{\bm{u}}^{(1)}(\bm{s})$, $\dots$,
$\dot{\bm{u}}^{(m_{\bm{s}})}(\bm{s})\sim\mathscr{S}(\bm{s})$ denote the sample
utilities produced by $m_{\bm{s}}>0$ queries to the simulator at strategy
profile $\bm{s}\in\bm{S}$. We define the empirical utility function
$\hat{\bm{u}}:\bm{S}\to\mathbb{R}^{|P|}$ by
$\hat{\bm{u}}(\bm{s})\doteq\frac{1}{m_{\bm{s}}}\sum_{i=1}^{m_{\bm{s}}}\dot{\bm{u}}^{(i)}(\bm{s})$
for all $\bm{s}\in\bm{S}$, and the ensuing empirical normal-form game by
$\hat{\Gamma}_{\mathscr{S}}\doteq\langle P,\bm{S},\hat{\bm{u}}\rangle$.
From here onwards, let $\Gamma_{\mathscr{S}}\doteq\langle
P,\bm{S},\mathscr{S}\rangle$ be an arbitrary simulation-based game with
underlying utility function $\bm{u}$, and let
$\hat{\Gamma}_{\mathscr{S}}\doteq\langle P,\bm{S},\hat{\bm{u}}\rangle$ be a
corresponding empirical game. Using our formalization, we can now present one
of the foundational results of EGTA (Tuyls et al. 2020).
###### Lemma 1.
If
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$, then
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}_{4\varepsilon}(\bm{u})\textrm{
and
}\textup{E}^{\diamond}(\bm{u})\subseteq\textup{E}^{\diamond}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}^{\diamond}_{4\varepsilon}(\bm{u})\enspace,$
or more generally,
$\textup{E}_{\gamma}(\bm{u})\subseteq\textup{E}_{2\varepsilon+\gamma}(\hat{\bm{u}})$
and
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$
for all $\gamma\geq 0$ (resp. for mixed equilibria).
This result can be understood as stating that given a sufficiently strong
approximation $\hat{\Gamma}_{\mathscr{S}}$ of a simulation-based game
$\Gamma_{\mathscr{S}}$, we can approximate pure (resp. mixed) Nash equilibria
in $\Gamma_{\mathscr{S}}$ with perfect recall – all pure (resp. mixed) Nash
equilibria in $\Gamma_{\mathscr{S}}$ are approximate Nash equilibria in
$\hat{\Gamma}_{\mathscr{S}}$ – and with _approximately_ perfect precision –
all approximate pure (resp. mixed) Nash equilibria in
$\hat{\Gamma}_{\mathscr{S}}$ are approximate pure (resp. mixed) Nash
equilibria in $\Gamma_{\mathscr{S}}$. Lemma 1 is one of the primary
motiviations for EGTA’s pursuit of designing efficient algorithms for learning
strong approximations of simulation-based games, as it guarantees that the
better an approximation an empirical game is of a simulation-based game, the
more strategically representative the empirical game will be of the underlying
simulation-based game.
### Tail Bounds
Next, we state the tail bounds upon which our novel regret pruning techniques
and progressive sampling algorithms depend. These are the same bounds derived
and used by Cousins et al.; for a more thorough discussion of them see Cousins
et al. (2022). For all subsequent results, we make the following “bounded
utilities” assumption.
###### Assumption 1 (Bounded Utilities).
For each strategy profile $\bm{s}\in\bm{S}$, the sample utilities produced via
$\mathscr{S}(\bm{s})$ lie on the bounded interval $[a_{\bm{s}},b_{\bm{s}}]$
for some fixed $a_{\bm{s}},b_{\bm{s}}\in\mathbb{R}$. We define
$c:=\sup_{\bm{s}\in\bm{S}}(b_{\bm{s}}-a_{\bm{s}})$.
The most straight-forward tail bound for mean-estimation is Hoeffding’s
Inequality, which was used by Tuyls et al. (2020). We use Hoeffding’s
inequality to bound each individual utility, combined with a union bound to
yield a guarantee for all utilities.
###### Theorem 1 (Hoeffding’s Inequality).
Let $\hat{\Gamma}_{\mathscr{S}}\doteq\langle P,\bm{S},\hat{\bm{u}}\rangle$ be
an empirical game. Then, with probability at least $1-\updelta$, for all
$(p,\bm{s})\in\bm{\mathcal{I}}$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq
c\sqrt{\frac{\ln\left(\nicefrac{{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}}{{\updelta}}\right)}{2m_{\bm{s}}}}\doteq\varepsilon^{\textup{H}}_{p}(\bm{s})\enspace.$
Let $\bm{v}_{p}(\bm{s})$ denote
$\operatorname*{\mathbb{V}}_{\dot{\bm{u}}(\bm{s})\sim\mathscr{S}(\bm{s})}\left[\dot{\bm{u}}_{p}(\bm{s})\right]$
for all $(p,\bm{s})\in P\times\bm{S}$. Since Hoeffding’s Inequality assumes a
worst-case variance on the utilities (i.e.,
$\bm{v}_{p}(\bm{s})=\nicefrac{{c^{2}}}{{4}}$), when variances are small, it
yields a very loose bound. When the variances of utilities are known,
Bennett’s inequality provides a non-uniform, variance-sensitive guarantee.
###### Theorem 2 (Bennett’s Inequality).
Let $\hat{\Gamma}_{\mathscr{S}}\doteq\langle P,\bm{S},\hat{\bm{u}}\rangle$ be
an empirical game. Then, with probability at least $1-\updelta$, for all
$(p,\bm{s})\in\bm{\mathcal{I}}$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\frac{c\ln\left(\nicefrac{{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}}{{\updelta}}\right)}{3m_{\bm{s}}}+\sqrt{\frac{2\bm{v}_{p}(\bm{s})\ln\left(\nicefrac{{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}}{{\updelta}}\right)}{m_{\bm{s}}}}$
Of course, since our only access to the utilities of the simulation-based game
are via the simulator, we do not know their variances. Cousins et al. (2022)
circumvent this limitation by deriving an “empirical Bennett’s inequality”
depending on empirical estimates of the true utility variances.
###### Theorem 3 (Empirical Bennett’s Inequality).
Let $\hat{\Gamma}_{\mathscr{S}}\doteq\langle P,\bm{S},\hat{\bm{u}}\rangle$ be
an empirical game. Let
$\kappa_{\updelta}\doteq\left(\frac{1}{3}+\frac{1}{2\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}\right)$.
For all $(p,\bm{s})\in\bm{\mathcal{I}}$, define
$\displaystyle{\hat{\bm{v}}_{p}(\bm{s})}$
$\displaystyle\doteq{\frac{1}{m-1}}{\sum}\limits_{j=1}^{m}\left(\bm{u}_{p}(\bm{s};y_{j})-\hat{\bm{u}}_{p}(\bm{s};\bm{Y})\right)^{2};$
$\displaystyle\varepsilon^{\hat{\bm{v}}}_{p}(\bm{s})$
$\displaystyle\doteq{\frac{2c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{3m}+\sqrt{\kappa_{\updelta}\Bigl{(}\frac{c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m-1}\Bigr{)}^{\smash{2}}+\frac{2c^{2}\hat{\bm{v}}_{p}(\bm{s})\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}}};$
$\displaystyle\varepsilon^{\hat{\textup{B}}}_{p}(\bm{s})$
$\displaystyle\doteq{\frac{c\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{3m}+\sqrt{\frac{2\left(\hat{\bm{v}}_{p}(\bm{s})+\varepsilon^{\hat{\bm{v}}}_{p}(\bm{s})\right)\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}}}.$
Then, with probability at least $1-\updelta$, for all
$(p,\bm{s})\in\bm{\mathcal{I}}$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon^{\hat{\textup{B}}}_{p}(\bm{s})$.
This empirical Bennett guarantee forms the basis for our progressive sampling
algorithms.
### Progressive Sampling Algorithms
1:procedure
PSP($\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},c,\updelta,\varepsilon$)
2: input: Conditional game $\Gamma_{\mathcal{Y}}$, condition distribution
$\mathscr{D}$, index set $\bm{\mathcal{I}}$, failure probability
$\updelta\in(0,1)$, target error $\varepsilon>0$
3: Initialize empirical utilities $\hat{\bm{u}}^{(0)}_{p}(\bm{s})=0$ for
$(p,\bm{s})\in\bm{\mathcal{I}}$
4: Initialize utility deviation bounds
$\hat{\varepsilon}^{(0)}_{p}(\bm{s})=\infty$ for
$(p,\bm{s})\in\bm{\mathcal{I}}$
5: Initialize active utility index set
$\bm{\mathcal{I}}^{(0)}\leftarrow\bm{\mathcal{I}}$
6: Initialize a sampling schedule $m_{1},\dots,m_{T}$ and cumulative sample
size $M_{0}\leftarrow 0$
7: for $t\in 1,\dots,T$ do
8: for $\bm{s}\in\bm{S}$ do
9: Determine unpruned player indices
$P(\bm{s};\bm{\mathcal{I}}^{(t)})\leftarrow\\{p\in
P\mid(p,\bm{s})\in\bm{\mathcal{I}}^{(t)}\\}$ at strategy profile $\bm{s}$
10: Query simulator for utilities of unpruned players:
$\\{\dot{\bm{u}}_{p}(\bm{s})\mid p\in
P(\bm{s};\bm{\mathcal{I}}^{(t)})\\}\leftarrow\mathscr{S}(\bm{s};P(\bm{s};\bm{\mathcal{I}}^{(t)}))$
11: Update empirical utilities
$\hat{\bm{u}}^{(t)}_{p}(\bm{s})\leftarrow\frac{M_{t-1}}{M_{t-1}+m_{t}}\cdot\hat{\bm{u}}^{(t-1)}_{p}(\bm{s})+\frac{m_{t}}{M_{t-1}+m_{t}}\cdot\dot{\bm{u}}_{p}(\bm{s})$
for $p\in P(\bm{s};\bm{\mathcal{I}}^{(t)})$
12: Compute new utility deviation bounds $\hat{\varepsilon}^{(t)}_{p}(\bm{s})$
for $p\in P(\bm{s};\bm{\mathcal{I}}^{(t)})$, each with failure probability
$\nicefrac{{\updelta}}{{\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}}$
13: end for
14: Update cumulative sample size $M_{t}\leftarrow M_{t-1}+m_{t}$
15: Prune any indices in $\bm{\mathcal{I}}^{(t)}$ which do not require further
estimation (e.g., well-estimated pruning:
$\bm{\mathcal{I}}^{(t)}\leftarrow\smash{\\{(p,\bm{s})\in\bm{\mathcal{I}}^{(t-1)}\mid\hat{\varepsilon}^{(t)}_{p}(\bm{s})>\varepsilon\\}}$,
i.e., prune indices that have met the target $\varepsilon$ error guarantee)
16: if all indices in $\bm{\mathcal{I}}^{(t)}$ are pruned (i.e.,
$\bm{\mathcal{I}}^{(t)}=\emptyset$) then
17: return empirical utilities $\hat{\bm{u}}^{(t)}$
18: end if
19: end for
20:end procedure
Algorithm 1 General Progressive Sampling Algorithm
Finally, we present the general class of progressive sampling (PS) algorithms
(see Algorithm 1) for learning simualation-based games. As the name suggests,
progressive sampling algorithms work by progressively sampling utilities,
pruning those which are sufficiently estimated for the relevant learning goal
at hand. One core component of progressive sampling algorithms is the sampling
schedule. On each iteration, a progressive sampling algorithm will collect the
number of samples dictated by the sampling schedule for each active (i.e.,
unpruned) utility index. It will then use the samples to update the empirical
game and to compute new utility deviation bounds (which in our case will be
dependent on the tail bounds introduced earlier). Notice that the utility
deviation bounds must each have individual failure probability
$\frac{\updelta}{\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}$, as opposed to
just $\frac{\updelta}{\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}$ as in
Theorem 3. This is to ensure that, via an additional union bound, all pruned
indices will, with high probability (w.h.p.), have been pruned justifiably
with respect to the true game. Finally, the empirical game and utility
deviation bounds will be used to inform which utility indices can be pruned on
the current iteration. Progressive sampling algorithms terminate once either
all utility indices are pruned, or the sampling schedule is exhausted.
In this paper, we focus on PS algorithms for learning equilibria. On the basis
of Lemma 1, one sufficient condition for pruning an index
$(p,\bm{s})\in\bm{\mathcal{I}}$ is that it is estimated (w.h.p.) to within
some target error $\varepsilon$. If this is the only pruning criteria used,
then upon termination of the algorithm, if all indices have been pruned, the
resulting empirical game will (w.h.p.) satisfy
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$, and will thus (w.h.p.) satisfy the
pure and mixed dual Nash containment results in Lemma 1. Cousins et al. (2022)
design precisely this algorithm, using Hoeffding (Theorem 1) and empirical
Bennett (Theorem 3) bounds to inform their pruning (pruning once the guarantee
corresponding to an index is tighter than the target error $\varepsilon$), and
carefully crafting a sampling schedule which guarantees that all indices will
be pruned prior to its exhaustion. They refer to this pruning approach as
_well-estimated pruning_ (Cousins et al. 2022). We show an example of how the
pruning criteria may be implemented in Algorithm 1 15.
## Regret Pruning
Having covered the requisite background, we can now begin our discussion of
regret pruning and present our novel regret-pruning variations. As discussed
earlier, Lemma 1 immediately suggests well-estimated pruning as a pruning
approach, and this approach was used to design the PS algorithm presented in
Cousins et al. (2022) (which we henceforth refer to as PS-WE for Progressive
Sampling with Well-Estimated Pruning).
###### Theorem 4.
If PS-
WE$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon)$
returns an empirical utility function $\hat{\bm{u}}$, then with probability at
least $1-\updelta$, for all $\gamma\geq 0$, it holds that
1. 1.
$\textup{E}_{\gamma}(\bm{u})\subseteq\textup{E}_{2\varepsilon+\gamma}(\hat{\bm{u}})$
and
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$
2. 2.
$\textup{E}^{\diamond}_{\gamma}(\bm{u})\subseteq\textup{E}^{\diamond}_{2\varepsilon+\gamma}(\hat{\bm{u}})$
and
$\textup{E}^{\diamond}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}^{\diamond}_{2\varepsilon+\gamma}(\bm{u})$
.
The condition presented in Lemma 1 (i.e.,
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$), however, is not the only condition
to yield these kinds of Nash containment results. Consider the PS algorithm
designed in Areyan Viqueira, Cousins, and Greenwald (2020) (which we refer to
as PS-REG-0 for Progressive Sampling with Regret Pruning; the 0 will
distinguish this algorithm from our new variations). Unlike PS-WE, PS-REG-0
uses uniform utility deviation bounds – they simply bound all utility
deviations by
$\sup_{(p,\bm{s})\in\bm{\mathcal{I}}}\varepsilon^{\hat{\textup{B}}}_{p}(\bm{s})$
($\varepsilon^{\hat{\textup{B}}}$ defined as in Theorem 3)111Areyan Viqueira,
Cousins, and Greenwald (2020) use a version of the empirical Bennett tail
bounds that is a constant factor looser than the one presented here, as the
tighter bound had not been derived until Cousins et al. (2022). PS-REG-0 uses
well-estimated pruning (though the authors do not explicitly call it that),
but it additionally uses what the authors call “regret pruning” (Areyan
Viqueira, Cousins, and Greenwald 2020). The authors claim that though using
this pruning approach does _not_ guarantee that the condition in Lemma 1 is
met upon termination of the algorithm, it nonetheless guarantees that (w.h.p.)
the pure Nash containment result
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}_{4\varepsilon}(\bm{u})$
is satisfied. In this section, we present a counterexample which shows that
their pruning approach does not in fact guarantee this pure Nash containment
result. We show that, instead, their pruning approach is only able to
guarantee a weaker Nash containment result. Furthermore, we also present 3
novel variations of the regret pruning criterion presented in Areyan Viqueira,
Cousins, and Greenwald (2020) which each have varying benefits and costs in
comparison to the original. The first variation is a generalization of the
original regret pruning criterion with respect to a hyper-parameter
$\gamma^{*}\geq 0$. We show that when $\gamma^{*}=0$, this variation is
identical to the regret pruning criterion from PS-REG-0. When
$\gamma^{*}=2\varepsilon$, however, we show that, at the cost of taking
slightly longer to prune indices, this new variation yields the stronger pure
Nash containment guarantee which Areyan Viqueira, Cousins, and Greenwald
(2020) originally intended their pruning criterion to meet (i.e.,
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}_{4\varepsilon}(\bm{u})$).
The second variation takes advantage of non-uniform utility deviation bounds
to yield the same guarantee as the first variation, while pruning indices
significantly sooner in practice than otherwise. Finally, the third variation
modifies the second variation, yielding a mixed Nash containment guarantee in
addition to the same pure Nash guarantee, at the cost of a slightly tighter
pruning criterion than variation 2.
### Old Regret Pruning
We begin by presenting the regret pruning criterion used in PS-REG-0 from
Areyan Viqueira, Cousins, and Greenwald (2020). Since PS-REG-0 uses uniform
utility deviation bounds we simply use $\hat{\varepsilon}^{(t)}$ to denote the
utility deviation bound on $\hat{\bm{u}}^{(t)}$ at iteration $t$. PS-REG-0
prunes an index $(p,\bm{s})\in\bm{\mathcal{I}}$ on an iteration $t$ if any of
the following holds:
1. 1.
$\hat{\varepsilon}^{(t)}\leq\varepsilon$ (well-estimated pruning)
2. 2.
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})\geq
2\hat{\varepsilon}^{(t)}$ (regret pruning).
Areyan Viqueira, Cousins, and Greenwald (2020) claim that PS-REG-0 satisfies
the following guarantee.
###### Claim 1.
If PS-
REG-0$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon)$
returns an empirical utility function $\hat{\bm{u}}$, then with probability at
least $1-\updelta$, it holds that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}_{4\varepsilon}(\bm{u})$.
We present a counter-example that shows that the second inclusion
$\textup{E}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}_{4\varepsilon}(\bm{u})$
does not necessarily hold.
###### Counterexample 1.
Consider a two-player game $\Gamma$ in which player $A$ has two pure
strategies, $a_{1}$ and $a_{2}$, while player $B$ has just one pure strategy
$b$. Define player $A$’s utility function by $\bm{u}_{A}(a_{1},b)=2$ and
$\bm{u}_{A}(a_{2},b)=1$. Suppose we run PS-REG-0 with target error
$\varepsilon\doteq 0.2$ and get the following:
$\displaystyle\textup{Iteration 1:}\quad\boxed{\begin{aligned}
&\hat{\bm{u}}^{(1)}_{A}(a_{1},b)=2.5\\\
&\hat{\bm{u}}^{(1)}_{A}(a_{2},b)=1.45\\\ &\hat{\varepsilon}^{(1)}=0.5\\\
\end{aligned}}$ $\displaystyle\quad\Longrightarrow\quad\begin{aligned}
&\textup{Index $(A,(a_{2},b))$ regret pruned}\\\ &\textup{(since
$\hat{\varepsilon}^{(1)}=0.5<\frac{\operatorname{Reg}_{A}((a_{2},b);\hat{\bm{u}}^{(1)})}{2}=0.525$)}\end{aligned}$
$\displaystyle\textup{Iteration 2:}\quad\boxed{\begin{aligned}
&\hat{\bm{u}}^{(2)}_{A}(a_{1},b)=1.8\\\
&\hat{\bm{u}}^{(2)}_{A}(a_{2},b)=1.45\\\ &\hat{\varepsilon}^{(2)}=0.2\\\
\end{aligned}}$ $\displaystyle\quad\Longrightarrow\quad\begin{aligned}
&\textup{Index $(A,(a_{1},b))$ well-estimated pruned;}\\\ &\textup{PS-REG-0{}
terminates with $\hat{\bm{u}}_{A}=\hat{\bm{u}}^{(2)}_{A}$}\end{aligned}$
We have that $(a_{2},b)\in\textup{E}_{2\varepsilon}(\hat{\bm{u}})$, since
$\operatorname{Reg}_{A}((a_{2},b);\hat{\bm{u}})=1.8-1.45=0.35<2\varepsilon=0.4$
and $\operatorname{Reg}_{B}((a_{2},b);\hat{\bm{u}})=0$. But we have
$(a_{2},b)\not\in\textup{E}_{4\varepsilon}(\bm{u})$, since
$\operatorname{Reg}_{A}((a_{2},b);\bm{u})=2-1=1>4\varepsilon=0.8$. Hence, we
have
$\textup{E}_{2\varepsilon}(\hat{\bm{u}})\not\subseteq\textup{E}_{4\varepsilon}(\bm{u})$.
Since all utility deviation guarantees have been held, this is not a failure
case. Thus, 1 cannot be true.
Though the second inclusion in 1 cannot be guaranteed, we show that an
alternative inclusion does hold (w.h.p.).
###### Theorem 5.
If PS-
REG-0$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon)$
returns an empirical utility function $\hat{\bm{u}}$, then with probability at
least $1-\updelta$, we have that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})$ and
$\textup{E}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon}(\bm{u})$.
From this guarantee, we see that if a game analyst is using PS-REG-0 to learn
an approximate pure Nash equilibrium of the simulation-based game, they will
need to compute a (true) pure Nash equilibrium of the resulting empirical
game. Of course, in many games, computing a pure Nash equilibrium is
computationally intractable (e.g., it is NP-complete in graphical games
(Gottlob, Greco, and Scarcello 2005)); the best that can be hoped for is an
approximate pure Nash equilibrium. Furthermore, even if the simulation-based
game has a pure Nash equilibrium, it is not then guaranteed that the empirical
game will also have a pure Nash equilibrium, but rather only that it will have
a $2\varepsilon$-pure Nash equilibrium. A limitation of PS-REG-0 is that it is
not able to provide any guarantees regarding the efficacy of approximate pure
Nash equilibria from the empirical game in the true game.
### New Regret Pruning Variations
We now introduce our novel regret pruning criteria. We begin by presenting a
variation which resolves the limitation observed in PS-REG-0 of lacking
guarantees regarding empirical approximate Nash equilibria. This variation is
derived on the basis of a stronger version of Lemma 1.
###### Lemma 2.
Let $\gamma^{*}\geq 0$. If
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$ satisfying
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$ or
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq\gamma^{*}$, then
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})$ and
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$
for all $0\leq\gamma\leq\gamma^{*}$.
Whereas Lemma 1 required all indices to be well-estimated, Lemma 2 requires
only indices with sufficiently low regret in both the true game and empirical
game to be well-estimated. This gives room for indices with provably high
regret to be regret-pruned. Of course, as Areyan Viqueira, Cousins, and
Greenwald (2020) also observed, this potential for regret pruning seems to
come at the cost of any guarantees regarding mixed Nash equilibria. On the
basis of Lemma 2, we design our first regret pruning variation.
###### Theorem 6.
Consider a PS algorithm, PS-REG, using uniform utility deviation bounds, which
conducts well-estimated pruning and on each iteration $t$, also _regret-
prunes_ any index $(p,\bm{s})\in\bm{\mathcal{I}}$ which satisfies
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>\max\\{2\hat{\varepsilon}^{(t)},\gamma^{*}+\varepsilon+\hat{\varepsilon}^{(t)}\\}\enspace.$
If PS-
REG$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon,\gamma^{*})$
returns an empirical utility function $\hat{\bm{u}}$, then with probability at
least $1-\updelta$, it holds that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})$ and
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$
for all $0\leq\gamma\leq\gamma^{*}$.
Notice that when $\gamma^{*}=0$, PS-REG is identical to PS-REG-0, even
yielding the same exact guarantees. PS-REG is thus a generalization of PS-
REG-0 to cases where $\gamma^{*}>0$. When $\gamma^{*}>0$, PS-REG yields, at
the cost of potentially reduced pruning, a stronger guarantee upon termination
than PS-REG-0, ensuring that even an approximate empirical pure Nash
equilibrium (so long as it is at worst a $\gamma^{*}$-pure Nash equilibrium)
will be an approximate pure Nash equilibrium in the true game. In practice,
the parameter $\gamma^{*}$ can be set to the smallest value for which the game
analyst is still certain they will be able to compute a $\gamma^{*}$-pure Nash
equilibria of the empirical game. If we set $\gamma^{*}=2\varepsilon$, we get
the dual pure Nash containment guarantee,
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\subseteq\textup{E}_{4\varepsilon}(\bm{u})$,
which PS-REG-0 was originally designed to meet.
One limitation of both PS-REG and PS-REG-0 is that they both depend only on
uniform utility deviation bounds. The next algorithm and regret pruning
variation takes advantage of non-uniform utility deviation bounds (Theorem 3)
to prune potentially more (and in practice significantly more; see Figure 1)
indices via both well-estimated pruning and regret-pruning, while yielding the
same guarantees as PS-REG. In the following result, we use
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}})$ to denote the high-
probability lower-bound on $\operatorname{Reg}_{p}(\bm{s};\bm{u})$ defined by
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}})\doteq\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}(\bm{u}_{p}(\bm{s}^{\prime})-\varepsilon_{p}(\bm{s}^{\prime}))-(\bm{u}_{p}(\bm{s})+\varepsilon_{p}(\bm{s})).$
###### Theorem 7.
Consider a PS algorithm, PS-REG+, using non-uniform utility deviation bounds,
which conducts well-estimated pruning and on each iteration $t$, also _regret-
prunes_ any index $(p,\bm{s})\in\bm{\mathcal{I}}$ which satisfies
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}_{p}(\bm{s})\\}\enspace.$
If PS-
REG+$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon,\gamma^{*})$
returns an empirical utility function $\hat{\bm{u}}$, then with probability at
least $1-\updelta$, it holds that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})$ and
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$
for all $0\leq\gamma\leq\gamma^{*}$.
Notice again that when $\gamma^{*}=0$, the pruning criterion becomes simply
$\operatorname{Reg}_{p}^{\downarrow}(\bm{s};\hat{\bm{u}}^{(t)})>0$ and yields
the same guarantees as PS-REG-0. Thus, when $\gamma^{*}=0$, PS-REG+ can also
reasonably be called PS-REG-0+.
Since all the aforementioned regret pruning variations derive from the result
presented in Lemma 2, they are only able to provide guarantees regarding pure
equilibria. This is the most glaring limitation of the new PS algorithms
presented so far, since many games do not even have strong approximate pure
Nash equilibria which the algorithms could potentially be used to learn. In
contrast, PS-WE derives from Lemma 1, and thus yields mixed Nash containment
guarantees, but of course does not allow for regret-pruning. We now present a
lemma which serves as a middle ground between Lemma 2 and Lemma 1.
###### Lemma 3.
If
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$, then for all $0\leq\gamma\leq
2\varepsilon$, it holds that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\textup{
and
}\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u}),$
and for all $\gamma\geq 0$, it holds that
$\textup{E}^{\diamond}(\bm{u})\subseteq\textup{E}^{\diamond}_{4\varepsilon}(\hat{\bm{u}})\textup{
and
}\textup{E}^{\diamond}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}^{\diamond}_{2\varepsilon+\frac{3\gamma}{2}}(\bm{u}).$
The condition in Lemma 3 is looser than that in Lemma 1, opening up the
potential for regret-pruning, but is (strictly) tighter than the condition in
Lemma 2 when $\gamma^{*}=2\varepsilon$ (i.e., empirical games which satisfy
the condition in Lemma 3 necessarily satisfy the condition in Lemma 2 when
$\gamma^{*}=2\varepsilon$, but the converse does not hold; see Appendix for
proof), which allows for the additional mixed Nash containment guarantee.
Notice further that unlike Lemma 2, Lemma 3 does not depend on any additional
parameter $\gamma^{*}$ and yields guarantees for all $\gamma\geq 0$, rather
than just $\gamma\in[0,\gamma^{*}]$. We use Lemma 3 to derive yet another
regret pruning variation.
###### Theorem 8.
Consider a PS algorithm, PS-REG-M, using non-uniform utility deviation bounds,
which conducts well-estimated pruning and on each iteration $t$, also _regret-
prunes_ any index $(p,\bm{s})\in\bm{\mathcal{I}}$ which satisfies
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>\varepsilon+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\enspace.$
If PS-
REG-M$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon)$
returns an empirical utility function $\hat{\bm{u}}$, then with probability at
least $1-\updelta$, for all $0\leq\gamma\leq 2\varepsilon$, it holds that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})\textup{
and
}\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u}),$
and for all $\gamma\geq 0$, it holds that
$\textup{E}^{\diamond}(\bm{u})\subseteq\textup{E}^{\diamond}_{4\varepsilon}(\hat{\bm{u}})\textup{
and
}\textup{E}^{\diamond}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}^{\diamond}_{2\varepsilon+\frac{3\gamma}{2}}(\bm{u}).$
The mixed Nash containment guarantee achieved by PS-REG-M is a
$\frac{\gamma}{2}$ factor looser than that achieved by PS-WE (Lemma 1). As a
result, slightly stronger approximate empirical mixed Nash equilibria will
need to be computed when using PS-REG-M in order to guarantee an equally
strong approximate true mixed Nash equilibrium as in PS-WE.
### Efficiency Bounds and Correctness
Using similar proof techniques to those used in Cousins et al. (2022) to
derive efficiency bounds for PS-WE, we derive upper bounds on the number of
samples each utility index requires prior to being pruned by PS-REG+ and PS-
REG-M, respectively. In the following results, suppose that the utility
deviation bounds being used by the mentioned PS algorithms are the minimum of
Hoeffding bounds (Theorem 1) and empirical Bennett bounds (Theorem 3).
###### Theorem 9.
When running PS-
REG+$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon,\gamma^{*})$,
with probability at least $1-\frac{\updelta}{3}$, the index
$(p,\bm{s})\in\bm{\mathcal{I}}$ will be pruned prior to the first iteration
$t$ with cumulative sample size $M_{t}\geq$
$2+2\ln\smash{\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}}\min\begin{cases}\frac{10c}{\operatorname{Reg}_{p}(\bm{s};\bm{u})-\gamma^{*}}+\frac{25\left\lVert{}\bm{v}_{p}(\operatorname{Adj}_{p,\bm{s}})\right\rVert{}_{\infty}}{(\operatorname{Reg}_{p}(\bm{s};\bm{u})-\gamma^{*})^{2}}\\\
\frac{5c}{2\varepsilon}+\frac{\bm{v}_{p}(\bm{s})}{\varepsilon^{2}}\end{cases}$
(defaulting to the second option when
$\operatorname{Reg}_{p}(\bm{s};\bm{u})\leq\gamma^{*}$).
When running PS-
REG-M$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon)$,
with probability at least $1-\frac{\updelta}{3}$, the index
$(p,\bm{s})\in\bm{\mathcal{I}}$ will be pruned prior to the first iteration
$t$ with cumulative sample size $M_{t}\geq$
$2+2\ln\smash{\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}}\min\begin{cases}\frac{12.5c}{\operatorname{Reg}_{p}(\bm{s};\bm{u})-\varepsilon}+\frac{25\left\lVert{}\bm{v}_{p}(\operatorname{Adj}_{p,\bm{s}})\right\rVert{}_{\infty}}{(\operatorname{Reg}_{p}(\bm{s};\bm{u})-\varepsilon)^{2}}\\\
\frac{5c}{2\varepsilon}+\frac{\bm{v}_{p}(\bm{s})}{\varepsilon^{2}}\end{cases}$
queries at profile $\bm{s}$ (defaulting to the second option when
$\operatorname{Reg}_{p}(\bm{s};\bm{u})\leq\varepsilon$).
The above result reinforces the idea that care is required when choosing a
sampling schedule for these algorithms. If the marginal sample size $m_{t}$ is
small at each iteration $t$, then very few queries will be wasted from when an
index is ready to be pruned to when it is actually pruned by the algorithm. On
the other hand, if very small marginal sample sizes are used, then a very
large schedule length $T$ will be required to reach a sufficiently large
cumulative sample size to prune all utility indices. But a larger schedule
length $T$ yields looser utility deviation bounds, thus resulting in _all_
indices requiring more queries to be pruned than otherwise. Hence, there is a
trade-off in designing a sampling schedule between keeping marginal sample
sizes small and keeping the schedule length small.
We discuss our particular choice of sampling schedule in the next section, and
in greater detail in the Appendix. In the following result, we use Hoeffding’s
Inequality to derive an upper bound on the requisite total cumulative sample
size a sampling schedule needs to ensure that all utility indices will be
pruned prior to its exhaustion.
###### Theorem 10.
Suppose that the total samples $M_{T}$ allocated in the sampling schedule is
greater than or equal to the maximum number of samples needed to prune an
arbitrary index, i.e.,
$M_{T}\doteq\sum_{t=1}^{T}m_{t}\geq\frac{c^{2}\ln\frac{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}}{2\varepsilon^{2}}\enspace.$
Then for each algorithm $\textup{PS}\in$
$\\{\textup{PS-WE{}, PS-REG-0{}, PS-REG{}, PS-REG+{}, PS-REG-M{}}\\},$
it is guaranteed that
PS$(\Gamma_{\mathcal{Y}},\mathscr{D},\bm{\mathcal{I}},\updelta,\varepsilon)$
will terminate and return an empirical game with the guarantees corresponding
to the respective algorithm.
## Experiments
Figure 1: Average query complexity vs $\nicefrac{{1}}{{\varepsilon}}$ for 10
runs of each algorithm for each target error
$\varepsilon\in\\{\frac{c}{2},\frac{c}{3},\dots,\frac{c}{100}\\}$. All
algorithm runs are conducted on a single randomly-generated two-player zero-
sum game with non-uniform additive noise.
In this section, we experimentally explore the behavior of all the
aforementioned PS algorithms. When choosing a sampling schedule for each
algorithm, we follow Cousins et al. (2022) in designing sampling schedules
which begin at sample size $\alpha$, a lower bound on the minimum number of
samples needed to prune an arbitrary index, and end with a cumulative sample
size $\omega$, an upper bound on the maximum number of samples needed to prune
an arbitrary index. For all algorithms, we use the upper bound from Theorem 10
for $\omega$, i.e.,
$\omega\doteq\frac{c^{2}\ln\frac{2\left\lvert{}\Gamma\right\rvert{}T}{\updelta}}{2\varepsilon^{2}}$,
thus guaranteeing that each algorithm will return an empirical game satisfying
the respective guarantees of the algorithm upon termination. For algorithms
using regret pruning, we set $\alpha$ to be a lower bound on the number of
samples to estimate a zero-variance utility to (w.h.p.) within $\frac{c}{2}$
error (since no regret pruning can occur prior to at least one index achieving
such an error guarantee; see Appendix for proof). For PS-WE, we follow Cousins
et al. (2022) in setting $\alpha$ to be a lower bound on the number of samples
needed to estimate a zero-variance utility to (w.h.p.) within a target error
$\varepsilon$, though we improve their lower bound by a small constant factor.
Finally, while our PS-WE sampling schedule has a geometric sampling schedule
(i.e., geometrically increasing cumulative sample size) as in Cousins et al.
(2022), using such a geometric schedule for our regret pruning algorithms
results in too many iterations spent on very small sample sizes, yielding
looser bounds with very few additional queries saved in return. To correct for
this, our regret pruning algorithms use a sampling schedule which is linear
until it reaches the corresponding $\alpha$ derived for PS-WE, and then
follows the same geometric sampling schedule used in PS-WE. For more details
regarding the sampling schedules, see the Appendix.
### Query Complexity vs. Target Error
In the following experiments, we test our algorithms on two-player random
zero-sum games (generated via the game-generator GAMUT (Nudelman et al. 2004))
with 40 actions for each player and utility values in the range $[-2,2]$. In
order to emulate a noisy simulator, we add noise to each sample utility value.
For each utility index $(p,\bm{s})\in\bm{\mathcal{I}}$, we sample a variance
modifier $\nu_{p,\bm{s}}\sim\textup{Beta(1.5, 3)}$, and then each time the
simulator is queried for $\dot{\bm{u}}(\bm{s})\sim\mathscr{S}(\bm{s})$, we set
$\dot{\bm{u}}_{p}(\bm{s})\doteq\bm{u}_{p}(\bm{s})+\mathcal{N}(\nu_{p,\bm{s}})$
where $\mathcal{N}(\nu_{p,\bm{s}})$ is a scaled and shifted Bernoulli random
variable, generating either $10\nu_{p,\bm{s}}$ or $-10\nu_{p,\bm{s}}$ with
equal probability. Notice then that our final utility range for these random
zero-sum simulation-based games is $c=24$. Sampling variance modifiers from
$\textup{Beta}(1.5,3)$ ensures that our utility indices have a wide range of
noise variables with mostly moderate variance, but with some noise variables
having particularly high variance and some particularly low variance.
For our first experiment (Figure 1), we compare the query complexities (i.e.,
the number of simulation queries placed prior to termination) of our
algorithms for varying target errors $\varepsilon$. We begin by generating a
two-player random zero-sum game $\Gamma_{\mathscr{S}}$ (and variance modifiers
$\nu_{p,\bm{s}}$ for each utility index $(p,\bm{s})\in\bm{\mathcal{I}}$). For
each target error
$\varepsilon\in\\{\frac{c}{2},\frac{c}{3},\dots,\frac{c}{100}\\}$, we then run
each of our aforementioned PS algorithms (with failure probability
$\updelta\doteq 0.05$) on $\Gamma_{\mathscr{S}}$ a total of 10 times and plot
the average query complexity across those runs. For each algorithm, we then
connect these average query complexities for each target error by a line plot
in Figure 1.
In Figure 1, we see that, as expected, PS-REG obtains its guarantees for
approximate pure equilibria of the empirical game at the cost of a slightly
greater query complexity than PS-REG-0. In a similar vein, PS-REG+ with
$\gamma^{*}=2\varepsilon$ also requires a greater number of queries than PS-
REG+ with $\gamma^{*}=0$ (i.e, PS-REG-0+) in order to yield its guarantees for
$\gamma^{*}$-pure equilibria of the empirical game. We also observe that PS-
REG-0 and PS-REG with $\gamma^{*}=2\varepsilon$, which both use uniform
utility deviation bounds, consume significantly more queries than PS-WE, which
takes advantage of non-uniform bounds but conducts no regret pruning. This
suggests that utilizing non-uniform utility deviation bounds is crucial for
designing query efficient progressive sampling algorithms. This idea is
further reinforced when looking at PS-REG+ and PS-REG-M, both of which use
regret pruning criteria that take advantage of non-uniform utility deviation
bounds, and outperform PS-WE by a very significant margin, especially when the
target error $\varepsilon$ is small. Another particularly surprising result is
that PS-REG-M only consumes marginally more queries than PS-REG+ with
$\gamma^{*}=2\varepsilon$, despite yielding strong mixed Nash containment
guarantees in return.
Figure 2: Average ratio of query complexity to query complexity of PS-WE vs
$\nicefrac{{1}}{{\varepsilon}}$ for a single run of each algorithm on each of
10 randomly generated two-player zero-sum games with non-uniform additive
noise for each target error
$\varepsilon\in\\{\frac{c}{2},\frac{c}{3},\dots,\frac{c}{100}\\}$. Standard
deviation bands are plotted and are just barely visible due to low variation.
Since our first experiment only tests on a single randomly generated
simulation-based game, it is possible that the generated game was just
particularly amenable to regret pruning. For our second experiment (Figure 2),
we observe the proportion of additional queries our new PS algorithms are able
to save on average (across 10 random zero-sum simulation-based games) in
comparison to PS-WE. This time, we run each PS algorithm (again with failure
probability $\updelta\doteq 0.05$) only once for each generated game.
In Figure 2, we observe that the comparative performance of the algorithms in
Figure 1 remain consistent across many different random two-player zero-sum
games. We further observe that past a certain turning point, progressive
sampling algorithms which use regret pruning techniques save a greater
proportion of queries with respect to PS-WE as smaller target errors
$\varepsilon$ are used. When the target error $\varepsilon$ is very small, PS-
REG+ (both $\gamma^{*}=0$ and $\gamma^{*}=2\varepsilon$) and PS-REG-M are able
to obtain their respective guarantees while saving more than 50% and 40%,
respectively, of the queries used by PS-WE.
## Conclusion
In this paper, we address a serious limitation of Areyan Viqueira, Cousins,
and Greenwald’s progressive sampling algorithm with regret pruning – it is
only able to yield guarantees regarding _true_ pure Nash equilibria of the
empirical game. We design two primary novel progressive sampling algorithms
for practitioners to use to learn equilibria in simulation-based games: PS-
REG+ and PS-REG-M. PS-REG+ combines well-estimated pruning with a novel regret
pruning variation which is modified to ensure the algorithm yields pure Nash
containment guarantees for approximate $\gamma^{*}$-pure Nash equilibria of
the empirical game and to take advantage of non-uniform utility deviation
bounds to prune utility indices as soon as possible. When using PS-REG+, a
game analyst will set $\gamma^{*}$ according to the weakest approximate pure
Nash equilibria for which they desire guarantees. PS-REG-M also incorporates
well-estimated pruning and a novel regret pruning variation, except unlike PS-
REG+, it yields strong Nash containment guarantees for all approximate pure or
mixed Nash equilibria of the empirical game, at the cost of a slightly greater
query complexity. Both PS-REG+ and PS-REG-M significantly outperform PS-WE,
the prior state-of-the-art algorithm for learning equilibria in simulation-
based games. In light of this, game analysts seeking such an algorithm should
use PS-REG+ if they only aim to learn pure Nash equilibria and should
otherwise use PS-REG-M.
In this work, we have only applied our progressive sampling algorithms and
pruning techniques to normal-form games. In future work, we aim to extend this
methodology to other game models such as extensive-form games. Additionally,
EGTA algorithms for learning equilibria of simulation-based game have thus far
been completely detached from algorithms for computing equilibria. A game
analyst must first use EGTA algorithms to learn a sufficiently strong
approximation of the simulation-based game, and then must compute equilibria
of this empirical game. Future work can incorporate variance-sensitive and
regret-sensitive progressive sampling techniques into an existing game-solving
algorithm to make an EGTA-aware game-solving algorithm.
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## Appendix A Appendix
### Notation
To improve readability of the following proofs, we introduce a few notational
short-hands.
1. 1.
Given a utility index $(p,\bm{s})\in\bm{\mathcal{I}}$ and a utility function
$\bm{u}$, let $\bm{s}^{*}_{\bm{u}}$ denote the best response of player $p$ to
the opponents’ strategies in $\bm{s}$, i.e.,
$\bm{s}^{*}_{\bm{u}}\doteq\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\bm{u}_{p}(\bm{s}^{\prime})$.
2. 2.
Given a mixed strategy profile $\bm{s}\in\bm{{S}^{\diamond}}$, player $p\in
P$, and strategy $t\in S_{p}$, we let $\bm{s}|_{t}$ denote the strategy
profile $\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}$ satisfying
$\bm{s}^{\prime}_{p}=t$, and let $\operatorname*{\mathbb{P}}[t|\bm{s}]$ denote
the probability that mixed strategy profile $\bm{s}$ assigns to strategy $t$.
### Nash Containment Lemmas
We begin by proving the three different Nash containment lemmas.
See 1
###### Proof.
Suppose that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$. Let $\gamma\geq 0$ be arbitrary, and
suppose that $\bm{s}\in\textup{E}_{\gamma}(\bm{u})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s})$
$\displaystyle\leq(\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})+\varepsilon)-(\bm{u}_{p}(\bm{s})-\varepsilon)$
By assumption
$\displaystyle\leq(\bm{u}_{p}(\bm{s})+\gamma+\varepsilon)-(\bm{u}_{p}(\bm{s})-\varepsilon)$
Since $\bm{s}\in\textup{E}_{\gamma}(\bm{u})$, it holds that
$\bm{u}_{p}(\bm{s})+\gamma\geq\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})$
$\displaystyle=2\varepsilon+\gamma,$
and hence $\bm{s}\in\textup{E}_{2\varepsilon+\gamma}(\hat{\bm{u}})$. Since
$\bm{s}$ was arbitrary, we have that
$\textup{E}_{\gamma}(\bm{u})\subseteq\textup{E}_{2\varepsilon+\gamma}(\hat{\bm{u}})$.
By completely analogous reasoning, we see that
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$.
∎
See 2
###### Proof.
Suppose that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$ satisfying
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$ or
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq\gamma^{*}$. Suppose that
$\bm{s}\in\textup{E}(\bm{u})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s})$
$\displaystyle\leq(\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})+\varepsilon)-(\bm{u}_{p}(\bm{s})-\varepsilon)$
$\displaystyle\begin{aligned}
&\textsf{$\operatorname{Reg}_{p}(\bm{s}^{*}_{\hat{\bm{u}}};\hat{\bm{u}})=0\leq\gamma^{*}$,
so $(p,\bm{s}^{*}_{\hat{\bm{u}}})$ is well-estimated (WE)}\\\
&\textsf{$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$ (since
$\bm{s}\in\textup{E}(\bm{u})$), so $(p,\bm{s})$ is WE}\end{aligned}$
$\displaystyle\leq(\bm{u}_{p}(\bm{s})+\varepsilon)-(\bm{u}_{p}(\bm{s})-\varepsilon)$
Since $\bm{s}\in\textup{E}(\bm{u})$, it holds that
$\bm{u}_{p}(\bm{s})\geq\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})$
$\displaystyle=2\varepsilon,$
and hence $\bm{s}\in\textup{E}_{2\varepsilon}(\hat{\bm{u}})$. Since $\bm{s}$
was arbitrary, we have that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})$. Now let
$\gamma\in[0,\gamma^{*}]$ and suppose instead that
$\bm{s}\in\textup{E}_{\gamma}(\hat{\bm{u}})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\bm{u})$
$\displaystyle=\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})-\bm{u}_{p}(\bm{s})$
$\displaystyle\leq(\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})+\varepsilon)-(\hat{\bm{u}}_{p}(\bm{s})-\varepsilon)$
$\displaystyle\begin{aligned}
&\textsf{$\operatorname{Reg}_{p}(\bm{s}^{*}_{\bm{u}};\bm{u})=0$ (by
definition), so $(p,\bm{s}^{*}_{\bm{u}})$ is WE}\\\
&\textsf{$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq\gamma^{*}$ (since
$\bm{s}\in\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{\gamma^{*}}(\hat{\bm{u}})$),
so $(p,\bm{s})$ is WE}\end{aligned}$
$\displaystyle\leq(\hat{\bm{u}}_{p}(\bm{s})+\gamma+\varepsilon)-(\hat{\bm{u}}_{p}(\bm{s})-\varepsilon)$
Since $\bm{s}\in\textup{E}_{\gamma}(\hat{\bm{u}})$, it holds that
$\hat{\bm{u}}_{p}(\bm{s})+\gamma\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})$
$\displaystyle=2\varepsilon+\gamma,$
and hence $\bm{s}\in\textup{E}_{2\varepsilon+\gamma}(\bm{u})$. Since $\bm{s}$
was arbitrary, we have that
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$
and we are done. ∎
Before proving the third Nash containment lemma, we prove an accessory lemma.
###### Lemma 4.
Let $\varepsilon>0$ be arbitrary. Suppose that for all
$(p,\bm{s})\in\bm{\mathcal{I}}$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}\enspace.$
Then for all $(p,\bm{s})\in\bm{\mathcal{I}}$ satisfying
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$, it must hold that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$.
###### Proof.
Suppose that $(p,\bm{s})\in\bm{\mathcal{I}}$ satisfies
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$. Since
$\operatorname{Reg}_{p}(\bm{s}^{*}_{\hat{\bm{u}}};\hat{\bm{u}})=0$, we have
that
$\left\lvert{}\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})\right\rvert{}\leq\varepsilon$,
which implies that
$\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\varepsilon$.
But by definition, we have that
$\bm{u}_{p}(\bm{s})\geq\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})$, and hence, it
must hold that
$\bm{u}_{p}(\bm{s})\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\varepsilon$.
By hypothesis, we also have that
$\bm{u}_{p}(\bm{s})\leq\hat{\bm{u}}_{p}(\bm{s})+\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}$.
Chaining these two inequalities, we have that
$\displaystyle\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\varepsilon$
$\displaystyle\leq\hat{\bm{u}}_{p}(\bm{s})+\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}$
$\displaystyle\Longleftrightarrow\quad$
$\displaystyle\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s})$
$\displaystyle\leq\varepsilon+\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}$
$\displaystyle\Longleftrightarrow\quad$
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$
$\displaystyle\leq\varepsilon+\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}\enspace.$
If $\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}>\varepsilon$, then
the second term in the max wins out, but solving for
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$ immediately yields that
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq 2\varepsilon$, a
contradiction. Hence, it must hold that
$\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\leq\varepsilon$, and
hence that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$.
∎
See 3
###### Proof.
Suppose that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$. We will first show the pure Nash
containment result, and then show the mixed Nash containment result. Suppose
that $\bm{s}\in\textup{E}({\bm{u}})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s})$
By definition
$\displaystyle\leq(\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})+\varepsilon)-(\bm{u}_{p}(\bm{s})-\varepsilon)$
$\displaystyle\begin{aligned} &\textsf{Since
$\operatorname{Reg}_{p}(\bm{s}^{*}_{\hat{\bm{u}}};\hat{\bm{u}})=0$, by
hypothesis $(p,\bm{s}^{*}_{\hat{\bm{u}}})$ is WE}\\\ &\textsf{Since
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$, by
\lx@cref{creftypecap~refnum}{lem:zero-regret} $(p,\bm{s})$ is
WE.}\end{aligned}$
$\displaystyle\leq(\bm{u}_{p}(\bm{s})+\varepsilon)-(\bm{u}_{p}(\bm{s})-\varepsilon)$
Since $\bm{s}\in\textup{E}(\bm{u})$, it holds that
$\bm{u}_{p}(\bm{s})\geq\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})$
$\displaystyle=2\varepsilon,$
and hence $\bm{s}\in\textup{E}_{2\varepsilon}(\hat{\bm{u}})$. Since $\bm{s}$
was arbitrary, we have that
$\textup{E}(\bm{u})\subseteq\textup{E}_{2\varepsilon}(\hat{\bm{u}})$.
Now let $\gamma\in[0,2\varepsilon]$ suppose instead that
$\bm{s}\in\textup{E}_{\gamma}(\hat{\bm{u}})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\bm{u})$
$\displaystyle=\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})-\bm{u}_{p}(\bm{s})$ By
definition
$\displaystyle\leq(\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})+\varepsilon)-(\hat{\bm{u}}_{p}(\bm{s})-\varepsilon)$
$\displaystyle\begin{aligned} &\textsf{Since
$\operatorname{Reg}_{p}(\bm{s}^{*}_{\bm{u}};\bm{u})=0$, by
\lx@cref{creftypecap~refnum}{lem:zero-regret} $(p,\bm{s}^{*}_{\bm{u}})$ is
WE}\\\ &\textsf{Since
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq\gamma\leq 2\varepsilon$ (by
def. of $\textup{E}_{\gamma}(\hat{\bm{u}})$), by hypothesis $(p,\bm{s})$ is
WE}\end{aligned}$
$\displaystyle\leq(\hat{\bm{u}}_{p}(\bm{s})+\gamma+\varepsilon)-(\hat{\bm{u}}_{p}(\bm{s})-\varepsilon)$
Since $\bm{s}\in\textup{E}_{\gamma}(\hat{\bm{u}})$, it holds that
$\hat{\bm{u}}_{p}(\bm{s})+\gamma\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})$
$\displaystyle=2\varepsilon+\gamma,$
and hence $\bm{s}\in\textup{E}_{2\varepsilon+\gamma}(\bm{u})$. Since $\bm{s}$
was arbitrary, we have that
$\textup{E}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}_{2\varepsilon+\gamma}(\bm{u})$,
and we have shown the pure Nash containment result.
Now we will prove the mixed Nash containment result. Suppose that
$\bm{s}\in\textup{E}^{\diamond}(\bm{u})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s})$
By definition
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\sum_{t\in
S_{p}}\hat{\bm{u}}_{p}(\bm{s}|_{t})\operatorname*{\mathbb{P}}[t|\bm{s}]$ By
definition (of utility at mixed strategy profile)
$\displaystyle\leq(\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})+\varepsilon)-\sum_{t\in
S_{p}}\left(\bm{u}_{p}(\bm{s}|_{t})-\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right\\}\right)\operatorname*{\mathbb{P}}[t|\bm{s}]$
By hypothesis $\displaystyle\leq(\bm{u}_{p}(\bm{s})+\varepsilon)-\sum_{t\in
S_{p}}\left(\bm{u}_{p}(\bm{s}|_{t})-\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right\\}\right)\operatorname*{\mathbb{P}}[t|\bm{s}]$
Since $\bm{s}\in\textup{E}^{\diamond}(\bm{u})$, we have
$\bm{u}_{p}(\bm{s})\geq\bm{u}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})$
$\displaystyle=\varepsilon+\sum_{t\in
S_{p}}\operatorname*{\mathbb{P}}[t|\bm{s}]\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right\\}$
By definition (of utility at mixed strategy profile)
$\displaystyle\leq\varepsilon+\sum_{t\in
S_{p}}\operatorname*{\mathbb{P}}[t|\bm{s}]\left(\varepsilon+\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right)$
$\max\\{A,B\\}\leq A+B$
$\displaystyle=2\varepsilon+\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}.$
By definition (of utility at mixed strategy profile)
Solving the above for $\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})$, we get
that $\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq 4\varepsilon$, and hence
$\bm{s}\in\textup{E}^{\diamond}_{4\varepsilon}(\hat{\bm{u}})$. Since $\bm{s}$
was arbitrary, we have that
$\textup{E}^{\diamond}(\bm{u})\subseteq\textup{E}^{\diamond}_{4\varepsilon}(\hat{\bm{u}})$.
Now instead let $\gamma\geq 0$, and suppose that
$\bm{s}\in\textup{E}^{\diamond}_{\gamma}(\hat{\bm{u}})$. Then we have that
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\bm{u})$
$\displaystyle=\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})-\bm{u}_{p}(\bm{s})$ By
definition $\displaystyle=\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})-\sum_{t\in
S_{p}}\bm{u}_{p}(\bm{s}|_{t})\operatorname*{\mathbb{P}}[t|\bm{s}]$ By
definition (of utility at mixed strategy profile)
$\displaystyle\leq(\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})+\varepsilon)-\sum_{t\in
S_{p}}\left(\hat{\bm{u}}_{p}(\bm{s}|_{t})-\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right\\}\right)\operatorname*{\mathbb{P}}[t|\bm{s}]$
Hypothesis + Lemma 4
$\displaystyle\leq(\hat{\bm{u}}_{p}(\bm{s})+\gamma+\varepsilon)-\sum_{t\in
S_{p}}\left(\hat{\bm{u}}_{p}(\bm{s}|_{t})-\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right\\}\right)\operatorname*{\mathbb{P}}[t|\bm{s}]$
Since $\bm{s}\in\textup{E}^{\diamond}_{\gamma}(\hat{\bm{u}})$, we have
$\hat{\bm{u}}_{p}(\bm{s})+\gamma\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})$
$\displaystyle=\gamma+\varepsilon+\sum_{t\in
S_{p}}\operatorname*{\mathbb{P}}[t|\bm{s}]\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right\\}$
By definition (of utility at mixed strategy profile)
$\displaystyle\leq\gamma+\varepsilon+\sum_{t\in
S_{p}}\operatorname*{\mathbb{P}}[t|\bm{s}]\left(\varepsilon+\frac{\operatorname{Reg}_{p}(\bm{s}|_{t};\hat{\bm{u}})}{2}\right)$
$\max\\{A,B\\}\leq A+B$
$\displaystyle=\gamma+2\varepsilon+\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}$
By definition (of utility at mixed strategy profile)
$\displaystyle\leq\frac{3\gamma}{2}+2\varepsilon\enspace,$ Since
$\bm{s}\in\textup{E}_{\gamma}(\hat{\bm{u}})$, we have
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq\gamma$
and hence
$\bm{s}\in\textup{E}^{\diamond}_{2\varepsilon+\frac{3\gamma}{2}}(\bm{u})$.
Since $\bm{s}$ was arbitrary, we have that
$\textup{E}^{\diamond}_{\gamma}(\hat{\bm{u}})\subseteq\textup{E}^{\diamond}_{2\varepsilon+\frac{3\gamma}{2}}(\bm{u})$.
∎
In the text, we claim that the condition in Lemma 3 is strictly tighter than
the condition in Lemma 2 when $\gamma^{*}=2\varepsilon$. We show this by
proving the forward implication from Lemma 3’s condition to Lemma 2’s
condition, and then showing the reverse implication to be false.
Suppose the condition from Lemma 3, i.e., that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$. Then by Lemma 4, we have that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$
for all $(p,\bm{s})\in\bm{\mathcal{I}}$ with
$\operatorname{Reg}_{p}(\bm{s};\bm{u})=0$. But by the hypothesis, we directly
have that for all $(p,\bm{s})\in\bm{\mathcal{I}}$ with
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})\leq 2\varepsilon$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})}{2}\right\\}\leq\max\\{\varepsilon,\varepsilon\\}=\varepsilon$.
Hence, the condition from Lemma 2 is satisfied.
But obviously the converse does not hold, since indices with
$\operatorname{Reg}_{p}(\bm{s};\bm{u})>0$ and
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}})>2\varepsilon$ can have
arbitrarily bad approximations (e.g., the empirical utilities can be
arbitrarily low), even if the condition in Lemma 2 holds.
### Algorithm Correctness Proofs
A crucial component to all of our progressive sampling algorithm correctness
proofs is the use of a union bound to ensure that (w.h.p.) all pruning that
occurs is justified with respect to the true game. We see this line of
reasoning in the following correctness proof for PS-WE.
See 4
###### Proof.
Recall that for each index $(p,\bm{s})\in\bm{\mathcal{I}}$ and iteration
$t\in\\{1,\dots,T\\}$, the scalar $\hat{\varepsilon}^{(t)}_{p}(\bm{s})$ is an
upper bound on
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})\right\rvert{}$
with probability at least
$1-\frac{\updelta}{\left\lvert{}\Gamma\right\rvert{}T}$. Thus, via a union
bound, with probability $1-\updelta$, it holds _for all_ indices
$(p,\bm{s})\in\bm{\mathcal{I}}$ and iterations $t\in\\{1,\dots,T\\}$ that if
$\hat{\bm{u}}_{p}^{(t)}(\bm{s})$ has been computed (i.e., the index hasn’t
been pruned on a prior iteration), it satisfies
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})\right\rvert{}\leq\hat{\varepsilon}^{(t)}_{p}(\bm{s})$.
Now suppose that PS-WE returns an empirical utility function $\hat{\bm{u}}$.
This implies that PS-WE managed to prune all utility indices prior to the
exhaustion of its sampling schedule. Combining this with the result from the
previous paragraph, we get that with probability at least $1-\updelta$, for
all $(p,\bm{s})\in\bm{\mathcal{I}}$, if we let $t$ denote the iteration on
which $(p,\bm{s})$ was well-estimated pruned, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}=\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})\right\rvert{}\leq\hat{\varepsilon}^{(t)}_{p}(\bm{s})\leq\varepsilon.$
The correctness guarantee then follows from Lemma 1 ∎
As the same union bound argument is applied in all of the following
correctness proofs, we do not repeat it again. We begin by proving the
correctness of our PS-REG+ algorithm, and show PS-REG and PS-REG-0 to simply
be special cases of this algorithm. Before proving the correctness of PS-REG+,
we prove another accessory lemma.
###### Lemma 5.
Suppose that for all $(p,\bm{s})\in\bm{\mathcal{I}}$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon_{p}(\bm{s})$.
Then for all $(p,\bm{s})\in\bm{\mathcal{I}}$, we have that
$\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}_{p}(\bm{s}^{\prime})-\varepsilon_{p}(\bm{s}^{\prime})\right)\leq\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})\leq\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}_{p}(\bm{s}^{\prime})+\varepsilon_{p}(\bm{s}^{\prime})\right)$
###### Proof.
Let $(p,\bm{s})\in\bm{\mathcal{I}}$. We have that
$\displaystyle\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})=\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\bm{u}_{p}(\bm{s}^{\prime})\geq\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}(\hat{\bm{u}}_{p}(\bm{s}^{\prime})-\varepsilon_{p}(\bm{s}^{\prime})).$
The second inequality holds by analogous reasoning. ∎
See 7
###### Proof.
Suppose that PS-REG+ returns an empirical utility function $\hat{\bm{u}}$,
thus guaranteeing that all indices have been pruned. Suppose that an index
$(p,\bm{s})\in\bm{\mathcal{I}}$ is regret-pruned on iteration $t$. Then, we
have (w.h.p.) that
$\displaystyle\operatorname{Reg}_{p}(\bm{s},\bm{u})$
$\displaystyle=\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})-\bm{u}_{p}(\bm{s})$
$\displaystyle\geq\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})\right)-\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s})+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
Lemma 5 \+ Definition of $\varepsilon^{(t)}_{p}(\bm{s})$
$\displaystyle=\operatorname{Reg}^{\downarrow}_{p}(\bm{s},\hat{\bm{u}}^{(t)})>0$
$\displaystyle\textsf{Definition of $\operatorname{Reg}^{\downarrow}$ +
Pruning Criterion}.$
We also have (w.h.p.) that
$\displaystyle\operatorname{Reg}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}})$
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}_{p}(\bm{s})$
$\displaystyle=\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})$
Since $(p,\bm{s})$ was pruned on iteration $i$
$\displaystyle\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})$
$\hat{\bm{u}}_{p}(\bm{s}^{*}_{\hat{\bm{u}}})\geq\hat{\bm{u}}_{p}(\bm{s}^{*}_{\bm{u}})$
by definition
$\displaystyle\geq(\bm{u}_{p}(\bm{s}^{*}_{\bm{u}})-\varepsilon)-\hat{\bm{u}}^{(t)}_{p}(\bm{s})$
$\displaystyle\begin{aligned} &\textsf{$(p,\bm{s}^{*}_{\bm{u}})$ cannot have
been regret pruned since
$\operatorname{Reg}_{p}(\bm{s}^{*}_{\bm{u}},\Gamma_{\mathscr{S}})=0$.}\\\
&\textsf{Hence, it must have been WE pruned.}\end{aligned}$
$\displaystyle\geq\left(\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\varepsilon}_{p}^{(t)}(\bm{s}^{\prime})\right)-\varepsilon\right)-\hat{\bm{u}}^{(t)}_{p}(\bm{s})$
$\displaystyle=\operatorname{Reg}^{\downarrow}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}}^{(t)})+\hat{\varepsilon}^{(t)}_{p}(\bm{s})-\varepsilon$
By definition
$\displaystyle>(\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}_{p}(\bm{s}))+\hat{\varepsilon}^{(t)}_{p}(\bm{s})-\varepsilon$
Pruning criterion $\displaystyle=\gamma^{*}.$
Since $(p,\bm{s})$ was arbitrary, we have (w.h.p.) that all regret-pruned
indices have positive corresponding regret in the true game and greater than
$\gamma^{*}$ corresponding regret in the empirical game. But since all indices
have been pruned, this implies that, with probability at least $1-\updelta$,
all indices $(p,\bm{s})\in\bm{\mathcal{I}}$ with
$\operatorname{Reg}_{p}(\bm{s},\Gamma_{\mathscr{S}})=0$ or
$\operatorname{Reg}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}})\leq\gamma^{*}$ will
be well-estimated pruned by PS-REG+ prior to termination, and will hence
satisfy
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\varepsilon$.
The correctness result then follows from Lemma 2. ∎
See 6
###### Proof.
Since PS-REG uses uniform utility deviation bounds, we have that for each
index $(p,\bm{s})\in\bm{\mathcal{I}}$ and iteration $t$, it holds that
$\hat{\varepsilon}^{(t)}_{p}(\bm{s})=\hat{\varepsilon}^{(t)}$. If we applied
the regret pruning technique from PS-REG+ to such an algorithm, the pruning
criterion would simplify as follows:
$\displaystyle\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})$
$\displaystyle>\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}_{p}(\bm{s})\\}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})\right)-\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s})+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
$\displaystyle>\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}_{p}(\bm{s})\\}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})\right]-2\hat{\varepsilon}^{(t)}$
$\displaystyle>\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}\\}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\bm{u}}^{(t)}_{p}(\bm{s})$
$\displaystyle>2\hat{\varepsilon}^{(t)}+\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}\\}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})$
$\displaystyle>\max\\{2\hat{\varepsilon}^{(t)},\gamma^{*}+\varepsilon+\hat{\varepsilon}^{(t)}\\}\enspace.$
But this is precisely the pruning criterion of PS-REG. Thus, PS-REG+ is simply
a generalization of PS-REG to cases with non-uniform utility deviation bounds,
and PS-REG must then yield the same guarantees as PS-REG+. ∎
See 5
###### Proof.
Consider the regret pruning criterion in PS-REG when $\gamma^{*}=0$. We know
that the first iteration $t$ on which
$\hat{\varepsilon}^{(t)}\leq\varepsilon$, all indices will be well-estimated
pruned and the algorithm will terminate. Thus, regret pruning will only occur
on iterations on which $\hat{\varepsilon}^{(t)}>\varepsilon$. But then the
regret pruning criterion simplifies to
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>\max\\{2\hat{\varepsilon}^{(t)},\varepsilon+\hat{\varepsilon}\\}=2\hat{\varepsilon}^{(t)}$,
which is precisely the pruning condition of PS-REG-0. Thus, PS-REG is simply a
generalization of PS-REG-0 to $\gamma^{*}>0$, and hence PS-REG-0 must yield
the same guarantees as PS-REG when $\gamma^{*}=0$. ∎
See 8
###### Proof.
Suppose an index $(p,\bm{s})\in\bm{\mathcal{I}}$ is regret-pruned on iteration
$i$. In the proof of Theorem 7, we see that
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}}^{(i)})>0$
implies that $\operatorname{Reg}_{p}(\bm{s},\Gamma_{\mathscr{S}})$. Hence,
this pruning criteria is also guaranteed not to regret prune any index
$(p,\bm{s}^{\prime})\in\bm{\mathcal{I}}$ satisfying
$\operatorname{Reg}_{p}(\bm{s},\Gamma_{\mathscr{S}})=0$. Further following the
proof of Theorem 7, we have that
$\operatorname{Reg}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}})\geq\operatorname{Reg}^{\downarrow}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}}^{(i)})+\hat{\varepsilon}^{(i)}_{p}(\bm{s})-\varepsilon$,
which when combined with our pruning criteria, yields (w.h.p.) that
$\operatorname{Reg}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}})\geq
2\hat{\varepsilon}^{(i)}_{p}(\bm{s})\geq
2\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}.$
Since $(p,\bm{s})$ was arbitrary, we have that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\frac{\operatorname{Reg}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}})}{2}$
for each index $(p,\bm{s})\in\bm{\mathcal{I}}$ that is regret-pruned. Since
the remaining indices must all be well-estimated, we have (w.h.p.) that for
all $(p,\bm{s})\in\bm{\mathcal{I}}$, it holds that
$\left\lvert{}\bm{u}_{p}(\bm{s})-\hat{\bm{u}}_{p}(\bm{s})\right\rvert{}\leq\max\left\\{\varepsilon,\frac{\operatorname{Reg}_{p}(\bm{s},\hat{\Gamma}_{\mathscr{S}})}{2}\right\\}.$
The conclusion then follows from Lemma 3. ∎
### Efficiency Bounds
(Cousins et al. 2022) derive high-probability sample complexity bounds for
their empirical Bennett tail bounds. We state these sample complexity results
below, and use them to derive our efficiency bounds for PS-REG+ and PS-REG-M.
###### Lemma 6.
Consider an index $(p,\bm{s})\in\bm{\mathcal{I}}$. If the sample size
$m_{\bm{s}}\geq
2+2\ln\frac{3}{\updelta}\left(\frac{5c}{2\varepsilon}+\frac{\bm{v}_{p}(\bm{s})}{\varepsilon^{2}}\right)$,
then with probability at least $1-\frac{\updelta}{3}$, it will hold that
$\varepsilon^{\hat{\textup{B}}}_{p}(\bm{s})\leq\varepsilon$.
See 9
###### Proof.
Each of our efficiency bounds is presented as a minimum over two bounds, the
first corresponding to regret pruning and the second to well-estimated
pruning. It is clear that the second is a direct consequence of Lemma 6. We
show the regret pruning bounds, beginning with PS-REG+.
Recall that PS-REG+ prunes an index $(p,\bm{s})\in\bm{\mathcal{I}}$ on an
iteration $t$ if
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}_{p}(\bm{s})\\}$.
We have (w.h.p.) that
$\displaystyle\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})$
$\displaystyle=\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})\right)-\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s})+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
$\displaystyle>\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})\right]-\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}_{p}^{(t)}(\bm{s}^{\prime})\right]-\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s})+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
$\displaystyle>\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\bm{u}_{p}(\bm{s})-\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)\right]-\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}_{p}^{(t)}(\bm{s}^{\prime})\right]-\left(\bm{u}_{p}(\bm{s})+2\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
$\displaystyle>\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\bm{u}_{p}(\bm{s}^{\prime})\right]-2\left[\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}_{p}^{(t)}(\bm{s}^{\prime})\right]-\left(\bm{u}_{p}(\bm{s})+2\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
$\displaystyle=\operatorname{Reg}_{p}(\bm{s};\bm{u})-2\hat{\varepsilon}^{(t)}_{p}(\bm{s})-2\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})\enspace.$
Hence, a strictly tighter pruning criterion for PS-REG+ would be pruning an
index $(p,\bm{s})\in\bm{\mathcal{I}}$ when
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\bm{u})-\gamma^{*}$
$\displaystyle>2\hat{\varepsilon}^{(t)}_{p}(\bm{s})+2\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})-\gamma^{*}+\max\\{0,\gamma^{*}+\varepsilon-\hat{\varepsilon}^{(t)}_{p}(\bm{s})\\}$
$\displaystyle=2\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})+\max\left\\{2\hat{\varepsilon}^{(t)}_{p}(\bm{s})-\gamma^{*},\varepsilon+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right\\}\enspace.$
We can make the pruning criterion even tighter by increasing the right-hand
side:
$\displaystyle
2\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})+\max\left\\{2\hat{\varepsilon}^{(t)}_{p}(\bm{s})-\gamma^{*},\varepsilon+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right\\}$
$\displaystyle\leq
4\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime}).$
Hence, the latest (w.h.p.) an index $(p,\bm{s})\in\bm{\mathcal{I}}$ will be
regret-pruned by PS-REG+ is when
$\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})<\frac{\operatorname{Reg}_{p}(\bm{s};\bm{u})-\gamma^{*}}{4}.$
Our result then follows via Lemma 6. By analogous reasoning, we have that a
strictly tighter regret pruning criterion than the one in PS-REG-M would be
pruning an index $(p,\bm{s})\in\bm{\mathcal{I}}$ when
$\displaystyle\operatorname{Reg}_{p}(\bm{s};\bm{u})-\varepsilon>3\hat{\varepsilon}^{(t)}_{p}(\bm{s})+2\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime}).$
Similar to before, we can make the pruning criterion even tighter by
increasing the right hand side:
$3\hat{\varepsilon}^{(t)}_{p}(\bm{s})+2\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})\leq
5\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})$
Hence, the latest (w.h.p.) an index $(p,\bm{s})\in\bm{\mathcal{I}}$ will be
regret pruned by PS-REG-M is when
$\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})<\frac{\operatorname{Reg}_{p}(\bm{s};\bm{u})-\varepsilon}{5}\enspace.$
Once again, our result follows from Lemma 6. ∎
See 10
###### Proof.
Recall that all the aforementioned progressive sampling algorithms use well-
estimated pruning. By Hoeffding’s Inequality (Theorem 1), we have that on
iteration $T$ of algorithm PS, for all $(p,\bm{s})\in\bm{\mathcal{I}}$, it
holds that
$\displaystyle\hat{\varepsilon}^{(T)}_{p}(\bm{s})\leq
c\sqrt{\frac{\ln\left(\frac{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}\right)}{M_{T}}}\leq
c\sqrt{\ln\left(\frac{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}\right)\cdot\frac{2\varepsilon^{2}}{c^{2}\ln\left(\frac{2\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}\right)}}=\varepsilon,$
and thus all indices which remain active until iteration $T$ will be pruned on
iteration $T$. Hence, the aforementioned algorithms are guaranteed to prune
all indices prior to the exhaustion of the sampling schedule, and thus will
return an empirical game satisfying the respective guarantees of the
algorithm. ∎
### Sampling Schedule
Our sampling schedule is derived via a sample complexity lower bound for the
empirical Bennett tail bounds presented in Theorem 3. (Cousins et al. 2022)
lower bound the empirical Bennett bounds via the zero-variance case of
Bennett’s inequality (Theorem 2). We, however, use a tighter lower bound which
we derive below.
###### Lemma 7.
Consider an index $(p,\bm{s})\in\bm{\mathcal{I}}$. If
$\varepsilon^{\hat{\textup{B}}}_{p}(\bm{s})\leq\varepsilon$, then it must hold
that the sample size
$m_{\bm{s}}>\left(\frac{1}{3}+\sqrt{\frac{4+2\sqrt{3}}{3}}\right)\cdot\frac{c\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{\varepsilon}$.
###### Proof.
We have that
$\displaystyle\varepsilon^{\hat{\bm{v}}}_{p}(\bm{s})$
$\displaystyle=\frac{2c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{3m}+\sqrt{\left(\frac{1}{3}+\frac{1}{2\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}\right)\left(\frac{c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m-1}\right)^{2}+\frac{2c^{2}\hat{\bm{v}}_{p}(\bm{s})\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}}$
$\displaystyle>\frac{2c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{3m}+\sqrt{\frac{1}{3}\left(\frac{c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m-1}\right)^{2}}$
$\displaystyle>\frac{2+\sqrt{3}}{3}\cdot\frac{c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}\enspace.$
We further have that
$\displaystyle\varepsilon^{\hat{\textup{B}}}_{p}(\bm{s})$
$\displaystyle=\frac{c\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{3m}+\sqrt{\frac{2\left(\hat{\bm{v}}_{p}(\bm{s})+\varepsilon^{\hat{\bm{v}}}_{p}(\bm{s})\right)\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}}$
$\displaystyle>\frac{c\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{3m}+\sqrt{\frac{2\cdot\frac{2+\sqrt{3}}{3}\cdot\frac{c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}\cdot\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}}$
$\displaystyle=\left(\frac{1}{3}+\sqrt{\frac{4+2\sqrt{3}}{3}}\right)\cdot\frac{c\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}}{\updelta}\right)}{m}$
The conclusion follows directly. ∎
Thus, our sampling schedule for PS-WE begins at
$\alpha=\left(\frac{1}{3}+\sqrt{\frac{4+2\sqrt{3}}{3}}\right)\cdot\frac{c\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}\right)}{\varepsilon}$
and ends at a cumulative sample size that is at least
$\omega\doteq\frac{c^{2}\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}\right)}{2\varepsilon^{2}}$
to satisfy Theorem 10. Following (Cousins et al. 2022), we then use a schedule
with a geometrically increasing cumulative sample size with a geometric factor
$\beta$ (for our experiments, we use $\beta=1.1$). Our schedule length is then
$T\doteq\lceil\log_{\beta}\left(\frac{\omega}{\alpha}\right)\rceil$. The first
sample size is defined by $m_{1}\doteq\alpha\beta$, and each following sample
size is defined by $m_{t}\doteq\alpha\beta^{t}-m_{t-1}$.
Of course, for all of our regret pruning algorithms, it may be possible for
regret pruning to occur prior to at least one index
$(p,\bm{s})\in\bm{\mathcal{I}}$ achieving
$\hat{\varepsilon}^{(t)}_{p}(\bm{s})\leq\varepsilon$. Regret pruning cannot,
however, happen prior to at least one index $(p,\bm{s})\in\bm{\mathcal{I}}$
achieving $\hat{\varepsilon}^{(t)}_{p}(\bm{s})\leq\frac{c}{2}$. This can be
seen by looking at the loosest regret pruning criterion we discuss, that used
in PS-REG+ with $\gamma^{*}=0$. An index $(p,\bm{s})\in\bm{\mathcal{I}}$ is
regret-pruned on iteration $t$ if
$\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>0$. Notice
that if $\hat{\varepsilon}^{(t)}_{p}(\bm{s})>\frac{c}{2}$ for all
$(p,\bm{s})\in\bm{\mathcal{I}}$, then for any given index
$(p,\bm{s})\in\bm{\mathcal{I}}$, we have that
$\displaystyle\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})$
$\displaystyle=\sup_{\bm{s}^{\prime}\in\operatorname{Adj}_{p,\bm{s}}}\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s}^{\prime})-\hat{\varepsilon}^{(t)}_{p}(\bm{s}^{\prime})\right)-\left(\hat{\bm{u}}^{(t)}_{p}(\bm{s})+\hat{\varepsilon}^{(t)}_{p}(\bm{s})\right)$
$\displaystyle<\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})-c,$
and hence $(p,\bm{s})$ will be regret-pruned only if it holds that
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})-c>\operatorname{Reg}^{\downarrow}_{p}(\bm{s};\hat{\bm{u}}^{(t)})>0$.
But the latter is impossible since
$\operatorname{Reg}_{p}(\bm{s};\hat{\bm{u}}^{(t)})\leq c$ by definition.
Hence, no index can be regret-pruned (by any of our regret pruning criteria)
prior to at least one index being achieving an estimation guarantee of at
least $\frac{c}{2}$.
Based on the above, we start our sampling schedule for all our regret pruning
algorithms on
$\alpha^{\prime}\doteq\left(\frac{1}{3}+\sqrt{\frac{4+2\sqrt{3}}{3}}\right)\cdot
2\ln\left(\frac{3\left\lvert{}\bm{\mathcal{I}}\right\rvert{}T}{\updelta}\right)$.
But, as argued in the text, using a schedule with a strictly geometrically
increasing schedule, we waste too many iterations on small sample sizes and
yield a schedule length that is too large. Hence, we instead fix the schedule
length to be $1.5$ times the schedule length used for PS-WE. We then occupy
the final two-thirds of our schedule with the same sample sizes used in the
sampling schedule for PS-WE, and occupy the first third of our schedule with
one that has linearly increasing cumulative sample size beginning at
$\alpha^{\prime}$ and ending at $\alpha$ (from above).
|
# Double layers in the Earth’s bow shock
Jiepu Sun
Space Sciences Laboratory
University of California at Berkeley
CA 94720, USA
<EMAIL_ADDRESS>
&Ivan Y. Vasko
Space Sciences Laboratory
University of California at Berkeley
CA 94720, USA
<EMAIL_ADDRESS>
&Stuart D. Bale
Space Sciences Laboratory
University of California at Berkeley
CA 94720, USA
&Rachel Wang
Space Sciences Laboratory
University of California at Berkeley
CA 94720, USA
&Forrest S. Mozer
Space Sciences Laboratory
University of California at Berkeley
CA 94720, USA
###### Abstract
We present Magnetospheric Multiscale observations of electrostatic double
layers in quasi-perpendicular Earth’s bow shock. These double layers have
predominantly parallel electric field with amplitudes up to 100 mV/m, spatial
widths of 50–700 m, and plasma frame speeds within 100 km/s. The potential
drop across a single double layer is 2–7% of the cross-shock potential in the
de Hoffmann-Teller frame and occurs over the spatial scale of ten Debye
lengths or one tenth of electron inertial length. Some double layers can have
spatial width of 70 Debye lengths and potential drop up to 30% of the cross-
shock potential. The electron temperature variation observed across double
layers is roughly consistent with their potential drop. While electron heating
in the Earth’s bow shock occurs predominantly due to the quasi-static electric
field in the de Hoffmann-Teller frame, these observations show that electron
temperature can also increase across Debye-scale electrostatic structures.
Key points:
1. 1.
The first identification of ion-acoustic double layers and electron
temperature variation across them in the Earth’s bow shock.
2. 2.
Double layers have typical spatial width around ten Debye lengths or one tenth
of electron inertial length.
3. 3.
Typical potential drop across a double layer is 2–7% of the cross-shock
potential in the de Hoffmann-Teller frame.
## 1 Introduction
Electron heating remains one of the not entirely resolved problems in the
physics of collisionless shock waves [1, 2, 3]. Previous spacecraft
measurements showed that the electron heating across quasi-perpendicular
Earth’s bow shock is mainly determined by the quasi-static cross-shock
electrostatic field in the de Hoffmann-Teller frame [1, 4, 5]. This
electrostatic field is rarely possible to be measured directly, but based on
the observed electron heating we expect it to have typical amplitude of a few
mV/m and spatial scale intermediate between electron and ion inertial lengths
[6, 7, 2, 8]. The major effect of high-frequency fluctuations produced by
plasma instabilities in the Earth’s bow shock is assumed to be a smoothing of
the electron distribution function shaped by quasi-static magnetic and
electric fields [1, 4, 5]. There are currently experimental [9, 8, 10] and
theoretical [3, 11, 12] indications that high-frequency fluctuations not only
smooth the electron distribution function, but can also contribute to the
electron heating. In this study we present experimental evidence that a
fraction of the electron heating in the Earth’s bow shock can occur across
Debye-scale electrostatic double layers. Note that the term heating used in
this letter is equivalent of temperature increase, because we observe spatial
profiles of the electron temperature, rather than its temporal evolution.
It is noteworthy that double layers in the form of more or less unipolar
spikes in the parallel (magnetic field-aligned) electric field are universally
observed during the energy release processes in space plasma [13]. Double
layers were originally reported in the auroral region [14, 15], while later
studies showed they had plasma frame speeds around local ion-acoustic speed
and spatial widths of about ten Debye lengths [16, 17]. Similar ion-acoustic
double layers were observed around plasma injection fronts in the inner
magnetosphere [18, 19], fast plasma flows in the plasma sheet [20, 21], and
reconnection current sheets in the magnetosphere [22, 23]. Parker Solar Probe
measurements have recently revealed double layers, presumably of ion-acoustic
type, in the Venusian bow shock [24]. The spacecraft measurements showed that
ion-acoustic double layers can indeed accelerate or heat electrons [20, 21].
The presence of electrostatic double layers in the Earth’s bow shock was
pointed out in the past [25, 26], but no in-depth analysis of their properties
and the associated electron heating was carried out.
In this letter we present Magnetospheric Multiscale measurements of
electrostatic double layers in the Earth’s bow shock and demonstrate that they
are of ion-acoustic type. We show that electron temperature variation across
the double layers is roughly consistent with the potential drop across them.
The origin of these double layers is discussed.
## 2 Observations
We first consider a quasi-perpendicular Earth’s bow shock crossing by
Magnetospheric Multiscale (MMS) spacecraft around 07:56:30 UT on 4th November
2015 that was previously used to study electrostatic solitary and ion-acoustic
waves [27, 10]. We use burst mode measurements of the DC-coupled magnetic
field at 128 S/s (samples per second) resolution provided by Digital and
Analogue Fluxgate Magnetometers [28], electron and ion moments and velocity
distribution functions provided at 0.03 and 0.15 s cadence respectively by the
Fast Plasma Investigation instrument [29], electric field fluctuations along
with voltage signals of individual voltage-sensitive probes at 8,192 S/s
resolution provided by Axial Double Probe [30] and Spin-Plane Double Probe
[31], and magnetic field fluctuations at 8,192 S/s resolution provided by the
Search Coil Magnetometer [32]. The electric field was measured by two pairs of
probes in the spin plane, with opposing probes mounted on tips of 60 m
antennas, and a pair of axial probes mounted on tips of 14.6 m antennas along
the spin axis. The electric field components in the orthogonal coordinate
system associated with the probes, whose schematic can be found in Supporting
Materials (SM), were computed as $E_{12}=(V_{2}-V_{1})/120\;{\rm m}$,
$E_{34}=(V_{4}-V_{3})/120\;{\rm m}$, $E_{56}=(V_{6}-V_{5})/29.2\;{\rm m}$, and
then multiplied by frequency response factors to be specified below. The
electric and magnetic fields will also be presented in the Geocentric Solar
Ecliptic (GSE) coordinate system whose $z-$axis essentially coincides with the
spacecraft spin axis.
Figure 1 presents MMS4 measurements in the aforementioned Earth’s bow shock
crossing. Panel (a) presents the magnetic field magnitude and highlights a
region around the ramp, where double layers were observed. The expanded view
of that region is shown in following panels. Panel (b) demonstrates that the
magnetic field is predominantly along the $y-$axis and, thus, almost within
the spacecraft spin plane. Panel (c) shows that the plasma flows at about 100
km/s parallel to the local magnetic field and 200 km/s perpendicular to it.
Three electric field components in panels (d)–(f) and their omnidirectional
wavelet spectrum in panel (g) show a sporadic occurrence of electric field
fluctuations at frequencies above a few hundred Hz. These are electrostatic
ion-acoustic waves considered in the previous study [10]. The intriguing,
though not universal, feature is that each burst of ion-acoustic waves is
preceded by a solitary structure, whose typical frequency is below 200 Hz.
These solitary structures are electrostatic, since no magnetic counterparts
were observed, with the electric field predominantly along the $y-$axis which
is parallel to the local magnetic field. We only present in-depth analysis of
the solitary structure with the largest electric field amplitude, because
similar analysis of other structures revealed they were of identical nature.
Figure 1: MMS4 observation of double layers in a quasi-perpendicular Earth’s
bow shock crossing on 4th November, 2015. Panel (a) presents the magnetic
field magnitude observed across the shock along with a highlighted region,
whose expanded view is shown in the following panels: (b), (c) magnetic field
and plasma flow velocity in the GSE coordinate system; (d)–(f) three electric
field components measured at 8,192 S/s in the GSE coordinate system and (g)
their omnidirectional continuous Morse wavelet spectrum in arbitrary units.
There are sporadic bursts of high-frequency electrostatic fluctuations (above
about 200 Hz) preceded by lower-frequency electrostatic structures (below 200
Hz). These electrostatic structures are highlighted in panels (d)–(f) with the
largest-amplitude one highlighted in red.
Figure 2 presents an interferometry analysis of the largest-amplitude solitary
structure highlighted in Figure 1. The electric field in the magnetic field-
aligned coordinate in panel (a) demonstrates that the solitary structure has a
full temporal width of 20 ms and predominantly parallel electric field with
amplitude of 100 mV/m. Note electrostatic fluctuations with the temporal scale
of about 1 ms, whose electric field is rather oblique to the electric field of
the solitary structure. These high-frequency fluctuations can be excluded by a
low-pass filter with the cutoff frequency of 200 Hz. Panel (b) presents
electric field components $E_{12}$, $E_{34}$, and $E_{56}$ computed using low-
pass filtered voltage signals and frequency response factors of 1.35 and 1.2
for spin plane and axial antennas respectively. The axial antenna’s factor may
actually vary between 0.8 and 1.6 [27], but our results are not sensitive to
its specific value, because the electric field of the solitary structure is
essentially in the spin plane; moreover, it is directed from probe 2 to probe
1. Applying the Maximum Variance Analysis [33] to the electric field in panel
(b), we determined a unit vector $\hat{\bf k}$ along the electric field
polarization direction; in the coordinate system associated with the probes we
have $\hat{\bf k}\approx(-0.91,0.12,0.39)$ that is within 20∘ of the local
magnetic field. Low-pass filtered voltage signals are presented in panels
(c)–(e) along with correlation coefficients and time delays between signals of
opposing probes. The highest correlation and the largest time delay were
observed between $V_{1}$ and $-V_{2}$. The speed of a locally planar
electrostatic structure can be estimated as follows [34, 27]:
$V_{s}=\hat{k}_{ij}l_{ij}/\Delta t_{ij}$, where $\Delta t_{ij}$ is the time
delay between signals $V_{i}$ and $-V_{j}$ of a pair of opposing probes, while
$l_{ij}$ is the corresponding antenna length. Taking into account that
$\hat{k}_{12}\approx-0.91$, $\Delta t_{12}\approx-2.93$ ms, and $l_{12}=60$ m,
we obtained the speed of the solitary structure in the spacecraft frame,
$V_{s}\approx 19$ km/s. Note that the electric field polarization direction
$\hat{\bf k}$, intrinsically ambiguous by 180∘, was chosen to be consistent
with propagation from probe 1 to probe 2 as observed in panel (c). Voltage
signals of other pairs of opposing probes are not perfectly correlated and
corresponding time delays are less reliable. The estimated speed allowed
translating temporal profiles into spatial ones and computing the
electrostatic potential of the solitary structure, $\Phi=\int E_{k}V_{s}dt$,
where $E_{k}$ is the electric field along the polarization direction. Panel
(f) shows that the solitary structure has a net potential drop across itself
of $\Delta\Phi\approx 25$ V; thus, it is actually a double layer.
The spatial width of the double layer defined as $w=\Delta\Phi/{\rm
max}(E_{k})$ is about 240 m, which is 15$\lambda_{D}$ or 0.2$\lambda_{e}$ in
units of local Debye length or electron inertial length. With the spatial
width defined that way, the electrostatic potential of the double layer can be
approximated as $\Phi(X)\approx 0.5\Delta\Phi\left(1+{\rm tanh}(2X/w)\right)$,
where $X$ is the spatial coordinate along the electric field polarization
direction. Note that the spatial extent of the double layer in the plane
perpendicular to the polarization direction is less than 10 km, since the
double layer was not observed aboard the other MMS spacecraft spatially
separated from MMS4 by at least 10 km in that plane. We estimated the speed of
the double layer in the plasma frame as $V_{s}^{*}=|V_{s}-\hat{\bf k}\cdot{\bf
V}_{i}|$, where ${\bf V}_{i}$ is the plasma flow velocity. We found
$V_{s}^{*}\approx 90$ km/s or about 0.9$c_{\rm IA}$, where $c_{\rm
IA}=(T_{e\parallel}/m_{p})^{1/2}\approx 100$ km/s is a rough estimation of the
local ion-acoustic speed ($m_{p}$ is the proton mass). Thus, this double layer
is of ion-acoustic type. The potential drop of 25 V is about 24% of the local
electron parallel temperature and about 6% of the entire cross-shock potential
$\Delta\Phi_{\rm HT}$ corresponding to the electric field in the de Hoffmann-
Teller frame; the cross-shock potential of $\Delta\Phi_{\rm HT}\approx 400$ V
was estimated using the electron momentum balance (SM).
Figure 2: The interferometry analysis of the double layer highlighted in red
in Figure 1: (a) the electric field measured at 8,192 S/s in the magnetic
field-aligned coordinate system (components parallel and perpendicular to the
local magnetic field are shown); (b) the electric field in the coordinate
system associated with voltage-sensitive probes and computed using voltage
signals $V_{1}$–$V_{6}$ of probes in panels (c)–(e):
$E_{12}=1.35\cdot(V_{2}-V_{1})/120\;{\rm m}$,
$E_{34}=1.35\cdot(V_{4}-V_{3})/120\;{\rm m}$, and
$E_{56}=1.2\cdot(V_{6}-V_{5})/29.2\;{\rm m}$; (c)–(e) low-pass filtered
(cutoff frequency of 200 Hz) and offset eliminated voltage signals of four
probes in the spacecraft spin plane ($V_{1}$–$V_{4}$) and of two probes
mounted on the axial antenna along the spin axis ($V_{5}$ and $V_{6}$); cross-
correlation coefficients and time delays between voltage signals of opposing
probes are indicated in panels; (f) the electric field $E_{k}$ along the
polarization direction $\hat{\bf k}$ computed by applying the Maximum Variance
Analysis to the electric field in panel (b) and the electrostatic potential
computed as $\Phi=\int E_{k}V_{s}dt$, where $V_{s}\approx 19$ km/s is the
double layer speed in the spacecraft frame estimated using the time delay
between $V_{1}$ and $-V_{2}$. The estimated speed $V_{s}$ allowed translating
the temporal axis into the spatial axis shown at the bottom. Figure 3: The
analysis of the electron temperature variation across the double layer shown
in Figure 2: (a) the electrostatic potential $\Phi=\int E_{k}V_{s}dt$ obtained
by integrating the low-pass filtered (with cutoff frequency of 200 Hz)
electric field $E_{k}$ over an interval about twice as long as the interval
shown in Figure 2; (b) and (c) the electron density and the parallel
temperature measured at 30 ms cadence; (d) and (e) the phase space density
(PSD) of electrons collected upstream and downstream of the double layer over
30 ms intervals highlighted in panels (a)–(c). PSDs correspond to electrons
with pitch angles within $(0^{\circ},30^{\circ})$ and
$(150^{\circ},180^{\circ})$ in panel (d) and within $(60^{\circ},120^{\circ})$
in panel (e). The dashed line in panel (d) corresponds to the expected PSD
downstream of the double layer that was computed using Liouville mapping of
the upstream PSD (the energy of upstream electrons was increased by 25 eV,
while their phase space density was conserved). The fluxes of electrons below
about 30 eV were contaminated by secondary and photoelectrons.
Figure 3 presents an analysis of the electron temperature variation and
momentum balance across the double layer. Panel (a) shows the electrostatic
potential obtained by integrating the low-pass filtered electric field over
about 120 ms time interval, which is twice longer than that in Figure 2.
Panels (b) and (c) present electron density $n_{e}$ and parallel temperature
$T_{e||}$ over the same interval. Because the double layer can only be
considered isolated over this interval (Figure 1e), there is merely one point
of plasma measurements available upstream and downstream of it. The 30 ms time
intervals where upstream and downstream electron distribution functions were
collected are highlighted in panels (a)–(c). The projection of the Ohm’s law
onto the local magnetic field indicates that the electrostatic field of the
double layer must be balanced by the electron pressure gradient (e.g., [35])
$\displaystyle
e\nabla_{{\color[rgb]{0,0,0}||}}\Phi\approx\nabla_{{\color[rgb]{0,0,0}||}}(n_{e}T_{e\parallel})/n_{e},$
(1)
where we omitted negligible inertia terms and neglected anomalous resistivity;
the latter is a reasonable assumption in the Earth’s bow shock [36, 37, 5].
The electron density increases across the double layer from 18 to 25 cm-3,
while parallel electron temperature increases from 87 to 107 eV. Therefore,
the expected potential drop is
$\Delta\Phi\approx\Delta(n_{e}T_{e\parallel})/e\langle n_{e}\rangle\approx 50$
V, where $\langle n_{e}\rangle\approx 21.5$ cm-3 is the averaged electron
density. The Ohm’s law prediction is consistent with the observed potential
drop of 25 V within a factor of two. The analysis of electron distribution
functions upstream and downstream of the double layer reveals the reason for
the quantitative disagreement.
Panel (d) presents the distribution function of upstream and downstream
electrons streaming quasi-parallel to the local magnetic field line. In
accordance with the presence of a net potential drop, the downstream
distribution function is wider than the upstream one. It is wider however than
the expected distribution computed via Liouville mapping of the upstream
distribution. The expected distribution in panel (d) was obtained by
increasing the energy of upstream electrons by 25 eV, while keeping their
phase space density conserved. Note that the distribution of downstream
electrons could not be computed below about 50 eV, because upstream electron
fluxes below about 30 eV were contaminated by secondary and photoelectrons
(e.g., Gershman et al. [38]). In addition, the distribution of downstream
electrons below 25 eV would not be possible to compute using the Liouville
mapping, because these electrons are trapped downstream of the double layer;
see a schematics of the electron phase space in the SM. We do not expect any
widening of the distribution function of electrons streaming quasi-
perpendicular to the local magnetic field, but observations in panel (e)
demonstrate the opposite. This is a strong indication of temporal variations
that occurred within 90 ms between observations upstream and downstream of the
double layer. The temporal variations are not necessarily local, because each
distribution function was collected over 30 ms, which translates into spatial
resolution of a few hundred kilometers ($>50$ eV electrons cover at least 100
km over 30 ms). We believe that the quantitative inconsistency within a factor
of two between the Ohm’s law prediction and the observed potential drop is
caused by temporal variations, which resulted in parallel temperature increase
and widening of the distribution function larger than expected for the local
potential drop of 25 V.
Table 1: The table presents various properties of double layers shown in Figures 2 and 4: spacecraft frame speed $V_{s}$ and plasma frame speed $V_{s}^{*}$ along with its value in units of local ion-acoustic speed estimation, $c_{\rm IA}=(T_{e\parallel}/m_{p})^{1/2}$; spatial width $w$ in physical units as well as in units of local Debye length $\lambda_{D}$ and electron inertial length $\lambda_{e}$; potential drop $\Delta\Phi$ in physical units as well as in units of local parallel electron temperature $T_{e\parallel}$ and in units of cross-shock potential $\Delta\Phi_{\rm HT}$ in the de Hoffmann-Teller frame. | $V_{s}$ & $V_{s}^{*}$ [km/s] | $w$ [m] | $\Delta\Phi$ [V] | $w/\lambda_{D}$ | $w/\lambda_{e}$ | $e\Delta\Phi/T_{e\parallel}$ | $V_{s}^{*}/c_{\rm IA}$ | $\Delta\Phi/\Delta\Phi_{\rm HT}$
---|---|---|---|---|---|---|---|---
Figure 2 | 19 & 90 | 240 | 25 | 15 | 0.23 | 24% | 0.9 | 6%
Figure 4a | 52 & 104 | 90 | 7.5 | 8 | 0.1 | 8% | 1.1 | 2%
Figure 4b | 51 & 62 | 670 | 32 | 67 | 0.9 | 33% | 0.6 | 32%
Figure 4c | 27 & 50 | 55 | 3 | 9 | 0.07 | 9% | 0.9 | 5%
Figure 4d | 12 & 95 | 65 | 2.2 | 13 | 0.1 | 7% | 1.8 | 7%
Figure 4 demonstrates double layers observed in several other Earth’s bow
shock crossings considered previously by Wang et al. [27]. The upper panel in
each panel set (a)–(d) presents the magnetic field magnitude along with a
highlighted region where the double layer was observed, while the middle panel
presents the parallel electric field over that interval. These panels
demonstrate that double layers can occur within or close to the shock ramp as
well as in the downstream region. The electric fields of all double layers are
parallel to the local magnetic field within 20∘ (SM). We carried out
interferometry analysis to estimate speeds and electrostatic potentials of the
double layers (SM). Bottom panels in (a)–(d) demonstrate electrostatic
potentials, while various parameters of the double layers are presented in
Table 1. These double layers have plasma and spacecraft frame speeds within
100 km/s, spatial widths within 50–700 m, and potential drops up to 30 V. They
are of ion-acoustic type, because their plasma frame speeds are comparable
with local ion-acoustic speed, $V_{s}^{*}/c_{\rm IA}\approx 0.5$–$2$. The
typical spatial width of the double layers is 8–15$\lambda_{D}$ or
0.07–0.23$\lambda_{e}$, though some double layers can have spatial width up to
70$\lambda_{D}$ or 0.9$\lambda_{e}$. The potential drop across a single double
layer is typically 2–7%, though can be up to 30%, of the cross-shock potential
in the de Hoffmann-Teller frame. In units of local parallel electron
temperature, these potential drops vary between 7% and 30%. For all the double
layers the prediction of the Ohm’s law (1) is consistent within 30% with the
observed potential drop; the only exception is the double layer in panel (d),
whose potential drop is relatively small and the Ohm’s law estimation is most
likely dominated by temporal variations (SM).
Figure 4: Double layers observed in other quasi-perpendicular Earth’s bow
shock crossings. In each set of panels (a)–(d), the upper panel presents the
magnetic field magnitude observed across a shock, the middle panel shows the
parallel electric field over the interval highlighted in the upper panel, and
the bottom panel presents the electrostatic potential computed using the
double layer speed estimated using interferometry (SM). Electrostatic
potentials are shown over time intervals highlighted in the middle panels; the
spatial axis corresponding to the temporal axis is at the top of the bottom
panels. Table 1 summarizes various parameters of the double layers.
.
## 3 Discussion and Conclusion
Previous spacecraft measurements showed that electron heating in the Earth’s
bow shock was determined by the quasi-static electric field in the de
Hoffmann-Teller frame [1, 4, 5], which arises from different electron and ion
dynamics [39]. The typical spatial scale of the quasi-static electric field is
between electron and ion inertial length [6, 7, 2, 8]. While we believe that
electron heating in the Earth’s bow shock is indeed dominated by the quasi-
static electric field, in this letter we demonstrated that the electron
temperature can also increase across electrostatic double layers. The electron
temperature variation across a single double layer can be 2–7% of the cross-
shock potential in the de Hoffmann-Teller frame and occurs over spatial scale
of only about ten Debye lengths or one tenth of electron inertial length. Some
double layers can have spatial width of 70 Debye lengths and potential drops
up to 30% of the cross-shock potential. Importantly, double layers can have
their high-potential side facing magnetosheath (Figures 2 and 4b) as well as
solar wind (Figures 4a,c,d). In the former the electron temperature across
double layers increases toward magnetosheath, while in the latter it increases
toward solar wind. Note that the quasi-static electric field always results in
electron temperature increase toward the magnetosheath.
Double layers presented in this letter are electrostatic and they
fundamentally differ from electric field spikes reported in the foot of a
quasi-perpendicular Earth’s bow shock [9]; those spikes were nonlinear
whistler mode structures [40, 41]. Similar electrostatic double layers were
previously reported in the auroral region [16, 17], plasma sheet [20, 21],
inner magnetosphere [18, 19], reconnection current sheet [22, 23], and
Venusian bow shock [24]. In rare cases the double layer propagation velocity
was revealed by interferometry [16, 17, 21], while typically it was assumed to
coincide with the local ion-acoustic speed. Our interferometry analysis showed
that double layers in the Earth’s bow shock have plasma frame speed in the
range of 0.5–2$c_{\rm IA}$, where we used a rough estimation of the local ion-
acoustic speed, $c_{\rm IA}=(T_{e\parallel}/m_{p})^{1/2}$, because its exact
value depends on non-Maxwellian features of electron and ion distribution
functions (e.g., Vasko et al. [42]). These double layers are of ion-acoustic
type and, thus, identical with double layers in the auroral region [16, 17]
and plasma sheet [21]. Similarly to previous observations, the double layers
are associated with high-frequency electric field fluctuations (Figures 1 and
4), though that is not a universal feature (Figure 4a). The origin of these
high-frequency fluctuations is not apparent, because the double layers are
rather weak to produce electron beams at the high-potential side (Figure 3).
The causal relation between the double layers and the temperature variation
across them is not obvious. Double layers formed by a current-driven ion-
acoustic instability can indeed result in electron heating [43, 44, 45], but
double layers formed at a contact boundary between cold and hot electron
populations result in no actual electron heating [46, 47]. In the case of a
current-driven ion-acoustic instability in a uniform plasma [43], a double
layer is associated with a negative potential spike (ion hole). Although ion
holes are abundant in the Earth’s bow shock [48, 49, 27], the observed double
layers are not associated with them (Figures 2 and 4) and, hence,
fundamentally different from ones observed in simulations by Sato and Okuda
[43]. The double layers are similar to those produced by a current-driven
instability developing around a localized plasma density perturbation [44],
but they are also not different from double layers formed at a contact
boundary between cold and hot electron populations, the latter produced by
some other heating mechanism [46, 47]. The origin of the double layers need to
be established in the future to reveal their ability in actual electron
heating.
In conclusion, we showed that the electron temperature in the Earth’s bow
shock can increase across Debye-scale electrostatic structures. The estimation
of net contribution of these structures to the electron heating remains a
challenge and the mechanism producing these double layers remains to be
revealed.
## 4 Open Research
All the additional information on double layers considered in this letter can
be found in the Supporting Materials. We thank the MMS team for excellent data
available at https://lasp.colorado.edu/mms/sdc/public/about/browse-wrapper/.
The work of J.S. and R.W. was supported by National Science Foundation grants
No. 2026680. The work of I.V. and F.M. was supported by NASA Heliophysics
Guest Investigator grant No. 80NSSC21K0730. I.V. also thanks the International
Space Science Institute, Bern, Switzerland for supporting the working group
"Resolving the Microphysics of Collisionless Shock Waves". I.V. thanks Anton
Artemyev and Sergey Kamaletdinov for valuable contributions.
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|
# High order asymptotic preserving well-balanced finite difference WENO
schemes for all Mach full Euler equations with gravity
###### Abstract.
In this paper, we propose a high order semi-implicit well-balanced finite
difference scheme for all Mach Euler equations with a gravitational source
term. To obtain the asymptotic preserving property, we start from the
conservative form of full compressible Euler equations and add the evolution
equation of the perturbation of potential temperature. The resulting system is
then split into a (non-stiff) nonlinear low dynamic material wave to be
treated explicitly, and (stiff) fast acoustic and gravity waves to be treated
implicitly. With the aid of explicit time evolution for the perturbation of
potential temperature, we design a novel well-balanced finite difference WENO
scheme for the conservative variables, which can be proven to be both
asymptotic preserving and asymptotically accurate in the incompressible limit.
Extensive numerical experiments were provided to validate these properties.
###### Key words and phrases:
compressible Euler equations; all Mach numbers; gravity; finite difference
WENO; high order method; asymptotic preserving; well-balanced.
Guanlan Huang
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, PR
China
<EMAIL_ADDRESS>
Yulong Xing
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
<EMAIL_ADDRESS>
Tao Xiong***Corresponding author.
School of Mathematical Sciences, Fujian Provincial Key Laboratory of
Mathematical Modeling
and High-Performance Scientific Computing, Xiamen University
Xiamen, Fujian 361005, PR China
<EMAIL_ADDRESS>
## 1\. Introduction
In this paper, we are interested in the compressible Euler equations with a
gravitational source
(1.1) $\left\\{\begin{array}[]{ll}\rho_{t}+\nabla\cdot(\rho{\bf
u})=0,\\\\[2.84526pt] (\rho{\bf u})_{t}+\nabla\cdot(\rho{\bf u}\otimes{\bf
u})+\nabla p=-\rho\nabla\Phi,\\\\[2.84526pt] E_{t}+\nabla\cdot\left((E+p){\bf
u}\right)=-\rho{\bf u}\cdot\nabla\Phi,\end{array}\right.$
where $\rho$ is the density, ${\bf u}$ is the velocity, $p$ is the pressure,
and $\rho{\bf u}$ is the momentum. $\Phi=\Phi({\bf x})$ is the gravitational
potential which is assumed to be time-independent. $E=\frac{1}{2}\rho|{\bf
u}|^{2}+\rho e$ is the total non-gravitational energy, with $e$ being the
specific internal energy. The system (1.1) needs to be closed by providing an
equation of state (EOS), which is usually given in the form
$e=\mathcal{E}(\rho,p)$. For an ideal gas, the EOS can be written as
$e=p/(\gamma-1)/\rho$, and the energy $E$ becomes
(1.2) $E=\frac{1}{2}\rho|{\bf u}|^{2}+\frac{p}{\gamma-1},$
with $\gamma>1$ being the ratio of specific heat.
On one hand, for hyperbolic systems with a source term, one important feature
is that they admit equilibrium state solutions, and well-balanced numerical
methods are desirable to exactly preserve such equilibrium states, so that
small perturbations around an equilibrium state can be well captured on
relatively coarse mesh sizes. In recent years, well-balanced schemes are very
attractive for shallow water equations with source terms, see [8, 29, 1, 43,
45, 44, 33], the review papers [42, 28] and the references therein. The
compressible Euler equations with a gravitational source (1.1) have a zero-
velocity hydrostatic equilibrium state [46] of the form
(1.3) ${\bf u}={\bf 0},\qquad\nabla p=-\rho\nabla\Phi.$
Numerically, similar to the shallow water equations, it is essential to
develop well-balanced schemes for the Euler equations (1.1) to exactly
preserve such an equilibrium state, especially for long-time simulations. Many
well-balanced schemes have been studied, including finite difference schemes
[46, 32], finite volume schemes [24, 15, 5, 17, 30, 40, 7, 6, 20, 26], and
discontinuous Galerkin finite element methods [41, 16, 31]. These schemes are
usually based on explicit time discretizations for compressible flows.
On the other hand, the Euler equations (1.1) have many applications with a
wide range of Mach numbers. We can rewrite (1.1) into a dimensionless form, by
introducing a set of dimensionless variables with some suitable reference
values
(1.4) $\hat{{{\bf x}}}=\frac{{{\bf
x}}}{\ell_{ref}},\,\hat{t}=\frac{t}{t_{ref}},\,\hat{\rho}=\frac{\rho}{\rho_{ref}},\,\hat{{\bf
u}}=\frac{{\bf
u}}{U_{ref}},\,\hat{p}=\frac{p}{p_{ref}},\,\hat{E}=\frac{E}{p_{ref}},\,\hat{\Phi}=\frac{\Phi}{\Phi_{ref}},$
where $U_{ref}=\ell_{ref}/t_{ref}$. Under these dimensionless variables, the
system (1.1) becomes [38, 9, 4]
(1.5a) $\displaystyle\rho_{t}+\nabla\cdot(\rho{\bf u})=0,$ (1.5b)
$\displaystyle(\rho{\bf u})_{t}+\nabla\cdot(\rho{\bf u}\otimes{\bf
u})+\frac{1}{\varepsilon^{2}}\nabla
p=-\frac{1}{\varepsilon^{2}}\rho\nabla\Phi,$ (1.5c) $\displaystyle
E_{t}+\nabla\cdot\left((E+p){\bf u}\right)=-\rho{\bf u}\cdot\nabla\Phi.$
Here we drop the hats for those dimensionless variables for clarity. The
parameter $\varepsilon={U_{ref}}/{c_{ref}}$ is a referenced global Mach
number, with $c_{ref}=\sqrt{{p_{ref}}/{\rho_{ref}}}$ being the acoustic
velocity depending on the background flow. The EOS (1.2) becomes
(1.6) $E=\frac{1}{2}\varepsilon^{2}\rho|{\bf u}|^{2}+\frac{p}{\gamma-1}.$
The dimensionless system (1.5) with the EOS (1.6) is still hyperbolic, and its
eigenvalues along the normal direction ${\bf{n}}$ are
(1.7) $\lambda_{1}={\bf u}\cdot{\bf{n}}-c/\varepsilon,\quad\lambda_{2}={\bf
u}\cdot{\bf{n}},\quad\lambda_{3}={\bf u}\cdot{\bf{n}}+c/\varepsilon,$
where $c=\sqrt{\gamma p/\rho}$ is the scaled speed of sound. The model
(1.5)-(1.6) with the global Mach number $\varepsilon$ ranging from $0$ to
$\mathcal{O}(1)$ is widely used for all-speed flows [10, 38].
For the system (1.5), due to the stiffness of eigenvalues given in (1.7),
shock capturing schemes with explicit time discretizations are subject to a
very strict time step restriction [3]
(1.8) $\Delta t=\text{CFL}\frac{\Delta x}{\max(|{\bf
u}|+c/\varepsilon)}\sim\varepsilon\Delta x,$
when $\varepsilon$ is small. Here $\Delta t$ is the time step, $\Delta x$ is
the mesh size and CFL denotes the CFL number. Besides, for schemes with
numerical viscosities depending on the eigenvalues (1.7), e.g. schemes with a
Lax-Friedrichs numerical flux, their numerical viscosities would be inversely
proportional to $\varepsilon$. As $\varepsilon$ goes to $0$, either the time
step is too small, or the numerical viscosities are too large, leading to very
inefficient numerical schemes [46, 32]. To avoid that, several semi-implicit
schemes have been developed for all-Mach Euler systems [3, 27, 38, 10, 4],
which devote to efficient schemes of easy implementation and uniform time
stabilities with respect to $\varepsilon$, and avoid nonlinear iterative
solvers as used in the fully implicit schemes.
In this work, we aim to provide an efficient high order solver for all-Mach
full Euler equations with a gravitational source (1.5), which enjoy the
benefit of being high order, well-balanced and asymptotic preserving (AP)
simultaneously. The design of high order AP method for Euler equations with
gravity is a nontrivial task. The target model (1.5) is different from all-
Mach homogeneous Euler equations considered in [13], where the stiff acoustic
wave only connects to the energy equation, so that only the gradient of
pressure in the momentum equation and the flux in the energy equation deserve
implicit treatments. All Froude number shallow water equations with a non-flat
bottom topography were recently studied in [22], where the source term (the
gradient of the bottom function) can be merged with the pressure gradient, due
to the polytropic EOS $p=p(h)$ with $h$ being the water height. For our model
(1.5), the stiff acoustic wave is balanced by the gravity wave, therefore both
of them require implicit treatments, which in turn leads to the requirement of
implicit treatments for both the density equation (1.5a) and the energy
equation (1.5c). As a result, one new challenge is that, due to the implicit
discretizations for these conservative variables, it is very hard to
numerically derive a consistent discretization for the perturbation of the
potential temperature $\theta_{2}$, which appears in the incompressible limit,
see (2.8). This is mainly due to the highly nonlinear relationships between
conservative and primitive variables, and the nonlinear EOS (1.6). Therefore,
the existing approaches in [13, 22] cannot be directly applied.
To address such an issue, one novel contribution of this paper is to add the
evolution of the perturbation $\theta_{2}$ of the potential temperature to
(1.5) in the numerical discretization. The original energy equation
corresponds to the updating of hydrostatic pressure in the compressible regime
for conservational purpose, and the added equation of $\theta_{2}$ extracts
the contribution of the energy equation in the scale of
$\mathcal{O}(\varepsilon^{2})$, and is corresponding to the update of the
hydrodynamic pressure $p_{2}$ in the incompressible regime. The added equation
of $\theta_{2}$ has simplified the treatment of the nonlinear EOS (1.6), which
helps us to ensure that the correct asymptotic limit is achieved. Then, we
split the system (1.5) into a (non-stiff) nonlinear low dynamic material wave
to be treated explicitly, and (stiff) fast acoustic and gravity waves to be
treated implicitly. A high order implicit-explicit (IMEX) method is utilized
as the temporal discretization. With the aid of explicit time evolution for
the perturbation of potential temperature $\theta_{2}$, we design a novel
well-balanced finite difference WENO scheme for the spatial discretization.
The temporal update of the conservative variables $(\rho,\rho{\bf u},E)$ is
through the conservative scheme, and $\theta_{2}$ is only an auxiliary
variable to aid with the design of AP method. The proposed method can be
formally proved to be AP and asymptotically accurate (AA). In addition, in the
AP and AA analyses, we can show that, as $\varepsilon\rightarrow 0$, (1.5c) is
a consistent updating of hydrostatic pressure, while $\theta_{2}$ contributes
to the hydrodynamic pressure. Extensive one- and two-dimensional numerical
experiments demonstrate the high order accuracy, well-balanced, AP and AA
properties of our proposed approach, as well as good performances in both high
and low Mach regimes.
The rest of the paper is organized as follows. In Section 2, a low Mach limit
of all-Mach Euler equations with gravity is reviewed. AP well-balanced
numerical methods are described in Section 3, with AP and AA analyses given in
Section 4. Numerical experiments are performed in Section 5, followed by
concluding remarks in Section 6. The detailed step-by-step flowchart of the
fully discrete high order scheme with a multi-stage IMEX time discretization
and a corresponding detailed proof of its AA property are presented as
supplementary materials.
## 2\. Low Mach limit for full Euler equations with gravity
In this section, let us briefly review the low Mach limit of Euler equations
with gravity (1.5). To derive such a limit, it would be more convenient to
start with an equation for the pressure $p$, instead of (1.5c) for the total
energy. By utilizing the EOS (1.6), (1.5c) can be replaced by
(2.1) $p_{t}+{\bf u}\cdot\nabla p+\gamma p\nabla\cdot{\bf u}=0.$
If we further define a potential temperature $\theta$ from
$p=(\rho\theta)^{\gamma}$, it yields
(2.2) $(\rho\theta)_{t}+\nabla\cdot(\rho\theta{\bf u})=0,$
which reduces to (after combined with (1.5a))
(2.3) $\theta_{t}+{\bf u}\cdot\nabla\theta=0,$
namely, the potential temperature $\theta$ satisfies a simple transport
equation with the velocity ${\bf u}$. Now the model (1.5) can be rewritten as
(2.4a) $\displaystyle\rho_{t}+\nabla\cdot(\rho{\bf u})=0,$ (2.4b)
$\displaystyle(\rho{\bf u})_{t}+\nabla\cdot(\rho{\bf u}\otimes{\bf
u})+\frac{1}{\varepsilon^{2}}\nabla
p=-\frac{1}{\varepsilon^{2}}\rho\nabla\Phi,$ (2.4c)
$\displaystyle\theta_{t}+{\bf u}\cdot\nabla\theta=0.$
Let us perform a Chapman-Enskog expansion in the following form [10]
(2.5) $\left\\{\begin{array}[]{ll}\rho({\bf x},t)=\rho_{0}({\bf
x},t)+\varepsilon^{2}\rho_{2}({\bf x},t)+\cdots,\\\\[2.84526pt] {\bf u}({\bf
x},t)={\bf u}_{0}({\bf x},t)+\varepsilon{\bf u}_{1}({\bf
x},t)+\cdots,\\\\[2.84526pt] p({\bf x},t)=p_{0}({\bf
x},t)+\varepsilon^{2}p_{2}({\bf x},t)+\cdots,\\\\[2.84526pt] \theta({\bf
x},t)=\theta_{0}({\bf x},t)+\varepsilon^{2}\theta_{2}({\bf
x},t)+\cdots.\end{array}\right.$
By substituting them into (2.4b), we can collect the leading order
$\mathcal{O}(1/\varepsilon^{2})$ terms
(2.6) $\nabla p_{0}=-\rho_{0}\nabla\Phi,$
where $p_{0}$ is the hydrostatic pressure. The $\mathcal{O}(1)$ terms of
(2.4b) yield
(2.7) $(\rho_{0}{\bf u}_{0})_{t}+\nabla\cdot(\rho_{0}{\bf u}_{0}\otimes{\bf
u}_{0})+\nabla p_{2}=-\rho_{2}\nabla\Phi,$
where $p_{2}$ corresponds to the hydrodynamic pressure.
To obtain a closed system for the low Mach limit, a further assumption is
needed due to the extra source term $-\rho_{2}\nabla\Phi$ appearing in (2.7),
which is different from the homogeneous all Mach full Euler system without
gravity [13]. One commonly used assumption [25, 10] is a constant background
potential temperature $\theta_{0}$ with hydrostatic background states
$\rho_{0}({\bf x})$ and $p_{0}({\bf x})$ satisfying (2.6). With such
assumption, from (2.4a), (2.7), (2.4c) and the expansion (2.5), we obtain the
following limiting incompressible equations
(2.8) $\left\\{\begin{array}[]{ll}\nabla\cdot\left(\rho_{0}{\bf
u}_{0}\right)=0,\\\\[5.69054pt] \left(\rho_{0}{\bf
u}_{0}\right)_{t}+\nabla\cdot(\rho_{0}{\bf u}_{0}\otimes{\bf u}_{0})+\nabla
p_{2}=-\rho_{2}\nabla\Phi,\\\\[5.69054pt] (\theta_{2})_{t}+{\bf
u}_{0}\cdot\nabla\theta_{2}=0,\end{array}\right.$
which is known as the Ogura and Phillips model [34]. The model (2.8) contains
four unknowns ${\bf u}_{0}$, $\rho_{2}$, $p_{2}$, $\theta_{2}$, and is closed
by the relation $p=(\rho\,\theta)^{\gamma}$, which yields
(2.9) $p_{2}=\lim_{\varepsilon\rightarrow
0}\frac{p-p_{0}}{\varepsilon^{2}}=\lim_{\varepsilon\rightarrow
0}\frac{(\rho\,\theta)^{\gamma}-(\rho_{0}\,\theta_{0})^{\gamma}}{\varepsilon^{2}}=\frac{\gamma
p_{0}}{\rho_{0}\theta_{0}}(\rho_{0}\,\theta_{2}+\theta_{0}\,\rho_{2}),$
where the Chapman-Enskog expansion (2.5) is used.
As compared to the homogeneous all Mach full Euler system without gravity
[13], an extra equation of $\theta_{2}$ exists in the incompressible system
(2.8). Due to the existence of the source term in the momentum equation of
(2.8), the limit model (2.8) requires (2.9) to close the system, which will
lead to additional numerical challenges as described in the next section.
###### Remark 2.1.
Another model assumes a stratified background potential temperature
$\theta_{0}({\bf x})$ with $\nabla\theta_{0}({\bf
x})=\mathcal{O}(\varepsilon^{2})$, which is typically used in a nearly
incompressible regime. Similarly if we have hydrostatic background states
$\rho_{0}({\bf x})$ and $p_{0}({\bf x})$ satisfying (2.6), the nearly
incompressible system is given as
(2.10) $\left\\{\begin{array}[]{ll}\nabla\cdot\left(\rho_{0}{\bf
u}_{0}\right)=0,\\\\[5.69054pt] \left(\rho_{0}{\bf
u}_{0}\right)_{t}+\nabla\cdot(\rho_{0}{\bf u}_{0}\otimes{\bf u}_{0})+\nabla
p_{2}=-\rho_{2}\nabla\Phi,\\\\[5.69054pt] (\theta_{2})_{t}+{\bf
u}_{0}\cdot\nabla\theta_{2}+\frac{1}{\varepsilon^{2}}{\bf
u}_{0}\cdot\nabla\theta_{0}=0.\end{array}\right.$
This is known as the Bannon’s anelastic model [2]. As $\varepsilon\rightarrow
0$, $\theta_{0}({\bf x})$ converges to a constant and (2.10) becomes (2.8).
## 3\. Numerical scheme
In this section, we propose a class of semi-implicit high order finite
difference weighted essentially non-oscillatory (WENO) schemes, with both AP
and well-balanced properties, for the all-Mach Euler system with gravity
(1.5). A novel new framework is presented to design a well-balanced high order
semi-implicit scheme, that is AP and AA in the incompressible limit, and is
conservative and robust in the compressible regime.
### 3.1. Model problem and semi-implicit splitting
For the model (1.5)-(1.6), the gravitational source appears both in the second
momentum and third energy equations. It couples all the conservative unknowns
together, which is different from the source term due to a non-flat bottom
topography in the shallow water equations [22]. Compared to all Mach
homogeneous Euler equations without gravity in [13], this introduces two extra
variables $\rho_{2}$ and $\theta_{2}$ in the limiting equations (2.8). As a
result, a fully nonlinear coupling of all conservative variables exists, and
it is highly nontrivial to extract an elliptic equation for the extra
variables $\rho_{2}$ or $p_{2}$ from (1.5) through the nonlinear EOS (1.6).
Note that such elliptic equation is very important to ensure a correct low
Mach limit (2.8). Therefore, the frameworks developed in [22, 13] cannot be
directly applied to (1.5).
In order to address the difficulty to ensure a correct asymptotic limit from
(1.5), one novel idea of this paper is to include the evolution of
perturbation of the potential temperature $\theta_{2}$ to the system
(3.1) $\left\\{\begin{array}[]{ll}\rho_{t}+\nabla\cdot(\rho{\bf
u})=0,\\\\[5.69054pt] (\rho{\bf u})_{t}+\nabla\cdot(\rho{\bf u}\otimes{\bf
u})+\frac{1}{\varepsilon^{2}}\nabla
p=-\frac{1}{\varepsilon^{2}}\rho\nabla\Phi,\\\\[5.69054pt]
E_{t}+\nabla\cdot\left(\left(E+p\right){\bf u}\right)=-\rho{\bf
u}\cdot\nabla\Phi,\\\\[5.69054pt] (\theta_{2})_{t}+{\bf
u}\cdot\nabla\theta_{2}+\frac{1}{\varepsilon^{2}}{\bf
u}\cdot\nabla\theta_{0}=0.\end{array}\right.$
Note that the equation of $\theta_{2}$ is taken from (2.4c) by assuming an
expansion $\theta({\bf x},t)=\theta_{0}({\bf
x})+\varepsilon^{2}\theta_{2}({\bf x},t)$, mimicking the one in (2.5). Here
$\theta_{0}({\bf x})$ is assumed to be a pre-determined time independent
background potential temperature and $\nabla\theta_{0}({\bf
x})=\mathcal{O}(\varepsilon^{2})$. This equation can ensure the right
incompressible limit (2.8), and more generally is consistent with (2.10) in
the nearly incompressible regime. The advantage of (3.1), compared with (1.5),
is that the third energy equation corresponds to the updating of hydrostatic
pressure $p_{0}$ (in the compressible regime) for conservational purpose,
while the fourth equation of $\theta_{2}$ extracts the contribution of the
energy equation in the scale of $\mathcal{O}(\varepsilon^{2})$, and is
corresponding to the hydrodynamic pressure $p_{2}$ which plays the role as a
Lagrangian multiplier in the incompressible regime. The fourth equation of
$\theta_{2}$, which is derived from (2.4c), has simplified the treatment of
the nonlinear EOS (1.6) as compared to the third energy equation, and this
helps us to easily form an elliptic equation for $\rho_{2}$ as described in
the design of our scheme below.
We first follow the idea in [21, 37] to split the system (3.1) into a stiff
part and a non-stiff component
(3.2)
$\frac{dU}{dt}=-\nabla\cdot\mathcal{F}_{E}-\nabla\cdot\mathcal{F}_{I}-\mathcal{S}_{E}-\mathcal{S}_{I},$
where $U=(\rho,\rho{\bf u},E,\theta_{2})^{T}$ and
(3.3a)
$\displaystyle\nabla\cdot\mathcal{F}_{E}=\alpha\Big{(}\nabla\cdot(\rho{\bf
u})_{E},\nabla\cdot(\rho{\bf u}\otimes{\bf
u}+p\,{\mathbb{I}})_{E},\nabla\cdot((E+p){\bf u})_{E},0\Big{)}^{T}$
$\displaystyle+\Big{(}0,(1-\alpha)\,\nabla\cdot(\rho{\bf u}\otimes{\bf
u})_{E}+\alpha\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\nabla p_{E},0,{\bf
u}_{E}\cdot\nabla\theta_{2,E}+\frac{1}{\varepsilon^{2}}{\bf
u}_{E}\cdot\nabla\theta_{0}\Big{)}^{T},$ (3.3b)
$\displaystyle\nabla\cdot\mathcal{F}_{I}=(1-\alpha)\left(\nabla\cdot(\rho{\bf
u})_{I},\frac{1}{\varepsilon^{2}}\nabla p_{I},\nabla\cdot((E+p){\bf
u})_{I},0\right)^{T},$ (3.3c)
$\displaystyle\mathcal{S}_{E}=\alpha\left(0,\frac{1}{\varepsilon^{2}}\rho_{E}\nabla\Phi,(\rho{\bf
u})_{E}\cdot\nabla\Phi,0\right)^{T},$ (3.3d)
$\displaystyle\mathcal{S}_{I}=(1-\alpha)\left(0,\frac{1}{\varepsilon^{2}}\rho_{I}\nabla\Phi,(\rho{\bf
u})_{I}\cdot\nabla\Phi,0\right)^{T}.$
The subscripts $E$ and $I$ represent non-stiff and stiff parts of the flux
function $\mathcal{F}$ and the source term $S$, which will be discretized
explicitly and implicitly, respectively. In the splitting, we take the
splitting parameter $\alpha=\min(\varepsilon^{2},1)$, that is,
$\alpha=\varepsilon^{2}$ when $\varepsilon\leq 1$ and (3.2) turns to be fully
explicit if $\varepsilon\geq 1$. In the following, we only consider
$\alpha=\varepsilon^{2}$ with $\varepsilon\leq 1$ for ease of presentation.
### 3.2. First order IMEX scheme
We start with presenting the first order IMEX AP time discretization for
(3.1). We focus on time discretization only and keep space to be continuous.
The spatial discretization will be discussed afterward. Following the
splitting in (3.2), the semi-discrete (in time) scheme takes the form
(3.4a) $\displaystyle\frac{\rho^{n+1}-\rho^{n}}{\Delta
t}+\varepsilon^{2}\,\nabla\cdot(\rho{\bf
u})^{n}+(1-\varepsilon^{2})\,\nabla\cdot(\rho{\bf u})^{n+1}=0,$ (3.4b)
$\displaystyle\frac{(\rho{\bf u})^{n+1}-(\rho{\bf u})^{n}}{\Delta
t}+\nabla\cdot(\rho{\bf u}\otimes{\bf
u}+p\,{\mathbb{I}})^{n}+\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\nabla
p^{n+1}$ $\displaystyle\hskip
99.58464pt=-\rho^{n}\nabla\Phi-\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\rho^{n+1}\nabla\Phi,$
(3.4c) $\displaystyle\frac{E^{n+1}-E^{n}}{\Delta
t}+\varepsilon^{2}\,\nabla\cdot\left((E+p){\bf
u}\right)^{n}+(1-\varepsilon^{2})\,\nabla\cdot\left((E+p){\bf u}\right)^{n+1}$
$\displaystyle\hskip 99.58464pt=-\varepsilon^{2}\,\left(\rho{\bf
u}\right)^{n}\cdot\nabla\Phi-(1-\varepsilon^{2})\,\left(\rho{\bf
u}\right)^{n+1}\cdot\nabla\Phi,$ (3.4d)
$\displaystyle\frac{\theta_{2}^{n+1}-\theta_{2}^{n}}{\Delta t}+{\bf
u}^{n}\cdot\nabla\theta_{2}^{n}+\frac{1}{\varepsilon^{2}}{\bf
u}^{n}\cdot\nabla\theta_{0}=0.$
Introducing the variables
(3.5a) $\displaystyle\widetilde{\rho}=\rho^{n}-\Delta
t\,\varepsilon^{2}\,\nabla\cdot(\rho{\bf u})^{n},$ (3.5b)
$\displaystyle\widetilde{\rho{\bf u}}=(\rho{\bf u})^{n}-\Delta
t\Big{(}\nabla\cdot(\rho{\bf u}\otimes{\bf
u}+p{\mathbb{I}})^{n}+\rho^{n}\nabla\Phi\Big{)},$ (3.5c)
$\displaystyle\widetilde{E}=E^{n}-\Delta
t\,\varepsilon^{2}\,\Big{(}\nabla\cdot\left((E+p){\bf
u}\right)^{n}+\,\left(\rho{\bf u}\right)^{n}\cdot\nabla\Phi\Big{)},$
we can rewrite (3.4) as
(3.6a) $\displaystyle\rho^{n+1}=\widetilde{\rho}-\Delta
t\,(1-\varepsilon^{2})\,\nabla\cdot(\rho{\bf u})^{n+1},$ (3.6b)
$\displaystyle(\rho{\bf u})^{n+1}=\widetilde{\rho{\bf u}}-\Delta
t\,\frac{1-\varepsilon^{2}}{\varepsilon^{2}}\left(\nabla
p^{n+1}+\rho^{n+1}\nabla\Phi\right),$ (3.6c) $\displaystyle
E^{n+1}=\widetilde{E}-\Delta
t\,(1-\varepsilon^{2})\,\left(\nabla\cdot\left((E+p){\bf
u}\right)^{n+1}+\,\left(\rho{\bf u}\right)^{n+1}\cdot\nabla\Phi\right),$
(3.6d) $\displaystyle\theta_{2}^{n+1}=\theta_{2}^{n}-\Delta t\,\left({\bf
u}^{n}\cdot\nabla\theta_{2}^{n}\right)-\Delta
t\,\left(\frac{1}{\varepsilon^{2}}{\bf u}^{n}\cdot\nabla\theta_{0}\right),$
which will be solved with the following three steps:
* •
Step 1: Combining (3.6a) and (3.6b) leads to
(3.7) $\rho^{n+1}=\widetilde{\rho}-(1-\varepsilon^{2})\,\Delta
t\,\nabla\cdot(\widetilde{\rho{\bf u}})+\Delta
t^{2}\,\frac{(1-\varepsilon^{2})^{2}}{\varepsilon^{2}}\nabla\cdot\left(\nabla
p^{n+1}+\rho^{n+1}\nabla\Phi\right),$
We now introduce $p_{2}$ and $\rho_{2}$ from
(3.8) $p^{n+1}=p_{0}({\bf
x})+\varepsilon^{2}\,p_{2}^{n+1},\quad\rho^{n+1}=\rho_{0}({\bf
x})+\varepsilon^{2}\,\rho_{2}^{n+1},$
according to (2.5), where $p_{0}({\bf x})$ and $\rho_{0}({\bf x})$ satisfy
$\nabla p_{0}=-\rho_{0}\nabla\Phi$ as given in (2.6). Therefore (3.7) becomes
(3.9)
$\varepsilon^{2}\rho_{2}^{n+1}=\widetilde{\rho}-(1-\varepsilon^{2})\,\Delta
t\,\nabla\cdot(\widetilde{\rho{\bf u}})+\Delta
t^{2}(1-\varepsilon^{2})^{2}\nabla\cdot\left(\nabla
p_{2}^{n+1}+\rho_{2}^{n+1}\nabla\Phi\right)-\rho_{0},$
which cannot be solved due to two unknowns $p^{n+1}_{2}$ and $\rho_{2}^{n+1}$.
From the relation $p=(\rho\theta)^{\gamma}$, we have
(3.10)
$p_{2}^{n+1}=(p-p_{0})/\varepsilon^{2}=\left[\Big{(}(\rho_{0}+\varepsilon^{2}\,\rho_{2}^{n+1})(\theta_{0}+\varepsilon^{2}\,\theta_{2}^{n+1})\Big{)}^{\gamma}-p_{0}\right]/\varepsilon^{2}.$
Substituting it into (3.9) leads to a nonlinear system for $\rho_{2}^{n+1}$,
which can be solved iteratively. To simplify the computation, we approximate
(3.10) with a linearization up to an approximation error
$\mathcal{O}(\varepsilon^{2})$,
(3.11) $p_{2}^{n+1}=\frac{\gamma
p_{0}}{\rho_{0}\theta_{0}}(\rho_{0}\theta_{2}^{n+1}+\rho^{n+1}_{2}\,\theta^{n+1}),$
where we denote $\theta^{n+1}=\theta_{0}+\varepsilon^{2}\,\theta_{2}^{n+1}$.
Substituting (3.11) into (3.9) yields
(3.12) $\varepsilon^{2}\rho_{2}^{n+1}=\widetilde{\widetilde{\rho}}+\Delta
t^{2}(1-\varepsilon^{2})^{2}\nabla\cdot\left(\nabla\left(\frac{\gamma
p_{0}}{\rho_{0}\theta_{0}}\theta^{n+1}\,\rho_{2}^{n+1}\right)+\rho_{2}^{n+1}\nabla\Phi\right),$
where
(3.13)
$\widetilde{\widetilde{\rho}}=\widetilde{\rho}-(1-\varepsilon^{2})\,\Delta
t\,\nabla\cdot(\widetilde{\rho{\bf u}})+\Delta
t^{2}(1-\varepsilon^{2})^{2}\Delta\left(\frac{\gamma
p_{0}}{\theta_{0}}\theta^{n+1}_{2}\right)-\rho_{0}.$
The purpose of adding the fourth equation on $\theta_{2}$ in (3.1) is to
evaluate $\theta^{n+1}_{2}$ from (3.6d) explicitly. With such information,
$\theta^{n+1}$ can be computed and (3.12) is only a linear system for
$\rho_{2}^{n+1}$. Otherwise, without (3.6d), (3.12) will be coupled to (3.6c),
which is very complicated to solve for $\rho_{2}$ and $\theta_{2}$. After
obtaining $\rho^{n+1}_{2}$, $p_{2}^{n+1}$ is available from (3.11).
* •
Step 2: $(\rho{\bf u})^{n+1}$ directly follows from (3.6b) and (3.8). In order
to ensure exact mass and energy conservation, especially in the compressible
regime, we further update $\rho^{n+1}$ from (3.6a), and finally $E^{n+1}$ from
(3.6c). We note that in (3.6c), the implicit term $((E+p){\bf u})^{n+1}$ is
computed based on
(3.14) $((E+p){\bf u})^{n+1}=H^{n+1}\,(\rho{\bf u})^{n+1},$
where the enthalpy $H^{n+1}$ is given by
(3.15)
$H^{n+1}=\left(\frac{E+p}{\rho}\right)^{n+1}=\frac{\gamma}{\gamma-1}\frac{p^{n+1}}{\rho^{n+1}}+\frac{1}{2}\varepsilon^{2}\frac{|(\rho{\bf
u})^{n+1}|^{2}}{(\rho^{n+1})^{2}},$
following the EOS (1.6). To compute $H^{n+1}$, we use $\rho^{n+1}$ and
$p^{n+1}$ from (3.8) available after step 1. In this way, $(\rho{\bf
u})^{n+1}$, $\rho^{n+1}$ and $E^{n+1}$ are updated in a conservative form and
explicitly via a sequential way.
###### Remark 3.1.
In the momentum equation (3.4b), the stiff gravitational source
$\rho\nabla\Phi$ which depends on the unknown variable $\rho$ needs to be
discretized implicitly. Correspondingly, we need an implicit discretization
for the density equation (3.4a) in the low Mach regime. On the other hand, the
implicit discretization of $\nabla p$ needs to be coupled with an implicit
discretization of the energy equation (3.4c). In this way, the mass, momentum
and energy equations in (3.4) all have components needing implicit
discretizations, and uniform time stability independent of $\varepsilon$ can
be obtained. This can be viewed as, in some sense, a combination of two
approaches for the isentropic and full Euler equations [12, 13]. We refer to
[12] for the Fourier analysis with respect to the rationale behind these
implicit discretizations.
###### Remark 3.2.
In [10], the authors used (2.2) instead of (1.5c), and designed an AP scheme
in the low Mach regime. $\rho\theta$ in (2.2) is not a physically conservative
variable. In this work, we keep the energy equation (1.5c), which is important
for shock capturing purpose in the high Mach regime. Besides, a numerical
discretization of (2.2) does not straightforwardly result in a consistent
discretization of $\theta_{2}$ in the limiting equation (2.8) as
$\varepsilon\rightarrow 0$. It requires $\nabla\cdot(\rho{\bf u})=0$ to be
satisfied at the discrete level as $\varepsilon\rightarrow 0$, which is not an
easy task.
### 3.3. High order IMEX Runge-Kutta (IMEX-RK) scheme
We now generalize the first-order semi-implicit scheme (3.4) to high-order, by
utilizing a high order IMEX-RK scheme proposed in [11], which has already been
used in [12, 13]. We first denote the split system (3.2) as
(3.16) $U_{t}=\mathcal{H}(U_{E},U_{I}),\qquad U(t_{0})=U_{0}.$
where
$\mathcal{H}(U_{E},U_{I})=-\nabla\cdot\mathcal{F}_{E}-\nabla\cdot\mathcal{F}_{I}-\mathcal{S}_{E}-\mathcal{S}_{I}$,
with $U_{E}=(\rho_{E},(\rho{\bf u})_{E},E_{E},\theta_{2,E})^{T}$ and
$U_{I}=(\rho_{I},(\rho{\bf u})_{I},E_{I},\theta_{2,I})^{T}$ representing the
terms that will be discretized explicitly and implicitly. The solutions
$U_{E}$ and $U_{I}$ will be updated separately, that is
(3.17)
$\left\\{\begin{array}[]{l}(U_{E})_{t}=\mathcal{H}(U_{E},U_{I}),\\\\[5.69054pt]
(U_{I})_{t}=\mathcal{H}(U_{E},U_{I}),\end{array}\right.$
with the same initial condition
(3.18) $U_{E}(t_{0})=U_{I}(t_{0})=U_{0}.$
Following [12, 13], we apply an IMEX-RK scheme to (3.17) with a double Butcher
tableau [14]
(3.19) $\begin{array}[]{c|c}\tilde{c}&\tilde{A}\\\
\hline\cr\vspace{-0.25cm}\hfil\\\ &\tilde{b}^{T}\end{array},\ \ \ \ \
\begin{array}[]{c|c}{c}&{A}\\\ \hline\cr\vspace{-0.25cm}\hfil\\\
&{b^{T}}\end{array},$
where $\tilde{A}=(\tilde{a}_{ij})$ is an $s\times s$ matrix for an explicit
scheme with $\tilde{a}_{ij}=0$ for $j\geq i$, and $A=({a}_{ij})$ is an
$s\times s$ matrix for an implicit scheme. A diagonally implicit RK (DIRK)
scheme with $a_{ij}=0$ for $j>i$ is used, which is simple and efficient for
implicit discretization. Other vectors are
$\tilde{c}=(\tilde{c}_{1},...,\tilde{c}_{s})^{T}$,
$\tilde{b}=(\tilde{b}_{1},...,\tilde{b}_{s})^{T}$, and
$c=(c_{1},...,c_{s})^{T}$, $b=(b_{1},...,b_{s})^{T}$, where $\tilde{c}$ and
$c$ satisfy the relation $\tilde{c}_{i}=\sum_{j=1}^{i-1}\tilde{a}_{ij}$,
$c_{i}=\sum_{j=1}^{i}a_{ij}$. For the high order IMEX-RK scheme, as discussed
in [12, 13], a stiffly accurate property with $b^{T}={\bf e}_{s}^{T}A$ where
${\bf e}_{s}=(0,\cdots,0,1)^{T}$ for the implicit part is preferred, which can
yield the AA property of the scheme. A sketched procedure of the IMEX-(DI)RK
scheme is given as follows:
1. (1)
Set $U_{E}^{(0)}=U_{I}^{(0)}=U^{n}$. For each inner stage $i=1\text{ to }s$,
* •
Update the explicit solution $U_{E}^{(i)}$
(3.20) $U_{E}^{(i)}=U^{n}+\Delta
t\sum^{i-1}_{j=1}\tilde{a}_{ij}\mathcal{H}(U_{E}^{(j)},U_{I}^{(j)});$
* •
Compute $U_{\star}^{(i)}$ based on known solutions at previous stages
(3.21) $U_{\star}^{(i)}=U^{n}+\Delta
t\sum^{i-1}_{j=1}a_{ij}\mathcal{H}(U_{E}^{(j)},U_{I}^{(j)}),$
then update the implicit solution $U_{I}^{(i)}$ via
(3.22) $U_{I}^{(i)}=U_{\star}^{(i)}+\Delta
ta_{ii}\mathcal{H}(U_{E}^{(i)},U_{I}^{(i)});$
2. (2)
Update the solution $U^{n+1}$ at the next time level $t^{n+1}$ by
(3.23) $U^{n+1}=U^{n}+\Delta
t\sum_{i=1}^{s}b_{i}\mathcal{H}(U_{E}^{(i)},U_{I}^{(i)}).$
### 3.4. Fully discrete scheme with high order spatial discretizations
In this part, we introduce high order WENO spatial discretizations for (3.2).
We expect the following desirable properties from the spatial discretization.
Firstly, it should be suitable for all Mach numbers, that is, for
$\varepsilon$ ranging from $0$ to $\mathcal{O}(1)$. Secondly, it can preserve
the well-balanced property, namely, the zero-velocity steady state solution
(1.3) can be maintained exactly on the discrete level. To achieve both
purposes, we adopt the spatial discretizations proposed in [13], and well-
balanced reconstruction strategies from [46, 32].
Without loss of generality, we present (3.2) in the two-dimensional spatial
setting. Taking $\alpha=\varepsilon^{2}$, we denote $U=(\rho,\rho u,\rho
v,E,\theta_{2})^{T}$ and
(3.24a)
$\displaystyle\nabla\cdot\mathcal{F}_{E}=\partial_{x}\mathcal{F}_{E}^{x}+\partial_{y}\mathcal{F}_{E}^{y},\quad\mathcal{S}_{E}=\mathcal{S}_{E}^{x}+\mathcal{S}_{E}^{y},$
(3.24b)
$\displaystyle\nabla\cdot\mathcal{F}_{I}=(1-\varepsilon^{2})(\partial_{x}\mathcal{F}_{I}^{x}+\partial_{y}\mathcal{F}_{I}^{y}),\quad\mathcal{S}_{I}=(1-\varepsilon^{2})(\mathcal{S}_{I}^{x}+\mathcal{S}_{I}^{y}),$
with
(3.25a)
$\displaystyle\partial_{x}\mathcal{F}_{E}^{x}=\varepsilon^{2}\Big{(}\partial_{x}(\rho
u),\,\partial_{x}(\rho u^{2}+p),\,\partial_{x}(\rho
uv),\,\partial_{x}((E+p)u),\,0\Big{)}_{E}^{T}$ $\displaystyle\hskip
36.98866pt+(1-\varepsilon^{2})\Big{(}0,\,\partial_{x}(\rho
u^{2}+p),\,\partial_{x}(\rho uv),\,0,\,0\Big{)}_{E}^{T}$ $\displaystyle\hskip
36.98866pt+\Big{(}0,\,0,\,0,\,0,\,u\partial_{x}\theta_{2}+\frac{1}{\varepsilon^{2}}u\partial_{x}\theta_{0}\Big{)}_{E}^{T}$
(3.25b)
$\displaystyle\partial_{y}\mathcal{F}_{E}^{y}=\varepsilon^{2}\Big{(}\partial_{y}(\rho
v),\,\partial_{y}(\rho uv),\,\partial_{y}(\rho
v^{2}+p),\,\partial_{y}((E+p)v),\,0\Big{)}_{E}^{T}$ $\displaystyle\hskip
36.98866pt+(1-\varepsilon^{2})\Big{(}0,\,\partial_{y}(\rho
uv),\,\partial_{y}(\rho v^{2}+p),\,0,\,0\Big{)}_{E}^{T}$ $\displaystyle\hskip
36.98866pt+\Big{(}0,\,0,\,0,\,0,\,v\partial_{y}\theta_{2}+\frac{1}{\varepsilon^{2}}v\partial_{y}\theta_{0}\Big{)}_{E}^{T},$
(3.25c) $\displaystyle\partial_{x}\mathcal{F}_{I}^{x}=\Big{(}\partial_{x}(\rho
u),\,\partial_{x}p_{2},\,0,\,\partial_{x}((E+p)u),\,0\Big{)}_{I}^{T},$ (3.25d)
$\displaystyle\partial_{y}\mathcal{F}_{I}^{y}=\Big{(}\partial_{y}(\rho
v),\,0,\,\partial_{y}p_{2},\,\partial_{y}((E+p)v),\,0\Big{)}_{I}^{T},$
and
(3.26a)
$\displaystyle\mathcal{S}_{E}^{x}=\Big{(}0,\rho\,\partial_{x}\Phi,0,\varepsilon^{2}\rho
u\partial_{x}\Phi,0\Big{)}_{E}^{T},\quad\mathcal{S}_{E}^{y}=\Big{(}0,0,\rho\,\partial_{y}\Phi,\varepsilon^{2}\rho
v\partial_{y}\Phi,0\Big{)}_{E}^{T},$ (3.26b)
$\displaystyle\mathcal{S}_{I}^{x}=\Big{(}0,\rho_{2}\,\partial_{x}\Phi,0,\rho
u\partial_{x}\Phi,0\Big{)}_{I}^{T},\quad\mathcal{S}_{I}^{y}=\Big{(}0,0,\rho_{2}\,\partial_{y}\Phi,\rho
v\partial_{y}\Phi,0\Big{)}_{I}^{T}.$
Here (3.8) with $\nabla p_{0}=-\rho_{0}\nabla\Phi$ is used in (3.25c), (3.25d)
and (3.26b), by replacing $p$ and $\rho$ with $p_{2}$ and $\rho_{2}$
respectively.
For a high order finite difference discretization, we take a uniform cartesian
mesh with mesh sizes $\Delta x$ and $\Delta y$ along $x$ and $y$ directions
respectively. The grid points are $(x_{i},y_{j})$ for $i=1,2,\dots,N_{x}$,
$j=1,2,\dots,N_{y}$. We denote $w_{ij}=w(x_{i},y_{j})$ for short with any
variable $w$, and the interface values are denoted as $w_{i\pm\frac{1}{2},j}$
and $w_{i,j\pm\frac{1}{2}}$ respectively.
We now briefly describe our spatial discretizations. Denoting ${\bf
q}=\rho{\bf u}$, the high order WENO approximation of the spatial operator
$\mathcal{H}(U_{E},U_{I})$ defined in (3.16) is given by
(3.27) $\mathcal{H}_{WENO}(U_{E},U_{I})=-\left(\begin{aligned}
&\varepsilon^{2}\nabla_{CW}\cdot{\bf
q}_{E}+(1-\varepsilon^{2})\nabla_{W}\cdot{\bf q}_{I}\\\
&\nabla^{WB}_{CW}\cdot\left(\frac{{\bf q}\otimes{\bf
q}}{\rho}+p\,{\mathbb{I}}\right)_{E}-\frac{\rho_{E}}{\rho_{0}}\nabla^{WB}_{CW}p_{0}\\\
&\hskip
34.14322pt+(1-\varepsilon^{2})\left(\nabla_{W}p_{2,I}+\rho_{2,I}\nabla\Phi\right)\\\
&\varepsilon^{2}\left(\nabla_{CW}\cdot(H{\bf q})_{E}+{\bf
q}_{E}\cdot\nabla\Phi\right)\\\ &\hskip
34.14322pt+(1-\varepsilon^{2})\left(\nabla_{W}\cdot(H{\bf q})_{I}+{\bf
q}_{I}\cdot\nabla\Phi\right)\\\ &{\bf
u}_{E}^{(j)}\cdot\nabla_{UW}\theta_{2,E}+\frac{1}{\varepsilon^{2}}{\bf
u}_{E}\nabla\theta_{0}\end{aligned}\right),$
where the WENO operators $\nabla_{CW}$, $\nabla_{W}$, $\nabla_{UW}$ and
$\nabla^{WB}_{CW}$ will be defined below. The fully discrete scheme can be
obtained by combining the spatial discretization (3.27) with the high order
IMEX temporal discretization outlined in (3.20)-(3.23). Note that this
involves an implicit step in (3.22). Following the approach to solve the first
order IMEX method (3.6) in Section 3.2, we can convert it into an elliptic
equation and then update the solution in a sequential way. For the high order
scheme in (3.22), we form the following elliptic equation for $\rho_{2}$
(3.28) $\varepsilon^{2}\rho_{2,I}^{(i)}=\rho_{\star\star\star}^{(i)}+(\Delta
ta_{ii})^{2}(1-\varepsilon^{2})^{2}\left(\Delta\Big{(}\frac{\gamma
p_{0}}{\theta_{0}}\,\theta_{2,E}^{(i)}+\frac{\gamma
p_{0}}{\rho_{0}\theta_{0}}\theta^{(i)}\,\rho_{2,I}^{(i)}\Big{)}+\nabla\cdot\Big{(}\rho_{2,I}^{(i)}\nabla\Phi\Big{)}\right),$
with
(3.29) $\rho_{\star\star\star}^{(i)}=\rho_{\star\star}^{(i)}-\Delta
ta_{ii}(1-\varepsilon^{2})\nabla_{W}\cdot{\bf q}_{\star\star}^{(i)}-\rho_{0},$
where
(3.30a) $\displaystyle\rho_{\star\star}^{(i)}=\rho_{\star}^{(i)}-\Delta
ta_{ii}\varepsilon^{2}\nabla_{CW}\cdot{\bf q}_{E}^{(i)},$ (3.30b)
$\displaystyle{\bf q}_{\star\star}^{(i)}={\bf q}_{\star}^{(i)}-\Delta
ta_{ii}\left(\nabla^{WB}_{CW}\cdot\left(\frac{{\bf q}\otimes{\bf
q}}{\rho}+p{\mathbb{I}}\right)_{E}^{(i)}-\frac{\rho_{E}^{(i)}}{\rho_{0}}\nabla^{WB}_{CW}p_{0}\right).$
Here $\rho_{\star}^{(i)}$ and ${\bf q}_{\star}^{(i)}$ are updated from (3.21).
For this elliptic equation, all the derivative operators in (3.28) are
discretized with high order central differences, e.g., the compact central
difference for the Laplacian operator as in [12]. After obtaining
$\rho_{2,I}^{(i)}$, we can update $U_{I}^{(i)}$ from (3.22) explicitly in a
sequential way. The detailed step-by-step flowchart of the fully discrete high
order scheme is provided in the supplementary material.
Taking the two dimensional model (3.24)-(3.26) described above as an example,
these WENO operators $\nabla_{CW}$, $\nabla_{W}$, $\nabla_{UW}$ and
$\nabla^{WB}_{CW}$ are defined as follows.
* •
Characteristic-wise WENO reconstruction $\nabla_{CW}$.
The traditional characteristic-wise WENO reconstruction is applied to those
conservative variables of the explicit part. Let us denote $U_{EC}=(\rho,\rho
u,\rho v,E)^{T}_{E}$ and
(3.31) $\mathcal{F}_{EC}^{x}=(\rho u,\rho u^{2}+p,\rho
uv,(E+P)u)^{T}_{E},\quad\mathcal{F}_{EC}^{y}=(\rho v,\rho uv,\rho
v^{2}+p,(E+P)v)^{T}_{E},$
with
(3.32) $\mathcal{S}_{EC}^{x}=(0,-\rho\partial_{x}\Phi,-\rho
u\partial_{x}\Phi,0)^{T}_{E},\quad\mathcal{S}_{EC}^{y}=(0,-\rho\partial_{y}\Phi,-\rho
v\partial_{y}\Phi,0)^{T}_{E}.$
We note that $\mathcal{F}_{EC}^{x}$ and $\mathcal{F}_{EC}^{y}$ are the fluxes
of 2D compressible Euler equations in a conservative form. They are exactly
the first terms on the right-hand side of (3.25a) and (3.25b) (with a factor
$\varepsilon^{2}$), corresponding to the conserved variable $U_{EC}$. We take
$\mathcal{F}_{EC}^{x}$ and $\mathcal{F}_{EC}^{y}$ to form our Jacobian
matrices and perform a 2D characteristic-wise finite difference WENO
reconstruction, as detailed in [23, 36], to obtain their numerical
approximation. The same reconstructed values are also used to evaluate the
second terms, e.g., $\partial_{x}(\rho u^{2}+p)$ and $\partial_{x}(\rho uv)$
in the second term on the right side of (3.25a) (with a factor
$(1-\varepsilon^{2})$). Such a characteristic-wise WENO reconstruction is
denoted by $\nabla_{CW}$. $\nabla_{CW}$ is very important for essentially non-
oscillatory shock capturing, compared with component-wise WENO
reconstructions, especially for multi-dimensional cases [13].
* •
Component-wise WENO reconstructions $\nabla_{W}$ and $\nabla_{UW}$.
When the characteristic direction is unclear, the component-wise finite
difference WENO reconstruction is applied to some first-order derivative
terms, e.g., the third terms in (3.25a) and (3.25b) and those in (3.25c) and
(3.25d). We refer to [23, 36] for detailed reconstruction procedures. Such
reconstructions with a Lax-Friedrichs flux is frequently used. Denoting
(3.33) $\mathcal{F}_{IC}^{x}=(\rho
u,p_{2},0,(E+p)u)^{T}_{I},\quad\mathcal{F}_{IC}^{y}=(\rho
v,0,p_{2},(E+p)v)^{T}_{I},$
and taking $\mathcal{F}_{IC}^{x}$ as an example, a Lax-Friedrichs flux
splitting is given by
(3.34)
$(\mathcal{F}_{IC}^{x})^{\pm}_{ij}=\frac{1}{2}\Big{(}(\mathcal{F}_{IC}^{x})_{ij}\pm\Lambda(U_{EC})_{ij}\Big{)},$
here we take $\Lambda=\max\limits_{\rho,{\bf u},p}\\{|{\bf
u}|+\min(1,1/\varepsilon)\sqrt{\frac{\gamma p}{\rho}}\\}$ and use $U_{EC}$ to
obtain numerical viscosities. For component-wise WENO reconstructions with a
Lax-Friedrichs numerical flux, we denote them as $\nabla_{W}$. On the other
hand, the component-wise WENO reconstruction with an upwind numerical flux is
used in the equation of $\theta_{2}$, and we denote it as $\nabla_{UW}$, e.g.
the $\nabla_{UW}\theta_{2,E}$ term in (3.27).
* •
Well-balanced WENO reconstruction.
For all Mach Euler equations with static gravitational fields (1.5), it admits
an equilibrium steady-state solution (1.3). Namely, if the equilibrium (1.3)
holds initially, it will hold for all later times. Numerically, well-balanced
schemes are designed in order to preserve such a steady-state solution on the
discrete level. Here we follow the approach proposed in [46, 32] to construct
well-balanced WENO approximation. The main idea is to use exactly the same
reconstructions to the flux and source terms, and revise the numerical
viscosity to a prebalanced form so that the numerical viscosities vanish for
equilibrium solutions. For the system (1.5), an equilibrium static solution
would have $\rho=\rho_{0}({\bf x})$ and $p=p_{0}({\bf x})$, which satisfies
(3.35) $\nabla p_{0}=-\rho_{0}\nabla\Phi.$
This is consistent with the asymptotic limit (2.6), namely, the equilibrium
solution is also a solution to (2.8) [10]. Here we still take
$\partial_{x}\mathcal{F}^{x}_{EC}$ as an example to demonstrate how such well-
balanced WENO scheme is designed. First, a well-balanced Lax-Friedrichs flux
splitting is based on
(3.36)
$(\mathcal{F}_{EC}^{x})^{\pm}_{ij}=\frac{1}{2}\Big{(}(\mathcal{F}_{EC}^{x})_{ij}\pm\Lambda(\tilde{U}_{EC})_{ij}\Big{)},$
where $\tilde{U}_{EC}=(\rho-\rho_{0},\rho{\bf
u},E-\frac{p_{0}}{\gamma-1})^{T}_{E}$. For the source term (3.26), we have
$\nabla\Phi=-\nabla p_{0}/\rho_{0}$ from (3.35), and can rewrite (3.26) as
(3.37) $\mathcal{S}_{EC}^{x}=(0,-\frac{\rho}{\rho_{0}}\partial_{x}p_{0},\rho
u\partial_{x}\Phi,0)^{T},\quad\mathcal{S}_{EC}^{y}=(0,-\frac{\rho}{\rho_{0}}\partial_{y}p_{0},0,\rho
v\partial_{y}\Phi)^{T}.$
To approximate $\mathcal{S}_{EC}^{x}$ numerically, in analogous to (3.36), we
split $p_{0}$ as
(3.38) $(p_{0})^{\pm}_{ij}=\frac{1}{2}(p_{0})_{ij}.$
The finite difference characteristic-wise WENO reconstructions used to
approximate $\\{(\mathcal{F}_{EC}^{x})^{\pm}_{ij}\\}$ will then be applied to
$\\{(p_{0})^{\pm}_{ij}\\}$ with exactly the same weights (evaluated based on
the smoothness indicators of $\\{(\mathcal{F}_{EC}^{x})^{\pm}_{ij}\\}$). In
this way, we can obtain the exact balance of these two terms on the discrete
level, when the equilibrium state is reached. We refer to [46, 32] for more
detailed discussions. Similar approximations of
$\partial_{y}\mathcal{F}^{y}_{EC}$ and $\mathcal{S}_{EC}^{y}$ can be applied.
For those terms with above described well-balanced WENO reconstructions, we
denote them as $\nabla_{CW}^{WB}$.
## 4\. Asymptotic preserving and asymptotically accurate properties
In this section, we will prove that our first order IMEX scheme (3.4) is AP
and the high order IMEX-RK scheme in Section 3.3 is AA. For any variable $w$,
let us denote $w^{n}({\bf x})=w({\bf x},t^{n})$. During the discussion, we
focus on time discretization and keep the space continuous. The initial
solutions are assumed to be well-prepared, namely, the following expansions
(4.1a) $\displaystyle\rho^{n}({\bf x})=\rho^{n}_{0}({\bf
x})+\varepsilon^{2}\rho^{n}_{2}({\bf x}),$ (4.1b) $\displaystyle(\rho{\bf
u})^{n}({\bf x})=(\rho{\bf u})^{n}_{0}({\bf x})+\varepsilon(\rho{\bf
u})^{n}_{1}({\bf x}),$ (4.1c) $\displaystyle p^{n}({\bf x})=p^{n}_{0}({\bf
x})+\varepsilon^{2}p^{n}_{2}({\bf x}),$
hold at $n=0$. The relation $p=(\rho\theta)^{\gamma}$ yields
(4.2) $\theta^{n}({\bf x})=\theta^{n}_{0}({\bf
x})+\varepsilon^{2}\theta^{n}_{2}({\bf x}).$
In addition, the initial conditions should satisfy
(4.3) $\nabla\cdot(\rho{\bf u})_{0}^{0}=0,\quad\nabla
p^{0}_{0}=-\rho_{0}^{0}\nabla\Phi.$
### 4.1. Asymptotic preserving
We start with proving the AP property of the first order IMEX scheme (3.4).
###### Theorem 4.1.
With well-prepared initial conditions (4.1)-(4.2) at $n=0$ which satisfy
(4.3), and assuming the expansion (4.1)-(4.2) at all later times, the first
order IMEX scheme (3.4) with (3.9), under the assumptions of (3.8), $\nabla
p_{0}=-\rho_{0}\nabla\Phi$ and $\theta_{0}$ is a constant, is asymptotic
preserving, namely, as $\varepsilon\rightarrow 0$, the limiting scheme of
(3.4) is a consistent discretization of (2.8).
###### Proof.
As $\varepsilon\rightarrow 0$, under the assumptions that $\nabla
p_{0}=-\rho_{0}\nabla\Phi$, $\theta_{0}$ is a constant, the scheme (3.4)
becomes
(4.4a) $\displaystyle\frac{\rho_{0}^{n+1}-\rho_{0}^{n}}{\Delta
t}+\nabla\cdot(\rho{\bf u})_{0}^{n+1}=0,$ (4.4b) $\displaystyle\frac{(\rho{\bf
u})_{0}^{n+1}-(\rho{\bf u})_{0}^{n}}{\Delta t}+\nabla\cdot(\rho{\bf
u}\otimes{\bf u}+p{\mathbb{I}})_{0}^{n}+\nabla
p^{n+1}_{2}=-\rho^{n}_{0}\nabla\Phi-\rho^{n+1}_{2}\nabla\Phi,$ (4.4c)
$\displaystyle\frac{E_{0}^{n+1}-E_{0}^{n}}{\Delta
t}+\nabla\cdot\left((E_{0}+p_{0}){\bf u}_{0}\right)^{n+1}=-\left(\rho{\bf
u}\right)_{0}^{n+1}\cdot\nabla\Phi,$ (4.4d)
$\displaystyle\frac{\theta_{2}^{n+1}-\theta_{2}^{n}}{\Delta t}+{\bf
u}_{0}^{n}\cdot\nabla\theta_{2}^{n}=0,$
and the EOS (1.6) reduces to (at the time levels $t^{n}$ and $t^{n+1}$)
(4.5) $E_{0}^{n}=\frac{p_{0}^{n}}{\gamma-1},\qquad
E_{0}^{n+1}=\frac{p_{0}^{n+1}}{\gamma-1}.$
We now prove that starting from the well-prepared initial conditions
(4.1)-(4.3) at $n=0$, the limiting scheme (4.4) is a consistent discretization
to (2.8) after one time step. First of all, due to $\nabla
p_{0}^{0}=-\rho_{0}^{0}\nabla\Phi$, (4.4b) becomes
(4.6) $\frac{(\rho{\bf u})_{0}^{1}-(\rho{\bf u})_{0}^{0}}{\Delta
t}+\nabla\cdot(\rho{\bf u}\otimes{\bf u})_{0}^{0}+\nabla
p^{1}_{2}=-\rho^{1}_{2}\nabla\Phi.$
Taking divergence on both sides of (4.6) and using $\nabla\cdot(\rho{\bf
u})^{0}_{0}=0$, we have
(4.7) $\frac{\nabla\cdot(\rho{\bf u})_{0}^{1}}{\Delta
t}+\nabla\cdot(\nabla\cdot(\rho{\bf u}\otimes{\bf
u})_{0}^{0})+\nabla\cdot(\nabla p^{1}_{2}+\rho^{1}_{2}\nabla\Phi)=0.$
As $\varepsilon\rightarrow 0$, the elliptic equation (3.9) becomes
(4.8) $0=\Delta t^{2}\,\nabla\cdot(\nabla\cdot(\rho{\bf u}\otimes{\bf
u})^{0}_{0})+\Delta t^{2}\nabla\cdot\left(\nabla
p_{2}^{1}+\rho_{2}^{1}\nabla\Phi\right),$
since $\widetilde{\rho}-(1-\varepsilon^{2})\,\Delta
t\,\nabla\cdot(\widetilde{\rho{\bf u}})-\rho^{0}_{0}$ in (3.9) approaches
$\Delta t^{2}\,\nabla\cdot(\nabla\cdot(\rho{\bf u}\otimes{\bf u})^{0}_{0})$,
due to (3.5a) and (3.5b). Substituting (4.8) into (4.7) leads to
(4.9) $\nabla\cdot(\rho{\bf u})_{0}^{1}=0,$
and from (4.4a), we obtain $\rho^{1}_{0}=\rho^{0}_{0}$.
Utilizing (4.5), the energy equation (4.4c) becomes
(4.10) $\frac{1}{\gamma-1}\frac{p_{0}^{1}-p_{0}^{0}}{\Delta
t}+\nabla\cdot\left(\frac{\gamma\,p^{1}_{0}}{(\gamma-1)\rho^{1}_{0}}(\rho{\bf
u})^{1}_{0}\right)=-\left(\rho{\bf u}\right)_{0}^{1}\cdot\nabla\Phi.$
If we assume a constant background potential temperature $\theta_{0}$ as in
Section 2, from $p=(\rho\theta)^{\gamma}$, we have
$p_{0}=C\,\rho^{\gamma}_{0}$ with $C=\theta_{0}^{\gamma}$, which leads to
(4.11)
$\nabla\left(\frac{p^{1}_{0}}{\rho_{0}^{1}}\right)=C\nabla(\rho^{1}_{0})^{(\gamma-1)}=C(\gamma-1)(\rho^{1}_{0})^{\gamma-2}\nabla\rho_{0}^{1},$
and using (4.9), we have
(4.12)
$\displaystyle\nabla\cdot\left(\frac{\gamma\,p^{1}_{0}}{(\gamma-1)\rho^{1}_{0}}(\rho{\bf
u})^{1}_{0}\right)$ $\displaystyle=\frac{\gamma}{\gamma-1}\left((\rho{\bf
u})^{1}_{0}\cdot\nabla\left(\frac{p^{1}_{0}}{\rho^{1}_{0}}\right)+\frac{p^{1}_{0}}{\rho^{1}_{0}}\nabla\cdot(\rho{\bf
u})^{1}_{0}\right)$ $\displaystyle=C\gamma(\rho^{1}_{0})^{\gamma-2}(\rho{\bf
u})^{1}_{0}\cdot\nabla\rho_{0}^{1}.$
And $\nabla p^{1}_{0}=-\rho^{1}_{0}\nabla\Phi$ yields
(4.13) $-\left(\rho{\bf u}\right)_{0}^{1}\cdot\nabla\Phi=(\rho{\bf
u})^{1}_{0}\cdot\frac{\nabla
p^{1}_{0}}{\rho^{1}_{0}}=C\gamma(\rho^{1}_{0})^{\gamma-2}(\rho{\bf
u})^{1}_{0}\cdot\nabla\rho_{0}^{1},$
therefore, the equation (4.10) leads to $p^{1}_{0}=p^{0}_{0}$. Note that the
update of density equation in (4.4a) yields $\rho^{1}_{0}=\rho^{0}_{0}$, and
the update of energy equation in (4.4c) yields $p^{1}_{0}=p^{0}_{0}$, which is
consistent with the assumption $\nabla p_{0}^{1}=-\rho^{1}_{0}\nabla\Phi$ at
the next time level.
From (4.9), (4.6) and (4.4d), we observe that the limiting scheme (4.4) is a
consistent discretization to (2.8) at the next time level, as well as
$p^{1}_{0}=p^{0}_{0}$ and $\rho^{1}_{0}=\rho^{0}_{0}$. By the mathematical
induction, we can prove that the same conclusion holds for all later time
levels in a similar way, under the assumptions of the expansion (4.1)-(4.2) at
any later time level $t^{n}$, which confirms the AP property of our scheme. ∎
### 4.2. Asymptotically accurate
The AP property can ensure the first-order scheme (3.4) to be a consistent
discretization of the limiting equation (2.8) when $\varepsilon\rightarrow 0$.
The high order scheme (3.20)-(3.23) is named AA [35, 12, 13], if it is AP and
preserves the high order temporal accuracy as $\varepsilon$ goes to zero.
###### Theorem 4.2.
Consider the high order semi-implicit IMEX-RK scheme (3.20)-(3.23) with space
continuous, and use a $k$-th order stiffly accurate IMEX-RK scheme (3.19).
Under the same assumptions for the AP property, for the solutions after one
time step, we have
$\lim_{\varepsilon\to 0}\theta^{1}({\bf
x};\varepsilon)=\theta_{0},\,\lim_{\varepsilon\to
0}\nabla\cdot\Big{(}\rho^{1}({\bf x};\varepsilon){\bf u}^{1}({{\bf
x}};\varepsilon)\Big{)}=0,\,\lim_{\varepsilon\to 0}(\nabla p^{1}({\bf
x};\varepsilon)+\rho^{1}({\bf x};\varepsilon)\nabla\Phi)=0.$
If we denote ${\bf V}^{1}({\bf x};\varepsilon)=\big{(}\rho^{1}({\bf
x};\varepsilon),\rho^{1}({\bf x};\varepsilon){\bf u}^{1}({\bf
x};\varepsilon),p^{1}({\bf x};\varepsilon),\theta_{2}^{1}({\bf
x};\varepsilon)\big{)}$ and let ${\bf V}^{e}({\bf x},t)=\big{(}\rho^{e}({\bf
x},t),\rho^{e}({\bf x},t){\bf u}^{e}({\bf x},t),p^{e}({\bf
x},t),\theta_{2}^{e}({\bf x},t)\big{)}$ be the exact solutions of (2.8), one
has the one-step error estimate
(4.14) $\lim_{\varepsilon\to 0}{\bf V}^{1}({\bf x};\varepsilon)={\bf
V}^{e}({\bf x},\Delta t)+\mathcal{O}(\Delta t^{k+1}),$
namely, the high order semi-implicit scheme is AA.
The proof follows from the mathematical induction and similar derivations of
the AP analysis. A detailed proof is provided in the supplementary material.
## 5\. Numerical tests
In this section, we will perform some numerical tests for the all-Mach full
Euler equations with gravity, where the Mach number ranges from $0$ to
${\mathcal{O}}(1)$. For one-dimensional (1D) problems, we consider the
vertical direction with coordinate denoted by $x$. For two-dimensional (2D)
problems, we use the coordinate $(x,y)$ with $y$ being the vertical direction,
unless specified otherwise.
The proposed method was outlined in Section 3.4. The third order SA IMEX-RK
scheme in [13, Section 3.2.2] will be used as temporal discretization, and the
fifth order finite difference WENO scheme [36] is adopted in space. The time
step is taken to be $\Delta t=\text{CFL}\,\Delta x/\Lambda$ with
$\Lambda=\max\\{|{\bf u}|+\min(1,1/\varepsilon)\sqrt{{\gamma p}/{\rho}}\\}$,
and we set $\text{CFL}=0.2$. Our proposed scheme is denoted as the “IMEX”
scheme in the following. We will compare our scheme to an explicit fifth order
well-balanced finite difference WENO scheme, as was developed in [32], which
is denoted as “WB-Xing”. We will demonstrate that our approach is high order
accurate, AP and AA, as well as that it is more computationally efficient as
compared to WB-Xing in the low Mach regime.
### 5.1. 1D accuracy test
We first consider the following 1D example with a linear gravitational field
$\Phi_{x}=1$. The hydrostatic steady state satisfying (2.6) is given by
(5.1)
$\rho_{0}(x)=\left(1-\frac{\gamma-1}{\gamma}x\right)^{\frac{1}{\gamma-1}},\quad
p_{0}(x)=\left(1-\frac{\gamma-1}{\gamma}x\right)^{\frac{\gamma}{\gamma-1}},\quad\theta_{0}(x)=1,$
with the adiabatic index $\gamma=1.4$. The corresponding perturbations are
also at a steady state, where
$\rho_{2}(x)=1+0.2\sin(\pi x),\qquad p_{2}(x)=4.5-x+0.2\cos(\pi x)/\pi,$
so that we have steady-state well-prepared solutions for (3.1)
$\rho(x)=\rho_{0}(x)+\varepsilon^{2}\rho_{2}(x),u(x)=0,p(x)=p_{0}(x)+\varepsilon^{2}p_{2}(x),\theta_{2}(x)=\frac{1}{\rho}\left(\frac{\rho_{0}\theta_{0}}{\gamma
p_{0}}p_{2}-\rho_{2}\theta_{0}\right).$
Here the computational domain is $\Omega=[0,2]$. We divide $\Omega$ into $N$
uniform cells, where $N=8\times 2^{i}\,(i=1,2,\ldots,5)$. The errors are
computed by comparing the numerical solution to the exact solution at a final
time $T=0.1$. We show the $L_{1}$ errors and orders for the density $\rho$ in
Table 5.1, by taking four different global Mach numbers
$\varepsilon=1,10^{-2},10^{-4}$ and $0$. We observe that our scheme has at
least fifth order accuracy for all $\varepsilon$’s, as the spatial error is
dominant and the solutions reach the steady state. The convergence orders of
$\rho u$ and $E$ are similar, and we omit them to save space.
Table 5.1. Example 5.1. $L_{1}$ errors and orders of $\rho$ with $\varepsilon=1,10^{-2},10^{-4},0$. $T=0.1$. N | $\varepsilon=1$ | $\varepsilon=10^{-2}$ | $\varepsilon=10^{-4}$ | $\varepsilon=0$
---|---|---|---|---
error | order | error | order | error | order | error | order
16 | 9.19E-05 | – | 1.20E-08 | – | 1.33E-08 | – | 1.33E-08 | –
32 | 2.94E-06 | 4.96 | 2.94E-10 | 5.35 | 9.63E-11 | 7.11 | 9.65E-11 | 7.11
64 | 8.90E-08 | 5.05 | 5.80E-12 | 5.66 | 4.25E-12 | 4.50 | 4.25E-12 | 4.51
128 | 2.74E-09 | 5.02 | 8.99E-14 | 6.01 | 6.03E-14 | 6.14 | 6.04E-14 | 6.14
256 | 8.52E-11 | 5.01 | 1.64E-15 | 5.77 | 7.30E-16 | 6.37 | 7.33E-16 | 6.36
### 5.2. 1D shock tube problem
In this example, we consider a 1D shock tube problem in a high Mach regime
where we take $\varepsilon=0.9$, and the initial conditions are given by
$(\rho,u,p)=(1,0,1),\quad\text{ if }x<0.5;\qquad(0.125,0,0.1),\quad\text{
otherwise; }$
which has been studied in [13] without the gravitational source. Here we
consider a linear gravity with $\Phi_{x}=1$. The background steady state is
still (5.1). The computational domain is $[0,1]$, with inflow and outflow
boundary conditions on the left and right boundaries respectively. We take
$N=200$ and show the results at $T=0.1$ in Fig. 5.1. We can observe that the
IMEX numerical solutions match those of WB-Xing very well, and both schemes
work well in the high Mach regime when the shock exists.
Figure 5.1. Example 5.2. Numerical solutions of $\rho$, $\rho u$ and $p$ with
$N=200$ at $T=0.1$.
### 5.3. 2D accuracy test
In this example, we consider a smooth 2D problem with a linear gravitational
field $(\Phi_{x},\Phi_{y})=(1,1)$. The steady state is given by
$\rho_{0}(x,y)=\left(1-\frac{\gamma-1}{\gamma}(x+y-2)\right)^{\frac{1}{\gamma-1}},\,\,p_{0}(x,y)=\left(1-\frac{\gamma-1}{\gamma}(x+y-2)\right)^{\frac{\gamma}{\gamma-1}},$
with $\theta_{0}(x,y)=1$ and the adiabatic index $\gamma=1.4$. Similar to
Example 5.1, the perturbations are also at a steady state
$\rho_{2}(x,y)=1+0.2\sin(\pi(x+y)),\quad
p_{2}(x,y)=4.5-x+0.2\cos(\pi(x+y))/\pi,$
so that the steady-state exact solutions are
$\rho(x,y)=\rho_{0}(x,y)+\varepsilon^{2}\rho_{2}(x,y),p(x,y)=p_{0}(x,y)+\varepsilon^{2}p_{2}(x,y),\theta_{2}=\frac{1}{\rho}\left(\frac{\rho_{0}\theta_{0}}{\gamma
p_{0}}p_{2}-\rho_{2}\theta_{0}\right).$
Here ${\bf u}=(u_{0},-u_{0})$ which is perpendicular to the gravitational
field, and we take $u_{0}=1$. The computational domain is $\Omega=[0,2]^{2}$,
which is divided into $N^{2}$ uniform cells, with $N=8\times
2^{i}\,(i=1,2,\ldots,5)$. The errors are computed by comparing the numerical
solution to the exact solution at the final time $T=0.5$. Similarly we show
the $L_{1}$ errors and orders of $\rho$ in Tables 5.2, with four different
global Mach numbers $\varepsilon=1,10^{-2},10^{-4}$ and $0$. We can observe
the expected convergence rates for all $\varepsilon$’s, similar to the 1D
case.
Table 5.2. Example 5.3. $L_{1}$ errors and orders of $\rho$ with $\varepsilon=1,10^{-2},10^{-4},0$. $T=0.05$. $N^{2}$ uniform cells. $N$ | $\varepsilon=1$ | $\varepsilon=10^{-2}$ | $\varepsilon=10^{-4}$ | $\varepsilon=0$
---|---|---|---|---
error | order | error | order | error | order | error | order
$16$ | 4.86E-04 | – | 1.83E-06 | – | 1.83E-06 | – | 1.83E-06 | –
$32$ | 3.70E-05 | 3.72 | 8.17E-08 | 4.49 | 8.14E-08 | 4.49 | 8.14E-08 | 4.49
$64$ | 1.48E-06 | 4.65 | 2.32E-09 | 5.14 | 2.31E-09 | 5.14 | 2.31E-09 | 5.14
$128$ | 5.51E-08 | 4.74 | 5.48E-11 | 5.41 | 5.41E-11 | 5.42 | 5.41E-11 | 5.42
$256$ | 1.38E-09 | 5.32 | 1.09E-12 | 5.66 | 1.06E-12 | 5.67 | 1.06E-12 | 5.67
### 5.4. Traveling vortex
This test has been considered for the 2D shallow water equations [22]. In
(3.1), if we assume $\theta=1$ and take $c_{p}=2,c_{v}=1$, then the specific
gas constant is $R=c_{p}-c_{v}=1$, and $\gamma=c_{p}/c_{v}=2$, it is
equivalent to the shallow water equations and in this case the pressure is
$p=\rho^{2}/2$ with $\varepsilon=0.05$. Following the settings in [22], for
the source term we take
$\Phi(x,y)=e^{-5(x-1)^{2}},$
namely, the gravity is along the $x$-direction for this example. Other initial
conditions are given by
$\displaystyle\rho(x,y,0)=110-\Phi+\left\\{\begin{aligned}
&\left(\frac{\varepsilon\Gamma}{\omega}\right)^{2}(k(\omega\Gamma_{c})-k(\pi)),\quad&\text{if
}\omega\Gamma_{c}\leq\pi;\\\ &0,\quad&\text{otherwise},\end{aligned}\right.$
$\displaystyle u(x,y,0)=2+\left\\{\begin{aligned}
&\Gamma(1+\cos(\omega\Gamma_{c}))(0.5-y),\quad&\text{if
}\omega\Gamma_{c}\leq\pi;\\\ &0,\quad&\text{otherwise},\end{aligned}\right.$
$\displaystyle v(x,y,0)=\left\\{\begin{aligned}
&\Gamma(1+\cos(\omega\Gamma_{c}))(x-0.5),\quad&\text{if
}\omega\Gamma_{c}\leq\pi;\\\ &0,\quad&\text{otherwise},\end{aligned}\right.$
with $\Gamma_{c}=\sqrt{(x-0.5)^{2}+(y-0.5)^{2}},\,\Gamma=8,\,\omega=4\pi,$ and
$k(\xi)=2\cos(\xi)+2\xi\sin(\xi)+\frac{1}{8}\cos(2\xi)+\frac{\xi}{4}\sin(2\xi)+\frac{3}{4}\xi^{2}.$
The hydrostatic steady state takes the form
$\rho_{0}(x,y)=110-\Phi,\qquad\theta_{0}=1,\qquad
p_{0}(x,y)=\frac{1}{2}\left(110-\Phi\right)^{2}.$
Here the computational domain is $[0,2]\times[0,1]$, with periodic boundary
conditions. We use this example to demonstrate that the new modeling (3.1) and
our proposed scheme can have similar good performances as in [22] for this
special case. We strictly follow our scheme in Section 3.4 with a mesh grid
$200\times 100$ and show the perturbation of density $\rho-\rho_{0}$ in Fig.
5.2. We can observe that the numerical results are very similar to those
obtained in [22], which demonstrates the nice performance of the proposed AP
method in the low Mach regime.
Figure 5.2. Example 5.4. Numerical solutions for the perturbation of density
$\rho-\rho_{0}$. Mesh $200\times 100$. From left to right: $t=0.3,0.6,1.0$,
respectively.
### 5.5. 2D isothermal equilibrium
This example is designed to test the well-balanced property of a given scheme,
as well as its ability to capture small disturbances around an equilibrium
state [41]. We consider an isothermal equilibrium state
(5.2)
$\rho(x,y)=\bar{\rho}\exp\left(-\frac{\bar{\rho}g}{\bar{p}}(x+y)\right),\,{\bf
u}=0,\,p(x,y)=\bar{p}\exp\left(-\frac{\bar{\rho}g}{\bar{p}}(x+y)\right),$
with $\bar{\rho}=1.21$, $\bar{p}=1$ and $g=1$. The linear gravitational field
with $(\Phi_{x},\Phi_{y})=(1,1)$ is taken. The adiabatic index is $\gamma=1.4$
and the global Mach number is set to be $\varepsilon=0.9$. The problem is
defined on a unit square $[0,1]^{2}$. We run the simulation up to the final
time $T=1$, on three different meshes $50\times 50$, $100\times 100$ and
$200\times 200$. We compare the numerical solutions to the steady-state exact
solutions. It was observed that both IMEX and WB-Xing can preserve the
solutions up to machine errors, and a well-balanced property is achieved for
both schemes. We omit the results here to save space.
Next, we add a small perturbation to the pressure
$p_{2}(x,y)=\frac{1}{810}\exp\left(-\frac{100\bar{\rho}g}{\bar{p}}((x-0.3)^{2}+(y-0.3)^{2}\right)$
which is a small Gaussian hump centered at $(0.3,0.3)$. The other terms are
kept to be the same. We take a mesh grid $100\times 100$ and compute the
numerical solution up to $T=0.15$. Here a transmissive boundary condition [39]
is used. The numerical results are shown in Fig. 5.3, and again we can observe
that both schemes can capture the small perturbations very well in this high
Mach regime.
Figure 5.3. Example 5.5. Numerical solutions of the pressure perturbation and
the density perturbation at $T=0.15$ with $N\times N=100\times 100$. Left: WB-
Xing; middle: IMEX; right: cuts along the line $y=0.305\,m$. Top: pressure
perturbation; bottom: density perturbation.
###### Remark 5.1.
For this 2D isothermal equilibrium example, numerical boundary conditions are
treated as follows. We split our main variable $U=(\rho,\rho{\bf
u},E,\theta)^{T}$ into two parts: a hydrostatic one $U^{0}$ and a perturbation
$U^{\prime}$, that is, $U=U^{0}+U^{\prime}$ with
$U^{0}=\left(\rho_{0},{\bf
0},\frac{p_{0}}{\gamma-1},\theta_{0}\right)^{T},\qquad
U^{\prime}=\left(\varepsilon^{2}\rho_{2},\rho{\bf
u},E-\frac{p_{0}}{\gamma-1},\varepsilon^{2}\theta_{2}\right)^{T}.$
We apply a transmissive boundary condition [39] to the perturbation
$U^{\prime}$, i.e., ghost values of $U^{\prime}$ are assigned by mirror
symmetry. Ghost values of the hydrostatic component $U^{0}$ are obtained via
extrapolation. The same boundary treatments are also applied to the following
two examples.
### 5.6. Rising thermal bubble
This is a benchmark test problem for atmospheric flows, simulating the
dynamics of a warm bubble, which has been studied in [18, 19, 41]. Here we
consider a linear gravitational field in the vertical direction with
$\Phi_{x}=0$ and $\Phi_{y}=g$, and the gravitational constant is
$g=9.8m/s^{2}$. The Exner pressure takes the form
$\Pi=1-(\gamma-1)gy/\gamma/R/\bar{T}$, with $\bar{T}=300\,K$, $\gamma=1.4,\,$
$R=287.058\,\text{J/kg K}$ being the gas constant, and the potential
temperature is
$\theta(x,y,0)=\theta_{0}+\Delta\theta(x,y,0)=300\,K+\left\\{\begin{aligned}
&0,&r>r_{c};\\\ &\frac{\theta_{c}}{2}(1+cos(\pi\,r/r_{c}))&r\leq
r_{c}.\end{aligned}\right.$
Here $r=\sqrt{(x-x_{c})^{2}+(y-y_{c})^{2}}$, $\theta_{c}=0.5\,K$,
$(x_{c},y_{c})=(500,350)\,m$ and $r_{c}=250\,m$, so that the initial condition
can be denoted as
$\rho(x,y,0)=\frac{\bar{p}}{R\theta}\,\Pi^{\frac{1}{\gamma-1}},\quad{\bf
u}(x,y,0)=(0,0),\quad p(x,y,0)=\bar{p}\,\Pi^{\frac{\gamma}{\gamma-1}},$
with the hydrostatic steady state
$\rho_{0}(x,y)=\frac{\bar{p}}{R\,\theta_{0}}\,\Pi^{\frac{1}{\gamma-1}},\qquad\theta_{0}(x,y)=300\,K,\qquad
p_{0}(x,y)=\bar{p}\,\Pi^{\frac{\gamma}{\gamma-1}}.$
where $\bar{p}=10^{5}N/m^{2}$ is a reference pressure at $y=0\,m$, and the
computational domain is $[0,1000]\times[0,1000]\,m^{2}$.
This problem can be rewritten into a dimensionless form by choosing the
reference values
$\displaystyle p_{ref}=10^{5}\,N/m^{2},\quad\rho_{ref}=10\,kg/m^{3},\quad
t_{ref}=10^{3}\,s,\quad l_{ref}=10^{3}\,m,$ $\displaystyle
U_{ref}=\frac{l_{ref}}{t_{ref}}=1\,m/s,\qquad\theta_{ref}=\frac{p_{ref}}{R\rho_{ref}}=\frac{10^{4}}{R}\,K,$
so that the global Mach number is
$\varepsilon=\frac{U_{ref}}{\sqrt{p_{ref}/\rho_{ref}}}=10^{-2}.$
We run the simulation for a very long time until the final stopping time
$t=700s$. We show the perturbation of potential temperature $\Delta\theta$ on
a mesh gird of $200\times 200$ in Fig. 5.4. Inviscid wall boundary conditions
[41] are used. We can see that both schemes can correctly capture the shear
movement of the rising bubble, which is along the opposite direction of
gravity, and eventually, it forms a shape of a mushroom cloud. For this
example with smooth solutions, we only use component-wise WENO reconstruction
with linear weights for saving computational cost. However, comparing the WB-
Xing scheme with our IMEX scheme, our scheme is much less diffusive on the
same mesh grid, as the explicit WB-Xing scheme has a numerical viscosity
inversely proportional to the Mach number $\varepsilon$, while our IMEX scheme
does not. We also compare the CPU cost of two schemes in Table 5.3, from which
we can observe that our IMEX scheme is much more efficient for this low Mach
problem.
Figure 5.4. Example 5.6. Numerical results for the perturbation of potential
temperature $\Delta\theta$. From left to right, $t=400s,600s,700s$,
respecively. Top: WB-Xing; Bottom: IMEX. Mesh: $200\times 200$.
### 5.7. Inertia-gravity wave
This test arises from atmospheric flows, and has also been studied in [18, 19,
41]. The computational domain is set as $[0,300000]\times[0,10000]\,m^{2}$,
with a periodic boundary condition in the $x$ direction and an inviscid wall
boundary condition in the $y$ direction. The linear gravitational field with
$(\Phi_{x},\Phi_{y})=(0,g)$ and $g=9.8\,m/s^{2}$ is considered. The Exner
pressure is
$\Pi=1+\frac{(\gamma-1)g^{2}}{\gamma
RT_{0}\mathscr{N}^{2}}\biggl{[}\exp\biggl{(}-\frac{\mathscr{N}^{2}}{g}y\biggr{)}-1\biggr{]},$
and the initial potential temperature is denoted as
$\theta(x,y,0)=\theta_{0}(y)+\Delta\theta(x,y,0),$
where
$\theta_{0}(y)=\bar{T}\exp\biggl{(}\frac{\mathscr{N}}{g}y\biggr{)}\,K,\quad\Delta\theta(x,y,0)=\theta_{c}\sin\left(\frac{\pi
y}{h_{c}}\right)\biggl{[}1+(x-x_{c})^{2}/a_{c}^{2}\biggr{]}^{-1},$
with the Brunt–Väisälä frequency $\mathscr{N}=0.01/s$, $\bar{T}=300\,K$,
$\theta_{c}=0.01\,K$, $h_{c}=10000\,m$, $x_{c}=100000\,m$ and $a_{c}=5000\,m$.
The initial conditions are given by
$\rho(x,y,0)=\frac{\bar{p}}{R\theta}\,\Pi^{\frac{1}{\gamma-1}},\quad{\bf
u}(x,y,0)=(20,0)\,m/s,\quad p(x,y,0)=\bar{p}\,\Pi^{\frac{\gamma}{\gamma-1}},$
and the equilibrium state is
$p_{0}=\bar{p}\,\Pi^{\frac{\gamma}{\gamma-1}},\qquad\theta_{0}=\bar{T}\exp\biggl{(}\frac{\mathscr{N}}{g}y\biggr{)}\,K,\qquad\rho_{0}=\frac{\bar{p}}{R\theta_{0}}\Pi^{\frac{1}{\gamma-1}},$
with $\bar{p}=10^{5}\,N/m^{2}$. As in Example 5.6, we consider the problem in
a dimensionless form, by choosing the following reference values
$\displaystyle p_{ref}=10^{5}N/m^{2},\quad\rho_{ref}=10^{-1}kg/m^{3},\quad
t_{ref}=10^{5}m/s^{2},\quad l_{ref}=10^{5}m,$ $\displaystyle
U_{ref}=\frac{l_{ref}}{t_{ref}}=1\,m/s,\qquad\theta_{ref}=\frac{p_{ref}}{R\rho_{ref}}=\frac{10^{6}}{R}\,K,$
and the corresponding global Mach number is
$\varepsilon=\frac{U_{ref}}{\sqrt{p_{ref}/\rho_{ref}}}=10^{-3}.$
In this example, for the WB-Xing scheme, an HLLC flux which is less diffusive
than the Lax-Friedrichs flux, is used in the simulation. We run the solution
up to $t=3000s$ for both schemes. For this example, we take CFL=$0.01$ for our
IMEX scheme, and CFL=$0.005$ for the WB-Xing scheme. We show the perturbation
of potential temperature $\Delta\theta$ on the mesh grid $400\times 50$ and
$800\times 50$ in Fig. 5.5 and Fig. 5.6. From these numerical solutions,
especially the cuts in Fig. 5.6, we can see the IMEX scheme has better
resolution than the WB-Xing scheme on the coarse grid. We also compare the CPU
cost of two schemes in Table 5.3, and observe that the IMEX scheme is much
more efficient in this low Mach regime.
Figure 5.5. Example 5.7. Numerical results for the perturbation of potential
temperature $\Delta\theta$ at $t=3000s$. Mesh grid: $400\times 50$ (left);
$800\times 50$ (right). Top: WB-Xing; bottom: IMEX.
Figure 5.6. Example 5.7. Numerical results for the perturbation of potential temperature $\Delta\theta$ along the line $y=4900\,m$ at $t=3000s$. Mesh grid: $400\times 50$ (left); $800\times 50$ (right). Table 5.3. CPU cost (seconds) for the WB-Xing and IMEX schemes. | $N_{x}\times N_{y}$ | WB-Xing | IMEX
---|---|---|---
Example 5.6 | $100\times 100$ | 7,328.77 | 1,354.02
$\varepsilon=10^{-2}$ | $200\times 200$ | 55,719.46 | 11,712.07
Example 5.7 | $400\times 50$ | 300,512.70 | 12,578.01
$\varepsilon=10^{-3}$ | $800\times 50$ | 585,356.01 | 32,064.40
## 6\. Conclusion
In this work, we designed a high order semi-implicit AP well-balanced finite
difference WENO scheme for the all Mach full Euler system with a gravitational
field. It is much more challenging than the shallow water equations with a
non-flat bottom [22], due to the existence of gravity which couples all
equations together in the limit. We proposed to add the evolution for the
perturbation of potential temperature in the design of our scheme, which
provides a general and easy framework for the development of AP schemes to
ensure the correct incompressible limit. The AP and AA properties are formally
analyzed. Numerical tests on both 1D and 2D problems have demonstrated the
high order accuracy, well-balanced, AP and AA properties of our proposed
scheme. Our IMEX scheme has also been shown to yield better resolution and
higher efficiency in the low Mach regime when compared with the explicit
scheme.
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|
# FREDSR: Fourier Residual Efficient Diffusive GAN for Fast Single Image Super
Resolution
Kyoungwan Woo1 * Achyuta Rajaram 2 *
###### Abstract
_FREDSR is a GAN variant that aims to outperform traditional GAN models in
specific tasks such as Single Image Super Resolution with extreme parameter
efficiency at the cost of per-dataset generalizeability. FREDSR integrates
fast Fourier transformation, residual prediction, diffusive discriminators,
etc to achieve strong performance in comparisons to other models on the
UHDSR4K dataset for Single Image 3x Super Resolution from 360p and 720p with
only 37000 parameters. The model follows the characteristics of the given
dataset, resulting in lower generalizeability but higher performance on tasks
such as real time up-scaling._
## 1 Introduction
Image to Image Translation(I2I) is a highly general meta-task that can
summarize most application-specific subtasks in image generation or
processing. Broadly, we use the definition from [1], where we seek to convert
an an input image from domain A to domain B, while preserving ”intrinsic
source content”, i.e. the meaningful semantic features of the image, while
converting only the ”extrinsic style”, pertaining to the specific domain. This
process can be defined as a mapping that synthesizes images that are
indistinguishable from a sample from the target distribution B, using the
semantic information present from a given sample of distribution A. This
definition can generalize to many problems in image processing, with past work
being used for image synthesis [2], image segmentation [3], style transfer
[4], image inpainting [5], image super-resolution [6], among many other
applications.
Given the formulation of the I2I problem stated above, it seems natural that
Generative Adversarial Networks(GAN) would be applicable for these tasks. GANs
seek to generate images indistinguishable from those in a given distribution
by playing a mini-max game between a discriminator and generator, where the
generator seeks to decrease the discriminator’s accuracy in determining if the
outputs of the generator belong to a specified distribution of images, while
the discriminator seeks to maximise said accuracy [7]. This methodology
circumvents the problems of using a pixel-level loss, or other naive
supervised losses, as when they are used, blurry images are creates due to
averaging of the multiple plausible solutions for a given translation.
As a result of this natural applicability to GANs, past research has sought to
apply GAN architectures to I2I translation problems, such as the pix2pix
network, which pioneered the development of GANs for I2I translation [8].
Pix2pix allowed for high-quality low-resolution I2I translation to be computed
without the need for problem-specific hand crafted loss functions. However,
the initial formulation has not been crafted for high-resolutions, with [9]
stating that conditional GANs such as pix2pix can struggle to generate high
definition images, due to training instability. Furthermore, they indicate
that including Perceptual Loss, an error function based on the outputs of
classification networks, that seek to perform large-scale feature extraction,
has significant benefits. Using this as a supervised objective allowed for
stable images and was an improvement over raw pixel-level objectives.
More recently, an improvement to pix2pix has been suggested, referred to as
pix2pixHD, combining classifier-based perceptual loss with discriminator based
loss in order to allow for high resolution image generation [10]. However,
this model requires 3 discriminators to allow for the processing of
information at every scale, which creates large increases to the computational
resources required during training. Furthermore, using several discriminators
increases learning instability, as matching gradient updates between the
Generator and Discriminator is one of the largest challenges in GAN training,
as not doing so properly leads to common GAN failure modes such as
Catastrophic Forgetting and Mode Collapse [11].
An alternative to this architecture, optimized for large mask high resolution
image inpainting, uses Fast Fourier Convolutions (FFCs) to allow for efficient
processing of global-scale information along with the local-scale details
[5],[12]. With this prior, we contribute:
i. Using the previously introduced FFC block, we create a novel model
architecture for I2I translation, adapting the original encoder-decoder stack
from pix2pix. The use of the FFC allows the model to quickly gain access to
global information. This allows for the model to operate on extrinsic global
information throughout its entirety, improving its parameter efficiency for
translation tasks. Additionally, this architecture is easily adaptable for
Single Image Super Resolution(SISR) problems, and the maintaining of global
information proves crucial to the performance of this architecture.
ii. We extend the diffusive discriminator from [13], which seeks to combat
discriminator overfitting by using a tool from the highly successful de-
noising Diffusion Probabilistic Model architecture [14], adding diffusion from
a Gaussian mixture distribution to the discriminator inputs, by using a
residual discriminator design.
Overall, we use all of these techniques to create a novel GAN variant, which
experimentation has shown to be performant for specialized SISR tasks, while
having much higher parameter efficiency in orders of magnitude. We motivate
the existence of such a technique in the application of real-time 3D
rendering, such as those found in modern video games. Modern gaming requires
high resolution rendering of complicated 3D meshes with potentially billions
of polygons, as well as expensive processes such as ray-traced per pixel
lighting, ambient occlusion, and motion blur. Performance in this application
has traditionally scaled harshly with output resolution, due to a quadratic
increase in total calculations. Thus, a real-time application of SISR can
provide efficiency benefits. One of the examples of utilizing unsupervised
learning techniques for this task was introduced by NVIDIA with their Deep
Learning Super Sampling application. FREDSR follows a similar notion of
extremely performant, yet specialized SISR problem task, and was optimized for
high performance on a specific dataset with extreme parameter efficiency.
## 2 Method
We seek to develop a novel Single Image Super Resolution(SISR) architecture
where we take a three channel input image high resolution and low resolution
pair $q_{hr}$ and $q_{lr}$. We propose a model architecture that takes in
$q_{lr}$ and outputs a three channel image $q_{h}$, which has the same
resolution as $q_{hr}$.
Figure 2: The model architecture of the FREDSR generator. This network uses
the recently proposed FFCs [12], along with a complex loss function that
combines adversarial, perceptual, and pixel-level losses. Graphic style was
inspired by [5].
### 2.1 Maintaining global information through all layers
Challenging problems in I2I translation, including that of large-scale
modification to high-resolution data, requires processing of global
information. We argue that for successful high-resolution I2I translation, a
high receptive field that encapsulates global information, and maintains it
throughout the network is necessary in order to efficiently process global
extrinsic parameters on an image, such as the time of day, lighting
conditions, the artistic style associated with a painter, etc. Traditional
fully convolutional models suffer from slow growth of effective receptive
field, due to the small kernels used [15]. To improve on this, [16] proposed
the dilated convolution, which introduces ”gaps” between the values of a
convolution, ”spreading out” a 3x3 convolution, for example over an 8x8 area,
while only using 9 parameters. However, this loses some local information
present, meaning that a complex architecture that combines these with
traditional convolutions is necessary in order to achieve usable results, as
was seen in the I2I density map generation network proposed in [17].
Additionally, even the introduction of such dilated convolutions only slightly
increases the receptive field. Thus, as a result, improvement is possible,
which comes by way of the Fast Fourier Convolution (FFC)[12], based on the
Fast Fourier Transform [18].
FFC splits information into two main channels, global and local, where the
local information is processed using traditional convolutions, and the global
information is processed using a real FFT2d. During each FCC block, the
information of both branches are combined to allow for more efficient use of
parameters. An illustration of the exact construction of the FFC block can be
seen in Figure 2. This allows for the efficient use of parameters. This unit
are fully differentiable, and we show them in used as a drop-in replacement
for traditional convolutions. Additionally, FFCs have been showed to be well
suited to capture periodic structures, which are extremely common both in
natural and artificial environments, such as brick buildings or waves on the
surface of water [5].
### 2.2 Model Architecture
Here, we introduce our novel I2I architecture, and the modifications necessary
for high resolution SISR tasks with an extremely efficient network. We have
already described the foundational units of this architecture, the FFC block.
This architecture is illustrated in Figure 2.
#### SISR Generator
A traditional I2I network cannot perform SISR, due to the differing
resolutions of the input and output. However, our architecture is easily
adaptable to perform this task. We perform this adaption based off of the
foundation of the VDSR model proposed in [19]. to begin, we upscale the low-
resolution input by using a bicubic upscaler. Then we use our model to predict
the difference in value between the input bicubic upscaled image, and the true
image. This process is visualized in figure 2. This model has several
advantages over other super-resolution networks, as the FFC module allow for
the more efficient use of parameters, preservation of low-frequency features
maintained through the upscaling. Additionally the residual learning allows
for much more efficient use of parameters and faster training, as no training
time or parameters are ”wasted” learning to copy the original image. This
allows our model to achieve competitive performance to SOTA methods with with
only 37115 total parameters on the specialized tasks.
Figure 3: The model architecture of the FREDSR discriminator. This was based
on a traditional discriminator, with special modifications for the super-
resolution task. This discriminator uses diffusion applied to its inputs,
among other tactics to improve discriminator performance and prevent
overfitting. Graphic style was inspired by [5].
#### SISR Discriminator
We allow the discriminator to train to distinguish between the residuals, also
known as the differences between the upscaled images and the true images, and
our model’s predicted residuals. This architecture is visualized in full in
figure 3.
### 2.3 Generative Loss Functions
In FREDSR, we combine various L1 and L2 losses. The benefits of combining
various level losses have been previously discussed in papers such as [8].
#### SSIM loss
In the case of L2 loss, we utilize the Structural Similarity Index Metric for
the SISR model. In conjunction with the residual discriminator, SSIM works to
deblurr the image. Compared to common L1/L2 losses such as MAE and MSE, SSIM
has been proved to outperform in super resolution tasks, such as in [20]. The
loss function is as follows:
$\mathcal{L}_{SSIM}=-SSIM(x,y)=-\frac{(2\mu_{x}\mu_{y}+C_{1})+(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})}$
(1)
With x,y defined as the generated and real images respectively.
#### Charbonnier Loss
As a method of enforcing color similarity, as well as mixing L1 and L2 losses,
we use the Charbonnier Loss function. This is a variation of MAE which is
differentiable and has shown to mix beneficial properties of L1 and L2 losses,
and has shown significant success.
$\mathcal{L}_{CHARB}=E(\sqrt{(x-y)^{2}+\epsilon})$ (2)
With x,y defined as the generated and real images respectively, and $\epsilon$
a constant chosen.
#### MGE loss
To improve the performance of the model for SISR tasks, we add an additional
loss on top of the global feature reconstruction loss. This loss is focused on
the maintaining of edges, and was introduced for SISR in [21]. The maintaining
of sharp edges allows for a more accurate image to that captured at high
resolution. MGE uses a classical sobel operator, first introduced in [22].
After this, we can compute the gradients for each pixel, then define MGE as
follows.
$\mathcal{L}_{MGE}=E((G(x)-G(y))^{2})$ (3)
With x,y defined as the generated and real images respectively.
#### Adversarial Loss
In the case of adversarial losses, as the model implementation follows a GAN
structure, we utilize the modified MiniMax loss from [7].The loss function is
as follows.
$\mathcal{L}_{adv}=E_{(y_{gen})}[log(D(y_{gen}))]$ (4)
where $y_{gen}$ are corresponding images taken from the original and high-
resolution generated images.
#### Final Generative loss
Additionally to the previous losses, we add a Perceptual Loss, as they have
shown incredible benefits within the training and evaluation of image
generation networks [23],[24],[25],[26]. Combining this with the above loss
functions, we end up with the final generative loss function to be minimized
by gradient descent as:
$\mathcal{L}_{final}=\lambda_{1}*\mathcal{L}_{adv}+\lambda_{2}*\mathcal{L}_{PL}+\lambda_{3}*\mathcal{L}_{MGE}+\lambda_{4}*\mathcal{L}_{SSIM}+\\\
{\lambda_{5}*\mathcal{L}_{CHARB}}$ (5)
with $\lambda_{1}$,$\lambda_{2}$,$\lambda_{3}$,$\lambda_{4}$, $\lambda_{5}$ as
scaling parameters.
### 2.4 Discriminative Loss Functions
#### Modified Minimax Loss
In the case of discriminative losses, as the model implementation follows a
GAN structure, we train the model to recognize real images as greater than
0.5, and fake images as less than 0.5. Thus, our loss is given by:
$\mathcal{L}_{disc}=E_{(x,y)}[log(D(y)]+E_{(y)}[log(1-D(G(x))]$ (6)
where x,y are corresponding images taken from the original and high-resolution
images respectively.
## 3 Experiments
FREDSR was developed on Tensorflow [27] and Keras [28], then trained on
UHDSR4K, a SIRS benchmark dataset from [29]. The images were downscaled using
bicubic algorithms, consistent with the original dataset.
Along with the techniques outlined in Sections 1 and 2, we utilize various
traditional and novel GAN training methods, such as discriminator restart
learning from [30] and variable Gaussian noise, an improved version of
adaptive blur and control from [31]. It is important to note that FREDSR
utilized minimal to no hyperparameter tuning: all $\lambda$ values were set to
equalize the importance of each loss function.
### 3.1 Diffusive Discriminator
A common problem with GAN stability is combating discriminator overfitting,
and several methods have been used to address this, including the use of
random data augmentations which adapt with the discriminator’s performance.
[32] For stable training of GANs, we implement the solution proposed in [13],
using adaptive noise samples from the forward chain of a Gaussian Mixture
distribution. This method has been shown both theoretically and experimentally
to provide stable and data-efficient GAN training, and improve over baselines
for high-quality image generation [13]. In this work, we implement the
diffusive inputs along with our residual discriminator.
### 3.2 Discriminator Restart Learning
Even with diffusion applied to the discriminator inputs, the discriminator
tends to converge faster than the generator, or simply learn the wrong weights
after an extended period of training. By resetting the discriminator learning
rate mid training, or a harsh full weight reinitialization, we ensure that the
discriminator continues learning based on a more converged generator output.
In this reset step, the discriminator receives a learning rate increase. The
generator’s focus on the adversarial loss is lowered as well, to allow the
discriminator to learn more freely. Once the discriminator is near convergence
again, the changes to the coefficients are reversed.
### 3.3 Gaussian Random Noise
Adding randomized noise to each convolution of the generator is a commonly
used tactic in deep learning to increase the generalizability of a model while
decreasing the chance of overfitting. [33] Although our datasets are near the
larger scale, to improve the higher-resolution generalizability of our model
as well as speed up training, we implement Gaussian random noise generations
in between each of our fast fourier convolutions. The gaussian noise levels
should decrease similarly to the generator loss, as in the later phases of
training, randomized data can lead to less favorable outputs.
### 3.4 AdamW Optimizer
We utilize the AdamW optimizer proposed by [34] to combat the weight
regularization introduced by Adam. Adam tends to regularize larger weights
less than smaller weights, and thus does not converge as well as traditional
methods such as stochastic gradient descent. AdamW has been proposed to
improve this issue, and we notice a significant improvement over the Adam
optimizer.
### 3.5 Cosine Annealing Decay Learning Rate
We utilize a cosine annealing decay learning rate scheduler to combine both
decaying learning rate schedulers and warm restart learning rate schedulers
proposed by [35]. Restart learning has been used commonly with stochastic
gradient descent to speed up training processes, and exponential decay has
been used to help models converge.
Figure 4: A graph plot of the learning rate for the SISR generator: Each peak
is five percent lower than the previous, and the learning rate will decay
until halved prior to the next reset.
### 3.6 UHDSR4K Image Super Resolution
For our main analysis, we focus on the super-resolution objective of the
UHDSR4K dataset introduced in [29]. This dataset is the largest-scale UHD
dataset in the field of 4K image super-resolution. For training, we downsample
the original 4K images to 1920x1080. From here, we construct our 3x upsampling
dataset, derived from the original UHDSR4K, by further downsampling to 640x360
resolution, and training our model to perform 3x super-resolution. This
dataset is composed of diverse environments including city scenes, people,
animals, buildings, cars, natural landscapes, and sculptures scraped from the
internet in order to diversify the camera processing applied, along with
allowing for high resolution images from many surroundings. In order to
compute our downsampling, we follow [29] by using bicubic interpolation. For
our metrics, we use standard super-resolution metrics, specifically SSIM and
PSNR [36],[37]. Drawing comparisons using all of these methods allows for
verifying that our model can both perform the super-resolution task and
successfully match pixels, as well as is globally closer to the original
image. It is important to note that our model’s hyperparameters were not fully
tuned, yielding suboptimal performance. We visualize generator loss with
respect to training steps in figure 5.
Figure 5: A partial graph plot of the generator loss over train steps. Note
than our model converges much slower than other models such as VDSR[19]. We
perform training on 12x Nvidia Tesla V100s for over 300 epochs.
#### Comparisons to Baselines
For our comparison, we use several baseline models, which allow for us to
directly benchmark our model’s performance on its original training dataset.
We note that although it would be preferred to train our baseline models on
the UHDSR4K dataset, as we have done with our model, due to time and resource
constraints we were unable to do so. However, as this dataset has high
diversity, we believe that it has coverage of the training distributions of
the baseline models. We test all models at 3x super-resolution on the UHDSR4K
dataset, with our training-time downscaling applied. Thus, we test 3x super
resolution from 640 pixels by 360 pixels to 1920 pixels by 1080 pixels on the
test dataset of UHDSR4K. For our baseline comparison models, we use only
publically available pretrained models. We select 3 strong deep learning
baselines with diverse techniques used. These baselines are EDSR[38],
ESPCN[39], and FSRCNN [40]. For completeness, we also report bilinear and
bicubic upscaling as a control. The results, calculated using the SSIM, and
PSNR metrics as described previously are tabulated in Table 1.
UHDSR4K 640x320 3x Upscaling
---
Method | Params | SSIM $\uparrow$ | PSNR$\uparrow$
Bicubic | 0 | 0.856 | 26.75
Bilinear | 0 | 0.848 | 26.49
EDSR | 43000k | 0.879 | 27.61
ESPCN | 20k | 0.858 | 27.02
FSRCNN | 12k | 0.857 | 27.03
FREDSR | 37k | 0.883 | 27.776
Table 1: Quantitative evaluation of 3x SISR performance on downscaled UHDSR4K
dataset. We report Structural Similarity Index Measure(SSIM) and Peak Signal
to Noise Ratio(PSNR) metrics. FREDSR compares favorably against these
baselines across all metrics.
As shown by this, our method can outperforms the baselines in the task of SISR
on its original training resolution and dataset, strictly outperforming
significantly larger models, indicating high overall performance and parameter
efficiency. Further ablation analysis is necessary in order to determine the
exact root of this high performance, but we believe it can be attributed to
the poor generalization of the other models, high receptive field of FREDSR,
and ability to take advantage of the information present in repetitive
structures in order to perform more accurate prediction.
#### High resolution Performance Across Resolutions
Figure 6: Positive samples for high resolution performance. These are examples
of our model’s performance on the UHDSR4K dataset, in order to display our
model’s high-resolution super-resolution capabilities. The top image is our
model’s output, while the bottom is the true high resolution image. As shown
here, along with our results from table 2, show an interesting improvement in
our models as the resolution improves. This can be attributed to the task of
super-resolution becoming easier at higher resolutions.
Along with our model’s performance on low resolutions, we seek to find out how
it performs on high resolution upscaling. To this end, we study performance on
high resolution upscaling. We point out the high amount of repetitive patches
present in urban scenes, and study the ability of our network to handle
repeated signals deformed by changes in perspective. This signal deformation
has been shown to cause challenges to past networks implementing FFCs [5],
thus we study our model’s performance in this domain. We compare our model
against a variety of strong baseline solutions for upscaling, and train a
model, using the pretrained 360p model as a base, to do 3x super resolution
from 720p to 4k.
#### UHDSR4K
For our full resolution UHDSR4K analysis, we perform 3x upscaling from the
standard resolutions 720p to 4k resolutions. For our comparison, we use the
PNSR and SSIM results from [29], testing our models against SRCNN[Dong2015],
FSRCNN [40],VSDR [19], EDSR[38], RCAN[41], RDN[42], HAN[43], DRLN[44], and
MANet [29]. We report PSNR and SSIM results from [29] along with our results
and compare our models generalization performance at higher resolution against
models that have been natively trained on upscaling the higher resolution. We
also include Bilinear and Bicubic upscaling for control.
UHDSR4K 1280x720 3x Upscaling
---
Method | Params | SSIM $\uparrow$ | PSNR$\uparrow$
Bilinear | 0 | 0.9240 | 30.781
Bicubic | 0 | 0.9375 | 31.984
SRCNN | 57l | 0.9503 | 34.082
FSRCNN | 12k | 0.9462 | 33.614
VDSR | 665k | 0.9575 | 35.115
EDSR | 43000k | 0.9608 | 35.674
RCAN | 16000k | 0.9608 | 35.576
RDN | 21900k | 0.9614 | 35.769
HAN | 20000k | 0.9601 | 35.547
DRLN | 34000k | 0.9617 | 35.808
MANet | 27000k | 0.9618 | 35.842
FREDSR | 37k | 0.9645 | 35.3344
Table 2: Quantitative evaluation of 3x SISR generalization performance on
original UHDSR4K dataset. We report Structural Similarity Index Measure(SSIM),
and Peak Signal to Noise Ratio(PSNR) metrics. We report metrics for other deep
learning based methods from [29]. FREDSR compares favorably against these
baselines across all metrics, even though it is comprised of far fewer
parameters. This can be attributed to the scale invariance caused by the
inductive biases in the FFC modules.
As demonstrated here, we outperform several strong baselines, despite having
orders of magnitudes of fewer parameters. This indicates an ability to
efficiently capture the requisite information, which we can attribute to the
use of FFC and the process of residual learning. Additionally, we see an
increase in performance as resolution increases, where a model trained on
multiple resolutions outperforms a model trained on only one. This is
potentially due to the nature of the task of super-resolution itself, however
further analysis is required to determine the root cause of this phenomenon.
#### Other datasets
However, we note that as expected, FREDSR does not perform as highly across
other datasets, such as Manga109 or Urban100, from [45] and [46] respectively.
This is visualized in Figure 7, where we show our poor generalization
performance. The reasoning behind this is clear: bicubic upscaling does not
perform as well on such datasets at all, and the learned residual images are
generally lacking of color and of a much higher resolution, compared to the
low resolution. This shows that FREDSR is highly specialized, losing
generalizability at the benefit of extreme parameter efficiency. However, for
the main real-world application of video game upscaling, this should not be a
major issue, due to the consistent resolution and environment present.
Figure 7: Poor performance, with low clarity in edges, can be seen on the
Manga109 or Urban100 datasets, from [45] and [46] respectively. This is
potentially caused by the large distributional shift present, namely the
change in resolution and color in residual images.
## 4 Related Work
Within the domain of super-resolution, several traditional algorithms and
data-driven models have been designed to accurately upscale images. Such
classical algorithms are aptly covered in [47], and will not be discussed
further here, as most state of the art solutions to SISR are based off of
learning based techniques [48]. The use of convolutional networks for super
resolution was first introduced in [Dong2015], which also initiated the use of
deep learning for this task. Skip connections, integral to stabilizing
training, improving parameter efficiency, and increasing performance [49] were
first introduced with the development of DRCN [19]. The use of GANs for this
task, which have the major benefit of generating high-quality images when
compared to more traditional methods, was introduced in [50]. The main
challenge faced with GANs, their low mode coverage, or output diversity, is
not applicable to the super-resolution objective, meaning that the main
advantages of GANs, namely their comparatively faster sampling and high
quality outputs than diffusion networks and VAEs respectively, can be utilized
[51]. In a similar fashion to [50], [10] combines adversarial and perceptual
loss functions in order to train a GAN for super-resolution. Other works have
noted the lack of high receptive field present in most super-resolution
networks, pointing out that redundant patches within an image, common in both
artificial and natural landscapes, allow for global context to be more
informative to the computation of local upscaling. As a result, several works
have taken advantage of the increased receptive field of the dilated, or
atrous, convolution [52], [21]. This further motivates our use of Fourier
convolutions, where we take advantage of their theoretically unbounded
receptive fields. Other works have also used recurrent and attention
mechanisms to allow for global cross-patch information usage [41], [43], [53],
[54]. These attention-based mechanisms are an extremely interesting novel
field for further works to explore.
## 5 Discussion
In this study, we investigate the use of the Fast Fourier convolution for
solving the specialized SISR objective. Using these convolutions, we have
constructed a novel GAN architecture that has shown to outperform several
baselines, while being significantly smaller. Our model arguably shows good
performance in urban settings due to the highly repeated sectors present,
which seems challenging to other methods, as seen in figure 1 and 5. Our model
seems to have extremely high parameter efficiency, outperforming models which
are comprised of more than a thousand times as many parameters. In exchange
for this, we lose generalizability outside of a given image distribution. We
believe that this tradeoff can be harmful, but mitigated under a circumstance
where we believe our method is strongest-video game upscaling. This opinion is
corroborated by the fact that video games have similar highly repeating
sectors to urban scenes, due to their general usage of built-in textures for
three-dimensional models. This model is highly efficient, which is necessary
for real-time application on consumer graphics hardware, and can be trained
for a given game, as within this narrow domain significant generalization is
not required. This would allow for rendering at a low resolution and
upscaling. Further work would be required in this domain to allow for enforced
temporal coherence.
However, convolutions are not the only approach to increasing the receptive
field of image processing networks, and models like MLP Mixer models [55] as
well as vision transformers [56] are exciting topics for future study in this
field. Especially in the case of super resolution, higher receptive fields
will unlock possibilities for the development of deep learning for high-
resolution image processing, allowing for an increase in fidelity, diversity,
and processing speed. Improved super resolution will allow for advancements in
medical imaging [57] and compression of image and video formats [58], among
many other applications.
## Acknowledgments
The authors acknowledge the MIT SuperCloud and Lincoln Laboratory
Supercomputing Center for providing (HPC, database, consultation) resources
that have contributed to the research results reported within this paper. We
additionally thank Professor Leslie Kaelbling and Ge Yang from the MIT
Computer Science and Artificial Intelligence Laboratory for their valuable
discussions and providing us with access to resources to complete this
research. Lastly, we acknowledge Tensorflow and Horovod as the development
language and platform of choice for this project.
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|
# Trace the Accretion Geometry of H 1743–322 with Type C Quasi-periodic
Oscillations in Multiple Outbursts
Qing C. Shui Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 100049, Beijing, China University of
Chinese Academy of Sciences, Chinese Academy of Sciences, 100049, Beijing,
China S. Zhang Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 100049, Beijing, China Yu P.
Chen Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 100049, Beijing, China Shuang N. Zhang
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, 100049, Beijing, China University of Chinese
Academy of Sciences, Chinese Academy of Sciences, 100049, Beijing, China Ling
D. Kong Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 100049, Beijing, China Institut für
Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics,
Eberhard Karls, Universität, Sand 1, D-72076 Tübingen, Germany Peng J. Wang
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, 100049, Beijing, China University of Chinese
Academy of Sciences, Chinese Academy of Sciences, 100049, Beijing, China L.
Ji School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082,
China Hong X. Yin Shandong Key Laboratory of Optical Astronomy and Solar-
Terrestrial Environment, School of Space Science and Physics, Institute of
Space Sciences, Shandong University, Weihai, Shandong 264209, China Jin L. Qu
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, 100049, Beijing, China L. Tao Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 100049, Beijing, China Ming Y. Ge Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
100049, Beijing, China Jing Q. Peng Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 100049,
Beijing, China University of Chinese Academy of Sciences, Chinese Academy of
Sciences, 100049, Beijing, China Z. Chang Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
100049, Beijing, China J. Li CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei 230026, China School of Astronomy and Space Science, University
of Science and Technology of China, Hefei 230026, China P. Zhang College of
Science, China Three Gorges University, Yichang 443002, China
###### Abstract
We present a systematic analysis of type C quasi-periodic oscillation (QPO)
observations of H 1743–322 throughout the _Rossi X-ray Timing Explorer_
(_RXTE_) era. We find that, while different outbursts have significant flux
differences, they show consistent positive correlations between the QPO
fractional root-mean-square (rms) amplitude and non-thermal fraction of the
emission, which indicate an independence of the intrinsic QPO rms on
individual outburst brightness in H 1743–322. However, the dependence of the
QPO rms on frequency is different between the outburst rise and decay phases,
where QPO fractional rms of the decay phase is significantly lower than that
of the rise phase at low frequencies. The spectral analysis also reveals
different ranges of coronal temperature between the two outburst stages. A
semi-quantitative analysis shows that the Lense-Thirring precession model
could be responsible for the QPO rms differences, requiring a variable coronal
geometric shape. However, the variable-Comptonization model could also account
for the findings. The fact that the rms differences and the hysteresis traces
in the hardness-intensity diagram (HID) accompany each other indicates a
connection between the two phenomena. By correlating the findings with QPO
phase lags and the quasi-simultaneous radio flux previously published, we
propose there could be corona-jet transitions in H 1743–322 similar to those
that have been recently reported in GRS 1915+105.
black hole physics – accretion, accretion disc – binaries, close – X-rays:
binaries – X-rays: individual (H 1743–322)
## 1 Introduction
Undergoing outbursts occasionally after staying with faint luminosity for a
long time in quiescence is the primary feature of low mass black hole X-ray
binaries (BHXRBs) (Remillard & McClintock, 2006; Done et al., 2007). Most
complete outbursts are observed to have four typical states: the low/hard
state (LHS), hard intermediate state (HIMS), soft intermediate state (SIMS)
and high soft state (HSS), characterized by different X-ray spectral and
variability properties (Belloni et al., 2005; Fender et al., 2009). In the
outburst rise phase, the system starts from quiescence, increases luminosity
with strong variability and non-thermal dominated spectra in the LHS,
experiences the hard-to-soft transition in the HIMS and SIMS, and then stays
in the HSS with the thermal dominated spectra and weakest variability for
weeks. With the decreasing accretion rate, the system goes through the soft-
to-hard transition back to the LHS, and fades in quiescence. The different
state transition luminosity between the rise and decay phases directly leads
to the trace of canonical ‘q’ shape in the hardness-intensity diagram (HID,
Homan et al., 2001; Homan & Belloni, 2005), and the so-called hysteresis
effect which is still not well understood in BHXRBs (Maccarone, 2003;
Zdziarski et al., 2004; Weng et al., 2021). In addition to these features in
the X-ray energy bands, BHXRBs are also characterized by radio/infrared
emission, which is generally believed to be associated with relativistic jets
(Fender, 2001; Fender & Belloni, 2004; Méndez et al., 2022, and references
therein). In the hard state, radio emission with a flat spectrum
($S_{\nu}\propto\nu^{\alpha}$ with $\alpha\sim 0$) is interpreted as self-
absorbed synchrotron emission from an optically thick, steady and compact jet
(Blandford & Königl, 1979; Fender, 2001).During the hard-to-soft transition,
the steady and compact jet is gradually quenched, where the radio emission, if
present, is thought to be attributed to optically thin synchrotron emission
from transient ejected plasma clouds with relativistic speeds ($v\sim c$)
(Fender et al., 2004).
The X-ray spectrum of a BHXRB usually consists of a thermal and a non-thermal
component, the former is believed to come from a geometrically thin and
optically thick disc (Shakura & Sunyaev, 1973; Lynden-Bell & Pringle, 1974),
while the latter is produced by the Comptonization of soft photons. However,
the geometry of the Comptonizing medium is relatively less clear, which could
be an extended cloud consisting of hot electrons ($\sim 100$ keV) called
“corona” (Bisnovatyi-Kogan & Blinnikov, 1976; Sunyaev & Titarchuk, 1980;
Titarchuk, 1994; Zdziarski et al., 1996, 1999; Życki et al., 1999) or/and the
jet-base (Markoff et al., 2005; You et al., 2021). A portion of Comptonized
photons can irradiate the disc and then end up as the reflection component
which has abundant features, like broad emission lines and Compton hump, etc.
(Dauser et al., 2010; García et al., 2014, 2015b, and references therein).
Low frequency quasi-periodic oscillations (LFQPOs, roughly 0.1–30 Hz, van der
Klis, 1989) are the most prominent features observed in the power density
spectrum (PDS) of BHXRBs, with the classification of type A, B and C based on
the centroid frequency, quality factor and root-mean-square (rms) amplitude
(Wijnands et al., 1999; Casella et al., 2005; Remillard & McClintock, 2006).
The appearance of type C QPOs is frequent in the LHS and HIMS with strong
amplitudes (fractional rms $\sim 10\%$) and flat-top noise components in the
PDS. In the past few decades, several models has been proposed to explain the
dynamical origin of QPOs based either on the geometric or the intrinsic
properties of the accretion flow. Some examples of the _intrinsic_ models are
trapped corrugation modes (Kato, 1990; Wagoner, 1999), the Accretion-ejection
instability model (AEI; Tagger & Pellat, 1999) and the Two-Component Advection
Flow model (TCAF; Molteni et al., 1996), etc.. For the _geometric_ models,
most of them are related to the relativistic Lense-Thirring (L-T) precession,
which was originally invoked to be the dynamic mechanism of QPOs by Stella &
Vietri (1998). As an extension of the relativistic precession model (RPM;
Stella et al., 1999), the L-T precession model proposed by Ingram et al.
(2009) assumes the entire hot flow precesses within the inner radius of the
truncated disc (Esin et al., 1997). We refer readers to Ingram & Motta (2019)
for recent reviews of observations and theories of LFQPOs.
In addition to the frequency, the radiative properties of QPOs, e.g. rms
amplitudes and time lags, etc., also provide extra useful information. Since
several observational studies have found that, for most type C QPOs, the
variability increases with the photon energy (Zhang et al., 2017; Huang et
al., 2018; Kong et al., 2020; Zhang et al., 2020) and no prominent disc-like
component exists in the rms spectra (Sobolewska & Życki, 2006; Axelsson et
al., 2013; Axelsson & Done, 2016), the radiative mechanism of the type C QPO
should be strongly related to the Comptonized emission. The inclination
dependence of amplitudes and time lags (see Motta et al., 2015; van den
Eijnden et al., 2017) and reflection variability extracted from the phase-
resolved spectroscopy (see Ingram & van der Klis, 2015; Ingram et al., 2016,
2017) add support to a geometrical origin, especially the L-T precession model
(Ingram et al., 2009). However, these observational findings from different
sources show very diverse lacking quantitative explanations, and most
recently, Nathan et al. (2022) applied the L-T precession model to fit the
phase-resolved spectroscopy of GRS 1915+105 and found an unexpectedly long
thermalization time-scale of $\sim 70$ ms, which is incompatible with the soft
lags ($\sim 1$ ms) found in other BHXRBs, e.g. MAXI J1820+070 (Kara et al.,
2019) and GX 339–4 (Uttley et al., 2011). These inconsistent findings indicate
L-T precession model, especially its radiative part, could be incomplete.
Karpouzas et al. (2020) have recently proposed a time-dependent Comptonization
model following the work of Lee & Miller (1998) and Kumar & Misra (2014) to
quantitatively explain the energy-dependence of phase lags and rms amplitudes
of kilohertz QPOs in neutron-star systems. This model has been successfully
applied to BHXRBs, e.g. GRS 1915+105 and MAXI J1348–630 (see Karpouzas et al.,
2021; García et al., 2021; Méndez et al., 2022; García et al., 2022). The
updated version of the model (Bellavita et al., 2022), incorporating a disc-
blackbody as the seed-photon source, has been applied to fit the rms amplitude
and phase lag spectra of type C QPOs in MAXI J1535–571, using _Insight_ -HXMT
data in the 1–100 keV energy range (Zhang et al., 2022).
The transient BHXRB H 1743–322 was first observed by _Ariel_ -V satellite in
1977 August (Kaluzienski & Holt, 1977), located at $\rm
RA=17^{h}46^{m}15^{s}.596$ and $\rm
Dec=-32^{\circ}14^{\prime}00^{\prime\prime}.860$. Based on the X-ray/radio
observations of the two-sided jet in the 2003 outburst, Steiner et al. (2012)
determined a distance of $8.5\pm 0.8\ \rm kpc$, and an inclination angle of
$75^{\circ}\pm 3^{\circ}$, respectively. By applying the relativistic
accretion disc model in the spectral fitting, they also estimated a black hole
spin of $a_{*}=0.2\pm 0.3$. H 1743–322 is an active black hole transient
source and was monitored by _RXTE_ to undergo outbursts frequently between
2003 and 2011. The 2003 outburst is the brightest one with observations in
multiple wavelengths (see Parmar et al., 2003; Homan et al., 2005; McClintock
et al., 2009), then two much fainter outbursts in 2004 and 2005, respectively,
followed behind (Bhattacharjee et al., 2017; Coriat et al., 2011). In 2008,
two outbursts (2008a and 2008b) were observed in both of X-ray and radio
bands, but the 2008a outburst is classified as failed-transition outburst for
the short outburst cycle without experiencing soft states (Capitanio et al.,
2009; Jonker et al., 2010; Coriat et al., 2011). Then this source entered into
a new outburst during 2009 (Chen et al., 2010; Motta et al., 2010), and
exhibited the last three outbursts in 2010 (2010a and 2010b) and 2011 (Zhou et
al., 2013). Recently, Aneesha & Mandal (2020) systematically analysed the
spectral evolution of the outbursts of H 1743–322 in the _RXTE_ era. In the
post _RXTE_ era, there were 2012, 2014, 2016 and 2018 outburst monitored by
other instruments (Shidatsu et al., 2014; Stiele & Yu, 2016; Chand et al.,
2020; Williams et al., 2020; Wang et al., 2022).
In this work, we systematically investigate type C QPO data born out of _RXTE_
observations in both of the rise and decay phases of seven outbursts from H
1743–322. We introduce observations and data reductions in Section 2, present
the data analysis and results in Section 3 , discuss these in Section 4 and
finally summarize in Section 5.
## 2 Observations and Data reductions
Fig. 1 shows the _RXTE_ /All Sky Monitor (ASM) light-curve of H 1743–322 in
the 1.5–12 keV energy band. Between 2003 and 2011, this source experienced
nine outbursts, and we focus on the 2003, 2008a, 2008b, 2009, 2010a, 2010b and
2011 outburst which contain abundant type C QPO observations carried out by
_RXTE_. The `HEASOFT` software package version 6.28 is used for the data
analysis. We generate good time internals (GTIs) with the constraints that the
elevation angle (ELV) is larger than $10^{\circ}$ and angular distance between
the pointing position and source (OFFSET) is less than $0.02^{\circ}$. The
standard data products of e.g. light curve, spectrum and background are
produced from observational data of Proportional Counter Array (PCA). Since
combining large spectral data sets with different PCA configurations can
produce large systematic errors (see Smith et al., 2009), we only use data
from Proportional Counter Unit (PCU) 2 in our subsequent spectral analysis
because it is the only unit which was 100% on during the observations. Data of
standard 2 mode are adopted for the spectral analysis in the energy range of
3–30 keV without any groupings and binnings. To account for calibration
uncertainties, a systematic error of 0.5 per cent is added to spectral
fittings. We generate the power density spectrum (PDS) using the light curve
born out of ‘Event’ and ‘Binned’ modes in the energy range of 3–30 keV (PCA
channels 7–71 of the calibration epoch
5111https://heasarc.gsfc.nasa.gov/docs/xte/e-c_table.html), which is the same
as that of the spetral fittings. Then both of the spectral and timing analyses
are performed using XSPEC version 12.12.0.
Figure 1: The _RXTE_ /ASM light curve of H 1743–322 in the 1.5–12 keV energy
band with a resolution of one point per day. In the _RXTE_ era, there are nine
outbursts between 2003 and 2011, where we focus on the 2003, 2008a, 2008b,
2009, 2010a, 2010b and 2011 outburst which contain abundant type C QPO
observations.
## 3 Analysis and results
### 3.1 Power Density Spectra
For each observation in the present study, we produce the PDS in the 1/64–-64
Hz frequency range with the 8-ms time resolution by taking Miyamoto
normalization (Belloni & Hasinger, 1990; Miyamoto et al., 1991). All the PDS
are fitted with several Lorentzian functions (see Fig. 2). In the PDS
fittings, we use at least two Lorentzian functions to fit the broad band noise
and QPO signal, respectively. The centriod frequency of the Lorentzian fitted
the low-frequency broad band noise is fixed at zero. However, if there are any
other significant residual structures (e.g. the second QPO harmonic), more
Lorentzian functions would be added to the fitting model. Since we primarily
focus on the QPO signal, whether to add more Lorentzians depends on whether
the Lorentzian functions used to fit the QPO harmonics describe the QPO
components well in the case of the current total model. To do this, after
fitting with the current model, we retain the best-fit parameters and remove
the Lorentzian components fitted the QPO harmonics to check whether the
residual structures from the data/model ratio plots are Lorentzian-like shapes
and broad band noises are well fitted. Following Motta et al. (2015), if a QPO
consists of multiple harmonic peaks, the QPO fractional rms is computed by
adding in quadrature the rms of the harmonic peaks (i.e. QPO fractional rms is
computed by $\sqrt{\sum{P_{i}}}$, where $P_{i}$ is the power calculated with
integration of the $i$-th QPO harmonic Lorentzian function). Additionally,
considering the background contribution, the QPO fractional rms reported in
the present study is finally computed by
${\rm rms}=\sqrt{\sum_{i}{P_{i}}}\times[(S+B)/S],$ (1)
where $S$ and $B$ are the source and background average rate, respectively
(see Bu et al., 2015). The best-fit QPO parameters of the present study are
presented in Appendix D.
Figure 2: Representative power density spectra from the outburst rise (plotted
in black) and decay phase (plotted in blue). Power density spectra are
produced in the 3–30 keV energy range and 1/64–64 Hz frequency range, then
fitted with the model consisting of multiple Lorentzian functions (dotted
lines). For panel (a), strong QPO components appear at high frequencies
($f_{\rm QPO}>2$ Hz), where QPO amplitudes of the two outburst phases are
comparable. For panel (b), the QPO components appear at low frequencies
($f_{\rm QPO}<2$ Hz), where QPO amplitude of the rise phase is significantly
larger than that of the decay phase.
### 3.2 Energy Spectra
In our spectral fittings, the Galactic absorption effect is always included in
our models by implementing $tbabs$ model (Wilms et al., 2000). The hydrogen
column density ($N_{\rm H}$) is fixed to $2.3\times 10^{22}\ {\rm cm^{-2}}$
following Miller et al. (2006). Firstly, we start with a simple model consists
of a power-law and a disc blackbody component (Model 1:
$tbabs\times(diskbb+powerlaw)$) and obtain large reduced-$\rm\chi^{2}$ ($\gg
2$) in the most observations. Then we replace $powerlaw$ with $nthcomp$, a
physically motivated thermal Comptonization model which describes the spectrum
of Compton upscattering photons (Zdziarski et al., 1996; Życki et al., 1999).
The spectral fitting with the second model (Model 2:
$tbabs\times(diskbb+nthcomp)$) gives a smaller reduced-$\rm\chi^{2}$ but, as
shown in Fig. 3a and b, the residuals still show obvious structures at
energies around the iron line ($\sim 6.4$ keV) and Compton hump ($\sim 10$–30
keV), providing evidence of a relativistic reflection component (García et
al., 2013, and references therein). Hence we add a reflection model
$relxillcp$ (Dauser et al., 2014; García et al., 2014) in our spectral
fittings (Model 3: $tbabs\times(diskbb+nthcomp+relxillcp)$).
For the spectral fitting with Model 3, we link the seed photon temperature
($kT_{\rm bb}$) of $nthcomp$ with the inner disk temperature ($kT_{\rm in}$)
of $diskbb$ and choose the flavor of disk-blackbody seed photons for
$nthcomp$. Since $relxillcp$ calculates the reflection component using the
$nthcomp$ continuum, it is self-consistent to link the relevant parameters
like the photon index ($\Gamma$) and electron temperature ($kT_{\rm e}$)
between two models. In the reflection model $relxillcp$, we assume the
canonical power-law emissivity profile, $\epsilon\propto r^{-3}$, for the disc
(Fabian et al., 1989). The parameter $R_{\rm out}$ is the outer radius of the
accretion disc which turns out to be not sensitive to the overall fitting and
hence is frozen at the maximum value ($1000R_{\rm g}$, where $R_{\rm
g}=GM/c^{2}$ is the gravitational radius). In order to make $relxillcp$ only
calculate the reflection component, we fix the reflection fraction ($R_{\rm
f}$) to $-1$. Referring to previous studies, the disc inclination ($i$), spin
parameter of the black hole ($a_{*}$) and iron abundance ($A_{\rm Fe}$, in
solar units) are set to $75^{\circ}$, 0.2 and 3.0, respectively (Steiner et
al., 2012; Chand et al., 2020). The shape of $nthcomp$ continuum is set by the
combination of the electron temperature ($kT_{\rm e}$) and scattering optical
depth ($\tau_{\rm s}$), where the higher cut-off energy is parameterized by
$kT_{\rm e}$ (Życki et al., 1999). However, in many observations, the cut-off
energy is beyond the energy band of our spectral analysis (3–30 keV). Although
$kT_{\rm e}$ can influence the reflection hump at energies around 20–40 keV
which gives a chance to estimate it beyond the spectral coverage (see García
et al., 2015a), we note that $kT_{\rm e}$ is not completely reliable in our
spectral fitting. For a less constrained $kT_{\rm e}$, we fix it to 300 keV.
Additionally, we notice the $diskbb$ contribution is marginal in where its
parameters are not well constrained by using only PCA data and the spectral
fitting does not even need this component in some observations (see also Plant
et al., 2014; Aneesha et al., 2019). For these observations, we fix the
parameter $kT_{\rm in}$ and normalization of $diskbb$ at 0, then let $kT_{\rm
bb}$ of $nthcomp$ as a free parameter in the fitting. We calculate the
unabsorbed flux for each component in the energy range 3–30 keV by convolving
$cflux$ model with these required models. After fitting energy spectra with
Model 3, we obtain a reasonable reduced-$\chi^{2}$ ($\sim 1$) for most
observations (see Appendix A) and compute the 90 per cent confident-level
uncertainties using the Markov Chain Monte Carlo (MCMC) technique, with length
40 000. The best-fit parameters of the two representative spectra presented in
Fig. 3 with Model 3 are summarized in Table 1.
Figure 3: Representative _RXTE_ /PCA energy spectra (the top panels) and fitting residuals (the three bottom narrow panels are plotted for Model 1, 2 and 3, respectively) for both of the rise phase (the left panel, obs.ID: 80146-01-30-00) and the decay phase (the right panel, obs.ID: 93427-01-04-00). The $diskbb$, $nthcomp$ and $relxillcp$ model are plotted in red, orange and blue, respectively. Table 1: Best-fit Parameters and the Corresponding 90% Confidence Intervals Obtained from Fitting the Representative _RXTE_ /PCA Spectra Presented in Fig. 3 Using the Model 3: $tbabs\times(diskbb+nthcomp+relxillcp)$. Model | Parameters | Rise Phase | Decay Phase
---|---|---|---
$tbabs$ | $N_{\rm H}$ ($10^{22}\rm cm^{-2}$) | 2.3 (fixed)
$relxillcp$ | $i$ (Deg) | 75 (fixed)
| $a_{*}$ | 0.2 (fixed)
| $R_{\rm out}\ (R_{\rm g})$ | 1000 (fixed)
| $q^{\rm a}$ | 3 (fixed)
| $A_{\rm Fe}$ (Solar Units) | 3 (fixed)
| $R_{\rm f}$ | $-1$ (fixed)
$diskbb$ | $T_{\rm in}$ (keV) | $0.76^{+0.05}_{-0.05}$ | $0.62^{+0.08}_{-0.18}$
| $N_{\rm disk}^{\rm b}$ | $866^{+280}_{-201}$ | $99^{+842}_{-53}$
$nthcomp$ | $\Gamma$ | $2.29^{+0.04}_{-0.03}$ | $1.83^{+0.03}_{-0.05}$
| $kT_{\rm e}$ (keV) | $8.6^{+1.0}_{-0.5}$ | 300(fixed)
| $N_{\rm nth}^{\rm c}$ | $2.52^{+0.29}_{-0.21}$ | $0.15^{+0.05}_{-0.08}$
$relxillcp$ | $R_{\rm in}\ (R_{\rm ISCO})$ | $17.9^{+76.8}_{-8.1}$ | 100 (fixed)
| $\log_{10}(\xi)^{\rm d}$ | $2.81^{+0.13}_{-0.17}$ | $3.53^{+0.67}_{-0.31}$
| $(\rm erg\ cm\ s^{-1})$
| $N_{\rm rel}^{\rm e}\ (10^{-3})$ | $42.0^{+17.1}_{-9.7}$ | $2.3^{+4.50}_{-0.10}$
| $\chi^{2}/{\rm d.o.f}$ | 50.18/46 | 38.29/48
### 3.3 Timing-spectral Joint Analysis
The timing and spectral analyses presented in Section 3.1 and 3.2,
respectively, provide the essential inputs for a spectral-timing joint
diagnostic of the outburst what is shown in follows.
#### 3.3.1 Correlations between the QPO rms and non-thermal component
Fig. 4 presents the relations between the QPO fractional rms and non-thermal
component during the rise and decay phases, separating the different
outbursts. For clarity’s sake, data points of the rise phase from different
outbursts are displayed in different colors and shapes in Fig. 4a and b, and
the data points of the decay phase are plotted in gray without separating
different outbursts, while in the Fig. 4c and d, we display data sets in the
opposite way. As shown in Fig. 4a, in the rise phase, there are no remarkable
correlations between the fractional rms and non-thermal fluxes ($F_{\rm
nthcomp}$), and data points from different outbursts distribute widely in
$F_{\rm nthcomp}$ which is consistent with the large differences in the
outburst peak fluxes shown in Fig. 1. However, if we display the QPO
fractional rms as a function of the non-thermal fraction ($F_{\rm
nthcomp}/F_{\rm total}$), the positive correlations between the fractional rms
and non-thermal component become significant and consistent among different
outbursts (see Fig. 4b). In the decay phase, different outbursts show the
similar non-thermal fluxes ($\rm\sim 10^{-9}\ ergs\ cm^{-2}\ s^{-1}$).
However, compared with the rise phase, the positive correlation between the
fractional rms and non-thermal fraction is not clear within the decay phase
because of the relatively larger error bar of the non-thermal fraction (see
Fig. 4c and d).
Figure 4: The relations between QPO fractional rms and the non-thermal
component. For clarity’s sake, data of the different outburst rise phases are
displayed in different colors in the panels (a) and (b), and data points of
the decay phases are plot in gray without separating different outbursts,
while the panels (c) and (d) display the data points of the rise and decay
phases in the opposite way.
#### 3.3.2 QPO rms dependence on frequency
Type C QPOs usually appear in the HIMS, a stage shows the significant
transition in both of the timing and spectral domains (Homan & Belloni, 2005;
Remillard & McClintock, 2006), where properties of the type C QPO, like
central frequency and fractional rms, evolve in a large value range. Previous
studies have presented that the fractional rms of type C QPOs varies with the
frequency (see Motta et al., 2015; van Doesburgh & van der Klis, 2020; Zhang
et al., 2020; Wang et al., 2022). Here, we analyse a number of type C QPO
samples across different outbursts of H 1743–322 and present the QPO
fractional rms dependence on fundamental frequency in both the rise and decay
phases in Fig. 5a for comparisons. Data points displayed are not be
distinguished among different outbursts, but only between the rise and decay
phases. As one can see, during the rise phase, the QPO fractional rms
increases slightly with frequency below 1 Hz and then remains roughly flat
around 1–2 Hz, while shows a significant drop at higher frequencies. However,
the presented relation of the decay phase deviates from that of the rise phase
at low frequencies, where the deviation becomes larger for the lower
frequency. Also, these relations between the QPO fractional rms and frequency
in both of the rise and decay phases are consistent among different outbursts.
The fractional rms presented in the study is computed by adding in quadrature
the rms of the harmonic peaks. However, we have checked the case that taking
the rms of QPO fundamental only, and find the dependence of fractional rms on
frequency deviates only slightly with respect to the results presented in Fig.
5, where the two outburst stages remain different branches in the relations
between QPO fractional rms and frequency clearly.
Since the fractional rms is defined as the fractional variability of the flux,
the fractional rms we compute in the Section 3.1 and presented in Fig. 5a can
be described by
${\rm rms}=\frac{F_{\rm var}}{F_{\rm total}}=\sigma\times\frac{F_{\rm
c}}{F_{\rm total}},$ (2)
where $F_{\rm var}$ is the variable flux (absolute rms), $F_{\rm total}$ is
the total time-averaged flux, $\sigma$ is a function for the intrinsic rms,
and $F_{\rm c}$ represents the flux contributing to the QPO variability (see
also Kong et al., 2020; Shui et al., 2021). If we assume that only the non-
thermal component contributes to the type C QPO, while the fractional rms is
diluted by the other components, i.e. the thermal and reflected components,
then $F_{\rm c}$ is equal to $F_{\rm nthcomp}$. So the intrinsic rms is the
variability amplitude of non-thermal emission, which can be computed by
$\sigma={\rm rms}\times\frac{F_{\rm total}}{F_{\rm nthcomp}}.$ (3)
The dependence of the intrinsic rms on frequency is presented in Fig. 5b, and
we find it is similar to that of the fractional rms displayed in Fig. 5a. It
is clear to see the two separate branches of the rise and decay phases.
Figure 5: QPO fractional rms (the left panel) and intrinsic rms (the right
panel) plotted as a function of frequency. The rise and decay phase are
distinguished in different colors and shapes: data points from the rise phase
are plotted as blue circles, while those from the decay phase are plotted as
black squares.
### 3.4 Coronal Parameters and Radio Emission
We have shown that the dependence of the QPO intrinsic rms (hereafter QPO rms)
on frequency shows two branches: the QPO rms in the outburst rise phase is
significantly larger than that in the decay phase at low frequencies. To
investigate this bi-modality in more detail, we present the dependence of the
rms-$f_{\rm QPO}$ relation on radio fluxes and spectral parameters in Fig. 6.
The quasi-simultaneous measurements of the radio flux density at $\sim 8.5$
GHz ($S_{\nu=8.5{\rm GHz}}$) are taken from McClintock et al. (2009); Jonker
et al. (2010); Coriat et al. (2011); Miller-Jones et al. (2012). For details
of the radio observations used in the present study, see Appendix D. We note
that not all _RXTE_ /PCA observations in this study have quasi-simultaneous
radio measurements. However, for clearly showing the radio dependence of the
two different branches, we plot the entire QPO data set from our timing
analysis in gray in the top two panels of Fig. 6, while coloring these data
points which have radio observations. Furthermore, since 2003 outburst is much
brighter than the other outbursts, we present the radio data of 2003 outburst
in Fig. 6a alone, while those of the other outbursts are shown in Fig. 6b. The
dependence of the coronal temperature ($kT_{\rm e}$) and photon index
($\Gamma$) are presented in the bottom two panels of Fig. 6, respectively. The
shade of the data points in each panel indicate the measured values of the
parameters. Radio emission during the outburst rise phase of 2003 outburst is
the strongest ($S_{\nu=8.5{\rm GHz}}>10\ {\rm mJy}$), while that of the other
outbursts is relatively weaker ($S_{\nu=8.5{\rm GHz}}\sim 2.5\ {\rm mJy}$).
For an individual outburst, there is a marginally decreasing trend of the
radio flux density from low QPO frequency ($f_{\rm QPO}<2$ Hz) to high
frequency ($f_{\rm QPO}\sim 8$ Hz). Additionally, radio emission of the decay
phase ($S_{\nu=8.5{\rm GHz}}<1\ {\rm mJy}$) is significantly weaker than that
of the rise branch. In contrast to the radio flux, the electron temperature is
higher in the decay branch. These triangles plotted in Fig. 6c indicate that
$kT_{\rm e}$ are too high to be constrained using _RXTE_ /PCA energy spectra,
hence fixed at 300 keV (see Section 3.2 for details). However, for the photon
index, there are no apparent differences between the two branches in the
similar frequency range.
Figure 6: The QPO intrinsic rms plotted as a function of frequency. The shade
of the data points indicate the quasi-simultaneous measurements of the radio
flux density (a and b), $kT_{\rm e}$ (c) and $\Gamma$ (d), respectively. In
panels (a) and (b), we plot the entire QPO data set from our timing analysis
in gray, only these data points with radio observations are colored.
Furthermore, since the outburst 2003 is much brighter than the other
outbursts, we present the radio flux density of the 2003 outburst in panel (a)
alone, while those of the other outbursts are shown in panel (b). In panel
(c), these triangles indicate that $kT_{\rm e}$ is too high to be constrained
using _RXTE_ /PCA energy spectra, hence fixed at 300 keV.
### 3.5 Phenomenological Analysis with the L-T Precession Model
The phenomenon that the QPO rms is dependent on the frequency and outburst
stage (see Fig. 5) indicates that the intrinsic properties of QPOs changed
during the outburst. If type C QPOs are produced by the L-T precession of the
entire corona (see Ingram et al., 2009; You et al., 2018), a simplified
geometrical model presented as follows can be used to estimate the intrinsic
rms semi-quantitatively.
Figure 7: Schematic diagram illustrating the cross-section of the corona.
$\hat{\bm{n}}$, $\hat{\bm{o}}$ and $\hat{\bm{a}}$ represent the normal of the
corona, unit vector pointing from the black hole to the observer and unit
vector of black hole spin. respectively. $\beta$ is the angle between binary
orbit and black hole spin, and $\theta$ is the included angle between
$\hat{n}$ and $\hat{o}$. The coronal shape is described by the height scale
$h/r$, the ratio of the semiminor and semimajor axis of the ellipse. We note
this presented representative geometry is a case of $\Phi=0$.
Using the coordinate system described by Veledina et al. (2013); Ingram et al.
(2015), the unit vector pointing from the central black hole to the observer
is given by
$\hat{\bm{o}}=\left(\sin{i}\cos{\Phi},\sin{i}\sin{\Phi},\cos{i}\right),$ (4)
where $i$ is the binary inclination and $\Phi$ is the azimuth of the observer.
The instantaneous normal of the corona is $\hat{\bm{n}}$, changing with the
precession phase angle $\omega$:
$\displaystyle\hat{\bm{n}}=($
$\displaystyle\sin{\beta}\cos{\beta}\left(1+\cos{\omega}\right),$
$\displaystyle\sin{\beta}\sin{\omega},\cos^{2}{\beta}-\sin^{2}{\beta}\cos{\omega}),$
(5)
where $\beta$ is the angle between binary orbit and black hole spin. We note
that $\omega$ changes from 0 to $2\pi$ on the precession period, hence leads
to the changes of $\hat{\bm{n}}$. Then cosine of the included angle ($\theta$)
between $\hat{\bm{n}}$ and $\hat{\bm{o}}$ can be written by
$\displaystyle\cos{\theta}=$ $\displaystyle\hat{\bm{n}}\cdot\hat{\bm{o}}$
$\displaystyle=$
$\displaystyle\sin{\beta}\cos{\beta}\sin{i}\cos{\Phi}\left(1+\cos{\omega}\right)$
$\displaystyle+\sin{\beta}\sin{\omega}\sin{i}\sin{\Phi}$
$\displaystyle+\cos{i}\left(\cos^{2}{\beta}-\sin^{2}{\beta}\cos{\omega}\right).$
(6)
We consider a simplified problem with assuming that the coronal shape is a
crushed sphere, i.e. viewed from the coronal normal ($\hat{\bm{n}}$) is a
circle while the cross-section is an ellipse (see Fig. 7). The shape of the
corona can be described by defining the scale height, $h/r$, where $h$ is the
semiminor axis and $r$ is the semimajor axis of the ellipse, respectively. In
the geometry described above, the projected area of the corona to the observer
is
$\displaystyle S_{\rm ob}$ $\displaystyle=\pi
r^{2}\left[\sin^{2}{\theta}\cdot\left(h/r\right)^{2}+\cos^{2}{\theta}\right]^{1/2}$
$\displaystyle=\pi
r^{2}\left\\{\left(h/r\right)^{2}+\left[1-(h/r)^{2}\right]\cos^{2}{\theta}\right\\}^{1/2}.$
(7)
For simplicity, we assume the radiation of the corona is homogeneous and
isotropic, then the observed flux from the corona is a function of $\theta$
and $h/r$:
$\displaystyle F$ $\displaystyle=\bar{I}\cdot\Delta\Omega=\bar{I}\cdot S_{\rm
ob}/D^{2}$
$\displaystyle\propto\left\\{\left(h/r\right)^{2}+\left[1-(h/r)^{2}\right]\cos^{2}{\theta}\right\\}^{1/2}\left(1-e^{-\tau}\right),$
(8)
where $\bar{I}$ is the average radiation intensity, $\Delta\Omega$ is the
solid angle of the corona in the view of the observer, $D$ is the distance
from the source to the observer, and $\tau$ is the effective optical depth,
also a function of $\theta$ and $h/r$. We estimate $\tau$ as a simple case:
$\tau=\tau_{0}\left\\{\left(h/r\right)^{2}+\left[1-\left(h/r\right)^{2}\right]\cos^{2}{\theta}\right\\}^{-1/2},$
(9)
where $\tau_{0}$ is the minimum optical depth of the corona, i.e. viewed from
the coronal normal (see Appendix B for details). Combining Equations 3.5 and
9, we find that the flux from the corona changes with $\theta$ on the
precession period. On the basis of the above, the minimum and maximum coronal
flux could be calculated in the $\omega$ value range $0-2\pi$. Then the QPO
rms can be estimated by
${\rm rms}=\frac{F_{\rm max}-F_{\rm min}}{F_{\rm max}+F_{\rm min}},$ (10)
where $F_{\rm max}$ and $F_{\rm min}$ are the maximum and minimum value of the
coronal flux, respectively, on the precession period.
Figure 8: The calculated results using the geometrical model. In the
calculation, we assume a translucent corona with $\tau_{0}=1$ which is
consistent with the previous hypotheses and simulations (see You et al., 2018;
Ingram & Motta, 2019; You et al., 2020). Panel (a) displays the intrinsic rms
dependence on the entire range of viewing angles with a fixed $h/r$ value
(0.2), panel (b) displays the dependence of intrinsic rms on $h/r$ with
different inclinations ($\Phi$ is set to $90^{\circ}$).
The calculated results using the simplified geometrical model are presented in
Fig. 8. In our calculation, we assume a translucent corona with $\tau_{0}=1$
which is consistent with the previous hypotheses and simulations (see Ingram
et al., 2015; You et al., 2018, 2020). Fig. 8a displays the intrinsic rms
dependence on the entire range of viewing angles ($h/r=0.2$), where it shows
that rms increases with the inclination ($i$), while gets the peak values at
$\Phi=180^{\circ}$. However, $i$ and $\Phi$ are constant for a specific black
hole source, so the scale height ($h/r$) is the only parameter to influence
intrinsic rms during the outburst evolution. In Fig. 8b, we find that QPO rms
is highly dependent on $h/r$, especially in the high inclination sources
($i\sim 60^{\circ}$–$90^{\circ}$). We note that the general relativity
effects, e.g. light bending, are not considered in the simplified L-T
precession model, but the results presented in Fig. 8 are consistent with
previous studies which took the general relativity into account (see Veledina
et al., 2013; Ingram et al., 2015; You et al., 2018). This fact indicates that
GR effects, e.g. light bending, could be the second-order effects to affect
the QPO rms in the L-T precession model.
Figure 9: Fitting results of the relations between QPO intrinsic rms and
frequency that presented in Fig. 5b with the simplified L-T precession model,
where the trend lines and colored shaded regions represent the median and
3$\sigma$ confidence intervals of the fittings: the rise phase is displayed in
red and decay phase is display in blue. The data points plotted in the left
panel are same as that presented in Fig. 5b.
Calculating with different values of $h/r$ can obtain different intrinsic rms,
which allows us to fit the relations between QPO intrinsic rms and frequency
presented in Fig. 5b with the model. From a phenomenological motivation, we
assume $h/r$ as a quadratic function of $f_{\rm QPO}$:
$h/r=k_{1}\cdot f_{\rm QPO}^{2}+k_{2}\cdot f_{\rm QPO}+k_{3},$ (11)
for both of the rise and decay phases, where $f_{\rm QPO}$ is the QPO
frequency, $k_{1}$, $k_{2}$ and $k_{3}$ are the model parameters. In this
fitting, we fix the inclination, $i$, to $75^{\circ}$ which is consistent with
our spectral analysis and the misalignment angle, $\beta$, to $10^{\circ}$
following Veledina et al. (2013) and Ingram et al. (2015). Since the azimuth
of observer, $\Phi$, is constant for a specific BH source, it only affects our
estimation for the value of $h/r$, not the evolution trend, we fix it to a
median ($90^{\circ}$). The MCMC technique is used to carry out the parameter
estimation with the uniform prior distribution. The posterior probability
distributions of the model parameters are presented in Appendix C. From the
fitting, we get $k_{1}=(5.4\pm 0.3)\times 10^{-3}$, $k_{2}=(-1.96\pm
0.27)\times 10^{-2}$, $k_{3}=(3.39\pm 0.05)\times 10^{-1}$ and $\chi^{2}/{\rm
d.o.f}=65.90/53$ for the rise phase, while $k_{1}=(7.5\pm 2.9)\times 10^{-3}$,
$k_{2}=(-5.0\pm 1.9)\times 10^{-2}$, $k_{3}=(4.70\pm 0.27)\times 10^{-1}$ and
$\chi^{2}/{\rm d.o.f}=28.02/47$ for the decay phase. The fitting results are
displayed in Fig. 9, where the trend lines and colored shaded regions
represent the median and 3$\sigma$ confidence intervals of the fitting, and
the data points shown in the left panel are the same as those presented in
Fig. 5b. In Fig. 9b, we display the $h/r$ dependence on $f_{\rm QPO}$
(described by Eq. 11). The evolution trends of $h/r$ are consistent between
the two phases at high frequencies ($f_{\rm QPO}>4$ Hz), which show the
increase with the increasing frequency. At low frequencies, $h/r$ of both the
two phases are negatively correlated to the frequency, but $h/r$ values and
the gradient of the decay phase are significantly larger than those of the
rise phase.
## 4 Discussion
We have systematically investigated the properties of type C QPOs by analysing
one hundred and six _RXTE_ /PCA observations across multiple outbursts of H
1743–322 from 2003 to 2011. Fig. 4 shows large variances of non-thermal fluxes
among different outbursts, which distort the correlation between the non-
thermal flux and the QPO fractional rms. However, the QPO fractional rms of
different outbursts is positively correlated to the non-thermal fraction
($F_{\rm nthcomp}/F_{\rm total}$) consistently. Additionally, the co-evolution
between the QPO intrinsic rms and frequency keeps similar traces among
different outbursts (see Fig. 5). Since different outbursts could have very
different luminosity levels, these consistent behaviours across outbursts
indicate that the QPO intrinsic rms (hereafter QPO rms) is independent on the
individual outburst brightness. However, the dependence of the QPO rms on
frequency can be classified into two branches, where QPO rms in the outburst
rise phase is significantly higher than that in the decay phase at low
frequencies. Radio observations and X-ray spectral analyses reveal more
differences between the two branches.
### 4.1 Trace the Coronal Geometry with the Simplified L-T Precession Model
This phenomenon that the QPO rms is independent on the outburst brightness has
been also reported in other sources (e.g. GX 339–4, Shui et al., 2021). If
type C QPOs are produced by L-T precession of the corona, the X-ray QPO
variability owes to the geometric wobble of the corona, which changes the
projection area of the corona with respect to the observer to modulate the
X-ray flux (Ingram et al., 2009, 2015; You et al., 2018, 2020). Calculations
presented in Section 3.5 give that the intrinsic amplitude is highly dependent
on the coronal shape, $h/r$, for a specific BH source (assuming a translucent
corona, $\tau\sim 1$). Accordingly, the similar QPO rms amplitudes with
different outburst intensities indicate the coronal shape may not depend on
the individual outburst brightness.
The left panel of Fig. 9 shows that, in the rise phase, the QPO rms increases
slightly at low frequencies, while decreases sharply after reaching the peak
value at $\sim 2$ Hz. In the dynamic part of the L-T precession model, $f_{\rm
QPO}$ is negatively correlated to the outer radius, $r$, of the corona (Ingram
et al., 2009), so the increasing $f_{\rm QPO}$ indicates a decreasing $r$. A
possible explanation for the dependence of the QPO rms on frequency is that
when the coronal outer radius ($r$) evolves, the coronal shape, i.e. $h/r$,
changes synchronously. The fitting results of the rms-$f_{\rm QPO}$ relation
with the simplified L-T precession model show that $h/r$ of the rise and decay
phases are consistent at high frequencies ($f_{\rm QPO}>3$ Hz), while $h/r$ of
the decay phase is larger than that of the rise phase in the low frequency
range (see Fig. 9). The spectral analysis shows that the coronal temperature
($kT_{\rm e}$) of the decay phase is obviously higher than that of the rise
phase (see Fig. 6c). If the corona is a hot accretion flow (e.g. Advection-
Dominated Accretion Flow, ADAF), it is mainly supported by the gas pressure
(Narayan & Yi, 1994; Yuan & Narayan, 2014; Liu & Qiao, 2022), and the height
($h$) could hence be lager with the higher coronal temperature. Accordingly,
the corona in the decay phase could have relatively larger $h/r$ at a specific
frequency, and then precesses with a lower variability amplitude. This is
because the variability of the coronal projection area with respect to the
observer is smaller with a higher $h/r$ value (see Fig. 8).
### 4.2 Qualitative Interpretation with the Time-dependent Comptonization
Model
Time-dependent Comptonization models can explain quantitatively the rms
spectrum and the phase lag spectrum of QPOs, by requiring coupled oscillations
of the physical quantities: coronal temperature, $kT_{\rm e}$, temperature of
the source of seed photons, $kT_{\rm s}$ and the external heating rate,
$\dot{H}_{\rm ext}$ (see Karpouzas et al., 2020; Bellavita et al., 2022).
Although the QPO dynamic origin is not specified in these models, the recent
proposal of Mastichiadis et al. (2022) that the QPO frequency arises from a
resonance between the hot Comptonizing corona and the colder accretion disc
via the coupling of the energy gains and losses in the system, could provide
the dynamic part to the time-dependent Comptonization models, since the disc-
corona coupling is also the mechanism suggested by these models to explain the
QPO radiative properties. In the time-dependent Comptonization models, the
fractional variability amplitude of QPOs (QPO fractional rms) is normalized by
the variability amplitude of the external hating rate, $\delta\dot{H}_{\rm
ext}$, which is a fitting parameter. The energy-averaged rms in a specific
energy range is therefore dependent on both the shape and the normalization
($\delta\dot{H}_{\rm ext}$) of the rms spectrum. Karpouzas et al. (2021) and
García et al. (2022) fitted the rms spectra and phase lag spectra of type C
QPO observations of GRS 1915+105 and found $\delta\dot{H}_{\rm ext}$ is
dependent on QPO frequency, where $\delta\dot{H}_{\rm ext}$ is positively
correlated strongly to $f_{\rm QPO}$ at low frequencies ($f_{\rm QPO}<1$ Hz),
and negatively correlated to $f_{\rm QPO}$ in the narrow frequency range of
1–1.8 Hz, then decreases slightly from $f_{\rm QPO}\sim 2$ Hz to $f_{\rm
QPO}\sim 6$ Hz. Since the rms dependence on $f_{\rm QPO}$ of H 1743–322 in the
rise phase is similar to that of GRS 1915+105, the $\delta\dot{H}_{\rm ext}$
dependence on $f_{\rm QPO}$ could also affect the rms-$f_{\rm QPO}$ relation
of H 1743–322 in the rise phase. It may also work in the decay phase of
H1743–322 if one considers in decay phase the positive and negative
correlations at lower and higher frequencies, respectively. In Fig. 6, we show
that the coronal temperature of the decay phase is higher than that of the
rise phase, while there are no significant differences in photon index
($\Gamma$). In this case, as shown in Fig. 3 of Bellavita et al. (2022), the
energy-averaged QPO rms in the energy range of 3–30 keV could be relatively
lower in the decay phase. We propose the possible reason is that for the
escaping photons in the specific energy range (3–30 keV), the higher coronal
temperature indicates the less scatterings of the photons before escaping the
Comptonizing medium, hence the effect of the variability amplitude
amplification is weaker, if a balance between Compton cooling and external
heating is at work for QPO amplification within the corona. We note that the
above proposal is only qualitative, while the detailed investigation of the
radiative properties of QPOs requires fitting the rms, phase lag, and time-
averaged spectra simultaneously with the time-dependent Comptonization model
(see Karpouzas et al., 2021; Méndez et al., 2022; García et al., 2022; Zhang
et al., 2022).
Figure 10: The hardness-intensity diagram (HID) of the 2018 failed-transition
outburst monitored by HXMT/LE (the top panel) and correlation between QPO rms
and frequency using data from both of _RXTE_ /PCA and HXMT/ME (the bottom
panel). The data of _Insight_ -HXMT are taken from Wang et al. (2022). The top
panel displays the HID with separating the rise and decay phases: the rise
data are plotted as red hollow squares, while the decay data are plotted as
green hollow triangles. The bottom panel displays the correlations with
separating data sets from different instruments and outburst phases.
### 4.3 Possible Relations Between Rms Differences and Hysteresis Effect
The most recent failed-transition outburst of H 1743–322, monitored by
_Insight_ -HXMT in 2018, has been reported by Wang et al. (2022). Different
from the outbursts presented above, this outburst remained in the LHS and
never showed the significant hysteresis trace in the HID throughout the
observed stage (see Fig. 10a). The _Insight_ -HXMT data of the 2018 outburst
are taken from Wang et al. (2022). We plot the fractional rms dependence on
$f_{\rm QPO}$ using both _Insight_ -HXMT/ME and _RXTE_ /PCA data in Fig. 10b.
As one can see, in the 2018 outburst, the QPO fractional rms remains roughly
constant with a value $\sim 12\%$ without significant differences between the
rise and decay phases. The 2008b outburst is also classified into failed-
transition outbursts, however, the system experienced the LHS-HIMS transition
and a hysteresis trace in the HID during this outburst (see Coriat et al.,
2011), with the rms differences between rise and decay phases which are
consistent with those of other complete outbursts (see Fig. 5). On the basis
of the above, the different rms-$f_{\rm QPO}$ relations between the rise and
decay phases seem to be associated with the hysteresis trace in the HID,
because these two phenomena accompany each other in H 1743–322. However,
details about the relations between the rms differences and hysteresis effect
need further investigations using more observational samples from more
sources.
### 4.4 Possible Scenario of the Corona-jet Coupling in H 1743–322
Radio emission of BHXRBs is thought to be strongly related to relativistic
jets (Fender, 2001), which can also produce the Comptonization of soft photons
from the disc (see Band & Grindlay, 1986; Georganopoulos et al., 2002; Reig &
Kylafis, 2021). The jet is believed to be coupled to the accretion flow, but
the nature of this connection is still not well understood. Based on a large
dataset of _RXTE_ observations, frequent radio observations and the time-
dependent Comptonization model, a series of studies have revealed a possible
picture of the corona-jet coupling in GRS 1915+105 (see Zhang et al., 2020;
Karpouzas et al., 2021; Méndez et al., 2022; García et al., 2022). In H
1743–322, although quasi-simultaneous radio observations are not as abundant
as that in GRS 1915+105, for each individual outburst, there is a marginally
decreasing trend of the radio flux density from low QPO frequency ($<2$ Hz) to
high QPO frequency ($\sim 8$ Hz) in the rise phase (see Fig. 6), which
indicates a quenching compact jet during the hard-to-soft state transition.
van den Eijnden et al. (2017) found that the QPO hard phase lag became
negative at high frequencies. Following the idea of the time-dependent
Comptonization model (see Karpouzas et al., 2020; Bellavita et al., 2022), the
nature of the hard lag (positive lag) is Comptonization, in which hard photons
could experience more scatterings than soft ones before escaping the medium.
The phase lag becomes negative due to the _feedback_ mechanism: the hard
Comptonized photons may return back to the disc and be re-emitted later, so
the softer photons could arrive later (Lee & Miller, 1998). Based on the
above, we propose that the scenario of the corona-jet coupling in GRS 1915+105
can be applied to the rise phase of H 1743–322. At low QPO frequencies where
the radio emission is strong, the Comptonizing medium is dominated by the jet-
like corona, where the phase lag is positive (see van den Eijnden et al.,
2017). However, at high QPO frequencies where the radio emission is weak, the
jet is quenched and replaced by an extended corona, which covers the inner
parts of the thin disc. Since the disc has a large solid angle to receive the
returning back photons, the _feedback_ effect could be strong enough to
produce the observed negative phase lag (van den Eijnden et al., 2017). We
refer readers to Méndez et al. (2022) for details about the physical picture
of the coupling between the corona and the jet. In the outburst decay phase,
the coronal temperature is relatively higher and the radio emission is weaker.
If the corona is powered by the magnetic energy (Merloni & Fabian, 2001),
especially as the case that the jet and the coronal power are tapped from the
common magnetic energy reservoir (see Malzac et al., 2004), the higher coronal
temperature in the decay phase could be associated with the relatively weaker
radio emission. On the basis of the above, we propose for H 1743–322 a similar
scenario as the one proposed for GRS 1915+105 (Méndez et al., 2022) which, as
in that case, would account for the differences of the radio emission and
coronal temperature between the two outburst stages. During the outburst decay
phase, the magnetic field lines are disorganized and the magnetic energy is
mainly dissipated stochastically in the corona, probably via magnetic
reconnection, hence the jet is quenched and the coronal temperature is high.
However, the magnetic configuration could be very different during the earlier
stage of the rise phase ($f_{\rm QPO}<2$ Hz and the phase lag is positive),
where the magnetic field lines are spatially coherent with a large scale
poloidal component, which could channel materials out of the corona and
collimate them in the direction perpendicular to the disc, then the
Comptonization mainly occurs in the jet. During the hard-to-soft transition,
the magnetic field lines vary from the coherent to the disorganized
configuration, the jet is quenched and replaced by the extended corona. Such a
different journey experienced by the accretion flow in the rise and decay
phases may play a role in the difference seen in the rms-frequency relation
(see Fig. 5).
## 5 Conclusions
We performed systematic analyses of type C QPO observations from seven
outbursts of H 1743–322 caught in the _RXTE_ era. With a number of
observational samples, we confirm the independence of the type C QPO intrinsic
rms on the individual outburst brightness which has been reported as well in
GX 339–4. However, the dependence of QPO rms on frequency shows two branches
in the outburst rise and decay phases, where the radio flux and coronal
temperature are also different between the two phases. Both of the L-T
precession model and the time-dependent Comptonization model can account for
the rms difference, where the former needs a variable coronal geometric shape.
Combining the recent _Insight_ -HXMT observations of this source during its
failed-transition outburst, we suggest such the rms difference between the two
outburst stages could be also related to the hysteresis effect in the HID. The
co-evolution among the radio flux, coronal temperature and phase lags
indicates there could be corona-jet transitions in H 1743–322 which have been
recently reported in GRS 1915+105.
We are grateful to the anonymous referee for constructive comments that helped
us improve this paper. This research has made use of data obtained from the
High Energy Astrophysics Science Archive Research Center (HEASARC), provided
by NASA’s Goddard Space Flight Center, and the _Insight_ -HXMT mission, a
project funded by China National Space Administration (CNSA) and the Chinese
Academy of Sciences (CAS). This work is supported by the National Key R&D
Program of China (2021YFA0718500) and the National Natural Science Foundation
of China under grants, U1838201, U1838202, 12173103, U2038101 and U1938103.
This work is partially supported by International Partnership Program of
Chinese Academy of Sciences (Grant No.113111KYSB20190020).
## Appendix A The distribution of reduced chi-square in the Spectral fitting
In this section, we show the reduced chi-square ($\chi^{2}_{\rm red}$)
distribution of the spectral fitting with the Model 3 presented in Section 3.2
(see Fig. 11). There are one hundred and six observational samples in our
spectral analysis.
Figure 11: The reduced chi-square ($\chi^{2}_{\rm red}$) distribution of the
spectral fitting with the Model 3 presented in Section 3.2.
## Appendix B Calculation details of the coronal optical depth
We consider a simplified coronal geometry with assuming the coronal shape is a
crushed sphere (see Section 3.5). As a simplified model, the general
relativity effects and Comptonization processes are neglected in our
calculations. The effective optical depth of the corona is
$\tau=\alpha L,$ (B1)
where $\alpha$ is the absorption coefficient, $L$ is the average size of the
corona in the view of the observer. We can define the three-dimensional
Cartesian coordinates $xyz$, where the $z$-axis is consistent with the normal
of the corona ($\hat{\bm{n}}$), and the unit vector of the observer,
$\hat{\bm{o}}$, is in the $yz$ plane. Then the ellipsoidal equation of the
coronal surface is written by
$\frac{x^{2}+y^{2}}{r^{2}}+\frac{z^{2}}{h^{2}}=1,$ (B2)
where $r$ is the coronal radius and $h$ is the coronal height. We consider
that the distance between the source and observer is far larger than the
coronal radius (i.e. $D\gg r$), so the photon trajectories from the corona to
the observer can be described as a cluster of parallel lines which are
perpendicular to the $x$-axis. Then the linear equation of a representative
photon trajectory to the observer is
$\displaystyle z$ $\displaystyle=\cot{\theta}\cdot y+z_{0}$ $\displaystyle x$
$\displaystyle=x_{0},$ (B3)
where $\theta$ is the include angle between $\hat{\bm{n}}$ and $\hat{\bm{o}}$,
$z_{0}$ is the $z$-intercept of the projection of the line in the $yz$ plane,
and $x_{0}$ is the $x$-intercept of the projection of the line in the $xy$
plane, respectively. Then the length of the corona which contributes to the
flux in the direction of the representative photon trajectory is
$l=\sqrt{1+\cot^{2}{\theta}}\frac{2hr\sqrt{\left(1-\frac{x_{0}^{2}}{r^{2}}\right)h^{2}+\left(1-\frac{x_{0}^{2}}{r^{2}}\right)r^{2}\cot^{2}{\theta}-z_{0}^{2}}}{h^{2}+r^{2}\cot^{2}{\theta}},$
(B4)
i.e. the length of the line cut by the ellipsoidal surface. The average size
of the corona in the view of the observer can be calculated by
$\displaystyle L$
$\displaystyle=\frac{\int_{-r}^{r}\int_{-s}^{s}l(x_{0},z_{0}){\rm d}z_{0}{\rm
d}x_{0}}{\int_{-r}^{r}\int_{-s}^{s}{\rm d}z_{0}{\rm d}x_{0}}$
$\displaystyle=\frac{4}{3}h\left\\{\left(h/r\right)^{2}+\left[1-\left(h/r\right)^{2}\right]\cos^{2}{\theta}\right\\}^{-1/2},$
(B5)
where
$s\equiv\sqrt{\left[1-(x_{0}/r)^{2}\right]h^{2}+\left[1-(x_{0}/r)^{2}\right]r^{2}\cot^{2}{\theta}}$.
When $z_{0}=\pm s$, the line of Eq. B is tangent to the ellipsoidal surface.
Based on Eq. B1 and B, the optical depth can be written by
$\tau=\frac{4}{3}\alpha
h\left\\{\left(h/r\right)^{2}+\left[1-\left(h/r\right)^{2}\right]\cos^{2}{\theta}\right\\}^{-1/2}.$
(B6)
We define $\tau_{0}\equiv\frac{4}{3}\alpha h$, where $\tau_{0}$ is the minimum
optical depth of the corona, i.e. viewed from the coronal normal ($\theta=0$),
then the optical depth can be written by
$\tau=\tau_{0}\left\\{\left(h/r\right)^{2}+\left[1-\left(h/r\right)^{2}\right]\cos^{2}{\theta}\right\\}^{-1/2}.$
(B7)
## Appendix C MCMC Parameter Probability Distributions
In this section, we show the contour maps and probability distributions for
the set of model parameters derived using the MCMC analysis of the relation
between the QPO intrinsic rms and frequency. The MCMC analysis is preformed
using `emcee` package (Foreman-Mackey et al., 2013), and the contour maps and
probability distributions are plotted using `corner` package (Foreman-Mackey,
2016). For each map, we show the 0.16, 0.5, and 0.84 quantiles (see Fig. 12).
Figure 12: One- and two-dimensional projections of the posterior probability
distributions, and the 0.16, 0.5 and 0.84 quantile contours derived from the
MCMC analysis for the model parameters $k_{1}$, $k_{2}$ and $k_{3}$ described
in Eq. 11. The left panel is plotted for the rise phase and the right panel is
plotted for the decay phase.
## Appendix D QPO parameters and radio observations
In this study, we preform a systematic analysis of type C QPOs with one
hundred and six _RXTE_ /PCA observational samples of black hole X-ray binary H
1743–322. This source also exhibit significant radio emission, so we take the
quasi-simultaneous radio observational results from previously published
studies to add to our joint analysis. In this section, we present the QPO
parameters of our analysis and the quasi-simultaneous radio flux measurements
(if present) at $\sim 8.5$ GHz. The parameters, QPO frequency ($f_{\rm QPO}$),
full width at half maximum (FWHM), fractional rms amplitude and radio flux
density at $\sim 8.5$ GHz ($S_{\nu=8.5{\rm GHz}}$), etc., are presented in
Table 2.
Table 2: QPO Parameters and the Quasi-simultaneous Radio Flux Density
ObsIDa | Outburst Phaseb | X-ray MJDc | $f_{\rm QPO}$ | FWHMd | Fractional rmse | Radio MJDf | $S_{\nu=8.5{\rm GHz}}^{\rm g}$ | Notes
---|---|---|---|---|---|---|---|---
| | | (Hz) | (Hz) | (%) | | (mJy) |
80138-01-06-00 | Rise | 52739.66 | $3.22_{-0.01}^{+0.01}$ | $0.35_{-0.02}^{+0.02}$ | $12.66_{-0.21}^{+0.22}$ | 52739.46 | $20.68\pm 0.06$ | VLAh
80138-01-07-00 | Rise | 52741.83 | $7.17_{-0.03}^{+0.03}$ | $1.13_{-0.07}^{+0.07}$ | $5.49_{-0.13}^{+0.13}$ | 52741.56 | $7.71\pm 0.12$ | VLAh
80146-01-01-00 | Rise | 52743.22 | $8.51_{-0.02}^{+0.02}$ | $0.56_{-0.04}^{+0.04}$ | $3.71_{-0.09}^{+0.09}$ | 52742.52 | $3.87\pm 0.13$ | VLAh
80146-01-02-00 | Rise | 52744.20 | $5.62_{-0.02}^{+0.02}$ | $0.85_{-0.04}^{+0.04}$ | $8.11_{-0.13}^{+0.13}$ | 52745.43 | $37.15\pm 0.13$ | VLAh
80146-01-03-00 | Rise | 52746.18 | $4.74_{-0.01}^{+0.01}$ | $0.55_{-0.03}^{+0.03}$ | $10.52_{-0.21}^{+0.22}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-03-01 | Rise | 52747.61 | $7.01_{-0.03}^{+0.03}$ | $1.19_{-0.08}^{+0.08}$ | $5.52_{-0.14}^{+0.14}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-29-00 | Rise | 52766.56 | $5.60_{-0.02}^{+0.02}$ | $0.74_{-0.04}^{+0.05}$ | $7.87_{-0.15}^{+0.15}$ | 52765.42 | $11.12\pm 0.13$ | VLAh
80146-01-30-00 | Rise | 52767.81 | $4.43_{-0.01}^{+0.01}$ | $0.53_{-0.02}^{+0.02}$ | $10.92_{-0.13}^{+0.13}$ | 52767.51 | $23.02\pm 0.12$ | VLAh
80146-01-31-00 | Rise | 52768.53 | $5.41_{-0.02}^{+0.02}$ | $0.83_{-0.05}^{+0.05}$ | $9.41_{-0.18}^{+0.19}$ | 52768.49 | $30.14\pm 0.16$ | VLAh
80146-01-32-00 | Rise | 52769.72 | $4.90_{-0.01}^{+0.01}$ | $0.55_{-0.03}^{+0.03}$ | $10.74_{-0.21}^{+0.23}$ | 52769.51 | $23.80\pm 0.14$ | VLAh
80146-01-33-01 | Rise | 52770.37 | $6.02_{-0.03}^{+0.03}$ | $0.85_{-0.06}^{+0.07}$ | $7.66_{-0.20}^{+0.20}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-33-00 | Rise | 52770.65 | $6.22_{-0.02}^{+0.02}$ | $0.85_{-0.05}^{+0.06}$ | $7.16_{-0.15}^{+0.15}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-34-00 | Rise | 52771.74 | $2.82_{-0.01}^{+0.01}$ | $0.33_{-0.02}^{+0.02}$ | $13.22_{-0.17}^{+0.19}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-35-00 | Rise | 52771.97 | $2.27_{-0.01}^{+0.01}$ | $0.19_{-0.03}^{+0.03}$ | $12.06_{-0.52}^{+0.52}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-36-00 | Rise | 52772.67 | $1.84_{-0.01}^{+0.01}$ | $0.25_{-0.01}^{+0.01}$ | $13.92_{-0.26}^{+0.26}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-37-00 | Rise | 52773.66 | $1.90_{-0.01}^{+0.01}$ | $0.26_{-0.02}^{+0.02}$ | $13.71_{-0.25}^{+0.25}$ | 52773.36 | $35.76\pm 0.23$ | VLAh
80146-01-39-00 | Rise | 52775.57 | $2.15_{-0.01}^{+0.01}$ | $0.29_{-0.02}^{+0.02}$ | $14.11_{-0.24}^{+0.24}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-40-00 | Rise | 52776.62 | $1.72_{-0.01}^{+0.01}$ | $0.32_{-0.02}^{+0.02}$ | $16.21_{-0.27}^{+0.27}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-41-00 | Rise | 52777.61 | $2.50_{-0.01}^{+0.01}$ | $0.31_{-0.02}^{+0.02}$ | $13.83_{-0.23}^{+0.22}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-42-00 | Rise | 52778.46 | $3.21_{-0.01}^{+0.01}$ | $0.46_{-0.02}^{+0.02}$ | $13.83_{-0.18}^{+0.19}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-43-01 | Rise | 52779.53 | $3.82_{-0.02}^{+0.02}$ | $0.41_{-0.03}^{+0.04}$ | $11.87_{-0.33}^{+0.34}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-43-00 | Rise | 52779.58 | $3.81_{-0.01}^{+0.01}$ | $0.37_{-0.02}^{+0.02}$ | $12.79_{-0.21}^{+0.22}$ | 52779.44 | $11.99\pm 0.17$ | VLAh
80146-01-44-00 | Rise | 52780.57 | $3.82_{-0.01}^{+0.01}$ | $0.37_{-0.02}^{+0.02}$ | $12.59_{-0.21}^{+0.22}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-45-00 | Rise | 52781.55 | $3.64_{-0.01}^{+0.01}$ | $0.45_{-0.02}^{+0.02}$ | $13.31_{-0.17}^{+0.18}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-46-00 | Rise | 52782.67 | $4.70_{-0.01}^{+0.01}$ | $0.45_{-0.03}^{+0.03}$ | $11.18_{-0.19}^{+0.21}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-47-00 | Rise | 52783.46 | $6.99_{-0.02}^{+0.02}$ | $1.40_{-0.05}^{+0.06}$ | $6.13_{-0.10}^{+0.10}$ | $\cdots$ | $\cdots$ | $\cdots$
80146-01-50-00 | Rise | 52786.29 | $9.43_{-0.04}^{+0.04}$ | $0.85_{-0.08}^{+0.09}$ | $2.09_{-0.08}^{+0.08}$ | 52786.36 | $16.06\pm 0.11$ | VLAh
80137-01-25-00 | Decay | 52937.02 | $7.11_{-0.15}^{+0.21}$ | $0.91_{-0.41}^{+1.51}$ | $6.89_{-1.17}^{+1.18}$ | 52939.99 | $0.14\pm 0.04$ | VLAh
80137-01-26-00 | Decay | 52938.00 | $5.93_{-0.39}^{+0.44}$ | $1.89_{-0.77}^{+1.32}$ | $9.05_{-1.66}^{+1.93}$ | $\cdots$ | $\cdots$ | $\cdots$
80137-02-01-00 | Decay | 52944.11 | $2.48_{-0.11}^{+0.15}$ | $0.97_{-0.38}^{+0.68}$ | $10.03_{-1.75}^{+2.11}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-03-01 | Decay | 54492.18 | $7.03_{-0.10}^{+0.10}$ | $0.67_{-0.26}^{+0.41}$ | $5.81_{-0.87}^{+1.19}$ | 54493.32 | $0.44\pm 0.09$ | ATCAi
93427-01-04-00 | Decay | 54498.84 | $3.90_{-0.08}^{+0.08}$ | $0.65_{-0.22}^{+0.33}$ | $8.00_{-1.10}^{+1.71}$ | 54499.74 | $0.52\pm 0.06$ | VLAj
93427-01-04-02 | Decay | 54500.80 | $2.60_{-0.10}^{+0.13}$ | $1.50_{-0.52}^{+0.82}$ | $13.42_{-1.77}^{+1.98}$ | 54501.64 | $0.48\pm 0.08$ | VLAj
93427-01-04-03 | Decay | 54502.83 | $2.36_{-0.16}^{+0.14}$ | $0.77_{-0.35}^{+0.39}$ | $8.86_{-1.44}^{+1.74}$ | 54502.56 | $0.45\pm 0.09$ | VLAj
93427-01-09-00 | Rise | 54742.98 | $0.33_{-0.01}^{+0.01}$ | $0.03_{-0.01}^{+0.01}$ | $15.19_{-1.67}^{+1.68}$ | 54744.21 | $1.74\pm 0.07$ | ATCAi
93427-01-09-01 | Rise | 54746.51 | $0.38_{-0.01}^{+0.01}$ | $0.03_{-0.01}^{+0.01}$ | $15.69_{-0.89}^{+0.91}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-09-03 | Rise | 54747.49 | $0.43_{-0.01}^{+0.01}$ | $0.03_{-0.01}^{+0.01}$ | $15.00_{-0.89}^{+0.91}$ | 54747.44 | $2.54\pm 0.08$ | ATCAi
93427-01-09-02 | Rise | 54748.21 | $0.47_{-0.01}^{+0.01}$ | $0.04_{-0.01}^{+0.01}$ | $14.31_{-0.98}^{+0.99}$ | 54748.44 | $2.43\pm 0.09$ | ATCAi
93427-01-10-00 | Rise | 54750.37 | $0.58_{-0.01}^{+0.01}$ | $0.06_{-0.02}^{+0.02}$ | $14.19_{-1.31}^{+1.31}$ | 54749.36 | $2.38\pm 0.11$ | ATCAi
93427-01-10-01 | Rise | 54752.26 | $0.68_{-0.01}^{+0.01}$ | $0.07_{-0.01}^{+0.01}$ | $15.59_{-0.91}^{+0.92}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-10-02 | Rise | 54755.23 | $0.75_{-0.02}^{+0.02}$ | $0.12_{-0.02}^{+0.03}$ | $13.88_{-1.24}^{+1.22}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-11-00 | Rise | 54756.25 | $0.86_{-0.01}^{+0.01}$ | $0.10_{-0.01}^{+0.01}$ | $15.44_{-0.67}^{+0.68}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-11-01 | Rise | 54758.15 | $1.02_{-0.01}^{+0.01}$ | $0.12_{-0.01}^{+0.02}$ | $14.89_{-0.62}^{+0.62}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-11-03 | Rise | 54762.14 | $5.60_{-0.06}^{+0.06}$ | $0.49_{-0.17}^{+0.23}$ | $5.43_{-0.66}^{+0.75}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-12-04 | Decay | 54767.84 | $3.87_{-0.03}^{+0.03}$ | $0.47_{-0.08}^{+0.09}$ | $10.48_{-0.62}^{+0.65}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-12-02 | Decay | 54768.64 | $3.30_{-0.03}^{+0.03}$ | $0.35_{-0.07}^{+0.09}$ | $11.45_{-0.81}^{+0.84}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-12-05 | Decay | 54769.14 | $2.91_{-0.04}^{+0.04}$ | $0.38_{-0.08}^{+0.10}$ | $12.88_{-1.05}^{+1.10}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-13-00 | Decay | 54770.12 | $2.61_{-0.02}^{+0.02}$ | $0.42_{-0.06}^{+0.06}$ | $12.40_{-0.61}^{+0.63}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-13-05 | Decay | 54770.40 | $2.26_{-0.02}^{+0.02}$ | $0.25_{-0.04}^{+0.05}$ | $11.90_{-0.68}^{+0.70}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-13-04 | Decay | 54771.76 | $2.38_{-0.02}^{+0.02}$ | $0.29_{-0.05}^{+0.05}$ | $11.89_{-0.68}^{+0.70}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-13-01 | Decay | 54772.15 | $2.25_{-0.03}^{+0.04}$ | $0.36_{-0.12}^{+0.15}$ | $12.44_{-1.27}^{+1.37}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-13-02 | Decay | 54773.27 | $2.27_{-0.03}^{+0.03}$ | $0.33_{-0.07}^{+0.08}$ | $12.21_{-0.86}^{+0.90}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-13-06 | Decay | 54774.64 | $2.12_{-0.02}^{+0.02}$ | $0.38_{-0.06}^{+0.07}$ | $12.26_{-0.75}^{+0.79}$ | 54774.43 | $0.94\pm 0.12$ | ATCAi
93427-01-13-03 | Decay | 54775.56 | $1.83_{-0.03}^{+0.03}$ | $0.32_{-0.06}^{+0.07}$ | $11.92_{-0.92}^{+0.96}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-14-00 | Decay | 54777.86 | $1.89_{-0.04}^{+0.04}$ | $0.29_{-0.07}^{+0.08}$ | $10.59_{-1.04}^{+1.10}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-14-01 | Decay | 54778.77 | $1.82_{-0.03}^{+0.04}$ | $0.29_{-0.09}^{+0.12}$ | $10.99_{-1.21}^{+1.32}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-14-02 | Decay | 54779.04 | $1.52_{-0.02}^{+0.02}$ | $0.21_{-0.05}^{+0.06}$ | $11.68_{-1.09}^{+1.14}$ | 54779.35 | $0.94\pm 0.08$ | ATCAi
93427-01-14-03 | Decay | 54780.02 | $1.48_{-0.03}^{+0.03}$ | $0.20_{-0.06}^{+0.07}$ | $9.89_{-1.16}^{+1.22}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-14-04 | Decay | 54781.78 | $1.72_{-0.02}^{+0.02}$ | $0.17_{-0.05}^{+0.07}$ | $9.88_{-1.01}^{+1.06}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-14-05 | Decay | 54782.89 | $2.21_{-0.04}^{+0.04}$ | $0.32_{-0.08}^{+0.10}$ | $11.15_{-1.11}^{+1.16}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-14-06 | Decay | 54783.81 | $2.07_{-0.05}^{+0.06}$ | $0.42_{-0.14}^{+0.20}$ | $11.16_{-1.44}^{+1.73}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-00 | Decay | 54784.45 | $1.79_{-0.04}^{+0.05}$ | $0.26_{-0.09}^{+0.12}$ | $9.74_{-1.22}^{+1.36}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-01 | Decay | 54785.70 | $1.95_{-0.08}^{+0.09}$ | $0.52_{-0.19}^{+0.27}$ | $10.83_{-1.74}^{+2.16}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-02 | Decay | 54786.48 | $1.76_{-0.03}^{+0.04}$ | $0.28_{-0.08}^{+0.11}$ | $9.31_{-1.07}^{+1.18}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-03 | Decay | 54787.73 | $1.54_{-0.06}^{+0.06}$ | $0.27_{-0.11}^{+0.16}$ | $9.15_{-1.57}^{+1.77}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-04 | Decay | 54788.64 | $0.97_{-0.03}^{+0.04}$ | $0.18_{-0.07}^{+0.14}$ | $9.29_{-1.67}^{+2.06}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-06 | Decay | 54788.84 | $0.99_{-0.07}^{+0.08}$ | $0.23_{-0.14}^{+0.19}$ | $8.09_{-2.08}^{+2.52}$ | $\cdots$ | $\cdots$ | $\cdots$
93427-01-15-05 | Decay | 54789.49 | $0.88_{-0.05}^{+0.05}$ | $0.22_{-0.11}^{+0.24}$ | $8.20_{-1.82}^{+2.68}$ | $\cdots$ | $\cdots$ | $\cdots$
94413-01-02-00 | Rise | 54980.40 | $0.91_{-0.01}^{+0.01}$ | $0.04_{-0.01}^{+0.01}$ | $17.96_{-0.96}^{+0.96}$ | 54978.38 | $2.24\pm 0.03$ | VLAk
94413-01-02-02 | Rise | 54980.85 | $1.00_{-0.01}^{+0.01}$ | $0.03_{-0.01}^{+0.01}$ | $17.47_{-0.75}^{+0.75}$ | 54978.38 | $2.24\pm 0.03$ | VLAk
94413-01-02-01 | Rise | 54981.95 | $1.19_{-0.01}^{+0.01}$ | $0.03_{-0.01}^{+0.01}$ | $17.16_{-1.07}^{+1.08}$ | 54981.95 | $2.73\pm 0.10$ | VLAk
94413-01-02-05 | Rise | 54982.28 | $1.28_{-0.01}^{+0.01}$ | $0.04_{-0.01}^{+0.01}$ | $17.47_{-0.81}^{+0.82}$ | $\cdots$ | $\cdots$ | $\cdots$
94413-01-02-04 | Rise | 54983.33 | $2.02_{-0.01}^{+0.01}$ | $0.09_{-0.01}^{+0.01}$ | $17.43_{-0.66}^{+0.67}$ | $\cdots$ | $\cdots$ | $\cdots$
94413-01-02-03 | Rise | 54984.37 | $3.58_{-0.01}^{+0.01}$ | $0.33_{-0.04}^{+0.04}$ | $14.97_{-0.69}^{+1.21}$ | 54984.35 | $1.8\pm 0.3$ | VLBAl
94413-01-07-00 | Decay | 55016.32 | $4.94_{-0.12}^{+0.06}$ | $0.26_{-0.26}^{+0.45}$ | $6.22_{-1.46}^{+2.10}$ | $\cdots$ | $\cdots$ | $\cdots$
94413-01-07-01 | Decay | 55019.45 | $3.40_{-0.10}^{+0.09}$ | $0.39_{-0.18}^{+0.29}$ | $7.49_{-1.54}^{+2.04}$ | 55019.46 | $0.592\pm 0.055$ | VLAk
94413-01-07-02 | Decay | 55021.42 | $3.81_{-0.11}^{+0.17}$ | $0.62_{-0.29}^{+0.85}$ | $10.16_{-2.10}^{+3.87}$ | 55021.42 | $0.410\pm 0.074$ | VLAk
95405-01-02-06 | Decay | 55223.39 | $3.82_{-0.04}^{+0.04}$ | $0.43_{-0.10}^{+0.13}$ | $9.55_{-0.84}^{+0.91}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-02-02 | Decay | 55224.67 | $3.05_{-0.04}^{+0.04}$ | $0.39_{-0.09}^{+0.12}$ | $10.54_{-0.95}^{+1.04}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-03-00 | Decay | 55226.53 | $2.02_{-0.03}^{+0.04}$ | $0.30_{-0.08}^{+0.10}$ | $11.23_{-1.11}^{+1.19}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-03-04 | Decay | 55227.77 | $2.24_{-0.04}^{+0.03}$ | $0.30_{-0.07}^{+0.09}$ | $11.36_{-1.07}^{+1.12}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-03-01 | Decay | 55228.61 | $2.20_{-0.03}^{+0.03}$ | $0.26_{-0.06}^{+0.07}$ | $10.21_{-0.86}^{+0.89}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-03-05 | Decay | 55229.54 | $1.71_{-0.04}^{+0.04}$ | $0.24_{-0.07}^{+0.09}$ | $10.10_{-1.22}^{+1.31}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-03-02 | Decay | 55230.59 | $1.35_{-0.04}^{+0.04}$ | $0.32_{-0.10}^{+0.14}$ | $10.75_{-1.40}^{+1.62}$ | $\cdots$ | $\cdots$ | $\cdots$
95405-01-04-01 | Decay | 55233.40 | $0.90_{-0.03}^{+0.03}$ | $0.15_{-0.09}^{+0.17}$ | $8.15_{-1.69}^{+2.44}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-01-00 | Rise | 55418.41 | $1.00_{-0.01}^{+0.01}$ | $0.05_{-0.01}^{+0.01}$ | $16.56_{-0.60}^{+0.60}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-02-01 | Rise | 55419.09 | $1.04_{-0.01}^{+0.01}$ | $0.04_{-0.01}^{+0.01}$ | $17.38_{-1.45}^{+1.46}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-02-00 | Rise | 55420.23 | $1.17_{-0.01}^{+0.01}$ | $0.04_{-0.01}^{+0.01}$ | $16.66_{-0.61}^{+0.61}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-03-00 | Rise | 55421.28 | $1.48_{-0.01}^{+0.01}$ | $0.05_{-0.01}^{+0.01}$ | $17.02_{-0.64}^{+0.64}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-02-03 | Rise | 55422.02 | $1.76_{-0.01}^{+0.01}$ | $0.08_{-0.01}^{+0.01}$ | $17.30_{-0.89}^{+0.89}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-02-02 | Rise | 55423.20 | $2.96_{-0.01}^{+0.02}$ | $0.20_{-0.03}^{+0.04}$ | $14.88_{-0.80}^{+0.87}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-03-01 | Rise | 55424.06 | $4.80_{-0.04}^{+0.04}$ | $0.37_{-0.07}^{+0.09}$ | $9.44_{-0.70}^{+0.71}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-23-00 | Decay | 55456.67 | $3.34_{-0.07}^{+0.08}$ | $0.58_{-0.22}^{+0.38}$ | $11.00_{-1.77}^{+2.30}$ | $\cdots$ | $\cdots$ | $\cdots$
95360-14-23-01 | Decay | 55457.12 | $2.57_{-0.04}^{+0.05}$ | $0.25_{-0.11}^{+0.18}$ | $9.34_{-1.47}^{+1.77}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-01-00 | Rise | 55663.67 | $0.43_{-0.01}^{+0.01}$ | $0.03_{-0.01}^{+0.01}$ | $14.61_{-0.87}^{+0.88}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-02-00 | Rise | 55667.59 | $0.67_{-0.01}^{+0.01}$ | $0.05_{-0.01}^{+0.01}$ | $15.72_{-0.53}^{+0.54}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-02-01 | Rise | 55668.98 | $0.89_{-0.01}^{+0.01}$ | $0.08_{-0.01}^{+0.02}$ | $16.05_{-0.96}^{+0.99}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-02-02 | Rise | 55670.62 | $1.36_{-0.01}^{+0.01}$ | $0.14_{-0.01}^{+0.01}$ | $15.80_{-0.52}^{+0.53}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-02-05 | Rise | 55671.53 | $1.82_{-0.01}^{+0.01}$ | $0.21_{-0.03}^{+0.03}$ | $15.61_{-0.67}^{+0.67}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-02-03 | Rise | 55672.84 | $3.61_{-0.03}^{+0.03}$ | $0.29_{-0.06}^{+0.07}$ | $12.21_{-0.86}^{+0.87}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-05-01 | Decay | 55690.13 | $2.98_{-0.07}^{+0.07}$ | $0.54_{-0.17}^{+0.27}$ | $10.99_{-1.35}^{+1.49}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-05-02 | Decay | 55691.51 | $2.01_{-0.03}^{+0.03}$ | $0.24_{-0.11}^{+0.16}$ | $11.00_{-1.31}^{+1.51}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-05-03 | Decay | 55693.00 | $1.81_{-0.04}^{+0.04}$ | $0.28_{-0.10}^{+0.14}$ | $10.88_{-1.34}^{+1.47}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-06-00 | Decay | 55694.00 | $1.47_{-0.03}^{+0.04}$ | $0.24_{-0.11}^{+0.16}$ | $9.48_{-1.39}^{+1.59}$ | $\cdots$ | $\cdots$ | $\cdots$
96425-01-06-01 | Decay | 55695.42 | $1.20_{-0.04}^{+0.04}$ | $0.19_{-0.08}^{+0.14}$ | $8.43_{-1.47}^{+1.64}$ | $\cdots$ | $\cdots$ | $\cdots$
Note. — a The power-law index of the emissivity profile ($\epsilon\propto
r^{-q}$). b Normalization of $diskbb$ model. c Normalization of $nthcomp$
model. d Log of the ionization parameter ($\xi$) of the accretion disc, where
$\xi=L/nR^{2}$, with $L$ as the ionizing luminosity, $n$ as the gas density,
and $R$ as the distance to the ionizing source. e Normalization of $relxillcp$
model.
Note. — a Observational ID of _RXTE_ data. b Outburst phase (rise phase or
decay phase). c Modified Julian Day of the _RXTE_ observation. d Full width at
half maximum of the QPO. e The QPO fractional rms calculated by Eq. 1. f
Modified Julian Day of the radio observation. g The radio flux density at
$\sim$8.5 GHz. h The radio flux measurements of Very Large Array (VLA) are
taken from McClintock et al. (2009). i The radio flux measurements of
Australia Telescope Compact Array (ATCA) are taken from Coriat et al. (2011).
j The radio flux measurements of Very Large Array (VLA) are taken from Jonker
et al. (2010). k The radio flux measurements of Very Large Array (VLA) are
taken from Coriat et al. (2011). l The radio flux measurements of Very Long
Baseline Array (VLBA) are taken from Miller-Jones et al. (2012).
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|
# High-Throughput Ab Initio Design of Atomic Interfaces using InterMatch
Eli Gerber School of Applied and Engineering Physics, Cornell University,
Ithaca, NY 14853, USA Steven B. Torrisi Department of Physics, Harvard
University, Cambridge, MA 02138, USA Energy & Materials Division, Toyota
Research Institute, Los Altos, CA 94022, USA Sara Shabani Department of
Physics, Columbia University, New York, NY, USA Eric Seewald Department of
Physics, Columbia University, New York, NY, USA Jordan Pack Department of
Physics, Columbia University, New York, NY, USA Jennifer E. Hoffman
Department of Physics, Harvard University, Cambridge, MA 02138, USA John A.
Paulson School of Engineering and Applied Sciences,Harvard University,
Cambridge, MA 02138, USA Cory R. Dean Department of Physics, Columbia
University, New York, NY, USA Abhay N. Pasupathy Department of Physics,
Columbia University, New York, NY, USA Eun-Ah Kim Department of Physics,
Cornell University, Ithaca, NY 14853, USA
( [file: main])
###### Abstract
Forming a hetero-interface is a materials-design strategy that can access an
astronomically large phase space. However, the immense phase space
necessitates a high-throughput approach for an optimal interface design. Here
we introduce a high-throughput computational framework, InterMatch, for
efficiently predicting charge transfer, strain, and superlattice structure of
an interface by leveraging the databases of individual bulk materials.
Specifically, the algorithm reads in the lattice vectors, density of states,
and the stiffness tensors for each material in their isolated form from the
Materials Project. From these bulk properties, InterMatch estimates the
interfacial properties. We benchmark InterMatch predictions for the charge
transfer against experimental measurements and supercell density-functional
theory calculations. We then use InterMatch to predict promising interface
candidates for doping transition metal dichalcogenide MoSe2. Finally, we
explain experimental observation of factor of 10 variation in the supercell
periodicity within a few microns in graphene/$\alpha$-RuCl3 by exploring low
energy superlattice structures as a function of twist angle using InterMatch.
We anticipate our open-source InterMatch algorithm accelerating and guiding
ever-growing interfacial design efforts. Moreover, the interface database
resulting from the InterMatch searches presented in this paper can be readily
accessed through MPContribs.
Figure 1: (a) Role of InterMatch in the materials discovery process. (b) Input
from the bulk database. Lattice vectors $\vec{a}_{\alpha}$,
$\vec{b}_{\alpha}$, density-of-states $g_{\alpha}$ of systems $\alpha=1,2$.
Elastic tensors are additional inputs. (c) $E_{F}^{\alpha}$ are bulk Fermi
levels and $E_{F}^{\prime}$ is new equilibrium Fermi level. (d) $\Delta n$ is
the transferred charge density and $d$ is the interlayer separation between
the two systems, taken to be the sum of the largest van der Waals radii of the
species in each system 1 and 2. (e) Superlattice vectors $\vec{v_{i}}$ (orange
arrows) and their near-equivalent vectors $\vec{u_{i}}$ (magenta arrows).
Candidate supercells, formed in each basis by combining
$\\{(\vec{v}_{i},\vec{u}_{i}),(\vec{v}_{j},\vec{u}_{j})\\}$ pairs specify the
strain $\varepsilon_{i}$ (blue arrows). (f) Optimal supercell minimizes the
elastic energy and the number of atoms in the cell.
With increasing control in interface fabrication, interfacial systems form an
arena of limitless possibilitiesGeim and Grigorieva (2013). Recent
developments with moiré heterostructuresKennes et al. (2021) further enlarged
the phase space to include the twist angle. However, the vast space of
possibilities also implies it is crucial to go beyond serendipitous
discoveries and empirical explorations to effectively harness the intrinsic
potential of interfacial systems. The traditional approach to theoretically
studying interfaces is to carry out density-functional theory (DFT)
calculations on a supercell system consisting of two materialsKomsa and
Krasheninnikov (2013); Terrones and Terrones (2014); Ebnonnasir et al. (2014);
Bokdam et al. (2014); Zhou et al. (2015). While such approaches are rigorous,
computational limitations regularly require imposing unnatural strain to form
a periodic structure. Moreover, the $\mathcal{O}(N^{3})$ scaling of DFT in the
number of electrons $N$ for each such calculation prohibits a comprehensive
exploration. Some of us recently proposed an “intermediate scale” approach
called Mismatched INterface Theory (MINT)Gerber et al. (2020), which can
predict charge transfer and natural strain approximating one layer of the
interface using finite-size scaling of atomic clusters. While MINT
calculations are much more computationally affordable, they are not fast
enough for an exhaustive survey in real-time. Therefore, a comprehensive and
fast approach to scanning the relevant phase space of interfacial combinations
is greatly needed.
With advancements in widely available comprehensive materials databasesHorton
et al. (2021); Levin (2020); Landis et al. (2012); Curtarolo et al. (2012);
Jain et al. (2013); Saal et al. (2013); de Jong et al. (2015); Ashton et al.
(2017); Haastrup et al. (2018); Zhou et al. (2019); Draxl and Scheffler
(2019); Talirz et al. (2020); Choudhary et al. (2020a), it is timely to
establish a high-throughput approach to interface design that can leverage the
information contained in these databases to make predictions of interface
physics. While it is possible to look up a pair of materials in the database
and try to match their properties, the phase space of all possible
combinations amounts easily to $>\mathcal{O}(10^{6})$ possibilities, which is
beyond the scale of what a manual search can reasonably accomplish. At the
same time, it is desirable to benefit from extensive existing databases and
search exhaustively for ideal combinations to reach a given objective. Indeed,
efforts to make interfacial predictions using bulk material databases are
beginning to emergeMathew et al. (2016); Ding et al. (2016); Choudhary et al.
(2020b); Boland and Singh (2022). However, so far, the existing approaches aid
growth and calculation decisions for a specific pair of materials rather than
allowing for a comprehensive query to yield fast, approximate predictions over
a wide range of possible interfaces.
In this paper, we introduce InterMatch, which uses information readily
available from preexisting materials databases such as the Materials Project
and 2DMatPedia databases to predict charge transferMilnes and Feucht (1972),
strainLandau et al. (1986), stabilityMuller (1994), and optimal
superlatticeKaxiras (2003) of an atomic interface. Using these predictions,
InterMatch can narrow the candidate pool from $C>\mathcal{O}(10^{6})$ to
$\mathcal{O}(10)$ that can then be investigated in greater detail using MINT
or supercell DFT (see Fig. 1(a)). We first illustrate how two branches of
InterMatch predict the charge transfer and optimal superlattice after querying
the entries of the Materials Project for each of the constituents of the
interface. We then benchmark InterMatch predictions for the charge transfer
against experimental measurements and supercell DFT predictions. We then
employ InterMatch’s branches to address two bottleneck problems obstructing
design of interfaces towards the goal of discovering new physics: the problem
of doping transition metal dichalcogenides and the problem of predicting
stable interface structure, applied to the graphene/$\alpha$-RuCl3 system. We
comment on many other classes of interfaces that can be optimized using
InterMatch.
Starting with ab initio materials data, InterMatch performs high-throughput
screening of possible heterostructures via pairwise calculation of desired
interface properties including charge transfer $\Delta n$, strain tensor
$\tilde{\varepsilon}$, optimized superlattice vectors $\vec{v_{1}},$
$\vec{v_{2}}$, and number of atoms $N$. Once InterMatch identifies a promising
pool of candidate combinations, one can make a more in-depth analysis of the
smaller pool using MINT or supercell DFT (See Fig 1(a)). The InterMatch
algorithm has two branches to predict two key electronic and mechanical
characteristics of candidate interfaces: charge transfer and optimized
superlattice structure (See Fig. 1). One branch is devoted to calculating
charge transfer (Fig. 1(c)-(d)), and the other is devoted to optimizing
supercell structure by minimizing the number of atoms and elastic energy (Fig.
1(e)-(f)). For the first branch that estimates the direction and magnitude of
charge transfer, we use a simple model to describe the Fermi level shifts
occurring in each material when they are brought together in proximityRuan and
Ching (1987). Fig. 1(c) shows how the Fermi level shifts are determined:
systems 1 and 2 are designated as “donor” or “acceptor” based on their
relative Fermi levels $E_{F}^{1}$ and $E_{F}^{2}$, and the difference of
integrals over $g_{1}(E)$ and $g_{2}(E)$
$\int_{E_{F}^{{}^{\prime}}}^{E_{F}^{1}}dE\;g_{1}(E)=\int_{E_{F}^{2}}^{E_{F}^{{}^{\prime}}}dE\;g_{2}(E)$
(1)
is minimized to determine the equilibrium Fermi level $E_{F}^{{}^{\prime}}$.
We take interaction between the two systems at the interface into account in
the estimation of the charge transfer $\Delta n$ using a simple capacitor
modelLowell and Rose-Innes (1980). Specifically, we model the interface as a
parallel plate capacitor with the separation $d$ given by the sum of the
largest van der Waals radii of the species in each system 1 and 2 (See Fig.
1(d)). The charge transfer depends on the equilibrium Fermi level
$E_{F}^{\prime}$ and the distance $d$ as $e\Delta
n=\varepsilon_{0}E_{F}^{{}^{\prime}}/d$.
Figure 2: Benchmarking InterMatch. (a) Comparison of charge transfer predicted
by InterMatch with measured experimental values for interfaces in RefsAnnadi
et al. (2013); Wang et al. (2020); Sutter et al. (2009); Zheng et al. (2021a).
(b) InterMatch screening of over 10,000 2D materials in heterostructure with
monolayer MoSe2, ranked in descending order of charge transfer
$\absolutevalue{\Delta n}$. (c) Substrate selection based on InterMatch
screening results from red box in (a) according to $\Delta n$, elastic energy
$\mathcal{E}$, and energy above-hull. (d) Comparison of InterMatch predictions
for $\Delta n$ (solid symbols) with supercell DFT calculations (open symbols).
The second branch of the InterMatch algorithm sketched in Fig. 1(e-f)
constructs optimal supercells from a pair of queried systems by calculating
strain and elastic energy at their interface over a series of supercell
configurations. Given the lattice vectors of system 1, $\vec{a}_{1}$,
$\vec{b}_{1}$, and those of system 2, $\vec{a}_{2}$, $\vec{b}_{2}$, the
algorithm searches for pairs of near-equivalent superlattice vectors
$\\{(\vec{u}_{1},\vec{v}_{1}),(\vec{u}_{2},\vec{v}_{2})\\}$:
$\displaystyle\vec{u}_{i}$
$\displaystyle=\mathbf{M}^{i}_{11}\vec{a}_{i}+\mathbf{M}^{i}_{12}\vec{b}_{i}$
(2) $\displaystyle\vec{v}_{i}$
$\displaystyle=\mathbf{M}^{i}_{21}\vec{a}_{i}+\mathbf{M}^{i}_{22}\vec{b}_{i}$
where $\mathbf{M}^{i}$ is a $2\times 2$ matrix of integer coefficients for the
system $i$ and the “near-equivalence” is defined by
$\mathbf{M}^{1}=(\tilde{\varepsilon}^{2}+1)\mathcal{R}_{\theta}\mathbf{M}^{2}.$
(3)
Here $\mathcal{R}_{\theta}$ is an in-plane rotation matrix by angle $\theta$
and $\tilde{\varepsilon}^{2}$ is the strain tensor resulting from straining
the Bravais lattice of system 2 to match that of system 1.
We choose system 2 to be the material with the smallest elements of the
stiffness tensor $\mathbf{C}$ (queried from the Materials Project) in the
strain direction. InterMatch then computes the elastic energy
$\mathcal{E}=\frac{1}{2}C_{ijkl}\varepsilon_{ij}\varepsilon_{kl}$ for the
superlattice candidate according to classical elastic plate theoryLandau et
al. (1986). The optimal supercell is determined by simultaneously minimizing
the elastic energy $\mathcal{E}$ and the cell area
$\absolutevalue{\vec{u}_{i}\crossproduct\vec{v}_{i}}$. The use of elastic
energy goes beyond previous approaches for finding the superlatticeStradi et
al. (2017); Lazić (2015) which only consider geometric strain.
Figure 3: Superlattice structure prediction optimizing elastic energies.
(a)-(b) (Top view) Diagonal stiffness tensor components $C_{11}$ and $C_{22}$
of primitive MoSe2 and ZrTe3 unit cells in Voigt notation. (c)-(d) (Top view)
ZrTe3 layer of two candidate MoSe2/ZrTe3 supercells with the same number of
atoms and average strain $\varepsilon_{av}^{\textnormal{ZrTe}_{3}}$. The solid
black boxes denote the strained ZrTe3 unit cells and the dashed red boxes are
the original unstrained primitive cells. (e) Average strain values
$\varepsilon_{av}$ of the MoSe2/ZrTe3 interfaces in Cells 1 and 2. (f) Elastic
energies $\mathcal{E}$ of the interfaces.
Now we demonstrate how elastic energy considerations can make a difference in
optimization of the superlattice. Consider the MoSe2/ZrTe3 interface. The
primitive unit cells of MoSe2 and ZrTe3 are shown in Fig. 3 (a) and (b)
respectively, along with the diagonal components of their stiffness tensors
$C_{ii}$ in Voigt notation. The anisotropy of the ZrTe3 stiffness tensor is
such that the energetic cost to deforming ZrTe3 along the direction of
$C_{22}$ far exceeds the cost of an equivalent deformation along the direction
of $C_{11}$. Cells 1 and 2 in Fig. 3 (c) and (d) result from an InterMatch
search for low-area, low-strain MoSe2/ZrTe3 supercells. The two cells are
identical in number of atoms and geometric strain $\varepsilon_{av}$ (shown in
Fig. 3 (e)), however, cell 1 is favored energetically due to the different
strains required to make each supercell commensurate with MoSe2 (Fig. 3 (f)).
We now benchmark charge transfer predictions by InterMatch against
experimentally measured charge transfer in known interfaces. Fig. 2 (a) shows
a comparison of InterMatch predictions of charge transfer with experimentally
obtained values for several interfaces: LaAlO3/SrTiO3(1 1 0)Annadi et al.
(2013), GR/$\alpha$-RuCl3Wang et al. (2020), GR/Pt$(1\,1\,1)$Sutter et al.
(2009), and MoS2/MgAl2O4Zheng et al. (2021a). The magnitudes of the $\Delta n$
predicted with InterMatch are at the same order of magnitude as the measured
values, especially given experimental error bars, with the exception of
GR/$\alpha$-RuCl3. However, spin-orbit coupling effects (absent from our
calculations) are known to affect the band structure of $\alpha$-RuCl3Kim et
al. (2015), altering the band alignment with GR and the resulting charge
transfer.
Next we turn to the application of charge transfer prediction to the problem
of doping transition metal dichalcogenides (TMDs). TMDs have emerged as an
exciting van der Waals material platform at the intersection of semiconductor
physics and strong correlation physicsManzeli et al. (2017). Due to the spin-
valley locking Ising spin-orbit coupling, an exotic $p$-wave superconducting
state was proposed for hole-doped TMDsHsu et al. (2017). Recent developments
in TMD moiré systems have further extended the phase space of possibilities.
However, a major bottleneck against testing these proposals is the difficulty
of establishing a good contact. Empirically, it has been established that
doping the contact area can significantly improve the contact resistanceZheng
et al. (2021b). However, gate-based doping does not scale well. While
successful modulation doping using work function difference was established in
graphene/$\alpha$-RuCl3 heterostructuresWang et al. (2020), it is desirable to
perform an exhaustive search of interface possibilities.
We seek 2D substrates for controlling carrier concentration in MoSe2. We
screen all entries of the 2DMatPedia database and 3000 entries from the
Materials Project for stable 2D materials composed of elements making up the
majority of commercially available semiconductors, semimetals, and metals. We
use InterMatch to down-sample from 10,000 candidate 2D substrates based on the
magnitude of the predicted charge transfer $\absolutevalue{\Delta n}$ to MoSe2
in the desired range $\gtrsim\mathcal{O}(10^{13})$ cm-2 (Fig. 2 (b)). We then
select from these the compounds with maximum $\absolutevalue{\Delta n}$,
minimal strain, and minimal above-hull energy (Fig. 2 (c)). Finally, we choose
a small subset of the top interfaces (2H-TaS2, $\beta$-GaSe, and ZrTe3, in the
case of our example) and benchmark InterMatch predictions against supercell
DFT calculations using the optimized supercells generated by InterMatch. Fig.
2 (d) shows a comparison of InterMatch predictions with the results from DFT
for the top three interfaces (for computational details, see the Supplemental
Material). The magnitude of the $\Delta n$ prediction from the two approaches
are within $10^{13}$ cm-2. Moreover, both approaches find consistent relative
magnitude of charge transfer. Given the high-throughput nature of InterMatch,
these agreements encourage using InterMatch as the first pass in searches for
optimal heterostructures.
As an example of the power of Intermatch to understand superlattice structure,
we consider the graphene/$\alpha$-RuCl3 heterostructure (GR/$\alpha$-RuCl3).
This system has attracted great interest due to the presence of strong
modulation dopingWang et al. (2020) and enhancement of $\alpha$-RuCl3’s
proximity to the Kitaev spin liquid phase. However, relatively little
attention has been paid to the atomic scale structure of the heterostructure
and the possible influence on electronic properties. In order to study this
experimentally, we used scanning tunneling microscopy (STM) to investigate the
properties of GR/$\alpha$-RuCl3 heterostructures created by mechanical
exfoliation and colamination, as shown in Fig. 4 (a). The angle between the
$\alpha$-RuCl3 substrate and graphene was not intentionally controlled. Shown
in Fig. 4 (b)-(d) are a set of STM topographs taken at various locations of
the GR/$\alpha$-RuCl3 heterostructure. The locations are within a few microns
of each other on the sample shown in Fig. 4 (a). Intriguingly, all three of
the regions show moiré patterns with large wavelengths - 2.7 nm in Fig. 4 (b),
11.7 nm in Fig. 4 (c) and 25.7 nm in Fig. 4 (d). All three of these
wavelengths are much larger than the wavelength set by the difference in
lattice constants.
Using InterMatch, we perform a comprehensive mapping of the space of
superlattice configurations spanned by $(\theta,L,\mathcal{E})$ where $\theta$
is the twist angle, $L$ is the moiré period, and $\mathcal{E}$ is the elastic
energy of the interface. The resulting spectrum of low-energy superlattice
configurations for $0^{\circ}\leq\theta\leq 30^{\circ}$ and
$0\;\textnormal{nm}\leq L\leq 30\;\textnormal{nm}$ is shown in Fig. 4 (e). We
identify four prominent moiré length scales (blue boxes in Fig. 4 (e))
occurring within a $5^{\circ}$ range between $15^{\circ}$-$20^{\circ}$. Three
of the four length scales coincide with those observed in STM at
$L=2.7,\;11.7,\;25.7$ nm, shown in Fig. 4 (b)-(d). Correctly identifying
energetically favorable GR/$\alpha$-RuCl3 superlattices over a narrow range of
twist angles showcases InterMatch’s capability to predict interfacial
structure of complex (e.g extremely lattice-mismatched) systems.
The presence of an atomic reconstruction at the interface of GR/$\alpha$-RuCl3
can have dramatic consequences for the spectroscopic properties of the
material. Shown in Fig. 4 (f)-(h) are scanning tunneling spectra averaged over
the regions shown in Fig. 4 (b)-(d). These spectra show dramatic differences
from the simple expectation for a doped Dirac spectrum as might be expected
from charge transfer alone. Instead, we observe strong resonances in all three
regions, with the spacing between resonances following the expectation from
Landau levels on a Dirac spectrum. Previously, such spectra have been observed
when graphene has a periodic bucklingMao et al. (2020), where it was ascribed
to periodic strain in the material. In our case, apart from the strain
associated with the moiré latticeShabani et al. (2021), we expect that there
will also be strong periodic variations in the dopingRizzo et al. (2022) that
contribute to the formation of resonances.
Figure 4: (a) Optical image of the measured device of GR/$\alpha$-RuCl3
contacted with bismuth indium tin for the STM measurements. The blue and green
dashed lines show the boundary of $\alpha$-RuCl3 and graphene, respectively.
(b)-(d) STM topographic images (in pm) of GR/$\alpha$-RuCl3 on 2.7 nm (set
points of -100 mV and -100 pA), 11.7 nm and 25.7nm (set points of -1 V and -50
pA) moiré patterns due to atomic reconstruction. (e) InterMatch predictions
for low-energy GR/$\alpha$-RuCl3 superlattice configurations as a function of
period $L$, twist angle $\theta$, and elastic energy $\mathcal{E}$. Left panel
is a projection onto the $L$-axis, bottom panel is a projection onto the
$\theta$-axis. Dashed box indicates interval of $\theta$ containing largest
range of stable superlattice periods, shaded blue boxes indicate regions of
likely superlattice configurations. Purple stars denote the periodicities
extracted from the experiment. (f)-(h) dI/dV measurements corresponding to the
three moiré patterns in (b)-(d) showing strong resonances dependent on moire
wavelengths.
In summary, we introduce and demonstrate InterMatch, a high-throughput
computational framework and database for predicting charge transfer, strain,
and superlattice of an interface between two arbitrary materials. Charge
transfer allows heterostructure-based modulation dopingWang et al. (2020),
which can guide device fabrication and contact design Zheng et al. (2021b).
Efficiently determining the smallest energetically favorable commensurate
supercells from a wide variety of interface configurations is crucial for
accelerating ab initio studies. We showcase the use of InterMatch by
identifying high-charge transfer substrates for doping TMDs, and by predicting
equilibrium moiré superlattice configurations for the lattice-mismatched
GR/$\alpha$-RuCl3 interface that are validated by STM measurements. The
presence of such long-wavelength superlattice modulations at van der Waals
interfaces present new opportunities to tailor bandstructure using materials
that do not have a close match in lattice constants. The evolving interface
database provides open access to InterMatch results which we hope will help
guide future exploration of interfacial systems.
To broadly benefit the community, we made the InterMatch code openly
accessible at https://doi.org/10.5281/zenodo.6823973Gerber and Torrisi (2022).
Moreover, we tabulate InterMatch results in an open-access “interface
database” directly integrated with the Materials Project via the MPContribs
platform. At the time of writing, the database contains $\sim 200,000$
interfaces (and counting) in simple JavaScript Object Notation (JSON) that are
queryable and sortable according to the chemical composition of either
constituent system, charge transfer, strain, and optimized supercell size. In
addition, we generate crystallographic information files (CIF) of interface
supercells with InterMatch which may be readily accessed from the database and
used as inputs for DFT or other first principles studies.
###### Acknowledgements.
The authors thank Kin Fai Mak, Jie Shan, Stephen Carr, Patrick Huck, Matthew
Horton, Jason Munro, and Vidya Madhavan for helpful discussions. EAK and JH
were supported by MURI grant FA9550-21-1-0429. EG was supported by the Cornell
Center for Materials Research with funding from the NSF MRSEC program
(DMR-1719875). SBT was supported by the Department of Energy Computational
Science Graduate Fellowship under grant DE-FG02-97ER25308. The computation was
done using the high powered computing cluster WALLE2 that was established
through the support of Gordon and Betty Moore Foundation’s EPiQS Initiative,
Grant GBMF10436 to EAK and the New Frontier Grant from Cornell University’s
College of Arts and Sciences and hosted and maintained by Cornell Center for
Advanced Computing. STM experiments were supported by NSF DMR-2004691 (ANP)
and AFOSR via grant FA9550-21-1-0378 (SS, ES). Sample synthesis for STM
measurements was supported by the NSF MRSEC program through Columbia in the
Center for Precision-Assembled Quantum Materials (PAQM), grant number
DMR-2011738.
## Supplemental Material
In Section I we present additional InterMatch results identifying optimal
substrate candidates for forming high-charge-transfer interfaces with group-VI
transition metal dichalcogenides and putative spin liquid materials
$\alpha$-RuCl3 and TbInO3. In Section II we provide details of the InterMatch
code. In Section III we provide computational details of the ab initio
density-functional theory (DFT) calculations of MoSe2 supercells in Fig. 2(d)
of the main text.
## I Charge transfer substrates for WTe2, WSe2, and $\alpha$-RuCl3
We further demonstrate InterMatch by applying it to two types of materials:
group-VI TMDs, and putative spin liquid $\alpha$-RuCl3Plumb et al. (2014).
TMDs’ unique combination of properties makes them highly attractive for
nanoelectronics applications and fundamental studies of novel physical
phenomenaManzeli et al. (2017). However, the realization of many such high-
performance devices and exotic phases is limited by the availability of
systems with high carrier mobility and low contact resistances between metal
contacts and the semiconductor. Quantum spin liquids (QSLs) are interacting
quantum systems in which spins do not order at low temperatures, and have been
theorized to offer insights into high-temperature superconductivity upon
doping. $\alpha$-RuCl3, for example, has been intensively discussed as a
possible candidate for Kitaev physics; however, it orders
antiferromagnetically at low temperatures due to the presence of additional
magnetic couplings extending beyond the pure Kitaev interaction. Doping
$\alpha$-RuCl3 with charge carriers has been predicted to enhance Kitaev
interactions and push $\alpha$-RuCl3 closer to the spin liquid phase. We use
InterMatch to identify stable, high-charge-transfer interfaces for electron-
and hole-doping the TMDs WTe2, WSe2, and MoSe2, and the putative QSL
$\alpha$-RuCl3. Fig. S1 shows sample InterMatch results of substrate
candidates for interfaces with WTe2, WSe2, and $\alpha$-RuCl3, highlighting
those that minimize elastic energy $\mathcal{E}$ and maximize charge transfer
$\Delta n$. In Fig. S1 (a)-(b), we survey all $\sim 70,000$ oxides from the
Materials Project. Oxides have a wide range of charge neutrality levels and
therefore constitute a powerful addition to electrostatic gating or chemical
doping for controlling carrier concentration in heterostructures. In Fig. S1
(c) we survey all entries in the 2DMatpedia database, and determine two-
dimensional (2D) substrate candidates to maximally dope $\alpha$-RuCl3.
Figure S1: Sample InterMatch results for different charge transfer interfaces.
The solid green line indicates the pareto-optimal frontier of substrate
candidates that minimize elastic energy $\mathcal{E}$ and maximize charge
transfer $\absolutevalue{\Delta n}$. Calculated quantities in (a)-(b) use ab
initio data from the Materials Project as inputs, while those in (c) use data
from the 2DMatpedia database.
## II InterMatch Code
The InterMatch code is written in Python 3.7 and makes extensive use of
pymatgenOng et al. (2013), an open-source Python package of the Materials
Project, for the manipulation and analysis of various structures of interest.
The code is continuously being developed, and the latest version can be
obtained at https://doi.org/10.5281/zenodo.6823973Gerber and Torrisi (2022).
We aim to provide an efficient scheme for computing interface properties
capable of screening a significant fraction of combinations of existing
Materials Project structure entries, returning the results in real-time
(typical run time for the calculation of a single interface is $10\pm 5$
seconds on a 1.7 GHz Intel Core i5 processor at 1333 MHz using 4 GB of RAM,
running macOS Sierra 10.12.6).
## III Computational Details
All ab initio DFT calculations were carried out within the total-energy plane
wave density-functional pseudopotential approach, using Perdew-Burke-Ernzerhof
generalized gradient approximation functionalsPerdew et al. (1996) and
optimized norm-conserving Vanderbilt pseudopotentials in the SG15
familySchlipf and Gygi (2015). Plane wave basis sets with energy cutoffs of 30
hartree were used to expand the electronic wave functions. We used fully
periodic boundary conditions and a $8\times 8\times 1$ $k$-point mesh to
sample the Brillouin zone. Electronic minimizations were carried out using the
analytically continued functional approach starting with a LCAO initial guess
within the DFT$++$ formalismFreysoldt et al. (2009), as implemented in the
open-source code JDFTxSundararaman et al. (2017) using direct minimization via
the conjugate gradients algorithmPayne et al. (1992). All unit cells were
constructed to be inversion symmetric about $z=0$ with a distance of $\sim 60$
bohr between periodic images of the MoSe2 surface, using coulomb truncation to
prevent image interaction.
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|
# Computationally Efficient Reinforcement Learning:
Targeted Exploration leveraging Simple Rules
Loris Di Natale, Bratislav Svetozarevic, Philipp Heer, and Colin N. Jones This
research was supported by the Swiss National Science Foundation under NCCR
Automation, grant agreement 51NF40_180545, and in part by the Swiss Data
Science Center, grant agreement C20-13.L. Di Natale, B. Svetozarevic, and P.
Heer are with the Urban Energy Systems Lab, Empa, Dübendorf, Switzerland.
{loris.dinatale, bratislav.svetozarevic<EMAIL_ADDRESS>Di Natale
and C. N. Jones are with the Laboratoire d’Automatique, EPFL, Lausanne,
Switzerland<EMAIL_ADDRESS>
###### Abstract
Model-free Reinforcement Learning (RL) generally suffers from poor sample
complexity, mostly due to the need to exhaustively explore the state-action
space to find well-performing policies. On the other hand, we postulate that
expert knowledge of the system often allows us to design simple rules we
expect good policies to follow at all times. In this work, we hence propose a
simple yet effective modification of continuous actor-critic frameworks to
incorporate such rules and avoid regions of the state-action space that are
known to be suboptimal, thereby significantly accelerating the convergence of
RL agents. Concretely, we saturate the actions chosen by the agent if they do
not comply with our intuition and, critically, modify the gradient update step
of the policy to ensure the learning process is not affected by the saturation
step. On a room temperature control case study, it allows agents to converge
to well-performing policies up to $6-7\times$ faster than classical agents
without computational overhead and while retaining good final performance.
## I Introduction
Despite its success in many applications [1, 2], model-free Reinforcement
Learning (RL), and in particular deep RL (DRL), usually suffer from data
inefficiency, i.e., they require a significant number of interactions with the
environment to converge [3, 4]. This stems from the necessity to explore the
state-action space to find optimal policies and leads to significant
computational costs. It also limits the deployment of DRL methods on physical
systems without pretraining in simulation: learning a building temperature
control policy from scratch can for example take years of data [5, 6].
To speed up the training of DRL agents, researchers have for example
investigated how to leverage expert demonstrations [7, 8], but this requires
access to an expert policy which is not always available in practice. Instead,
we postulate that prior knowledge of physical systems often allows us to
design simple rules that agents should follow a priori, such as “Do not heat
the room if it is already $26\text{\,}\mathrm{\SIUnitSymbolCelsius}$”; we
indeed know this action will always be suboptimal in that state, there is no
need for agents to explore its consequences.
In this paper, we hence propose modifications of actor-critic algorithms to
encode simple rules in RL agents, introducing state-dependent constraints on
the agents’ actions to restrict exploration to interesting regions of the
state-action space. In other words, the key idea is to avoid visiting state-
action pairs that are known to be suboptimal by the expert to accelerate the
convergence towards meaningful solutions and thus increase the efficiency of
(D)RL. Note that while state-dependent bounds were concurrently introduced in
[9], they are enforced a posteriori in the environment instead of directly
modifying the agent’s behavior and do not necessarily improve convergence. In
another line of work, prior knowledge successfully accelerated learning in
[10], but relying on fuzzy rather than direct rule integration.
### I-A Constraining RL agents
To bound the decisions taken by RL agents, one typically defines some
constrained set of actions and either project the actions of the agents on
this set at each time step or switch to a fallback controller when needed [11,
12, 13, 14]. The main challenge with these operations is that they are usually
not differentiable and hence cannot be learned by RL agents, with the notable
exceptions of [15, 16, 17], who leveraged differentiable optimization layers
[15], modified the policy updates to account for projections [16], or derived
a closed-form solution of the projection step [17]. However, these methods
either entail additional computational burden [15, 16] or rely on a learned
linearization of the constraints [17]. Alternatively, one could also apply
tools from the safe RL literature [18, 19, 20], typically relying on
constrained policy optimization [21, 22]. However, this would again introduce
both engineering and computational overhead.
The complexity of the methods discussed above often stems from the fact that
they are designed to impose state constraints on DRL agents, which is a more
challenging problem in general since it leads to complex action bounds for the
agent at each step. Here, however, we argue that prior knowledge can
straightforwardly be used to accelerate the training of DRL agents through
simple state-dependent box constraints on their actions, which allows us to
leverage less computationally expensive tools.
To alleviate the issue of non-differentiability without increasing either the
engineering or the computational burden, Reward Shaping (RS) heuristics might
be used in various forms to penalize agents when constraints are violated, let
them know when a fallback controller was used or they were saturated, or
introduce prior knowledge about the task to solve [14, 23, 24]. While such
methods might accelerate the learning process to some extent, they are however
indirect, i.e., they only influence the learned policies through the reward
function that the agent will learn to optimize over time. Moreover, shaping
the reward function simultaneously impacts the learning process of both the
actor and the critic.
### I-B Contribution
In this work, we propose to clip DRL agents’ actions according to simple
expert-designed state-dependent bounds and then modify the gradient update
step of the actor to let agents learn from their mistakes and accelerate their
convergence to expected actions. To explain the effectiveness of the gradient
modification, we provide an intuitive analytical analysis of its impact on the
learning of DRL agents. Importantly, contrary to RS, our modifications only
affect the actor, allowing the critic to learn the true Q-values.
Remarkably, our method bypasses the need for complex projection steps and does
not require access to a fallback controller or an expert policy. Moreover and
critically, the proposed modifications do not impact the computational
complexity of the algorithm, are straightforward to design and implement, and
can be coupled with any actor-critic algorithms. Note that, in contrast with
[10], where the expert knowledge is potentially overridden by the policy, our
method enforces the wanted behaviors on agents at all times.
The effectiveness of the proposed Efficient Agents (EAs) is demonstrated in
simulation on a room temperature control case study, where they converge up to
$6$ – $7$ times faster than classical ones and $2$ – $3$ times faster than RS-
based agents while retaining good final performance. This hints at how the
proposed modifications can provide a simple yet effective and computationally
inexpensive mean to leverage expert knowledge to accelerate DRL algorithms.
## II Preliminaries
### II-A Reinforcement Learning
At each time step $t$, given an observation $s_{t}$ of the state of the
environment, an RL agent chooses an action $a_{t}$. The environment then
transitions to $s_{t+1}$ according to the transition probabilities
$P(s_{t},a_{t})$ and sends the new state and the reward signal
$r(s_{t},a_{t})$ to the agent. The objective of any deterministic RL
algorithm111While the presented analyses deal with deterministic actor-critic
agents for clarity, the results can easily be extended to the stochastic case.
is to find a policy $\pi(s_{t})$ that maximizes the expected discounted
cumulative returns:
$\displaystyle J(\pi)=\mathbb{E}_{a_{t}\sim\pi(s_{t}),s_{t+1}\sim
P(s_{t},a_{t})}\left[\sum_{t=0}^{\infty}{\gamma^{t}r(s_{t},a_{t})}\right],$
(1)
where $\gamma$ is the discount factor trading off near- and long-term rewards,
and the initial state $s_{0}\sim\rho$ is sampled from the corresponding
initial distribution. With a slight abuse of notation on the expectation for
clarity, we can define the Q-function of any state-action pair $(s,a)$:
$\displaystyle
Q^{\pi}(s,a)=\mathbb{E}_{\pi}\left[\sum_{t=0}^{\infty}{\gamma^{t}r(s_{t},a_{t})|s_{0}=s,a_{0}=a}\right],$
(2)
which captures the expected returns when action $a$ is chosen in state $s$ and
the policy $\pi$ is followed thereafter.
In our simulations, we let agents explore the environment with the
$\epsilon$-greedy exploration strategy, which means we apply the following
action to the environment:
$a(s)=\text{clip}(\pi(s)+\epsilon,a^{\textit{low}},a^{\textit{up}}),\qquad\epsilon\sim\mathcal{N}(0,\sigma),$
(3)
where the noisy actions are clipped elementwise between $a^{\textit{low}}$ and
$a^{\textit{up}}$, the predefined action bounds from the environment, and
$\epsilon$ is the Gaussian exploration noise with a standard deviation of
$\sigma$. All the transition tuples $(s,a,r,s^{\prime})$ observed by the agent
are stored in a replay buffer.
### II-B Actor-critic algorithms
In practice, policies and Q-functions are often parametrized with Neural
Networks (NNs) as $\pi_{\theta}$ and $Q_{\phi}$, respectively, leading to DRL,
and numerous algorithms have been developed to maximize (1) [25]. In this
work, we are interested in deterministic actor-critic methods stemming from
[26], where both the actor $\pi_{\theta}$ (also referred to as the policy) and
the critic $Q_{\phi}$ are optimized in parallel leveraging gradient descent.
While different flavors exist, most algorithms compute the gradient of the
critic using the Temporal Difference (TD) loss [26]:
$\displaystyle\hat{\nabla}_{\phi}Q_{\phi}$
$\displaystyle=\nabla_{\phi}\left[\frac{1}{|B|}\sum_{b=(s,a,r,s^{\prime})\in
B}{\left(Q_{\phi}(s,a)-y(b)\right)^{2}}\right],$ (4)
with
$y(b)=\left(r+\gamma\max_{a^{\prime}}Q_{\phi}(s^{\prime},a^{\prime})\right)$
and where a batch $B$ of past transitions is sampled from the replay buffer
and used to estimate expectations. Leveraging the policy gradient theorem
[27], one can similarly use the critic to estimate the actor gradient as:
$\displaystyle{\color[rgb]{0,0,0}\hat{\nabla}_{\theta}\pi_{\theta}}$
$\displaystyle=-{\color[rgb]{0,0,0}\nabla_{\theta}\left[\frac{1}{|B|}\sum_{s\in
B}{Q_{\phi}(s,\pi_{\theta}(s))}\right].}$ (5)
Note that these gradients are easily computed using automatic differentiation
when the actor and the critic are parametrized with NNs.
In this paper, we rely on the Twin Delayed Deep Deterministic (TD3) policy
gradient algorithm, which introduces a few modifications to limit the well-
known overestimation bias of Q-functions plaguing vanilla actor-critic
algorithms [28]. Remarkably, however, these adjustments do not impact the
actor gradient in (5), allowing us to seamlessly integrate the proposed
modifications detailed in Section III.
## III Methods
### III-A State-dependent action saturation
In many cases, prior knowledge allows us to design state-dependent upper and
lower bounds $a^{\textit{max}}(s)$ and $a^{\textit{min}}(s)$, respectively, on
the actions we expect well-performing control policies to take in a given
state $s$, with:
$\displaystyle\quad a^{\textit{low}}\leq a^{\textit{min}}(s)\leq
a^{\textit{max}}(s)\leq a^{\textit{up}}.$ (6)
To limit the exploration of known suboptimal state-action pairs, we can then
modify (3) accordingly to:
$\displaystyle a(s)$
$\displaystyle=\text{clip}(\pi_{\theta}(s)+\epsilon,a^{\textit{min}}(s),a^{\textit{max}}(s)).$
(7)
Note that these bounds stemming from prior knowledge are also enforced at test
time when $\epsilon=0$ to ensure an agent would never turn left if whenever
there is a wall in that direction, for example, neither during the training
nor the deployment phase.
### III-B Actor gradient modification
The major problem with the clipping operation in (7) is its non-
differentiability. Worse yet, its subdifferentials go to zero whenever agents
are saturated (see (9) in Section III-C), making any backward flow of
information on the overriding process impossible. As a countermeasure, to let
agents learn from their mistakes, we also modify the actor gradient (5) to:
$\displaystyle\hat{\nabla}^{\textit{EA}}_{\theta}\pi_{\theta}=-\nabla_{\theta}\Bigg{(}$
$\displaystyle\frac{1}{|B|}\sum_{(s,a)\in
B}\bigg{[}{Q_{\phi}(s,\pi_{\theta}(s))}$
$\displaystyle-\frac{\lambda}{2}\left(\pi_{\theta}(s)-a(s)\right)^{2}\bigg{]}\Bigg{)},$
(8)
where $\lambda$ is a hyperparameter. The last term in (8) penalizes actions
chosen by the policy $\pi_{\theta}(s)$ if they deviate from the constrained
action $a(s)$ that was applied to the environment, thus steering the agent’s
decisions towards expected actions.222In another line of work, this penalty
was also used in [15] to improve the robustness of differentiable layer-based
RL for state-constrained problems.
Alternatively, we note here that one could instead modify the reward function
to include this penalty (RS) and then maximize (1). Remarkably, however, the
latter also impacts the learning process of the critic in (4) when applied to
actor-critic frameworks, contrary to our method. We will show empirical
benefits of the proposed modification (8) over RS-based penalties in terms of
convergence speed in Section V.
### III-C Implications of the modified gradients
Let $C(s)=\left\\{a\in\mathbb{R}:a^{\textit{min}}(s)\leq a\leq
a^{\textit{max}}(s)\right\\}$ for any given state $s$.333Without loss of
generality, we assume that $a\in\mathbb{R}$ in this section for clarity. This
assumption can easily be lifted for multi-dimensional problems. Grouping all
the parameters $\theta$ in a vector and recalling the definition of the action
$a(s)$ applied to the environment in state $s$ from (7), we can define its
subgradient $\nabla_{\theta}a(s)$ as:
$\displaystyle a(s)$ $\displaystyle=\begin{cases}a^{min}(s),&\text{if
}\pi(s)<a^{min}(s),\\\ \pi_{\theta}(s)+\epsilon,&\text{if }\pi_{\theta}(s)\in
C(s),\\\ a^{max}(s),&\text{if }\pi(s)>a^{max}(s).\end{cases}$
$\displaystyle\implies\nabla_{\theta}a(s)$
$\displaystyle=\begin{cases}\nabla_{\theta}\pi_{\theta}(s),&\text{if
}\pi_{\theta}(s)\in C(s),\\\ 0,&\text{else,}\end{cases}$ (9)
where $\nabla_{\theta}\pi_{\theta}(s)$ is the actor gradient. We can then
rewrite the gradient of EAs (8) as:
$\displaystyle\hat{\nabla}^{\textit{EA}}_{\theta}\pi_{\theta}$
$\displaystyle=-\frac{1}{|B|}\sum_{(s,a)\in
B}\bigg{[}{\nabla_{\theta}Q_{\phi}(s,\pi_{\theta}(s))}$
$\displaystyle\qquad\quad-\nabla_{\theta}\left(\frac{\lambda}{2}\left(\pi_{\theta}(s)-a(s)\right)^{2}\right)\bigg{]}$
$\displaystyle=-\frac{1}{|B|}\sum_{(s,a)\in
B}\bigg{[}\nabla_{\theta}Q_{\phi}(s,\pi_{\theta}(s))$
$\displaystyle\qquad\quad-\left(\lambda\left(\nabla_{\theta}\pi_{\theta}(s)-\nabla_{\theta}a(s)\right)^{\top}\left(e_{\theta}(s)\right)\right)\bigg{]},$
where we introduce the error term $e_{\theta}(s)=\pi_{\theta}(s)-a(s)$. We
hence get the following modified actor gradient, where we omit $(s,a)\in B$
for clarity:
$\displaystyle\hat{\nabla}^{\textit{EA}}_{\theta}\pi_{\theta}$
$\displaystyle=\begin{cases}-\frac{1}{|B|}\sum_{B}\Big{[}\nabla_{\theta}Q_{\phi}(s,\pi_{\theta}(s))\Big{]},\
\text{if }\pi_{\theta}(s)\in C(s),\\\
-\frac{1}{|B|}\sum_{B}\Big{[}\nabla_{\theta}Q_{\phi}(s,\pi_{\theta}(s))\\\
\qquad\quad\
-\lambda\nabla_{\theta}\pi_{\theta}(s)^{\top}e_{\theta}(s)\Big{]},\
\text{else.}\end{cases}$
Remarkably, the additional penalty term in (8) hence allows us to solve the
issue of the subdifferentials of the clipping operator being zero when actions
are saturated, modifying the gradients only when the constraints are not met.
Indeed, as long as the action chosen by the agent respects the constraints
provided by the expert, the classical gradient (5) is used. On the other hand,
as soon as the constraints are not met, the gradient is modified in the
direction $e_{\theta}(s)$ to accelerate the convergence of $\pi_{\theta}(s)$
to $C{\color[rgb]{0,0,0}(s)}$ despite the subdifferential of the clipped
action being zero, confirming the graphical intuition from [15, Fig. 2]. This
allows EAs to learn from their mistakes and — we hypothesize — helps them
rapidly converge to meaningful policies.
## IV Room temperature control case study
To assess the effectiveness of the proposed method, we apply it to a
temperature control case study, where the objective is to minimize the energy
consumption of a room while maintaining the comfort of the occupants,
represented by predefined temperature bounds that should not be exceeded.
### IV-A Reinforcement Learning framework
The continuous action space of the agents corresponds to how much heating or
cooling power, should be applied at each time step, normalized between
$a^{\textit{low}}=-1$ and $a^{\textit{up}}=1$. During the heating season,
$a^{\textit{low}}$ corresponds to the heating being turned off and
$a^{\textit{up}}$ to full heating, and the contrary in the cooling case.
Physically Consistent Neural Networks (PCNNs) [29] are used to simulate one
bedroom in the NEST building [30] and $s_{t}$ gathers time, weather,
temperature, and comfort bound information (see [6] for details). The reward
function is defined as the negative weighted sum of energy consumption $E_{t}$
and comfort violations, i.e. how far from the designed bounds the temperature
inside the room is:
$\displaystyle r(s_{t},a_{t})$
$\displaystyle=-\max{\\{L_{t}-T_{t},T_{t}-U_{t},0\\}}-\alpha E_{t},$ (10)
$\displaystyle E_{t}$ $\displaystyle=\begin{cases}\frac{a_{t}+1}{2}\
E^{\textit{max}}_{\textit{heat}},&\text{in the heating season},\\\
\frac{1-a_{t}}{2}\ E^{\textit{max}}_{\textit{cool}},&\text{in the cooling
season}.\end{cases}$
where $L_{t}$ and $U_{t}$ represent the lower and upper comfort bounds on the
temperature $T_{t}$ at time $t$, respectively, $\alpha$ is a weighting factor,
and $E^{\textit{max}}_{\textit{heat}}$ and $E^{\textit{max}}_{\textit{cool}}$
stand for the maximal heating and cooling power, respectively.
### IV-B Design of the saturation rules
In the context of room temperature control, we intuitively know that an
optimal policy should gradually stop heating when the temperature reaches the
upper comfort bound and gradually start heating as soon as the lower bound is
not met (and vice versa for cooling). To encode these simple rules, we design
state-dependent action bounds as follows:
$\displaystyle a^{\textit{min}}(s_{t})$
$\displaystyle=\text{clip}\left(\frac{(L_{t}-m)-T_{t}}{n-m},\ 0,\
1\right)^{2}*2-1$ (11) $\displaystyle a^{\textit{max}}(s_{t})$
$\displaystyle=1-2*\text{clip}\left(\frac{T_{t}-(U_{t}+m)}{n-m},\ 0,\
1\right)^{2},$ (12)
with $n\geq m\geq 0$ representing design parameters to leave more or less
freedom to the agents. In words, we start constraining the action of the
agents as soon as the temperature deviates from the bounds for more than $m$
degrees and then quadratically increase the constraint until $n$ degrees have
been reached, where the agent is forced to use the maximum or minimum power,
as pictured in Fig. 1.
Figure 1: Representation of the action bounds used in this work.
## V Results
To investigate the influence of $m$ and $n$, which measure how much prior
knowledge is transmitted to DRL agents, we train different EAs (EA $m$ / $n$).
For comparison purposes, we also train agents with the classical actor
gradient (5), introducing the additional squared penalty
$\frac{\lambda}{2}\left(\pi_{\theta}(s)-a(s)\right)^{2}$ in the reward
function instead as another computationally inexpensive means to incorporate
prior knowledge in agents (RS $m$ / $n$). Finally, we also analyze two
classical DRL agents with different random seeds (Classical $1$ and
$2$).444The code and data are available on https://gitlab.nccr-
automation.ch/loris.dinatale/efficient-drl.
### V-A Final performance
All the agents were trained on up to three-day-long episodes randomly sampled
from three years of data. They were evaluated after each $96$ steps of
$15\text{\,}\mathrm{min}$, i.e. one day’s worth of data, hereafter also
referred to as an epoch, on a testing set of $50$ unseen sequences of three
days. They all use the same hyperparameters as in [6]. We manually set
$\lambda=100$ for EAs to ensure the constraints are enforced as fast as
possible and $\lambda=10$ for RSs since larger penalties led to instability.
While we empirically observed more robust performance of EAs with respect to
$\lambda$ compared to RSs, a complete sensitivity analysis is left for future
work.
The best reward obtained by all the trained agents over the first $500$ epochs
can be found in Table I, and the corresponding trade-off between energy
consumption and comfort violations is plotted in Fig. 2. These results
illustrate how tighter parameters $m$ and $n$, i.e., higher levels of prior
knowledge, allow EAs and RSs to converge to better solutions in this limited
training regime. In particular, it allows EAs to reduce the amount of comfort
violations without significantly increasing the energy consumption. Classical
DRL agents on the other hand usually use less energy at the cost of additional
comfort violations in this early phase of learning before converging to near-
optimal solutions after longer training times [6].
TABLE I: Best reward obtained by each agent on the test set over the first $500$ epochs. Agent | Rew. | Agent | Rew. | Agent | Rew.
---|---|---|---|---|---
Classical 1 | -2.64 | Classical 2 | -2.75 | |
RS $0$ / $1$ | -2.58 | EA $0$ / $1$ | -2.85 | EA $0.5$ / $1$ | -2.83
RS $0$ / $0.5$ | -2.44 | EA $0$ / $0.5$ | -2.58 | EA $0.25$ / $0.5$ | -2.74
RS $0$ / $0.25$ | -2.37 | EA $0$ / $0.25$ | -2.51 | EA $0.2$ / $0.25$ | -2.62
RS $0$ / $0.1$ | -3.69 | EA $0$ / $0.1$ | -2.46 | EA $0.075$ / $0.1$ | -2.46
Figure 2: Average energy consumption (En.) and comfort violations (Vio.) over
the test set corresponding to each agent in Table I. The performance of two
industrial baselines, of an agent trained for 125,000 epochs (Best Agent), and
the optimal performance achievable (Optimum), all computed as in [6], are
reported in gray.
### V-B Visualization of the impact of prior knowledge
Figure 3: Behavior of a classical agent and EAs with various $m$ and $n$
parameters (ours) minimizing the heating power consumption (bottom) while
maintaining the temperature in the grey dotted bounds (top). Left: Performance
before training, where EAs are saturated once they exceed the bounds by $n$
degrees. Right: Performance after training, showing how all agents converged
to similar solutions (Table II).
To intuitively understand the effect of action saturation, we visualize its
impact on some EAs in Fig. 3, where the behavior of all agents is plotted
before training on the left, and after on the right, for the same three days
during the heating season. Focusing on the left plot, we see the untrained
classical DRL agent in black letting the temperature diverge to an
uncomfortably high range (out of the bounds of the plot) as it starts
exploring the state space using roughly constant heating power. On the other
hand, all the EAs are forced to stop heating once they are $n$ degrees out of
bounds. Consequently, even before training, such agents will not overheat the
room and keep it at acceptable temperatures for the occupants, corresponding
to what we expect from good controllers. However, note that EAs can present
control input oscillations due to the impact of external disturbances, mainly
the solar gains around noon, triggering the saturation mechanism on and off.
On the right plot, after training, one can observe that all EAs generally take
comparable decisions — still being sometimes saturated, which ensures
compliance with prior expert knowledge — leading to similar temperature
patterns. On the other hand, the classical agent presents a slightly different
behavior, with smoother decision patterns. Interestingly, this agent is the
only one heating in the early afternoon, while the EAs wait until the end of
the afternoon to heat the room with high power and meet the comfort bound
tightening at $8$pm. This allows the classical agent to use less energy than
EAs over these three days but can incur additional comfort violations (Table
II), as expected from Fig. 2.
TABLE II: Reward, sum of comfort violations (Vio.), and energy consumption (En.) of each agent over the three days depicted on the right of Fig. 3. Agent | Classical 1 | EA (ours)
---|---|---
$m$ / $n$ | - | $0$ / $1$ | $0$ / $0.5$ | $0$ / $0.25$
Reward | -0.68 | -0.69 | -0.89 | -0.60
Vio. [Kh] | 1.28 | 1.18 | 2.23 | 0.65
En. [kWh] | 5.03 | 5.38 | 5.54 | 5.46
### V-C Data efficiency of the proposed gradient modification
Figure 4: Convergence speed of various EAs with different $m$ and $n$
parameters (ours), compared to agents using RS and two classical agents in
black (one in the top plots, one in the bottom ones). The vertical lines and
annotations specify the number of days of data required to obtain a reward of
$-2.95$ for each agent, which corresponds to the performance of two industrial
rule-based baselines.
A comparison of the convergence speed of various agents over the first $300$
epochs is plotted in Fig. 4, where the vertical lines and annotations
illustrate the number of days required to attain performance on par with rule-
based on-off industrial baselines from [6]. In general, we observe that all
the EAs attain returns on par with the baselines significantly earlier than
classical DRL agents, in as little as $29$ days instead of roughly $200$, an
improvement of almost an order of magnitude. In particular, the smaller $n$ is
chosen (from left to right in Fig. 4), the faster the convergence of the EAs
in green and blue. Intuitively, this makes sense, as tighter constraints
introduce more prior knowledge to the EAs, thereby allowing them to find
interesting solutions faster, without losing time exploring suboptimal state-
action pairs. On the other hand, the influence of $m$ is less marked, with
$m\neq 0$ (blue) and $m=0$ (green) leading to very similar convergence
patterns in the bottom row of plots in Fig. 4.
Remarkably, RS does not seem to drastically speed the training up in this case
study (red). While RS $0$ / $0.25$ does converge twice as fast as the
classical DRL agents, RS $0$ / $0.1$ does not converge at all, hinting at the
fragility of RS in general. Even when they converge, RSs are still two to
three times slower than their EA counterparts. On the other hand, RS seems to
lead to more consistent performance than classical agents and EAs after a few
hundred epochs, which is confirmed by their good final performance in Fig. 2.
Overall, these results support our claim that, as long as the rules provided
to the agents are well-defined and correspond to expected behaviors, the
proposed modifications can indeed greatly accelerate the convergence of DRL
agents. Critically, this does not significantly impact the quality of the
final solution (Sec. V-A). Interestingly, incorporating more specific expert
knowledge in EAs — through smaller $m$ and $n$ — further accelerates their
convergence. This corresponds to our intuition: better-defined rules help
agents more. Remarkably, the modifications proposed in Sec. III provide the
desired speedup for a wide variety of parameters $m$ and $n$, contrary to RS,
hinting at the robustness of the proposed scheme.
## VI Conclusion
Starting from the postulate that prior expert knowledge often gives us an
intuition of how good control policies should behave, we presented a scheme to
encode it in actor-critic frameworks through simple rules to accelerate
learning and decrease the associated computational load. These rules take the
form of bounds on the agent’s actions that can directly be enforced during
training and online operations. To ensure agents learn from their mistakes, we
also modified the actor gradients to steer control policies towards expected
behaviors, limiting the exploration of known suboptimal state-action pairs.
Critically, both these operations are computationally inexpensive, ensuring
the gains in sample complexity positively impact the training time of the
agents.
On a room temperature control case study, this scheme allowed us to accelerate
the convergence speed of DRL agents by up to $6$ – $7\times$. Furthermore,
modifying actor gradients proved to be $2$ – $3$ times more effective and more
robust than the widespread reward shaping method. Remarkably, this was done
without suffering from a significant drop in the final performance of the
control policies, illustrating how prior knowledge can help alleviate the
computational burden of DRL. This represents an interesting first step towards
efficient agents that can be deployed and learned from scratch on physical
systems, potentially bypassing the need for complex simulators. In future
work, it would be interesting to investigate annealing strategies on $\lambda$
or leverage primal-dual optimization tools to adaptively tune the influence of
the additional penalty in the actor gradient and let agents learn more
expressive policies after the initial exploration phase.
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|
# ACFlow: An open source toolkit for analytical continuation of quantum Monte
Carlo data
Li Huang Science and Technology on Surface Physics and Chemistry Laboratory,
P.O. Box 9-35, Jiangyou 621908, China
###### Abstract
The purpose of analytical continuation is to establish a real frequency
spectral representation of single-particle or two-particle correlation
function (such as Green’s function, self-energy function, and dynamical
susceptibilities) from noisy data generated in finite temperature quantum
Monte Carlo simulations. It requires numerical solutions of a family of
Fredholm integral equations of the first kind, which is indeed a challenging
task. In this paper, an open source toolkit (dubbed ACFlow) for analytical
continuation of quantum Monte Carlo data is presented. We at first give a
short introduction to the analytical continuation problem. Next, three primary
analytical continuation algorithms, including maximum entropy method,
stochastic analytical continuation, and stochastic optimization method, as
implemented in this toolkit are reviewed. And then we elaborate major
features, implementation details, and basic usage of this toolkit. Finally,
four representative examples are shown to demonstrate usefulness and
flexibility of the ACFlow toolkit.
###### keywords:
Quantum Monte Carlo simulation , Analytical continuation problem , Maximum
entropy method , Stochastic analytical continuation , Stochastic optimization
method
††journal: Computer Physics Communications
PROGRAM SUMMARY
Program Title: ACFlow
CPC Library link to program files: (to be added by Technical Editor)
Developer’s repository link: https://github.com/huangli712/ACFlow
Code Ocean capsule: (to be added by Technical Editor)
Licensing provisions (please choose one): GPLv3
Programming language: Julia
Supplementary material:
Journal reference of previous version:*
Does the new version supersede the previous version?:*
Reasons for the new version:*
Summary of revisions:*
Nature of problem (approx. 50-250 words): Most of the quantum Monte Carlo
methods work on imaginary axis. In order to extract physical observables and
compare them with the experimental results, analytical continuation must be
done in the post-processing stage to convert the quantum Monte Carlo simulated
data from imaginary axis to real axis.
Solution method (approx. 50-250 words): Three established analytical
continuation methods, including maximum entropy method, stochastic analytical
continuation, and stochastic optimization method, have been implemented in the
ACFlow toolkit.
Additional comments including restrictions and unusual features (approx.
50-250 words): The ACFlow toolkit is written by pure Julia language. It is
highly optimized and parallelized. It can be executed interactively in a
Jupyter notebook environment.
## 1 Introduction
It is well-known that quantum Monte Carlo (QMC) method is a powerful and exact
numerical approach, and has been widely used in many research fields, such as
nuclear physics [1], condense matter physics [2], and many-body physics [3].
In this paper, we just focus on the finite temperature QMC algorithms, which
are used to solve the interacting lattice models or quantum impurity models
[4]. Generally speaking, the simulated results of QMC methods are some sorts
of single-particle or two-particle correlation functions, which are usually
defined on imaginary time axis ($\tau\equiv-it$) or Matsubara frequency axis
($i\omega_{n}$). Therefore, they can’t be compared directly with the
correspondingly experimental results, including but not limited to the
electronic density of states $A(\omega)$, optical conductivity
$\sigma(\omega)$, dynamical structure factor $S(\mathbf{q},\omega)$, and so
on. It is necessary to convert the QMC simulated results from imaginary time
axis or Matsubara frequency axis to real axis (i.e. $\tau\to\omega$ or
$i\omega_{n}\to\omega$), which is the origin of the analytical continuation
problem.
Let’s concentrate on the following Fredholm integral equation of the first
kind:
$g(y)=\int K(y,x)f(x)~{}dx.$ (1)
Here, $K(y,x)$ is the known kernel function, $f(x)$ is the model function, and
$g(y)$ denotes the raw data. Given $f(x)$, it is quite easy to get $g(y)$ via
numerical integration. However, given $g(y)$, solving the Fredholm integral
equation reversely to get $f(x)$ is not as easy as expected. There is no
universal solution. Notice that the so-called analytical continuation problem
can be reformulated in terms of the Fredholm integral equation. Thus, its
objective is to seek a reasonable $f(x)$ to satisfy the above equation. The
QMC simulated data $g(y)$ are noisy and the kernel function $K(y,x)$ is ill
conditioned, which make analytical continuation of QMC simulated data a huge
challenge. In order to solve this problem, peoples have developed numerous
methods in the past decades. These methods include the least square fitting
method, singular value decomposition [5, 6], Padé approximation [7, 8, 9, 10],
Tikhonov-Philips regularization method, maximum entropy method [11, 12],
stochastic analytical continuation [13, 14], stochastic optimization method
[15, 16], sparse modelling method [17], and machine learning method [18, 19,
20], etc. However, each method has its pros and cons. None of these methods
can override the others. The analytical continuation problem is still far away
from being completely solved.
In recent years, quite a few analytical continuation codes have been released,
including maxent (by Mark Jarrell) [12], maxent (in ALPSCore) [21],
$\Omega$Maxent [22], ana_cont [23], SOM (in TRIQS) [24, 25], Stoch (in ALF)
[26], just to name a few. We note that the maximum entropy method has
dominated this field for quite a long time. Thus most of these codes only
implement the maximum entropy method [12, 21, 22, 23]. It is rather difficult
to crosscheck the simulated results obtained by various analytical
continuation methods. In addition, the features of the available codes are
quite limited and hard to examine new algorithms. In order to fill in this
gap, we would like to present a new open source toolkit, called ACFlow, for
analytical continuation. This toolkit implements three primary analytical
continuation methods, including the maximum entropy method, stochastic
analytical continuation, and stochastic optimization method, within an united
framework. It provides an easy-to-used library and application interface. Some
diagnostic and analytical tools are also available. With ACFlow, the users can
easily setup and execute analytical continuation calculations, and validate
the obtained results. We believe that this toolkit will play a vital role in
solving analytical continuation problems.
The rest of this paper is organized as follows. In section 2, background of
the analytical continuation problem is introduced. In section 3, basic
principles and key ingredients of the three analytical continuation methods as
implemented in the ACFlow toolkit are summarized. Section 4 gives a brief
overview about ACFlow’s main features and structures. Section 5 is the major
part of this paper, it explains basic usage, input and output files of ACFlow.
In order to demonstrate usefulness of this toolkit, four typical examples are
illustrated in section 6. Finally, section 7 serves as a short conclusion.
## 2 Problem
### 2.1 Finite temperature Green’s functions
Under the Wick’s rotation $t\to i\tau$, the time evolution operator in the
Heisenberg picture $e^{itH}$ will be replaced by $e^{-\tau H}$. Such a
transformation will increase efficiency of QMC random walking and suppress
numerical oscillation (when $t$ is large, the periodic oscillation of
$e^{itH}$ is quite obvious). This is an important reason why most of the
finite temperature QMC algorithms are formulated in imaginary time axis. The
outputs of finite temperature QMC simulations are usually single-particle or
two-particle correlation functions. For example, the single-particle Green’s
function $G(\tau)$ is defined as follows:
$G(\tau)=\langle\mathcal{T}_{\tau}d(\tau)d^{\dagger}(0)\rangle,$ (2)
where $\tau$ denotes imaginary time, $\mathcal{T}_{\tau}$ denotes time-ordered
operator, and $d$ and $d^{\dagger}$ are annihilation and creation operators,
respectively. The Matsubara Green’s function $G(i\omega_{n})$ can be measured
by QMC simulations or constructed from $G(\tau)$ via direct Fourier
transformation:
$G(i\omega_{n})=\int^{\beta}_{0}d\tau~{}e^{-i\omega_{n}\tau}G(\tau),$ (3)
$G(\tau)=\frac{1}{\beta}\sum_{n}e^{i\omega_{n}\tau}G(i\omega_{n}).$ (4)
Here, $\beta$ means the inverse temperature ($\beta\equiv 1/T$) and
$\omega_{n}$ is the Matsubara frequency. Note that $\omega_{n}$ is equal to
$(2n+1)\pi/\beta$ for fermions and $2n\pi/\beta$ for bosons ($n$ is an
integer).
### 2.2 Spectral density
Clearly, neither $G(\tau)$ nor $G(i\omega_{n})$ can be observed
experimentally. We have to extract dynamical response function, i.e., the
spectral density $A(\omega)$, from them. $A(\omega)$ is indeed an observable
quantity. It is related to $G(\tau)$ via the following Laplace transformation:
$G(\tau)=\int^{+\infty}_{-\infty}d\omega\frac{e^{-\tau\omega}}{1\pm
e^{-\beta\omega}}A(\omega),$ (5)
where +(-) in the denominator is for fermionic (bosonic) system.
$G(i\omega_{n})$ and $A(\omega)$ manifest similar relation:
$G(i\omega_{n})=\int^{+\infty}_{-\infty}d\omega\frac{A(\omega)}{i\omega_{n}-\omega}.$
(6)
It is obvious that Eq. (5) and Eq. (6) are indeed two special forms of the
Fredholm integral equation of the first kind [see Eq. (1)]. So, the central
problem of analytical continuation is to search optimal $A(\omega)$ for given
$G(\tau)$ or $G(i\omega_{n})$.
Sometimes the spectral density $A(\omega)$ is called spectral function in the
references. It is tied to the imaginary part of real frequency Green’s
function $G(\omega)$:
$A(\omega)=-\frac{1}{\pi}\rm{Im}G(\omega).$ (7)
From Im$G(\omega)$, Re$G(\omega)$ could be calculated via the Kramers-Kronig
transformation:
$\mathrm{Re}G(\omega)=\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}d\omega^{\prime}~{}\frac{\mathrm{Im}G(\omega^{\prime})}{\omega^{\prime}-\omega},$
(8)
where $\mathcal{P}$ means Cauchy principal value. Besides Eq. (5) and Eq. (6),
$A(\omega)$ has to obey some additional constraints or sum-rules. For
fermionic systems, the spectral functions must be positive:
$A(\omega)\geq 0.$ (9)
While for bosonic systems, the constraint becomes:
$\text{sign}(\omega)A(\omega)\geq 0.$ (10)
In addition, the spectral function $A(\omega)$ is always bounded,
$\int^{+\infty}_{-\infty}d\omega~{}A(\omega)<\infty.$ (11)
It can be utilized to normalize the resulting spectral function.
### 2.3 Kernel functions
Eq. (5) and Eq. (6) can be reformulated as follows:
$G(\tau)=\int^{+\infty}_{-\infty}d\omega~{}K(\tau,\omega)A(\omega),$ (12)
and
$G(i\omega_{n})=\int^{+\infty}_{-\infty}d\omega~{}K(\omega_{n},\omega)A(\omega),$
(13)
where $K(\tau,\omega)$ and $K(\omega_{n},\omega)$ are the so-called kernel
functions. Their definitions are as follows:
$K(\tau,\omega)=\frac{e^{-\tau\omega}}{1\pm e^{-\beta\omega}},$ (14)
and
$K(\omega_{n},\omega)=\frac{1}{i\omega_{n}-\omega},$ (15)
where +(-) in the denominator of Eq. (14) stands for fermions (bosons).
As mentioned above, the kernel function is quite strange. The values of
$K(\tau,\omega)$ could change by tens of orders of magnitude. Especially, at
large positive and negative frequencies, $K(\tau,\omega)$ is exponentially
small. It implies that at large $|\omega|$ the features of $A(\omega)$ are
sensitive to the fine structures of $G(\tau)$. However, the data of $G(\tau)$
provided by QMC simulations are always fluctuant and noisy [27]. Tiny
deviations in $G(\tau)$ from its expected values can lead to enormous changes
in $A(\omega)$. Thus, analytical continuation is often characterized as an
ill-posed problem [12].
In principle, for incomplete and noise $G(\tau)$ or $G(i\omega_{n})$, the
number of spectral functions $A(\omega)$ that satisfy Eq. (12) and Eq. (13) is
infinite. So the question becomes which $A(\omega)$ should be chosen. Now
there are two different strategies to solve this problem. The first one is to
choose the most likely $A(\omega)$. The second one is to evaluate the average
of all the candidate spectral functions. In next section, we will introduce
three primary analytical continuation methods that follow the two strategies
and have been implemented in the ACFlow toolkit. For the sake of simplicity,
we will concentrate on analytical continuation of imaginary time Green’s
functions in main text.
## 3 Methods
### 3.1 Maximum entropy method
Perhaps the maximum entropy method is the most frequently used approach for
analytical continuation problems in the last decades [11, 12] because of its
high computational efficiency. Next, we will discuss the basic principle and
several variants of it.
#### 3.1.1 Bayesian inference
Bayes’s theorem is the cornerstone of the maximum entropy method. Given two
events $a$ and $b$, Bayes’s theorem says:
$P[a|b]P[b]=P[b|a]P[a],$ (16)
where $P[a]$ is the probability of event $a$, $P[a|b]$ is the conditional
probability of event $a$ with given event $b$. In the scenario of analytical
continuation problem, $\bar{G}(\tau)$ and $A(\omega)$ are treated as two
events, where $\bar{G}(\tau)$ denotes the measured value of $G(\tau)$. So the
best solution for $A(\omega)$ is of course the one that maximizes
$P[A|\bar{G}]$, which is called the posterior probability. According to the
Bayes’s theorem, we get
$P[A|\bar{G}]=\frac{P[\bar{G}|A]P[A]}{P[\bar{G}]},$ (17)
where $P[\bar{G}|A]$ is the likelihood function, $P[A]$ is the prior
probability, and $P[\bar{G}]$ is the evidence. Since the evidence is a
normalization constant depending on the prior probability and the likelihood
function only, it is ignored in the following discussions. Thus,
$P[A|\bar{G}]\propto P[\bar{G}|A]P[A].$ (18)
#### 3.1.2 Posterior probability
In the maximum entropy method, the likelihood function $P[\bar{G}|A]$ is
assumed to be in direct proportion to $e^{-\chi^{2}/2}$. Here, $\chi^{2}$ is
named as goodness-of-fit function. It measures the distance between
$\bar{G}(\tau)$ and reconstructed imaginary time Green’s function
$\tilde{G}(\tau)$:
$\chi^{2}=\sum^{L}_{i=1}\left[\frac{\bar{G}_{i}(\tau)-\tilde{G}_{i}(\tau)}{\sigma_{i}}\right]^{2},$
(19) $\tilde{G}_{i}=\sum_{j}K_{ij}A_{j}.$ (20)
Here, $L$ is number of imaginary time points, $\sigma$ denotes the error bar
(standard deviation) of $\bar{G}(\tau)$. $K_{ij}$ and $A_{j}$ are discrete
kernel and spectral functions, respectively. On the other hand, the prior
probability $P[A]$ is supposed to be in direct proportion to $e^{\alpha S}$,
where $\alpha$ is a regulation parameter and $S$ means entropy. Sometimes $S$
is also known as the Kullback-Leibler distance. Its formula is as follows:
$S=\int
d\omega\left(A(\omega)-m(\omega)-A(\omega)\log\left[\frac{A(\omega)}{m(\omega)}\right]\right),$
(21)
where $m(\omega)$ is the default model function.
According to the Bayes’s theorem, the posterior probability
$P[A|\bar{G}]\propto e^{Q}$ and
$Q=\alpha S-\frac{\chi^{2}}{2}.$ (22)
#### 3.1.3 Algorithms of maximum entropy method
Now the original analytical continuation problem becomes how to figure out the
optimal $A(\omega)$ that maximizes $Q$. In other words, we have to solve the
following equation:
$\frac{\partial Q}{\partial A}\bigg{|}_{A=\hat{A}}=0,$ (23)
where $\hat{A}(\omega)$ is the optimal $A(\omega)$. Eq. (23) can be easily
solved by using standard Newton method. However, the obtained
$\hat{A}(\omega)$ is $\alpha$-dependent. That is to say, for a given $\alpha$,
there is always a $\hat{A}(\omega)$ that satisfies Eq. (23). So, new problem
arises because we have to figure out a way to construct the final spectral
function from these $\alpha$-resolved $\hat{A}(\omega)$. Now there exist four
algorithms, namely “historic”, “classic”, “bryan”, and “$\chi^{2}$kink”. Next
we will introduce them one by one.
_Historic algorithm_. The historic algorithm is quite simple. The $\alpha$
parameter will be adjusted iteratively to meet the following criterion:
$\chi^{2}=N,$ (24)
where $N$ is the number of mesh points for spectral density $A(\omega)$.
_Classic algorithm_. The basic equation for the classic algorithm reads:
$-2\alpha S(A_{\alpha})=\text{Tr}\left[\frac{\Lambda(A_{\alpha})}{\alpha
I+\Lambda(A_{\alpha})}\right],$ (25)
where $I$ is an identity matrix. The elements of $\Lambda$ matrix are
calculated as follows:
$\Lambda_{ij}=\sqrt{A_{i}}\left(\sum_{kl}K_{ki}[C^{-1}]_{kl}K_{lj}\right)\sqrt{A_{j}},$
(26)
where $C$ is the covariance matrix. Eq. (25) will be iteratively solved until
the optimal $\alpha$ and $\hat{A}(\omega)$ are determined.
_Bryan algorithm_. In both historic and classic algorithms, the spectral
function $\hat{A}(\omega)$ is always related to an optimal $\alpha$ parameter.
However, the spirit of the bryan algorithm [28] is completely different. It
tries to generate a series of $\alpha$ parameters and yield the corresponding
$A_{\alpha}(\omega)$. Then the final spectral function $A(\omega)$ is obtained
by evaluating the following integration:
$\overline{A(\omega)}=\int d\alpha~{}A_{\alpha}(\omega)P[\alpha|\bar{G}].$
(27)
_$\chi^{2}$ kink algorithm_. This algorithm was proposed by Bergeron and
Tremblay [22] recently. The first step is to generate a series of $\alpha$
parameters, and evaluate the corresponding spectral functions
$A_{\alpha}(\omega)$ and the goodness-of-fit functions $\chi^{2}[A_{\alpha}]$.
Then we plot $\log_{10}(\chi^{2})$ as a function of $\log_{10}(\alpha)$.
Usually this plot is split into three different regions: (1) Default model
region. In the limit of $\alpha\to\infty$, $\chi^{2}$ goes to a constant high
value. It means that the likelihood function $e^{-\chi^{2}/2}$ has negligible
weight, such that the prior probability $e^{\alpha S}$ becomes dominant and
minimizes $Q[A]$. At that time, the calculated $A(\omega)$ resembles the
default model function $m(\omega)$. (2) Noise-fitting region. In the limit of
$\alpha\to 0$, $\chi^{2}$ is relatively flat and approaches its global
minimum. In this region, the minimization algorithm tends to fit the noise in
$G(\tau)$. (3) Information-fitting region. $\alpha S$ is comparable with
$\chi^{2}/2$, so that $\chi^{2}$ is strongly dependent on $\alpha$. Bergeron
_et al._ suggested that the optimal $\alpha$ parameter situates in the
crossover between noise-fitting region and information-fitting region [22]. So
the second derivative of $\chi^{2}$ with respect to $\alpha$ is calculated,
and the maximum value in the resulting curve indicates the optimal value of
$\alpha$. Quite recently, Kaufmann and Held proposed a more numerically stable
and flexible approach to compute the optimal $\alpha$ [23]. They use the
following empirical function to fit dataset
$\\{\log_{10}(\alpha),\log_{10}(\chi^{2})\\}$:
$\phi(x;a,b,c,d)=a+\frac{b}{1+e^{-d(x-c)}},$ (28)
where $a$, $b$, $c$, and $d$ are fitting parameters. Then the optimal $\alpha$
is approximated by $10^{c-f/d}$, where $f$ is a numerical constant (Its
favorite value lies in $[2,2.5]$).
### 3.2 Stochastic analytical continuation
In principle, for given Green’s function $G$, there exists infinitely many
spectral densities $A(\omega)$ that can be used to reconstruct $G$ via Eq.
(12) and Eq. (13). The maximum entropy method tries to pick up the most likely
spectral function which maximizes $P[A|\bar{G}]$ (It actually maximizes $Q$)
[11, 12]. Here, we would like to introduce an alternative approach, namely the
stochastic analytical continuation [13, 14, 29, 30, 31, 32, 33, 34]. It is
argued that the weights for all the possible spectral densities are the same
if they can give rise to the same $\chi^{2}$. At first, a sequence of spectral
densities will be generated by stochastic method. Then an unbiased thermal
average of all possible spectra, Boltzmann weighted according to goodness-of-
fit function $\chi^{2}$, produces an average spectrum. Thus sometimes the
method was named as average spectrum method or stochastic sampling method in
the references [35, 36, 37, 38]. There are several variants for the stochastic
analytical continuation. Next we will introduce two representative algorithms
as proposed by A. W. Sandvik [13] and K. S. D. Beach [29], respectively.
#### 3.2.1 Sandvik’s algorithm
Figure 1: Typical Monte Carlo field configurations for the stochastic
analytical continuation (A. W. Sandvik’s algorithm) [13]. Here, the $\delta$
functions reside at unrestricted frequencies $\\{\omega_{i}\\}$, but their
amplitudes $\\{\gamma_{i}\\}$ are equal and fixed. Note that different
parameterizations are also possible [34].
It was early on realized that a different way to achieve a smooth spectrum is
to average over many solutions with reasonable $\chi^{2}$ values [33]. Several
years later, A. W. Sandvik introduced the stochastic analytical continuation
in a slightly different form [13]. He suggested that the spectral function
$A(\omega)$ can be parameterized using $N$ $\delta$ functions (Please see
Figure 1 for a schematic diagram):
$A(\omega)=\sum^{N}_{i=1}\gamma_{i}\delta(\omega-\omega_{i}),$ (29)
where $\gamma_{i}$ and $\omega_{i}$ denote the amplitude and position of the
$i$-th $\delta$ function, respectively. Next, the Metropolis important
sampling algorithm is employed to sample the configuration space
$\mathcal{C}=\\{\omega_{i},\gamma_{i}\\}$. In practice, there are two
elementary Monte Carlo updates. One is to change the amplitudes of a pair of
$\delta$ functions under the constraint $\sum_{i}\gamma_{i}=1$. Another one is
to shift position of a randomly chosen $\delta$ function. Of course, block or
global updates can be implemented to improve ergodicity and sampling
efficiency [35, 36].
The transition probability of Monte Carlo updates reads:
$p(\mathcal{C}\to\mathcal{C}^{\prime})=\exp\left(-\frac{\Delta\chi^{2}}{2\Theta}\right),$
(30)
where the goodness-of-fit function $\chi^{2}$ can be evaluated by Eq. (19),
$\Theta$ is a regulation parameter which is similar to the $\alpha$ parameter
appeared in the maximum entropy method. Well, the remaining problem is how to
fix $\Theta$. Sandvik suggested to measure the following entropic term for a
series of $\Theta$:
$S(\Theta)=-\sum^{N}_{i=1}\gamma_{i}\log(\gamma_{i})K(0,\omega_{i}),$ (31)
where $K$ is the kernel function as defined above [13]. Then make a plot of
$S$ with respect to $\log(\Theta^{-1})$. Overall, when $\Theta$ is large, $S$
exhibits large fluctuations. When $\Theta$ is small, $S$ will approach its
global minimum steadily. A sharp drop in $S$ before the approach to a constant
value has been observed, and there is a local maximum at some
$\Theta=\hat{\Theta}$ preceding the drop. Thus, Sandvik postulated that
$\hat{\Theta}$ was the optimum value at which to accumulate and average the
spectral function. Syljuåsen _et al._ [38] suggested that let $\Theta=1$.
Fuchs _et al._ tried to fix $\Theta$ by using Bayesian inference. Such that
their approach was named as stochastic analytical inference [14]. Very
recently, Shao and Sandvik _et al._ proposed a smart method to determine the
optimal value of $\Theta$ [32, 30]. $\Theta$ is adjusted so that
$\langle\chi^{2}(\Theta)\rangle\approx\chi^{2}_{\text{min}}+c\sqrt{2\chi^{2}_{\text{min}}},$
(32)
where $c$ is a constant of order 1, $\chi^{2}_{\text{min}}$ is the minimum
value of $\chi^{2}$ at given $\Theta$. Note that $\chi^{2}_{\text{min}}$ can
be obtained in a simulated annealing process [39] to very low $\Theta$.
#### 3.2.2 Beach’s algorithm
Figure 2: Typical Monte Carlo field configurations for the stochastic
analytical continuation (K. S. D. Beach’s algorithm) [29]. Note that the
amplitudes $\\{\gamma_{i}\\}$ of all the $\delta$ functions are not identical.
Both amplitudes $\\{\gamma_{i}\\}$ and positions $\\{r_{i}\\}$
($0.0<r_{i}<1.0$) can be sampled by using Monte Carlo method.
K. S. D. Beach proposed another variant of stochastic analytical continuation
in 2004 [29]. In his proposal, the analytical continuation problem is mapped
into a system of interacting classic fields at first. Then the classic field
is sampled using Monte Carlo method to obtain the final solution. He concluded
that the maximum entropy method is simply the mean field limit of the
stochastic analytical continuation. Next, this algorithm will be explained
concisely.
_Classic fields_. Recalled that the goodness-of-fit functional $\chi^{2}[A]$
measures how closely the Green’s function generated from $A(\omega)$ matches
the raw input data. Its expression is rewritten as follows:
$\chi^{2}[A]=\int^{\beta}_{0}\frac{1}{\sigma(\tau)^{2}}\left|\int
d\omega~{}K(\tau,\omega)A(\omega)-\bar{G}(\tau)\right|^{2}d\tau.$ (33)
At first, a new variable $x$ is introduced. The relation between $x$ and
$\omega$ is:
$x=\phi(\omega)=\int^{\omega}_{-\infty}d\omega^{\prime}~{}m(\omega^{\prime}),$
(34)
where $m(\omega)$ denotes the default model function. Clearly, the
$\phi(\omega)$ function defines a smooth mapping from $\mathbf{R}\to[0,1]$.
Since $\omega=\phi^{-1}(x)$, a dimensionless classic field $n(x)$ is created:
$n(x)=\frac{A(\phi^{-1}(x))}{m(\phi^{-1}(x))}.$ (35)
It is easy to prove that both $n(x)$ and $A(\omega)$ obey similar
normalization condition:
$\int d\omega~{}A(\omega)=\int^{1}_{0}dx~{}n(x)=1.$ (36)
Next, in analogy with the goodness-of-fit functional $\chi^{2}[A]$, the
Hamiltonian for the system of classic field $\\{n(x)\\}$ can be defined as
follows:
$H[n(x)]=\int^{\beta}_{0}\frac{d\tau}{\sigma(\tau)^{2}}\left|\int^{1}_{0}dx~{}K(\tau,x)n(x)-\bar{G}(\tau)\right|.$
(37)
Supposing $\alpha$ is an inverse temperature of the system, then the partition
function $Z$ is:
$Z=\int\mathcal{D}n~{}e^{-\alpha H[n]},$ (38)
where
$\int\mathcal{D}n=\int^{\infty}_{0}\left[\prod_{x}dn(x)\right]\delta\left(\int^{1}_{0}dx~{}n(x)-1\right).$
(39)
The thermally averaged value of the classic field is:
$\langle n(x)\rangle=\frac{1}{Z}\int\mathcal{D}n~{}n(x)e^{-\alpha H[n]}.$ (40)
Finally, according to the definition of the classic field, the averaged
spectral density $\langle A(\omega)\rangle$ can be expressed as:
$\langle A(\omega)\rangle=\langle n(\phi(\omega))\rangle m(\omega).$ (41)
So, by introducing the classic field $\\{n(x)\\}$, the analytical continuation
problem is converted into a statistical sampling of the classic field, which
is easily solved by using Monte Carlo method.
_Monte Carlo sampling_. Next we clarify how to sample the classic field.
Similar to Sandvik’s algorithm [13, 34], $n(x)$ is parameterized as a
superposition of many $\delta$ functions (see Figure 2 for a schematic
diagram):
$n_{\mathcal{C}}(x)=\sum_{i}\gamma_{i}\delta(x-r_{i}),$ (42)
where $\gamma_{i}$ and $r_{i}$ denote amplitude (weight) and position of the
$i$-th $\delta$ function, respectively. And $\mathcal{C}$ means a
configuration space formed by a set of $r_{i}$ and $\gamma_{i}$,
$\mathcal{C}=\\{r_{i},\gamma_{i}\\}.$ (43)
Note that $\gamma_{i}$ and $r_{i}$ satisfy the following constraints:
$\forall i,~{}\gamma_{i}>0,~{}\sum_{i}\gamma_{i}=1,~{}0\leq r_{i}\leq 1.$ (44)
Supposed that there is a transition from $\mathcal{C}$ to
$\mathcal{C}^{\prime}$
($\\{r_{i},\gamma_{i}\\}\to\\{r^{\prime}_{i},\gamma^{\prime}_{i}\\}$):
$r_{i}\to r^{\prime}_{i}=r_{i}+\sum_{\lambda\in\Lambda}\delta_{i\lambda}\Delta
r_{\lambda},$ (45)
$\gamma_{i}\to\gamma^{\prime}_{i}=\gamma_{i}+\sum_{\lambda\in\Lambda}\delta_{i\lambda}\Delta\gamma_{\lambda},$
(46)
where $\Lambda$ means a subset of the $\delta$ functions, then the Hamiltonian
of the system is changed from $H_{\mathcal{C}}$ to $H_{\mathcal{C}^{\prime}}$.
According to Eq. (37), $H_{\mathcal{C}}$, $H_{\mathcal{C}^{\prime}}$, and
their difference $\Delta H$ can be calculated by:
$H_{\mathcal{C}}=\int^{\beta}_{0}d\tau~{}h_{\mathcal{C}}(\tau)^{2},$ (47)
$H_{\mathcal{C}^{\prime}}=\int^{\beta}_{0}d\tau\left[h_{\mathcal{C}}(\tau)+\Delta
h(\tau)\right]^{2},$ (48) $\Delta
H=H_{\mathcal{C}^{\prime}}-H_{\mathcal{C}}=\int^{\beta}_{0}d\tau~{}\Delta
h(\tau)[2h_{\mathcal{C}}(\tau)+\Delta h(\tau)].$ (49)
Here,
$h(\tau)=\frac{1}{\sigma(\tau)}\left[\int^{1}_{0}dx~{}K(\tau,x)n(x)-\bar{G}(\tau)\right],$
(50)
and
$\Delta
h(\tau)=\frac{1}{\sigma(\tau)}\sum_{\lambda\in\Lambda}\left[\gamma^{\prime}_{\lambda}K(\tau,r^{\prime}_{\lambda})-\gamma_{\lambda}K(\tau,r_{\lambda})\right].$
(51)
Finally, the transition probability from $\mathcal{C}$ to
$\mathcal{C}^{\prime}$ reads
$p(C\to C^{\prime})=\exp(-\alpha\Delta H).$ (52)
_Parallel tempering_. The parallel tempering trick [40] is adopted to improve
the Monte Carlo algorithm as described above. It is possible to proceed
multiple simulations simultaneously for a sequence of inverse temperature
parameters $\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{N}\\}$. The ratio for two
adjacent $\alpha$ parameters is a constant: $\alpha_{p+1}/\alpha_{p}=R$. Note
that the field configurations in all simulations evolve in parallel but not
independently. We can swap the field configurations between two adjacent
layers. Of course, the detailed balance is always preserved, and each
simulation will eventually settle into thermal equilibrium at given $\alpha$.
The transition probability of such a global Monte Carlo update is:
$p(\mathcal{C}\to\mathcal{C}^{\prime})=\exp[(\alpha_{p}-\alpha_{q})(H_{p}-H_{q})],$
(53)
where $p$ and $q$ are layer indices, and $p=q\pm 1$. Parallel tempering
eliminates the need for an initial annealing stage. Another advantage of
parallel tempering is that it yields a complete temperature profile of all the
important thermodynamic variables (such as specific heat and internal energy),
which can be used to estimate the critical $\alpha$ and the final spectral
function $\langle A(\omega)\rangle$.
_Critical inverse temperature_. Clearly, $\langle n(x)\rangle$ strongly
depends on the inverse temperature $\alpha$. How to use these
$\alpha$-dependent $\langle n(x)\rangle$ to construct the final spectral
function? Beach suggested a novel method [29]. During parallel tempering
process, the internal energy of the system is also measured in addition to
$\langle n(x)\rangle$:
$U(\alpha_{p})=\langle H[n]\rangle_{\alpha_{p}}.$ (54)
Let us plot $\log_{10}[U(\alpha)]$ as a function of $\log_{10}(\alpha)$. We
find that $\log_{10}[U(\alpha)]$ drops quickly at first when
$\log_{10}(\alpha)$ increases, and then it approaches to a constant value
slowly. The knee in $\log_{10}[U(\alpha)]$ function, occurring in the vicinity
of $\alpha=\alpha^{*}$ (the corresponding layer index $p=p^{*}$), signals a
jump in specific heat (a thermodynamic phase transition). Then the averaged
spectral function is constructed by:
$\langle\langle
n(x)\rangle\rangle=\frac{\sum^{N-1}_{p=p*}[U(\alpha_{p})-U(\alpha_{p+1})]\langle
n(x)\rangle_{\alpha_{p}}}{U(\alpha_{p*})-U(\alpha_{N})},$ (55)
where $N$ is the total number of $\alpha$, and $\alpha_{p*}$
($\equiv\alpha^{*}$) is the critical inverse temperature.
_Likelihood function_. Neither of the Sandvik’s and Beach’s algorithms needs
extra entropic term to regulate the spectral densities [13, 29]. All the
stochastically generated spectra are treated on the same footing. Thus, the
calculated spectral function retains more subtle structures than that obtained
by the maximum entropy method. Actually, in the stochastic analytical
continuation,
$\langle A\rangle=\int\mathcal{D}A~{}P[A|\bar{G}]A.$ (56)
The weight of the candidate spectral function $A$ is given by the likelihood
function $P[A|\bar{G}]$. Eq. (30) and Eq. (52) can be viewed as likelihood
functions in the stochastic analytical continuation.
### 3.3 Stochastic optimization method
Figure 3: Typical Monte Carlo field configurations for the stochastic
optimization method [15]. The spectral function is parameterized by multiple
rectangle functions. Here, $c_{i}$, $w_{i}$, and $h_{i}$ denote the center,
width, and height of the $i$-th rectangle, respectively.
A. O. Mishchenko _et al._ [15] proposed the stochastic optimization method.
Though it looks like the stochastic analytical continuation [13, 29], their
differences are quite apparent. The stochastic optimization method does not
need any likelihood function or Boltzmann distribution to weight the candidate
spectral functions. It generates a lot of spectral functions through Monte
Carlo samplings. For each candidate spectral function, the deviation $D$
between the reconstructed Green’s function $\tilde{G}$ and original Green’s
function $\bar{G}$ is measured. Those spectral functions with small deviations
$D$ are selected and averaged. Such that the desired spectral function is
obtained.
_Deviation function_. In the stochastic optimization method, the deviation
between reconstructed data $\tilde{G}$ and input data $\bar{G}$ is described
by:
$D[A]=\sum^{M}_{m=1}|\Delta(m)|,$ (57)
where $M$ is the number of input data, and $\Delta(m)$ is the deviation
function,
$\Delta(m)=\frac{\bar{G}(m)-\tilde{G}(m)}{S(m)}.$ (58)
Here, $S(m)=|G(m)|^{d}$ (where $0\leq d\leq 1$). Recently, Krivenko _et al._
suggested that it would be better to use the goodness-of-fit functional
$\chi^{2}[A]$ to replace $D[A]$ [24, 25].
_Spectral density_. The stochastic optimization method will try to accumulate
the candidate spectral functions that manifest small $D[A]$. Supposed the
Monte Carlo simulations are repeated for $L$ times. For the $i$-th Monte Carlo
simulation, the spectral density $A_{i}(\omega)$ and deviation $D[A_{i}]$ are
recorded. The minimum value of deviation is $\min\\{D[A_{i}]\\}$. Thus, the
final spectral density reads:
$A(\omega)=\frac{1}{L_{\text{good}}}\sum^{L}_{i=1}\theta(\alpha_{\text{good}}\min\\{D[A_{i}]\\}-D[A_{i}])A_{i}(\omega).$
(59)
Here, $\theta(x)$ is the Heaviside step function, and $\alpha_{\text{good}}$
is a adjustable parameter. $L_{\text{good}}$ denotes the number of “good”
spectral functions:
$L_{\text{good}}=\sum^{L}_{i=1}\theta(\alpha_{\text{good}}\min\\{D[A_{i}]\\}-D[A_{i}]).$
(60)
That is to say, only those spectral functions who satisfy the following
condition will be selected:
$D[A_{i}]\leq\alpha_{\text{good}}\min\\{D[A_{i}]\\}.$ (61)
Clearly, the larger $\alpha_{\text{good}}$ is, the more spectral functions are
included. It is usually set to 2.
_Rectangle representation_. Similar to the stochastic analytical continuation
[13, 29], the stochastic optimization method usually employs a few rectangle
functions to parameterize the spectral function:
$A(\omega)=\sum_{i}R_{\\{c_{i},w_{i},h_{i}\\}}(\omega),$ (62)
where $i$ is the index of rectangle function. The definition of rectangle
function $R_{\\{c_{i},w_{i},h_{i}\\}}(\omega)$ reads:
$R_{\\{c_{i},w_{i},h_{i}\\}}(\omega)=h_{i}\theta[\omega-(c_{i}-w_{i}/2)]\theta[(c_{i}+w_{i}/2)-\omega],$
(63)
where $c_{i}$, $w_{i}$, $h_{i}$ denote the center, width, and height of the
$i$-th rectangle, respectively. Pay attention to that the area of all
rectangles must be normalized to 1:
$\sum_{i}h_{i}w_{i}=1.$ (64)
_Monte Carlo sampling_. The parameters of all rectangle functions create a
configuration space:
$\mathcal{C}=\\{c_{i},w_{i},h_{i}\\}.$ (65)
Then the Metropolis algorithm is utilized to sample this configuration space.
Mishchenko _et al._ introduces seven Monte Carlo updates [15, 24], including:
(a) Insert a new rectangle, change width and height of another rectangle; (b)
Remove an existing rectangle, change width and height of another rectangle;
(c) Shift position of any rectangles; (d) Change widths of any two rectangles;
(e) Change heights of any two rectangles; (f) Split a rectangle into two new
rectangles; (g) Merge two adjacent rectangles into a new rectangle. The
transition probability of these Monte Carlo updates is:
$p(\mathcal{C}\to\mathcal{C}^{\prime})=\left(\frac{D[A_{\mathcal{C}}]}{D[A_{\mathcal{C}^{\prime}}]}\right)^{1+d}$
(66)
As compared to the maximum entropy method [11, 12], the likelihood function,
entropic term, and model function are absent in the stochastic optimization
method. As compared to the stochastic analytical continuation [13, 29], there
are no adjustable parameters, such as $\Theta$ in Sandvik’s algorithm and
$\alpha$ in Beach’s algorithm. Thus, the simulated results of the stochastic
optimization method are less affected by artificial parameters.
## 4 Overview
### 4.1 Major features
Now the ACFlow toolkit supports three analytical continuation methods as
introduced above. It includes four different analytical continuation solvers,
namely MaxEnt, StochAC, StochSK, and StochOM. Just as their names suggested,
the MaxEnt solver implements the maximum entropy method [12]. The StochAC and
StochSK solvers implement the K. S. D. Beach’s algorithm [29] and A. W.
Sandvik’s algorithm [13] of the stochastic analytical continuation,
respectively. The StochOM solver implements the stochastic optimization method
[15]. The ACFlow toolkit also provides a convenient library, which can be used
to prepare and carry out analytical continuation calculations flexibly. The
major features of the ACFlow toolkit are summarized in Table 1.
Features | MaxEnt | StochAC | StochSK | StochOM
---|---|---|---|---
Matrix-valued Green’s function | Y | N | N | N
Imaginary time grid | Y | Y | Y | Y
Matsubara frequency grid | Y | Y | Y | Y
Linear mesh | Y | Y | Y | Y
Nonlinear mesh | Y | Y | Y | Y
Fermionic kernel | Y | Y | Y | Y
Bosonic kernel | Y | Y | Y | Y
Self-defined model function | Y | N | N | N
Constrained analytical continuation | N | Y | Y | Y
Regeneration of input data | Y | Y | Y | Y
Kramers-Kronig transformation | Y | Y | Y | Y
Parallel computing | N | Y | Y | Y
Parallel tempering | N | Y | N | N
Interactive mode | Y | Y | Y | Y
Script mode | Y | Y | Y | Y
Standard mode | Y | Y | Y | Y
Table 1: Major features of the ACFlow toolkit. MaxEnt, StochAC, StochSK, and
StochOM are the four analytical continuation solvers as implemented in this
toolkit.
In Table 1, “Y” means yes while “N” means no. “Interactive mode”, “Script
mode”, and “Standard model” are the three running modes supported by the
ACFlow toolkit. We will introduce them in next section. The MaxEnt solver
supports the “historic”, “classic”, “bryan”, and “chi2kink” algorithms to
determine the $\alpha$ parameter. The StochAC solver is only compatible with a
flat model function, while the StochSK and StochOM solvers don’t rely on any
default model functions. The StochOM solver does not support analytical
continuation of fermionic imaginary time Green’s function for the moment.
### 4.2 Implementations
The ACFlow toolkit is developed with pure Julia language. Thanks to powerful
type system and multiple dispatch paradigm of the Julia language, the four
different analytical continuation solvers are integrated into an united
software architecture. Redundant codes are greatly reduced. It is quite easy
to implement new analytical continuation solver or add new features to the
existing solvers in the future. Distributed computing is a built-in feature of
Julia. So, it is straightforward to realize parallel calculations in the
ACFlow toolkit. Now except for the MaxEnt solver, all the other solvers are
parallelized.
Filename | Description
---|---
ACFlow.jl | Entry of the ACFlow module.
maxent.jl | Maximum entropy method.
sac.jl | Stochastic analytical continuation (K. S. D. Beach’s algorithm).
san.jl | Stochastic analytical continuation (A. W. Sandvik’s algorithm).
som.jl | Stochastic optimization method.
global.jl | Numerical and physical constants.
types.jl | Basic data structures and computational parameters.
base.jl | Driver for analytical continuation simulation.
inout.jl | Read input data and write calculated results.
config.jl | Parse configuration file and extract computational parameters.
math.jl | Root finding, numerical integration, interpolation, Einstein summation, and curve fitting.
util.jl | Some utility functions.
mesh.jl | Meshes for spectral density.
grid.jl | Grids for input data.
model.jl | Default model functions.
kernel.jl | Kernel functions.
Table 2: List of source codes of the ACFlow toolkit.
The source codes of the ACFlow toolkit are placed in the acflow/src folder.
Their functions are summarized in Table 2. The documentation of the ACFlow
toolkit is written by using the Markdown language and Documenter.jl package.
The source codes are placed in the acflow/docs folder. The users can build
documentation by themselves. Please see section 5 for how to do that. Or they
can read the latest documentation in the following website:
https://huangli712.github.io/projects/acflow/index.html
Ten tests and four tutorials are also shipped with the ACFlow toolkit. Their
source codes are placed in the acflow/test folder. See acflow/test/test.md and
acflow/test/tutor.md for more details. The code repository of the ACFlow
toolkit is:
https://github.com/huangli712/ACFlow
## 5 Getting started
In this section, we will discuss how to install and use the ACFlow toolkit.
### 5.1 Installation
It is an easy task to install the ACFlow toolkit. First, since it is written
in pure Julia language, it is necessary to install the Julia runtime
environment at first. The newest version of Julia is always preferred (version
$>$ 1.60). Since the core codes only rely on Julia’s built-in standard
library, no the third-party packages are needed. Second, just download source
codes of the ACFlow toolkit from its github repository. It should be a
compressed file, such as acflow.zip or acflow.tar.gz. Please uncompress it in
your favorite directory by using the following commands:
$ unzip acflow.zip
or
$ tar xvfz acflow.tar.gz
Third, the users have to declare a new environment variable ACFLOW_HOME.
Supposed that the root directory of the ACFLow toolkit is
/home/your_home/acflow, then ACFLOW_HOME should be setup as follows:
$ export ACFLOW_HOME=/home/your_home/acflow/src
Finally, in order to generate the documentation, the users should type the
following commands in the terminal:
$ pwd
/home/your_home/acflow
$ cd docs
$ julia make.jl
After a few seconds, the documentation is built and saved in the
acflow/docs/build directory if everything is OK. The home page of the
documentation is acflow/docs/build/index.html. We can open it with any web
browsers.
### 5.2 Run
The ACFlow toolkit is designed to be flexible and easy-to-use. It provides
three running modes to facilitate analytical continuation calculations, namely
the interactive, script, and standard modes.
_Interactive mode_. With the ACFlow toolkit, the users can setup and carry out
analytical continuation simulations interactively in Julia’s REPL (Read-Eval-
Print Loop) environment. For example,
julia> push!(LOAD_PATH, ENV["ACFLOW_HOME"])
julia> using ACFlow
julia> setup_args("ac.toml")
julia> read_param()
julia> mesh, Aout, Gout = solve(read_data())
Here, ac.toml is a configuration file, which contains essential computational
parameters. The return values of the solve() function (i.e., mesh, Aout, and
Gout) are mesh at real axis $\omega$, spectral density $A(\omega)$, and
reproduced Green’s function $\tilde{G}$, respectively. They can be further
analyzed or visualized by the users.
_Script mode_. The core functionalities of the ACFlow toolkit are exposed to
the users via a simple application programming interface. So, the users can
write Julia scripts easily by themselves to perform analytical continuation
simulations. A minimal Julia script (acrun.jl) is listed as follows:
#!/usr/bin/env julia
push!(LOAD_PATH, ENV["ACFLOW_HOME"])
using ACFlow
setup_args("ac.toml")
read_param()
mesh, Aout, Gout = solve(read_data())
Of course, this script can be extended to finish complex tasks. In section
6.1, a realistic example is provided to show how to complete an analytical
continuation of Matsubara self-energy function via the script mode.
_Standard mode_. In the standard mode, the users have to prepare the input
data manually. In addition, a configuration file must be provided. Supposed
that the configuration file is ac.toml, then the analytical continuation
calculation is launched as follows:
$ /home/your_home/acflow/util/acrun.jl ac.toml
or
$ /home/your_home/acflow/util/Pacrun.jl ac.toml
Noted that the acrun.jl script runs sequentially, while the Pacrun.jl script
supports parallel and distributed computing. As we can conclude from the
filename extension of configuration file (ac.toml), it adopts the TOML
specification. The users may edit it with any text-based editors. Next we will
introduce syntax and format of the input data files and configuration files.
### 5.3 Input files
The input files for the ACFlow toolkit can be divided into two groups: data
files and configuration files.
_Data files_. The input data should be store in CSV-like text files. For
imaginary time Green’s function, the data file should contain three columns.
They represent $\tau$, $\bar{G}(\tau)$, and standard deviation of
$\bar{G}(\tau)$. For fermionic Matsubara Green’s function, the data file
should contain five columns. They represent $\omega_{n}$, Re$G(i\omega_{n})$,
Im$G(i\omega_{n})$, standard deviation of Re$G(i\omega_{n})$, and standard
deviation of Im$G(i\omega_{n})$. For bosonic correlation function
$\chi(i\omega_{n})$, the data file should contain four columns. They represent
$\omega_{n}$, Re$\chi(i\omega_{n})$, and standard deviation of
Re$\chi(i\omega_{n})$.
_Configuration files_. The configuration file adopts the TOML format. It is
used to setup the computational parameters. It consists of one or more blocks.
Possible blocks (or sections) of the configuration file include [BASE],
[MaxEnt], [StochAC], [StochSK], and [StochOM]. The [BASE] block is mandatory,
while the other blocks are optional. A schematic configuration file (ac.toml)
is listed as follows:
⬇
[BASE]
finput = "giw.data"
solver = "StochOM"
...
[MaxEnt]
method = "chi2kink"
...
[StochAC]
nfine = 10000
...
[StochSK]
method = "chi2min"
...
[StochOM]
ntry = 100000
...
In the [BASE] block, the analytical continuation problem is defined. The
solver used to solve the problem must be assigned. The types of mesh, grid,
default model function, and kernel function are also determined. The [MaxEnt],
[StochAC], [StochSK], and [StochOM] blocks are used to customize the
corresponding analytical continuation solvers further. In Table 3-Table 6, all
the possible input parameters for these blocks are collected and summarized.
As for detailed explanations of these parameters, please refer to the user
guide of the ACFlow toolkit. The uses can find it in the acflow/docs
directory.
[BASE] block
---
Parameter | Type | Default | Description
finput | string | “green.data” | Filename for input data.
solver | string | “MaxEnt” | Solver for the analytical continuation problem.
ktype | string | “fermi” | Type of kernel function.
mtype | string | “flat” | Type of default model function.
grid | string | “ffreq” | Grid for input data (imaginary axis).
mesh | string | “linear” | Mesh for output data (real axis).
ngrid | integer | 10 | Number of grid points.
nmesh | integer | 501 | Number of mesh points.
wmax | float | 5.0 | Right boundary (maximum value) of mesh.
wmin | float | -5.0 | Left boundary (minimum value) of mesh.
beta | float | 10.0 | Inverse temperature.
offdiag | bool | false | Treat the off-diagonal part of matrix-valued function?
pmodel | array | N/A | Additional parameters for customizing the default model.
pmesh | array | N/A | Additional parameters for customizing the mesh.
exclude | array | N/A | Restriction of energy range of the spectrum.
Table 3: Possible parameters for the [BASE] block. [MaxEnt] block
---
Parameter | Type | Default | Description
method | string | “chi2kink” | How to determine the optimized $\alpha$ parameter?
nalph | integer | 12 | Total number of the chosen $\alpha$ parameters.
alpha | float | 1e9 | Starting value for the $\alpha$ parameter.
ratio | float | 10.0 | Scaling factor for the $\alpha$ parameter.
blur | float | -1.0 | Shall we preblur the kernel and spectrum?
Table 4: Possible input parameters for the [MaxEnt] block, which are used to
setup the solver based on the maximum entropy method [12, 11]. [StochAC]
block
---
Parameter | Type | Default | Description
nfine | integer | 10000 | Number of points of a very fine linear mesh.
ngamm | integer | 512 | Number of $\delta$ functions.
nwarm | integer | 4000 | Number of Monte Carlo thermalization steps.
nstep | integer | 4000000 | Number of Monte Carlo sweeping steps.
ndump | integer | 40000 | Intervals for monitoring Monte Carlo sweeps.
nalph | integer | 20 | Total number of the chosen $\alpha$ parameters.
alpha | float | 1.0 | Starting value for the $\alpha$ parameter.
ratio | float | 1.2 | Scaling factor for the $\alpha$ parameter.
[StochSK] block
Parameter | Type | Default | Description
method | string | “chi2min” | How to determine the optimized $\Theta$ parameter?
nfine | integer | 100000 | Number of points of a very fine linear mesh.
ngamm | integer | 1000 | Number of $\delta$ functions.
nwarm | integer | 1000 | Number of Monte Carlo thermalization steps.
nstep | integer | 20000 | Number of Monte Carlo sweeping steps.
ndump | integer | 200 | Intervals for monitoring Monte Carlo sweeps.
retry | integer | 10 | How often to recalculate the goodness-of-fit function.
theta | float | 1e6 | Starting value for the $\Theta$ parameter.
ratio | float | 0.9 | Scaling factor for the $\Theta$ parameter.
Table 5: Possible input parameters for the [StochAC] and [StochSK] blocks,
which are used to setup the two solvers based on the stochastic analytical
continuation (Beach’s and Sandvik’s algorithms) [13, 29]. [StochOM] block
---
Parameter | Type | Default | Description
ntry | integer | 2000 | Number of attempts to figure out the solution.
nstep | integer | 1000 | Number of Monte Carlo steps per try.
nbox | integer | 100 | Number of boxes to construct the spectrum.
sbox | float | 0.005 | Minimum area of the randomly generated rectangles.
wbox | float | 0.02 | Minimum width of the randomly generated rectangles.
norm | float | -1.0 | Is the norm calculated?
Table 6: Possible input parameters for the [StochOM] block, which are used to
setup the solver based on the stochastic optimization method [15].
### 5.4 Output files
Once the analytical continuation simulation is finished, the final spectral
function $A(\omega)$ is outputted to Aout.data. As is shown in Eq. (7),
$A(\omega)$ is equivalent to the imaginary part of real frequency Green’s
function Im$G(\omega)$. Then the ACFlow toolkit will automatically calculate
the corresponding real part Re$G(\omega)$ via the Kramers-Kronig
transformation [see Eq. (8)]. The full Green’s function at real axis
$G(\omega)$ is stored in Gout.data. The spectral function is also used to
reconstruct the imaginary time or Matsubara Green’s functions
[$\tilde{G}(\tau)$ or $\tilde{G}(i\omega_{n})$], which is stored in repr.data.
Besides the three output files, the ACFlow toolkit will generate quite a few
output files, which can be used to analyze and diagnose the calculated
results. All of the possible output files of the ACFlow toolkit are collected
and explained in Table 7.
Filename | Description
---|---
Aout.data | Final spectral function $A(\omega)$.
Gout.data | Full Green’s function at real axis $G(\omega)$.
repr.data | Reproduced Green’s function $\tilde{G}$ at imaginary time or frequency axis.
model.data | Default model function $m(\omega)$.
chi2.data | $\log_{10}(\chi^{2})$ vs $\log_{10}(\alpha)$.
prob.data | $P[\alpha|\bar{G}]$ vs $\alpha$ for the MaxEnt solver (bryan algorithm).
Aout.data.alpha_$i$ | $\alpha$-resolved spectral function $A_{\alpha}(\omega)$ for the StochAC solver.
hamil.data | $U(\alpha)$ vs $\alpha$ for the StochAC solver.
goodness.dat | $\log_{10}(\chi^{2})$ vs $\log_{10}(\Theta)$ for the StochSK solver.
stat.data | Monte Carlo statistical information for stochastic sampling method.
Table 7: Possible output files of the ACFlow toolkit.
## 6 Examples
In order to demonstrate usefulness of the ACFlow toolkit, four examples are
illustrated in this section. These examples cover typical application
scenarios of the ACFlow toolkit, including analytical continuations of
Matsubara self-energy function, Matsubara Green’s function, imaginary time
Green’s function, and current-current correlation function within the script
mode or standard mode. All of the necessary source codes and data files, which
can be used to reproduce the results as shown in this section, are placed in
the /home/your_home/acflow/test/T* folders.
### 6.1 Matsubara self-energy function
Now let us consider the following single-band Hubbard model on a Bethe lattice
at first:
$H=-t\sum_{\langle
ij\rangle\sigma}c^{\dagger}_{i\sigma}c_{j\sigma}-\mu\sum_{i}n_{i}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},$
(67)
where $t$ is the hopping parameter, $\mu$ is the chemical potential, $U$ is
the Coulomb interaction, $n$ is the occupation number, $\sigma$ denotes the
spin, $i$ and $j$ are site indices. This model is solved by using the
dynamical mean-field theory (dubbed DMFT) [41] with the hybridization
expansion continuous-time quantum Monte Carlo solver (dubbed CT-HYB) [3] as
implemented in the $i$QIST package [42, 43]. The parameters used in the DMFT +
CT-HYB calculation are $t=0.5$, $U=2.0$, $\mu=1.0$, and $\beta=10.0$. Once the
DMFT self-consistent calculation is finished, the Matsubara self-energy
function $\Sigma(i\omega_{n})$ is obtained. We are going to convert it to real
frequency self-energy function $\Sigma(\omega)$. The data of Matsubara self-
energy function $\Sigma(i\omega_{n})$ have been preprocessed and stored in
siw.data. This file contains five columns, which are used to record the
Matsubara frequency $\omega_{n}$, Re$\Sigma(i\omega_{n})$,
Im$\Sigma(i\omega_{n})$, error bar of Re$\Sigma(i\omega_{n})$, error bar of
Im$\Sigma(i\omega_{n})$, respectively. Only the first twenty Matsubara
frequency points are kept, because the high-frequency data are somewhat noisy.
The purpose of this example is to demonstrate usage of the MaxEnt solver and
the script mode of the ACFlow toolkit. Next we will explain the key steps in
detail. As for the complete Julia script, please refer to sigma.jl and
gendata.jl in the /home/your_home/acflow/test/T01/ folder.
First, we have to load the essential Julia packages. Both the DelimitedFiles
and Printf packages belong to Julia’s standard library. They are used to read
input data and write calculated results, respectively.
⬇
#!/usr/bin/env julia
push!(LOAD_PATH, ENV["ACFLOW_HOME"])
using DelimitedFiles
using Printf
using ACFlow
welcome() # Print welcome message only
Next, the data of Matsubara self-energy function are read from siw.data. The
Hartree term $\Sigma_{H}$ should be subtracted from its real part:
$\Sigma(i\omega_{n})\to\Sigma(i\omega_{n})-\Sigma_{H}.$ (68)
Note that $\Sigma_{H}$ is approximately equal to the asymptotic value of real
part of $\Sigma(i\omega_{n})$ when $n$ goes to infinite.
⬇
# Deal with self-energy function
#
# Read self-energy function
dlm = readdlm("siw.data")
#
# Get grid
grid = dlm[:,1]
#
# Get self-energy function
Sinp = dlm[:,2] + im * dlm[:,3] # Value
Serr = dlm[:,4] + im * dlm[:,5] # Error bar
#
# Subtract hartree term
Sh = 1.0
@. Sinp = Sinp - Sh
Next, the computational parameters are encapsulated into two dictionaries. The
dictionary B is for the [BASE] block, while the dictionary S is for the MaxEnt
solver. Then the setup_param() function is called, so that these parameters
take effect. Here, the MatEnt solver [12, 11] is employed to tackle the
analytical continuation problem. But the other stochastic sampling solvers are
also applicable. The default model function is gaussian. The mesh for spectral
density is non-uniform (A tangent mesh). The number of used $\alpha$
parameters is 15, and the optimal $\alpha$ parameter is determined by the
$\chi^{2}$kink algorithm [22].
⬇
# Setup parameters
#
# For [BASE] block
# See types.jl/_PBASE for default setup
B = Dict{String,Any}(
"solver" => "MaxEnt", # Choose MaxEnt solver
"mtype" => "gauss", # Default model function
"mesh" => "tangent", # Mesh for spectral density
"ngrid" => 20, # Number of input points
"nmesh" => 801, # Number of output points
"wmax" => 8.0, # Right boundary of mesh
"wmin" => -8.0, # Left boundary of mesh
"beta" => 10.0, # Inverse temperature
)
#
# For [MaxEnt] block
# See types.jl/_PMaxEnt for default setup
S = Dict{String,Any}(
"nalph" => 15, # Number of alpha
"alpha" => 1e12, # Starting value of alpha
"blur" => -1.0, # Enable preblur or not
)
#
# Let the parameters take effect
setup_param(B, S)
It is quite easy to start the analytical continuation calculation. Just call
the solve() function and pass the grid, input data, and error bar data to it.
The return values of this function call are real frequency mesh, spectral
density, and reconstructed Matsubara self-energy function.
⬇
# Call the solver
mesh, Aout, Sout = solve(grid, Sinp, Serr)
Finally, the real frequency self-energy function must be supplemented with the
Hartree term, and then the final results are written into sigma.data.
⬇
# Calculate final self-energy function on real axis
#
# Add hartree term
@. Sout = Sout + Sh
#
# Write self-energy function to sigma.data
open("sigma.data", "w") do fout
for i in eachindex(mesh)
z = Sout[i]
@printf(fout, "%20.16f␣%20.16f␣%20.16f\n",
mesh[i], real(z), imag(z))
end
end
Figure 4: Analytical continuation of Matsubara self-energy function by using
the maximum entropy method. (a) Real part of real frequency self-energy
function. (b) Imaginary part of real frequency self-energy function. (c)
$\chi^{2}$ as a function of $\alpha$. The vertical bar indicates the optimal
$\alpha$ parameter chosen by the $\chi^{2}$kink algorithm. (d) Reproduced and
original data for imaginary part of the Matsubara self-energy functions.
The calculated results are displayed in Fig. 4. Fig. 4(a) and (b) show the
real and imaginary parts of the real frequency self-energy function,
respectively. Near the Fermi level, Re$\Sigma(\omega)$ exhibits quasi-linear
behavior, with which the quasiparticle weight $Z$ and effective mass of
electron $m^{*}$ can be easily evaluated [41]. As for the imaginary part,
Im$\Sigma(0)$ is finite, which indicates that the electron-electron scattering
is not trivial. Fig. 4(c) shows the $\alpha$-dependent $\chi^{2}$. The
vertical bar in this figure indicates the optimal $\alpha$ is around 102.154.
In Fig. 4(d), the reproduced and raw Matsubara self-energy functions are
compared. It is apparent that they are consistent with each other.
### 6.2 Matsubara Green’s function
The purpose of the second example is to treat the Matsubara Green’s function
by using the StochOM solver.
At first, please consider the following spectral density with two gaussian
peaks:
$A(\omega)=A_{1}\exp\left[\frac{-(\omega-\epsilon_{1})^{2}}{2\Gamma^{2}_{1}}\right]+A_{2}\exp\left[\frac{-(\omega-\epsilon_{2})^{2}}{2\Gamma^{2}_{2}}\right],$
(69)
with $A_{1}=1.0$, $A_{2}=0.3$, $\epsilon_{1}=0.5$, $\epsilon_{2}=-2.5$,
$\Gamma_{1}=0.2$, and $\Gamma_{2}=0.8$. Then the Matsubara Green’s function
$G(i\omega_{n})$ is evaluated by using Eq. (13) with $\beta=10.0$. Random
noises, generated by the formula $0.0001r_{1}\exp(i2\pi r_{2})$ where $r_{1}$
and $r_{2}$ are pseudo random numbers in ($0.0<r_{1}$, $r_{2}<1.0$), are added
to $G(i\omega_{n})$. The error bar of $G(i\omega_{n})$ is fixed to 1e-4. The
generated data for $G(i\omega_{n})$ are written in giw.data.
Next, we are going to use the standard mode, such that a configure file
(ac.toml) must be prepared. It is listed as follows. Since the StochOM solver
is chosen, the [BASE] and [StochOM] blocks must be present.
⬇
[BASE]
finput = "giw.data"
solver = "StochOM"
ktype = "fermi"
mtype = "flat"
grid = "ffreq"
mesh = "linear"
ngrid = 10
nmesh = 501
wmax = 5.0
wmin = -5.0
beta = 10.0
offdiag = false
[StochOM]
ntry = 100000
nstep = 1000
nbox = 100
sbox = 0.005
wbox = 0.02
norm = -1.0
Figure 5: Analytical continuation of Matsubara Green’s function by using the
stochastic optimization method. (a) Simulated and exact spectral functions.
(b) Reconstructed and synthetic Matsubara Green’s functions. Only the
imaginary parts are presented in this figure.
Then we use the acrun.jl or Pacrun.jl script to perform analytical
continuation simulation. The calculated results are shown in Fig. 5. As is
seen in Fig. 5(a), both the sharp peak around 0.5 eV and the broad peak around
-2.5 eV are correctly reproduced by the StochOM solver. In Fig. 5(b), the
reconstructed Matsubara Green’s function agrees quite well with the raw input
data.
### 6.3 Imaginary time Green’s function
In this example, analytical continuation of imaginary time Green’s function
will be tested. Note that this example is borrowed from Reference [29]
directly.
The exact spectral function reads:
$A(\omega)=\begin{cases}\frac{1}{W}\frac{|\omega|}{\sqrt{\omega^{2}-\Delta^{2}}},~{}\quad&\text{if}~{}\Delta<|\omega|<W/2.\\\
0,&\text{otherwise}.\end{cases}$ (70)
Here, $W$ denotes bandwidth, and $\Delta$ is used to control size of the
energy gap. Let $W=6$ and $2\Delta=1$. This spectrum should exhibit flat
shoulders, steep peaks, and sharp gap edges at the same time. Actually, it is
a typical spectrum of a BCS superconductor.
First, the imaginary time Green’s function $G(\tau)$ is generated using Eq.
(12). Then a normally-distributed random noise is add to $G(\tau)$. Maximum
amplitude of the noise is 1e-4. The error bar of $G(\tau)$ is fixed to 1e-3.
The data are written in gtau.data.
Next, we try to prepare the configure file (ac.toml). In this case, we would
like to benchmark the StochAC solver, so the solver parameter is set to
“StochAC” and the grid parameter is set to “ftime”. Furthermore, the exclude
parameter is enabled to impose some _a priori_ constraints to the spectrum.
The full ac.toml is listed as follows:
⬇
[BASE]
finput = "giw.data"
solver = "MaxEnt"
ktype = "fermi"
mtype = "flat"
grid = "ffreq"
mesh = "linear"
ngrid = 10
nmesh = 501
wmax = 5.0
wmin = -5.0
beta = 10.0
offdiag = false
exclude = [[-5.0,-3.0], [-0.5,0.5], [3.0,5.0]]
[StochAC]
nfine = 10000
ngamm = 512
nwarm = 4000
nstep = 10000000
ndump = 40000
nalph = 40
alpha = 1.00
ratio = 1.20
We perform analytical continuation simulation by using the acrun.jl or
Pacrun.jl script. In order to obtain smooth spectral density, it is useful to
increase number of $\delta$ functions (See ngamm parameter) and number of
Monte Carlo sampling steps (See nstep parameter).
Figure 6: Analytical continuation of imaginary time Green’s function by using
the stochastic analytical continuation (Beach’s algorithm). (a) Simulated and
exact spectral functions. (b) $\alpha$-dependent spectral functions. (c)
Internal energy $U$ as a function of $\alpha$. The vertical bar indicates the
optimal $\alpha$ parameter. (d) Simulated and exact imaginary time Green’s
functions.
Figure 6 shows the calculated results. In Fig. 6(a), the exact spectral
function is compared with the simulated spectrum. Note that besides the
StochAC solver, the other three solvers are also tested. Their results are
also plotted in this figure for a direct comparison. It is remarkable that the
StochAC and StochSK solvers do a superior job of modelling the spectrum. The
major characteristics of the spectrum, including flat regions, steep peaks,
and sharp gap edges, are well captured by the two solvers. Especially, we have
finished more tests without any constraints on the spectral density. The gap
in the spectrum can be reproduced as well. On the other hand, the spectra
obtained by the MaxEnt and StochOM solvers are much too smooth, and show extra
shoulder peaks around $\pm$2.0. Figure 6(b) shows $\alpha$-resolved spectral
functions $A_{\alpha}(\omega)$ for selected $\alpha$ parameters. Fluctuation
in the flat regions of the calculated spectral density grows when $\alpha$
increases. Figure 6(c) shows internal energy $U$ as a function of $\alpha$.
From this figure, the critical $\alpha$ is estimated, which is indicated by
the vertical bar. Finally, the reproduced Green’s function $\tilde{G}(\tau)$
agrees quite well with the raw input data, as is shown in Fig. 6(d).
### 6.4 Current-current correlation function
The previous three examples only involve fermionic correlators. How about
bosonic correlation functions? In this example, we will demonstrate how to
perform analytical continuation simulation for a typical bosonic correlator,
the current-current correlation function $\Pi(\tau)$, to obtain the optical
conductivity $\sigma(\omega)$. Note that this example is taken from Reference
[5] directly.
The exact optical conductivity $\sigma(\omega)$ reads:
$\sigma(\omega)=\left\\{\frac{W_{1}}{1+(\omega/\Gamma_{1})^{2}}+\frac{W_{2}}{1+[(\omega-\epsilon)/\Gamma_{2}]^{2}}+\frac{W_{2}}{1+[(\omega+\epsilon)/\Gamma_{2}]^{2}}\right\\}\frac{1}{1+(\omega/\Gamma_{3})^{6}},$
(71)
where $W_{1}=0.3$, $W_{2}=0.2$, $\Gamma_{1}=0.3$, $\Gamma_{2}=1.2$,
$\Gamma_{3}=4.0$, and $\epsilon=3.0$. The current-current correlation function
$\Pi(\tau)$ can be evaluated from $\sigma(\omega)$ by using the following
equation:
$\Pi(\tau)=\int^{\infty}_{-\infty}K(\tau,\omega)\sigma(\omega)~{}d\omega,$
(72)
where the kernel function $K(\tau,\omega)$ is different from Eq. (14). It
reads:
$K(\tau,\omega)=\frac{1}{\pi}\frac{\omega
e^{-\tau\omega}}{1-e^{-\beta\omega}}.$ (73)
In this case, $\beta$ is fixed to be 20.0.
At first, we use Eq. (71) $\sim$ Eq. (73) to prepare $\Pi(\tau)$. A normally-
distributed random noise is add to $\Pi(\tau)$. Maximum amplitude of the noise
is 1e-4. The error bar of $\Pi(\tau)$ is fixed to 1e-4. The data of
$\Pi(\tau)$ are written in chit.data.
Next, we conduct analytical continuation simulation as usual. The used
configuration file is attached as follows. Here, the StochSK solver is
adopted, so the solver parameter is “StochSK” and the grid parameter is
“btime”. And the Shao-Sandvik algorithm [30] is applied to seek optimal
$\Theta$, so the method parameter is “chi2min”. The users can further increase
the values of nfine, ngamm, and nstep parameters to improve computational
accuracy.
⬇
[BASE]
finput = "chit.data"
solver = "StochSK"
ktype = "bsymm"
mtype = "flat"
grid = "btime"
mesh = "linear"
ngrid = 501
nmesh = 801
wmax = 8.0
wmin = 0.0
beta = 20.0
offdiag = false
[StochSK]
method = "chi2min"
nfine = 40000
ngamm = 1000
nwarm = 1000
nstep = 1000000
ndump = 200
retry = 10
theta = 1e+6
ratio = 0.90
Figure 7: Analytical continuation of current-current correlation function by
using the stochastic analytical continuation (Sandvik’s algorithm). (a)
Simulated and exact optical conductivities $\sigma(\omega)$. (b) Simulated and
exact current-current correlation functions $\Pi(\tau)$.
The calculated results are illustrated in Fig. 7. From Fig. 7(a), it is clear
that the main features of optical conductivity are successfully captured by
the StochSK solver. Both the sharp Drude peak at $\omega=0$ and a broad
satellite peak around $\omega=3.0$ are well reproduced. As is seen in Fig.
7(b), the reconstructed $\tilde{\Pi}(\tau)$ coincides with the original
$\Pi(\tau)$.
## 7 Concluding remarks
In this paper, a full-fledged analytical continuation toolkit named ACFlow is
presented. It implements several primary analytical continuation methods,
including the maximum entropy method, stochastic analytical continuation, and
stochastic optimization method. It provides quite a few validation and
diagnostic tools. It can be used with great flexibility for the analytical
continuations of arbitrary fermionic and bosonic correlation functions
generated by finite-temperature quantum Monte Carlo simulations.
Note that analytical continuation problem is a hotspot in computational
physics and many-body physics all the time. Many efforts have been devoted to
solve it in recent years. Noticeable achievements include maximum quantum
entropy method [44], Nevanlinna analytical continuation [45, 46], blocked-mode
sampling and grid point sampling in stochastic analytical continuation [35,
36], constrained stochastic analytical continuation [31, 34], machine learning
assisted analytical continuation [18, 19], and so on. We would like to
incorporate these new progresses into the ACFlow toolkit in the near future.
Declaration of competing interest
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.
Data availability
The data that support the findings of this study will be made available upon
reasonable requests to the corresponding author.
Acknowledgement
This work is supported by the CAEP Foundation (under Grant No. CX100000) and
the National Natural Science Foundation of China (under Grants No. 11874329
and No. 11934020).
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# A fully asymptotic preserving decomposed multi-group method for the
frequency-dependent radiative transfer equations
Xiaojiang Zhang<EMAIL_ADDRESS>Peng Song<EMAIL_ADDRESS>Institute of Applied Physics and Computational Mathematics, Beijing, 100094,
P.R. China. HEDPS, Center for Applied Physics and Technology, College of
Engineering, Peking University, Beijing, 100871, P.R. China. Yi Shi
<EMAIL_ADDRESS>School of Mathematics, Shandong University, Jinan, 250100,
P.R. China. Min Tang<EMAIL_ADDRESS>
###### Abstract
The opacity of FRTE depends on not only the material temperature but also the
frequency, whose values may vary several orders of magnitude for different
frequencies. The gray radiation diffusion and frequency-dependent diffusion
equations are two simplified models that can approximate the solution to FRTE
in the thick opacity regime. The frequency discretization for the two limit
models highly affects the numerical accuracy. However, classical frequency
discretization for FRTE considers only the absorbing coefficient. In this
paper, we propose a new decomposed multi-group method for frequency
discretization that is not only AP in both gray radiation diffusion and
frequency-dependent diffusion limits, but also the frequency discretization of
the limiting models can be tuned. Based on the decomposed multi-group method,
a full AP scheme in frequency, time, and space is proposed. Several numerical
examples are used to verify the performance of the proposed scheme.
## 1 Introduction
The frequency-dependent radiative transfer equation (FRTE) describes the time
evolution of the probability density function of photons that transport and
interact with the background material, which plays an important role in
astrophysics, inertial confinement fusion (ICF), and high energy physics. Due
to the complexity of the FRTE and its wide applications, FRTE has attracted
the attention of many physicists and mathematicians.
The numerical simulation of FRTE is challenging due to the following
difficulties: 1) The high dimensionality. The photon probability distribution
function depends on space, time, moving velocity and photon frequency, which
is a 7 dimensional function. For the simplest slab geometry, the dimension is
4, which is hard to find an efficient solver due to the ”curse of
dimensionality”; 2) Multiscale parameters. Absorbing and scattering
coefficients are two important material properties that influence the photons’
movements. The absorbing coefficient $\sigma_{a}$ (scattering coefficient
$\sigma_{s}$) gives the probability that photons will be absorbed (scattered)
by the background material. Both $\sigma_{a}$ and $\sigma_{s}$ may depend on
the material type, material temperature and the photon frequency. When
$\sigma_{a}+\sigma_{s}$ is large, the mean free path (the average distance
between two successive scattering or absorption) of the photon is small, the
material is referred to as ”optical thick”, while when $\sigma_{a}+\sigma_{s}$
events is at $O(1)$, the material is ”optical thin”. For photons with
different frequencies, the values of $\sigma_{a}$ and $\sigma_{s}$ may vary
several orders of magnitude, which induces the multiscale property of the
system. To guarantee both accuracy and stability, classical schemes require
very fine space and time steps, which leads to high computational costs. 3)
The strong nonlinearity. The material opacity and the Planck function which
gives the equilibrium radiation frequency distribution depend nonlinearly on
the material temperature. If one needs an efficient solver that allows for
large time steps, some terms have to be treated implicitly and thus nonlinear
iterations are required. However, the nonlinear iterations for high
dimensional equations are extremely expensive.
One popular strategy for the FRTE simulation is the stochastic method
[steinberg2022multi, densmore2012hybrid] based on the Fleck-Cummings’ implicit
Monte Carlo (IMC) method [FC71]. However, it requires a large number of
sampling particles and suffers from unavoidable statistical fluctuations. On
the other hand, there are deterministic solvers for the radiative transfer
equation (RTE), which is a simplified version of FRTE that ignores the
frequency dependence. Deterministic solvers include Discontinuous Galerkin
method [lcs, bailey2008piecewise], photon free method [chang2007incorporation,
chang2007deterministic], diffusion-synthetic acceleration method (DSA) [H1993,
Larsen1988, ALarsen2002] etc. and some of them have been extended to FRTE.
In order to deal with the multiscale parameters in FRTE, two strategies are
proposed recently. One is designing asymptotic preserving (AP) schemes. The AP
scheme provides a general framework of solving the difficulty coming from
multiscale parameters. A scheme is AP when the asymptotic limit of the
discretization becomes a stable solver for the limit model as the scaling
parameter in the microscopic equation goes to zero [jin20101,
hu2017asymptotic, 2018Tang]. In last decade, research on the AP schemes has
been developed rapidly for various applications and we refer interesting
readers to a recent review paper [jin_2022]. The other approach is the moment-
based acceleration scheme which is called High-Order/Low-Order (HOLO)
algorithm [chacon2017multiscale, yee2017stable, park2019multigroup]. In the
HOLO algorithm, a low-order system which consists of the first two moments of
the RTE and the material temperature equation is solved. Then the RTE is
solved by determining the Planckian emission source term using the temperature
obtained in the lower-order system. The benefit of HOLO algorithms is
nonlinear elimination, but their long-term accuracy and nonlinear stability
require discrete consistency between HO and LO formulations
[chacon2017multiscale]. Both approaches are well developed for RTE
simulations.
Designing AP schemes for the nonlinear radiation hydrodynamic system is a
popular topic recently. For the gray radiative transfer equations (GRTE) in
the gray radiative diffusion limit, several AP schemes can be found in the
literature. An AP scheme is constructed in [2001Klar] by decomposing the
distribution function into the equilibrium and non-equilibrium parts; in
[2020park], the authors developed an AP-HOLO algorithm based on the linear
discontinuous Galerkin HO transport solver and the corner-balance LO solver;
in [sun2015asymptotic], an unified gas kinetics scheme (AP-UGKS) was proposed.
However, in order to use an opacity independent time step, a linearized
iterative solver involving the radiation equation has to be employed for the
time-implicit part. In [tang2021accurate], an AP accurate front capturing
scheme was proposed that allows both space and time steps being independent of
the opacity and only requires a scalar Newton’s solver. Later on, in order to
get AP schemes for more complex applications, UGKS has been extended to FRTE
[sunj] and the radiation magnetohydrodynamics (RMHD) system [jin2022spatial,
sun2020multiscale]; The AP linear-discontinuous spatial differencing scheme
has been extended to FRTE in [MWTS]; The idea of intensity decomposition used
in [tang2021accurate] has been extended to the RMHD system in
[jin2022spatial]. However, none of the above mentioned work considered about
the improvement of the frequency discretization.
[terh] separates particles into several groups. It is assumed that particles
in the same group have the same frequency and the same absorption and
scattering coefficients. However, due to the limitation of computational
resources, the number of groups is usually not enough to resolve all variances
in the frequency distribution. On the other hand, when the opacity is thick,
there are two approximating models, one is the gray radiation diffusion model
where the particles are at the equilibrium Planck distribution in frequency;
the other is the frequency-dependent diffusion limit (FDDL), where particles
with different frequencies diffuse at different speeds. In order to use a
small number of groups to provide a correct solution behavior, different
frequency discretizations of the limit models are proposed in the literature,
for example, the piece-constant approximation [chang2007deterministic,
chang2007incorporation], the Rosseland mean approximation [MWTS,
yee2017stable], the Planck approximation [sunj, sunj18, 2017An]. However, it
is not clear how to construct a group discretization for the FRTE whose
diffusion limits can coincide with the group discretization of the limiting
models.
In this paper, by decomposing the intensity into three parts, we obtain a new
decomposed multi-group method whose asymptotic limit, when the mean free path
goes to zero, can be consistent with the required frequency discretization of
both limiting models. To the best of our knowledge, the decomposed multi-group
method is the first work that gives an AP discretization in the frequency
domain.
Based on the new decomposed multi-group method, a full AP scheme in frequency,
space, and time is proposed. The scheme accuracy is guaranteed in both
optically thin and thick regimes when large spatial and time steps are used.
Moreover, nonlinear iterations are employed to solve a system with only
macroscopic quantities. Then linear transport equations with decoupled
direction and energy groups are solved. Thus the computational cost is similar
to HOLO [park2019multigroup].
The organization of this paper is as follows. Section 2 shows two different
asymptotic limits of the FRTE, one is the gray radiative diffusion limit, and
the other is the FDDL. The classical multi-group method and our decomposed
multi-group method are compared in the frequency-dependent optical thick
regime in Sections 3 and 4. The semi-discretization that is AP in frequency
and time is proposed in section 5, and the fully discretized scheme that is AP
in frequency, time, and space is displayed in section 6. Their AP properties
in both gray radiation and frequency-dependent diffusion limits are shown by
asymptotic analysis. In section 7, some numerical tests are given to
illustrate the stability and accuracy of our proposed scheme. Finally, the
paper is concluded with some discussions in Section 8.
## 2 The FRTE and its limit models
### 2.1 The model and nondimensionalization
The FRTE under consideration is as in [2017An], that writes:
$\left\\{\begin{aligned} &\frac{1}{c}\partial_{t}I+\Omega\cdot\nabla
I=\sigma_{a}\left(B(\nu,T)-I\right)+\sigma_{s}\left(\rho-I\right),\qquad\rho=\frac{1}{4\pi}\int_{4\pi}I\text{d}\Omega:=\left\langle
I\right\rangle,\\\
&C_{v}\partial_{t}T\equiv\partial_{t}U=\int\limits_{4\pi}\int\limits_{0}^{\infty}\sigma_{a}\left(I-B(\nu,T)\right)\text{d}\nu\text{d}\Omega,\
\ \\\ \end{aligned}\right.$ (2.1)
where $x$ is the spatial variable; $\Omega$ is the angular variable;
$\nu\in(0,+\infty)$ is the frequency variable; $I(t,x,\Omega,\nu)$ is the
radiation intensity; $T(t,x)$ is the material temperature;
$\sigma_{a}(x,\nu,T)$ and $\sigma_{s}(x,\nu,T)$ are the absorbing and
scattering coefficient respectively; $c$ and $C_{v}$ represent respectively
the light speed and the specific heat capacity; the Planck function $B(\nu,T)$
is defined by
$B(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/kT}-1},$
with Boltzmann’s constant $k$ and Planck’s constant $h$. The Planck function
satisfies
$\int\limits_{0}^{\infty}4\pi B\
\text{d}\nu=a_{r}cT^{4},\qquad\int\limits_{0}^{\infty}4\pi\frac{\partial
B}{\partial T}\ \text{d}\nu=4a_{r}cT^{3},$
with the radiation constant $a_{r}$. The FRTE system (2.1) is composed of two
equations, the first one describes the time evolution of radiation intensity
and the second one gives the time evolution of background material
temperature. If we integrate the first equation in system (2.1) with respect
to the frequency $\nu$ from $0$ to $+\infty$ and assume that the absorbing and
scattering coefficients are independent of the frequency, the GRTE can be
obtained. GRTE is a good approximation to FRTE when the media is assumed to be
gray, i.e. its properties are independent of the radiation spectrum
[traugott1968grey]. However, in reality, media opacity varies a lot for
different temperatures, densities and particle frequencies. FRTE has to be
considered in some situations [vaytet2011numerical, vaytet2013influence].
We then consider the dimensionless form of (2.1). Similar as in [LREJ, HFJM],
we consider the following nondimensionalization:
$x=\widehat{x}\ell_{\infty},\quad
t=\widehat{t}t_{\infty},\quad\nu=\widehat{\nu}\frac{kT_{\infty}}{h},\quad\sigma_{a}=\widehat{\sigma}_{a}/\ell_{\infty},\quad\sigma_{s}=\widehat{\sigma}_{s}/\ell_{\infty},$
$I=\widehat{I}\frac{a_{r}chT_{\infty}^{3}}{k},\quad
B=\widehat{B}\frac{a_{r}chT_{\infty}^{3}}{k},\quad\rho=\widehat{\rho}\frac{a_{r}chT_{\infty}^{3}}{k},\quad
U=\widehat{U}\frac{\ell_{\infty}^{2}}{t_{\infty}^{2}},$
where variables with a hat denote nondimensional quantity and variables with
$\infty-$subscript are the characteristic value with unit. More precisely,
$\ell_{\infty}$, $t_{\infty}$ and $T_{\infty}$ are respectively the reference
length, time and temperature, and $a_{r}$ is the radiation constant.
Then the full dimensionless FRTE becomes
$\left\\{\begin{aligned}
&\frac{1}{\mathcal{C}}\partial_{\widehat{t}}\widehat{I}+\Omega\cdot\nabla_{\widehat{x}}\widehat{I}=\widehat{\sigma}_{a}\mathscr{L}_{a}\left(\widehat{B}-\widehat{I}\right)+\widehat{\sigma}_{s}\mathscr{L}_{s}\left(\widehat{\rho}-\widehat{I}\right),\\\
&\widehat{C}_{v}\partial_{\widehat{t}}\widehat{T}\equiv\partial_{\widehat{t}}\widehat{U}=\mathcal{C}\mathcal{P}_{0}\int\limits_{4\pi}\int\limits_{0}^{\infty}\widehat{\sigma}_{a}\mathscr{L}_{a}\left(\widehat{I}-\widehat{B}\right)\text{d}\widehat{\nu}\text{d}\Omega,\
\ \\\ \end{aligned}\right.$ (2.2)
where $\mathcal{C}=\frac{ct_{\infty}}{\ell_{\infty}}$,
$\mathcal{P}_{0}=\frac{a_{r}t_{\infty}^{2}T_{\infty}^{4}}{\ell^{2}_{\infty}}$,
$\widehat{C}_{v}$ is a positive constant and
$\widehat{\sigma}_{a}\mathscr{L}_{a}$, $\widehat{\sigma}_{s}\mathscr{L}_{s}$
are respectively $\sigma_{a}$ and $\sigma_{s}$. According to [LMH],
$\mathcal{P}_{0}$ is a nondimensional parameter that measures radiation
effects on the matter, which is always considered as
$\mathcal{P}_{0}=\mathcal{O}(1)$ when a moderate amount of radiation in the
matter is considered. $\mathcal{C}=\mathcal{O}(1/\varepsilon)$ for
nonrelativistic case, which indicates that the speed of light travels much
faster than the material velocity when $\varepsilon\ll 1$. Thus, in this
paper, we fix the values of $\mathcal{P}_{0}$ and $\mathcal{C}$ as
$\mathcal{P}_{0}=1,\qquad\mathcal{C}=1/\varepsilon.$ (2.3)
Depending on the scales of $\mathscr{L}_{a}$ and $\mathscr{L}_{s}$, the
solution of system (2.1) tends to solutions of different limit systems in the
diffusive regime [GPG]. How the different scales of $\mathscr{L}_{a}$ and
$\mathscr{L}_{s}$ lead to different limits will be discussed in the subsequent
part. For simplicity, the hat in (2.2) will be dropped later on.
### 2.2 The diffusion limits of the FRTE
We focus on two different regimes for the FRTE, one is the gray radiation
diffusion regime and the other is the frequency dependent diffusion regime,
whose derivations are similar as in [GPG, LMH]. The gray radiation diffusion
limit is valid when collisions are so frequent that the frequency distribution
depends predominantly on the local temperature. In the FDDL, the determination
of the frequency distribution becomes complicated, it depends on both the
local temperature and the state the radiation field [Holland1969,
manuel2001non]. Both limiting models are widely used in the simulations of
RMHD problems [CHENG2020109724, CD20, jin2022spatial]. For the convenience of
readers, the details of the derivation are in Appendix A.
Using (2.3), the FRTE (2.1) becomes:
$\displaystyle\varepsilon\partial_{t}I+\Omega\cdot\nabla
I=\mathscr{L}_{a}\sigma_{a}\left(B(\nu,T)-I\right)+\mathscr{L}_{s}\sigma_{s}\left(\rho(t,x,\nu)-I\right),$
(2.4a) $\displaystyle\varepsilon
C_{v}\partial_{t}T=\int\limits_{4\pi}\int\limits_{0}^{\infty}\mathscr{L}_{a}\sigma_{a}\left(I-B(\nu,T)\right)\text{d}\nu\text{d}\Omega.$
(2.4b)
* •
The gray radiation diffusion regime. In this regime, the radiation intensity
$I$ adapts to the material temperature. The following three different scalings
yield the gray radiation diffusion equation:
$\displaystyle\mathscr{L}_{a}=1/\varepsilon,\quad\mathscr{L}_{s}=\varepsilon,$
(2.5a) $\displaystyle\mathscr{L}_{a}=1,\quad\mathscr{L}_{s}=1/\varepsilon,$
(2.5b)
$\displaystyle\mathscr{L}_{a}=1/\varepsilon,\quad\mathscr{L}_{s}=1/\varepsilon.$
(2.5c)
When $\varepsilon\to 0$ in (2.4), the material temperature $T$ can be
approximated by the solution of the following gray radiation diffusion
equation
$\partial_{t}(T^{(0)})^{4}+C_{v}\partial_{t}T^{(0)}=\nabla\cdot\left(\frac{D_{d}}{\sigma}\nabla(T^{(0)})^{4}\right)\,,$
(2.6)
with $D_{d}=\frac{1}{3}I_{d}$ (where $I_{d}$ denotes the 3 by 3 identity
matrix) and the mean opacity $\sigma$ being given by
$\frac{1}{\sigma(x,T)}\equiv\frac{\int_{0}^{\infty}\frac{4\pi\mathcal{C}}{\mathscr{L}_{a}\sigma_{a}(x,\nu,T)+\mathscr{L}_{s}\sigma_{s}(x,\nu,T)}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{\int_{0}^{\infty}4\pi\frac{\partial
B^{(0)}(\nu,T)}{\partial
T}\text{d}\nu}=\frac{\int_{0}^{\infty}\frac{4\pi\mathcal{C}}{\mathscr{L}_{a}\sigma_{a}(x,\nu,T)+\mathscr{L}_{s}\sigma_{s}(x,\nu,T)}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{4(T^{(0)})^{3}}.$ (2.7)
The radiation intensity $I$ can then be approximated by
$B(\nu,T^{(0)})-\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B(\nu,T^{(0)})}{\partial T}\Omega\cdot\nabla T^{(0)}.$ (2.8)
* •
The frequency dependent diffusion regime. In this regime, the radiation
intensity $I$ tends to be independent of $\Omega$, but the frequency
distribution can no longer be determined by the material temperature $T$.
Similar as in [GPG], we consider the following scaling
$\mathscr{L}_{a}=\varepsilon,\quad\mathscr{L}_{s}=1/\varepsilon.$
In this regime, the FRTE (2.4) writes:
$\displaystyle\varepsilon\partial_{t}I+\Omega\cdot\nabla
I=\varepsilon\sigma_{a}\left(B(\nu,T)-I\right)+\frac{\sigma_{s}}{\varepsilon}\left(\rho(t,x,\nu)-I\right),$
(2.9a) $\displaystyle
C_{v}\partial_{t}T=\int\limits_{4\pi}\int\limits_{0}^{\infty}\sigma_{a}\left(I-B(\nu,T)\right)\text{d}\nu\text{d}\Omega.$
(2.9b)
When $\varepsilon\to 0$ in (2.9), the solution
$I(t,x,\Omega,\nu)\approx\rho^{(0)}(t,x,\nu)$ where $\rho^{(0)}$ and $T$
satisfies the following frequency dependent diffusion system:
$\left\\{\begin{aligned}
&\partial_{t}\rho^{(0)}-\nabla\cdot\left(\frac{D_{d}}{\sigma_{s}}\nabla\rho^{(0)}\right)=\sigma_{a}(B^{(0)}-\rho^{(0)}),\\\
&C_{v}\partial_{t}T^{(0)}=\int_{0}^{\infty}4\pi\sigma_{a}(\rho^{(0)}-B^{(0)})\text{d}\nu.\
\ \\\ \end{aligned}\right.$ (2.10)
## 3 Extension of the classical multi-group method and the diffusive limits
### 3.1 The classical multi-group method and its extension
We first consider the frequency discretization of the system (2.2). In the
multi-group method, the continuous frequency space $(0,\infty)$ is divided
into $G$ groups, and each frequency interval is denoted by
$(\nu_{g-1/2},\nu_{g+1/2})$ ($g=1,\cdots,G$), with $\nu_{\frac{1}{2}}=0$,
$\nu_{G+\frac{1}{2}}=+\infty$. By integrating the first equation in system
(2.2) over the frequency interval $(\nu_{g-1/2},\nu_{g+1/2})$ yields
$\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{1}{\mathcal{C}}\partial_{t}I+\Omega\cdot\nabla
I\text{d}\nu=\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\mathscr{L}_{a}\sigma_{a}\left(B(\nu,T)-I\right)+\mathscr{L}_{s}\sigma_{s}\left(\rho-I\right)\text{d}\nu.\\\
$ (3.1)
Then the multi-group discretization of FRTE becomes:
$\displaystyle\frac{1}{\mathcal{C}}\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\mathscr{L}_{a}\sigma_{a}\left(B-I\right)-\mathscr{L}_{s}\sigma_{s}\left(\rho-I\right)\text{d}\nu,\quad(g=1,\cdots,G),$
(3.2a) $\displaystyle
C_{v}\partial_{t}T=\mathcal{C}\mathcal{P}_{0}\int\limits_{4\pi}\sum\limits_{g^{\prime}=1}^{G}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\left[\mathscr{L}_{a}\sigma_{a}\left(I-B\right)\right]\text{d}\nu\text{d}\Omega,$
(3.2b)
in which the radiation intensity $I(t,x,\Omega,\nu)$ and Planck function
$B(\nu,T)$ in different groups are given by
$I_{g}=\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}I(t,x,\Omega,\nu)\text{d}\nu,\qquad
B_{g}=\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}B(\nu,T)\text{d}\nu.$
(3.3)
After frequency discretization, only $I_{g}$ in each group and the Planck
function $B$ are known, while the full frequency distribution of $I$ is not
known. Thus the main difficulty is how to approximate the two terms
$\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}\left(B-I\right)\text{d}\nu\quad\mbox{and}\quad\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{s}\left(\rho-I\right)\text{d}\nu.$
(3.4)
In order to approximate the two terms on the right hand side of (3.1) by
$I_{g}$, $B_{g}$ and $T$, some approximations can be found in the literature.
We review some classical multi-group frequency discretization for FRTE in the
subsequent part. Most works in the literature consider only the case when
$\sigma_{s}=0$. In this subsection, we extend similar idea of approximating
the term
$\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\mathscr{L}_{a}\sigma_{a}\left(B(\nu,T)-I\right)$
in [yee2017stable, MWTS] to the case when $\sigma_{s}\neq 0$.
In the classical multi-group frequency discretization in [terh], by defining
$\sigma_{a,g}$ and $\sigma_{s,g}$ as follows
$\sigma_{a,g}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}(B-I)\mathrm{d}\nu}{B_{g}-I_{g}},\quad\sigma_{s,g}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{s}(\rho-I)\mathrm{d}\nu}{\rho_{g}-I_{g}},$
(3.5)
the multi-group frequency discretization for FDDL writes
$\displaystyle\frac{1}{\mathcal{C}}\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\mathscr{L}_{a}\sigma_{a,g}\left(B_{g}-I_{g}\right)+\mathscr{L}_{s}\sigma_{s,g}\left(\rho_{g}-I_{g}\right),\qquad(g=1,\cdots,G),$
(3.6a) $\displaystyle
C_{v}\partial_{t}T=\mathcal{C}\mathcal{P}_{0}\int\limits_{4\pi}\sum\limits_{g^{\prime}=1}^{G}\left[\mathscr{L}_{a}\sigma_{a,g}\left(I_{g}-B_{g}\right)\right]\text{d}\Omega.$
(3.6b)
Here $\sigma_{a,g}$ and $\sigma_{s,g}$ are respectively the mean absorption
and scattering coefficients that have to be approximated by $I_{g}$, $B_{g}$
and $T$. In the diffusive regime, different approximations yield different
coefficients in the limit equations. Two classical approximations are
discussed in the subsequent part.
The most straight forward way is to use piece-wise constant approximation and
let $B(\nu)\approx B_{g}$ and $\rho(\nu)\approx\rho_{g}$, $I(\nu)\approx
I_{g}$ when $\nu\in(\nu_{g-\frac{1}{2}},\nu_{g+\frac{1}{2}})$. Then
$\sigma_{a,g}\approx\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}\mathrm{d}\nu,\qquad\sigma_{s,g}\approx\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{s}\mathrm{d}\nu.$
(3.7)
In [yee2017stable, MWTS] and some related work later on, the authors
considered the Rosseland mean for which $\sigma_{a,g}$ are determined from the
approximation in (2.8) which yields
$B-I\approx\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}(T)\Omega\cdot\nabla T,$
and the Rosseland mean is
$\sigma_{a,g}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{a}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial
T}(T)\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}(T)\mathrm{~{}d}\nu}.$ (3.8)
Extending similar idea to the approximation of $\sigma_{s,g}$, the most
straight forward way is to use the following approximation
$\rho-I\approx-\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\Omega\cdot\nabla\rho.$
If the intensity $I$ has Planck distribution in the frequency domain, i.e.
$\rho\approx I\approx B(\nu,T_{r})$, where the radiation temperature $T_{r}$
is given by
$T_{r}^{4}=\int_{4\pi}\int_{0}^{+\infty}I\mathrm{d}\Omega\mathrm{d}\nu=4\pi\int_{0}^{\infty}\rho\mathrm{d}\nu=4\pi\sum\limits_{g^{\prime}=1}^{G}\rho_{g},$
(3.9)
the scattering coefficient $\sigma_{s,g}$ can then be approximated by
$\displaystyle\sigma_{s,g}$
$\displaystyle=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{s}(\rho-I)\mathrm{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}(\rho-I)\mathrm{~{}d}\nu}\approx\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{s}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\Omega\cdot\nabla\rho\mathrm{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\Omega\cdot\nabla\rho\mathrm{~{}d}\nu}$
(3.10)
$\displaystyle=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{s}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}(T_{r})\Omega\cdot\nabla
T_{r}\mathrm{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}(T_{r})\Omega\cdot\nabla
T_{r}\mathrm{~{}d}\nu}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{s}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial
T}(T_{r})\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{a}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}(T_{r})\mathrm{~{}d}\nu}.$
It is easy to see that the Rosseland mean is an approximation to the average
of $\sigma_{a}$, $\sigma_{s}$ inside the interval
$(\nu_{g+\frac{1}{2}},\nu_{g+\frac{1}{2}})$ as well.
### 3.2 The diffusion limit of the classical multi-group method
#### 3.2.1 The gray radiation diffusion regime
When $\mathscr{L}_{a}=1/\varepsilon$, $\mathscr{L}_{s}=1/\varepsilon$, the
multi-group frequency discretization in (3.6) becomes
$\displaystyle\varepsilon\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\frac{\sigma_{a,g}}{\varepsilon}\left(B_{g}-I_{g}\right)+\frac{\sigma_{s,g}}{\varepsilon}\left(\rho_{g}-I_{g}\right),\qquad(g=1,\cdots,G),$
(3.11a) $\displaystyle\varepsilon
C_{v}\partial_{t}T=4\pi\sum\limits_{g^{\prime}=1}^{G}\left[\frac{\sigma_{a,g^{\prime}}}{\varepsilon}\left(\rho_{g^{\prime}}-B_{g^{\prime}}\right)\right].$
(3.11b)
We derive its gray radiation diffusion limit equation in the subsequent part.
By using Chapman-Enskog expansion in (3.11a), one has
$\displaystyle I_{g}$
$\displaystyle=\frac{\sigma_{a,g}}{\sigma_{a,g}+\sigma_{s,g}}B_{g}+\frac{\sigma_{s,g}}{\sigma_{a,g}+\sigma_{s,g}}\rho_{g}-\frac{\varepsilon}{\sigma_{a,g}+\sigma_{s,g}}\left[\varepsilon\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}\right]$ (3.12)
$\displaystyle=\frac{\sigma_{a,g}}{\sigma_{a,g}+\sigma_{s,g}}B_{g}+\frac{\sigma_{s,g}}{\sigma_{a,g}+\sigma_{s,g}}\rho_{g}-\frac{\varepsilon}{\sigma_{a,g}+\sigma_{s,g}}\Omega\cdot\nabla\left(\frac{\sigma_{a,g}}{\sigma_{a,g}+\sigma_{s,g}}B_{g}+\frac{\sigma_{s,g}}{\sigma_{a,g}+\sigma_{s,g}}\rho_{g}\right)+\mathcal{O}\left(\varepsilon^{2}\right).$
Taking the integral with respect to $\Omega$ of equation (3.11a), we can
obtain
$\rho_{g}=B_{g}+\mathcal{O}(\varepsilon)\,,$
which implies that $\rho_{g}\approx B_{g}$. Taking the integral with respect
to $\Omega$ of the equation (3.11a), adding up all groups and (3.11b), one can
get
$4\pi\partial_{t}\left(\sum_{g=1}^{G}\rho_{g}\right)+\frac{4\pi}{\varepsilon}\sum_{g=1}^{G}\nabla\cdot\left\langle\Omega
I_{g}\right\rangle+C_{v}\partial_{t}T=0\,.$ (3.13)
Plugging (3.12) into the equation (3.13), using the condition
$\rho_{g}=B_{g}+\mathcal{O}(\varepsilon)$, letting $\varepsilon\to 0$, then
(3.13) reduces to
$\partial_{t}\left(\sum_{g=1}^{G}4\pi
B_{g}\right)+C_{v}\partial_{t}T=\sum_{g=1}^{G}\nabla\cdot\left(\frac{4\pi}{3(\sigma_{a,g}+\sigma_{s,g})}\nabla
B_{g}\right)\,,$ (3.14)
which implies that
$\partial_{t}\left(\sum_{g=1}^{G}4\pi
B_{g}\right)+C_{v}\partial_{t}T=\sum_{g=1}^{G}\nabla\cdot\left(\frac{4\pi}{3(\sigma_{a,g}+\sigma_{s,g})}\frac{\partial
B_{g}}{\partial T}\frac{1}{4T^{3}}\nabla T^{4}\right)\,,$ (3.15)
When the group discretization of the Planck equilibrium $B$ satisfies the
following relations
$4\pi\sum_{g=1}^{G}B_{g}=\int\limits_{0}^{\infty}4\pi B\ \text{d}\nu=T^{4},$
(3.16)
(3.15) then reduces to
$\partial_{t}T^{4}+C_{v}\partial_{t}T=\nabla\cdot\left(\frac{1}{3\widehat{\sigma}_{t}}\nabla
T^{4}\right)\,,$ (3.17)
When one approximates $\sigma_{a,g}$ and $\sigma_{s,g}$ as in (3.7), the
diffusion coefficient of the above equation is determined by
$\frac{1}{\widehat{\sigma}^{c}_{t}}=\frac{1}{4T^{3}}\left(\sum\limits_{g=1}^{G}\frac{4\pi}{\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int_{\nu_{g-1/2}}^{\nu_{g+1/2}}{\sigma_{a}+\sigma_{s}}\text{d}\nu}\frac{\partial
B_{g}}{\partial T}\right).$ (3.18)
On the other hand, when one uses the Rosseland mean as in (3.8) and (3.10),
one has
$\frac{1}{\sigma_{a,g}+\sigma_{s,g}}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial
T}(T)\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial T}(T)\mathrm{~{}d}\nu},$
which implies
$\frac{1}{\widehat{\sigma}^{r}_{t}}=\frac{1}{4T^{3}}\left(\sum\limits_{g=1}^{G}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{4\pi}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu\right).$ (3.19)
Both $\widehat{\sigma}_{t}^{c}$ and $\widehat{\sigma}_{t}^{r}$ provide
reasonable approximations to $\sigma$ in (2.7) when the $\sigma_{a}$,
$\sigma_{s}$ are at $O(1)$ and when there are enough groups. However, we will
see in the next subsection that $\widehat{\sigma}^{c}_{t}$ determined by
piece-wise constant approximation as in (3.18) can not provide a good
approximation to the the diffusion coefficient as in (2.7), while the
Rosseland mean $\widehat{\sigma}^{r}_{t}$ can. Moreover, When $\sigma_{s}=0$,
$\frac{1}{\widehat{\sigma}^{r}_{t}}$ becomes
$\frac{1}{\widehat{\sigma}^{r}_{a}}=\frac{1}{4T^{3}}\left(\sum\limits_{g=1}^{G}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{4\pi}{\sigma_{a}}\frac{\partial
B}{\partial T}\text{d}\nu\right).$
which is the same as in [MWTS].
#### 3.2.2 The frequency dependent diffusion regime
When $\mathscr{L}_{a}=\varepsilon$, $\mathscr{L}_{s}=1/\varepsilon$, the
multi-group frequency discretization in (3.6) becomes
$\displaystyle\varepsilon\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\varepsilon\sigma_{a,g}\left(B_{g}-I_{g}\right)+\frac{\sigma_{s,g}}{\varepsilon}\left(\rho_{g}-I_{g}\right),\qquad(g=1,\cdots,G),$
(3.20a) $\displaystyle\varepsilon
C_{v}\partial_{t}T=4\pi\sum\limits_{g^{\prime}=1}^{G}\left[\varepsilon\sigma_{a,g^{\prime}}\left(\rho_{g^{\prime}}-B_{g^{\prime}}\right)\right].$
(3.20b)
We derive its radiation diffusion limit equation in the subsequent part. By
using Chapman-Enskog expansion in (3.20a), one has
$\displaystyle I_{g}$
$\displaystyle=\frac{\varepsilon^{2}\sigma_{a,g}}{\varepsilon^{2}\sigma_{a,g}+\sigma_{s,g}}B_{g}+\frac{\sigma_{s,g}}{\varepsilon^{2}\sigma_{a,g}+\sigma_{s,g}}\rho_{g}-\frac{\varepsilon}{\varepsilon^{2}\sigma_{a,g}+\sigma_{s,g}}\left[\varepsilon\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}\right]$ (3.21)
$\displaystyle=\frac{\sigma_{s,g}}{\varepsilon^{2}\sigma_{a,g}+\sigma_{s,g}}\rho_{g}-\frac{\varepsilon}{\varepsilon^{2}\sigma_{a,g}+\sigma_{s,g}}\Omega\cdot\nabla\left(\frac{\sigma_{s,g}}{\varepsilon^{2}\sigma_{a,g}+\sigma_{s,g}}\rho_{g}\right)+\mathcal{O}\left(\varepsilon^{2}\right)$
$\displaystyle=\rho_{g}-\frac{\varepsilon}{\sigma_{s,g}}\Omega\cdot\nabla\rho_{g}+\mathcal{O}\left(\varepsilon^{2}\right).$
Taking the integral with respect to $\Omega$ of the equation (3.20a), one can
get
$\partial_{t}\rho_{g}+\frac{1}{\varepsilon}\nabla\cdot\left\langle\Omega
I_{g}\right\rangle=\sigma_{a,g}(B_{g}-\rho_{g})\,.$ (3.22)
Plugging (3.21) into the equation (3.22), letting $\varepsilon\to 0$, then
(3.22) reduces to
$\partial_{t}\rho_{g}-\nabla\cdot\left(\frac{1}{3\sigma_{s,g}}\nabla\rho_{g}\right)=\sigma_{a,g}(B_{g}-\rho_{g})\,.$
(3.23)
Moreover, by letting $\varepsilon\to 0$ in (3.20b), one has
$C_{v}\partial_{t}T=4\pi\sum\limits_{g^{\prime}=1}^{G}\left[\sigma_{a,g^{\prime}}\left(\rho_{g^{\prime}}-B_{g^{\prime}}\right)\right],$
and the above two equations give a group discretization of the frequency
dependent diffusion limit (2.10). When one approximates $\sigma_{a,g}$ and
$\sigma_{s,g}$ as in (3.7), the coefficients in the frequency dependent
diffusion limit are determined by
$\sigma^{c}_{a,g}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}\mathrm{d}\nu,\quad\frac{1}{\sigma_{s,g}^{c}}=\frac{1}{\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{s}\mathrm{d}\nu}$
When (3.8) and (3.10) are used, $\sigma^{r}_{a,g}$ and $1/\sigma_{s,g}^{r}$
become
$\sigma^{r}_{a,g}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{a}}{\sigma_{s}}\frac{\partial
B}{\partial
T}(T)\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s}}\frac{\partial
B}{\partial
T}(T)\mathrm{~{}d}\nu},\quad\frac{1}{\sigma^{r}_{s,g}}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s}}\frac{\partial
B}{\partial
T}(T)\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial T}(T)\mathrm{~{}d}\nu}.$
We emphasize that, different approximations provide different coefficient
values for different groups in the limiting models, which is crucial to get
the correct solution behavior.
### 3.3 Frequency discretization of the FDDL
From the above discussion, we can see that different group discretizations for
FRTE yield different approximations of the group diffusion and absorption
coefficients in the FDDL. In the frequency dependent diffusion regime,
different group diffusion and absorption coefficients can affect the
propagation speed of heat waves under different physical conditions. Moreover,
which approximation is better depends on different physical conditions
[yee2017stable, park2019multigroup].
In this subsection, we would like to compare several different approximations
of the coefficients $\sigma$ in (2.6) and $\sigma_{s}$, $\sigma_{a}$ in
(2.10). We do not discuss about which coefficient approximation is better in
the limiting model, but to show that they can be quite different from each
other.
Three different approximations of $\sigma_{a,g}$ are considered: the piece-
wise constant approximation $\sigma^{c}_{a,g}$ in (3.7), the Rosseland mean
approximation $\sigma_{a,g}^{r}$ in (3.8) and the Planck approximation. In the
Planck approximation as in [sunj], the group absorptions to discretize (2.10)
are defined as follows
$\sigma^{e}_{a,g}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}B(\nu,T)\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}B(\nu,T)\mathrm{~{}d}\nu},\quad\sigma^{a}_{a,g}=\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}\rho\mathrm{~{}d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\rho\mathrm{~{}d}\nu}.$
(3.24)
It should be pointed out that $\sigma^{a}_{a,g}$ is a weighted integration
with the unknown function $\rho$. Usually, the unknown function $\rho$ is
replaced by the Planck function with the radiation temperature $T_{r}$, and
the multi-group FDDL writes:
$\displaystyle\partial_{t}\rho_{g}+\nabla\cdot\left(\frac{1}{3\sigma_{s,g}}\nabla\rho_{g}\right)=\sigma^{e}_{a,g}B_{g}-\sigma^{a}_{a,g}\rho_{g},\qquad(g=1,\cdots,G),$
(3.25a) $\displaystyle
C_{v}\partial_{t}T=4\pi\sum\limits_{g^{\prime}=1}^{G}\left(\sigma^{a}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{e}_{a,g^{\prime}}B_{g^{\prime}}\right).$
(3.25b)
Similar as in [densmore2012hybrid, sunj], we consider a simple dependence of
the frequency such that $\sigma_{a}=\nu^{-3}$ and $\sigma_{s}=\nu^{-3}$. Here
we ignore the scale parameters $\mathscr{L}_{a}$ and $\mathscr{L}_{s}$ since
they not affect the results of the comparison. The frequency domain is from
$0.1\ eV$ to $100\ keV$ and divided into 30 frequency groups, and the groups
are logarithmically spaced.
The following quantities are compared:
* •
The two different approximations of $1/\sigma_{t}$ in (3.18) and (3.19) at
different temperatures $T$. The results are displayed in Table 1 and Figure.
1, and the reference solution is calculated by dividing the frequency domain
into 600 groups.
* •
The diffusion coefficients $1/\sigma_{s}$ in the FDDL (2.10) are approximated
by the $1/\sigma^{c}_{s,g}$ and two different $1/\sigma^{r}_{s,g}$ at $T=1\
keV$ and $T=16\ keV$. In Figure. 1, From Figure. 1, the Rosseland mean
approximation is closer to the reference.
* •
For the absorbing coefficient $\sigma_{a}$, we compare the piece-wise constant
approximation $\sigma^{c}_{a,g}$, the Rosseland mean approximation
$\sigma_{a,g}^{r}$ and the Plank approximation $\sigma^{e}_{a,g}$ in Figure.
2. We observe that they can be very different.
Table 1: The different approximations to $1/\sigma$ in the gray radiation diffusion limit (2.7). The reference solution is calculated by dividing the frequency domain into 600 groups. T | Reference | $1/\sigma_{t}^{r}$ | relative error | $1/\sigma_{t}^{c}$ | relative error
---|---|---|---|---|---
$1\ keV$ | 25.3901 | 25.3920 | $7.48\times 10^{-5}$ | 61.7458 | 1.43
$2\ keV$ | 203.1204 | 203.1042 | $7.97\times 10^{-5}$ | 493.8879 | 1.43
$4\ keV$ | $1.6250\times 10^{3}$ | $1.6251\times 10^{3}$ | $6.15\times 10^{-5}$ | $3.9516\times 10^{3}$ | 1.43
$8\ keV$ | $1.2574\times 10^{4}$ | $1.2284\times 10^{4}$ | $2.31\times 10^{-2}$ | $2.8811\times 10^{4}$ | 1.29
$16\ keV$ | $5.1674\times 10^{4}$ | $5.4364\times 10^{4}$ | $5.21\times 10^{-2}$ | $9.2914\times 10^{4}$ | $7.98\times 10^{-1}$
Figure 1: Left: comparison of the different approximations of the mean free
path $1/\sigma$ in the gray radiation diffusion limit (2.7) at the different
temperatures; Right: comparison of the different approximations of the
diffusion coefficient $1/\sigma_{s}$ in the FDDL (2.10) at $T=1\ keV$ and
$T=16\ keV$.
Figure 2: (a)comparison of the different approximations of $\sigma_{a}$ at
$T=1\ keV$ and $T=16\ keV$; (b)zoom in part of figure (a), here the some red
dots and green dots overlap; (c)zoom in, here the blue dots and green dots
overlap.
## 4 The decomposed multi-group method and its asymptotic limits
### 4.1 The decomposed multi-group method
A new decomposed multi-group method is proposed in this section. We decompose
the radiation intensity $I(t,x,\Omega,\nu)$ into three parts:
$I(t,x,\Omega,\nu)=\left\langle I\right\rangle+3\Omega\cdot\left\langle\Omega
I\right\rangle+Q(t,x,\Omega,\nu),$ (4.1a) $\hskip
91.04872pt:=\rho(t,x,\nu)+\Omega\cdot R(t,x,\nu)+Q(t,x,\Omega,\nu),$ (4.1b)
with $\left\langle\cdot\right\rangle=\frac{1}{4\pi}\int_{4\pi}\cdot\
\text{d}\Omega$ and 3$\left\langle\Omega I\right\rangle=R$. Then taking
$\left\langle\Omega\cdot\right\rangle$ on both sides of (4.1a) and using the
condition $\left\langle\Omega\Omega\right\rangle:=D_{d}=\frac{1}{3}I_{d}$
(where $I_{d}$ denotes the 3 by 3 identity matrix), one gets
$\left\langle\Omega I\right\rangle=\left\langle\Omega
I\right\rangle+\left\langle\Omega Q\right\rangle,$
which indicates that
$\left\langle\Omega Q\right\rangle=0.$ (4.2)
Moreover, taking $\left\langle\cdot\right\rangle$ on both sides of (4.1), one
gets
$\left\langle I\right\rangle=\left\langle\rho+\Omega\cdot
R+Q\right\rangle=\left\langle I\right\rangle+\left\langle Q\right\rangle,$
which implies
$\left\langle Q\right\rangle=0.$ (4.3)
(4.2) and (4.3) are two properties that $Q$ has to satisfy.
We reconstruct the intensity $I$ in each interval $(\nu_{g-1/2},\nu_{g+1/2})$
in the frequency domain as follows:
$I(t,x,\Omega,\nu)=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}{\color[rgb]{0,0,0}\Big{(}\omega_{1}\rho_{g}+\Omega\cdot
R_{g}+Q_{g}+s_{\rho_{g}}+s_{\Omega\cdot R_{g}}\Big{)}}$ (4.4)
where
$\rho_{g}=\left\langle I_{g}\right\rangle,\qquad R_{g}=3\left\langle\Omega
I_{g}\right\rangle,\qquad Q_{g}=I_{g}-\rho_{g}-\Omega\cdot R_{g},$
and
$s_{\rho_{g}}={\color[rgb]{0,0,0}(\nu_{g+1/2}-\nu_{g-1/2})B(\nu,T)-\omega_{2}B_{g}},\qquad
s_{\Omega\cdot
R_{g}}=\frac{\frac{\nu_{g+1/2}-\nu_{g-1/2}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial
T}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu}\Omega\cdot R_{g}-\Omega\cdot R_{g}.$ (4.5)
Here $\omega_{1}$ and $\omega_{2}$ are two given weight functions that depend
on $\nu$. We will see later on that by tuning $\omega_{1}$, $\omega_{2}$, one
can get the required approximations of the group absorption coefficients in
FDDL. The first three terms on the right hand side of (4.4) $\rho_{g}$,
$\Omega\cdot R_{g}$, $Q_{g}$ depend only on $g$ not in $\nu$. Moreover, the
zeroth and first moments of $Q_{g}$ equal to 0, which is the same as $Q$. The
two additional terms $s_{\rho_{g}}$, $s_{\Omega\cdot R_{g}}$ depend on $\nu$
but satisfy
$\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}s_{\rho_{g}}\text{d}\nu=\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}s_{\Omega\cdot
R_{g}}\text{d}\nu=0$. Thus the reconstructed $I$ in (4.4) satisfies
$\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}I\text{d}\nu=I_{g}$. The two
additional terms $s_{\rho_{g}}$ and $s_{\Omega\cdot R_{g}}$ are important to
get the correct limits in the gray radiation and frequency dependent diffusion
regime. The detailed derivations will be displayed in section 3.3.
It is important to note that, as far as $I_{g},T$ are given, $\rho_{g}$,
$R_{g}$, $Q_{g}$ and $s_{\rho_{g}}$, $s_{\Omega\cdot R_{g}}$ are known. Then
$I(t,x,\Omega,\nu)$ in (4.4) can be determined by $I_{g}$ and the two terms in
(3.4) are approximated by
$\displaystyle\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}\left(B-I\right)\text{d}\nu=\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}\Big{[}B-\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}{\color[rgb]{0,0,0}\big{(}\omega_{1}\rho_{g}+\Omega\cdot
R_{g}+Q_{g}+s_{\rho_{g}}+s_{\Omega\cdot R_{g}}\big{)}}\Big{]}\text{d}\nu$
$\displaystyle=\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}\Big{[}B-\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}{\color[rgb]{0,0,0}\Big{(}\omega_{1}\rho_{g}+(\nu_{g+1/2}-\nu_{g-1/2})B-\omega_{2}B_{g}+\frac{\frac{\nu_{g+1/2}-\nu_{g-1/2}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial
T}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu}\Omega\cdot R_{g}+Q_{g}\Big{)}}\Big{]}\text{d}\nu$
$\displaystyle{\color[rgb]{0,0,0}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}(\omega_{2}B_{g}-\omega_{1}\rho_{g})\text{d}\nu-\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{a}\text{d}\nu
Q_{g}}$ $\displaystyle\hskip
199.16928pt-\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{a}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu\frac{\Omega\cdot
R_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu},$
and
$\displaystyle\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{s}\left(\rho-I\right)\text{d}\nu$
$\displaystyle=\frac{-1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{s}\big{(}\Omega\cdot
R_{g}+Q_{g}+s_{\Omega\cdot R_{g}}\big{)}\text{d}\nu$
$\displaystyle=\frac{-1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\sigma_{s}\text{d}\nu
Q_{g}-\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\sigma_{s}}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu\frac{\Omega\cdot
R_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu}.$
Then we can get the decomposed multi-group radiative transfer equation:
$\displaystyle\frac{1}{\mathcal{C}}\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\mathscr{L}_{a}\sigma^{2}_{a,g}B_{g}-\mathscr{L}_{a}\sigma^{1}_{a,g}\rho_{g}-\big{(}\mathscr{L}_{a}+\mathscr{L}_{s}\big{)}\sigma_{t,g}\Omega\cdot
R_{g}-(\mathscr{L}_{a}\sigma_{a,g}+\mathscr{L}_{s}\sigma_{s,g})Q_{g},$
$\displaystyle\hskip 341.43306pt(g=1,\cdots,G),$ (4.6a) $\displaystyle
C_{v}\partial_{t}T=4\pi\mathcal{C}\mathcal{P}_{0}\sum\limits_{g^{\prime}=1}^{G}\left[\mathscr{L}_{a}\left(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}}\right)\right],$
(4.6b)
where
$\displaystyle\sigma^{1}_{a,g}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\omega_{1}\sigma_{a}\text{d}\nu,\quad\sigma^{2}_{a,g}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\omega_{2}\sigma_{a}\text{d}\nu,$
(4.7)
$\displaystyle\sigma_{a,g}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}\text{d}\nu,\quad\sigma_{s,g}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{s}\text{d}\nu,$
$\displaystyle\hskip
85.35826pt\sigma_{t,g}=\frac{1/(\mathscr{L}_{a}+\mathscr{L}_{s})}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\mathscr{L}_{a}\sigma_{a}+\mathscr{L}_{s}\sigma_{s}}\frac{\partial
B}{\partial
T}\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial T}\text{d}\nu}.$
It is important to use the reconstruction as in (4.4) to get the required
absorption coefficients for group discretizations of the FDDL. One can
consider the following possible $\omega_{1}$ and $\omega_{2}$:
* •
When $\omega_{1}=\omega_{2}=1$, then the absorbing coefficients are
$\sigma^{1}_{a,g}=\sigma^{2}_{a,g}=\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}\text{d}\nu,$
which will yield piece-constant approximations of the group absorption
coefficients in FDDL.
* •
When $\omega_{1}=\frac{\frac{\partial B}{\partial
T}(T_{r})}{\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\frac{\partial B_{g}}{\partial
T}(T_{r})},\quad\omega_{2}=\frac{\frac{\partial B}{\partial
T}(T)}{\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}\frac{\partial B_{g}}{\partial
T}(T)}$, then the absorbing coefficients are
$\sigma^{1}_{a,g}=\frac{\int_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}\frac{\partial
B}{\partial T}(T_{r})\text{d}\nu}{\frac{\partial B_{g}}{\partial
T}(T_{r})},\quad\sigma^{2}_{a,g}=\frac{\int_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}\frac{\partial
B}{\partial T}(T)\text{d}\nu}{\frac{\partial B_{g}}{\partial T}(T)},$
which gives the Rosseland mean approximations of the group absorption
coefficients in FDDL.
* •
When
$\omega_{1}=\frac{B(T_{r})}{\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}B_{g}(T_{r})},\quad\omega_{2}=\frac{B(T)}{\frac{1}{\nu_{g+1/2}-\nu_{g-1/2}}B_{g}(T)}$,
then the absorbing coefficients are
$\sigma^{1}_{a,g}=\frac{\int_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}B(T_{r})\text{d}\nu}{B_{g}(T_{r})},\quad\sigma^{2}_{a,g}=\frac{\int_{\nu_{g-1/2}}^{\nu_{g+1/2}}\sigma_{a}B(T)\text{d}\nu}{B_{g}(T)},$
which corresponds to the Planck approximation of the absorption coefficient in
FDDL.
When the different $\omega_{1}$ and $\omega_{2}$ are chosen, the different
approximations of the absorbing coefficients in FDDL are determined. Comparing
(4.6) with (3.6), we can see that the scattering and absorbing coefficients
for $\rho_{g}$, $R_{g}$ and $Q_{g}$ are different in the decomposed multi-
group method, while $\rho_{g}$, $R_{g}$ and $Q_{g}$ share the same values of
scattering and absorbing coefficients in the classical multi-group method.
### 4.2 The decomposed multi-group method in the diffusion regime
#### 4.2.1 The gray radiation diffusion regime
For the scaling $\mathscr{L}_{a}=1/\varepsilon,\mathscr{L}_{s}=1/\varepsilon$,
the decomposed multi-group radiative transfer system (4.6) can be rewritten
into
$\displaystyle\varepsilon\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\frac{1}{\varepsilon}\left(\sigma_{a,g}^{2}B_{g}-\sigma_{a,g}^{1}\rho_{g}\right)-\frac{\Omega\cdot
R_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\varepsilon}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial
T}\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}\text{d}\nu}-\frac{\sigma_{a,g}+\sigma_{s,g}}{\varepsilon}Q_{g},$
$\displaystyle\hskip 298.75394pt(g=1,\cdots,G),$ (4.8a)
$\displaystyle\varepsilon
C_{v}\partial_{t}T=4\pi\sum\limits_{g^{\prime}=1}^{G}\left[\frac{1}{\varepsilon}\left(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}}\right)\right].$
(4.8b)
At the leading order of (4.8a), one has
$\displaystyle\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}-\frac{\Omega\cdot
R_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial
T}\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}\text{d}\nu}-(\sigma_{a,g}+\sigma_{s,g})Q_{g}=\mathcal{O}(\varepsilon),$
(4.9)
Integrating with respect to $\Omega$ in equation (4.9) and using $\left\langle
Q_{g}\right\rangle=0$, we can obtain
$\sigma^{1}_{a,g}\rho_{g}=\sigma^{2}_{a,g}B_{g}+\mathcal{O}(\varepsilon)\,.$
(4.10)
Summing the above equation over the group index $g$, and by the definitions
(3.5), one can get
$\int\limits_{0}^{\infty}\sigma_{a}\rho\text{d}\nu=\int\limits_{0}^{\infty}\sigma_{a}B\text{d}\nu+\mathcal{O}(\varepsilon),$
(4.11)
which implies that
$\rho_{g}=B_{g}+\mathcal{O}(\varepsilon)\,.$ (4.12)
Integrating (4.8a) with respect to $\Omega$ and adding up all groups and
equation (4.8b), one gets
$4\pi\partial_{t}\left(\sum_{g=1}^{G}\rho_{g}\right)+\frac{4\pi}{3\varepsilon}\sum_{g=1}^{G}\nabla\cdot
R_{g}+C_{v}\partial_{t}T=0\,.$ (4.13)
From (4.12) and the conditions that $B_{g}$ satisfies in (3.16),
$4\pi\sum_{g=1}^{G}\rho_{g}=T^{4}+\mathcal{O}(\varepsilon)$. In order to get
the diffusion limit equation, one has to determine the relationship between
$\rho_{g}$ and $R_{g}$. Taking the first moment of (4.8a), and using
$I_{g}=\rho_{g}+\Omega\cdot R_{g}+Q_{g}$, (4.12) and $\left\langle\Omega
Q_{g}\right\rangle$=0, one has
$-\frac{1}{\varepsilon}R_{g}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}\text{d}\nu}\left[\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\nabla\rho_{g}}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial
T}\text{d}\nu+\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\nabla\left\langle\Omega\Omega
Q_{g}\right\rangle}{\sigma_{a}+\sigma_{s}}\frac{\partial B}{\partial
T}\text{d}\nu\right]+\mathcal{O}(\varepsilon).$ (4.14)
Taking the second moment of $\Omega$ in equation (4.9) and noting (4.12)
yields
$\left\langle\Omega\Omega
Q_{g}\right\rangle:=K_{Q_{g}}=\mathcal{O}(\varepsilon),$
which implies
$-\frac{1}{\varepsilon}R_{g}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}\text{d}\nu}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\nabla\rho_{g}}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu+\mathcal{O}(\varepsilon).$ (4.15)
Plugging (4.12), (4.15) into the equation (4.13), when $\varepsilon\to 0$,
(4.13) reduces to
$\partial_{t}\left(\sum_{g=1}^{G}4\pi
B_{g}\right)+C_{v}\partial_{t}T=\sum_{g=1}^{G}\nabla\cdot\left[\left(\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{4\pi}{3(\sigma_{a}+\sigma_{s})}\frac{\partial
B}{\partial T}\text{d}\nu\right)\nabla T\right]\,.$ (4.16)
Due to (3.16), (4.16) becomes
$\partial_{t}T^{4}+C_{v}\partial_{t}T=\nabla\cdot\left(\frac{1}{3\widehat{\sigma}_{R}}\nabla
T^{4}\right)\,,$ (4.17)
where
$\frac{1}{\widehat{\sigma}_{R}}=\frac{1}{4T^{3}}\left(\sum\limits_{g=1}^{G}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{4\pi}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu\right),$
which is consistent with equation (2.6).
#### 4.2.2 The frequency dependent diffusion regime
In the frequency dependent diffusion regime, the scaling is
$\mathscr{L}_{a}=\varepsilon$ and $\mathscr{L}_{s}=1/\varepsilon$, then the
multi-group radiative transfer system (4.6) can be rewritten to
$\displaystyle\varepsilon\partial_{t}I_{g}+\Omega\cdot\nabla
I_{g}=\varepsilon\left(\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}\right)-\frac{\Omega\cdot
R_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\varepsilon\sigma_{a}+\frac{\sigma_{s}}{\varepsilon}}\frac{\partial
B}{\partial
T}\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}\text{d}\nu}-\left(\varepsilon\sigma_{a,g}+\frac{\sigma_{s,g}}{\varepsilon}\right)Q_{g},$
$\displaystyle\hskip 341.43306pt(g=1,\cdots,G),$ (4.18a)
$\displaystyle\varepsilon
C_{v}\partial_{t}T=4\pi\sum\limits_{g^{\prime}=1}^{G}\left[\varepsilon\left(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}}\right)\right].$
(4.18b)
The asymptotic analysis for the FDDL begins to find the relation between
$\rho_{g}$ and $R_{g}$. From (4.18a), one has
$\displaystyle-\frac{\Omega\cdot
R_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s}}\frac{\partial
B}{\partial
T}\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial T}\text{d}\nu}-\sigma_{s,g}Q_{g}=\mathcal{O}(\varepsilon).$ (4.19)
Taking the second moment in $\Omega$ of equation (4.19) yields
$K_{Q_{g}}=\mathcal{O}(\varepsilon).$ (4.20)
Moreover, using the relation (4.20), the leading order of the first moment of
equation (4.18a) is
$-\frac{1}{\varepsilon}R_{g}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}\text{d}\nu}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\nabla\rho_{g}}{\sigma_{s}}\frac{\partial
B}{\partial T}\text{d}\nu+\mathcal{O}(\varepsilon),$ (4.21)
which gives the relation between $R_{g}$ and $\rho_{g}$. Integrating (4.18a)
with respect to $\Omega$, one can get
$\partial_{t}\rho_{g}+\frac{1}{3\varepsilon}\nabla\cdot
R_{g}=\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}\,.$ (4.22)
Plugging (4.21) into (4.22), and letting $\varepsilon\to 0$, (4.22) gives
$\partial_{t}\rho_{g}-\nabla\cdot\left(\frac{1}{3\sigma^{\prime}_{s,g}}\nabla\rho_{g}\right)=\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}\,,$
(4.23)
where
$\frac{1}{\sigma^{\prime}_{s,g}}=\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s}}\frac{\partial
B}{\partial
T}\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial T}\text{d}\nu.$ From equation (4.18b), one has
$C_{v}\partial_{t}T=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}})\right].$
In summary, the diffusion limit of the decomposed multi-group radiative
transfer system (4.18) is
$\displaystyle\partial_{t}\rho_{g}-\nabla\cdot\left(\frac{1}{3\sigma^{\prime}_{s,g}}\nabla\rho_{g}\right)=\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}\,,$
(4.24a) $\displaystyle
C_{v}\partial_{t}T=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}})\right].$
(4.24b)
The decomposed multi-group radiative transfer equations (3.2) is AP in both
gray radiation diffusion limit and the FDDL. Moreover, one can choose
different approximations of $\sigma_{a,g}^{1}$, $\sigma_{a,g}^{2}$ in the
limiting frequency discretization.
## 5 AP time discretization of the decomposed multi-group method
### 5.1 The time discretization
Based on the decomposition in (4.1) for radiation intensity $I$ and the
decomposed multi-group method in (4.6), we will present a time discretization
for the FRTE and show its AP property.
Let
$t^{n}=n\Delta t,\qquad n=0,1,2,\cdots,$
and
$I^{n}\approx
I(t^{n},x,\Omega,\nu),\qquad\rho^{n}\approx\rho(t^{n},x,\nu),\qquad
T^{n}\approx T(t^{n},x),\qquad B^{n}\approx B(\nu,T(t^{n},x)),$
with
$I_{g}^{n}\approx\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}I(t^{n},x,\Omega,\nu)\text{d}\nu,\qquad\rho_{g}^{n}\approx\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\rho(t^{n},x,\nu)\text{d}\nu,\qquad
B_{g}^{n}\approx\int\limits_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}B(\nu,T(t^{n},x))\text{d}\nu.$
We use the following semi-discrete scheme in time for (3.2) in each group:
$\displaystyle\frac{1}{\mathcal{C}}\frac{I_{g}^{n+1}-I_{g}^{n}}{\Delta
t}+\Omega\cdot\nabla(\rho^{n+1}_{g}+\Omega\cdot R^{n+1}_{g}+Q^{n}_{g})=$
$\displaystyle\hskip
14.22636pt\mathscr{L}_{a}\left(\sigma^{2}_{a,g}B^{n+1}_{g}-\sigma^{1}_{a,g}\rho^{n+1}_{g}\right)-(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g}\Omega\cdot
R^{n+1}_{g}-(\mathscr{L}_{a}\sigma_{a,g}+\mathscr{L}_{s}\sigma_{s,g})Q^{n}_{g},$
(5.1a)
$\displaystyle\frac{C_{v}}{\mathcal{C}\mathcal{P}_{0}}\frac{T^{n+1}-T^{n}}{\Delta
t}=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\mathscr{L}_{a}\left(\sigma^{1}_{a,g^{\prime}}\rho^{n+1}_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B^{n+1}_{g^{\prime}}\right)\right],$
(5.1b)
Instead of solving system (5.1) directly, we take the zeroth and first moments
of (5.1a), and then couple them together with (5.1b). More precisely,
$\displaystyle\frac{1}{\mathcal{C}}\frac{\rho_{g}^{n+1}-\rho_{g}^{n}}{\Delta
t}+\left\langle\Omega\cdot\nabla(\Omega\cdot
R^{n+1}_{g})\right\rangle=\mathscr{L}_{a}\left(\sigma^{2}_{a,g}B^{n+1}_{g}-\sigma^{1}_{a,g}\rho^{n+1}_{g}\right),$
(5.2a) $\displaystyle\frac{1}{3\mathcal{C}}\frac{R_{g}^{n+1}-R_{g}^{n}}{\Delta
t}+\left\langle\Omega\Omega\cdot\nabla(\rho^{n+1}_{g}+Q^{n}_{g})\right\rangle=-\frac{(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g}R^{n+1}_{g}}{3},$
(5.2b)
$\displaystyle\frac{C_{v}}{\mathcal{C}\mathcal{P}_{0}}\frac{T^{n+1}-T^{n}}{\Delta
t}=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\mathscr{L}_{a}\left(\sigma^{1}_{a,g^{\prime}}\rho^{n+1}_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B^{n+1}_{g^{\prime}}\right)\right].$
(5.2c)
The macroscopic quantities $\rho_{g},\ R_{g}$ and $T$ can be updated by
solving equation (5.2), then $Q_{g}$ can be updated implicitly by equation
(5.1a), i.e.,
$\displaystyle\frac{1}{\mathcal{C}}\frac{\rho_{g}^{n+1}-\rho_{g}^{n}}{\Delta
t}+\frac{\Omega}{\mathcal{C}}\cdot\frac{R_{g}^{n+1}-R_{g}^{n}}{\Delta
t}+\frac{1}{\mathcal{C}}\frac{Q_{g}^{n+1}-Q_{g}^{n}}{\Delta
t}+\Omega\cdot\nabla(\rho^{n+1}_{g}+\Omega\cdot R^{n+1}_{g}+Q^{n+1}_{g})=$
(5.3) $\displaystyle\hskip
28.45274pt\mathscr{L}_{a}\left(\sigma^{2}_{a,g}B^{n+1}_{g}-\sigma^{1}_{a,g}\rho^{n+1}_{g}\right)-(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g}\Omega\cdot
R^{n+1}_{g}-(\mathscr{L}_{a}\sigma_{a,g}+\mathscr{L}_{s}\sigma_{s,g})Q^{n+1}_{g},$
in which only a linear transport equation is solved with all direction and
energy group being decoupled. Next, we will discuss the diffusion limit of
system (5.2).
### 5.2 The gray radiation diffusion regime of (5.2)
The asymptotic analysis of the time discretization is similar to the analysis
in Section 3.3. In the gray radiation diffusion regime,
$\mathscr{L}_{a}=1/\varepsilon$ and $\mathscr{L}_{s}=1/\varepsilon$. At the
leading order of (5.2a), one gets
$\sigma^{1}_{a,g}\rho^{n+1}_{g}=\sigma^{2}_{a,g}B^{n+1}_{g}+\mathcal{O}(\varepsilon)\,,$
(5.4)
Summing the above equation over the group index $g$, and by the definitions
(3.5), one can get
$\int\limits_{0}^{\infty}\sigma_{a}\rho^{n+1}\text{d}\nu=\int\limits_{0}^{\infty}\sigma_{a}B^{n+1}\text{d}\nu+\mathcal{O}(\varepsilon),$
(5.5)
which implies that
$\rho^{n+1}_{g}=B^{n+1}_{g}+\mathcal{O}(\varepsilon)\,.$ (5.6)
In (5.2a), adding up all the equations in each group and (5.2c), we have
$4\pi\frac{\sum_{g=1}^{G}\rho^{n+1}_{g}-\sum_{g=1}^{G}\rho^{n}_{g}}{\Delta
t}+C_{v}\frac{T^{n+1}-T^{n}}{\Delta
t}=\frac{4\pi}{3\varepsilon}\sum_{g=1}^{G}\left(\nabla\cdot
R_{g}^{n+1}\right),$ (5.7)
From (5.6) and the conditions that $B_{g}$ satisfies in (3.16),
$4\pi\sum_{g=1}^{G}\rho^{n+1}_{g}=(T^{n+1})^{4}+\mathcal{O}(\varepsilon)$. To
get the diffusion limit of equation (5.7), the relation between
$\rho_{g}^{n+1}$ and $R^{n+1}_{g}$ has to be analyzed. By using Champan-Enskog
expansion in equation (5.3), one can get
$\displaystyle-\frac{\Omega\cdot
R^{n+1}_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial
T}(T^{n+1})\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T^{n+1})\text{d}\nu}-(\sigma_{a,g}+\sigma_{s,g})Q^{n+1}_{g}=\mathcal{O}(\varepsilon).$
(5.8)
Taking the second moment in $\Omega$ of equation (5.8) yields
$\left\langle\Omega\Omega
Q^{n+1}_{g}\right\rangle:=K^{n+1}_{Q_{g}}=\mathcal{O}(\varepsilon).$ (5.9)
Taking the first moment of (5.1a), and using the conditions
$\rho^{n+1}_{g}\approx B^{n+1}_{g}$, $K^{n}_{Q_{g}}=\mathcal{O}(\varepsilon)$
and $\left\langle\Omega Q^{n}_{g}\right\rangle$=0, one has
$-\frac{1}{\varepsilon}R^{n+1}_{g}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T^{n+1})\text{d}\nu}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\nabla\rho^{n+1}_{g}}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial T}(T^{n+1})\text{d}\nu+\mathcal{O}(\varepsilon),$ (5.10)
which gives the relation between $R_{g}$ and $\rho_{g}$, then putting it into
(5.7), and using (5.6), when $\varepsilon\to 0$, (5.7) reduces to
$4\pi\frac{\sum\limits_{g=1}^{G}B^{n+1}_{g}-\sum\limits_{g=1}^{G}B^{n}_{g}}{\Delta
t}+C_{v}\frac{T^{n+1}-T^{n}}{\Delta
t}=\sum_{g=1}^{G}\nabla\cdot\left[\left(\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{4\pi}{3(\sigma_{a}+\sigma_{s})}\frac{\partial
B}{\partial T}(T^{n+1})\text{d}\nu\right)\nabla T^{n+1}\right]\,.$ (5.11)
Simultaneously, we can obtain
$\frac{(T^{n+1})^{4}-(T^{n})^{4}}{\Delta t}+C_{v}\frac{T^{n+1}-T^{n}}{\Delta
t}=\nabla\cdot\left(\frac{1}{3\widehat{\sigma}_{R}}\nabla(T^{n+1})^{4}\right)\,,$
(5.12)
with
$\frac{1}{\widehat{\sigma}_{R}}=\frac{1}{4(T^{n+1})^{3}}\left(\sum\limits_{g=1}^{G}\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}\frac{4\pi}{\sigma_{a}+\sigma_{s}}\frac{\partial
B}{\partial T}(T^{n+1})\text{d}\nu\right),$
which is a semi-discretization for equation (4.17).
### 5.3 The frequency dependent diffusion regime of (5.2)
In the frequency dependent diffusion regime, $\mathscr{L}_{a}=\varepsilon$ and
$\mathscr{L}_{s}=1/\varepsilon$. By using Champan-Enskog expansion in equation
(5.3), one can get
$\displaystyle-\frac{\Omega\cdot
R^{n+1}_{g}}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s}}\frac{\partial
B}{\partial
T}(T^{n+1})\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T^{n+1})\text{d}\nu}-\sigma_{s,g}Q^{n+1}_{g}=\mathcal{O}(\varepsilon).$
(5.13)
Taking the second moment for $\Omega$ in equation (5.13) yields
$\left\langle\Omega\Omega Q^{n+1}_{g}\right\rangle=\mathcal{O}(\varepsilon).$
(5.14)
From the second equation of system (5.2), the leading order is
$-\frac{1}{\varepsilon}R^{n+1}_{g}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T^{n+1})\text{d}\nu}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\nabla\rho^{n+1}_{g}}{\sigma_{s}}\frac{\partial
B}{\partial T}(T^{n+1})\text{d}\nu+\mathcal{O}(\varepsilon),$ (5.15)
then putting it into the zeroth moment of (5.1a), and sending $\varepsilon\to
0$, one gets
$\frac{\rho_{g}^{n+1}-\rho_{g}^{n}}{\Delta
t}-\nabla\cdot\left(\frac{1}{3\sigma^{\prime}_{s,g}}\nabla\rho^{n+1}_{g}\right)=\sigma^{2}_{a,g}B_{g}^{n+1}-\sigma^{1}_{a,g}\rho_{g}^{n+1},$
(5.16)
where
$\frac{1}{\sigma^{\prime}_{s,g}}=\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s}}\frac{\partial
B}{\partial
T}(T_{r}^{n+1})\text{d}\nu\big{/}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial T}(T_{r}^{n+1})\text{d}\nu.$ Moreover, sending $\varepsilon\to 0$
in equation (5.1b) gives
$C_{v}\frac{T^{n+1}-T^{n}}{\Delta
t}=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\left(\sigma^{1}_{a,g^{\prime}}\rho^{n+1}_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B^{n+1}_{g^{\prime}}\right)\right].$
(5.17)
In summary, the diffusion limit of system (5.2) in the frequency dependent
diffusion regime is
$\left\\{\begin{aligned} &\frac{\rho_{g}^{n+1}-\rho_{g}^{n}}{\Delta
t}-\nabla\cdot\left(\frac{1}{3\sigma^{\prime}_{s,g}}\nabla\rho^{n+1}_{g}\right)=\sigma^{2}_{a,g}B_{g}^{n+1}-\sigma^{1}_{a,g}\rho_{g}^{n+1}\,,\\\
&C_{v}\frac{T^{n+1}-T^{n}}{\Delta
t}=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\left(\sigma^{1}_{a,g^{\prime}}\rho^{n+1}_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B^{n+1}_{g^{\prime}}\right)\right],\end{aligned}\right.$
which is a semi-discretization for system (4.24).
## 6 AP full discretization for FRTE
For the ease of exposition, we will explain our spatial discretion in 1D. That
is, $x\in[0,L]$, $\Omega\in[-1,1]$, and $\left\langle
f(\Omega)\right\rangle=\frac{1}{2}\int_{-1}^{1}f(\Omega)\mathrm{d}\Omega$, and
the boundary conditions for the FRTE are
$\displaystyle
I_{g}(t,0,\Omega)=b_{g,\text{L}}(t,\Omega)=\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}b_{\text{L}}(t,\Omega,\nu)\text{d}\nu,~{}\text{
for }~{}\Omega>0;$ (6.1) $\displaystyle
I_{g}(t,L,\Omega)=b_{g,\text{R}}(t,\Omega)=\int\limits_{\nu_{g-1/2}}^{\nu_{g+1/2}}b_{\text{R}}(t,\Omega,\nu)\text{d}\nu,~{}\text{
for }~{}\Omega<0\,.$
Higher dimensions can be treated in the dimension by dimension manner. Let
$\Delta x=L/N_{x}$, and we consider the uniform mesh as follows
$x_{i}=(i-1)\Delta x,\quad i=1,2\cdots,N_{x}+1,$
and let
$x_{i+\frac{1}{2}}=\left(x_{i}+x_{i+1}\right)/2,\qquad\mbox{for
$i=1,\cdots,N_{x}.$}$
### 6.1 Even-odd space discretization
To get a consistent stencil in spatial discretization, we use the even-odd
parity method [k2016asymptotic, tang2021accurate, jin2000uniformly]. Let
$E_{g}(\Omega)=\frac{1}{2}\big{(}I_{g}(\Omega)+I_{g}(-\Omega)\big{)},\qquad
O_{g}(\Omega)=\frac{1}{2}(I_{g}(\Omega)-I_{g}(-\Omega)),\qquad\Omega>0$ (6.2)
be respectively the even and odd part of $I_{g}$, and we consider the even
part on half spatial grid and odd part on regular grid, i.e.,
$\displaystyle E_{g,i+1/2}(\Omega)\approx E_{g}(x_{i+1/2},\Omega),\quad
i=1,\cdots,N_{x}\,;$ (6.3) $\displaystyle O_{g,i}(\Omega)\approx
O_{g}(x_{i},\Omega),\quad i=1,\cdots,N_{x}+1\,.$ (6.4)
Moreover, plugging the decomposition (4.1) into (6.2), we have
$E_{g}=\rho_{g}+\big{(}Q_{g}(\Omega)+Q_{g}(-\Omega)\big{)}:=\rho_{g}+E_{Q_{g}},\quad
O_{g}=\Omega\cdot
R_{g}+\frac{1}{2}\big{(}Q_{g}(\Omega)-Q_{g}(-\Omega)\big{)}:=\Omega\cdot
R_{g}+O_{Q_{g}}\,.$ (6.5)
Then the full discretization for the macroscopic equation (5.2) and the
microscopic equation (5.3) are described below.
Space discretization for the first term (5.2). System (5.1) can be written
into a system for $\rho_{g},\ R_{g}$ and $E_{Q_{g}},\ O_{Q_{g}}$:
$\left\\{\begin{aligned} &\frac{1}{\mathcal{C}}\partial_{t}(\Omega\cdot
R_{g}+O_{Q_{g}})+\Omega\cdot\nabla_{x}(\rho_{g}+E_{Q_{g}})=-(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g}\Omega\cdot
R_{g}-(\mathscr{L}_{a}\sigma_{a,g}+\mathscr{L}_{s}\sigma_{s,g})O_{Q_{g}}\,;\\\
&\frac{1}{\mathcal{C}}\partial_{t}(\rho_{g}+E_{Q_{g}})+\Omega\cdot\nabla_{x}(\Omega\cdot
R_{g}+O_{Q_{g}})=\mathscr{L}_{a}\left(\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}\right)-(\mathscr{L}_{a}\sigma_{a,g}+\mathscr{L}_{s}\sigma_{s,g})E_{Q_{g}}\,;\\\
&\frac{C_{v}}{\mathcal{C}\mathcal{P}_{0}}\partial_{t}T=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\mathscr{L}_{a}\left(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}}\right)\right]\,.\end{aligned}\right.$
(6.6)
In terms of $\rho_{g},\ R_{g}$ and $E_{Q_{g}},\ O_{Q_{g}}$, the system (5.2)
becomes
$\left\\{\begin{aligned}
&\frac{1}{3\mathcal{C}}\partial_{t}R_{g}+\left\langle\Omega\Omega\cdot\nabla(\rho_{g}+E_{Q_{g}})\right\rangle=-\frac{(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g}R_{g}}{3},\\\
&\frac{1}{\mathcal{C}}\partial_{t}\rho_{g}+\left\langle\Omega\cdot\nabla(\Omega\cdot
R_{g})\right\rangle=\mathscr{L}_{a}\left(\sigma^{2}_{a,g}B_{g}-\sigma^{1}_{a,g}\rho_{g}\right),\\\
&\frac{C_{v}}{\mathcal{C}\mathcal{P}_{0}}\partial_{t}T=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\mathscr{L}_{a}\left(\sigma^{1}_{a,g^{\prime}}\rho_{g^{\prime}}-\sigma^{2}_{a,g^{\prime}}B_{g^{\prime}}\right)\right].\end{aligned}\right.$
(6.7)
The spatial discretization then takes the following form:
$\displaystyle\frac{1}{3\mathcal{C}}\frac{R_{g,i}^{n+1}-R_{g,i}^{n}}{\Delta
t}+\frac{1}{3\Delta
x}(\rho^{n+1}_{g,i+1/2}-\rho^{n+1}_{g,i-1/2})+\frac{1}{\Delta
x}\left(\int_{0}^{1}\Omega\Omega
E^{n}_{Q_{g},i+1/2}\text{d}\Omega-\int_{0}^{1}\Omega\Omega
E^{n}_{Q_{g},i-1/2}\text{d}\Omega\right)$ $\displaystyle\hskip
213.39566pt=-\frac{(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g,i}R^{n+1}_{g,i}}{3},\quad
2\leq i\leq N_{x}\,;$ (6.8a)
$\displaystyle\frac{1}{\mathcal{C}}\frac{\rho_{g,i+1/2}^{n+1}-\rho_{g,i+1/2}^{n}}{\Delta
t}+\frac{R^{n+1}_{g,i+1}-R^{n+1}_{g,i}}{3\Delta
x}=\mathscr{L}_{a}\left(\sigma^{2}_{a,g,i+1/2}B^{n+1}_{g,i+1/2}-\sigma^{1}_{a,g,i+1/2}\rho^{n+1}_{g,i+1/2}\right),$
$\displaystyle\hskip 341.43306pt\quad 1\leq i\leq N_{x}\,;$ (6.8b)
$\displaystyle\frac{C_{v}}{\mathcal{C}\mathcal{P}_{0}}\frac{T_{i+1/2}^{n+1}-T_{i+1/2}^{n}}{\Delta
t}=\sum\limits_{g^{\prime}=1}^{G}\left[4\pi\mathscr{L}_{a}\left(\sigma^{1}_{a,g^{\prime},i+1/2}\rho^{n+1}_{g^{\prime},i+1/2}-\sigma^{2}_{a,g^{\prime},i+1/2}B^{n+1}_{g^{\prime},i+1/2}\right)\right],1\leq
i\leq N_{x}\,.$ (6.8c)
Space discretization for the second term (5.3). The variables
$\rho_{g}^{n+1}$, $R_{g}^{n+1}$ and $T^{n+1}$ will be updated by solving
system (6.8), and this is a nonlinear system that has to be solved by Newton
iteration. Then the full discretization of (5.3) to update $O_{Q_{g}}$,
$E_{Q_{g}}$ is:
$\displaystyle\frac{1}{\mathcal{C}}\frac{O^{n+1}_{Q_{g},i}-O^{n}_{Q_{g},i}}{\Delta
t}+\frac{\Omega}{\mathcal{C}}\frac{R^{n+1}_{g,i}-R^{n}_{g,i}}{\Delta
t}+\Omega\frac{\rho^{n+1}_{g,i+1/2}-\rho^{n+1}_{g,i-1/2}}{\Delta
x}+\Omega\frac{E^{n+1}_{Q_{g},i+1/2}-E^{n+1}_{Q_{g},i-1/2}}{\Delta x}$
$\displaystyle\hskip
42.67912pt=-(\mathscr{L}_{a}+\mathscr{L}_{s})\sigma_{t,g,i}\Omega\cdot
R^{n+1}_{g,i}-(\mathscr{L}_{a}\sigma_{a,g,i}+\mathscr{L}_{s}\sigma_{s,g,i})O^{n+1}_{Q_{g},i},\quad
2\leq i\leq N_{x}\,;$ (6.9a)
$\displaystyle\frac{1}{\mathcal{C}}\frac{\rho_{g,i+1/2}^{n+1}-\rho_{g,i+1/2}^{n}}{\Delta
t}+\frac{1}{\mathcal{C}}\frac{E^{n+1}_{Q_{g},i+1/2}-E^{n}_{Q_{g},i+1/2}}{\Delta
t}+\Omega\Omega\frac{R^{n+1}_{g,i+1}-R^{n+1}_{g,i}}{\Delta
x}+\Omega\frac{O^{n+1}_{Q_{g},i+1}-O^{n+1}_{Q_{g},i}}{\Delta x}$
$\displaystyle\hskip
28.45274pt=\mathscr{L}_{a}\left(\sigma^{2}_{a,g,i+1/2}B^{n+1}_{g,i+1/2}-\sigma^{1}_{a,g,i+1/2}\rho^{n+1}_{g,i+1/2}\right)-(\mathscr{L}_{a}\sigma_{a,g,i+1/2}+\mathscr{L}_{s}\sigma_{s,g,i+1/2})E^{n+1}_{Q_{g},i+1/2},$
$\displaystyle\hskip 312.9803pt\quad 1\leq i\leq N_{x}\,.$ (6.9b)
Here only a linear transport equation is solved with all direction and energy
group being decoupled.
Boundary conditions To cope with (6.1), using the relation (6.2), we have
$E_{g,3/2}^{n+1}+\frac{1}{2}\left(O_{g,2}^{n+1}+O_{g,1}^{n+1}\right)=b_{\text{g,L}}(\Omega),~{}\Omega>0;\quad
E_{g,N_{x}+1/2}^{n+1}-\frac{1}{2}\left(O_{g,N_{x}}^{n+1}+O_{g,N_{x}+1}^{n+1}\right)=b_{\text{g,R}}(-\Omega),~{}\Omega>0\,.$
(6.10)
Hence,
$O_{g,1}^{n+1}(\Omega)=2\left(b_{\text{g,L}}(\Omega)-E_{g,3/2}^{n+1}\right)-O_{g,2}^{n+1},\quad
O_{g,N_{x}+1}^{n+1}(\Omega)=2\left(E_{g,N_{x}+\frac{1}{2}}^{n+1}-b_{\text{g,R}}(-\Omega)\right)-O_{g,N_{x}}^{n+1}.$
(6.11)
To summarize, we have the following one time step update of the fully discrete
version of FRTE.
Input: $T_{i+1/2}^{n}$, $E_{g,i+1/2}^{s}$,$O_{g,i}^{n}$
Output: $T_{i+1/2}^{n+1}$,$E_{g,i+1/2}^{n+1}$,$O_{g,i}^{n+1}$
1 get $R_{g,i}^{n+1}$, $\rho_{g,i+1/2}^{n+1}$, $T_{i+1/2}^{n+1}$ from the
system (6.8);
2 get $O_{Q_{g},i}^{n+1}$ and $E_{Q_{g},i+1/2}^{n+1}$ from the system (6.9);
obtain $E_{g,i+1/2}^{n+1}$,$O_{g,i}^{n+1}$ from (6.5).
Algorithm 1 one step of fully discrete update for FRTE
### 6.2 The gray radiation diffusion regime of (6.8)
In this subsection, the gray radiation diffusion regime is considered, i.e.,
$\mathscr{L}_{a}=1/\varepsilon$ and $\mathscr{L}_{s}=1/\varepsilon$. From
(6.8b), one can get
$\sigma^{1}_{a,g,i+1/2}\rho^{n+1}_{g,i+1/2}=\sigma^{2}_{a,g,i+1/2}B^{n+1}_{g,i+1/2}+\mathcal{O}(\varepsilon),$
(6.12)
Summing the above equation over the group index $g$, and by the definitions
(3.5), one can get
$\int\limits_{0}^{\infty}\sigma_{a}\rho_{g,i+1/2}^{n+1}\text{d}\nu=\int\limits_{0}^{\infty}\sigma_{a}B_{g,i+1/2}^{n+1}\text{d}\nu+\mathcal{O}(\varepsilon),$
(6.13)
which implies that
$\rho^{n+1}_{g,i+1/2}=B^{n+1}_{g,i+1/2}+\mathcal{O}(\varepsilon).$ (6.14)
Next, we will find the relation between $\rho^{n+1}_{g,i+1/2}$ and
$R^{n+1}_{g,i}$. By using Champan-Enskog expansion in equation (6.9b), and
using (6.14), one can get
$\displaystyle-\sigma_{a,g,i+1/2}E_{Q_{g},i+1/2}^{n+1}=\mathcal{O}(\varepsilon).$
(6.15)
Taking the second moment for $\Omega$ in equation (6.15) yields
$\left\langle\Omega\Omega
E_{Q_{g},i+1/2}^{n+1}\right\rangle=\mathcal{O}(\varepsilon).$ (6.16)
By using (6.16), and letting $\varepsilon\to 0$, equation (6.8a) reduces to
$-\frac{1}{\varepsilon}R^{n+1}_{g,i}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T_{i}^{n+1})\text{d}\nu}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{a,i}+\sigma_{s,i}}\frac{\partial
B}{\partial
T}(T_{i}^{n+1})\text{d}\nu\frac{\rho^{n+1}_{g,i+1/2}-\rho^{n+1}_{g,i-1/2}}{\Delta
x}.$ (6.17)
Substituting (6.17) into equation (6.8b), adding up all the equations in each
group, then adding it to equation (6.8c), using the equation (6.14) and
letting $\varepsilon\to 0$, one can get
$\displaystyle
4\pi\frac{\sum_{g=1}^{G}B^{n+1}_{g,i+1/2}-\sum_{g=1}^{G}B^{n}_{g,i+1/2}}{\Delta
t}+C_{v}\frac{T_{i+1/2}^{n+1}-T_{i+1/2}^{n}}{\Delta
t}=\sum_{g=1}^{G}\frac{1}{3\Delta x}$ (6.18)
$\displaystyle\left[\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{4\pi}{\sigma_{a,i+1}+\sigma_{s,i+1}}\frac{\partial
B}{\partial
T}(T_{i+1}^{n+1})\text{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T_{i+1}^{n+1})\text{d}\nu}\frac{B^{n+1}_{g,i+3/2}-B^{n+1}_{g,i+1/2}}{\Delta
x}-\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{4\pi}{\sigma_{a,i}+\sigma_{s,i}}\frac{\partial
B}{\partial
T}(T_{i}^{n+1})\text{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T_{i}^{n+1})\text{d}\nu}\frac{B^{n+1}_{g,i+1/2}-B^{n+1}_{g,i-1/2}}{\Delta
x}\right]\,.$
Moreover, we have
$4\pi\sum_{g=1}^{G}B^{n+1}_{g,i+1/2}=\int_{0}^{\infty}B(\nu,T^{n+1}_{i+1/2})\text{d}\nu=(T^{n+1}_{i+1/2})^{4},$
which is is consistent with equation (4.17), and the calculations of the other
two gray radiation diffusion states are similar.
### 6.3 The frequency dependent diffusion regime of (6.8)
In the frequency dependent diffusion regime, $\mathscr{L}_{a}=\varepsilon$ and
$\mathscr{L}_{s}=1/\varepsilon$. By using Champan-Enskog expansion in equation
(6.9b), one can get
$\displaystyle-\sigma_{s,g,i+1/2}E_{Q_{g},i+1/2}^{n+1}=\mathcal{O}(\varepsilon).$
(6.19)
Taking the second moment for $\Omega$ in equation (6.19) yields
$\left\langle\Omega\Omega
E_{Q_{g},i+1/2}^{n+1}\right\rangle=\mathcal{O}(\varepsilon).$ (6.20)
By using (6.20), and letting $\varepsilon\to 0$, equation (6.8a) reduces to
$-\frac{1}{\varepsilon}R^{n+1}_{g,i}=\frac{1}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T_{r,i}^{n+1})\text{d}\nu}\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s,i}}\frac{\partial
B}{\partial
T}(T_{r,i}^{n+1})\text{d}\nu\frac{\rho^{n+1}_{g,i+1/2}-\rho^{n+1}_{g,i-1/2}}{\Delta
x}.$ (6.21)
Substituting (6.21) into equation (6.8b), and letting $\varepsilon\to 0$, one
can get
$\displaystyle\frac{\rho_{g,i+1/2}^{n+1}-\rho_{g,i+1/2}^{n}}{\Delta t}$
$\displaystyle-\frac{1}{3\Delta
x}\left[\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s,i+1}}\frac{\partial
B}{\partial
T}(T_{r,i+1}^{n+1})\text{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T_{r,i+1}^{n+1})\text{d}\nu}\frac{\rho^{n+1}_{g,i+3/2}-\rho^{n+1}_{g,i+1/2}}{\Delta
x}\right.$ (6.22)
$\displaystyle\left.-\frac{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{1}{\sigma_{s,i}}\frac{\partial
B}{\partial
T}(T_{r,i}^{n+1})\text{d}\nu}{\int_{\nu_{g-\frac{1}{2}}}^{\nu_{g+\frac{1}{2}}}\frac{\partial
B}{\partial
T}(T_{r,i}^{n+1})\text{d}\nu}\frac{\rho^{n+1}_{g,i+1/2}-\rho^{n+1}_{g,i-1/2}}{\Delta
x}\right]=\sigma^{2}_{a,g,i+\frac{1}{2}}B^{n+1}_{g,i+\frac{1}{2}}-\sigma^{1}_{a,g,i+\frac{1}{2}}\rho^{n+1}_{g,i+\frac{1}{2}}.$
Moreover, letting $\varepsilon\to 0$, equation (6.8c) reduces to
$C_{v}\frac{T_{i+1/2}^{n+1}-T_{i+1/2}^{n}}{\Delta
t}=\sum\limits_{g^{\prime}=1}^{G}4\pi\left(\sigma^{1}_{a,g^{\prime},i+\frac{1}{2}}\rho^{n+1}_{g^{\prime},i+\frac{1}{2}}-\sigma^{2}_{a,g^{\prime},i+\frac{1}{2}}B^{n+1}_{g^{\prime},i+\frac{1}{2}}\right).$
(6.23)
which is consistent with equation (4.24).
## 7 Numerical Examples
In this section, we conduct several numerical examples to test the performance
of the AP scheme for the FRTE. The units of the length, time, temperature and
energy are respectively $cm,\ ns,\ keV$ and $Jk$ ($1\ Jk=10^{9}\ J$). With the
above units, let the speed of light $c$ be 29.98 $cm\ ns^{-1}$, the specific
heat capacity $C_{v}$ be 0.1 $Jk\ keV^{-1}\ cm^{-3}$ and the radiation
constant $a_{r}$ be 0.01372 $Jk\ keV^{-4}\ cm^{-3}$. As in
[densmore2012hybrid, sunj], in all of these examples, the opacities take the
following forms :
$\sigma_{a}=\frac{\sigma_{a0}(x)}{(h\nu)^{3}\sqrt{kT}},\qquad\sigma_{s}=\frac{\sigma_{s0}(x)}{(h\nu)^{3}\sqrt{kT}}.$
(7.1)
In all numerical examples in this section, similar as in [densmore2012hybrid,
sunj], we divide the frequency domain from 0.1 $eV$ to 100 $keV$ into 30
frequency groups, and the groups are logarithmically spaced. In each group, we
determine the group absorption, scattering and total cross section by using
(4.7) and (7.1). Moreover, from Figure 2, the piece-wise constant
approximations $\sigma_{a,g}^{c}$ are closer to the reference solutions, thus
we use $\omega_{1}=\omega_{2}=1$ in all examples.
### 7.1 Example 1
In this example, we will show the convergence order and stability of the
proposed scheme. The thickness of the computational domain is $2\ cm$ and the
initial temperature is
$T(0,x)=\max\big{\\{}1-20(x-1)^{2},10^{-3}\big{\\}}\ keV.$ (7.2)
The initial radiation intensity is set to be a Planck distribution evaluated
at the temperature (7.2). Zero Dirichlet boundary condition is used in this
example. Four different sets of $\sigma_{a0}$, $\sigma_{s0}$ are tested: 1)
$\sigma_{a0}=1\ keV^{7/2}\ cm^{-1}$, $\sigma_{s0}=1000\ keV^{7/2}\ cm^{-1}$;
2) $\sigma_{a0}=1000\ keV^{7/2}\ cm^{-1}$, $\sigma_{s0}=1\ keV^{7/2}\
cm^{-1}$; 3) $\sigma_{a0}=1000\ keV^{7/2}\ cm^{-1}$, $\sigma_{s0}=1000\
keV^{7/2}\ cm^{-1}$; 4) $\sigma_{a0}=1\ keV^{7/2}\ cm^{-1}$, $\sigma_{s0}=1\
keV^{7/2}\ cm^{-1}$. In the four cases, the first case belongs to the
frequency dependent diffusion regime, and the second case and the third case
are the gray radiation diffusion regime, and $\sigma_{a}$ and $\sigma_{s}$ are
both optical thin in the last case.
In Fig. 3, we plot the errors
$\text{error}_{T}=\|T_{\Delta x}(\cdot,t_{\text{max}})-T_{\Delta
x/2}(\cdot,t_{\text{max}})\|_{l_{1}}$ (7.3)
with $\Delta x=2*10^{-2}\ cm$, $10^{-2}\ cm$, $5*10^{-3}\ cm$, $2.5*10^{-3}\
cm$, $1.25*10^{-3}\ cm$ and $\Delta t=2*10^{-2}\ ns$, $10^{-2}\ ns$,
$5*10^{-3}\ ns$, $2.5*10^{-3}\ ns$, $1.25*10^{-3}\ ns$, and $t_{\text{max}}=1\
ns$. A uniform first order accuracy can be observed for all four different
cases. Moreover, it is important to point out that the scheme is stable when
the time step $\Delta t=8\Delta x$ for all the cases.
Figure 3: Example 1. Errors of different $\Delta x$, $\Delta t$ for the four
cases. Here different $\Delta x=2*10^{-2}\ cm$, $10^{-2}\ cm$, $5*10^{-3}\
cm$, $2.5*10^{-3}\ cm$, $1.25*10^{-3}\ cm$ are tested and $\Delta t$ are
chosen to be $\Delta t=2*10^{-2}\ ns$, $10^{-2}\ ns$, $5*10^{-3}\ ns$,
$2.5*10^{-3}\ ns$, $1.25*10^{-3}\ ns$ respectively.
At last, we compare the results in the gray radiation diffusion regime and
frequency dependent diffusion regime to the reference results. For the four
cases, we use a uniform spatial mesh and the cell size $\Delta x=0.01\ cm$,
and the time step $\Delta t=0.04\ ns$. Moreover, the reference solution of all
the cases are computed with $\Delta x=0.001\ cm$ and $\Delta t=0.00001\ ns$.
The numerical results at $t=2\ ns$ are displayed in Fig. 4. The material
temperature from our AP method agrees well with the reference solution in all
the four cases.
(a) $\sigma_{a0}=1\ keV^{7/2}\ cm^{-1},\ \sigma_{s0}=1000\ keV^{7/2}\ cm^{-1}$
(b) $\sigma_{a0}=1000\ keV^{7/2}\ cm^{-1},\ \sigma_{s0}=1\ keV^{7/2}\ cm^{-1}$
(c) $\sigma_{a0}=1000\ keV^{7/2}\ cm^{-1},\sigma_{s0}=1000\ keV^{7/2}\
cm^{-1}$
(d) $\sigma_{a0}=1\ keV^{7/2}\ cm^{-1},\sigma_{s0}=1\ keV^{7/2}\ cm^{-1}$
Figure 4: Example 1. Comparison of the material temperature using our AP
scheme for equilibrium regime and the reference solution at $t=2\ ns$.
### 7.2 Example 2
We test an example in [densmore2012hybrid, sunj, steinberg2022multi] where
particles are injected from the left boundary. Three space homogeneous
problems with $\sigma_{a0}=10\ keV^{7/2}\ cm^{-1},\ 100\ keV^{7/2}\ cm^{-1}$,
$1000\ keV^{7/2}\ cm^{-1}$, and $\sigma_{s0}=0\ keV^{7/2}\ cm^{-1}$ are
tested. The thickness of the computational domain is $5\ cm$, the initial
temperature is $10^{-3}\ keV$ and the initial radiation intensity is a Planck
distribution evaluated at the initial temperature. On the left boundary, the
incident radiation intensity is given by a Planck distribution with the
temperature being 1 $keV$, and we use a reflective boundary condition on the
right boundary. For all three tests, we use a uniform spatial mesh with
$\Delta x=0.005\ cm$ and $\Delta t=0.005\ $ns to ensure the scheme stable. The
reference solution is calculated using a finer temporal mesh such that $\Delta
t=0.00001\ $ns. Moreover, in the example , the time size $\Delta t=0.005$ ns
used in our AP method is much bigger than the time step $\Delta t=0.004/c$ ns
used in [sunj]. The numerical results at $t=1\ ns$ are displayed in Fig. 5
(Note that we only show the part where the material temperature differs
significantly from the initial value in the second and third figure). The good
agreement between our solution and the reference solution indicates that our
method works well in the optical thin and thick regime.
(a) $\sigma_{a0}=10\ keV^{7/2}\ cm^{-1}$
(b) $\sigma_{a0}=100\ keV^{7/2}\ cm^{-1}$
(c) $\sigma_{a0}=1000\ keV^{7/2}\ cm^{-1}$
Figure 5: Example 2. Plot of three different cases at $t=1\ ns$. The red
curves are reference solutions computed in the finer mesh. Here $\Delta
x=0.005\ cm$, in our AP scheme $\Delta t=0.005\ ns$, and in the reference
solution, $\Delta t=0.00001\ ns$.
### 7.3 Example 3
A 1D Marshak wave problem similar as in [steinberg2022multi,
densmore2012hybrid, sunj] is tested, where both optical thin and thick regions
coexist. The opacities take the forms of (7.1) with
$\sigma_{a0}=\begin{cases}10\quad\mathrm{keV}^{7/2}\ cm^{-1},&0<x<2.0\ cm,\\\
1000\quad\mathrm{keV}^{7/2}\ cm^{-1},&2.0<x<3.0~{}cm,\end{cases}$
and $\sigma_{s0}=0$. Moreover, the width of the computational domain is $3$cm
and we divide it uniformly into $150$ spatial cells. The time step is $\Delta
t=0.02\ ns$. The problem begins with flat material and radiation temperature
profiles at 1 $eV$. A boundary source of 1 $keV$ is placed at the left
boundary, while the right boundary is vacuum. In this example, the reference
solution is computed with $\Delta x=0.001\ cm$ and $\Delta t=0.00001\ ns$. The
numerical results at $t=1\ ns$ are displayed in Fig. 6. Moreover, the time
size $\Delta t=0.02$ ns of our AP method in the example is much bigger than
the time step $\Delta t=0.016/c$ ns used in [sunj]. The material temperature
from our AP method agrees well with the reference solution in the co-exit
regime.
Figure 6: Example 3. Plot of multi-region problem at $t=1\ ns$. The red curves
are reference solutions computed in the finer mesh. Here $\Delta x=0.02\ cm$,
$\Delta t=0.02\ ns$ in our AP scheme, and in the reference solution, $\Delta
x=0.005\ cm$ and $\Delta t=0.00001\ ns$.
## 8 Conclusion
In this paper, we propose an AP decomposed multi-group scheme for the FRTE in
the gray radiation diffusion regime and the frequency-dependent diffusion
regime. The idea is to decompose the intensity into three parts: the zeroth
moment of the intensity, the first moment of the intensity, and the residual
term. The main contribution is the new decomposed multi-group energy
discretization, which allows tuning of the approximations of the coefficients
in the radiation diffusion regime.
In the full AP discretization, the two macroscopic moments and the material
temperature are updated first by treating them implicitly with the explicit
residual term. After the zeroth and first-order moments are obtained, we
update the residual term by solving implicitly a linear transport equation for
each decoupled direction and energy group. The AP property is proved by
asymptotic analysis. Numerical results of both optically thin and thick
regions are presented. We can see that coarse meshes in energy, space, and
time can be used to get the correct results. We will focus on the extension of
the algorithm to high-dimensional problems and curvilinear geometrical
problems in the future.
Acknowledgement: X. J. Zhang and M. Tang were supported by NSFC12031013,
Shanghai pilot innovation project 21JC1403500 and the Strategic Priority
Research Program of Chinese Academy of Sciences Grant No.XDA25010401; X. J.
Zhang was supported by China Postdoctoral Science Foundation No.2022M722107;
P. Song was supported by NSFC12031001; Y. Shi was supported by NSFC12001052.
## References
## Appendix A A The derivation of the two limiting models
In the scaling of (2.5a), we assume that the intensity $I$ and material
temperature $T$ can be expanded as in [MWTS]:
$\displaystyle I=I^{(0)}+\varepsilon I^{(1)}+\varepsilon^{2}I^{(2)}+\cdots,$
(A.1) $\displaystyle T=T^{(0)}+\varepsilon
T^{(1)}+\varepsilon^{2}T^{(2)}+\cdots,$
and the Planck function follows that
$B=B^{(0)}+\varepsilon B^{(1)}+\cdots,$ (A.2)
where
$B^{(0)}=B|_{\varepsilon=0}=B|_{T=T^{{(0)}}},$ $B^{(1)}=\frac{\partial
B}{\partial\varepsilon}\Bigg{|}_{\varepsilon=0}=\frac{\partial B}{\partial
T}\frac{\partial
T}{\partial\varepsilon}\Bigg{|}_{\varepsilon=0}=\frac{\partial B}{\partial
T}\Bigg{|}_{T=T^{{(0)}}}T^{(1)},$
By substituting ansatzes (A.1) and (A.2) into equation (2.4a) and collecting
the terms of the same order in $\varepsilon$, we have
$\displaystyle\mathcal{O}\left(\frac{1}{\varepsilon}\right)$
$\displaystyle:I^{(0)}=B^{(0)},$ (A.3a)
$\displaystyle\mathcal{O}\left(1\right)$ $\displaystyle:\Omega\cdot\nabla
I^{(0)}=\sigma_{a}\left(B^{(1)}-I^{(1)}\right).$ (A.3b)
Taking (2.4a) integral with respect to $\nu$ from 0 to $\infty$, and taking
its integral with respect to $\Omega$, then adding the obtained equation to
(2.4b), and dividing both sides of the equation by $\varepsilon$, one has
$4\pi\partial_{t}\int\limits_{0}^{\infty}\rho\text{d}\nu+\frac{1}{\varepsilon}\int\limits_{4\pi}\int\limits_{0}^{\infty}\Omega\cdot\nabla
I\text{d}\nu\text{d}\Omega+C_{v}\partial_{t}T=0.$
By using (A.3a), (A.3b) and condition
$\left\langle\Omega\Omega\right\rangle:=D_{d}=\frac{1}{3}I_{d}$ (where $I_{d}$
denotes the 3 by 3 identity matrix), and sending $\varepsilon\to 0$, which
give
$4\pi\partial_{t}\int\limits_{0}^{\infty}B^{(0)}\text{d}\nu+C_{v}\partial_{t}T^{(0)}=\int\limits_{0}^{\infty}\nabla\cdot\left(\frac{4\pi
D_{d}}{\sigma_{a}}\nabla B^{(0)}\right)\text{d}\nu.$ (A.4)
by using the following relations
$\int\limits_{0}^{\infty}4\pi B\
\text{d}\nu=T^{4},\qquad\int\limits_{0}^{\infty}4\pi\frac{\partial B}{\partial
T}\ \text{d}\nu=4T^{3},$
then (A.4) reduces to
$\partial_{t}(T^{(0)})^{4}+C_{v}\partial_{t}T^{(0)}=\nabla\cdot\left(\frac{D_{d}}{\sigma_{R}}\nabla(T^{(0)})^{4}\right)\,,$
(A.5)
with the Rosseland mean opacity $\sigma_{R}$ which is given by
$\frac{1}{\sigma_{R}(x,T)}\equiv\frac{\int_{0}^{\infty}\frac{4\pi}{\sigma_{a}(x,\nu,T)}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{\int_{0}^{\infty}4\pi\frac{\partial
B^{(0)}(\nu,T)}{\partial
T}\text{d}\nu}=\frac{\int_{0}^{\infty}\frac{4\pi}{\sigma_{a}(x,\nu,T)}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{4(T^{(0)})^{3}}.$
In the scaling of (2.5b), the same expansions (A.1) and (A.2) are used,
moreover, substitute (A.1) and (A.2) into the radiation transfer equation and
collect the terms of the same order in $\varepsilon$, which gives
$\displaystyle\mathcal{O}\left(\frac{1}{\varepsilon}\right)$
$\displaystyle:I^{(0)}=\rho^{(0)},$ (A.6a)
$\displaystyle\mathcal{O}\left(1\right)$ $\displaystyle:\Omega\cdot\nabla
I^{(0)}=\sigma_{a}\left(B^{(0)}-I^{(0)}\right)+\sigma_{s}\left(\rho^{(1)}-I^{(1)}\right).$
(A.6b)
Taking (A.6b) integral with respect to $\Omega$, one has
$B^{(0)}=\rho^{(0)},$
by similar calculations in the first part, one can get the following
equilibrium system:
$\partial_{t}(T^{(0)})^{4}+C_{v}\partial_{t}T^{(0)}=\nabla\cdot\left(\frac{D_{d}}{\sigma_{S}}\nabla(T^{(0)})^{4}\right)\,,$
(A.7)
and the mean opacity $\sigma_{S}$ is given by
$\frac{1}{\sigma_{S}}\equiv\frac{\int_{0}^{\infty}\frac{4\pi}{\sigma_{s}}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{\int_{0}^{\infty}4\pi\frac{\partial
B^{(0)}(\nu,T)}{\partial
T}\text{d}\nu}=\frac{\int_{0}^{\infty}\frac{4\pi}{\sigma_{s}}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{4(T^{(0)})^{3}}.$
In the scaling of (2.5c), the same expansions (A.1) and (A.2) are used,
moreover, substitute (A.1) and (A.2) into the radiation transfer equation and
collect the terms of the same order in $\varepsilon$, which gives
$\displaystyle\mathcal{O}\left(\frac{1}{\varepsilon}\right)$
$\displaystyle:(\sigma_{a}+\sigma_{s})I^{(0)}=\sigma_{a}B^{(0)}+\sigma_{s}\rho^{(0)},$
(A.8a) $\displaystyle\mathcal{O}\left(1\right)$
$\displaystyle:\Omega\cdot\nabla
I^{(0)}=\sigma_{a}\left(B^{(1)}-I^{(1)}\right)+\sigma_{s}\left(\rho^{(1)}-I^{(1)}\right).$
(A.8b)
Taking (A.8a) and (A.8b) integral with respect to $\Omega$, one has
$B^{(0)}=\rho^{(0)},\qquad B^{(1)}=\rho^{(1)},$
by similar calculations in the first part, one can get the following
equilibrium system:
$\partial_{t}(T^{(0)})^{4}+C_{v}\partial_{t}T^{(0)}=\nabla\cdot\left(\frac{D_{d}}{\sigma_{T}}\nabla(T^{(0)})^{4}\right)\,,$
(A.9)
and the mean opacity $\sigma_{T}$ is given by
$\frac{1}{\sigma_{T}}\equiv\frac{\int_{0}^{\infty}\frac{4\pi}{(\sigma_{a}+\sigma_{s})}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{\int_{0}^{\infty}4\pi\frac{\partial
B^{(0)}(\nu,T)}{\partial
T}\text{d}\nu}=\frac{\int_{0}^{\infty}\frac{4\pi}{(\sigma_{a}+\sigma_{s})}\frac{\partial
B^{(0)}(\nu,T)}{\partial T}\text{d}\nu}{4(T^{(0)})^{3}}.$
In the frequency-dependent radiative diffusion regime, the same expansions
(A.1) and (A.2) are used, moreover, substitute (A.1) and (A.2) into equation
(2.9a) and collect the terms of the same order in $\varepsilon$, which gives
$\displaystyle\mathcal{O}\left(\frac{1}{\varepsilon}\right)$
$\displaystyle:I^{(0)}=\rho^{(0)},$ (A.10a)
$\displaystyle\mathcal{O}\left(1\right)$ $\displaystyle:\Omega\cdot\nabla
I^{(0)}=\sigma_{s}\left(\rho^{(1)}-I^{(1)}\right).$ (A.10b)
Taking (2.9a) integral with respect to $\Omega$, and dividing both sides of
the equation by $\varepsilon$, one has
$4\pi\partial_{t}\rho+\frac{4\pi}{\varepsilon}\left\langle\Omega\cdot\nabla
I\right\rangle=4\pi\sigma_{a}(B-\rho),$
by similar calculations in the first part, one can get the following non-
equilibrium system:
$\left\\{\begin{aligned}
&\partial_{t}\rho^{(0)}-\nabla\cdot\left(\frac{D_{d}}{\sigma_{s}}\nabla\rho^{(0)}\right)=\sigma_{a}(B^{(0)}-\rho^{(0)}),\\\
&C_{v}\partial_{t}T^{(0)}=\int_{0}^{\infty}4\pi\sigma_{a}(\rho^{(0)}-B^{(0)})\text{d}\nu.\
\ \\\ \end{aligned}\right.$ (A.11)
|
# Automated anomaly-aware 3D segmentation of bones and cartilages in knee MR
images from the Osteoarthritis Initiative
Boyeong Woo<EMAIL_ADDRESS>Craig Engstrom William Baresic Jurgen Fripp
Stuart Crozier Shekhar S. Chandra School of Information Technology and
Electrical Engineering, The University of Queensland, Australia School of
Human Movement and Nutrition Sciences, The University of Queensland, Australia
Australian e-Health Research Centre, Commonwealth Scientific and Industrial
Research Organisation, Australia
###### Abstract
In medical image analysis, automated segmentation of multi-component
anatomical structures, which often have a spectrum of potential anomalies and
pathologies, is a challenging task. In this work, we develop a multi-step
approach using U-Net-based neural networks to initially detect anomalies (bone
marrow lesions, bone cysts) in the distal femur, proximal tibia and patella
from 3D magnetic resonance (MR) images of the knee in individuals with varying
grades of osteoarthritis. Subsequently, the extracted data are used for
downstream tasks involving semantic segmentation of individual bone and
cartilage volumes as well as bone anomalies. For anomaly detection, the U-Net-
based models were developed to reconstruct the bone profiles of the femur and
tibia in images via inpainting so anomalous bone regions could be replaced
with close to normal appearances. The reconstruction error was used to detect
bone anomalies. A second anomaly-aware network, which was compared to anomaly-
naïve segmentation networks, was used to provide a final automated
segmentation of the femoral, tibial and patellar bones and cartilages from the
knee MR images containing a spectrum of bone anomalies. The anomaly-aware
segmentation approach provided up to 58% reduction in Hausdorff distances for
bone segmentations compared to the results from the anomaly-naïve segmentation
networks. In addition, the anomaly-aware networks were able to detect bone
lesions in the MR images with greater sensitivity and specificity (area under
the receiver operating characteristic curve [AUC] up to 0.896) compared to the
anomaly-naïve segmentation networks (AUC up to 0.874).
###### keywords:
Anomaly detection, Segmentation, U-Net, Knee osteoarthritis, MRI
††journal: Medical Image Analysis
## 1 Introduction
Deep learning methods, particularly convolutional neural networks (CNNs), have
shown promising results in image recognition tasks and are a rapidly evolving
area of research in the field of automated medical image analysis. Common deep
learning-based computer vision tasks that have been applied to medical imaging
include classification, detection, and segmentation [24]. For example, U-Net,
proposed by Ronneberger et al. [23], has become a popular CNN model for
medical image segmentation. There are now several publicly available medical
image databases, such as the Osteoarthritis Initiative [19], which provide the
basis for deep learning research in medical imaging.
Major challenges in automated analysis of medical images using deep learning
include lack of annotated data and variable presence of anomalies. Manual
annotation of tomographic images from techniques such as computed tomography
(CT) and magnetic resonance (MR) imaging is typically expertise- and time-
intensive. This makes efficient automated volumetric image processing
particularly desirable in medical imaging but also presents a major challenge
to deep learning research in medical imaging because there are often not
enough annotated data for training a deep learning model. In addition,
incidental anomalies (e.g. lesions, anatomical variations, imaging artefacts,
etc.) apart from any “primary” pathoanatomy of interest are frequently present
in clinical settings, making automated segmentation of multi-component
anatomical structures difficult. A method to identify and quantify “secondary”
anomalies will be required for robust segmentation performance.
This work is an extension to preliminary experiments presented in Woo et al.
[26] involving segmentation of the distal femur and proximal tibia using
U-Net-based networks. In the current work, we develop an approach based on CNN
to detect bone anomalies (bone marrow lesions, bone cysts) in the distal
femur, proximal tibia and patella from knee MR images from individuals with
varying levels of osteoarthritis. The rational was that the output from the
anomaly detector can also be utilized for improving segmentation of the knee
with osteoarthritis containing bone lesions. The main contributions of this
work include:
1. 1.
The use of 3D U-Net-based CNNs for anomaly detection via inpainting in medical
imaging. While there have been previous works that used U-Nets for inpainting
and visual anomaly detection [15, 29], they have focused on natural images or
2D image slices. This work extends such ideas to 3D medical images using fully
3D inpainting.
2. 2.
Using information learned from anomaly detection to improve segmentation of
the distal femur, proximal tibia and patella from knee MR images using an
anomaly-aware CNN approach. The current work shows that the proposed approach
is capable of significantly improving the segmentation of femur and tibia on
osteoarthritic knee MR images with bone anomalies.
3. 3.
Demonstrating that the anomaly-aware approach has an advantage when the size
of the training dataset is limited. We added additional labels to the
segmentation task, including the patella and bone lesions, which are
relatively more difficult to segment due to their smaller volumes and greater
variability. The proposed anomaly-aware approach is able to detect and segment
these new labels with moderate accuracy despite the limited size of the
training dataset.
4. 4.
Evaluation of the effect of anomaly-aware approach on segmentation performance
using two different categories of CNNs: 3D U-Net and 3D context aggregation
network (CAN). The current work shows that the anomaly-aware approach can be
readily applied to different segmentation CNNs to improve their segmentation
performance as well as transfer learning.
## 2 Background
In recent years, several machine learning algorithms have been proposed for
automatic anomaly detection. Unsupervised methods using generative models were
shown to be promising [3, 4] and are particularly useful in medical imaging
for which labeled data can be extremely difficult to obtain. A recent work by
Pinaya et al. [20] showed that leading-edge techniques such as transformers
can also be utilized for anomaly detection and segmentation. However,
application of transformers to 3D images is currently challenging due to their
very high demands on data and computational resources. Convolutional
autoencoders such as U-Net are computationally less intensive than
transformers. Pinaya et al. [20] also used vector quantized variational
autoencoder (VQ-VAE) for dimensionality reduction which allowed them to apply
transformers.
U-Net, developed by Ronneberger et al. [23], is a popular CNN model for
semantic segmentation. It is a fully convolutional network with an autoencoder
structure, consisting of a contracting path (encoder) and an expansive path
(decoder). The distinguishing feature of U-Net is the skip connections that
transfer feature maps at each resolution from the encoder to the decoder. The
skip connections recover spatial information lost during downsampling, which
is critical for segmentation tasks. The original U-Net proposed was a 2D
convolutional network for segmentation of 2D images. Çiçek et al. [5] extended
this work to dense volumetric segmentation using 3D U-Net. V-Net [16] is
another autoencoder-based model similar to U-Net that is trained end-to-end on
image volumes.
Some later works on volumetric segmentation using 3D CNNs (typically U-Net-
like) have employed a technique referred to as “deep supervision” [12, 10,
22]. The main idea of deep supervision is to provide integrated direct
supervision to the hidden layers, rather than providing supervision only at
the output layer [14]. Deep supervision has been found to speed up convergence
likely because it encourages deeper layers to produce improved segmentation
results [12]. The skip connections in U-Net force aggregation only at the
same-scale feature maps, and it was shown that redesigned skip connections and
deep supervision in U-Net enable a significantly higher level of segmentation
performance [30].
In addition to segmentation, CNNs with U-Net architecture have been applied
for different tasks, such as image-to-image translation [11]. More recently,
Liu et al. [15] adopted a U-Net-based model to perform image inpainting and
generate synthetic brain tissue intensities for a tumor region. Zavrtanik et
al. [29] proposed an anomaly detection method whereby a U-Net-based model was
used to reconstruct images from partial inpaintings which were then used to
localize visual anomalies in the images. The current work is similar to this
idea and perform unsupervised anomaly detection on unlabeled 3D knee MR images
through inpainting and lossy reconstruction. The approach is to “erase” a
region of interest (ROI) from an image, which may or may not contain
anomalies, and then let the network generate synthetic tissue intensities in
the region without the potential anomalies. The difference between the
original image and the reconstructed image can then be used to detect the
anomalies. Here, we use a version of 3D U-Net for efficient volumetric image
processing.
Fig. 1: Overall pipeline for our method including anomaly detection and the
downstream segmentation task. Components 1 and 2 are the anomaly detection
models (Sections 3.1 and 3.2). Component 3 is the downstream segmentation task
using the anomaly-aware segmentation approach (Section 3.3).
As an alternative to autoencoder-based CNNs, context aggregation network (CAN)
was proposed by Yu and Koltun [27] for semantic segmentation. It is
structurally different from U-Net in that it is not based on an encoder-
decoder architecture. In U-Net, the progressive downsampling achieves the
effect of integrating contextual information at multi-scale, and the lost
resolution is recovered through upsampling. However, since semantic
segmentation task requires full-resolution output, there is the question of
whether such downsampling and upsampling are truly necessary, which is why Yu
and Koltun [27] proposed using dilated convolutions rather than downsampling
for multi-scale context aggregation. As with U-Net, CAN was proposed as a 2D
network, but Dai et al. [6] extended it to a 3D version for volumetric
segmentation.
Accurate segmentation of knee bones and cartilages is an important step in
processing and analyzing knee MR images in the context of osteoarthritis. The
use of CNNs has grown in knee tissue segmentation to automatically learn image
features [7]. The first published work in this area is Prasoon et al. [21],
which used a triplanar patch-based approach where three CNNs were trained to
classify the central voxel as cartilage or background. Although this triplanar
approach has now become obsolete, the finding that it achieved a better
performance than the previous state-of-the-art showed that a CNN is capable of
learning image features from knee MR images.
In our preliminary experiments using 3D U-Net and 3D CAN for knee MR image
segmentation, the CNNs were able to achieve mean Dice similarity coefficients
(DSCs) for bones and cartilages in the knee joint similar to Ambellan et al.
[2] for the OAI ZIB dataset (dataset described in Section 4.1). However, it
was observed that the CNN-based segmentation of images was difficult for cases
with visible coexisting abnormalities. Without shape regularization, the
surface distance errors tended to be high because CNN outputs often contained
“holes” and “noises” (false negatives/positives) due to the localized nature
of the CNN-based classification. Indeed, Ambellan et al. [2] used a
combination of U-Nets and statistical shape models (SSMs), and the authors
explicitly state that SSM regularization of the CNN outputs was needed to
attain “anatomically plausible” segmentations, which is consistent with our
preliminary finding. In particular, it was difficult to achieve a good
segmentation with a U-Net or CAN when there was a large anomaly in the image
(see Figure 6). Therefore, we also propose an anomaly-aware segmentation
mechanism for CNNs which is more robust in the presence of anomalies.
## 3 Methods
The overall pipeline for the current work has 3 major components (Figure 1).
In Component 1, the anatomical regions of interest—here, the distal femur and
proximal tibia profiles—were erased from the images using the reference
segmentation masks, and then these regions were inpainted using a 3D U-Net-
based model ($G$). In Component 2, the outputs from Component 1 and another 3D
U-Net-based model ($A$) were used to change anomalous bone regions in the
original images to close to normal appearances. In Component 3, a 3D CNN-based
segmentation network ($S$), which utilizes the information extracted from
Component 2, was used to guide the automated segmentation of bone and
cartilage volumes compared to the vanilla segmentation networks, specifically
aiming to improve segmentation of the structures from images containing
visible bone anomalies.
### 3.1 Anomaly detection using masked images
Using the reference segmentation masks available in the OAI ZIB MR image
dataset (see Section 4.1), the profiles of the femur and tibia were erased
from the MR images. To ensure that the boundaries are not missed and to
include some periarticular areas, the masks were dilated by 50 pixels in all
directions before being applied to the images. The masked images were then
used as input to the 3D U-Net-based model referred to as $G$, which was
trained to reconstruct the original image, i.e. to inpaint the erased area.
(a)
(b)
Fig. 2: The two anomaly detection networks with a 3D U-Net-based architecture.
Blue boxes represent feature maps, with the number of channels denoted above
each box. (a) Network $G$ regenerates the original images from masked images
through inpainting and decoding of compressed images. The compressed images
are provided by a small network $C$ trained concurrently with $G$. (b) Network
$A$ is trained using the images generated by $G$ as the target output given
the original images. The compressed images from $C$ are added to the decoder.
IN: Instance Normalization.
Since the erased profile was relatively large, an image compressor $C$ was
added here to assist with the inpainting. This $C$ was trained simultaneously
with $G$ to compress the original MR image, and the compressed image was then
fed into the decoder part of $G$. The compressor $C$ had to be a very small
network because the bone anomalies must not be recovered in the output; a
larger network is likely to restore the anomalies in the inpainted images.
Figure 2(a) shows the overall structure of the model.
The loss function for training $G$ and $C$ was the mean squared error (MSE)
between the original image $x$ and the regenerated image $G(x)$:
$\mathcal{L}_{G}=||x-G(x)||_{2}^{2}.$ (1)
It was expected that the model $G$ would recover most of the image but not the
anomalous bone regions, and therefore, the squared differences between the
original image and the inpainted image $\mathrm{E}(G(x))=(x-G(x))^{2}$ were
used to highlight the anomalies. The region with a larger difference is more
likely to be an anomalous region.
### 3.2 Anomaly detection using the original images
A main limitation of the above model $G$ is that it requires a segmentation
mask to start with, in order to generate input for the network. Therefore,
another network $A$ was trained, which takes the original images (without any
masking) as its input and the output images of $G$ as its target output. The
network architecture of $A$ was the same as $G$, and the image compressor $C$,
trained previously with $G$, was added here as well. Figure 2(b) shows the
overall structure of the model.
The loss function for training $A$ was the mean squared error (MSE) between
the output from the previous network $G(x)$ and the output from the current
network $A(x)$, plus the MSE between the respective error images
$\mathrm{E}(G(x))=(x-G(x))^{2}$ and $\mathrm{E}(A(x))=(x-A(x))^{2}$ to further
guide the model:
$\mathcal{L}_{A}=||G(x)-A(x)||_{2}^{2}+||\mathrm{E}(G(x))-\mathrm{E}(A(x))||_{2}^{2}.$
(2)
Since the outputs of $G$ were used as the target output, it was expected that
the model $A$ would produce outputs that are very similar to the outputs from
$G$, but it has the main advantage of not requiring segmentation masks to
generate the input. This model can be used to detect bone anomalies when given
the original images only. Again, the squared differences between the original
image and the generated image $\mathrm{E}(A(x))=(x-A(x))^{2}$ were used to
highlight the bone anomalies.
### 3.3 Downstream task: Anomaly-aware segmentation
(a) (b)
Fig. 3: The anomaly-aware segmentation network $S$ based on (a) 3D U-Net and
(b) 3D CAN with deep supervision. The information extracted from the anomaly
detector $A$ was utilized to inform the segmentation of the distal femur and
proximal tibia from the knee MR images containing bone abnormalities.
To demonstrate the utility of the model $A$, the error images
$\mathrm{E}(A(x))$ were utilized to construct segmentation models which can
manage anomalies. The anomaly-aware mechanism is a generalized method that can
be applied to a CNN-based segmentation network. In this study, the proposed
method was tested on two different types of CNNs: 3D U-Net [5] and 3D CAN [6].
Figure 3(a) shows the 3D U-Net architecture, with 4 downsamplings and
upsamplings. Figure 3(b) shows the modified 3D CAN. Ideally, CAN would have no
downsampling, but it was impossible to use a reasonable number of filters (at
least 32) in the full-resolution (160 slices $\times$ 384 $\times$ 384) layers
due to the memory limitation of the graphics card, so two downsampling (and
upsampling) blocks were added as a trade-off; the CAN module is applied after
the two downsamplings. To help stabilize convergence, both networks were
modified with deep supervision (see Section 2) by producing secondary
segmentation maps at deeper levels of the network and combining them with the
final segmentation map via upsampling and element-wise summation.
Although the original U-Net [23] used the conventional categorical cross-
entropy as the loss function, a multi-class Dice loss is now often used in
medical image segmentation because it intrinsically addresses the class
imbalance problem commonly seen with medical images [10]. The multi-class Dice
loss function is defined as:
$\mathcal{L}_{DSC}=1.0-\frac{2}{|K|}\sum_{k\in
K}{\frac{\sum_{i}{u_{i,k}v_{i,k}}}{\sum_{i}{u_{i,k}}+\sum_{i}{v_{i,k}}}}.$ (3)
Here, $u$ is the softmax output of the network and $v$ is the one-hot encoded
ground truth segmentation map; $K$ is the number of classes, and $u_{i,k}$ and
$v_{i,k}$ denote the softmax output and ground truth label, respectively, for
class $k$ at voxel $i$.
While the conventional U-Net is able to produce plausible segmentation results
for most images without anomalies, it can easily fail when there are anomalies
in the images since anomalies were not taken into account. To address this
problem, the error images $\mathrm{E}(A(x))$ were added as an additional input
(Figure 3) along with an additional loss function. Inspired by the work of Nie
and Shen [17], which used a difficulty-aware attention mechanism, a focal
cross-entropy loss was added to the loss function, where the focal weights
were given by the error images:
$\mathcal{L}_{FCE}=-\sum_{i}{\sum_{k\in K}{F_{i}v_{i,k}\log{u_{i,k}}}},$ (4)
where $F=1.0+\beta\mathrm{E}(A(x))$ and $F_{i}$ denotes the focal weight at
voxel $i$. Here, a weighting factor of $\beta=99.0$ was used so that the
values of $F$ ranges from 1 to 100. (If $\beta=0.0$, Equation 4 would be the
same as the usual categorical cross-entropy.)
The total loss for the segmentation network was then:
$\mathcal{L}_{S}=\mathcal{L}_{DSC}+\alpha\mathcal{L}_{FCE},$ (5)
where $\alpha$ is another weighting factor. Here, $\alpha=10.0$ was used.
Using this loss, the segmentation network can be trained to pay more attention
to the bone voxels that were found to be anomalous by the network $A$ and
hence likely to be difficult for the segmentation network to classify. For a
segmentation task, it is assumed that segmentation masks are not available for
test images, so only the outputs from $A$ (not $G$), which did not require
segmentation masks, would be used for training and testing the segmentation
network.
Table 1: Summary of the datasets used in the current study. | OAI ZIB | OAI ZIB–UQ | OAI AKOA
---|---|---|---
Number of subjects | 507 | 20 | 24
Timepoints | baseline | baseline | baseline & last follow-up
Manual segmentations | FB, FC, TB, TC | FB, FC, TB, TC, PB, PC, FL, TL PL
Used for | training $G$ | training $S^{T}$ | testing $G$ & $A$
training & testing (CV) $A$ and $S$ | training & testing (CV) $S^{T}$
FB: femoral bone; FC: femoral cartilage; TB: tibial bone; TC: tibial
cartilage; PB: patellar bone; PC:
patellar cartilage; FL: femoral lesion; TL: tibial lesion; PL: patellar
lesion; CV: cross-validation
In this paper, we will refer to our 3D U-Net and 3D CAN modified with deep
supervision and anomaly-aware mechanism as $UNET$-$S$ and $CAN$-$S$,
respectively.
## 4 Experiments
We trained and tested our method on subsets of knee MR images from the
publicly available Osteoarthritis Initiative (OAI) database [19]. Three
different subsets were used: OAI ZIB, OAI ZIB–UQ, and OAI AKOA (summarized in
Table 1).
### 4.1 MR image datasets
This study used the publicly available knee MR image dataset OAI ZIB,
generated by researchers at Zuse Institute Berlin (ZIB) [2]. The dataset
consists of 507 MR examinations from the OAI for which manual reference
segmentations of femoral and tibial bones and cartilages were produced by
experienced analysts starting from a model-based auto-segmentation. The images
are 160 sagittal slices $\times$ 384 $\times$ 384 voxels, and they are all
images of the right knee at baseline. The MR imaging sequence is 3D DESS
(double-echo steady state) with water excitation. Segmentation labels consist
of the background (0), femoral bone (1), femoral cartilage (2), tibial bone
(3), and tibial cartilage (4). (Background refers to all voxels that were not
labeled specifically.) The dataset covers the full spectrum of osteoarthritis
grades. Further details of the dataset can be found in Ambellan et al. [2].
The public OAI ZIB dataset is relatively large, but it has segmentation labels
for femoral and tibial bones and cartilages only. As a pilot study, we added
some additional segmentation labels to 20 MR examinations from the OAI ZIB
dataset with varying osteoarthritis severity. The additional segmentation
labels included patellar bone (5) and patellar cartilage (6). In addition,
bone marrow lesions (BMLs) or subchondral cysts for each bone—i.e. femoral
lesion (7), tibial lesion (8), and patellar lesion (9)—were also segmented. We
will refer to these 20 MR examinations as the OAI ZIB–UQ dataset.
A new dataset called OAI AKOA with 9 segmentation labels (10 including image
background) was also produced. The dataset contains images from the OAI
database with “accelerated” knee osteoarthritis, defined as patients who had
Kellgren–Lawrence (KL) grade $\leq$ 1 at baseline but progressed to KL grade
$\geq$ 2 within the follow-up period. The OAI AKOA dataset consists of 48 MR
examinations from 24 patients acquired at 2 timepoints, (baseline and last
imaging follow-up at a maximum of 96 months). The segmentations for the OAI
ZIB–UQ and OAI AKOA datasets were carried out manually using ITK-SNAP [28] by
an analyst (WB) with supervision from another analyst (CE) with expertise in
segmentation of the human musculoskeletal system.
The manual segmentations in the OAI ZIB–UQ and OAI AKOA datasets as well as
the cross-validation data splits (vide infra) are released in
https://github.com/wooboyeong/Anomaly-Aware-3D-Segmentation. Ethics approval
for collection of human data was obtained by the OAI and the participating
clinical sites [19].
### 4.2 Experimental setup
Fig. 4: Transfer learning for further segmentation of knee MR images. Here,
$N$ refers to the number of MR examinations while $K$ refers to the number of
segmentation classes (including the background) in the dataset. See Table 1
for the list of segmentation classes. Fig. 5: Example outputs from the bone
anomaly detection networks $G$ and $A$. Figures (a) and (b) are images from
the OAI ZIB dataset, and Figure (c) is an image from the OAI AKOA dataset. The
last column shows the error images (color-mapped and overlaid on the input
images) highlighting the difference between the input image and the output
from $A$. Regions of BMLs (blue arrows) in the femur and patella and part of
an osteophyte (yellow arrow) on the femur had high reconstruction errors. Fig.
6: Example segmentation outputs for the femoral and tibial bones (purple) and
cartilages (yellow) generated by the individual network implementation with
the OAI ZIB dataset. The examples show (a) images with little to no visible
bone anomalies where all networks produced good segmentation masks and (b,c)
images with visible bone anomalies where segmentation networks tended to fail
to produce plausible segmentation masks. The anomaly-aware networks,
especially $UNET$-$S$, were better able to correctly segment the images with
anomalies.
To evaluate the anomaly-aware segmentation method, the segmentation networks
$UNET$-$S$ and $CAN$-$S$ were compared with the standard 3D U-Net and 3D CAN
without any modification, as well as the models with deep supervision only
(without the anomaly-aware mechanism). The loss function for these networks
was multi-class Dice loss only and there was no additional input. The network
architecture and the training setup were otherwise the same as $UNET$-$S$ and
$CAN$-$S$. For convenience, the models without deep supervision will be
referred to as $UNET$ and $CAN$, and those with deep supervision will be
referred to as $UNET+$ and $CAN+$. See A for implementation and training
details.
Table 2: Mean DSC, ASD, and HD values for segmentations of the femoral and tibial bone and cartilage volumes from the proposed anomaly-aware method ($UNET$-$S$ and $CAN$-$S$) with their baseline networks, evaluated using 5-fold cross-validation on the OAI ZIB dataset ($N$ = 507). Class | Metric | UNET | UNET+ | UNET-S | CAN | CAN+ | CAN-S
---|---|---|---|---|---|---|---
| DSC (%) | 98.6$\pm$0.33 | 98.7$\pm$0.33 | 98.7$\pm$0.30 | 98.6$\pm$0.37† | 98.6$\pm$0.35† | 98.7$\pm$0.30
| ASD (mm) | 0.23$\pm$0.06 | 0.24$\pm$0.08† | 0.22$\pm$0.05 | 0.25$\pm$0.14†‡ | 0.25$\pm$0.09†‡ | 0.23$\pm$0.07
FB | HD (mm) | 7.05$\pm$6.01†‡ | 9.96$\pm$7.06†‡ | 3.40$\pm$2.67 | 10.07$\pm$7.53†‡ | 11.33$\pm$7.67†‡ | 4.22$\pm$3.96
HD∗ (mm) | 55.54$\pm$35.29†‡ | 24.18$\pm$19.96†‡ | 4.05$\pm$5.94 | 71.78$\pm$26.49†‡ | 28.28$\pm$19.94†‡ | 5.82$\pm$8.36
| HD0 (mm) | 5.97$\pm$5.39†‡ | 9.46$\pm$7.33†‡ | 2.82$\pm$1.63 | 8.98$\pm$7.14†‡ | 10.24$\pm$7.76†‡ | 3.46$\pm$2.71
| HD1 (mm) | 7.74$\pm$6.29†‡ | 10.29$\pm$6.87†‡ | 3.78$\pm$3.10 | 10.76$\pm$7.71†‡ | 12.03$\pm$7.54†‡ | 4.71$\pm$4.52
| DSC (%) | 89.6$\pm$2.79‡ | 89.7$\pm$2.91‡ | 89.5$\pm$2.67 | 89.2$\pm$2.71 | 89.3$\pm$2.69 | 89.0$\pm$2.46
| ASD (mm) | 0.26$\pm$0.07 | 0.26$\pm$0.07‡ | 0.26$\pm$0.07 | 0.27$\pm$0.07† | 0.27$\pm$0.07 | 0.27$\pm$0.07
FC | HD (mm) | 7.44$\pm$4.49†‡ | 6.71$\pm$3.99†‡ | 5.29$\pm$2.39 | 8.46$\pm$4.89†‡ | 7.17$\pm$4.15†‡ | 5.57$\pm$2.63
HD∗ (mm) | 14.38$\pm$14.85†‡ | 14.06$\pm$15.04†‡ | 5.58$\pm$3.69 | 34.82$\pm$30.24†‡ | 16.74$\pm$16.18†‡ | 6.05$\pm$4.31
| HD0 (mm) | 6.51$\pm$4.25†‡ | 5.59$\pm$3.06† | 4.52$\pm$1.60 | 7.48$\pm$4.59†‡ | 5.92$\pm$3.51†‡ | 4.85$\pm$1.95
| HD1 (mm) | 8.04$\pm$4.54†‡ | 7.43$\pm$4.35†‡ | 5.78$\pm$2.68 | 9.09$\pm$4.98†‡ | 7.98$\pm$4.33†‡ | 6.03$\pm$2.89
| DSC (%) | 98.7$\pm$0.35 | 98.7$\pm$0.33 | 98.7$\pm$0.32 | 98.6$\pm$0.35† | 98.6$\pm$0.36† | 98.7$\pm$0.33
| ASD (mm) | 0.22$\pm$0.10 | 0.22$\pm$0.06 | 0.21$\pm$0.06 | 0.23$\pm$0.08† | 0.24$\pm$0.10†‡ | 0.22$\pm$0.08
TB | HD (mm) | 6.29$\pm$5.72†‡ | 7.03$\pm$5.23†‡ | 3.23$\pm$1.86 | 9.26$\pm$7.03†‡ | 10.09$\pm$7.23†‡ | 3.85$\pm$3.16
HD∗ (mm) | 76.04$\pm$45.14†‡ | 27.63$\pm$26.23†‡ | 3.82$\pm$5.63 | 65.28$\pm$37.62†‡ | 37.06$\pm$24.76†‡ | 5.90$\pm$10.58
| HD0 (mm) | 5.59$\pm$4.97†‡ | 5.85$\pm$4.37†‡ | 3.01$\pm$1.82 | 8.44$\pm$6.81†‡ | 9.94$\pm$7.64†‡ | 3.12$\pm$1.94
| HD1 (mm) | 6.73$\pm$6.11†‡ | 7.78$\pm$5.59†‡ | 3.38$\pm$1.87 | 9.78$\pm$7.13†‡ | 10.19$\pm$6.96†‡ | 4.32$\pm$3.67
| DSC (%) | 85.9$\pm$4.21 | 85.8$\pm$4.16 | 86.0$\pm$4.00 | 85.3$\pm$4.11 | 85.2$\pm$4.24† | 85.3$\pm$4.17
| ASD (mm) | 0.27$\pm$0.10 | 0.27$\pm$0.10 | 0.26$\pm$0.09 | 0.28$\pm$0.10† | 0.28$\pm$0.10 | 0.27$\pm$0.11
TC | HD (mm) | 5.83$\pm$3.23†‡ | 5.43$\pm$2.83† | 4.74$\pm$2.03 | 7.00$\pm$4.15†‡ | 6.09$\pm$3.36†‡ | 4.89$\pm$2.20
HD∗ (mm) | 11.47$\pm$16.26†‡ | 8.75$\pm$12.13†‡ | 4.74$\pm$2.03 | 16.38$\pm$17.96†‡ | 14.23$\pm$17.93†‡ | 4.99$\pm$2.81
| HD0 (mm) | 4.70$\pm$2.18 | 4.69$\pm$2.29 | 4.19$\pm$1.40 | 6.01$\pm$3.42†‡ | 5.22$\pm$2.78†‡ | 4.32$\pm$1.52
| HD1 (mm) | 6.56$\pm$3.58†‡ | 5.91$\pm$3.04† | 5.09$\pm$2.27 | 7.63$\pm$4.45†‡ | 6.65$\pm$3.58†‡ | 5.26$\pm$2.48
Bold with underline represents the best value within each metric. Bold without
underline represents the second best value.
HD∗ refers to Hausdoff distances before post-processing. All other metrics
refer to results after post-processing.
HD0 refers to HDs (after post-processing) for cases with mild to no
osteoarthritis (radiographic grade $\leq$ 2; $N_{0}$ = 198).
HD1 refers to HDs (after post-processing) for cases with moderate to severe
osteoarthritis (radiographic grade $\geq$ 3; $N_{1}$ = 309).
† represents significant difference (p-value $<$ 0.05) compared to UNET-S with
Tukey’s HSD test.
‡ represents significant difference (p-value $<$ 0.05) compared to CAN-S with
Tukey’s HSD test.
The OAI ZIB dataset was randomly and evenly split into 5-fold cross-validation
sets, i.e. 102/102/101/101/101 MR examinations in each test set, and the split
was maintained over all parts of Components 2 ($A$) and 3 ($S$) in Figure 1,
as well as over all different segmentation models being evaluated. ($S$ refers
to either $UNET$–$S$ or $CAN$–$S$.) Segmentation performance was evaluated
using Dice similarity coefficient (DSC), average surface distance (ASD), and
Hausdorff distance (HD) values. See B for the definitions of these evaluation
metrics.
### 4.3 Transfer learning for further segmentation
As an extension to the above experiments, the segmentation networks were
trained further with the expanded number of segmentation labels in the OAI
ZIB–UQ and OAI AKOA datasets. Since these datasets are relatively small, the
segmentation networks were first trained on the entire OAI ZIB dataset, and
then the learned weights were transferred to new networks (Figure 4). The new
networks have 10 channels in the output layer instead of 5 since there are now
10 segmentation classes (including image background). The new networks will be
denoted with the superscript T.
The Dice loss function (see Equation 3) was modified so that higher weights
were given to the new classes that the network now has to learn:
$\mathcal{L}_{wDSC}=\frac{1}{|K|}\sum_{k\in
K}w_{k}\left(1.0-\frac{2\sum_{i}{u_{i,k}v_{i,k}}}{\sum_{i}{u_{i,k}}+\sum_{i}{v_{i,k}}}\right),$
(6)
where $w_{k}=1.0$ for the first 5 classes that were already learned, and
$w_{k}=10.0$ for the new 5 classes. In addition, since many images do not
contain lesions, $w_{k}$ was changed to 0 if
$\sum_{i}{u_{i,k}}+\sum_{i}{v_{i,k}}=0.0$; this means the image did not
contain label $k$ and the network successfully predicted that there is no
label $k$, so the loss in this case would be 0. The loss function for the
baseline networks ($UNET^{T}$, $UNET+^{T}$, $CAN^{T}$, $CAN+^{T}$) was this
weighted multi-class Dice loss.
The focal cross-entropy loss for the anomaly-aware networks (see Equation 4)
was the same as before. Therefore, the total loss for $UNET$-$S^{T}$ and
$CAN$-$S^{T}$ was:
$\mathcal{L}_{S^{T}}=\mathcal{L}_{wDSC}+\alpha\mathcal{L}_{FCE}.$ (7)
The new networks were then trained on the entire OAI ZIB–UQ dataset to learn
to segment the 10 classes. Lastly, in order to test the pipeline $A\rightarrow
S^{T}$, the anomaly information for images in the OAI AKOA dataset was
obtained by running the images through the network $A$ that was trained on the
OAI ZIB dataset. Since $A$ only requires the original images for
reconstruction, it did not need any further training or transfer learning.
However, since the segmentation networks only had 20 images (the OAI ZIB–UQ
dataset) for learning the new labels, it required some further training, so
the OAI AKOA dataset was randomly and evenly split into 5-fold cross-
validation sets on a patient basis, i.e.
5$\times$2/5$\times$2/5$\times$2/5$\times$2/4$\times$2 testing images in each
set. Due to the limited size of the datasets, online data augmentation was
applied to reduce overfitting.
Figure 4 shows a summary of the transfer learning process. For all of the
networks, weights for the first two convolutional layers were frozen during
training (see Figure 10 in the Appendix). See A for implementation and
training details.
## 5 Results
### 5.1 Anomaly detection
Figure 5 shows example output images from the anomaly detection networks $G$
and $A$. The input MR images of the knee have some visible bone anomalies
including BMLs and osteophytes. The network outputs are lossy reconstructions
of the input images with the bright signal bone anomalies within the
cancellous bone mostly removed from the images. Some of the osteophytes were
incompletely reconstructed. The network $A$ only had the original images as
inputs, but the outputs from $A$ still have most of the anomalies blurred out.
The last column of Figure 5 shows the reconstruction error images from $A$ in
which the anomalous regions detected within the cancellous bones are
highlighted.
Note that the MR image in Figure 5(c) is from the OAI AKOA dataset, which was
not used for training either $G$ or $A$, but only the masks of femur and tibia
were used to generate input for $G$ since the model was only trained with the
OAI ZIB dataset. However, since the masks had been dilated before erasing the
images (see Section 3.1), the subchondral patellar lesions were also detected,
as can be seen in Figure 5(c). Also note that masks are unnecessary for $A$ in
any case.
### 5.2 Anomaly-aware segmentation on OAI ZIB dataset
Figure 6 shows example outputs from the anomaly-aware segmentation networks
$UNET$-$S$ and $CAN$-$S$ and also the outputs from their baseline networks. As
noted in Section 2, the CNNs performed well in terms of DSCs for most of the
OAI ZIB images (Figure 6(a)), but failed on some cases with severe
abnormalities. The anomaly-aware method was found to be more robust against
these difficult cases, resulting in a noticeable improvement in the quality of
the segmentation of bone volume for the femur and tibia (Figures 6(b) and
6(c)). The image in Figure 6(b) was the most difficult from the OAI ZIB
dataset for the segmentation CNNs due to the presence of a large femoral BML,
but the segmentation error was fixed with $UNET$-$S$ and partially fixed with
$CAN$-$S$. The image in Figure 6(c) also has a notable femoral BML, but both
$UNET$-$S$ and $CAN$-$S$ were able to correctly segment the femoral bone.
Table 2 shows the quantitative results for the segmentation task. Tukey’s
honestly significant difference (HSD) test was used to compare all possible
pairs of means for each metric; this test is is similar to the t-test, except
it corrects for family-wise error rate. There was not a significant
improvement in the mean DSCs, but HDs were substantially reduced for all of
the segmentation classes for both $UNET$-$S$ and $CAN$-$S$ compared to their
baselines. The HDs for cases with moderate to severe osteoarthritis (HD1 in
Table 2) were higher than the HDs for cases with mild to no osteoarthritis
(HD0) for all models, but the HD1 was still relatively low with the anomaly-
aware models compared to the other models.
Note that the DSCs, ASDs, and HDs were calculated after post-processing the
CNN outputs. The post-processing was performed because most of the CNN outputs
for non-anomaly-aware models contained random stray voxels which resulted in
extremely high HDs (see HD∗ in Table 2). As mentioned in Section 2, this is
one of the main limitations of CNN-based segmentations. The random stray
voxels were removed by extracting the largest components and removing smaller
objects that are distant from the main structures. See C for description of
the post-processing method. The mean HDs were significantly reduced after
post-processing for the non-anomaly-aware models. The CNN outputs from the
anomaly-aware models did not contain as many stray voxels to start with, so
the effect of post-processing was much less obvious.
Even after the post-processing, the mean HDs for the anomaly-aware models were
significantly less than the mean HDs for the non-anomaly-aware models. See
also the boxplots in Figure 7 to assess the distributions of HDs (calculated
after post-processing). It can be seen that the HDs for the anomaly-aware
method are much smaller overall.
Fig. 7: Boxplots of Hausdorff distance (HD) values for the proposed anomaly-
aware segmentation approach ($UNET$-$S$ and $CAN$-$S$) and baseline networks,
evaluated on the OAI ZIB dataset using 5-fold cross-validation. Note that
these HDs are results after post-processing.
### 5.3 Anomaly-aware segmentation on OAI AKOA dataset with transfer learning
Fig. 8: Example outputs from the segmentation networks for images from the OAI
AKOA dataset, segmenting the patella and visible bone lesions in addition to
the femur and tibia. Note that the images are zoomed in to view the patella
more closely. The masks are overlaid on the input images (purple: bones;
yellow: cartilages; red: bone lesions). (a) When the images had no visible
anomalies, all networks except $UNET^{T}$ produced good segmentation of the
patella. The network $UNET^{T}$ failed to converge for the patellar cartilage
label. (b,c) The anomaly-aware networks were able to detect and segment most
of the visible lesions along with the anatomical structures on images that the
other segmentation networks had difficulty with.
Figure 8 shows example outputs from the anomaly-aware segmentation networks
$UNET$-$S^{T}$ and $CAN$-$S^{T}$ for the OAI AKOA test sets and also the
outputs from their baseline networks with transfer learning. As with Section
5.2, images with noticeable abnormalities tended to be more difficult to
segment. For images with no visible anomalies, all networks except $UNET^{T}$
produced good segmentation of the patella (Figure 8(a)). The network
$UNET^{T}$ did not converge for the patellar cartilage label for any of the
training sets in 5-fold cross-validation. For images with visible
abnormalities, $UNET$-$S^{T}$ and $CAN$-$S^{T}$ correctly detected most of the
visible lesions and produced acceptable segmentations while the other networks
occasionally failed (Figures 8(b) and 8(c)).
Table 3 shows the quantitative results for segmentation of the bones and
cartilages. Again, there were statistically significant improvements in HDs
for the anomaly-aware method compared to the baseline. In this case, the DSCs
of the femoral bone and tibial bone volumes were also significantly improved.
Note that the same post-processing method as in Section 5.2 was applied here
as well. The boxplots in Figure 9 shows the distributions of HDs (calculated
after post-processing). It can be observed that the HDs for the anomaly-aware
method are much smaller than their baseline networks.
Table 3: Mean DSC, ASD, and HD values for segmentation of the femoral, tibial, and patellar bone and cartilage volumes from the proposed anomaly-aware method ($UNET$-$S^{T}$ and $CAN$-$S^{T}$) with their baseline networks with transfer learning, evaluated using 5-fold cross-validation on the OAI AKOA dataset ($N$ = 24$\times$2). Class | Metric | UNETT | UNET+T | UNET-ST | CANT | CAN+T | CAN-ST
---|---|---|---|---|---|---|---
| DSC (%) | 96.9$\pm$1.63†‡ | 97.3$\pm$1.04†‡ | 98.4$\pm$0.41 | 96.5$\pm$1.75†‡ | 97.0$\pm$1.30†‡ | 98.4$\pm$0.55
FB | ASD (mm) | 0.55$\pm$0.39†‡ | 0.47$\pm$0.26 | 0.24$\pm$0.08 | 0.97$\pm$0.91†‡ | 0.54$\pm$0.36†‡ | 0.26$\pm$0.12
HD (mm) | 16.36$\pm$9.49† | 18.21$\pm$6.56†‡ | 8.96$\pm$6.08 | 41.78$\pm$22.94†‡ | 15.84$\pm$8.06† | 10.24$\pm$7.62
| HD∗ (mm) | 41.16$\pm$24.56†‡ | 34.00$\pm$15.91†‡ | 10.70$\pm$9.96 | 87.78$\pm$5.74†‡ | 30.71$\pm$21.26†‡ | 16.28$\pm$16.02
| DSC (%) | 85.3$\pm$3.13 | 85.9$\pm$2.67 | 86.6$\pm$2.48 | 84.5$\pm$3.10†‡ | 85.0$\pm$3.22 | 86.4$\pm$2.53
FC | ASD (mm) | 0.32$\pm$0.07†‡ | 0.29$\pm$0.05 | 0.26$\pm$0.04 | 0.34$\pm$0.10†‡ | 0.31$\pm$0.06†‡ | 0.26$\pm$0.04
HD (mm) | 19.75$\pm$5.58†‡ | 9.43$\pm$5.17†‡ | 5.73$\pm$2.25 | 7.64$\pm$3.66 | 10.41$\pm$4.64†‡ | 6.09$\pm$3.96
| HD∗ (mm) | 29.98$\pm$16.98†‡ | 14.07$\pm$10.51† | 5.73$\pm$2.25 | 15.99$\pm$17.43†‡ | 20.52$\pm$15.33†‡ | 7.22$\pm$6.70
| DSC (%) | 97.4$\pm$0.93†‡ | 97.6$\pm$0.73†‡ | 98.4$\pm$0.31 | 97.4$\pm$1.04†‡ | 97.3$\pm$1.40†‡ | 98.3$\pm$0.31
TB | ASD (mm) | 0.51$\pm$0.27†‡ | 0.37$\pm$0.14 | 0.24$\pm$0.10 | 0.42$\pm$0.20†‡ | 0.46$\pm$0.58†‡ | 0.24$\pm$0.08
HD (mm) | 26.50$\pm$17.89†‡ | 12.55$\pm$7.12 | 6.98$\pm$7.13 | 16.61$\pm$8.97†‡ | 9.64$\pm$6.48 | 7.06$\pm$5.42
| HD∗ (mm) | 107.47$\pm$6.61†‡ | 46.55$\pm$23.83†‡ | 7.96$\pm$7.96 | 99.87$\pm$21.99†‡ | 39.56$\pm$25.70†‡ | 15.94$\pm$17.27
| DSC (%) | 84.5$\pm$4.24 | 84.7$\pm$3.66 | 85.0$\pm$3.16 | 83.2$\pm$4.55 | 83.6$\pm$4.50 | 84.7$\pm$3.44
TC | ASD (mm) | 0.31$\pm$0.14 | 0.30$\pm$0.11 | 0.28$\pm$0.11 | 0.43$\pm$0.74 | 0.32$\pm$0.13 | 0.29$\pm$0.12
HD (mm) | 11.08$\pm$7.35†‡ | 6.07$\pm$3.47 | 4.84$\pm$2.53 | 13.59$\pm$6.91†‡ | 5.66$\pm$2.97 | 5.19$\pm$2.36
| HD∗ (mm) | 30.21$\pm$23.06†‡ | 19.61$\pm$18.21†‡ | 4.84$\pm$2.53 | 36.27$\pm$21.11†‡ | 12.97$\pm$17.77 | 9.19$\pm$13.59
| DSC (%) | 96.0$\pm$1.29 | 96.2$\pm$0.86 | 96.6$\pm$0.80 | 96.0$\pm$1.48 | 95.8$\pm$1.29† | 96.3$\pm$1.02
PB | ASD (mm) | 0.33$\pm$0.12 | 0.30$\pm$0.07 | 0.26$\pm$0.07 | 0.41$\pm$0.59 | 0.42$\pm$0.31 | 0.32$\pm$0.18
HD (mm) | 10.22$\pm$9.78† | 8.27$\pm$5.86† | 3.69$\pm$1.97 | 7.64$\pm$7.17 | 11.86$\pm$8.50†‡ | 7.43$\pm$6.69
| HD∗ (mm) | 78.45$\pm$20.18†‡ | 75.03$\pm$19.35†‡ | 10.33$\pm$21.00‡ | 94.30$\pm$27.55†‡ | 72.00$\pm$26.59†‡ | 29.04$\pm$34.58†
| DSC (%) | 0.0$\pm$0.00§ | 85.1$\pm$5.48 | 85.7$\pm$5.02 | 83.9$\pm$6.02 | 84.5$\pm$5.43 | 84.5$\pm$6.15
PC | ASD (mm) | N/A§ | 0.29$\pm$0.11 | 0.27$\pm$0.08 | 0.33$\pm$0.13 | 0.30$\pm$0.12 | 0.31$\pm$0.12
HD (mm) | N/A§ | 4.20$\pm$3.90 | 3.49$\pm$2.36 | 4.93$\pm$3.99 | 5.13$\pm$4.44 | 4.29$\pm$3.70
| HD∗ (mm) | N/A§ | 24.23$\pm$30.61†‡ | 7.54$\pm$16.23 | 38.96$\pm$36.00†‡ | 28.89$\pm$33.77†‡ | 5.98$\pm$12.17
Bold with underline represents the best value within each metric. Bold without
underline represents the second best value.
HD∗ refers to Hausdoff distances before post-processing. All other metrics
refer to results after post-processing.
† represents significant difference (p-value $<$ 0.05) compared to UNET-ST
with Tukey’s HSD test.
‡ represents significant difference (p-value $<$ 0.05) compared to CAN-ST with
Tukey’s HSD test.
§ UNETT failed to converge, so it was excluded from Tukey’s HSD test (for the
PC label only).
Fig. 9: Boxplots of Hausdorff distance (HD) values for the proposed anomaly-
aware segmentation approach ($UNET$-$S^{T}$ and $CAN$-$S^{T}$) and baseline
networks with transfer learning, evaluated on the OAI AKOA dataset using
5-fold cross-validation. These HDs are results after post-processing. Note
also that $UNET^{T}$ failed to converge for the PC label, so it was excluded
from the plot.
Performance for segmentation of bone lesions was less straightforward to
evaluate because (a) there are a variable number of lesions (some images have
none while some images have many) and (b) some lesions are too small to be
reliably detected. In this study, we assumed “bone-wise” lesion detection for
simplicity, in which each bone was classified as either a positive or a
negative case:
* 1.
For positive cases (bone has a lesion), a prediction was considered to be
“true positive” if the segmentation DSC $\geq$ 5% and “false negative” if DSC
$<$ 5%.
* 2.
For negative cases (bone has no lesion), a prediction was considered to be
“true negative” if the network successfully predicted that there is no lesion
in the bone and “false positive” if the network predicted that there is one.
Note there are 3 bones (i.e. 3 “cases”) in each image: femur, tibia, and
patella. In the 48 images from the OAI AKOA dataset, there were 76 positive
cases and 68 negative cases in total.
Using these criteria, sensitivity (true positive rate; TPR) was $>$ 90% for
most models while specificity (true negative rate; TNR) was $<$ 50% (Table 4).
The specificity is low mainly due to the presence of subtle lesions in many
images. See Figure 11 in the Appendix showing example images with very small
lesions that often resulted in “false positives” or “false negatives”. In most
cases, the neural networks were highly sensitive and detected those small
lesions, resulting in high false positive rates. However, such lesions are not
consistently detected by human observers either and often are not clinically
important. Therefore, we applied various size thresholds to the lesion masks
to see if the specificity is higher for larger lesions. Using progressively
increasing thresholds up to 6.0 mm3, isolated lesions smaller than the
threshold were removed from the output masks, and the same post-processing was
applied to the manual masks for consistency. See C for details on the post-
processing for bone lesion detection.
Table 4 shows the bone lesion detection and segmentation performance of the
different models in terms of accuracy and mean DSC. By progressively
increasing the threshold, sensitivity was decreased but specificity was
increased. The highest accuracy achieved with the post-processing was 84.7%
using $CAN$-$S^{T}$. The highest mean DSC (segmentation performance for
positive cases only) was 53.6% using $UNET$-$S^{T}$. The area under the
receiver operating characteristic curve (AUC) was 0.896 for $CAN$-$S^{T}$ and
0.892 for $UNET$-$S^{T}$, both of which were higher than their baselines. Note
in Table 4 that $UNET^{T}$ did not converge for the patellar lesion label. Its
specificity is paradoxically high because it did not detect any patellar
lesion, but the accuracy is still low due to low sensitivity. Another notable
finding is that while $UNET$-$S^{T}$ performed the best in the segmentation
tasks overall (Tables 2 and 3), $CAN$-$S^{T}$ performed the best in the bone
lesion detection task (Table 4).
## 6 Discussion
### 6.1 Anomaly detection
The current study presented a method to apply 3D U-Net-based CNNs for visual
anomaly detection in volumetric medical images. There were two steps to this
end, involving two separate networks $G$ and $A$. Network $G$ (the first step)
could be sufficient if you already have the appropriate segmentation masks and
you are only interested in highlighting anomalies in the images. However,
segmentation masks are initially unavailable in most cases, so network $A$
(the second step) was added to carry out the same task without having to
obtain segmentation masks first. The last column of Figure 5 shows that the
anomaly detection network $A$ can visually highlight anomalies on MR images of
the knee.
These autoencoder-based networks also reconstruct the images with most of the
visible lesions removed and therefore, they can theoretically be used to show
what the images likely looked like if the bones had no lesions. The main
limitation for this is that the quality of the images reconstructed from the
current networks are rather poor. This is complex because it is actually easy
for convolutional autoencoders to reconstruct images with minimal
reconstruction error, but for anomaly detection, the reconstruction needs to
be lossy since the anomalies should be removed, which means we might need to
compromise on image quality. Nevertheless, a useful future work would include
improving the model to make the images look more anatomically realistic. A
possible approach would be to modify the image compressor, for example by
using a more sophisticated model such as a conditional encoder or vector
quantization [18].
### 6.2 Anomaly-aware segmentation
This study also presented a generalized method for CNNs in which the
information from the anomaly detectors is utilized to improve image
segmentation. Although the main purpose was to compare the segmentation
performance of the anomaly-aware networks to that of the baseline networks,
there were also some incidental findings from the experiments such as the
effect of deep supervision and the difference between U-Net and CAN.
First of all, it was demonstrated that the anomaly-aware mechanism is capable
of improving the segmentation of overall bone volume on MR images of
osteoarthritic knees with visible anomalies. The networks $UNET$ and $CAN$
were already capable of achieving high DSCs, but it was difficult to achieve
good HDs without any shape regularization or post-processing. While deep
supervision ($UNET+$ and $CAN+$) helped improve convergence and overall
performance, it was found that the anomaly-aware method ($UNET$-$S$ and
$CAN$-$S$) was able to further improve these for both U-Net and CAN. The
anomaly-aware attention mechanism provided a substantial improvement in HDs in
addition to a visible improvement in the quality of segmentation (Figures
6–9). Although none of the CNNs were completely error-proof, as can be seen
with the few outliers in HDs (Figures 7 and 9), the significant drop in mean
HDs with the anomaly-aware models indicates that these models make errors much
less frequently. This is probably because the additional information provided
to the network guided the network to focus its attention to the ROI, and the
model was less vulnerable to anomalies and noise in the images.
The differences between HD∗s before post-processing and HDs after post-
processing (Tables 2 and 3) are also notable. The post-processing decreased
HDs for most images for the non-anomaly-aware models, whereas it had an effect
only for some images for the anomaly-aware models. Indeed, the post-processing
made no difference in HDs for 94% of all images (524/555) for $UNET$-$S$ and
for 84% (464/555) for $CAN$-$S$. In contrast, the post-processing made no
difference in HDs for no images (0/555) for $UNET$ and $CAN$ and for 20%
(110/555) and 10% (58/555), respectively, for $UNET+$ and $CAN+$. The
inference time for the CNNs (without post-processing) was about 1 second per
image, with the post-processing taking additional 4 seconds per image.
An additional finding from Table 2 is that the U-Net-based models performed
slightly better than the CAN-based models in the segmentation of femur and
tibia on the OAI ZIB dataset. However, $UNET^{T}$ failed to learn the patellar
cartilage and patellar lesion labels on the OAI AKOA dataset (Tables 3 and 4)
despite using weighted Dice loss. Some possible reasons for the difficulty in
learning these labels include (a) not enough training images, (b) highly
imbalanced classes, (c) spatial heterogeneity, and (d) larger proportional
impact of BMLs in the patella. An interesting observation is that the other
models were still able to converge for all segmentation classes. Since the
only difference between $UNET^{T}$ and $UNET+^{T}$ was deep supervision, it
can be inferred that the deep supervision helped stabilize convergence. The
more stable convergence behavior in $CAN^{T}$ compared to $UNET^{T}$ was
initially unexpected but explainable; two possible explanations are as
follows:
1. 1.
The CAN-based models are relatively “simple”, having a much less number of
parameters than the U-Net-based models. For example, $CAN$ has 852,565
learnable parameters while $UNET$ has 5,887,765 learnable parameters. Having
many parameters is not necessarily helpful because it can increase the chance
of overfitting and the model might converge slower due to the model
complexity.
2. 2.
Since the CAN-based models used minimal downsampling, they might have been
better than $UNET^{T}$ at detecting very small structures such as the patellar
cartilage and lesions. The U-Net-based models, on the other hand, rely on
downsampling for aggregating multi-scale contextual information, which can be
advantageous when segmenting larger anatomical structures but less desirable
when having to analyze smaller structures. This is also consistent with the
finding that although $UNET$-$S^{T}$ performed the best as a segmentation
method, $CAN$-$S^{T}$ gave the best performance when used as a bone lesion
detection method (Table 4). Indeed, all CAN-based methods performed better
than the U-Net-based methods on average in terms of accuracy and AUC.
Table 4: Bone lesion detection and segmentation performance on the OAI AKOA dataset in terms of accuracy and mean DSC. Here, Acc. refers to the accuracy with no post-processing while $\lceil$Acc.$\rceil$ refers to the highest accuracy achieved with post-processing. Both are reported with the corresponding sensitivity (TPR) and specificity (TNR). $\lceil$DSC$\rceil$ is the highest mean DSC (averaged over all bone lesions) achieved with post-processing. AUC is the area under the receiver operating characteristic (ROC) curve. Note that $UNET^{T}$ failed to converge for the patellar lesion label. Results for each bone can be found in the supplementary material. Model | Acc. (TPR, TNR) | $\lceil$Acc.$\rceil$ (TPR, TNR) | $\lceil$DSC$\rceil$ | AUC
---|---|---|---|---
UNETT | 0.597 (0.632, 0.559) | 0.729 (0.592, 0.882) | 0.349 | 0.713
UNET+T | 0.563 (0.921, 0.162) | 0.764 (0.750, 0.779) | 0.501 | 0.795
UNET-ST (Ours) | 0.681 (0.947, 0.382) | 0.819 (0.803, 0.838) | 0.536 | 0.892
CANT | 0.681 (0.921, 0.412) | 0.833 (0.750, 0.926) | 0.462 | 0.871
CAN+T | 0.611 (0.934, 0.250) | 0.806 (0.697, 0.926) | 0.464 | 0.874
CAN-ST (Ours) | 0.694 (0.921, 0.441) | 0.847 (0.855, 0.838) | 0.493 | 0.896
Bold with underline represents the best value within each metric.
Bold without underline represents the second best value.
Lastly, it can also be noted that the models with deep supervision
($UNET+^{T}$ and $CAN+^{T}$) did not outperform those without deep supervision
($UNET^{T}$ and $CAN^{T}$) in terms of detection accuracy; the models with
deep supervision had higher sensitivity but also lower specificity, so the
overall accuracy was not improved (Table 4). The benefit of the anomaly-aware
method is more apparent since the two models ($UNET$-$S^{T}$ and
$CAN$-$S^{T}$) performed better in terms of accuracy, mean DSC, as well as AUC
when compared to their baselines. The anomaly-aware models were able to
maintain both high sensitivity and specificity. This is likely because these
models already had information about where the lesions are likely to be
whereas the baseline networks had to learn the information from scratch.
According to these findings, our proposed method is expected to be helpful in
the analysis of medical images with visible anomalies, but the main limitation
is that the current pipeline is rather involved. In Figure 1, it can be noted
that the models $A$ and $S$ actually form a linear pipeline. Therefore, future
work could investigate combining $A$ and $S$ into a single multi-task model to
perform anomaly detection and segmentation simultaneously. In addition, since
we can detect and segment pathologies, a future work may also include another
downstream task such as classification of osteoarthritis grades. Osteophytes
are also a major feature of osteoarthritis, but detection of osteophytes was
not included in the current work because manual segmentation of osteophytes
was challenging. A method to evaluate osteophyte detection may help with the
development of the automated classification of osteoarthritis grades. Finally,
since the anomaly-aware approach is a generalized method for CNNs, one could
look into combining it with another CNN-based method such as nnU-Net [9] to
further enhance the model.
## 7 Conclusion
In summary, this work demonstrated how simple U-Net-like neural networks can
be used for detecting bone lesions in knee MR images through reconstruction
via inpainting. Moreover, it showed how the detected anomalies can be further
utilized for downstream tasks such as segmentation. The anomaly-aware networks
gave a better performance on average than their baseline networks in the
segmentation tasks as well as in the detection of bone lesions. The stable
convergence behavior and performance with the new labels in the OAI ZIB–UQ and
OAI AKOA datasets are promising and suggest that the proposed method has an
advantage when there are relatively few training images and/or the classes are
highly imbalanced. It is hoped that future works will show additional
improvements and further applications of the anomaly detection and anomaly-
aware segmentation models in medical imaging.
## Acknowledgments
We would like to thank Dr Jessica Bugeja for curating the OAI AKOA dataset.
## Appendix A Implementation details
All of the neural networks in this study were implemented using Tensorflow [1]
version 2.4 with Keras API (http://tensorflow.org/guide/keras) and were
trained on a high-performance computer with NVIDIA Tesla V100-SXM2-32GB.
The two anomaly detection networks $G$ and $A$ (Sections 3.1 and 3.2; Figure
2) were both based on 3D U-Net [5], consisting of a contracting path (encoder)
and an expansive path (decoder) with skip connections. Overall, the networks
had 5 levels of resolution with 4 progressive downsamplings followed by 4
progressive upsamplings using strided convolutions. In the contracting path,
there were two convolution blocks at each level, where each convolution block
was a 3D convolution layer with a kernel size of 3 $\times$ 3 $\times$ 3
followed by an instance normalization [25] and leaky rectified linear unit
(ReLU) activation function with a negative slope coefficient of 0.1. Instance
normalization was used because the large image size only allowed for a batch
size of 1. The number of feature maps was progressively increased as the
resolution was decreased.
(a)
(b)
Fig. 10: The anomaly-aware segmentation network $S^{T}$ for transfer learning
based on (a) 3D U-Net and (b) 3D CAN. The network $S^{T}$ is a slight
modification from $S$ (Figure 3) where 5 more channels were added to the
output layer. During training, the first two convolution blocks were frozen.
The image compressor $C$ was a very small network, again with 4 progressive
downsamplings, but with the number of feature maps progressively decreasing as
the resolution was decreased. To create a bottleneck, the downsampled image
was flattened, and then a dense layer with 100 units was applied. Then, after
another dense layer with 5760 units, the flattened image was reshaped into a
3D image of 10 $\times$ 24 $\times$ 24 to be added to the decoder part of the
U-Net.
In the expansive path of U-Net, the resolution was progressively recovered
using transposed convolutions, and the feature maps from the encoder at the
corresponding level were transferred using element-wise summation. The
combined feature maps then passed through two more convolution blocks before
another transposed convolution for upsampling. After reaching the original
image resolution, the final output convolution layer with a kernel size of 1
$\times$ 1 $\times$ 1 and sigmoid activation function was applied. The MR
images were Z-normalized (centered to 0 mean, unit standard deviation),
clipped at [-5, 5], and subsequently rescaled to [0, 1] for network input. The
models were trained using the Adam optimizer [13] with a learning rate of
0.0005 for 50 epochs for each training set.
For the downstream segmentation task (Section 3.3; Figure 3), two different
types of CNNs were used: 3D U-Net and 3D CAN. The architecture of the U-Net-
based models ($UNET$, $UNET+$ and $UNET$-$S$) was similar to $G$ and $A$,
except that the final activation function was softmax and that $UNET+$ and
$UNET$-$S$ were modified with deep supervision.
The CAN-based models ($CAN$, $CAN+$ and $CAN$-$S$) had 3 levels of resolution,
where the first two levels were basically the same as those of U-Net. After
two downsamplings, the CAN module was applied instead of further downsampling.
The CAN module consisted of convolution blocks with progressively increasing
dilation rates and then a non-dilated convolution block as the final block of
the CAN module. A convolution block was almost the same as that defined above
for U-Net except its dilation rates and kernel initializer. While non-dilated
convolution layers were initialized using the Glorot uniform initializer [8],
dilated convolutions were initialized with the identity initializer which was
found to be more effective for context aggregation [27]. After the CAN module,
the resolution was recovered with transposed convolutions and skip connections
as with U-Net, and then the final output layer was applied.
For the segmentation networks, the “mixed precision” policy in Keras API was
used to enable the use of a larger number of feature maps in 3D CNNs. Mixed
precision refers the use of 16-bit floating-point type in parts of the model
during training to make it use less memory. The output layer (softmax layer)
was kept in the 32-bit type for numerical stability. The MR images were
Z-normalized for network input, and the models were trained using the Adam
optimizer with a learning rate of 0.0005 for 50 epochs for each training set.
A few modifications were made to all segmentation networks for transfer
learning (Section 4.3; Figure 10). Firstly, 5 more channels were added to the
output layer to segment the additional classes. In addition, since the initial
layers in CNNs tend to extract general image features, weights for the first
two convolution blocks were frozen during training, and the rest of the
network was trained with a slightly lower learning rate of 0.00025 for 50
epochs for each training set. Due to the limited size of the datasets, online
data augmentation was applied to reduce overfitting. The data augmentation
consisted of random scaling (scaling factor within [0.9, 1.1]), random
rotation (angle within [-10, 10] degrees in a random direction), and random
translation (within [-10, 10] pixels in each direction).
## Appendix B Evaluation metrics
Segmentation performance was evaluated using Dice similarity coefficient
(DSC), which is defined as:
$DSC=\frac{2|B\cap A|}{|B|+|A|}.$ (8)
Here, $A$ and $B$ denote the set of positive voxels in the ground truth
segmentation map and the predicted segmentation map, respectively. DSC is a
volumetric measure usually expressed as a percentage. Although it is useful
for general assessment of the overall segmentation results, it provides
limited sensitivity to errors on the boundaries of the segmentation if the
segmented volume is large. We therefore also included surface distance
measures for the anatomical structures, to evaluate segmentation errors on
their boundaries. The average surface distance (ASD) and Hausdorff distance
(HD; also known as maximum surface distance) are defined as:
$ASD=\frac{1}{n_{\partial(A)}+n_{\partial(B)}}\left(\sum_{a\in\partial(A)}d(a,B)+\sum_{b\in\partial(B)}d(b,A)\right),$
(9)
$HD=\max\left(\max_{a\in\partial(A)}d(a,B),\max_{b\in\partial(B)}d(b,A)\right).$
(10)
Here, $\partial(\cdot)$ denotes the boundary of the segmentation set,
$n_{\partial(\cdot)}$ denotes the number of voxels on the boundary
$\partial(\cdot)$, and
$d(p,Q)=\min_{q\in\partial(Q)}||p-q||_{2}$ (11)
is the minimum distance from point $p$ to the boundary of a segmentation set
$Q$.
## Appendix C Post-processing of CNN outputs
For the bone and cartilage labels, random stray voxels in the CNN outputs were
removed by extracting the largest component for each label and removing
smaller components that are more than 50 voxels away from the largest
component. Simply extracting the largest component only (i.e. removing all
smaller components) was invalid because sometimes that also removed correctly
segmented voxels. For example, in some images with very severe cartilage
degeneration, there might be smaller fragments of cartilages, so removing all
smaller components sometimes makes the segmentation worse. Therefore, some
allowance had to be made for the stray voxels that are relatively close to the
main structures even though that means not all stray voxels could be removed.
The post-processing still removed most of the very extremely stray voxels,
significantly reducing mean HDs for the non-anomaly-aware models.
For the bone lesion labels, a different post-processing method was used since
there are a variable number of lesions and many of the lesions are quite small
in size. Figure 11 shows example images with very small lesions that often
resulted in “false positives” or “false negatives”, decreasing the specificity
of the segmentation networks for the bone lesion detection task (Section 5.3).
To see if the specificity is higher for larger lesions, various size
thresholds were applied to the lesion masks. Using progressively increasing
size thresholds of 0.0 (no post-processing), 0.5, 1.0, …, 6.0 mm3, isolated
lesions smaller than the threshold were removed from the output masks, with
the same post-processing applied to the manual masks for consistency.
Fig. 11: Example of (a) “false positive” and (b) “false negative” case, which
mostly occurred with very small lesions. The orange boxes highlight the small
lesions.
For $>$ 6.0 mm3 size threshold, some positive cases changed to negative cases,
paradoxically increasing the sensitivity of the detection method. (It means
all positive cases had an isolated lesion at least 6.0 mm3.) At 6.0 mm3
threshold, the specificities of the models ranged from 68–88%. In order to
complete the receiver operating characteristic (ROC) curves, the size
threshold was fixed at 6.0 mm3, and then the specificity of the detection
method was increased by increasing the softmax output threshold. The voxels
with softmax output (probability) less than the threshold were removed from
the output masks, using thresholds of 0.5, 0.6, 0.7, …, 1.0.
The areas under the ROC curves (AUCs) along with the highest accuracies and
the highest mean DSCs achieved with post-processing are reported in Table 4.
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# SGDraw: Scene Graph Drawing Interface Using Object-Oriented Representation
Tianyu Zhang
JAIST
Ishikawa,Japan
<EMAIL_ADDRESS>Xusheng Du
JAIST
Ishikawa,Japan
<EMAIL_ADDRESS>Chia-Ming Chang
The University of Tokyo
Tokyo,Japan
<EMAIL_ADDRESS>Xi Yang
Jilin University
Jilin,China
<EMAIL_ADDRESS>Haoran Xie
JAIST
Ishikawa,Japan
<EMAIL_ADDRESS>
###### Abstract
Scene understanding is an essential and challenging task in computer vision.
To provide the visually fundamental graphical structure of an image, the scene
graph has received increased attention due to its powerful semantic
representation. However, it is difficult to draw a proper scene graph for
image retrieval, image generation, and multi-modal applications. The
conventional scene graph annotation interface is not easy to use in image
annotations, and the automatic scene graph generation approaches using deep
neural networks are prone to generate redundant content while disregarding
details. In this work, we propose SGDraw, a scene graph drawing interface
using object-oriented scene graph representation to help users draw and edit
scene graphs interactively. For the proposed object-oriented representation,
we consider the objects, attributes, and relationships of objects as a
structural unit. SGDraw provides a web-based scene graph annotation and
generation tool for scene understanding applications. To verify the
effectiveness of the proposed interface, we conducted a comparison study with
the conventional tool and the user experience study. The results show that
SGDraw can help generate scene graphs with richer details and describe the
images more accurately than traditional bounding box annotations. We believe
the proposed SGDraw can be useful in various vision tasks, such as image
retrieval and generation.
## 1 Introduction
Scene graph is a common and popular way to describe scene understanding, first
proposed for image retrieval tasks to search for images with similar
descriptions in image datasets [8]. In addition, scene graphs are used for a
wide range of vision applications, such as image retrieval [8, 16], image
captioning [3], visual reasoning [17], visual answering [5], and robotics [1].
Among these previous works, the generation of scene graphs from images has
seldom been discussed. In this work, we focus on the user interface for
generating the scene graph.
The task of scene graph generation is to generate a corresponding graph-
structure representation from an image, and then to abstract the objects and
object relationships from the images. Scene graphs can facilitate the
understanding of scenes in images, especially complex ones. The conventional
approaches for scene graph creation have been mainly based on a manually
annotated image detection dataset, and the annotation tasks are usually time-
consuming and laborious. Currently, the visual genome dataset [10] is the most
common and popular dataset for scene graphs, with around $10^{5}$ annotated
images. However, the scene graph drawing with a bounding box may lose accuracy
due to missing detection results. To solve this issue, we aim to provide a
novel scene graph drawing interface with users in the loop.
Figure 1: Illustration of the structural unit (a) as an object-oriented
representation and (b) its example in this work. Taking two objects as an
example, the attributes as objects can be directly manipulated, and attached
to the objects with user operations. The relationships can then be constructed
among the objects.
In this work, we provide the interactive drawing user interface for scene
graphs using object-oriented representation, as shown in Figure 1, which can
achieve more comprehensive and detailed results than using the conventional
annotation with a bounding box. Therefore, we propose SGDraw, an object-
oriented approach for scene graph annotation by considering objects, object
relationships, and object attributes. SGDraw can take advantage of the user’s
cognitive ability in the drawing process from target images. To verify the
proposed interface, we conducted a comparison study between the previous
annotation method using bounding boxes and our proposed approach. The
evaluation results demonstrated the superiority of the proposed approach in
generating comprehensive and detailed scene graphs.
We list our main contributions as follows:
* •
We propose an object-oriented annotation approach of scene graphs, which
adopts the structural unit of the object, its properties, and the relationship
among objects. In this way, we can describe the image in detail and generate a
comprehensive and detailed scene graph.
* •
We design a web-based lightweight annotation tool that is easy to deploy and
operate so that the proposed SGDraw can generate scene graphs interactively.
* •
We conduct user experiments to verify the performance of our proposed object-
oriented annotation approach for annotation, especially the effectiveness of
annotation on complex images.
## 2 Related works
### 2.1 Scene Graph Generation
Earlier works on scene graph generation have constructed several datasets,
such as RW-SGD [8] and VRD [13]. Visual Genome (VG) [10] became the standard
scene graph dataset due to its large size and challenging nature. Scene graphs
can be generated in two different approaches [11]. The conventional methods of
scene graph generation adopt both object detection and pairwise predicate
estimation. The objects are first detected given a bounding box, and then the
predicates are predicted using conditional random fields [4, 8] or a
classification approach [9, 15]. The other approach is to jointly infer the
objects and their relationships based on the suggestions of object regions
[24]. These approaches focus on objects and their relations without attention
to object attributes that play a significant weight of the image description.
We consider that equalizing attributes over objects and their relations can
improve the quality of scene graph generation.
### 2.2 User Interface for Scene Graph
The scene graph can be drawn with an designed annotation tool with the
colleced data from Amazon’s Mechanical Turk [8]. This tool can obtain
instances of objects and their relationships by users drawing bounding boxes
to identify objects and describing relationships between pairs of objects
through text box input. GeneAnnotator [23] provided a semi-automatic
annotation tool for scene graphs, which provides rule-based relationship
recommendation algorithms that can reduce the annotation effort. SceneSketcher
[12] generated scene graphs from sketches to retrieve scene images satisfying
the user’s specific requirements via a freehand sketch. In this work, we aim
to provide the interactive drawing interface for scene graphs.
### 2.3 Interactive Drawing Interface
Interactive interface design can make drawing activities easy and effective to
common users. Dualface [7] proposed a drawing interface with global and local
guidance to enable novices to draw face portraits. DualSmoke [20] provided a
two-stage global-to-local generation framework for the interactive smoke
illustration design, which guides ordinary users without knowledge in the
field of fluid simulation to interactively complete various smoke illustration
designs through sketches. An interactive user interface [6] was proposed to
support sketch-based fluid design with a perceptual understanding of human
sketches. In contrast to these drawing interfaces, we focus on the user
interface for scene graphs.
### 2.4 Object-Oriented Interface
Object-oriented methods have been widely used in design tasks. An object-
oriented interface [14] was proposed to generate and manipulate polyhedral
data flow graphs. An integrated set of object-oriented pixel-based
vectorization algorithms [18] was then presented for various classes of
graphic objects in engineering drawings. These interfaces performed
interactions that were previously tedious or even impossible with a coherent
and consistent interaction experience throughout the entire interface. Next,
an object-oriented drawing approach [19] was proposed to represent attributes
as objects that can be manipulated directly. Considering the basic components
of the scene graph, the objects, object properties, and their relationships
could be manipulated directly by the user in this work.
## 3 Scene Graph Drawing
In this work, we aim to propose a drawing interface for scene graph annotation
and generation. The basic structural unit of the proposed approach is
represented in Figure 1.
Figure 2: Workflow of the proposed interface. (A) With a task image inputted
to the SGDraw, the user starts (B) adding objects, and then they can (C) add
attributes or (D) relationships. After that, they can choose to (E) use the
auxiliary operations (like cloning, as shown in the figure) to change the
structure of the graph. Repeat these steps until (F) obtaining the desired
scene graph.
### 3.1 Scene Graph
In this work, scene graph generation is achieved by objects without bounding
boxes, and we provide the open-ended and free-form manipulation of scene graph
nodes. The scene graph from an image is defined as follows by a tuple of
$(O^{\prime},R)$:
$P(G|I)=P(O^{\prime},R|I),$ (1)
where $I$ represents the input image, and $G$ is the desired target scene
graph containing objects with attributes $O^{\prime}$ and relations $R$.
Specifically, the image is partitioned into a set of objects
$O^{\prime}=\\{o^{\prime}_{1},...,o^{\prime}_{n}\\}$ corresponding to each of
the $n$ objects with attributes in the image. $R=\\{r_{1},...,r_{l}\\}$ is the
set of relations between pairs of objects.
$P(o^{\prime})=P(o,A),$ (2)
where $o^{\prime}$ is the target object with attributes to be generated,
containing the input target object $o$ and its attribute set $A$. Each object
comes with its unique set of attributes $A=\\{a_{1},...,a_{m}\\}$
corresponding to the $m$ attributes of the target object.
In this work, we handle the object and its attributes as a manipulable object
unit. By linking the pairwise relationships between different object units, we
can perform the task of generating the original image to the scene graph. In
contrast to the conventional scene graph generation task that focuses on the
objects and their relationships, we also focus on the object attributes that
are easily ignored and that accounts for a large proportion of the images. The
generated scene graph can be more consistent with the image content and
represent the image scene in more details.
### 3.2 Object-Oriented Representation
Inspired by the object-oriented drawing approach [19] [22], we aim to propose
an object-oriented representation of scene graphs for image annotation and
graph editing tasks. In this representation, we can extend attributes to
objects directly for manipulation, as shown in Figure 2. Specifically, the
proposed object-oriented approach has the following features:
Abstraction. The input images may contain much repetitive content, such as
many people. For repetitive objects, the concrete objects may contain similar
properties. Therefore, we can abstract them into a class, such as the class of
people. Through the process of class abstraction, we can reuse the scene graph
data to enhance the versatility and scalability of the proposed system.
Uniqueness. Each object has its unique identifier by which the corresponding
object can be located. The identity of the annotated objects will not change
during the whole task. For example, different human objects in the category
are given different identifiers, such as Man1 and Man2. By assigning the
identifiers to different objects of the same category, we can correspond scene
graph nodes to unique objects in the image to generate detailed and higher-
quality scene graphs.
Polymorphism. Polymorphism refers to the fact that objects have diverse
properties. Multiple different attributes can be assigned to the same object
at the same time, and the same attribute can be assigned to multiple different
objects at the same time. For example, in the case of several men and women,
different attributes like black color for clothes and blue color for pants can
be assigned to Man1, and attributes like short hair and black hair can be
assigned to Man2 and Woman1. The diversity of attributes allows each object to
correspond to the same or different attributes , which enhances the
flexibility and reusability of the system.
Inheritance. Inheritance refers to the ability to define and implement objects
on the basis of objects that already exist. The properties and relationships
associated with existing objects can be used as their own content, and new
properties and relationships can be added. Specifically, the user can select
objects to store after drawing the scene graph. All properties and
relationships of the selected object can be stored in the drop-down box in the
left corner of the interactive interface. In subsequent operations, the user
can select the object at any time to inherit all previously stored properties
and relationships. The use of inheritance provides a hierarchical structure
between objects, attributes, and relationships. Inheritance makes it possible
to share common features and increase the reusability of the system.
Figure 3: Screenshot of the SGDraw interface. On the left is the input image
display area, and on the right is the scene graph operation and real-time
generation area. Meanwhile, on the right side of the interactive interface, we
provide some common attributes and relationships for users to choose and
reference.
### 3.3 SGDraw Interface
In order to explore the effectiveness of the proposed SGDraw, we developed a
web-based user interface, as shown in Figure 3 (Please refer to the
introduction video of SGDraw demos111https://youtu.be/acy0SNLfahg). The
objects, attributes, and pairwise relationships between objects are considered
as objects that can be manipulated directly. We also provide an open and fully
free annotation tool that takes full advantage of the user’s cognitive ability
to perform the task of annotating any kind of images and generating their
corresponding scene graphs in real-time. In contrast to the previous approach
that requires drawing the bounding box on the image first and then describing
the objects inside the box, SGDraw can help users easily add/delete and drag
nodes depending on their own knowledge of the image content until a
satisfactory scene graph is generated. Specifically, we provide the SGDraw
with the following interaction features:
Figure 4: Examples of scene graphs completed by SGDraw with (a) an image
input, and (b) a completed scene graph. (c) shows the scene graph freely
designed from the text input “a couple drive to the forest to have a picnic.”
Direct operation of the scene graph. In previous interfaces [8, 23], the users
cannot observe the visualized scene graphs in real time and modify the input
content. To solve this issue, we consider the input of the scene graph as
drawing. We provide three kinds of nodes: red nodes for objects; yellow nodes
for relationships; blue nodes for attributes. The user can add a new object
and attribute by simply right-clicking on the background and then the object
respectively. A relationship node can be generated when clicking two different
objects. We also provided common attributes and relationships for reference,
which were derived from the common ranking in the scene dataset [21]. We
belive that common attributes and relationships can simplify users’ input and
facilitate creative thinking.
Various auxiliary operations. We designed the auxiliary operations to modify
the graph structure and simplify the input process, including removing,
cloning, undoing, zooming, collapsing, and dragging (see supplementary
material for details). We implemented the undo function to reduce the loss
caused by user error operations, and the removing function to help users
change the graph structure freely. For complex scene graph drawing, we
implemented the zooming and collapsing functions to help users understand the
overall graph structure. The dragging function allows users to optimize the
layout to achieve visual results that are easier to operate and observe. We
also added the cloning function for similar attributes and relationships to
increase the efficiency of the drawing process. These operations are directly
conducted on the scene graphs with real-time feedback to help users clearly
understand the changes in the scene graphs.
Flexible data archive. SGDraw has designed two ways to save the scene graph:
JSON files to facilitate subsequent research, and SVG files so that the visual
scene graphs becomes available to study and understand the scene images more
intuitively. For SVG files, users can draw the images directly and make the
image interactive by changing part of the code, and then inserting it into
HTML while viewing through the browser. SGDraw allows the users to load the
graph and modify it for further usage, which can improve the reusability and
maintainability of the scene graph data.
## 4 User study
In the user study, we confirmed SGDraw for scene graphs generations and
collected feedback from potential users about the effectiveness and usability
of the interface. We asked 14 participants (aged 25 to 30, eight males and six
females) who are graduate students with knowledge of machine learning.
Especially, we designed two experiments: a comparison study and a user
experience study. We reproduced the previous interface for scene graphs [8] in
the comparison study, and seven task images used in the experiments were
selected from the Visual Genome dataset [10]. The evaluation process consisted
of three stages:
1) Introduction and Training (10–15 minutes). We first introduced the
background of the scene graphs. The experimenter then guided the participants
to explore the functions and workflow of the previous interface and SGDraw,
and we intentionally provided a simple image for participants as an example.
This was done for two reasons: First, is ensured that participants had a full
understanding of the interfaces, and second, it allowed participants the
opportunity to explore and demonstrate understanding.
2) Experience and Usage (20–30 minutes). In order to avoid the proficiency
effect caused by the interface order for usage, participants were divided into
two groups. One group used the previous interface first and then used SGDraw.
The other group used SGDraw first and then used the previous interface. Each
participant was asked to draw the scene graphs in two interfaces based on two
different images. The system automatically counted time without user
knowledge. We provided seven images for participants to complete the scene
graph. When the participants thought they had finished describing the content
of the images, the experimenter collected the results.
We divided the results into seven tasks based on seven provided images. Each
task included four results for an image, two results from the SGDraw, and two
results from the previous interface. We collected the objects, attributes, and
relationships that participants described on the scene graphs. After
collecting the data, we calculated the instances (including objects,
attributes, and relationships) per minute for each interface to compare the
efficiency of the two interfaces.
3) Questionnaire and Interview (10–15 minutes). The participants next
completed questionnaires about the system. The questionnaires were composed of
a 5-point Likert scale (1 for strongly disagree and 5 for strongly agree) to
collect the participants’ experience of SGDraw. The questions on the first
questionnaire, based on the System Usability Scale(SUS), were used for the
usability measurement of an interface. The second questionnaire collected
usability feedback on every function. The interview then consisted of open-
ended questions that were asked to gain the users’ feedback on workflow,
utility, and usability.
## 5 Results
### 5.1 Generated Scene Graph
Figure 4 shows that SGDraw could help users complete the scene graph drawing
task. Without the given ground truth image, the users would use SGDraw to
express the image they desired. Users were not limited to using the scene
graphs in the dataset for subsequent tasks such as generation and retrieval.
In addition, compared with the results of previous research that did not
visualize [8] or whose layout was messy [23], the generated scene graphs in
SGDraw are constructed based on an automatic tree layout, which enhances the
readability and aesthetic of the users’ results. The automatic tree layout
helps but does not limits users. The results will be expanded automatically
and hierarchically, and users are allowed to change the graph structure with
any node or link. In the results, red nodes represent objects, blue nodes
represent attributes, and yellow nodes represent relationships. The nodes
classified by colors strengthen the user’s understanding of specific details
and further enhance the comprehensibility of the results on top of the clear
layout. The joint storage of the visual scene graphs and the text results are
more conducive to the user’s subsequent use and modification.
Figure 5: We conducted the comparison study on Pre-UI [8] and the propsed
SGDraw. (a) shows the Pre-UI that we reproduced, and (b) shows the
visualization result of the graph. (c) is the screen shot of SGDraw, and (d)
is the drawn result.
### 5.2 Comparison Study
Figure 6: Scene graph generation results from user study. One participant used
SGDraw to complete the complex scene graph for the task image by the auxiliary
tool. The similar attributes and relationships of objects in the image made
our interface facilitate the scene graph drawing process. Figure 7: The time
cost of tasks. The system counts time automatically and implicitly.
In the comparison study, participants are asked to draw scene graphs with the
previous interface (pre-UI) and SGDraw respectively, with two given images, as
shown in Figure 5. We recorded the number of objects, attributes, and
relationships and the time cost. As shown in Figure 7, SGDraw does not achieve
a time advantage on all drawings, but it does achieve the advantage of
instance quantity(including objects, attributes, and relationships.) as shown
in Figure 8. To verify efficiency, we calculated the number of instances that
could be completed for each task per minute. In all experiments, SGDraw
achieved the advantage of efficiency.
For Task 1, SGDraw had a larger time cost but better performance in instances
quantity (4.9 instances per minute, and 2.1 instances per minute for pre-UI).
As shown in Figure 6, the objects of the task have the same attributes and
relationships. After completing the scene graph of one person, the
participants only needed to use the cloning function with a few modifications
using SGDraw to complete the whole task. For Task 7, SGDraw achieved higher
scores in both time cost and instance quantity than pre-UI. After the
interview and analysis, we found that the recommendation part played a
significant role. For images with blurry objects, participants had to think
about and describe the scene for a long time. The common part helped
participants facilitate the input that similar attributes can be added. This
part enlightened the participants’ minds and reduced their thinking pressure.
Task 6 had the smallest difference in the efficiency of the two interfaces. In
this task, the image composition was simple and clear; thus, the participants
understood the main objects explicitly in the image. Based on less similar
attributes and relationships, scene graph was completely dependent on the
participants to finish it, so that SGDraw achieved fair efficiency for this
task.
### 5.3 User Experience Study
Workflow. Participants were interviewed about how the SGDraw integrated with
their workflow. Almost all responded positively; the specific feedback from
the participants included the following: “After learning the simple
operations, my workflow in this interface was fluid.” (P2). “In traditional
painting, the overall composition of the picture is firstly carried out to
determine the position of the objects. In addition, some people have the habit
of proceeding to the next object after describing one object. Either of these
two workflows can be well implemented in this interface, and both have
specific functional assistance.” (P5). Participants also pointed out that
there is some disadvantage to the SGDraw. “The system is smooth and can
complete tasks well, but a lot of use of the right click will be unfriendly to
people who are used to the left click.” (P10).
According to the interviews, SGDraw obtained a good performance in the
workflow. The participants agreed that SGDraw is smooth and helpful for users
with different backgrounds. We also found that few function operations may
cause discomfort for users with specific computer habits.
Figure 8: For each task, experimenters calculated the instances (including
objects, attributes, and relationships) in two interfaces. The number of
instances that are completed by the SGDraw is more than pre-UI [8].
Utility. Participants were also asked to rate the usefulness of each of the
functions. The functions of SGDraw received a high agreement, as shown in
Figure 9. All of the functions obtained more than four points, which verified
that the proposed functions could achieve simple operation and good user
experience. Among the various functions, SGDraw’s ability to delete the link
and node with their subtrees was strongly praised by participants. In
addition, the scores of some functions were underrated due to the limitations
of usage fields. For example, in the scenes with complex and repetitive
objects (such as stadiums, and parking), the cloning function always brought a
good experience to users. However, in other simple scenes, cloning was not so
important.
Figure 9: Result of five-point Likert-scale responses to “Do you agree that
technique is useful for your scene graph drawing tasks?”
Based on the various functions, that SGDraw could save and reuse scene graphs
was strongly favored by participants. Users could thus build their datasets in
turn to facilitate later operations. Some specific feedback included the
following: “I like the Cloning function; it frees me from repetitive typing
work. Being able to replicate attributes and relationships together is simply
awesome.” (P5). “The Removing function combines delete and rollback, which is
a sensible approach. I can easily do what I want, and reduce the cost of my
mistakes.” (P7). The functions in SGDraw have high approval that indicates
that the various functions are valued and desired. Users can achieve the
desired results through the functions we provided in SGDraw, which makes the
input process more convenient and quick.
Usability. Participants were asked to complete questionnaires about interface
usability. The questionnaire was designed with the question items using SUS
(System Usability Scale), which provides an overall usability assessment
measure consisting of 10 items. We use a five-Point Likert scale ( one for
strongly disagree and five for strongly agree).
Figure 10: The result of the SUS questionnaire.
As shown in Figure 10, SGDraw achieved a good performance with high scores in
positive items and low scores in negative items. Almost all participants
agreed that SGDraw is easy to use (Scored 4.36) and that most people learned
to use it quickly (Scored 4.57). They strongly disagreed that there were too
many inconsistencies in SGDraw (Scored 1.29) and many things to learn before
usage (Scored 1.57). The total SUS availability score of SGDraw was 78.9 out
of 100. Based on the results of previous research [2], this total score shows
that SGDraw should be judged to be acceptable, and the Adjective Rating should
be “excellent.”
All participants commented that the interface is intuitive and easy to learn.
The participants noted the following: “The common attributes part is great! It
helps me spread my mind when I don’t know what to write, and it’s usually
hidden without making me rely on it.” (P11). “No need for more knowledge, it
seems friendly for most people.” (P1). “It is interesting to operate directly
on the results. You can intuitively see the changes in the graphs. Compared
with the textual results, the graphs are always easier to understand.” (P4).
Several participants pointed out that more functions are needed, such as
selecting the common attributes by themselves: “Quite smart and consistent
functions it has created. For people with different tasks, the common
attributes required are also different. Perhaps it is a better way to let
users set the common attributes themselves.”(P6).
Based on the interviews, the visualization results and direct operations on
the graphs have received favorable reviews. We noted that the common area on
the right side of SGDraw sparked discussion. Participants agreed that it can
simplify operations and inspire users’ minds, but it should also be able to be
smarter. At the moment, we only display some common attributes fixedly.
However, different users need different common attributes. Allowing users to
change the common area by themselves will improve the interface’s
interactivity.
## 6 Conclusion
This work presented SGDraw, a scene graph drawing interface based on the
object-oriented method, to help users draw scene graphs succinctly and
conveniently. We designed a set of complete functions and presented common
tools to ease user operations. Finally, we conducted a user study to verify
our proposed interface. We found that the proposed interface can especially
work well with complex scene graphs.
For limitations of this work, the frequent right clicks may make users who are
accustomed to left clicks uncomfortable. In addition, the function design can
be improved to help users obtain free and better experiences. As future work,
we envision combining SGDraw with the scene graph generation algorithms, which
can help users semi-automatically complete the scene graph drawing.
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|
# AirCon: Over-the-Air Consensus for Wireless Blockchain Networks
Xin Xie,Cunqing Hua, Pengwenlong Gu, and Wenchao Xu, X. Xie and C. Hua are
with the School of Cyber Science and Engineering, Shanghai Jiao Tong
University, Shanghai 200240, China.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>P. Gu is with the
Department INFRES, LTCI, Telecom Paris, Institut Polytechnique de Paris,
France. E-mail<EMAIL_ADDRESS>Xu is with the Department of Computing, The Hong
Kong Polytechnic University, Hong Kong, China. E-mail:
<EMAIL_ADDRESS>
###### Abstract
Blockchain has been deemed as a promising solution for providing security and
privacy protection in the next-generation wireless networks. Large-scale
concurrent access for massive wireless devices to accomplish the consensus
procedure may consume prohibitive communication and computing resources, and
thus may limit the application of blockchain in wireless conditions. As most
existing consensus protocols are designed for wired networks, directly apply
them for wireless users may exhaust their scarce spectrum and computing
resources. In this paper, we propose AirCon, a byzantine fault-tolerant (BFT)
consensus protocol for wireless users via the over-the-air computation. The
novelty of AirCon is to take advantage of the intrinsic characteristic of the
wireless channel and automatically achieve the consensus in the physical layer
while receiving from the end users, which greatly reduces the communication
and computational cost that would be caused by traditional consensus
protocols. We implement the AirCon protocol integrated into an LTE system and
provide solutions to the critical issues for over-the-air consensus
implementation. Experimental results are provided to show the feasibility of
the proposed protocol, and simulation results to show the performance of the
AirCon protocol under different wireless conditions.
###### Index Terms:
Consensus protocol, Over-the-air computation, Wireless blockchain network,
Lattice coding
## 1 Introduction
The next-generation wireless networks are expected to provide ubiquitous
access to heterogeneous devices with ultra-high throughput, reliability, and
extremely low latency [1]. This will bring great challenges to the security
management in current 4G/5G mobile systems. Blockchain, a generic distributed
ledger technology (DLT), has received extensive attention due to the
significant advantages from decentralization, immutability, and security in
recent years. The blockchain technology enables registering and updating
transactions in a decentralized fashion via consensus among participants,
which has become the foundation of new security architecture for future
wireless networks.
The consensus protocol plays an important role in the blockchain system. The
efficiency of the consensus protocol determines blockchain system security
bounds (fault tolerances) and performance such as transaction throughput,
delay, and node scalability[2]. Among many consensus protocols, the proof-
based algorithm (PoX) is often used in a public chain, such as proof-of-work
(PoW)[3], proof-of-stake (PoS)[4], whereby participants can join/leave the
network without authentication. These PoX-based consensus protocols achieve
good scalability with the cost of high resource consumption. For instance, the
PoW protocol employed in Bitcoin consumes a huge amount of power resources to
compute a meaningless hash value. And the confirmation delay and throughput of
these consensus protocols also limit the application scenario. The voting-
based consensus protocols, such as PBFT[5], RAFT[6], relies heavily on inter-
participant communications to achieve consensus. Therefore, the communication
resource is critical for the voting-based protocols, which can only be
applicable for small/middle scale networks.
Most of the existing consensus protocols are primarily designed over wired
communication networks, some new challenges may arise when they are deployed
in wireless networks. First, wireless devices usually have limited resources
(energy, storage, computation, etc.), so the resource-consuming PoX protocols
are not suitable. Second, the wireless communication channel condition varies
dynamically, not to mention that users often suffer from limited bandwidth
resources, which significantly affects the consensus performance, especially
for the voting-based protocols.
As a solution, the wireless channel features can be utilized to offer a
solution to this dilemma. For the voting-based protocols, a block is generated
based on the decision from the majority participants[7]. For instance, in the
PBFT protocol, each participant determines the consensus results by counting
the number of messages with consistent hash from other participants. For
traditional “communicate-then-consensus” solutions, the consensus process can
be conducted only after all messages from other participants are successful
received and decoded. Instead of collecting messages from all participants,
the consensus process only require the result of a function of these messages,
e.g., the number of messages with consistent hash from other participants,
rather than the details bits of each message. Therefore, a “communicate-and-
consensus” scheme can be adopted to reduce the communication and computational
complexity. Specifically, the communicating and consensus tasks can be
accomplished simultaneously by leveraging the over-the-air computation
(AirComp) technology[8] in wireless networks, which is a novel technology that
exploits the superposition property of wireless multiple access channel (MAC)
to aggregate signals from multiple participants over the same wireless
channel. In this way, the destructive interferences from different
participants are turned into a constructive one by appropriately matching the
structure of the wireless channel and the function of the signals can be
computed from the aggregated signal.
In this paper, we propose “AirCon”, an over-the-air consensus protocol for
wireless networks, which significantly reduces the communication and
computational overhead in a wireless network by using the AirComp technique.
The novelty of this protocol is two-folded. First, the hash symbols of all
participating users are encoded using lattice codes and transmitted to the
base station (BS) simultaneously over the same wireless spectrum via the
AirComp technique, which greatly reduces the wireless resource usage. Second,
by leveraging the structural property of the lattice codes, each user can
verify the consistency of its hash value with respect to the aggregated hash
symbols without decoding the signal, whereby the consensus can be achieved
with extreme low computational complexity.
To show the feasibility of the AirCon protocol, we implement AirCon on an LTE
system, where two key issues are solved: 1) Synchronization problem, which is
the key to ensure that all participants transmit data in the same frequency
simultaneously such that the signals can be accurately aligned at the BS. We
will discuss the details of LTE synchronization mechanisms and show that these
intrinsic mechanisms are sufficient for satisfying the synchronization
requirement for the implementation of AirComp in an LTE system. 2) Uplink
channel estimation and feedback problem. The participants need to learn the
uplink channel state information (CSI) so that the channel fading can be pre-
compensated. To this end, we propose a flexible reference symbol assignment
scheme, which achieves a better tradeoff between estimation accuracy and
estimation latency.
The main contributions of the paper are summarized as follows:
* •
We propose a novel AirComp-based consensus protocol (AirCon), which enables
all participants to transmit their hash symbols to the BS simultaneously using
the same wireless channel, and thus significantly reduces the communication
cost of consensus messages.
* •
We propose a hash consistency verification scheme in the physical layer,
wherein each participant determines whether its hash value is consistent with
the aggregated hash symbols or not without decoding the signal, which
significantly reduces the complexity of consensus computation.
* •
We implement AirCon based on srsLTE111The latest version of srsLTE has been
renamed as srsRAN and to support 5G standards.[14], an open-source LTE soft
defined radio (SDR) platform. Extensive experiments are conducted to validate
the practical performance of AirCon in a real-world LTE testbed. We also
provide simulation results to evaluate the performance of the proposed schemes
under more general network conditions.
The rest of this paper is organized as follows. In Section 2, we discuss the
research progress in blockchain-enabled wireless networks and the AirComp
technique. In Section 3, we introduce the system model and propose a modified
PBFT-based over-the-air consensus (AirCon) procedure for wireless blockchain
systems. The detailed design of AirCon protocol is presented in Section 4 and
the details for implementation of AirCon in the LTE system are discussed in
Section 5. Experimental and simulation results are provided in Section 6.
Finally, we summarize this work in Section 7.
## 2 Related Works
In this section, we give a brief introduction to the research progress on
blockchain-enabled wireless networks and AirComp technique.
### 2.1 Blockchain-enabled wireless networks
Blockchain has been studied extensively in literature as a security and
privacy protection scheme for various application scenarios in wireless
networks. In [15], the authors examined the application of blockchain in the
Internet of things (IoT). They concluded that the blockchain-IoT combination
is powerful and can lead to significant transformations across several
industries. Blockchain technology also can be applied in the internet of
vehicles[16] to provide cybersecurity protection for vehicular communications,
including the dynamic control of source reliability, and the integrity and
validity of the information exchanged. The incorporation of blockchain into
the next-generation radio access network (RAN) also attracted great interest.
A unified framework of the blockchain radio access network (B-RAN) was
proposed in[17] as a trustworthy and secure paradigm for upcoming 6G
networking. Some critical elements of B-RAN, such as the deployment of smart
contract[18], trustworthy access[19, 20], mathematical modeling[21] were also
explored.
Due to the resource-limited feature of wireless devices, the blockchain system
needs to be improved for the wireless environment. The authors in[17] proposed
proof-of-devices (PoD) as a low-cost consensus protocol, which is based on the
fact that B-RAN is comprised of a tremendous number of devices and attackers
cannot control $51\%$ devices of the whole network. In an edge computing
scenario, the authors in [22] consider constructing a collaborative mining
network (CMN) to execute mining tasks for mobile blockchain. Miners can
offload their mining tasks to non-mining devices within a CMN when the
resources are insufficient. The authors in [23] proposed a lightweight
blockchain system to reduce the computational cost and speed up the block
generation rate in the industrial IoT (IIoT).
On the other hand, some research works were concerned with the impact of a
wireless channel on blockchain performance. In[24], the authors analyzed the
trade-off between communication reliability and computing power in blockchain
security and presented a lower bound to the computing power that is needed to
conduct an attack with given communication reliability. Based on the widely
used CSMA/CA mechanism, the impact of communication transmission delay on the
confirmation delay, transaction per second, and transaction loss probability
is analyzed in [25]. The authors in [26] analyzed the impact of node
geographical distribution in the spatial domain and designed an algorithm to
determine the optimal full-function node deployment. In [2], the authors
evaluated the impact of scarce frequency spectrum resources on blockchain
performance.
### 2.2 Over-the-air computation
In this paper, we design a novel consensus protocol based on the AirComp
technology, which exploits the intrinsical superposition property of the
wireless channel to achieve efficient transmissions of multiple users. AirComp
was proposed by B. Nazer and M. Gastpar in their cornerstone work[8], where an
optimal joint source-channel strategy was developed to reliably reconstruct a
function of sources over a multiple-access channel. Early research works of
AirComp took on the information-theoretic view and focuses on the achievable
computation rate under structured coding schemes. The transmission strategy
under different source distribution was explored in [27, 28, 29]. The authors
in [27, 28] employed lattice codes to efficiently compute the sum of source
signals over MACs. Some recent research efforts have been made to study
AirComp from other aspects. The authors in [30, 31] considered a more general
power-limited scenario where some devices cannot fully compensate for channel
fading, then a threshold-based power-control policy was proposed to minimize
the mean square error (MSE) of the whole system. Multifunction computation was
achieved by using AirComp in a multiple antenna system, beamforming strategies
have been designed in [32, 33] to achieve optimal performance. In[34], the
blind AirComp technique was proposed for low-complexity and low-latency
wireless networks, which does not require channel state information for
AirComp.
The AirComp technique can be applied in a wide range of application scenarios.
In[10, 11], AirComp was proposed for fast data fusion, whereby the fusion
center (FC) attempts to compute a specific function from the data of all
sensor nodes (e.g., average reading). AirComp also has been extended for model
aggregation in the distributed edge learning system [35, 36, 37], which is
shown to reduce the communication latency without significant loss of the
learning accuracy[38]. In a multi-agent system, AirComp can be used for
distributed consensus control[39, 40] to reach an agreement over a set of
variables of common interest, such as velocity, acceleration, and trajectory
in vehicular platooning. Our work extends the application of AirComp to the
consensus protocol in wireless blockchain systems.
Despite the extensive theoretical work in literature, the research on the
implementation issues for AirComp is relatively scarce. In[10] and [11], the
authors proposed to modulate the value to be computed through AirComp as the
mean value or power of a random sequence, so the accurate synchronization and
CSI is not necessary for implementing the AirComp technique. In [12], the
authors proposed to achieve system synchronization via the “AirShare” scheme
[41], and the uplink CSI was obtained by using channel reciprocity between
uplink and downlink transmissions. In [13], the uplink CSI was obtained by
feedback from the fusion center. These implementation schemes were designed
for analog function computation (AFC) in a wireless network. In this paper, we
implement the AirComp technique based on the universal LTE system, which
demonstrate that AirComp can be realized in the current digital communication
systems.
## 3 System model
Figure 1: Blockchain-enabled wireless network model. Figure 2: Diagram of the
over-the-air consensus protocol. Hash verification for the generated block is
achieved through the BS using the multiple access and broadcast channels.
We consider a blockchain-enabled wireless network as illustrated in Fig. 1,
where a set of $K$ users are served by a BS. These users maintain the same
blockchain. The transaction blocks are generated periodically based on a
consensus protocol, and stored in a distributed manner in all users.
In a blockchain system, the block appending procedure can be separated into
three stages: Request, Consensus and Confirm, as shown in Fig. 2. Our work
focus on the consensus step and the other two steps are the same as the
traditional blockchain systems.
Request: the client sends a request to the primary user to trigger a
transaction. After collecting enough transactions, the primary user generates
a candidate block and broadcasts it to other replica users and requests to
start the consensus procedure 222We assume that the primary user is always
honest. If other users or clients detect some malicious behavior from the
primary user (such as broadcasting some illegal transactions), view change
procedures[5] can be triggered to replace the primary user.
Consensus: The consensus protocol in Fig. 2 is a revision of the PBFT protocol
for wireless networks, which consists of four phases as follows:
* •
Pre-prepare: After receiving the candidate block and request from the primary
user, all replica users verify the transactions of the candidate block, then
send the hash symbols of candidate blocks to the BS in the same uplink channel
and enter the prepare phase. Note that user-specific information (such as user
ID) should not be included in the generation of the hash value.
* •
Prepare and Commit: The BS obtains the superposed hash symbols and broadcasts
them to all users. All replica users adopt a threshold-based two-round hash
verification method to determine the consensus result.
* •
Reply: Only those users satisfying the threshold return the reply message to
the primary user through the BS. If more than $\lceil{K/2}\rceil$ users return
the message, the primary user determines that the consensus has been achieved.
Confirm: If the consensus has been achieved, the primary user sends a confirm
message to the BS, which is broadcast to all replica users and the client,
then the candidate block is appended in each blockchain which is
distributively stored at each replica user.
The bottleneck of the consensus protocol shown in Fig. 2 lies in the _Prepare_
and the _Commit_ phases. In the traditional hash cross-validation, the hash
value generated by different users should be transmitted to each other via
orthogonal wireless channels (such as TDMA, FDMA, or OFDMA in 4G/5G networks).
Therefore, the wireless resource usage in these two phases increases with the
number of participating users, which cannot guarantee reliable and low latency
consensus for larger network size.
## 4 AirCon Protocol Design
In this section, we propose a novel over-the-air consensus (AirCon) protocol
to address the transmission bottleneck problem in the _Prepare_ and the
_Commit_ phases based on the AirComp and the lattice coding techniques. The
idea is to modulate the hash bits of different users using the lattice codes,
which can be transmitted simultaneously to the BS over the same wireless
channel. Due to the structural property of the lattice codes, the BS can adopt
the AirComp technique to aggregate the linear combination of all codes (which
is a codeword itself), and forwards it back to all users, which can be used by
each user to make a consensus decision without decoding the signal.
In the following, we first introduce the preliminary background on the AirComp
and lattice coding techniques. We then present the hash consistency
verification scheme based on these two techniques, with which a two-round
consensus procedure for the AirCon protocol is designed.
### 4.1 Over-the-air computation
We consider an OFDM-based wireless system, which consists of a single BS and a
set of $K$ users. All users transmit signals to the BS through the same uplink
channel. The received signal at the BS is expressed as:
$\mathbf{y}=\sum_{k=0}^{K-1}\mathbf{H}_{k}\mathbf{B}_{k}\mathbf{x}_{k},$ (1)
where $\mathbf{x}_{k}\in\mathbb{C}^{N\times 1}$ is the symbol vector
transmitted by user $k$ with the covariance matrix
$\mathbb{E}\\{\mathbf{x}_{k}\mathbf{x}_{k}^{H}\\}=\mathbf{I}$, where
$\mathbf{I}$ is the identity matrix.
$\mathbf{B}_{k}=\texttt{Diag}\\{b_{k0},b_{k1},\cdots,b_{kN-1}\\}\in\mathbb{C}^{N\times
N}$ is the pre-processing matrix for user $k$.
$\mathbf{H}_{k}=\texttt{Diag}\\{h_{k0},h_{k1},\cdots,h_{kN-1}\\}\in\mathbb{C}^{N\times
N}$ is the channel matrix from user $k$ to the BS. $N$ is the total number of
sub-carriers. $K$ is the total number of users. The superscript $(\cdot)^{H}$
denotes Hermitian transpose.
Under the condition that each user $k$ has perfect knowledge of
$\mathbf{H}_{k}$, and its transmit power is not bounded, then the channel
fading can be pre-compensated by a simple channel-inversion operation as
$\mathbf{B}_{k}=\mathbf{H}_{k}^{-1}$. In this way, the signals transmitted by
all users will be aggregated at the BS as follows:
$\mathbf{y}=\sum_{k=0}^{K-1}\mathbf{x}_{k},$ (2)
which is equivalent to the computation of the summation of all
$\mathbf{x}_{k}$s over the wireless channel.
Motivated by this property, the AirComp technique can be generalized to
support a wide range of mathematical operations, which are based on the
property of nomographic function in the following form:
$f(\mathbf{s}_{0},\mathbf{s}_{1},...,\mathbf{s}_{K-1})=\psi{\bigg{(}}\sum_{k=0}^{K-1}\phi{(\mathbf{s}_{k})}{\bigg{)}},$
(3)
where $\phi(\cdot)$ and $\psi(\cdot)$ denote pre- and post-processing
functions, respectively.
Some well-known nomographic functions include:
1. 1.
Arithmetic Mean:
$f(s_{0},s_{1},...,s_{K-1})=\frac{1}{K}\sum_{k=0}^{K-1}s_{k}$, with
$\phi(s_{k})=s_{k}$ and $\psi(y)=y/N$;
2. 2.
Euclidean Norm: $f(s_{0},s_{1},...,s_{K-1})=\sqrt{\sum_{k=0}^{K-1}s_{k}^{2}}$,
with $\phi(s_{k})=s_{k}^{2}$ and $\psi(y)=\sqrt{y}$;
3. 3.
Number of Active Node: $f(s_{0},s_{1},...,s_{K-1})$ is the number of active
node, with $\phi(s_{k})=1\,(\text{active})\,\text{or}\,0\,(\text{inactive)}$
and $\psi(y)=y$.
### 4.2 Nested lattice code
A lattice is an infinite discrete set of points in the Euclidean space that
are regularly arranged and are closed under addition[42]. As an important
channel coding technique, the structural properties of nested lattice coding
are well suited for multiple access channel in wireless networks, which allows
multiple transmitters to effectively share the same radio resources and can
protect against channel noise. Specifically, a $d$-dimensional lattice
$\Lambda$ in the Euclidean space $\mathbb{R}^{d}$ can be generated as follows:
$\Lambda=\\{\mathbf{Gu}\ :\ \textbf{u}\in\mathbb{Z}^{d}\\},$ (4)
where $\mathbf{G}=[\mathbf{g}_{1},\mathbf{g}_{2},...,\mathbf{g}_{d}]$ is a
full-rank generator matrix.
A lattice $\Lambda_{C}$ is _nested_ in some lattice $\Lambda_{F}$ if
$\Lambda_{C}\subseteq\Lambda_{F}$, i.e., $\Lambda_{C}$ is a sublattice of
$\Lambda_{F}$. In this case, $\Lambda_{F}$ is denoted as the fine lattice,
which defines the codewords, while $\Lambda_{C}$ is denoted as the coarse
lattice, which is used for shaping. Specifically, the _nested lattice code_
$\mathcal{L}$ is the set of all points of a fine lattice $\Lambda_{F}$ that is
within the fundamental Voronoi region $\mathcal{V}_{C}$ of a coarse lattice
$\Lambda_{C}$:
$\mathcal{L}=\Lambda_{F}\cap\mathcal{V}_{C}=\\{\textbf{x}\ :\
\textbf{x}=\lambda\ \text{mod}\ \Lambda_{C},\lambda\in\Lambda_{F}\\},$ (5)
where the _fundamental Voronoi region_ , $\mathcal{V}_{C}$, of the lattice
$\Lambda_{C}$, is the set of all points in $\mathbb{R}^{d}$ that are closest
to the zero vector:
$\mathcal{V}_{C}=\\{\textbf{z}\ :\
||\textbf{z}||\leq||\textbf{z}-\lambda||,\forall\lambda\in\Lambda_{C},\textbf{z}\in\mathbb{R}^{d}\\},$
(6)
For each user $k$, a $d$-dimension nested lattice codeword
${\bf{x}}_{k}\in\mathcal{L}$ can be generated based on its hash value
${\bf{s}}_{k}\in\mathbb{F}^{l}_{p}$ by the encoding function $\phi(\cdot)$ as
follows:
$\displaystyle\phi({\bf{s}}_{k}):\mathbb{F}^{l}_{p}$
$\displaystyle\rightarrow\mathbb{R}^{d}$ (7) $\displaystyle{\bf{s}}_{k}$
$\displaystyle\rightarrow{\bf{x}}_{k}$
where $\mathbb{F}_{p}=\\{0,1,...,p-1\\}$ forms a finite field under integer
arithmetic modulo $p$. Then for a nested lattice codebook $\mathcal{L}$, the
following property is held for all ${\bf{x}}_{k}\in\mathcal{L}$:
${\bigg{[}}\sum_{k=0}^{K-1}{\bf{x}}_{k}\mod{\Lambda}_{C}{\bigg{]}}\in\mathcal{L},$
(8)
that is, the sum of lattice codewords modulo the shaping lattice is a codeword
itself. Due to this linearity preserving characteristic, there exists a post-
processing function $\psi(\cdot)$ that satisfies[9]:
$\psi{\bigg{(}}\sum_{k=0}^{K-1}\phi({\bf{s}}_{k})\mod{\Lambda_{C}}{\bigg{)}}=\underset{k=0}{\overset{K-1}{\bigoplus}}{\bf{s}}_{k},$
(9)
which is a nomographic function, that is,
$f({\bf{s}}_{0},{\bf{s}}_{1},...,{\bf{s}}_{K-1})=\underset{k=0}{\overset{K-1}{\bigoplus}}{\bf{s}}_{k}$.
Based on this property, AirComp can be used to transmit and compute the
superposition of hash values from all users. Specifically, each user $k$ maps
its hash value $\mathbf{s}_{k}$ to a lattice codeword $\mathbf{x}_{k}$ using
the mapping operation in Eq.(7), then all users can transmit their lattice
codes simultaneously to the BS using the AirComp technique. At the BS, the
received signal is $\mathbf{y}=\sum_{k=0}^{K-1}\mathbf{x}_{k}+\mathbf{w}$,
which is a $d$-dimension linear combination of the lattice codes of all users
with noise of $\mathbf{w}$.
In theory, there are many choices for the lattice codes with different
codeword dimensions. However, in practice, the transmitted symbol can only
represent information in a 2-dimensional space (in-phase and quadrature
dimension). Therefore, for the high dimensional hash value, such as the
128-bits hash generated by the MD5 algorithm[43], it needs $2^{64}$ codewords
at each dimension to represent the hash, which is impossible in a real
communication system.
Figure 3: Nested $\mathbb{Z}^{2}$ cubic lattice.
In this work, we consider an $\mathbb{Z}^{2}$ cubic lattice
$\tilde{\Lambda}_{F}$ as illustrated in Fig. 3, where the points are
$\tilde{\Lambda}_{F}$ points and the Voronoi region of the coarse lattice
$\tilde{\Lambda}_{C}$ is drawn in dashed lines. Only the eight codes in the
outer tier of the Voronoi region are used (shown in red color), so the code
rate is $B=\log_{2}8=3$ bits/symbol. Therefore, for an $L$-bits hash value, it
can be partitioned into $N=\lceil{L/3}\rceil$ symbols.
### 4.3 Hash consistency verification
Based on the $\mathbb{Z}^{2}$ cubic lattice in Fig. 3, each user $k$ can
construct a hash symbol vector
$\tilde{\bf{x}}_{k}=[\tilde{x}_{k0},\tilde{x}_{k1},...,\tilde{x}_{kN-1}]^{T}$,
where $\tilde{x}_{kn}$ is mapped to one of the eight codes. Then each element
of $\tilde{\bf{x}}_{k}$ is transmitted in an OFDM sub-carrier. The received
symbol at the BS in sub-carrier $n$ is
$\tilde{y}_{n}=\sum_{k=0}^{K-1}\tilde{x}_{kn}+w_{n}$.
The received symbol at sub-carrier $n$ can be quantized to the nearest point
$t_{n}\in\mathbb{Z}^{2}$ in the $\tilde{\Lambda}_{F}$ as follows:
${t_{n}}=\arg\underset{\lambda\in{\tilde{\Lambda}_{F}}}{\min}||\tilde{y}_{n}-\lambda||,\,n\in[0,N-1],$
(10)
which are then broadcasted to all users through the downlink channel. A vector
$\mathbf{t}=[t_{0},t_{1},...,t_{N-1}]^{T}\in\mathbb{Z}^{2N}$ is constructed by
each user, which is the complete linear combination of the lattice codes of
all users.
Based on the received vector $\mathbf{t}$, we design a consensus algorithm by
exploiting the geometric characteristics of lattice codes. Let $C_{m}$ denote
a set of $m$ users that transmit the consistent hash symbol vector, and
$C_{K-m}$ denote the rest $K-m$ users that transmit different hash symbol
vectors. Then $\mathbf{t}$ can be re-written as follows:
$\mathbf{t}=\sum_{k\in C_{m}}\tilde{\mathbf{x}}_{k}+\sum_{k\in
C_{K-m}}\tilde{\mathbf{x}}_{k}+\mathbf{e},$ (11)
where $\mathbf{e}$ is the random error pattern after symbols quantization in
(10) with $\mathbb{E}\\{\mathbf{e}\\}=\mathbf{0}$.
We assume that the hash symbol vector elements are uniformly mapped from the
lattice codewords as shown in Fig. 3. For each user $k\in C_{m}$, we assume
the transmitted hash symbol vector is $\bar{\mathbf{x}}$, i.e.,
$\tilde{\mathbf{x}}_{k}=\bar{\mathbf{x}},\forall k\in C_{m}$. For each user
$k\in C_{K-m}$, we firstly consider the case that the hash symbol vector is
independent of each other. Then for $k\neq j$, we have
$\mathbb{E}\\{\tilde{\mathbf{x}}_{k}^{T}\tilde{\mathbf{x}}_{j}\\}=\begin{cases}N\sigma_{s}^{2},&{\text{If}~{}\forall
k,j\in C_{m}}\\\ 0,&{\text{Otherwise}}\end{cases}$ (12)
where $\sigma_{s}^{2}$ is the variance of codewords. Furthermore, we define
$\tilde{I}_{k}$ as the hash consistency factor (HCF) of user $k$, which is the
normalized inner product of its hash symbol vector $\tilde{\mathbf{x}}_{k}$
and vector $\mathbf{t}$, that is:
$\displaystyle\tilde{I}_{k}=\frac{\mathbf{t}^{T}\tilde{\mathbf{x}}_{k}}{K|\tilde{\mathbf{x}}_{k}|^{2}},$
(13)
From (12), we can obtain the expectation of $\tilde{I}_{k}$ as follows:
$\mathbb{E}\\{\tilde{I}_{k}\\}=\begin{cases}\frac{m}{K},&{\forall k\in
C_{m}}\\\ \frac{1}{K}.&{\forall k\in C_{K-m}}\end{cases}$ (14)
Figure 4: $\mathbb{E}\\{\tilde{I}_{k^{\prime}}\\}$ and
$\mathbb{E}\\{\tilde{I}_{k^{\prime\prime}}\\}$ vs. $m$ ($K=21$).
In Fig. 4, we plot $\mathbb{E}\\{\tilde{I}_{k}\\}$ with 21 users ($K=21$) for
$m$ varying from 1 to 20. It can be seen that for $m\geq 2$,
$\mathbb{E}\\{\tilde{I}_{k^{\prime}}\\}>\mathbb{E}\\{\tilde{I}_{k^{\prime\prime}}\\},\forall
k^{\prime}\in C_{m},k^{\prime\prime}\in C_{K-m}$. Thus $\tilde{I}_{k}$ can be
used as a reliable metric for each user to determine whether its hash symbol
vector is consistent with the majority of hash vectors in $\mathbf{t}$ or not,
that is, it belongs to one of the $m$ users sending the same hash or not.
Therefore, this metric can be used to check if the consensus is reached or
not. Specifically, in the proposed AirCon protocol, the consensus can be
reached if $m>\lfloor{K/2}\rfloor$ users generate the same hash vector.
According to (14), we can set a threshold as $T_{h}=0.5$. After receiving the
superposed hash vector $\mathbf{t}$, each user $k$ calculates $\tilde{I}_{k}$
as defined in (13). If $\tilde{I}_{k}\geq T_{h}$, the user assumes that the
consensus can be achieved and sends the result to the primary user.
However, if some users are controlled by an attacker, the consensus
performance will be degraded based on this ideal threshold. For instance,
consider the case where $m$ users are honest and the rest of $K-m$ users are
controlled by a malicious attacker. The hash symbol vector transmitted by
honest users is $\bar{\mathbf{x}}$, while the hash symbol vector transmitted
by malicious users is $\hat{\mathbf{x}}=-\bar{\mathbf{x}}$, then the
aggregated vector at the BS is
$\mathbf{t}=m\bar{\mathbf{x}}+(K-m)\hat{\mathbf{x}}=(2m-K)\bar{\mathbf{x}}$.
In this case, $\tilde{I}_{k}=(2m-K)/K$ for a honest user $k$, so the consensus
cannot be achieved with a threshold of $T_{h}=0.5$ unless
$m>\lfloor{3K/4}\rfloor$. Therefore, this one-round consensus procedure is
vulnerable to this kind of conspiracy attack.
As a solution, we propose a two-round consensus procedure to improve the
security of the proposed consensus procedure. Specifically, in the first
round, malicious users can be filtered by setting a proper threshold and only
honest users enter the second round to decide the consensus results. In the
following, we will analyze the optimal threshold for the first round and the
corresponding fault tolerance of the proposed consensus protocol.
Before getting into the details, we first make some assumptions about the
capabilities of an attacker:
1. 1.
_AS_ 1: An attacker knows the hash symbols transmitted by each honest user.
This is feasible because the attacker can calculate the legal hash by acting
as an honest user;
2. 2.
_AS_ 2: An attacker cannot manipulate the parameters of the physical layer,
such as the transmitting power.
3. 3.
_AS_ 3: An attacker cannot manipulate the consensus protocol, such as the
consensus threshold.
We assume that an attack is successful if the consensus cannot be achieved
even if the number of honest users is more than half of the total number of
users (i.e., $m>\lfloor{K/2}\rfloor$).
Based on the aforementioned attacking model, we assume all honest users
transmit $\bar{\mathbf{x}}$ and all malicious users transmit
$\hat{\mathbf{x}}$. Then $\mathbf{t}$ can be re-written as follows:
$\mathbf{t}=m\bar{\mathbf{x}}+(K-m)\hat{\mathbf{x}}+\mathbf{e},$ (15)
and the expectation of $\tilde{I}_{k}$ can be re-written by
$\mathbb{E}\\{\tilde{I}_{k}\\}=\begin{cases}1-\alpha+\alpha\rho,&{\forall k\in
C_{m}}\\\ \alpha+(1-\alpha)\rho,&{\forall k\in C_{K-m}}\end{cases}$ (16)
where $\alpha=\frac{K-m}{K}$ is the percentage of malicious users and
$\rho=\mathbb{E}\\{\frac{\hat{\mathbf{x}}^{T}\bar{\mathbf{x}}}{|\bar{\mathbf{x}}|^{2}}\\}=\mathbb{E}\\{\frac{\bar{\mathbf{x}}^{T}\hat{\mathbf{x}}}{|\hat{\mathbf{x}}|^{2}}\\}$
denote the correlation coefficient between $\hat{\mathbf{x}}$ and
$\bar{\mathbf{x}}$. The attacker can change $\tilde{I}_{k}$ of the honest
users by manipulating $\rho$ between $[-1,1]$ with properly setting of
$\hat{\mathbf{x}}$.
In the first round of consensus, a threshold should be set such that all
honest users can get into the second round while the malicious users should be
filtered as many as possible. To this end, the threshold should be set to
$T_{h1}=\underset{\rho\in[-1,1]}{\text{min}}(1-\alpha+\alpha\rho)=1-2\alpha$
(17)
From the attacker’s perspective, it must find a $\rho$ satisfying
$\alpha+(1-\alpha)\rho>T_{h1}=1-2\alpha$ such that malicious users also can
get into the second round. Otherwise, all malicious users will be filtered in
the first round. Therefore, the range of $\rho$ can be given by
$\rho>\frac{1-3\alpha}{1-\alpha}$ (18)
In the second round of consensus, the HCF $\tilde{I}_{k}$ of honest users
should be more than $T_{h2}=0.5$ such that the consensus can be achieved even
if all malicious users satisfy the threshold in the first round of consensus
by setting $\rho$ as in (18), that is
$\tilde{I}_{k}=1-\alpha+\alpha\rho=1-\alpha+\alpha\frac{1-3\alpha}{1-\alpha}>\frac{1}{2},~{}\forall
k\in C_{m}$ (19)
Then we can obtain the range of $\alpha$ as follows:
$\alpha<\frac{\sqrt{17}-1}{8}\approx 0.39:=\alpha^{*}$ (20)
Substituting (20) into (17), we can obtain the optimal value of $T_{h1}$ as
follows:
$T_{h1}^{*}:=0.22$ (21)
The range of $\alpha$ in (20) specifies the fault tolerance of the two-round
consensus protocol. If the percentage of malicious users does not exceed 0.39,
then the consensus can be achieved in the second round. Otherwise, the
consensus cannot be achieved even if $m>\lfloor{K/2}\rfloor$.
We show two examples in Fig. 5 and Fig. 6. In Fig. 5, we plot the HCF for all
users with $\alpha=0.35$, which is under the threshold $\alpha^{*}$. The range
of $\rho$ can be divided into two parts by the critical point $\rho_{0}$ that
satisfies $\alpha+(1-\alpha)\rho_{0}=T_{h1}=1-2\alpha$. When $\rho<\rho_{0}$,
the region is denoted as _Safety Region_ , all malicious users will be
filtered by $T_{h1}$ and cannot enter the second round. When $\rho>\rho_{0}$,
the region is denoted as _Consensus Region_ , malicious users can enter the
second round, but they cannot affect the consensus results because the HCF of
all honest users always exceeds $T_{h2}=0.5$ such that all honest users can
enter _Reply Phase_ and send the reply message to the primary user. Note that
in the second round, if $\rho_{0}<\rho<\rho^{\prime}_{0}$, the HCF of
malicious users is less than $T_{h2}$ such that they cannot send the reply
message. If $\rho>\rho^{\prime}_{0}$, malicious users also send a reply
message. However, regardless of whether malicious users can send a reply
message or not, it will not affect the consensus result because all honest
users always send the reply message. Therefore, malicious users cannot affect
consensus results whatever $\rho$ is set when $\alpha=0.35$.
In Fig. 6, we plot the HCF of all users with $\alpha=0.45$, which is over the
threshold $\alpha^{*}$. In this case, we can divide the range of $\rho$ into
three parts by two critical point $\rho_{0}$ and $\rho_{1}$, where $\rho_{0}$
satisfies $\alpha+(1-\alpha)\rho_{0}=T_{h1}=1-2\alpha$ and $\rho_{1}$
satisfies $1-\alpha+\alpha\rho_{1}=T_{h2}=0.5$. Similarly, when
$\rho<\rho_{0}$, the consensus can be achieved because malicious users cannot
enter the second round. When $\rho>\rho_{1}$, the consensus also can be
achieved because the HCF of honest users always exceeds $T_{h2}$. However,
when $\rho_{0}<\rho<\rho_{1}$, we denote it as _Attacking Region_ , malicious
users can enter the second round, and the HCF of all honest users does not
exceed $T_{h2}$, so the consensus cannot be achieved and the attack is
successful in this region.
Figure 5: Consensus performance under fault tolerance, $\alpha=0.35$ Figure 6:
Consensus performance over fault tolerance, $\alpha=0.45$
### 4.4 Two-round consensus procedure
Based on the hash consistency verification scheme discussed in the previous
subsection, we propose a two-round consensus procedure as follows.
In the first round (_Prepare phase_), all users send their hash symbols to the
BS via the AirComp technique. The BS feeds back the superposed hash symbol
vector $\mathbf{t}$ to all users. Each user $k$ calculates the HCF
$\tilde{I}_{k}$ according to (13) and compares it with the threshold
$T_{h1}=0.22$. If $\tilde{I}_{k}>T_{h1}$, then the user changes to the
prepared state and enters the second round of the consensus process.
In the second round (_Commit phase_), only users who are in the prepared state
send the hash symbols to the BS via the AirComp technique. The BS feeds back
the new superposed hash symbol vector $\mathbf{t}^{\prime}$ to all users. Each
prepared user $k$ calculates the HCF $\tilde{I}_{k}$ once again using
$\mathbf{t}^{\prime}$ and compares with the threshold $T_{h2}=0.5$. Only those
users with $\tilde{I}_{k}>T_{h2}$ enter the _Reply Phase_ and return the reply
message to the BS via the AirComp technique. The superposed reply message is
forwarded to the primary user by the BS.
The final consensus result is determined by the primary user based on the
superposed reply message from the BS. The primary user also can calculate the
HCF in (13) between its reply message and the superposed message. If the HCF
exceeds $T_{h}=0.5$, which suggests that more than half of the total users
return the consistent reply message, then the consensus is achieved.
## 5 AirCon implementation based on LTE system
In this section, we consider the implementation of AirCon protocol based on
the open-source srsLTE platform, which provides the standard LTE protocols. We
present solutions to some of the critical issues for practical AirComp
implementation. Specifically, two problems are considered for AirComp
implementation: 1) Synchronization problem; 2) Uplink channel estimation and
feedback problem.
### 5.1 Synchronization
The first challenging problem for AirComp implementation is to achieve strict
timing/frequency synchronization across different users so that accurate
signal superposition can be obtained at the BS.
Fortunately, the LTE system has a complete set of synchronization mechanisms
that can meet the synchronization requirement of AirComp. Specifically, in the
downlink of the LTE system, the BS broadcasts synchronization channels
(including PSS and SSS) periodically. If a user wants to connect to the BS,
the first step is to search for these two channels to get timing and frequency
synchronization. As long as the user is connected to the BS, it will keep
tracking the timing/frequency synchronization via PSS/SSS. The user also
compensates for the frequency offset of uplink channels according to the
estimated value from the downlink channels.
In the uplink, a “timing advance” mechanism is adopted by all users for timing
synchronization with the BS. Specifically, when the user connects to the BS,
the BS measures the propagation delay and calculates a timing offset, named
“Timing Advance (TA)”, which is fed back to the corresponding user. Whenever
the user transmits data to the BS, it should transmit at the time with an
advance of TA so that the signals from different users arrive at the BS
without timing offset.
In addition, the physical layer of the LTE system is based on the OFDM
technique, which adopts a cyclic prefix (CP) to cope with the multipath
interference problem. Due to the cyclic property, the CP can also be used for
channel synchronization. That is, as long as the timing offset is within a CP,
the impact of the timing offset can be treated as a channel phase shift and
compensated as a part of channel fading.
In summary, the existing synchronization mechanisms in the LTE system are
sufficient for the implementation of AirComp, so there is no need to design a
dedicated AirComp synchronization scheme in an LTE system, which is part of
the reason we choose the LTE system as the implementation platform of the
AirCon protocol.
### 5.2 Channel estimation and feedback
The purpose of uplink channel estimation is to cope with the channel fading
problem via pre-compensation on the user side. In practice, there are two
different ways to obtain uplink CSI. One way is that the BS estimates the
uplink channel and sends feedback to the corresponding user, the other way is
that a user infers the uplink CSI from the downlink signals by leveraging the
channel reciprocity of the TDD channel. In this work, we choose the first
solution since it can be used for both FDD and TDD systems.
To get uplink CSI at the BS, a user needs to transmit reference symbols in the
uplink channel. Ideally, the reference symbols should be transmitted at each
sub-carrier so that the channel can be estimated accurately for all sub-
carriers. However, this will consume too many channel resources and take a
longer time for channel estimation for a large number of participating users.
To this end, we exploit the property that the sub-carriers within coherence
bandwidth may have a similar fading coefficient, so the reference symbols are
only needed for every $M$ sub-carriers333$M$ is a parameter depending on the
coherence bandwidth. Note that this scheme is also adopted by the uplink
sounding reference signal (SRS) in the LTE system.. As a result, at most $M$
users can share an OFDM symbol, therefore a total of $\lceil{K/M}\rceil$ OFDM
symbols are needed for reference symbol transmissions. Fig. 7 illustrates the
assignment of the reference symbols for different users.
Figure 7: Assignment of reference symbols.
Upon receiving the reference symbols, the BS extracts the received symbols and
estimates the CSI in the order of user index one by one. The simplest
estimation method is the least square (LS) method. Taking user 0 as a example,
the BS firstly constructs the received symbol vector
$\mathbf{y}=[y_{0},y_{M},\cdots,y_{\lfloor{N/M}\rfloor{M}}]$ from symbol 0,
the corresponding transmitted reference signal for user 0 is
$\mathbf{x}=[x_{0},x_{1},\cdots,x_{\lfloor{N/M}\rfloor}]$, then the LS
estimation for uplink channel knowledge of user 0 can be expressed as
$\tilde{\mathbf{h}}_{\text{LS}}=\left[\frac{y_{0}}{x_{0}},\frac{y_{M}}{x_{1}},\cdots,\frac{y_{\lfloor{N/M}\rfloor{M}}}{x_{\lfloor{N/M}\rfloor}}\right]^{T}$
(22)
However, the LS method does not consider the noise in the reference symbols.
The LMMSE method[46] can be employed to further reduce the noise impact by
utilizing the cross-correlation between sub-carriers, which is given by [46]:
$\tilde{\mathbf{h}}_{\text{LMMSE}}=\mathbf{R}_{\mathbf{h}\mathbf{h}}(\mathbf{R}_{\mathbf{h}\mathbf{h}}+\frac{\beta}{\text{SNR}}\mathbf{I})^{-1}\tilde{\mathbf{h}}_{\text{LS}},$
(23)
where $\beta=\mathbb{E}\\{|x_{n}|^{2}\\}\mathbb{E}\\{|1/x_{n}|^{2}\\}$ is a
constant depending on the reference symbol,
$\mathbf{R}_{\mathbf{h}\mathbf{h}}=\mathbb{E}\\{\mathbf{h}\mathbf{h}^{H}\\}$
is the channel autocorrelation matrix444The matrix
$\mathbf{R}_{\mathbf{h}\mathbf{h}}$ can be obtained from either a typical
channel model[46] or the channel LS estimation..
Based on the channel estimation $\tilde{h}_{kn}$ for each subcarrier, the BS
can set the coefficient of $\mathbf{B}_{k}$ as
$b_{kn}={\tilde{h}^{\dagger}_{kn}}/{|\tilde{h}_{kn}|^{2}}$, which can
compensate the channel fading and achieve the ideal signal aggregation as
shown in Eq.(1) and Eq.(2). This parameter should be fed back to the users as
shown in Fig. 8. Similar to the uplink channel estimation, two OFDM symbols
are shared by every $M$ users, where the uplink CSI is transmitted in the
_Data Symbol_ and downlink reference symbols are transmitted in the _Pilot
Symbol_ for downlink channel estimation. Therefore, totally
2$\lceil{K/M}\rceil$ OFDM symbols are needed for downlink feedback
transmissions. We still take user 0 as example: the coefficient vector
$\mathbf{b}_{0}=[b_{00},b_{0M},\cdots,b_{0~{}\lfloor{N/M}\rfloor{M}}]$ is
transmitted at the corresponding sub-carrier in the _Data Symbol_ 0\. In the
_Pilot Symbol_ 0, the downlink reference symbol $s_{0}$ is transmitted at the
corresponding sub-carrier. After receiving these two symbols, user $0$ firstly
estimates the downlink channel using the received symbols from _Pilot Symbol_
0\. Then the user can obtain the coefficient vector $\mathbf{b}_{0}$ from
_Data Symbol_ 0 by canceling the impact of downlink channel fading. Since each
user only utilizes $1/M$ sub-carriers, the coefficients for the rest of sub-
carriers can be estimated by interpolation[45, 47] at the users.
Figure 8: Assignment of downlink feedback symbols.
In the low-SNR scenario, in addition to the channel estimation algorithm, a
retransmission scheme (i.e., the uplink reference symbols and downlink
feedback symbols are transmitted multiple times) can be adopted to further
improve the performance. The impact of retransmission times on the consensus
accuracy will be studied in section 6.
### 5.3 AirCon implementation
The transmission procedure of the AirComp protocol is summarized in Fig. 9,
which consists of all procedures discussed in previous subsections. Firstly,
the synchronization between the users and the BS will be established. Then the
BS estimates uplink CSI for all users based on the uplink reference signals.
The coefficients of $\mathbf{B}_{k}$ are computed based on the channel
estimation results and fed back to all users. Finally, all users transmit data
using the AirComp technique.
Figure 9: AirComp signaling and data transmission procedures.
We implement the AirCon protocol based on the above AirComp procedures in the
LTE system. The AirCon protocol is started when all users are in a stable
state, such as the RRC IDLE state. In our implementation, we mainly focus on
the feasibility of AirComp technology on hash verification of blockchain
consensus, so only hash bits are generated in each user, rather than
broadcasting an entire block from the primary user. In this case, we do not
appoint a specific primary user. The consensus request is generated in the BS,
which is a dedicated system information block (SIB) to notify all users that
the BS is ready for consensus and users can transmit an uplink reference
signal for AirComp uplink channel estimation. Since a user usually does not
receive SIBs when it is in RRC IDLE state, the BS can set the
_systemInfoModification_ field in the paging message (all users must receive
the paging message periodically) to ensure all users receive the SIB (i.e.,
the consensus request). Similarly, the consensus result is determined by the
BS rather than the primary user.
### 5.4 Complexity analysis of the AirCon protocol
The complexity of the consensus protocol in wireless networks mainly comes
from two aspects: communication complexity and computation complexity. As a
benchmark, the traditional PBFT protocol is also illustrated in Fig. 10 for
the convenience of complexity comparison. We mainly focus on the complexity in
the _Prepare_ and _Commit_ phases.
Figure 10: Traditional PBFT consensus diagram.
The total number of messages in the _Prepare_ and _Commit_ phases of the
traditional PBFT protocol is
$\text{Num}^{\text{message}}_{\text{PBFT}}=(K-1)(K-1)+K(K-1)=(2K-1)(K-1)$ (24)
Assuming each message consumes $N$ wireless resource blocks (RBs), then the
number of wireless RBs consumed by the PBFT protocol is
$\text{Num}^{\text{RBs}}_{\text{PBFT}}=2N\cdot\text{Num}^{\text{message}}_{\text{pbft}}=2N(2K-1)(K-1)$
(25)
where the RE consumption in both the uplink and downlink directions is
considered.
For the AirCon protocol, the total number of wireless RBs consumed in the
_Prepare_ and _Commit_ phases is
$\text{Num}^{\text{REs}}_{\text{AirCon}}=4N$ (26)
Therefore, the communication complexity of the traditional PBFT protocol is
$\mathbb{O}(K^{2})$, while the communication complexity of the AirCon protocol
is $\mathbb{O}(1)$.
If the overhead of channel estimation (CE) and feedback are also considered,
the extra overhead is $4\lceil{K/M}\rceil N$, which has a complexity of
$\mathbb{O}(K)$. In this case, the total number of consumed RBs is
$\text{Num}^{\text{REs}}_{\text{AirCon\\_CE}}=4N+4\lceil{K/M}\rceil N$ (27)
Note that the complexity of channel estimation depends on the system. If
channel reciprocity is utilized to estimate the uplink channel, the complexity
of channel estimation can be reduced to $\mathbb{O}(1)$. The RB consumption in
both consensus protocols under different users number is illustrated in
Fig.11.
Figure 11: Consensus protocol REs consuming.
As for the computational complexity, the hash verification procedures in the
traditional consensus protocol are carried out in the application layer.
Therefore, demodulation, decoding in the physical layer, and decryption in the
upper layer are required. However, in our AirCon protocol, the consensus
procedures are conducted in the physical layer, and the demodulation/decoding
procedures are not required using the proposed hash consistency verification
method. Therefore, the computational complexity only comes from
$\mathbb{O}(K)$ multiplication operations.
In summary, compared with the traditional PBFT protocol, the AirCon protocol
has much lower communication overhead and computational complexity.
## 6 Performance Evaluation
In this section, we show the implementation of the AirCon testbed and provide
experimental results to demonstrate the performance under the real-world
testbed. Due to the limitation of testbed conditions, we also present
simulation results to evaluate the performance of the AirCon protocol under
more general channel conditions.
### 6.1 AirCon testbed design
We implement the AirCon protocol based on srsLTE (version 20.04), an open-
source LTE platform consisting of complete UE/eNodeB protocol stacks and a
lightweight core network (CN) protocol stack. The software defines radio (SDR)
board USRP B210[48] is used to transmit and receive RF signals. Based on the
srsLTE platform, the following modules are re-used for AirComp implementation:
(1) user attaching procedures, which guarantees all users are synchronized to
the eNodeB; (2) The OFDM waveform generation, which can generate waveform
based on the LTE frame structure. We then develop all other modules related to
the AirCon protocol, which include: (1) reference symbols, hash bits
generation, and lattice code modulation; (2) channel estimation module; (3)
downlink feedback module; (4) hash consistency verification module and (5)
AirCon protocol control module.
All devices of the testbed and setup of the test environment are shown in Fig.
12. Totally eight USRP boards are used in our experiment, where one board is
used as BS and the other seven boards are used as users. All these USRP boards
are driven by four PCs via USB3.0 interfaces.
For each experiment setup, the consensus procedures are repeated 1000 times.
For a given number of $m$ users with consistent hash symbols, we define
consensus error ratio ($\text{CER}(m)$) as a measure of consensus performance,
which is given by:
$\text{CER}(m)=\frac{\text{Number of consensus errors}}{\text{Total number of
consensuses}}$ (28)
whereby a consensus error corresponds to one of the following two cases: (1)
The consensus is achieved when $m\leq\lfloor{\frac{K}{2}}\rfloor$; (2) The
consensus is not achieved when $m>\lfloor{\frac{K}{2}}\rfloor$.
Figure 12: Testbed components and environment
### 6.2 Testbed results
Firstly, we demonstrate the symbol superposition results of the AirComp
technique. As shown in Fig. 13, if all users transmit the same hash symbols,
the received symbols at the BS are scaled with the number of users. On the
contrary, as shown in Fig. 14, if all users transmit different hash symbols,
the superposed symbols should be distributed on all lattice nodes randomly.
From these two figures, it can be seen that the channel fading is effectively
eliminated using the proposed AirComp implementation schemes (in particular
the channel estimation, feedback, and pre-compensation schemes), and the
desired channel superposition is achieved. Therefore, it is feasible to apply
the AirComp technology in a digital communication system.
Figure 13: Symbol superposition when all users transmit same hash symbols, the
received symbols are scaled with the number of users Figure 14: Symbol
superposition when three users transmit different hash symbols, the received
symbols fall on all fine lattice points
Secondly, we show the superposed hash symbols at the BS at different consensus
stages. In Fig. 15, we show the case that all users transmit consistent hash
symbols. It can be seen that all users can get into the Commit and Reply
stages. In the second case, only four users transmit consistent hash symbols,
while the other three users transmit random hash symbols. As shown in Fig. 16,
these three users with random hash symbols are filtered in the Commit stage,
and only four users with consistent hash symbols enter the Reply stage.
(a) _Prepare_ stage
(b) _Commit_ stage
(c) _Reply_ stage
Figure 15: Seven users transmit consistent hash symbols, all users determine
the final consensus result
(a) _Prepare_ stage
(b) _Commit_ stage
(c) _Reply_ stage
Figure 16: Four users transmit consistent hash symbols, three users transmit
random hash symbols, those users with random hash symbols are filtered
The performance of the AirCon protocol are shown in Fig. 17 and Fig. 18 with
different users number ($K$). For $K=5$, UE5 and UE6 in Fig. 12 are excluded
in the experiments. It can be observed that when the number of users with
consistent hash symbols ($m$) is close to the threshold for reaching a
consensus ($m=3$ for $K=5$ and $m=4$ for $K=7$), it may lead to consensus
error. However, even in the worst case, the consensus error is lower than
$1\%$, which demonstrates that the proposed AirCon protocol is reliable in
practice. By carefully examining the experimental results, we observe that the
consensus error is mainly caused by inaccurate downlink feedback.
Figure 17: Consensus error ratio for real testbed result with $K=5$ Figure 18:
Consensus error ratio for real testbed result with $K=7$
### 6.3 AirCon simulation setup
We further evaluate the performance of the proposed AirCon protocol under more
general conditions using simulations. In the simulation, the frame structure
is the same as the practical LTE system with a 1.92MHz sampling rate. The
channel gains for different users are generated independently.
In our simulation, three wireless channels are considered: 1) AWGN; 2) flat
fading; 3) EPA (Extended Pedestrian A model), which is a multi-path channel
model defined in the 3GPP TS 36.104[50] for the typical pedestrian wireless
environments. The path delay and the corresponding delay power of the EPA
channel are shown in Table I. We introduce the average CER (ACER) as a metric
to compare the performance difference among different simulation setups, which
is given by
$\text{ACER}=\frac{1}{K}\sum_{m=1}^{K}\text{CER}(m)$ (29)
TABLE I: Parameters of EPA channel Delay (ns) | 0 | 30 | 70 | 90 | 110 | 190 | 410
---|---|---|---|---|---|---|---
Power (dB) | 0 | -1 | -2 | -3 | -8 | -17.2 | -20.8
### 6.4 Simulation results
In Fig. 19, we show the ACER of the AirCon protocol under different channel
conditions. Firstly, it can be seen that the consensus performance is degraded
among all types of channels when SNR decreases. Secondly, the performance gap
between different types of channels is increasing in the low-SNR region, which
suggests that the channel estimation accuracy loss is the dominant factor of
the consensus performance loss in the low-SNR region. If the channel fading is
well pre-compensated (e.g., an AWGN case), the average consensus error ratio
of AirCon is about $1\%$ even if the SNR is only 0 dB.
Figure 19: Consensus error ratio under different SNR ($K=11$)
In the low-SNR region, the retransmission scheme can help improve consensus
accuracy. In Fig. 20, we show the consensus performance improvement under
different retransmission numbers. It can be observed that the consensus error
ratio decreases gradually with the increase of the retransmission number. Note
that the retransmission will consume more wireless resources. Therefore, the
trade-off between retransmission number and consensus performance needs to be
considered in practice.
Figure 20: Consensus error ratio under different repetition number (SNR = 0
dB)
We also study the influence of total user number ($K$) on consensus accuracy.
Due to space limitation, we only show the AWGN channel case. It can be
observed from Fig. 21 that the AirCon protocol performs better when $K$
increases. The improvement of consensus performance comes from two aspects:
(1) The consensus error only appears when $m$ is around the threshold for
reaching the consensus (namely, $m=\lceil{\frac{K}{2}}\rceil$ and
$m=\lceil{\frac{K}{2}}\rceil+1$ in our simulation). Therefore, the proportion
of cases that will not incur consensus error increases when the total number
of users increases. (2) Even in the cases that $m=\lceil{\frac{K}{2}}\rceil$
and $m=\lceil{\frac{K}{2}}\rceil+1$, the consensus error ratio also decreases,
which is illustrated in Fig. 22. That is because when more users participate
in the AirComp process, the average effect of noise on each user in the
superposed signal is reduced, so the consensus performance becomes better.
Figure 21: Averaging consensus error ratio under different $K$ (AWGN channel)
Figure 22: Consensus error ratio around threshold under different $K$ (SNR = 0
dB)
## 7 Conclusion
In this paper, we have proposed the AirCon to achieve low complexity consensus
for blockchain-enabled wireless networks, which is novel in that the hash
symbols of all users are transmitted to the BS simultaneously over the same
wireless spectrum via the AirComp and lattice coding techniques, and the
consensus can be done in the physical layer without decoding the hash symbols.
We have shown that the AirCon protocol can significantly reduce the
transmission and computational complexity during the consensus procedure. We
have shown the design of AirCon based on the universal LTE system and provided
solutions for the critical issues involved in the AirComp implementation, such
as channel estimation and feedback. We also have implemented AirCon based on a
srsLTE testbed, and real-world measurement results demonstrate that the
proposed AirCon protocol is feasible in a practical LTE system. In addition,
we have also conducted extensive simulation experiments and shown the accuracy
and robustness of the proposed consensus protocol under different network
conditions. The proposed AirCon is inspiring for designing efficient consensus
schemes over wireless networks, where a more efficient channel status
acquiring scheme can bring significant performance gain for AirCon.
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# A $3\times 3$ Lax form for the $q$-Painlevé equation of type $E_{6}$
Kanam Park
National Institude of Technology, Toba College,
1-1, Ikegami, Toba, Mie, 517-0012, Japan
Email<EMAIL_ADDRESS>
###### Abstract
For the $q$-Painlevé equation with affine Weyl group symmetry of type
$E_{6}^{(1)}$, a $2\times 2$ matrix Lax form and a second order scalar lax
form were known. We give a new $3\times 3$ matrix Lax form and a third order
scalar equation related to it. Continuous limit is also discussed.
Key Words and Phrases : Lax formalism, $q$-Painlevé equation,
2020 Mathematics Subject Classification Numbers: 14H70, 34M56, 39A13.
## 1 Introduction
The $q$-Painlevé equation with affine Weyl group symmetry of type
$E_{6}^{(1)}$ was first discovered in [8]. The well-known form of it is as
follows
$\begin{array}[]{c}\cfrac{(\overline{f}g-1)(fg-1)}{f\overline{f}}=\cfrac{(g-1/a_{5})(g-1/a_{6})(g-1/a_{7})(g-1/a_{8})}{(g-a_{3})(g-a_{4})},\vspace{12pt}\\\
\cfrac{(\overline{f}g-1)(\overline{f}\overline{g}-1)}{g\overline{g}}=\cfrac{(\overline{f}-a_{5})(f-a_{6})(f-a_{7})(f-a_{8})}{(\overline{f}-a_{1}/q)(\overline{f}-a_{2}/q)},\vspace{12pt}\\\
q=\cfrac{a_{1}a_{2}}{a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}},\end{array}$ ( 1.1)
where $f$ and $g$ are dependent variables, $a_{1}$, $a_{2}$, $\cdots$, $a_{8}$
are parameters, $q$ is a constant and the overline symbol $"\ \bar{}\ "$
denotes the discrete time evolution.
In previous works, the following Lax forms for the $q$-Painlevé equation for
type $E_{6}^{(1)}$ has been obtained. In [11], a $2\times 2$ matrix Lax pair
was derived as a reduction of the $q$-Garnier system. In [15], a second order
scalar Lax form was obtained as a reduction from the $q$-Painlevé equation of
type $E_{8}$. The relation between these Lax forms was given in [14].
In this article, we give a new Lax form with $3\times 3$ matrix Lax pair. We
derive such a Lax form as a special case of the system investigated in our
previous work [7]. As a result, we derive an equation which is equivalent to
the $q$-Painlevé equation of type $E_{6}^{(1)}$ [3], [9].
In the previous work [7], we defined a nonlinear $q$-difference system as a
connection preserving deformation of the following linear equation
$\begin{array}[]{l}\Psi(qz)=\Psi(z)A(z),\quad
A(z)=DX_{1}^{\varepsilon_{1}}(z)X_{2}^{\varepsilon_{2}}(z)\cdots
X_{M}^{\varepsilon_{M}}(z),\end{array}$ ( 1.2)
$\begin{array}[]{c}D=\mathrm{diag}[d_{1},d_{2},\cdots,d_{N}],\\\
X_{i}(z)=\mathrm{diag}[u_{1,i},u_{2,i},\cdots
u_{N,i}]+\Lambda,\quad\Lambda=\begin{bmatrix}0&1&&{\bf O}\\\
&\ddots&\ddots&\\\ &&&1\\\ z&&&0\end{bmatrix},\end{array}$ ( 1.3)
where the exponents are $\varepsilon_{i}=\pm 1$ $(1\leq i\leq M)$, $u_{j,i}$
($1\leq j\leq N$) are dependent variables and $c_{i},d_{j}$ are parameters
which satisfy
$\prod_{j=1}^{N}u_{j,i}=c_{i}.$ ( 1.4)
Since one can exchange the order of matrices $X_{i}^{\pm 1}$ by suitable
rational transformations of variables $u_{j,i}$, the equation (1.2)
essentially depends on $M_{+}$, $M_{-}$, where
$M_{\pm}=\\#\\{\varepsilon_{i}|\varepsilon_{i}=\pm 1\\}$.
The contents of this paper is as follows. In section 2, we set up a linear
$q$-difference equation (2.1), which is a case of $(M_{+},M_{-},N)=(3,0,3)$
for the equation (1.2). And we discuss about its two deformations. One
deformation gives rise to a well known form of the $q$-Painlevé equation of
type $E_{6}^{(1)}$, and the other deformation gives an equation for a non-
standard direction. In section 3, we derive a scalar equation from the
$3\times 3$ matrix equation (2.1) and consider its characteristic properties.
In section 4, we study continuous limit of our constructions and its relation
to the Boalch’s Lax pair [1].
###### Remark 1.1.
The equation (1.2) in case $(M_{+},M_{-},N)=(2n+2,0,2)$ is known that it is
equivalent to the linear $q$-difference equation related to the $2n$
dimensional $q$-Garnier system [10]. And in case of
$(M_{+},M_{-},N)=(2,0,2n+2)$, the equation (1.2) is also equivalent to the
linear $q$-difference equation related to the $2n$ dimensional system
$q$-$P_{(n+1,n+1)}$ [12]. In both cases, when $n=1$, they give rise to the
$q$-$P_{\rm VI}$ equation [4]. We note that the equation (2.1) is coincide to
the $q$-difference linear equation of the Lax form in [13] in case of ($m$,
$n$)=($3$, $1$).
## 2 A $3\times 3$ matrix Lax form
In this section, we consider two types of deformations for the linear
$q$-difference equation (1.2) in a case $(M_{+},M_{-},N)=(3,0,3)$.
We consider the connection preserving deformation for the following
$q$-difference equation for an unknown function
$\Psi(z)=[\Psi_{1}(z),\Psi_{2}(z),\Psi_{3}(z)]$,
$\begin{array}[]{c}\Psi(qz)=\Psi(z)A(z)=\Psi(z)A(z,t),\vspace{6pt}\\\
A(z)=DX_{1}(z)X_{2}(z)X_{3}(z)=\begin{bmatrix}b_{1}d_{1}&*&*\\\
0&b_{2}d_{2}&*\\\ 0&0&b_{3}d_{3}\end{bmatrix}+\begin{bmatrix}d_{1}&0&0\\\
*&d_{2}&0\\\ *&*&d_{3}\end{bmatrix}z,\end{array}$ ( 2.1)
where the matrices $D$ and $X_{i}(z)$ $(1\leq i\leq 3)$ stand for
$\begin{array}[]{c}D=\mathrm{diag}[d_{1},d_{2},d_{3}],\vspace{6pt}\\\
X_{i}(z)=\mathrm{diag}[u_{1,i},u_{2,i},u_{3,i}]+\Lambda,\vspace{6pt}\\\
\Lambda=\begin{bmatrix}0&1&0\\\ 0&0&1\\\ z&0&0\end{bmatrix},\end{array}$ (
2.2)
and $u_{j,i}$ ($1\leq i$, $j\leq 3$) are dependent variables and $c_{i},d_{j}$
are parameters which satisfy
$\begin{array}[]{c}\prod_{i=1}^{3}u_{j,i}=b_{j}\quad(1\leq j\leq
3),\quad\prod_{j=1}^{3}u_{j,i}=c_{i}\quad(1\leq i\leq 3),\vspace{6pt}\\\
b_{1}b_{2}b_{3}=c_{1}c_{2}c_{3}.\end{array}$ ( 2.3)
The first equation in (2.3) is equivalent to that the characteristic exponents
at $z=0$ of the equation (2.1) are $b_{j}d_{j}$. The second equation in (2.3)
is equivalent to the following condition
$|A(z)|=d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}).$ ( 2.4)
Through a gauge transformation by a $3\times 3$ diagonal matrix we can take
two components in (2.1) as 1. In the following, we use this kind of gauge
fixings in case by case. By the condition (2.4), two of the remaining four
components are determined by other components and parameters
$b_{j},c_{i},d_{j}$.
In this article, we will consider two deformations $T_{1}$ and $T_{2}$ for the
equation (2.1) which act on parameters $b_{j},\ c_{i},\ d_{j}$ as:
$\begin{array}[]{l}T_{1}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}})\to(b_{1},qb_{2},\cfrac{b_{3}}{q},c_{1},c_{2},c_{3},qd_{1},d_{2},qd_{3}),\vspace{6pt}\\\
T_{2}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}})\to(\cfrac{b_{1}d_{1}}{d_{2}},\cfrac{b_{2}d_{2}}{d_{3}},b_{3},c_{1},c_{2},c_{3}q,d_{2},d_{3},\cfrac{d_{1}}{q}).\end{array}$
( 2.5)
### 2.1 The deformation $T_{1}$
We will show that the deformation $T_{1}$ gives rise to the standard form of
the $q$-$E_{6}^{(1)}$. We consider the following $q$-difference linear
equation
$\begin{array}[]{c}\Psi(qz)=\Psi(z)A(z),\quad
A(z)=\begin{bmatrix}b_{1}d_{1}&1&v_{1}\\\ 0&b_{2}d_{2}&1\\\
0&0&b_{3}d_{3}\end{bmatrix}+\begin{bmatrix}d_{1}&0&0\\\ v_{2}&d_{2}&0\\\
v_{3}&v_{4}&d_{3}\end{bmatrix}z,\end{array}$ ( 2.6)
$|A(z)|=d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}).$ ( 2.7)
We take a deformation equation for the equation (2.6), (2.7) as follows:
$\begin{array}[]{c}T_{1}\Psi(z)=\Psi(z)B(z),\quad
B(z)=\begin{bmatrix}0&0&w_{1}\\\ 0&0&w_{2}\\\
0&0&w_{3}\end{bmatrix}+\begin{bmatrix}w_{4}&0&0\\\ w_{5}&0&0\\\
w_{6}&1&w_{7}\end{bmatrix}z,\vspace{6pt}\\\
|B(z)|=(w_{1}w_{5}-w_{2}w_{4})z^{2},\end{array}$ ( 2.8)
$T_{1}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}};v_{1},v_{4})\to(b_{1},qb_{2},\cfrac{b_{3}}{q},c_{1},c_{2},c_{3},qd_{1},d_{2},qd_{3};\overline{v_{1}},\overline{v_{4}}),$
( 2.9)
###### Theorem 2.1.
Through a compatibility condition of the equations (2.6), (2.9), (2.8)
$B(z)\overline{A(z)}=A(z)B(qz),$ ( 2.10)
we obtain the following equations
$\begin{array}[]{c}\cfrac{(fg-1)(\overline{f}g-1)}{f\overline{f}}=\cfrac{(g-1/c_{1})(g-1/c_{2})(g-1/c_{3})(g-d_{2}/b_{3}d_{3})}{(g-1/b_{2})(g-d_{2}/b_{1}d_{1})},\vspace{6pt}\\\
\cfrac{(\overline{f}g-1)(\overline{f}\overline{g}-1)}{g\overline{g}}=\cfrac{(\overline{f}-c_{1})(\overline{f}-c_{2})(\overline{f}-c_{3})(\overline{f}-b_{3}d_{3}/d_{2})}{(\overline{f}-b_{3}d_{3}/d_{1})(\overline{f}-b_{3}/q)},\end{array}$
( 2.11)
where,
$\begin{array}[]{c}f=\cfrac{b_{3}d_{3}}{d_{3}-v_{1}v_{4}},\quad
g=\cfrac{d_{2}v_{1}}{b_{2}d_{2}v_{1}-1},\end{array}$ ( 2.12)
and $\overline{*}$ stands for $T_{1}(*)$.
The equation (2.11) is the well known form of the $q$-Painlevé equation of
type $E_{6}$ [3], [9] (See also [5]).
###### Proof.
The result is obtained by a direct computation of the compatibility condition
of (2.10). Since the computation is rather heavy, we will give a comment how
to do it efficiently. Though the $2$ variables $v_{2}$, $v_{3}$ can be
represented by the rational functions of the remaining two variables $v_{1}$,
$v_{4}$ by the relation (2.7), it is more efficient to do this elimination
after the calculation of the compatibility condition (2.10) in $4$ variables,
and then reduce it to 2 variables. In this way, we get the following time
evolutions for $v_{1}$, $v_{4}$ as rational functions of $v_{1}$, $v_{4}$:
$\begin{array}[]{c}\overline{v_{1}}=\cfrac{C_{1}(v_{1},v_{4})C_{2}(v_{1},v_{4})}{D_{1}(v_{1},v_{4})D_{2}(v_{1},v_{4})},\quad\overline{v_{4}}=\cfrac{C_{3}(v_{1},v_{4})}{D_{3}(v_{1},v_{4})D_{4}(v_{1},v_{4})},\end{array}$
( 2.13)
where
$\begin{array}[]{rl}C_{3}(v_{1},v_{4}),D_{2}(v_{1},v_{4})&:\text{polynomials
in }v_{1},v_{4}\text{ of degree }(4,\ 3),\\\ C_{2}(v_{1},v_{4})&:\text{a
polynomial in }v_{1},v_{4}\text{ of degree }(3,\ 1),\\\
C_{1}(v_{1},v_{4}),D_{3}(v_{1},v_{4}),D_{4}(v_{1},v_{4})&:\text{polynomials in
}v_{1},v_{4}\text{ of degree }(2,\ 1),\\\ D_{1}(v_{1},v_{4})&:\text{a
polynomial in }v_{1},v_{4}\text{ of degree }(1,\ 0).\end{array}$ ( 2.14)
The remaining task is to rewrite the equation (2.13) to (2.11). A useful way
to solve it is to look at the singularities of the equations [2]. Namely, we
investigate the points at which the right hand side of the equations (2.13)
are indeterminate. We focus on the equation $D_{3}(v_{1},\ v_{4})$ of them
$D_{3}(v_{1},v_{4})=d_{3}\left(-b_{2}d_{2}v_{1}+b_{3}d_{2}v_{1}+1\right)+v_{1}v_{4}\left(b_{2}d_{2}v_{1}-1\right).$
( 2.15)
Investigating common zero points of the equation (2.15) and the other
polynomials $C_{k}(v_{1},\ v_{4})$, $D_{l}(v_{1},\ v_{4})$, we find 4
indeterminate points as follows:
$\begin{array}[]{c}(v_{1},v_{4})=\big{(}-\cfrac{1}{(u^{-1}-b_{2})d_{2}},-\cfrac{(u^{-1}-b_{3})(u^{-1}-b_{2})d_{3}d_{2}}{u^{-1}}\big{)},\vspace{6pt}\\\
(u=c_{1}^{-1},c_{2}^{-1},c_{3}^{-1},d_{2}/b_{3}d_{3}).\end{array}$ ( 2.16)
In view of the form of the points in (2.16), we define the variables $f$, $g$
as follows
$(v_{1},v_{1}v_{4})=\big{(}-\cfrac{1}{(g^{-1}-b_{2})d_{2}},\cfrac{(f-b_{3})d_{3}}{f}\big{)}.$
( 2.17)
By the transformation (2.17), the equation $D_{3}(v_{1},v_{4})=0$ is
transformed to an equation $fg=1$. Through the correspondence (2.17) and the
time evolution equations for the variables $v_{1}$, $v_{4}$ (2.13), we have
time evolution equations for $f$, $g$ (2.11). ∎
### 2.2 The deformation $T_{2}$
In this subsection, we take a deformation equation which is one of that
considered in the previous work [7]. It corresponded to the permutations of
the matrices $X_{i}(z)^{\pm 1}$. We consider the following Lax pair
$\begin{array}[]{c}\Psi(qz)=\Psi(z)A(z),\quad
A(z)=DX_{1}(z)X_{2}(z)X_{3}(z)=\begin{bmatrix}b_{1}d_{1}&1&*\\\
0&b_{2}d_{2}&1\\\ 0&0&b_{3}d_{3}\end{bmatrix}+\begin{bmatrix}d_{1}&0&0\\\
*&d_{2}&0\\\ *&*&d_{3}\end{bmatrix}z,\end{array}$ ( 2.18)
$\begin{array}[]{c}T_{2}{\Psi}(z)=\Psi(z)B(z),\quad
B(z)=X_{3}(z/q)^{-1},\vspace{6pt}\\\
|B(z)|=\cfrac{q}{z+qc_{3}},\vspace{6pt}\\\
T_{2}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}};x,y)\to(\cfrac{b_{1}d_{1}}{d_{2}},\cfrac{b_{2}d_{2}}{d_{3}},b_{3},c_{1},c_{2},c_{3}q,d_{2},d_{3},\cfrac{d_{1}}{q};\overline{x},\overline{y}),\end{array}$
( 2.19)
where we define variables $x$, $y$ and gauge freedom $w_{1}$, $w_{2}$ with the
variables $u_{j,i}$ in (2.18) as follows:
$x=\frac{u_{1,1}u_{1,2}u_{2,1}u_{2,2}}{u_{1,1}+u_{2,2}},\quad
y=\frac{1}{u_{1,1}u_{1,2}\left(u_{2,1}+u_{3,2}\right)},\quad
w_{1}=u_{1,1},\quad w_{2}=u_{1,3}.$ ( 2.20)
###### Theorem 2.2.
Through a compatibility condition of the equations (2.18), (2.19), (2.20),
$B(z)\overline{A(z)}=A(z)B(qz),$ ( 2.21)
we obtain the following equations
$\begin{array}[]{c}\cfrac{\overline{x}+c_{1}}{\overline{x}+c_{2}}=\cfrac{E_{1}E_{2}}{E_{3}E_{4}},\quad\cfrac{c_{1}\overline{y}+1}{c_{2}\overline{y}+1}=\cfrac{F_{1}F_{2}}{F_{3}F_{4}},\end{array}$
( 2.22)
where,
$\begin{array}[]{l}E_{1}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{1}d_{3}y\left(c_{2}+x\right)\right)+c_{1}d_{2}d_{3}x\left(c_{2}y+1\right),\\\
E_{2}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{3}d_{3}qy\left(c_{1}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{1}y+1\right),\\\
E_{3}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{2}d_{3}y\left(c_{1}+x\right)\right)+c_{2}d_{2}d_{3}x\left(c_{1}y+1\right),\\\
E_{4}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{3}d_{3}qy\left(c_{2}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{2}y+1\right),\\\
F_{1}=E_{1},F_{2}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{2}d_{3}y\left(c_{1}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{1}y+1\right),\\\
F_{3}=E_{3},F_{4}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{1}d_{3}y\left(c_{2}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{2}y+1\right),\end{array}$
( 2.23)
and $\overline{*}$ stands for $T_{2}(*)$.
###### Proof.
Solving a compatibility condition (2.21) with (2.20) for the variables
$\overline{x}$, $\overline{y}$, we obtain the following equations
$\overline{x}=\cfrac{(xy-1)G_{1}(x,y)}{H_{1}(x,y)},\quad\overline{y}=\cfrac{xG_{2}(x,y)}{H_{2}(x,y)},$
( 2.24)
where $G_{k}(x,y)$, $H_{k}(x,y)$ $(k=1,2)$ are polynomials in variables $x$,
$y$. The polynomial $G_{1}(x,y)$ is of degree $(1,1)$, $G_{2}(x,y)$ is of
degree $(1,2)$ and $H_{k}(x,y)$ are of degree $(2,2)$. Using a method [5] for
finding point configuration, a configuration of points for the equation (2.22)
is as follows:
$\begin{array}[]{rl}(x,y)=&(-c_{1},-\cfrac{1}{c_{1}}),(-c_{2},-\cfrac{1}{c_{2}}),(-\cfrac{b_{1}d_{1}}{d_{2}},-\cfrac{d_{2}}{b_{1}d_{1}}),(-\cfrac{b_{1}b_{2}}{c_{3}},-\cfrac{c_{3}}{b_{1}b_{2}}),\\\
&(-b_{2},0),(-\cfrac{b_{1}b_{2}d_{1}}{c_{3}d_{3}q},0),\\\
&(0,-\cfrac{1}{b_{1}}),(0,-\cfrac{b_{2}d_{2}}{c_{1}c_{2}d_{3}}).\end{array}$ (
2.25)
Calculating $\cfrac{\overline{x}+c_{1}}{\overline{x}+c_{2}}$ and
$\cfrac{c_{1}\overline{y}+1}{c_{2}\overline{y}+1}$ respectively, we have
equations (2.22), (2.23). ∎
###### Remark 2.1.
Here, the time evolution equations $\overline{u_{j,i}}$ are derived by solving
the compatibility condition (2.21). For another derivation using the
transformations which correspond to permutations of the matrices
$X_{i}^{\varepsilon_{i}}(z)$, see [7] Proposition 2.1.
Before ending this subsection, we show a relation between pairs of the
variables $(f,g)$ in (2.11) and $(x,y)$ in (2.22).
###### Proposition 2.1.
The Lax equations (2.6) with (2.12) and (2.18) with (2.20) are equivalent with
each other if the variables $f$, $g$ and $x$, $y$ are related as:
$\begin{array}[]{l}f=\cfrac{c_{1}c_{2}c_{3}\left(b_{2}+x\right)(xy-1)}{b_{1}b_{2}y\left(c_{1}+x\right)\left(c_{2}+x\right)+c_{3}x\left(b_{2}(1-xy)+c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)},\vspace{6pt}\\\
g=-\cfrac{y\left(b_{1}b_{2}+c_{3}x\right)}{c_{3}\left(b_{2}(1-xy)+x\right)+b_{1}b_{2}xy}\end{array}$
( 2.26)
###### Proof.
Comparing the coefficient matrix $A(z)$ of the equation (2.6), (2.12) to the
coefficient matrix $A(z)$ of the equation (2.18), (2.20), we can solve for the
variables $x$, $y$ and gauge freedom $w_{1}$, $w_{2}$ in terms of variables
$f$, $g$ and parameters $b_{j}$, $c_{i}$, $d_{j}$ ($1\leq i,j\leq 3$). Then we
have the desired relation between pairs of variables $(f,g)$ and $(x,y)$
(2.26). ∎
## 3 A scalar equation related to the $q$-$E_{6}^{(1)}$
In this section, we derive a scalar $q$-difference equation from the matrix
$q$-difference equation (2.18), (2.20) for an unknown function
$\Psi(z)=[\Psi_{1}(z),\Psi_{2}(z),\Psi_{3}(z)]$ and its geometric properties.
Eliminating functions $\Psi_{2}(z)$, $\Psi_{3}(z)$ in the equation (2.18),
(2.20), we obtain the following third linear $q$-difference equation for
$\Psi_{1}(z)=:\Phi(z)$
$L(z):=P_{3}(z)\Phi(q^{3}z)+P_{2}(z)\Phi(q^{2}z)+P_{1}(z)\Phi(qz)+P_{0}(z)\Phi(z)=0,$
( 3.1)
where
$\begin{array}[]{l}P_{3}(z)=p_{31}(z-u),\\\
P_{2}(z)=p_{22}z^{2}+p_{21}z+p_{20},\\\
P_{1}(z)=p_{13}z^{3}+p_{12}z^{2}+p_{11}z+p_{10},\\\
P_{0}(z)=-P_{3}(qz)d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}).\end{array}$ (
3.2)
Here, the coefficients $p_{k,l}$ $(1\leq k\leq 3,0\leq l\leq 3)$ in the
polynomials $P_{k}(z)$ (3.2) depend on parameters $b_{j}$, $c_{i}$, $d_{j}$,
and a variable $u$ defined as the zero of $P_{3}(z)$. The variable $u$ is
expressed in terms of $x$, $y$ as follows
$\begin{array}[]{l}u=\cfrac{y\left(b_{1}\left(c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)+c_{1}c_{2}(1-xy)\right)}{b_{2}\left(b_{1}y+1\right)(xy-1)}\vspace{6pt}\\\
\cfrac{\left(b_{2}d_{3}y\left(b_{1}\left(c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)+c_{1}c_{2}(1-xy)\right)+b_{2}^{2}d_{2}\left(b_{1}y+1\right)(1-xy)+c_{3}d_{3}x\left(c_{1}y+1\right)\left(c_{2}y+1\right)\right)}{\left(d_{2}\left(b_{2}\left(b_{1}y+1\right)(xy-1)-x\left(c_{1}y+1\right)\left(c_{2}y+1\right)\right)-d_{3}y\left(b_{1}\left(c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)+c_{1}c_{2}(1-xy)\right)\right)}.\end{array}$
( 3.3)
Explicit forms of the polynomials $P_{j}(z)$ ($0\leq j\leq 3$) (3.2) are given
in Appendix.
Then we have
###### Lemma 3.1.
The equation $L(z)=0$ (3.1) has the following properties:
1. (i)
It is a linear four term equation between $\Phi(q^{j}z)$ ($0\leq j\leq 3$) and
its coefficients $P_{j}(z)$ are polynomials for $z$ of degree $4-j$,
2. (ii)
A polynomial $P_{0}(z)$ has four zero points at $z=-c_{i}$ ($1\leq i\leq 3$),
$u/q$,
3. (iii)
The exponents of solutions $\Phi(z)$ are $b_{1}d_{1}$, $qb_{2}d_{2}$,
$qb_{3}d_{3}$ (at $z=0$) and $d_{1}$, $d_{2}$, $d_{3}$ (at $z=\infty$),
4. (iv)
A point $z=u$ such that $P_{3}(z)=0$ is an apparent singularity, namely we
have
$v:=\cfrac{P_{0}(u)}{P_{1}(u/q)}=\cfrac{P_{1}(u)}{P_{2}(u/q)}=\cfrac{P_{2}(u)}{P_{3}(u/q)}.$
( 3.4)
Conversely, the equation $L(z)=0$ (3.1) is uniquely characterized by these
properties (i)-(iv) up to normalization.
###### Proof.
The properties (i)-(iv) follows by computation through eliminating
$\Psi_{2}(z)$, $\Psi_{3}(z)$ in (2.18), (2.20). The converse can be confirmed
that coefficients $P_{j}(z)$ are defined uniquely by (i)-(iv) up to a
normalization. To see this, we consider the following equation which satisfies
the properties (i), (ii):
$L^{\prime}(z)=P_{3}^{\prime}(z)\Phi(q^{3}z)+P_{2}^{\prime}(z)\Phi(q^{2}z)+P_{1}^{\prime}(z)\Phi(qz)+P_{0}^{\prime}(z)\Phi(z)=0,$
( 3.5) $\begin{array}[]{l}P_{3}^{\prime}(z)=p_{31}^{\prime}(z-u),\\\
P_{2}^{\prime}(z)=p_{22}^{\prime}z^{2}+p_{21}^{\prime}z+p_{20}^{\prime},\\\
P_{1}^{\prime}(z)=p_{13}^{\prime}z^{3}+p_{12}^{\prime}z^{2}+p_{11}^{\prime}z+p_{10}^{\prime},\\\
P_{0}^{\prime}(z)=p_{04}^{\prime}(z-u/q)(z+c_{1})(z+c_{2})(z+c_{3}).\end{array}$
( 3.6)
From the property (iii), the condition of the exponents of solutions $\Phi(z)$
at $z=0$ determines the coefficients $p_{31}^{\prime}$, $p_{20}^{\prime}$,
$p_{10}^{\prime}$ and the condition of the exponents of solutions $\Phi(z)$ at
$z=\infty$ determines the coefficients $p_{22}^{\prime}$, $p_{13}^{\prime}$.
The remaining coefficients except for $p_{04}^{\prime}$ are determined by the
property (iv). If we put the normalization factor $p_{04}^{\prime}$ as
$q^{5}uvc_{1}c_{2}d_{1}d_{2}d_{3}$, the function $L^{\prime}(z)$ (3.5) equals
to the function $L(z)$ (3.1). ∎
In the following, viewing $\Phi(q^{i}z)$ $(0\leq i\leq 3)$ as parameters, we
regard the scalar $q$-difference equation $L(z)=0$ (3.1) as an algebraic curve
in variables $(u,v)$ $\in$ $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We represent
as the curve as $P(u,v)=0$. The features of the curve are the following.
###### Lemma 3.2.
The algebraic curve $P(u,v)=0$ has the following properties:
1. (i)
The polynomial $P(u,v)$ has the following form
$P(u,v)=\displaystyle\sum_{0\leq j\leq 3\atop 0\leq i+j\leq
4}c_{i,j}u^{i}v^{j},\quad c_{0,0}:=c_{0}z^{2}\Phi(qz).$ ( 3.7)
The coefficients $c_{i,j}$ depend on $b_{i}$, $c_{i}$, $d_{i}$, $q$, $z$,
$\Phi(q^{i}z)$.
2. (ii)
It passes the following $8$ points
$\begin{array}[]{c}(u,v)=(0,qb_{1}d_{1}),\quad(0,q^{2}b_{2}d_{2}),\quad(0,q^{2}b_{3}d_{3}),\quad(qz,\infty),\vspace{6pt}\\\
(z,0),\quad(-c_{1},0),\quad(-c_{2},0),\quad(-c_{3},0),\end{array}$ ( 3.8)
and 3 points in the coordinate $(r,s)=(u,v/u)$
$(r,s)=(\infty,q^{2}d_{1}),\quad(\infty,q^{2}d_{2}),\quad(\infty,q^{2}d_{3}).$
( 3.9)
3. (iii)
At $u=z$ the equation $P(u,v)=0$ has the following property
$\begin{array}[]{c}\displaystyle\big{(}\sum_{0\leq i\leq
3-j}c_{i,j+1}z^{i}\big{)}\Big{|}_{z\to
qz,\Phi(q^{k}z)\to\Phi(q^{k+1}z)}=\sum_{0\leq i\leq
4-j}c_{i,j}(qz)^{i}\quad(j=0,1,2).\end{array}$ ( 3.10)
Conversely, the equation $P(u,v)=0$ is uniquely characterized by these
properties (i)-(iii) up to normalization factor $c_{0}$.
###### Proof.
The properties (i)-(iii) follow for the polynomial $P(u,v)$. Conversely, we
consider a polynomial $P^{\prime}(u,v):=\displaystyle\sum_{0\leq j\leq 3\atop
0\leq i+j\leq 4}c_{i,j}^{\prime}u^{i}v^{j}$,
$c_{0,0}^{\prime}(z)=c_{0}^{\prime}\Phi(qz)$. The polynomial $P^{\prime}(u,v)$
has $14$ coefficients. From the property (ii), $10$ coefficients are described
in terms of parameters $b_{j}$, $c_{j}$, $d_{j}$ and the coefficient
$c_{0}^{\prime}$.
$\begin{array}[]{l}P^{\prime}(u,v)=-\cfrac{c_{0}^{\prime}z(qz-u)\Phi(qz)}{b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{6}}v^{3}\vspace{6pt}\\\
+\left(uc_{1,2}^{\prime}+\cfrac{c_{0}^{\prime}z\Phi(qz)\left(b_{2}d_{2}qz+b_{3}d_{3}qz+b_{1}d_{1}z-d_{1}u^{2}-d_{2}u^{2}-d_{3}u^{2}\right)}{b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{4}}\right)v^{2}\vspace{6pt}\\\
+\left(u^{2}c_{2,1}^{\prime}+uc_{1,1}^{\prime}-\cfrac{c_{0}^{\prime}z\Phi(qz)\left(b_{2}b_{3}d_{2}d_{3}qz+b_{1}b_{2}d_{1}d_{2}z+b_{1}b_{3}d_{1}d_{3}z-d_{1}d_{2}u^{3}-d_{1}d_{3}u^{3}-d_{2}d_{3}u^{3}\right)}{b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{2}}\right)v\vspace{6pt}\\\
+\cfrac{c_{0}^{\prime}z\left(c_{1}+u\right)\left(c_{2}+u\right)(z-u)\left(c_{3}+u\right)\Phi(qz)}{b_{1}b_{2}b_{3}}.\end{array}$
( 3.11)
The remaining $3$ parameters $c_{1,1}^{\prime}$, $c_{1,2}^{\prime}$,
$c_{2,1}^{\prime}$ are determined in terms of parameters $b_{i}$, $c_{i}$,
$d_{i}$, $q$, $z$, $\Phi(q^{i}z)$, $c_{0}^{\prime}$ by the property (iii).
Namely the property (iii) gives $3$ linear inhomogeneous equations among
$c_{1,1}^{\prime}$, $c_{1,2}^{\prime}$, $c_{2,1}^{\prime}$,
$c_{1,1}^{\prime}\big{|}_{z\to qz}$, $c_{1,2}^{\prime}\big{|}_{z\to qz}$,
$c_{2,1}^{\prime}\big{|}_{z\to qz}$. Though these relations are apparently
$q$-difference equations, we can solve them algebraically. For example, in the
equation (3.10), we solve $c_{12}^{\prime}$ when $j=2$. Then when $j=0$, we
solve $\left.c_{21}^{\prime}\right|_{z\to qz,\Phi(q^{k}z)\to\Phi(q^{k+1}z)}$.
And finally solving $c_{11}^{\prime}$ when $j=1$, they algebraically can be
solved. ∎
Explicit forms of the coefficients $c_{i,j}(z)$ of the polynomial $P(u,v)$ are
in Appendix.
### 3.1 Relations among pairs of variables $(f,g)$, $(x,y)$ and $(u,v)$
###### Proposition 3.1.
Under the relations among variables ($f$, $g$), ($x$, $y$) and ($u$, $v$):
$\begin{array}[]{l}(f,g)=\vspace{6pt}\\\
\left(\cfrac{c_{1}c_{2}c_{3}\left(b_{2}+x\right)(xy-1)}{b_{1}b_{2}y\left(c_{1}+x\right)\left(c_{2}+x\right)+c_{3}x\left(b_{2}(1-xy)+c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)},-\cfrac{y\left(b_{1}b_{2}+c_{3}x\right)}{c_{3}\left(b_{2}(1-xy)+x\right)+b_{1}b_{2}xy}\right),\end{array}$
( 3.12) $\begin{array}[]{l}(x,y)=\vspace{6pt}\\\
\Big{(}\Big{(}b_{2}v\left(-b_{2}d_{2}q^{2}-d_{3}q^{2}u+v\right)\\\
\quad\quad\left(c_{1}c_{2}d_{3}q^{2}\left(c_{3}\left(b_{2}d_{2}q^{2}-v\right)+b_{2}d_{2}q^{2}u\right)+b_{1}b_{2}\left(d_{2}q^{2}\left(-b_{2}v+d_{3}q^{2}u\left(c_{3}+u\right)-uv\right)+v\left(v-d_{3}q^{2}u\right)\right)\right)\Big{)}\\\
\cdot\Big{(}q^{2}(b_{2}^{2}d_{2}(d_{2}q^{2}\left(d_{3}q^{2}\left(c_{1}+u\right)\left(c_{3}+u\right)-uv\right)\left(d_{3}q^{2}\left(c_{2}+u\right)\left(c_{3}+u\right)-uv\right)\\\
\quad-v\left(-d_{3}q^{2}v\left(b_{1}\left(2c_{3}+u\right)+u\left(c_{3}+2u\right)\right)+d_{3}^{2}q^{4}u\left(b_{1}+u\right)\left(c_{3}+u\right)+uv^{2}\right))\\\
\quad+b_{2}^{3}d_{2}^{2}\left(-q^{2}\right)v\left(d_{3}q^{2}\left(b_{1}+u\right)\left(c_{3}+u\right)-uv\right)-b_{2}c_{3}d_{3}v(b_{1}v\left(v-d_{3}q^{2}u\right)\\\
\quad+d_{2}q^{2}\left(c_{2}u\left(d_{3}q^{2}\left(c_{3}+u\right)-v\right)+c_{1}\left(d_{3}q^{2}\left(2c_{2}+u\right)\left(c_{3}+u\right)-uv\right)\right))+c_{1}c_{2}c_{3}^{2}d_{3}^{2}q^{2}v^{2})\Big{)}^{-1},\\\
\cfrac{b_{2}d_{2}q^{2}u\left(d_{3}q^{2}\left(c_{3}+u\right)-v\right)}{c_{1}c_{2}d_{3}q^{2}\left(c_{3}\left(b_{2}d_{2}q^{2}-v\right)+b_{2}d_{2}q^{2}u\right)+b_{1}b_{2}v\left(-b_{2}d_{2}q^{2}-d_{3}q^{2}u+v\right)}\Big{)},\end{array}$
( 3.13)
the corresponding Lax equations (2.6) with (2.17), (2.18) with (2.20) and
(2.18) with (3.3), (3.4) are equivalent. Conversely, such relations among
$(f,g)$, $(x,y)$ and $(u,v)$ are uniquely determined as (3.12), (3.13).
###### Proof.
Using the relation (3.12), we can check that the equation (2.6) with (2.17) is
equivalent to (2.18) with (2.20). Similarly, using the relation (3.13), we can
check that the equation (2.18) with (2.20) is equivalent to (2.18) with with
(3.3), (3.4). The converse is obvious from the form of the Lax matrix. ∎
## 4 Continuous limit
In this section, we describe a relation between our result and the result of
Boalch [1]. in [1], a Lax pair for the additional-difference Painlevé equation
with affine Weyl symmetry group of type $E_{6}$ was described. The linear
differential equation of the Lax pair is as follows
$\begin{array}[]{c}\cfrac{d}{dz}\Psi(z)=\Psi(z)\left(\cfrac{A_{1}^{b}}{z}+\cfrac{A_{2}^{b}}{z-1}\right),\quad
A_{3}^{b}:=-(A_{1}^{b}+A_{2}^{b}),\\\ \end{array}$ ( 4.1)
where the matrices $A_{i}^{b}$ ($1\leq i\leq 3$) are $3\times 3$ matrices with
different eigenvalues.
We show that the linear $q$-difference equation (2.1) reduces to the equation
(4.1) via a continuous limit $q\to 1$. The equation (2.1) takes the following
form after a scale transformation $z\to-z$ and gauge transformations
$\begin{array}[]{c}(1-z)\Psi(qz)=\Psi(z)\mathcal{A}(z),\quad\mathcal{A}(z)=\begin{bmatrix}b_{1}d_{1}&k_{1}&v_{1}\\\
0&b_{2}d_{2}&k_{2}\\\ 0&0&b_{3}d_{3}\end{bmatrix}-\begin{bmatrix}d_{1}&0&0\\\
v_{2}&d_{2}&0\\\ v_{3}&v_{4}&d_{3}\end{bmatrix}z,\vspace{6pt}\\\
|\mathcal{A}(z)|=d_{1}d_{2}d_{3}(z-c_{1})(z-c_{2})(z-c_{3}),\end{array}$ (
4.2)
where $k_{j}$ ($j=1$, $2$) are constants. We put $q=e^{h}$ and consider the
limit $h\to 0$. We set
$\begin{array}[]{c}b_{i}=q^{\beta_{i}},\quad c_{i}=q^{\gamma_{i}},\quad
d_{i}=q^{\delta_{i}}\quad(1\leq i\leq 3),\vspace{6pt}\\\
k_{j}=hl_{j}\quad(j=1,2),\quad v_{m}=hu_{m}\quad(1\leq m\leq 4),\end{array}$ (
4.3)
where $l_{j}$ are constants. By using Taylor’s expansion for (4.2), (4.3)
$\begin{array}[]{rl}(LHS)&=(1-z)\Psi(z)+h(1-z)z\cfrac{d}{dz}\Psi(z)+\mathcal{O}(h^{2}),\vspace{6pt}\\\
(RHS)&=(1-z)\Psi(z)+h\Psi(z)\begin{bmatrix}\beta_{1}-\delta_{1}(z-1)&l_{1}&u_{1}\\\
-u_{2}z&\beta_{2}-\delta_{2}(z-1)&l_{2}\\\
-u_{3}z&-u_{4}z&\beta_{3}-\delta_{3}(z-1)\end{bmatrix}+\mathcal{O}(h^{2}),\end{array}$
( 4.4)
we find the following limit as $h\to 0$
$\begin{array}[]{c}\cfrac{d}{dz}\Psi(z)=\Psi(z)\left(\cfrac{A_{1}}{z}+\cfrac{A_{2}}{z-1}\right),\vspace{6pt}\\\
A_{1}=\begin{bmatrix}\beta_{1}+\delta_{1}&l_{1}&u_{1}\\\
0&\beta_{2}+\delta_{2}&l_{2}\\\ 0&0&\beta_{3}+\delta_{3}\end{bmatrix},\quad
A_{2}=\begin{bmatrix}-\beta_{1}&-l_{1}&-u_{1}\\\ u_{2}&-\beta_{2}&-l_{2}\\\
u_{3}&u_{4}&-\beta_{3}\end{bmatrix},\vspace{6pt}\\\
A_{3}:=-(A_{1}+A_{2})=\begin{bmatrix}\delta_{1}&0&0\\\ u_{2}&\delta_{2}&0\\\
u_{3}&u_{4}&\delta_{3}\end{bmatrix},\end{array}$ ( 4.5)
where eigenvelues of the matrix $A_{2}$ are $\gamma_{i}$ ($1\leq i\leq 3$) by
the condition of the determinant of the matrix $\mathcal{A}(z)$ (4.2).
Therefore the linear $q$-difference equation (2.1) reduces to the equation
(4.1) via a continuous limit $q\to 1$.
In the following, we consider a continuous limit $q\to 1$ of the result in
§2.1. In the subsection §2.1, through a compatibility condition of the
equations (2.10), we derived a standard $q$-Painlevé equation of type $E_{6}$.
We take the following equation as a deformation equation for the differential
equation (4.5) which is rewritten version of (2.9)
$\begin{array}[]{c}T\Psi(z)=B(z)\Psi(z),\quad
B(z)=\begin{bmatrix}w_{1}&w_{2}&w_{3}\\\ 0&0&0\\\
0&0&0\end{bmatrix}+\begin{bmatrix}w_{4}&0&0\\\ 1&0&0\\\
w_{5}&w_{6}&w_{7}\end{bmatrix}z,\vspace{6pt}\\\
|B(z)|=(w_{3}w_{6}-w_{2}w_{7})z^{2},\vspace{6pt}\\\
T:(\beta_{1},\beta_{2},\beta_{3},\gamma_{1},\gamma_{2},\gamma_{3},\delta_{1},\delta_{2},\delta_{3})\to(\beta_{1}-1,\beta_{2}+1,\beta_{3},\gamma_{1},\gamma_{2},\gamma_{3},\delta_{1}+1,\delta_{2},\delta_{3}+1).\end{array}$
( 4.6)
Solving a compatibility condition for the equation (4.5), (4.6), we obtain the
following additional-difference Painlevé equation of type $E_{6}$ [5], [9]
$\begin{array}[]{c}(f+g)(\overline{f}+g)=\cfrac{\left(g+\gamma_{1}\right)\left(g+\gamma_{2}\right)\left(g+\gamma_{3}\right)\left(g+\beta_{1}+\delta_{1}-\delta_{2}\right)}{\left(g+\beta_{2}+1\right)\left(g+\beta_{3}-\delta_{2}+\delta_{3}+1\right)},\vspace{6pt}\\\
(\overline{f}+\overline{g})(\overline{f}+g)=\cfrac{(\overline{f}-\gamma_{1})(\overline{f}-\gamma_{2})(\overline{f}-\gamma_{3})(\overline{f}-\beta_{1}-\delta_{1}+\delta_{2})}{(\overline{f}-\beta_{1}+\beta_{2}-1)(\overline{f}-\beta_{1}-\delta_{1}+\delta_{3}+1)}.\end{array}$
( 4.7)
where,
$f=\beta_{3}+\frac{u_{1}u_{4}}{l_{1}},\quad
g=\frac{l_{1}l_{2}}{u_{1}}-\beta_{2},$ ( 4.8)
and $\overline{*}$ stands for $T(*)$. From the above, we derive the
additional-difference Painlevé equation of type $E_{6}$ solving a
compatibility condition of the Lax pair via a continuous limit $q\to 1$.
## Acknowledgement
The author would like to express her gratitude to Professor Yasuhiko Yamada
for valuable suggestions and encouragement. She also thanks supports from JSPS
KAKENHI Grant Numbers 17H06127 and 26287018 for the travel expenses in
accomplishing this study.
## Appendix
In the appendix, we give an explicit forms of $P_{j}(z)$ (3.2)
$\begin{array}[]{l}P_{3}(z)=p_{31}(z-u),\\\
P_{2}(z)=p_{22}z^{2}+p_{21}z+p_{20},\\\
P_{1}(z)=p_{13}z^{3}+p_{12}z^{2}+p_{11}z+p_{10},\\\
P_{0}(z)=-P_{3}(qz)d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}),\end{array}$ (
4.9)
where the coefficients $p_{j,k}$ $(1\leq j\leq 3,0\leq k\leq 3)$ are
$\begin{array}[]{l}p_{31}=-q^{2}uvc_{1}c_{2},\vspace{6pt}\\\
p_{22}=q^{4}uvc_{1}c_{2}\left(d_{1}+d_{2}+d_{3}\right),\vspace{6pt}\\\
p_{21}=-quvc_{1}c_{2}\left(ud_{3}q^{3}+\left(qu-
b_{2}\right)d_{2}q^{2}-b_{3}d_{3}q^{2}-vq+v+\left(q^{3}u-qb_{1}\right)d_{1}\right),\vspace{6pt}\\\
p_{20}=-q^{2}u^{2}vc_{1}c_{2}\left(b_{1}d_{1}+qb_{2}d_{2}+qb_{3}d_{3}\right),\vspace{6pt}\\\
p_{13}=-q^{5}uvc_{1}c_{2}\left(d_{1}d_{2}+d_{2}d_{3}+d_{3}d_{1}\right),\vspace{6pt}\\\
p_{12}=-q(ub_{1}b_{2}b_{3}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{5}+uc_{1}^{2}c_{2}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{5}\vspace{6pt}\\\
\qquad+c_{1}(u^{2}c_{2}^{2}d_{1}d_{2}d_{3}q^{5}+ub_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{5}\vspace{6pt}\\\
\qquad-
c_{2}(qd_{1}(qu\left(qud_{2}\left(-ud_{3}q^{2}+vq+v\right)+v\left(q(q+1)ud_{3}-v\right)\right)\vspace{6pt}\\\
\qquad-
b_{1}\left(v-q^{2}b_{2}d_{2}\right)\left(v-q^{2}b_{3}d_{3}\right))+v(d_{2}(u\left(q(q+1)ud_{3}-v\right)\vspace{6pt}\\\
\qquad+b_{2}\left(q^{2}b_{3}d_{3}-v\right))q^{2}+v\left(v-q^{2}\left(u+b_{3}\right)d_{3}\right))))),\vspace{6pt}\\\
p_{11}=u(ub_{1}b_{2}b_{3}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{6}+uc_{1}^{2}c_{2}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{6}\vspace{6pt}\\\
\qquad+c_{1}(u^{2}c_{2}^{2}d_{1}d_{2}d_{3}q^{6}+ub_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{6}\vspace{6pt}\\\
\qquad-
c_{2}(qd_{1}(b_{1}\left(b_{2}d_{2}\left(-b_{3}d_{3}q^{3}+vq+v\right)q^{2}+v\left(q^{2}(q+1)b_{3}d_{3}-v\right)\right)\vspace{6pt}\\\
\qquad-
qu\left(v-q^{2}ud_{2}\right)\left(v-q^{2}ud_{3}\right))+v(d_{2}(u\left(q^{2}ud_{3}-v\right)\vspace{6pt}\\\
\qquad+b_{2}\left(q^{2}(q+1)b_{3}d_{3}-v\right))q^{2}+v\left(v-q^{2}\left(u+b_{3}\right)d_{3}\right))))),\vspace{6pt}\\\
p_{10}=q^{3}u^{2}vc_{1}c_{2}\left(qb_{2}b_{3}d_{2}d_{3}+b_{1}d_{1}\left(b_{2}d_{2}+b_{3}d_{3}\right)\right).\end{array}$
( 4.10)
We give also an explicit forms of the coefficients $c_{ij}$ $(0\leq j\leq
3,0\leq i+j\leq 4)$ of the polynomial $P(u,v)$ in variables $u$, $v$:
$\begin{array}[]{l}c_{01}=c_{0}\cfrac{z^{2}}{q^{2}}\left(-\cfrac{q}{b_{1}d_{1}}-\cfrac{1}{b_{2}d_{2}}-\cfrac{1}{b_{3}d_{3}}\right)\Phi(qz),\vspace{6pt}\\\
c_{02}=c_{0}\cfrac{z^{2}\left(b_{1}d_{1}+qb_{2}d_{2}+qb_{3}d_{3}\right)\Phi(qz)}{q^{4}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{03}=-c_{0}\cfrac{z^{2}\Phi(qz)}{q^{5}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{10}=c_{0}\left(\cfrac{c_{1}c_{2}z^{2}}{b_{1}b_{2}b_{3}}+\cfrac{z^{2}}{c_{1}}+\cfrac{z^{2}}{c_{2}}-z\right)\Phi(qz),\vspace{6pt}\\\
c_{11}=-c_{0}\cfrac{z}{q^{4}b_{1}b_{2}b_{3}c_{1}c_{2}d_{1}d_{2}d_{3}}(b_{1}d_{1}(qb_{2}d_{2}\left(q^{2}b_{3}\left(z+c_{1}\right)\left(z+c_{2}\right)d_{3}\Phi(z)-(q+1)c_{1}c_{2}\Phi(qz)\right)\vspace{6pt}\\\
\quad\qquad+c_{1}c_{2}\left(\Phi\left(q^{2}z\right)-q(q+1)b_{3}d_{3}\Phi(qz)\right))\vspace{6pt}\\\
\quad\qquad+c_{1}c_{2}(zd_{3}\Phi\left(q^{2}z\right)q^{2}+zd_{1}\left(-qzd_{3}\Phi(qz)+qd_{2}\left(\left(z+c_{1}\right)\left(z+c_{2}\right)d_{3}\Phi(z)-z\Phi(qz)\right)+\Phi\left(q^{2}z\right)\right)q^{2}\vspace{6pt}\\\
\quad\qquad+b_{3}d_{3}\Phi\left(q^{2}z\right)q-d_{2}\left(q\left(qz^{2}+(q+1)b_{2}b_{3}\right)d_{3}\Phi(qz)-\left(qz+b_{2}\right)\Phi\left(q^{2}z\right)\right)q-\Phi\left(q^{3}z\right))),\vspace{6pt}\\\
c_{12}=c_{0}\cfrac{z}{q^{5}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}}(zd_{3}\Phi(qz)q^{2}+\left(qz-
b_{2}\right)d_{2}\Phi(qz)q-b_{3}d_{3}\Phi(qz)q-\Phi\left(q^{2}z\right)q\vspace{6pt}\\\
\qquad\qquad+\left(q^{2}z-b_{1}\right)d_{1}\Phi(qz)+\Phi\left(q^{2}z\right)),\vspace{6pt}\\\
c_{13}=c_{0}\cfrac{z\Phi(qz)}{q^{6}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{20}=c_{0}\left(\cfrac{c_{1}z\left(z-c_{2}\right)z+z^{2}c_{2}}{b_{1}b_{2}b_{3}}-\cfrac{-z^{2}+c_{1}z+c_{2}z}{c_{1}c_{2}}\right)\Phi(qz),\vspace{6pt}\\\
c_{21}=c_{0}\cfrac{b_{1}d_{1}}{q^{4}b_{1}b_{2}b_{3}c_{1}c_{2}d_{1}d_{2}d_{3}}(qb_{2}d_{2}\left(qb_{3}\left(z+c_{1}\right)\left(z+c_{2}\right)d_{3}\Phi(z)-c_{1}c_{2}\Phi(qz)\right)\vspace{6pt}\\\
\qquad\quad+c_{1}c_{2}\left(\Phi\left(q^{2}z\right)-qb_{3}d_{3}\Phi(qz)\right))-c_{0}c_{1}c_{2}(zd_{1}(-\left(z+c_{1}\right)\left(z+c_{2}\right)d_{2}d_{3}\Phi(z)+(q+1)zd_{2}\Phi(qz)\vspace{6pt}\\\
\qquad\quad+(q+1)zd_{3}\Phi(qz)-\Phi\left(q^{2}z\right))q^{2}-zd_{3}\Phi\left(q^{2}z\right)q^{2}-b_{3}d_{3}\Phi\left(q^{2}z\right)q+d_{2}(q\left((q+1)z^{2}+b_{2}b_{3}\right)d_{3}\Phi(qz)\vspace{6pt}\\\
\qquad\quad-\left(qz+b_{2}\right)\Phi\left(q^{2}z\right))q+\Phi\left(q^{3}z\right)),\vspace{6pt}\\\
c_{22}=-c_{0}\cfrac{z\left(d_{1}+d_{2}+d_{3}\right)\Phi(qz)}{q^{4}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{30}=c_{0}\left(-\cfrac{-z^{2}+c_{1}z+c_{2}z}{b_{1}b_{2}b_{3}}-\cfrac{1}{c_{1}c_{2}}\right)\Phi(qz),\vspace{6pt}\\\
c_{31}=c_{0}\cfrac{\left(d_{1}d_{2}+d_{2}d_{3}+d_{3}d_{1}\right)z\Phi(qz)}{q^{2}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{40}=-c_{0}\cfrac{z\Phi(qz)}{b_{1}b_{2}b_{3}}.\end{array}$ ( 4.11)
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|
2022
# Entropy fluctuation formulas of fermionic Gaussian states
Youyi Huang Lu Wei Department of Computer Science
Texas Tech University
Lubbock, 79409, Texas, USA
###### Abstract
We study the statistical behaviour of quantum entanglement in bipartite
systems over fermionic Gaussian states as measured by von Neumann entropy. The
formulas of average von Neumann entropy with and without particle number
constrains have been recently obtained, whereas the main results of this work
are the exact yet explicit formulas of variances for both cases. For the
latter case of no particle number constrain, the results resolve a recent
conjecture on the corresponding variance. Different than existing methods in
computing variances over other generic state models, the key ingredient in
proving the results of this work relies on a new simplification framework. The
framework consists of a set of new tools in simplifying finite summations of
what we refer to as dummy summation and re-summation techniques. As a
byproduct, the proposed framework leads to various new transformation formulas
of hypergeometric functions.
###### keywords:
von Neumann entropy, fermionic Gaussian states, quantum entanglement, random
matrix theory, orthogonal polynomials, special functions
## 1 Introduction and main results
Quantum entanglement is the most important feature in quantum mechanics. The
understanding of the phenomenon of entanglement is crucial in realizing the
revolutionary advances of quantum science. In the emerging field of quantum
information processing, quantum entanglement is also the resource and medium
that enable the underlying quantum technologies.
In this work, we study the statistical behaviour of entanglement over the
fermionic Gaussian states. In the past decades, considerable effort has been
devoted to investigating the degree of entanglement as measured by different
entanglement entropies over the well-known Hilbert-Schmidt ensemble HLW06 ;
Page93 ; Foong94 ; Ruiz95 ; VPO16 ; Wei17 ; Wei20 ; HWC21 ; Lubkin78 ;
Sommers04 ; Giraud07 ; MML02 ; Wei19T . In particular, these studies focus on
the statistical behaviour of entanglement entropies such as von Neumann
entropy HLW06 ; Page93 ; Foong94 ; Ruiz95 ; VPO16 ; Wei17 ; Wei20 ; HWC21 ,
quantum purity Lubkin78 ; Sommers04 ; Giraud07 , and Tsallis entropy MML02 ;
Wei19T . Driven by the recent breakthrough in probability theory on the Bures-
Hall ensemble Bortola09 ; Bortola10 ; Bortola14 ; FK16 , considerable progress
has been made in understanding the von Neumann entropy SK2019 ; Wei20BHA ;
Wei20BH ; LW21 and quantum purity Sommers04 ; Osipov10 ; Borot12 ; LW21 over
the Bures-Hall ensemble. Similar investigations are now being carried out over
the fermionic Gaussian ensemble, which is a generic state model relevant for
different quantum information processing tasks BHK21 ; BHKRV22 ; LRV20 ;
OGMM19 ; ST21 . Very recently, the mean values of von Neumann entropy with and
without particle number constrains over the fermionic Gaussian ensemble are
obtained respectively in BHK21 and BHKRV22 . As an important step towards
characterizing the statistical distribution of von Neumann entropy, we aim to
derive the corresponding variances, which describe the fluctuation of the
entropy around their mean values. The exact variance of von Neumann entropy
over fermionic Gaussian states without particle number constrain has been
conjectured in a previous work of the authors HW22 . In the current work, we
prove the conjecture as well as derive a variance formula for the case of a
fixed particle number.
### 1.1 Problem formulation
We introduce the formulation that leads to the fermionic Gaussian states with
and without particle number constrains. A system of $N$ fermionic degrees of
freedom can be decomposed into two subsystems $A$ and $B$ of the dimensions
$m$ and $n$, respectively, with $m+n=N$. Without loss of generality, we assume
$m\leq n$ . In the present work, we consider two scenarios of fermionic
Gaussian states – the fermionic Gaussian states with arbitrary number of
particles and the fermionic Gaussian states with a fixed number of particles.
#### Case A: Arbitrary number of particles
A system of $N$ fermionic modes can be formulated in terms of a set of
fermionic creation and annihilation operators $\hat{a}_{i}$ and
$\hat{a}_{i}^{{\dagger}}$, $i=1,\dots,N$. Since the modes are fermionic, these
operators obey the canonical anti-commutation relation OGMM19 ; BHKRV22 ,
$\\{\hat{a}_{i},\hat{a}_{j}^{{\dagger}}\\}=\delta_{ij}\mathbb{I},\qquad\\{\hat{a}_{i},\hat{a}_{j}\\}=0=\\{\hat{a}_{i}^{{\dagger}},\hat{a}_{j}^{{\dagger}}\\},$
(1)
where $\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{B}\hat{A}$ denotes the anti-
commutation relation and $\mathbb{I}$ is an identity operator. Equivalently,
one can also describe these fermionic modes in terms of the Majorana operators
$\gamma_{l}$, $l=1,\dots,2N$, and
$\hat{\gamma}_{2i-1}=\frac{\hat{a}_{i}^{\dagger}+\hat{a}_{i}}{\sqrt{2}},\qquad\hat{\gamma}_{2i}=\imath\frac{\hat{a}_{i}^{\dagger}+\hat{a}_{i}}{\sqrt{2}}$
(2)
with $\imath=\sqrt{-1}$ being the imaginary unit. Note that the Majorana
operators are Hermitian satisfying the anti-commutation relation
$\\{\hat{\gamma}_{l},\hat{\gamma}_{k}\\}=\delta_{lk}\mathbb{I}.$ (3)
By collecting the Majorana operators into a $2N$ dimensional operator-valued
column vector
${\gamma}=(\hat{\gamma}_{1},\dots,\hat{\gamma}_{2N})^{{\dagger}}$, a fermionic
Gaussian state is then written as the density operator of the form ST21 ;
BHKRV22
$\rho(\gamma)=\frac{\mathrm{e}^{-\gamma^{\dagger}Q\gamma}}{\tr(\mathrm{e}^{-\gamma^{\dagger}Q\gamma})},$
(4)
where the coefficient matrix $Q$ is a $2N\times 2N$ imaginary anti-symmetric
matrix as the consequence of the anti-communication relation (3). There always
exsits an orthogonal matrix $M$ that diagnoses the coefficient matrix $Q$ by
transforming ${\gamma}$ into another Majorana basis
$\mu=(\hat{\mu}_{1},\dots,\hat{\mu}_{2N})^{\dagger}=M\gamma$ BHKRV22 . A
fermionic Gaussian state is labelled by its anti-symmetric covariance matrix
BHKRV22
$J=-\imath\tanh(Q)=M^{T}J_{0}M,$ (5)
where $\tanh(x)$ denotes the hyperbolic tangent function AS72 , the matrix
$J_{0}$ takes the block diagonal form
$J_{0}=\left(\begin{array}[]{ccc}\tanh(\lambda_{1})\mathbb{A}&\dots&0\\\
\vdots&\ddots&\vdots\\\ 0&\dots&\tanh(\lambda_{N})\mathbb{A}\\\
\end{array}\right),$ (6)
and
$\mathbb{A}=\left(\begin{array}[]{cc}0&1\\\ -1&0\\\ \end{array}\right).$ (7)
We consider the von Neumann entropy as the measure of entanglement between the
two subsystems. By restricting the matrix $J$ to the entries from subsystems
$A$, the restricted matrix $J_{A}$ becomes the $2m\times 2m$ left-upper block
of $J$. The von Neumann entropy of a fermionic Gaussian state of case A can be
represented in terms of the real positive eigenvalues $x_{i},i=1,\dots,m$ of
$\imath J_{A}$ as BHK21 ; BHKRV22 ; HW22
$S=-\sum_{i=1}^{m}v(x_{i}),$ (8)
where
$v(x)=\frac{1-x}{2}\ln\frac{1-x}{2}+\frac{1+x}{2}\ln\frac{1+x}{2}.$ (9)
The resulting joint probability density of the eigenvalues $x_{i},i=1,\dots,m$
is proportional to BHK21
$\prod_{1\leq i<j\leq
m}\left(x_{i}^{2}-x_{j}^{2}\right)^{2}\prod_{i=1}^{m}\left(1-x_{i}^{2}\right)^{n-m},\qquad
x_{i}\in[0,1],$ (10)
which is obtained by recursively applying the result (KFI19, , Proposition
A.2).
#### Case B: Fixed number of particles
For a fermionic Gaussian state $\ket{F}$ with a fixed particle number $p$,
$m\leq p\leq n$, the corresponding covariance matrix $H$ can be expressed via
the commutator of fermionic creation and annihilation operators as BHKRV22 ;
LRV20
$H_{ij}=-\imath\bra{F}\hat{a}_{i}^{{\dagger}}\hat{a}_{j}-\hat{a}_{j}\hat{a}_{i}^{{\dagger}}\ket{F}.$
(11)
Recall the canonical anti-commutation relation (1), the entries of the matrix
$H$ are then of the form
$H_{ij}=-2\imath G_{ij}+\imath\delta_{ij}\mathbb{I},$ (12)
where $G_{ij}=\bra{F}\hat{a}_{i}^{{\dagger}}\hat{a}_{j}\ket{F}$ denotes the
entries of an $N\times N$ matrix $G$ of a fermionic system of $N$ modes. There
exists a unitary transformation $U$ that diagonalizes $G$ into the form
$U^{{\dagger}}GU$, where the first $p$ diagonal elements are equal to $1$ and
the rest are $0$. Therefore, one can write
$G=U_{N\times p}U_{N\times p}^{\dagger}.$ (13)
A fermionic Gaussian state of dimension $N=m+n$ with $p$ particles can be
fully characterized by the matrices $H$ and $G$. The von Neumann entropy of
the fermionic system in the case B can be represented as BHKRV22 ; LRV20
$~{}S=-\sum_{i=1}^{m}v(2y_{i}-1),\qquad y_{i}\in[0,1],$ (14)
where $y_{i}$, $i=1,\dots,m$ are the eigenvalues of the restricted $m\times m$
matrix $G_{A}=U_{m\times p}U_{m\times p}^{\dagger}$. The eigenvalue
distribution of the random matrix $U_{m\times p}U_{m\times p}^{\dagger}$ is
the well-known Jacobi unitary ensemble Mehta ; Forrester . We denote $x_{i}$,
$i=1,\dots,m$ the eigenvalues of the $m\times m$ left-upper block of matrix
$\imath H$. Changing the variables $x_{i}=2y_{i}-1$ in (14) leads to the von
Neumann entropy (8) of case B. The resulting joint probability density of the
eigenvalues $x_{i},i=1,\dots,m$ is proportional to BP21
$\prod_{1\leq i<j\leq
m}\left(x_{i}-x_{j}\right)^{2}\prod_{i=1}^{m}\left({1+x_{i}}\right)^{p-m}\left(1-x_{i}\right)^{n-p},\qquad
x_{i}\in[-1,1].$ (15)
It is important to point out that the two joint probability densities (10) and
(15) can be compactly represented by a single joint density as
$f_{\mathrm{FG}}(x)\propto\prod_{1\leq i<j\leq
m}\left(x_{i}^{\gamma}-x_{j}^{\gamma}\right)^{2}\prod_{i=1}^{m}\left(1-x_{i}\right)^{a}\left(1+x_{i}\right)^{b},$
(16)
where for the case A we have
$\gamma=2,~{}~{}~{}a=b=n-m\geq 0,~{}~{}~{}x\in[0,1],$ (17)
and for the case B we have
$\gamma=1,~{}~{}~{}a=n-p\geq 0,~{}~{}~{}b=p-m\geq 0,~{}~{}~{}x\in[-1,1].$ (18)
We omit the normalizations in the density (16) as they will not be made use of
in the subsequent calculations. Note that the variance computation is
difficult for an arbitrary $\gamma$ in (16), and one has to consider the cases
$\gamma=2$ and $\gamma=1$ separately.
### 1.2 Main results
We now introduce the exact mean and variance formulas of von Neumann entropy
for both case A and case B. The mean values have been recently computed BHK21
; BHKRV22 as summarized in Proposition 1 and Proposition 2 for case A and
case B, respectively. The corresponding variance formulas are presented in
Proposition 3 and Proposition 4 below, which are the main results of the work.
###### Proposition 1 (BHK21 ).
For subsystem dimensions $m\leq n$, the mean value of the von Neumann entropy
(8) of fermionic Gaussian states with arbitrary number of particles (17) is
given by
$\displaystyle\mathbb{E}\\!\left[S\right]=$
$\displaystyle\left(m+n-\frac{1}{2}\right)\psi_{0}(2m+2n)+\left(\frac{1}{4}-m\right)\psi_{0}(m+n)+\left(\frac{1}{2}-n\right)\psi_{0}(2n)$
$\displaystyle-\frac{1}{4}\psi_{0}(n)-m,$ (19)
where
$\psi_{0}(x)=\frac{\differential\ln\Gamma(x)}{\differential x}$ (20)
is the digamma function.
###### Proposition 2 (BHKRV22 ).
For subsystem dimensions $m\leq n$, the mean value of the von Neumann entropy
(8) of fermionic Gaussian states with a fixed particle number (18) is given by
$\displaystyle\mathbb{E}\\!\left[S\right]=$
$\displaystyle-\frac{m(m+n-p)}{m+n}\psi_{0}(m+n-p)+(m+n)\psi_{0}(m+n+1)$
$\displaystyle-\frac{mp}{m+n}\psi_{0}(p+1)-n\psi_{0}(n+1)-m.$ (21)
###### Proposition 3.
For subsystem dimensions $m\leq n$, the variance of the von Neumann entropy
(8) of fermionic Gaussian states with arbitrary number of particles (17) is
given by
$\displaystyle\mathbb{V}\\!\left[S\right]=~{}\\!\\!$
$\displaystyle\\!\left(\frac{1}{2}-m-n\right)\psi_{1}(2m+2n)+\left(n-\frac{1}{2}\right)\psi_{1}(2n)+\left(\frac{m(2m+n-1)}{2m+2n-1}-\frac{1}{8}\right)$
$\displaystyle\times\psi_{1}(m+n)+\frac{\psi_{1}(n)}{8}-\frac{1}{2}(\psi_{0}(2m+2n)-\psi_{0}(2n)),$
(22)
where
$\psi_{1}(x)=\frac{\differential^{2}\ln\Gamma(x)}{\differential x^{2}}$ (23)
is the trigamma function.
###### Proposition 4.
For subsystem dimensions $m\leq n$, the variance of the von Neumann entropy
(8) of fermionic Gaussian states with a fixed particle number (18) is given by
$\displaystyle\mathbb{V}\\!\left[S\right]=~{}\\!\\!$ $\displaystyle
c_{0}\psi_{1}(m+n-p)-(m+n)\psi_{1}(m+n)+n\psi_{1}(n)+c_{1}\psi_{1}(p)+c_{2}$
$\displaystyle\times(\psi_{0}(m+n-p)-\psi_{0}(p))^{2}+c_{3}\left(\psi_{0}(m+n-p)-\psi_{0}(p)\right)-\psi_{0}(m+n)$
$\displaystyle+\psi_{0}(n)+c_{4},$ (24)
where the coefficients $c_{i}$ are summarized in Table 1 below with
$(a)_{n}=\Gamma(a+n)/\Gamma(a)$ denoting the Pochhammer symbol.
Table 1: Coefficients of von Neumann entropy variance in Proposition 4 $c_{0}~{}=$ | $\displaystyle\frac{m(m+n-p)\left(m^{2}+2mn+n^{2}-np-1\right)}{(m+n-1)_{3}}$
---|---
$c_{1}~{}=$ | $\displaystyle\frac{mp\left(m^{2}+mn+np-1\right)}{(m+n-1)_{3}}$
$c_{2}~{}=$ | $\displaystyle\frac{mnp(m+n-p)}{(m+n)(m+n-1)_{3}}$
$c_{3}~{}=$ | $\displaystyle-\frac{m(m+1)(m+n-2p)}{(m+n)(m+n)_{2}}$
$c_{4}~{}=$ | $\displaystyle-\frac{m(2m+n+2)}{(m+n)(m+n)_{2}}$
The proof to Proposition 3 and Proposition 4 will be presented in Section 2.
Note that a special case of equal subsystem dimensions ($m=n$) of the result
in Proposition 3 has been established very recently HW22 by utilizing an
existing simplification framework developed in Wei17 ; Wei20 ; HWC21 ; Wei22 ;
Wei20BH ; LW21 ; HW22 . However, for the case of arbitrary subsystem
dimensions, the existing framework is not sufficient to simplify some of the
summations in the variance calculation, where a key technical contribution of
the work is to develop a new simplification framework.
Figure 1: Variance of von Neumann entropy: analytical results versus
simulations. The black lines represent the obtained analytical result (3) for
the cases $n=m$, $n=3m$, and $n=4m$. The red lines are drawn by the result (4)
for the cases $p=m,n=3m$; $p=2m,n=3m$; and $p=2m,n=4m$. The diamond scatters
represent numerical simulations.
To illustrate the derived results (3) and (4), we plot in Figure 1 the exact
variance of von Neumann entropy as compared with the simulations111The
simulations performed in figures 1–3 utilize the Mathematica codes provided by
Santosh Kumar based on the log-gas approach as discussed in (SK2019, ,
Appendix B).. In Figure 1, we observe that the variance in case A approaches
to a constant when system dimensions increase with a fixed ratio
$f_{1}=\frac{m}{n+m},$ (25)
where the variance in case B follows the same behavior with fixed $f_{1}$ and
$f_{2}=\frac{p}{n+m}.$ (26)
This phenomenon can be analytically established by the asymptotic results of
variances in the literature. For case A, in the asymptotic regime BHK21
$m\to\infty,\qquad n\to\infty,\qquad 0<f_{1}\leq\frac{1}{2},$ (27)
one has BHK21
$\mathbb{V}\\!\left[S\right]=\frac{1}{2}\left(f_{1}+f_{1}^{2}+\ln(1-f_{1})\right)+o\left(\frac{1}{m+n}\right),$
(28)
whereas for case B, in the asymptotic regime BHKRV22
$m\to\infty,\qquad p\to\infty,\qquad n\to\infty,\qquad 0<f_{1}\leq
f_{2}\leq\frac{1}{2},$ (29)
one has BHKRV22
$\displaystyle\mathbb{V}\\!\left[S\right]=\\!~{}$ $\displaystyle
f_{1}+f_{1}^{2}+\ln\left(1-f_{1}\right)+f_{1}f_{2}\left(1-f_{1}\right)\left(1-f_{2}\right)\ln^{2}\frac{1-f_{2}}{f_{2}}$
$\displaystyle+f_{1}^{2}\left(2f_{2}-1\right)\ln\frac{1-f_{2}}{f_{2}}+o\left(\frac{1}{(m+n)^{2}}\right).$
(30)
The above asymptotic variances (28) and (1.2) can be directly recovered by the
results in Proposition 3 and Proposition 4, respectively. Moreover, the
correction terms of any order can be simply obtained from our exact variance
formulas upon using the asymptotic behaviour of polygamma functions
$\displaystyle\psi_{0}(x)=$
$\displaystyle\ln(x)-\frac{1}{2x}-\sum_{l=1}^{\infty}\frac{B_{2l}}{2lx^{2l}},\qquad
x\to\infty,$ (31) $\displaystyle\psi_{1}(x)=$
$\displaystyle\frac{1+2x}{2x^{2}}+\sum_{l=1}^{\infty}\frac{B_{2l}}{x^{2l+1}},\qquad
x\to\infty,$ (32)
where $B_{k}$ is the $k$-th Bernoulli number AS72 . For example, utilizing the
next order of correction, the asymptotic result (1.2) is refined to
$\displaystyle\mathbb{V}\\!\left[S\right]=$ $\displaystyle
f_{1}^{2}+f_{1}+\ln\left(1-f_{1}\right)+f_{1}f_{2}\left(1-f_{1}\right)\left(1-f_{2}\right)\ln^{2}\frac{1-f_{2}}{f_{2}}+f_{1}^{2}$
$\displaystyle\times\left(2f_{2}-1\right)\ln\frac{1-f_{2}}{f_{2}}+\frac{1}{12(m+n)^{2}}\Bigg{(}\frac{f_{1}^{2}}{\left(f_{2}-1\right){}^{2}}+\frac{f_{1}^{2}}{f_{2}^{2}}+12f_{1}^{2}$
$\displaystyle-12f_{1}+\frac{1}{\left(f_{1}-1\right){}^{2}}+\frac{f_{1}-3f_{1}^{2}}{f_{2}-1}+\frac{3f_{1}^{2}-f_{1}}{f_{2}}-1$
$\displaystyle+\frac{2\left(f_{1}-1\right)f_{1}\left(12f_{2}^{3}-18f_{2}^{2}+4f_{2}+1\right)}{\left(f_{2}-1\right)f_{2}}\ln\frac{1-f_{2}}{f_{2}}$
$\displaystyle+12\left(f_{1}-1\right)f_{1}\left(f_{2}-1\right)f_{2}\ln^{2}\frac{1-f_{2}}{f_{2}}\\!\Bigg{)}+o\left(\frac{1}{(m+n)^{4}}\right).$
(33)
Figure 2: Probability densities of standardized von Neumann entropy for case A
in (17): a comparison of Gaussian density (35) to the simulation results. The
dash-dot line in blue and the dashed line in red refer to the standardized von
Neumann entropy (34) of subsystem dimensions $m=2$, $n=4$, and $m=16$, $n=32$,
respectively. The solid black line represents the Gaussian density (35).
Figure 3: Probability densities of standardized von Neumann entropy for case B
in (18): a comparison of Gaussian density (35) to the simulation results. The
dash-dot line in blue and the dashed line in red refer to the standardized von
Neumann entropy (34) of dimensions $m=2$, $p=4$, $n=6$, and $m=6$, $p=12$,
$n=18$, respectively. The solid black line represents the Gaussian density
(35).
To understand the distribution of the von Neumann entropy, simple
approximations can now be constructed by using the obtained mean and variance
formulas. We first standardize the von Neumann entropy as
$X=\frac{S-\mathbb{E}\\!\left[S\right]}{\sqrt{\mathbb{V}\\!\left[S\right]}},$
(34)
where the random variable $X$ is of zero mean and unit variance. We now
compare the distribution of $X$ with a standard Gaussian distribution
$\phi(x)=\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{1}{2}x^{2}},\qquad
x\in(-\infty,\infty).$ (35)
In figures 2–3, we plot the simulation results of the standardized von Neumann
entropy $X$ as compared with a standard Gaussian. Specifically, the ratios are
fixed to $f_{1}=1/3$ for case A in Figure 2 and $f_{1}=1/4$, $f_{2}=1/2$ for
case B in Figure 3. It is observed from the figures that the Gaussian density
captures accurately the distribution of the standardized von Neumann entropy
$X$ for moderately large dimensions. We also observe that the true
distribution of $X$ is non-symmetric and appears to be left-skewed when the
subsystem dimensions are small as seen from the dash-dot blue curves. In
comparison, when the subsystem dimensions become larger, the distributions of
$X$ appear to be closer to the Gaussian distribution. In fact, the Gaussian
density as a limiting behavior of von Neumann entropy has been conjectured
over different random matrix models of Hilbert-Schmidt ensemble Wei20 , Bures-
Hall ensemble Wei20BH , and fermionic Gaussian ensemble of an arbitrary number
of particles HW22 . Here, one is also attempting to conjecture that under the
asymptotic regime (29), the standardized von Neumann entropy (34) of fermionic
Gaussian states with a fixed particle number (18) converges in distribution to
a standard Gaussian.
The rest of the paper is organized as follows. The detailed calculations of
the main results in Proposition 3 and Proposition 4 are provided in Section 2.
In Appendix A, we list the summation representations of the integrals involved
in the variance computations. Some additional finite sum identities utilized
in the simplification are listed in Appendix B.
## 2 Variance calculation
In this section, we prove the results in Proposition 3 and Proposition 4. In
Section 2.1, we obtain the summation representations of the variances. Tools
in simplifying these summations are introduced in Section 2.2, where the new
simplification framework consisting of six lemmas is presented first. The
detailed simplification procedures that lead to the claimed results (3) and
(4) are discussed in Section 2.3.
### 2.1 Correlation functions and integral calculations
Recall the definition (8) of von Neumann entropy
$S=-\sum_{i=1}^{m}v(x_{i}),$ (36)
with
$v(x)=\frac{1-x}{2}\ln\frac{1-x}{2}+\frac{1+x}{2}\ln\frac{1+x}{2},$ (37)
computing its variance requires one and two arbitrary eigenvalue densities of
the fermionic Gaussian ensemble (16). Denoting $g_{l}(x_{1},\dots,x_{l})$ as
the joint density of $l$ arbitrary eigenvalues, the variance of von Neumann
entropy is written as
$\mathbb{V}\\!\left[S\right]=\mathbb{E}\\!\left[S^{2}\right]-\mathbb{E}^{2}\\!\left[S\right],$
(38)
where
$\displaystyle\mathbb{E}\\!\left[S^{2}\right]$
$\displaystyle=m\int_{x}v^{2}(x)g_{1}(x)\differential
x+m(m-1)\iint_{x,y}v(x)v(y)g_{2}\left(x,y\right)\differential x\differential
y$ (39) $\displaystyle\mathbb{E}\\!\left[S\right]$
$\displaystyle=m\int_{x}v(x)g_{1}(x)\differential x.$ (40)
In (39) and (40), the support is $x,y\in[0,1]$ for case A, and the support is
$x,y\in[-1,1]$ for case B.
For fermionic Gaussian ensemble (16), it is a well-known result in random
matrix theory that the joint density $g_{l}(x_{1},\dots,x_{l})$ can be written
in terms of an $l\times l$ determinant as Mehta ; Forrester
$g_{l}(x_{1},\dots,x_{l})=\frac{(m-l)!}{m!}\det\left(K\left(x_{i},x_{j}\right)\right)_{i,j=1}^{l}.$
(41)
The determinant in (41) is known as the $l$-point correlation function
Forrester , where
$K\left(x,y\right)=\sqrt{w(x)w(y)}\sum_{k=0}^{m-1}\frac{J_{k}^{(a,b)}(x)J_{k}^{(a,b)}(y)}{h_{k}}$
(42)
is the correlation kernel with the weight function
$w(x)=\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}.$ (43)
For convenience, we summarize in Table 2 the parameters $a$, $b$, and the
order $k$ of the Jacobi polynomial $J_{k}^{(a,b)}(x)$ along with the
normalization constants $h_{k}$ of case A and case B.
Table 2: Parameters of correlation kernel (42) in case A and case B. | Case A: Arbitrary number of particle (17) | Case B: Fixed number of particle (18)
---|---|---
$a$ | $n-m$ | $n-p$
$b$ | $n-m$ | $p-m$
$k$ | $2k$ | $k$
$h_{k}$ | $\displaystyle\frac{(4k+2a+1)^{-1}\Gamma^{2}(2k+a+1)}{\Gamma(2k+2a+1)\Gamma(2k+1)}$ | $\displaystyle\frac{2\Gamma(k+a+1)\Gamma(k+b+1)}{(2k+a+b+1)\Gamma(k+1)\Gamma(k+a+b+1)}$
| |
The constants $h_{k}$ in Table 2 are obtained as follows. For case B, the
orthogonality relation Forrester
$\displaystyle\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}J^{(a,b)}_{k}(x)J^{(a,b)}_{l}(x)\differential
x$
$\displaystyle=\frac{2\Gamma(k+a+1)\Gamma(k+b+1)}{(2k+a+b+1)\Gamma(k+1)\Gamma(k+a+b+1)}\delta_{kl},\quad\real(a,b)>-1,$
(44)
directly leads to the normalization constant $h_{k}$ of Jacobi polynomials
$J^{(a,b)}_{k}$. For case A, by using the parity property of Jacobi
polynomials Szego
$J^{(a,b)}_{k}(-x)=(-1)^{k}J^{(b,a)}_{k}(x),$ (45)
the orthogonality relation (44) can also be written as
$\displaystyle\int_{0}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{a}J^{(a,a)}_{2k}(x)J^{(a,a)}_{2l}(x)\differential
x$
$\displaystyle=\frac{\Gamma(2k+a+1)\Gamma(2k+a+1)}{(4k+2a+1)\Gamma(2k+1)\Gamma(2k+2a+1)}\delta_{kl},\quad\real(a)>-1,$
(46)
which gives the normalization constant $h_{k}$ of the polynomials
$J^{(a,a)}_{2k}(x)$ as shown in Table 2.
By using the joint density (41) and the results (39)–(40), the variance (38)
now boils down to computing two integrals involving the $1$-point and
$2$-point correlation functions as, cf. Wei17 ; Wei20BH ; BHK21 ; HW22 ,
$\mathbb{V}\\!\left[S\right]=\mathrm{I_{A}}-\mathrm{I_{B}},$ (47)
where
$\displaystyle\mathrm{I_{A}}$
$\displaystyle=\int_{x}v^{2}(x)K(x,x)\differential x$ (48)
$\displaystyle\mathrm{I_{B}}$
$\displaystyle=\iint_{x,y}v(x)v(y)K^{2}\left(x,y\right)\differential
x\differential y$ (49)
with $x,y\in[0,1]$ for case A and $x,y\in[-1,1]$ for case B.
We now compute the above two integrals $\mathrm{I_{A}}$ and $\mathrm{I_{B}}$
into the corresponding summation representations. Note that the subsequent
calculations of the case A in (17) and case B in (18) are different, which
will be performed separately in the following.
#### Case A: Arbitrary number of particles
By the definition of the correlation kernel (42) and keeping in mind the
parity property (45) of Jacobi polynomials, the integral
$\mathrm{I_{A}}=\int_{0}^{1}v^{2}(x)K(x,x)\differential x$ (50)
of fermionic Gaussian states with arbitrary number of particles boils down to
computing the two parts
$\mathrm{I_{A}}=\mathrm{A_{1}}+\mathrm{A_{2}},$ (51)
where
$\displaystyle\mathrm{A_{1}}$
$\displaystyle=\sum_{k=0}^{m-1}\frac{1}{h_{k}}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{a+2}\ln^{2}\frac{1+x}{2}J_{2k}^{(a,a)}(x)^{2}\differential
x$ (52) $\displaystyle\mathrm{A_{2}}$
$\displaystyle=\sum_{k=0}^{m-1}\frac{1}{h_{k}}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a+1}\left(\frac{1+x}{2}\right)^{a+1}\ln\frac{1-x}{2}\ln\frac{1+x}{2}J_{2k}^{(a,a)}(x)^{2}\differential
x.$
Here, we recall that $a=n-m\geq 0$ denotes the subsystem difference of
fermionic Gaussian states with arbitrary number of particles.
Similarly, the integral
$\mathrm{I_{B}}=\int_{0}^{1}\int_{0}^{1}v(x)v(y)K^{2}\left(x,y\right)\differential
x\differential y$ (54)
can be written in terms of the following two integrals
$\mathrm{I_{B}}=\mathrm{B_{1}}+\mathrm{B_{2}},$ (55)
where
$\displaystyle\mathrm{B_{1}}=$
$\displaystyle\sum_{k=0}^{m-1}\frac{1}{h_{k}^{2}}\left(\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{a+1}\ln\frac{1+x}{2}J_{2k}^{(a,a)}(x)^{2}\differential
x\right)^{2}$ (56) $\displaystyle\mathrm{B_{2}}=$
$\displaystyle\sum_{j=1}^{m-1}\sum_{k=0}^{m-j-1}\frac{2}{h_{k+j}h_{k}}\Bigg{(}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{a+1}\ln\frac{1+x}{2}$
$\displaystyle\times J_{2k+2j}^{(a,a)}(x)J_{2k}^{(a,a)}(x)\differential
x\Bigg{)}^{2}.$ (57)
Computing the above integrals $\mathrm{A_{1}}$, $\mathrm{A_{2}}$,
$\mathrm{B_{1}}$, and $\mathrm{B_{2}}$ requires the following two integral
identities. The first one is
$\displaystyle\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a_{1}}\left(\frac{1+x}{2}\right)^{c}J^{(a_{1},b_{1})}_{k_{1}}(x)J^{(a_{2},b_{2})}_{k_{2}}(x)\differential
x$
$\displaystyle=\frac{2\left(k_{1}+1\right)_{a_{1}}}{\left(b_{2}+k_{2}+1\right)_{a_{2}}}\sum_{i=0}^{k_{2}}\frac{(-1)^{i+k_{2}}(i+1)_{c}\left(i+b_{2}+1\right)_{a_{2}+k_{2}}}{\Gamma\left(k_{2}-i+1\right)\Gamma\left(a_{1}+c+i+k_{1}+2\right)}$
$\displaystyle~{}~{}~{}\\!~{}\times\left(c+i-b_{1}-k_{1}+1\right)_{k_{1}},\quad\real(a_{1},a_{2},b_{1},b_{2},c)>-1.$
(58)
To show this identity, we first note that the Jacobi polynomial
$J_{k}^{(a,b)}(x)$ supported in $x\in[-1,1]$ admits different representations
Szego ; Forrester
$\displaystyle J^{(a,b)}_{k}(x)$
$\displaystyle=\frac{(-1)^{k}(b+1)_{k}}{k!}\sum_{i=0}^{k}\frac{(-k)_{i}(k+a+b+1)_{i}}{(b+1)_{i}\Gamma(i+1)}\left(\frac{1+x}{2}\right)^{i}$
(59)
$\displaystyle=\sum_{i=0}^{k}\frac{(-1)^{i}\Gamma(a+k+1)(k+b-i+1)_{i}}{\Gamma(i+1)\Gamma(a+i+1)\Gamma(k-i+1)}\left(\frac{1-x}{2}\right)^{i}\left(\frac{1+x}{2}\right)^{k-i}.$
The identity (58) is then obtained by using the definition (59) for the
polynomial $J^{(a_{2},b_{2})}_{k_{2}}$ before applying the well-known integral
identity Szego ; Forrester
$\displaystyle\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{c}J^{(a,b)}_{k}(x)\differential
x$
$\displaystyle=\frac{2\Gamma(c+1)(k+1)_{a}(c-b-k+1)_{k}}{\Gamma(a+c+k+2)},\quad\real(a,b,c)>-1.$
(61)
The second identity is
$\displaystyle\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{d}\left(\frac{1+x}{2}\right)^{c}J^{(a_{1},b_{1})}_{k_{1}}J^{(a_{2},b_{2})}_{k_{2}}(x)\differential
x$
$\displaystyle=\frac{2\Gamma\left(a_{2}+k_{2}+1\right)\Gamma\left(b_{2}+k_{2}+1\right)}{\Gamma\left(c+d+k_{1}+k_{2}+2\right)}\sum_{i=0}^{k_{2}}\frac{(-1)^{i}\Gamma\left(d-a_{1}+i+1\right)}{\Gamma(i+1)\Gamma\left(a_{2}+i+1\right)}$
$\displaystyle~{}~{}~{}\\!~{}\times\frac{\Gamma\left(c-b_{1}-i+k_{2}+1\right)}{\Gamma\left(k_{2}-i+1\right)\Gamma\left(b_{2}-i+k_{2}+1\right)}\sum_{j=0}^{k_{1}}\frac{(-1)^{j}\left(k_{1}-j+1\right)_{d+i}}{\Gamma(j+1)}$
$\displaystyle~{}~{}~{}\\!~{}\times\frac{\left(c-i+j-b_{1}-k_{1}+k_{2}+1\right)_{b_{1}+k_{1}}}{\Gamma\left(d-a_{1}+i-j+1\right)},\quad\real(a_{1},a_{2},b_{1},b_{2},c,d)>-1,$
(62)
which is obtained by using the definition (LABEL:eq:J2) for the polynomial
$J^{(a_{2},b_{2})}_{k_{2}}$ before applying the identity (HW22, , Equation
(62))
$\displaystyle\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{d}\left(\frac{1+x}{2}\right)^{c}J^{(a,b)}_{k}(x)\differential
x$
$\displaystyle=\frac{2\Gamma(c-b+1)\Gamma(d-a+1)}{\Gamma(c+d+k+2)}\sum_{i=0}^{k}\frac{(-1)^{i}\Gamma(c+i+1)\Gamma(d-i+k+1)}{\Gamma(i+1)\Gamma(k-i+1)}$
$\displaystyle~{}~{}~{}\\!~{}\times\frac{1}{\Gamma(d-a-i+1)\Gamma(c-b+i-k+1)},\quad\real(a,b,c,d)>-1.$
(63)
The integral $\mathrm{A_{1}}$ is now calculated by applying the identity (58),
where we need to assign
$a_{1}=b_{1}=a_{2}=b_{2}=a,~{}~{}~{}k_{1}=k_{2}=2k,$ (64)
and take twice derivatives of $c$ before setting $c=a+2$. Under the same
specialization (64), the integral $\mathrm{A_{2}}$ is calculated by taking
derivatives of both $c$ and $d$ of the identity (62) before setting $c=d=a+1$,
while the integral $\mathrm{B_{1}}$ is calculated by taking derivative of $c$
of the identity (58) before setting $c=a+1$. According to the result (57), the
integral $\mathrm{B_{2}}$ is calculated by specializing
$a_{1}=b_{1}=a_{2}=b_{2}=a,~{}~{}~{}k_{1}=2k+2j,~{}~{}~{}k_{2}=2k,$ (65)
in the identity (58), and taking derivative of $c$ before setting $c=a+1$.
In writing down the summation forms of $\mathrm{A_{1}}$, $\mathrm{A_{2}}$,
$\mathrm{B_{1}}$, and $\mathrm{B_{2}}$, one will have to resolve the
indeterminacy by using the following asymptotic expansions of gamma and
polygamma functions of negative arguments AS72 when $\epsilon\rightarrow 0$,
$\displaystyle\Gamma(-l+\epsilon)$
$\displaystyle=\frac{(-1)^{l}}{l!\epsilon}\left(1+\psi_{0}(l+1)\epsilon+o\left(\epsilon^{2}\right)\right)$
(66) $\displaystyle\psi_{0}(-l+\epsilon)$
$\displaystyle=-\frac{1}{\epsilon}+\psi_{0}(l+1)+\left(2\psi_{1}(1)-\psi_{1}(l+1)\right)\epsilon+o\left(\epsilon^{2}\right)$
(67) $\displaystyle\psi_{1}(-l+\epsilon)$
$\displaystyle=\frac{1}{\epsilon^{2}}-\psi_{1}(l+1)+\psi_{1}(1)+\zeta(2)+o\left(\epsilon\right).$
(68)
The resulting summation forms of $\mathrm{A_{1}}$, $\mathrm{A_{2}}$,
$\mathrm{B_{1}}$, and $\mathrm{B_{2}}$ are summarized in (A.1)–(A.1) in
Appendix A.
#### Case B: Fixed number of particles
By the definition of correlation kernel (42), the $\mathrm{I_{A}}$ integral
(48) of case B boils down to computing the two parts
$\mathrm{I_{A}}=\mathcal{A}_{1}+\mathcal{A}_{2},$ (69)
where
$\displaystyle\mathcal{A}_{1}=$
$\displaystyle\sum_{k=0}^{m-1}\frac{1}{h_{k}}\int_{-1}^{1}\left(\left(\frac{1+x}{2}\right)^{2}\ln^{2}\frac{1+x}{2}+\left(\frac{1-x}{2}\right)^{2}\ln^{2}\frac{1-x}{2}\right)$
$\displaystyle\times\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}J_{k}^{(a,b)}(x)^{2}\differential
x$ (70) $\displaystyle\mathcal{A}_{2}=$
$\displaystyle\sum_{k=0}^{m-1}\frac{2}{h_{k}}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a+1}\left(\frac{1+x}{2}\right)^{b+1}\ln\frac{1-x}{2}\ln\frac{1+x}{2}J_{k}^{(a,b)}(x)^{2}\differential
x.$
Due to the parity property (45), $\mathcal{A}_{1}$ admits a symmetric
structure as
$\mathcal{A}_{1}=\mathcal{A}_{1}^{(a,b)}+\mathcal{A}_{1}^{(b,a)},$ (72)
where
$\mathcal{A}_{1}^{(a,b)}=\sum_{k=0}^{m-1}\frac{1}{h_{k}}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b+2}\ln^{2}\frac{1+x}{2}J_{k}^{(a,b)}(x)^{2}\differential
x.$ (73)
The summations in (LABEL:eq:fpA2) and (73) can be evaluated by using the
following confluent form of Christoffel-Darboux formula Forrester
$\sum_{k=0}^{m-1}\frac{J_{k}^{(a,b)}(x)^{2}}{h_{k}}=\alpha_{1}J_{m-1}^{(a+1,b+1)}(x)J_{m-1}^{(a,b)}(x)-\alpha_{2}J_{m-2}^{(a+1,b+1)}(x)J_{m}^{(a,b)}(x),$
(74)
where
$\displaystyle\alpha_{1}$
$\displaystyle=\frac{m(a+b+m)(a+b+m+1)}{h_{m-1}(a+b+2m-1)_{2}}$ (75)
$\displaystyle\alpha_{2}$
$\displaystyle=\frac{m(a+b+m)^{2}}{h_{m-1}(a+b+2m-1)_{2}}.$ (76)
Consequently, we have
$\displaystyle\mathcal{A}_{1}^{(a,b)}=~{}$
$\displaystyle\alpha_{1}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b+2}\ln^{2}\frac{1+x}{2}J_{m-1}^{(a+1,b+1)}(x)J_{m-1}^{(a,b)}(x)\differential
x$
$\displaystyle\\!-\alpha_{2}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b+2}\ln^{2}\frac{1+x}{2}J_{m-2}^{(a+1,b+1)}(x)J_{m}^{(a,b)}(x)\differential
x$ $\displaystyle\mathcal{A}_{2}=~{}$ $\displaystyle
2\alpha_{1}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a+1}\left(\frac{1+x}{2}\right)^{b+1}\ln\frac{1-x}{2}\ln\frac{1+x}{2}$
$\displaystyle\\!\times J_{m-1}^{(a+1,b+1)}(x)J_{m-1}^{(a,b)}(x)\differential
x$
$\displaystyle\\!-2\alpha_{2}\int_{-1}^{1}\left(\frac{1-x}{2}\right)^{a+1}\left(\frac{1+x}{2}\right)^{b+1}\ln\frac{1-x}{2}\ln\frac{1+x}{2}$
$\displaystyle\\!\times J_{m-2}^{(a+1,b+1)}(x)J_{m}^{(a,b)}(x)\differential
x.$ (78)
Similarly, the $\mathrm{I_{B}}$ integral (49) can be written in terms of the
following two integrals
$\mathrm{I_{B}}=\mathcal{B}_{1}+\mathcal{B}_{2},$ (79)
where
$\displaystyle\mathcal{B}_{1}=$
$\displaystyle\sum_{k=0}^{m-1}\frac{1}{h_{k}^{2}}\Bigg{(}\int_{-1}^{1}\left(\frac{1-x}{2}\ln\frac{1-x}{2}+\frac{1+x}{2}\ln\frac{1+x}{2}\right)\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}$
$\displaystyle\times J_{k}^{(a,b)}(x)^{2}\differential x\Bigg{)}^{2}$ (80)
$\displaystyle\mathcal{B}_{2}=$
$\displaystyle\sum_{j=1}^{m-1}\sum_{k=0}^{m-j-1}\frac{2}{h_{k+j}h_{k}}\Bigg{(}\int_{-1}^{1}\left(\frac{1-x}{2}\ln\frac{1-x}{2}+\frac{1+x}{2}\ln\frac{1+x}{2}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\times\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}J_{k+j}^{(a,b)}(x)J_{k}^{(a,b)}(x)\differential
x\Bigg{)}^{2}.$ (81)
Note that the integrals in $\mathcal{B}_{1}$ or $\mathcal{B}_{2}$ also admit
the same symmetric structure as (72) due to the parity property of Jacobi
polynomials. Because of the symmetry, the remaining computation process is
similar to that of case A. Specifically, the integrals in
$\mathcal{A}_{1}^{(a,b)}$ are calculated by taking twice derivative of $c$ of
the identity (58), where one needs to assign
$a_{1}=a,~{}~{}~{}b_{1}=b,~{}~{}~{}a_{2}=a+1,~{}~{}~{}b_{2}=b+1,~{}~{}~{}k_{1}=k_{2}=m-1,$
(82)
and
$a_{1}=a,~{}~{}~{}b_{1}=b,~{}~{}~{}a_{2}=a+1,~{}~{}~{}b_{2}=b+1,~{}~{}~{}k_{1}=m,~{}~{}~{}k_{2}=m-2,$
(83)
respectively corresponding to the first and second integrals in
(LABEL:eq:A1cd) before setting $c=b+2$. By the result (78), the two integrals
in $\mathcal{A}_{2}$ are calculated by taking derivatives of $c$ and $d$ of
identity (62) respectively with the specializations (82) and (83), before
setting $c=b+1$, $d=a+1$. The integral
$\int_{-1}^{1}\frac{1+x}{2}\ln\frac{1+x}{2}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}J_{k}^{(a,b)}(x)^{2}\differential
x$ (84)
in $\mathcal{B}_{1}$ is calculated by specializing
$a_{1}=a_{2}=a,~{}~{}~{}b_{1}=b_{2}=b,~{}~{}~{}k_{1}=k_{2}=k$ (85)
in the identity (58), and taking derivative of $c$ before setting $c=b+1$. The
integral
$\int_{-1}^{1}\frac{1+x}{2}\ln\frac{1+x}{2}\left(\frac{1-x}{2}\right)^{a}\left(\frac{1+x}{2}\right)^{b}J_{k+j}^{(a,b)}(x)J_{k}^{(a,b)}(x)\differential
x$ (86)
in $\mathcal{B}_{2}$ is calculated by specializing
$a_{1}=a_{2}=a,~{}~{}~{}b_{1}=b_{2}=b~{}~{}~{}k_{1}=k+j,~{}~{}~{}k_{2}=k$ (87)
in the identity (58), and taking derivative of $c$ before setting $c=b+1$.
After resolving the indeterminacy of gamma and polygamma functions by using
(66)–(68), one arrives at the summation representations (281)–(A.2) as listed
in Appendix A.
### 2.2 Tools for simplification of summations
The major task in obtaining the variance formulas in propositions 3 and 4 is
to simplify the summation representations (A.1)–(A.2) into the closed-form
results (3) and (4). We summarize the necessary simplification tools in this
section, where the existing simplification framework is briefly reviewed in
Section 2.2.1 and the new simplification framework is introduced in Section
2.2.2.
#### 2.2.1 Existing simplification framework
The existing simplification tools have been utilized in various moments
calculations over different ensembles, including the Hilbert-Schmidt ensemble
Wei17 ; Wei20 ; HWC21 ; Wei22 , the Bures-Hall ensemble Wei20BH ; LW21 , and
the fermionic Gaussian ensemble HW22 .
In the existing framework, we mainly have two types of finite sum identities.
The first type is of the form
$\sum_{i=1}^{m}i^{c}\psi_{j_{1}}^{b_{1}}(i+a_{1})\psi_{j_{2}}^{b_{2}}(i+a_{2})\cdots\psi_{j_{m}}^{b_{m}}(i+a_{m}),$
(88)
where $a$, $b$, $c$, $j$ are non-negative integers. The main idea in deriving
the identities of this type of sums is to change the summation orders and make
use of the obtained lower order summation formulas in a recursive manner. For
example, the summation
$\sum_{i=1}^{m}\psi_{0}^{2}(i+a)$ (89)
will be derived by using the identity (287) of the lower order summation
$\sum_{i=1}^{m}\psi_{0}(i+a)$. Specifically, by using the finite sum form of
the digamma function
$\psi_{0}(l)=-\gamma+\sum_{k=1}^{l-1}\frac{1}{k},$ (90)
the summation (89) can be rewritten as
$\psi_{0}(a)\sum_{i=1}^{m}\psi_{0}(i+a)+\sum_{j=1}^{m}\frac{1}{a+j-1}\sum_{i=j}^{m}\psi_{0}(i+a),$
(91)
where we have changed the summation order of the double sum. The remaining
sums can be simplified by using the lower order identity (287), leading to the
result in (B.1).
The second type of summations is of the form
$S_{f}(m,n)=\sum_{i=1}^{m}\frac{(n-i)!}{(m-i)!}f(i),\qquad m\leq n,$ (92)
where $f(i)$ can be the product of polygamma and rational functions in $i$. A
well-known result of the second type summation is the identity
$\sum_{i=1}^{m}\frac{(n-i)!}{(m-i)!}=\frac{n!}{(m-1)!(n-m+1)},$ (93)
which is a special case of the Chu-Vandermonde identity Luke . In the existing
simplification framework, the identity (93) is a fundamental result in
obtaining several other second type summations. For example, when
$f(i)=\frac{1}{i},$ (94)
the summation
$S_{1/i}(m,n)=\sum_{i=1}^{m}\frac{(n-i)!}{(m-i)!}\frac{1}{i}$ (95)
is computed by first obtaining the recurrence relation
$S_{1/i}(m,n)=\frac{n}{m}S_{1/i}(m-1,n-1)+\frac{n-m}{m}\sum_{i=1}^{m}\frac{(n-1-i)!}{(m-i)!}.$
(96)
After recurring $m$ times, and using the existing result (93), one obtains the
closed-form result of the summation (95) as listed in (305).
#### 2.2.2 New simplification framework
The existing simplification framework is useful in simplifying the summations
in (A.1), (A.1), (281), and (A.2). However, the framework is insufficient to
simplify some of the summations in (A.1), (A.1), (A.2), and (A.2). The main
technique difficulty is that these summations, after exhausting all the
possibilities of changing the order of summations, are not related to the
first or second type summations of the existing framework. To convert these
sums into the ones of the existing framework, the following new simplification
framework is needed.
The first technique in the new framework is what we refer to as “dummy
summation". The idea of the dummy summation is as follows. For a summation
$F$,
$F=\sum_{i}f_{i},$ (97)
where
$\sum_{i}=\sum_{i_{1}}\sum_{i_{2}}\dots\sum_{i_{a}}$ (98)
denotes a finite nested sum of $a$ indexes $i=\\{i_{1},i_{2},\dots,i_{a}\\}$.
Each index $i_{k}$, $k=1,2,\dots,a$ may depend on its previous ones
$i_{1},i_{2},\dots,i_{k-1}$. For the summation $F$ that is not simplifiable
under the existing framework by any changes of the order of summations, one
may interpret the summand $f_{i}$ into an additional nested finite sum over a
set of new indexes $j=\\{j_{1},j_{2},\dots,j_{b}\\}$ as
$f_{i}=\sum_{j}g_{i,j},$ (99)
such that the resulting representation
$F=\sum_{i,j}g_{i,j}$ (100)
admits further simplifications by using the existing tools of first and second
type sums (88) and (92) after appropriately changing some of the summation
orders. Note that the indexes in the set $j$ in (99) may depend on the ones in
the set $i$.
To illustrate this new technique, we consider the following two examples that
will be utilized in the simplifications in Section 2.3. The first example is
the summation
$F=\sum_{i_{1}=1}^{m}f_{i_{1}},$ (101)
where
$f_{i_{1}}=i_{1}\psi_{0}(i_{1}+a).$ (102)
In this case, we have $i=\\{i_{1}\\}$ in the definition (97). By interpreting
$i_{1}$ in the summand (102) as a dummy sum
$i_{1}=\underbrace{1+1+\dots+1}_{i_{1}},$ (103)
the original single sum in (101) now becomes a double sum
$F=\sum_{i_{1}=1}^{m}\sum_{j_{1}=1}^{i_{1}}g_{i_{1},j_{1}},$ (104)
where
$g_{i_{1},j_{1}}=\psi_{0}(i_{1}+a)$ (105)
with $j=\\{j_{1}\\}$ in the definition (100). After changing the summation
order in (104) as
$F=\sum_{j_{1}=1}^{m}\sum_{i_{1}=j_{1}}^{m}\psi_{0}(i_{1}+a),$ (106)
the sum over $i_{1}$ can be simplified by using an existing identity (287),
leading to
$F=\frac{1-a}{2}\sum_{j_{1}=1}^{m}\psi_{0}\left(j_{1}+a\right)+\frac{m}{4}\left(2(a+m)\psi_{0}(a+m+1)-m-1\right).$
(107)
Simplifying the single summation in (107) by the identity (287) directly gives
the result listed in (B.1). In the above example, the dummy summation (103)
converts the original sum (101) into a double sum (104), which appears more
complicated but is simplifiable using an available identity (287) of the
existing framework. Note also that the dummy summation technique may not be
critical in this example, which may be derived in other ways.
In the second example, we introduce a dummy summation (99) that is crucial as
a subroutine in simplifying the summations in (A.1) and (A.2). The essential
idea in creating this dummy summation is to interpret a Gamma ratio
$\frac{\Gamma(i)}{\Gamma(c+i)}=\frac{1}{i(i+1)\cdots(i+c-1)},\qquad
c,i\in\mathbb{Z}^{+}$ (108)
as its partial fraction decomposition
$a_{1}\frac{1}{i}+a_{2}\frac{1}{i+1}+\dots+a_{c}\frac{1}{i+c-1}$ (109)
with $a_{j},j=1,2,\dots,c$ denoting the coefficients of the decomposition. The
dummy summation that corresponds to the above interpretation is
$\frac{\Gamma(i)}{\Gamma(c+i)}=\sum_{j=1}^{c}a_{j}\frac{1}{i+j-1},$ (110)
where
$a_{j}=\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)}.$ (111)
The coefficients $a_{j}$ are computed by evaluating a unit-argument
hypergeometric function Brychkov08
$\sum_{j=1}^{c}a_{j}\frac{1}{i+j-1}=\frac{\\!{}_{2}F_{1}(-c,i-1\mathchar
24635\relax\;i\mathchar 24635\relax\;1)}{\Gamma(c+1)},$ (112)
where one utilizes the well-known identity Brychkov08
$_{2}F_{1}(a,b\mathchar 24635\relax\;c\mathchar
24635\relax\;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},\quad\real(c)>\real(a+b).$
(113)
The dummy summation in (110) is critical in proving the lemmas 2–4 as
discussed later in this section. By utilizing these lemmas, the summations in
(A.1) and (A.2) involving ratios of Gamma functions then become simplifiable
under the existing simplification framework as will be shown in Section 2.3.
The second technique in the new framework is what we refer to as the “re-
summation" technique. The idea of this new technique is as follows. For a
finite summation $G_{m}$ that appears not summable under the existing
framework with $m$ denoting one of its parameters or the finite upper limit of
the summation. The re-summation technique aims to find alternative forms of
the summation $G_{m}$ to reveal the potential cancellations with other sums,
such that the remaining terms can be simplified by using the existing tools.
Specifically, the re-summation of $G_{m}$ can be generated by iterating a
suitably chosen recurrence relation
$G_{m}=c_{m-1}G_{m-1}+r_{m-1},$ (114)
where $c_{i}$ denotes the coefficient and $r_{i}$ is the remainder of the
recurrence relation. Here, the remainder can contain summations, see the
example (121) below. Each iteration is to replace the term $G_{m-i}$ with its
previous one $G_{m-i-1}$. Keep iterating until $G_{m-i}$ vanishes, we then
obtain an alternative form of $G_{m}$, which is considered a useful one if it
facilitates the cancellation with other sums.
To illustrate the re-summation technique, we consider the difference of two
summations
$\mathcal{G}=G_{m}-G^{\prime}_{m},$ (115)
where
$\displaystyle G_{m}$
$\displaystyle=\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+1)(a-i+2m+1)}$
(116) $\displaystyle G^{\prime}_{m}$
$\displaystyle=\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+1)i}.$
(117)
We first notice that the two summations in (116) and (117) above can not be
simplified individually by using the identities of the existing framework. In
addition, these two summations do not appear to cancel directly in the
combination (115). By iterating $m$ times the following tailor made recurrence
relations (in terms of the various choices of coefficients and reminders), cf.
(114),
$\displaystyle G_{m}$ $\displaystyle=c_{m-1}G_{m-1}+r_{m-1}$ (118)
$\displaystyle G^{\prime}_{m}$
$\displaystyle=c_{m-1}G^{\prime}_{m-1}+r^{\prime}_{m-1},$ (119)
where
$\displaystyle c_{m-1}$ $\displaystyle=\frac{a+m}{m}$ (120) $\displaystyle
r_{m-1}$
$\displaystyle=-\frac{a}{m}\sum_{i=1}^{m+1}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+2)}+\frac{a+m}{m}\left(\frac{\Gamma(a+m)}{\Gamma(m)(a+2m-1)}\right.$
$\displaystyle~{}~{}~{}\\!~{}\\!\left.-\frac{\Gamma(a+m+1)}{\Gamma(m+1)(a+2m)}\right)$
(121) $\displaystyle r^{\prime}_{m-1}$
$\displaystyle=\frac{a}{m}\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m)}{\Gamma(m-i+1)},$
(122)
we obtain the re-summation results
$\displaystyle G_{m}=~{}\\!$
$\displaystyle\frac{a\Gamma(a+m+1)}{\Gamma(m+1)}\sum_{i=1}^{m}\sum_{j=1}^{i+1}\frac{(-1)^{j+1}\Gamma(i)\Gamma(a+i-j+1)}{\Gamma(i-j+2)\Gamma(a+i+1)}$
$\displaystyle+\frac{\Gamma(a+m+1)}{\Gamma(m+1)}\sum_{i=1}^{m}\left(\frac{1}{a+2i-1}-\frac{a+i}{i(a+2i)}\right)$
(123) $\displaystyle G^{\prime}_{m}=~{}\\!$
$\displaystyle\frac{a\Gamma(a+m+1)}{\Gamma(m+1)}\sum_{i=1}^{m}\sum_{j=2}^{i+1}\frac{(-1)^{j+1}\Gamma(i)\Gamma(a+i-j+1)}{\Gamma(i-j+2)\Gamma(a+i+1)}.$
(124)
The relationship between the two summations now become obvious. After
inserting the results (123) and (124) into (115), the cancellation occurs
directly between the two double summations in (123) and (124). The remaining
terms give the closed-form result
$\displaystyle\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+1)}\left(\frac{1}{a-i+2m+1}-\frac{1}{i}\right)$
$\displaystyle=\frac{\Gamma(a+m+1)}{\Gamma(m+1)}\left(\psi_{0}(a+2m+1)-\psi_{0}(a+m+1)\right),$
(125)
which is a special case of Lemma 5 when $c=0$. This result is also utilized in
simplifying the double summations in (A.2). Note that the emphasis here is the
inner cancellation among multiple sums when utilizing re-summation technique.
On the other hand, this technique concerning only one summation has been
studied in Wei17 ; Wei20 ; HWC21 .
Using the two techniques in the new simplification framework, we obtain the
following six lemmas. More precisely, proving the Lemma 1 utilizes the re-
summation technique. The lemmas 2–4 are obtained by using the dummy summation
together with the result of Lemma 1. The Lemma 5 and Lemma 6 are established
by using the re-summation technique. In the six lemmas obtained by the new
framework, each single sum is converted to another one, which will be applied
in Section 2.3 to convert summations involving multiple ratios of Gamma
functions into the ones that are simplifiable by the existing framework.
###### Lemma 1.
For any complex numbers $a,b,c\notin\mathbb{Z}^{-}$, we have
$\displaystyle\sum_{i=1}^{m}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m+1-i)\Gamma(m+b+1-i)(c+i)}$
$\displaystyle=\frac{1}{\Gamma(b+m)\Gamma(c+m+1)\Gamma(a+b+m)}\sum_{i=1}^{m}\frac{\Gamma(c-i+m+1)\Gamma(a+b-i+2m)}{\Gamma(m-i+1)\Gamma(a-i+m+1)}.$
(126)
Proof: Proving the Lemma 1 uses the re-summation technique. The left side of
the identity (1) can be rewritten as
$G_{m}=\sum_{i=1}^{m}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m-i)\Gamma(m+b+1-i)(m-i)(c+i)}.$
(127)
In (127), by performing a partial fraction decomposition
$\frac{1}{(m-i)(c+i)}=\frac{1}{c+m}\left(\frac{1}{c+i}+\frac{1}{m-i}\right),$
(128)
we obtain the recurrence relation, cf. (114),
$G_{m}=c_{m-1}G_{m-1}+r_{m-1},$ (129)
where
$\displaystyle c_{m-1}=$ $\displaystyle\frac{1}{c+m}$ (130) $\displaystyle
r_{m-1}=$
$\displaystyle\frac{1}{c+m}\sum_{i=1}^{m}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(m+b+1-i)}.$
(131)
The summation in (131) is related to a hypergeometric function
$\,{}_{2}F_{1}(1-b-m,1-m\mathchar 24635\relax\;a+1\mathchar 24635\relax\;1)$
that admits a closed-form representation by using the identity (113).
Therefore, we have
$\displaystyle\sum_{i=1}^{m}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(m+b+1-i)}$
$\displaystyle=\frac{\Gamma(a+b+2m-1)}{\Gamma(m)\Gamma(a+m)\Gamma(b+m)\Gamma(a+b+m)},$
(132)
and
$r_{m-1}=\frac{\Gamma(a+b+2m-1)}{(c+m)\Gamma(m)\Gamma(a+m)\Gamma(b+m)\Gamma(a+b+m)}.$
(133)
Iterate $m$ times the recurrence relation (129), the desired identity (1) is
established. This completes the proof of Lemma 1.
###### Lemma 2.
For any complex numbers $a,b\notin\mathbb{Z^{-}}$, and any
$c\in\mathbb{Z^{+}}$, we have
$\displaystyle\sum_{i=1}^{m}\frac{1}{\Gamma(c+i)\Gamma(a+i)\Gamma(m+1-i)\Gamma(m+b+1-i)}$
$\displaystyle=\frac{1}{\Gamma(m+b)\Gamma(m+a+b)\Gamma(c)\Gamma(m+c)}$
$\displaystyle~{}~{}~{}~{}\times\sum_{i=1}^{m}\frac{\Gamma(m+a+b+i-1)\Gamma(m+c-i)}{\Gamma(a+i)\Gamma(m-i+1)}.$
(134)
Proof: We prove Lemma 2 by using the dummy summation technique. Denote
$F=\sum_{i=1}^{m}f_{i}$ (135)
with
$f_{i}=\frac{1}{\Gamma(c+i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b+m-i+1)}.$ (136)
We first introduce an additional summation in (135) by replacing
$\Gamma(i)/\Gamma(c+i)$ with the summation form in (110). The summation $F$
now becomes
$F=\sum_{i=1}^{m}\sum_{j=1}^{c}g_{i,j}$ (137)
with
$g_{i,j}=\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)(i+j-1)}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b+m-i+1)}.$
(138)
In the double summation (137), we apply the Lemma 1 to first evaluate the sum
over $i$, leading to
$\displaystyle F=$
$\displaystyle\frac{1}{\Gamma(b+m)\Gamma(a+b+m)}\sum_{i=1}^{m}\frac{\Gamma(a+b-i+2m)}{\Gamma(m-i+1)\Gamma(a-i+m+1)}$
$\displaystyle\times\sum_{j=1}^{c}\frac{(-1)^{j+1}\Gamma(m-i+j)}{\Gamma(j)\Gamma(c-j+1)\Gamma(j+m)},$
(139)
where it is now possible to evaluate the sum over j as
$\displaystyle\sum_{j=1}^{c}\frac{(-1)^{j+1}\Gamma(m-i+j)}{\Gamma(j)\Gamma(c-j+1)\Gamma(j+m)}$
$\displaystyle=\frac{\Gamma(m-i+1)}{\Gamma(c)\Gamma(m+1)}\,_{2}F_{1}(1-c,m-i+1\mathchar
24635\relax\;m+1\mathchar 24635\relax\;1).$ (140)
$\displaystyle=\frac{\Gamma(c+i-1)\Gamma(m-i+1)}{\Gamma(c)\Gamma(i)\Gamma(c+m)}.$
(141)
The result (141) is obtained by using the identity (113) to evaluate the unit
argument hypergeometric function in (140). Inserting (141) into (139) and
shifting the index $i\to m-i+1$, Lemma 2 is proved.
###### Lemma 3.
For any complex numbers $a,b\notin\mathbb{Z^{-}}$, and any
$c\in\mathbb{Z^{+}}$, we have
$\displaystyle\sum_{i=1}^{m}\frac{1}{\Gamma(c+i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)i}$
$\displaystyle=\frac{1}{\Gamma(a)\Gamma(a+m)\Gamma(1+b+m)\Gamma(b+c+m)}\sum_{i=1}^{m}\frac{\Gamma(a-i+m)\Gamma(b+c+i+m)}{\Gamma(c+i)\Gamma(m-i+1)i}$
$\displaystyle~{}~{}~{}\\!~{}+\frac{\psi_{0}(a)-\psi_{0}(a+m)}{\Gamma(a)\Gamma(c)\Gamma(m+1)\Gamma(b+m+1)}.$
(142)
Proof: Similar to the proof of Lemma 2, we introduce an additional summation
in
$F=\sum_{i=1}^{m}\frac{1}{\Gamma(c+i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)i}$
(143)
by replacing the term $\Gamma(i)/\Gamma(c+i)$ with the dummy summation (110).
Consequently, one has
$F=\sum_{i=1}^{m}\sum_{j=1}^{c}g_{i,j}$ (144)
with
$g_{i,j}=\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)(i+j-1)}\frac{1}{\Gamma(i+1)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)},$
(145)
which is the same form as the summand in (138). Therefore, one uses the
results (137)–(140) to obtain the identity (142), and Lemma 3 is proved.
###### Lemma 4.
For any complex numbers $a,b\notin\mathbb{Z^{-}}$, and any
$c,d\in\mathbb{Z^{+}}$, we have
$\displaystyle\sum_{i=1}^{m}\frac{1}{\Gamma(c+i)\Gamma(a+i)\Gamma(d+m-i+1)\Gamma(b+m-i+1)}$
$\displaystyle=\frac{1}{\Gamma(d)\Gamma(a+m)\Gamma(a+b+m)\Gamma(c+d+m)}\sum_{i=1}^{m}\frac{\Gamma(c+d+i-1)\Gamma(a+b-i+2m)}{\Gamma(c+i)\Gamma(b-i+m+1)}$
$\displaystyle~{}~{}~{}+\frac{1}{\Gamma(c)\Gamma(b+m)\Gamma(a+b+m)\Gamma(c+d+m)}\sum_{i=1}^{m}\frac{\Gamma(c+d+i-1)\Gamma(a+b-i+2m)}{\Gamma(d+i)\Gamma(a-i+m+1)}.$
Proof: Proving the Lemma 4 also utilizes the dummy summation technique. Denote
the left side of the identity (LABEL:eq:lemma4) as
$F=\sum_{i=1}^{m}f_{i},$ (147)
with
$f_{i}=\frac{1}{\Gamma(c+i)\Gamma(a+i)\Gamma(d+m-i+1)\Gamma(b+m-i+1)}.$ (148)
By interpreting respectively the gamma ratios $\Gamma(i)/\Gamma(c+i)$ and
$\Gamma(m-i+1)/\Gamma(d+m-i+1)$ as the dummy summations (110) and
$\sum_{k=1}^{d}\frac{(-1)^{k+1}}{\Gamma(k)\Gamma(d-k+1)(m-i+k)},$ (149)
the summation $F$ can be written as
$F=\sum_{i=1}^{m}\sum_{j=1}^{c}\sum_{k=1}^{d}g_{i,j,k},$ (150)
where
$\displaystyle g_{i,j,k}=~{}\\!\\!$
$\displaystyle\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle\times\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)(i+j-1)}\frac{(-1)^{k+1}}{\Gamma(k)\Gamma(d-k+1)(m-i+k)}.$
(151)
After taking a partial fraction decomposition
$\frac{1}{(m-i+k)(i+j-1)}=\frac{1}{(i+j-1)(j+k+m-1)}+\frac{1}{(k+m-i)(j+k+m-1)}$
(152)
in (2.2.2), the summation (150) is split into two sums
$F=F_{1}+F_{2},$ (153)
where
$\displaystyle F_{1}=~{}\\!\\!$
$\displaystyle\sum_{j=1}^{c}\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)}\sum_{k=1}^{d}\frac{(-1)^{k+1}}{\Gamma(k)\Gamma(d-k+1)(j+k+m-1)}$
$\displaystyle\times\sum_{i=1}^{m}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)(i+j-1)}$
(154) $\displaystyle F_{2}=~{}\\!\\!$
$\displaystyle\sum_{k=1}^{d}\frac{(-1)^{k+1}}{\Gamma(k)\Gamma(d-k+1)}\sum_{j=1}^{c}\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)(j+k+m-1)}$
$\displaystyle\times\sum_{i=1}^{m}\frac{1}{\Gamma(i)\Gamma(b+i)\Gamma(m-i+1)\Gamma(a-i+m+1)(i+k-1)}.$
(155)
Specialize $c=d$ and $i=j+m$ in the identity (110), one has
$\sum_{k=1}^{d}\frac{(-1)^{k+1}}{\Gamma(k)\Gamma(d-k+1)(j+k+m-1)}=\frac{\Gamma(j+m)}{\Gamma(d+j+m)}.$
(156)
Insert the above result into (154), and evaluate the resulting summation over
$i$ by the identity (1), one arrives at
$\displaystyle F_{1}=~{}\\!\\!$
$\displaystyle\frac{1}{\Gamma(b+m)\Gamma(a+b+m)}\sum_{j=1}^{c}\frac{(-1)^{j+1}}{\Gamma(j)\Gamma(c-j+1)\Gamma(d+j+m)}$
$\displaystyle\times\sum_{i=1}^{m}\frac{\Gamma(m+j-i)\Gamma(a+b-i+2m)}{\Gamma(m-i+1)\Gamma(a-i+m+1)},$
(157)
where the summation over $j$ is simplified, by using the identity (113), to
$\displaystyle\sum_{j=1}^{c}\frac{(-1)^{j+1}\Gamma(m-i+j)}{\Gamma(j)\Gamma(c-j+1)\Gamma(d+j+m)}$
$\displaystyle=\frac{\Gamma(m-i+1)}{\Gamma(c)\Gamma(d+m+1)}\,_{2}F_{1}(1-c,m-i+1\mathchar
24635\relax\;d+m+1\mathchar 24635\relax\;1)$ (158)
$\displaystyle=\frac{\Gamma(m-i+1)\Gamma(c+d+i-1)}{\Gamma(c)\Gamma(d+i)\Gamma(c+d+m)}.$
(159)
Consequently, we have
$\displaystyle F_{1}=\\!\\!~{}$
$\displaystyle\frac{1}{\Gamma(c)\Gamma(b+m)\Gamma(a+b+m)\Gamma(c+d+m)}$
$\displaystyle\times\sum_{i=1}^{m}\frac{\Gamma(c+d+i-1)\Gamma(a+b-i+2m)}{\Gamma(d+i)\Gamma(a-i+m+1)}.$
(160)
In the same manner, we obtain
$\displaystyle F_{2}=\\!\\!~{}$
$\displaystyle\frac{1}{\Gamma(d)\Gamma(a+m)\Gamma(a+b+m)\Gamma(c+d+m)}$
$\displaystyle\times\sum_{i=1}^{m}\frac{\Gamma(c+d+i-1)\Gamma(a+b-i+2m)}{\Gamma(c+i)\Gamma(b-i+m+1)}.$
(161)
Insert the results (160)–(161) into (153), we complete the proof of Lemma 4.
###### Lemma 5.
Denote
$\Phi_{a,b,c,d}^{(x)}=\frac{\Gamma(x+c+1)\Gamma(x+d+1)}{\Gamma(x+a+1)\Gamma(x+b+1)},$
(162)
we have
$\displaystyle\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+1)}\left(\frac{1}{a+c-i+2m+1}-\frac{1}{c+i}\right)$
$\displaystyle=\frac{\Gamma(a+c+m+1)}{\Gamma(c+m+1)}\sum_{i=1}^{m}\left(\frac{\Phi_{0,a+c,a,c}^{(m-i)}}{a+c-2i+2m+1}+a\Phi_{1,a+c+1,a,c}^{(m-i)}\right.$
$\displaystyle~{}~{}~{}~{}\left.\\!-\frac{\Phi_{1,a+c,a+1,c}^{(m-i)}}{a+c-2i+2m+2}\right),$
(163)
where it is sufficient to consider $\real(a,c)\geq 0$.
Proof: We prove the Lemma 5 by using the re-summation technique. We denote the
left side of the identity (163) as
$\mathcal{G}=G_{m}-G^{\prime}_{m},$ (164)
where
$\displaystyle G_{m}=$
$\displaystyle\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+1)(a+c-i+2m+1)}$
(165) $\displaystyle G^{\prime}_{m}=$
$\displaystyle\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m+1)}{\Gamma(m-i+1)(c+i)}.$
(166)
Here, we choose the recurrence relations
$\displaystyle G_{m}=$ $\displaystyle c_{m-1}G_{m-1}+r_{m-1}$ (167)
$\displaystyle G^{\prime}_{m}=$ $\displaystyle
c_{m-1}G^{\prime}_{m-1}+r^{\prime}_{m-1},$ (168)
where
$\displaystyle c_{m-1}=~{}\\!\\!$ $\displaystyle\frac{a+c+m}{c+m}$ (169)
$\displaystyle r_{m-1}=~{}\\!\\!$
$\displaystyle\frac{a}{c+m}\sum_{i=0}^{m}\frac{(-1)^{i}\Gamma(a-i+m)}{\Gamma(m-i+1)}+\frac{(a+c+m)\Gamma(a+m)}{(c+m)\Gamma(m+1)}$
$\displaystyle\times\left(\frac{m}{a+c+2m-1}-\frac{a+m}{a+c+2m}\right)$ (170)
$\displaystyle r^{\prime}_{m-1}=~{}\\!\\!$
$\displaystyle\frac{a}{c+m}\sum_{i=1}^{m}\frac{(-1)^{i}\Gamma(a-i+m)}{\Gamma(m-i+1)}.$
(171)
Iterating $m$ times the recurrence relations (167) and (168), we obtain
$\displaystyle G_{m}=~{}\\!\\!$
$\displaystyle\frac{\Gamma(a+c+m+1)}{\Gamma(c+m+1)}\left(a\sum_{i=1}^{m}\\!\right.\sum_{j=0}^{m-i+1}(-1)^{j}\Phi_{a+c+1,1-j,c,a-j}^{(m-i)}$
$\displaystyle+\\!\left.\sum_{i=1}^{m}\left(\frac{\Phi_{0,a+c,c,a}^{(m-i)}}{a+c-2i+2m+1}-\frac{\Phi_{1,a+c,c,a+1}^{(m-i)}}{a+c-2i+2m+2}\right)\right)$
(172) $\displaystyle G^{\prime}_{m}=~{}\\!\\!$
$\displaystyle\frac{a\Gamma(a+c+m+1)}{\Gamma(c+m+1)}\sum_{i=1}^{m}\sum_{j=1}^{m-i+1}(-1)^{j}\Phi_{a+c+1,1-j,c,a-j}^{(m-i)}.$
(173)
It is noticed that the double summation in (172) is the same form as the one
in (173). By inserting the re-summations (172) and (173) into (164), we obtain
the desired identity (163). This proves Lemma 5.
###### Lemma 6.
Using the notation (162), we have
$\displaystyle\sum_{i=1}^{m}\Phi_{0,a,b,a+b}^{(m-i)}\left(\frac{1}{a+b+c-i+2m+1}-\frac{1}{c+i}\right)$
$\displaystyle=\Phi_{0,a,b,a+b}^{(m+c)}\sum_{i=1}^{m}\Phi_{0,a,b,a+b}^{(i-1)}\Phi_{b,a+b,0,a}^{(c+i-1)}\left(\frac{1}{a+b+c+2i-1}-\frac{b(a+b)}{ai(a+b+c+i)}\right.$
$\displaystyle~{}~{}~{}~{}\\!\left.-\frac{b(a-b)}{a(a+i)(b+c+i)}+\frac{(b+i)(a+b+i)}{i(a+i)(a+b+c+2i)}-\frac{a+b+2i-2}{(b+i-1)(a+b+i-1)}\right)$
$\displaystyle~{}~{}~{}~{}+\frac{1}{b}\Phi_{0,a,b,a+b}^{(m-1)},$ (174)
where it is sufficient to consider $a,b,c,a+b,a+b+c\notin\mathbb{Z}^{-}$.
Proof: The proof of Lemma 6 uses the re-summation technique. The left side of
the identity (174) can be written as
$\mathcal{G}=G_{m}-G^{\prime}_{m},$ (175)
where
$\displaystyle G_{m}$
$\displaystyle=\sum_{i=1}^{m}\Phi_{0,a,b,a+b}^{(m-i)}\frac{1}{a+b+c-i+2m+1}$
(176) $\displaystyle G^{\prime}_{m}$
$\displaystyle=\sum_{i=1}^{m}\Phi_{0,a,b,a+b}^{(m-i)}\frac{1}{c+i}.$ (177)
In relating the two summations above, we choose the following recurrence
relations
$\displaystyle G_{m}$ $\displaystyle=c_{m-1}G_{m-1}+r_{m-1}$ (178)
$\displaystyle G^{\prime}_{m}$
$\displaystyle=c_{m-1}G^{\prime}_{m-1}+r^{\prime}_{m-1},$ (179)
where
$\displaystyle c_{m-1}=~{}\\!\\!$
$\displaystyle\frac{(b+c+m)(a+b+c+m)}{(c+m)(a+c+m)}$ (180) $\displaystyle
r_{m-1}=~{}\\!\\!$ $\displaystyle
c_{m-1}\left(\frac{\Phi_{0,a,b,a+b}^{(m-1)}}{a+b+c+2m-1}+\frac{\Phi_{0,a,b,a+b}^{(m)}}{a+b+c+2m}\right)-\frac{b}{a}c_{m-1}$
$\displaystyle\times\left(\frac{(a-b)}{b+c+m}\sum_{i=1}^{m}\Phi_{0,a+1,b,a+b}^{(m-i)}+\frac{(a+b)}{a+b+c+m}\sum_{i=1}^{m+1}\Phi_{1,a,b,a+b}^{(m-i)}\right)$
(181) $\displaystyle r^{\prime}_{m-1}=~{}\\!\\!$
$\displaystyle\frac{b(a-b)}{a(a+c+m)}\sum_{i=1}^{m-1}\Phi_{0,a+1,b,a+b}^{(m-i-1)}+\frac{b(a+b)}{a(c+m)}\sum_{i=1}^{m}\Phi_{1,a,b,a+b}^{(m-i-1)}.$
(182)
Iterating $m$ times the above relations (178) and (179), one obtains
$\displaystyle G_{m}=~{}\\!\\!$
$\displaystyle\Phi_{0,a,b,a+b}^{(m+c)}\left(\rule{0.0pt}{19.0633pt}\\!-\frac{b(a+b)}{a}\sum_{i=1}^{m}\Phi_{1,a,b,a+b}^{(i-2)}\sum_{j=1}^{m-i+1}\Phi_{a+b+1,b,0,a}^{(m-j+c)}-\frac{b(a-b)}{a}\right.$
$\displaystyle\times\sum_{i=1}^{m}\Phi_{0,a+1,b,a+b}^{(i-1)}\sum_{j=1}^{m-i+1}\Phi_{b+1,a+b,a,0}^{(m-j+c)}+\sum_{i=1}^{m}\Phi_{b,a+b,0,a}^{(m-i+c)}$
$\displaystyle\times\left(\frac{\Phi_{0,a,b,a+b}^{(m-i+1)}}{a+b+c-2i+2m+2}+\frac{\Phi_{0,a,b,a+b}^{(m-i)}}{a+b+c-2i+2m+1}\right.$
$\displaystyle\left.\\!-\frac{b(a+b)\Phi_{1,a,b,a+b}^{(m-i)}}{a(a+b+c-i+m+1)}\right)\\!\left.\rule{0.0pt}{19.0633pt}\right)$
(183) $\displaystyle G^{\prime}_{m}=~{}\\!\\!$
$\displaystyle\Phi_{0,a,b,a+b}^{(m+c)}\left(\frac{b(a+b)}{a}\sum_{i=1}^{m}\Phi_{1,a,b,a+b}^{(i-2)}\sum_{j=1}^{m-i+1}\Phi_{a+b+1,b+1,0,a+1}^{(m-j+c)}\right.$
$\displaystyle+\\!\left.\frac{b(a-b)}{a}\sum_{i=1}^{m}\Phi_{0,a+1,b,a+b}^{(i-2)}\sum_{j=1}^{m-i+1}\Phi_{b+1,a+b+1,a,1}^{(m-j+c)}\right).$
(184)
Note that in order to reveal the potential cancellations between the two re-
summations above, one will also need the identity
$\displaystyle(c_{1}-b_{1})\sum_{j=1}^{m}\Phi_{a_{1},b_{1}+1,c_{1},d_{1}+1}^{(m-j)}+(d_{1}-a_{1}+1)\sum_{j=1}^{m}\Phi_{a_{1},b_{1},c_{1},d_{1}}^{(m-j)}$
$\displaystyle=\Phi_{a_{1}-1,b_{1},c_{1},d_{1}}^{(m)}-\Phi_{a_{1}-1,b_{1},c_{1},d_{1}}^{(0)},\qquad
a_{1},b_{1},c_{1},d_{1}\notin\mathbb{Z}^{-}.$ (185)
This identity is obtained by iterating $m$ times the following recurrence
relation
$H_{m,a_{1},d_{1}}=H_{m-1,a_{1}+1,d_{1}+1}+s_{m-1},$ (186)
where
$H_{m,a_{1},d_{1}}=\sum_{j=1}^{m}\Phi_{a_{1},b_{1},c_{1},d_{1}}^{(m-j)}$ (187)
$s_{m-1}=H_{m,a_{1},d_{1}}-H_{m-1,a_{1}+1,d_{1}+1}=(c_{1}-b_{1})\sum_{j=1}^{m-1}\Phi_{a_{1},b_{1},c_{1}-1,d_{1}}^{(m-j)}+\Phi_{a_{1},b_{1},c_{1},d_{1}}^{(0)}.$
(188)
We now use the identity (185) with the specializations
$\displaystyle a_{1}=a+b+c+i,~{}~{}~{}b_{1}=b+c+i-1,~{}~{}~{}c_{1}=c+i-1,$
$\displaystyle d_{1}=a+c+i-1,~{}~{}~{}m\to m-i+1$ (189)
and
$\displaystyle a_{1}=b+c+i,~{}~{}~{}b_{1}=a+b+c+i-1,~{}~{}~{}c_{1}=a+c+i-1,$
$\displaystyle d_{1}=c+i-1,~{}~{}~{}m\to m-i+1,$ (190)
the result (184) becomes
$\displaystyle G^{\prime}_{m}=$
$\displaystyle~{}\Phi_{0,a,b,a+b}^{(c+m)}\Bigg{(}\\!-\frac{b(a+b)}{a}\sum_{i=1}^{m}\Phi_{1,a,b,a+b}^{(i-2)}\sum_{j=1}^{m-i+1}\Phi_{a+b+1,b,0,a}^{(m-j+c)}-\frac{b(a-b)}{a}$
$\displaystyle\\!~{}\times\sum_{i=1}^{m}\Phi_{0,a+1,b,a+b}^{(i-2)}\sum_{j=1}^{m-i+1}\Phi_{b+1,a+b,a,0}^{(m-j+c)}+\sum_{i=1}^{m}\left(\Phi_{a+b,b,0,a}^{(c+i-1)}-\Phi_{a+b,b,0,a}^{(c+m)}\right)$
$\displaystyle\\!~{}\times\left(\frac{a-b}{a}\Phi_{0,a+1,b,a+b}^{(i-2)}+\frac{a+b}{a}\Phi_{1,a,b,a+b}^{(i-2)}\right)\\!\Bigg{)},$
(191)
where the double summations are of the same form as the ones in (183).
Therefore, all the double summations are cancelled completely in (175). The
remaining terms are the desired result (174). This completes the proof of
Lemma 6.
Note that the results in lemmas 2–4 are analytically continued to any complex
number $c$ or $d$ except for negative integers. This fact allows us to take
derivatives of the formulas (134), (142), and (LABEL:eq:lemma4) in obtaining
the identities (B.2)–(B.2) listed in Appendix B.2. For example, the identity
(B.2) is established by taking derivatives of $c$ and $d$ of the formula
(LABEL:eq:lemma4) before setting $c=d=0$. The lemmas 1–4 along with the
identities (B.2)–(B.2) are useful in simplifying the summations in (A.1) and
(A.2). It is also worth mentioning that the identities of Lemma 5 and Lemma 6
admit closed-form representations for some special cases, e.g., when the
parameter $c$ is fixed to non-negative integers, since all the gamma functions
involved are reduced to rational functions. These identities as well as their
derivatives of $c$ are the key tools in simplifying the summations in (A.1)
and (A.2).
Note also that in lemmas 1–4, by rewriting the summations involved as
hypergeometric functions, the corresponding identities induce the following
new transformation formulas of unit arguments (192)–(195), which may be of
independent interest.
$\,{}_{3}F_{2}(c+1,1-m,1-b-m\mathchar 24635\relax\;a+1,c+2\mathchar
24635\relax\;1)$
$\displaystyle=\frac{(c+1)\Gamma(a+1)\Gamma(a+b+2m-1)}{(c+m)\Gamma(a+m)\Gamma(a+b+m)}$
$\displaystyle~{}~{}~{}\times\,_{3}F_{2}(1,1-m,1-a-m\mathchar
24635\relax\;2-a-b-2m,1-c-m\mathchar 24635\relax\;1)$ (192)
$\,{}_{3}F_{2}(1,1-m,1-b-m\mathchar 24635\relax\;a+1,c+1\mathchar
24635\relax\;1)$
$\displaystyle=\frac{c}{c+m-1}\,_{3}F_{2}(1,1-m,a+b+m\mathchar
24635\relax\;a+1,2-c-m\mathchar 24635\relax\;1)$ (193)
$\,{}_{4}F_{3}(1,1,1-m,1-b-m\mathchar 24635\relax\;2,a+1,c+1\mathchar
24635\relax\;1)$
$\displaystyle=\frac{a(b+c+m)}{(a+m-1)(b+m)}\,_{4}F_{3}(1,1,1-m,b+c+m+1\mathchar
24635\relax\;2,c+1,2-a-m\mathchar 24635\relax\;1)$
$\displaystyle~{}~{}~{}+\frac{ac(\psi_{0}(a)-\psi_{0}(a+m))}{m(b+m)}$ (194)
$\displaystyle\frac{\,{}_{3}F_{2}(1,1-b-m,1-d-m\mathchar
24635\relax\;a+1,c+1\mathchar
24635\relax\;1)}{\Gamma(a+1)\Gamma(c+1)\Gamma(b+m)\Gamma(d+m)}$
$\displaystyle=\frac{\,{}_{3}F_{2}(1,1-b,1-d\mathchar
24635\relax\;a+m+1,c+m+1\mathchar
24635\relax\;1)}{\Gamma(b)\Gamma(d)\Gamma(a+m+1)\Gamma(c+m+1)}-\frac{1}{a+b+m-1}$
$\displaystyle~{}~{}~{}\times\left(\frac{\,{}_{3}F_{2}(1,1-a,c+d+m\mathchar
24635\relax\;2-a-b-m,d+m+1\mathchar
24635\relax\;1)}{\Gamma(a)\Gamma(c)\Gamma(b+m)\Gamma(d+m+1)}\right.$
$\displaystyle~{}~{}~{}+\left.\\!\frac{\,{}_{3}F_{2}(1,1-b,c+d+m\mathchar
24635\relax\;2-a-b-m,c+m+1\mathchar
24635\relax\;1)}{\Gamma(b)\Gamma(d)\Gamma(a+m)\Gamma(c+m+1)}\right)$
$\displaystyle~{}~{}~{}+\frac{\Gamma(c+d)\Gamma(a+b+2m-1)}{\Gamma(c)\Gamma(d)\Gamma(a+m)\Gamma(b+m)\Gamma(a+b+m)\Gamma(c+d+m)}$
$\displaystyle~{}~{}~{}\times\left(\frac{1}{d}\,_{3}F_{2}(1,c+d,1-a-m\mathchar
24635\relax\;d+1,2-a-b-2m\mathchar 24635\relax\;1)\right.$
$\displaystyle~{}~{}~{}+\\!\left.\frac{1}{c}\,_{3}F_{2}(1,c+d,1-b-m\mathchar
24635\relax\;c+1,2-a-b-2m\mathchar 24635\relax\;1)\right).$ (195)
### 2.3 Simplification of summations
In this section, we compute the integrals $\mathrm{I_{A}}$ and
$\mathrm{I_{B}}$ in obtaining the variance formulas in Proposition 3 and
Proposition 4. For convenience, we summarize in Table 3 below the
corresponding integrals of the variance (47) for case A and case B.
Table 3: Integrals of the variance (47) for case A and case B. | $\mathrm{I_{A}}$ | $~{}~{}~{}~{}\mathrm{I_{B}}$
---|---|---
Case A: Arbitrary number of particle (17) | (51) | (55)
Case B: Fixed number of particle (18) | (69) | (79)
| |
#### Case A: Arbitrary number of particles
In case A, the variance calculation boils down to simplifying the summation
representations of $\mathrm{A_{1}}$, $\mathrm{A_{2}}$, $\mathrm{B_{1}}$, and
$\mathrm{B_{2}}$ in (A.1)–(A.1) as listed in Appendix A.1. The corresponding
simplification procedures are shown below.
We first discuss the simplifications of summation representations (A.1) and
(A.1) in computing the integral $\mathrm{I_{A}}$ in (51). The summations in
(A.1) can be simplified by appropriately changing the summation orders and
using the first type identities (287)–(B.1) of the existing framework, where
the closed-form identities are (287)–(294) and the semi closed-form ones are
(295)–(B.1). Semi closed-form identities that we referred to represent the
relation between two single summations. These sums are what we refer to as
unsimplifiable basis, which will cancel completely in obtaining the closed-
form results. On the other hand, whether the considered unsimplifiable basis
can be computed into closed-form formulas is not of primary importance.
The simplification of the summations in (A.1) is as follows. In (A.1), the
first double summation is
$\displaystyle\sum_{k=0}^{m-1}\sum_{j=2k-2}^{2k}\frac{2(-1)^{j}(2a+4k+1)(j+1)_{2}(a+j+1)_{2}}{\Gamma(2k-j+1)\Gamma(j-2k+3)(2a+j+2k+1)_{3}}\bigg{(}\\!(\psi_{0}(j+3)$
$\displaystyle-\psi_{0}(2a+j+2k+4)-\psi_{0}(j-2k+3)+\psi_{0}(a+j+3))^{2}$
$\displaystyle-\psi_{1}(2a+j+2k+4)+\psi_{1}(a+j+3)-\psi_{1}(j-2k+3)+\psi_{1}(j+3)\\!\bigg{)},$
(196)
which is directly reduced to a single sum after evaluating the sum of $j$. By
using the identities (287)–(B.1) along with the results AS72
$\displaystyle\psi_{0}(mk)$ $\displaystyle=\ln
m+\frac{1}{m}\sum_{i=0}^{m-1}\psi_{0}\\!\left(k+\frac{i}{m}\right)$ (197)
$\displaystyle\psi_{1}(mk)$
$\displaystyle=\frac{1}{m^{2}}\sum_{i=0}^{m-1}\psi_{1}\\!\left(k+\frac{i}{m}\right),\qquad
m\in\mathbb{Z^{+}},$ (198)
the remaining single sum is computed to the unsimplifiable basis of the form
$\sum_{k=1}^{m}\frac{\psi_{0}(k+c)}{k+d},\qquad c\neq d.$ (199)
The other double summation in (A.1) is
$\displaystyle\sum_{k=0}^{m-1}\sum_{j=0}^{2k-3}\frac{4(2a+4k+1)(j+1)_{2}(a+j+1)_{2}}{(2k-j-2)_{3}(2a+j+2k+1)_{3}}\Psi(j),$
(200)
where
$\Psi(j)=\psi_{0}(2a+j+2k+4)-\psi_{0}(a+j+3)+\psi_{0}(2k-j-2)-\psi_{0}(j+3).$
(201)
To process the summation (200), we first perform the partial fraction
decomposition of the term
$\displaystyle\frac{4(2a+4k+1)(j+1)_{2}(a+j+1)_{2}}{(2k-j-2)_{3}(2a+j+2k+1)_{3}}$
$\displaystyle=-\frac{2(a+k)(2a+2k+1)}{2a+j+2k+2}+\frac{2(a+k)(a+2k-1)(2a+2k-1)}{(2a+4k-1)(2a+j+2k+1)}$
$\displaystyle~{}~{}~{}\\!~{}+\frac{2(a+k+1)(a+2k+2)(2a+2k+1)}{(2a+4k+3)(2a+j+2k+3)}-\frac{2(k+1)(2k+1)(a+2k+2)}{(2a+4k+3)(j-2k)}$
$\displaystyle~{}~{}~{}\\!~{}-\frac{2k(2k-1)(a+2k-1)}{(2a+4k-1)(j-2k+2)}+\frac{2k(2k+1)}{j-2k+1}.$
(202)
The sum (200) now boils down to
$\displaystyle\sum_{k=0}^{m-1}\mathrm{p}_{c}(k)\sum_{j=0}^{2k-3}\frac{1}{j+2k+2a+c}\Psi(j)$
(203)
$\displaystyle\sum_{k=0}^{m-1}\mathrm{p}^{\prime}_{c}(k)\sum_{j=0}^{2k-3}\frac{1}{j-2k-1+c}\Psi(j),$
(204)
where the parameter $c$ takes the values $c=1,2,3$, $\mathrm{p}_{c}(k)$ and
$\mathrm{p}^{\prime}_{c}(k)$ denote the polynomials in $k$ from the partial
fraction decomposition (2.3). When considering $c=2$ and the term
$\psi_{0}(j+3)$ in (201), the double summation
$-\sum_{k=0}^{m-1}2(a+k)(2a+2k+1)\sum_{j=0}^{2k-3}\frac{\psi_{0}(j+3)}{j+2k+2a+2}$
(205)
is simplified to a single sum as
$\displaystyle\sum_{j=0}^{m-3}\Bigg{(}\psi_{0}(2j+3)((j+1)(2j+1)(\psi_{0}(1)-\psi_{0}(a+j+m+1))+(j-m+2)$
$\displaystyle\times(2a-j+m)+(j+1)(2j+1)(\psi_{0}(a+2j+3)-\psi_{0}(1)))+\psi_{0}(2j+4)$
$\displaystyle\times\left((j+1)(2j+3)\left(\psi_{0}(1)-\psi_{0}\\!\left(a+j+m+\frac{3}{2}\right)\right)+(j+1)(2j+3)\right.$
$\displaystyle\times\\!\left.\left(\psi_{0}\\!\left(a+2j+\frac{7}{2}\right)-\psi_{0}(1)\right)+(2a-j+m-1)(j-m+2)\right)\\!\Bigg{)}.$
(206)
The above result is obtained as follows. In (205), it is noticed that directly
evaluating the inner summation does not permit further simplifications.
Instead, we first have to separate the inner summation that depends on an even
index $2k$ into two as
$-2\sum_{k=0}^{m-1}(a+k)(2a+2k+1)\left(\sum_{j=0}^{k-2}\frac{\psi_{0}(2j+3)}{2a+2k+2j+2}+\sum_{j=0}^{k-2}\frac{\psi_{0}(2j+4)}{2a+2k+2j+3}\right),$
(207)
which allows us to change the summation order as
$\displaystyle-2\sum_{j=0}^{m-3}\psi_{0}(2j+3)\sum_{k=j+2}^{m-1}\frac{(a+k)(2a+2k+1)}{2a+2k+2j+2}$
$\displaystyle-2\sum_{j=0}^{m-3}\psi_{0}(2j+4)\sum_{k=j+2}^{m-1}\frac{(a+k)(2a+2k+1)}{2a+2k+2j+3}.$
(208)
Evaluating the inner summations over $k$ in (2.3) directly gives the result
(2.3). Other double summations in (203) and (204) are similarly simplified
into single sums. The resulting single sums, cf. (2.3), are further
manipulated into the unsimplifiable basis of the form (199) by using the first
type identities (287)–(B.1) along with the result (197).
Different from the summations in (A.1), the simplification of the summations
in (A.1) will need new simplification techniques before the existing framework
is applicable. The summations in (A.1) include three double sums and one
triple sum. We first simplify the inner summations over $j$ of the three
double sums by using the identities (B.2) and (B.2) of the new simplification
framework. Specifically, the inner summation
$\sum_{j=0}^{2k}\frac{2(j+1)(2k-j+1)\psi_{0}(j+2)\psi_{0}(2k-j+2)}{\Gamma(j+1)\Gamma(a+j+1)\Gamma(2k-j+1)\Gamma(a-j+2k+1)}$
(209)
of the first double sum in (A.1) is simplified to a semi-closed form
representation that can be compactly written as
$\displaystyle\frac{1}{\Gamma(2k)\Gamma(a+2k+1)\Gamma(2a+2k+1)}\Bigg{(}\\!\left(\frac{1-4a^{2}}{4a+8k-2}+a-8k^{2}-14k-\frac{15}{2}\right)$
$\displaystyle\times\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k)}{\Gamma(a-j+2k)j^{2}}+\left(\left(\frac{1-4a^{2}}{4a+8k-2}+a-8k^{2}-14k-\frac{15}{2}\right)\psi_{0}(2k)\right.$
$\displaystyle+\\!\left.\frac{1-4a^{2}}{2(2a+4k-1)^{2}}+\frac{3-4a^{2}}{4a+8k-2}+\frac{4(a-1)}{a+2k}+a-14k-\frac{4}{k}-19\right)$
$\displaystyle\times\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k)}{\Gamma(a-j+2k)j}\\!\Bigg{)}+\mathrm{CF},$
(210)
where the shorthand notation $\mathrm{CF}$ denotes the closed-form terms
omitted. The result (2.3) is obtained by using the identities (132), (B.2),
and (B.2) after one rewrites the summation (209) as
$\displaystyle
8\sum_{j=2}^{2k+1}\frac{1}{\Gamma(j)\Gamma(a+j)\Gamma(2k-j+2)\Gamma(a-j+2k+2)}$
$\displaystyle+8\sum_{j=1}^{2k+1}\frac{\psi_{0}(j)}{\Gamma(j)\Gamma(a+j)\Gamma(2k-j+2)\Gamma(a-j+2k+2)}$
$\displaystyle+8\sum_{j=1}^{2k}\frac{\psi_{0}(j)}{\Gamma(j)\Gamma(a+j+1)\Gamma(2k-j+1)\Gamma(a-j+2k+1)}$
$\displaystyle+2\sum_{j=1}^{2k+1}\frac{\psi_{0}(j)\psi_{0}(2k-j+2)}{\Gamma(j)\Gamma(a+j)\Gamma(2k-j+2)\Gamma(a-j+2k+2)}$
$\displaystyle+4\sum_{j=1}^{2k}\frac{\psi_{0}(j)\psi_{0}(2k-j+1)}{\Gamma(j)\Gamma(a+j)\Gamma(2k-j+1)\Gamma(a-j+2k+2)}$
$\displaystyle+2\sum_{j=1}^{2k-1}\frac{\psi_{0}(j)\psi_{0}(2k-j)}{\Gamma(j)\Gamma(a+j+1)\Gamma(2k-j)\Gamma(a-j+2k+1)}$
$\displaystyle-\frac{4}{\Gamma(a+1)\Gamma(a+2k+1)}\left(\frac{\psi_{0}(2k+1)}{\Gamma(2k+1)}+\frac{\psi_{0}(2k)}{\Gamma(2k)}\right).$
(211)
Using the same approach, the other inner summations over $j$ of the first
three double summations in (A.1) are simplified to a semi closed-form
representation, cf. (2.3), where the resulting unsimplifiable basis are
$\displaystyle\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k)}{\Gamma(a-j+2k)j}$ (212)
$\displaystyle\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k)}{\Gamma(a-j+2k)j^{2}}.$
(213)
Now we simplify the inner summations over the indexes $i$ and $j$ of the
triple sum in (A.1). After the partial fraction decomposition
$\frac{1}{(j)_{3}}=-\frac{1}{j+1}+\frac{1}{2(j+2)}+\frac{1}{2j},$ (214)
the corresponding inner summation is written as
$\displaystyle\sum_{j=1}^{2k-1}\sum_{i=1}^{2k-j}\frac{4i(2k-i+2)\Gamma(a+2k+j+3)\Gamma(a+2k-j+1)}{\Gamma(j+i+1)\Gamma(2k-j-i+1)\Gamma(a+i)\Gamma(a+2k-i+2)}$
$\displaystyle\times(\psi_{0}(a+2k+j+3)-\psi_{0}(2a+4k+4)+\psi_{0}(2k-i+3)-\psi_{0}(j+3))$
$\displaystyle\times\left(-\frac{1}{j+1}+\frac{1}{2(j+2)}+\frac{1}{2j}\right).$
(215)
We first consider the simplification of the summation
$\displaystyle\sum_{j=1}^{2k-1}\sum_{i=1}^{2k-j}\frac{4i(2k-i+2)\Gamma(a+2k+j+3)\Gamma(a+2k-j+1)}{(j+1)\Gamma(j+i+1)\Gamma(2k-j-i+1)\Gamma(a+i)\Gamma(a+2k-i+2)}$
$\displaystyle\times(\psi_{0}(a+2k+j+3)-\psi_{0}(2a+4k+4)+\psi_{0}(2k-i+3)-\psi_{0}(j+3)),$
(216)
which involves the term $1/(j+1)$ from the decomposition (214), and the
remaining parts in (2.3) can be simplified in the same manner. For
convenience, we divide (2.3) into the three sums
$\displaystyle\sum_{j=1}^{2k-1}\sum_{i=1}^{2k-j}\frac{4i(2k-i+2)\Gamma(a+2k+j+3)\Gamma(a+2k-j+1)}{(j+1)\Gamma(j+i+1)\Gamma(2k-j-i+1)\Gamma(a+i)\Gamma(a+2k-i+2)}$
$\displaystyle\times(-\psi_{0}(2a+4k+4)-\psi_{0}(j+3))$ (217)
$\displaystyle\sum_{j=1}^{2k-1}\sum_{i=1}^{2k-j}\frac{4i(2k-i+2)\Gamma(a+2k+j+3)\Gamma(a+2k-j+1)}{(j+1)\Gamma(j+i+1)\Gamma(2k-j-i+1)\Gamma(a+i)\Gamma(a+2k-i+2)}$
$\displaystyle\times\psi_{0}(2k-i+3)$ (218)
$\displaystyle\sum_{j=1}^{2k-1}\sum_{i=1}^{2k-j}\frac{4i(2k-i+2)\Gamma(a+2k+j+3)\Gamma(a+2k-j+1)}{(j+1)\Gamma(j+i+1)\Gamma(2k-j-i+1)\Gamma(a+i)\Gamma(a+2k-i+2)}$
$\displaystyle\times\psi_{0}(a+2k+j+3).$ (219)
The digamma functions in the summation (217) is independent to the index $i$.
This fact allows us to evaluate the summation over $i$ by using the identity
(134) of Lemma 2. Specifically, we first rewrite (217) as
$\displaystyle\sum_{j=1}^{2k-1}\frac{\Gamma(a-j+2k+1)\Gamma(a+j+2k+3)(-\psi_{0}(2a+4k+4)-\psi_{0}(j+3))}{j+1}$
$\displaystyle\times\sum_{i=1}^{2k-j}\frac{1}{\Gamma(a+i)\Gamma(i+j)\Gamma(a-i+2k+2)\Gamma(2k-i-j)}\frac{4i(2k-i+2)}{(i+j)(2k-j-i)}.$
(220)
We now partial fraction decompose the rational function with respect to $i$ in
(220). The summation (217) becomes
$\displaystyle\sum_{j=1}^{2k-1}\frac{\Gamma(a-j+2k+1)\Gamma(a+j+2k+3)(-\psi_{0}(2a+4k+4)-\psi_{0}(j+3))}{j+1}$
$\displaystyle\times\left(4\sum_{i=1}^{2k-j-1}\frac{1}{\Gamma(a+i)\Gamma(i+j)\Gamma(a-i+2k+2)\Gamma(2k-j-i)}-\frac{j+2k+2}{k}\right.$
$\displaystyle\times
2j\sum_{i=1}^{2k-j-1}\frac{1}{\Gamma(a+i)\Gamma(i+j+1)\Gamma(a-i+2k+2)\Gamma(2k-j-i)}+\frac{2k-j}{k}$
$\displaystyle\times\\!\left.2(j+2)\sum_{i=1}^{2k-j}\frac{1}{\Gamma(a+i)\Gamma(i+j)\Gamma(a-i+2k+2)\Gamma(2k-j+1-i)}\right),$
(221)
where the inner summations over $i$ can be evaluated by the identity (134)
with the specializations
$\displaystyle a=j,\qquad~{}~{}~{}~{}~{}~{}\\!b=a+j+2,\qquad c=a,\qquad
m=2k-j-1$ (222) $\displaystyle a=j+1,\qquad b=a+j+2,\qquad c=a,\qquad
m=2k-j-1$ (223) $\displaystyle a=j,\qquad~{}~{}~{}~{}~{}~{}\\!b=a+j+1,\qquad
c=a,\qquad m=2k-j.$ (224)
The summation (221) becomes
$\displaystyle\sum_{j=1}^{2k-1}\left(j+1-\frac{(a+2k+1)^{2}}{j+1}\right)\frac{\psi_{0}(2a+4k+4)+\psi_{0}(j+3)}{\Gamma(a)\Gamma(a+2k+1)}$
$\displaystyle\times\left(4(a-j+2k-1)(a+j+2k+1)\sum_{i=1}^{2k-j-1}\frac{\Gamma(a+i-1)\Gamma(a-i+4k)}{\Gamma(i)\Gamma(2k-i)}\right.$
$\displaystyle-\frac{2j(j+2k+2)(a-j+2k-1)}{k}\sum_{i=1}^{2k-j-1}\frac{\Gamma(a+i-1)\Gamma(a-i+4k+1)}{\Gamma(i)\Gamma(2k-i+1)}$
$\displaystyle+\\!\left.\frac{2(j+2)(2k-j)(a+j+2k+1)}{k}\sum_{i=1}^{2k-j}\frac{\Gamma(a+i-1)\Gamma(a-i+4k+1)}{\Gamma(i)\Gamma(2k-i+1)}\right).$
(225)
In (225), we further change the orders of summations to simplify the sums over
$j$. These summations admit closed-form expressions by using the identities
(B.1)–(B.1) and (293). The remaining summations only consist of single sums,
which can be further simplified by using the identities (B.2)–(B.2) into the
unsimplifiable basis (212) and
$\displaystyle\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k-1)\psi_{0}(j)}{\Gamma(a-j+2k+1)}$
(226)
$\displaystyle\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k-1)\psi_{0}(j)}{\Gamma(a-j+2k)j}$
(227)
$\displaystyle\sum_{j=1}^{2k}\frac{\Gamma(2a-j+4k)\psi_{0}(2a-j+4k)}{\Gamma(a-i+2k)j}.$
(228)
We now move on to simplifying the summation (218). In (218), we change the
summation order to evaluate the sum over $j$ by the identity (142) in Lemma 3.
The summation then becomes
$\displaystyle\Gamma^{2}(a+2k+2)\sum_{i=1}^{2k-1}\sum_{j=1}^{2k-i}\frac{4(i+j+1)(a+i+j+1)(2k-i-j+1)}{j\Gamma(i+1)\Gamma(a+i+2)\Gamma(2k-i+1)\Gamma(a-i+2k+2)}$
$\displaystyle\times(a-i-j+2k+1)\psi_{0}(i+j+2)-\Gamma^{2}(a+2k+2)\sum_{i=1}^{2k-1}4i(2k-i+2)$
$\displaystyle\times\frac{\psi_{0}(2k-i+3)\left(\psi_{0}(a+i+1)-\psi_{0}(a+2k+2)\right)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}-\Gamma(a+2k+1)$
$\displaystyle\times\Gamma(a+2k+3)\sum_{i=1}^{2k-1}\frac{4(2k-i+2)(2k-i+1)\psi_{0}(2k-i+3)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}+\frac{1}{\Gamma(a+2)}$
$\displaystyle\times\frac{\Gamma(a+2k+2)}{\Gamma(2k+1)}\sum_{i=1}^{2k-1}\frac{4i(a+i)(2k-i+2)(a-i+2k+2)\psi_{0}(2k-i+3)}{2k-i+1}.$
(229)
We now shift the index $i\to 2k-i$ of the three single sums in the above
result (229). The first two single sums are simplified into the unsimplifiable
basis (212) by using the identities (B.2)–(B.2). The last single sum in (229)
is simplified directly into a closed-form representation by using the
identities (B.1)–(B.1) and (293). For the double summation in (229), we take
the partial fraction decomposition
$\displaystyle\frac{4(i+j+1)(a+i+j+1)(2k-i-j+1)(a-i-j+2k+1)}{j}$
$\displaystyle=\frac{4(i+1)(a+i+1)(2k-i+1)(a-i+2k+1)}{j}+4j^{3}+16(i-k)j^{2}$
$\displaystyle~{}~{}~{}\\!~{}-4\left(a^{2}+2a(k+1)-6i^{2}+12ik-4k^{2}+4k+2\right)j$
$\displaystyle~{}~{}~{}\\!~{}+8(i-k)\left(-a^{2}-2a(k+1)+2(i+1)(i-2k-1)\right).$
(230)
In (230), the polynomial part can be simplified similarly approach as in
(225), while the rational part is simplified as follows. The corresponding
summation is
$\displaystyle\sum_{i=1}^{2k-1}\frac{4(i+1)(2k-i+1)\Gamma^{2}(a+2k+2)}{\Gamma(i+1)\Gamma(a+i+1)\Gamma(2k-i+1)\Gamma(a-i+2k+1)}\sum_{j=1}^{2k-i}\frac{\psi_{0}(i+j+2)}{j}.$
(231)
To evaluate (231), the sum over $j$ is computed by using the identity (B.1)
with the specialization
$i=j,\qquad a=i+1,\qquad b=1,\qquad m=2k-i,$ (232)
then we shift the summation index $i\to 2k-i$ of the outer sum. Consequently,
(231) is simplified to
$\displaystyle-\sum_{i=1}^{2k-1}\frac{4(i+1)(2k-i+1)\Gamma^{2}(a+2k+2)}{\Gamma(i+1)\Gamma(a+i+1)\Gamma(2k-i+1)\Gamma(a-i+2k+1)}$
$\displaystyle\times\Bigg{(}\sum_{j=1}^{2k-i+1}\frac{\psi_{0}(i+j+1)}{j}-\frac{1}{2}(\left(\psi_{0}(2k-i+2)+\psi_{0}(i+1)-2\psi_{0}(1)\right)$
$\displaystyle\times\left(\psi_{0}(2k-i+2)+\psi_{0}(i+1)\right)\left.-\psi_{1}(2k-i+2)-\psi_{1}(i+1)+2\psi_{1}(1)\right)\\!\Bigg{)}.$
(233)
In (233), the double sum is the same form as in (231) but with a negative
sign. Therefore, by adding up (231) and (233), and dividing the result by two,
we reduce the double summation in (231) to a single sum as
$\displaystyle\sum_{i=1}^{2k-1}\frac{4(i+1)(2k-i+1)\Gamma^{2}(a+2k+2)}{\Gamma(i+1)\Gamma(a+i+1)\Gamma(2k-i+1)\Gamma(a-i+2k+1)}\sum_{j=1}^{2k-i}\frac{\psi_{0}(i+j+2)}{j}$
$\displaystyle=\sum_{i=1}^{2k-1}\frac{2\Gamma^{2}(a+2k+2)}{\Gamma(i+1)\Gamma(a+i+1)\Gamma(2k-i+1)\Gamma(a-i+2k+1)}\Big{(}\frac{1}{2}(2k-i+1)$
$\displaystyle~{}~{}~{}\\!~{}\times(i+1)((\psi_{0}(2k-i+2)+\psi_{0}(i+1))(\psi_{0}(2k-i+2)+\psi_{0}(i+1)$
$\displaystyle~{}~{}~{}\\!~{}-2\psi_{0}(1))-\psi_{1}(2k-i+2)-\psi_{1}(i+1)+2\psi_{1}(1))+(2k-i+1)$
$\displaystyle~{}~{}~{}\\!~{}\times\left(\psi_{0}(2k-i+2)+\psi_{0}(i+2)-\psi_{0}(2k+3)-\psi_{0}(1)\right)-(i+1)\psi_{0}(2k+3)\Big{)},$
(234)
which is further simplified into a semi closed-form expression involving the
unsimplifiable basis (212)–(213) by using the identities (B.2), and (B.2).
We now simplify the summation (219) as a last piece in (A.1). We first change
the summation order in (219) to evaluate the sum over $j$ by the identity
(B.2). As a result, (219) is simplified to
$\displaystyle\Gamma^{2}(a+2k+2)\sum_{i=1}^{2k}4i(a+i)(2k-i+2)(a-i+2k+2)$
$\displaystyle\times\sum_{j=1}^{2k-i+1}\frac{\psi_{0}(a-i+2k+3)-\psi_{0}(a-i-j+2k+3)+\psi_{0}(a+2k+2)}{j\Gamma(i+j)\Gamma(a+i+j+1)\Gamma(2k-i-j+2)\Gamma(a-i-j+2k+3)}$
$\displaystyle-\Gamma^{2}(a+2k+2)\sum_{i=1}^{2k-1}\frac{4i(2k-i+2)\psi_{0}(a+2k+2)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}$
$\displaystyle\times\left(\psi_{0}(a+i+1)-\psi_{0}(a+2k+2)\right)-\Gamma(a+2k+1)\Gamma(a+2k+3)$
$\displaystyle\times\sum_{i=1}^{2k-1}\frac{4(-i+2k+2)\psi_{0}(a+2k+3)}{\Gamma(i)\Gamma(a+i)\Gamma(-i+2k+1)\Gamma(a-i+2k+2)}-16(a+2)k(a+2k)$
$\displaystyle\times\frac{\Gamma(a+2k+2)}{\Gamma(a+2)\Gamma(2k+1)}\left(\psi_{0}(a+2k+2)-\psi_{0}(a+2)+\psi_{0}(a+3)\right),$
(235)
where the single summations are simplified similarly to the ones in (229). To
simplify the double summation in (235), we shift the summation index $i\to
2k-i-j+2$ as
$\displaystyle\sum_{i=1}^{2k}\frac{\Gamma^{2}(a+2k+2)}{\Gamma(i)\Gamma(a+i+1)\Gamma(2k-i+2)\Gamma(a-i+2k+3)}$
$\displaystyle\times\sum_{j=1}^{2k-i+1}\frac{4(i+j)(a+i+j)(2k-i-j+2)(a-i-j+2k+2)}{j}$
$\displaystyle\times\left(\psi_{0}(a+i+j+1)-\psi_{0}(a+i+1)+\psi_{0}(a+2k+2)\right).$
(236)
In (236), the inner summations over $j$ are simplified into a closed-form
representation except for the sum
$\sum_{j=1}^{2k-i+1}\frac{\psi_{0}(a+i+j+1)}{j}.$ (237)
Therefore, the resulting summations consist of several single sums as well as
a double sum. These single sums are simplified into closed-form expressions by
using the identity (132) and its derivative of $a$. The double summation
involves the sum in (237), and is written as
$\displaystyle\sum_{i=1}^{2k}\frac{4i(2k-i+2)\Gamma^{2}(a+2k+2)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}\sum_{j=1}^{2k-i+1}\frac{\psi_{0}(a+i+j+1)}{j}.$
(238)
To process (238), we first evaluate the inner summation by the identity (B.1)
with the specialization
$i=j,\qquad a=a+1,\qquad b=i,\qquad m=2k-i+1,$ (239)
and the sum (238) becomes
$\displaystyle\sum_{i=1}^{2k}\frac{4i(2k-i+2)\Gamma^{2}(a+2k+2)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}\Bigg{(}\sum_{j=1}^{2k-i+1}\frac{\psi_{0}(i+j)}{j}$
$\displaystyle-\sum_{j=1}^{a+1}\frac{\psi_{0}(j+2k+1)}{i+j-1}+\frac{1}{2}(-\psi_{1}(a+i+1)+\psi_{1}(i)+\left(\psi_{0}(a+i+1)-\psi_{0}(i)\right)$
$\displaystyle\times\left(\psi_{0}(a+i+1)+2\left(\psi_{0}(2k-i+2)-\psi_{0}(1)\right)+\psi_{0}(i)\right))\\!\Bigg{)}.$
(240)
In comparison with the summations in (233), the summations in (240) can be
simplified similarly except for a double sum
$\displaystyle\sum_{i=1}^{2k}\frac{4i(2k-i+2)\Gamma^{2}(a+2k+2)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}\sum_{j=1}^{a+1}\frac{\psi_{0}(j+2k+1)}{i+j-1},$
(241)
where it does not permit further simplifications by directly evaluating the
inner sum. To proceed further (241), we change the summation order and take
the partial fraction decomposition
$\frac{4i(2k-i+2)}{i+j-1}=-\frac{4(j-1)(j+2k+1)}{i+j-1}-4(i-1)+4(j+2k).$ (242)
The summation (241) is now written as
$\displaystyle-\Gamma^{2}(a+2k+2)\sum_{j=1}^{a+1}4(j-1)(j+2k+1)\psi_{0}(j+2k+1)$
$\displaystyle\times\sum_{i=1}^{2k}\frac{1}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}\frac{1}{i+j-1}$
$\displaystyle-4\sum_{j=1}^{a+1}\psi_{0}(j+2k+1)\sum_{i=2}^{2k}\frac{\Gamma^{2}(a+2k+2)}{\Gamma(i-1)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}$
$\displaystyle+4\sum_{j=1}^{a+1}\psi_{0}(j+2k+1)\sum_{i=1}^{2k}\frac{(j+2k)\Gamma^{2}(a+2k+2)}{\Gamma(i)\Gamma(a+i)\Gamma(2k-i+2)\Gamma(a-i+2k+2)}.$
(243)
In (243), the second and third double summations are simplified directly into
closed-form expressions by using the identities (132), and (287)–(B.1). The
first double summation in (243), after applying the identity (1) with the
specialization
$b=a,\qquad c=j-1,\qquad m=2k+1,$ (244)
is simplified to
$\displaystyle\frac{(a+2k+1)\Gamma(a+2k+2)}{\Gamma(a+1)\Gamma(2k+1)}\sum_{j=1}^{a+1}\frac{4(j-1)(j+2k+1)}{(j+2k)}\psi_{0}(j+2k+1)-$
$\displaystyle\frac{(a+2k+1)\Gamma(a+2k+2)}{\Gamma(2a+2k+1)}\sum_{j=1}^{a+1}\sum_{i=1}^{2k+1}4(j-1)(j+2k+1)\psi_{0}(j+2k+1)$
$\displaystyle\times\frac{\Gamma(2a-i+4k+2)}{\Gamma(2k-i+2)\Gamma(a-i+2k+2)}\frac{\Gamma(2k-i+j+1)}{\Gamma(j+2k+1)}.$
(245)
The single sum over $j$ in (245) admits a closed-form representation by
applying the identity (287), (B.1), and (293). The double sum in (245) is
simplified by first using the identities (307)–(B.1) to compute the sum over
$j$, before using the identities (304)–(B.1) to evaluate the sum over $i$.
Consequently, we obtain a semi closed-form result of (245), and the
corresponding unsimplifiable basis are (212) and
$\sum_{i=1}^{2k}\frac{\psi_{0}(2a+i+2k)}{i}.$ (246)
We have so far completed the simplification of the inner summations over
indexes $i$ and $j$ in (A.1). Summing up these results, we observe the
complete cancellation among the unsimplifiable basis (212)-(213) and
(226)-(228). The only survived term is (246). In the resulting outer sum over
index $k$ in (A.1), all the gamma functions are reduced to rational ones.
Therefore, (A.1) can now be simplified into a similar form as (A.1) in terms
of the unsimplifiable basis (199).
Inserting the resulting summations of (A.1) and (A.1) into (51), we obtain
$\displaystyle\mathrm{I_{A}}=$
$\displaystyle\sum_{k=1}^{m}\left(\left(\frac{2a-1}{2k}-\frac{2a+1}{2(a+k)}+\frac{2a+1}{2k+1}-\frac{2a-1}{2a+2k+1}\right)\psi_{0}(2a+4k)\right.$
$\displaystyle+\left(\frac{2(a+m)(2a+3m-1)}{2a+4m-1}\left(\frac{1}{a+2k+1}+\frac{1}{a+2k}\right)-\frac{2a}{2k+1}\right.$
$\displaystyle+\\!\left.\frac{1-2a}{2k}+\frac{1}{2(a+k)}\right)\psi_{0}(2a+2k)+\left(\frac{2am-2a+6m^{2}-6m+1}{(2k+1)(2a+4m-1)}\right.$
$\displaystyle+\\!\left.\frac{4am+2a+12m^{2}-1}{4k(2a+4m-1)}+\frac{1}{4(a+k)}\right)\psi_{0}(a+2k)-\left(\frac{2a+2m-1}{2(2k+1)}\right.$
$\displaystyle+\\!\left.\frac{a+m}{2k}\right)\psi_{0}(a+k+m)-\left(\frac{2a+2m-1}{4k}+\frac{a+m}{2k+1}\right)$
$\displaystyle\times\\!\left.\psi_{0}\\!\left(a+k+m+\frac{1}{2}\right)-\left(\frac{1}{2(2k+1)}+\frac{1}{4k}\right)\psi_{0}(a+k)\right)+\mathrm{CF}.$
(247)
We remind the readers that the omitted closed-form terms are denoted by the
abbreviation $\mathrm{CF}$, which in general is different in each use. Now we
discuss the simplification of the summations in (A.1) and (A.1) in computing
the integral $\mathrm{I_{B}}$ in (55). The single summation in (A.1) are
computed into the unsimplifiable basis of the form (199) by using the
identities (287)–(B.1) along with the result (197).
The simplification of the double summation in (A.1) will utilize Lemma 6. By
partial fraction decomposing the rational functions in $j$ and shifting the
summation index $j\to m-k-j$, the simplification of (A.1) boils down to
computing the summations
$\displaystyle\sum_{k=1}^{m-1}\mathrm{p}_{c,\lambda}(k)\frac{\Gamma(2m-2k+1)}{\Gamma(2a-2k+2m+1)}\sum_{j=1}^{m-k}\frac{\Gamma(2m+2a-2j-2k+1)}{\Gamma(2m-2j-2k+1)}$
$\displaystyle\times\left(\frac{1}{\left(a-j-2k+2m+\frac{1}{2}+c\right)^{\lambda}}-\frac{1}{(j+c)^{\lambda}}\right),$
(248)
where the parameter $c$ and $\lambda$ take the values $c=-1/2,0,1/2$,
$\lambda=1,2$, and $\mathrm{p}_{c,\lambda}(k)$ denotes the rational functions
in $k$. These summations are simplified by using Lemma 6 to evaluate the inner
sums over $j$, which reduces the gamma ratio
$\frac{\Gamma(2m-2k+1)}{\Gamma(2a-2k+2m+1)}$ (249)
into a rational function. Specifically, in the case when $c=1/2$ and
$\lambda=1$ in (248), the corresponding inner summation is
$\sum_{j=1}^{m-k}\frac{\Gamma(2m+2a-2j-2k+1)}{\Gamma(2m-2j-2k+1)}\left(\frac{1}{1+a-j-2k+2m}-\frac{1}{j+\frac{1}{2}}\right).$
(250)
By using the relation Brychkov08
$\Gamma(2k)=\frac{2^{2k-1}}{\sqrt{\pi}}\Gamma(k)\Gamma\\!\left(k+\frac{1}{2}\right),$
(251)
the summation (250) is written as
$2^{2a}\sum_{j=1}^{m-k}\Phi_{0,-\frac{1}{2},a,a-\frac{1}{2}}^{(m-k-j)}\left(\frac{1}{1+a-j-2k+2m}-\frac{1}{j+\frac{1}{2}}\right).$
(252)
Here, we recall the notation (162)
$\Phi_{a,b,c,d}^{(x)}=\frac{\Gamma(x+c+1)\Gamma(x+d+1)}{\Gamma(x+a+1)\Gamma(x+b+1)}.$
(253)
By using Lemma 6 with the specialization
$a=-\frac{1}{2},~{}~{}~{}b=a,~{}~{}~{}c=\frac{1}{2},~{}~{}~{}m\to m-k,$ (254)
(250) is simplified to
$\displaystyle\sum_{j=1}^{m-k}\frac{\Gamma(2m+2a-2j-2k+1)}{\Gamma(2m-2j-2k+1)}\left(\frac{1}{1+a-j-2k+2m}-\frac{1}{j+\frac{1}{2}}\right)$
$\displaystyle=\frac{\Gamma(2a-2k+2m+2)}{\Gamma(2m-2k+2)}\Bigg{(}\psi_{0}(a+1)-\psi_{0}(a-2k+2m+1)$
$\displaystyle~{}~{}~{}\\!~{}-\frac{2(2a+1)}{2a-2k+2m+1}+2\Bigg{)}.$ (255)
To simplify the summation (248) for $c=1/2$ and $\lambda=2$, we again use the
relation (251), and the corresponding inner summation
$\sum_{j=1}^{m-k}\frac{\Gamma(2m+2a-2j-2k+1)}{\Gamma(2m-2j-2k+1)}\left(\frac{1}{(1+a-j-2k+2m)^{2}}-\frac{1}{\left(j+\frac{1}{2}\right)^{2}}\right)$
(256)
becomes
$2^{2a}\sum_{j=1}^{m-k}\Phi_{0,-\frac{1}{2},a,a-\frac{1}{2}}^{(m-k-j)}\left(\frac{1}{(1+a-j-2k+2m)^{2}}-\frac{1}{(j+\frac{1}{2})^{2}}\right),$
(257)
which is evaluated by taking derivative of the parameter $c$ of the identity
(174) with the specialization
$a=-\frac{1}{2},~{}~{}~{}b=a,~{}~{}~{}m\to m-k,$ (258)
before setting $c=1/2$. As a result, (256) is simplified to
$\displaystyle\sum_{j=1}^{m-k}\frac{\Gamma(2m+2a-2j-2k+1)}{\Gamma(2m-2j-2k+1)}\left(\frac{1}{(1+a-j-2k+2m)^{2}}-\frac{1}{\left(j+\frac{1}{2}\right)^{2}}\right)$
$\displaystyle=\frac{\Gamma(2m+2a-2k+2)}{\Gamma(2m-2k+2)}\sum_{j=1}^{m-k}\Bigg{(}\\!\left(-\frac{4a^{2}-1}{a+j-1}-\frac{1-4a^{2}}{a+j}-\frac{4a^{2}-6a+2}{-2a-2j+3}\right.$
$\displaystyle~{}~{}~{}\\!~{}-\\!\left.\frac{2a(2a+1)}{2a+2j+1}+\frac{1}{a+2j}-\frac{2-8a}{2a+2j-1}-\frac{1}{-a-2j+1}\right)\left(\psi_{0}\\!\left(j+\frac{1}{2}\right)\right.$
$\displaystyle~{}~{}~{}\\!~{}-\psi_{0}\\!\left(a+j+\frac{1}{2}\right)+\psi_{0}(a-k+m+1)+\psi_{0}\\!\left(a-k+m+\frac{3}{2}\right)+\psi_{0}(j)$
$\displaystyle~{}~{}~{}\\!~{}-\\!\left.\psi_{0}(a+j)-\psi_{0}(m-k+1)-\psi_{0}\\!\left(m-k+\frac{3}{2}\right)\right)+\frac{a+j}{j(a+2j)^{2}}$
$\displaystyle~{}~{}~{}\\!~{}+\frac{1}{2a+2j-1}\left(\frac{a(2a-1)(2j-1)}{j(a+j)^{2}}-\frac{2a(2a+1)}{\left(a+j+\frac{1}{2}\right)^{2}}+\frac{2j-1}{(a+2j-1)^{2}}\right)\\!\Bigg{)}.$
(259)
The other cases of $c$ and $\lambda$ combinations are obtained similarly.
Inserting the results into (248), the gamma ratio (249) of the outer sum over
$k$ is reduced to a rational function. The resulting summations are further
computed into unsimplifiable basis of the form (199) by using the identities
(287)–(B.1).
Inserting the resulting summations of (A.1) and (A.1) into (55), we obtain
$\displaystyle\mathrm{I_{B}}=$
$\displaystyle\sum_{k=1}^{m}\left(\left(\frac{2(a+m)(2a+3m-1)}{2a+4m-1}\left(\frac{1}{a+2k+1}+\frac{1}{a+2k}\right)+\frac{m}{k}+\frac{2m-1}{2k+1}\right)\right.$
$\displaystyle\times\psi_{0}(2a+2k)+\left(\frac{m}{2a+2k+1}+\frac{2m-1}{4(a+k)}-\frac{2m-1}{2(2k+1)}-\frac{m}{2k}\right)\times$
$\displaystyle\psi_{0}\\!\left(a+2k+\frac{1}{2}\right)+\left(\frac{m}{2(a+k)}+\frac{m}{2a+2k+1}+\frac{m(2m-1)}{2a+4m-1}\right.$
$\displaystyle\times\left.\left.\left(\frac{1}{2k+1}+\frac{1}{2k}\right)\right)\psi_{0}(a+2k)\right)+\mathrm{CF}.$
(260)
Now inserting the $\mathrm{I_{A}}$ expression (247) and $\mathrm{I_{B}}$
expression (260) into (47), we obtain
$\displaystyle\mathbb{V}\\!\left[S\right]=$
$\displaystyle\frac{-2a-2m+1}{4}\Omega_{1}^{\left(a,a+\frac{1}{2}\right)}+\frac{-2a-2m+1}{4}\left(\Omega_{2}^{\left(a+\frac{1}{2},0\right)}+\Omega_{2}^{\left(a+\frac{1}{2},\frac{1}{2}\right)}\right)$
$\displaystyle-\frac{a+m}{2}\left(\Omega_{2}^{\left(a,0\right)}+\Omega_{2}^{\left(a,\frac{1}{2}\right)}\right)+\mathrm{CF},$
(261)
where
$\displaystyle\Omega_{1}^{\left(a,b\right)}=$
$\displaystyle\sum_{k=1}^{m}\left(\frac{\psi_{0}(a+k)}{b+k}+\frac{\psi_{0}(b+k)}{a+k}\right)$
(262) $\displaystyle\Omega_{2}^{\left(a,b\right)}=$
$\displaystyle\sum_{k=1}^{m}\left(\frac{\psi_{0}(a+b+k+m)}{b+k}+\frac{\psi_{0}(a+k)}{b+k}+\frac{\psi_{0}(a+b+2k)}{a+k}\right.$
$\displaystyle-\\!\left.\frac{\psi_{0}(a+b+k)}{a+k}-\frac{\psi_{0}(a+b+2k)}{b+k}\right).$
(263)
Simplifying the single sums in (261) by using the closed-form identities (B.1)
and (B.1) directly leads to the desired result (3). This completes the proof
of Proposition 3.
#### Case B: Fixed number of particles
In case B, the variance calculation boils down to simplifying the summation
representations of $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{B}_{1}$,
$\mathcal{B}_{2}$ as summarized in (281)–(A.2) in Appendix A.2.
Note that the simplification procedure in case A also works for the majority
of the summations in case B. The only new summation in case B is the double
summation
$\displaystyle\sum_{k=1}^{m-2}4(a+b-2k+2m+1)\frac{\Gamma(m-k+1)}{\Gamma(a+b-k+m+1)}\sum_{j=1}^{m-k-1}\frac{\Gamma(a+b-j-k+m)}{\Gamma(m-k-j)}$
$\displaystyle\times(-1)^{j}(2j+2k-2m-a-b+1)\left(\frac{1}{(j)_{3}(a+b-j-2k+2m-1)_{3}}\right)^{2}$
$\displaystyle\times\left((1-a)j(a+b-j-2k+2m-1)-2(a-k+m)(a+b-k+m)\right)$
$\displaystyle\times\left((1-b)j(a+b-j-2k+2m-1)-2(b-k+m)(a+b-k+m)\right),$
(264)
which is obtained after opening the bracket of the double summation in (A.2)
and shifting the index $k\to m-j-k$. To process (264), we take the partial
fraction decomposition of the rational functions (starting from the second to
the last line) in $j$, the summation (264) now boils down to
$\displaystyle\sum_{k=1}^{m-2}\mathrm{p}_{c,\lambda}(k)\frac{\Gamma(m-k+1)}{\Gamma(a+b-k+m+1)}\sum_{j=1}^{m-k-1}\frac{(-1)^{j}\Gamma(m-k+a+b-j)}{\Gamma(m-k-j)}$
$\displaystyle\times\left(\frac{1}{(2m-2k-1+a+b-j+c)^{\lambda}}-\frac{1}{(j+c)^{\lambda}}\right),$
(265)
where the parameters $c$ and $\lambda$ take the values $c=0,1,2$,
$\lambda=1,2$, and $\mathrm{p}_{c,\lambda}(k)$ denotes the rational
polynomials in $k$. The simplification of these summations will utilize the
Lemma 5. For the case when $c=0$ and $\lambda=1$ in (265), the corresponding
inner summation can be simplified into a closed-form expression by directly
using the result (2.2.2), which is a special case of the identity (163) in
Lemma 5. When $c=0$ and $\lambda=2$ in (265), the inner summation
$\sum_{j=1}^{m-k-1}\frac{(-1)^{j}\Gamma(a+b-j-k+m)}{\Gamma(m-k-j)}\left(\frac{1}{(a+b-j-2k+2m-1)^{2}}-\frac{1}{j^{2}}\right)$
(266)
is simplified to
$\displaystyle\frac{\Gamma(a+b-k+m)}{\Gamma(m-k)}\sum_{j=1}^{m-k-1}\Bigg{(}\\!\left(\frac{1}{a+b-j-k+m}-\frac{1}{a+b-2j-2k+2m}\right.$
$\displaystyle-\\!\left.\frac{1}{a+b-2j-2k+2m-1}\right)(\psi_{0}(m-k)-\psi_{0}(a+b-j-k+m)$
$\displaystyle-\psi_{0}(a+b-k+m)-\psi_{0}(m-k-j))+\frac{1}{(a+b-2j-2k+2m-1)^{2}}\Bigg{)}.$
(267)
The result (267) is obtained by taking derivative of $c$ of the identity (163)
with the specialization
$a\to a+b,~{}~{}~{}m\to m-k-1,$ (268)
before setting $c=0$. For other combinations of $c$ and $\lambda$, the
corresponding inner summations in (265) can be simplified similarly as the two
cases above.
Inserting the simplification results of the inner summations into (264), the
resulting sums only consist of polygamma and rational functions, which are
further computed into unsimplifiable basis of the form (199) by using the
identities (287)–(B.1).
For the integrals $\mathrm{I_{A}}$ and $\mathrm{I_{B}}$ in case B, the
corresponding summations (281)–(A.2) are now simplified to the results shown
below. For $\mathrm{I_{A}}$, one has
$\displaystyle\mathrm{I_{A}}=~{}\\!\\!$ $\displaystyle
d_{1}\sum_{k=1}^{m}\frac{\psi_{0}(a+k)}{k}+d_{2}\sum_{k=1}^{m}\frac{\psi_{0}(b+k)}{k}-2m\sum_{k=1}^{m}\frac{\psi_{0}(a+b+k+m)}{k}$
$\displaystyle+d_{3}\sum_{k=1}^{m}\frac{\psi_{0}(a+b+k+m)}{a+k}+\left(d_{3}+d_{4}\right)\sum_{k=1}^{m}\frac{\psi_{0}(a+b+k+m)}{b+k}+\mathrm{CF},$
(269)
where the coefficients $d_{i}$ are
$\displaystyle d_{1}=$
$\displaystyle\frac{2m(a+m)\left(a^{2}+a(b+3m)+2bm+3m^{2}-1\right)}{(a+b+2m-1)_{3}}$
(270) $\displaystyle d_{2}=$
$\displaystyle\frac{2m(b+m)\left(a(b+2m)+b^{2}+3bm+3m^{2}-1\right)}{(a+b+2m-1)_{3}}$
(271) $\displaystyle d_{3}=$
$\displaystyle\frac{2(b+m)(a+b+m)\left(m(3a+4b)+(a+b)^{2}+3m^{2}-1\right)}{(a+b+2m-1)_{3}}$
(272) $\displaystyle d_{4}=$ $\displaystyle\frac{2(a-b)(a+b+m)}{a+b+2m}.$
(273)
For $\mathrm{I_{B}}$, one has
$\displaystyle\mathrm{I_{B}}=~{}\\!\\!$ $\displaystyle
d_{1}\sum_{k=1}^{m}\frac{\psi_{0}(a+k)}{k}+d_{2}\sum_{k=1}^{m}\frac{\psi_{0}(b+k)}{k}+2m\sum_{k=1}^{m}\frac{\psi_{0}(a+b+k)}{k}$
$\displaystyle-2m\sum_{k=1}^{m}\frac{\psi_{0}(a+b+2k)}{k}+d_{3}\sum_{k=1}^{m}\frac{\psi_{0}(a+b+k)}{b+k}+d_{4}\sum_{k=1}^{m}\frac{\psi_{0}(b+k)}{a+k}$
$\displaystyle-
d_{4}\sum_{k=1}^{m}\frac{\psi_{0}(a+b+2k)}{a+k}+d_{4}\sum_{k=1}^{m}\frac{\psi_{0}(a+b+2k)}{b+k}$
$\displaystyle+\left(d_{3}+d_{4}\right)\sum_{k=1}^{m}\frac{\psi_{0}(a+b+k)}{a+k}+\mathrm{CF}.$
(274)
Inserting the $\mathrm{I_{A}}$ expression (269) and $\mathrm{I_{B}}$
expression (274) into (47), we arrive at
$\displaystyle\mathbb{V}\\!\left[S\right]=$
$\displaystyle-2m\Omega_{2}^{(a+b,0)}-d_{4}\Omega_{2}^{(b,a)}+\left(d_{3}+d_{4}\right)\Omega_{3}^{(a,b)}+\mathrm{CF},$
(275)
where the summation $\Omega_{2}^{(a,b)}$ is defined in (263), and
$\displaystyle\Omega_{3}^{(a,b)}=~{}\\!\\!$
$\displaystyle\sum_{k=1}^{m}\left(\frac{\psi_{0}(a+b+k+m)}{a+k}+\frac{\psi_{0}(a+b+k+m)}{b+k}-\frac{\psi_{0}(a+b+k)}{a+k}\right.$
$\displaystyle-\\!\left.\frac{\psi_{0}(a+b+k)}{b+k}\right).$ (276)
By using the identities (B.1) and (B.1), the result (275) is simplified to the
variance formula (4), which completes the proof of Proposition 4.
## 3 Conclusions
In this work, we compute the exact yet explicit variance formulas of von
Neumann entanglement entropy over fermionic Gaussian states with and without
particle number constrains. The obtained formulas provide insights into the
fluctuations of von Neumann entropy. An essential ingredient in obtaining the
results is a new simplification framework of dummy summation and re-summation
techniques. The new framework may also be useful in computing higher order
moments of von Neumann entropy as well as other entanglement indicators over
the fermionic Gaussian ensemble.
Acknowledgments The work of Lu Wei is supported in part by the U.S. National
Science Foundation ($\\#$2150486).
## Appendix A Summation representations
In this appendix, we list the summation representations of the integrals in
$\mathrm{I_{A}}$ and $\mathrm{I_{B}}$ as summarized in Table 3. The summation
representations for case A are listed in Appendix (A.1), the ones for case B
are listed in Appendix (A.2).
### A.1 Summation representations of case A
0pt
$\displaystyle\mathrm{A_{1}}=$
$\displaystyle\sum_{k=0}^{m-1}\sum_{j=2k-2}^{2k}\frac{2(-1)^{j}(2a+4k+1)(j+1)_{2}(a+j+1)_{2}}{\Gamma(2k-j+1)\Gamma(j-2k+3)(2a+j+2k+1)_{3}}\bigg{(}(\psi_{0}(j+3)$
$\displaystyle-\psi_{0}(2a+j+2k+4)-\psi_{0}(j-2k+3)+\psi_{0}(a+j+3))^{2}$
$\displaystyle-\psi_{1}(2a+j+2k+4)+\psi_{1}(a+j+3)-\psi_{1}(j-2k+3)+\psi_{1}(j+3)\bigg{)}$
$\displaystyle+\sum_{k=0}^{m-1}\sum_{j=0}^{2k-3}\frac{4(2a+4k+1)(j+1)_{2}(a+j+1)_{2}}{(2k-j-2)_{3}(2a+j+2k+1)_{3}}(\psi_{0}(2a+j+2k+4)$
$\displaystyle-\psi_{0}(a+j+3)+\psi_{0}(2k-j-2)-\psi_{0}(j+3))$ (277)
$\displaystyle\mathrm{A_{2}}=$
$\displaystyle\sum_{k=0}^{m-1}\frac{(2a+4k+1)\Gamma(2k+1)\Gamma(2a+2k+1)}{\Gamma(2a+4k+4)}\left(\rule{0.0pt}{19.0633pt}\sum_{j=0}^{2k}\frac{2(j+1)(2k-j+1)}{\Gamma(j+1)\Gamma(a+j+1)}\right.$
$\displaystyle\times\frac{\Gamma^{2}(a+2k+2)}{\Gamma(2k-j+1)\Gamma(a+2k-j+1)}((\psi_{0}(a+2k+2)-\psi_{0}(2a+4k+4)$
$\displaystyle-\psi_{0}(2)+\psi_{0}(2k-j+2))(\psi_{0}(a+2k+2)-\psi_{0}(2a+4k+4)+\psi_{0}(j+2)$
$\displaystyle-\psi_{0}(2))-\psi_{1}(2a+4k+4))-\sum_{j=0}^{2k}\frac{(j+1)\Gamma(a+2k+1)\Gamma(a+2k+3)}{\Gamma(j)\Gamma(a+j+1)\Gamma(a-j+2k+1)}$
$\displaystyle\times\frac{1}{\Gamma(2k-j+1)}((\psi_{0}(a+2k+1)-\psi_{0}(2a+4k+4)+\psi_{0}(2k-j+2)$
$\displaystyle-\psi_{0}(1))(\psi_{0}(a+2k+3)-\psi_{0}(2a+4k+4)+\psi_{0}(j+2)-\psi_{0}(3))$
$\displaystyle-\psi_{1}(2a+4k+4))-\sum_{j=0}^{2k}\frac{(2k-j+1)\Gamma(a+2k+1)\Gamma(a+2k+3)}{\Gamma(a+j+1)\Gamma(2k-j)\Gamma(2k-j+a+1)}$
$\displaystyle\times\frac{1}{\Gamma(j+1)}((\psi_{0}(a+2k+3)-\psi_{0}(2a+4k+4)+\psi_{0}(2k-j+2)$
$\displaystyle-\psi_{0}(3))(\psi_{0}(a+2k+1)-\psi_{0}(2a+4k+4)+\psi_{0}(j+2)-\psi_{0}(1))$
$\displaystyle-\psi_{1}(2a+4k+4))+\sum_{j=1}^{2k-1}\sum_{i=1}^{2k-j}\frac{4i(2k-i+2)\Gamma(a+2k-j+1)}{\Gamma(a+i)\Gamma(a+2k-i+2)}$
$\displaystyle\times\frac{\Gamma(a+2k+j+3)}{(j)_{3}\Gamma(j+i+1)\Gamma(2k-j-i+1)}(\psi_{0}(a+2k+j+3)$
$\displaystyle-\psi_{0}(2a+4k+4)+\psi_{0}(2k-i+3)-\psi_{0}(j+3))\left.\\!\\!\rule{0.0pt}{19.0633pt}\right)$
(278)
$\displaystyle\mathrm{B_{1}}=$
$\displaystyle\sum_{k=0}^{m-1}\left(\psi_{0}(a+2k)+\psi_{0}(2a+2k)-2\psi_{0}(2a+4k)-\frac{1}{2}\left(\frac{a}{a+2k+1}\right.\right.~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\\!\left.\left.+\frac{a}{a+2k}+\frac{2}{2a+4k+1}\right)+1\right)^{2}$
(279)
$\displaystyle\mathrm{B_{2}}=$
$\displaystyle\sum_{k=1}^{m-1}\sum_{j=1}^{m-k}\frac{\Gamma(2a+2k-1)\Gamma(2j+2k-1)}{2(2j-1)^{2}j^{2}(2j+1)^{2}\Gamma(2k-1)\Gamma(2a+2j+2k-1)}$
$\displaystyle\times\frac{(2a+4k-3)(2a+4j+4k-3)}{(a+j+2k-2)^{2}(a+j+2k-1)^{2}(2a+2j+4k-3)^{2}}\left(a^{2}(2j+1)\right.$
$\displaystyle+\\!\left.a(j+1)(2j+4k-3)+2j^{2}+j(4k-3)+4k^{2}-6k+2\right)^{2}$
(280)
### A.2 Summation representations of case B
$\mathcal{A}_{1}=\mathcal{A}_{1}^{(a,b)}+\mathcal{A}_{1}^{(b,a)},$ (281)
0pt
$\displaystyle\mathcal{A}_{1}^{(a,b)}=$
$\displaystyle-\frac{2(m(b+m))}{a+b+2m}\sum_{i=1}^{m-3}\frac{(b+i+1)(i)_{2}}{(m-i-2)_{3}(a+b+i+m+1)}(\psi_{0}(b+i+2)$
$\displaystyle-\psi_{0}(a+b+i+m+2)-\psi_{0}(m-i-2)+\psi_{0}(i+2))+\frac{a+b+m}{a+b+2m}$
$\displaystyle\times
2(a+m)\sum_{i=1}^{m-2}\frac{(b+i+1)(i)_{2}}{(m-i-1)(a+b+i+m)_{3}}(-\psi_{0}(a+b+i+m+3)$
$\displaystyle+\psi_{0}(b+i+2)-\psi_{0}(m-i-1)+\psi_{0}(i+2))-\frac{m(b+m)}{a+b+2m}$
$\displaystyle\sum_{i=m-3}^{m-1}\frac{(b+i+2)(-1)^{i+m}(i+1)_{2}}{\Gamma(m-i)\Gamma(i-m+4)(a+b+i+m+2)}\big{(}\psi_{1}(b+i+3)$
$\displaystyle+\psi_{1}(i+3)-\psi_{1}(i-m+4)-\psi_{1}(a+b+i+m+3)+(\psi_{0}(i+3)$
$\displaystyle-\psi_{0}(a+b+i+m+3)-\psi_{0}(i-m+4)+\psi_{0}(b+i+3))^{2}\big{)}$
$\displaystyle-\frac{(a+m)(a+b+m)(b+m)(m-1)_{2}}{(a+b+2m)(a+b+2m-1)_{3}}\big{(}\\!-\psi_{1}(a+b+2m+2)$
$\displaystyle+\psi_{1}(b+m+1)+\psi_{1}(m+1)-\psi_{1}(1)+\psi_{0}^{2}(1)+(\psi_{0}(b+m+1)$
$\displaystyle-\psi_{0}(a+b+2m+2)+\psi_{0}(m+1))(\psi_{0}(b+m+1)+\psi_{0}(m+1)$
$\displaystyle-\psi_{0}(a+b+2m+2)-2\psi_{0}(1))\big{)}$ (282)
$\displaystyle\mathcal{A}_{2}=$
$\displaystyle-\frac{2(a+m)(b+m)\Gamma(m+1)}{(a+b+2m)(a+b+m+2)_{m}}\sum_{i=0}^{m-1}(-1)^{i}\frac{(i+1)(m-i)}{\Gamma(a+i+2)}$
$\displaystyle\times\sum_{j=i-1}^{i+1}(-1)^{j}\frac{\Gamma(a+i-j+m+1)(b-i+m+1)_{j}}{\Gamma(j+1)\Gamma(i-j+2)\Gamma(j-i+2)\Gamma(m-j)}$
$\displaystyle\times(\psi_{1}(a+b+2m+2)+(\psi_{0}(a+i-j+m+1)-\psi_{0}(a+b+2m+2)$
$\displaystyle-\psi_{0}(i-j+2)+\psi_{0}(i+2))(\psi_{0}(a+b+2m+2)+\psi_{0}(j-i+2)$
$\displaystyle-\psi_{0}(b-i+j+m+1)-\psi_{0}(m-i+1)))-2\Gamma(m+1)\Gamma(a+m+1)$
$\displaystyle\times\frac{(a+b+m)\Gamma(b+m+1)}{(a+b+2m)(a+b+m+1)_{m+1}}\sum_{i=0}^{m-2}\frac{1}{\Gamma(i+1)\Gamma(a+i+2)}$
$\displaystyle\times\frac{1}{\Gamma(m-i-1)\Gamma(b-i+m)}(\psi_{1}(a+b+2m+2)+(\psi_{0}(a+m+1)$
$\displaystyle-\psi_{0}(a+b+2m+2)+\psi_{0}(i+2)-\psi_{0}(1))(\psi_{0}(a+b+2m+2)$
$\displaystyle-\psi_{0}(b+m+1)-\psi_{0}(m-i)+\psi_{0}(1)))+\mathcal{A}_{2}^{(a,b)}+\mathcal{A}_{2}^{(b,a)},$
(283) $\displaystyle\mathcal{A}_{2}^{(a,b)}=$
$\displaystyle\frac{2\Gamma(m+1)\Gamma(a+b+m+1)}{\Gamma(a+b+2m+2)}\Bigg{(}\frac{(a+m)(b+m)(a+b+m+1)}{a+b+2m}$
$\displaystyle\times\sum_{i=1}^{m-2}\frac{i(m-i+1)}{\Gamma(b+i+1)\Gamma(a-i+m+2)}\sum_{j=1}^{m-i-1}\frac{\Gamma(a+j+m+2)}{(j)_{3}\Gamma(i+j+1)}$
$\displaystyle\times\frac{\Gamma(b-j+m)}{\Gamma(m-i-j)}(\psi_{0}(m-i+2)-\psi_{0}(a+b+2m+2)-\psi_{0}(j+3)$
$\displaystyle+\psi_{0}(a+j+m+2))-\frac{a+b+m}{a+b+2m}\sum_{i=1}^{m-1}\frac{i(m-i)}{\Gamma(b+i+1)}$
$\displaystyle\times\frac{1}{\Gamma(a-i+m+1)}(\psi_{0}(a+j+m+1)-\psi_{0}(a+b+2m+2)$
$\displaystyle+\psi_{0}(m-i+1)-\psi_{0}(j+1))\Bigg{)}$ (284)
$\displaystyle\mathcal{B}_{1}=$
$\displaystyle\sum_{k=0}^{m-1}\left(\left(\frac{a^{2}-b^{2}}{4(a+b+2k)}+\frac{b^{2}-a^{2}}{4(a+b+2k+2)}+\frac{1}{2}\right)\psi_{0}(a+k+1)\right.$
$\displaystyle+\left(\frac{a^{2}-b^{2}}{4(a+b+2k+2)}+\frac{b^{2}-a^{2}}{4(a+b+2k)}+\frac{1}{2}\right)\psi_{0}(b+k+1)$
$\displaystyle+\psi_{0}(a+b+k+1)-2\psi_{0}(a+b+2k+2)-\frac{a+b}{2(a+b+2k)}$
$\displaystyle-\\!\left.\frac{a+b}{2(a+b+2k+2)}+\frac{1}{a+b+2k+1}+1\right)^{2}$
(285) $\displaystyle\mathcal{B}_{2}=$
$\displaystyle\sum_{k=1}^{m-1}\frac{k(a+b+k)}{2(a+k)(b+k)(a+b+2k)(a+b+2k-1)_{3}}\Bigg{(}2(a+k)(b+k)$
$\displaystyle\times(\psi_{0}(b+k+1)-\psi_{0}(a+k+1))+\frac{(k-1)(a-b)(a+b+2k+1)}{a+b+k}\\!\Bigg{)}^{2}$
$\displaystyle+\sum_{k=1}^{m-2}\sum_{j=1}^{m-k-1}\frac{2(a+b+2k-1)\Gamma(j+k+1)(a+b+2j+2k+1)}{\Gamma(k)\Gamma(a+k)\Gamma(b+k)\Gamma(a+j+k+1)\Gamma(b+j+k+1)}$
$\displaystyle\times\frac{\Gamma(a+b+k)}{\Gamma(a+b+j+k+1)}\Bigg{(}\frac{\Gamma(a+k)\Gamma(b+j+k+1)}{(j)_{3}(a+b+j+2k-1)_{3}}\left(a^{2}(j+2)+a(j+2)\right.$
$\displaystyle\times\left.\\!(b+j+2k)+j(b+2k+1)+2k(b+k)+j^{2}\right)$
$\displaystyle-\frac{(-1)^{j}\Gamma(b+k)\Gamma(a+j+k+1)}{(j)_{3}(a+b+j+2k-1)_{3}}(2(k-1)(a+(b+1)(j+2)+2k-2)$
$\displaystyle+(b+1)(j+2)(a+b+j+1))\\!\Bigg{)}^{2}$ (286)
## Appendix B Summation identities
In this appendix, we list the finite sum identities useful in simplifying the
summations in Appendix A that are not listed in the main part of the paper.
The identities of the existing simplification framework are listed in Appendix
B.1 and the identities of the new simplification framework are listed in
Appendix B.2.
### B.1 Summation identities of the existing simplification framework
We list below the identities of the existing simplification framework. Here,
it is sufficient to assume that $a,b\geq 0,a\neq b$ in identities (287)–(293),
(B.1)–(B.1), (B.1), (B.1)–(B.1), $a>m$ in (B.1), and $a,b\geq 1,n\geq m$ in
(B.1), (304)–(B.1). 0pt
$\displaystyle\sum_{i=1}^{m}\psi_{0}(i+a)=(m+a)\psi_{0}(m+a+1)-a\psi_{0}(a+1)-m$
(287)
$\displaystyle\sum_{i=1}^{m}i\psi_{0}(i+a)=$
$\displaystyle-\frac{1}{2}(a-m-1)(a+m)\psi_{0}(a+m+1)$
$\displaystyle+\frac{1}{2}(a-1)a\psi_{0}(a+1)-\frac{1}{4}m(-2a+m+3)$ (288)
$\displaystyle\sum_{i=1}^{m}i^{2}\psi_{0}(i+a)=$
$\displaystyle\frac{1}{6}\left(2a^{3}-3a^{2}+a+2m^{3}+3m^{2}+m\right)\psi_{0}(a+m+1)$
$\displaystyle-\frac{1}{6}a\left(2a^{2}-3a+1\right)\psi_{0}(a+1)$
$\displaystyle-\frac{1}{36}m\left(12a^{2}-6am-24a+4m^{2}+15m+17\right)$ (289)
$\displaystyle\sum_{i=1}^{m}i^{3}\psi_{0}(i+a)=$
$\displaystyle-\frac{1}{4}\left(a^{4}-2a^{3}+a^{2}-m^{4}-2m^{3}-m^{2}\right)\psi_{0}(a+m+1)$
$\displaystyle+\frac{1}{4}(a-1)^{2}a^{2}\psi_{0}(a+1)-\frac{1}{48}m\left(-12a^{3}+6a^{2}m+30a^{2}\right.$
$\displaystyle-\\!\left.4am^{2}-18am-26a+3m^{3}+14m^{2}+21m+10\right)$ (290)
$\displaystyle\sum_{i=1}^{m}\psi_{1}(i+a)=$
$\displaystyle(m+a)\psi_{1}(m+a+1)-a\psi_{1}(a+1)$
$\displaystyle+\psi_{0}(m+a+1)-\psi_{0}(a+1)$ (291)
$\displaystyle\sum_{i=1}^{m}\psi_{0}^{2}(i+a)=$
$\displaystyle(a+m)\psi_{0}^{2}(a+m){}-(2a+2m-1)\psi_{0}(a+m)-a\psi_{0}^{2}(a){}$
$\displaystyle+(2a-1)\psi_{0}(a)+2m.$ (292)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(i+a)}{i+a}=\frac{1}{2}\left(\psi_{1}(m+a+1)-\psi_{1}(a+1)+\psi_{0}^{2}(m+a+1)-\psi_{0}^{2}(a+1)\right)$
(293)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(m+1-i)}{i}=\psi_{0}^{2}(m+1)-\psi_{0}(1)\psi_{0}(m+1)+\psi_{1}(m+1)-\psi_{1}(1)$
(294)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(m+1+i)}{i}=\psi_{0}^{2}(m+1)-\psi_{0}(1)\psi_{0}(m+1)-\frac{1}{2}\psi_{1}(m+1)+\frac{1}{2}\psi_{1}(1)$
(295)
0pt
$\displaystyle\sum_{i=1}^{m}\psi_{0}(i+a)\psi_{0}(i+b)$
$\displaystyle=(b-a)\sum_{i=1}^{m-1}\frac{\psi_{0}(a+i)}{b+i}-a\psi_{0}(a+1)\psi_{0}(b+1)+(m+a)\psi_{0}(m+a)$
$\displaystyle~{}~{}~{}\\!~{}\times\psi_{0}(m+b)+a\psi_{0}(a+1)-(m+a-1)\psi_{0}(m+a)-(m+b)\psi_{0}(m+b)$
$\displaystyle~{}~{}~{}\\!~{}+(b+1)\psi_{0}(b+1)+2m-2$ (296)
$\displaystyle\sum_{i=1}^{m}i\psi_{0}(i+a)\psi_{0}(i+b)$
$\displaystyle=\frac{1}{2}(b-a+1)(a-b)\sum_{i=1}^{m-1}\frac{\psi_{0}(a+i)}{b+i}-\frac{1}{4}a(a+2b-3)\psi_{0}(a+1)-\frac{1}{4}(b+1)$
$\displaystyle~{}~{}~{}\\!~{}\times(2a+b-2)\psi_{0}(b+1)+\frac{1}{2}(a-1)a\psi_{0}(a+1)\psi_{0}(b+1)+\frac{1}{4}(a+m-1)$
$\displaystyle~{}~{}~{}\\!~{}\times(a+2b-m-2)\psi_{0}(a+m)+\frac{1}{4}(b+m)(2a+b-m-1)\psi_{0}(b+m)-\frac{1}{2}$
$\displaystyle~{}~{}~{}\\!~{}\times\left(a^{2}-a-m(m+1)\right)\psi_{0}(a+m)\psi(b+m)-\frac{1}{4}(m-1)(3a+3b-m-4)$
(297)
$\displaystyle\sum_{i=1}^{m}i^{2}\psi_{0}(i+a)\psi_{0}(i+b)$
$\displaystyle=\frac{1}{6}(a-b)\left(3a^{2}+2ab-4a-2b^{2}-b+1\right)\sum_{i=1}^{m-1}\frac{\psi_{0}(a+i)}{b+i}-\frac{1}{6}\left(-a^{2}(5b+2)\right.$
$\displaystyle~{}~{}~{}\\!~{}+\\!\left.3a^{3}+a(5b-1)-m\left(2m^{2}+3m+1\right)\right)\psi_{0}(a+m)\psi_{0}(b+m)+\frac{1}{6}(a-1)a$
$\displaystyle~{}~{}~{}\\!~{}\times(3a-5b+1)\psi_{0}(a+1)\psi_{0}(b+1)-\Bigg{(}\\!-\frac{1}{12}(2b-1)m^{2}+\frac{1}{36}m\left(24a^{2}\right.$
$\displaystyle~{}~{}~{}\\!~{}-\left.\\!24ab-24a+12b^{2}+12b-1\right)+\frac{1}{36}(a-1)\left(28a^{2}-18ab-5a+6b\right.$
$\displaystyle~{}~{}~{}\\!~{}+\\!\left.12b^{2}+6\right)+\frac{(a-1)(a-b)}{3(b+m-1)}+\frac{m^{3}}{9}\Bigg{)}\psi_{0}(a+m)-\frac{1}{36}\left(4m^{3}12a^{2}b\right.$
$\displaystyle~{}~{}~{}\\!~{}-3(2a-1)m^{2}+\left(12a^{2}-12a-1\right)m-30a^{2}+6ab^{2}-12ab+30a+4b^{3}$
$\displaystyle~{}~{}~{}\\!~{}-\\!\left.3b^{2}-b\right)\psi_{0}(b+m)+\frac{1}{36}a\left(28a^{2}-9a(2b+1)+12b^{2}-13\right)\psi_{0}(a+1)$
$\displaystyle~{}~{}~{}\\!~{}+\frac{1}{36}\left(6a^{2}(2b-3)+6a\left(b^{2}-2b+2\right)+4b^{3}-3b^{2}-b+6\right)\psi_{0}(b+1)$
$\displaystyle~{}~{}~{}\\!~{}+\frac{2m^{3}}{27}-\frac{5}{36}m^{2}(a+b-1)+\frac{1}{36}\left(-40a^{2}+12ab+51a-16b^{2}+3b-16\right)$
$\displaystyle~{}~{}~{}\\!~{}+\frac{1}{108}m\left(120a^{2}-36ab-138a+48b^{2}+6b+25\right)+\frac{a-1}{3(b+m-1)}$
$\displaystyle~{}~{}~{}\\!~{}-\frac{a-1}{3(a+m-1)}$ (298)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(a+1-i)}{i}=$
$\displaystyle-\sum_{i=1}^{m}\frac{\psi_{0}(i+a-m)}{i}+\frac{1}{2}\left(\psi_{1}(a+1)-\psi_{1}(a-m)\right)$
$\displaystyle+(\psi_{0}(a-m)+\psi_{0}(a+1))(\psi_{0}(m+1)-\psi_{0}(1))$
$\displaystyle+\frac{1}{2}(\psi_{0}(a-m)-\psi_{0}(a+1))^{2}$ (299)
$\displaystyle\sum_{i=1}^{m}\left(\frac{\psi_{0}(i+b)}{i+a}+\frac{\psi_{0}(i+a)}{i+b}\right)$
$\displaystyle=\psi_{0}(m+a+1)\psi_{0}(m+b+1)-\psi_{0}(a+1)\psi_{0}(b+1)+\frac{1}{a-b}(\psi_{0}(m+a+1)$
$\displaystyle~{}~{}~{}-\psi_{0}(m+b+1)-\psi_{0}(a+1)+\psi_{0}(b+1))$ (300)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(a+b+i)}{i}=$
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(b+i)}{i}-\sum_{i=1}^{a}\frac{\psi_{0}(b+i+m)}{b+i-1}+\frac{1}{2}\big{(}(\psi_{0}(a+b)$
$\displaystyle-\psi_{0}(b))\times(\psi_{0}(a+b)+\psi_{0}(b)+2(\psi_{0}(m+1)-\psi_{0}(1)))$
$\displaystyle-\psi_{1}(a+b)+\psi_{1}(b)\big{)}$ (301)
$\displaystyle\sum_{i=1}^{m}\left(\frac{\psi_{0}(a+b+i+m)}{b+i}+\frac{\psi_{0}(a+i)}{b+i}+\frac{\psi_{0}(a+b+2i)}{a+i}-\frac{\psi_{0}(a+b+i)}{a+i}\right.$
$\displaystyle-\\!\left.\frac{\psi_{0}(a+b+2i)}{b+i}\right)$
$\displaystyle=\frac{\psi_{0}\left(\frac{a}{2}+\frac{b}{2}+m\right)}{b-a}-\frac{(a+b+m)\psi_{0}(a+b+m)}{b(a+m)}+\frac{\psi_{0}(a+b+2m)}{a+m}$
$\displaystyle~{}~{}~{}\\!~{}+\psi_{0}(a+m)\left(\psi_{0}(b+m+1)-\psi_{0}(b)-\frac{1}{b-a}\right)+\frac{a\psi_{0}(a)}{b(b-a)}+\frac{\psi_{0}(a+b)}{b}$
$\displaystyle~{}~{}~{}~{}\\!-\frac{\psi_{0}\left(\frac{a}{2}+\frac{b}{2}\right)}{b-a}$
(302)
$\displaystyle\sum_{i=1}^{m}\left(\frac{\psi_{0}(a+b+i+m)}{a+i}+\frac{\psi_{0}(a+b+i+m)}{b+i}-\frac{\psi_{0}(a+b+i)}{a+i}\right.$
$\displaystyle-\\!\left.\frac{\psi_{0}(a+b+i)}{b+i}\right)$
$\displaystyle=\frac{(a+b+2m)\psi_{0}(a+b+2m)}{(a+m)(b+m)}-\left(\frac{1}{a+m}+\frac{1}{a}+\frac{1}{b+m}+\frac{1}{b}\right)\psi_{0}(a+b+m)$
$\displaystyle~{}~{}~{}~{}+\left(\frac{1}{a}+\frac{1}{b}\right)\psi_{0}(a+b)+\left(\psi_{0}(a+m)-\psi_{0}(a)\right)\left(\psi_{0}(b+m)-\psi_{0}(b)\right)$
(303)
$\displaystyle\sum_{i=1}^{m}\frac{(n-i)!}{(m-i)!}=\frac{n!}{(m-1)!(n-m+1)}$
(304)
$\displaystyle\sum_{i=1}^{m}\frac{(n-i)!}{(m-i)!i}=\frac{n!}{m!}(\psi_{0}(n+1)-\psi_{0}(n-m+1))$
(305)
$\displaystyle\sum_{i=1}^{m}\frac{(n-i)!}{(m-i)!i^{2}}=$
$\displaystyle\frac{n!}{m!}\Bigg{(}\sum_{i=1}^{m}\frac{\psi_{0}(i+n-m)}{i}+\frac{1}{2}\big{(}\psi_{1}(n-m+1)-\psi_{1}(n+1)$
$\displaystyle+\psi_{0}(n-m+1){}^{2}-\psi_{0}(n+1){}^{2}\big{)}+\psi_{0}(n-m)$
$\displaystyle\times\left(-\psi_{0}(n-m+1)+\psi_{0}(n+1)-\psi_{0}(m+1)+\psi_{0}(1)\right)\\!\Bigg{)}$
(306)
$\displaystyle\sum_{i=1}^{m}\frac{(n-i)!}{(m+a-i)!}=\frac{1}{n-m-a+1}\left(\frac{n!}{(a+m-1)!}-\frac{(n-m)!}{(a-1)!}\right)$
(307)
$\displaystyle\sum_{i=1}^{m}\frac{(n-i)!}{(m+a-i)!}\psi_{0}(m+a-i+1)$
$\displaystyle=\frac{1}{1-a-m+n}\left(\frac{n!}{(a+m-1)!}\left(\psi_{0}(a+m)-\frac{1}{1-a-m+n}\right)\right.$
$\displaystyle~{}~{}~{}\\!~{}-\\!\left.\frac{(n-m)!}{(a-1)!}\left(\psi_{0}(a)-\frac{1}{1-a-m+n}\right)\right)$
(308)
The derivation of the identities (287)–(B.1) and (304)–(B.1) can be found in
Wei17 ; Wei20 ; Wei20BH ; HWC21 ; HW22 ; Milgram . In the same manner as
obtaining (B.1) in Section 2.2.1, the three identities (B.1)–(B.1) are derived
by first rewriting respectively the summations
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(a+b+i)}{i}=\sum_{i=1}^{m}\frac{\psi_{0}(b+i)}{i}+\sum_{j=1}^{a}\sum_{i=1}^{m}\frac{1}{i}\frac{1}{b+i+j-1}$
(309)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(a+b+2i)}{a+i}=\sum_{i=1}^{m}\frac{\psi_{0}(a+b+i)}{a+i}+\sum_{j=1}^{m}\sum_{i=j}^{m}\frac{1}{(a+i)(a+b+i+j-1)}$
(310)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(a+b+i+m)}{b+i}=\sum_{i=1}^{m}\frac{\psi_{0}(a+b+i)}{b+i}+\sum_{j=1}^{m}\sum_{i=1}^{m}\frac{1}{(b+i)(a+b+i+j-1)}.$
(311)
For the identity (307), it is obtained by first considering
$\displaystyle\sum_{i=1}^{m}\frac{(n-i)!}{(m+a-i)!}=\sum_{i=1}^{a+m}\frac{(n-i)!}{(m+a-i)!}-\sum_{i=1}^{a}\frac{(n-m-i)!}{(a-i)!}$
(312)
before applying (304). Note that the identity (307) is analytically continued
to any complex number $a$, where, by taking an derivative of $a$, the identity
(B.1) is established.
### B.2 Additional summation identities of the new simplification framework
Here, we list the additional summation identities of the new simplification
framework useful in the simplification process in Section 2.3. These
identities are obtained by taking appropriate derivatives of the formulas in
lemmas 1–4. It is sufficient to assume that $a,b,c\geq 0$ in (B.2)–(B.2).
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}^{2}(a+i)-\psi_{1}(a+i)}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle=\frac{\Gamma(a+b+2m-1)}{\Gamma(m)\Gamma(a+m)\Gamma(b+m)\Gamma(a+b+m)}\big{(}\psi_{1}(a+b+2m-1)-\psi_{1}(a+b+m)$
$\displaystyle~{}~{}\\!~{}~{}-\psi_{1}(a+m)+\left(\psi_{0}(a+b+m)-\psi_{0}(a+b+2m-1)+\psi_{0}(a+m)\right){}^{2}\big{)}$
(313)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(i)}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle=-\frac{1}{\Gamma(m)\Gamma(b+m)\Gamma(a+b+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(a-i+m)i}$
$\displaystyle~{}~{}~{}\\!~{}+\frac{\psi_{0}(m)\Gamma(a+b+2m-1)}{\Gamma(m)\Gamma(a+m)\Gamma(b+m)\Gamma(a+b+m)}$
(314)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(i)\psi_{0}(b-i+m+1)}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle=\frac{1}{\Gamma(m)\Gamma(b+m)\Gamma(a+b+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(a-i+m)i}(-\psi_{0}(a+b+m)$
$\displaystyle~{}~{}~{}\\!~{}+\psi_{0}(a+b-i+2m-1)-\psi_{0}(b+m))+\frac{\Gamma(a+b+2m-1)\psi_{0}(m)}{\Gamma(m)\Gamma(a+m)\Gamma(b+m)}$
$\displaystyle~{}~{}\\!~{}~{}\times\frac{1}{\Gamma(a+b+m)}(\psi_{0}(a+b+m)-\psi_{0}(a+b+2m-1)+\psi_{0}(b+m))$
(315)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(i)\psi_{0}(a+i)}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle=\frac{1}{\Gamma(m)\Gamma(b+m)\Gamma(a+b+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(a-i+m)i}(-\psi_{0}(a+b+m)$
$\displaystyle~{}~{}\\!~{}~{}+\psi_{0}(a+b-i+2m-1)-\psi_{0}(a-i+m))+\frac{\psi_{0}(m)\Gamma(a+b+2m-1)}{\Gamma(m)\Gamma(b+m)\Gamma(a+b+m)}$
$\displaystyle~{}~{}\\!~{}~{}\times\frac{\psi_{0}(a+b+m)-\psi_{0}(a+b+2m-1)+\psi_{0}(a+m)}{\Gamma(a+m)}$
(316)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}^{2}(i)-\psi_{1}(i)}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle=\frac{2}{\Gamma(m)\Gamma(b+m)\Gamma(a+b+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(a-i+m)i}(\psi_{0}(i)-\psi_{0}(m)$
$\displaystyle~{}~{}~{}\\!~{}-\psi_{0}(1))+\frac{\Gamma(a+b+2m-1)}{\Gamma(m)\Gamma(a+m)\Gamma(b+m)\Gamma(a+b+m)}\left(\psi_{0}^{2}(m)-\psi_{1}(m)\right)$
(317)
$\displaystyle\sum_{i=1}^{m}\frac{\psi_{0}(i)\psi_{0}(m-i+1)}{\Gamma(i)\Gamma(a+i)\Gamma(m-i+1)\Gamma(b-i+m+1)}$
$\displaystyle=\frac{1}{\Gamma(m)\Gamma(a+b+m)}\left(-\frac{1}{\Gamma(a+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(b-i+m)i^{2}}-\frac{1}{\Gamma(b+m)}\right.$
$\displaystyle~{}~{}\\!~{}~{}\times\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(a-i+m)i^{2}}-\frac{\psi_{0}(m)}{\Gamma(a+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(b-i+m)i}$
$\displaystyle~{}~{}\\!~{}~{}-\left.\\!\frac{\psi_{0}(m)}{\Gamma(b+m)}\sum_{i=1}^{m-1}\frac{\Gamma(a+b-i+2m-1)}{\Gamma(a-i+m)i}\right)+\frac{\Gamma(a+b+2m-1)}{\Gamma(m)\Gamma(a+m)\Gamma(b+m)}$
$\displaystyle~{}~{}\\!~{}~{}\times\frac{1}{\Gamma(a+b+m)}\left(\psi_{0}^{2}(m)-\psi_{1}(m)+\psi_{1}(1)\right)$
(318)
$\displaystyle\sum_{i=1}^{m}\frac{\Gamma(c-i+m)\Gamma(a+b+i+m)}{\Gamma(i)\Gamma(m-i+1)}\psi_{0}(i)$
$\displaystyle=-\frac{\Gamma(b+1)\Gamma(c)\Gamma(c+m)}{\Gamma(m)\Gamma(b+c+1)}\sum_{i=1}^{m-1}\frac{\Gamma(b+c-i+m)}{\Gamma(c-i+m)i}+\frac{\Gamma(b+1)\Gamma(c)\Gamma(b+c+m)}{\Gamma(m)\Gamma(b+c+1)}$
$\displaystyle~{}~{}~{}\\!~{}\times\left(\psi_{0}(b+c+m)-\psi_{0}(b+c+1)+\psi_{0}(m)\right)$
(319)
$\displaystyle\sum_{i=1}^{m}\frac{\Gamma(c-i+m)\Gamma(a+b+i+m)}{\Gamma(a+i)\Gamma(m-i+1)}\left(\psi_{0}^{2}(i)-\psi_{1}(i)\right)$
$\displaystyle=\frac{2\Gamma(b+1)\Gamma(c)\Gamma(c+m)}{\Gamma(m)\Gamma(b+c+1)}\left(-\sum_{i=1}^{m-1}\frac{\Gamma(b+c-i+m)\psi_{0}(b+c-i+m)}{\Gamma(c-i+m)i}\right.$
$\displaystyle~{}~{}~{}\\!~{}+\sum_{i=1}^{m-1}\frac{\Gamma(b+c-i+m)\psi_{0}(i)}{\Gamma(c-i+m)i}-\left(-\psi_{0}(b+c+1)+\psi_{0}(m)+\psi_{0}(1)\right)$
$\displaystyle~{}~{}~{}\\!~{}\times\\!\left.\sum_{i=1}^{m-1}\frac{\Gamma(b+c-i+m)}{\Gamma(c-i+m)i}\right)+\frac{\Gamma(b+1)\Gamma(c)\Gamma(b+c+m)}{\Gamma(m)\Gamma(b+c+1)}\big{(}\psi_{1}(b+c+m)$
$\displaystyle~{}~{}~{}\\!~{}-\psi_{1}(b+c+1)-\psi_{1}(m)+\left(\psi_{0}(b+c+m)-\psi_{0}(b+c+1)+\psi_{0}(m)\right){}^{2}\big{)}$
(320)
$\displaystyle\sum_{i=1}^{m}\frac{\Gamma(c-i+m)\Gamma(a+b+i+m)}{\Gamma(a+i)\Gamma(m-i+1)i}\psi_{0}(a+b+i+m)$
$\displaystyle=\Gamma(c+m)\Gamma(a+b+m)\Bigg{(}\\!\frac{\psi_{0}(a+b+m)}{\Gamma(a)\Gamma(m+1)}\left(\psi_{0}(c+m)-\psi_{0}(c)\right)+\Gamma(b+m+1)$
$\displaystyle~{}~{}~{}\\!~{}\times\Gamma(c)\sum_{i=1}^{m}\frac{\psi_{0}(a+b+m)-\psi_{0}(b-i+m+1)+\psi_{0}(b+m+1)}{\Gamma(a+i)\Gamma(c+i)\Gamma(m-i+1)\Gamma(b-i+m+1)i}\Bigg{)}$
(321)
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|
# Extracting Semantic Knowledge from GANs with Unsupervised Learning
Jianjin Xu, Zhaoxiang Zhang, Xiaolin Hu J. Xu is with the School of
Mathematics and Computer Science, Panzhihua University and the Department of
Computer Science and Technology, Tsinghua University.
E-mail<EMAIL_ADDRESS>Z. Zhang is with with the Insitute of Automation,
Chinese Academy of Sciences and the Centre for Artificial Intelligence and
Robotics, Hong Kong Innovation and Science Insitute, Chinese Academy of
Sciences.
E-mail<EMAIL_ADDRESS>X. Hu is with the Department of Computer
Science and Technology, Institute for Artificial Intelligence, State Key
Laboratory of Intelligent Technology and Systems, THU-Bosch JCML Center, THBI,
IDG/McGovern Institute for Brain Research, BNRist, Tsinghua University,
Beijing, China. He is also with the Chinese Institute for Brain Research
(CIBR), Beijing, China.
E-mail<EMAIL_ADDRESS>work was supported in part by the National
Natural Science Foundation of China (Nos. 62061136001, 61836014, and U19B2034)
and THU-Bosch JCML center. Corresponding author: Xiaolin Hu.
###### Abstract
Recently, unsupervised learning has made impressive progress on various tasks.
Despite the dominance of discriminative models, increasing attention is drawn
to representations learned by generative models and in particular, Generative
Adversarial Networks (GANs). Previous works on the interpretation of GANs
reveal that GANs encode semantics in feature maps in a linearly separable
form. In this work, we further find that GAN’s features can be well clustered
with the linear separability assumption. We propose a novel clustering
algorithm, named KLiSH, which leverages the linear separability to cluster
GAN’s features. KLiSH succeeds in extracting fine-grained semantics of GANs
trained on datasets of various objects, _e.g_., car, portrait, animals, and so
on. With KLiSH, we can sample images from GANs along with their segmentation
masks and synthesize paired image-segmentation datasets. Using the synthesized
datasets, we enable two downstream applications. First, we train semantic
segmentation networks on these datasets and test them on real images,
realizing unsupervised semantic segmentation. Second, we train image-to-image
translation networks on the synthesized datasets, enabling semantic-
conditional image synthesis without human annotations.
###### Index Terms:
GAN, Unsupervised Learning, Semantic Segmentation, Conditional Image Synthesis
## 1 Introduction
Representation is at the core of modern deep learning practice. Bengio _et
al_. [1] propose that a good representation should have (1) _explanatory
factors organized in hierarchy_ , and (2) _natural clustering of categories_.
Chan _et al_. [2] claim that the objective of a deep network is to learn a
linearly discriminative representation of the data. Researchers have found
that the representation learned by GANs [3] not only explains the variation of
images [4, 5, 6] but also is linearly discriminative [7], indicating that GANs
learn a good representation.
Given a dataset of images, a GAN learns to model the distribution of data and
synthesizes images unseen in the dataset. The image synthesis process of GAN
is typically a mapping from noise vectors to a series of intermediate feature
maps and then to RGB images. From the perspective of causality, GAN’s
representations contain all the information in the image. In other words, the
variation factors (_e.g_., the attributes of objects) present in the image
must also exist in GAN’s representation. Previous works have identified two
types of variations captured by GANs. First, the image-level attributes
(_e.g_., the gender of face images) are separable by linear hyperplanes in
GAN’s latent space [4, 8, 5, 9, 10]. Second, the object-level semantics
(_e.g_., the presence and location of a window in bedroom images) are encoded
in GAN’s feature maps [6] in a linearly separable form [7]. We term the rich
semantic information in GAN’s features as the semantic knowledge learned by
the GANs.
Figure 1: Semantic segmentation obtained by using the average feature of each
class compared to using a trained linear classifier. The prediction from a
pretrained DeepLabV3 [11] is regarded as the groundtruth. The linear
classifier is supervised by a DeepLabV3 network pretrained on CelebAMask-HQ
dataset [12] according to Xu _et al_. [7].
Existing works [13, 14, 15] often train few-shot learning classifiers to
reveal GAN’s semantic knowledge. The classifiers predict each spatial location
of GAN’s feature maps. Training the classifiers needs only a few labeled
examples. The trained classifier is subsequently used for image editing [13,
14] and semantic segmentation [15]. However, the labeling cost for the few-
shot learning method is still high. As reported in [13, 14], the number of
annotations ranges from 16 to 40 and labeling each image takes around 10
minutes even for a skillful user. Therefore, if one can design an unsupervised
segmentation algorithm based on GAN’s feature maps for image editing, it would
enable users to upload their own images, train their own GANs, and edit those
images without much labor cost. This is the aim of this study. Specifically,
we study how to reveal GAN’s semantic knowledge with unsupervised learning
methods.
Existing unsupervised learning methods mostly apply conventional clustering
algorithms on GAN’s features, _e.g_., K-means [16, 17] and Agglomerative
Hierarchical Clustering (AHC) [18]. They use clustering results as an
intermediate step for editing the attribute [17] or swapping the style [16] in
a localized region. However, the obtained clusters are not satisfactory.
Please refer to their results [16, 17, 18] and also our reproduction in Fig.
5. For example, for face images, we observed the oversegmentation of the hair
class, and the missing of tiny classes like brow, upper lip, and lower lip
(see Fig. 5, the last column).
Figure 2: An illustration of clustering on linearly separable data. Top-left, the groundtruth cluster assignments, and the decision boundaries of the linear SVM trained on the groundtruth. Other panels, results of different clustering algorithms. Decision boundaries are visualized as solid lines, except for AHC which has no such boundary. “x” denotes the centroids of K-means. GAN | StyleGAN2 | StyleGAN | PGGAN
---|---|---|---
Dataset | FFHQ | CelebAHQ | CelebAHQ
Linear Classifier [7] | 82.7 | 73.8 | 69.3
Centroids of Features | 49.6 | 46.6 | 41.8
TABLE I: The semantics segmentation performance (measured in mIoU) of linear
classifier or centroids of features. Please see Sec. 4.2 for details.
The reason might be that K-means and AHC require data to be centered around
their class centroids, which is not well satisfied in GAN’s features. To
illustrate this, we first took a pretrained GAN and segmented the generated
images with a pretrained DeepLabV3 [11] model. The features from the GAN were
averaged into a centroid for each semantic class. Then, we segmented another
set of generated images by assigning the closest centroid to GAN’s features on
each spatial location. It was observed that the segmentation results were
rather coarse (Fig. 1).
Interestingly, a previous work [7] found that GAN’s features of a semantic
class were linearly separable from other classes. This observation is referred
to as the GAN’s linear separability in semantics. We also found that the
linear classifier produced better segmentation than using the centroid method
(Tab. I). Therefore, we propose to cluster GAN’s features by leveraging their
linear separability property, instead of the conventional centering
assumption.
We note that linearly separable data does not necessarily satisfy the
centering assumption required by conventional clustering algorithms. To
illustrate this, we generated three sets of points in a 2D plane, which were
linearly separable from each other (Fig. 2, top-left). In this synthetic
dataset, each cluster had a complex structure and its centroid had limited
information about the structure. It was observed that both AHC and K-means
assigned a part of the blue cluster to the red cluster.
We propose K-means with Linear Separability Heuristic (KLiSH) to find
semantically meaningful clusters in GAN’s features. In brief, KLiSH starts
with a large number of initial K-means clusters and iteratively merges the
clusters according to their degree of linear separability. KLiSH successfully
clustered the three sets of points on the toy dataset described above (Fig. 2,
bottom-right). We compared KLiSH to several clustering algorithms, AHC,
K-means, and K-means based Approximate SPectral clustering (KASP) [19] on
several GANs and datasets. Results showed that KLiSH outperformed all
baselines in most cases.
One can use KLiSH to obtain fine-grained semantic segmentation for the
generated images. The process is straightforward. First, sample images from
GANs and apply the classifier returned by KLiSH on GAN’s feature maps. Then,
train downstream models on the resulting image-segmentation dataset. We
demonstrate two applications in this paper: Unsupervised Fine-Grained
Segmentation (UFGS) and Unsupervised Semantic-Conditional Synthesis (USCS).
UFGS seeks to reduce the labeling cost of fine-grained semantic annotation on
a new dataset. USCS aims to offer semantic-controllable image synthesis and
editing on datasets without semantic annotation. Both UFGS and USCS can be
realized easily with the semantic knowledge extracted from pretrained GANs.
Figure 3: The pipeline of re-purposing pretrained GANs for downstream tasks,
UFGS and USCS. First, we propose the K-means with Linear Separability
Heuristic (KLiSH) to cluster GAN’s features. Then, we generate datasets with
synthesized images and segmentation masks together from GANs. Finally, we
train different models on the synthesized datasets and perform semantic
segmentation on _real_ images and semantic-conditional image synthesis. Both
applications require no human annotations at all.
The pipeline of UFGS and USCS is illustrated in Fig. 3. Given a dataset of
images, we first train a GAN on it and use KLiSH to parse the semantic
clusters learned by GANs. Then, we generate a paired image-segmentation
dataset by sampling images from GANs along with the segmentation masks.
Finally, we train corresponding downstream models on the generated dataset.
In summary, our contributions are as follows. First, we propose KLiSH and
improve the quality of clustering on GAN’s features. Second, we provide
evidence for GAN’s linear separability in semantics. Third, we propose two
novel applications, UFGS and USCS, marking a successful attempt at fine-
grained semantic segmentation and semantic-conditional synthesis in an
unsupervised learning setting.
## 2 Related work
#### Generative Adversarial Networks
Originally put forward by Goodfellow _et al_. [3], GANs go through many
improvements in training [20, 21], architecture [22, 23, 24, 25] and objective
[26, 27, 28, 29]. Nowadays, GANs can map latent vectors to diverse and photo-
realistic images on various datasets. Typical data domains are face [30, 31,
32], car [33, 34, 20], animal [35, 36], ImageNet [37, 38], _etc_.. Among
various GANs, StyleGAN2 [34] is one of the state-of-the-art models. Another
genre of GANs translates images to images [39, 40, 41]. When the input image
is a semantic mask, the task is termed Semantic-Conditional Synthesis (SCS).
Typical models capable of SCS include pix2pix [42], pix2pixHD [43], GauGAN
[44] and so on [45, 12, 46, 47]. Most of them heavily rely on large-scale
semantically-annotated datasets. Sun _et al_. [48] propose to synthesize
objects in an image from bounding boxes, which lowers the label requirement,
but still needs a large-scale dataset. In contrast, the USCS proposed in this
work is fully unsupervised – it is realized by making use of the
representations learned by pretrained GANs.
#### GAN interpretation
Increasing attention is placed on studying GAN’s representation. Most existing
works focus on studying GAN’s latent vectors and feature maps. For the first
aspect, prior works [4, 5] find that there are linear boundaries in latent
space that separate the positive and negative attributes of the sampled
images. Some other works focus on the unsupervised discovery of those
boundaries [8, 10, 9]. For the second aspect, GAN Dissection [6] identifies
convolution units that have causal relationships with semantics in the
generated images. Xu and Zheng [7] propose and validate that GANs encode high-
resolution semantics linearly in feature maps. Other works study the
parameters of GANs [49, 50]. This work proposes an unsupervised approach to
extract learned semantic classes from GANs, providing strong evidence that the
GAN learns a naturally clustered and linearly separable semantic
representation.
#### Repurposing GAN for semantic segmentation
Recently, increasing attention is directed towards semantic segmentation using
GAN’s representation. Existing works either fall into few-shot learning or
unsupervised learning. The few-shot learning methods typically apply
classifiers on a pretrained GAN’s feature maps to get semantic segmentation
for the generated image [7, 51, 15, 14]. These methods can segment the images
well with a limited number of image annotations (ranging from 1 to 40
depending on the complexity of the target data).
For unsupervised learning, the majority of methods [52, 53, 54, 55, 56, 57]
focus on decoupling the foreground and background in GANs without external
labeling. Though the segmentation is accurate, the practical usage is limited
as only two clusters are segmented. Methods on obtaining more clusters mostly
involve applying K-means++ [16] or AHC [18, 17] directly on GAN’s features. He
_et al_. [58] modify K-means by re-weighting channels. Different from existing
methods, we propose a new clustering algorithm, KLiSH, by leveraging a strong
characteristic of GAN’s features – linear separability. We conducted
quantitative evaluations and showed the superiority of KLiSH to previous
clustering algorithms on GAN’s features.
Aside from GAN, other works [59, 60, 61] also study unsupervised semantic
segmentation on large-scale datasets, but mainly focus on object-level
semantics instead of part-level semantics and their results are not directly
comparable to GAN-based methods.
## 3 Clustering Generator’s Features
### 3.1 Linear Separability of GAN’s Features
#### Classical linear separability
KLiSH is based on GAN’s linear semantics [7], thus we first introduce this
characteristic to make our paper self-contained.
When the generator synthesizes an image, it also produces a set of feature
maps. A $1\times 1$ convolutional layer is applied to the upsampled and
concatenated feature maps, resulting in a score map of semantic classes for
the generated image. As the extraction process is fully linear, the method is
named Linear Semantic Extractor (LSE). The LSE is compared to two nonlinear
semantic extractors (NSEs) with more complicated architectures. The LSE and
NSEs are trained under similar settings and evaluated by the accuracy of the
semantic masks they extracted compared to the pretrained segmentation network.
Results show that the relative performance gaps between LSE and NSEs are
within $3.5\%$ across various datasets and models. Therefore, it is shown that
linear classifiers suffice to separate GAN’s representations of different
semantics. For more detailed settings and results, we refer readers to the
original paper [7].
The phenomenon that a linear classifier can classify a given dataset is
usually described as linearly separable. Given a dataset
$\mathcal{A}=\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$ where
$\mathbf{x}_{i}\in\mathbb{R}^{D}$ and $y_{i}\in\\{1,\ldots,M\\}$, the
classical linear separability states that there exists a set of hyperplanes
$\mathbf{W}\in\mathbb{R}^{M\times D}$ and $\mathbf{b}\in\mathbb{R}^{D}$ such
that
$\forall(\mathbf{x},y)\in\mathcal{A},k\neq
y,\mathbf{W}_{y}\mathbf{x}+\mathbf{b}_{y}>\mathbf{W}_{k}\mathbf{x}+\mathbf{b}_{k},$
(1)
where $\mathbf{W}_{k}\in\mathbb{R}^{1\times D}$ is a row vector, and
$\mathbf{W}_{y}$ denotes the classifier weight for the ground-truth class of a
data point $\mathbf{x}$.
#### Binary linear separability
Instead of using classical linear separability, we find that binary linear
separability is more useful for building our clustering algorithm, which is
defined below.
###### Definition 1
Given a dataset $\mathcal{A}=\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$, we say
$\mathcal{A}$ satisfies the binary linear separability if there exists
$\mathbf{W}\in\mathbb{R}^{M\times D}$ and $\mathbf{b}\in\mathbb{R}^{D}$ such
that $\forall(\mathbf{x}_{i},y_{i})\in\mathcal{A}$, $k\in\\{1,\dots,M\\}$,
$\mathcal{I}(y_{i}=k)\cdot(\mathbf{W}_{k}\mathbf{x}_{i}+\mathbf{b}_{k})>1,$
(2)
where $\mathcal{I}(y_{i}=k)$ returns 1 if $y_{i}=k$ and -1 otherwise.
In other words, if for each class in the dataset, there is a two-class SVM
separating itself from others, then the dataset satisfies the binary linearly
separability. Unless otherwise specified, linear separability refers to
Definition 1 throughout the rest of the paper.
Clearly, Eq. 2 is harder to satisfy than Eq. 1. Clusters separable with a
classical linear classifier might not be separable with binary SVMs. Do
features in GAN satisfy the binary linear separability? We empirically found
that the answer is yes. The experiment is presented in Sec. 4.2.
### 3.2 Clustering with Linear Separability Heuristic
Now, we describe the algorithm for clustering GAN’s features by linear
separability.
#### Initialize from K-means
We choose K-means for three reasons. First, K-means is easily scalable to
large datasets with time complexity of $O(N)$. Second, the decision boundaries
between clusters are linear and thus are suitable to implement linear
separability. Third, K-means with a large number of clusters provides a good
initialization for our algorithm. More specifically, most clusters found by
K-means do not span across multiple semantic classes, see the K-means@100
column in Fig. 5. This simplifies our problem as we only need to merge these
clusters to recover the ground-truth classes.
#### Merge clusters
Ideally, the merged clusters should belong to the same semantic class.
However, we have no access to semantic information as the algorithm is fully
unsupervised. We make a reasonable assumption: the more linearly separable a
cluster is, the more likely it is a true semantic class. Under this
assumption, our goal is to maximize the linear separability of clusters.
How to quantify the degree of linear separability? According to Definition 1,
it is implied that how accurately the SVM classifies a cluster indicates how
well its linear separability is. Therefore, we use the performance of the SVM
trained on the target dataset to indicate its linear separability.
First, a pseudo-labeled dataset $\mathcal{A}$ with cluster assignments is
constructed. Second, binary SVMs are trained on $\mathcal{A}$ using the
L2-regularized L2-loss Support Vector Classification objective [62]:
$\mathcal{L}=\frac{\lambda_{1}}{M|\mathcal{A}|}\sum_{(\mathbf{x},y)\in\mathcal{A}}\bigg{[}(1-\mathbf{W}_{y}\mathbf{x}-\mathbf{b}_{y})_{+}^{2}\\\
+\sum_{k\neq
y}(1+\mathbf{W}_{k}\mathbf{x}+\mathbf{b}_{k})_{+}^{2}\bigg{]}+\frac{1}{2M}\left\lVert\mathbf{W}\right\rVert_{F}^{2},$
(3)
where $(\cdot)_{+}$ denotes $max(0,\cdot)$ and
$\left\lVert\cdot\right\rVert_{F}$ denotes the Frobenius norm. We use L-BFGS
[63] to minimize $\mathcal{L}$ due to the convexity of the objective. The SVM
is considered as converged if its $L_{\infty}$ norm of weight changing is
below $10^{-4}$. Third, as in semantic segmentation, we measure the
performance of SVM by IoU (Intersection-over-Union) defined as
$\displaystyle\text{IoU}_{k}(\mathbf{W},\mathcal{A})=\frac{|\mathcal{S}_{k}\cap\mathcal{Y}_{k}|}{|\mathcal{S}_{k}\cup\mathcal{Y}_{k}|},$
(4)
where
$\mathcal{S}_{k}=\\{\mathbf{x}_{i}|\mathbf{W}_{k}\mathbf{x}_{i}+\mathbf{b}_{k}>0\\}$
is the prediction given by the SVM,
$\mathcal{Y}_{k}=\\{\mathbf{x}_{i}|y_{i}=k\\}$ is the set of positive samples
for class $k$, and $(\mathbf{x}_{i},y_{i})\in\mathcal{A}$. Therefore, we can
use IoU to find the cluster with the lowest linear separability.
Suppose that the identified cluster is $p$. The next step is to find another
cluster $q$ to merge with $p$. As implied by Eq. 4, the cluster $p$ is not
classified accurately by the SVM, _i.e_., the SVM might confuse $p$ with other
clusters. Then a reasonable strategy is to merge $p$ with another cluster that
confuses it the most. To quantify the degree of confusion, we propose the
Effective Cosine Similarity (ECoS) metric as defined below.
###### Definition 2
Consider a dataset $\mathcal{A}=\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$.
Suppose a set of binary SVMs with weight $\mathbf{W}\in\mathbb{R}^{M\times D}$
and bias $\mathbf{b}\in\mathbb{R}^{D}$ are trained on $\mathcal{A}$ until
convergence. The ECoS between cluster $i$ and cluster $j$ is defined as
$\begin{split}&\text{ECoS}_{i,j}(\mathbf{W},\mathcal{A})=\cos\langle\mathbf{s}_{i}^{\dagger},\mathbf{s}_{j}^{\dagger}\rangle\\\
&\mathbf{s}_{k}^{\dagger}=\text{CLAMP}_{[0,1]}\left(\frac{\mathbf{s}_{k}+1}{2}\right)\\\
&\mathbf{s}_{k}=\left[\mathbf{x}_{1}^{T}\mathbf{W}_{k}+\mathbf{b}_{k},\ldots,\mathbf{x}_{N}^{T}\mathbf{W}_{k}+\mathbf{b}_{k}\right]\end{split}$
(5)
where $\text{CLAMP}_{[a,b]}$ denotes clamping the input within $[a,b]$.
Here, $\mathbf{s}_{k}$ is the cluster scores predicted by the SVM.
$\mathbf{s}_{k}^{\dagger}$ scales and truncates $\mathbf{s}_{k}$ within [0,
1], representing the confidence of SVM predicting positive data points. As
each feature vector corresponds to a pixel, $\mathbf{s}_{k}^{\dagger}$ can be
resized into a cluster confidence map. In other words, ECoS measures the
cosine similarity between two cluster confidence maps.
#### Filter initial clusters
We observed in experiments that a small portion of initial clusters did span
across multiple semantic classes. For example, see K-means@100 of
StyleGAN2-FFHQ in Fig. 5, the boundary around ear (visualized as a yellow
region) contains part of “ear”, “hair”, and “face”. This type of initial
clusters would harm the accuracy of merged clusters. Fortunately, we
empirically found that these clusters were not linearly separable, _i.e_.,
could be filtered by setting a threshold of IoU. The threshold could also be
determined adaptively as described below.
First, train SVMs on the initial clusters and calculate their IoU, resulting
in a vector $\mathbf{m}$. Second, $\mathbf{m}$ is converted from [0, 1] to
$(-\infty,\infty)$ by using the inverse sigmoid function $h^{-1}(\cdot)$.
Third, any clusters below $\mu_{\mathbf{m}}-\sigma_{\mathbf{m}}$ are filtered,
where $\mu$, $\sigma$ are the mean and standard deviation of
$h^{-1}(\mathbf{m})$, respectively. Finally, K-means is run again with the
filtered centroids as initialization.
Input: $\mathcal{D}=\\{\mathbf{x}_{i}\\}_{i=1}^{N}$; $K_{0}$
Output: $\Theta=\\{(\mathbf{W}^{t},\mathbf{b}^{t})\\}_{t=1}^{K_{0}-1}$
$\mathbf{W}^{0},\mathbf{b}^{0},\mathcal{A}^{0}$ = K-means($\mathcal{D},K_{0}$)
// Filter initial centriods
$\mathbf{W}^{\prime},\mathbf{b}^{\prime}$ =
TrainSVM($\mathbf{W}^{0},\mathbf{b}^{0},\mathcal{A}^{0}$)
$\mathbf{m}=\left[h^{-1}(\text{IoU}_{p}(\mathbf{W}^{\prime},\mathbf{b}^{\prime},\mathcal{A}^{0}))\right]_{p=1}^{K_{0}}$
$\mathbf{W}^{0},\mathbf{b}^{0},\mathcal{A}^{0}\leftarrow\text{K-means}(\mathcal{D},\\{\mathbf{W}_{p}^{0}|\mathbf{m}_{p}>\mu_{\mathbf{m}}-\sigma_{\mathbf{m}}\\})$
for _$t=1,\ldots,K_{0}-1$_ do
$\mathbf{W}^{t},\mathbf{b}^{t}$ =
TrainSVM($\mathbf{W}^{t-1},\mathbf{b}^{t-1},\mathcal{A}^{t-1}$)
$\Theta\leftarrow\Theta\cup\\{(\mathbf{W}^{t},\mathbf{b}^{t})\\}$
$p^{t},\phi^{t}=\min_{p}\text{IoU}_{p}(\mathbf{W}^{t},\mathcal{A}^{t-1})$
$q^{t},\psi^{t}=\max_{q}\text{ECoS}_{p^{t},q}(\mathbf{W}^{t},\mathcal{A}^{t-1})$
// Merge cluster $p^{t}$ into $q^{t}$
$\mathcal{A}^{t}=\\{(\mathbf{x}_{i},q^{t})|y_{i}=p^{t}\\}\cup\\{(\mathbf{x}_{i},y_{i})|y_{i}\neq
p^{t}\\}$
$\mathbf{W}^{t}\leftarrow\text{cat}(\ldots,\mathbf{W}_{p^{t}-1}^{t},\mathbf{W}_{p^{t}+1}^{t},\ldots)$
$\mathbf{b}^{t}\leftarrow\text{cat}(\ldots,\mathbf{b}_{p^{t}-1}^{t},\mathbf{b}_{p^{t}+1}^{t},\ldots)$
Algorithm 1 The KLiSH clustering algorithm.
#### KLiSH algorithm
The full KLiSH algorithm is presented in Algorithm 1, where $\mathcal{D}$ is
the dataset of GAN’s features, $K_{0}$ is the initial cluster number, and the
output $\Theta=\\{(\mathbf{W}^{t},\mathbf{b},\phi^{t})\\}_{t=1}^{K_{0}-1}$
records the SVM and IoU at each merging step.
The initialization stage has two steps. First, K-means is applied to the
datasets. Second, filtering is performed as described above, resulting in the
initial cluster assignments $\mathcal{A}^{0}$. Then, clusters are iteratively
merged for $K_{0}-1$ steps. In step $t$, an SVM is first trained on dataset
$\mathcal{A}^{t-1}$ , which is obtained from step $t-1$. Next, the IoU and
ECoS are calculated as defined in Eq. 4 and Eq. 5. The class with the lowest
IoU, denoted by $p^{t}$, will be merged with the class with the highest ECoS
with it, denoted by $q^{t}$. The process is repeated until only one class is
left. Without additional information, KLiSH cannot identify a proper number of
classes, just like K-means. We can either slide through cluster numbers and
select manually or set up a threshold $\phi^{*}$ and stop when the IoU reaches
the threshold.
## 4 Evaluation of KLiSH
### 4.1 Experiment Setup
#### Pretrained GANs
We experimented on four widely used GANs, PGGAN [20], StyleGAN [33], StyleGAN2
[34], and StyleGAN2-ADA [35]. These GANs were pretrained on various datasets,
including FFHQ [34], CelebAHQ [64], AFHQ [35], MetFace [35], and LSUN [65]
Car, Bedroom and Church split. The pretrained models were obtained from the
official release 111https://github.com/NVlabs/stylegan3.
GANs | layer indices | channels
---|---|---
StyleGAN2-Bedroom | 9, 11, 13 | 896
StyleGAN2-Church
PGGAN-[Bedroom, Church] | 7, 9, 11, 13 | 960
StyleGAN-[Bedroom, Church]
StyleGAN2-Car | 9, 11, 13, 15 | 960
ADA-[Cat, Dog, Wildlife]
PGGAN-CelebAHQ | 9, 11, 13, 15, 17 | 496
StyleGAN-CelebAHQ
StyleGAN2-FFHQ | 9, 11, 13, 15, 17 | 992
ADA-MetFace
TABLE II: The indices of GAN layers selected for clustering.
#### GAN feature selection
We only collected features from a subset of GAN’s layers mainly due to GPU
memory constraints. To select the layers, we scanned through a generator’s
layers from high resolution to low and stopped adding new layers if the
accumulated number of channels exceeded 1,000. If multiple layers were having
the same resolution, only the last layer of that resolution was selected. The
selected layers for all GANs are shown in Tab. II. The feature maps from the
selected layers are then bilinearly interpolated into the same resolution and
concatenated along the channel axis. For clustering, we use $256\times 256$
resolution. For generating the segmentation (in UFGS and USCS), we use the
full image resolution.
#### Semantic segmentation networks
For face images (CelebAHQ and FFHQ), we trained a DeepLabV3 model with
ResNet50 [66] backbone on CelebAMask-HQ [12]. Following [7], we merged
duplicate classes like “left eye” and “right eye”, resulting in 15 classes.
The model was trained on the training split of CelebAMask-HQ for 20 epochs. We
used Adam with learning rate $10^{-3}$, $\beta_{1}=0.9$, $\beta_{2}=0.999$ and
batch size 32. The learning rate was lowered to $10^{-4}$ after 10 epochs. We
found that 20 epochs were sufficient for the convergence of the DeepLabV3
model. For bedroom and church images, we used a DeepLabV3 model
222https://github.com/zhanghang1989/PyTorch-Encoding pretrained on ADE20K
[67].
#### Baseline clustering algorithms
We compared KLiSH with K-means, AHC, and KASP [19]. K-means and AHC have been
used to cluster GAN’s features [16, 58, 18, 17], but they implicitly require
the centroid assumption (described in Tab. I). Spectral clustering does not
have such a constraint, however, it is not practical for our task as it has an
$O(N^{3})$ time complexity [19]. Therefore, we consider using KASP, a fast
approximate algorithm for spectral clustering. Interestingly, KASP also
performs K-means first and then merges the initial clusters. Different from
KLiSH, KASP uses spectral clustering on the initial centroids to merge them
and does not use linear separability.
We implemented a multi-GPU version of K-means++. KASP was implemented
following the original paper [19]. For AHC, we made two modifications based on
the scikit-learn implementation 333https://scikit-
learn.org/stable/modules/generated/sklearn.cluster.AgglomerativeClustering.html.
First, we trained a linear classifier on the cluster assignments obtained by
AHC for prediction; otherwise, AHC would rebuild the whole merging tree to
predict new data, which was impractical. Second, we gave AHC less number of
features due to its expensive memory cost. More optimization of AHC might be
possible, but would be beyond the scope of this work.
#### Evaluation of clustering
For each GAN, we sampled a total of 256 images, resulting in a block of
features with shape $(256,256,256,D)$. The only exception is AHC, where we
used 16 images and $64\times 64$ resolution, due to the expensive memory cost
mentioned above. Then, we applied KLiSH, K-means, AHC, and KASP to the sampled
features and evaluated them on the same set of 10k sampled images (different
from those used for clustering).
For evaluation, we consider two widely used metrics, Adjusted Mutual
Information (AMI) and Adjusted Rand Index (ARI). Both AMI and ARI are designed
for general-purpose clustering tasks and may not reflect the performance of
segmentation tasks faithfully. In the context of (supervised) semantic
segmentation, mIoU is the dominant metric. Therefore, we also propose the
Maximum matching mIoU (MIoU) metric defined below,
$\displaystyle\displaystyle\text{MIoU}(\\{\mathcal{Y}_{m}\\}_{m=1}^{M},\\{\mathcal{C}_{k}\\}_{k=1}^{K})$
$\displaystyle=\max_{\mathbf{a}\in\\{0,\ldots,M\\}^{K}}\frac{1}{M}J(\mathbf{a}),$
(6) $\displaystyle J(\mathbf{a};\\{\mathcal{Y}_{m}\\},\\{\mathcal{C}_{k}\\})$
$\displaystyle=\sum_{m=1}^{M}\text{IoU}(\mathcal{Y}_{m},\bigcup_{\\{k|\mathbf{a}_{k}=m\\}}\mathcal{C}_{k}),$
(7)
where $M$ is the number of groundtruth classes, $K$ is the number of clusters,
$\mathcal{Y}_{m}$ is the groundtruth class $m$, and $\mathcal{C}_{r}$ is the
predicted cluster $r$. $\mathbf{a}$ is a $K$-dimensional vector that encodes
the matched class for each cluster. If the $r$-th element of $\mathbf{a}$,
$\mathbf{a}_{r}$ equals 0, then the cluster $r$ does not match any groundtruth
class. Otherwise, $\mathbf{a}_{r}>0$ indicates cluster $r$ is matched to
groundtruth class $\mathbf{a}_{r}$. In short, MIoU measures the segmentation
performance of the predicted clusters after they have been matched correctly
to the groundtruth classes. MIoU should only be compared when two clustering
algorithms produce the same number of clusters, _i.e_., when the cost of label
permutation is identical. This constraint is also required by other clustering
metrics such as AMI and ARI.
Figure 4: Demonstration of MIoU-permuted labels.
The MIoU is calculated by a greedy algorithm as described in Algorithm 2, in
which $(\mathbf{a}|_{\mathbf{a}_{k}\leftarrow m})$ denotes that the
$k^{\text{th}}$ element of $\mathbf{a}$ is set to $m$. To demonstrate the
effectiveness of MIoU, we tested Algorithm 2 on a few images. As shown in Fig.
4, the label permutations found by Algorithm 2 well matched the predicted
clusters to the groundtruth classes.
Input: $\\{\mathcal{Y}_{m}\\}_{m=1}^{M},\\{\mathcal{C}_{k}\\}_{k=1}^{K}$
Output: MIoU, $\mathbf{a}^{t}\in\\{0,1,\ldots,M\\}^{K}$
$\mathbf{a}^{0}=\mathbf{0}$
for _$t=1,\ldots,K$_ do
$\mathcal{I}^{t}=\\{k|\mathbf{a}_{k}^{t-1}=0\\}\times\\{1,\ldots,M\\}$
$\mathcal{G}^{t}=\\{\mathbf{a}^{\prime}|\mathbf{a}^{\prime}=(\mathbf{a}^{t-1}|_{\mathbf{a}_{k}^{t-1}\leftarrow
m}),(k,m)\in\mathcal{I}^{t}\\}$
$\mathbf{a}^{t}=\max_{\mathbf{a}^{\prime}\in\mathcal{G}^{t}}J(\mathbf{a}^{\prime};\\{\mathcal{Y}_{m}\\},\\{\mathcal{C}_{k}\\})$
MIoU = $J(\mathbf{a}^{t};\\{\mathcal{Y}_{m}\\},\\{\mathcal{C}_{k}\\})$
Algorithm 2 Maximum matching mIoU.
All experiments were conducted using PyTorch [68]. KLiSH required 8 GeForce
RTX 2080 Ti GPUs to run and took less than 30 minutes per GAN model.
Figure 5: Results of K-means, AHC, KASP, and KLiSH on face images and car images. The first row shows the generated images and the other rows show the clustering results obtained with different cluster numbers, denoted by the number after “@”. “final” refers to the selected number of clusters for downstream tasks, which is 26, 11, 30, and 26 from left to right, respectively. GAN | StyleGAN2 | StyleGAN | PGGAN
---|---|---|---
Dataset | FFHQ | Bedroom | Church | CelebAHQ | Bedroom | Church | CelebAHQ | Bedroom | Church
SVM | 81.7 | 43.5 | 33.7 | 72.2 | 30.9 | 31.4 | 66.4 | 27.7 | 45.4
LSE | 82.7 | 45.2 | 35.2 | 73.8 | 37.3 | 34.8 | 69.3 | 31.1 | 47.8
$\Delta\%$ | -1.3 | -3.8 | -4.3 | -2.2 | -9.8 | -17.2 | -4.2 | -10.9 | -5.0
TABLE III: The semantic extraction performance (in mIoU%) of SVM and LSE on
various GANs and datasets. $\Delta$ denotes the relative difference,
$\Delta=\frac{\text{SVM}-\text{LSE}}{\text{LSE}}$.
### 4.2 Binary Linear Separability Results
As described in Sec. 3.1, KLiSH relies on the binary linear separability of
GAN’s features (Definition 1). To test if this assumption is satisfied, we
compared the performance of semantic segmentation using GAN’s features using
either the SVM or the LSE. The LSE shared the same architecture with the SVM
but was trained with cross-entropy loss, reflecting the classical linear
separability. For each GAN, we sampled 50k images, collected their features,
and obtained their segmentation masks using the pretrained DeepLabV3 model.
Next, we trained LSE and SVM on the features and segmentation masks for 1
epoch using the Adam optimizer with $10^{-3}$ learning rate. The SVM used Eq.
3 with $\lambda_{1}=5000$, which was identical to the value used in KLiSH. For
evaluation, we sampled another different 10k images and measured the mIoU
between the segmentation of LSE or SVM and the segmentation of the DeepLabV3
model.
Their performance is presented in Tab. III. SVM achieved close performance
with LSE in most cases. The relative differences between SVM and LSE were
within 5% on 6 models. Therefore, in general, the classical linear
separability of GAN’s features can be relaxed to the binary linear
separability.
K | metric | K-means | KASP | AHC | KLiSH (ours)
---|---|---|---|---|---
euclidean | euclidean | ward | arccos | euclidean
StyleGAN2-FFHQ
20 | AMI | 64.8 | 61.9 | 62.9 | 65.2 | 70.6
ARI | 57.1 | 54.4 | 60.7 | 65.1 | 70.5
MIoU | 50.2 | 40.3 | 46.4 | 37.7 | 52.7
$26^{\dagger}$ | AMI | 56.3 | 59.1 | 58.0 | 68.0 | 68.3
ARI | 25.1 | 37.2 | 37.7 | 69.2 | 65.4
MIoU | 52.6 | 48.9 | 46.9 | 43.8 | 56.3
30 | AMI | 56.3 | 59.3 | 57.7 | 65.9 | 67.6
ARI | 25.1 | 38.5 | 35.7 | 64.2 | 64.6
MIoU | 52.6 | 53.2 | 49.9 | 45.5 | 58.0
StyleGAN-CelebAHQ
20 | AMI | 53.0 | 55.5 | 53.1 | 62.5 | 69.0
ARI | 38.3 | 38.1 | 49.1 | 67.0 | 73.1
MIoU | 30.9 | 36.8 | 30.5 | 25.4 | 41.5
$30^{\dagger}$ | AMI | 51.9 | 49.7 | 43.4 | 57.9 | 63.6
ARI | 28.2 | 30.5 | 21.4 | 54.8 | 61.8
MIoU | 38.5 | 33.5 | 31.7 | 27.7 | 45.7
PGGAN-CelebAHQ
20 | AMI | 37.9 | 39.3 | 39.4 | 37.1 | 51.0
ARI | 18.5 | 19.4 | 20.5 | 30.1 | 53.5
MIoU | 15.2 | 16.9 | 16.4 | 19.1 | 23.1
$26^{\dagger}$ | AMI | 35.1 | 37.9 | 37.6 | 36.0 | 51.0
ARI | 11.5 | 15.3 | 15.6 | 28.7 | 53.5
MIoU | 17.4 | 19.0 | 16.4 | 19.2 | 29.0
30 | AMI | 35.1 | 36.2 | 38.0 | 38.1 | 50.2
ARI | 11.5 | 12.4 | 15.4 | 30.7 | 52.4
MIoU | 17.4 | 18.6 | 18.4 | 19.6 | 29.8
TABLE IV: Quantitative evaluation of clustering algorithms on face image
generators. The final number of clusters used is indicated by a $\dagger$. For
all metrics, the larger the better. The clustering results are reported as the
best one among the 5 trials.
### 4.3 KLiSH Clustering Results
#### Quantitative evaluations
We evaluated the performance of clustering algorithms on GANs trained on face
images. We report the metrics at cluster numbers K=20, K=30, and the finally
selected number in Tab. IV.
It was clear that KLiSH outperformed existing clustering algorithms. For AMI
and MIoU, KLiSH was superior to all baselines in all cases. Notably, at the
finally selected number of clusters, KLiSH achieved significantly higher MIoU
scores. It indicated that KLiSH achieved the best segmentation performance at
the cluster number that would be used by downstream tasks. For ARI, KLiSH also
achieved the highest score in almost every case, except for K=26 in
StyleGAN2-FFHQ, where AHC (arccos) achieved a better score. Even though,
qualitative results showed that AHC was not better than KLiSH in this case
(see row “final” of column “StyleGAN2-FFHQ” of Fig. 5). AHC failed to cluster
“eye”, “nose”, “upper lip”, “lower lip”, but KLiSH successfully identified all
those clusters. KLiSH’s lower ARI might result from leaving two clusters for
“face” and “neck”. However, in this case, it would be rather easy to merge
them manually. In contrast, dividing the undersegmented clusters obtained by
AHC would be more difficult.
In summary, KLiSH was better than K-means, AHC, and KASP on GAN’s features.
#### Qualitative results
We visualize the clusters obtained by all the cluster algorithms on various
GANs and datasets in Fig. 5. The results were from the same trial (best out of
5 trials) as in Tab. IV. We had two main observations.
First, KLiSH was significantly more accurate than K-means, AHC, and KASP. In
StyleGAN2-FFHQ, K-means@30 failed to cluster “upper lip”, “lower lip”,
“teeth”, “brow” classes and still undersegmented “background” and “hair”.
AHC@30 already failed to cluster “nose” and the classes in mouth region. In
contrast, KLiSH@30 had no undersegmentation of groundtruth classes. Similarly,
for car images, KLiSH@final successfully segmented the “headlight”, “wheel”,
and even two “side glasses”. In contrast, the most competitive baseline,
AHC@final not only undersegmented “background” and “wheel” classes, but also
completely missed the “headlight”.
Second, the advantage of KLiSH over existing clustering algorithms was
consistent across GANs and datasets. For StyleGAN-CelebAHQ, AHC@final missed
“nose”, “lip”, and “teeth” classes and undersegmented “hair”, while
KLiSH@final did not. For PGGAN-CelebAHQ, although all the clustering
algorithms did not obtain good results – probably because PGGAN had learned
the dataset well. Still, KLiSH@final successfully segmented “ear” and “hair”
classes, but AHC@final missed “ear” and mixed “hair” with “background” (see
the rightmost column). We provide more visualization in Fig. 7, which is also
supportive of the observations described above.
Figure 6: Demonstration of a merging step. The first two columns show the
images and clusters. The confidence maps
$\\{\mathbf{s}_{k}^{\dagger}\\}_{k=1}^{M}$ of the image are visualized
(partially) in other columns. $\\{\mathbf{s}_{k}^{\dagger}\\}_{k=1}^{M}$ take
value within [0, 1] and are visualized using the color bar on the right. The
cluster in yellow box (say cluster $p$) has the smallest IoU (Eq. 4) with
K=35. The cluster in green box (say cluster $q$) has the largest ECoS (Eq. 5)
with cluster $p$. Cluster $p$ and $q$ are then merged in this step. Figure 7:
Comparison of clustering algorithms on various datasets and GANs. “final”
refers to 18, 9, 16, 15, and 28 from left to right.
#### Visualization of a merging step
To help readers gain a concrete picture of how KLiSH merged the clusters, we
also visualized a single merging step in Fig. 6. See the cluster confidence
map of “nose” and “brow” (Rows 2 and 3, Column 4 in the right figure), the red
regions are dominated by deep red, indicating that the SVM predicts them with
high confidence. On the contrary, the heatmap denoted by the yellow box has a
large proportion of shallow red, indicating that the confidence of SVM to
predict that cluster was low. This cluster had the lowest IoU at K=35 and was
subsequently merged with another oversegmented cluster of hair, resulting in
better cluster assignments.
## 5 Applications
Method | #shots | #classes | mIoU(%)
---|---|---|---
UFGS (KLiSH) | 0 | 15 | 47.2
UFGS (AHC) | 0 | 15 | 36.6
LSE[7] | 1 | 15 | 47.1 $\pm$ 2.8
10 | 58.5 $\pm$ 0.9
RepurposeGAN [15] | 10 | 9 | 68.0
DatasetGAN [14] | 16 | 8 | 70.0
TABLE V: The performance of two UFGS methods and several few-shot learning
methods.
method | #samples | skin | nose | eye-g | eye | brow | ear | teeth | u-lip | l-lip | hair | hat | ear-r | neck | cloth | mouth*
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
UFGS (KLiSH) | (K=26) | 82.2 | 76.0 | 25.7 | 50.4 | 60.2 | 50.3 | 32.1 | 52.2 | 51.0 | 76.5 | 0.0 | 0.0 | 70.1 | 34.0 | -
UFGS (AHC) | (K=26) | 62.0 | 19.3 | 27.6 | 62.6 | 30.6 | 49.3 | 0.00 | 47.1 | 60.3 | 80.5 | 0.1 | 11.0 | 34.6 | 27.2 | -
LSE | N=1 | 83.8 | 81.7 | 1.9 | 72.9 | 51.0 | 45.6 | 32.7 | 64.8 | 68.2 | 69.0 | 0.6 | 1.7 | 68.8 | 16.1 | -
N=10 | 89.3 | 85.5 | 25.7 | 72.2 | 61.4 | 58.9 | 60.5 | 70.1 | 74.2 | 85.2 | 6.3 | 11.5 | 74.7 | 43.6 | -
RepurposeGAN | N=1 | - | 76 | - | 58 | - | - | - | - | - | - | - | - | - | - | 71
N=10 | 90 | 85 | - | 75 | 68 | 37 | - | - | - | 84 | - | - | 73 | 16 | 87
TABLE VI: IoU(%) per class of DeepLabV3 trained on synthetic datasets
generated with different annotation methods. The evaluation is conducted on
real images from the test split of CelebAMask-HQ. “eye-g”, “u-lip”, “l-lip”
denotes “eyeglasses”, “upper lip”, and “lower lip”, respectively. The mouth
class is a union of “teeth”, “u-lip”, and “l-lip”. Our UFGS achieved close
performance with one-shot LSE on most classes and outperformed one-shot LSE on
hard classes like “eyeglasses” and “cloth”.
Figure 8: UFGS results on _real_ images from CelebAMask-HQ (rows 1 and 2) and
PASCAL VOC (rows 3 and 4). Figure 9: The qualitative results of USCS on
various GANs and datasets. The semantic masks are drawn by users and the
modifications on the masks are indicated by dashed boxes. The image on the
right of each mask is generated by our USCS model.
### 5.1 Unsupervised Fine-Grained Segmentation
Semantic segmentation labeling is very expensive. As described by Lee _et al_.
in CelebAMask-HQ [12], annotating fine-grained semantic segmentation is not
only time-consuming but also involves multiple iterations of refinement. To
reduce the labeling cost, we propose an approach for Unsupervised Fine-grained
Segmentation (UFGS).
Existing methods on unsupervised segmentation [52, 54, 55, 56] only segment
foreground from background, thus are not applicable to UFGS. A more closely
related task to UFGS is few-shot learning-based segmentation [7, 15, 14].
Although they reduce the number of annotations, the cost of fine-grained
semantic annotations is still significant. Zhang _et al_. [14] reports that
annotating a single face image cost on average 20 minutes _per image_. In
contrast, the only labeling cost involved in UFGS is naming and merging a few
labels, which could take less than one minute for a _whole dataset_.
#### Method
The proposed method for UFGS is as follows: First, train a GAN on the target
dataset, run KLiSH on the GAN, and select a proper number of semantic
clusters. Next, generate an image-segmentation dataset by sampling images from
the GAN and using the SVM returned by KLiSH to obtain the segmentation masks
of the image. We apply the SVM directly on the feature maps upsampled to the
resolution of images. Then, train a segmentation network, _e.g_., DeepLabV3
[11], on the synthetic dataset. Finally, segment _real_ images in the original
dataset.
#### Evaluation
We evaluated the performance of our UFGS method on face images using the test
split of the CelebAMask-HQ dataset. The cluster assignments were matched to
the ground-truth classes by maximizing MIoU (Algorithm 2) using 100 images
from the validation split. Note that the images and labels are not used to
train the model. In practice, users can match clusters to labels with just
several clicks. The performance of our method was compared to few-shot
learning methods [7, 15, 14].
To the best of our knowledge, except for face images, there are no large-scale
semantic annotations on the datasets described in Sec. 4.1. PASCAL VOC [69]
contains car images, but they have a significant domain gap with the car
images from the GAN’s dataset, LSUN-Car. In two relevant works, DatasetGAN[14]
and RepurposeGAN [15], the authors make their own test sets by filtering
PASCAL VOC and have not released them as of now. Therefore, we only present
qualitative results on PASCAL VOC and report quantitative results on
CelebAMask-HQ.
#### Experiment setup
We used StyleGAN2-FFHQ and StyleGAN2-Car for UFGS on CelebAMask-HQ and PASCAL
VOC cars, respectively. The baseline for comparison was UFGS with AHC, as it
was the most competitive method with KLiSH on StyleGAN2-FFHQ according to Tab.
IV. The settings were identical to UFGS with KLiSH except that the datasets
were generated with clusters found by AHC. We also compared UFGS with KLiSH to
a few-shot semantic extractor, LSE. We reproduced LSE on StyleGAN2-FFHQ and
StyleGAN2-Car following the settings in [7]. In brief, we randomly sampled 1
or 10 images (for one-shot or 10-shot experiments) and obtained their
segmentation using the pretrained DeepLabV3 model. Then, we trained the LSE on
the features and labels of those images for 6,400 iterations. To address the
variation of few-shot experiments, we trained LSE five times, each time with a
different set of image annotations.
After training the LSE and obtaining the KLiSH clusters, we then compared
their performances on benefiting the downstream segmentation task. We first
sampled a synthesized image segmentation dataset for each of the LSE models
and KLiSH results, which consisted of 50k images and segmentation masks. Then,
we trained a DeepLabV3 model on each dataset following the training settings
described in Sec. 4.1. Finally, we tested the DeepLabV3 models on the real
images from the test split of the CelebAMask-HQ dataset.
#### Results
We present the qualitative results of our UFGS method in Fig. 8. As shown in
Row 1 and 2, UFGS model segmented facial classes like “eye”, “brow”, “lips”,
“teeth” and “nose” well. For car images in Row 3 and 4, our UFGS method also
successfully segmented “headlight”, “body”, “glasses”, and “wheel”. However,
we found there was a domain gap between synthesized images and real images.
See Row 1, Col 3 and 4 in Fig. 8. The hand in front of the face is a rare case
in the dataset. Therefore, the segmentation model cannot deal with this case
well.
The quantitative results are reported in Tab. V and Tab. VI. As shown in Tab.
V, KLiSH surpassed UFGS with AHC by a large margin. In comparison with few-
shot learning methods, UFGS achieved close performance with the one-shot
method, LSE, but lagged behind the 10-shot methods.
In summary, though the performance of our UFGS method was limited by the
domain gap as a result of using GANs, we for the first time showed accurate
fine-grained semantic segmentation in an unsupervised learning setting.
### 5.2 Unsupervised Semantic-Conditional Synthesis
Enhancing controllability over the generated images is important for
commercial applications of GANs. Most GANs do not allow the user to control
the spatial structure of generated images precisely. Semantic-Controllable
Synthesis (SCS) aims to provide controllability over precise spatial
structures of generated images. Though achieving greater controllability for
image synthesis, existing methods for SCS, _e.g_., pix2pix [42], SEAN [45] and
GauGAN [44], rely heavily on large-scale human annotated semantic masks.
Recently, the labeling requirements of SCS have been lowered to a few
annotations [7, 13]. Yet, the user still has to label a few multi-class fine-
grained segmentation masks, which limits the application in practice. For
example, users may want to enable SCS on images of tigers. If there are no
semantic segmentation masks on these images, the user needs to annotate 16 to
40 images. The annotation process would take around 2 to 5 hours, according to
Zhang _et al_. [14].
To lower the label dependency of SCS methods, we propose the task of
Unsupervised SCS, which aims to achieve semantic-controllable image synthesis
on arbitrary datasets without human annotations.
#### Method
Our USCS application works as follows. First, the user shall provide a target
image dataset. Then, we train an unconditional GAN and generate an image-
segmentation dataset using the same method as described in Sec. 5.1. Finally,
we train a GauGAN [44] on the generated dataset. The above process can be done
offline. Then users can draw and edit on the semantic mask and the GauGAN will
generate desired images.
We developed a web-based interface to help users generate images with desired
structures. Before the user starts painting, a few images and their semantic
masks are shown as examples. Next, the user can click on a palette and paint
with selected semantic classes. Then, the user can draw freely and submit the
painting to the server, which will return an image synthesized given the
semantic mask.
#### Experiment setup
We conducted the USCS experiment on StyleGAN2 models trained on FFHQ and Car
datasets and StyleGAN2-ADA models trained on MetFaces, AFHQ [40] Cat, Dog, and
Wildlife split. Similar to the setting of UFGS, we also synthesized a dataset
with 50k paired images and segmentation masks from each GAN and trained a
GauGAN on each dataset. The GauGAN was trained using the official release
444https://www.github.com/nvlabs/spade for 10 epochs on 8 GPUs. The training
took around 100 hours for a GauGAN model.
#### Results
The editing results on different datasets and GAN models are presented in Fig.
9. We observed that USCS supported precise, diverse, and even out-of-
distribution image editing operations. First, USCS respected the semantic mask
well, accurately translating the user’s semantic input into corresponding
images while maintaining good photo-realisticity. See the “big round eye”
example in ADA-Cat. The enlarged eyes had a matched shape as specified by the
user. Second, our USCS method well demonstrated the diversity and degree of
freedom of editing operations. As shown in the left part of StyleGAN2-FFHQ,
the shape of the sunglasses could even be changed to triangle. Third, the out-
of-distribution editing operations could also lead to photorealistic synthesis
results. See the “elf ear” examples in StyleGAN2-FFHQ and ADA-Wild, the
synthesized sharp ear is not present in the training set of the model.
Besides, the “blink eye” in ADA-MetFace and ADA-Wild, and “more wheels” in
StyleGAN2-Car are all out-of-distribution semantic masks.
To the best of our knowledge, SCS cannot be applied to datasets of wildlife
and artistic portraits, _etc_., because so far no semantic annotations on such
datasets are available. EditGAN [13] relies on 16 to 30 fine-grained
segmentation masks to enable semantic editing on Cat, Dog, _etc_., but the
cost of labeling is still high (around 10 minutes per image). In contrast, the
USCS model is free from the constraint of annotations. Please find a
demonstration of our USCS web application in Supplementary Video 1 and have a
try on our online demonstration555https://atlantixjj.github.io/KLiSH/.
## 6 Limitations
Our method is generally constrained by the performance of GANs. First, the
clustering performance of KLiSH is dependent on the degree of linear
separability of GAN’s features. If the GAN does not encode semantics into _one
vs. rest_ linearly separable clusters, then the KLiSH might fail to find
meaningful clusters. Second, the performance of UFGS and USCS is limited by
the synthesis quality of the original GAN. When the GAN fails to generate
realistic images, the UFGS model will have a domain gap when applied to real
images and USCS will also generate unrealistic images.
## 7 Conclusion
In this paper, we propose KLiSH to find semantic clusters in GAN’s features.
By making use of GAN’s linear separability in semantics, KLiSH achieves better
clustering results than conventional clustering algorithms including K-means,
AHC, and KASP. The clusters found by KLiSH accurately correspond to the
semantic parts of generated images. Therefore, for each pretrained GAN, we can
generate an image-segmentation dataset for downstream tasks. One can easily
enable semantics-based applications including UFGS and USCS with the dataset
generated by GANs. The surprising performance of UFGS and USCS make us believe
that GANs can play a more important role in representation learning.
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| Jianjin Xu (S’01-M’08-SM’13) received a B.E. degree in computer science
from the Tsinghua University, Beijing, China, in 2019, and a M.S. degree in
computer science from the Columbia University, New York City, NY, U.S., in
2021. He is currently an Assistant Research Scientist at the School of
Mathematics and Computer Science, Panzhihua University, Sichuan, China. His
current research interests include generative models, neural network
interpretation, and computer vision. Previously he was a research assistant at
the Department of Computer Science and Technology, Tsinghua University.
---|---
| Zhaoxiang Zhang received his bachelor’s degree in Circuits and Systems from
the University of Science and Technology of China (USTC) in 2004. In 2004, he
joined the National Laboratory of Pattern Recognition (NLPR), Institute of
Automation, Chinese Academy of Sciences, under the supervision of Professor
Tieniu Tan, and he received his Ph.D. degree in 2009. In October 2009, he
joined the School of Computer Science and Engineering, Beihang University, as
an Assistant Professor (2009-2011), an Associate professor (2012-2015) and the
vise-director of the Department of Computer application technology
(2014-2015). In July 2015, he returned to the Institute of Automation, Chinese
Academy of Sciences. He is now a full Professor in the Center for Research on
Intelligent Perception and Computing (CRIPAC) and the National Laboratory of
Pattern Recognition (NLPR). His research interests include Computer Vision,
Pattern Recognition, and Machine Learning. Recently, he specifically focuses
on biologically inspired intelligent computing and its applications on human
analysis and scene understanding. He has published more than 150 papers in the
international journals and conferences, including reputable international
journals such as IEEE TPAMI, IJCV, JMLR and top level international
conferences like CVPR, ICCV, ECCV, ICLR, NeurIPS, AAAI and IJCAI. He served as
the associated editor of IEEE TCSVT, PR, and Frontiers of Computer Science. He
served as the area chair of international conferences like CVPR, ICCV, AAAI,
IJCAI. He is a senior member of IEEE.
---|---
| Xiaolin Hu (S’01-M’08-SM’13) received B.E. and M.E. degrees in automotive
engineering from the Wuhan University of Technology, Wuhan, China, in 2001 and
2004, respectively, and a Ph.D. degree in automation and computer-aided
engineering from the Chinese University of Hong Kong, Hong Kong, in 2007. He
is currently an Associate Professor at the Department of Computer Science and
Technology, Tsinghua University, Beijing, China. His current research
interests include deep learning and computational neuroscience. At present, he
is an Associate Editor of the IEEE Transactions on Pattern Analysis and
Machine Intelligence, IEEE Transactions on Image Processing, and Cognitive
Neurodynamics. Previously he was an Associate Editor of the IEEE Transactions
on Neural Networks and Learning Systems.
---|---
|
# Coordinating Cross-modal Distillation for Molecular Property Prediction
††thanks: * These authors contributed equally.
Hao Zhang2,* Nan Zhang1,*, Ruixin zhang2, Lei Shen2, Yingyi Zhang2, Meng Liu3
1 Academy for Engineering and Technology, Fudan University, China
2 Tencent Youtu Lab
3 Shanghai Jiao Tong University
###### Abstract
In recent years, molecular graph representation learning (GRL) has drawn much
more attention in molecular property prediction (MPP) problems. The existing
graph methods have demonstrated that 3D geometric information is significant
for better performance in MPP. However, accurate 3D structures are often
costly and time-consuming to obtain, limiting the large-scale application of
GRL. It is an intuitive solution to train with 3D to 2D knowledge distillation
and predict with only 2D inputs. But some challenging problems remain open for
3D to 2D distillation. One is that the 3D view is quite distinct from the 2D
view, and the other is that the gradient magnitudes of atoms in distillation
are discrepant and unstable due to the variable molecular size. To address
these challenging problems, we exclusively propose a distillation framework
that contains global molecular distillation and local atom distillation. We
also provide a theoretical insight to justify how to coordinate atom and
molecular information, which tackles the drawback of variable molecular size
for atom information distillation. Experimental results on two popular
molecular datasets demonstrate that our proposed model achieves superior
performance over other methods. Specifically, on the largest MPP dataset
PCQM4Mv2 served as an ”ImageNet Large Scale Visual Recognition Challenge” in
the field of graph ML, the proposed method achieved a 6.9% improvement
compared with the best works. And we obtained fourth place with the MAE of
0.0734 on the test-challenge set for OGB-LSC 2022 Graph Regression Task. We
will release the code soon.
## I Introduction
In recent years, molecular property prediction (MPP) has played a crucial role
in new materials and drug discovery. Since atoms and bonds in a molecule are
naturally in a graph structure, graph neural networks (GNN) [1] have shown
great vitality in MPP. Many works [2, 3, 4] take 2D molecular structures as
input of GNN for training and testing. Their performances are limited because
they do not consider the 3D atom coordinates that determine certain chemical
and physical functionalities of molecules. Methods [5, 6, 7] that exploit 3D
structures exhibit advantages in accuracy, especially properties related to
quantum mechanics, such as single-point energy, atomic forces, and dipole
moments. However, obtaining accurate 3D structures is expensive and time-
consuming. Calculating 3D information with DFT takes ${10^{5}}$ times longer
than model feedforward [2], which seriously impedes the application of
3D-based methods.
Some works attempt to augment 2D information with 3D information through
training to enhance the prediction accuracy while keeping inference speed. One
direction is developing 2D-3D contrastive tasks. Another way is to generate
original conformations through reconstruction tasks. Usually, these tasks
require several different 3D conformers to create positive and negative pairs
for self-supervised learning, but obtaining different conformers is time-
consuming for large-scale data. Also, different conformers from one SMILES[8]
code may represent different property values, which lower the performance of
contrastive-based methods. Lastly, some applications do not support the
requirement of multi conformers. With the force field given, one SMILES code
usually can only generate one 3D conformer structure.
Distillation is another intuitive solution that utilizes 3D knowledge in
training while keeping inference speed. Also, distillation methods do not have
the limitation of multiple conformers requirements. But some challenging
problems remain for distillation methods. On the one hand, the 3D view is
quite distinct from the 2D view, making the top-level-feature-only
distillation less effective. On the other hand, which tokens participate in
distillation is still an open question for the MPP. As self-attention
calculates information of the virtual token from all-atom features, it is
natural to include all-atom tokens together in distillation. But simply
distilling atom tokens and virtual token together leads to poor performance.
We found the gradient magnitudes of atom tokens distillation are inconsistent
and unstable due to the variable molecular size. So distilling local atom
information without correction will cause degradation problems.
Motivated by the above observation, we propose a novel coordinating cross-
modal distillation (CCMD) framework. We formulate a coordinating weight, which
dynamically scales and balances the gradients of global molecular token and
local atom tokens according to the number of atoms. Also, the distillation of
virtual token and atom tokens is carried in all layers to enhance the 2D-3D
distillation performance. Lastly, we introduce absolute position encoding to
integrate adjacent edge information, achieving better performance in both
views of 2D and 3D. Experimental results on two popular molecular datasets
demonstrate that our proposed model achieves state-of-the-art performance.
In conclusion, our main contributions to this work can be summarized as
follows:
* •
We exclusively develop an enhanced cross-modal distillation on all layers for
molecular property prediction, which distills global molecular information and
local atom information to transfer 3D geometric information to 2D view.
* •
For coordinating global molecular and local atom information, we formulate a
novel weight according to the number of atoms in a molecular to scale gradient
magnitudes, which boosts the distillation performance. Specific, we
demonstrate the coordinating weight is $f(N)=\frac{1}{{N}^{2}}$ for
transformer and $f(N)=\frac{1}{{N}}$ for GIN. N is the number of atoms in a
molecular.
* •
We design an absolute position encoding integrated with adjacent edge
information, which could distinguish atoms same atom id in the input layer and
increase the performance of both 2D and 3D.
* •
CCMD can enable molecular attention to more atoms in the 2D view. Extensive
experiments on the PCQM4Mv2 dataset [9] demonstrate that our proposed method
consistently outperforms other state-of-the-art methods in most situations.
## II Related Work
### II-A Molecular Property Prediction
According to different dimensions, molecular expressions contain 1D SMILES
code, 2D graph, and 3D graph. Due to the natural capability for representing
the molecular structure, graph neural networks (GNN) have attracted attention
in molecular property prediction, especially with the input of 2D or 3D
expressions.
#### II-A1 2D Methods
Many works focus on applying GNN to 2D graphs. For example, Gilmer et al.[2]
proposed Message Passing Neural Networks, which combine message passing and
aggregating algorithms with GNN, forming a successful framework for MPP. Liu
et al.[3] introduced a simple unsupervised representation for molecules that
embeds the vertices in the molecule graph and constructs a compact feature by
assembling the vertex embeddings. Ying et al.[10] proposed Graphormer, which
utilizes a Transformer [11] in the GNN by effectively encoding the graph
structure information. Although these methods have shown promising
capabilities, their potential is limited as they do not use 3D coordinates
that better represent molecular structure and energy.
#### II-A2 3D Methods
3D molecular structures contain spatial information important to molecular
property prediction, such as bond angles and lengths. Many recent methods try
to answer how to utilize this 3D information in the GNN model. For instance,
Klicpera et al.[5] proposed to let message embeddings interact based on the
distance between atoms and the angle between directions to improve quantum
mechanical property prediction. Lu et al.[12] proposed a Multilevel Graph
Convolutional neural Network that extracts features from the conformation and
spatial information with multilevel interactions. Liu et al.[7] proposed
SphereNet, a 3D graph network framework with spherical message passing.
3D-based methods achieve significant performance improvements. But obtaining
3D structures is time-costing, which reduces the value of 3D-based methods in
large-scale applications.
### II-B Cross Modality Training Methods
Since training is relatively insensitive to time-consuming, an intuitive
approach is to use 2D and 3D information in training and only use 2D in
prediction.
#### II-B1 Contrastive Methods
These methods use 3D conformers and 2D structures to construct positive and
negative pairs while applying a contrastive loss during training. For example,
Liu et al.[13] proposed the Graph Multi-View Pretraining framework, in which
they perform self-supervised learning by leveraging the correspondence and
consistency between 2D topological structures and 3D geometric views. Li et
al.[14] developed GeomGCL, a dual-view geometric message passing network that
utilizes the molecular geometry across 2D and 3D views. Hu et al.[15] proposed
a strategy of pre-training GNN at the level of individual nodes as well as
entire graph, so the GNN can learn useful local and global information. Stark
et al.[16] proposed 3D Infomax, which pre-trains a GNN by maximizing the
mutual information between its 2D and 3D embeddings. The drawbacks of
contrastive methods lie in the requirements of multiple conformers. Large-
scale conformer generation is costly, while one SMILES code usually generates
only one 3D conformer with a specific force field.
#### II-B2 Knowledge Distilling Methods
Knowledge distilling (KD) is an intuitive solution for 3D to 2D cross-modality
training. As our work relies on Transformer-based methods such as Graphormer,
we only discuss Transformer related KD methods. Many works [17, 18, 19, 20,
21] have attempted to apply KD to the Transformer-based model. Upon molecular
property prediction, Zhu et al.[22] proposed ST-KD, an end-to-end Transformer
KD framework, bridging the knowledge transfer between graph-based and SMILES-
based models. But ST-KD only KD the atom to atom attentions, neglecting the
informative embeddings in atom tokens. Based on our experiment, gradient
magnitudes of atom-token distillation are discrepant and unstable due to the
variable molecular size. Distilling atom-token embeddings without correction
will cause degradation problems.
To address this issue, we propose a theoretical insight to justify how to
coordinate atom token embeddings and molecular features and formulate an auto-
correcting distilling weight to dynamically tune and balance the gradients
according to the number of atoms.
## III Method
Figure 1: The illustration of of our CCMD framework with global molecular and
local atom information for molecular property prediction. $\mathcal{L}_{m}$
represents the global molecular distillation, $\mathcal{L}_{a}$ represents the
local atom information distillation, and coordinating denotes the coordinating
weight scaling gradient magnitudes.
### III-A Overview
In this section, we present the key components of CCMD. As illustrated in Fig.
1, the whole graph transformer framework consists of a 3D model and a 2D
model, and the backbone is based on the graphormer [10]. Firstly, the 3D model
is trained with 3D information as a teacher model. Then, frozing the
parameters of the teacher, we train the 2D student model with a distilling
loss and a supervised target loss. The distilling module consists of the
global molecular distillation and local atom distillation, which can further
improve the performance of 2D molecular graphs. At last, for coordinating
global and local information, we formulate a novel coordinating weight to
scale and balance the gradients optimally.
### III-B Preliminary
For convenience, let $\mathcal{D}=\\{(X,Y,E)\\}$ denotes a 2D molecular graph,
the $X$, $Y$ and $E$ denote the input of atoms, property labels and edges
respectively. In addition, we use $\widetilde{}$ mark the 3D view, so 3D
molecular graph can be denoted as
$\widetilde{\mathcal{D}}=\\{(\widetilde{X},\widetilde{Y},\widetilde{E})\\}$,
where $\widetilde{E}=f(\widetilde{P})$, and $\widetilde{P}$ represents the 3D
coordinates of atoms. For instance, the $i$-th atom embedding in $l$-th layer
can be denoted as $X_{i}^{l}$, where $i\in[0,N+1]$. $i=0$ denotes the virtual
token representing the molecular representation, $N$ denotes the total number
of atoms. $l\in[0,L]$ denotes the order number of the layer.
### III-C Input Representations
#### III-C1 Absolute Position Encoding
Following the previous works (Park et al. 2022; Ying et al. 2021), we
integrate an absolute position encoding (APE) into graphormer [10] illustrated
in Fig.2. The APE is the adjacent edge information of the node, which could
distinguish some atoms with the same atom id easily in the input layer. Let
$X^{0}$ denote the input tokens which are integrates with the APE, $X^{0}$ is
calculated as:
$X^{0}=X\oplus E,$ (1)
where $\oplus$ is used to gather the information from neighbors. In specific,
the $i$-th input token is calculated as:
$X_{i}^{0}=MLP(\sum_{i=1}^{N+1}e_{i,j}).$ (2)
For the 2D view, ${e}_{i,j}$ is the feature of original Chemical bonds. For
the 3D view, we utilize radial basis functions (RBF) [23] as the transformed
function $\widetilde{e}_{i,j}=RBF(d_{ij})$ to obtain the edge information,
where ${d}_{ij}$ is the distance between atom i and atom j. This 3D embedding
can encode diverse geometric factors
### III-D Backbone
The backbone used is graphormer [10]. All tokens in the $l+1$ layer is updated
by layer normalization (LN) and multi-head attention (MHA) on the tokens in
the $l$ layer
$X^{{}^{\prime}(l+1)}=MHA(LN(X^{l}))+X^{l},$ (3)
then the feed-forward blocks (FFN) is applied to the tokens in the $l+1$
layer:
$X^{l+1}=FFN(LN(X^{l+1}))+X^{{}^{\prime}(l+1)}.$ (4)
Moreover, the final molecular representation $R$ is obtained from the virtual
token of the last layer by an MLP:
$R=MLP(X_{0}^{L}).$ (5)
At last, for the supervised task, we use $L1$ loss. So the supervised term
$\mathcal{L}_{3D}$ is defined as:
$\mathcal{L}_{3D}(\widetilde{R},\widetilde{Y})=L_{1}Loss(\widetilde{R},\widetilde{Y}),$
(6)
The supervised term $\mathcal{L}_{2D}$ is defined as:
$\mathcal{L}_{2D}(R,Y)=L_{1}Loss({R},{Y}),$ (7)
### III-E Distillation
#### III-E1 Challenges
Usually, for most of the existing distillation methods, the total loss is
defined as:
$\mathcal{L}(R,Y,\widetilde{R})={L}_{2D}(R,Y)+\mathcal{W}*L_{1}Loss({X},\widetilde{X}),$
(8)
${L}_{2D}(R,Y)$ is the task-related loss, and $L_{1}Loss({X},\widetilde{X})$
is the distilling loss on all tokens and ${W}$ is the hyper-parameter.
However, the virtual token ${X}_{0}$ is the target, which represents the
global representation of the molecular, and the other tokens ${X}_{i}(i>0)$
are the atom representations that are auxiliary to the molecular
representation. So the strategy of distillation should be divided into two
modules: global molecular distillation and local atom distillation. In
addition, We should select a proper weight to balance the global molecular
distillation and local atom distillation.
#### III-E2 Global molecular distilling
The virtual token ${X}_{0}$ represents the molecular embedding and usually be
used to predict molecular property, so distilling the virtual token embedding
provides a global view of learning on 3D geometry structural knowledge.
Considering that 2D graphs are seriously distinct from 3D conformers, we
conduct the global molecular distilling from 3D to 2D on all layers which
could learn more geometric information at different levels step by step. At
last, the target is conducted by minimizing the following loss function given
any pairs of $\widetilde{X}_{l,0}$ and $X_{l,0}$ in the training batch:
$\mathcal{L}_{m}=\sum_{l=1}^{L}L_{1}Loss(\widetilde{X}_{0}^{l},X_{0}^{l})$ (9)
#### III-E3 Local Atom Distilling
The virtual token ${X}_{0}$ is the global view of molecular, and the token
${X}_{i}(i>0)$ is the local view of molecular. The virtual token is calculated
from atom tokens by self-attention with the updating process. Considering the
relationship between the virtual token and atom tokens, we use the local atom
distilling to further boost the performance of the virtual token. One possible
way is to calculate the similarity (distance) of each pairwise atom embedding,
then the local atom calibration distillation loss is defined as:
$\mathcal{L}_{a}=\sum_{l=1}^{L}\sum_{j=1}^{N+1}L_{1}Loss(\widetilde{X}_{j}^{l},X_{j}^{l})$
(10)
Figure 2: The illustration of the backbone with absolute position encoding and
relative position encoding.
#### III-E4 Coordinating weight
For boosting the performance of distilling with global and local information,
we formulate a novel coordinating weight according to the number of molecular
size to scale gradient magnitudes. The analyzing and reducing process from two
perspectives is as follows.
Coordinating the total loss Comparing the global molecular loss
$\mathcal{L}_{m}$ and local atom loss $\mathcal{L}_{a}$, we can draw the
conclusion that $\mathcal{L}_{a}$ is unstable due to the sum operation on all
tokens. This means that when the molecular is very large(with big N), the
final loss ${L}$ is exploding. So, we should add the function $f(N)$ according
to the variable number of atoms to scale and calibrate the local atom loss
$\mathcal{L}_{a}$. The formulation is as follows:
$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{2D}+(\sum_{l=1}^{L}L_{1}Loss(\widetilde{X}_{0}^{l},X_{0}^{l})$
(11)
$\displaystyle+f(N)*\sum_{l=1}^{L}\sum_{j=1}^{N+1}L_{1}Loss(\widetilde{X}_{j}^{l},X_{j}^{l}),$
$L$ is fixed and $N$ is variable which represents the number of atoms. In
order to eliminate the effect of sum operation on all tokens, it’s intuitive
to set $f(N)=\frac{1}{N}$. In addition, we couldn’t add $\frac{1}{N}$ to the
molecular loss, which will dissipate the loss of the virtual token as N
increases, so it’s necessary to separate global and local loss, and only scale
the atom loss ${{L}_{a}}$. Generally, to make the total loss not explode, we
should add a function $f(N)\propto{O}(\frac{1}{N})$ or
$f(N)\propto{o}(\frac{1}{N})$. ${O}$ means “is of the same order as” and ${o}$
means “is ultimately smaller than”.
Coordinating the molecular gradient Since the virtual token is the target
embedding, its gradient is much important. So we will calculate the
relationship between its gradient and N. To simplify the forward of
transformer, $X_{i}^{l+1}$ in $(l+1)$ layer is formed as:
$X_{i}^{l+1}=MHA(X^{l})\\\ \&=\sum_{j=0}^{N+1}({W}^{l}_{ji}*{X}^{l}_{j})$ (12)
The ${W}^{l}_{ji}$ is the attention weight. In the back-propagation, the
gradient of the virtual token in $l$th layer is as formulated:
$\displaystyle{grad(X_{0}^{l})}$ $\displaystyle=\nabla_{X_{0}^{l}}{L}$ (13)
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*(\nabla_{X_{0}^{l}}{X_{k}^{l+1}}))$
Finally, for transformer, the virtual token gradient $grad(X_{0}^{l+1})$ in
${l}$ layer can be described in follow:
$\displaystyle{grad(X_{0}^{l})}$ $\displaystyle\propto{{O}({N}^{2})},$ (14)
Different from multi-head attention in transformer, a simple GNN always use
${MLP}$ to calculate ${X}^{l+1}_{i}$ with ${X}^{l}_{0}$ . So
$(\nabla_{X_{0}^{l}}{X_{k}^{l+1}})$ is constant, then the result is:
$\displaystyle{grad(X_{0}^{l})}$ $\displaystyle\propto{{O}({N})},$ (15)
For detailed proof, please refer to the Section1 in Supplementary Material.
For transformer, the formula indicates that the gradient of virtual token
$grad(X_{0}^{l})$ will be scaled by ${N}^{2}$, and N is the number of atoms.
So scaled with $\frac{1}{N}$ couldn’t solve the unstable gradient of the
virtual token. However, for a simple GNN, $\frac{1}{N}$ is enough.
Conclusion To this end, for transformer, we proved the weight
$f(N)=\frac{1}{{N}^{2}}$ is suitable to scale the losses and balance the
gradient magnitudes of the global molecular information and local atom
information. So the total objective function is formulated as:
$\mathcal{L}=\mathcal{L}_{2D}+(\mathcal{L}_{m}+\frac{1}{{N}^{2}}*\mathcal{L}_{a}).$
(16)
## IV Experiments
To evaluate our method and investigate the effectiveness of the proposed
components, we carry out extensive experiments on two popular molecular
property prediction tasks, especially in PCQM4Mv2 [9]. In this section, we
introduce the detailed configuration in our experiments and present the
detailed experimental results.
### IV-A Datasets
#### IV-A1 PCQM4Mv2 dataset
PCQM4Mv2 dataset [9] served as an ”ImageNet Large Scale Visual Recognition
Challenge” in the field of graph ML is a quantum chemistry dataset originally
curated under the PubChemQC project. It contains 3.8 million molecular graphs.
The 3D structure for training molecules is calculated by DFT, which is
equipped with only one conformer for each SMILES code.
#### IV-A2 MolHIV dataset
MolHIV dataset [24] is adopted from the MOLECULENET [25] with 41K molecular
graphs for molecular property prediction. All the molecules are pre-processed
using RDKIT [26]. Each graph represents a molecule, where nodes are atoms, and
edges are chemical bonds. Input node features are 9-dimensional, containing
atomic number and chirality, as well as other additional atom features such as
formal charge and whether the atom is in the ring. Input edge features are
3-dimensional, containing bond type, bond stereochemistry as well as an
additional bond feature indicating whether the bond is conjugated.
### IV-B Experimental Setup
Our experiments are conducted on eight NVIDIA Tesla V100 GPUs. The
implementation is based on PyTorch [27]. We use the Graphormer architecture
[10] as the based model (L = 12, d = 768) for both 2D and 3D branches. Our
CCMD method contains improved baseline with APE, global molecular
distillation, local atom distillation, searched weight and coordinating
weight, denoted as baseline, $\mathcal{L}_{m}$, $\mathcal{L}_{a}$,
${W}_{search}$ and ${W}_{coordinating}$ respectively. In order to prevent the
explosion of a total loss, the local atom distillation loss is calculated by
means denoted as $\overline{\mathcal{L}_{a}}$, where
$\overline{\mathcal{L}_{a}}=\frac{1}{N}*\mathcal{L}_{a}$. So we set the
coordinating weight as $\frac{1}{N}$ for experiments on graphormer and $1$ for
experiments on GIN [28].
For the settings and metrics on the PCQM4Mv2 dataset, the previous methods use
the 2D view for training and validation. We add the 3D structure calculated by
DFT for training and distillation, and also evaluate on the validation with 2D
view (the test labels are no longer available and results are given over the
validation set). And split of dataset is followed by previous work [10]. We
set the batch size as 512 with 70 epochs and use the Adam algorithm [29] with
an initialized learning rate of 0.0002 and a momentum of 0.9. The results are
evaluated in terms of Mean Absolute Error (MAE), and a lower MAE indicates
better performance.
For the settings and metrics on the MolHIV dataset, we add the 3D structure
calculated by RDKIT for 3D view training and distillation. And we evaluate on
the test dataset with 2D view. And split of dataset is followed by previous
work [10]. We pretrain 3D and 2D branches on the PCQM4Mv1 dataset following
the previous works [10]. When finetune on the MolHIV dataset, we set the batch
size as 64 with 10 epochs and use Adam algorithm [29] with an initialized
learning rate of 0.0002 and a momentum of 0.9. The results are evaluated in
terms of Area Under the ROC Curve (AUC), and a higher AUC indicates better
performance.
### IV-C Experimental Results
#### IV-C1 The comparison on PCQM4Mv2 dataset
Table I presents the results for molecular property prediction task on the
OGB-LSC PCQM4Mv2 datasets [9]. What’s worth mentioning is that contrastive
methods are hard applicable because the PCQM4Mv2 is equipped with only one
conformer for each SMILES code. Compared with established methods, our model
is currently the best on the PCQM4Mv2 leaderboard. As shown in Table I, our
approach achieves the significant MAE improvement by 6% compared with the
state-of-the-art method, indicating the effectiveness of our proposed method.
Figure 3: The learning curves of 2D supervised performance and distilling performance during training process. TABLE I: The comparison with state-of-the-art methods on the validation set of PCQM4Mv2 dataset. Our approach outperforms the best model on average by 6%. Method | MAE
---|---
GCN [30] | 0.1379
GCN-VN[2, 30] | 0.1153
Graphormer [10] | 0.0864
EGT [31] | 0.0869
GRPE-Standard [32] | 0.0890
GPS [33] | 0.0858
Ours | 0.0809
#### IV-C2 The comparison on MOLHIV dataset
We further investigate our proposed method on the MOLHIV dataset. The
comparison results are shown in Table II. From Table II, we can see that our
method outperforms other methods, achieving prominent improvement by 1.39% on
AUC compared with baseline. In addition, the origin MOLHIV dataset is not
attached with 3D information, so we use the rdkit to generate the 3D position
as the 3D coordinates of atoms. It should be noted that the generated 3D
coordinates with randomly selected 3D conformers are not matched the real
coordinates, so the generated force fields are different from the real force
fields. The weakness of the 3D teacher model limits the performance of the
student. Our approach can still obtain improvement with biased conformers,
outperforming state-of-the-art methods and software (e.g.rdkit) generated 3D
coordinates-based results. These phenomenon demonstrate the effectiveness of
our distilling framework.
TABLE II: The results of our method compared with other methods on the test set of MOLHIV dataset. Method | AUC
---|---
DeeperGCN-FLAG[30] | 79.42 ± 1.20
PNA [30] | 79.05 ± 1.32
DGN [28] | 79.70 ± 0.97
PHC-GNN[30] | 79.34 ± 1.16
(baseline)Graphormer [10] | 80.51 ± 0.53
GraphMVP [13] | 77.0
GeomGCL[14] | 80.6 ± 0.009
EGT [31] | 80.60 ± 0.65
GRPE-Standard [32] | 81.39 ± 0.49
GPS [33] | 78.80 ± 0.0101
Ours(baseline+$\mathcal{L}_{m}$) | 81.4±0.11
Ours(baseline+$\overline{\mathcal{L}_{a}}$) | 79.0±0.01
Ours(baseline+$\mathcal{L}_{m}$ +($\mathcal{W}_{search}*\overline{\mathcal{L}_{a}})$) | 81.47±0.03
Ours(baseline+$\mathcal{L}_{m}$ +($\mathcal{W}_{coordinating}*\overline{\mathcal{L}_{a}})$) | 81.9±0.01
### IV-D Ablation Study
We investigated the impact of these components of our proposed method on the
validation set of PCQM4Mv2 dataset, and results are shown in Table III.
TABLE III: The ablation study of our method on the validation set of the PCQM4Mv2 dataset. L denotes the conduction on the last layer and ”all” denotes distilling with the features, which are not divided into global and local respectively. Our contributions have a significant impact on performance. Method | MAE
---|---
Graphormer [10] | 0.0864
baseline(reproduce+APE) | 0.0845
baseline+3D | 0.040
(L)baseline+all | 0.0853
(L)baseline+$\mathcal{L}_{m}$ | 0.0842
(L)baseline+$\overline{\mathcal{L}_{a}}$ | 0.0874
(L)baseline $+$ $\mathcal{L}_{m}$ $+$ $\mathcal{W}_{coordinating}$ $*$($\overline{\mathcal{L}_{a}}$) | 0.0837
baseline+$\mathcal{L}_{m}$ | 0.0822
baseline+$\overline{\mathcal{L}_{a}}$ | 0.087
baseline$+$ $\mathcal{L}_{m}$ $+$ $\mathcal{W}_{search}*$($\overline{\mathcal{L}_{a}}$) | 0.0818
baseline$+$ $\mathcal{L}_{m}$ $+$ $\mathcal{W}_{coordinating}*$($\overline{\mathcal{L}_{a}}$) | 0.0809
##### The validation of absolute position encoding (APE)
We train our graphormer with APE for 80 epochs as the baseline in Table III.
Compared to the original graphormer training with 300 epochs, our baseline
obtains a 1.52% improvement in the second row.
##### The validation of dividing local atom and global molecular
We observed that distilling the feature encoded with virtual token and atom
token in the fourth row in Table III generates a negative gain decreased by
0.94% compared with baseline. The phenomenon indicates binding virtual token
and atom tokens damages the performance of the distillation.
##### The validation of global molecular
Compared to the baseline model, the global molecular distillation
$\mathcal{L}_{m}$ slightly increases the performance by distilling the virtual
token embedding which is encoded with the structure information, reaching
0.36% improvement in the fifth row in Table III.
##### The validation of layers
In order to expand the effect of distilling molecular representation, we
conduct the distillation on all layers, which generates 2.72% improvement
compared with conducting on the last layer. This result demonstrates that
distilling on all layers superiors to distilling on the last layer.
##### The validation of local atoms
In addition, we employ local atom loss $\mathcal{L}_{a}$ on all layers. We
have observed one interesting phenomenon: when only leveraging the local atom
information distillation, the network decreased by 3.43% in sixth row. This
phenomenon demonstrates that distilling local atom information without
correction will cause degradation problems.
##### The validation of search weight
In the seventh row, the network further increases 3.2% compared with baseline
by distilling local atom and global molecular representation with coordinating
weight searched manually denoted as $\mathcal{W}_{search}$. The result
indicates that local atoms with coordinating weight can boost the performance
of global molecular. But the optimal coordinating weight is hard to search
manually due to the variable molecular size.
##### The validation of coordinating weight
Finally, when we incorporate the global molecular and local atoms with
coordinating weight $\mathcal{W}_{coordinating}$ to balance the grads
according to the number of atoms, our method in the eleventh row achieves the
best result with 0.809 MAE which achieves large improvement by 4.26% compared
with baseline. These results validate that our proposed coordination
distillation with global molecular and local atom information is effective for
molecular property prediction.
### IV-E Visionlization of effectiveness
To understand and analyze the availability of our approach, we show the
variation of attention weight in Fig. 4. We observed that the correlation
between points is more prominent and notable compared with the baseline model.
Compared with the baseline that all atoms only focus on the 2-hop nearest
neighbors, both the global molecular token and local atom tokens could focus
on almost all the atoms with CCMD. Thus, CCMD enriched the student model with
more structured information. This phenomenon promulgates the significance of
distillation. To reveal the significance of coordinating weight, we
certificate the validation of coordinating distilling strategy in Fig. 5,
which shows that coordinating weight is superior to any manually searched
weight. To analyze the behavior of the model after adding the distillation
loss term during training, we present the learning curves of 2D supervised
performance and distilling performance in Fig. 3. We can observe that the
convergence of MAE is more stable compared with only 2D supervised loss.
Figure 4: The visionlization of student model according to the attention
weight. We compare the attention of atom tokens in yellow dashed fields and
virtual token in the blue dashed boxes.
Figure 5: The comparison between coordinating weight and manually searched
weights. The purple points are the results with manually searched weights, and
the green point is the result with coordinating weight according to the number
of atoms.
### IV-F Effectiveness with Other Backbone
To evaluate the effectiveness of our approach, we change the gragh transformer
backbone of 2D branch into GIN [28] architecture (L = 3, d = 768) on the
PCQM4Mv2 dataset in Table IV. The 3D model based on gragh transformer is a
heavy model which has 12 blocks and 57.65 MB parameters, and the 2D model
based on GIN is a light GNN model which has 12 blocks and 1.923 MB parameters.
GIN and gragh transformer are very different in structure, making 2D-3D
distilling much more challenging, while our method in the fourth row still
increases MAE by 3.36% compared with baseline in the first row. What’s more,
the distillation with coordinating weight as $\frac{1}{N}$ in the fourth row
increases 0.67% compared with coordinating weight as $\frac{1}{{N}^{2}}$ in
the third row. This phenomenon indicates that coordinating weight with
$\frac{1}{N}$ is enough to scale the gradients for the GNN network. These
results in Table IV confirm the scalability and effectiveness of the
distilling strategy.
TABLE IV: The results with GIN backbone on the validation set of PCQM4Mv2 dataset. Method | MAE
---|---
baseline | 0.122
baseline+$\mathcal{L}_{m}$ | 0.1198
baseline+$\mathcal{L}_{m}$\+ $\frac{1}{N^{2}}*(\mathcal{L}_{a})$ | 0.1187
baseline+$\mathcal{L}_{m}$ \+ $\frac{1}{N}*\mathcal{L}_{a}$ | 0.1179
## V Conclusion
In this work, we exclusively propose a novel CCMD framework for molecular
property prediction. In particular, we integrate global molecular token
distillation and local atom information distillation with a novel coordinating
weight to balance and scale the gradient magnitudes. In addition, we establish
an absolute position encoding to integrate adjacent edge information.
Experimental results on two popular molecular property prediction tasks
demonstrate the superiority of our model. In the future, we will further
experiment on the datasets from other areas to demonstrate the generalization
ability of our method and explore more superior distilling strategy.
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## VI section 1
In this part, we will prove the relation of gradient on the virtual token and
atom number ${N}$.
For transformer, to simplify the proof, $X_{i}^{l+1}$ in $l$ layer is formed
as:
$X_{i}^{l+1}=MHA(X^{l})=\sum_{j=0}^{N+1}({W}^{l}_{ji}*{X}^{l}_{j})$ (17)
The ${W}^{l}_{ji}$ is the attention weight, and ${W}^{l}_{ji}$ is formulated
as:
${W}^{l}_{ji}=(\frac{e^{{X}^{l}_{i}*{X}^{l}_{j}}}{\sum_{j=0}^{N+1}(e^{{X}^{l}_{i}*{X}^{l}_{j})}})$
(18)
So,
$\displaystyle{grad(X_{0}^{l})}$ $\displaystyle=\nabla_{X_{0}^{l}}{L}$ (19)
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*(\nabla_{X_{0}^{l}}{X_{k}^{l+1}}))$
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*(\nabla_{X_{0}^{l}}(\sum_{j=0}^{N+1}{W}_{jk}*{X}^{l}_{j}))$
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*(\nabla_{X_{0}^{l}}((\sum_{j=1}^{N+1}{W}_{jk}*{X}^{l}_{j})+{W}_{0k}*{X}^{l}_{0}))$
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*((\sum_{j=1}^{N+1}\nabla_{X_{0}^{l}}({{W}_{jk})*{X}^{l}_{j}})+{W}_{0k})$
$\displaystyle\propto(\sum_{k=0}^{N+1}(\nabla_{X_{k}^{l+1}}{L})*(\sum_{j=1}^{N+1}\nabla_{X_{0}^{l}}({{W}_{jk})*{X}^{l}_{j}})+{O}({N})$
$\displaystyle\propto{O}(N)*((\sum_{j=1}^{N+1}\nabla_{{X}_{0}^{l}}{{W}_{jk}})+1)$
Now, we calculate ${\nabla_{{X}_{0}}{{W}_{jk}}}$ of the l-th layer, for
convenience, ${a}$ and ${b}$ denote constants independent of the variable N
and ${X}_{0}$:
$\displaystyle\nabla_{{X}_{0}}{{W}_{jk}}$
$\displaystyle=\nabla_{{X}_{0}}(\frac{e^{{X}_{j}*{X}_{k}}}{\sum_{j=0}^{N+1}e^{{X}_{j}*{X}_{k}}})$
(20)
$\displaystyle\propto(\nabla_{{X}_{0}}(\frac{a}{\sum_{j=0}^{N+1}e^{{X}_{j}*{X}_{k}}}))$
$\displaystyle\propto(\nabla_{{X}_{0}}(\frac{a}{\sum_{j=1}^{N+1}e^{{X}_{j}*{X}_{k}}+e^{{X}_{0}*{X}_{k}}}))$
$\displaystyle\propto(\nabla_{{X}_{0}}(\frac{a}{b+e^{{X}_{0}*{X}_{k}}}))$
$\displaystyle\propto({O}(1))$
Finally, the virtual token gradient $grad(X_{0}^{l})$ in $l$ layer can be
described in follow:
$\displaystyle{grad(X_{0}^{l})}$
$\displaystyle\propto{{O}({N})}*\sum_{j=1}^{N+1}({O}(1))$ (21)
$\displaystyle\propto{{O}({N}^{2})},$
For a simple GNN like GIN:
$\displaystyle{X}^{l+1}_{i}$ $\displaystyle={W}^{l+1}_{0i}*{X}^{l}_{0}$ (22)
So,
$\displaystyle{grad(X_{0}^{l})}$ $\displaystyle=\nabla_{X_{0}^{l}}{L}$ (23)
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*(\nabla_{X_{0}^{l}}{X_{k}^{l+1}}))$
$\displaystyle=\sum_{k=0}^{N+1}((\nabla_{X_{k}^{l+1}}{L})*({W}^{l+1}_{0k}))$
$\displaystyle\propto{O}(N)$
|
# Uniformizations of compact Sasakian manifolds
Hisashi Kasuya Department of Mathematics, Graduate School of Science, Osaka
University, Osaka, Japan<EMAIL_ADDRESS>and Natsuo Miyatake
Institute of Mathematics for Industry, Kyushu University 744 Motooka, Fukuoka
819-0395, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
We give a criterion for compact Sasakian manifolds to be deformed to Sasakian
manifolds which are locally isomorphic to circle bundles of anti-canonical
bundles over Hermitian symmetric spaces as a Sasakian analogue of Simpson’s
uniformization results related to variations of Hodge structure and Higgs
bundles.
###### Key words and phrases:
Sasakian manifold, Uniformization, Variation of Hodge structures, Higgs bundle
###### 2010 Mathematics Subject Classification:
14D07, 32Q30, 32L05, 53C25,53C07
This work was partially supported by JSPS KAKENHI Grant Number 19H01787.
## 1\. Introduction
We are interested in classifications of compact Sasakian manifolds. Sasakian
geometry is an odd-dimensional counterpart of Kähler geometry. However, in
contrast to classifications of compact Kähler manifolds up to biholomorphism,
the existence of a finite dimensional local moduli space of a compact Sasakian
manifold is not given by deformation theory. Sasakian structures are usually
defined by contact CR-structures $(T^{1,0}_{M},\eta)$ on odd-dimensional
manifolds $M$ satisfying the “normality” condition. It is known that a space
of locally non-isomorphic CR structures on a compact strongly pseudo-convex CR
manifold can have arbitrary many parameters (see [BSW]). For classifying
$3$-dimensional compact Sasakian manifolds, it is a great idea to classify
Sasakian structures up to deformations of certain types. In [Bel1, Bel2],
Belgun proves that any $3$-dimensional compact Sasakian manifold is a
deformation of a standard one; locally isomorphic to one of canonical
homogeneous (left-invariant) Sasakian structures on $S^{3}$,
$\widetilde{SL}_{2}({\mathbb{R}})$, $Nil_{3}$ (see also [Ge]).
Belgun’s result is an analogue of uniformizations of compact Riemannian
surfaces. It is natural to expect that we can prove a Sasakian version of
uniformization results on higher dimensional compact Kähler manifolds in terms
of Belgun’s arguments.
The most important part of uniformizations of compact Riemannian surfaces is
to prove that every Riemannian surface of genus $\geq 2$ admits a metric of
constant negative curvature. In [Hi], Hitchin reproves this statement as a
corollary of the Kobayashi-Hitchin correspondence of Higgs bundles. In [Si1],
Simpson gives a uniformization theorem of higher dimensional compact Kähler
manifolds by generalizing Hitchin’s idea. Precisely, Simpson shows that the
existence of a uniformizing variation of Hodge structure is a criterion for
uniformizing a compact Kähler manifold by a Hermitian symmetric space. By
using this, he shows that the stability of a certain Higgs bundle is a
criterion for uniformizing a compact Kähler manifold by the unit ball.
The main result of this paper is to give a Sasakian version of Simpson’s
uniformization theorem. For a Hermitian symmetric space $D$ of non-compact
type, we can define the homogeneous Sasakian structure
$(T_{D}^{1,0},\eta_{D})$ on the circle bundle
$S^{1}(\bigwedge^{m}T^{1,0}_{D})$ of the determinant line bundle of the
holomorphic tangent bundle of $D$. We give a criterion for uniformizing
compact Sasakian manifolds by the homogeneous Sasakian structure
$(T_{D}^{1,0},\eta_{D})$.
###### Theorem 1.1.
Let $(M,T^{1,0}_{M},\eta)$ be a compact Sasakian manifold. If
$(M,T^{1,0}_{M},\eta)$ admits a uniformizing $G$-variations of Hodge structure
$(G,\rho,h)$ and $c_{1,B}(T_{M})=-C[d\eta]$ for a positive constant $C$, then
the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$ where
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ is the universal covering of
$S^{1}(\bigwedge^{m}T^{1,0}_{D})$.
$c_{1,B}(T_{M})$ is the first basic Chern class of $(M,T^{1,0}_{M},\eta)$
which is also very important for studying Sasaki-Einstein geometry (see [BG]).
Note that the condition $c_{1,B}(T_{M})=C[d\eta]$ for a positive constant $C$
is a criterion for the existence of a Sasaki-Einstein metric on a compact
Sasakian manifold in contrast to our result. An $A^{1}_{B}$-deformation is a
slight modification of a second type deformation in the sense of Belgun [Bel2]
(see also a deformation of type II in [BG, Definition 7.5.9]). An almost
isomorphism is a diffeomorphism commuting with almost contact structures
associated with Sasakian manifolds equivalently Sasakian isomorphism up to
rescalings.
Combining this criterion and the Sasakian version of Simpson’s correspondence
between flat bundles and Higgs bundles ([Si1, Si]) proved in [BK, BK2], we
obtain a criterion for uniformizing compact Sasakian manifolds in terms of
stablities of Higgs bundles. For a Sasakian manifold $(M,T^{1,0}_{M},\eta)$,
we have the canonical transverse holomorphic foliation ${\mathcal{F}}_{\xi}$
and hence we can define holomorphic structures and Higgs bundle structures on
“basic” vector bundles over $(M,{\mathcal{F}}_{\xi})$. We have a canonical
Basic Higgs bundle structure $(E_{M},\,\theta_{M})$ on the vector bundle
$E_{M}=T^{1,0}_{M}\oplus{\mathbb{C}}_{M}$ where $T^{1,0}_{M}$ is the
holomorphic tangent bundle associated with a CR structure and
${\mathbb{C}}_{M}$ is the trivial line bundle.
###### Theorem 1.2.
We assume:
1. (1)
The basic Higgs bundle $(E_{M},\,\theta_{M})$ is stable.
2. (2)
The equality
$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0$
holds.
3. (3)
$c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for some positive constant $C$.
Then the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to the
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$ for
$D=SU(n,1)/S(U(n)\times U(1))$.
This result is based on the existence of Hermitian-Einstein metrics on stable
basic Higgs bundles over compact Sasakian manifolds. In [Zh], by using the
existence of Sasaki-Einstein metrics, the result of uniformization by the odd-
dimensional sphere is proved.
As a corollary of Theorem 1.2, we obtain an analogue of the existence of a
metric of constant negative curvature on a Riemannian surface of genus $\geq
2$.
###### Corollary 1.3.
Let $(M,T^{1,0}_{M},\eta)$ be a compact $3$-dimensional Sasakian manifold with
$c_{1,B}(T^{1,0})=-C[d\eta]$. Then the universal covering $\widetilde{M}$ with
the lifting of some $A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost
isomorphic to $\widetilde{PU}(1,1)$ with a left-invariant Sasakian structure.
We have $\widetilde{PU}(1,1)=\widetilde{SL}_{2}({\mathbb{R}})$. This result is
a part of Belgun’s classification in [Bel1, Bel2]. Belgun uses the non-Kähler
complex structure on $M\times S^{1}$ for a compact Sasakian manifold $M$. On
the other hand, we do not use such a complex structure.
Acknowledgement. We are grateful to Professor Takuro Mochizuki for his
valuable comments on the earlier works [BK, BK2] which led to the beginning of
this project.
## 2\. Sasakian Geometry
### 2.1. Sasakian structures on CR manifolds
Let $M$ be a $(2n+1)$-dimensional real smooth manifold. A CR-structure on $M$
is an $n$-dimensional complex involutive sub-bundle $T^{1,0}_{M}$ of the
complexified tangent bundle
$TM_{{\mathbb{C}}}\,=\,TM\otimes_{\mathbb{R}}{{\mathbb{C}}}$ such that
$T^{1,0}_{M}\cap T^{0,1}_{M}=\\{0\\}$ where
$T^{0,1}_{M}=\overline{T^{1,0}_{M}}$. Define the real $2n$-dimensional sub-
bundle $S=TM\cap(T^{1,0}_{M}\oplus T^{0,1}_{M})\subset TM$. We have the almost
complex structure $I:S\to S$ associated with the decomposition
$S_{{\mathbb{C}}}=T^{1,0}_{M}\oplus T^{0,1}_{M}$.
A strongly pseudo-convex CR structure on $M$ is a pair $(T^{1,0}_{M},\eta)$ of
a CR structure $T^{1,0}_{M}$ and a contact $1$-form $\eta$ such that
$\ker\eta=S$ and the bilinear form defined by $L_{\eta}(X,Y)=d\eta(X,IY)$ on
$S$ is a Hermitian metric on $(S,I)$. Take the Reeb vector field $\xi$
associated with the contact $1$-form $\eta$. $(T^{1,0}_{M},\eta)$ is Sasakian
if for any smooth section $X$ of $T^{1,0}_{M}$ the Lie bracket $[\xi,X]$ is
also a smooth section of $T^{1,0}_{M}$. We consider the $1$-dimensional
foliation ${\mathcal{F}}_{\xi}$ on $M$ generated by $\xi$. If
$(T^{1,0}_{M},\eta)$ is Sasakian, then $T^{1,0}_{M}$ defines a transverse
holomorphic structure on ${\mathcal{F}}_{\xi}$ and $d\eta$ is a transverse
Kähler form on ${\mathcal{F}}_{\xi}$. We say that a Sasakian structure is
quasi-regular if every leaf of ${\mathcal{F}}_{\xi}$ is closed. If $M$ is
compact and a Sasakian structure on $M$ is quasi-regular, then the flow of
$\xi$ induces an action $S^{1}\times M\to M$ (see [Wa]).
Let $(M,T^{1,0}_{M},\eta)$ be a Sasakian manifold. A differential form
$\omega$ on $M$ is called basic if the equations
$i_{\xi}\omega=0=i_{\xi}d\omega$
hold. We denote by $A^{\ast}_{B}(M)$ the subspace of basic forms in the de
Rham complex $A^{\ast}(M)$. Then $A^{\ast}_{B}(M)$ is a sub-complex of the de
Rham complex $A^{\ast}(M)$. Denote by $H_{B}^{\ast}(M)$ the cohomology of the
basic de Rham complex $A^{\ast}_{B}(M)$. We note that $d\eta\in A^{2}_{B}(M)$
and $[d\eta]\not=0\in H_{B}^{2}(M)$ if $M$ is compact. We have the bigrading
$A^{r}_{B}(M)_{{\mathbb{C}}}=\bigoplus_{p+q=r}A^{p,q}_{B}(M)$ as well as the
decomposition of the exterior differential
$d_{|A^{r}_{B}(M)_{{\mathbb{C}}}}=\partial_{B}+\overline{\partial}_{B}$ on
$A^{r}_{B}(M)_{{\mathbb{C}}}$, so that
$\partial_{B}:A^{p,q}_{B}(M)\to A^{p+1,q}_{B}(M)\ \text{ and }\
\overline{\partial}_{B}:A^{p,q}_{B}(M)\to A^{p,q+1}_{B}(M)\,.$
We note that $d\eta\in A^{1,1}_{B}(M)$. For a strongly pseudo-convex CR-
manifold $(M,T^{1,0}_{M},\eta)$ there exists a unique affine connection
$\nabla^{TW}$ on $TM$ such that the following conditions hold ([Ta, We]):
1. (1)
$S$ is parallel with respect to $\nabla^{TW}$.
2. (2)
$\nabla^{TW}I\,=\nabla^{TW}d\eta=\nabla^{TW}\eta\,=\nabla^{TW}\xi=\,0$.
3. (3)
The torsion $T^{TW}$ of the affine connection $\nabla^{TW}$ satisfies the
equation
$T^{TW}(X,\,Y)\,=\,-d\eta(X,\,Y)\xi$
for all $X,\,Y\,\in\,S_{x}$ and $x\,\in\,M$.
This affine connection $\nabla^{TW}$ is called the Tanaka–Webster connection.
It is known that $(T^{1,0},\eta)$ is a Sasakian manifold if and only if
$T^{TW}(\xi,\,v)\,=\,0$ for all $v\,\in\,TM$.
Define the homomorphism $\Phi_{\xi}:TM\to TM$ by the extension of $I:S\to S$
satisfying $\Phi_{\xi}(\xi)=0$. Then, $\Phi^{2}_{\xi}=-{\rm
Id}+\xi\otimes\eta$. We call $\Phi_{\xi}$ the almost contact structure
associated with a Sasakian structure $(T^{1,0}_{M},\eta)$. We define the
Riemannian metric $g_{\eta}(X,Y)=\eta(X)\eta(Y)+d\eta(X,\Phi_{\xi}Y)$. We call
$g_{\eta}$ the Sasakian metric associated with a Sasakian structure
$(T^{1,0}_{M},\eta)$.
Let $(N,J)$ be a complex manifold. A smooth map $f:M\to N$ is a holomorphic if
$df\circ\Phi_{\xi}=J\circ df$. A holomorphic map $f:M\to N$ satisfies
$df(\xi)=0$. We note that the holomorphic condition of a smooth map $f:M\to N$
depends only on the transverse holomorphic foliation ${\mathcal{F}}_{\xi}$. A
smooth function $f:M\to{\mathbb{C}}$ is holomorphic if and only if $f\in
A_{B}^{0}(M)$ and $\bar{\partial}_{B}f=0$.
### 2.2. Morphisms, equivalences and deformations
Let $(M_{1},T^{1,0}_{M_{1}},\eta_{1})$ and $(M_{2},T^{1,0}_{M_{2}},\eta_{2})$
be two Sasakian manifolds. A smooth map $f:M_{1}\to M_{2}$ is a CR map if
$df(T^{1,0}_{M_{1}})\subset T^{1,0}_{M_{2}}$. a CR map $f:M_{1}\to M_{2}$
satisfies $f^{\ast}\eta_{2}=g(x)\eta_{1}$ for some function $g(x)$ on $M_{1}$.
By the standard arguments (see [BaK, Lemma 2.3]), we have:
###### Lemma 2.1.
Let $f:M_{1}\to M_{2}$ be a diffeomorphism. Then the following two conditions
are equivalent:
* •
$f$ is a CR map and $f^{\ast}\eta_{2}=\eta_{1}$.
* •
$f$ is isometry between $(M_{1},g_{\eta_{1}})$ and $(M_{2},g_{\eta_{2}})$ and
$df(\xi_{1})=\xi_{2}$.
A diffeomorphism satisfying the equivalent condition in this lemma is a
Sasakian isomorphism.
A smooth map $f:M_{1}\to M_{2}$ is a almost contact map if
$df\circ\Phi_{\xi_{1}}=\Phi_{\xi_{2}}\circ df$. A almost contact map is a CR-
map but the converse is not true. We can easily check that an almost contact
map $f:M_{1}\to M_{2}$ satisfies $f^{\ast}\eta_{2}=c\eta_{1}$ and
$df(\xi_{1x})=c\xi_{2f(x)}$ for some constant $c$. It is sufficient to check
the above function $g(x)$ is constant. By the basicness
${\mathcal{L}}_{\xi_{1}}d\eta_{1}={\mathcal{L}}_{\xi_{2}}d\eta_{2}=0$ and
$df(\xi_{1x})=c(x)\xi_{2f(x)}$, we have
${\mathcal{L}}_{\xi_{1}}(g)\eta_{1}={\mathcal{L}}_{\xi_{1}}(f^{\ast}\eta_{2})=(i_{\xi_{1}}d+di_{\xi_{1}})(f^{\ast}\eta_{2})=di_{\xi_{1}}f^{\ast}\eta_{2}=dg$
and so $dg=0$ on the sub-bundle $S={\rm ker}\eta_{1}\subset TM$. It is
sufficient to prove $dg(\xi_{1x})=0$ for any $x\in M$. Since $d\eta$ is non-
degenerate on $S$, we can take local sections $X,Y$ of $S$ such that
$d\eta_{1}(X_{x},Y_{x})\not=0$. By the property of the Tanaka-Webster
connection $\nabla^{TW}$, we have
$\nabla^{TW}_{X}Y-\nabla^{TW}_{Y}X-[X,Y]=-d\eta(X,Y)\xi.$
Since $\nabla^{TW}_{X}Y-\nabla^{TW}_{Y}X$ is a local section of $S$,
$-d\eta_{1}(X,Y)dg(\xi_{1})=dg(\nabla^{TW}_{X}Y-\nabla^{TW}_{Y}X)-X(Y(g))+Y(X(g))=0.$
This implies $dg(\xi_{1x})=0$.
Let $(M,T^{1,0}_{M},\eta)$ be a Sasakian manifold. A rescaling of
$(T^{1,0}_{M},\eta)$ is a Sasakian structure $(T^{1,0}_{M\tau},R\eta)$ for a
real number $R>0$. The Reeb vector field of $R\eta$ is $\frac{1}{R}\xi$. A
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is a Sasakian structure
$(T^{1,0}_{M\tau},\eta^{\tau})$ such that $\eta^{\tau}=\eta+\tau$ for $\tau\in
A^{1}_{B}$ and $T^{1,0}_{M\tau}=\\{X-\tau(X)\xi|X\in T^{1,0}_{M}\\}$. The Reeb
vector field of $\eta^{\tau}$ is $\xi$ and the transverse holomorphic
structure on ${\mathcal{F}}_{\xi}$ induced by $T^{1,0}_{M\tau}$ is same as the
one induced by $T^{1,0}_{M}$.
By the above argument on almost contact maps and the standard arguments (see
[BG, Proposition 8.1.1]), we have:
###### Lemma 2.2.
Let $f:M_{1}\to M_{2}$ be a diffeomorphism. Then the following two conditions
are equivalent:
* •
$f$ is a Sasakian isomorphism up to rescalings.
* •
$f$ is an almost contact map.
A diffeomorphism satisfying the equivalent condition in this lemma is an
almost isomorphism. Obviously, an almost isomorphism is a CR-diffeomorphism.
## 3\. Basic vector bundles over Sasakian manifolds
### 3.1. Basic vector bundles
We define structures on smooth vector bundles over Sasakian manifolds
$(M,T_{M}^{1,0},\eta)$ related to ${\mathcal{F}}_{\xi}$ in terms of Rawnsley’s
partial flat connections in [Ra].
A structure of basic vector bundle on a $C^{\infty}$ vector bundle $E$ is a
linear operator
$\nabla_{\xi}\,:\,{\mathcal{C}}^{\infty}(E)\,\longrightarrow\,{\mathcal{C}}^{\infty}(E)$
such that
$\nabla_{\xi}(fs)\,=\,f\nabla_{\xi}s+\xi(f)s.$
Consider the flow ${\mathbb{R}}\times M\to M$ generated by $\xi$. Then if we
have a lifting action ${\mathbb{R}}\times E\ni(t,v)\mapsto\phi_{t}(v)\in E$ of
the flow ${\mathbb{R}}\times M\to M$ on a smooth vector bundle $E$, then we
define the basic bundle structure on $E$ by
$\nabla_{\xi}(s)_{x}=\frac{d}{dt}_{|t=0}\phi_{t}(v)_{x}.$
A differential form $\omega\,\in\,A^{\ast}(M,\,E)$ with values in $E$ is
called basic if the equations
$i_{\xi}\omega=0=\nabla_{\xi}\omega$
hold where we extend $\nabla_{\xi}:A^{\ast}(M,\,E)\to A^{\ast}(M,\,E)$. Let
$A^{\ast}_{B}(M,\,E)\,\subset\,A^{\ast}(M,\,E)$ denote the subspace of basic
forms in the space $A^{\ast}(M,\,E)$ of differential forms with values in $E$.
A Hermitian metric on $E$ is called basic if it is $\nabla_{\xi}$-invariant.
Unlike the usual Hermitian metric, a basic vector bundle may not admits a
basic Hermitian metric in general. For a connection operator
$\nabla:A^{\ast}(M,\,E)\to A^{\ast+1}(M,\,E)$ such that the covariant
derivative of $\nabla$ along $\xi$ is $\nabla_{\xi}$, the curvature
$R^{\nabla}\in A^{2}(M,\,{\rm End}(E))$ is basic i.e. $R^{\nabla}\in
A^{2}_{B}(M,\,{\rm End}(E))$. For the Chern forms $c_{i}(E,\nabla)\in
A^{2i}(M)$ associated with $\nabla$, we have $c_{i}(E,\nabla)\in
A_{B}^{2i}(M)$ and we define the basic Chern classes $c_{i,B}(E)\in
H^{2i}_{B}(M)$ by the cohomology classes of $c_{i}(E,\nabla)$.
A structure of basic holomorphic bundle on a $C^{\infty}$ vector bundle $E$ is
a linear differential operator
$\nabla^{\prime\prime}\,:\,{\mathcal{C}}^{\infty}(E)\,\longrightarrow\,{\mathcal{C}}^{\infty}(E\otimes(\langle\xi\rangle\oplus
T^{0,1})^{\ast})$ such that
* •
for any $X\in\langle\xi\rangle\oplus T^{0,1}$, and any smooth function $f$ on
$M$, the equation
$\nabla^{\prime\prime}_{X}(fs)\,=\,f\nabla^{\prime\prime}_{X}s+X(f)s$
holds for all smooth sections $s$ of $E$, and
* •
if we extend $\nabla^{\prime\prime}$ to
$\nabla^{\prime\prime}\,:\,{\mathcal{C}}^{\infty}(E\otimes\bigwedge^{k}(\langle\xi\rangle\oplus
T^{0,1})^{\ast})\,\longrightarrow\,{\mathcal{C}}^{\infty}(E\otimes\bigwedge^{k+1}(\langle\xi\rangle\oplus
T^{0,1})^{\ast})$, then
$\nabla^{\prime\prime}\circ\nabla^{\prime\prime}\,=\,0$.
A basic holomorphic vector bundle $E$ has a canonical basic bundle structure
corresponding to the derivative $\nabla^{\prime\prime}_{\xi}$.
$\nabla^{\prime\prime}$ defines the linear operator
$\bar{\partial}_{E}:A^{p,q}_{B}(M,\,E)\to A^{p,q+1}_{B}(M,\,E)$ so that
$\bar{\partial}_{E}(f\omega)=\bar{\partial}_{B}f\wedge\omega+f\bar{\partial}_{E}(\omega)$
for $f\in A^{0}_{B}(M),\omega\in A^{p,q}_{B}(M,\,E)$. If a basic holomorphic
vector bundle $E$ admits a basic Hermitian metric $h$, as complex case, we
have a unique unitary connection $\nabla^{h}$ such that for any
$X\in\langle\xi\rangle\oplus T^{0,1}$,
$\nabla^{h}_{X}=\nabla^{\prime\prime}_{X}$.
###### Example 3.1.
Consider the ${\mathcal{C}}^{\infty}$-trivial complex line bundle
$E\,=\,M\times{\mathbb{C}}\,\longrightarrow\,M$. For any $C\in{\mathbb{R}}$,
we define the connection $\nabla^{C}\,=\,d-2\pi\sqrt{-1}C\eta$ on $E$. Then,
the curvature of $\nabla^{C}$ is $Cd\eta$. Since $d\eta\,\in\,A^{1,1}_{B}(M)$,
this $\nabla^{C}$ induces a structure of a basic holomorphic bundle.
Consequently, we have a non-trivial holomorphic vector bundle structure
$E_{C}$ on $E$ that depends on $C\,\in\,{\mathbb{C}}$. The basic cohomology
class of $Cd\eta$ is the basic first Chern class of the basic vector bundle
$E_{C}$. Thus $\\{E_{C}\\}_{C\in{\mathbb{R}}}$ is a family of basic vector
bundles such that $E_{C}\,\not\cong\,E_{C^{\prime}}$ for every
$C\,\not=\,C^{\prime}$. The standard constant Hermitian metric $h$ on the
${\mathcal{C}}^{\infty}$ trivial line bundle $E\,=\,M\times{\mathbb{C}}$ is
basic on $E_{C}$. The connection $\nabla^{C}$ is unitary for $h$, and hence
$\nabla^{C}$ is the canonical connection for the basic Hermitian metric $h$.
Consider the flow ${\mathbb{R}}\times M\to M$ generated by $\xi$. Define the
lifting action ${\mathbb{R}}\times
M\times{\mathbb{C}}\ni(t,x,v)\mapsto(x_{t},e^{-2\pi C\sqrt{-1}t}v)\in E$ of
the flow ${\mathbb{R}}\times M\to M$. Then this induces the basic bundle
structure $\nabla^{C}_{\xi}$ of $E_{C}$.
###### Example 3.2.
The Tanaka-Webster connection $\nabla^{TW}$ defines a structure of a basic
holomorphic bundle on $T^{1,0}$. Since we have $\nabla^{TW}_{\xi}(X)=[\xi,X]$,
the basic bundle structure $\nabla^{TW}_{\xi}$ is given by the action
${\mathbb{R}}\times T^{1,0}X\to T^{1,0}X$ which is the differential of the
action ${\mathbb{R}}\times M\to M$ generated by the flow of $\xi$.
### 3.2. Equivariant cohomology
Assume that $M$ is compact and a Sasakian structure on $M$ is quasi-regular.
Consider the equivariant cohomology
$H^{\ast}_{S^{1}}(M,R)=H^{\ast}(M\times_{S^{1}}ES^{1},R)$ for the action
$S^{1}\times M\to M$ with coefficients in a commutative ring $R$.
$H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ is the cohomology of the Cartan model
$A^{\ast}(M)^{S^{1}}\otimes{\mathbb{R}}[u]$ with the differential $d_{S^{1}}$
so that $d_{S^{1}}u=0$, $d_{S^{1}}\omega=d\omega-i_{\xi}\omega u$ where $u$ is
of degree $2$. The natural inclusion $A^{\ast}_{B}(M)\to
A^{\ast}(M)^{S^{1}}\otimes{\mathbb{R}}[u]$ is a cochain complex homomorphism.
Since the action $S^{1}\times M\to M$ is locally free, the induced map
$H_{B}^{\ast}(M)\to H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ is an isomorphism ([GS,
Section 5]).
We use the result in [MuR] which says that equivariant first Chern classes in
$H^{\ast}_{S^{1}}(M,{\mathbb{Z}})$ classify lifts of the action $S^{1}\times
M\to M$ on line bundles. For our case, basic bundle structures $\nabla_{\xi}$
on line bundles can be considered as infinitesimal lifts of the action
$S^{1}\times M\to M$. Let $L_{1}\to M,L_{2}\to M$ be line bundles admitting
lifts of the action $S^{1}\times M\to M$ and consider them as basic line
bundles. Assume that we have basic Hermitian metrics $h_{1},h_{2}$ and basic
unitary connections $\nabla_{1},\nabla_{2}$ on $L_{1}$ and $L_{2}$
respectively. We note that the equivariant first Chern classes in
$H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ are images of the basic first Chern classes
via the canonical map $H_{B}^{\ast}(M)\to H^{\ast}_{S^{1}}(M,{\mathbb{R}})$.
Since $H_{B}^{\ast}(M)\to H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ is an isomorphism,
we can say that if $c_{1,B}(L_{1})=c_{1,B}(L_{2})$, then $L_{1}^{l}$ and
$L_{2}^{l}$ are equivariantly isomorphic for some positive integer $l$.
###### Proposition 3.3.
If $c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for some positive constant $C$, then for
some positive integer $l$, the $l$-th power of the anti-canonical bundle
$\bigwedge^{n}T^{1,0}_{M}$ with the $S^{1}$-action given by the differential
of the action $S^{1}\times M\to M$ is equivariantly isomorphic to $E_{-lC}$.
###### Remark 3.4.
If $c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for a positive constant $C$, then the
Sasakian structure $(T^{1,0}_{M},\eta)$ is quasi-regular ([BG, Theorem
8.1.14]).
## 4\. Variations of Hodge structure and uniformizations
### 4.1. Variations of Hodge structure over Sasakian manifolds
Let $G$ be a connected semi-simple Lie group. Assume that $G$ has the finite
center and no compact factor. A Hodge structure on $G$ is a real Hodge
structure
$\mathfrak{g}_{{\mathbb{C}}}=\bigoplus_{p\in{\mathbb{Z}}}\mathfrak{g}^{p,-p}$
of weight $0$ on the Lie algebra $\mathfrak{g}$ of $G$ such that the Lie
bracket on $\mathfrak{g}$ is a morphism of real Hodge structures and
$-B_{\mathfrak{g}}$ is a polarization where $B_{\mathfrak{g}}$ is the Killing
form on $\mathfrak{g}$. Define the subalgebra
$\mathfrak{v}=\mathfrak{g}^{0,0}\cup\mathfrak{g}\subset\mathfrak{g}$ and the
subgroup $V=\exp(\mathfrak{v})\subset G$. Then $V$ is compact. The subalgebra
$\bigoplus_{p>0}\mathfrak{g}^{p.-p}\subset\mathfrak{g}_{{\mathbb{C}}}$ defines
a complex structure on $D=G/V$. The complex manifold $D$ is called the period
domain. Define the subalgebra $\mathfrak{p}=\bigoplus_{p\geq
0}\mathfrak{g}^{p.-p}$ and the subgroup $P=\exp(\mathfrak{p})\subset
G_{{\mathbb{C}}}$ in the complexification $G_{{\mathbb{C}}}$ of $G$. Then
$\mathfrak{p}$ is a parabolic subalgebra in $\mathfrak{g}_{{\mathbb{C}}}$ and
$G/V$ is a dual manifold of $G/P$ in the sense of [GrSc]. The holomorphic
tangent bundle $T^{1,0}_{D}$ is a homogeneous vector bundle given by
$(G\times\bigoplus_{p>0}\mathfrak{g}^{p.-p})/V$ and hence we have the
decomposition $T^{1,0}_{D}=\bigoplus_{p>0}E^{p,-p}$. Consider the connection
form $\lambda$ on the anti-canonical bundle $\bigwedge^{m}T^{1,0}_{D}$ given
by the Maurer-Cartan form where $m=\dim_{{\mathbb{C}}}D$. Let
$\eta_{D}=-\lambda$. Then $d\eta$ is a non-degenerate $2$-form on $D$ which is
positive on $\bigoplus_{p\,odd}E^{p,-p}$ and negative on
$\bigoplus_{p\,even}E^{p,-p}$ see [GrSc, (4.23)]. Let $(M,T^{1,0}_{M},\eta)$
be a Sasakian manifold. Denote by $\widetilde{M}$ the universal cover of $M$
at $x\in M$ and take the lifting Sasakian structure
$(T^{1,0}_{\tilde{M}},\tilde{\eta})$. A $G$-variations of Hodge structure is
$(G,\rho,h)$ such that:
1. (1)
$G$ is a connected semi-simple Lie group equipped with a Hodge structure.
2. (2)
$\rho:\pi_{1}(M,x)\to G$ is a representation.
3. (3)
$h:\widetilde{M}\to D$ is a $\rho$-equivariant holomorphic map satisfying
$dh(T^{1,0}_{\widetilde{M}})\subset E^{1,-1}$.
### 4.2. Uniformizing Variations of Hodge structure
A Hodge structure on $G$ is Hermitian if
$\mathfrak{g}_{{\mathbb{C}}}=\mathfrak{g}^{1,-1}\oplus\mathfrak{g}^{0,0}\oplus\mathfrak{g}^{-1,1}$.
In this case, $D$ is a Hermitian symmetric manifold of non-compact type and
$d\eta_{D}$ is an invariant Kähler form on $D$. Regarding $\eta$ as a contact
$1$-form, $(T^{1,0}_{D},\eta_{D})$ is a $G$-invariant Sasakian structure on
the circle bundle $S^{1}(\bigwedge^{m}T^{1,0}_{D})$. We have
$S^{1}(\bigwedge^{m}T^{1,0}_{D})=(G\times S^{1})/V=G/V^{\prime}$ where
$V^{\prime}$ is the kernel of the non-trivial representation $V\to
GL(\bigwedge^{m}\mathfrak{g}^{1,-1})$. $S^{1}(\bigwedge^{m}T^{1,0}_{D})$ is
also $G$-homogeneous and the Sasakian structure $(T^{1,0}_{D},\eta_{D})$ is
homogeneous We can take the Reeb vector field $\xi_{D}$ as a fundamental
vector field of the action $S^{1}\times(G\times
S^{1})/V\ni(a,[g,t])\mapsto[g,a^{-1}t]\in(G\times S^{1})/V$. Since $D$ is
contractible, by trivializing $S^{1}(\bigwedge^{m}T^{1,0}_{D})\cong
S^{1}\times D$, we have the universal covering
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})\to
S^{1}(\bigwedge^{m}T^{1,0}_{D})$ with the infinite cyclic covering group.
Let $\Gamma\subset G$ be a discrete subgroup which acts freely and cocompactly
on $S^{1}(\bigwedge^{m}T^{1,0}_{D})=G/V^{\prime}$. We consider the compact
Sasakian manifold $(\Gamma\backslash
S^{1}(\bigwedge^{m}T^{1,0}_{D}),T^{1,0}_{D}\eta_{D})$. We observe that:
* •
By the construction, we have $c_{1,B}(M)=-[d\eta_{D}]$.
* •
Defining $h:\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})\to D$ by the
composition of the covering $\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})\to
S^{1}(\bigwedge^{m}T^{1,0}_{D})$ and the projection
$S^{1}(\bigwedge^{m}T^{1,0}_{D})\to D$ and $\rho:\pi_{1}(\Gamma\backslash
S^{1}(\bigwedge^{m}T^{1,0}_{D}))\to G$ by the composition the quotient
$\pi_{1}(\Gamma\backslash S^{1}(\bigwedge^{m}T^{1,0}_{D}))\to\Gamma$ by the
covering group and the inclusion $\Gamma\to G$, we have the $G$-variation of
Hodge structure $(G,\rho,h)$.
The main purpose of this section is to prove that these properties uniformize
compact Sasakian manifolds by the homogeneous Sasakian structure
Let $(M,T^{1,0}_{M},\eta)$ be a Sasakian manifold. A $G$-variation of Hodge
structure $(G,\rho,h)$ is uniformizing if a Hodge structure on $G$ is
Hermitian and the differential $dh:T^{1,0}_{\widetilde{M}}\to T^{1,0}_{D}$ is
an isomorphism at every fiber.
###### Theorem 4.1.
Let $(M,T^{1,0}_{M},\eta)$ be a compact Sasakian manifold. If
$(M,T^{1,0}_{M},\eta)$ admits a uniformizing $G$-variations of Hodge structure
$(G,\rho,h)$ and $c_{1,B}(T_{M}^{1,0})=-C[d\eta]$ for a positive constant $C$,
then the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$.
###### Proof.
By Proposition 3.3 and Remark 3.4, for some positive integer $l$, the $l$-th
power of $\bigwedge^{n}T^{1,0}_{M}$ is equivariantly isomorphic to $E_{-lC}$.
Consider the basic holomorphic line bundle $L$ over $M$ defined by the pull-
back $h^{\ast}\bigwedge^{m}T^{1,0}_{D}$. Then, Since $f$ is $\rho$-equivariant
holomorphic and $dh:T^{1,0}_{\tilde{M}}\to T^{1,0}_{D}$ is an isomorphism at
every fiber, $L$ is identified with $\bigwedge^{m}T^{1,0}_{M}$ via the
differential $dh$. We regard the $\rho$-equivariant map $h:\widetilde{M}\to D$
as a $V$-reduction $P$ of the flat bundle $(\widetilde{M}\times
G)/\pi_{1}(M,x)$. Then $L$ is induced by $P$ associated with the
representation $V\to GL(\bigwedge\mathfrak{g}^{1,-1})$. Take an equivariant
isomorphism $L^{l}\cong E_{-lC}$. By $E_{-lC}=M\times{\mathbb{C}}$,
corresponding to $M\ni x\mapsto(x,1)\in M\times{\mathbb{C}}$, we have a
nowhere vanishing section of $L^{l}$ and it induces an $\rho$-equivariant map
$f:\widetilde{M}\to(G\times S^{1})/V=(G\times S^{1})/V=G/V^{\prime\prime}$
where $V^{\prime\prime}$ is the kernel of the representation $\Lambda^{l}$.
$f$ is a lift of $h:\widetilde{M}\to D$. $G/V^{\prime}$ is a $l$-sheeted
covering of $G/V^{\prime\prime}$. $(T^{1,0}_{D},\eta_{D})$ defines a Sasakian
structure on $G/V^{\prime\prime}$. By the equivariance of $L^{l}\cong
E_{-lC}$, we have $df(\tilde{\xi})=lC\xi_{D}$. Since $M$ is compact and $f$ is
$\rho$-equivariant, the Riemann metric $f^{\ast}g_{\eta_{D}}$ is complete on
$\widetilde{M}$. $f:\widetilde{M}\to G/V^{\prime\prime}$ is a local isometry
and hence covering map.
Let $\tau=-\tilde{\eta}+\frac{1}{C}f^{\ast}\eta_{D}$. Then, by
$\tau(\tilde{\xi})=0$ and $d\tau\in A^{2}_{B}(\widetilde{M})$, we have
$\tau\in A^{1}_{B}(\widetilde{M})$. The $A^{1}_{B}$-deformation
$(T^{1,0}_{\widetilde{M}\tau},\tilde{\eta}+\tau)$ of
$(T^{1,0}_{\widetilde{M}},\tilde{\eta})$ is almost isomorphic to
$(\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D}),T^{1,0}_{D},\eta_{D})$. Since
$f$ is $\rho$-equivariant, $(T^{1,0}_{\widetilde{M}\tau},\tilde{\eta}+\tau)$
is defined on $M$. ∎
###### Remark 4.2.
In this proof, $f:\widetilde{M}\to G/V^{\prime}$ satisfies
$df(T^{1,0}_{\widetilde{M}})=T^{1,0}_{D}$ modulo $\langle\xi\rangle$ but may
not be a CR-map. Hence, the $A_{B}^{1}$-deformation
$(T^{1,0}_{\widetilde{M}\tau},\tilde{\eta}+\tau)$ is essential.
## 5\. Higgs bundles and uniformizations
### 5.1. Higgs bundles over Sasakian manifolds
Let $(M,T^{1,0}_{M},\eta)$ be a compact Sasakian manifold. A basic Higgs
bundle over $M$ is a pair $(E,\,\theta)$ consisting of a basic holomorphic
vector bundle $E$ and $\theta\,\in\,A^{1,0}_{B}(M,\,{\rm End}(E))$ satisfying
the following two conditions:
$\overline{\partial}_{{\rm End}(E)}\theta\,=\,0\ \ \text{ and }\ \
\theta\wedge\theta\,=\,0\,.$
We define the degree of a basic holomorphic vector bundle $E$ by
${\rm deg}(E)\,:=\,\int_{M}c_{1,B}(E)\wedge(d\eta)^{n-1}\wedge\eta\,.$
Denote by ${\mathcal{O}}_{B}$ the sheaf of holomorphic functions on $M$, and
for a holomorphic vector bundle $E$ on $M$, denote by ${\mathcal{O}}_{B}(E)$
the sheaf of holomorphic sections of $E$. Consider ${\mathcal{O}}_{B}(E)$ as a
coherent ${\mathcal{O}}_{B}$-sheaf.
For a basic Higgs bundle $(E,\,\theta)$, a sub-Higgs sheaf of $(E,\,\theta)$
is a coherent ${\mathcal{O}}_{B}$-subsheaf $\mathcal{V}$ of
${\mathcal{O}}_{B}(E)$ such that
$\theta({\mathcal{V}})\,\subset\,{\mathcal{V}}\otimes\Omega_{B}$, where
$\Omega_{B}$ is the sheaf of basic holomorphic $1$-forms on $M$. By [BH,
Proposition 3.21], if ${\rm rk}(\mathcal{V})\,<\,{\rm rk}(E)$ and
${\mathcal{O}}_{B}(E)/\mathcal{V}$ is torsion-free, then there is a
transversely analytic sub-variety $S\,\subset\,M$ of complex co-dimension at
least 2 such that ${\mathcal{V}}\big{|}_{M\setminus S}$ is given by a basic
holomorphic sub-bundle $V\,\subset\,E\big{|}_{M\setminus S}$. The degree ${\rm
deg}(\mathcal{V})$ can be defined by integrating
$c_{1,B}(V)\wedge(d\eta)^{n-1}\wedge\eta$ on this complement $M\setminus S$.
###### Definition 5.1.
We say that a basic Higgs bundle $(E,\,\theta)$ is stable if $E$ admits a
basic Hermitian metric and for every sub-Higgs sheaf ${\mathcal{V}}$ of
$(E,\,\theta)$ such that ${\rm rk}(\mathcal{V})\,<\,{\rm rk}(E)$ and
${\mathcal{O}}_{B}(E)/\mathcal{V}$ is torsion-free, the inequality
$\frac{{\rm deg}(\mathcal{V})}{{\rm rk}(\mathcal{V})}\,<\,\frac{{\rm
deg}(E)}{{\rm rk}(E)}$
holds.
A basic Higgs bundle $(E,\,\theta)$ is called polystable if
$(E,\,\theta)\,=\,\bigoplus_{i=1}^{k}(E_{i},\,\theta_{i})\,,$
where each $(E_{i},\,\theta_{i})$ is a stable Higgs bundle with
$\frac{{\rm deg}(E_{i})}{{\rm rk}(E_{i})}\,=\,\frac{{\rm deg}(E)}{{\rm
rk}(E)}\,.$
Let $(E,\,\theta)$ be a basic Higgs bundle over a compact Sasakian manifold
$M$. Assume that $E$ admits a basic Hermitian metric $h$. Define
$\overline{\theta}_{h}\,\in\,A^{0,1}_{B}(M,\,{\rm End}(E))$ by
$h(\theta(e_{1}),\,e_{2})\,=\,h(e_{1},\,\overline{\theta}_{h}(e_{2}))$ for all
$e_{1},\,e_{2}\,\in\,E_{x}$ and all $x\,\in\,M$. Define the canonical
connection $D^{h}\,=\,\nabla^{h}+\theta+\overline{\theta}_{h}$ on $E$.
###### Theorem 5.2 ([BK, BK2]).
If a basic Higgs bundle $(E,\,\theta)$ is stable and satisfies
$c_{1,B}(E)\,=\,0\qquad{\rm
and}\qquad\int_{M}c_{2,B}(E)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0,$
then there exists a basic Hermitian metric $h$ so that the canonical
connection $D^{h}$ is flat and such Hermitian metric is unique up to a
positive constant. Moreover, such metric $h$ is a Harmonic metric on the flat
bundle $(E,D^{h})$ with respect to the Sasakian metric $g_{\eta}$ and hence
the flat bundle $(E,D^{h})$ is semi-simple ([Co]).
This hermitian metric is said to be a harmonic metric on $(E,\theta)$.
A polarized complex basic variation of Hodge structure of weight $w$ over $M$
is $(E=\bigoplus_{p+q=w}E^{p,q},D,H)$ so that
1. (1)
$(E,D)$ is a complex flat vector bundle.
2. (2)
$\bigoplus_{p+q=w}E^{p,q}$ is a direct sum of $C^{\infty}$-subbundles such
that
$D_{\xi}{\mathcal{C}}^{\infty}(E^{p,q})\subset{\mathcal{C}}^{\infty}(E^{p,q})$.
3. (3)
(The Griffiths transversality conditions) For any $X\in T^{1,0}_{M}$ (resp.
$X\in T^{0,1}_{M}$),
$D_{X}{\mathcal{C}}^{\infty}(E^{p,q})\subset{\mathcal{C}}^{\infty}(E^{p,q})\oplus{\mathcal{C}}^{\infty}(E^{p-1,q+1})$
(resp.
$D_{X}{\mathcal{C}}^{\infty}(E^{p,q})\subset{\mathcal{C}}^{\infty}(E^{p,q})\oplus{\mathcal{C}}^{\infty}(E^{p+1,q-1})$)
4. (4)
$H$ is a parallel Hermitian form so that decomposition
$\bigoplus_{p+q=w}E^{p,q}$ is orthogonal and $h$ is positive on $E^{p,q}$ for
even $p$ and negative for odd $p$.
Define the basic bundle structure $\nabla_{\xi}$ on $E$ and each $E^{p,q}$. By
the flatness $D^{2}=0$ and the Griffiths transversality conditions, we have:
* •
We have $D=\nabla+\theta+\bar{\theta}$ such that $\nabla$ is a connection on
each $E^{p,q}$ and $\theta\in A^{1}(M,{\rm Hom}(E^{p,q},E^{p-1,q+1}))$,
$\bar{\theta}\in A^{1}(M,{\rm Hom}(E^{p,q},E^{p+1,q-1}))$.
* •
$\nabla$ gives a basic holomorphic bundle structure $\nabla^{\prime\prime}$,
$\theta\in A^{1,0}(M,{\rm Hom}(E^{p,q},E^{p-1,q+1}))$, $\bar{\partial}_{{\rm
End}(E)}\theta=0$ and $\theta\wedge\theta=0$. Thus $(E,\theta)$ is a basic
Higgs bundle.
* •
Define the basic Hermitian metric $h$ on $E$ by $(-1)^{p}H$ on each $E^{p,q}$.
Then $\nabla$ is the canonical connection associated with $h$ and
$\bar{\theta}=\overline{\theta}_{h}$ as the above sense. Thus $h$ is a
Harmonic metric on a basic Higgs bundle $(E,\theta)$.
Let $G=SU(E_{x},H_{x})$ and $V=S(\Pi_{p+q=w}U(E^{p,q}_{x},H_{x}))$. Take the
monodromy $\rho:\pi_{1}(M,x)\to GL(E_{x})$ of the flat bundle $(E,D)$. Since
$h$ is parallel, we can assume $\rho(\pi_{1}(M,x))\subset G$. Let
$\mathfrak{g}$ be the Lie algebra of $G$. The decomposition
$E=\bigoplus_{p+q=w}E^{p,q}$ gives the decomposition
$\mathfrak{g}_{{\mathbb{C}}}=\bigoplus_{r}\mathfrak{g}^{r,-r}$. This
decomposition is a Hodge structure on $G$ and the period domain is $G/V$ (cf.
[Sch]). Considering $h$ as a $V$-reduction of the flat bundle
$(\tilde{M}\times G)/\pi_{1}(M,x)$, we have a $\rho$-equivariant map
$h:\tilde{M}\to G/V$ such that $dh=\theta$ on $T^{1,0}_{\tilde{M}}$. By the
Griffiths transversality conditions, $(G,\rho,h)$ is a $G$-variation of Hodge
structure.
Let $(E,\,\theta)$ be a polystable basic Higgs bundle satisfying
$c_{1,B}(E)\,=\,0\qquad{\rm
and}\qquad\int_{M}c_{2,B}(E)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0.$
By Theorem 5.2, we have a harmonic metric $h$ on $(E,\theta)$. For any $t\in
U(1)$, we have the new polystable basic Higgs bundle $(E,\,t\theta)$ and we
can easily check that $h$ is also a harmonic metric on $(E,\,t\theta)$. Assume
that $U(1)$ acts on $E$ by holomorphic automorphisms via $\mu:U(1)\to{\rm
Aut}(E)$ so that $\mu(t)\theta\mu(t)^{-1}=t\theta$. Then, $\mu(t)$ is an
isomorphism between basic Higgs bundles $(E,\,\theta)$ and $(E,\,t\theta)$. By
the harmonicity of $h$ on both $(E,\,\theta)$ and $(E,\,t\theta)$ and the
uniqueness of harmonic metrics, we can say that the action $\mu:U(1)\to{\rm
Aut}(E)$ preserves a harmonic metric $h$. Fix $w\in{\mathbb{Z}}$, define the
sub-bundle $E^{p,w-p}$ consisting of $e\in E$ such that for any $t\in U(1)$,
$\mu(t)e=t^{p}e$. The polarized complex basic variation of Hodge structure of
weight $w$ $(E=\bigoplus_{p+q=w}E^{p,q},D^{h},H)$ is defined by $H=(-1)^{p}h$
on each $E^{p,q}$.
### 5.2. Uniformizations associated with the canonical Higgs bundles
Define the Higgs bundle $(E_{M},\,\theta_{M})$ by the following way:
* •
$E_{M}\,=\,{\mathbb{C}}_{M}\oplus T^{1,0}_{M}$ where ${\mathbb{C}}_{M}$ is the
trivial basic holomorphic line bundle on $M$, and
* •
$\theta_{M}\,=\,\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right)$ where
$1$ is the identity element in ${\rm
End}(T^{1,0}_{M})=(T^{1,0}_{M})^{\ast}\otimes T^{1,0}_{M}$ regarded as an
element in $A^{1,0}_{B}(M,{\rm Hom}({\mathbb{C}}_{M},T^{1,0}_{M}))$.
###### Theorem 5.3.
We assume:
1. (1)
The basic Higgs bundle $(E_{M},\,\theta_{M})$ is stable.
2. (2)
The equality
$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0$
holds.
3. (3)
$c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for some positive constant $C$.
Then the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$ for
$D=SU(n,1)/S(U(n)\times U(1))$.
###### Proof.
By $c_{1,B}(T^{1,0}_{M})=-C[d\eta]$, $c_{1,B}(E\otimes E_{C^{\prime}})=0$ for
some $C^{\prime}\in{\mathbb{R}}$. By
$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0,$
we have $\int_{M}c_{2,B}(E_{M}\otimes
E_{C^{\prime}})\wedge(d\eta)^{n-2}\wedge\eta\,=\,0.$
$\nabla^{\prime\prime}=\nabla_{|\langle\xi\rangle\oplus T^{0,1}}$ defines a
holomorphic bundle structure on $L$. Thus, $(E_{M}\otimes
E_{C^{\prime}},\,\theta\otimes{\rm Id}_{L})$ is a basic Higgs bundle. Since
$(E_{M},\,\theta_{M})$ is stable, $(E_{M}\otimes
E_{C^{\prime}}\,\theta_{M}\otimes{\rm Id}_{E_{C^{\prime}}})$ is also stable.
Let $h$ be a harmonic metric on $(E\otimes E_{C^{\prime}},\,\theta\otimes{\rm
Id}_{E_{C^{\prime}}})$. Define $E^{0,1}=T^{1,0}\otimes E_{C^{\prime}}$ and
$E^{1,0}=E_{C^{\prime}}$. We have the polarized complex basic variation of
Hodge structure of weight $1$ $(E_{M}\otimes E_{C^{\prime}}=E^{1,0}\oplus
E^{0,1},D^{h},H)$. $SU(E_{x},H_{x})\cong SU(n,1)$ and the Hodge structure on
$SU(n,1)$ is Hermitian and the period domain $D$ is the unit ball
$SU(n,1)/S(U(n)\times U(1))$. We obtain a $SU(n,1)$-variation of Hodge
structure $(SU(n,1),\rho,h)$. By the definition, for the map $h:\tilde{M}\to
SU(n,1)/S(U(n)\times U(1))$, $dh=\theta$ is an isomorphism on each fiber of
$T^{1,0}_{\widetilde{M}}$. Hence, $(SU(n,1),\rho,h)$ is uniformizing. ∎
###### Remark 5.4.
If the basic Higgs bundle $(E_{M},\theta_{M})$ is stable, then the Miyaoka-Yau
type inequality
$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,\geq\,0\,.$
holds. Thus the second condition means that the equality of the Miyaoka-Yau
type inequality holds.
In [Zh], Zhang proves that if $c_{1,B}(T^{1,0}_{M})=C[d\eta]$ for some
positive constant $C$ and the certain condition in terms of Sasaki-Einstein
geometry holds, then the Miyaoka-Yau type inequality holds and the equality is
a criterion for uniformizing a higher dimensional compact Sasakian manifold by
the odd-dimensional sphere.
###### Remark 5.5.
In case $D=SU(n,1)/S(U(n)\times U(1))$, the line bundle
$\bigwedge^{n}T^{1,0}_{D}$ is a homogeneous vector bundle corresponding to the
representation $S(U(n)\times U(1))\ni\left(\begin{array}[]{cc}U&0\\\ 0&{\rm
det}(U)^{-1}\end{array}\right)\mapsto{\rm det}(U)^{n+1}$. Define the subgroup
$U^{\prime}(n)=\left\\{\left(\begin{array}[]{cc}U&0\\\ 0&{\rm
det}(U)^{-1}\end{array}\right):{\rm det}(U)^{n+1}=1\right\\}$ in $SU(n,1)$. We
have $S^{1}(\bigwedge^{n}T^{1,0}_{D})=SU(n,1)/U^{\prime}(n)=PU(n,1)/SU(n)$.
Taking the universal covering $\widetilde{PU}(n,1)$ of $PU(n,1)$, we have
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})=\widetilde{PU}(n,1)/SU(n)$. The
lifting of the Sasakian structure $(T_{D}^{1,0},\eta_{D})$ is
$\widetilde{PU}(n,1)$-invariant.
Assume $n=1$. Then $H^{2}_{B}(M)=\langle[d\eta]\rangle$. In this case, if
$c_{1,B}(T^{1,0})=-C[d\eta]$ for some positive constant $C$, then the Higgs
bundle $(E_{M},\,\theta_{M})$ is stable. By $H^{4}_{B}(M)=0$, the equality
$\int_{M}\left(2c_{2,B}(T^{1,0})-\frac{n}{n+1}c_{1,B}(T^{1,0})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0$
is trivial.
###### Corollary 5.6.
Let $(M,T^{1,0}_{M},\eta)$ be a compact $3$-dimensional Sasakian manifold with
$c_{1,B}(T^{1,0})=-C[d\eta]$. Then the universal covering $\widetilde{M}$ with
the lifting of some $A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost
isomorphic to $\widetilde{PU}(1,1)$ with a left-invariant Sasakian structure.
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|
# Dynamically Finding Optimal Observer States to Minimize Localization Error
with Complex State-Dependent Noise
Troi Williams1, Po-Lun Chen2, Sparsh Bhogavilli2, Vaibhav Sanjay1, and Pratap
Tokekar1 *This work was supported by the Computing Innovation Fellows Project
2021 (NSF Award #2127309).1These authors are with the Department of Computer
Science, University of Maryland, College Park, MD 20742, USA.
<EMAIL_ADDRESS>authors are with the A. James Clark
School of Engineering, University of Maryland, College Park, MD 20742, USA.
<EMAIL_ADDRESS>
###### Abstract
We present DyFOS, an active perception method that Dynamically Finds Optimal
States to minimize localization error while avoiding obstacles and occlusions.
We consider the scenario where a ground target without any exteroceptive
sensors must rely on an aerial observer for pose and uncertainty estimates to
localize itself along an obstacle-filled path. The observer uses a downward-
facing camera to estimate the target’s pose and uncertainty. However, the pose
uncertainty is a function of the states of the observer, target, and
surrounding environment. To find an optimal state that minimizes the target’s
localization uncertainty, DyFOS uses a localization error prediction pipeline
in an optimization search. Given the states mentioned above, the pipeline
predicts the target’s localization uncertainty with the help of a trained,
complex state-dependent sensor measurement model (which is a probabilistic
neural network in our case). Our pipeline also predicts target occlusion and
obstacle collision to remove undesirable observer states. The output of the
optimization search is an optimal observer state that minimizes target
localization uncertainty while avoiding occlusion and collision. We evaluate
the proposed method using numerical and simulated (Gazebo) experiments. Our
results show that DyFOS is almost 100x faster than yet as good as brute force.
Furthermore, DyFOS yielded lower localization errors than random and heuristic
searches.
Figure 1: This figure illustrates a high-level representation of the observer
and target state machines (right) and one iteration of our motivating example
(left). The target (grey rover) with no exteroceptive sensors moves along a
path (solid purple line) with obstacles (green trees). The target’s
localization uncertainty is drawn as an orange dashed ellipse. At time $i$,
the target sends a rendezvous state to the observer (blue drone), predicting
that it (the target) will arrive at time $j$. The observer finds and moves to
an optimal state that will minimize the corrected localization uncertainty of
the target (see Figure 2 for a detailed view of the optimization module). When
the target arrives (at approximately time $j$), the observer estimates the
target’s pose and uncertainty and sends them to the target. The target updates
its pose and uncertainty and sends another rendezvous state to the observer.
This process repeats until the target arrives at the goal. This paper focuses
on the modules in rounded rectangles with solid lines.
## I Introduction
Navigating through known or unknown environments is challenging when an
autonomous robot is ill-equipped to perceive its surrounding environment. For
example, many works explore how adverse weather affects sensors typically used
on autonomous cars, such as cameras, LiDAR, radar, and GPS [1, 2, 3, 4, 5]. We
note that sensor performance degradation or malfunction also occurs in other
situations outside of adverse weather. Small robots may not even have
exteroceptive sensing capable of supporting robust localization. We are
interested in the situation where a robot can no longer access its
exteroceptive sensors for an extended period. The ground robot (termed
_target_) wants to navigate to a goal along a path that contains obstacles
(Figure 1). We aim to minimize the target’s localization uncertainty by
leveraging cooperation from an aerial robot (termed _observer_). The target
receives pose and uncertainty measurements from the observer to update its
pose estimate.
The localization uncertainty is a function of the measurement uncertainty
(from the observer). We consider the case where the measurement uncertainty is
a function of the states of the observer, target, and surrounding environment
(e.g., the ambient lighting conditions). Although we traditionally assume
measurement uncertainty is constant, other works have shown that the
measurement uncertainty can be state-dependent (some include [6, 7, 8, 9,
10]). Therefore, we aim to find an optimal observer state that minimizes the
target’s localization uncertainty.
To localize the target accurately, we propose DyFOS, a novel active perception
method that Dynamically Finds Optimal States to minimize localization error
(uncertainty) (Figure 2). The main contribution of DyFOS is a localization
error prediction pipeline that is used in an optimization search. Given the
state of the observer, target, and surrounding environment, the pipeline
predicts the target’s corrected localization error with the help of a complex
State-Dependent Sensor Measurement Model (SDSMM). The pipeline also allows
constraints to avoid (predicted) obstacle collision or occlusion. Together,
the optimization method and the pipeline select optimal observer poses that
would minimize localization error and allow us to forego selecting optimal
states in a heuristic manner (for example, requiring the target to be in the
center of the sensor’s field of view [11]).
Our paper is organized as follows. Section II places our work in the context
of the literature. We formulate our problem in Section III. Section IV
describes the DyFOS algorithm. We discuss our experiments and their results in
Section V. Finally, Section VII concludes the paper.
## II Related Work
Active perception is a process in which an agent performs a set of strategic
actions that allow it to gather more insightful information about some
phenomenon [12, 13, 14]. Active perception has been applied to various cases
including autonomous scientific information gathering [15], foreground
segmentation [16], object detection and target tracking [17, 9, 18, 19, 11],
ocean flow and vehicle states [20], and searching for individuals [21, 22]. Of
these related work, several are similar to our proposed method and motivating
example [23, 11, 19].
Gürcüoglu et al. [23] proposed an active target tracking approach for a team
of quadrotors that were equipped with 3D range sensors. Their approach
contributes (1) a hierarchical controller that tracks desired optimal
trajectories for each quadcopter and (2) a method that uses a Kalman Filter to
fuse the position estimates of a target from each drone. The optimal
trajectories attempt to minimize the fused position error of the target.
Similar to [23], we compute optimal drone states that minimize the perceived
error of a target. However, unlike [23], we focused on developing a complex,
non-analytical measurement noise model that influenced the observer’s 2D pose.
Therefore, our method must dynamically search for optimal solutions. We can
also use any controller with our observer.
Falanga et al. [11] proposed the first perception-aware, model predictive
control framework for quadrotors. Their method used numerical optimization to
compute trajectories that simultaneously optimized action and camera-specific,
human-specified perception objectives, both of which can conflict. The human-
specified perception objectives required the point of interest to be near the
center of the image. Like Falanga et al., we optimize for perceptual
objectives (namely, minimizing the target’s pose error). However, our
localization error prediction pipeline influences the (output) observer pose
rather than human-specified objectives. Furthermore, our method handles
collision and occlusion constraints.
Tallamraju et al. [19] proposed an active approach to a cooperative detection
and tracking algorithm (in [24]) that tracked a person performing activities.
Their method aimed to minimize the 3D position error of the person by ensuring
optimal viewpoint configurations of the drones. Although we both perform
target tracking, our methods differ. First, We employ a learned noise model to
compute an optimal pose and, by extension, a viewpoint (via heading) instead
of using heuristics and an analytical noise model. Second, we output an
optimal pose (instead of a motion plan), allowing a flexible choice of motion
planners.
## III Problem Formulation
### III-A Preliminaries
Consider a world with a planar surface, an observer, a target, and a
collection of static obstacles. The target is an autonomous ground vehicle
(AGV) that wants to navigate along a path. Let
${\boldsymbol{\bar{x}}^{t}_{i}\in\text{SE}(2)}$,
${\boldsymbol{P}^{t}_{i}\in\mathbb{R}^{3\times 3}}$ denote an estimate of the
target’s 2D pose and its covariance (uncertainty) of the state error at
timestep $i$, respectively. The target has a map $\boldsymbol{M}$ that
contains the locations and dimensions of obstacles.
Due to damage or failure of the target’s exteroceptive sensors, the target
cannot localize or avoid obstacles. Therefore, it relies on an observer to
estimate its state and uncertainty. The observer is a rotorcraft that can
localize itself accurately. Let
${\boldsymbol{\bar{x}}^{o}_{i}\in\text{SE}(3)}$ represent the 3D pose of the
observer. The observer uses a downward, front-facing camera (with pitch
$\beta$) to estimate the target’s state and uncertainty. We define the
measurement model as
$\boldsymbol{z}_{i}=\boldsymbol{x}^{t}_{i}+\boldsymbol{\sigma}_{i}$. Here,
$\boldsymbol{x}^{t}_{i}$ is the target’s true 2D pose and
$\boldsymbol{\sigma}_{i}\in\mathbb{R}^{3}$ is the state-dependent measurement
noise [6] at time $i$. We assume each state-dependent noise
$\boldsymbol{\sigma}_{i}\sim\mathcal{N}({0,\boldsymbol{\Sigma}_{i}})$ is
normally distributed with covariance $\boldsymbol{\Sigma}_{i}$, and each
covariance is a function of the states of the observer, target, and
surrounding environment. Given the observer, target, and environment states,
we also assume each covariance can be predicted or estimated dynamically by a
state-dependent sensor measurement model (SDSMM). Finally, we assume the
(predicted or estimated) states are available offline and online.
The observer requires time to compute an optimal state, move to such a state,
and observe the target. To account for this time, the observer rendezvouses
with the target at rendezvous states $\boldsymbol{x}^{t}_{j}$ that are
dynamically selected along the target’s path. Each rendezvous state is at a
timestep $j$ in the future (where $i<j$) and is selected based on the
approximate compute and travel times of the observer.
### III-B Problem Statement
Our goal is two-fold. First, we find a future, optimal observer state
$\boldsymbol{x}^{o\ast}_{j}$ that minimizes the predicted localization
uncertainty of the target as it navigates along a path. Second, we seek
observer states that avoid collisions and occlusions due to obstacles. To
accomplish these goals, we propose an algorithm called DyFOS, which
dynamically finds an optimal observer state that will 1) avoid obstacles, 2)
avoid occlusions, and 3) minimize the target’s predicted localization
uncertainty after a measurement update at timestep $j$.
We search an optimal state $\boldsymbol{x}^{o\ast}_{j}$ for time $j$ using the
following optimization problem:
$\begin{split}\boldsymbol{x}^{o\ast}_{j}=&\operatorname*{arg\,max}_{\boldsymbol{x}^{o}_{j}\in\boldsymbol{\mathcal{X}}^{o}_{j}}\mathcal{O}\big{(}\boldsymbol{P}^{t}_{j}\big{)}\\\
\text{subject
to}&~{}c_{l}\\!\left(\boldsymbol{x}^{o}_{j},\boldsymbol{\hat{x}}^{t}_{j},\boldsymbol{M}\right)=0,\forall
l\in[1,L].\end{split}$ (1)
Here, $(\boldsymbol{\hat{x}}^{t}_{j},\boldsymbol{\hat{P}}^{t}_{j})$ is the
target’s rendezvous state and predicted state uncertainty at timestep $j$.
$\boldsymbol{x}^{o}_{j}$ is a sampled observer state from
$\boldsymbol{\mathcal{X}}^{o}$, the set of all observer states.
${\mathcal{C}=\\{c_{l}\\}_{l=1}^{L}}$ are a set of constraints that predict
obstacle collision or occlusion. $\boldsymbol{P}^{t}_{j}$ is the predicted,
corrected localization uncertainty after a measurement update at time $j$.
Moreover, $\boldsymbol{M}$ is our environment model. Inspired by [25, 26], we
choose $\boldsymbol{\hat{x}}^{t}_{j}$ as a waypoint along the target’s path.
Furthermore, we compute $\boldsymbol{\hat{P}}^{t}_{j}$ by propagating the
target’s current localization uncertainty $\boldsymbol{\hat{P}}^{t}_{i}$ from
timestep $i$ to timestep $j$ using the Kalman Filter equations. Section IV
discusses this optimization problem.
## IV Dynamically Finding Optimal States to Minimize Localization Error
Figure 2: DyFOS is composed of an optimization method (dashed green) and a
localization error prediction pipeline (LEPP) (solid orange). Together, the
optimizer and pipeline find an optimal observer state
$\boldsymbol{x}^{o\ast}_{j}$ that predicts the lowest target localization
error $\hat{E}_{\textbf{P}}$ after a measurement update at time $j$. Then the
observer flies to the optimal state to estimate the target’s pose and
uncertainty. See Figure 1 for the entire state machine.
We propose a method that searches for an optimal observer state
$\boldsymbol{x}^{o\ast}_{j}$ that minimizes the predicted, corrected
localization uncertainty $\boldsymbol{P}^{t}_{j}$ of a target at time $j$.
Without the loss of generality, we describe our proposed method in the context
of a multi-robot application, where a target (with no exteroceptive sensors)
receives pose measurements $\boldsymbol{z}_{j}$ and their estimated
uncertainties $\boldsymbol{\hat{\Sigma}}_{j}$ from an aerial observer.
Figure 2 illustrates the proposed algorithm. The algorithm is composed of 1) a
localization error prediction pipeline (LEPP) and 2) an optimization method.
The pipeline contains four main steps. First, we sample an observer state
$\boldsymbol{x}^{o}_{j}$ from a time-varying state space
$\boldsymbol{\mathcal{X}}^{o}_{j}$ (Section IV-A). Next, we construct a
feature vector $\boldsymbol{\lambda}_{j}$ using the sampled state
$\boldsymbol{x}^{o}_{j}$, the rendezvous state $\boldsymbol{\hat{x}}^{t}_{j}$
at time $j$, and our environment model $\boldsymbol{M}$. Then an SDSMM
predicts a covariance of the measurement noise $\boldsymbol{\hat{\Sigma}}_{j}$
given the feature vector $\boldsymbol{\lambda}_{j}$ (Section IV-B). The
pipeline also evaluates the constraints given states of the observer, target,
and environment (Section IV-C). Finally, the pipeline predicts a localization
error $\hat{E}_{\textbf{P}}$ given the predicted measurement uncertainty and
the output of our constraints (Section IV-D).
The optimizer executes the pipeline many times to find one optimal observer
state $\boldsymbol{x}^{o\ast}_{j}$ for a future time $j$. Discussions on
selecting an optimizer and finding the appropriate parameters are beyond the
scope of this paper. However, Section V mentions the optimizer used in our
experiments.
### IV-A The Spatio-Temporal, Observer State Space
The pipeline begins with sampling observer states from an observer state space
that varies with space and time. Let $\boldsymbol{\mathcal{X}}^{o}$ define the
set of all observer states in a map. Searching for an optimal pose
$\boldsymbol{x}^{o\ast}_{j}$ within an arbitrarily large map with an
arbitrarily long state vector $\boldsymbol{x}^{o}$ requires a significant
amount of computation. Therefore, we define a new search space
$\boldsymbol{\mathcal{X}}^{o}_{j}$ that reduces the computational complexity.
First, we seek a 2D pose for the observer. We assume the observer’s altitude
is near constant and its velocities, accelerations, pitch, and roll are near
zero at time $j$. Second, we seek observer poses where the target (at the
rendezvous state $\boldsymbol{\hat{x}}^{t}_{j}$) is in range and field-of-view
of the observer. Using both statements, we define the set
$\boldsymbol{\mathcal{X}}^{o}_{j}$ as:
$\begin{split}\boldsymbol{\mathcal{X}}^{o}_{j}=&\\{\boldsymbol{x}^{o}_{j}\in\boldsymbol{\mathcal{X}}^{o}_{j},\boldsymbol{x}^{o}_{j}\subset\text{SE}(2)~{}|~{}\rho_{\text{min}}\leq\rho_{j}\leq\rho_{\text{max}},\\\
&|\psi_{j}|\leq\frac{\text{FoV}_{\psi}}{2},~{}|\phi_{j}|\leq\frac{\text{FoV}_{\phi}}{2}\\}.\end{split}$
(2)
Here, $\rho_{\text{min}}$ and $\rho_{\text{max}}$ denote the minimum and
maximum sensing range of the camera. $\text{FoV}_{\psi}$ and
$\text{FoV}_{\phi}$ represent the camera’s horizontal and vertical field of
views (FoVs). Finally, $\rho_{j}$ is the distance between the camera and the
target, while $\psi_{j}$ and $\phi_{j}$ are the horizontal and vertical angles
of the target relative to the camera (respectively). We note that
$\boldsymbol{\mathcal{X}}^{o}_{j}$ also imposes an “in sensing range”
constraint that varies with time and space due to the motion of the target and
the positions of obstacles in the observer’s workspace. Figure 4 illustrates
sample states from $\boldsymbol{\mathcal{X}}^{o}_{j}$, where each sampled
state is within range and view of the target. Collision and occlusion
predicting is handled in Section IV-C.
Selecting poses from $\boldsymbol{\mathcal{X}}^{o}_{j}$ is user-specific. For
example, one can sample uniformly and use gradient-based methods to select
subsequent poses.
### IV-B State-Dependent Sensor Measurement Model
Figure 3: We implement a state-dependent sensor measurement model (SDSMM)
using a multi-layer perception (MLP) and a non-trainable Gaussian layer. We
use the states of the observer, target, and surrounding environment to
construct a feature vector $\boldsymbol{\lambda}_{k}$. Then, given
$\boldsymbol{\lambda}_{k}$, the MLP outputs a free parameter vector
$\boldsymbol{\upsilon}_{k}$. Afterward, the Gaussian layer computes the
covariance of the measurement noise $\boldsymbol{\hat{\Sigma}}{k}$ using the
parameter vector $\boldsymbol{\upsilon}_{k}$.
A state-dependent sensor measurement model (SDSMM) is a trainable model that
estimates the state-dependent covariance of the measurement noise [6, 27].
Following Williams and Sun [6, 7], we implement an SDSMM using a probabilistic
neural network (Figure 3). The input to the SDSMM is a set of features
$\boldsymbol{\lambda}\in\mathbb{R}^{d_{\lambda}{}}$ that correlate with the
measurement noise111Williams and Sun [6, 27] refer to such features as a
combined state. However, we avoid using the term to alleviate confusion
between a combined state and states of robots.. The output of our SDSMM is an
estimate of the covariance of the measurement noise
$\boldsymbol{\hat{\Sigma}}{}$ of the target’s 2D pose.
We train an SDSMM using a dataset
${\mathcal{D}=\\{(\boldsymbol{e}_{k},\boldsymbol{\lambda}_{k})\\}}_{k=1}^{K}$
that contains a collection of $K$ measurement errors $\boldsymbol{e}_{k}$ and
their corresponding features $\boldsymbol{\lambda}_{k}$. The measurement error
is defined as
${\boldsymbol{e}_{k}=(\boldsymbol{z}_{k}-\boldsymbol{x}^{t}_{k})}$, where
$\boldsymbol{z}_{k}$ is a measurement of the target’s pose using a pose
estimation algorithm and $\boldsymbol{x}^{t}_{k}$ is the ground truth pose of
the target. Generally speaking, the feature vector $\boldsymbol{\lambda}_{k}$
for a given sensor and estimation problem varies. For instance, a feature
vector for a camera and target localization problem may contain the ambient
lighting of the area, the velocity of the camera, and the distance and
relative velocity between the camera and the target. We note that these
features can be computed given the ground truth, estimates, or predictions of
the states of the observer, target, and surrounding environment. For example,
we compute the features using ground truth states during training. However, we
compute the feature vector $\boldsymbol{\lambda}_{k}$ using estimated or
predicted states during runtime.
Given the dataset $\mathcal{D}$, we use the negative log-likelihood loss
function $\mathcal{L}$ to train our model. Since our measurement noise is
normally-distributed, our loss function simplifies to
$\mathcal{L}=-\frac{d_{\sigma}}{2}\ln{(2\pi)}-\frac{1}{2}\sum_{k=1}^{K}\left(\ln{(|\boldsymbol{\hat{\Sigma}}_{k}|)}+\boldsymbol{e}_{k}^{\top}\boldsymbol{\hat{\Sigma}}_{k}^{-1}\boldsymbol{e}_{k}\right),$
(3)
where $\boldsymbol{\hat{\Sigma}}_{k}$ is the output of the SDSMM and
$d_{\sigma}$ is the dimension of $\boldsymbol{\hat{\Sigma}}_{k}$.
### IV-C Observer State Constraints
Figure 4: We show an example of predicting feasible and non-feasible observer
states via our constraints (Section IV-C). For example, a sampled, future
observer state $\boldsymbol{x}^{o}_{j}$ is not feasible if we predict the
observer will collide with an obstacle (state 2) or an obstacle will occlude
the target at the rendezvous state $\boldsymbol{\hat{x}}^{t}_{j}$ (state 1
with a partial FoV). Otherwise, we predict collision and occlusion will not
occur (state 3 with the solid green FoV).
We use constraints to predict which states are feasible during our online
search (Figure 4). Let ${\mathcal{C}=\\{c_{{}_{l}}\\}_{l=1}^{L}}$ contain a
set of constraints that determines if an observer state
${\boldsymbol{x}^{o}_{j}\in\boldsymbol{\mathcal{X}}^{o}_{j}}$ is a feasible
solution. Collectively, the set of constraints $\mathcal{C}$ defines a
feasible region that varies with time and space, and this feasible region
contains the optimal observer state we seek.
We use two constraints: collision avoidance and line-of-sight. Each constraint
is defined as a function. The input to each function is a set of variables
that describe a sampled observer, the target, or the surrounding environment.
Each function returns $1$ if the prediction is true and $0$ otherwise. For
example, we predict a sampled observer will collide with an obstacle if the
two objects intersect. Likewise, we predict an obstacle will occlude the
target if an obstacle is hit by rays emitting from the observer to the target.
Our numerical and simulated experiments employed these constraints using
simple geometric shapes [28] and an Octomap [29], respectively.
### IV-D Localization Error Prediction
The final step in our pipeline predicts a localization error for timestep $j$.
We remind the reader that we seek to minimize the target’s corrected
localization uncertainty $\boldsymbol{P}^{t}_{j}$ using the measurement
$\boldsymbol{z}_{j}$ and its estimated uncertainty
$\boldsymbol{\hat{\Sigma}}_{j}$. Therefore, we define the localization error
$\hat{E}_{\textbf{P}}$ as
$\hat{E}_{\textbf{P}}=\mathcal{O}\big{(}\boldsymbol{P}^{t}_{j}\big{)}=\ln\det\left(\boldsymbol{P}^{t}_{j}\right)+\eta\sum_{l=1}^{L}\mathbf{1}(c_{l}),$
(4)
where $\boldsymbol{P}^{t}_{j}$ is the uncertainty after a measurement update
at time $j$, $\eta$ is a user-defined constraint penalty, and $c_{l}$ is the
output of a constraint (Section IV-C). Suppose we use a Gaussian distribution
to estimate the target’s pose and uncertainty. Then we can predict the
corrected localization uncertainty $\boldsymbol{P}^{t}_{j}$ using the Riccati
equations of the Kalman Filter [23, 9]. In the Riccati equations, we compute
the target’s predicted localization uncertainty $\boldsymbol{\hat{P}}^{t}_{j}$
(prior to the measurement update) by propagating the target’s localization
uncertainty from time $i$ up to time $j$. Furthermore, the predicted
measurement uncertainty $\boldsymbol{\hat{\Sigma}}_{j}$ comes from the SDSMM
(Section IV-B). In addition to varying with time, we note that
$\boldsymbol{P}^{t}_{j}$ and, ultimately, $\hat{E}_{\textbf{P}}$ also vary
with each sampled state because the predicted measurement uncertainty
$\boldsymbol{\hat{\Sigma}}_{j}$ is a function of the sampled observer states
(Section IV-A).
## V Experiments
We performed numerical experiments and a simulated experiment in Gazebo. Each
experiment used one of three algorithms to compute an observer pose. Our
numerical experiments compared the pose to a brute-force solution. The
algorithms used were random placement, heuristic placement, and DyFOS. The
random algorithm arbitrarily placed the observer within view of the target.
The heuristic algorithm arbitrarily placed the observer within $0.5$ to $3.1$
m of the target, which embodied a “sense close” rule of thumb. Finally, DyFOS
used the method described in Section IV and the differential evolution global
optimizer [30] in SciPy [31]. All algorithms employed the constraints (Section
IV-C). We also denoted obstacles using 2D shapes and an Octomap in the
numerical and simulated experiments, respectively.
DyFOS used an SDSMM to predict the measurement noise. To train the SDSMM, we
defined our camera’s ground truth uncertainty model as
$\boldsymbol{\Sigma}=\text{diag}(\sigma_{x},\sigma_{y},\sigma_{\theta})$,
where $\sigma_{x}=0.05(x+1)^{2}+0.05$, $\sigma_{y}=0.05(y-1)^{2}+0.05$, and
$\sigma_{\theta}=\frac{4\pi}{180}$. The minimum uncertainty was at $(-1,1)$
(in the observer’s body frame). Here, $x$, $y$ were the target’s estimated
position relative to the observer. These equations were chosen arbitrarily.
During training and online, the input to the SDSMM was a noisy 2D position of
the target relative to the observer. Each predicted covariance
$\boldsymbol{\hat{\Sigma}}_{k}$ represented the uncertainty in the target’s 2D
pose. Our training set was defined as
$\mathcal{D}=\\{(\boldsymbol{\lambda}_{k},\boldsymbol{e}_{k})\\}_{k=1}^{K}$,
where $K=10,000$. Here, $\boldsymbol{\lambda}_{k}$ is the position of the
target relative to the observer,
${\boldsymbol{e}_{k}\sim\mathcal{N}({0,\boldsymbol{\Sigma}_{k}})}$ is the
corresponding measurement error, and $\boldsymbol{\Sigma}_{k}$ is the
measurement noise from the ground truth model.
We used a brute force method to approximate ground truth poses that 1)
satisfied all constraints and 2) had the lowest navigation loss using the
ground truth noise model. The predicted localization error was computed using
(4).
### V-A Numerical Placement in Random 2D Environments
The placement experiment compared the output observer pose from each algorithm
with the brute force solution. We generated 1,000 arbitrary 2D maps, which
were comprised of one target and 25 obstacles. We selected arbitrary poses for
the target and obstacles. The observer and target were 0.75 meters and 1.5
meters long, respectively. We sampled the length (in meters) of each obstacle
using the uniform distribution $\mathcal{U}(0.5,2.5)$. The goal of each
algorithm was to minimize the localization uncertainty
$\boldsymbol{P}^{t}_{j}=\boldsymbol{I}\in\mathbb{R}^{3x3}$.
### V-B Numerical Target Localization
The mobile target localization experiment featured a ground target that
navigated along a path while receiving pose estimates and their uncertainty
from the observer. The observer used the three algorithms to determine an
optimal pose at each time step. We used three paths: a line, a circle, and a
figure-$\infty$. The length or radius of each path was five meters. We ran
each algorithm on a path 25 times to compute the mean performance. A Kalman
Filter was used to estimate the target’s pose and incorporate pose estimates
and their uncertainties from the observer. Finally, an LQG controller was used
to compute control commands for the target.
Figure 5: This figure shows a drone (the observer) localizing the Turtlebot
(the target). The drone detects the Turtlebot via an ArUco marker.
### V-C Static Target Localization in Gazebo
The target localization experiment in Gazebo featured a stationary Turtlebot
(target) and an Iris drone (observer) with a 45-degree downward-facing camera.
We performed nine trials where the drone used one of the three algorithms to
find an observer pose, flew to that pose, and estimated the localization
uncertainty of the Turtlebot (Figure 5).
We performed the experiments in a world with a $30$ m2 planar surface. The
world contained 25 static obstacles that were 10 m tall, had a diameter of one
meter, and were in the workspace of both robots.
## VI Results and Discussion
We used four metrics to evaluate each algorithm: 1) the predicted localization
error, 2) each observer’s distance from the optimal solution, 3) each
algorithm’s run times, and 4) the Kalman filter convergence. We computed the
localization error using (4). The distance from the optimal solution was the
2D distance between the target and the observer.
Figure 6: This figure shows the distributions for the localization errors
(top) and distances from the optimal solution (bottom) for the placement
experiment with $1,000$ randomly-generated worlds. DyFOS performed on par with
the brute force method. Lower is better.
Localization Error. No observer collided with any obstacle, and no obstacle
occluded the target for any algorithm or world in our numerical placement
experiment. We also observed that DyFOS had a mean error of ${-8.997\pm
0.046}$, performing on par with the brute force algorithm (with a mean error
of $-8.815$) (Figure 6 top). By contrast, the localization error for the
heuristic and random placement algorithms varied significantly, yielding mean
errors of ${-8.101\pm 0.525}$ and ${-7.459\pm 0.738}$ (respectively).
DyFOS also outperformed the random and heuristic algorithms in both
localization experiments. Among the three algorithms, our method achieved the
lowest loss the quickest and had the least variation for all three paths in
the numerical experiment. Likewise, DyFOS also had the lowest localization
errors in the simulated experiment. The mean localization errors for DyFOS,
heuristic, and random were $-8.768\pm 0.002$, $-8.385\pm 0.299$, and
$-7.699\pm 0.675$, respectively. We believe that the low loss and small
variation are because DyFOS actively reduces the localization error. Finally,
although the random and heuristic algorithms produced low errors, their
respective errors plateaued higher than DyFOS (see Figure 7 for numerical
results).
Distance from the Optimal Solution. We observed similar performance trends as
the above analyses for the numerical experiment (Figure 6). The mean distance
from the optimal pose for the brute force algorithm and DyFOS were ${2.6\pm
12.7}$ cm and ${9.8\pm 17.6}$ cm, respectively. For $94.1\%$ of the
(${1,000}$) worlds, DyFOS placed the observer within $25$ centimeters of the
optimal pose. (Recall that the ground truth measurement uncertainty is least
when the target is at $(1,-1)$ meters relative to the observer). By contrast,
the heuristic and random algorithms placed the observer within $488$ and $768$
cm of the optimal pose (respectively) and yielded significant variation for
the same number ($941$) of worlds. For perspective, a DJI Mini 3 PRO has a
width of 24.5 cm unfolded and without propellers [32]. For the other $5.9\%$
of the worlds, DyFOS produced poses that were $25$ cm to $1.26$ m away from
the optimal pose. These higher distances were due to fixing the observer’s
heading, avoiding obstacles, and preventing occlusions. DyFOS and the brute
force algorithm generally output similar poses in such cases.
DyFOS also performed better than the other two searches in the simulated
experiment. The minimum of the noise model was $(3,0.5)$ for this setup. The
mean difference between the optimal solution and DyFOS was $2.6\pm 0.9$ cm.
Meanwhile, the mean differences between the optimal solution and the heuristic
and random searches were $80.4\pm 40.8$ cm and $165.3\pm 81.8$ cm,
respectively. Finally, we note that DyFOS performed better in simulated
experiments because we did not fix the observer’s heading in the Gazebo.
Run Times. The run times vary due to the complexity of each algorithm. The
heuristic and random placement algorithms took less than $0.01$ s to find an
observer pose since they did not use any sensor model. The run time for DyFOS
was ${0.90\pm 0.24}$ s with a maximum of $2.46$ s using one core, which is
reasonable for real-time usage. Finally, the brute force algorithm took
${75.52\pm 2.34}$ s with a maximum of $83.82$ s using $32$ cores to find a
solution, which is not reasonable for real-time usage.
| Line | Circle | Figure-8
---|---|---|---
Random | $11.1\pm 5.4$ | $16.4\pm 8.2$ | $24.9\pm 11.0$
Heuristic | $8.6\pm 2.8$ | $12.8\pm 6.1$ | $13.9\pm 6.0$
DyFOS | $\boldsymbol{5.1\pm 1.7}$ | $\boldsymbol{9.9\pm 3.7}$ | $\boldsymbol{12.6\pm 5.2}$
TABLE I: RMSE (cm) for each algorithm. Bolded means best.
Figure 7: This figure shows the mean and standard deviation statistics for a
Kalman Filter on the circle path over time. With DyFOS, the target’s estimated
position converged the quickest (top). Furthermore, DyFOS had the lowest
navigation loss and variation (bottom).
Kalman Filter Convergence. We examined the time each algorithm took for the
filter to converge with the target’s true pose and the error over time (Figure
7 top). For the Figure-$\infty$ and line paths, the three algorithms converged
within a similar time and had a consistent error for the remainder of the
trajectory. However, DyFOS converged significantly quicker for the circle path
than the random and heuristic algorithms (within $15$ s for DyFOS versus at
least $80$ s for the other algorithms). The results also showed that DyFOS had
the lowest RMSEs for all three paths (Table I).
## VII Conclusion
We proposed DyFOS, a novel active perception method that Dynamically Finds
Optimal States to minimize localization error while avoiding obstacles and
occlusions. DyFOS relies on user-defined constraints, a learned state-
dependent sensor measurement model, and a loss function to evaluate the
quality of each candidate state with respect to the localization task. DyFOS
exploits a learned SDSMM to minimize the target’s localization uncertainty in
a hazard-filled world. Our experiments showed that DyFOS performs on par with
brute force methods but is almost 100x faster. The average runtime was less
than one second using one CPU core, which may be reasonable for some real-time
applications. DyFOS also achieves lower localization errors than random and
heuristic search algorithms.
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|
# JWST NIRCam Defocused Imaging: Photometric Stability Performance and How it
Can Sense Mirror Tilts
Everett Schlawin Steward Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA Thomas Beatty Department of Astronomy
University of Wisconsin
Madison, Madison, WI 53706 Brian Brooks Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Nikolay K. Nikolov Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Thomas P. Greene NASA Ames Research Center, Space Science
and Astrobiology Division, MS 245-6, Moffett Field, CA, 94035, USA Néstor
Espinoza Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Department of Physics & Astronomy, Johns Hopkins
University, Baltimore, MD 2121818, USA Kayli Glidic Steward Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA Keith Baka Anton Pannekoek Institute, University of
Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Eiichi Egami
Steward Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA John Stansberry Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Martha Boyer Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Mario Gennaro Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Department of Physics and Astronomy
Johns Hopkins University
3400 North Charles Street, Baltimore, MD 21218, USA Jarron Leisenring Steward
Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA Bryan Hilbert Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Karl Misselt Steward Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA Doug Kelly Steward Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA Alicia Canipe Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Charles Beichman NASA Exoplanet Science Institute/IPAC
Jet Propulsion Laboratory, California Institute of Technology
1200 E California Blvd
Pasadena, CA 91125 Matteo Correnti Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 J. Scott Knight Ball Aerospace & technologies, Corp,
1600 Commerce Street, Boulder, CO 80301 Alden Jurling NASA Goddard Space
Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA Marshall D. Perrin
Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218 Lee D. Feinberg NASA Goddard Space Flight Center, 8800
Greenbelt Rd, Greenbelt, MD 20771, USA Michael W. McElwain NASA Goddard Space
Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA Nicholas Bond
ADNET Systems, Inc./Code 550, NASA’s Goddard Space Flight Center, Greenbelt,
MD 20771 David Ciardi NASA Exoplanet Science Institute/IPAC
Jet Propulsion Laboratory, California Institute of Technology
1200 E California Blvd
Pasadena, CA 91125 Sarah Kendrew European Space Agency, Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore MD 21218 Marcia Rieke
Steward Observatory
933 North Cherry Avenue
Tucson, AZ 85721, USA
###### Abstract
We use JWST NIRCam short wavelength photometry to capture a transit lightcurve
of the exoplanet HAT-P-14 b to assess performance as part of instrument
commissioning. The short wavelength precision is 152 ppm per 27 second
integration as measured over the full time series compared to a theoretical
limit of 107 ppm, after corrections to spatially correlated 1/f noise.
Persistence effects from charge trapping are well fit by an exponential
function with short characteristic timescales, settling on the order of 5-15
minutes. The short wavelength defocused photometry is also uniquely well
suited to measure the realtime wavefront error of JWST. Analysis of the images
and reconstructed wavefront maps indicate that two different hexagonal primary
mirror segments exhibited “tilt events” where they changed orientation rapidly
in less than $\sim$1.4 seconds. In some cases, the magnitude and timing of the
flux jumps caused by tilt events can be accurately predicted with a telescope
model. These tilt events can be sensed by simultaneous longer-wavelength
NIRCam grism spectral images alone in the form of changes to the point spread
function, diagnosed from the FWHM. They can also be sensed with the FGS
instrument from difference images. Tilt events possibly from sudden releases
of stress in the backplane structure behind the mirrors were expected during
the commissioning period because they were found in ground-based testing. Tilt
events have shown signs of decreasing in frequency but have not disappeared
completely. The detectors exhibit some minor (less than 1%) deviations from
linear behavior in the first few groups of each integration, potentially
impacting absolute fluxes and transit depths on bright targets where only a
handful of groups are possible. Overall, the noise is within 50% of the
theoretical photon noise and read noise. This bodes well for high precision
measurements of transiting exoplanets and other time variable targets.
stars: atmospheres — stars: individual (HAT-P-14 (catalog )) — stars:
variables: general
††facilities: JWST(NIRCam), JWST(FGS), Palomar††software: astropy (Astropy
Collaboration et al., 2013), photutils v0.3 (Bradley et al., 2016), matplotlib
(Hunter, 2007), numpy (van der Walt et al., 2011), scipy (Virtanen et al.,
2020), starry (Luger et al., 2019), pynrc (Leisenring, 2019), webbpsf (Perrin
et al., 2014), pysiaf (Sahlmann et al., 2019) jwst (Bushouse et al., 2022)
## 1 Introduction
JWST will provide transformational new studies of exoplanet atmospheres with
its unprecedented view in the infrared (Ahrer et al., 2022; Greene et al.,
2016; Morley et al., 2017; Schlawin et al., 2018; Bean et al., 2018). The
NIRCam instrument’s imaging, coronagraphic and spectrographic modes are
providing powerful new insights into a range of astrophysical environments
from the Solar System to the most distant galaxies from 0.6 µm to 5.0 µm
(Rieke et al., 2005). The NIRCam instrument has two wavelength channels that
simultaneously observe the same field. NIRCam’s grism time series mode
produces simultaneous spectroscopy on the long wavelength channel and
photometry on the short wavelength channel. The long wavelength spectroscopy
employs a grism that enables spectroscopy from 2.4 to 5.0 µm at R=1100 to
1700, with the wavelength coverage determined by the filter selected on the
filter wheel (Greene et al., 2017). This wavelength range includes many
prominent features in exoplanet atmospheres including H2O, CH4, CO, CO2 and
NH3 (Greene et al., 2016). The molecular, atomic and ionic spectral features,
when detected in exoplanet atmospheres, can give clues about the formation of
planets, composition, dynamics, structure and cloud and haze composition (e.g.
Bitsch et al., 2021; Mordasini et al., 2016; Powell et al., 2019; Kempton et
al., 2017).
Given the small scale heights of planetary atmospheres ($\lesssim
10^{-3}R_{\odot}$) compared to their host star radii ($\sim 1R_{\odot}$) and
their low temperatures compared to their host star temperatures
($T_{p}^{4}/T_{*}^{4}\lesssim 10^{-2}$, where $T_{p}$ and $T_{*}$ are the
temperatures of the planet and star respectively), exoplanet transit
spectroscopy requires extreme precision. The largest signals from even the
most favorable planets are still very small. The ‘prominent’ 1.4 $\mu$m water
vapor feature strength varies from 100 ppm in the prototypical hot Jupiter HD
209458 b (Deming et al., 2013) down to less than 30 ppm for the super-Earth GJ
1214 b (Kreidberg et al., 2018). The most favorable CO2 signatures are
predicted to be 50 ppm and below in the temperate Earth-sized TRAPPIST-1 d and
e (Barstow & Irwin, 2016; Lustig-Yaeger et al., 2019). Lessons from previous
space telescopes such as Spitzer, Kepler and the Hubble Space Telescope (HST)
have taught us that many instrument systematics can be present including
intrapixel sensitivity, thermal breathing, charge trapping, and pointing
errors (Beichman et al., 2014).
Here we present the performance of the NIRCam short-wavelength time-series
photometry based on commissioning observations of the HAT-P-14 system. A
companion paper, Beatty et al. (2022) describes the simultaneous long-
wavelength grism spectroscopy. The light that enters NIRCam is split by a
dichroic beamsplitter so that the short wavelengths (0.6 to 2.3 µm) and long
wavelengths (2.4 to 5.0 µm) are directed to different channels’ optics and
detectors. The pupil wheel and filter wheel selections determine the
wavelength extents of the observations ( a short wavelength F210M filter
paired with the long wavelength filter F322W2 will cover approximately 1.99 to
2.23 µm on the short wavelength channel and 2.43 to 4.01 µm on the long
wavelength channel). The flight software is configured to run the detectors
with the same size subarrays and readout patterns so the two exposures are
commanded to be simultaneous to within one frame time. For the grism time
series mode, two short wavelength detectors (NRCA1 and NRCA3) centered on
nearly the same piece of sky as the long wavelength (NRCALONG) are read out by
detector electronics and downloaded from the observatory. Appendix A shows the
relative layouts of the detectors.111Also see https://jwst-
docs.stsci.edu/jwst-near-infrared-camera/nircam-instrumentation/nircam-
detector-overview for the detector layout (STScI, 2016).
When performing grism spectroscopy, the short wavelength channel can be
(currently) configured to use the Weak Lens +4 or Weak Lens +8 lens optics,
which are designed for mirror wavefront sensing (Greene et al., 2010). The
Weak Lens +4 and Weak Lens +8 lens optics are a subset of all defocusing
lenses, which also include Weak Lens -4 and Weak Lens -8. The Weak Lens +4 (on
the filter wheel) and Weak Lens +/- 8 (on the pupil wheel) can be used in
series simultaneously so that focuses of -8, -4, 0, +4 and +12 Peak-to-Valley
(P-V) waves can be configured (Greene et al., 2010). The lenses were tuned to
be sensitive to various spatial frequencies and levels of aberration in the
wavefront at 2.12 µm (Dean & Bowers, 2003; Perrin et al., 2016). The Weak Lens
+4 defocuses the light by 4.0 P-V units at 2.12 µm and results in a core PSF
of approximately 31 pixels and the Weak Lens +8 defocuses the light by 8.0 P-V
units at 2.12 µm and results in a core PSF of approximately 65 pixels.
Physically, the Weak Lens +8 defocuses the light by modifying the wavefront by
+8.0 waves P-V at 2.12 um at the exit pupil, where the positive convention is
that the phase at the edge of the wavefront leads the phase at the center of
the pupil. We assume that the equation for the phase in waves for the Weak
Lens +8 is $8*(r^{2}-0.5)$ where r is the exit pupil radius normalized to
unity at the edge and 0 in the center. This wavefront error displaces the best
focus along the chief ray.
The defocused PSF from a weak lens makes the short wavelength imaging very
sensitive to changes in the wavefront and it also spreads the light over more
pixels to decrease the rate of saturation for a given star brightness. While
not yet supported, there is also an active proposal to allow the use of the
Dispersed Hartmann Sensor (DHS) in the short wavelength channel instead of the
weak lenses. Utilizing the DHS in the short wavelength channel would allow
simultaneous spectroscopy across a large fraction of the full NIRCam
wavelength regime (Schlawin et al., 2017).
The target HAT-P-14 b (Torres et al., 2010) was selected for calibration
purposes because it has no strongly detectable atmospheric features and a flat
spectrum due to a high gravity (42 m s-2, calculated from Stassun et al.,
2017) and small atmospheric scale height (2$\times 10^{-4}$ R⊙). The 3.4 MJ
1.4 RJ 1600 K planet (Stassun et al., 2017; Southworth, 2012) orbits a
$K=8.85$, 6600 K F-type star with low chromospheric activity (Torres et al.,
2010). It was observed by JWST NIRSpec (Espinoza et al., 2022), NIRISS and
NIRCam to enable cross-instrument performance checks and to determine the
transit depths are consistent. The expected variations in the transmission
spectrum are 16 ppm, assuming a typical atmosphere that exhibits atmospheric
features that are equivalent to 0.9 atmospheric scale heights (Wakeford et
al., 2019), so the differences across wavelengths are expected to be about as
large as the theoretical ideal photon noise $\sim$9 ppm for these
observations. One option to characterize stability is to observe a known
quiescent star and verify that the lightcurves are flat with time. However, in
the event that the lightcurve is not perfectly flat, disentangling the
measured precision from the baseline trend curves is challenging. Any de-
trending done to the data, such as a Gaussian process regression, can remove
trends but could also remove astrophysical signal on a time variable source.
Therefore, we observed a known transiting planet with a known ephemeris to
determine the ability of JWST to measure the transit depth. While the target
was chosen to have no significant atmospheric signal (no significant change in
transit depth with wavelength), it still has a transit depth of 6480 $\pm$ 240
ppm (Torres et al., 2010) or about 0.6% which is readily detectable.
Figure 1: The Weak Lens +8 PSF spreads the light over a large region, which
reduces the chance of saturation and reduces intra-pixel sensitivity. The Weak
Lens +8 was designed for retrieving the primary mirror state and thus phase
retrieval with the Weak Lens enables real-time monitoring of the mirror
wavefront error. The above image is near the middle of a 2048$\times$128
SUBGRISM128 subarray with the F210M filter, a 27.0 second long integration
time on HAT-P-14 with a plate scale of about 31 mas /px. The background
subtraction annulus as shown as concentric circles and is the inner radius is
the same as the source aperture used for photometry. The subarray position was
adjusted downward after HAT-P-14 b commissioning to better center the
hexagonal PSF on the subarray.
Section 2 describes the observations during commissioning and how they were
analyzed. Section 3 describes systematic effects in the data including mirror
tilt events, tilt timescales, the charge trapping ramp and observed non-
linearity. Section 4 describes the noise performance, pointing performance and
time accuracy as measured from in-flight data and the transit depth precision
when accounting for contamination from a background source. We present our
conclusions in Section 5.
Table 1: Summary of Key Times During Observation
UT start time | Integration Number | Description
---|---|---
(YYYY-mm-ddThh:mm:ss) | |
2022-05-02T06:43:04 | 1 | Exposure Start
2022-05-02T09:21:34 | 344 | Transit Start (Contact 1)
2022-05-02T10:10:01 | 449 | HGA Move
2022-05-02T10:30:20 | 492 | Transit Midpoint
2022-05-02T10:30:41 | 493 | Tilt Event 1
2022-05-02T11:23:19 | 607 | Tilt Event 2
2022-05-02T11:39:05 | 641 | Transit End (Contact 4)
2022-05-02T12:43:25 | 780 | Exposure End
Note. — Times are in Coordinated Universal Time. The barycentric dynamical
times are approximately 4.63 minutes later. Integration numbers are given for
1-based counting
## 2 Observations and Image Processing
### 2.1 Observation Description
We observed the hot Jupiter HAT-P-14 b to test the instrument performance and
verify that the NIRCam Grism Time Series mode was suitable for science. The
planet was selected to have a short duration transit (2.3 hr), high brightness
that enables high precision, low stellar activity and high planet surface
gravity to have a small atmospheric signal. The short duration requirement was
selected to minimize the needed observing time but has the flip side of
preferring targets with near-grazing transits (high impact parameters).
HAT-P-14 b’s impact parameter is 0.91 (Fukui et al., 2016), so fitting the
limb darkening without the planet crossing through the stellar mid-point can
complicate the transit depth measurement and stellar models may be preferred
for the limb darkening law, as described in Section 4.1. The observation
exposure started at 2022-05-02T06:43 UTC, which was the earliest start time
allowed by the Astronomy Proposal Tool (APT) special requirements to ensure a
long baseline before transit ingress. The lightcurve was observation 1 of
Commissioning Program ID 1442, which is available on the Barbara A. Mikulski
Archive for Space Telescopes (MAST). Table 1 lists key times during the
observations.
The exposure duration was 6 hours, which enabled 2.6 hours of stabilization
time before planet ingress. The SW channel was configured with the Weak Lens+8
pupil, F210M filter which covers approximately 1.9 to 2.2 $\mu$m. This was
paired with the GRISMR pupil and F322W2 filter on the long wavelength channel,
discussed in Beatty et al. (2022). The exposure setup was a single exposure
with the BRIGHT2 readout pattern, 20 groups integrated up the ramp and 780
integrations for a 2048$\times$128 subarray (SUBGRISM128) with 4 output
amplifier channels. NIRCam readout electronics sample non-destructive reads
(ie. frames) to create a ramp and then reset the detector at the end of an
integration. Detector groups are a set of 1 or more detector image frames
where the frames have been either averaged or skipped by onboard electronics
and the average frame (ie. a group) is downloaded from the spacecraft. The
averaging or else skipping of frames between groups is done to reduce the data
volume on some observations. Observations when more than one frame is averaged
or skipped per group reduce the data volume as compared to saving every frame
per integration. In BRIGHT2, there are 2 frames (ie. detector samples)
averaged per group and zero frames skipped. This resulted in a frame time of
0.67597 seconds, a group time of 1.35194 seconds and a cadence of 27.72001
seconds per integration for 780 integrations.
The 6 hour observation was continuous with no interruptions. Pointing
performance was extremely stable as discussed in Section 4.3 below. There was
one high gain antenna (HGA) move during the exposure but no interruptions or
pauses to the integrations in the 6 hour exposure. This allows detector charge
trapping effects to approach a steady state, unlike with HST where each
satellite orbit around Earth results in a new charge trapping ramp effect
approximately every 90 minutes (e.g. Berta et al., 2012).
The point spread function (PSF) with the Weak Lens+8 pupil is shown in Figure
1. The Weak Lens +8 (along with the -8 wave and +4 wave lenses) is designed to
measure the optical path differences of the primary mirror surface. It was
enabled as an option in the grism time series and imaging time series modes to
spread out the light over more pixels and also reduce the saturation on bright
targets. Having the light spread out over more pixels reduces the intrapixel
sensitivity, which strongly affected Spitzer Space Telescope lightcurves
because the PSF was highly undersampled (Ingalls et al., 2016). It also allows
realtime monitoring of the mirror surface along with the long wavelength grism
spectroscopy, discussed in Section 3.1. We note that the full extent of the
PSF is asymmetrically truncated by the subarray. This will be mitigated in
Cycle 1 science observations because an update was made to better center the
subarray position. Even with better centering, the SUBGRISM128 subarray
truncates some of the wings of the PSF. In other words, the commissioning
observations (Figure 1) had the PSF is asymmetrically truncated and in science
observations it will be still truncated in the wings but more symmetrically.
Fortunately, JWST pointing is so stable (to be discussed in Section 4.3) that
there were no impacts from the PSF truncation on the lightcurve and zero-
padding the image for wavefront analysis resulted in high quality wavefront
phase retrievals (to be discussed in Section 3.1).
Figure 2: The default rateints product for integration number 83 with no extra
corrections (top) has large pre-amplifier offsets (rectangular-shaped offsets)
and 1/f noise (horizontal banding). A row-by-row, odd/even by amplifier
(ROEBA) correction, described in Section 2.2 uses background pixels to reduce
the pre-amplifier offset and 1/f noise. The background pixels are the ones
outside of the dashed black circle that has a radius of 201 pixels.
### 2.2 Data Analysis from Uncalibrated Data to Rates Per integration
We begin with the uncal data (also known as “MAST level 1 uncalibrated”, “User
Data Product Stage 0 Fully-Populated FITS file” or Data Processing Level 1b).
We then run Stage 1 of the Calwebb STScI pipeline (Bushouse et al., 2022) with
some modifications described below, to produce a rateints file which contains
the counts in DN/s for each integration.
We started with the STScI JWST Detector 1 Stage 1 pipeline version 1.6.0 and
the Calibration Reference Data System (CRDS) context jwst_0822.pmap to convert
the raw data values to signal slopes (DN/s) per integration. The Detector
Stage 1 pipeline begins by scaling the groups (which doesn’t affect NIRCam
readout modes), initializing the data quality flags, flagging saturated pixels
and then subtracting a superbias. When examining the superbias subtraction, we
note that every 4 integrations has a bias offset structure due to the fast
frame resets that reset pixels across the whole detector to ensure that charge
doesn’t migrate into the subarray. For this SUBGRISM128 subarray, the fast
frame reset occurs every 4 integrations. However, accounting for the different
superbias every 4 integrations resulted in very little change to the
lightcurve so we used a single superbias subtraction for all integrations.
Normally, detector stage 1 processing then does a reference pixel correction
step. The reference pixels are light-insensitive pixels that share the same
electronic noise sources as the regular light-sensitive pixels and thus allow
efficient subtraction of common-mode noise sources. Normally, the bottom and
side reference pixels are used to mitigate odd/even column effects, pre-
amplifier resets and 1/f noise. However, the SUBGRISM128 subarray for the
short-wavelength time series mode has no reference pixels along the top or
bottom detector rows.222The SW subarray is designed to be centered on the
defocused PSF when the same source is centered near the Y=34 long wavelength
pixel on the LW subarray. See section A for a diagram. The relative positions
of the short and long wavelength detectors when projected on the sky result in
the weak lens +8 image being centered at Y= 167.5 in 1-based counting absolute
coordinates on the short wavelength detector. The subarray position was
adjusted to better center the PSF after HAT-P-14 was observed. Thus, the
reference pixel step does not correct for large pre-amplifier offsets and
smaller odd/even offsets. We instead replace the reference pixel step with a
similar step that uses background instead of reference pixels to remove many
of the same effects.
We use a shorthand for this modified reference pixel step ROEBA, which stands
for row-by-row, odd/even by amplifier correction, which lists the steps in
reverse order. First, we select one of the four amplifiers for correction at a
time. In this case a block pixels from 1 to 512, 513 to 1024, 1025 to 1536 or
1537 to 2048 pixels in the X direction and the entire 128 pixel tall subarray
in the Y direction. For this correction, it is necessary to define a
background region to use as a proxy for reference pixels that has minimal
contamination from bright sources. We select all pixels that are more than 201
pixels away from the central PSF as shown in Figure 2. We then apply a slow-
read correction, which subtracts the median odd count level in the background
region from all odd numbered columns and the median even count level from in
the background region from all even numbered columns. This removes the pre-
amplifier offsets. We then calculate the row-by-row median count level from
the background region and subtract this from all pixels in that row. This
largely removes the longer time scale (low frequency) components of the 1/f
noise (e.g. Schlawin et al., 2020). The resulting correction from ROEBA shown
in Figure 2 (bottom) has substantially smaller pre-amplifier reset offsets in
each amplifier as well as less 1/f noise banding along the horizontal
direction. We note that the ROEBA method uses many more pixels across the
horizontal fast-read direction than the side reference pixels on the detector
so it can improve results even when reference pixels are available in a
subarray. Another advantage of the ROEBA correction is that many fewer pixels
are erroneously marked as jumps at the jump detection step (described below).
One assumption in the ROEBA step is that it relies on a clean background
region and could potentially over-subtract when there is faint scattered light
or background sources so inspection for neighboring sources and faint emission
is necessary before applying it to all NIRCam data.
We next apply the non-linearity correction with default parameters. The result
of the bias subtraction and ROEBA step is that the signal level is close to 0
DN at the background before applying the linearity correction step. The stage
1 pipeline normally has a dark current subtraction step after non-linearity
correction, but dark current on the NIRCam short wavelength detectors is less
than 0.05 e-/sec and there is currently too much signal from cosmic rays and
persistence to measure well and subtract on these detectors. We next run the
jump step correction with a threshold of 6 sigma. Without the ROEBA
correction, many pixels are erroneously flagged as “jumps” (e.g. cosmic rays)
because the pre-amplifier offsets and 1/f noise can appear as jumps in signal
from one detector group to another. With the ROEBA correction, we note that
these erroneous jumps are much less frequent and instead the jump step flags
true outliers such as cosmic rays. One exception where the jump step does not
flag a cosmic ray perfectly is the outer radii of ‘snowball“ events (Rigby et
al., 2022), as shown in Figure 2 near X=737,Y=101. Here, only the brighter
portions of the snowball are flagged with the jump detection algorithm used in
the analysis for this paper. As the final step of stage 1, we run the ramp fit
step with default parameters to calculate a count rate per pixel in Data
Numbers (DN) per second. There is another gain scaling step for NIRSpec data
but it does not affect the NIRCam data presented in this paper.
### 2.3 Photometric Extraction
We next use aperture photometry with photutils (Bradley et al., 2016) to
measure the flux for each integration to derive a flux as a function of time.
We fit the central peak of the PSF with a Gaussian and used this to determine
the aperture center for each integration. We use a circular source aperture
with a radius of 81 pixels, and the mean background rate from a background
annulus with an inner radius of 81 and an outer radius of 101 pixels. These
pixels were found to produce the minimum lightcurve scatter from an aperture
size grid search. Although the ROEBA step described above largely removes the
background, we found that the additional background annulus subtraction
improved the precision, likely because of residual 1/f not removed by ROEBA.
We found the same level of noise whether we did aperture centering or kept the
apertures fixed, with the exception of better photometry during the HGA move
affecting one integration to compensate for the small amount of image motion
within that one integration. We normalize the flux time series to a value near
1.0 (1000 parts per thousand, ppt) for lightcurve fitting and noise analysis,
which we describe in Section 4.1.
## 3 Lightcurve Systematics
The lightcurves overall exhibited very low level systematic errors, which we
describe in this section. There were changes to the PSF due to “tilt” events
as discussed in Section 3.1 through 3.3, an exponential ramp discussed in
Section 3.4 and some evidence for non-linearity discussed in Section 3.5.
### 3.1 Tilt Events
We originally noticed a large jump in the long wavelength time series (Beatty
et al., 2022) but also found one at the same time at some aperture sizes in
the short wavelength and a smaller jump later in the time series. The jumps in
flux correspond to changes in the mirror surface, known as “tilt events”
(Rigby et al., 2022). These were first noticed by choosing an arbitrary
reference image (in this case the rate from the 6th integration) and dividing
the rate from all other integrations by it. Figure 3 shows three example ratio
images from 1) before the tilt events, 2) after the first tilt event and 3)
after the second tilt event. The tilt events show up as clear wave patterns in
these differential images due to changes in the mirror surface. The short
wavelength Weak Lens +8 defocused PSF is part of the wavefront sensing tools
aboard JWST and is thus highly sensitive to changes in the mirror surface.
Figure 3: Weak Lens Ratio Images before and after both Tilt Events (top) and
Phase retrieval, oriented as if looking down at the mirror surface from the
secondary mirror (bottom plots). Figure 4: The jump in flux caused by the C6
tilt event is wavelength dependent. The black point and blue points with error
bars show the measured flux jump size for the short wavelength (SW) aperture
of 2.4″ and LW aperture of 0.630″, respectively. Our current model (green
circle) correctly predicts the SW channel’s jump with a 2.4″ aperture but
under-predicts the LW 0.63″ jump size (orange curve) for the grism spectrum,
especially at 2.5 to 3.0 µm.
We used a non-linear optimization based phase retrieval algorithm (Fienup,
1982, 1993) based on the algorithmic differentiation (Jurling & Fienup, 2014).
The coherent propagation model used a flexibly sampled discrete Fourier
transform (Jurling et al., 2018). We used a bias and gain invariant error
metric (Thurman & Fienup, 2009) as the cost function. The phase retrieval
model was initialized using the wavefront and pupil data calculated from the
JWST optical model for the nearest available field point, the NRCA3-FP2MIMF
multi-instrument multi-field point. Because PSF data were only available with
the +8 wave weak lens, this was a single image phase retrieval problem. We
configured the model with 128 pixels across the array storing the pupil and
256 pixels across the cropped and padded image. Because of the band-pass of
the F210M filter used for imaging, we used a broadband point-spread function
model (Fienup, 1999; Jurling et al., 2018). We ran the phase retrieval
optimization in two steps.
In the first step we used the first image in the time series to establish a
starting point. We sequentially estimated image position (using a sub-pixel
cross correlation method (Guizar-Sicairos et al., 2008)), global low order
Zernike aberrations, segment level piston, and tip/tilt errors (1st order
segment Zernike aberrations). Finally, we performed joint estimation over a
smoothly interpolated low resolution grid in wavefront and amplitude. With a
single image the ability to distinguish between phase and amplitude errors in
this last step is limited.
In the second step, we begin from the reference point established by the first
phase retrieval. In this case the optimization was done jointly over global
and segment Zernike aberrations without sequential bootstrapping. This
facilitates change detection by requiring the pupil amplitude and higher order
wavefront variation to be the same throughout the time series, but still
allowing low order and segment wavefront changes through time.
The phase retrieval results are shown in Figure 3 and it is clear that the C6
mirror segment changed orientation (tilted) at event 1 and that the C1 mirror
segment tilted at event 2. In event 1, the wavefront over C6 changed by 18 nm
RMS (evaluated over the segment) compared to the initial wavefront, and in
event 2, C1 changed by 26 nm (again evaluated over the segment). These changes
are much smaller than the wavelength of the light, so both the individual
segments and the overall telescope remain diffraction limited.
As shown in Figure 4, the C6 event produced a larger change with a magnitude
that varies from 200 ppm to 1000 ppm across the 2.4 µm to 4.0 µm wavelength
range for the long wavelength and 254 $\pm$ 22 ppm in the short wavelength
photometry using an aperture radius of 79 pixels or 2.45 arcsec. Using the
wavefront phase retrieval described in this section above (Jurling & Fienup,
2014) for the optical path differences, we make a prediction for the magnitude
of the flux jump by comparing the enboxed or encircled energy before and after
the jumps. We use enboxed energy for the long wavelength spectroscopic
extraction box and encircled energy for the short wavelength photometry. The
short wavelength (SW) model prediction of 260 ppm is very close to the
measured 254 $\pm$ 22 ppm for the large aperture used in the short wavelength
time series. The long wavelength (LW) jump model prediction under-predicts the
jump size from 2.5 to 3.5 $\mu$m but agrees from 2.7 to 4.1 $\mu$m. Some of
the difference between the model and measurement could be the difference
between an imaging and dispersed PSF and how they are extracted. Preliminary
models with a dispersed LW grism image can reproduce the 1000 ppm change at
2.5 $\mu$m.
### 3.2 Ways to Detect Tilt Events
Figure 5: The lightcurves exhibited jumps most noticeably in the long
wavelength spectroscopy, shown here as a broadband time series (plot A). These
jumps (marked as dash-dot vertical lines) can be detected in a variety of
ways. Apertures placed on the short wavelength weak lens (WL) PSF are highly
sensitive to mirror tilts on mirrors C6 and C1 (plot B). The time series are
shown on a group-by-group difference image (points in blue and orange, plot B)
as well as on the full jump-corrected integration (solid blue and orange
lines, plot B). The FGS instrument can also sense mirror tilts by a dot
product of a difference images, as defined in Equation 1 (plot C). The FWHM of
the long wavelength spectroscopy can also be used to sense mirror tilts (plot
D). The HGA move (vertical dotted line) also can create spikes in some tilt
statistics. The apertures used to sense the C6 and C1 mirror1 tilts (right
plot) are the ones used to generate the time series in plot B.
It is possible to quantitatively measure tilt events in addition to visually
inspecting the differential images or doing the phase retrieval shown in
Figure 3. The mirror changes can be evaluated with photometric apertures on
key parts of mirror surfaces, such as in Figure 5 (right plot). The time
series of these apertures can cleanly sense the two tilt events like step
functions as shown in Figure 5 (left second from the top). We also calculate
this photometry on the difference between pairs of adjacent detector groups to
assess the mirror changes at a higher time cadence (colored points in Figure
5).
The NIRCam grism time series and time series imaging modes are the only modes
currently supported that allows simultaneous high cadence wavefront sensing
along with the time series. In some time series modes such as the NIRSpec
Bright Object Time Series (Birkmann et al., 2022) or MIRI low resolution time
series (Kendrew et al., 2015), defocused imaging is not currently available
for real-time wavefront sensing of the primary mirrors, so other measures are
desirable to sense tilt events. NIRISS Single Object Slitless mode (SOSS)
(Doyon et al., 2012) has a de-focusing lens so this mode has some analogous
capabilities as the short wavelength weak lens.
We explored some ways to sense tilt events with the NIRCam grism time series
spectroscopy and the fine guidance sensor (FGS) fine guide data streams (Doyon
et al., 2012). We experimented with a few different statistics for the FGS
images. We found that the flux from the telemetry stream333See the
SA_ZFGINSTCT keyword https://jwst-docs.stsci.edu/methods-and-roadmaps/jwst-
time-series-observations/jwst-time-series-observations-noise-sources, (STScI,
2016) can sense the first tilt event (C6) but could not easily reveal the
second tilt event (C1). In addition, the first tilt event (C6) could be
detected with a dot product with the average PSF or the $3\times 3$ pixel flux
of the time series but the second tilt event from the C1 mirror was harder to
pull out of the noise. The FWHM of the FGS 8$\times$8 images also shows step-
like changes with the two tilt events but at low signal to noise. We found
that principal component analysis (PCA) could reveal evidence of both tilt
events but is also not easily distinguishable above the noise.
The noise-weighted “differential dot product” was most effective in detecting
tilt events with FGS.
$D_{i}\equiv\left((\vec{A_{i}}-\vec{R})/\vec{V}\right)\cdot\left(\vec{M}-\vec{R}\right),$
(1)
where $D_{i}$ is the differential dot product of 8$\times$8 images reshaped to
64-pixel linear vectors. In the expression, $\vec{A_{i}}$ is a single FGS cal
data image, $\vec{R}$ is a reference image after the tilt event, $\vec{V}$ is
the variance image, and $\vec{M}$ is a reference image before the tilt. This
method did use prior information about the timing of tilt events from NIRCam
data to find the average image from before the tilt $\vec{M}$ and the average
image after the tilt $\vec{R}$, but it may be possible to iteravely scan the
time series to determine $\vec{M}$ and $\vec{R}$ from FGS alone. This
statistic essentially magnifies the differences between the shape of the PSF
before and after the tilt event and weights by the noise. Figure 5 shows that
both tilt events can be sensed with FGS using a dot product, where the
$\vec{M}$ and $\vec{R}$ are chosen for the first tilt event (C6) and second
tilt event (C1).
Additionally, the full width at half maximum (FWHM, bottom left plot in Figure
5) of the long wavelength grism PSF is sensitive to tilt events because the
PSF changes with the mirror tilt. Thus, even when weak lens imaging is not
available, the FGS images and science instrument FWHM can be used in all time
series modes to give clues about tilt events. However, Equation 1 requires
some knowledge of the timing to find $\vec{M}$ and $\vec{R}$ unless they can
be found iteratively. Therefore, a more general-purpose indicator of tilt
events may be the FWHM of science data, which can be tracked in all time
series observations.
Tilt events are hypothesized to be structural microdynamics in the telescope
that may occur when stresses in the backplane structure behind the mirrors are
suddenly relaxed. As these stresses are released, the frequency of tilt events
is expected to decrease in time. Regular wavefront measurements indicate that
they are less frequent in the transition from commissioning to science
observations in July 2022 than earlier in commissioning. However, tilt events
were common even 4.5 months between launch and these observations, and similar
tilt events may occur during Cycle 1 science observations.
### 3.3 Tilt Timescale
The mirror changes appear to be step functions in Figure 5, so the change
happens on a timescale faster than the 27 second cadence of integrations. We
inspect the mirror changes at a higher cadence (but noisier) time series by
calculating the mirror tilt specific photometry on the group-by-group
difference images, which have a cadence of 1.351 seconds. In this analysis,
adjacent points are thus anti-correlated because they share a detector group
so if one sample is anomalously high, the next one will be anomalously low. As
shown in Figure 6, there is a rapid change that takes 2 sampling durations of
1.351 seconds each to go from low to high or high to low, or in other words
one sample between the low and high values. There are two causes for the
intermediate sample: 1) the tilt timescale is between 1.351 and 2.702 seconds
and 2) the timescale is shorter than 1.351 seconds and occurs midway through
the group. We consider the first scenario less likely as it would require some
fine-tuning to cause a tophat-like disturbance between these two time
intervals that shows little sign of change in the PSF beyond 2.702 seconds. We
therefore expect that the timescale is faster than the shortest measurable
time with the NIRCam photometry which is the 1.351 second group time. We also
note that these observations used the BRIGHT2 mode, which includes two frames
within a group so the tilt could happen between these two frames within a
group or within one of the frames. We do not see any discontinuities within
the image that would indicate a change within the frame, but the PSF only
covers a fraction of the field of view so it is not a strong constraint.
Figure 6: Mirror photometry on the pairwise-subtracted images, zoomed in on
events. We use the same apertures as shown in Figure 5 to sense the tilts on
the C1 and C6 primary mirror segments. These two happen on a timescale as fast
or faster than the 1.351 second time between BRIGHT2 detector groups, which
contain 2 frames each. For the C1 tilt event, the jump likely happened in the
middle of a detector group so it has a sample between the low and high flux
values.
### 3.4 Charge Trapping Ramp
The HST WFC3 IR detector, which is an earlier generation HgCdTe detector,
exhibits ramps with an exponential settling behavior (e.g. Berta et al., 2012;
Zhou et al., 2017). Ground-based laboratory tests showed likely negligible
effects for typical JWST detectors (e.g. Schlawin et al., 2021). However, the
NRCA3 short wavelength detector exhibits the most persistent charge and likely
charge trap density of the 10 NIRCam detectors that are used in flight
(Leisenring et al., 2016). NRCA3 is the same detector used for the short
wavelength component of the grism time-series observations when the long
wavelength filter is selected to be F277W, F322W2, F356W.444https://jwst-
docs.stsci.edu/jwst-near-infrared-camera/nircam-operations/nircam-target-
acquisition/nircam-grism-time-series-target-acquisition The NRCA1 detector
that is paired with the F444W filter, had about half the accumulated counts
from charge trap release as the NRCA3 detector in ground based tests
(Leisenring et al., 2016), so it is expected to have a smaller amplitude
exponential ramp.
We fit an exponential model to the charge trapping behavior, as is common with
HST charge trapping ramps (e.g. Berta et al., 2012) and show the results in
Figure 7. We first correct the small 260 ppm jump in the lightcurve using a
wavefront model described in Section 3.1. We fit the lightcurve with the
following planet and systematics model:
$f(t)=\left(A+Bx^{\prime}+Cx^{\prime
2}\right)\left(1-R\exp{(x-x_{0})/\tau}\right)f_{a}(t),$ (2)
where A, B and C are coefficients in the quadratic baseline trend and
normalization, $x$ is barycentric time, $R$ is the exponential amplitude,
$x_{0}$ is the exposure start time, $f_{a}(x)$ is the astrophysical limb
darkened lightcurve (starry, Luger et al., 2019), with a 4 parameter
polynomial limb darkening law from ExoCTK and $x^{\prime}$ is the normalized
time,
$x^{\prime}=\frac{x-x_{med}}{x_{max}-x_{min}},$ (3)
where $x_{med}$ is the median time, $x_{max}$ is the maximum time and
$x_{min}$ is the minimum time. We fit the lightcurves by fixing $C=0$ (ie. a
linear fit) and also letting it be a free parameter (ie a quadratic fit). As
will be shown in Section 4.2, significant correlated errors are seen in the
residuals if a linear baseline trend is adopted instead of a quadratic trend.
We find an exponential amplitude $R$ of 731 ppm and an exponential time $\tau$
of 5.1 minutes for a linear trend and $R$ of 656 ppm and $\tau=$15 of minutes
for a quadratic trend. The normalization and polynomial trend constants for
the quadratic trend are A=${1000.24\pm 0.01}$ parts per thousand (ppt),
B=${-0.911\pm 0.03}$ ppt and C=${0.80\pm 0.14}$ ppt. In absolute units, this
corresponds to a slope of -0.15 ppt/hr and derivate of the slope of 0.022
(ppt/hr2). We note that initially, we fit the baseline and exponential start
to just the initial part of the lightcurve and found an order-of-magnitude
exponential settling timescale of 11 minutes (Rigby et al., 2022). The 5.1 and
15 minutes minutes came from a MCMC Bayesian fit to the full lightcurve with a
linear and quadratic baseline respectively, whereas the 11 minutes comes from
least squared minimization of the first 300 points (before planet ingress) and
has a steeper linear slope. We have not determined the cause of the long
timescale trend, but note a that different linear trends were seen on the NRS1
and NRS2 detectors (Espinoza et al., 2022) so it may be detector-related. The
quadratic slope and exponential ramp terms in Equation 3 are correlated and
thus the change in slope is fit with a different exponential settling time. We
also looked through the previous JWST activities and found that NIRCam’s
previous use was 37 hours before for wavefront sensing. NIRCam detectors
remained in idle reset mode for those 37 hours so it is unlikely any long
timescale traps were filled and are being released to create the downward
slope seen in the data.
Figure 7: The lightcurve settling behavior is very fast with a fitted settling
timescale of 5-15 minutes but is somewhat degenerate with the time series
baseline.
### 3.5 Non-Linearity Effects
The H2RG detectors used in the near infrared instruments on JWST are never
strictly linear at any well filling fraction, ranging from sub-percent non-
linearity to tens of percent at 98% the hard saturation value (e.g. Canipe et
al., 2017). However, they become increasingly non-linear as the detector
approaches full well capacity and saturation (e.g. Plazas et al., 2017).
Correction polynomials are applied by the CalWebb JWST pipeline (Bushouse et
al., 2022) to linearize the counts as function of counts (DN). A different but
analogous method has been shown that linearity corrections are possible even
up to high well filling fractions ($\sim$97% Canipe et al., 2017).
We assessed the difference between pairs of groups up the ramp to look for
evidence of non-linearity. The stellar flux is highly stable within a 27
second long integration so it is expected that a linearity-corrected ramp
should have a constant difference between each successive group. We find
however that the first 4 group differences and the last 9 have a higher rate
of flux than the middle 6 group differences, as shown in Figure 8. The early
higher fluxes may be related to the release of trapped charge (e.g. Smith et
al., 2008; Leisenring et al., 2016) or a type of reset anomaly (Rauscher et
al., 2007). We do not perform a dark current subtraction, as discussed in
Section 2.2. However, the dark current is less than 0.05 e-/sec on the short
wavelength NIRCam detectors, and the 1% change in the differential samplings
(shown in Figure 8) on top of a representative rate of 400 e-/s would require
4 $e^{-}$/s of dark current. So the reset anomaly seen here may be different
from the anomaly seen on other H2RG devices (Rauscher et al., 2007).
Exoplanet time series observations with a small number of groups ($\lesssim
5$) could have an absolute photometric flux offsets but also systematic
differences in the transit depth as compared to observations with many groups
($\gtrsim 5$). This is because the non-linearity effects can change the
response of the detector to a differential signal (ie. a non-constant
derivative of count rate as a function of signal in e-/s/Mjy). With the
observed 1% change in signal from group 2-1 to group 4-3 in Figure 8, a 2%
deep transit of a hot Jupiter could have a measured transit depth of up to
$\sim$ 1% $\times$ 2% $\times$ 1/4 = 50 ppm deeper if just using group 2 and
1. This difference is below the measurement noise for a single transit of
HAT-P-14 b using only group 2 - 1 but could be assessed on a brighter target.
There is also a possibility of reciprocity failure with HgCdTe detectors that
can cause systematic changes in transit depth (Biesiadzinski et al., 2011;
Schlawin et al., 2021).
Figure 8: The difference in counts between groups up the ramp is not constant
across an integration, as would be expected after linearity correction curves
are applied. The violin-shaped points are the distributions of fluxes as
calculated by the pairwise difference image of two detector groups, using all
of the out-of-transit integrations . The median flux from all out-of-transit
integrations is shown as a short horizontal line. Future observations with
just 2 or 3 total groups may exhibit absolute flux differences and transit
depth differences due to this non-linearity soon after detector reset.
## 4 Lightcurve Performance
Figure 9: Lightcurve and residuals before and after corrections for an
exponential ramp and jump from the first tilt event. The blue symbols with
error bars are the fluxes for each integration at a cadence of 27.7 seconds,
while the orange symbols are 8 minute long time-integrated bins that aid in
viewing smaller-level changes. Red symbols are outlier points not used in the
fitting.
### 4.1 Lightcurve Fitting
We fit the lightcurve with a transit model to determine the lightcurve
performance, discussed in Section 4.2 and the timing accuracy, discussed in
Section 4.5. We remove a 260 ppm jump in the F210M (2.1 µm) time series due to
a tilt event using a wavefront model of the optics discussed in Section 3.1,
allow a quadratic trend with time and fit an exponential settling ramp
discussed in Section 3.4. We use the integration mid-times in the barycentric
reference frame and barycentric dynamical time standard int_mid_BJD_TDB
included in the INT_TIMES extension of the JWST data products.
We fit the corrected lightcurve with a starry transit lightcurve model (Luger
et al., 2019), an exponential ramp, a quadratic trend in time and the 260 ppm
jump correction. The planet’s near-grazing impact parameter (0.91 Fukui et
al., 2016)) means that the planet does not traverse near the stellar midpoint
and fitting the limb darkening parameters as free parameters can lead to large
uncertainties in the planet’s radius. We fix the limb darkening law to a 6600
K, [Fe/H]=0.11, $\log{g}$ = 4.25 (Stassun et al., 2017) ATLAS9 model (Kurucz,
2017) calculated with ExoCTK for the F210M filter for $\mu>0.05$ where $\mu$
is the cosine of the angle between the line of signt and the emergent
intensity (e.g. Kipping, 2013) for a 4 parameter non-linear law. We use the
starry lightcurve model, which uses a polynomial limb darkening law (Agol et
al., 2020), so we fit the intensity function from a 4 parameter nonlinear law
with a 6 parameter polynomial limb darkening law, which has been shown to be
accurate to $\lesssim 0.5$ ppm differences in flux (Agol et al., 2020). We
start with priors centered on the values from Bonomo et al. (2017) for the
period, inclination, eccentricity and argument of pericenter, but widened
these in case of systematic errors to 83.5 $\pm 0.3^{\circ}$, $e$=0.1071 $\pm$
0.01 and $\omega$=106.1 $\pm 5^{\circ}$. We use the ephemeris from publicly
vailable TESS data and a wide a/R∗ of 8.9 $\pm$ 1.0 centered on the value from
Stassun et al. (2017). The resulting lightcurve fits and resiuals are shown in
Figure 9. We show a model fit that does not account for two of the systematics
to better illustrate them as well as another that accounts for them (the
charge trap ramp and tilt event jump).
### 4.2 Photometric Performance
After fitting the lightcurve with a transit model that includes a jump
correction from a phase retrieval, an exponential charge trapping ramp and
quadratic baseline described in Section 4.1, we analyze the statistical
properties of the residuals. We include all points that are not marked as
outliers (all points within 5 $\sigma$ of the best fit model) which includes
775 out of 780 total integrations. The standard deviation (scatter) in all of
the non-outlier residuals is 152 ppm, compared to a theoretical limit of 107
ppm from photon and read noise. Thus, the measured noise is 42% larger than
the theoretical limit, some of which could be due to 1/f noise (Schlawin et
al., 2020). Even after ROEBA subtraction, there remains higher frequency
(shorter timescale than the 5.24 ms row read time) noise. We note that the
background annulus subtraction reduced the standard deviation of out-of-
transit flux in the lightcurve by 37% over skipping the background annulus
subtraction so there is likely residual 1/f noise. We next assess time-
correlations in the data by binning the residuals.
A key metric in the performance of the NIRCam photometry is how well the noise
scales as a function of bin size. This commonly is presented in the form of an
Allan Variance plot (e.g. Pont et al., 2006; Croll et al., 2011). In this
plot, the photometric scatter is computed as a standard deviation as a
function of bin size. In the case where each time sample is independent, the
noise drops as $1/\sqrt{N}$ and this is plotted as “white noise scaling”. When
the measured photometric scatter begins to diverge from the $1/\sqrt{N}$ power
law, this is a sign that the time samples are correlated and time-dependent
systematics are present.
Figure 10 (left) shows that the noise falls with $1/\sqrt{N}$. Thus, no noise
floor is measurable down to the 20 ppm level, but 1/f noise limits the
precision in our current analysis to 42% above the theoretical limit. It may
be possible to optimize the aperture to a hexagonal aperture, but previous
tests with a circular annulus had little difference in noise performance on
simulated data. For this Allan variance analysis, we have removed the first
tilt event from the lightcurve using a model discussed in Section 3.1, a
quadratic trend with time and an exponential settling ramp discussed in
Section 3.4. For comparison, we also calculated an Allan variance curve when
the baseline trend was fit with a linear polynomial instead of a quadratic
polynomial in time. In this curve, the noise begins to grow near 5 minutes
where the noise is about 60 ppm. Thus, there is some curvature to the
lightcurve either related to instrument or astrophysical trends.
Figure 10: The lightcurve errors per bin drop nearly as $1/\sqrt{N}$ for $N$
time points, after correcting for the exponential startup, quadratic trend and
transit (left plot curve). Thus, the noise is largely independent for each
integration. If we only include a linear baseline trend to fit the lightcurves
(right), there is excess noise beginning for bin sizes around 4 minutes. The
precision before time binning is about 42% worse than the ideal limit of
photon and read noise, likely because of 1/f noise correlations between pixels
within each frame.
The measured 20 ppm lightcurve scatter at a time bin size of 30 minutes can be
compared to the state-of-the-art best precision lightcurves from space-based
photometers. For the brightest targets observed by Kepler, the precision was
measured to be $\sim$15 ppm (Jenkins et al., 2010). For the bright exoplanet
system 55 Cnc, the precision with TESS was 10 ppm for 30 minutes (Meier Valdés
et al., 2022) and for CHEOPS the precision achievable on a 1.6 hour eclipse
observed 41 times using only out-of-eclipse data was 3 ppm (Demory et al.,
2022), which would scale to 3 ppm * sqrt(1.6 hours/0.5 hours * 41 eclipses
/2)=20 ppm for 30 minutes. The scaling uses the relative time of the eclipse
and the half hour benchmark, 41 separate eclipse events that were averaged and
finally a factor of 2 accounting for the fact that the out-of-transit to in-
transit ratio adds another factor of sqrt(2) in noise. Elsewhere, the
instrumental noise on hour-long timescales for CHEOPS is reported to be 15 to
80 ppm (Maxted et al., 2022). In the JWST photometry presented in this work,
precision is limited by photon counting statistics, so brighter sources with
more photons per minute will be needed to assess what is the highest possible
precision as compared to the state-of-the-art performance.
### 4.3 Pointing Performance
Target acquisition was successful and placed the target at X=1060.7 px,
Y=167.5 px on the NRCA3 detector in full frame 1-based Data Management System
coordinates. Pointing was stable to 0.01 pixels (0.3 mas) in the X direction
and 0.009 px (0.3 mas) in the Y direction. Some of the higher frequency jitter
is thus averaged over the 27 second long integrations. Thus, JWST attitude
control provides incredible pointing stability at the 10-2 pixel level and
this produces no noticeable changes in flux with position and jitter, likely
constituting a negligible part of the error budget. The guide star has an FGS
magnitude of 15.3 and we expect similarly high precision pointing performance
for guide stars from magnitude 12.5 to 15.5. Under the HgCdTe crosshatching
structures on the NIRCam ALONG/A5 detector, 2 mas pointing was expected to
produce 6 ppm changes of flux, so 0.3 mas jitter is likely to matter at the
single digit ppm level and flux changes at this level cannot be measured for
the HAT-P-14 target.
### 4.4 High Gain Antenna Move
JWST high gain antenna moves maintain pointing at the designated ground
stations on Earth’s Deep Space Network. While high gain antenna moves should
happen during slews or between exposures, the moves are permitted to occur
after 10,000 s during long time series observations with JWST. We had a high
gain antenna adjustment at 2022-05-02T10:10:01 UTC, which was measurable with
the Fine Guidance Sensor and NIRCam short wavelength time series centroids. As
shown in Figure 11, the pointing change from the HGA move settles very quickly
in less than 0.5 minutes. Furthermore, the position was returned back to the
original pointing to within 1 mas. The data around an HGA move can be
discarded and in the case of weak lens photometry that is spread over many
pixels, only produced a transient 500 ppm change in flux. Thus, HGA moves are
not expected to cause significant issues to most time series observations as
long as the Fine Guide mode does not lose a guide star from its subarray.
Figure 11: The High Gain Antenna moves are unlikely to have a big impact on
time series observations. The HGA move only affects about 0.5 minutes of data,
which is mostly corrected by shifting the aperture but could be excluded from
the lightcurve if it is an outlier. The normalized flux changes by a maximum
of about 0.5 parts per thousand (ppt). The plate scale of NIRCam’s Short
Wavelength detector is 31 mas/px
.
### 4.5 Timing Accuracy
We find a transit center time of 2022-05-02T10:34:55 +/- 7 seconds BJDTDB,
which is consistent (within 1.7 $\sigma$) with a prediction from the TESS
ephemeris of 2022-05-02T10:36:35.1 +/- 60 seconds BJDTDB. Other planets with
higher precision ephemerides will test JWST timing to higher accuracy. There
is also a dedicated GO program program identification number 1666 (PI Poshak
Gandhi) that will use a double white dwarf binary and is expected to calibrate
the absolute timing of an exposure to the 100 ms level.
### 4.6 HAT-P-14 B Contamination
The HAT-P-14 system was observed from the ground with Palomar adaptive optics
(AO) on 2021-08-08 UTC to assess the contamination and dilution of the transit
depth and find any unknown companions. No companions were detected with a
contrast less than $\Delta$mag = 7.0 within 0.5″. The nearby stellar companion
HAT-P-14 B (Ngo et al., 2015) was also imaged with Palomar AO to better
constrain the infrared colors and contamination on the transit depth. The
separation is 0.85″ at a position angle of 264 degrees, which is well within
the 2.5″ source aperture used in the lightcurves of the central (hexagonal)
portion of the PSF shown in Figure 1. With the Palomar AO imaging, we find a
delta K magnitude between the HAT-P-14 A and B of $\Delta$K=4.99, which is a
similar wavelength to this JWST F210M photometry. The JWST target acquisition
image shown in Figure 12 shows both HAT-P-14 A and HAT-P-14 B. Using the
target acquisition image, we find that the contrast is $\Delta$ MagF335M=5.03
$\pm$ 0.05 for the F335M filter (3.35 µm) and the separation is 0.83″. This
contrast is a similar to $K$ band, as expected for the Raleigh Jeans limit for
long wavelengths.
Using webbpsf, we simulate the overlap of two Weak Lens +8 PSFs for a 81 pixel
aperture and estimate the transit depth dilution. We use the flight wavefront
optical path difference evaluated at a time of 2022-05-02T10:30 UTC using
webbpsf and perform aperture photometry on the simulated PSFs of HAT-P-14 A
and HAT-P-14 B separately. For the F210M filter at 2.1 µm, we find that
HAT-P-14 B should dilute the fitted 6540 $\pm$ 22 ppm transit depth by 59 ppm
for an undiluted transit depth 6599 $\pm$ 22 ppm. This is within 3.2 $\sigma$
of the long wavelength F322W2 broadband result of 6670 ppm, but with
independent fitting methods and priors on orbital parameters and limb
darkening. This is also consistent with the NIRSpec time series broadband fit
with 6627 $\pm$ 8 ppm with an independent analysis (Espinoza et al., 2022),
with different priors on orbital parameters and limb darkening. We do not
expect the atmosphere to contribute much more than 15 ppm as described in
Section 1 for 0.9 atmospheric scale heights, but atmospheric gases could
contribute at 1.5$\sigma$ level for 2 atmospheric scale heights of absorption.
Figure 12: Long Wavelength Target Acquisition centered near HAT-P-14 A with a
faint ($\Delta$mag=5) stellar companion HAT-P-14 B (upper left) at 860 mas
separation that dilutes the transit depth.
## 5 Conclusions
Transiting exoplanet science and other time-variability studies will benefit
from JWST high precision time series photometry using the defocused photometry
that is collected simultaneously with the grism time series mode. Here, we
present performance results from the hot Jupiter HAT-P-14 b that was observed
to commission the instruments with a known flat transmission spectrum signal.
We detect the planet transit at high precision and find that the transit depth
is consistent with the NIRSpec result to within 28 ppm $\pm$ 20 ppm as
expected for this relatively high gravity planet.
The weak lens time series is particularly informative about small changes in
the PSF as a result of wavefront variations, and we found two clear tilt
events where two different hexagonal primary mirror segments quickly changed
orientation with a timescale $\lesssim 1.4$ seconds. Difference images clearly
show the tilt events associated with specific mirror segments. These tilt
events can cause jumps in time series’ signal levels but can be predicted
effectively for short wavelengths using a wavefront model. We also show how
tilt events can be sensed with other instrument modes that do not use
defocusing optics. FGS differential images and the FWHM of the PSF can both
reveal tilt events as step functions and will be good metrics to assess mirror
stability.
The NRCA3 detector exhibited a charge trapping persistence ramp that is
similar in shape to the ones seen on HST, but it settles out quickly on a 5-15
minutes decay timescale with an amplitude of 660 to 700 ppm and does not have
a big impact on the time series.
The overall precision is very high and has a low scatter over the residuals of
152 ppm compared to the photon and read noise value of 107 ppm per 27 second
integration, so it is only 41% above the theoretical limit for this quiescent
F-type $K$=8.85 star. Some of this excess noise may be due to residual 1/f
noise not corrected by the row-by-row subtraction. The noise bins down
approximately with the square root of the number of points in a time bin,
indicating that systematic noise is minimal after the exponential and jump
corrections and a quadratic polynomial as a function of time. Overall, the
prospects are good for high precision time series measurements of exoplanets
and other astrophysical phenomena. Defoucsed photometric monitoring will be
valuable for measuring mirror tilt events and mitigating their impact on high
precision measurements.
## acknowledgements
Funding for E Schlawin is provided by NASA’s Goddard Space Flight Center. T
Greene acknowledges funding from the JWST project via NASA WBS
411672.05.05.02.02. This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France; and NASA’s Astrophysics Data System
Bibliographic Services. Part of the research was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration (80NM0018D0004). We
respectfully acknowledge the University of Arizona is on the land and
territories of Indigenous peoples. Today, Arizona is home to 22 federally
recognized tribes, with Tucson being home to the O’odham and the Yaqui.
Committed to diversity and inclusion, the University strives to build
sustainable relationships with sovereign Native Nations and Indigenous
communities through education offerings, partnerships, and community service.
Thank you to Eddie Bergeron for sharing analysis on detecting tilt events with
FGS.
## Appendix A Subarray Positions
To help orient JWST users on the subarrays and relative positions of the short
wavelength and long wavelength detector images, we provide diagrams of the
locations as measured by in-flight data. Figure 13 displays the relative
locations, in telescope coordinates, of the NIRCam LW grism trace (dashed
lines) and the NIRCam SW hexagonal PSF (green hexagon) from the +8 wave
defocused pupil element (Weak Lens +8), which are observed simultaneously.
There is a small amount of refraction with the grism so there is a vertical
offset between the two positions as placed in the same coordinate system. The
edges of the full frame images for the A1 (short wavelength), A3 (short
wavelength) and A5 (long wavelength) detectors are all shown as dotted lines
while the edges of the SUBGRISM128 subarrays for the same 3 detectors are
shown as solid lines. The positions here are shown for Cycle 1 after
commissioning where small (8-10 SW px) tweaks were made to better-center the
A1 and A3 subarrays on the hexagonal PSF. The reference pixels on JWST NIRCam
detectors are at the full frame boundaries, so the relative positions of the
both SUBGRISM64 and SUBGRISM128 subarrays necessitate excluding the bottom 4
reference pixel rows on the short wavelength NRCA1 and NRCA3 detectors.
Significant improvements in performance can be made if background pixels are
used in the same manner as reference pixels to correct for amplifier offsets
and 1/f noise, as discussed in Section 2.2.
Figure 13: The NIRCam SW and LW channels observe approximately the same region
of the sky simultaneously, however the overlap of detectors is not exact. The
relative position of the SW defocused weak lens image (green hexagon) is shown
compared to the LW grism F322W2 (top figure) and F444W (bottom figure) central
traces (brown dashed lines). The SW SUBGRISM128 subarrays on the A1 and A3
detectors (blue solid lines) are aligned to capture light from the hexagonal
+8 wave PSF while the the LW SUBGRISM128 subarray (solid red line) is aligned
to capture light from the LW grism trace and surrounding background area. The
full frame detector boundaries (dotted lines) are where the 4 reference pixels
are located, so the bottom references are included with the LW SUBGRISM128
subarray (solid red line) but no bottom reference pixels are available in the
A1 and A3 SUBGRISM128 subarrays (solid blue lines). All positions are shown in
V2 and V3 Telescope coordinate system using pysiaf (Sahlmann et al., 2019)
after 8-10 SW pixel adjustments were made to the A1 and A3 subarrays following
HAT-P-14 commissioning observations to better center the A1 and A3 subarrays
on the hexagonal PSF.
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# Exact Controllability for a Refined Stochastic Plate Equation111This work is
supported by the NSF of China under grants 12025105, 11971334 and 11931011, by
the Chang Jiang Scholars Program from the Chinese Education Ministry, and by
the Science Development Project of Sichuan University under grants
2020SCUNL101 and 2020SCUNL201.
Qi Lü 222School of Mathematics, Sichuan University, Chengdu, P. R. China.
Email<EMAIL_ADDRESS> and Yu Wang 333School of Mathematics, Sichuan
University, Chengdu, P. R. China. Email<EMAIL_ADDRESS>
###### Abstract
A widely used stochastic plate equation is the classical plate equation
perturbed by a term of Itô’s integral. However, it is known that this equation
is not exactly controllable even if the controls are effective everywhere in
both the drift and the diffusion terms and also on the boundary. In some
sense, this means that some key feature has been ignored in this model. Then,
a one-dimensional refined stochastic plate equation is proposed and its exact
controllability is established in [28]. In this paper, by means of a new
global Carleman estimate, we establish the exact controllability of the
multidimensional refined stochastic plate equation with two interior controls
and two boundary controls. Moreover, we give a result about the lack of exact
controllability, which shows that the action of two interior controls and at
least one boundary control is necessary.
AMS Mathematics Subject Classification. 93B05, 93B07.
Keywords. Stochastic plate equation, exact controllability, observability
estimate, Carleman estimate.
## 1 Introduction
Let $T>0$ and $(\Omega,\mathcal{F},\mathbf{F},\mathbb{P})$ with
$\mathbf{F}=\\{\mathcal{F}_{t}\\}_{t\geq 0}$ be a complete filtered
probability space on which a one-dimensional standard Brownian motion
$\\{W(t)\\}_{t\geq 0}$ is defined and $\mathbf{F}$ is the natural filtration
generated by $W(\cdot)$, augmented by all the $\mathbb{P}$ null sets in
$\mathcal{F}$. Write $\mathbb{F}$ for the progressive $\sigma$-field with
respect to $\mathbf{F}$. Let $H$ be a Banach space. Denote by
$L^{2}_{\mathcal{F}_{t}}(\Omega;H)$ the space of all
$\mathcal{F}_{t}$-measurable random variables $\xi$ such that
$\mathbb{E}|\xi|_{H}^{2}<\infty$; by $L_{\mathbb{F}}^{2}(0,T;H)$ the space
consisting of all $H$-valued $\mathbf{F}$-adapted processes $X(\cdot)$ such
that $\mathbb{E}\bigl{(}|X(\cdot)|_{L^{2}(0,T;H)}^{2}\bigr{)}<\infty$; by
$L_{\mathbb{F}}^{\infty}(0,T;H)$ the space consisting of all $H$-valued
$\mathbf{F}$-adapted bounded processes; and by
$C_{\mathbb{F}}([0,T];L^{2}(\Omega;H))$ the space consisting of all $H$-valued
$\mathbf{F}$-adapted processes $X(\cdot)$ such that $X(\cdot):[0,T]\rightarrow
L^{2}_{\mathcal{F}_{\cdot}}(\Omega;H)$ is continuous. All these spaces are
Banach spaces with the canonical norms (e.g., [24, Section 2.6]).
Let $G\subset\mathbb{R}^{n}$ ($n\in\mathbb{N}$) be a bounded domain with a
$C^{4}$ boundary $\Gamma$. Set $Q=(0,T)\times G$ and
$\Sigma=(0,T)\times\Gamma$. Denote by $\nu(x)=(\nu^{1}(x),\cdots,\nu^{n}(x))$
the unit outward normal vector of $\Gamma$ at point $x$.
Consider the following refined stochastic plate equation:
$\left\\{\begin{aligned} &dy=\hat{y}dt+(a_{3}y+f)dW(t)&&\quad\text{ in }Q,\\\
&d\hat{y}+\Delta^{2}ydt=(a_{1}y+a_{2}\cdot\nabla
y+a_{5}g)dt+(a_{4}y+g)dW(t)&&\quad\text{ in }Q,\\\ &y=h_{1},\frac{\partial
y}{\partial\nu}=h_{2}&&\quad\text{ on }\Sigma,\\\
&(y(0),\hat{y}(0))=(y_{0},\hat{y}_{0})&&\quad\text{ in
}G.\end{aligned}\right.$ (1.1)
Here, $(y_{0},\hat{y}_{0})\in H^{-1}(G)\times(H^{3}(G)\cap H^{2}_{0}(G))^{*}$
(where $(H^{3}(G)\cap H^{2}_{0}(G))^{*}$ is the dual space of $H^{3}(G)\cap
H^{2}_{0}(G)$ with respect to the pivot space $L^{2}(G)$), the coefficients
$\displaystyle a_{1},a_{3},a_{4}\in
L^{\infty}_{\mathbb{F}}(0,T;W^{1,\infty}(G)),\quad a_{2}\in
L^{\infty}_{\mathbb{F}}(0,T;W^{2,\infty}(G;\mathbb{R}^{n})),\quad a_{5}\in
L^{\infty}_{\mathbb{F}}(0,T;W^{3,\infty}(G)).$
and the controls
$\displaystyle(f,g,h_{1},h_{2})$ $\displaystyle\in
L^{2}_{\mathbb{F}}(0,T;H^{-1}(G))\times L^{2}_{\mathbb{F}}(0,T;(H^{3}(G)\cap
H^{2}_{0}(G))^{*})\times L^{2}_{\mathbb{F}}(0,T;L^{2}(\Gamma))$
$\displaystyle\quad\times L^{2}_{\mathbb{F}}(0,T;H^{-1}(\Gamma)).$
###### Remark 1.1.
The term $a_{5}g$ reflects the influence of the control $g$ in the diffusion
term on the drift term, i.e., if one puts a control $g$ in the diffusion term
and $a_{5}g$ will appear as a side effect. This leads to some technical
difficulties in the study of the exact controllability of Eq. 1.1.
The control system Eq. 1.1 is a nonhomegeneous boundary value problem. Its
solution is understood in the sense of transposition. For the readers’
convenience, we recall it briefly below. A systematic introduction to that can
be found in [24, Section 7.2].
First, we introduce the following reference equation:
$\displaystyle\left\\{\begin{aligned}
&dz=\hat{z}dt+(Z-a_{5}z)dW(t)&&\quad\text{ in }Q_{\tau},\\\
&d\hat{z}+\Delta^{2}zdt=[(a_{1}-\operatorname{div}a_{2}-a_{4}a_{5})z-a_{2}\cdot\nabla
z-a_{3}\hat{Z}+a_{4}Z]dt+\hat{Z}dW(t)&&\quad\text{ in }Q_{\tau},\\\
&z=\frac{\partial z}{\partial\nu}=0&&\quad\text{ on }\Sigma_{\tau},\\\
&(z(\tau),\hat{z}(\tau))=(z^{\tau},\hat{z}^{\tau})&&\quad\text{ in
}G,\end{aligned}\right.$ (1.2)
where $\tau\in(0,T]$, $Q_{\tau}\mathop{\buildrel\Delta\over{=}}(0,\tau)\times
G$, $\Sigma_{\tau}\mathop{\buildrel\Delta\over{=}}(0,\tau)\times\Gamma$, and
$(z^{\tau},\hat{z}^{\tau})\in L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap
H^{2}_{0}(G))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{1}_{0}(G))$. By the
classical well-posedness result for backward stochastic evolution equations
(e.g.,[24, Section 4.2]), we know that (1.2) admits a unique weak solution
$\displaystyle(z,Z,\hat{z},\hat{Z})$ $\displaystyle\in
L^{2}_{\mathbb{F}}(\Omega;C([0,\tau];(H^{3}(G)\cap H^{2}_{0}(G))))\times
L^{2}_{\mathbb{F}}(0,\tau;(H^{3}(G)\cap H^{2}_{0}(G)))$
$\displaystyle\quad\times
L^{2}_{\mathbb{F}}(\Omega;C([0,\tau];H_{0}^{1}(G)))\times
L^{2}_{\mathbb{F}}(0,\tau;H_{0}^{1}(G)).$
Furthermore, for $0\leq s,t\leq\tau$, it holds that
$\displaystyle|(z(t),\hat{z}(t))|_{L^{2}_{\mathcal{F}_{t}}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathcal{F}_{t}}(\Omega;H_{0}^{1}(G))}$ (1.3)
$\displaystyle\leq
C(|(z(s),\hat{z}(s))|_{L^{2}_{\mathcal{F}_{s}}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathcal{F}_{s}}(\Omega;H_{0}^{1}(G))}$
$\displaystyle\qquad+|(Z,\hat{Z})|_{L^{2}_{\mathbb{F}}(0,\tau;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathbb{F}}(0,\tau;H_{0}^{1}(G))}).$
Here and in what follows, we denote by $C$ a generic positive constant
depending on $G$, $T$, $\tau$ and $a_{i}$, $i=1,\cdots,5$, whose value may
vary from line to line.
Next, we give the following hidden regularity for solutions to Eq. 1.2.
###### Proposition 1.1.
Let $(z^{\tau},\hat{z}^{\tau})\in
L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap H^{2}_{0}(G))\times
L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{1}_{0}(G))$. Then the solution
$(z,Z,\hat{z},\hat{Z})$ of Eq. 1.2 satisfies $|\nabla\Delta z||_{\Gamma}\in
L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))$. Furthermore,
$\displaystyle|\nabla\Delta z|_{L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))}$
$\displaystyle\leq
C|(z^{\tau},\hat{z}^{\tau})|_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap
H^{2}_{0}(G))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{1}_{0}(G))}.$
Proof of Proposition 1.1 is put in Section 2.
Now we are in a position to give the definition of the transposition solution
to Eq. 1.1.
###### Definition 1.1.
A pair of stochastic processes $(y,\hat{y})\in
C_{\mathbb{F}}([0,T];L^{2}(\Omega;H^{-1}(G)))\times$ $C_{\mathbb{F}}([0,T];$
$L^{2}(\Omega;(H^{3}(G)\cap H^{2}_{0}(G))^{*}))$ is a transposition solution
to Eq. 1.1 if for any $\tau\in(0,T]$ and $(z^{\tau},\hat{z}^{\tau})\in
L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap H^{2}_{0}(G))\times$
$L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{1}_{0}(G))$, we have
$\begin{array}[]{ll}\displaystyle\mathbb{E}\langle\hat{y}(\tau),z^{\tau}\rangle_{(H^{3}(G)\cap
H^{2}_{0}(G))^{*},H^{3}(G)\cap
H^{2}_{0}(G)}-\langle\hat{y}_{0},z(0)\rangle_{(H^{3}(G)\cap
H^{2}_{0}(G))^{*},H^{3}(G)\cap H^{2}_{0}(G)}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle-\mathbb{E}\langle
y(\tau),\hat{z}^{\tau}\rangle_{H^{-1}(G),H_{0}^{1}(G)}+\langle
y_{0},\hat{z}(0)\rangle_{H^{-1}(G),H_{0}^{1}(G)}\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle=-\mathbb{E}\int_{0}^{\tau}\langle
f,\widehat{Z}\rangle_{H^{-1}(G),H_{0}^{1}(G)}dt+\mathbb{E}\int_{0}^{\tau}\langle
g,Z\rangle_{(H^{3}(G)\cap H^{2}_{0}(G))^{*},H^{3}(G)\cap H^{2}_{0}(G)}dt\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\quad+\mathbb{E}\int_{0}^{\tau}\int_{\Gamma}\frac{\partial\Delta
z}{\partial\nu}h_{1}d\Gamma dt-\mathbb{E}\int_{0}^{\tau}\langle h_{2},\Delta
z\rangle_{H^{-1}(\Gamma),H^{1}(\Gamma)}dt.\end{array}$
Here, $(z,Z,\hat{z},\hat{Z})$ solves Eq. 1.2.
Combining Proposition 1.1 and the well-posedness for stochastic evolution
equation with unbounded control operator in the sense of transposition
solution (e.g., [24, Theorem 7.12]), we immediately get the following well-
posedness result for Eq. 1.1.
###### Proposition 1.2.
For each $(y_{0},\hat{y}_{0})\in H^{-1}(G)\times(H^{3}(G)\cap
H^{2}_{0}(G))^{*}$, the system Eq. 1.1 admits a unique transposition solution
$(y,\hat{y})$. Moreover,
$\displaystyle|(y,\hat{y})|_{C_{\mathbb{F}}([0,T];L^{2}(\Omega;H^{-1}(G)))\times
C_{\mathbb{F}}([0,T];L^{2}(\Omega;(H^{3}(G)\cap H^{2}_{0}(G))^{*}))}$
$\displaystyle\leq C\big{(}|y_{0}|_{H^{-1}(G)}+|\hat{y}_{0}|_{(H^{3}(G)\cap
H^{2}_{0}(G))^{*}}+|f|_{L^{2}_{\mathbb{F}}(0,T;H^{-1}(G))}+|g|_{L^{2}_{\mathbb{F}}(0,T;(H^{3}(G)\cap
H^{2}_{0}(G))^{*})}$
$\displaystyle\quad\quad+|h_{1}|_{L^{2}_{\mathbb{F}}(0,T;L^{2}(\Gamma))}+|h_{2}|_{L^{2}_{\mathbb{F}}(0,T;H^{-1}(\Gamma))}\big{)}.$
Now we give the definition of the exact controllability for Eq. 1.1.
###### Definition 1.2.
The system Eq. 1.1 is called exactly controllable at time $T$ if for any
$(y_{0},\hat{y}_{0})\in H^{-1}(G)\times(H^{3}(G)\cap H^{2}_{0}(G))^{*}$ and
$(y_{1},\hat{y}_{1})\in L^{2}_{\mathcal{F}_{T}}(\Omega;H^{-1}(G))\times
L^{2}_{\mathcal{F}_{T}}(\Omega;(H^{3}(G)\cap H^{2}_{0}(G))^{*})$, there exist
controls
$\displaystyle(f,g,h_{1},h_{2})$ $\displaystyle\in
L^{2}_{\mathbb{F}}(0,T;H^{-1}(G))\times L^{2}_{\mathbb{F}}(0,T;(H^{3}(G)\cap
H^{2}_{0}(G))^{*})\times L^{2}_{\mathbb{F}}(0,T;L^{2}(\Gamma))$
$\displaystyle\quad\times L^{2}_{\mathbb{F}}(0,T;H^{-1}(\Gamma))$
such that the solution $(y,\hat{y})$ to Eq. 1.1 satisfies that
$(y(T,\cdot),\hat{y}(T,\cdot))=(y_{1},\hat{y}_{1})$, ${\mathbb{P}}$-a.s.
###### Remark 1.2.
In the definition of the exact controllability for Eq. 1.1, we put the state
space to be $L^{2}_{\mathcal{F}_{T}}(\Omega;H^{-1}(G))\times
L^{2}_{\mathcal{F}_{T}}(\Omega;(H^{3}(G)\cap H^{2}_{0}(G))^{*})$. It is
natural to choose the state space as
$L^{2}_{\mathcal{F}_{T}}(\Omega;L^{2}(G))\times
L^{2}_{\mathcal{F}_{T}}(\Omega;H^{-2}(G))$. Further, the controls $g\in
L^{2}_{\mathbb{F}}(0,T;(H^{3}(G)\cap H^{2}_{0}(G))^{*})$ and $h_{2}\in
L^{2}_{\mathbb{F}}(0,T;H^{-1}(\Gamma))$ are very irregular. We expect to use
more regular controls to achieve the desired goal. However, we do not know how
to do that now. Indeed, even for the deterministic plate equation, to the best
of our knowledge, the existing results (e.g., [15]) can only prove the exact
controllability in the space $H^{-1}(G)\times(H^{3}(G)\cap H^{2}_{0}(G))^{*}$
with controls in the Dirichlet and Neumann boundary conditions.
The main result of this paper is the following.
###### Theorem 1.3.
The system Eq. 1.1 is exactly controllable at any time $T>0$.
###### Remark 1.3.
Similar to the derivation process in [28, 23], the refined stochastic plate
equation Eq. 1.1 can be obtained from the classical stochastic plate equation:
$\left\\{\begin{aligned} &dy_{t}+\Delta^{2}ydt=(a_{1}y+a_{2}\cdot\nabla
y+f)dt+(a_{4}y+g)dW(t)&&\quad\text{ in }Q,\\\ &y=h_{1},\frac{\partial
y}{\partial\nu}=h_{2}&&\quad\text{ on }\Sigma,\\\
&(y(0),y_{t}(0))=(y_{0},y_{1})&&\quad\text{ in }G.\end{aligned}\right.$ (1.4)
Here, $(y_{0},y_{1})$ are the initial data, and $f,g,h_{1},h_{2}$ are
controls.
The system Eq. 1.4 is widely used in structural engineering, and can be
applied to beams, bridges and other structures, see [3, 12, 4, 2]. In
particular, Eq. 1.4 can be used to characterize fluttering or large-amplitude
vibration of an elastic panel excited by aerodynamic forces which are
perturbed by random fluctuations (e.g., [3]). However, similar to Theorem 4.1
in [28] and Theorem 2.1 in [23], one can show that the system Eq. 1.4 is not
exactly controllable for any $T>0$. Inspired by the negative controllability
result, and similar to [28, 23], we study a refined stochastic plate equation
Eq. 1.1.
We put four controls in the system Eq. 1.1. Similarly to Theorem 2.3 in [23],
one can find that boundary controls $h_{1}$ and $h_{2}$ in Eq. 1.1 can not be
dropped simultaneously, and internal controls $f$ and $g$ must be acted on the
whole domain $G$. More precisely, we have the following result.
###### Theorem 1.4.
The system Eq. 1.1 is not exactly controllable at any time $T>0$ provided that
one of the following three conditions is satisfied:
1. (1).
$a_{3}\in C_{\mathbb{F}}([0,T];L^{\infty}(G))$,
$G\backslash\overline{G_{0}}\neq\emptyset$ and $\operatorname{supp}f\subset
G_{0}$;
2. (2).
$a_{4}\in C_{\mathbb{F}}([0,T];L^{\infty}(G))$,
$G\backslash\overline{G_{0}}\neq\emptyset$ and $\operatorname{supp}g\subset
G_{0}$;
3. (3).
$h_{1}=h_{2}=0$.
###### Remark 1.4.
It is worth studying whether one the of boundary controls can be removed. This
can be done for deterministic plate equation (e.g., [15, 18]). However, we
have no idea for how to do that.
###### Remark 1.5.
By letting $G=(0,1)$, we can deduce from Theorem 3.2 that the system Eq. 1.1
is exactly controllable with controls in any nonempty subset of the boundary
$\Gamma$, which has recently been proved in [28]. In fact, thanks to Remark
1.7, for any $(y_{0},\hat{y}_{0})$ and $(y_{1},\hat{y}_{1})$ satisfing
Definition 1.2, one can find $(f,g,h_{1},h_{2})\in
L^{2}_{\mathbb{F}}(0,T;H^{-1}(G))\times L^{2}_{\mathbb{F}}(0,T;(H^{3}(G)\cap
H^{2}_{0}(G))^{*})\times(L^{2}_{\mathbb{F}}(0,T))^{2}$ such that the solution
$(y,\hat{y})$ to Eq. 1.1, where the boundary conditions are
$\displaystyle y(\cdot,0)=h_{1},\quad y_{x}(\cdot,0)=h_{2},\quad
y(\cdot,1)=0,\quad y_{x}(\cdot,1)=0,\quad\text{~{} on }(0,T)$
satisfies that $(y(T,\cdot),\hat{y}(T,\cdot))=(y_{1},\hat{y}_{1})$. In the
multidimensional case, it is worth to studying whether Eq. 1.1 is still
exactly controllable under the assumption that
$(h_{1},h_{2})\in(L^{2}_{\mathbb{F}}(0,T;L^{2}(\Gamma_{0})))^{2}$, where
$\Gamma_{0}$ is a nonempty subset of $\Gamma$.
By a standard duality argument, Theorem 1.3 is equivalent to the following
observability estimate (e.g., [24, Theorem 7.17]).
###### Theorem 1.5.
There exists a constant $C>0$ such that for every $(z^{T},\hat{z}^{T})\in
L_{\mathcal{F}_{T}}^{2}(\Omega;H^{3}(G)\cap H^{2}_{0}(G))\times
L_{\mathcal{F}_{T}}^{2}(\Omega;H^{1}_{0}(G))$, it holds that
$\displaystyle|(z^{T},\hat{z}^{T})|^{2}_{L_{\mathcal{F}_{T}}^{2}(\Omega;H^{3}(G)\cap
H^{2}_{0}(G))\times L_{\mathcal{F}_{T}}^{2}(\Omega;H^{1}_{0}(G))}$
$\displaystyle\leq C\mathbb{E}\int_{\Sigma}(|\nabla\Delta z|^{2}+|\Delta
z|^{2})d\Gamma dt+C|(Z,\hat{Z})|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G)\cap
H^{2}_{0}(G))\times L^{2}_{\mathbb{F}}(0,T;H^{1}_{0}(G))},$
where $(z,Z,\hat{z},\hat{Z})$ is the solution to the equation Eq. 1.2 with
$\tau=T$, $z(T)=z^{T}$, and $\hat{z}(T)=\hat{z}^{T}$.
###### Remark 1.6.
A sharp trace estimate for the deterministic plate equation is established in
[17]. It suggests that we may need better regularity than $H_{0}^{2}(G)\times
L^{2}(G)$ for the initial datum of dual system Eq. 1.2 to estimate
$|\nabla\Delta z|$ on the boundary. This is the reason that we choose the
final data of Eq. 1.2 in $L_{\mathcal{F}_{T}}^{2}(\Omega;H^{3}(G)\cap
H^{2}_{0}(G))\times L_{\mathcal{F}_{T}}^{2}(\Omega;H^{1}_{0}(G))$.
###### Remark 1.7.
Let $G=(0,1)$. By choosing $x_{0}>1$ in Eq. 3.4, from the proof of Theorems
3.2 and 1.5, we can deduce that
$\displaystyle|(z^{T},\hat{z}^{T})|^{2}_{L_{\mathcal{F}_{T}}^{2}(\Omega;H^{3}(G)\cap
H^{2}_{0}(G))\times L_{\mathcal{F}_{T}}^{2}(\Omega;H^{1}_{0}(G))}$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T}\big{(}|z_{xxx}(0)|^{2}+|z_{xx}(0)|^{2}\big{)}dt+C|(Z,\hat{Z})|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G)\cap
H^{2}_{0}(G))\times L^{2}_{\mathbb{F}}(0,T;H^{1}_{0}(G))}.$
This is the main result in [28].
###### Remark 1.8.
One can also consider the exact controllability of a refined stochastic palte
equation when the boundary controls are as follows:
$\displaystyle y=h_{1}\quad\mbox{ and }\quad\Delta y=h_{2}\text{~{} on
~{}}\Sigma.$
By the standard duality argument and the technique of transforming the
controllability of the forward stochastic equation into the controllability of
a backward equation (e.g., [24, Section 7.5]), we only need to prove that for
any $(z_{0},\hat{z}_{0})\in\\{\eta\in H^{3}(G)|\Delta\eta\in
H^{1}_{0}(G)\\}\times H^{1}_{0}(G)$, it holds that
$\displaystyle|(\Delta z_{0},\hat{z}_{0})|^{2}_{H_{0}^{1}(G)\times
H_{0}^{1}(G)}\leq
C\mathbb{E}\int_{0}^{T}\int_{\Gamma}\bigg{(}\bigg{|}\frac{\partial\Delta
z}{\partial\nu}\bigg{|}^{2}+\bigg{|}\frac{\partial
z}{\partial\nu}\bigg{|}^{2}\bigg{)}d\Gamma dt,$ (1.5)
where $(z,\hat{z})$ is the solution to
$\left\\{\begin{aligned} &dz=\hat{z}dt-a_{5}zdW(t)&&\quad\text{ in }Q,\\\
&d\hat{z}+\Delta^{2}zdt=[(a_{1}-\operatorname{div}a_{2})z-a_{2}\cdot\nabla
z-a_{4}a_{5}z]dt&&\quad\text{ in }Q,\\\ &z=\Delta z=0&&\quad\text{ on
}\Sigma,\\\ &(z(0),\hat{z}(0))=(z_{0},\hat{z}_{0})&&\quad\text{ in
}G.\end{aligned}\right.$ (1.6)
Following the idea in [29], we can rewrite the equation (1.6) as two coupled
stochastic Schrödinger equations, and the desired observability estimate can
be obtained from the Carleman estimate for the latter. In fact, letting
$u=i\hat{z}+\Delta z$, we have
$\displaystyle idz+\Delta zdt=udt-ia_{5}zdW(t),$
and
$\displaystyle-idu+\Delta udt$
$\displaystyle=[(a_{1}-\operatorname{div}a_{2})z-a_{2}\cdot\nabla
z-a_{4}a_{5}z]dt+i\Delta(a_{5}z)dW(t).$
Clearly, it holds that $z=u=0$ on $\Sigma$. Thanks to Theorem 1.2 in [20], we
can obtain that
$\displaystyle|(\Delta z_{0},\hat{z}_{0})|^{2}_{H_{0}^{1}(G)\times
H_{0}^{1}(G)}\leq
C\mathbb{E}\int_{0}^{T}\int_{\Gamma}\bigg{(}\bigg{|}\frac{\partial\Delta
z}{\partial\nu}\bigg{|}^{2}+\bigg{|}\frac{\partial
z}{\partial\nu}\bigg{|}^{2}+\bigg{|}\frac{\partial\hat{z}}{\partial\nu}\bigg{|}^{2}\bigg{)}d\Gamma
dt.$ (1.7)
Then one may follow the rechnique in [16] to derive (1.5) from (1.7). On the
other hand, one can also follow the technique in this paper to prove the
desired observability (1.5).
There are a large number of published works ([1, 9, 10, 11, 15, 18, 19, 26,
29, 5] and the references therein) studying the exact controllability for
deterministic plate equations. However, as far as we know, [28] is the only
published work investigating the exact controllability of stochastic beam
equations, in which the authors show that Eq. 1.1 is exactly controllable when
$G$ is an interval. In this paper, we shall use a stochastic Carleman estimate
to prove Theorem 1.5. Such kind of estimate is one of the most useful tool in
studying controllability for stochastic partial differential equations (see
[6, 8, 21, 20, 24, 22, 27] and the references given there). Nevertheless, [28]
is the only published work using this method to study the exact
controllability of stochastic beam equations.
The rest of this paper is organized as follows. In Section 2, we provide some
preliminaries. Section 3 is devoted to establishing a Carleman estimate for
the adjoint equation Eq. 1.2. By means of that Carleman estimate, we prove
Theorem 1.5 in Section 4. At last, Section 5 is addressed to the proof of
Theorem 1.4.
## 2 Some preliminary results
This section provides some preliminary results. In the rest of this paper, the
notation $y_{x_{i}}\equiv y_{x_{i}}(x)=\partial y(x)/\partial x_{i}$ will be
used for simplicity, where $x_{i}$ is the $i$-th coordinate of a generic point
$x=(x_{1},\cdots,x_{n})$ in $\mathbb{R}^{n}$. In a similar manner, we use
notations $z_{x_{j}}$, $v_{x_{j}}$, etc. for the partial derivatives of $z$
and $v$ with respect to $x_{j}$.
We first prove the hidden regularity for solutions to Eq. 1.2.
###### Proof of Proposition 1.1.
For any $\rho\mathop{\buildrel\Delta\over{=}}(\rho^{1},\cdots,\rho^{n})\in
C^{2}(\mathbb{R}^{n+1};\mathbb{R}^{n})$, by Itô’s formula and Eq. 1.2, we have
$\begin{array}[]{ll}\displaystyle\sum_{j=1}^{n}(2\rho\cdot\nabla\Delta z\Delta
z_{x_{j}}-\rho^{j}|\nabla\Delta z|^{2})_{x_{j}}dt\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle=-\operatorname{div}\rho|\nabla\Delta
z|^{2}dt+2\rho\cdot\nabla\Delta
z\big{[}(a_{1}-\operatorname{div}a_{2}-a_{4}a_{5})z-a_{2}\cdot\nabla
z+a_{4}Z-a_{3}\hat{Z}\big{]}dt\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\quad+2\rho\cdot\nabla\Delta z\hat{Z}dW(t)+2\nabla\Delta
zD\rho\cdot\nabla\Delta zdt-2d(\rho\cdot\nabla\Delta
z\hat{z})+2\rho_{t}\cdot\nabla\Delta z\hat{z}dt\\\ \vskip 6.0pt plus 2.0pt
minus
2.0pt\cr\displaystyle\quad+2\rho\cdot\nabla\Delta(Z-a_{5}z)\hat{Z}dt+2\rho\cdot\nabla\Delta(Z-a_{5}z)\hat{z}dW(t)+2\operatorname{div}(\hat{z}\rho\nabla^{2}\hat{z})dt\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\quad-2\operatorname{div}(\hat{z}\nabla\hat{z}D\rho)dt+2\Delta\rho\cdot\nabla\hat{z}\hat{z}dt+2\nabla\hat{z}D\rho\cdot\nabla\hat{z}dt-\operatorname{div}(\rho|\nabla\hat{z}|^{2})dt+\operatorname{div}\rho|\nabla\hat{z}|^{2}dt.\end{array}$
(2.1)
Since $\Gamma\in C^{3}$, there exists a vector field $\zeta\in
C^{2}(\mathbb{R}^{n};\mathbb{R}^{n})$ such that $\zeta=\nu$ on $\Gamma$ (e.g.,
[13, Lemma 2.1]). Setting $\rho=\zeta$, integrating Eq. 2.1 in $Q_{\tau}$, and
taking expectation on $\Omega$, we have
$\displaystyle\mathbb{E}\int_{\Sigma_{\tau}}\biggl{(}2\bigg{|}\frac{\partial\Delta
z}{\partial\nu}\bigg{|}^{2}-|\nabla\Delta z|^{2}\biggr{)}d\Gamma dt$
$\displaystyle=\mathbb{E}\int_{\Sigma_{\tau}}\sum_{j=1}^{n}(2\rho\cdot\nabla\Delta
z\Delta z_{x_{j}}-\rho^{j}|\nabla\Delta z|^{2})\nu^{j}d\Gamma dt$
$\displaystyle=-2\mathbb{E}\int_{G}\rho\cdot\nabla\Delta
z^{\tau}\hat{z}^{\tau}dx+2\mathbb{E}\int_{G}\rho\cdot\nabla\Delta
z(0)\hat{z}(0)dx$
$\displaystyle\quad+\mathbb{E}\int_{Q_{\tau}}\bigl{\\{}-\operatorname{div}\rho|\nabla\Delta
z|^{2}+2\rho\cdot\nabla\Delta
z\big{[}(a_{1}-\operatorname{div}a_{2}-a_{4}a_{5})z-a_{2}\cdot\nabla
z+a_{4}Z-a_{3}\hat{Z}\big{]}$ $\displaystyle\qquad\qquad\quad+2\nabla\Delta
zD\rho\cdot\nabla\Delta z+2\rho_{t}\cdot\nabla\Delta
z\hat{z}+2\rho\cdot\nabla\Delta(Z-a_{5}z)\hat{Z}+2\Delta\rho\cdot\nabla\hat{z}\hat{z}$
$\displaystyle\qquad\qquad\quad+2\nabla\hat{z}D\rho\cdot\nabla\hat{z}+\operatorname{div}\rho|\nabla\hat{z}|^{2}\bigr{\\}}dxdt.$
This implies
$\displaystyle 2\bigg{|}\frac{\partial\Delta
z}{\partial\nu}\bigg{|}^{2}_{L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))}-|\nabla\Delta
z|^{2}_{L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))}\leq
C|(z^{\tau},\hat{z}^{\tau})|^{2}_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H_{0}^{1}(G))}.$ (2.2)
Denote by $\nabla_{\sigma}$ the tangential gradient on $\Gamma$. We have
$\displaystyle|\nabla\Delta z|^{2}=\bigg{|}\frac{\partial\Delta
z}{\partial\nu}\bigg{|}^{2}+|\nabla_{\sigma}\Delta z|^{2},$
which, together with Eq. 2.2, implies that
$\displaystyle\bigg{|}\frac{\partial\Delta
z}{\partial\nu}\bigg{|}^{2}_{L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))}\leq|\nabla_{\sigma}\Delta
z|^{2}_{L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))}+C|(z^{\tau},\hat{z}^{\tau})|^{2}_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H_{0}^{1}(G))}.$
Now we are going to prove
$\displaystyle|\nabla_{\sigma}\Delta
z|^{2}_{L^{2}_{\mathbb{F}}(0,\tau;L^{2}(\Gamma))}\leq
C|(z^{\tau},\hat{z}^{\tau})|^{2}_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H_{0}^{1}(G))}.$ (2.3)
As the proof of Theorem 2.2 in [14], we introduce the following operator
$\left\\{\begin{aligned}
\mathscr{B}&=\sum_{i=1}^{n}b_{i}(x)\frac{\partial}{\partial x_{i}}=\text{ a
first-order differential operator (time independent)}\\\ &\text{with
coefficients $b_{i}\in C^{4}(\overline{G})$ and such that $\mathscr{B}$ is
tangential to $\Gamma$, i.e.,}\\\ &\sum_{i=1}^{n}b_{i}(x)\nu^{i}=0\text{ on
}\Gamma.\end{aligned}\right.$ (2.4)
The operator $\mathscr{B}$ can be thought of as the pre-image, under the
diffeomorphism via partitions of unity from $G$ onto half-space
$\\{(x,y)\in\mathbb{R}^{n}\mid x>0,y\in\mathbb{R}^{n-1}\\}$ of the tangential
derivative on the boundary $x=0$.
Define
$\displaystyle p\mathop{\buildrel\Delta\over{=}}\mathscr{B}z\in
C_{\mathbb{F}}([0,\tau];L^{2}(\Omega;H^{2}(G))),\qquad\hat{p}\mathop{\buildrel\Delta\over{=}}\mathscr{B}\hat{z}\in
C_{\mathbb{F}}([0,\tau];L^{2}(\Omega;L^{2}(G))).$
From Eq. 1.2, we have
$\left\\{\begin{aligned} &dp=\hat{p}dt+g_{1}dW(t)&&\quad\text{ in
}Q_{\tau},\\\ &d\hat{p}+\Delta^{2}pdt=f_{2}dt+g_{2}dW(t)&&\quad\text{ in
}Q_{\tau},\\\ &p=0,\frac{\partial
p}{\partial\nu}=\bigg{[}\frac{\partial}{\partial\nu},\mathscr{B}\bigg{]}z&&\quad\text{
on }\Sigma_{\tau},\\\
&(p(\tau),\hat{p}(\tau))=(\mathscr{B}z^{\tau},\mathscr{B}\hat{z}^{\tau})&&\quad\text{
in }G,\end{aligned}\right.$ (2.5)
where
$g_{1}=\mathscr{B}(Z-a_{5}z),\quad g_{2}=\mathscr{B}\hat{Z},$
and
$f_{2}=\mathscr{B}\big{[}(a_{1}-\operatorname{div}a_{2}-a_{4}a_{5})z-a_{2}\cdot\nabla
z-a_{3}\hat{Z}+a_{4}Z\big{]}+[\Delta^{2},\mathscr{B}]z,$
and $[\cdot,\cdot]$ denotes the commutator of two operators.
From (2.4), we get that
$\displaystyle|[\Delta^{2},\mathscr{B}]z|_{L^{2}_{\mathbb{F}}(0,\tau;H^{-1}(G))}\leq
C|(z^{\tau},\hat{z}^{\tau})|_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;(H^{3}(G)\cap
H_{0}^{2}(G)))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H_{0}^{1}(G))}$ (2.6)
and that
$\begin{array}[]{ll}\displaystyle\mathbb{E}\int_{\Sigma_{\tau}}|\nabla_{\sigma}\Delta
z|^{2}d\Gamma dt=\mathbb{E}\int_{\Sigma_{\tau}}|\mathscr{B}\Delta
z|^{2}d\Gamma dt=\mathbb{E}\int_{\Sigma_{\tau}}|\Delta p|^{2}d\Gamma
dt+\mathbb{E}\int_{\Sigma_{\tau}}|[\mathscr{B},\Delta]z|^{2}d\Gamma
dt.\end{array}$ (2.7)
By (2.4) again, we find that
$\begin{array}[]{ll}\displaystyle\mathbb{E}\int_{\Sigma_{\tau}}|[\mathscr{B},\Delta]z|^{2}d\Gamma
dt&\displaystyle\leq C\mathbb{E}\int_{0}^{\tau}|z|_{H^{3}(G)}^{2}dt\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\leq
C|(z^{\tau},\hat{z}^{\tau})|^{2}_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times
L_{\mathcal{F}_{\tau}}^{2}(\Omega;H_{0}^{1}(G))}.\end{array}$ (2.8)
Combining Eqs. 2.7 and 2.8, to show Eq. 2.3, we only need to prove
$\displaystyle\mathbb{E}\int_{\Sigma_{\tau}}|\Delta p|^{2}d\Gamma dt\leq
C|(z^{\tau},\hat{z}^{\tau})|^{2}_{L_{\mathcal{F}_{\tau}}^{2}(\Omega;(H^{3}(G)\cap
H_{0}^{2}(G)))\times L_{\mathcal{F}_{\tau}}^{2}(\Omega;H_{0}^{1}(G))}.$ (2.9)
For any $\rho\mathop{\buildrel\Delta\over{=}}(\rho^{1},\cdots,\rho^{n})\in
C^{2}(\mathbb{R}^{n+1};\mathbb{R}^{n})$, by Itô’s formula and Eq. 2.5, we have
$\begin{array}[]{ll}\displaystyle\sum_{i,k,l=1}^{n}(\rho^{i}p_{x_{k}x_{k}}p_{x_{l}x_{l}})_{x_{i}}dt\\\\[-2.84526pt]
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle=\sum_{i,k,l=1}^{n}\big{[}\rho^{i}_{x_{i}}p_{x_{k}x_{k}}p_{x_{l}x_{l}}+(2\rho^{i}p_{x_{i}x_{k}x_{k}}p_{x_{l}})_{x_{l}}-(2\rho^{i}_{x_{l}}p_{x_{i}x_{k}}p_{x_{l}})_{x_{k}}+2\rho^{i}_{x_{l}x_{k}}p_{x_{i}x_{k}}p_{x_{l}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\quad~{}~{}+2\rho^{i}_{x_{l}}p_{x_{i}x_{k}}p_{x_{l}x_{k}}-(2\rho^{i}p_{x_{l}x_{k}x_{k}}p_{x_{l}})_{x_{i}}+(2\rho^{i}_{x_{i}}p_{x_{l}x_{k}}p_{x_{l}})_{x_{k}}-2\rho^{i}_{x_{i}x_{k}}p_{x_{l}x_{k}}p_{x_{l}}\\\\[8.53581pt]
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\quad~{}~{}-2\rho^{i}_{x_{i}}p_{x_{l}x_{k}}p_{x_{l}x_{k}}+(2\rho^{i}p_{x_{l}x_{k}x_{k}}p_{x_{i}})_{x_{l}}-(2\rho^{i}_{x_{l}}p_{x_{l}x_{k}}p_{x_{i}})_{x_{k}}+2\rho^{i}_{x_{l}x_{k}}p_{x_{l}x_{k}}p_{x_{i}}\\\\[8.53581pt]
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\quad~{}~{}+2\rho^{i}_{x_{l}}p_{x_{l}x_{k}}p_{x_{i}x_{k}}\big{]}dt-2\rho\cdot\nabla
p(f_{2}dt+g_{2}dW(t))+d(2\rho\cdot\nabla p\hat{p})-2\rho_{t}\cdot\nabla
p\hat{p}dt\\\\[8.53581pt] \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\quad+\operatorname{div}\rho\hat{p}^{2}dt-2\rho\cdot
d\nabla pd\hat{p}-\operatorname{div}(\rho\hat{p}^{2})dt-2\rho\cdot\nabla
g_{1}\hat{p}dW(t).\end{array}$ (2.10)
Setting $\rho=\zeta$, integrating Eq. 2.10 in $Q_{\tau}$, and taking
expectation on $\Omega$, we have
$\displaystyle\mathbb{E}\int_{\Sigma_{\tau}}|\Delta p|^{2}d\Gamma dt$ (2.11)
$\displaystyle=\mathbb{E}\int_{\Sigma_{\tau}}\sum_{i,k,l=1}^{n}\rho^{i}\nu^{i}p_{x_{k}x_{k}}p_{x_{l}x_{l}}d\Gamma
dt$ $\displaystyle=2\mathbb{E}\int_{G}\rho\cdot\nabla
p^{\tau}\hat{p}^{\tau}dx-2\mathbb{E}\int_{G}\rho\cdot\nabla p(0)\hat{p}(0)dx$
$\displaystyle\quad+\mathbb{E}\int_{Q_{\tau}}\big{(}\operatorname{div}\rho\hat{p}^{2}-2\rho\cdot\nabla
pf_{2}-2\rho_{t}\cdot\nabla p\hat{p}-2\rho\cdot\nabla g_{1}g_{2}\big{)}dxdt$
$\displaystyle\quad+\mathbb{E}\int_{Q_{\tau}}\sum_{i,k,l=1}^{n}\big{(}\rho^{i}_{x_{i}}p_{x_{k}x_{k}}p_{x_{l}x_{l}}+2\rho^{i}_{x_{l}x_{k}}p_{x_{i}x_{k}}p_{x_{l}}+2\rho^{i}_{x_{l}}p_{x_{i}x_{k}}p_{x_{l}x_{k}}-2\rho^{i}_{x_{i}x_{k}}p_{x_{l}x_{k}}p_{x_{l}}$
$\displaystyle\qquad\qquad\qquad\quad~{}~{}-2\rho^{i}_{x_{i}}p_{x_{l}x_{k}}p_{x_{l}x_{k}}+2\rho^{i}_{x_{l}x_{k}}p_{x_{l}x_{k}}p_{x_{i}}+2\rho^{i}_{x_{l}}p_{x_{l}x_{k}}p_{x_{i}x_{k}}\big{)}dxdt$
$\displaystyle\quad+2\mathbb{E}\int_{\Sigma_{\tau}}\sum_{i,k,l=1}^{n}\big{(}\rho^{i}_{x_{i}}p_{x_{l}x_{k}}p_{x_{l}}\nu^{k}-\rho^{i}_{x_{l}}p_{x_{i}x_{k}}p_{x_{l}}\nu^{k}+\rho^{i}p_{x_{l}x_{k}x_{k}}p_{x_{i}}\nu^{l}-\rho^{i}_{x_{l}}p_{x_{l}x_{k}}p_{x_{i}}\nu^{k}\big{)}d\Gamma
dt,$
which, together with Eqs. 2.6, 2.4 and 2.5, implies Eq. 2.9. Then we complete
the proof. ∎
Next, we give a pointwise weighted identity, which will play an important role
in the proof of Theorem 3.2.
We have the following fundamental identity.
###### Theorem 2.1.
Let $v$ be an $H^{4}(G)$-valued Itô process and $\hat{v}$ be an
$H^{2}(G)$-valued Itô process such that
$dv=(\hat{v}+f_{1})dt+g_{1}dW(t)$
for some $f_{1}\in L^{2}_{\mathbb{F}}(0,T;H^{2}(G))$ and $g_{1}\in
L^{2}_{\mathbb{F}}(0,T;H^{4}(G)\cap H_{0}^{2}(G)).$ Let $\eta\in
C^{2}({\mathbb{R}}\times{\mathbb{R}}^{n})$. Set $\theta=e^{\ell}$,
$\ell=s\xi$, $\xi=e^{\lambda\eta}$, $w=\theta v$, and
$\hat{w}=\theta\hat{v}+\ell_{t}w$. Then, for any $t\in[0,T]$ and a.e.
$(x,\omega)\in G\times\Omega$,
$\displaystyle 2\theta
I_{2}(d\hat{v}+\Delta^{2}vdt)-2\operatorname{div}(V_{1}+V_{2})dt$
$\displaystyle=2I_{2}^{2}dt+2I_{2}I_{3}+2(M_{1}+M_{2})dt+2\sum_{i,j,k,l=1}^{n}\Lambda^{ijkl}_{1}w_{x_{i}x_{j}}w_{x_{k}x_{l}}dt+2\sum_{i,j=1}^{n}\Lambda_{2}^{ij}w_{x_{i}}w_{x_{j}}dt$
(2.12)
$\displaystyle\quad+2\Lambda_{3}w^{2}dt+2\Lambda_{4}+2d(I_{2}\hat{w})-\sum_{i=1}^{n}(\Phi_{1}^{i}\hat{w}d\Delta
w)_{x_{i}}.$
Here
$I_{1}=\Delta^{2}wdt+\Psi_{2}\Delta
wdt+\sum_{i,j=1}^{n}\Psi_{3}^{ij}w_{x_{i}x_{j}}dt+\sum_{i=1}^{n}\Psi_{4}^{i}w_{x_{i}}dt+\sum_{i=1}^{n}\Psi_{5}^{i}w_{x_{i}}dt+\Psi_{6}wdt+d\hat{w},$
(2.13)
with
$\begin{cases}\Psi_{2}=2s^{2}\lambda^{2}\xi^{2}|\nabla\eta|^{2},\quad\Psi_{3}^{ij}=4s^{2}\lambda^{2}\xi^{2}\eta_{x_{i}}\eta_{x_{j}},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\Psi_{4}^{i}=12s^{2}\lambda^{3}\xi^{2}|\nabla\eta|^{2}\eta_{x_{i}}+4s^{2}\lambda^{2}\xi^{2}\Delta\eta\eta_{x_{i}},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\Psi_{5}^{i}=\sum_{j=1}^{n}8s^{2}\lambda^{2}\xi^{2}\eta_{x_{i}x_{j}}\eta_{x_{j}},\quad\Psi_{6}=s^{4}\lambda^{4}\xi^{4}|\nabla\eta|^{4},\end{cases}\quad
i,j=1,\cdots,n$ (2.14) $I_{2}=\sum_{i=1}^{n}\Phi_{1}^{i}\Delta
w_{x_{i}}+\Phi_{2}\Delta
w+\sum_{i,j=1}^{n}\Phi_{3}^{ij}w_{x_{i}x_{j}}+\sum_{i=1}^{n}\Phi_{4}^{i}w_{x_{i}}+\Phi_{5}w,$
(2.15)
with
$\begin{cases}\displaystyle\Phi_{1}^{i}=-4s\lambda\xi\eta_{x_{i}},\quad\Phi_{2}=-2s\lambda^{2}\xi|\nabla\eta|^{2}-2s\lambda\xi\Delta\eta-\lambda,\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\Phi_{3}^{ij}=4s\lambda\xi\eta_{x_{i}x_{j}}-4s\lambda^{2}\xi\eta_{x_{i}}\eta_{x_{j}},\quad\Phi_{4}^{i}=-4s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\eta_{x_{i}},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\Phi_{5}=-6s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}-12s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\quad\qquad-2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta-s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\nabla\eta|^{4},\end{cases}i,j=1,\cdots,n$
(2.16) $\displaystyle I_{3}$
$\displaystyle=-8s\lambda\xi\sum\limits_{i,j=1}^{n}\eta_{x_{i}x_{j}}w_{x_{i}x_{j}}dt-4\nabla\Delta\ell\cdot\nabla
wdt+4(\nabla\ell\cdot\nabla\Delta\ell)wdt+2|\nabla^{2}\ell|^{2}wdt$
$\displaystyle\quad-\Delta^{2}\ell
wdt+|\Delta\ell|^{2}wdt+8s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)wdt+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\nabla\eta|^{4}wdt+\lambda\Delta
wdt$ (2.17) $\displaystyle\quad-\ell_{t}\theta f_{1}dt-\ell_{t}\theta
g_{1}dW(t)+(\ell_{t}^{2}-\ell_{tt})wdt-2\ell_{t}\hat{w}dt,$ $\displaystyle
V_{1}$ $\displaystyle=[V_{1}^{1},V_{1}^{2},\cdots,V_{1}^{n}],\quad
V_{2}=[V_{2}^{1},V_{2}^{2},\cdots,V_{2}^{n}],$ $\displaystyle V_{1}^{j}$
$\displaystyle=\sum_{i,k=1}^{n}\Big{[}\sum_{l=1}^{n}\Phi_{1}^{l}w_{x_{k}x_{k}x_{l}}w_{x_{i}x_{i}x_{j}}-\frac{1}{2}\sum_{l=1}^{n}\Phi_{1}^{j}w_{x_{k}x_{k}x_{l}}w_{x_{i}x_{i}x_{l}}+\frac{1}{2}\Psi_{2}\Phi_{1}^{j}w_{x_{i}x_{i}}w_{x_{k}x_{k}}$
$\displaystyle\quad\quad\quad\ \
+\sum_{l=1}^{n}\Psi_{3}^{ik}\Phi_{1}^{l}w_{x_{i}x_{k}}w_{x_{l}x_{j}}-\frac{1}{2}\sum_{l=1}^{n}\Psi_{3}^{ij}\Phi_{1}^{l}w_{x_{i}x_{k}}w_{x_{k}x_{l}}+\Phi_{4}^{k}w_{x_{i}x_{i}x_{j}}w_{x_{k}}-\Phi_{4}^{k}w_{x_{i}x_{j}}w_{x_{i}x_{k}}$
$\displaystyle\quad\quad\quad\ \
+\frac{1}{2}\Phi_{4}^{j}w_{x_{i}x_{k}}^{2}+\Big{(}\Psi_{2}\Phi_{4}^{k}\delta_{ij}-\frac{1}{2}\Psi_{2}\Phi_{4}^{j}\delta_{ik}-\frac{1}{2}\Psi_{6}\Phi_{1}^{j}\delta_{ik}+\Psi_{3}^{ij}\Phi_{4}^{k}\Big{)}w_{x_{i}}w_{x_{k}}\Big{]},$
$\displaystyle V_{2}^{j}$
$\displaystyle=\sum_{i,l,r,m=1}^{n}\Theta_{1}^{ijlrm}w_{x_{i}x_{i}x_{l}}w_{x_{r}x_{m}}+\sum_{i,k,l,r=1}^{n}\Theta_{2}^{ijklr}w_{x_{i}x_{k}}w_{x_{l}x_{r}}+\sum_{i,k,l=1}^{n}\Theta_{3}^{ijkl}w_{x_{i}x_{k}}w_{x_{l}}$
$\displaystyle\quad+\sum_{i,k=1}^{n}\Theta_{4}^{ijk}w_{x_{i}}w_{x_{k}}+\sum_{i=1}^{n}\Theta_{5}w_{x_{i}x_{i}x_{j}}w+\sum_{i,k=1}^{n}\Theta_{6}^{ijk}w_{x_{i}x_{k}}w+\sum_{i=1}^{n}\Theta_{7}^{ij}w_{x_{i}}w+\Theta_{8}^{j}w^{2}+\Theta_{9}^{j},$
$\displaystyle M_{1}$ $\displaystyle=8s\lambda^{2}\xi|\nabla\Delta
w\cdot\nabla\eta|^{2}+32s^{3}\lambda^{4}\xi^{3}|\nabla^{2}w\nabla\eta\nabla\eta|^{2}+48s^{3}\lambda^{3}\xi^{3}\nabla^{2}\eta(\nabla^{2}w\nabla\eta)(\nabla^{2}w\nabla\eta)$
$\displaystyle\quad+16s^{3}\lambda^{3}\xi^{3}(\nabla^{2}w\nabla\eta\nabla\eta)\sum\limits_{i,j=1}^{n}\eta_{x_{i}x_{j}}w_{x_{i}x_{j}}-16s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{2}|\nabla^{2}w\nabla\eta|^{2}$
$\displaystyle\quad+6s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}|\nabla^{2}w|^{2}+4s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)|\nabla^{2}w|^{2}+2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta|\nabla^{2}w|^{2}$
$\displaystyle\quad+2s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}|\Delta
w|^{2}-4s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)|\Delta
w|^{2}-2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta|\Delta w|^{2}$
$\displaystyle\quad+40s^{5}\lambda^{6}\xi^{5}|\nabla\eta|^{4}|\nabla
w\cdot\nabla\eta|^{2}+64s^{5}\lambda^{5}\xi^{5}(\nabla^{2}\eta\nabla\eta\nabla\eta)|\nabla
w\cdot\nabla\eta|^{2}$
$\displaystyle\quad-16s^{5}\lambda^{6}\xi^{5}|\nabla\eta|^{6}|\nabla
w|^{2}+8s^{7}\lambda^{8}\xi^{7}|\nabla\eta|^{8}w^{2}+\lambda|\nabla\Delta
w|^{2}-s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\nabla\eta|^{4}|\Delta w|^{2}$
$\displaystyle\quad+4s^{5}\lambda^{\frac{11}{2}}\xi^{5}|\nabla\eta|^{4}|\nabla
w\cdot\nabla\eta|^{2}+2s^{5}\lambda^{\frac{11}{2}}\xi^{5}|\nabla\eta|^{6}|\nabla
w|^{2}-s^{7}\lambda^{\frac{15}{2}}\xi^{7}|\nabla\eta|^{8}w^{2},$
$\displaystyle M_{2}$
$\displaystyle=\biggl{(}-\frac{1}{2}\sum_{i=1}^{n}\Phi_{1x_{i}}^{i}+\Phi_{2}\biggr{)}|\nabla\hat{w}|^{2}+\sum_{i,j=1}^{n}\big{(}-\Phi_{1x_{j}}^{i}+\Phi_{3}^{ij}\big{)}\hat{w}_{x_{i}}\hat{w}_{x_{j}}$
$\displaystyle\quad+\frac{1}{2}\biggl{[}\sum_{i,j=1}^{n}\big{(}\Phi_{1x_{i}x_{j}x_{j}}^{i}-\Phi_{2x_{i}x_{i}}\delta_{ij}-\Phi_{3x_{i}x_{j}}^{ij}+\Phi_{4x_{i}}^{i}\delta_{ij}\big{)}-2\Phi_{5}\biggr{]}\hat{w}^{2}$
$\displaystyle\quad-\sum_{i,k,j=1}^{n}\big{(}\Psi_{5}^{i}\Phi_{1}^{k}\big{)}_{x_{j}}w_{x_{i}}w_{x_{k}x_{j}}-4s^{2}\lambda^{3}\xi^{2}|\nabla^{2}w\nabla\eta|-2s^{2}\lambda^{3}\xi^{2}|\nabla\eta|^{2}|\Delta
w|^{2}$ $\displaystyle\quad+s^{4}\lambda^{5}\xi^{4}|\nabla\eta|^{4}|\nabla
w|^{2},$
$\displaystyle\Lambda_{1}^{ijkl}=\Phi_{3x_{i}x_{j}}^{kl}+\Phi_{3x_{i}x_{l}}^{kj}+\sum_{r=1}^{n}\Big{(}\frac{1}{2}\Phi_{2x_{r}x_{r}}\delta_{ij}\delta_{kl}-\Phi_{3x_{j}x_{r}}^{kr}\delta_{il}-\Phi_{3x_{i}x_{r}}^{kr}\delta_{lj}+\frac{1}{2}\sum_{m=1}^{n}\Phi_{3x_{r}x_{m}}^{rm}\delta_{ik}\delta_{lj}\Big{)},$
$\displaystyle\Lambda_{2}^{ij}=\sum_{k=1}^{n}\Big{[}-\Phi_{4x_{i}x_{k}x_{k}}^{j}+\frac{1}{2}\Phi_{4x_{i}x_{j}x_{k}}^{k}-2\Phi_{5x_{i}x_{j}}+(\Psi_{2}\Phi_{3}^{jk})_{x_{i}x_{k}}-\frac{1}{2}(\Psi_{2}\Phi_{3}^{ij})_{x_{k}x_{k}}$
$\displaystyle\qquad\qquad\quad-\sum_{l=1}^{n}\frac{1}{2}(\Psi_{2}\Phi_{3}^{kl}\delta_{ij})_{x_{k}x_{l}}+(\Psi_{3}^{ik}\Phi_{2})_{x_{k}x_{j}}-\sum_{l=1}^{n}\frac{1}{2}(\Psi_{3}^{kl}\Phi_{2}\delta_{ij})_{x_{k}x_{l}}-\frac{1}{2}(\Psi_{3}^{ij}\Phi_{2})_{x_{k}x_{k}}$
$\displaystyle\qquad\qquad\quad+\sum_{l=1}^{n}(\Psi_{3}^{ik}\Phi_{3}^{jl})_{x_{k}x_{l}}-\sum_{l=1}^{n}\frac{1}{2}(\Psi_{3}^{ij}\Phi_{3}^{kl})_{x_{k}x_{l}}-\frac{1}{2}(\Psi_{3}^{kl}\Phi_{3}^{ij})_{x_{k}x_{l}}+\frac{1}{2}(\Psi_{4}^{i}\Phi_{1}^{j})_{x_{k}x_{k}}$
$\displaystyle\qquad\qquad\quad+(\Psi_{4}^{i}\Phi_{2})_{x_{j}}+\frac{1}{2}(\Psi_{4}^{k}\Phi_{2}\delta_{ij})_{x_{k}}-(\Psi_{4}^{i}\Phi_{3}^{jk})_{x_{k}}+\frac{1}{2}(\Psi_{4}^{k}\Phi_{3}^{ij})_{x_{k}}+(\Psi_{5}^{i}\Phi_{2})_{x_{j}}$
$\displaystyle\qquad\qquad\quad+\frac{1}{2}(\Psi_{5}^{k}\Phi_{2}\delta_{ij})_{x_{k}}-(\Psi_{5}^{i}\Phi_{3}^{jk})_{x_{k}}+\frac{1}{2}(\Psi_{5}^{k}\Phi_{3}^{ij})_{x_{k}}\Big{]},$
$\displaystyle\Lambda_{3}=\sum_{i=1}^{n}\Big{[}\sum_{j=1}^{n}\frac{1}{2}\Phi_{5x_{i}x_{i}x_{j}x_{j}}+\sum_{j=1}^{n}\frac{1}{2}(\Psi_{2}\Phi_{5})_{x_{j}x_{j}}+\sum_{j=1}^{n}\frac{1}{2}(\Psi_{3}^{ij}\Phi_{5})_{x_{i}x_{j}}-\frac{1}{2}(\Psi_{4}^{i}\Phi_{5})_{x_{i}}-\frac{1}{2}(\Psi_{5}^{i}\Phi_{5})_{x_{i}}$
$\displaystyle\quad\quad\quad\quad-\sum_{j=1}^{n}\frac{1}{2}(\Psi_{6}\Phi_{1}^{i})_{x_{i}x_{j}x_{j}}+\frac{1}{2}(\Psi_{6}\Phi_{2})_{x_{i}x_{i}}\Big{]},$
$\displaystyle\Lambda_{4}=\sum_{i,j=1}^{n}\big{[}-\Phi_{1t}^{i}w_{x_{i}x_{j}x_{j}}+\big{(}-\Phi_{2t}\delta_{ij}-\Phi_{3t}^{ij}\big{)}w_{x_{i}x_{j}}\big{]}\hat{w}dt-\sum_{i=1}^{n}\Phi_{4t}^{i}w_{x_{i}}\hat{w}dt-\Phi_{5t}w\hat{w}dt$
$\displaystyle\qquad+\sum_{i,j=1}^{n}\big{(}\Phi^{i}_{1x_{i}}\hat{w}+\Phi^{i}_{1}\hat{w}_{x_{i}}-\Phi_{2}\hat{w}\delta_{ij}\big{)}\big{[}(\theta
f_{1})_{x_{j}x_{j}}dt+(\theta
g_{1})_{x_{j}x_{j}}dW(t)\big{]}-\Phi^{i}_{5}\hat{w}\big{(}\theta
f_{1}dt+\theta g_{1}dW(t)\big{)}$
$\displaystyle\qquad-\sum_{i,j=1}^{n}\Phi^{ij}_{3}\hat{w}\big{[}(\theta
f_{1})_{x_{i}x_{j}}dt+(\theta
g_{1})_{x_{i}x_{j}}dW(t)\big{]}-\sum_{i=1}^{n}\Phi^{i}_{4}\hat{w}\big{[}(\theta
f_{1})_{x_{i}}dt+(\theta g_{1})_{x_{i}}dW(t)\big{]}$
$\displaystyle\qquad-\sum_{i,j=1}^{n}\big{(}\Phi^{i}_{1}dw_{x_{i}x_{i}x_{j}}+\Phi_{2}\delta_{ij}dw_{x_{i}x_{i}}+\Phi_{3}^{ij}dw_{x_{i}x_{j}}+\Phi^{i}_{4}\delta_{ij}dw_{x_{i}}\big{)}d\hat{w}+\Phi_{5}dwd\hat{w},$
and
$\displaystyle\Theta_{1}^{ijklrm}=\Phi_{2}\delta_{lj}\delta_{rm}+\Phi_{3}^{rm}\delta_{lj}-\Phi_{3}^{mj}\delta_{lr}+\Phi_{3}^{lr}\delta_{mj},$
$\displaystyle\Theta_{2}^{ijklr}=-\frac{1}{2}\Phi_{2x_{j}}\delta_{ik}\delta_{lr}-\Phi_{3x_{i}}^{lr}\delta_{kj}+\Phi_{3x_{k}}^{rj}\delta_{il}+\sum_{m=1}^{n}\Phi_{3x_{m}}^{rm}\delta_{kj}\delta_{il}-\frac{1}{2}\sum_{m=1}^{n}\Phi_{3x_{m}}^{jm}\delta_{il}\delta_{kr}-\Phi_{3x_{i}}^{kl}\delta_{rj},$
$\displaystyle\Theta_{3}^{ijkl}=-\Phi_{4x_{i}}^{l}\delta_{kj}-\Phi_{5}\delta_{ik}\delta_{lj}+\Psi_{2}\Phi_{3}^{ik}\delta_{lj}-\Psi_{2}\Phi_{3}^{ij}\delta_{kl}+\Psi_{3}\Phi_{2}\delta_{ik}-\Psi_{3}\Phi_{2}\delta_{ij}+\Psi_{3}^{ik}\Phi_{3}^{lj}$
$\displaystyle\qquad\quad\;-\Psi_{3}^{ij}\Phi_{3}^{kl}+\Psi_{4}^{l}\Phi_{1}^{k}\delta_{ij}+\Psi_{5}^{l}\Phi_{1}^{k}\delta_{ij},$
$\displaystyle\Theta_{4}^{ijk}=\Phi_{4x_{i}x_{j}}^{k}-\frac{1}{2}\Phi_{4x_{i}x_{k}}^{j}+2\Phi_{5x_{i}}\delta_{kj}-\Phi_{5x_{j}}\delta_{ik}-(\Psi_{2}\Phi_{3}^{jk})_{x_{i}}+\frac{1}{2}(\Psi_{2}\Phi_{3}^{ik})_{x_{j}}$
$\displaystyle\qquad\quad+\sum_{l=1}^{n}\frac{1}{2}(\Psi_{2}\Phi_{3}^{jl})_{x_{l}}\delta_{ik}-\sum_{l=1}^{n}(\Psi_{3}^{il}\Phi_{2})_{x_{l}}\delta_{kj}+\frac{1}{2}\sum_{l=1}^{n}(\Psi_{3}^{lj}\Phi_{2})_{x_{l}}\delta_{ik}+\frac{1}{2}(\Psi_{3}^{ik}\Phi_{2})_{x_{j}}$
$\displaystyle\qquad\quad-\sum_{l=1}^{n}(\Psi_{3}^{ij}\Phi_{3}^{kl})_{x_{l}}+\frac{1}{2}\sum_{l=1}^{n}(\Psi_{3}^{ik}\Phi_{3}^{jl})_{x_{l}}+\frac{1}{2}\sum_{l=1}^{n}(\Psi_{3}^{lj}\Phi_{3}^{ik})_{x_{l}}-\frac{1}{2}(\Psi_{4}^{i}\Phi_{1}^{k})_{x_{j}}+\Psi_{4}^{i}\Phi_{2}\delta_{kj}$
$\displaystyle\qquad\quad-\frac{1}{2}\Psi_{4}^{j}\Phi_{2}\delta_{ik}+\Psi_{4}^{i}\Phi_{3}^{kj}-\frac{1}{2}\Psi_{4}^{j}\Phi_{3}^{ik}+\Psi_{5}^{i}\Phi_{2}\delta_{kj}-\frac{1}{2}\Psi_{5}^{j}\Phi_{2}\delta_{ik}+\Psi_{5}^{i}\Phi_{3}^{kj}-\frac{1}{2}\Psi_{5}^{j}\Phi_{3}^{ik},$
$\displaystyle\Theta_{5}=\Phi_{5},$
$\displaystyle\Theta_{6}^{ijk}=-\Phi_{5x_{j}}\delta_{ik}+\Psi_{6}\Phi_{1}^{k}\delta_{ij},$
$\displaystyle\Theta_{7}^{ij}=\sum_{k=1}^{n}\Phi_{5x_{k}x_{k}}\delta_{ij}+\Psi_{2}\Phi_{5}\delta_{ij}+\Psi_{3}^{ij}\Phi_{5}-(\Psi_{6}\Phi_{1}^{i})_{x_{j}}+\Psi_{6}\Phi_{2}\delta_{ij}+\Psi_{6}\Phi_{3}^{ij},$
$\displaystyle\Theta_{8}^{j}=-\frac{1}{2}\sum_{i=1}^{n}\Phi_{5x_{i}x_{i}x_{j}}-\frac{1}{2}(\Psi_{2}\Phi_{5})_{x_{j}}-\frac{1}{2}\sum_{i=1}^{n}(\Psi_{3}^{ij}\Phi_{5})_{x_{i}}+\frac{1}{2}\Psi_{4}^{j}\Phi_{5}+\frac{1}{2}\Psi_{5}^{j}\Phi_{5}$
$\displaystyle\quad\quad\
+\frac{1}{2}\sum_{i=1}^{n}(\Psi_{6}\Phi_{1}^{j})_{x_{i}x_{i}}-\frac{1}{2}(\Psi_{6}\Phi_{2})_{x_{j}}-\frac{1}{2}\sum_{i=1}^{n}(\Psi_{6}\Phi_{3}^{ij})_{x_{i}}+\frac{1}{2}\Psi_{6}\Phi_{4}^{j}.$
$\displaystyle\Theta_{9}^{j}=\sum_{i=1}^{n}\biggl{(}\Phi^{i}_{1x_{i}}\hat{w}_{x_{j}}\hat{w}+\Phi_{1}^{i}\hat{w}_{x_{i}}\hat{w}_{x_{j}}-\frac{1}{2}\Phi_{1x_{i}x_{j}}^{i}\hat{w}^{2}-\frac{1}{2}\Phi_{1}^{j}\hat{w}_{x_{i}}^{2}-\Phi_{3}^{ij}\hat{w}\hat{w}_{x_{i}}+\frac{1}{2}\Phi_{3x_{i}}^{ij}\hat{w}^{2}\biggr{)}$
$\displaystyle\qquad~{}-\Phi_{2}\hat{w}\hat{w}_{x_{j}}+\frac{1}{2}\Phi_{2x_{j}}\hat{w}^{2}-\frac{1}{2}\Phi_{4}^{j}\hat{w}^{2}.$
###### Remark 2.1.
Since we do not put any further assumptions on $v$ and $\hat{v}$, the identity
(2.12) seems to be very complicated. For solutions to (1.2) or (1.6), many
terms, such as $V_{1}$ and $V_{2}$, will merge or vanish by means of the
boundary conditions. Furthermore, compared with energy terms, such as
$8s\lambda^{2}\xi|\nabla\Delta w\cdot\nabla\eta|^{2}$,
$2s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}|\Delta w|^{2}$,
$40s^{5}\lambda^{6}\xi^{5}|\nabla\eta|^{4}|\nabla w\cdot\nabla\eta|^{2}$,
$\lambda|\nabla\Delta w|^{2}$, many terms in (2.12) is of no great importance.
We only need to estimate their orders with respect to $s$ and $\lambda$.
Hence, an effective way to simplify (2.12) is that we do not write these terms
explicitly and just claim that the order of $s$ and $\lambda$ for them are
lower than the terms with a “good sign”. However, we do not do this since we
want to provide the full details for readers, and particularly for beginners.
Although the form of the identity (2.12) is very complex, its proof follows
from some basic computations. To avoid defocusing the main theme of this
paper, we put the proof of Theorem 2.1 in the appendix.
## 3 Carleman estimate for the adjoint equation
This section is devoted to establishing a Carleman estimate for a backward
stochastic plate equation. To this end, we first introduce the weight function
$\eta$.
For any $\delta>0,T>0$ and $0<\varepsilon_{1}<\frac{1}{2}$, we choose
$x_{0}\in\mathbb{R}^{n}\backslash\overline{G}$ such that
$R_{0}\mathop{\buildrel\Delta\over{=}}\min_{x\in\overline{G}}|x-x_{0}|^{2}>2\delta,$
(3.1)
and choose sufficiently large $\beta$ satisfing
$R_{1}\mathop{\buildrel\Delta\over{=}}\max_{x\in\overline{G}}|x-x_{0}|^{2}\leq\beta\varepsilon_{1}^{2}T^{2}-\delta.$
(3.2)
We also choose sufficiently small $\varepsilon_{0}$ with
$0<\varepsilon_{0}<\varepsilon_{1}$ such that
$R_{0}-\beta\varepsilon_{0}^{2}T^{2}\geq\delta.$ (3.3)
Let
$\eta(t,x)=|x-x_{0}|^{2}-\beta\bigg{(}t-\frac{T}{2}\bigg{)}^{2}.$ (3.4)
From Eqs. 3.4, 3.1, 3.2 and 3.3, it is easy to see that $\eta$ satisfies the
following conditions.
###### Condition 3.1.
1. (1).
$|\eta(t,x)|_{C^{2}(\overline{Q})}\leq C_{1}$.
2. (2).
$|\nabla\eta(t,x)|\geq C_{2}>0,\quad\forall~{}(t,x)\in\overline{Q}$.
3. (3).
For all $(t,x)$ in
$J_{1}\mathop{\buildrel\Delta\over{=}}[(0,T/2-\varepsilon_{1}T)\cup(T/2+\varepsilon_{1}T,T)]\times
G,$ it holds that $\eta(t,x)\leq-\delta$.
4. (4).
For all $(t,x)$ in
$J_{2}\mathop{\buildrel\Delta\over{=}}(T/2-\varepsilon_{0}T,T/2+\varepsilon_{0}T)\times
G,$ it holds that $\eta(t,x)\geq\delta$.
Recall that $\theta=e^{\ell}$, $\ell=s\xi$ and $\xi=e^{\lambda\eta}$. With
$\eta$ given by (3.4), the functions $\ell$ and $\theta$ are also defined.
We also need the following known result.
###### Lemma 3.1.
[7, Theorem 2.1] Let $q\in H^{2}_{0}(G)$. Then there exists a constant $C>0$
independent of $s$ and $\lambda$, and parameter $\widehat{\lambda}>1$ and
$\widehat{s}>1$ such that, for all $\lambda\geq\widehat{\lambda}$ and
$s\geq\widehat{s}$,
$\displaystyle
s^{4}\lambda^{6}\int_{Q}\xi^{4}\theta^{2}\big{(}s^{2}\lambda^{2}\xi^{2}|q|^{2}+|\nabla
q|^{2}\big{)}dxdt\leq C\int_{Q}s^{3}\lambda^{4}\xi^{3}\theta^{2}|\Delta
q|^{2}dxdt.$
We have the following Carleman estimate.
###### Theorem 3.2.
There exist constants $C>0$ and $\lambda_{0}>0$ such that for all
$\lambda\geq\lambda_{0}$, one can find $s_{0}=s_{0}(\lambda)>0$ so that for
any $s\geq s_{0}$,
$(v,\hat{v})\in L^{2}_{\mathbb{F}}(\Omega;C([0,T];H^{4}(G)\cap
H_{0}^{2}(G)))\times L^{2}_{\mathbb{F}}(\Omega;C([0,T];H_{0}^{2}(G))),$
and
$f_{1},f_{2},g_{2}\in L^{2}_{\mathbb{F}}(0,T;H_{0}^{2}(G)),\quad g_{1}\in
L^{2}_{\mathbb{F}}(0,T;H^{4}(G)\cap H_{0}^{2}(G))$
satisfying
$\left\\{\begin{array}[]{ll}\displaystyle
dv=(\hat{v}+f_{1})dt+g_{1}dW(t)&\text{ in }Q,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle d\hat{v}+\Delta^{2}vdt=f_{2}dt+g_{2}dW(t)&\text{ in
}Q,\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle v=\frac{\partial
v}{\partial\nu}=0&\text{ on }\Sigma,\end{array}\right.$ (3.5)
and
$v(0,\cdot)=v(T,\cdot)=\hat{v}(0,\cdot)=\hat{v}(T,\cdot)=0\quad\mbox{ in
}G,\quad\text{$\mathbb{P}$-\rm a.s.},$ (3.6)
it holds that
$\displaystyle\mathbb{E}\int_{Q}\theta^{2}\bigl{(}s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}+\lambda|\nabla\Delta
v|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}v|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
v|^{2}$ $\displaystyle\qquad\quad+s^{4}\lambda^{6}\xi^{4}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}|v|^{2}\bigr{)}dxdt$ $\displaystyle\leq
C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+f_{2}^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
(3.7) $\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$
$\displaystyle\quad+C\mathbb{E}\int_{\Sigma}\theta^{2}\bigr{(}s\lambda\xi|\nabla\Delta
v|^{2}+s^{3}\lambda^{3}\xi^{3}|\Delta v|^{2}\bigl{)}d\Gamma dt.$
###### Proof of Theorem 3.2.
In what follows, for a positive integer $r$, we denote by $O(\lambda^{r})$ a
function of order $\lambda^{r}$ for large $\lambda$. Similarly, we use the
notation $O(e^{C\lambda})$.
In order to shorten the formulae, we define
$\displaystyle\mathcal{A}_{1}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{Q}\big{(}s^{5}O(e^{C\lambda})+s^{6}\xi^{6}O(\lambda^{7})\big{)}|w|^{2}dxdt,$
$\displaystyle\mathcal{A}_{2}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{Q}\big{(}s^{3}O(e^{C\lambda})+s^{4}\xi^{4}O(\lambda^{5})\big{)}|\nabla
w|^{2}dxdt,$
$\displaystyle\mathcal{A}_{3}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{Q}\big{(}sO(e^{C\lambda})+s^{2}\xi^{2}O(\lambda^{3})\big{)}|\nabla^{2}w|^{2}dxdt,$
$\displaystyle\mathcal{A}_{4}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{Q}O(1)|\nabla\Delta
w|^{2}dxdt,$ (3.8)
$\displaystyle\mathcal{B}_{2}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{\Sigma}\big{(}s^{2}O(e^{C\lambda})+s^{3}\xi^{3}O(\lambda^{2})\big{)}|\nabla^{2}w|^{2}dxdt,$
$\displaystyle\hat{\mathcal{A}}_{1}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{Q}\big{(}s^{2}O(e^{C\lambda})+s^{3}\xi^{3}O(\lambda^{3})\big{)}|\hat{w}|^{2}dxdt,$
$\displaystyle\hat{\mathcal{A}}_{2}\mathop{\buildrel\Delta\over{=}}\mathbb{E}\int_{Q}\big{(}O(\lambda)+s\xi
O(1)\big{)}|\nabla\hat{w}|^{2}dxdt,$
and
$\mathcal{A}\mathop{\buildrel\Delta\over{=}}\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3}+\mathcal{A}_{4}+\hat{\mathcal{A}}_{1}+\hat{\mathcal{A}}_{2}.$
Integrating the equality Eq. 2.12 on $Q$, taking mathematical expectation in
both sides, and noting Eq. 3.6, we obtain
$\displaystyle 2\mathbb{E}\int_{Q}\theta
I_{2}(d\hat{v}+\Delta^{2}vdt)dx-2\mathbb{E}\int_{Q}\operatorname{div}(V_{1}+V_{2})dxdt$
$\displaystyle=2\mathbb{E}\int_{Q}I_{2}^{2}dxdt+2\mathbb{E}\int_{Q}I_{2}I_{3}dx+2\mathbb{E}\int_{Q}(M_{1}+M_{2})dxdt$
(3.9)
$\displaystyle\quad+2\sum_{i,j,k,l=1}^{n}\mathbb{E}\int_{Q}\Lambda^{ijkl}_{1}w_{x_{i}x_{j}}w_{x_{k}x_{l}}dxdt+2\sum_{i,j=1}^{n}\mathbb{E}\int_{Q}\Lambda_{2}^{ij}w_{x_{i}}w_{x_{j}}dxdt$
$\displaystyle\quad+2\mathbb{E}\int_{Q}\Lambda_{3}w^{2}dxdt+2\mathbb{E}\int_{Q}\Lambda_{4}dx.$
By Condition 3.1, Eqs. 2.14, 2.16 and 3, we have
$\sum_{i,j,k,l=1}^{n}\mathbb{E}\int_{Q}\Lambda^{ijkl}_{1}w_{x_{i}x_{j}}w_{x_{k}x_{l}}dxdt\geq-C\mathbb{E}\int_{Q}s\lambda^{4}\xi|\nabla^{2}w|^{2}dxdt\geq\mathcal{A}_{3},$
(3.10)
$\sum_{i,j=1}^{n}\mathbb{E}\int_{Q}\Lambda_{2}^{ij}w_{x_{i}}w_{x_{j}}dxdt\geq-C\mathbb{E}\int_{Q}\big{(}s^{3}\lambda^{6}\xi^{3}+s^{2}\lambda^{6}\xi^{2}\big{)}|\nabla^{2}w|^{2}dxdt\geq\mathcal{A}_{2},$
(3.11)
$\mathbb{E}\int_{Q}\Lambda_{3}w^{2}dxdt\geq-C\mathbb{E}\int_{Q}s^{3}\lambda^{8}\xi^{3}\big{(}1+s\xi+s^{2}\xi^{2}\big{)}|\nabla
w|^{2}dxdt\geq\mathcal{A}_{1}$ (3.12)
and
$\displaystyle\mathbb{E}\int_{Q}\Lambda_{4}dx$
$\displaystyle\geq-C\mathbb{E}\int_{Q}\big{[}|\nabla\Delta
w|^{2}+sO(e^{C\lambda})|\nabla^{2}w|^{2}+s^{3}O(e^{C\lambda})|\nabla
w|^{2}+s^{5}O(e^{C\lambda})|w|^{2}$
$\displaystyle\qquad\qquad\quad~{}+s^{2}O(e^{C\lambda})|\hat{w}|^{2}+s^{3}\lambda^{3}\xi^{3}|\hat{w}|^{2}+(s\xi+\lambda)|\nabla\hat{w}|^{2}\big{]}dxdt$
$\displaystyle\quad-C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
(3.13) $\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$
$\displaystyle\geq-C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
$\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt+\mathcal{A}.$
It follows from (3.9)–(3.13) that
$\displaystyle 2\mathbb{E}\int_{Q}\theta
I_{2}(d\hat{v}+\Delta^{2}vdt)dx-2\mathbb{E}\int_{Q}\operatorname{div}(V_{1}+V_{2})dxdt$
$\displaystyle\geq
2\mathbb{E}\int_{Q}I_{2}^{2}dxdt+2\mathbb{E}\int_{Q}I_{2}I_{3}dx+2\mathbb{E}\int_{Q}(M_{1}+M_{2})dxdt$
(3.14)
$\displaystyle\qquad-C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}$
$\displaystyle\qquad\qquad\qquad\quad~{}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt+\mathcal{A}.$
We will estimate the terms in Eq. 3.14 one by one. The procedure is divided
into three steps.
_Step 1._ In this step, we consider the divergence terms.
Thanks to the boundary conditions satisfied by $v$, it is easy to check that
$\displaystyle w$ $\displaystyle=0,\quad\nabla
w=0,\quad\nabla^{2}w=\theta\nabla^{2}v\quad\mbox{ on }\Sigma.$ (3.15)
We also have
$\displaystyle\frac{\partial
w_{x_{i}}}{\partial\nu}\nu^{j}=w_{x_{i}x_{j}}=w_{x_{j}x_{i}}=\frac{\partial
w_{x_{j}}}{\partial\nu}\nu^{i},$ (3.16)
which implies
$\displaystyle|\Delta w|^{2}$
$\displaystyle=\sum_{i,j=1}^{n}w_{x_{i}x_{i}}w_{x_{j}x_{j}}=\sum_{i,j=1}^{n}\frac{\partial
w_{x_{i}}}{\partial\nu}\nu^{i}\frac{\partial w_{x_{j}}}{\partial\nu}\nu^{j}$
$\displaystyle=\sum_{i,j=1}^{n}\frac{\partial
w_{x_{i}}}{\partial\nu}\nu^{j}\frac{\partial
w_{x_{j}}}{\partial\nu}\nu^{i}=\sum_{i,j=1}^{n}w_{x_{i}x_{j}}^{2}$ (3.17)
$\displaystyle=|\nabla^{2}w|^{2}=\sum_{i=1}^{n}\biggl{(}\frac{\partial
w_{i}}{\partial\nu}\biggr{)}^{2}\quad\mbox{ on }\Sigma.$
Thanks to Eq. 3.15, we obtain that
$\displaystyle V_{1}\cdot\nu$
$\displaystyle=\sum_{i,j,k,l=1}^{n}\Phi_{1}^{l}w_{x_{k}x_{k}x_{l}}w_{x_{i}x_{i}x_{j}}\nu^{j}-\frac{1}{2}\sum_{i,j,k,l=1}^{n}\Phi_{1}^{j}w_{x_{k}x_{k}x_{l}}w_{x_{i}x_{i}x_{l}}\nu^{j}$
$\displaystyle\quad+\frac{1}{2}\sum_{i,j,k=1}^{n}\Psi_{2}\Phi_{1}^{j}w_{x_{i}x_{i}}w_{x_{k}x_{k}}\nu^{j}+\sum_{i,j,k,l=1}^{n}\Psi_{3}^{ik}\Phi_{1}^{l}w_{x_{i}x_{k}}w_{x_{l}x_{j}}\nu^{j}$
(3.18)
$\displaystyle\quad-\frac{1}{2}\sum_{i,j,k,l=1}^{n}\Psi_{3}^{ij}\Phi_{1}^{l}w_{x_{i}x_{k}}w_{x_{k}x_{l}}\nu^{j}-\sum_{i,j,k=1}^{n}\Phi_{4}^{k}w_{x_{i}x_{j}}w_{x_{i}x_{k}}\nu^{j}$
$\displaystyle\quad+\frac{1}{2}\sum_{i,j,k=1}^{n}\Phi_{4}^{j}w_{x_{i}x_{k}}^{2}\nu^{j}\qquad\mbox{
on }\Sigma.$
By Conditions 3.1 and 2.16, we have that
$\displaystyle\sum_{i,j,k,l=1}^{n}\Phi_{1}^{l}w_{x_{k}x_{k}x_{l}}w_{x_{i}x_{i}x_{j}}\nu^{j}-\frac{1}{2}\sum_{i,j,k,l=1}^{n}\Phi_{1}^{j}\nu^{j}w_{x_{k}x_{k}x_{l}}w_{x_{i}x_{i}x_{l}}\geq-
Cs\lambda\xi|\nabla\Delta w|^{2}\quad\mbox{ on }\Sigma.$ (3.19)
From Eqs. 2.14 and 2.16, we get that
$\displaystyle\frac{1}{2}\sum_{i,j,k=1}^{n}\Psi_{2}\Phi_{1}^{j}\nu^{j}w_{x_{i}x_{i}}w_{x_{k}x_{k}}=-4s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\frac{\partial\eta}{\partial\nu}|\Delta
w|^{2}\quad\mbox{ on }\Sigma.$ (3.20)
Combining Eq. 2.14, Eq. 2.16, Eq. 3.16 and Eq. 3.17, we find that
$\displaystyle\sum_{i,j,k,l=1}^{n}\Psi_{3}^{ik}\Phi_{1}^{l}w_{x_{i}x_{k}}w_{x_{l}x_{j}}\nu^{j}$
$\displaystyle=-16\sum_{i,j,k,l=1}^{n}s^{3}\lambda^{3}\xi^{3}\eta_{x_{i}}\eta_{x_{k}}\eta_{x_{l}}\nu^{j}w_{x_{i}x_{k}}w_{x_{l}x_{j}}$
$\displaystyle=-16\sum_{i,j,k,l=1}^{n}s^{3}\lambda^{3}\xi^{3}\eta_{x_{i}}\eta_{x_{k}}\nu^{j}w_{x_{i}x_{k}}\frac{\partial
w_{j}}{\partial\nu}\frac{\partial\eta}{\partial\nu}$ (3.21)
$\displaystyle=-16\sum_{i,j,k,l=1}^{n}s^{3}\lambda^{3}\xi^{3}\biggl{(}\frac{\partial\eta}{\partial\nu}\biggr{)}^{3}\biggl{(}\frac{\partial
w_{j}}{\partial\nu}\biggr{)}^{2}$
$\displaystyle=-16\sum_{i,j,k,l=1}^{n}s^{3}\lambda^{3}\xi^{3}\biggl{(}\frac{\partial\eta}{\partial\nu}\biggr{)}^{3}|\Delta
w|^{2}\quad\mbox{ on }\Sigma$
and
$\displaystyle-\frac{1}{2}\sum_{i,j,k,l=1}^{n}\Psi_{3}^{ij}\Phi_{1}^{l}w_{x_{i}x_{k}}w_{x_{k}x_{l}}\nu^{j}-\sum_{i,j,k=1}^{n}\Phi_{4}^{k}w_{x_{i}x_{j}}w_{x_{i}x_{k}}\nu^{j}+\frac{1}{2}\sum_{i,j,k=1}^{n}\Phi_{4}^{j}w_{x_{i}x_{k}}^{2}\nu^{j}$
$\displaystyle=8\sum_{i,j,k,l=1}^{n}s^{3}\lambda^{3}\xi^{3}\biggl{(}\frac{\partial\eta}{\partial\nu}\biggr{)}^{3}|\Delta
w|^{2}+2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\frac{\partial\eta}{\partial\nu}|\Delta
w|^{2}\quad\mbox{ on }\Sigma.$ (3.22)
Hence, combining Conditions 3.1, 3.18, 3.19, 3.20, 3.21 and 3.22, we obtain
that
$\displaystyle V_{1}\cdot\nu$ $\displaystyle\geq-Cs\lambda\xi|\nabla\Delta
w|^{2}-8\sum_{i,j,k,l=1}^{n}s^{3}\lambda^{3}\xi^{3}\biggl{(}\frac{\partial\eta}{\partial\nu}\biggr{)}^{3}|\Delta
w|^{2}-2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\frac{\partial\eta}{\partial\nu}|\Delta
w|^{2}$ $\displaystyle\geq-Cs\lambda\xi|\nabla\Delta
w|^{2}-Cs^{3}\lambda^{3}\xi^{3}|\Delta w|^{2}\quad\mbox{ on }\Sigma.$
This implies that
$\displaystyle\mathbb{E}\int_{Q}\operatorname{div}V_{1}dxdt$
$\displaystyle\geq-C\mathbb{E}\int_{\Sigma}s^{3}\lambda^{3}\xi^{3}|\Delta
w|^{2}d\Gamma dt-C\mathbb{E}\int_{\Sigma}s\lambda\xi|\nabla\Delta
w|^{2}d\Gamma dt.$ (3.23)
Thanks to Condition 3.1, Eqs. 3.15, 2.16 and 3, we have
$\displaystyle\mathbb{E}\int_{Q}\operatorname{div}V_{2}dxdt=\mathbb{E}\int_{\Sigma}V_{2}\cdot\nu
d\Gamma dt$
$\displaystyle=\mathbb{E}\int_{\Sigma}\sum_{j=1}^{n}\Big{(}\sum_{i,k,l,r,m=1}^{n}\Theta_{1}^{ijklrm}w_{x_{i}x_{k}x_{l}}w_{x_{r}x_{m}}+\sum_{i,k,l,r=1}\Theta_{2}^{ijklr}w_{x_{i}x_{k}}w_{x_{l}x_{r}}+\Theta_{9}^{j}\Big{)}\nu^{j}d\Gamma
dt$ $\displaystyle\geq\mathbb{E}\int_{\Sigma}\Big{[}s\xi
O(\lambda)|\nabla\Delta w|^{2}+sO(e^{C\lambda})|\nabla^{2}w|^{2}\Big{]}d\Gamma
dt$ (3.24) $\displaystyle\geq-C\mathbb{E}\int_{\Sigma}s\lambda\xi|\nabla\Delta
w|^{2}d\Gamma dt+\mathcal{B}_{2}.$
Combining Eqs. 3.14, 3.23 and 3.24, we obtain that
$\displaystyle 2\mathbb{E}\int_{Q}\theta
I_{2}\big{(}d\hat{v}+\Delta^{2}vdt\big{)}dx+C\mathbb{E}\int_{\Sigma}s^{3}\lambda^{3}\xi^{3}|\Delta
w|^{2}d\Gamma dt$
$\displaystyle+C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}$
$\displaystyle\qquad\qquad\quad~{}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$ (3.25)
$\displaystyle+C\mathbb{E}\int_{\Sigma}s\lambda\xi|\nabla\Delta w|^{2}d\Gamma
dt+\mathcal{A}+\mathcal{B}_{2}$ $\displaystyle\geq
2\mathbb{E}\int_{Q}I_{2}^{2}dxdt+2\mathbb{E}\int_{Q}I_{2}I_{3}dx+2\mathbb{E}\int_{Q}(M_{1}+M_{2})dxdt.$
_Step 2._ In this step, we study $\displaystyle
2\mathbb{E}\int_{Q}(M_{1}+M_{2})dxdt$ via integration by parts.
From Condition 3.1, section 3 and Eq. 3.15, we get that
$\displaystyle-\mathbb{E}\int_{Q}16s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{2}|\nabla^{2}w\nabla\eta|^{2}dxdt$
$\displaystyle=-\mathbb{E}\int_{Q}16s^{3}\lambda^{4}\xi^{3}\sum_{i,j,k,l=1}^{n}\eta_{x_{i}}^{2}\eta_{x_{k}}\eta_{x_{l}}w_{x_{k}x_{j}}w_{x_{l}x_{j}}dxdt$
$\displaystyle=\mathbb{E}\int_{\Sigma}8s^{3}\lambda^{4}\sum_{i,j,k,l=1}^{n}[-2\xi^{3}\eta_{x_{i}}^{2}\eta_{x_{k}}\eta_{x_{l}}w_{x_{k}x_{j}}w_{x_{l}}+(\xi^{3}\eta_{x_{i}}^{2}\eta_{x_{k}}\eta_{x_{l}})_{x_{j}}w_{x_{k}}w_{x_{l}}]\nu^{j}d\Gamma
dt$ (3.26)
$\displaystyle\quad+\mathbb{E}\int_{Q}8s^{3}\lambda^{4}\sum_{i,j,k,l=1}^{n}[2\xi^{3}\eta_{x_{i}}^{2}\eta_{x_{k}}\eta_{x_{l}}w_{x_{k}x_{j}x_{j}}w_{x_{l}}-(\xi^{3}\eta_{x_{i}}^{2}\eta_{x_{k}}\eta_{x_{l}})_{x_{j}x_{j}}w_{x_{k}}w_{x_{l}}]dxdt$
$\displaystyle\geq-4\mathbb{E}\int_{Q}s\lambda^{2}\xi|\nabla\Delta
w\nabla\eta|^{2}dxdt-16\mathbb{E}\int_{Q}s^{5}\lambda^{6}\xi^{5}|\nabla\eta|^{4}|\nabla
w\nabla\eta|^{2}dxdt-\mathcal{A}_{2}.$
Thanks to Eqs. 3.4 and 3.1, we know that there esists $\lambda_{1}>0$ such
that for all $\lambda\geq\lambda_{1}$, it holds that
$\displaystyle
64\mathbb{E}\int_{Q}s^{5}\lambda^{5}\xi^{5}(\nabla^{2}\eta\nabla\eta\nabla\eta)|\nabla
w\cdot\nabla\eta|^{2}dxdt$
$\displaystyle=128\mathbb{E}\int_{Q}s^{5}\lambda^{5}\xi^{5}|\nabla\eta|^{2}|\nabla
w\cdot\nabla\eta|^{2}dxdt$ (3.27)
$\displaystyle\geq-\mathbb{E}\int_{Q}s^{5}\lambda^{6}\xi^{5}|\nabla\eta|^{4}|\nabla
w\cdot\nabla\eta|^{2}dxdt.$
Combining Condition 3.1, Eq. 3.4, section 3 and Eq. 3.15, we get that
$\displaystyle\mathbb{E}\int_{Q}\big{[}4s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)+2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta\big{]}(|\nabla^{2}w|^{2}-|\Delta
w|^{2})$
$\displaystyle=\frac{1}{2}\mathbb{E}\int_{Q}\Delta\big{[}4s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)+2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta\big{]}|\nabla
w|^{2}dxdt$
$\displaystyle\quad+\mathbb{E}\int_{Q}\nabla\big{[}4s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)+2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta\big{]}\cdot\nabla
w\Delta wdxdt$ (3.28)
$\displaystyle\geq-\mathbb{E}\int_{Q}s^{2}\xi^{2}O(\lambda^{3})|\nabla^{2}w|^{2}dxdt-\mathbb{E}\int_{Q}\big{(}s^{3}O(e^{C\lambda})+s^{4}\xi^{4}O(\lambda^{5})\big{)}|\nabla
w|^{2}dxdt$ $\displaystyle\geq-\mathcal{A}_{2}-\mathcal{A}_{3},$
and that
$\displaystyle
32\mathbb{E}\int_{Q}s^{3}\lambda^{3}\xi^{3}(\nabla^{2}w\nabla\eta\nabla\eta)\sum\limits_{i,j=1}^{n}\eta_{x_{i}x_{j}}w_{x_{i}x_{j}}dxdt$
$\displaystyle=32\sum_{i,j,k,l=1}^{n}\mathbb{E}\int_{Q}s^{3}\lambda^{3}\xi^{3}\eta_{x_{k}x_{l}}\eta_{x_{i}}\eta_{x_{j}}w_{x_{i}x_{j}}w_{x_{k}x_{l}}dxdt$
$\displaystyle=32\sum_{i,j,k,l=1}^{n}\mathbb{E}\int_{\Sigma}s^{3}\lambda^{3}\xi^{3}\eta_{x_{k}x_{l}}\eta_{x_{i}}\eta_{x_{j}}(w_{x_{i}}w_{x_{k}x_{l}}\nu^{j}-w_{x_{i}}w_{x_{j}x_{l}}\nu^{k})d\Gamma
dt$
$\displaystyle\quad-32\sum_{i,j,k,l=1}^{n}\mathbb{E}\int_{Q}s^{3}\lambda^{3}(\xi^{3}\eta_{x_{k}x_{l}}\eta_{x_{i}}\eta_{x_{j}})_{x_{j}}w_{x_{i}}w_{x_{k}x_{l}}dxdt$
(3.29)
$\displaystyle\quad+32\sum_{i,j,k,l=1}^{n}\mathbb{E}\int_{Q}s^{3}\lambda^{3}(\xi^{3}\eta_{x_{k}x_{l}}\eta_{x_{i}}\eta_{x_{j}})_{x_{k}}w_{x_{i}}w_{x_{j}x_{l}}dxdt$
$\displaystyle\quad+32\mathbb{E}\int_{Q}s^{3}\lambda^{3}\xi^{3}\nabla^{2}\eta(\nabla^{2}w\nabla\eta)(\nabla^{2}w\nabla\eta)dxdt$
$\displaystyle\geq
32\mathbb{E}\int_{Q}s^{3}\lambda^{3}\xi^{3}\nabla^{2}\eta(\nabla^{2}w\nabla\eta)(\nabla^{2}w\nabla\eta)dxdt-\mathcal{A}_{2}-\mathcal{A}_{3}$
$\displaystyle=64\mathbb{E}\int_{Q}s^{3}\lambda^{3}\xi^{3}|\nabla^{2}w\nabla\eta|^{2}dxdt-\mathcal{A}_{2}-\mathcal{A}_{3}.$
From sections 3 and 3.15, we see that
$\displaystyle\mathbb{E}\int_{Q}\bigl{[}\big{(}8s^{3}\lambda^{4}\xi^{3}-s^{3}\lambda^{\frac{7}{2}}\xi^{3}\big{)}|\nabla\eta|^{4}|\Delta
w|^{2}-2\big{(}8s^{3}\lambda^{4}\xi^{3}-s^{3}\lambda^{\frac{7}{2}}\xi^{3}\big{)}|\nabla\eta|^{6}s^{2}\lambda^{2}\xi^{2}|\nabla
w|^{2}$
$\displaystyle\qquad~{}+\big{(}8s^{3}\lambda^{4}\xi^{3}-s^{3}\lambda^{\frac{7}{2}}\xi^{3}\big{)}|\nabla\eta|^{8}s^{4}\lambda^{4}\xi^{4}|w|^{2}\bigr{]}dxdt$
$\displaystyle=\mathbb{E}\int_{Q}\bigl{[}\big{(}8s^{3}\lambda^{4}\xi^{3}-s^{3}\lambda^{\frac{7}{2}}\xi^{3})|\nabla\eta|^{4}(\Delta
w+s^{2}\lambda^{2}\xi^{2}|\nabla\eta|^{2}w\big{)}^{2}$ (3.30)
$\displaystyle\qquad\qquad-2\big{(}8s^{3}\lambda^{4}\xi^{3}-s^{3}\lambda^{\frac{7}{2}}\xi^{3}\big{)}|\nabla\eta|^{6}s^{2}\lambda^{2}\xi^{2}(|\nabla
w|^{2}+w\Delta w)\bigr{]}dxdt$
$\displaystyle\geq\mathbb{E}\int_{Q}\big{(}8s^{3}\lambda^{4}\xi^{3}-s^{3}\lambda^{\frac{7}{2}}\xi^{3}\big{)}|\nabla\eta|^{4}\big{(}\Delta
w+s^{2}\lambda^{2}\xi^{2}|\nabla\eta|^{2}w\big{)}^{2}dxdt-\mathcal{A}_{1}.$
Noting that $(a+b+c)^{2}\leq 3(a^{2}+b^{2}+c^{2})$ for $a,b,c\in\mathbb{R}$,
we find that
$\displaystyle\mathbb{E}\int_{Q}s^{3}\lambda^{4}\xi^{3}\theta^{2}|\nabla\eta|^{4}|\Delta
v|^{2}dxdt$
$\displaystyle=\mathbb{E}\int_{Q}s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}\big{(}\Delta
w-2s\lambda\xi\nabla\eta\nabla
w+s^{2}\lambda^{2}\xi^{2}|\nabla\eta|^{2}w-s\lambda^{2}\xi|\nabla\eta|^{2}w-s\lambda\xi|\Delta\eta|w\big{)}^{2}dxdt$
(3.31) $\displaystyle\leq
3\mathbb{E}\int_{Q}s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}\big{(}\Delta
w+s^{2}\lambda^{2}\xi^{2}|\nabla\eta|^{2}w\big{)}^{2}dxdt+12\mathbb{E}\int_{Q}s^{5}\lambda^{6}\xi^{5}|\nabla\eta|^{4}|\nabla
w\cdot\nabla\eta|^{2}+\mathcal{A}_{1}.$
Combining sections 3, 3.27, 3, 3, 3.30 and 3.31, we know there exists
$\lambda_{2}\geq\lambda_{1}$ such that for all $\lambda\geq\lambda_{2}$, it
holds that
$\displaystyle\mathbb{E}\int_{Q}M_{1}dxdt+\mathcal{A}\geq\mathbb{E}\int_{Q}\lambda|\nabla\Delta
w|^{2}dxdt+\mathbb{E}\int_{Q}s^{3}\lambda^{4}\xi^{3}\theta^{2}|\nabla\eta|^{4}|\Delta
v|^{2}dxdt.$ (3.32)
It follows from Condition 3.1, Eqs. 2.14, 2.16, 3.4 and 3 that
$\displaystyle\mathbb{E}\int_{Q}M_{2}dxdt+\mathcal{A}\geq\mathbb{E}\int_{Q}\big{(}16s\lambda\xi|\nabla\hat{w}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\nabla\eta|^{4}\hat{w}^{2}\big{)}dxdt.$
(3.33)
Thanks to sections 3, 3.32 and 3.33, for $\lambda\geq\lambda_{2}$, we have
that
$\displaystyle 2\mathbb{E}\int_{Q}\theta
I_{2}(d\hat{v}+\Delta^{2}vdt)dx+C\mathbb{E}\int_{\Sigma}s^{3}\lambda^{3}\xi^{3}|\Delta
w|^{2}d\Gamma dt$
$\displaystyle\quad+C\mathbb{E}\int_{\Sigma}s\lambda\xi|\nabla\Delta
w|^{2}d\Gamma dt+\mathcal{A}+\mathcal{B}_{2}$
$\displaystyle+C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
$\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$ (3.34)
$\displaystyle\geq
2\mathbb{E}\int_{Q}I_{2}^{2}dxdt+2\mathbb{E}\int_{Q}I_{2}I_{3}dx$
$\displaystyle\quad+\mathbb{E}\int_{Q}\big{(}\lambda|\nabla\Delta
w|^{2}+s^{3}\lambda^{4}\xi^{3}\theta^{2}|\Delta
v|^{2}+s\lambda\xi|\nabla\hat{w}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}\hat{w}^{2}\big{)}dxdt.$
_Step 3._ In this step, we get the estimate of $v$.
From (3.5), we have that
$2\mathbb{E}\int_{Q}\theta
I_{2}\big{(}d\hat{v}+\Delta^{2}vdt\big{)}dx\leq\mathbb{E}\int_{Q}I_{2}^{2}dxdt+\mathbb{E}\int_{Q}\theta^{2}f_{2}^{2}dxdt.$
(3.35)
Thanks to Condition 3.1, Eqs. 2.17 and 3, we find that
$\displaystyle
2\mathbb{E}\int_{Q}I_{2}I_{3}dx\geq-\mathbb{E}\int_{Q}I_{2}^{2}dxdt-\mathcal{A}-C\mathbb{E}\int_{Q}s^{2}\lambda^{2}\xi^{2}\theta^{2}f_{1}^{2}dxdt.$
(3.36)
Noting that
$\displaystyle|\nabla\Delta w|^{2}\leq C\theta^{2}\big{(}|\nabla\Delta
v|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}v|^{2}\big{)}\quad\mbox{ on }\Sigma,$
by Eqs. 3.17 and 3.15, there exists $\lambda_{3}>0$ such that for all
$\lambda\geq\lambda_{3}$, there is $s_{1}=s_{1}(\lambda)>0$, such that for all
$s\geq s_{1}$, we have that
$\begin{array}[]{ll}\displaystyle\mathcal{B}_{2}+\mathbb{E}\int_{\Sigma}s^{3}\lambda^{3}\xi^{3}|\Delta
w|^{2}d\Gamma dt+\mathbb{E}\int_{\Sigma}s\lambda\xi|\nabla\Delta w|^{2}d\Gamma
dt\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq
C\mathbb{E}\int_{\Sigma}s\lambda\xi\theta^{2}\big{(}|\nabla\Delta
v|^{2}+s^{2}\lambda^{2}\xi^{2}|\Delta v|^{2}\big{)}d\Gamma dt.\end{array}$
(3.37)
Thanks to Eqs. 3.34, 3.35, 3.36 and 3.37, for $\lambda\geq\lambda_{3}$ and
$s\geq s_{1}$, we get that
$\displaystyle
C\mathbb{E}\int_{\Sigma}s\lambda\xi\theta^{2}\big{(}|\nabla\Delta
v|^{2}+s^{2}\lambda^{2}\xi^{2}|\Delta v|^{2}\big{)}d\Gamma dt+\mathcal{A}$
$\displaystyle+C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+f_{2}^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
(3.38) $\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$
$\displaystyle\geq\mathbb{E}\int_{Q}\big{(}\lambda|\nabla\Delta
w|^{2}+s^{3}\lambda^{4}\xi^{3}\theta^{2}|\Delta
v|^{2}+s\lambda\xi|\nabla\hat{w}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}\hat{w}^{2}\big{)}dxdt.$
By Lemma 3.1, for $\lambda\geq\max\\{\lambda_{3},\widehat{\lambda}\\}$ and
$s\geq\max\\{s_{1},\widehat{s}\\}$, we obtain that
$\displaystyle\mathbb{E}\int_{Q}(s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}+s^{4}\lambda^{6}\xi^{4}\theta^{2}|\nabla
v|^{2})dxdt\leq C\mathbb{E}\int_{Q}s^{3}\lambda^{4}\xi^{3}\theta^{2}|\Delta
v|^{2}dxdt,$
which, together with section 3, implies that
$\displaystyle
C\mathbb{E}\int_{\Sigma}s\lambda\xi\theta^{2}\big{(}|\nabla\Delta
v|^{2}+s^{2}\lambda^{2}\xi^{2}|\Delta v|^{2}\big{)}d\Gamma dt+\mathcal{A}$
$\displaystyle+C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+f_{2}^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
(3.39) $\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$
$\displaystyle\geq\mathbb{E}\int_{Q}\big{(}\lambda|\nabla\Delta
w|^{2}+s^{3}\lambda^{4}\xi^{3}\theta^{2}|\Delta
v|^{2}+s^{4}\lambda^{6}\xi^{4}\theta^{2}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}$
$\displaystyle\qquad\qquad+s\lambda\xi|\nabla\hat{w}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}\hat{w}^{2}\big{)}dxdt.$
Let $\tilde{v}=s\lambda^{2}\xi e^{s\xi}v$. Then, we have that
$\displaystyle\mathbb{E}\int_{Q}s^{2}\lambda^{4}\xi^{2}\theta^{2}|\nabla^{2}v|^{2}dxdt$
$\displaystyle=\mathbb{E}\int_{Q}s^{2}\lambda^{4}\xi^{2}\theta^{2}|\nabla^{2}(s^{-1}\lambda^{-2}\xi^{-1}\theta^{-1}\tilde{v})|^{2}dxdt$
$\displaystyle\leq
C\mathbb{E}\int_{Q}s^{2}\lambda^{4}\xi^{2}(s^{-2}\lambda^{-4}\xi^{-2}|\nabla^{2}\tilde{v}|^{2}+\lambda^{-2}|\nabla\tilde{v}|^{2}+s^{2}\xi^{2}|\tilde{v}|^{2})dxdt$
$\displaystyle\leq
C|\tilde{v}|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{2}(G))}+C\mathbb{E}\int_{Q}s^{2}\lambda^{2}\xi^{2}|\nabla\tilde{v}|^{2}dxdt+C\mathbb{E}\int_{Q}s^{4}\lambda^{4}\xi^{4}|\tilde{v}|^{2}dxdt$
(3.40) $\displaystyle\leq
C|\tilde{v}|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{2}(G))}+C\mathbb{E}\int_{Q}s^{2}\lambda^{2}\xi^{2}(s^{4}\lambda^{6}\xi^{4}\theta^{2}v^{2}+s^{2}\lambda^{4}\xi^{2}\theta^{2}|\nabla
v|^{2})dxdt$
$\displaystyle\quad+C\mathbb{E}\int_{Q}s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}dxdt$
$\displaystyle\leq
C|\tilde{v}|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{2}(G))}+C\mathbb{E}\int_{Q}(s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}+s^{4}\lambda^{6}\xi^{4}\theta^{2}|\nabla
v|^{2})dxdt.$
It follows from $\tilde{v}=0$ on $\Sigma$ that
$\begin{array}[]{ll}\displaystyle|\tilde{v}|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{2}(G))}\\\\[2.84526pt]
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq
C|\Delta\tilde{v}|^{2}_{L^{2}_{\mathbb{F}}(0,T;L^{2}(G))}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\displaystyle\leq
C\mathbb{E}\int_{Q}\big{(}s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}+s^{4}\lambda^{6}\xi^{4}\theta^{2}|\nabla
v|^{2}+s^{2}\lambda^{4}\xi^{2}\theta^{2}|\Delta v|^{2}\big{)}dxdt.\end{array}$
(3.41)
Combining Eqs. 3.40 and 3.41, we obtain that
$\displaystyle\mathbb{E}\int_{Q}s^{2}\lambda^{4}\xi^{2}\theta^{2}|\nabla^{2}v|^{2}dxdt\leq
C\mathbb{E}\int_{Q}\big{(}s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}+s^{4}\lambda^{6}\xi^{4}\theta^{2}|\nabla
v|^{2}+s^{2}\lambda^{4}\xi^{2}\theta^{2}|\Delta v|^{2}\big{)}dxdt.$ (3.42)
From sections 3 and 3.42, there exists
$\lambda_{4}\geq\max\\{\lambda_{3},\widehat{\lambda}\\}$ such that for all
$\lambda\geq\lambda_{4}$, there is an
$s_{2}=s_{2}(\lambda)>\max\\{s_{1},\widehat{s}\\}$, such that for all $s\geq
s_{2}$, we have that
$\displaystyle
C\mathbb{E}\int_{\Sigma}s\lambda\xi\theta^{2}\big{(}|\nabla\Delta
v|^{2}+s^{2}\lambda^{2}\xi^{2}|\Delta v|^{2}\big{)}d\Gamma dt+\mathcal{A}$
$\displaystyle+C\mathbb{E}\int_{Q}\theta^{2}\big{(}s^{6}\lambda^{6}\xi^{6}f_{1}^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
f_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}f_{1}|^{2}+f_{2}^{2}+s^{6}\lambda^{6}\xi^{6}g_{1}^{2}$
$\displaystyle\qquad\qquad\quad~{}+s^{4}\lambda^{4}\xi^{4}|\nabla
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}g_{1}|^{2}+|\nabla\Delta
g_{1}|^{2}+s^{2}\lambda^{2}\xi^{2}|g_{2}|^{2}\big{)}dxdt$ (3.43)
$\displaystyle\geq\mathbb{E}\int_{Q}\big{(}\lambda|\nabla\Delta
w|^{2}+s^{3}\lambda^{4}\xi^{3}\theta^{2}|\Delta
v|^{2}+s^{2}\lambda^{4}\xi^{2}\theta^{2}|\nabla^{2}v|^{2}+s^{4}\lambda^{6}\xi^{4}\theta^{2}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}\theta^{2}|v|^{2}$
$\displaystyle\qquad\qquad+s\lambda\xi|\nabla\hat{w}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}\hat{w}^{2}\big{)}dxdt.$
Recalling $v=\theta^{-1}w$ and $\hat{v}=\theta^{-1}(\hat{w}-\ell_{t}w)$, we
get that
$\displaystyle\mathbb{E}\int_{Q}\theta^{2}\big{(}\lambda|\nabla\Delta
v|^{2}+s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}\big{)}dxdt$
$\displaystyle=\mathbb{E}\int_{Q}\theta^{2}\big{(}\lambda|\nabla\Delta(\theta^{-1}w)|^{2}+s\lambda\xi|\nabla[\theta^{-1}(\hat{w}-\ell_{t}w)]|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}\theta^{-2}|\hat{w}-\ell_{t}w|^{2}\big{)}dxdt$
(3.44) $\displaystyle\leq
C\mathbb{E}\int_{Q}\big{[}\lambda\big{(}|\nabla\Delta
w|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}w|^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
w|^{2}+s^{6}\lambda^{6}\xi^{6}|w|^{2}\big{)}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}\big{(}|\hat{w}|^{2}+s^{2}\lambda^{2}\xi^{2}|w|^{2}\big{)}$
$\displaystyle\qquad\qquad~{}+s\lambda\xi\big{(}|\nabla\hat{w}|^{2}+s^{2}\lambda^{2}\xi^{2}|\hat{w}|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla
w|^{2}+s^{4}\lambda^{4}\xi^{4}|w|^{2}\big{)}\big{]}dxdt$ $\displaystyle\leq
C\mathbb{E}\int_{Q}\big{(}\lambda|\nabla\Delta
w|^{2}+s\lambda\xi|\nabla\hat{w}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{w}|^{2}\big{)}dxdt+\mathcal{A}.$
Thanks to sections 3 and 3.1, there exists $\lambda_{5}>0$ such that for all
$\lambda\geq\lambda_{5}$, there is an $s_{3}=s_{3}(\lambda)>0$, such that for
all $s\geq s_{3}$, we have
$\displaystyle\mathcal{A}$
$\displaystyle\leq\frac{1}{C}\mathbb{E}\int_{Q}\theta^{2}\bigl{(}s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}v|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
v|^{2}$ $\displaystyle\qquad\qquad\quad\;+s^{4}\lambda^{6}\xi^{4}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}|v|^{2}\bigr{)}dxdt.$ (3.45)
Let us choose $\lambda_{0}\geq\max\\{\lambda_{4},\lambda_{5}\\}$. Combining
sections 3, 3.44 and 3.45, for all $\lambda\geq\lambda_{0}$, one can find
$s_{0}=s_{0}(\lambda)\geq\max\\{s_{2},s_{3}\\}$ so that for any $s\geq s_{0}$,
inequality Theorem 3.2 holds. ∎
## 4 Proof of the observability estimate
###### Proof of Theorem 1.5.
Let $\chi\in C_{0}^{\infty}([0,T])$ satisfy
$\chi=1\text{ in
}\bigg{(}\frac{T}{2}-\varepsilon_{1}T,\frac{T}{2}+\varepsilon_{1}T\bigg{)}.$
Put $v=\chi z$ and $\hat{v}=\chi\hat{z}+\chi_{t}z$ for $(z,\hat{z})$
satisfying Eq. 1.2, then $(v,\hat{v})$ fulfills
$v(0,\cdot)=v(T,\cdot)=\hat{v}(0,\cdot)=\hat{v}(T,\cdot)=0$ in $G$, and solves
$\left\\{\begin{array}[]{ll}\displaystyle
dv=\hat{v}dt+\chi(Z-a_{5}z)dW(t)&\text{ in }Q,\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle
d\hat{v}+\Delta^{2}vdt=\tilde{f}_{2}dt+\tilde{g}_{2}dW(t)&\text{ in }Q,\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle v=\frac{\partial
v}{\partial\nu}=0&\text{ on }\Sigma,\end{array}\right.$ (4.1)
where
$\tilde{f}_{2}=\chi[(a_{1}-\operatorname{div}a_{2}-a_{4}a_{5})z-a_{2}\nabla
z-a_{3}\hat{Z}+a_{4}Z]+2\chi_{t}\hat{z}+\chi_{tt}z$
and
$\tilde{g}_{2}=\chi\hat{Z}+\chi_{t}(Z-a_{5}z).$
By Theorem 3.2, for $\lambda\geq\lambda_{0}$ and $s\geq s_{0}$, we have
$\displaystyle\mathbb{E}\int_{Q}\theta^{2}\chi^{2}\bigl{(}\lambda|\nabla\Delta
z|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}z|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
z|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla
z|^{2}+s^{6}\lambda^{8}\xi^{6}|z|^{2}\bigr{)}dxdt$
$\displaystyle\leq\mathbb{E}\int_{Q}\theta^{2}\bigl{(}s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}+\lambda|\nabla\Delta
v|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}v|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
v|^{2}$ $\displaystyle\qquad\qquad~{}+s^{4}\lambda^{6}\xi^{4}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}|v|^{2}\bigr{)}dxdt$ $\displaystyle\leq
C\mathbb{E}\int_{Q}\theta^{2}\chi^{2}\big{(}s^{6}\lambda^{6}\xi^{6}z^{2}+s^{4}\lambda^{4}\xi^{4}|\nabla
z|^{2}+s^{2}\lambda^{2}\xi^{2}|\nabla^{2}z|^{2}+|\nabla\Delta
z|^{2}+|z|^{2}+|\nabla z|^{2}\big{)}dxdt$
$\displaystyle\quad+C\mathbb{E}\int_{J_{1}}\theta^{2}\bigr{(}|\hat{z}|^{2}+z^{2}+s^{2}\lambda^{2}\xi^{2}z^{2}\bigl{)}dxdt+C(s,\lambda)|(Z,\hat{Z})|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G))\times
L^{2}_{\mathbb{F}}(0,T;H^{1}(G))}$
$\displaystyle\quad+C\mathbb{E}\int_{\Sigma}\theta^{2}\bigr{(}s\lambda\xi|\nabla\Delta
z|^{2}+s^{3}\lambda^{3}\xi^{3}|\Delta z|^{2}\bigl{)}d\Gamma dt.$
This, together with Condition 3.1, implies that there exists
$\tilde{\lambda}_{1}\geq\lambda_{0}$ such that for all
$\lambda\geq\tilde{\lambda}_{1}$, there is
$\tilde{s}_{1}=\tilde{s}_{1}(\lambda)\geq s_{0}$, so that for any
$s\geq\tilde{s}_{1}$, it holds that
$\displaystyle\mathbb{E}\int_{Q}\theta^{2}\bigl{(}s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}+\lambda|\nabla\Delta
v|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}v|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
v|^{2}$ $\displaystyle\qquad\quad+s^{4}\lambda^{6}\xi^{4}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}|v|^{2}\bigr{)}dxdt$ (4.2) $\displaystyle\leq
C\mathbb{E}\int_{J_{1}}\theta^{2}\bigr{(}|\hat{z}|^{2}+z^{2}+s^{2}\lambda^{2}\xi^{2}z^{2}\bigl{)}dxdt+C\mathbb{E}\int_{\Sigma}\theta^{2}\bigr{(}s\lambda\xi|\nabla\Delta
z|^{2}+s^{3}\lambda^{3}\xi^{3}|\Delta z|^{2}\bigl{)}d\Gamma dt$
$\displaystyle\quad+C(s,\lambda)|(Z,\hat{Z})|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G))\times
L^{2}_{\mathbb{F}}(0,T;H^{1}(G))}.$
Thanks to Condition 3.1, we obtain
$\displaystyle e^{2se^{\lambda\delta}-C\lambda-6\ln
s}\mathbb{E}\int_{J_{2}}\bigl{(}|\nabla\Delta z|^{2}+|\nabla^{2}z|^{2}+|\nabla
z|^{2}+z^{2}+|\nabla\hat{z}|^{2}+|\hat{z}|^{2}\bigr{)}dxdt$ $\displaystyle\leq
e^{2se^{\lambda\delta}}\mathbb{E}\int_{J_{2}}\bigl{(}s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}+\lambda|\nabla\Delta
v|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}v|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
v|^{2}$ $\displaystyle\qquad\qquad\qquad+s^{4}\lambda^{6}\xi^{4}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}|v|^{2}\bigr{)}dxdt$ (4.3)
$\displaystyle\leq\mathbb{E}\int_{Q}\theta^{2}\bigl{(}s\lambda\xi|\nabla\hat{v}|^{2}+s^{3}\lambda^{\frac{7}{2}}\xi^{3}|\hat{v}|^{2}+\lambda|\nabla\Delta
v|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}v|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
v|^{2}$ $\displaystyle\qquad\qquad~{}+s^{4}\lambda^{6}\xi^{4}|\nabla
v|^{2}+s^{6}\lambda^{8}\xi^{6}|v|^{2}\bigr{)}dxdt.$
From Eqs. 1.3 and 3.1, we see that
$\displaystyle|(z^{T},\hat{z}^{T})|_{L^{2}_{\mathcal{F}_{T}}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathcal{F}_{T}}(\Omega;H_{0}^{1}(G))}$
$\displaystyle\leq C\mathbb{E}\int_{J_{2}}\bigl{(}|\nabla\Delta
z|^{2}+|\nabla^{2}z|^{2}+|\nabla
z|^{2}+z^{2}+|\nabla\hat{z}|^{2}+|\hat{z}|^{2}\bigr{)}dxdt$ (4.4)
$\displaystyle\quad+C(s,\lambda)|(Z,\hat{Z})|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathbb{F}}(0,T;H_{0}^{1}(G))},$
and that
$\displaystyle\mathbb{E}\int_{J_{1}}\theta^{2}\big{(}|\hat{z}|^{2}+z^{2}+s^{2}\lambda^{2}\xi^{2}z^{2}\big{)}dxdt$
$\displaystyle\displaystyle\leq e^{2se^{-\lambda\delta}+C\lambda+2\ln
s}\mathbb{E}\int_{J_{1}}\big{(}|\hat{z}|^{2}+z^{2}\big{)}dxdt$ (4.5)
$\displaystyle\leq Ce^{2se^{-\lambda\delta}+C\lambda+2\ln
s}|(z^{T},\hat{z}^{T})|_{L^{2}_{\mathcal{F}_{T}}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathcal{F}_{T}}(\Omega;H_{0}^{1}(G))}$
$\displaystyle\quad+C(s,\lambda)|(Z,\hat{Z})|^{2}_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathbb{F}}(0,T;H_{0}^{1}(G))}.$
Combining (4.2)–(4) choosing $\lambda\geq\tilde{\lambda}_{1}$ and
$s\geq\tilde{s}_{1}$ such that
$\displaystyle
C\exp\big{(}2se^{-\lambda\delta}-2se^{\lambda\delta}+C\lambda+8\ln
s\big{)}\leq\frac{1}{2},$
we get the desired observability estimate. ∎
## 5 Proof of Theorem 1.4
The proof of Theorem 1.4 is similar to that for [23, Theorem 2.3]. We provide
it here for the convenience of the readers. To begin with, we recall the
following result.
###### Lemma 5.1.
[25, Lemma 2.1] There exists a random variable $\zeta\in
L^{2}_{\mathcal{F}_{T}}(\Omega)$ such that it is impossible to find
$\varsigma_{1},\varsigma_{2}\in L^{2}_{\mathbb{F}}(0,T)\times
C_{\mathbb{F}}([0,T];L^{2}(\Omega))$ and $\alpha\in\mathbb{R}$ satisfying
$\displaystyle\zeta=\alpha+\int_{0}^{T}\varsigma_{1}(t)dt+\int_{0}^{T}\varsigma_{2}(t)dW(t).$
###### Proof of Theorem 1.4.
We employ a contradiction argument and divide the proof into three cases.
_Case 1: $a_{3}\in C_{\mathbb{F}}([0,T];L^{\infty}(G))$,
$G\backslash\overline{G_{0}}\neq\emptyset$ and $\operatorname{supp}f\subset
G_{0}$_.
Let $\rho\in C_{0}^{\infty}(G\backslash G_{0})$ satisfying
$|\rho|_{L^{2}(G)}=1$. Suppose that Eq. 1.1 was exactly controllable. By
Definition 1.2, for $(y_{0},\hat{y}_{0})=(0,0)$, there exist controls
$(f,g,h_{1},h_{2})$ with $\operatorname{supp}f\subset G_{0}$ a.e.
$(t,\omega)\in(0,T)\times\Omega$ such that the solution to Eq. 1.1 fulfills
$(y(T),\hat{y}(T))=(\rho\zeta,0)$, where $\zeta$ is given in Lemma 5.1. Hence,
$\displaystyle\rho\zeta=\int_{0}^{T}\hat{y}dt+\int_{0}^{T}(a_{3}y+f)dW(t).$
(5.1)
Multiplying Eq. 5.1 by $\rho$ and integrating it in $G$, we arrive that
$\displaystyle\zeta=\int_{0}^{T}\langle\hat{y},\rho\rangle_{(H^{3}(G)\cap
H_{0}^{2}(G))^{*},H^{3}(G)\cap H_{0}^{2}(G)}dt+\int_{0}^{T}\langle
a_{3}y,\rho\rangle_{H^{-1}(G),H_{0}^{1}(G)}dW(t),$
which contradicts Lemma 5.1.
_Case 2: $a_{4}\in C_{\mathbb{F}}([0,T];L^{\infty}(G))$,
$G\backslash\overline{G_{0}}\neq\emptyset$ and $\operatorname{supp}g\subset
G_{0}$_.
Choose $\rho$ as in Case 1. Assume that Eq. 1.1 was exactly controllable.
Then, for $(y_{0},\hat{y}_{0})=(0,0)$, there exist controls
$(f,g,h_{1},h_{2})$ with $\operatorname{supp}g\subset G_{0}$ a.e.
$(t,\omega)\in(0,T)\times\Omega$ such that the solution to Eq. 1.1 fulfills
$(y(T),\hat{y}(T))=(0,\zeta)$.
Clearly, $(\phi,\hat{\phi})\mathop{\buildrel\Delta\over{=}}(\rho
y,\rho\hat{y})$ satisfies
$\displaystyle\left\\{\begin{aligned} &d\phi=\hat{\phi}dt+(a_{3}\phi+\rho
f)dW(t)&&\quad\text{ in }Q,\\\ &d\hat{\phi}+\Delta^{2}\phi
dt=\tilde{f}_{2}dt+a_{4}\phi dW(t)&&\quad\text{ in }Q,\\\
&\phi=\frac{\partial\phi}{\partial\nu}=0&&\quad\text{ on }\Sigma,\\\
&(\phi(0),\hat{\phi}(0))=(0,0)&&\quad\text{ in }G,\end{aligned}\right.$
where $\tilde{f}_{2}=[\Delta^{2},\rho]y+a_{1}\phi+\rho a_{2}\cdot\nabla\phi$.
Furthermore, we have $(\phi(T),\hat{\phi}(T))=(0,\rho\zeta)$. Hence, we have
$\displaystyle\zeta=-\int_{0}^{T}(\langle\Delta^{2}\phi,\rho\rangle_{H^{-5}(G),H_{0}^{5}(G)}+\langle\tilde{f}_{2},\rho\rangle_{H^{-4}(G),H_{0}^{4}(G)})dt+\int_{0}^{T}\langle
a_{3}\phi,\rho\rangle_{H^{-1}(G),H_{0}^{1}(G)}dW(t),$
which contradicts Lemma 5.1.
_Case 3: $h_{1}=h_{2}=0$._
Assume that Eq. 1.1 was exactly controllable. Then, from the equivalence
between the exact controllability of Eq. 1.1 and the observability estimate of
Eq. 1.2, we get that for any $(z^{T},\hat{z}^{T})\in
L^{2}_{\mathcal{F}_{T}}(\Omega;H^{3}(G)\cap H_{0}^{2}(G))\times
L^{2}_{\mathcal{F}_{T}}(\Omega;H_{0}^{1}(G))$, the solution
$(z,Z,\hat{z},\hat{Z})$ to Eq. 1.2 ( with $\tau=T$ and
$(z(T),\hat{z}(T))=(z^{T},\hat{z}^{T})$) satisfies
$\displaystyle|(z^{T},\hat{z}^{T})|_{L^{2}_{\mathcal{F}_{T}}(\Omega;H^{3}(G)\cap
H_{0}^{2}(G))\times L^{2}_{\mathcal{F}_{T}}(\Omega;H_{0}^{1}(G))}\leq
C(|Z|_{L^{2}_{\mathbb{F}}(0,T;H^{3}(G)\cap
H^{2}_{0}(G))}+|\hat{Z}|_{L^{2}_{\mathbb{F}}(0,T;H_{0}^{1}(G))}).$ (5.2)
For any nonzero $(\Phi_{0},\Phi_{1})\in(H^{3}(G)\cap H_{0}^{2}(G))\times
H_{0}^{1}(G)$, let $(\Phi,\hat{\Phi})$ solve the equation
$\displaystyle\left\\{\begin{aligned} &d\Phi=\hat{\Phi}dt-a_{5}\Phi
dW(t)&&\quad\text{ in }Q,\\\ &d\hat{\Phi}+\Delta^{2}\Phi
dt=[(a_{1}-\operatorname{div}a_{2}-a_{4}a_{5})\Phi-
a_{2}\cdot\nabla\Phi]dt&&\quad\text{ in }Q,\\\
&\Phi=\frac{\partial\Phi}{\partial\nu}=0&&\quad\text{ on }\Sigma,\\\
&(\Phi(0),\hat{\Phi}(0))=(\Phi_{0},\Phi_{1})&&\quad\text{ in
}G.\end{aligned}\right.$
Clearly, $(\Phi,0,\hat{\Phi},0)$ solves Eq. 1.2 with the final datum
$(z^{T},\hat{z}^{T})=(\Phi(T),\hat{\Phi}(T))$, a contradiction to the
inequality Eq. 5.2. ∎
## Appendix A Proof of the weighted identity
###### Proof of Theorem 2.1.
It is clear that
$\displaystyle dw=d(\theta v)=\theta dv+\ell_{t}\theta vdt=\hat{w}dt+\theta
f_{1}dt+\theta g_{1}dW(t),$
and
$\displaystyle\theta d\hat{v}$ $\displaystyle=\theta
d[\theta^{-1}(\hat{w}-\ell_{t}w)]=d\hat{w}-\ell_{t}dw-\ell_{tt}wdt-\ell_{t}\hat{w}dt+\ell_{t}^{2}dt$
$\displaystyle=d\hat{w}-2\ell_{t}\hat{w}dt+(\ell_{t}^{2}-\ell_{tt})wdt-\ell_{t}\theta
f_{1}dt-\ell_{t}\theta g_{t}dW(t).$ (A.1)
We also have
$\displaystyle\theta\Delta^{2}v$
$\displaystyle=\Delta^{2}w-4s\lambda\xi\nabla\eta\cdot\nabla\Delta
w-4s\lambda^{2}\xi(\nabla^{2}w\nabla\eta\nabla\eta)-4s\lambda\xi\sum\limits_{i,j=1}^{n}\eta_{x_{i}x_{j}}w_{x_{i}x_{j}}$
$\displaystyle\quad+2s^{2}\lambda^{2}\xi^{2}|\nabla\eta|^{2}\Delta
w-2s\lambda^{2}\xi|\nabla\eta|^{2}\Delta w-2s\lambda\xi\Delta\eta\Delta
w+4s^{2}\lambda^{2}\xi^{2}(\nabla^{2}w\nabla\eta\nabla\eta)$
$\displaystyle\quad-4\nabla\Delta\ell\cdot\nabla
w+12s^{2}\lambda^{3}\xi^{2}|\nabla\eta|^{2}\nabla\eta\nabla
w+8s^{2}\lambda^{2}\xi^{2}(\nabla^{2}\eta\nabla\eta\nabla w)$ (A.2)
$\displaystyle\quad-4s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\nabla\eta\nabla
w+4s^{2}\lambda^{2}\xi^{2}\Delta\eta\nabla\eta\nabla
w+4(\nabla\ell\cdot\nabla\Delta\ell)w+2|\nabla^{2}\ell|^{2}w-\Delta^{2}\ell w$
$\displaystyle\quad-6s^{3}\lambda^{4}\xi^{3}|\nabla\eta|^{4}w-4s^{3}\lambda^{3}\xi^{3}(\nabla^{2}\eta\nabla\eta\nabla\eta)w+s^{4}\lambda^{4}\xi^{4}|\nabla\eta|^{4}w-2s^{3}\lambda^{3}\xi^{3}|\nabla\eta|^{2}\Delta\eta
w+|\Delta\ell|^{2}w.$
From Eqs. 2.13, 2.14, 2.15, 2.16, A, 2.17 and A.2, we have
$2\theta I_{2}(d\hat{v}+\Delta^{2}vdt)=2I_{2}(I_{1}+I_{2}dt+I_{3}).$
We will compute $I_{1}I_{2}$ under the form
$\sum\limits_{i=1}^{7}\sum\limits_{j=1}^{5}I_{ij}$, where $I_{ij}$ is the
product of the $i$-th term of $I_{1}$ with the $j$-th term of $I_{2}$. Note
that $I_{ij}$ are the same as Appendix A in [22] for $i=1,\cdots,6$ and
$j=1,\cdots,5$, except for $I_{13}$.
We have
$\displaystyle I_{13}$
$\displaystyle=\sum_{i,j,k,l=1}^{n}\Phi_{3}^{kl}w_{x_{i}x_{i}x_{j}x_{j}}w_{x_{k}x_{l}}dt$
$\displaystyle=\sum_{i,j,k,l=1}^{n}\biggl{(}\Phi_{3}^{kl}w_{x_{i}x_{i}x_{j}}w_{x_{k}x_{l}}-\Phi_{3}^{kj}w_{x_{i}x_{i}x_{l}}w_{x_{k}x_{l}}+\Phi_{3}^{kl}w_{x_{i}x_{i}x_{l}}w_{x_{k}x_{j}}-\Phi_{3_{x_{i}}}^{kl}w_{x_{i}x_{j}}w_{x_{k}x_{l}}$
$\displaystyle\qquad\qquad~{}~{}+\Phi_{3_{x_{l}}}^{kj}w_{x_{i}x_{l}}w_{x_{i}x_{k}}+\Phi_{3_{x_{l}}}^{kl}w_{x_{i}x_{j}}w_{x_{i}x_{k}}-\frac{1}{2}\Phi_{3_{x_{l}}}^{jl}w_{x_{i}x_{k}}^{2}-\Phi_{3_{x_{i}}}^{kl}w_{x_{j}x_{l}}w_{x_{i}x_{k}}\biggr{)}_{x_{j}}dt$
$\displaystyle\quad-\sum_{i,j,k,l=1}^{n}\Phi_{3}^{kl}w_{x_{i}x_{i}x_{l}}w_{x_{k}x_{j}x_{j}}dt+\sum_{i,j,k,l=1}^{n}\Phi_{3x_{i}x_{j}}^{kl}w_{x_{i}x_{j}}w_{x_{k}x_{l}}dt-\sum_{i,j,k,l=1}^{n}\Phi_{3x_{l}x_{j}}^{kl}w_{x_{i}x_{j}}w_{x_{i}x_{k}}dt$
$\displaystyle\quad-\sum_{i,j,k,l=1}^{n}\Phi_{3x_{i}x_{l}}^{kl}w_{x_{i}x_{j}}w_{x_{k}x_{j}}dt+\sum_{i,j,k,l=1}^{n}\Phi_{3x_{i}x_{j}}^{kl}w_{x_{i}x_{l}}w_{x_{k}x_{j}}dt+\frac{1}{2}\sum_{i,j,k,l=1}^{n}\Phi_{3x_{k}x_{l}}^{kl}w_{x_{i}x_{j}}^{2}dt.$
We have
$\displaystyle I_{71}$
$\displaystyle=\sum_{i,j=1}^{n}\Phi_{1}^{i}w_{x_{i}x_{j}x_{j}}d\hat{w}$
$\displaystyle=\sum_{i,j=1}^{n}\biggl{(}-\Phi_{1}^{j}\hat{w}dw_{x_{i}x_{i}}+\Phi_{1x_{i}}^{i}\hat{w}\hat{w}_{x_{j}}-\frac{1}{2}\Phi_{1x_{i}x_{j}}^{i}\hat{w}^{2}+\Phi_{1}^{i}\hat{w}_{x_{i}}\hat{w}_{x_{j}}-\frac{1}{2}\Phi_{1}^{j}\hat{w}_{x_{i}}^{2}\biggr{)}_{x_{j}}dt$
$\displaystyle\quad+\sum_{i,j=1}^{n}d(\Phi^{i}_{1}w_{x_{i}x_{j}x_{j}}\hat{w})+\frac{1}{2}\sum_{i,j=1}^{n}\Phi_{1x_{i}x_{j}x_{j}}^{i}\hat{w}^{2}dt-\frac{1}{2}\sum_{i,j=1}^{n}\Phi_{1x_{i}}^{i}\hat{w}_{x_{j}}^{2}dt-\sum_{i,j=1}^{n}\Phi_{1x_{j}}^{i}\hat{w}_{x_{i}}\hat{w}_{x_{j}}dt$
$\displaystyle\quad-\sum_{i,j=1}^{n}\Phi_{1t}^{i}w_{x_{i}x_{j}x_{j}}\hat{w}dt+\sum_{i,j=1}^{n}(\Phi_{1x_{i}}^{i}\hat{w}+\Phi_{1}^{i}\hat{w}_{x_{i}})[(\theta
f_{1})_{x_{j}x_{j}}dt+(\theta g_{1})_{x_{j}x_{j}}dW(t)]$
$\displaystyle\quad-\sum_{i,j=1}^{n}\Phi_{1}^{i}dw_{x_{i}x_{j}x_{j}}d\hat{w},$
$\displaystyle I_{72}$
$\displaystyle=\sum_{i=1}^{n}\Phi_{2}w_{x_{i}x_{i}}d\hat{w}$
$\displaystyle=\sum_{j=1}^{n}\biggl{(}-\Phi_{2}\hat{w}\hat{w}_{x_{j}}+\frac{1}{2}\Phi_{2x_{j}}\hat{w}^{2}\biggr{)}_{x_{j}}dt+\sum_{i=1}^{n}d(\Phi_{2}w_{x_{i}x_{i}}\hat{w})-\sum_{i=1}^{n}\Phi_{2t}w_{x_{i}x_{i}}\hat{w}dt$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{n}\Phi_{2x_{i}x_{i}}\hat{w}^{2}dt+\sum_{i=1}^{n}\Phi_{2}\hat{w}_{x_{i}}^{2}dt-\sum_{i=1}^{n}\Phi_{2}dw_{x_{i}x_{i}}d\hat{w}$
$\displaystyle\quad-\sum_{i=1}^{n}\Phi_{2}\hat{w}[(\theta
f_{1})_{x_{i}x_{i}}dt+(\theta g_{1})_{x_{i}x_{i}}dW(t)],$ $\displaystyle
I_{73}$ $\displaystyle=\sum_{i,j=1}^{n}\Phi_{3}^{ij}w_{x_{i}x_{j}}d\hat{w}$
$\displaystyle=\sum_{i,j=1}^{n}\biggl{(}-\Phi_{3}^{ij}\hat{w}\hat{w}_{x_{i}}+\frac{1}{2}\Phi_{3x_{i}}^{ij}\hat{w}^{2}\biggr{)}_{x_{j}}dt+\sum_{i,j=1}^{n}d(\Phi_{3}^{ij}w_{x_{i}x_{j}}\hat{w})-\sum_{i,j=1}^{n}\Phi_{3t}^{ij}w_{x_{i}x_{j}}\hat{w}dt$
$\displaystyle\quad-\frac{1}{2}\sum_{i,j=1}^{n}\Phi_{3x_{i}x_{j}}^{ij}\hat{w}^{2}dt-\sum_{i,j=1}^{n}\Phi_{3}^{ij}dw_{x_{i}x_{j}}d\hat{w}+\sum_{i,j=1}^{n}\Phi_{3}^{ij}\hat{w}_{x_{i}}\hat{w}_{x_{j}}dt$
$\displaystyle\quad-\sum_{i,j=1}^{n}\Phi_{3}^{ij}\hat{w}[(\theta
f_{1})_{x_{i}x_{j}}dt+(\theta g_{1})_{x_{i}x_{j}}dW(t)],$ $\displaystyle
I_{74}$ $\displaystyle=\sum_{i=1}^{n}\Phi_{4}^{i}w_{x_{i}}d\hat{w}$
$\displaystyle=\sum_{j=1}^{n}\biggl{(}-\frac{1}{2}\Phi_{4}^{j}\hat{w}^{2}\biggr{)}_{x_{j}}dt+\sum_{i=1}^{n}d(\Phi_{4}^{i}w_{x_{i}}\hat{w})-\sum_{i=1}^{n}\Phi_{4t}^{i}w_{x_{i}}\hat{w}dt+\frac{1}{2}\sum_{i=1}^{n}\Phi_{4x_{i}}^{i}\hat{w}^{2}dt$
$\displaystyle\quad-\sum_{i=1}^{n}\Phi^{i}_{4}dw_{x_{i}}d\hat{w}-\sum_{i=1}^{n}\Phi_{4}^{i}\hat{w}[(\theta
f_{1})_{x_{i}}dt+(\theta g_{1})_{x_{i}}dW(t)],$ $\displaystyle I_{75}$
$\displaystyle=\Phi_{5}wd\hat{w}=d(\Phi_{5}w\hat{w})-\Phi_{5t}w\hat{w}dt-\Phi_{5}\hat{w}^{2}dt-\Phi_{5}dwd\hat{w}-\Phi_{5}\hat{w}(\theta
f_{1}dt+\theta g_{1}dW(t)).$
By summing all the $I_{ij}$, we get Eq. 2.12. ∎
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[table]capposition=top
# Towards Training GNNs using Explanation Directed Message Passing
Valentina Giunchiglia
Imperial College London
<EMAIL_ADDRESS>Varun Shukla11footnotemark: 1
Ludwig-Maximilians Universität München
<EMAIL_ADDRESS>Gonzalez
Imperial College London
<EMAIL_ADDRESS>Agarwal
Adobe
<EMAIL_ADDRESS>Equal contribution.
###### Abstract
With the increasing use of Graph Neural Networks (GNNs) in critical real-world
applications, several post hoc explanation methods have been proposed to
understand their predictions. However, there has been no work in generating
explanations on the fly during model training and utilizing them to improve
the expressive power of the underlying GNN models. In this work, we introduce
a novel explanation-directed neural message passing framework for GNNs, Expass
(EXplainable message PASSing), which aggregates only embeddings from nodes and
edges identified as important by a GNN explanation method. Expass can be used
with any existing GNN architecture and subgraph-optimizing explainer to learn
accurate graph embeddings. We theoretically show that Expass alleviates the
oversmoothing problem in GNNs by slowing the layer-wise loss of Dirichlet
energy and that the embedding difference between the vanilla message passing
and Expass framework can be upper bounded by the difference of their
respective model weights. Our empirical results show that graph embeddings
learned using Expass improve the predictive performance and alleviate the
oversmoothing problems of GNNs, opening up new frontiers in graph machine
learning to develop explanation-based training frameworks.
## 1 Introduction
Graph Neural Networks (GNNs) are increasingly used as powerful tools for
representing graph-structured data, such as social, information, chemical, and
biological networks [1, 2]. With the deployment of GNN models in critical
applications (e.g., financial systems and crime forecasting [3, 4]), it
becomes essential to ensure that the relevant stakeholders understand and
trust their decisions. To this end, several approaches [5, 6, 7, 8, 9, 10, 11,
12, 13] have been proposed in recent literature to generate post hoc
explanations for predictions of GNN models.
In contrast to other modalities like images and texts, generating instance-
level explanations for graphs is non-trivial. In particular, it is more
challenging since individual node embeddings in GNNs aggregate information
using the entire graph structure, and, therefore, explanations can be on
different levels (i.e., node attributes, nodes, and edges). While several
categories of GNN explanation methods have been proposed: gradient-based [14,
10, 5], perturbation-based [13, 9, 11, 8, 15], and surrogate-based [7, 12],
their utility is limited to generating post hoc node- and edge-level
explanations for a given pre-trained GNN model. Thus, the capability of GNN
explainers to improve the predictive performance of a GNN model lacks
understanding as there is very little work on systematically analyzing the
reliability of state-of-the-art GNN explanation methods on model performance
[16].
To address this, recent works have explored the joint optimization of machine
learning models and explanation methods to improve the reliability of
explanations [17, 18]. Zhou et al. [18] proposed DropEdge as a technique to
drop random edges (similar to generating random edge explanations) during
training to reduce overfitting in GNNs. More recently, Spinelli et al. [17]
used meta-learning frameworks to generate GNN explanations and show an
improvement in the performance of specific GNN explanation methods. While
these works make an initial attempt at jointly optimizing explainers and
predictive models, they are neither generalizable nor exhaustive. They fail to
show improvement in the downstream GNN performance [17] and degree of
explainability [18] across diverse GNN architectures and explainers. Further,
there is little to no work done on either theoretically analyzing the effect
of GNN explanations on the neural message framework in GNNs or on important
GNN properties like oversmoothing [19].
Present work. In this work, we introduce a novel explanation-directed neural
message passing framework, Expass, which can be used with any GNN model and
subgraph-optimizing explainer to learn accurate graph representations. In
particular, Expass utilizes GNN explanations to steer the underlying GNN model
to learn graph embeddings using only important nodes and edges. Expass aims to
define local neighborhoods for neural message passing, i.e., identify the most
important edges and nodes, using explanation weights, in the $k$-hop local
neighborhood of every node in the graph. Formally, we augment existing message
passing architectures to allow information flow along important edges while
blocking information along irrelevant edges.
We present an extensive theoretical and empirical analysis to show the
effectiveness of Expass on the predictive, explainability, and oversmoothing
performance of GNNs. Our theoretical results show that the embedding
difference between vanilla message passing and Expass frameworks is upper-
bounded by the difference between their model weights. Further, we show that
embeddings learned using Expass relieve the oversmoothing problem in GNNs as
they reduce information propagation by slowing the layer-wise loss of
Dirichlet energy (Section 4.2). For our empirical analysis, we integrate
Expass into state-of-the-art GNN models and evaluate their predictive,
oversmoothing, and explainability performance on real-world graph datasets
(Section 5). Our results show that, on average, across five GNN models, Expass
improves the degree of explainability of the underlying GNNs by 39.68%. Our
ablation studies show that for an increasing number of GNN layers, Expass
achieves 34.4% better oversmoothing performance than its vanilla counterpart.
Finally, our results demonstrate the effectiveness of using explanations
during training, paving the way for new frontiers in GraphXAI research to
develop explanation-based training algorithms.
## 2 Related works
Graph Neural Networks. Graph Neural Networks (GNNs) are complex non-linear
functions that transform input graph structures into a lower dimensional
embedding space. The main goal of GNNs is to learn embeddings that reflect the
underlying input graph structure, i.e., neighboring nodes in the graph are
mapped to neighboring points in the embedding space. Prior works have proposed
several GNN models using spectral and non-spectral approaches. Spectral models
[20, 21, 22, 23, 24] leverage Fourier transform and graph Laplacian to define
convolution approaches for GNN models. However, non-spectral approaches [25,
26, 27, 28, 29] define the convolution operation by leveraging the local
neighborhood of individual nodes in the graph. Most modern non-spectral models
are message-passing frameworks [30, 31], where nodes update their embedding by
aggregating information from $k$-hop neighboring nodes.
Post hoc Explanations. With the increasing development of complex high-
performing GNN models [25, 26, 27, 28, 29], it becomes critical to understand
their decisions. Prior works have focused on developing several post hoc
explanation methods to explain the decisions of GNN models [5, 13, 7, 12, 11,
9, 32]. More specifically, these explanation methods can be broadly
categorized into i) gradient-based methods [5] that leverage the gradients of
the GNN model to generate explanations; ii) perturbation-based methods [13, 9,
11] that aim to generate explanations by calculating the change in GNN
predictions upon perturbations of the input graph structure (nodes, edges, or
subgraphs); and iii) surrogate-based methods [7, 12] that fit a simple
interpretable model to approximate the predictive behavior of the given GNN
model. Finally, recent works have introduced frameworks to theoretically and
empirically analyze the behavior of state-of-the-art GNN explanation methods
with respect to several desirable properties [33, 16].
## 3 Preliminaries
Notations. Let $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathbf{X})$ denote an
undirected graph comprising of a set of nodes $\mathcal{V}$ and a set of edges
$\mathcal{E}$. Let
$\mathbf{X}{=}\\{\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{N}\\}$ denote
the set of node feature vectors for all nodes in $\mathcal{V}$, where
$\mathbf{x}_{v}\in\mathbb{R}^{d}$ captures the attribute values of a node $v$
and $N{=}|\mathcal{V}|$ denotes the number of nodes in the graph. Let
$\mathbf{A}\in\mathbb{R}^{N\times N}$ be the graph adjacency matrix, where
element $\mathbf{A}_{uv}=1$ if there exists an edge $e\in\mathcal{E}$ between
nodes $u$ and $v$ and $\mathbf{A}_{uv}=0$ otherwise. We use
${\mathcal{N}}_{u}$ to denote the set of immediate neighbors of node $u$,
i.e., , ${\mathcal{N}}_{u}=\\{v\in\mathcal{V}|A_{uv}=1\\}$. Finally, the
function $\text{deg}:\mathcal{V}\mapsto\mathbb{Z}_{>0}$ is defined as
$\text{deg}(v)=|\mathcal{N}_{v}|$ and outputs the degree of a node
$v\in\mathcal{V}$
Graph Neural Networks (GNNs). Formally, GNNs can be formulated as message
passing networks [30] specified by three key operators Msg, Agg, and Upd.
These operators are recursively applied on a given graph $\mathcal{G}$ for a
$L$-layer GNN model defining how neural messages are shared, aggregated, and
updated between nodes to learn the final node representations in the
$L^{\text{th}}$ layer of the GNN. Commonly, a message between a pair of nodes
$(u,v)$ in layer $l$ is characterized as a function of their hidden
representations $\mathbf{h}_{u}^{(l-1)}$ and $\mathbf{h}_{v}^{(l-1)}$ from the
previous layer:
$\mathbf{m}_{uv}^{(l)}=\textsc{Msg}(\mathbf{h}_{u}^{(l-1)},\mathbf{h}_{v}^{(l-1)}).$
The Agg operator retrieves the messages from the neighborhood of node $u$ and
aggregates them as:
$\mathbf{m}_{u}^{(l)}=\textsc{Agg}(\mathbf{m}_{uv}^{(l)}\,|\,v\in\mathcal{N}_{u})$.
Next, the Upd operator takes the aggregated message $\mathbf{m}_{u}^{(l)}$ at
layer $l$ and combines it with $\mathbf{h}_{u}^{(l-1)}$ to produce node $u$’s
representation for layer $l$ as
$\mathbf{h}_{u}^{(l)}=\textsc{Upd}(\mathbf{m}_{u}^{(l)},\mathbf{h}_{u}^{(l-1)})$.
Lastly, the final node representation for node $u$ is given as
$\mathbf{z}_{u}=\mathbf{h}_{u}^{(L)}$.
Graph Explanations. In contrast to other modalities like images and texts, an
explanation method for graphs can formally generate multi-level explanations.
For instance, in a graph classification task, the explanations for a given
graph prediction can be with respect to the node attributes
$\mathbf{M}_{\text{x}}\in\mathbb{R}^{d}$, nodes
$\mathbf{M}_{n}\in\mathbb{R}^{N}$, or edges
$\mathbf{M}_{e}\in\mathbb{R}^{N\times N}$. Note that these explanation masks
are continuous but can be discretized using specific thresholding strategies
[33].
Oversmoothing. Cai et al. [34] and Zhou et al. [35] defined bounds for
analyzing oversmoothing for a GNN using Dirichlet Energy. For a graph
$\mathcal{G}$ with adjacency matrix $\mathbf{A}$ and degree matrix
$\mathbf{D}$, we define $\tilde{\mathbf{A}}{=}\mathbf{A}+\mathbf{I}_{N}$ and
$\tilde{\mathbf{D}}{=}\mathbf{D}+\mathbf{I}_{N}$ as the adjacency and degree
matrices respectively of the graph $\mathcal{G}$ with self-loops. We also
define the augmented normalized Laplacian of $\mathcal{G}$ as
$\tilde{\Delta}{=}\mathbf{I}_{N}-\tilde{\mathbf{D}}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}$,
and $\mathbf{P}{=}\mathbf{I}_{N}-\tilde{\Delta}$.
## 4 Our Framework: Expass
Here, we describe Expass, our proposed explainable message-passing framework
that aims to learn accurate and interpretable graph embeddings. In particular,
Expass incorporates explanations into the message-passing framework of GNN
models by only aggregating embeddings from key nodes and edges as identified
using an explanation method.
Problem formulation (Explanation Directed Message Passing). Given a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\textbf{X})$, Expass aims to generate a
$d$-dimensional embedding $\mathbf{z}_{u}\in\mathbb{R}^{d}$ for each node
$u\in\mathcal{V}$ using an explanation-directed message passing framework that
filters out the noise from unimportant edges and improves the expressive power
of GNNs.
### 4.1 Explanation Directed Message Passing
Figure 1: Overview of Expass: a) Expass investigates the problem of injecting
explanations into the message-passing framework to increase the expressive
power and performance of GNNs. b) Shown is the general message passing scheme
where, for node $u$, messages are aggregated from nodes
$v_{i}\in\mathcal{N}_{u}$ in the 1-hop neighborhood of $u$. c) Expass injects
explanations into the message passing framework translates by masking out
messages from neighboring nodes $v_{i}\in\mathcal{N}_{u}$ with explanation
scores $s_{uv_{i}}\approx 0$ when $u$ is correctly classified.
The central idea of Expass is to propose a novel method for improving the
neural message passing scheme of GNN models by utilizing explanations during
model training and aggregating important neural messages along edges in graph
neighborhoods. Next, we describe the existing message-passing scheme in GNNs
and our explainable counterpart.
Message Passing. As described in Section 3, each GNN layer can be described
using the $\textsc{Msg},\textsc{Agg},$ and Upd operators. For each node
$u\in\mathcal{V}$, the $(l+1)^{th}$ layer embeddings $\mathbf{h}^{(l+1)}_{u}$
is computed using a GNN operating on the node’s neighboring attributes.
Formally, the GNN layer can be formulated as:
where $\mathbf{h}^{(l+1)}_{u}$ represents the updated embedding of node $u$,
$\psi$ is the Msg operator, $\bigoplus$ is the Agg operator (e.g., summation),
$\phi$ is an Upd function (e.g., any non-linear activation function), and
$\mathbf{h}^{(l)}_{u}$ represents the embedding of node $u$ from the previous
layer. We obtain an embedding $\mathbf{z}_{u}$ for node $u$ by stacking $L$
GNN layers. Finally, the node embeddings $\mathbf{Z}\in\mathbb{R}$ are then
passed to a Readout function to obtain an embedding for the graph.
Expass. Here, we describe our proposed explainable message-passing scheme that
incorporates explanations into the message-passing step in individual GNN
layers on the fly during the training process. Given an explanation method,
which generates an importance score $s_{uv}\in\mathbf{M}^{e}_{u}$ for every
edge $e_{uv}\in\mathcal{E}$, we can weight the edge contribution in the
neighborhood $\mathcal{N}_{u}$ of node $u$ as:
Note that Expass is agnostic to explanation types and can also incorporate
explanations on node attributes and node level. For instance, the importance
scores for individual nodes can be computed by averaging the outgoing scores
$s_{uv}$ for all $v\in\mathcal{N}_{u}$. Subsequently, we can replace the
$s_{uv}$ score by using the average score $s_{u}$ to weight edges in the
Expass layers, and for node attributes, we can multiply the node attribute
explanation $\mathbf{M}^{a}_{u}$ to the original node attribute vector.
To enable explainable message passing and only retain the important embeddings
for node $u$, Expass removes the knowledge of irrelevant nodes and edges from
the local neighborhood $\mathcal{N}_{u}$ of node $u$ using its explanations.
For instance, if node $v$ is considered important to node $u$, Expass
transforms the aggregated messages of node $u$ using the node importance
scores $s_{uv}$. Note that since the explanations of node $u$ include
important nodes and edges in the $L$-hop neighborhood of node $u$, even though
node $u$ is only locally modified, the change will spread through all the
nodes in every GNN layer. Furthermore, to avoid spurious correlations, we
ensure that explanations are only generated for correctly classified nodes and
graphs. Explanation weights infuse information from higher-order neighborhoods
into each layer of the GNN model, specifically, from as many $L$-hop neighbors
because explanation weights within each layer are computed using the $L$-layer
GNN model. To illustrate this, we next show the weight computations for a GNN
explanation method.
Without loss of generality, let us consider GNNExplainer as our explanation
method whose mask for the selected graph is formulated as:
$\mathcal{G}_{\text{mask}}=(\mathbf{X}^{\prime},\mathbf{A}^{\prime})=(\mathbf{X}\odot\sigma(\mathbf{M}^{\text{x}}),\mathbf{A}\odot\sigma(\mathbf{M}^{\text{e}}))$,
where $W=[\mathbf{M}^{\text{x}},\mathbf{M}^{\text{e}}]$ are the explainers
parameters, $\sigma$ is the sigmoid function, and $\odot$ denotes element-wise
multiplication. Here, $s_{uv}$ represents the element in row $v$ and column
$u$ of $\mathbf{M}^{\text{e}}$. Gradient descent-based optimization is used to
find the optimal values for the masks minimizing the following objective:
$L_{e}=-\sum_{c=1}^{C}1[y=c]\log f_{\theta}(Y=y|\mathcal{G}_{\text{mask}})$,
where $f_{\theta}$ is the $L$-layer GNN model and $C$ is the total number of
classes. This shows that a $L$-hop neighborhood is used to compute $s_{uv}$.
Formally, it minimizes the uncertainty of the predictive model when the GNN
computation is limited to the explanation subgraph. This uncertainty is
minimized as a proxy of the maximization of the mutual information between the
prediction with the unmasked graph and masked graph.
### 4.2 Theoretical Analysis
Here, we provide a detailed theoretical analysis of our proposed Expass
framework. In particular, we (i) provide a theoretical upper bound on the
embedding difference obtained from a vanilla message passing and Expass
framework and (ii) show that graph embeddings learned using Expass relieves
the oversmoothing problem in GNNs by reducing information propagation.
###### Theorem 1 (Differences between Expass and Vanilla Message Passing).
Given a non-linear activation function $\sigma$ that is Lipschitz continuous,
the difference between the node embeddings between a vanilla message passing
and Expass framework can be bounded by the difference in their individual
weights, i.e.,
$\|\mathbf{h}_{u}^{(l)}-\mathbf{h^{\prime}}_{u}^{(l)}\|_{2}\leq\|\mathbf{W}_{a}^{(l)}{-}\mathbf{W^{\prime}}_{a}^{(l)}\|_{2}\|\mathbf{h}_{u}^{(l-1)}\|_{2}+\|\mathbf{W}_{n}^{(l)}{-}\mathbf{W^{\prime}}_{n}^{(l)}\|_{2}\sum_{\mathclap{v\in\mathcal{N}_{u}\cap
s_{v}=1}}\|\mathbf{h}_{v}^{(l-1)}\|_{2},$ (1)
where $\mathbf{W}_{a}^{(l)}$ and $\mathbf{W^{\prime}}_{a}^{(l)}$ are the
weights for node $u$ in layer $l$ of the vanilla message passing and Expass
framework and $\mathbf{W}_{n}^{(l)}$ and $\mathbf{W^{\prime}}_{n}^{(l)}$ are
their respective weight matrix with the neighbors of node $u$ at layer $l$.
###### Proof Sketch.
In Theorem 1, we prove that the $\ell_{2}$-norm of the differences between the
embeddings of vanilla message passing and Expass framework at layer $l$ is
upper bounded by the difference between their weights and the embeddings of
node $u$ and its subgraph. See Appendix A for more details. ∎
###### Definition 1 (Dirichlet Energy for a Node Embedding Matrix [35]).
Given a node embedding matrix
$\mathbf{H}^{(l)}=[\mathbf{h}_{1}^{(l)},\dots,\mathbf{h}_{n}^{(l)}]^{T}$
learned from the GNN model at the $l^{th}$ layer, the Dirichlet Energy
$E(\mathbf{H}^{(l)})$ is defined as:
$E(\mathbf{H}^{(l)})=tr(\mathbf{H}^{(l)^{T}}\tilde{\Delta}\mathbf{H}^{(l)})=\frac{1}{2}\sum_{i,j\in\mathcal{V}}a_{ij}||\frac{\mathbf{H}_{i}^{(l)}}{\sqrt{1+\text{deg}_{i}}}-\frac{\mathbf{H}_{j}^{(l)}}{\sqrt{1+\text{deg}_{j}}}||_{2}^{2}$
(2)
where $a_{ij}$ are elements in the adjacency matrix $\tilde{\mathbf{A}}$ and
$\text{deg}_{i},\text{deg}_{j}$ is the degree of node $i$ and $j$,
respectively.
Cai et al. [34] extensively show that higher Dirichlet energies correspond to
lower oversmoothing. Furthermore, they show that the removal of edges or,
similarly, the reduction of edge weights on graphs helps alleviate
oversmoothing.
###### Proposition 1 (Expass relieves Oversmoothing).
Expass alleviates oversmoothing by slowing the layer-wise loss of Dirichlet
energy.
The complete proof is provided in Appendix A.
## 5 Experiments
Next, we present experimental results for our Expass framework. More
specifically, we address the following questions: Q1) Does Expass enable GNNs
to learn more accurate embeddings and improve their degree of explainability?
Q2) How does Expass affect the oversmoothing and predictive performance of
GNNs with an increasing number of layers? Q3) Does Expass depend on the
quality of explanations for improving the predictive and oversmoothing
performance of GNNs and are they better than attention weights? Q4) How does
Expass helps in the evolution of explanation during the training of the GNN
model? 111Code to reproduce the results is available here
### 5.1 Datasets and Experimental setup
We first describe the datasets used to study the utility of our proposed
Expass framework and then outline the experimental setup.
Datasets. We use real-world molecular chemistry datasets to evaluate the
effectiveness of Expass w.r.t. the performance of the underlying GNN model and
understand the trade-off between explainability and accuracy for a graph
classification task. We consider four benchmark datasets, which includes Mutag
[36], Alkane-Carbonyl [37], DD [38], and Proteins [39]. See Appendix B.1 for a
detailed overview of the datasets.
GNN Architectures and Explainers. To investigate the flexibility of Expass, we
incorporate it into five different GNN models: GCN [40], GraphConv [41],
LEConv [42], GraphSAGE [28], GAT [43], and GIN [27]. We use GNNExplainer [13]
as our baseline GNN explanation method to generate edge-level explanations for
most of our experiments. In addition, we use Integrated Gradients [44] and
PGMExplainer [12], a node-level explanation method, to demonstrate Expass’s
sensitivity to the choice of explainers.
Implementation details. We consider DropEdge [45] as our baseline method for
comparing the oversmoothing performance of Expass as DropEdge randomly removes
edges from the input graph at each training epoch, acting like a message
passing reducer. Across all experiments, we use topK (k=40%) node
features/edges, and use them to generate explanations for all explanation
methods. All other hyperparameters of the explanation and baseline methods
were set following the author’s guidelines. For all our experiments (unless
mentioned otherwise), we use the baseline architectures with three GNN layers
followed by ReLU layers and set the hidden dimensionality to 32. Finally, we
use a single linear layer to transform the graph embeddings to their
respective classes. See Appendix B.2 for more details.
Performance metrics for GNN Explainers. To measure the reliability of GNN
explanation methods, we use the graph explanation faithfulness metric [16]:
$\text{GEF}(\hat{y}_{u},\hat{y}_{u^{\prime}})=1-\exp^{-\text{KL}(\hat{y}_{u}||\hat{y}_{u^{\prime}})}$,
where $\hat{y}_{u}$ is predicted probability vector using the whole subgraph
and $\hat{y}_{u^{\prime}}$ is the predicted probability vector using the
masked subgraph, where we generate the masked subgraph by only using the topK
features identified by an explanation and the Kullback-Leibler (KL) divergence
score (denoted by “$||$” operator) quantifies the distance between two
probability distributions. Note that GEF is a measure of the unfaithfulness of
the explanation. So, higher values indicate a higher degree of unfaithfulness.
Performance metrics for Oversmoothing. Zhou et al. [18] introduced the Group
Distance Ratio (GDR) metric to quantify oversmoothing in GNNs. It measures the
ratio between the average of pairwise representation distances between graphs
belonging to different (inter) and same (intra) groups. Formally, one would
prefer to reduce the intra-group class representations and increase the inter-
group distance to relieve the over-smoothing issue. Hence, lower GDR values
denote higher oversmoothing in GNNs.
Burn-in period. We defined the burn-in period as a number n of epochs during
training in which no explanations are used. The burn-in period is necessary to
avoid feeding spurious explanations to the model. The length of the burn-in
period (i.e., the number of epochs) was treated as a hyperparameter and fine-
tuned using the validation set. At the end of the burn-in period, a predefined
percentage of correctly predicted graphs per batch is randomly sampled and
their explanations are used in the model training. The percentage of correctly
predicted graphs sampled in each batch was set to 0.4 for all our experiments.
See Appendix C.2 for ablation on burn-in periods.
### 5.2 Results
Q1) Expass improves the predictive performance and explainability of GNNs. To
measure the predictive performance and degree of explainability of GNNs
trained using Expass, we compute their average predictive performance (using
AUROC and F1-score) and fidelity (using Graph Explanation Faithfulness) using
different GNN models and datasets. Across four datasets and five GNN
architectures, we find that Expass-augmented GNNs learn graph embeddings that
are more accurate (higher AUROC and F1-score) and result in more faithful
explanations (lower Graph Explanation Faithfulness score) than their vanilla
counterparts. On average, Expass improves the AUROC and F1-score by 1.51% and
1.05%, respectively. In particular, we observe that Expass improves the
predictive behavior of high-performing models like GIN (+2.06% in AUROC and
+2.50% in F1-score) but shows little to no improvement in the case of LeConv,
which utilizes a node-scoring mechanism through the similarity between a node
and its neighbors’ embeddings. Finally, we find that Expass-augmented GNNs
significantly improve the explainability of a GNN and achieve a 39.68% better
faithfulness score as compared to vanilla GNNs (Table 1). See Appendix C.1 for
results on node classification graph downstream tasks.
Table 1: Results of Expass for five GNNs and four graph datasets. Shown is average performance across three independent runs. Arrows ($\uparrow$, $\downarrow$) indicate the direction of better performance. Expass improves the predictive power (AUROC and F1-score) and degree of explainability (Graph Explanation Faithfulness) of original GNNs across multiple datasets (shaded area). Values corresponding to best performance are bolded. Dataset | Method | AUROC ($\uparrow$) | F1-score ($\uparrow$) | GEF ($\downarrow$)
---|---|---|---|---
Alkane-Carbonyl | | GCN
---
Expass-GCN
| 0.97$\pm$0.01
---
0.98$\pm$0.00
| 0.95$\pm$0.01
---
0.96$\pm$0.01
| 0.33$\pm$0.02
---
0.23$\pm$0.02
| GraphConv
---
Expass-GraphConv
| 0.97$\pm$0.01
---
0.98$\pm$0.00
| 0.94$\pm$0.00
---
0.97$\pm$0.00
| 0.38$\pm$0.05
---
0.22$\pm$0.03
| LeConv
---
Expass-LeConv
| 0.98$\pm$0.01
---
0.98$\pm$0.00
| 0.96$\pm$0.00
---
0.96$\pm$0.01
| 0.37$\pm$0.03
---
0.24$\pm$0.03
| GraphSAGE
---
Expass-GraphSAGE
| 0.98$\pm$0.00
---
0.99$\pm$0.00
| 0.96$\pm$0.00
---
0.97$\pm$0.01
| 0.40$\pm$0.12
---
0.18$\pm$0.06
| GIN
---
Expass-GIN
| 0.96$\pm$0.01
---
0.98$\pm$0.01
| 0.94$\pm$0.02
---
0.96$\pm$0.02
| 0.35$\pm$0.06
---
0.11$\pm$0.04
DD | | GCN
---
Expass-GCN
| 0.73$\pm$0.02
---
0.74$\pm$0.01
| 0.70$\pm$0.02
---
0.70$\pm$0.02
| 0.49$\pm$0.04
---
0.30$\pm$0.09
| GraphConv
---
Expass-GraphConv
| 0.75$\pm$0.03
---
0.77$\pm$0.03
| 0.73$\pm$0.03
---
0.73$\pm$0.03
| 0.25$\pm$0.10
---
0.19$\pm$0.04
| LeConv
---
Expass-LeConv
| 0.76$\pm$0.03
---
0.77$\pm$0.03
| 0.74$\pm$0.02
---
0.73$\pm$0.04
| 0.17$\pm$0.03
---
0.31$\pm$0.10
| GraphSAGE
---
Expass-GraphSAGE
| 0.74$\pm$0.02
---
0.76$\pm$0.03
| 0.70$\pm$0.02
---
0.71$\pm$0.02
| 0.21$\pm$0.04
---
0.20$\pm$0.03
| GIN
---
Expass-GIN
| 0.74$\pm$0.01
---
0.76$\pm$0.01
| 0.70$\pm$0.01
---
0.74$\pm$0.01
| 0.37$\pm$0.03
---
0.35$\pm$0.05
Mutag | | GCN
---
Expass-GCN
| 0.71$\pm$0.11
---
0.77$\pm$0.02
| 0.87$\pm$0.01
---
0.89$\pm$0.00
| 0.09$\pm$0.03
---
0.04$\pm$0.01
| GraphConv
---
Expass-GraphConv
| 0.91$\pm$0.02
---
0.93$\pm$0.01
| 0.94$\pm$0.02
---
0.94$\pm$0.01
| 0.66$\pm$0.03
---
0.24$\pm$0.03
| LeConv
---
Expass-LeConv
| 0.92$\pm$0.03
---
0.92$\pm$0.03
| 0.94$\pm$0.02
---
0.96$\pm$0.01
| 0.65$\pm$0.05
---
0.30$\pm$0.06
| GraphSAGE
---
Expass-GraphSAGE
| 0.76$\pm$0.02
---
0.76$\pm$0.02
| 0.86$\pm$0.03
---
0.87$\pm$0.03
| 0.24$\pm$0.08
---
0.11$\pm$0.03
| GIN
---
Expass-GIN
| 0.92$\pm$0.02
---
0.94$\pm$0.02
| 0.93$\pm$0.01
---
0.95$\pm$0.01
| 0.61$\pm$0.05
---
0.32$\pm$0.04
Proteins | | GCN
---
Expass-GCN
| 0.73$\pm$0.05
---
0.74$\pm$0.03
| 0.68$\pm$0.04
---
0.69$\pm$0.03
| 0.19$\pm$0.02
---
0.08$\pm$0.02
| GraphConv
---
Expass-GraphConv
| 0.75$\pm$0.03
---
0.75$\pm$0.03
| 0.70$\pm$0.03
---
0.70$\pm$0.04
| 0.49$\pm$0.06
---
0.10$\pm$0.03
| LeConv
---
Expass-LeConv
| 0.77$\pm$0.03
---
0.76$\pm$0.02
| 0.72$\pm$0.04
---
0.71$\pm$0.03
| 0.51$\pm$0.01
---
0.15$\pm$0.07
| GraphSAGE
---
Expass-GraphSAGE
| 0.73$\pm$0.04
---
0.73$\pm$0.04
| 0.69$\pm$0.04
---
0.69$\pm$0.04
| 0.17$\pm$0.07
---
0.06$\pm$0.01
| GIN
---
Expass-GIN
| 0.77$\pm$0.04
---
0.78$\pm$0.03
| 0.73$\pm$0.05
---
0.73$\pm$0.04
| 0.20$\pm$0.07
---
0.19$\pm$0.01
Q2) Expass relieves Oversmoothing in GNNs. We examine the oversmoothing (using
the Group Distance Ratio metric [18]) and predictive performance of GNNs
trained using Expass with their vanilla counterparts. The oversmoothing
problem in GNNs shows that the representations of nodes converge to similar
vectors as the number of layers increases. Therefore, we analyze the
oversmoothing of the GNNs for an increasing number of layers and find that, on
average, across two architectures, Expass improves the group distance ratio by
34.4% (Figure 2). Further, we also analyzed the oversmoothing behavior of
Expass for node classification tasks (in Appendix C.1) and our results
indicate an inherent trade-off between oversmoothing and predictive
performance of GNNs (Figures 5-7).
Figure 2: The effects of the number of GNN layers on the oversmoothing
performance of Expass (orange) and Vanilla (green) GCN (left column) and GIN
(right column) models trained on Alkane-Carbonyl dataset. Across models with
increasing number of layers, Expass achieves higher GDR performance without
sacrificing the predictive performance of the GCN model. See Figs. 5-7 for
predictive performance results.
Q3) Ablation studies. We conduct ablations on several components of Expass
with respect to its oversmoothing and predictive performance.
Expass for different TopK Explanations. We investigate the oversmoothing and
predictive performance of GNNs for different topK explanations (i.e., topK
edges identified by a GNN explanation) chosen in the message passing. Results
show that Expass alleviates oversmoothing by using only the topK edges to
learn graph embeddings and explicitly filter out the noise from unimportant
edges. In particular, we observe that the GDR values decrease (denoting higher
oversmoothing) with the increase in the use of topK edges (Figure 3). More
specifically, we find that the GDR value at topK=0.1 is 11.92% higher than
vanilla message passing (i.e., using all edges in the graph).
Expass vs. DropEdge. We compare the predictive and oversmoothing and
predictive performance of Expass and DropEdge. Here, we show that message
passing using optimized explanation-directed information outperforms random
edge removal. We find that Expass outperforms DropEdge across both
oversmoothing and accuracy metrics. In particular, on average, across
different topK values, Expass improves the oversmoothing, AUROC, and F1-score
performance of vanilla message passing by 71.16%, 9.53%, and 12.63%,
respectively (Figure 3).
(a) Group Distance Ratio (b) AUROC (c) F1-Score
Figure 3: The effects of choosing only the topK percent of important edges on
the (a) oversmoothing, (b) AUROC, and (c) F1-score performance of GCN model
trained on Alkane-Carbonyl dataset. Over a wide range of topK values
($0.1<\text{topK}<1.0$), Expass outperforms DropEdge [45] on all the three
metrics. Note that their performance converges for $\text{topK}=1.0$ as that
denotes using all the edges in the graph.
Expass using Node Explanations. We investigate the effect of the choice of the
baseline explanation method on the performance of Expass with respect to the
vanilla message passing framework. More specifically, we evaluate the
predictive and explainability performance of Expass-augmented GNNs when
trained using node explanations generated using Integrated Gradients (IG)
[44]. Similar to the results of Expass with GNNExplainer as the baseline
explanation method (Table 1), we find that Expass trained using IG
explanations also improves the AUROC (+2.80%), F1-score (+1.11%), and GEF
(+23.67%) of the vanilla GNN model. Our results show that the choice of
explainer can make a difference in the Expass performance, depending on the
dataset. For instance, IG is a node-masking explainer that is not considered a
strong explanation method and its effects are variable across datasets [33].
We recommend using graph-specific explainers that optimize for fidelity and
sparsity on the edges of the input graph, which would be a best fit to
increase the performance of the network. See Appendix C.3 for results using
PGMExplainer. Further, our results show that Expass is a model- and explainer-
agnostic framework that can improve the downstream task and explainability
performance across different GNN architectures using diverse GNN explainers.
Expass vs. Attention. We demonstrate the utility of using explanations vs.
attention weights in the message passing step using GAT [43] model
architecture. On average, across four datasets, we find that Expass achieves
higher AUROC (+3.85%) and F1-score (+2.24%) than the attention-based GAT model
(Table 3). In addition, GNNExplainer [13] demonstrated that post hoc GNN
explainers generate better explanations than attention weights, which further
highlights the benefits of Expass. In comparison to Expass, GAT can be
considered as a special case of our framework, where attention weights replace
explanations. On the other hand, Expass has larger benefits since it can be
applied to any existing GNN architectures that lack explainability.
Table 2: Results of Expass for GCN using the node explanations from Integrated Gradients [44] for message passing for various datasets. Shown is average performance across three independent runs. Arrows ($\uparrow$, $\downarrow$) indicate the direction of better performance. Expass improves the predictive power (AUROC and F1-score) and degree of explainability (Graph Explanation Faithfulness) of original GNNs across multiple datasets (shaded area). Dataset | Method | AUROC ($\uparrow$) | F1-score ($\uparrow$) | GEF ($\downarrow$)
---|---|---|---|---
| DD
---
| GCN
---
Expass-GCN
| 0.73$\pm$0.02
---
0.75$\pm$0.01
| 0.70$\pm$0.02
---
0.71$\pm$0.03
| 0.25$\pm$0.03
---
0.23$\pm$0.04
| Alkane
---
| GCN
---
Expass-GCN
| 0.97$\pm$0.01
---
0.97$\pm$0.01
| 0.95$\pm$0.01
---
0.95$\pm$0.01
| 0.09$\pm$0.01
---
0.1$\pm$0.01
| Mutag
---
| GCN
---
Expass-GCN
| 0.71$\pm$0.11
---
0.77$\pm$0.02
| 0.87 $\pm$0.01
---
0.88$\pm$0.01
| 0.09$\pm$0.02
---
0.04$\pm$0.02
| Proteins
---
| GCN
---
Expass-GCN
| 0.73$\pm$0.04
---
0.73$\pm$0.04
| 0.68$\pm$0.04
---
0.67$\pm$0.05
| 0.05$\pm$0.01
---
0.04$\pm$0.01
Table 3: Results of Expass and GAT for various datasets. Shown is the average performance across three independent runs. Arrows ($\uparrow$, $\downarrow$) indicate the direction of better performance. Expass improves the predictive power (AUROC and F1-score) and degree of explainability (Graph Explanation Faithfulness) of original GNNs across multiple datasets (shaded area). Dataset | Method | AUROC ($\uparrow$) | F1-score ($\uparrow$)
---|---|---|---
| DD
---
| GAT
---
Expass-GCN
| 0.72$\pm$0.02
---
0.74$\pm$0.01
| 0.68$\pm$0.03
---
0.70$\pm$0.01
| Alkane
---
| GAT
---
Expass-GCN
| 0.97$\pm$0.01
---
0.98$\pm$0.00
| 0.95$\pm$0.01
---
0.96$\pm$0.01
| Mutag
---
| GAT
---
Expass-GCN
| 0.69$\pm$0.10
---
0.77$\pm$0.02
| 0.86 $\pm$0.01
---
0.89$\pm$0.00
| Proteins
---
| GAT
---
Expass-GCN
| 0.74$\pm$0.04
---
0.74$\pm$0.03
| 0.68$\pm$0.04
---
0.69$\pm$0.03
Q4) Visualizing explanations. Here, we visualize how the explanation develops
over the training process of the GNN model. In particular, we visualize the
generated explanations from Expass-GCN with GNNExplainer trained on the MUTAG
dataset at different epochs during the training process and find that the
explanations converge to the ground-truth explanation of a non-mutagenic
molecule (i.e., the absence of a carbon ring alongside the highlighted NO2
molecules) as the training progresses (Figure 8). Further, we compare the
generated explanations for a vanilla GCN and its Expass counterpart and find
that the explanation for vanilla GCN falsely identifies the carbon-carbon
bonds as important (Figure 4). This qualitative analysis provides further
evidence for the observed higher faithfulness results (Table 1) of
explanations generated using our proposed Expass framework.
VanillaGCN Expass-GCN
Figure 4: Visualizing the explanation generated for a non-mutagenic molecule
prediction using Vanilla GCN (left) and Expass-GCN (right) with GNNExplainer
method. Note that the explanation from vanilla GCN falsely identifies the
carbon-carbon bonds as important. This qualitative analysis provides further
evidence for the observed higher faithfulness results of explanations
generated using our proposed Expass framework.
## 6 Conclusion
In this work, we propose the problem of learning graph embeddings using
explanation-directed message passing in GNNs. To this end, we introduce
Expass, a novel message-passing framework that can be used with any existing
GNN model and subgraph-optimizing explainer to learn accurate embeddings by
aggregating only embeddings from nodes and edges identified as important by a
GNN explainer. We perform an extensive theoretical analysis to show that
Expass relieves the oversmoothing problem in GNNs, and the embedding
difference between the vanilla message passing framework and Expass can be
upper bounded by the difference of their respective layer weights. Our
empirical results on benchmark datasets show that Expass improves the
explainability of the underlying GNN model without sacrificing its predictive
performance. However, the training of Expass depends on the choice of
explanation method, the number of data points to explain, and the dataset of
choice, which is computationally more expensive than its vanilla counterparts.
We find that the training time of Expass can be improved by using techniques
like batch processing and efficient sampling of correctly-classified nodes and
graphs. Further, adapting post-hoc explainers to generate subgraphs utilizing
the embedding space would also improve the computation time of Expass. Our
proposed method and findings open exciting new avenues to learn graph
representations by jointly training models and explanation methods. We
anticipate that Expass could open new frontiers in graph machine learning for
developing explanation-based training frameworks.
## Acknowledgements
The authors would like to thank the anonymous reviewers for their helpful
feedback that helped improve the work. CA would like to thank Lasse Mohr and
Samuele Firmani for the helpful discussions at the beginning of the project
and LOGML Summer School for connecting with the students. The views expressed
here are those of the authors and do not reflect the official policy or
position of the affiliated company.
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## Appendix A Proofs for Theorems in Section 4
Theorem 1. Given a non-linear activation function $\sigma$ that is Lipschitz
continuous, the difference between the node embeddings between a vanilla
message passing and Expass framework can be bounded by the difference in their
individual weights, i.e.,
$\|\mathbf{h}_{u}^{(l)}-\mathbf{h^{\prime}}_{u}^{(l)}\|_{2}\leq\|\mathbf{W}_{a}^{(l)}{-}\mathbf{W^{\prime}}_{a}^{(l)}\|_{2}\|\mathbf{h}_{u}^{(l-1)}\|_{2}+\|\mathbf{W}_{n}^{(l)}{-}\mathbf{W^{\prime}}_{n}^{(l)}\|_{2}\sum_{\mathclap{v\in\mathcal{N}_{u}\cap
s_{v}=1}}\|\mathbf{h}_{v}^{(l-1)}\|_{2},$ (3)
where $\mathbf{W}_{a}^{(l)}$ and $\mathbf{W^{\prime}}_{a}^{(l)}$ are the
weights for node $u$ in layer $l$ of the vanilla message passing and Expass
framework and $\mathbf{W}_{n}^{(l)}$ and $\mathbf{W^{\prime}}_{n}^{(l)}$ are
their respective weight matrix with the neighbors of node $u$ at layer $l$.
###### Proof.
For a given node $u$, the node representation output by layer $l$ of the GNN
is given by:
$\mathbf{h}_{u}^{(l)}=\sigma\Big{(}\mathbf{W}_{a}^{(l)}\mathbf{h}_{u}^{(l-1)}+\mathbf{W}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}\mathbf{h}_{v}^{(l-1)}\Big{)},$
(4)
where we consider the Agg operator as a fully-connected layer, Upd to be a
sigmoid activation function $\sigma(\cdot)$, $\mathbf{W}_{a}^{(l)}$ is the
weights for node $u$ in layer $l$ and $\mathbf{W}_{n}^{(l)}$ is the weight
matrix with the neighbors of node $u$ at layer $l$.
Let us consider an edge in-hoc explanation that generates a binary mask
highlighting the important edges for the prediction of node $u$. Note that
using the edge mask, we can also get a node-level mask signifying the
importance of neighboring nodes. Let us denote that node explanation mask as
$s_{v}$ where $s_{v}=1$ if the node is important, otherwise $s_{v}=0$.
Formally, the corresponding message passing equations for Expass can be
written as:
$\mathbf{h^{\prime}}_{u}^{(l)}=\sigma\Big{(}\mathbf{W^{\prime}}_{a}^{(l)}\mathbf{h^{\prime}}_{u}^{(l-1)}+\mathbf{W^{\prime}}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}s_{v}\mathbf{h^{\prime}}_{v}^{(l-1)}\Big{)},$
(5)
where $\mathbf{h^{\prime}}_{u}^{(l)}$ and $\mathbf{h^{\prime}}_{v}^{(l)}$
represents the embeddings of node $u$ and $v$ using the feedback explanation,
and $\mathbf{W^{\prime}}_{a}^{(l)}$ and $\mathbf{W^{\prime}}_{n}^{(l)}$
represents the corresponding weights at layer $l$ for GNN model trained using
Expass.
The difference between the node embeddings obtained after the message-passing
in layer $l$ from Equations 4-5 is given as:
$\mathbf{h}_{u}^{(l)}-\mathbf{h^{\prime}}_{u}^{(l)}=\sigma\Big{(}\mathbf{W}_{a}^{(l)}\mathbf{h}_{u}^{(l-1)}+\mathbf{W}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}\mathbf{h}_{v}^{(l-1)}\Big{)}-\sigma\Big{(}\mathbf{W^{\prime}}_{a}^{(l)}~{}\mathbf{h^{\prime}}_{u}^{(l-1)}+\mathbf{W^{\prime}}_{n}^{(l)}~{}\sum_{\mathclap{v\in\mathcal{N}_{u}}}s_{v}\mathbf{h^{\prime}}_{v}^{(l-1)}\Big{)},$
(6)
Taking the $\ell_{2}$-norm on both sides and assuming a normalized Lipschitz
non-linear sigmoid activation, i.e.,
$\|\sigma(b)-\sigma(a)\|_{2}\leq\|b-a\|_{2}$, we get:
$\displaystyle\|\mathbf{h}_{u}^{(l)}-\mathbf{h^{\prime}}_{u}^{(l)}\|_{2}$
$\displaystyle=\|\sigma\Big{(}\mathbf{W}_{a}^{(l)}\mathbf{h}_{u}^{(l-1)}{+}\mathbf{W}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}\mathbf{h}_{v}^{(l-1)}\Big{)}{-}\sigma\Big{(}\mathbf{W^{\prime}}_{a}^{(l)}~{}\mathbf{h^{\prime}}_{u}^{(l-1)}{+}\mathbf{W^{\prime}}_{n}^{(l)}~{}\sum_{\mathclap{v\in\mathcal{N}_{u}}}s_{v}\mathbf{h^{\prime}}_{v}^{(l-1)}\Big{)}\|_{2}$
$\displaystyle\leq\|\mathbf{W}_{a}^{(l)}\mathbf{h}_{u}^{(l-1)}+\mathbf{W}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}\mathbf{h}_{v}^{(l-1)}-\mathbf{W^{\prime}}_{a}^{(l)}~{}\mathbf{h^{\prime}}_{u}^{(l-1)}-\mathbf{W^{\prime}}_{n}^{(l)}~{}\sum_{\mathclap{v\in\mathcal{N}_{u}}}s_{v}\mathbf{h^{\prime}}_{v}^{(l-1)}\|_{2}$
$\displaystyle\leq\|\mathbf{W}_{a}^{(l)}\mathbf{h}_{u}^{(l-1)}-\mathbf{W^{\prime}}_{a}^{(l)}~{}\mathbf{h^{\prime}}_{u}^{(l-1)}+\mathbf{W}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}\mathbf{h}_{v}^{(l-1)}-\mathbf{W^{\prime}}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}}}s_{v}\mathbf{h^{\prime}}_{v}^{(l-1)}\|_{2}$
$\displaystyle\leq\|\mathbf{W}_{a}^{(l)}\mathbf{h}_{u}^{(l-1)}{-}\mathbf{W^{\prime}}_{a}^{(l)}~{}\mathbf{h}_{u}^{(l-1)}\|_{2}{+}\|\mathbf{W}_{n}^{(l)}\sum_{\mathclap{v\in\mathcal{N}_{u}\cap
s_{v}=0}}\mathbf{h}_{v}^{(l-1)}{+}(\mathbf{W}_{n}^{(l)}-\mathbf{W^{\prime}}_{n}^{(l)})\sum_{\mathclap{v\in\mathcal{N}_{u}\cap
s_{v}=1}}\mathbf{h}_{v}^{(l-1)}\|_{2}$ (Using Triangle Inequality and
Faithfulness property of explanations)
Given a faithful explanation, the node embeddings for node $u$ using the
vanilla message passing network are equivalent to that Expass since most
explainers optimize the mask to approximate the input embedding. More
specifically, for a given node embedding
$\mathbf{h^{\prime}}_{u}^{(l-1)}=\mathbf{h}_{u}^{(l-1)}+\epsilon_{u}$, a
faithful explanation bounds the $\epsilon_{u}$ to zero. In addition to
faithfulness, a GNN using vanilla message passing and Expass can predict a
node $u$ to the same class only if both frameworks generate similar node
embeddings (Proposition 1 in Agarwal et al. [4]).
Using Matrix-norm and Triangle Inequality for the sum in the neighborhood, we
get:
$\displaystyle\|\mathbf{h}_{u}^{(l)}-\mathbf{h^{\prime}}_{u}^{(l)}\|_{2}\leq\|(\mathbf{W}_{a}^{(l)}-\mathbf{W^{\prime}}_{a}^{(l)})~{}\mathbf{h}_{u}^{(l-1)}\|_{2}+\|\mathbf{W}_{n}^{(l)}\|_{2}\sum_{\mathclap{v\in\mathcal{N}_{u}\cap
s_{v}=0}}\|\mathbf{h}_{v}^{(l-1)}\|_{2}+$
$\displaystyle\|\mathbf{W}_{n}^{(l)}-\mathbf{W^{\prime}}_{n}^{(l)}\|_{2}\sum_{\mathclap{v\in\mathcal{N}_{u}\cap
s_{v}=1}}\|\mathbf{h}_{v}^{(l-1)}\|_{2}$
Again, using the faithfulness property of explanations, the contribution of
node embeddings from node $v\in\mathcal{N}(u)\cap s_{v}=0$ is irrelevant to
the final embedding and can be removed. Finally, using Matrix-norm inequality
on the first term, we get:
$\displaystyle\|\mathbf{h}_{u}^{(l)}-\mathbf{h^{\prime}}_{u}^{(l)}\|_{2}$
$\displaystyle\leq\|\mathbf{W}_{a}^{(l)}{-}\mathbf{W^{\prime}}_{a}^{(l)}\|_{2}\|\mathbf{h}_{u}^{(l-1)}\|_{2}+\|\mathbf{W}_{n}^{(l)}{-}\mathbf{W^{\prime}}_{n}^{(l)}\|_{2}\sum_{\mathclap{v\in\mathcal{N}(u)\cap
s_{v}=1}}\|\mathbf{h}_{v}^{(l-1)}\|_{2}$
Thus, we observe that the embedding difference at layer $l$ between a vanilla
message passing network and the Expass is purely based on the difference
between their weights and the embeddings of node $u$ and its subgraph. ∎
###### Definition 2 (Dirichlet Energy for a Node Embedding Matrix [35]).
Given a node embedding matrix
$\mathbf{h}^{(l)}=[\mathbf{h}_{1}^{(l)},\dots,\mathbf{h}_{n}^{(l)}]^{T}$
learned from the GNN model at the $l^{th}$ layer, the Dirichlet Energy
$E(\mathbf{h}^{(l)})$ is defined as:
$E(\mathbf{h}^{(l)})=tr(\mathbf{h}^{(l)^{T}}\tilde{\Delta}\mathbf{h}^{(l)})=\frac{1}{2}\sum_{i,j\in\mathcal{V}}a_{ij}||\frac{\mathbf{h}_{i}^{(l)}}{\sqrt{1+d_{i}}}-\frac{\mathbf{h}_{j}^{(l)}}{\sqrt{1+d_{j}}}||_{2}^{2}$
(7)
where $a_{ij}$ are elements in the adjacency matrix $\tilde{\mathbf{A}}$ and
$d_{i},d_{j}$ is the degree of node $i$ and $j$, respectively.
Cai et al. [34] extensively show that higher Dirichlet energies correspond to
lower oversmoothing. Furthermore, they show that the removal of edges or
,similarly, reduction of edge weights on graphs help alleviate oversmoothing.
Proposition 1 (Expass relieves Oversmoothing). Expass alleviates oversmoothing
by slowing the layer-wise loss of Dirichlet energy.
###### Proof Sketch.
Here, we show the capabilities of Expass as a framework that alleviates the
oversmoothing problem in GNNs. To this end, we utilize the bounds on the
Dirichlet energy of the Expass embeddings at the $l^{th}$ layer of the GNN
model by Zhou et al. [35]:
$(1-\lambda_{1})^{2}s^{(l)}_{min}E(\mathbf{h}^{(l-1)})\leq
E(\mathbf{h}^{(l)})\leq(1-\lambda_{0})^{2}s^{(l)}_{max}E(\mathbf{h}^{(l-1)}),$
(8)
where $\lambda_{1},\lambda_{0}$ are the non-zero eigenvalues of the symmetric
normalized Laplacian $\tilde{\Delta}$ that is closest to 1 and 0,
respectively, and $s^{(l)}_{min},s^{(l)}_{max}$ are the squares of the minimum
and maximum singular values of weight $\mathbf{W}^{(l)}$, respectively. Since
Expass reduces the input graph to its specific explanation, we argue that it
can alleviate oversmoothing by reducing the information propagation along
irrelevant nodes and edges. From the perspective of Dirichlet energy, we know
from [19] that, for Erdős-Rényi graphs, $\lambda_{0}$ converges to 1 as the
graph becomes denser. Oono et al. [19] state that GNNs oversmooth on
sufficiently large graphs (similar to Erdős-Rényi graphs). Under this
assumption, Expass, by definition introduces sparsity inside the
$\tilde{\Delta}$ of the input graph by using a smaller set of topK important
edges for learning embeddings and, thus, reduces $\lambda_{0}$ to tighten the
upper-bound in Equation 8. In practice, the choice of explainer used in Expass
can reduce $\lambda_{0}$ to varying degrees. More specifically, explainers
that promote sparsity would push $\lambda_{0}$ closer to zero and slow down
the decrease of Dirichlet energy in subsequent GNN layers. Finally, we know
from Cai et al. [34] that higher values of Dirichlet energy per layer
correspond to lower oversmoothing, we assert that Expass alleviates
oversmoothing. ∎
## Appendix B Experiment
### B.1 Datasets
Mutag. The MUTAG [36] dataset contains 188 graph molecules labeled into two
different classes according to their mutagenic properties, i.e., effect on the
Gram-negative bacterium S. Typhimuriuma. Kazius et al. [36] identifies several
toxicophores - motifs in the molecular graph - that correlate with
mutagenicity.
Alkane-Carbonyl. The Alkane-Carbonyl [37] dataset contains 1,125 molecular
graphs categorized into two classes where an instance in the positive group
indicates a molecule that contains an unbranched alkane and a carbonyl (C=O)
functional group.
DD. The DD [38] dataset was derived from [46] and contains 1,178 protein
graphs where nodes represent individual amino-acids and edges represent their
spatial proximity. The task is to predict whether a given protein is an enzyme
or not.
Proteins. The Proteins [39] dataset was derived from [46] and contains 1,113
protein graphs where nodes represent secondary structure elements and edges
indicate neighborhood in the amino-acid sequence or the 3D space. The task is
to predict whether a given protein is an enzyme or not.
PubMed. The PubMed dataset [47] is a citation network from the PubMed
database, with over 4 million nodes and edges respectively. It contains a bag-
of-words representation of documents and citation links between documents. The
task is to predict a node’s class among 3 classes.
### B.2 Implementation details
GNN libraries and models. All our models were implemented using PyTorch
Geometric (2.1.0) and PyTorch (1.11.0). For our experiments, we used baseline
GNN architectures with three layers followed by ReLU layers and set the hidden
dimensionality to 32. Finally, we used a single linear layer to transform the
graph embeddings to their respective classes. We selected Adam as our
optimizer and a weighted Cross Entropy Loss to train both vanilla and Expass
frameworks. All models were trained over three independent runs with a
learning rate of 0.01 for 200 epochs for DD and Proteins datasets and 150
epochs for Alkane and MUTAG datasets.
Expass. We define the burn-in period as a number of epochs during training in
which no explanations are used. The burn-in period is necessary to avoid
feeding spurious explanations to the model since an untrained model can lead
to unfaithful explanations. The length of the burn-in period was treated as a
hyperparameter and fine-tuned during the model fine-tuning phase. After fine-
tuning, we found that a burn-in period in the range [5, 15] worked best,
whereas most Expass models outperformed their vanilla counterparts using a
burn-in period of 5 and 10 epochs in our experiments. We generated
explanations for a specific percentage of correctly predicted graphs sampled
in each batch and were set to 0.4 for all our experiments. The generated
explanations are normalized to [0, 1] and hard-masked over the topK most
relevant edges, where topK is a percentage of the total number of edges in the
input graph and was set to $\text{topK}\in[0.3,0.4]$ for our experiments.
GNN explanation methods. At each epoch, the model weights were frozen to
generate explanations, which were calculated as the median over n independent
runs of GNNExplainer, in order to obtain consistent explanations. Then, the
model weights were trained using generated explanations. We chose the median
instead of the mean to prevent the individual outliers from significantly
changing the final explanations. The number of individual runs of an explainer
was treated as a hyperparameter and was set to five for GNNExplainer and one
for Integrated Gradients (as it generates consistent explanations over
multiple runs). In each run, the GNNExplainer was trained for 200 epochs (150
in the case of Alkane) with a learning rate of 0.01. All other hyperparameters
of the explanation methods were set using the author’s guidelines. Note that
these multiple iterations of the explainers are not required for Expass to
perform well when using other stable GNN explainers. To summarize, the
following parameters were treated as hyperparameters: the learning rate of the
model, the learning rate of the explanation method, the number of epochs the
explanation method was trained for, the number of times the GNNExplainer was
computed at each epoch for each sampled graph, the percentage of correctly
classified graphs that were randomly sampled to compute the explanations and
the percentage of top edges/nodes that were selected as the most relevant. On
the other hand, in the case of vanilla models, the learning rate was fine-
tuned during the tuning phase.
Dataset. The train, validation, and test split was at 80%, 10%, and 10% for
Alkane, Proteins, and DD following prior works. In the case of MUTAG, no
validation set was used due to the smaller dataset size, and the train and
test split was at 80% and 20%.
GNN performance metric. The GEF scores were evaluated as the mean over the
individual scores of all generated explanations on the test dataset, where the
explanations were hard-masked with $\textup{topK}=0.1$ for GNNExplainer, and
$\textup{topK}=0.25$ for Integrated Gradients/PGMExplainer. Note that, since
Integrated Gradients/PGMExplainer generates a node mask instead of an edge
mask, we required a higher topK value to generate a non-empty hard mask over
the input graphs, since we retain the topK most relevant nodes in the
explanation mask.
(a) Group Distance Ratio (b) AUROC (c) F1-Score
Figure 5: The effects of the number of GNN layers on the (a) oversmoothing,
(b) AUROC, and (c) F1-score performance of Expass-GCN and Vanilla-GCN trained
on Alkane-Carbonyl dataset. Across models with increasing number of layers,
Expass achieves higher GDR performance without sacrificing the predictive
performance of the GCN model.
(a) Group Distance Ratio (b) AUROC (c) F1-Score
Figure 6: The effects of the number of GNN layers on the (a) oversmoothing,
(b) AUROC, and (c) F1-score performance of Expass-GIN and Vanilla-GIN trained
on Alkane-Carbonyl dataset. We observe that, across models with an increasing
number of layers, Expass achieves higher GDR performance and there exists an
inherent trade-off between oversmoothing and predictive performance of GIN.
## Appendix C Additional results
### C.1 Node classification results
We extend our proposed framework to GNN models trained on different graph
downstream tasks. In particular, we conduct additional experiments to obtain
the over-smoothing and predictive behavior of Expass for node-level tasks. We
train five state-of-the-art GNN models and their Expass counterparts on the
PubMed node classification dataset. Our results show that Expass alleviates
the over-smoothing effect in GNNs for models with higher depths (Figure 7) and
achieves on-par or higher predictive performance (Table 4). We find that, on
average, Expass augmented GCN achieves 19.53% better over-smoothing
performance for node-classification GNN models with higher depths.
Table 4: Results of Expass for five GNNs using PubMed node classification dataset. Shown is the average performance across five independent runs. Arrows ($\uparrow$, $\downarrow$) indicate the direction of better performance. Expass improves the predictive power (testing accuracy) of original GNNs across multiple datasets (shaded area). Method | Testing Accuracy ($\uparrow$)
---|---
| GCN
---
Expass-GCN
| 0.7596$\pm$0.002
---
0.7616$\pm$0.002
| GraphConv
---
Expass-GraphConv
| 0.7652$\pm$0.002
---
0.7682$\pm$0.002
| LeConv
---
Expass-LeConv
| 0.7424$\pm$0.003
---
0.7244$\pm$0.010
| GraphSAGE
---
Expass-GraphSAGE
| 0.7462$\pm$0.004
---
0.7533$\pm$0.002
| GIN
---
Expass-GIN
| 0.7233$\pm$0.002
---
0.7310$\pm$0.009
(a) Group Distance Ratio (b) Accuracy
Figure 7: The effects of the number of GNN layers on the (a) oversmoothing,
(b) testing accuracy performance of Expass-GCN and Vanilla-GCN trained on
PubMed node classification dataset. We observe that, across models with an
increasing number of layers, Expass achieves higher GDR performance and
achieves on-par or better testing accuracy.
### C.2 Burn-in period
The burn-in period was treated as a hyperparameter and fine-tuned for each
dataset and architecture. An example of the effect of the burn-in period on
the AUROC and F1-score is reported in Table 5, where the change in performance
is evaluated for the Proteins dataset when using a lag of 5, 10, and 15.
Table 5: Results of Expass for various burn-in periods. Shown is the average performance across three independent runs (and standard error). Arrows ($\uparrow$, $\downarrow$) indicate the direction of better performance. Method | Burn-in period | AUROC ($\uparrow$) | F1-score ($\uparrow$)
---|---|---|---
| Expass-GIN
---
| 5
---
10
15
| 0.76 $\pm$0.04
---
0.78$\pm$0.03
0.78$\pm$0.03
| 0.72$\pm$0.05
---
0.73$\pm$0.04
0.73$\pm$0.03
| Expass-GraphSAGE
---
| 5
---
10
15
| 0.73 $\pm$0.04
---
0.73$\pm$0.04
0.73$\pm$0.04
| 0.68$\pm$0.04
---
0.69$\pm$0.04
0.69$\pm$0.04
| Expass-LeConv
---
| 5
---
10
15
| 0.76$\pm$0.02
---
0.74$\pm$0.04
0.75$\pm$0.03
| 0.71$\pm$0.03
---
0.69$\pm$0.04
0.71$\pm$0.03
### C.3 PGMExplainer results
Here, we showcase the flexibility of the Expass with respect to different GNN
explainers. In Table 6, we show the effectiveness of Expass with respect to
different explainers by further utilizing PGMExplainer [12] as the explanation
generator. We utilize the graph explanations of PGMExplainer as node masks on
the input graph (as they cannot generate edge-level masks) and incorporate
them using our explanation-aware message-passing scheme with GCN as our
underlying architecture. On average, across three datasets, we find that
Expass trained using PGM-Explainer achieves higher GEF (+36.56%) than their
vanilla counterparts (Table 6). Also, it achieves a boost of 11.27% in AUROC
for MUTAG and a 1% improvement in the F1-score for the Alkane dataset. We
observe that PGMExplainer, similar to Integrated Gradients, produces node
masks, which lack detail and do not provide finer changes to the underlying
GNN model, like edge masks. We hypothesize that this contributes to the large
variation in the predictive performance across datasets.
Table 6: Results of Expass for GCN using the node explanations from PGMExplainer [12] for message passing for various datasets. Shown is the average performance across three independent runs. Arrows ($\uparrow$, $\downarrow$) indicate the direction of better performance. Expass improves the predictive power (AUROC and F1-score) and degree of explainability (Graph Explanation Faithfulness) of original GNNs across multiple datasets (shaded area). Dataset | Method | AUROC ($\uparrow$) | F1-score ($\uparrow$) | GEF ($\downarrow$)
---|---|---|---|---
| Alkane
---
| GCN
---
Expass-GCN
| 0.97$\pm$0.01
---
0.97$\pm$0.01
| 0.95$\pm$0.01
---
0.96$\pm$0.01
| 0.31$\pm$0.02
---
0.28$\pm$0.03
| Mutag
---
| GCN
---
Expass-GCN
| 0.71$\pm$0.11
---
0.79$\pm$0.03
| 0.87$\pm$0.01
---
0.86$\pm$0.01
| 0.21$\pm$0.07
---
0.07$\pm$0.01
| Proteins
---
| GCN
---
Expass-GCN
| 0.73$\pm$0.04
---
0.66$\pm$0.02
| 0.68$\pm$0.04
---
0.67$\pm$0.05
| 0.03$\pm$0.00
---
0.02$\pm$0.00
Epoch 25 Epoch 50 Epoch 75
Epoch 100 Epoch 125 Epoch 149
Figure 8: Generated explanations from Expass-GCN trained on the MUTAG dataset
at different epochs during the training process and find that the explanations
does converge to the ground-truth explanation of a mutagenic molecule (i.e.,
the absence of a carbon ring) as the training progresses. This qualitative
analysis provides further evidence for the observed higher faithfulness
results of explanations generated using our proposed Expass framework.
|
# Determining Dust Properties in Protoplanetary Disks: SED-derived Masses and
Settling With ALMA
Anneliese M. Rilinger Department of Astronomy and Institute for Astrophysical
Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215
Catherine C. Espaillat Department of Astronomy and Institute for Astrophysical
Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215 Zihua
Xin Department of Astronomy and Institute for Astrophysical Research, Boston
University, 725 Commonwealth Avenue, Boston, MA 02215 Álvaro Ribas Institute
of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
European Southern Observatory (ESO), Alonso de Córdova 3107, Vitacura, Casilla
19001, Santiago de Chile, Chile Enrique Macías ESO Garching, Karl-
Schwarzschild-Str. 2, 85748, Garching bei Munchen, Germany Sarah Luettgen
Department of Astronomy and Institute for Astrophysical Research, Boston
University, 725 Commonwealth Avenue, Boston, MA 02215 Ann and H.J. Smead
Department of Aerospace Engineering Sciences, University of Colorado at
Boulder, 3775 Discovery Dr, Boulder, CO 80303 Anneliese M. Rilinger
<EMAIL_ADDRESS>
###### Abstract
We present spectral energy distribution (SED) modeling of 338 disks around T
Tauri stars from eleven star-forming regions, ranging from $\sim$0.5 to 10 Myr
old. The disk masses we infer from our SED models are typically greater than
those reported from (sub)mm surveys by a factor of 1.5-5, with the discrepancy
being generally higher for the more massive disks. Masses derived from (sub)mm
fluxes rely on the assumption that the disks are optically thin at all
millimeter wavelengths, which may cause the disk masses to be underestimated
since the observed flux is not sensitive to the whole mass in the disk; SED
models do not make this assumption and thus yield higher masses. Disks with
more absorbing material should be optically thicker at a given wavelength;
which could lead to a larger discrepancy for disks around massive stars when
the disk temperature is scaled by the stellar luminosity. We also compare the
disk masses and degree of dust settling across the different star-forming
regions and find that disks in younger regions have more massive disks than
disks in older regions, but a similar degree of dust settling. Together, these
results offer potential partial solutions to the “missing” mass problem: disks
around T Tauri stars may indeed have enough material to form planetary
systems, though previous studies have underestimated the mass by assuming the
disks to be optically thin; these planetary systems may also form earlier than
previously theorized since significant dust evolution (i.e., settling) is
already apparent in young disks.
protoplanetary disks - star formation
## 1 Introduction
The population of confirmed exoplanets detected by Kepler, TESS, radial
velocity surveys, and other methods, is diverse and growing (Dressing &
Charbonneau, 2013; Winn & Fabrycky, 2015). A complete picture of the formation
process that can explain the origin of known exoplanets is not yet completely
clear (see for example the recent review by Drazkowska et al., 2022). As the
location of and source of material for planet formation (Williams & Cieza,
2011), protoplanetary disks are a key piece of the puzzle. Studying the
properties of these disks can help inform and constrain various theories of
planet formation, including the timescales over which the formation can occur.
The Atacama Large Millimeter/submillimeter Array (ALMA) has been invaluable
for studying protoplanetary disks. In particular, large surveys of various
star-forming regions performed with ALMA have provided information about disk
masses (see Table 1, Eisner et al., 2018; van Terwisga et al., 2020, 2022;
Anderson et al., 2022) and dust substructures (e.g., Andrews et al., 2018) for
disks around T Tauri stars (TTS). The regions studied by these surveys span
from approximately $\sim$0.5–10 Myr, allowing for comparisons of how disk
properties evolve over time.
Table 1: (Sub)millimeter Surveys of Star-forming Regions Studied in this Work Region | Age | Approx. Dist. | Ref. | NtotalaaObjects in binary or multiple systems were excluded from these counts. | NdetaaObjects in binary or multiple systems were excluded from these counts. | Nsample
---|---|---|---|---|---|---
| Myr | pc | | | |
$\rho$ Ophiuchus | 0.5 - 2 | 140 | Cieza et al. (2019); Williams et al. (2019) | 165 | 112 | 77
Taurus | 1 - 2 | 140 | Andrews et al. (2013); Long et al. (2019) | 98 | 65 | 25
L1641 | 1.5 | 428 | Grant et al. (2021) | 101 | 89 | 56
Cha II | 1 - 2 | 198 | Villenave et al. (2021) | 27 | 20 | 14
Lupus | 1 - 3 | 160 | Ansdell et al. (2016, 2018) | 87 | 62 | 41
Cha I | 2 - 3 | 180 | Pascucci et al. (2016) | 55 | 41 | 23
IC348 | 2 - 3 | 310 | Ruíz-Rodríguez et al. (2018) | 131 | 36 | 24
Corona Australis | 3 | 160 | Cazzoletti et al. (2019) | 33 | 16 | 10
$\sigma$ Ori | 3 - 5 | 385 | Ansdell et al. (2017) | 88 | 37 | 31
$\lambda$ Ori | 5 | 400 | Ansdell et al. (2020) | 33 | 13 | 12
Upper Sco | 5 - 10 | 145 | Barenfeld et al. (2016) | 79 | 42 | 25
Two disk properties are especially informative for understanding the planet
formation process: disk dust mass and dust settling. As the mass reservoirs
for planet formation, the amount of dust and gas in a disk directly constrains
the number and size of planets that could potentially form in that system. The
review by Bergin & Williams (2018) summarizes the various methods used to
calculate disk masses (see also Miotello et al., 2022). One of the most
commonly applied methods calculates disk masses from millimeter-wavelength
flux measurements. The disk masses deduced from the ALMA surveys listed in
Table 1 are typically larger in younger regions (1–3 Myr) than in older
regions (see e.g., Barenfeld et al., 2016; van der Plas et al., 2016; van
Terwisga et al., 2022); these results can be used to constrain the evolution
of protoplanetary disks and the timescale of planet formation. Moreover, the
estimated masses of the young disks are often too small to account for
observed planetary systems (Manara et al., 2018) or require nearly 100%
efficiency in the planet formation process (Mulders et al., 2021). A few
possible solutions to this “missing mass” discrepancy have been offered.
First, the masses obtained from millimeter observations may underestimate the
actual disk masses (Ballering & Eisner, 2019; Ribas et al., 2020). Second,
planet formation may occur rapidly, within the first few Myr of a disk’s
lifetime (e.g., Najita & Kenyon, 2014). Third, as possibly observed by Ginski
et al. (2021), the disk may accrete additional material (in significant
amounts) from its surroundings, replenishing the disk (e.g., Throop & Bally,
2008; Kuffmeier et al., 2017); the amount of material we see in the disk would
thus be only a fraction of the total material available for planet formation,
alleviating the missing mass problem (Manara et al., 2018).
Dust settling is an important early step in the planet formation process. The
dust grains in disks originate from the interstellar medium (ISM) as small
($\sim$0.25$\mu$m) amorphous silicates. As the disk evolves, these dust grains
collide with each other; if the collisions are inelastic the grains can grow
to $\sim$micron sizes. These larger grains feel a greater drag force as they
decouple from the bulk gas motion and move through the gas in the disk. The
grains therefore experience a net force towards the disk midplane and settle
out of the disk atmosphere (Dullemond & Dominik, 2004). Since this process
takes time, it may be expected that older disks would be more settled than
younger disks. However, infrared (IR) observations have shown that this
process is already occurring as early as $\sim$1 Myr (Furlan et al., 2009;
Ribas et al., 2017; Grant et al., 2018). See the recent review by Miotello et
al. (2022) for an overview of dust settling and disk vertical structure.
To build a complete picture of how disk masses and dust settling change over
time – and therefore how the planet formation process is unfolding – the ideal
disk sample would include as many disks as possible from star-forming regions
spanning a wide range of ages. Until recently, such a study has been difficult
to implement, given the complexity and computational demands of the physical
models used to determine disk properties. Fortunately, we now have access to
machine-learning technology that can reproduce the results of a physical model
in a fraction of the computational time: Ribas et al. (2020) trained an
Artificial Neural Network (ANN) to replicate the output of the D’Alessio
Irradiated Accretion Disk (DIAD) models (D’Alessio et al., 1998, 1999, 2001,
2005, 2006). The ANN thus allows us to model hundreds of disks and perform a
robust statistical analysis on their parameters.
In this work, we use this modeling framework to consistently model a large
sample of protoplanetary disks in multiple star-forming regions in order to
probe how disk mass and dust settling vary over time. In Section 2, we present
our sample of protoplanetary disks. Our modeling process and results are
presented in Section 3, and we discuss our findings in Section 4. Finally,
Section 5 provides a summary.
## 2 Disk Sample
We present here our sample of consistently-modeled T Tauri stars and their
surrounding disks. These objects are located in eleven star-forming regions,
all of which have been studied at millimeter wavelengths: Ophiuchus (Cieza et
al., 2019; Williams et al., 2019), Taurus (Andrews et al., 2013; Long et al.,
2019), L1641 (Grant et al., 2021), Cha II (Villenave et al., 2021), Lupus
(Ansdell et al., 2016, 2018), Cha I (Pascucci et al., 2016), IC348 (Ruíz-
Rodríguez et al., 2018), Corona Australis (Cazzoletti et al., 2019), $\sigma$
Ori (Ansdell et al., 2017), $\lambda$ Ori (Ansdell et al., 2020), and Upper
Scorpius (Barenfeld et al., 2016). The results from the two recent ALMA
surveys of Orion A (van Terwisga et al., 2022) and Serpens (Anderson et al.,
2022) are not included in our sample but will be discussed in a future work.
See Table 1 for ages and distances for each region. Most of these surveys
report to be complete down to the substellar limit (i.e., for spectral types
$<$ M6). However, the objects surveyed in L1641, Cha II, IC 348, and Upper Sco
were selected based on 24$\mu$m or 70$\mu$m detections, meaning that these
surveys are likely incomplete at low stellar masses (spectral types M4 and
M5). Recent results using data from the Gaia surveys (Galli et al., 2020;
Luhman & Esplin, 2020; Galli et al., 2021; Grasser et al., 2021; Krolikowski
et al., 2021; Esplin & Luhman, 2022) confirm the incompleteness of these
samples.
The total number of single Class II TTS observed in each region, Ntotal, is
also reported in Table 1. Objects in binary or multiple systems with
separations $<$ 1000 au were not included because companion objects can alter
the disk properties and influence disk evolution (Harris et al., 2012; Kounkel
et al., 2016; Ruíz-Rodríguez et al., 2016; Cox et al., 2017; Akeson et al.,
2019; Barenfeld et al., 2019; Cazzoletti et al., 2019; Villenave et al., 2021;
Rota et al., 2022). See the recent review by Offner et al. (2022) for a
detailed discussion on the impact of binarity on disk systems.
In addition to being single objects, certain other criteria must be met for an
object to be included in our sample. We require the objects to have
millimeter-wavelength photometry detections in order to constrain our SED
models at long wavelengths. Though this requirement introduces a bias towards
more massive disks since those disks are easier to detect in the millimeter,
millimeter detections are crucial for constraining the model fit at long
wavelengths and therefore the disk mass. The number of single Class II objects
detected in the millimeter in each region, $N_{det}$ is reported in Table 1.
We also require objects to have significant photometric and/or spectroscopic
coverage in the infrared to constrain the rest of the SED. This requirement
excludes all disks in NGC2024 (0.5 Myr, 414 pc; van Terwisga et al., 2020) and
Orion (1 Myr, 400 pc; Eisner et al., 2018), despite their ALMA coverage, since
they lack sufficient photometry points.
No transitional disks (TDs) are included in our sample, since their inner
cavities can affect the shape of an object’s SED (see for example the review
by Andrews, 2020); this effect on the SED is not accounted for in our modeling
framework, so we exclude these objects to avoid that source of uncertainty.
TDs were determined by the shape of their SED (Cieza et al., 2010; Espaillat
et al., 2012; Luhman & Mamajek, 2012; Maucó et al., 2016; van der Marel et
al., 2016; Grant et al., 2018) or through resolved imaging (Isella et al.,
2010; Andrews et al., 2011; Canovas et al., 2015). As noted by Andrews (2020),
the current catalog of very high-resolution observations in which substructure
could be detected is biased towards larger, brighter disks around more massive
hosts. Excluding TDs from our sample may preferentially exclude more massive
disks and introduce a bias into our sample.
After applying these selection criteria, our modeling sample consists of 383
disks around T Tauri stars. Of these objects, 338 are used for our analysis;
Section 3.2 describes our reasons for excluding 45 objects from the analysis.
Table 1 lists the number of objects in our sample used for analysis per
region, Nsample.
## 3 Analysis and Results
### 3.1 SED Models
In order to determine disk properties for each of the TTS objects in our
sample, we fit models to each object’s SED. SEDs were constructed using
photometry points spanning visual to millimeter wavelengths, obtained using
the Vizier catalog access tool (Ochsenbein et al., 2000). We include
photometry from Gaia DR3, the American Association of Variable Star Observers
Photometric All-Sky Survey (AAVSO; Henden et al., 2015, 2016), the Guide Star
Catalog (GSC2.3; Lasker et al., 2008), 2MASS, WISE, Spitzer (labeled as c2d in
the figures), Herschel, and any available millimeter photometry (usually ALMA,
but also including the Submillimetre Common-User Bolometer Array on the James
Clerk Maxwell Telescope (SCUBA; Holland et al., 2013) and the Submillimeter
Array (SMA; Ho et al., 2004)). After visual inspection of the SED, we removed
a few photometry data which were in obvious disagreement with the rest of the
SED and which had apparent contamination in their images. We also include low-
resolution (R $\sim$ 60–130) spectra from the InfraRed Spectrograph (IRS) on
the Spitzer Space Telescope (Houck et al., 2004) for the 187 objects in our
sample that were observed with this instrument. In order to counteract uneven
weighting in the fitting process due to the much larger number of points in a
spectrum than in the photometry for a given object, we bin the spectrum into a
smaller number of points. Spectrum points are binned according to the nearest
wavelength at which our model fluxes are calculated, yielding approximately
10-20 binned points, depending on the wavelength span of the spectrum.
#### 3.1.1 The Artificial Neural Network
Previous studies (e.g., Rubinstein et al., 2018; Macías et al., 2018; Rilinger
et al., 2019; Rilinger & Espaillat, 2021) used the DIAD code (D’Alessio et
al., 1998, 1999, 2001, 2005, 2006) to successfully model disk SEDs. However,
these models are computationally expensive, and the ability to perform robust
statistical analysis is limited. In this work, we use the ANN created by Ribas
et al. (2020) to model the disk SEDs. The ANN was trained on tens of thousands
of DIAD models, and can mimic DIAD outputs in a fraction of the computational
time. A second ANN, ANNdiskmass is employed to determine the disk mass for
each object based on the best-fit model parameters. Neither the DIAD models
nor the ANN include disk mass as a free parameter; rather, disk masses are
inferred from the other best-fit parameters. See the description of $\alpha$
and $\dot{M}$ below as well as Ribas et al. (2020) for details.
DIAD takes various disk and stellar parameters as inputs to calculate the
emission from the system. The dust in the disk is assumed to be distributed in
two populations: smaller dust grains in the atmosphere of the disk and larger
dust grains in the disk midplane. These populations are parameterized by the
maximum dust grain size in each population (i.e., $a_{\rm max,upper}$ and
$a_{\rm max,midplane}$, respectively). The amount of dust settling that has
occurred is represented by the dimensionless parameter $\epsilon$ as described
by D’Alessio et al. (2006) and defined as follows. We assume the grain size
distributions and the dust-to-gas mass ratio ($\zeta$) to be constant in
radius and vary only with height above and below the disk midplane. In order
to keep the total $\zeta$ constant while still allowing for settling of large
grains, $\zeta_{small}$, the dust-to-gas ratio of small grains in the
atmosphere (with sizes up to $a_{\rm max,upper}$) decreases while
$\zeta_{big}$, the dust-to-gas ratio of larger grains in the midplane (with
sizes up to $a_{\rm max,midplane}$) increases. The settling parameter
$\epsilon$ is thus defined such that
$\epsilon=\zeta_{small}/\zeta_{std}$ (1)
where $\zeta_{std}$ is the standard assumed dust-to-gas mass ratio of 0.01.
Therefore, lower $\epsilon$ values correspond to larger $\zeta_{big}$,
implying more settled disks.
Disk viscosity is characterized by $\alpha$, following Shakura & Sunyaev
(1973). In the DIAD models, $\alpha$ and the mass accretion rate $\dot{M}$ are
the two main parameters that set the disk mass. The surface mass density
$\Sigma$ is proportional to $\dot{M}$$\alpha^{-1}$; integrating $\Sigma$ over
the disk radius yields disk mass. For a fixed mass accretion rate, smaller
values of alpha thus correspond to larger disk masses, and vice versa. As
discussed by Ribas et al. (2020), $\alpha$ and $\dot{M}$ can therefore be
correlated, but they also have other smaller effects on the SED that can help
constrain them. We allow alpha to vary between $10^{-4}$ and $10^{-1}$. These
values are consistent with those derived by Rafikov (2017) and Ansdell et al.
(2018); these works calculated $\alpha$ using measured mass accretion rates
and disk sizes. ALMA molecular line observations generally support low
turbulence values (on the order of 10-4 – 10${-3}$ Flaherty et al., 2015;
Pinte et al., 2016; Flaherty et al., 2017), though some objects show higher
values ($\sim$ 0.01 – 0.1 Flaherty et al., 2018, 2020). We vary $\dot{M}$
between $10^{-10}\ \rm{\it M}_{\odot}\,yr^{-1}{}$ and $10^{-6.5}\ \rm{\it
M}_{\odot}\,yr^{-1}{}$; this is consistent with typical accretion rates
reported for Class II disks (e.g., Valenti et al., 1993; Hartigan et al.,
1995; Gullbring et al., 1998; Ingleby et al., 2013; Manara et al., 2016a;
Simon et al., 2016).
We note that DIAD calculates the radial and vertical disk density and
temperature structure self-consistently. In general, the disk emission at a
given wavelength will depend on the optical depth along the line of sight and
the disk temperature structure. When the disk emission is optically thick, the
temperature at the surface where the line-of-sight optical depth is unity will
be the dominant component. As such, even when a disk is optically thick from
the optical to the mm wavelengths, DIAD can still provide strong constraints
on various disk parameters.
The geometry of the disk is also important for fitting the SED, so the disk
radius $R_{\rm disk}$ and inclination $i$ are also included as input
parameters. Furthermore, the inner edge of the disk is an important source of
disk emission, since the inner disk wall is assumed to be directly irradiated
by the central star. The location of the inner wall is defined by the dust
sublimation temperature, $T_{wall}$, and the height of the wall is scaled by a
factor $z_{\rm wall}$.
In addition to the nine disk parameters described above, the stellar
temperature $T_{*}$, radius $R_{*}$, and mass $M_{*}$ are also important input
parameters for DIAD (and therefore ANN). We ensure that we only allow for
combinations of stellar parameters that are consistent with models of stellar
evolution, following Ribas et al. (2020). We incorporate an $Age_{*}$
parameter into our fitting process (see Section 3.1.2); combined with $M_{*}$,
we use this parameter to calculate $T_{*}$ and $R_{*}$ based on the MESA
Isochrones and Stellar Tracks (MIST; Paxton et al., 2011, 2013, 2015; Dotter,
2016; Choi et al., 2016). The output $T_{*}$ and $R_{*}$ are therefore
consistent with the input $M_{*}$, and all three parameters form a consistent
set that can be used as input for DIAD. We generally find strong agreement
between the $T_{*}$ and $R_{*}$ calculated by our models and the values
reported in the literature. See Appendix A for a comparison with literature
values. We include these stellar parameters in our fitting procedure in order
to allow for a complete Bayesian analysis of the rest of the parameters. As
discussed in Ribas et al. (2020), the stellar parameters derived in this
fitting process are not intended to be precisely accurate values, as would be
expected from a complete treatment of photospheric spectra. Rather, our values
of $M_{*}$, $Age_{*}$, $T_{*}$, and $R_{*}$ are used to ensure that the models
are internally consistent and to account for the uncertainties they produce in
other parameters.
Finally, once an SED has been calculated using these input parameters, we
redden the output SED to the appropriate extinction value $A_{v}$ using the
McClure (2009) extinction law. We determine the $A_{v}$ values using the
Markov Chain Monte Carlo (MCMC) fitting process described below in Section
3.1.2. The reddened SEDs are ultimately scaled to the object’s distance, which
is also fit with the MCMC. Gaia parallaxes (Gaia Collaboration et al., 2016,
2018, 2021) are used as priors where possible; the approximate distances to
each region given in Table 1 are used for the 42 objects without Gaia
measurements.
#### 3.1.2 MCMC SED Modeling
Including all of the free parameters described in the previous section, DIAD
takes 13 input parameters. In addition to these parameters, we include four
additional parameters to account for photometric uncertainties and outliers.
One of these is the free parameter $f$ by which we scale uncertainties in flux
density to account for possible systematic uncertainties, as follows:
$s_{n}^{2}=\sigma_{n,model}^{2}+\sigma_{n,obs}^{2}+f^{2}\,y_{n,obs}^{2}\quad$
(2)
where index n corresponds to each measurement, $\sigma_{n,model}$ is the
adopted 10$\%$ uncertainty for the ANN prediction at the same wavelength,
$\sigma_{n,obs}$ is the uncertainty in the flux for the measurement,
$y_{n,obs}$ represents the observed fluxes of the measurement. The other three
are parameters in a mixture model used to account for possible photometric
outliers. Following Hogg et al. (2010), $P_{\rm out}$ is the probability that
any photometry point is an outlier; $y_{\rm out}$ and $V_{\rm out}$ are the
mean and variance of the outliers, respectively. Our model thus has a total of
17 free parameters.
In order to sufficiently probe the large parameter space and to estimate
posterior distributions for the model parameters, we implement a Bayesian
analysis framework in the form of a Markov Chain Monte Carlo (MCMC) fitting
process. We adopt the likelihood functions used by Xin et al. (in press), and
the ptemcee (Vousden et al., 2016) parallel-tempering version of the more
commonly used emcee MCMC code (Foreman-Mackey et al., 2013). Following Xin et
al. (in press), we separate the photometry points into two categories:
“critical”, which includes the 2MASS and millimeter-wavelength points, and
“general”, which includes all other photometry points. This categorization
allows us to ascribe a higher weighting to the 2MASS and ALMA photometry;
fitting these points is crucial for accurately scaling the stellar photosphere
and calculating the disk mass, respectively. We use four standard Gaussian
likelihood functions, one each for the critical photometry data, the spectral
data, $T_{*}$, and $R_{*}$, given as:
$\text{\mathcal{L}}_{n,data}=\frac{1}{\sqrt{2\pi(s_{n}^{2})}}\exp{\biggl{(}-\frac{(y_{n,data}-y_{n,{\rm
model}})^{2}}{2(s_{n}^{2})}\biggr{)}}$ (3)
where $data$ is the data set (critical photometry, IRS spectrum, $T_{*}$, or
$R_{*}$), index n corresponds to each measurement, $y_{n,x}$ represents the
observed values, $y_{n,{\rm model}}$ represents the values predicted by ANN,
and $s_{n}$ is the uncertainty scaling described above. We use the mixture
model outlined above for the general photometry data (all other photometry
points):
$\text{\mathcal{L}}_{n,gen}=\frac{1-P_{\text{out}}}{\sqrt{2\pi(s_{n}^{2})}}\exp{\biggl{(}-\frac{(y_{n,gen}-y_{n,{\text{model}}})^{2}}{2(s_{n}^{2})}\biggr{)}}+\frac{P_{\text{out}}}{\sqrt{2\pi[s_{n}^{2}+V_{\text{out}}^{2}]}}\exp{\biggl{(}-\frac{(y_{n,gen}-y_{\text{out}})^{2}}{2[s_{n}^{2}+V_{\text{out}}^{2}]}\biggr{)}}$
(4)
where $y_{n,gen}$ represents the observed fluxes of the photometry points. The
overall likelihood function is thus:
$\ln\mathcal{L}=\sum_{n=1}^{N_{\rm
gen}}\ln\mathcal{L}_{n,gen}+\sum_{n=1}^{N_{\rm
crit}}\ln\mathcal{L}_{n,crit}+\sum_{n=1}^{N_{\rm
spect}}\ln\mathcal{L}_{n,spect}+\ln\text{\mathcal{L}}_{T_{\rm
eff}}+\ln\text{\mathcal{L}}_{R_{*}}$ (5)
We adopt flat priors for most parameters, except those for which we use
reported values in the literature. The dust sublimation temperature $T_{wall}$
has a Gaussian prior centered around the typically-assumed value of 1400 K
with a standard deviation of 50 K. Gaussian priors are also used for parallax
and $M_{*}$ for objects with Gaia parallaxes, and reported $M_{*}$ values,
respectively, using their values as the centers and their uncertainties as the
standard deviations. For objects with reported disk inclinations, we adopt a
Gaussian prior, truncated at the limits of 0 and 70 degrees of inclination
(the range of values on which the ANN was trained). The maximum dust grain
sizes $a_{\rm max,\ upper}$ and $a_{\rm max,\ midplane}$ are both fit with
Jeffreys priors (Jeffreys, 1946, 1961), within their limits of 0.25 - 10
$\mu$m and 100 - 104 $\mu$m, respectively. All other parameters are assumed to
have flat, uniform priors, within their allowed ranges of values (see Table
C.1 in Ribas et al. (2020) for the ranges).
We run ptemcee with three temperatures and 102 walkers (the minimum number of
required walkers for three temperatures and 17 parameters) for 2 $\times$ 105
steps. This corresponds to $\sim$2000 times the autocorrelation time. We
visually inspected the chain to confirm that we had achieved convergence, and
discard the first 105 steps based on this inspection. The remaining 105 steps
are therefore converged and represent $\sim$1000 times the autocorrelation
time. Since using all 105 steps would be computationally expensive in our
analysis, we sample 104 steps from these converged steps, which is still
sufficient to explore the parameter space. These 104 steps were used to
construct posterior distributions for the properties of interest, and to
obtain the input parameters needed for ANNdiskmass to calculate the
corresponding distribution of disk masses. To determine the best-fit
parameters, we take the median of the $10^{4}$ steps for each parameter, and
report the $1\sigma$ value for the uncertainties.
### 3.2 Model Results
Using the MCMC fitting process described above, we obtain SED models for each
of the 383 objects in our sample. In order to fully visualize the resulting
fits for each object, we randomly sample 1000 steps from the converged MCMC
chain and plot the corresponding models along with the photometry and, where
available, an IRS spectrum. Some example SED fits are shown in Figure 1; SED
fits for the other objects are available in an online figure set. The corner
plots for all objects are available on https://zenodo.org/record/7235076
(catalog Zenodo).
Figure 1: The observed SED and models for one example disk from each region.
Results of the modeling process are shown by randomly selecting 1000 models
(blue lines) from the posterior distributions. Instrument abbreviations are
defined in Section 3.1. SEDs and models for all objects are available in an
online figure set. Some objects have upper limit photometry points; these are
represented by downward-facing triangles in the online figure set. We note
that the photometry has not been dereddened; we redden the models as described
in Section 3.1.1.
After visual inspection, we determined that we obtained successful SED models,
in which the models reproduce the observed SED, for 338 of the 383 objects in
our sample, approximately 88%. Three examples of unsuccessful fits are shown
in Figure 2. The unsuccessful fits can be attributed to a few explanations.
First, TTSs are well-known to be variable on timescales ranging from minutes
to decades (Siwak et al., 2018). This can explain discrepancies between
photometric observations that were taken at different times. Since our model
does not account for variability, differences between our model and the
photometry – particularly at shorter wavelengths, where variability is more
pronounced – are to be expected in some cases. Discrepancies between our model
and near-IR photometry can also occur for disks around cool, very low mass
stars, since the Pecaut & Mamajek (2013) colors used by DIAD are more
uncertain for late spectral types (see for example, the left panel in Figure
2).
Figure 2: Three examples of unsuccessful SED fits. In the left panel, the
models fail to fit the J, H, and K band photometry at the peak; in the middle
panel, the models fail to fit the millimeter photometry; in the right panel,
the shape of the SED indicates that this object might be highly inclined
(i.e., viewed nearly edge-on), which would require a scattered-light model
component to be fit well.
Another potential source of disagreement is our assumption regarding dust
composition. The DIAD models upon which the ANN is based assume a standard
mixture of olivine silicates and graphite and adopt standard opacities
calculated from Mie theory. The optical constants are taken from Dorschner et
al. (1995) and Draine & Lee (1984) for the silicates and graphite,
respectively. DIAD, and therefore the ANN, do not fit spectral features in
detail, but generally replicate the shape and size of the spectrum. In cases
where our model does not perfectly replicate the IRS spectrum, the disk may
have a different dust composition than assumed in our model.
Millimeter photometry points with large uncertainties can also cause issues
with the SED fit. As mentioned in Section 2, millimeter photometry is crucial
for fitting the SED at long wavelengths, especially since in many cases, the
millimeter point is the only photometry point beyond 24 $\mu$m. Points with
larger uncertainties are given lower weighting in the MCMC fitting process; if
the millimeter uncertainties are too large, the model can miss that point (see
for example the middle panel of Figure 2). Future millimeter observations of
these objects at higher sensitivities may result in lower uncertainties and
therefore better-constrained model fits.
Finally, the geometry and configuration of the system may add extra challenges
to fitting the SED. If a disk is highly inclined (i.e., viewed nearly edge-
on), a scattered light model component would be necessary for an accurate fit
(see Figure 2, right panel); though DIAD can include scattered light, this
feature is not yet available with the ANN. Additionally, though we attempted
to exclude all objects with known companions and/or disk substructures, it is
likely that some objects in our sample have companions or substructures that
are yet undetected. Such a system may affect the SED in ways that are not
accounted for by the DIAD models. Thus, some amount of unsuccessful SED fits
is expected.
Despite these challenges, we obtain successful model fits for 338 of the 383
objects in our sample. Moving forward, we consider only the 338 objects with
satisfactory model fits.
## 4 Discussion
### 4.1 The Role of Optical Depth on Derived Disk Masses
In this section, we compare the disk masses inferred from our modeling process
to the masses previously reported in the literature. Typically, the following
equation is used to convert an observed millimeter-wavelength flux to a dust
mass (Hildebrand, 1983):
$M_{d}=\frac{F_{\nu}d^{2}}{\kappa_{\nu}B_{\nu}(T_{d})}$ (6)
where $M_{d}$ is the mass of dust in the disk, $F_{\nu}$ is the millimeter
flux density, $\kappa_{\nu}$ is the dust opacity coefficient at frequency
$\nu$, and $B_{\nu}$($T_{d}$) is the Planck function evaluated at dust
temperature Td. This equation relies on some assumptions, namely, that the
disk is isothermal and optically thin at the reference frequency. Furthermore,
assumptions must be made regarding the value of $T_{d}$ (whether it is
constant for all disks, or if it scales with luminosity), and the value of
$\kappa_{\nu}$ and how it scales with frequency. Finally, many of the previous
millimeter surveys of disks were published prior to the Gaia data releases,
and therefore had to rely on constant distance assumptions for all objects, as
opposed to precise measurements.
Different studies chose different assumptions for the terms in Equation 6, so
to ensure that we made a fair comparison, we recalculated all of the disk
masses using Equation 6 according to a consistent set of assumptions. All
distances were calculated from Gaia parallaxes, where possible (the distances
in Table 1 were used for objects without Gaia data). In previous studies, dust
temperature was assumed either to be a constant value (typically 20 K; Ansdell
et al., 2016, 2017; Ruíz-Rodríguez et al., 2018; Cazzoletti et al., 2019;
Williams et al., 2019; Ansdell et al., 2020) or to vary with stellar
luminosity:
$T_{d}=25(L_{*}/L_{\odot})^{0.25}K$ (7)
as in Andrews et al. (2013), Barenfeld et al. (2016), Pascucci et al. (2016),
and Villenave et al. (2021). We set $\kappa_{\nu}$ to 2.3 cm2 g-1 at 230 GHz
(Beckwith et al., 1990). A common assumption is that $\kappa_{\nu}$ scales
with frequency as $\nu^{1.0}$ (i.e., the power law index $\beta$ = 1.0,
Ansdell et al., 2016, 2017; Cieza et al., 2019; Ansdell et al., 2020; Grant et
al., 2021). Other works (i.e., Pascucci et al., 2016; Barenfeld et al., 2016)
use $\beta$ = 0.4. We set $\nu$ to the frequency of the ALMA continuum
observations: 225 GHz (Band 6) for most regions, except Cha I, Lupus, and
Upper Sco which were observed at 338 GHz (Band 7). The choice of $\beta$ has
no effect on the value of $\kappa_{\nu}$ for the surveys done in Band 6. For
the surveys done in Band 7, using a $\beta$ of 0.4 yields a value of
$\kappa_{\nu}$ that is $\sim$ 0.8 times the value obtained using a $\beta$ of
1, so the effect is minimal. Following Manara et al. (2022), we adopt a
$\beta$ of 1 for this analysis. We calculated two sets of disk masses, one
using each dust temperature assumption. The left panel of Figure 3 shows the
recalculated values using the $T_{d}$ scaling in Equation 7; the right panel
of Figure 3 shows recalculated values using a constant $T_{d}$ of 20 K.
Figure 3: Comparison between our disk masses from SED modeling and disk masses
calculated from single millimeter-wavelength fluxes. In Panel (a), we scale
the dust temperature according to the stellar luminosity (calculated from our
modeled T∗ and R∗ values), as in Andrews et al. (2013). In Panel (b), we
assume a constant dust temperature of 20 K. In both panels, the black solid
line represents the best-fit linear relationship and the black dotted line
represents a one-to-one relationship. The linear fits presented here can be
used to correct disk masses calculated using Equation 6. Uncertainties in the
fitting parameters are provided in the text.
Generally, as shown in Figure 3, the masses inferred from SED modeling are
greater than the masses obtained via Equation 6. A growing number of studies
report that Equation 6 underestimates the dust disk mass. In their study of
disks in Taurus using radiative transfer models, Ballering & Eisner (2019)
found that Equation 6 underestimates disk masses by a factor of $\sim$1-5.
Similarly, using ANN, Ribas et al. (2020) found disk masses greater than those
reported in Andrews et al. (2013) (calculated from Equation 6) by a factor of
$\sim$3\. Macías et al. (2021) also found 3-5 times higher masses in TW Hya
when comparing previous mm-flux-based mass estimates with a dust mass obtained
from a radially resolved modelling of multi-wavelength ALMA observations that
accounted for the radial variations in the optical depth of the disk. Most
recently, Liu et al. (2022) investigated the effects of disk structure and
dust properties on disk mass estimates through a radiative transfer parameter
study; they find that mm-flux-based mass estimates typically underestimate
disk masses by a factor of a few.
For the majority of disks in our study, our model-derived masses are 1.5-5
times larger than those obtained using Equation 6, consistent with Ballering &
Eisner (2019), Ribas et al. (2020), Macías et al. (2021), and Liu et al.
(2022). Our result confirms the findings of these works: the assumptions
required by Equation 6 can lead to an underestimate of the disk mass. Since
radiative transfer models, and therefore ANN, do not rely on these
assumptions, our disk masses do not suffer the same underestimation.
Given the growing list of ALMA surveys of T Tauri disks, a method to determine
disk masses from millimeter photometry, such as Equation 6, is a useful tool.
We present here a correction to Equation 6, based on the relationships in
Figure 3. In each panel of Figure 3, we used the LinearRegression model in the
scikit-learn Python package to fit a line to the relationship between disk
masses from our SED models and disk masses calculated using Equation 6. We
repeated the linear fitting 500 times; the best-fit lines and their
uncertainties reported here represent the mean and standard deviation of the
coefficients from the 500 linear fits. Given a disk mass $M$ in Earth masses
calculated from millimeter photometry using Equation 6, the slopes and
intercepts of the best-fit lines in Figure 3 can be used to obtain a new
$M_{corrected}$ in Earth masses:
$log_{10}(M_{corrected})=1.09(\pm 0.02)*log_{10}(M)+0.19(\pm 0.02)$ (8)
if the disk temperature was scaled by the stellar luminosity, or
$log_{10}(M_{corrected})=0.97(\pm 0.02)*log_{10}(M)+0.29(\pm 0.01)$ (9)
if the disk temperature was assumed to be a constant 20 K.
While nearly all of our disk masses from SED modeling are larger than the
recalculated literature disk masses, the discrepancy is not uniform when disk
temperatures are scaled by stellar luminosity. As shown by the equations
presented above, we find that disks with greater dust masses tend to deviate
more from the recalculated literature values than disks with lower masses in
the case where disk temperatures are scaled by the stellar luminosity. More
luminous stars can supply more heat to the dust in the disk, increasing
$T_{d}$. Assuming a constant dust temperature may result in an underestimate
of $T_{d}$ for more massive stars, which, following Equation 6, would in
principle result in an overestimate of $M_{dust}$. This has the result of
flattening the linear relationship shown in Figure 3(b), since the more
massive disks tend to be around more massive stars (Andrews et al., 2013;
Manara et al., 2022). If we account for this effect by scaling $T_{d}$ with
stellar luminosity, we observe the trend of greater discrepancy at higher
masses (see Figure 3(a)). While it might seem that using the apparently
incorrect fixed temperature can provide better results for more massive disks,
we note that the trend is reversed for low mass disks: using a $T_{d}$ that
scales with stellar luminosity provides a much better agreement with our SED-
based masses. Regardless of the relationship between stellar luminosity and
disk temperature, we find that Equation 6 underestimates disk mass compared to
our SED-derived masses, regardless of the relationship between stellar
luminosity and disk temperature.
Another potential source of the discrepancy is the value of the reference dust
opacity, $\kappa_{230\ GHz}$. As mentioned above, masses are typically
calculated using a reference $\kappa_{230\ GHz}$ equal to 2.3 cm2 g-1,
regardless of the size of the dust grains in the disk. The DIAD models compute
different dust opacities depending on the size of the dust grains in the disk
midplane, $a_{max,\ midplane}$. These dust opacities have a value of $\sim$2.3
cm2 g-1 for 400 $\mu$m grains, $\sim$1.9 cm2 g-1 for a grain size of 1 mm, and
a minimum of 0.85 cm2 g-1 for 1 cm grains. In most disks in our sample,
$a_{max,\ midplane}$ is not well-constrained, and in fact the posteriors are
usually flat from 100 microns to 1 cm. The difference in opacity hence
enlarges (and mostly sets) our error bars in dust mass, making sure that the
uncertainties in dust opacity are properly accounted for. Even in objects
where $a_{max,\ midplane}$ might be relatively well constrained, the different
opacities would only be able to explain mass differences a factor of 1.5-2.
To probe the effect of the value of $\kappa_{230\ GHz}$ on dust mass, we again
recalculated masses using Equation 6. This time, we used the appropriate
$\kappa_{230\ GHz}$ for each object as calculated by DIAD for its median
$a_{max,\ midplane}$. All other assumptions were kept the same as described
above and in Manara et al. (2022). Figure 4 shows the ratio of our SED-derived
dust masses to these new recalculated dust masses plotted versus the new
recalculated dust masses. As can be seen in the Figure, our SED-derived dust
masses are still a few times greater than those calculated from Equation 6,
even when accounting for different $\kappa_{230\ GHz}$. We note that
uncertainties in dust composition and structure could affect the value of both
$\kappa_{230\ GHz}$ and $\beta$ by as much as an order of magnitude (Testi et
al., 2014; Miotello et al., 2022), but these would have the same effect both
on our dust masses and the millimeter-flux ones. While these uncertainties
could alter the dust masses, the differences in $\kappa_{230\ GHz}$ alone
cannot explain the observed discrepancy between our dust masses and those
derived from millimeter fluxes. Differences in $\kappa_{230\ GHz}$ cannot
alone explain the observed discrepancy between our dust masses and those
derived from millimeter fluxes.
Figure 4: Comparison between our disk masses from SED modeling and disk masses
calculated from single millimeter-wavelength fluxes, varying $\kappa_{230\
GHz}$ according to the median $a_{max,\ midplane}$ value of each object. The
y-axis shows the ratio of our SED-derived masses to the flux-derived masses.
The dashed black line shows a one-to-one relationship between the two masses,
and the solid black lines show the linear fits from Figure 3. As in Figure 3,
dust temperatures are scaled according to the stellar luminosity in Panel (a)
and held at a constant value in Panel (b). In both cases, even when correcting
for a more appropriate $\kappa_{230\ GHz}$ value, the dust masses we derive
from SED fitting are still generally larger than those calculated from
Equation 6.
An alternative explanation for the observed trend is related to the assumption
made in Equation 6 that the disk is optically thin at the reference frequency.
More massive disks are expected to be optically thicker at a given wavelength
due to the greater amount of absorbing material present; if these disks are
optically thick, Equation 6 will underestimate the disk mass since the
observed flux is not sensitive to the whole mass in the disk. Modeling by
Mohanty et al. (2013), for example, showed that the 850 micron flux density
becomes independent of disk mass in the optically thick limit. Ballering &
Eisner (2019), Ribas et al. (2020), and Liu et al. (2022) all cite optical
depth as a way to explain the discrepancy between disk mass estimates from
(sub)mm fluxes and those from disk SED modeling. As mentioned above in Section
3.1.1, the SED models calculate the full disk structure, which includes the
optical depth. Therefore, even when the observed millimeter emission is
partially optically thick, the simulated millimeter emission takes this into
account, and thus the total dust mass can be constrained.
To test this explanation, we ran DIAD models for a subset of our disk sample
in order to calculate optical depths. Using the median values for each model
parameter from our fits, we used DIAD to calculate the structure of the disk
and optical depth along the line of sight as a function of radius at 225 GHz
(1.3 mm). We then computed a flux-weighted mean optical depth using the flux
at each radius as the weight for the optical depth at that radius. In this
way, we obtained a single average optical depth of the disk that takes into
account the regions of the disk where most of the emission comes from. These
flux-weighted mean optical depths are plotted versus dust mass in Figure 5 for
each disk in Taurus, Lupus, Cha I and Upper Sco. (We selected these regions
since they are the four regions studied by Pascucci et al. (2016); we compare
our $M_{dust}$ – $M_{*}$ relationships to theirs in the following section.) In
each region, optical depth generally increases with disk mass. Many disks in
each region have a flux-weighted mean optical depth close to or greater than
one, indicating that the optically thin assumption is not valid for these
disks. Thus, disk masses calculated from Equation 6 are systematically
underestimated; disk masses obtained with DIAD and ANNdiskmass, which do not
assume the disks are optically thin, are more reliable for determining disk
masses.
We also show the disk size for each disk in Figure 5 by coloring each point
according to its median outer radius from the MCMC fitting. These outer radii
are larger than the FWHMs reported for resolved disks in Ansdell et al.
(2016), Pascucci et al. (2016), and Barenfeld et al. (2016), though larger
FWHMs generally correspond to larger outer radii in our sample. Disks with
lower masses but high optical depths have smaller radii, indicating that they
are compact and dense, while disks with higher masses but lower optical depths
have larger radii and are comparatively less dense.
#### 4.1.1 Impact on Planet Formation Models
The higher masses resulting from our SED models can help alleviate the tension
between the observed masses of exoplanetary systems and protoplanetary disks.
Previously, Manara et al. (2018) reported that the dust disks in Lupus and Cha
I are a factor of 3–5 too small to account for the typical mass of planetary
cores. A more recent study by Mulders et al. (2021) reports more comparable
masses between disks and exoplanetary systems, but requires nearly 100%
efficiency in the planet formation process to explain the observed masses. The
masses we report here are typically a factor of 1.5–5 greater than the
recalculated literature values. The assumption that disks are optically thin
can therefore result in a significant underestimation of their masses. Our
disk masses ease the requirement for extremely high planet formation
efficiency. We note, however, that increasing the available disk mass by a
factor of 1.5-5 would imply planet formation efficiencies between $\sim$20%
and $\sim$60%, to achieve the same exoplanet disk masses as reported by
Mulders et al. (2021). Given that planet formation models do not still have a
robust estimate of the planet formation efficiency, it is not fully clear if
the optical depth alone could solve the “missing” mass problem or if
alternative scenarios are still required (e.g., planetesimals already being
formed, late infall of material onto the disk).
Figure 5: Flux-weighted mean optical depth of each disk plotted versus disk
mass for four star-forming regions: (a) Taurus, (b) Lupus, (c) Cha I, and (d)
Upper Sco. Optical depths are flux-weighted averages of the 1.3 mm optical
depth at each radius in the disk as calculated by the DIAD models. Disk masses
are the median values from our models for each object. The horizontal dashed
line denotes an optical depth of 1. Colors represent the disk outer radii.
Some disks in each region have optical depth greater than 1, especially higher
mass disks.
### 4.2 Disk Mass Trends with Host Mass
Figure 6: The relationship between disk dust masses and host star masses for
four star-forming regions: (a) Taurus, (b) Lupus, (c), Cha I, and (d) Upper
Sco. Red points are median $M_{dust}$ and $M_{*}$ values from the MCMC output
and black points are $M_{dust}$ and $M_{*}$ values from (Pascucci et al.,
2016). Linear fits for these two groups are shown in their matching colors,
with a 1$\sigma$ uncertainty shown by the shaded region. The values listed in
the upper left corner of each panel are the Spearman correlation coefficient
and p value for our modeled $M_{dust}$ and $M_{*}$ values. Taurus has a lower
fraction of modeled objects than other regions, so we also include a fit to a
mixed sample (purple line): our modeled objects plus single objects and non-
detections not included in our sample, with masses recalculated from their mm
fluxes (blue symbols, see Section 4.1).
Disk masses have been shown to scale with the masses of their host stars in
individual star forming regions (e.g., Pascucci et al., 2016; Testi et al.,
2022), though the trend is murkier when multiple regions are considered
together (Manara et al., 2022). In Figure 6 we plot disk dust mass versus host
mass for the four star-forming regions studied by Pascucci et al. (2016).
Following Pascucci et al. (2016), we use the Bayesian linear fitting procedure
developed by Kelly (2007), which allows for measurement uncertainties as well
as upper limits, to fit the recalculated dust mass relationship. Our MCMC
fitting process results in asymmetric uncertainties, which are not supported
by the Kelly (2007) fitting procedure, so we instead use Python Abstract
Interfaces for Data Analysis (PAIDA)111paida.sourceforge.net to obtain a
linear fit to the modeled dust masses that incorporates the uncertainties.
For Lupus, Cha I, and Upper Sco, we find $M_{dust}$-$M_{*}$ relationships that
are consistent with those reported in Pascucci et al. (2016); i.e., the slopes
and intercepts of the fits reported in Table 2 for each region are within
1-2$\sigma$ of the fits reported by Pascucci et al. (2016). We generally find
slightly larger intercepts than Pascucci et al. (2016); this can be attributed
to the result reported above, that we find larger disk masses than previously
reported. The relationship we find for disks in Taurus is shallower than
previously reported. This difference in slope may be due to the inherent bias
in our sample introduced by the exclusion of upper limits. Since most the
disks in Taurus were observed at lower sensitivity by the SMA, many of these
objects only have upper limits for their disk masses; the linear fit for
Taurus reported by Pascucci et al. (2016) incorporates these points, but our
sample does not and is limited to higher mass objects.
To mediate the effect of this difference in sensitivity, we also present an
$M_{dust}$-$M_{*}$ relationship for a mixed sample of Taurus objects. This
mixed sample includes our modeled objects, plus any detected objects not known
to be in binary or multiple systems but excluded for other reasons, and single
non-detections. We adopt host and disk masses from our SED models where
possible; for the remaining objects, host masses were taken from Pascucci et
al. (2016) and disk masses were recalculated from literature fluxes using our
consistent method described in Section 4.1, with a constant dust temperature
of 20 K. The $M_{dust}$-$M_{*}$ relationship of this mixed sample, shown in
Figure 6(a) is more consistent with the relationship from Pascucci et al.
(2016), as well as the other three regions presented in Figure 6.
We note that the correlation of the $M_{dust}$-$M_{*}$ relationship in each
region is only moderate at best. Spearman correlation coefficients ($r_{s}$)
for our modeled $M_{dust}$ and $M_{*}$ values are given in the upper left
corner of each panel in Figure 6 and range between $\sim$ 0.3 and $\sim$ 0.5.
The $r_{s}$ value describes how well the relationship between $M_{dust}$ and
$M_{*}$ is described by a monotonic function; $r_{s}$ values closer to 1
indicate stronger correlation. For each region, we also report the p value,
which gives the probability that a random, uncorrelated sample would produce a
similar correlation. For Taurus, the $r_{s}$ and p values given in Figure 6
are for our sample only; for the mixed sample, $r_{s}$ = 0.425 and p = 0.001.
Though the $r_{s}$ values show only a moderate correlation, the low p values
for each of these four regions indicates that the moderate correlation is
statistically significant. Pascucci et al. (2016) also reported moderate
correlations, with dispersions of $\sim$0.8 dex in the $M_{dust}$ – $M_{*}$
relationship for each region.
The $M_{dust}$-$M_{*}$ relationships for the remaining seven regions in our
sample are shown with their $r_{s}$ and p values in Figure 7; we report the
slopes and intercepts of the relationships in Table 2. We note that for three
regions (Cha II, Corona Australis, and $\lambda$ Ori) the sample sizes may be
too small to accurately assess correlation. Of the four regions with
substantial samples (Ophiuchus, L1641, IC 348, and $\sigma$ Ori), only one,
$\sigma$ Ori, has a statistically significant trend (i.e., p $<$ 0.05). The
$r_{s}$ value for this region is comparable to the $r_{s}$ values for the
regions shown in Figure 6, showing a moderate correlation between $M_{dust}$
and $M_{*}$. Ophiuchus, L1641, and IC 348 do not show any significant
correlation. All eleven regions in this sample show a large dispersion in
$M_{dust}$ for a given $M_{*}$. As noted in the review by Manara et al.
(2022), the fact that this dispersion is observed in all regions suggests it
is due to inherent variation in disk properties, as opposed to environmental
effects in a specific region.
Figure 7: The relationship between disk dust masses and host star masses for
the star-forming regions not studied by Pascucci et al. (2016): (a) Ophiuchus,
(b) L1641, (c), Cha II, (d) IC348, (e) Corona Australis, (f) $\sigma$ Ori, and
(g) $\lambda$ Ori. Red points are median $M_{dust}$ and $M_{*}$ values from
the MCMC output. Linear fits for each region are shown in red, with a
1$\sigma$ uncertainty shown by the shaded region. The values listed in the
upper left corner of each panel are the Spearman correlation coefficient and p
value for our modeled $M_{dust}$ and $M_{*}$ values. Table 2: $M_{dust}$ –
$M_{*}$ Relationships
Region | This work | Pascucci et al. (2016)
---|---|---
| $m$ | $b$ | $m$ | $b$
Ophiuchus TTS | 0.6$\pm$0.1 | 1.1$\pm$0.1 | … | …
Ophiuchus TTS and BDs | 0.9$\pm$0.1 | 1.2$\pm$0.1 | … | …
Taurus TTS | 0.2$\pm$0.2 | 1.9$\pm$0.1 | 1.1$\pm$0.2 | 1.0$\pm$0.1
| 2.1$\pm$0.3aaThese values are the fit to a mixed sample: our modeled objects plus single objects and non-detections not included in our sample, with masses recalculated from their mm fluxes. | 1.4$\pm$0.2aaThese values are the fit to a mixed sample: our modeled objects plus single objects and non-detections not included in our sample, with masses recalculated from their mm fluxes. | |
Taurus TTS and BDs | 1.6$\pm$0.1 | 2.1$\pm$0.1 | … | …
L1641 | 0.2$\pm$0.1 | 1.43$\pm$0.1 | … | …
Cha II | 1.9$\pm$0.2 | 1.7$\pm$0.1 | … | …
Lupus TTS | 1.0$\pm$0.1 | 1.5$\pm$0.1 | 1.1$\pm$0.3 | 1.4$\pm$0.2
Lupus TTS and BDs | 1.1$\pm$0.1 | 1.3$\pm$0.1 | … | …
Cha I | 1.5$\pm$0.1 | 1.7$\pm$0.1 | 1.3$\pm$0.2 | 1.1$\pm$0.1
IC 348 | 0.0$\pm$0.2 | 0.9$\pm$0.1 | … | …
Corona Australis | 0.0$\pm$0.2 | 0.4$\pm$0.1 | … | …
$\sigma$ Ori | 1.1$\pm$0.2 | 1.5$\pm$0.1 | … | …
$\lambda$ Ori | 0.6$\pm$0.2 | 0.6$\pm$0.1 | … | …
Upper Sco TTS | 1.8$\pm$0.2 | 0.9$\pm$0.1 | 1.9$\pm$0.4 | 0.8$\pm$0.2
Upper Sco TTS and BDs | 0.8$\pm$0.1 | 0.6$\pm$0.1 | … | …
Note. — The slopes ($m$) and intercepts ($b$) reported here correspond to the
following linear relationship: log$(M_{dust}/M_{\oplus})$ = $m$ x
log$(M_{*}/M_{\odot})$ \+ $b$.
#### 4.2.1 TTS Compared to Brown Dwarfs
Figure 8: Disk dust masses plotted versus host masses for the four star-
forming regions studied in Rilinger & Espaillat (2021). TTS objects are shown
as red open circles and BD objects are shown as black filled circles. TTS disk
and host masses are from the SED models presented in this work. BD disk masses
were derived via SED modeling by Rilinger & Espaillat (2021) and BD host
masses were taken from the literature (see citations in Rilinger & Espaillat
(2021)). Linear fits for just the TTS are represented by red lines; linear
fits to the TTS and BDs together are shown in blue. As in Figure 6(a), we show
linear fits to both our sample and the mixed sample for Taurus.
Given the scaling of disk mass with host mass for TTS (e.g., Pascucci et al.,
2016; Ansdell et al., 2016, 2017, and this work), we explored whether this
trend extends to disks around lower mass brown dwarf (BD, $M_{*}\lesssim$ 0.08
$M_{\odot}$) hosts. Four of the star-forming regions presented in this work
were also studied by Rilinger & Espaillat (2021), who used DIAD to obtain SED
models for 49 disks around BDs in Ophiuchus, Taurus, Lupus, and Upper Sco.
Since the modeling process is consistent between that work and this, we can
directly compare the disk masses we infer for TTS to the disk masses Rilinger
& Espaillat (2021) inferred for BDs.
In Figure 8, we show how the inclusion of BD disks affects the $M_{dust}$ vs
$M_{*}$ relationships presented above. Using PAIDA, we obtained linear fits
for the combined sample of TTS and BDs in each region, which are shown in
Figure 8 by the blue lines. We also reproduce the linear fits to the TTS
sample alone from Figure 6 for comparison. In the three younger regions, the
the combined TTS and BD fit is generally in good agreement with the fit to the
TTS (see Table 2). In Upper Sco, however, including the BD disks in the fit
results in a significantly shallower slope. In other words, in this region, BD
disk dust masses are higher than would be expected from the TTS disk mass –
host mass trend. Instead of decreasing with time (see Section 4.3), BD disk
masses appear to remain consistent over a span of $\sim$ 10 Myr.
Studies of protoplanetary disk fractions in various star-forming regions
indicate that the fraction of stars that host disks increases for later
spectral types (Carpenter et al., 2006; Luhman & Mamajek, 2012; Ribas et al.,
2015; Luhman, 2022). In other words, lower-mass stars appear to retain their
disks longer than higher-mass stars. The results presented here add additional
evidence to this theory. BD disks are typically less massive than their TTS
counterparts in a given star-forming region, as expected based on the
established trend of decreasing disk mass with host mass. However, the BD disk
dust masses remain constant across regions spanning as much as 10 Myr in age
while the corresponding TTS disks exhibit a substantial decrease in disk dust
mass (Section 4.3).
One way in which disks are expected to lose mass is through photoevaporation
due to radiation from the central star (Ercolano et al., 2017; Weber et al.,
2020). Picogna et al. (2021) recently showed that the rate of photoevaporation
scales with stellar mass. Given the lower masses of BDs compared to pre-main-
sequence stars, photoevaporation rates should be lower in BD disks than TTS
disks. That BD disks appear to maintain their disk masses over millions of
years, while their TTS counterparts show a decrease in mass, may be the result
of slower disk dissipation occurring in these objects. The potential for
planet formation in a given disk is thus both enhanced by and inhibited by the
mass of its host: a more massive YSO is more likely to host a larger disk
early in its lifetime, but it will also likely emit more UV and X-ray
radiation, which will dissipate the disk more quickly.
### 4.3 Comparison Between Regions of Various Ages
In this work we studied disks in eleven star-forming regions, which vary in
age from $\sim$0.5 to $\sim$10 Myr. By consistently modeling the SEDs of the
disks in each of these regions, we can directly compare their properties and
probe how the population of disks changes with time. In particular, we
consider the disk masses and the degree of dust settling; both of these
properties have implications for the planet-forming potential of these disks.
#### 4.3.1 Disk Mass with Age
Figure 9: Probability distribution functions of dust mass for each region and
for all regions together (lower right panel). The red dashed line in each
panel represents the median dust mass for the overall sample ($\sim$ 8
$M_{\oplus}$) and the blue dotted lines represent the median dust mass in each
region.
Disk mass distributions for each object are constructed using 10000 random
samples from the object’s MCMC chain. We use these masses to construct
normalized histograms of disk mass for each region. These normalized
histograms are plotted in Figure 9 for each region, as well as the normalized
overall mass distribution including all regions. As shown in Figure 9, the
peak of the mass distribution tends to shift to lower masses as the age of the
region increases. Young regions such as Taurus have mass distributions that
peak above the median dust mass of the entire sample (represented by the red
dashed line in Figure 9), while older regions such as Upper Sco have mass
distributions that peak below the median dust mass. The mass distributions of
moderately-aged regions such as IC 348 peak near the overall median.
Two regions appear to deviate from this trend: Ophiuchus and Corona Australis.
Ophiuchus has a mass distribution that peaks near the median of the overall
sample, despite being the youngest region in our sample. This result is
consistent with the findings of Williams et al. (2019), who report Ophiuchus
disk dust masses to be smaller than the masses of disks in other, slightly
older regions. A possible explanation suggested by Williams et al. (2019) is
that the local cloud environment may significantly affect disk masses.
Kuffmeier et al. (2020), for example, report that star-forming regions with
higher ionization rates may contain less massive disks, since ionization can
decrease disk size. The mass distribution for disks in Corona Australis also
peaks at lower disk masses than may be expected for a moderately young star-
forming region. Another potential explanation for the low disk masses in these
regions is descibed by Testi et al. (2022). Dynamical modeling by Turrini et
al. (2012, 2019); Gerbig et al. (2019) indicates that formation of planets
within a disk can trigger collisional fragmentation of planetesimals,
increasing the total mass of grains detectable at millimeter wavelengths. If
planets can form within the first 0.5 – 1 Myr of a disk’s lifetime, the
formation of this secondary population of dust grains may not have started yet
in the disks in Ophiuchus and Corona Australis. Furthermore, recent studies of
Corona Australis (Cazzoletti et al., 2019; Esplin & Luhman, 2022) suggest that
the region may be comprised of two populations: the young ($<$ 3 Myr) Coronet
Cluster, and an older, more extended population. The lower disk masses we find
in this region support the existence of this older population.
Our finding that disk dust masses generally decrease as the age of the region
increases is consistent with previous studies (e.g., Ansdell et al., 2016;
Barenfeld et al., 2016; Pascucci et al., 2016; Cieza et al., 2019; van
Terwisga et al., 2019; Villenave et al., 2021; van Terwisga et al., 2022),
which have found that older regions typically contain less massive disks than
their younger counterparts. However, an important caveat is that we have only
included objects which have millimeter-wavelength detections. The different
ALMA surveys that studied each region had a range of sensitivities and thus
different detection limits; regions observed with lower sensitivities will
thus be biased towards higher masses, since the lower mass disks are likely to
be less luminous, smaller in radius, and harder to detect. In particular, the
survey of $\sigma$ Ori performed by Ansdell et al. (2017) was only sensitive
to dust masses down to $\sim$2 $M_{\oplus}$. This upper limit is much higher
than that achieved by surveys of other regions: Barenfeld et al. (2016),
Pascucci et al. (2016), Ansdell et al. (2016), and Ansdell et al. (2020) all
detect dust masses on the order of a few tenths of an Earth mass for Upper
Sco, Cha I, Lupus, and $\lambda$ Ori, respectively. The lower sensitivity of
the $\sigma$ Ori survey compared to that of other regions means that this
region’s mass distribution is biased towards higher dust masses.
In order to account for the different sensitivities of the ALMA surveys, we
show cumulative distributions of the dust mass in each region in Figure 10.
These distributions were constructed using a Kaplan-Meier esitmator from the
lifelines package (Davidson-Pilon et al.,
2021)222https://lifelines.readthedocs.io/. In order to ensure a fair
comparison between regions, we include masses for all single objects in each
region. We do not include any binary objects or objects in multiple systems,
since the presence of a companion can significantly affect the evolution of a
disk (e.g., Akeson et al., 2019; Barenfeld et al., 2019; Kounkel et al., 2019;
Zurlo et al., 2020, 2021; Offner et al., 2022). For objects in our sample, we
use the masses derived from DIAD. For single objects that were detected in the
millimeter but either unable to be fit or not included in our sample, we use
the dust mass calculated from the flux measurement (as in Section 4.1). For
non-detections, we compute the upper limit fluxes as three times the
uncertainty plus any positive measured flux density, as in Barenfeld et al.
(2016); mass upper limits were calculated from these flux upper limits. The
L1641 survey targets were selected based on Herschel 70 $\mu$m detections,
which biases the sample towards brighter, more massive disks as noted by Grant
et al. (2021). To counteract this bias we include objects from the more
complete Survey of Orion Disks with ALMA (SODA) project (van Terwisga et al.,
2022). Even when accounting for the sample biases and non-detections, the
results reported above still hold: Figure 10 shows a decrease of disk mass
with age, and Ophiuchus and Corona Australis still appear to have lower disk
masses than expected from their ages.
To further explore the decrease in disk mass with time, we plot in Figure 11
the median disk mass for each region versus age of the region. Median values
are determined from the cumulative distribution functions shown in Figure 10
by taking the value of $M_{dust}$ at which 50% of disks have masses $\geq
M_{dust}$. We do not include Corona Australis, $\lambda$ Ori, or $\sigma$ Ori
since these regions have $<$ 50% detections in the millimeter. Following Testi
et al. (2022), we also show the trend $M_{dust}\propto t^{-1}$. Testi et al.
(2022) normalized this trend to the median dust mass in Ophiuchus. As we have
argued above, disks in Ophiuchus tend to have unusually low disk masses for
their age; we opt instead to scale the $M_{dust}\propto t^{-1}$ trend to the
weighted average median dust mass at 1.5 Myr (i.e., the weighted average of
the median values for Taurus, Cha II, and L1641). This trend agrees reasonably
well with our median disk masses and indicates that the disk dust mass will be
reduced to $\sim$ one-third of its initial amount after 3 Myr, which is
consistent with typical disk dissipation timescales (Mamajek, 2009; Ribas et
al., 2014).
Figure 10: Cumulative distribution functions of dust mass for each region. We
include dust masses from our SED modeling for objects in our sample, as well
as consistently-recalculated dust masses (see Section 4.1) for objects not
included in our sample (including non-detections) in order to more fairly
compare between regions. Figure 11: Median disk dust masses for regions with
$>$50% mm-flux detections, plotted versus age of the region. The black dashed
line shows the relationship $M_{dust}\propto t^{-1}$, scaled to the weighted
mean mass at 1.5 Myr.
#### 4.3.2 Dust Settling with Age
Figure 12 shows the settling parameter $\epsilon$ probability distribution
functions for each of the eleven regions as well as the overall distribution
for all regions together. Larger values of $\epsilon$ correspond to lower
degrees of settling. As for the disk dust mass distributions, these normalized
histograms were constructed from 10000 random $\epsilon$ values drawn from the
MCMC chain for each object.
The $\epsilon$ distributions for all regions are remarkably similar. Nearly
every region peaks at the lowest $\epsilon$ value allowed by our SED modeling,
though some regions have a second, smaller peak at higher $\epsilon$ values.
We note that $\epsilon$ is defined as the ratio between the dust-to-gas mass
ratio in the disk atmosphere and the global dust-to-gas mass ratio of the disk
(assumed to be 0.01). Lower values of $\epsilon$ thus correspond to dust-
depleted disk atmospheres caused by a high degree of settling. Our observed
trend indicates that significant amounts of dust settling has occurred in
these disks, regardless of their age. One may expect the degree of settling in
a disk to increase with the age of the disk. However, our result is in
agreement with recent studies, which have shown that disks show evidence of
dust settling even at young ages ($\sim$ 1Myr; Furlan et al., 2009; Lewis &
Lada, 2016; Grant et al., 2018; Rilinger & Espaillat, 2021).
Large dust grains are expected to settle to the disk midplane faster than
small dust grains, since larger grains decouple more easily from the gas in
the disk, and feel a drag force as they orbit the central star (Dullemond &
Dominik, 2004). The high degrees of settling that we find in young disks may
therefore indicate that the dust grains in these disks grew significantly in
the first $\sim$1 Myr. Alternatively, high degrees of settling may be the
result of low turbulence in these disks. Turbulence causes the dust and gas in
a disk to mix, which counteracts the settling process. If the gas density is
sufficiently low (Dullemond & Dominik, 2004), or the gas velocity in the disk
atmosphere is low (e.g., Ciesla, 2007; Flaherty et al., 2015, 2017),
turbulence is less efficient and thus dust settling can occur more easily. The
high degrees of settling we find in the disks studied here may therefore be
the result of rapid dust grain growth and/or lack of turbulence in the upper
layers of the disks.
Figure 12: Probability distribution functions of the dust settling parameter
$\epsilon$ for each region, and for all regions together (lower right panel).
The red dashed line in each panel represents the median $\epsilon$ value for
the overall sample ($\sim 10^{-3}$) and the blue dotted lines represent the
median $\epsilon$ values for each region.
## 5 Summary and Conclusion
Using the physically motivated DIAD models in conjunction with an ANN, we have
obtained SED fits for 338 disks around T Tauri stars. These disks are located
in the following eleven star-forming regions, ranging in age from $\sim$ 0.5 –
10 Myr old: Ophiuchus, Taurus, L1641, Cha II, Lupus, Cha I, IC 348, Corona
Australis, $\sigma$ Ori, $\lambda$ Ori, and Upper Scorpius. Our main results
are as follows:
1. 1.
We confirm that the previously-reported mm-flux-based masses are a factor of
1.5-5 lower than the masses derived from physical models, consistent with
results in Ballering & Eisner (2019); Ribas et al. (2020). Masses derived from
millimeter fluxes depend on the assumption that the disk is optically thin at
that wavelength; if the disk is optically thick, this assumption results in an
underestimate of the disk mass. Our SED models do not rely on this assumption,
and thus yield more accurate masses. The discrepancy between the two methods
is higher for the more massive disks when the disk temperature is scaled by
the stellar luminosity. We present two equations (Equations 8 and 9) to be
used for correcting millimeter flux-derived disk masses to account for this
effect.
2. 2.
We find $M_{dust}$ – $M_{*}$ relationships that are generally consistent with
those reported in Pascucci et al. (2016). Five of the star-forming regions
studied here (Taurus, Lupus, Cha I, $\sigma$ Ori, and Upper Sco) show
statistically significant (p $<$ 0.05), moderate correlations. Ophiuchus,
L1641, and IC348 do not have a statistically significant trend; Cha II, Corona
Australis, and $\lambda$ Ori have too few points to accurately assess their
$M_{dust}$ – $M_{*}$ relationships. We do not find any trend of this
relationship with age.
3. 3.
Brown dwarfs are generally consistent with the $M_{dust}$ – $M_{*}$
relationships observed for TTS in younger regions. In the 5-10 Myr-old Upper
Sco region, however, brown dwarf disks are more massive than predicted from
their host masses. This result may indicate that brown dwarf disks dissipate
more slowly than their higher-mass companions; photoevaporation is more
efficient in higher-mass stars, which may explain this trend.
4. 4.
We find clear evolution in the disk masses with time, with older regions
having lower disk masses than younger regions, in agreement with previous
studies (e.g., Barenfeld et al., 2016; van der Plas et al., 2016; van Terwisga
et al., 2022). As an important caveat, different sensitivities of ALMA surveys
of various star-forming regions may have biased portions of our sample towards
higher masses. Future, more sensitive surveys may reveal a larger population
of low-mass disks, especially in older regions.
5. 5.
The degree of dust settling appears to be consistent across age, with even the
youngest regions showing appreciable settling. This result agrees with Ribas
et al. (2017) and Grant et al. (2018), who also reported appreciable levels of
dust settling in regions as young as 1 Myr.
In conclusion, these results may help to ease the reported “missing” mass
problem. By assuming the disks to be optically thin, previous studies may have
underestimated the masses of disks around T Tauri stars; our physical models
are not limited by this assumption and yield masses that are more consistent
with observed planetary systems. Furthermore, the high degrees of dust
settling we find in disks of all ages may indicate that dust processing and
evolution happens quickly; thus planetary systems may also form earlier than
previously theorized.
We thank the referee for the careful reading of the manuscript and helpful
feedback. AMR thanks A. Meredith Hughes and Philip Muirhead for their
insightful comments and discussion. This work was funded by NASA ADAP
80NSSC20K0451. We acknowledge support from the NRAO Student Observing Support
program through award SOSPA8-007. Á.R. has been supported by the UK Science
and Technology research Council (STFC) via the consolidated grant ST/S000623/1
and by the European Union’s Horizon 2020 research and innovation programme
under the Marie Sklodowska-Curie grant agreement No. 823823 (RISE DUSTBUSTERS
project).This work has made use of data from the European Space Agency (ESA)
mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data
Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This paper utilizes the
D’Alessio Irradiated Accretion Disk (DIAD) code. We wish to recognize the work
of Paola D’Alessio, who passed away in 2013. Her legacy and pioneering work
live on through her substantial contributions to the field.
## Appendix A Agreement with Reported Literature Values
Here we present a comparison between the stellar values obtained from our SED
modeling and values reported in the literature. As described in Section 3.1.1,
we incorporate literature values as priors for stellar parameters when
literature values exist, and we ensure our $T_{*}$ and $R_{*}$ values are
consistent with stellar evolution models using MIST (Paxton et al., 2011,
2013, 2015; Dotter, 2016; Choi et al., 2016). For a given $M_{*}$ and
$Age_{*}$, $T_{*}$ and $R_{*}$ are calculated with MIST; these four values are
then used in our MCMC fitting process.
Literature values for $M_{*}$, $T_{*}$ and $R_{*}$ are taken from: Walter et
al. (1997); Luhman & Rieke (1999); López Martí et al. (2005); Wilking et al.
(2005); Lada et al. (2006); Cieza et al. (2007); Forbrich & Preibisch (2007);
Luhman et al. (2007); Muench et al. (2007); Luhman et al. (2008); Sicilia-
Aguilar et al. (2008); Meyer & Wilking (2009); Cieza et al. (2010); Hernández
et al. (2010); McClure et al. (2010); Bayo et al. (2011); Erickson et al.
(2011); Rigliaco et al. (2012); Andrews et al. (2013); Spezzi et al. (2013);
Alcalá et al. (2014); Herczeg & Hillenbrand (2014); Hernández et al. (2014);
Manara et al. (2015); Barenfeld et al. (2016); Manara et al. (2016a, b);
Pascucci et al. (2016); Alcalá et al. (2017); Ansdell et al. (2017); Grant et
al. (2018); Ruíz-Rodríguez et al. (2018); Cazzoletti et al. (2019); Ansdell et
al. (2020). Stellar temperatures are determined from spectral types; for
objects with reported spectral types but no reported temperatures, we used the
Pecaut & Mamajek (2013) scaling to convert the reported spectral types to
$T_{*}$ values. Stellar radii were either obtained via comparison to a
reference SED (Spezzi et al., 2008, 2013), or calculated from the stellar
luminosity and $T_{*}$. Luminosities were obtained by applying a bolometric
correction to the observed $J$ band magnitude.
In Figure A.1, we compare the previously reported $M_{*}$, $T_{*}$ and $R_{*}$
values to those obtained via our SED modeling. For all three parameters, we
find very strong agreement. We also present a comparison of the best-fit
parallaxes we obtain for each object versus the parallaxes reported by Gaia in
Figure A.2, which also shows a strong agreement. As mentioned in Section
3.1.1, Gaia parallaxes are included as a prior in our model, so a strong
correlation is expected.
Figure A.3 shows a comparison between our modeled mass accretion rates
($\dot{M}$) and $\dot{M}$ values reported in the literature (scaled to Gaia
distances). Literature $\dot{M}$ values are taken from Natta et al. (2006) for
Ophiuchus, Antoniucci et al. (2011) for Cha II, Alcalá et al. (2017) for
Lupus, Manara et al. (2017) for Cha I, Natta et al. (2014) for Sigma Ori, and
Manara et al. (2020) for Upper Sco. These works calculate the accretion
luminosity, and hence $\dot{M}$, from Br$\gamma$, Pa$\beta$ and other emission
line equivalent widths (Natta et al., 2006; Antoniucci et al., 2011; Natta et
al., 2014) or UV excess (Alcalá et al., 2017; Manara et al., 2017, 2020). We
note that the ANN was trained on $log_{10}$($\dot{M}$) values between -10 and
-6.5. Some objects have reported $\dot{M}$ values outside this range, though
these objects have $\alpha$ values that are well-constrained, so this does not
introduce a bias in our disk masses.
The $\dot{M}$ error bars from DIAD are generally much larger than those
obtained by measuring $\dot{M}$ with emission lines or UV excess. In principle
these two accretion rates do not necessarily need to match, since the mass
accretion rate onto the star could be different than in the disk. The mass
accretion rate onto the star could also be variable (e.g., Robinson &
Espaillat, 2019), by as much as two orders of magnitude (Claes et al., 2022),
while the mass accretion rate probed by DIAD would be an average that sets the
surface density profile in an alpha-disk. Finally, we note that chromospheric
emission can create noise in UV observations, making it difficult to derive
accurate $\dot{M}$ values below the noise threshold (Ingleby et al., 2011).
Manara et al. (2013) report a noise threshold of $\sim 10^{-10.5}\ \rm{\it
M}_{\odot}\,yr^{-1}{}$ for objects with similar $M_{*}$ and ages as the
objects presented here. Despite these challenges, we find a general agreement
between $\dot{M}$ values from our modeling and $\dot{M}$ values from the
literature above the noise threshold.
Figure A.1: Comparison between reported values and values obtained via our SED
modeling for $M_{*}$ (panel a), $T_{*}$ (panel b) and $R_{*}$ (panel c). The
black dotted line in each panel represents a one-to-one correlation. Our
modeled values for each of these parameters agree very well with previously
reported values. Figure A.2: Comparison between parallax values obtained via
our SED modeling and those measured by Gaia. The black dotted line represents
a one-to-one correlation. Our parallax values agree very well with previously
reported values. Figure A.3: Comparison between mass accretion rate values
obtained via our SED modeling and those reported in the literature. The black
dotted line represents a one-to-one correlation. The black dashed line denotes
the approximate threshold below which chromospheric activity interferes with
derivations of $\dot{M}$ in the literature (Manara et al., 2013).
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|
# The Force Balance of Electrons During Kinetic Anti-parallel Magnetic
Reconnection
J. Egedal H. Gurram S. Greess Department of Physics, University of
Wisconsin-Madison, Madison, Wisconsin 53706, USA W. Daughton A. Lê Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
###### Abstract
Fully kinetic simulations are applied to the study of 2D anti-parallel
reconnection, elucidating the dynamics by which the electron fluid maintains
force balance within both the electron diffusion region (EDR) and the ion
diffusion region (IDR). Inside the IDR, magnetic field-aligned electron
pressure anisotropy (${p_{e\parallel}}\gg{p_{e\perp}})$ develops upstream of
the EDR. Compared to previous investigations, the use of modern computer
facilities allows for simulations at the natural proton to electron mass ratio
$m_{i}/m_{e}=1836$. In this high-$m_{i}/m_{e}$-limit the electron dynamics
changes qualitatively, as the electron inflow to the EDR is enhanced and
mainly driven by the anisotropic pressure. Using a coordinate system with the
$x$-direction aligned with the reconnecting magnetic field and the
$y$-direction aligned with the central current layer, it is well-known that
for the much studied 2D laminar anti-parallel and symmetric scenario the
reconnection electric field at the $X$-line must be balanced by the $\partial
p_{exy}/\partial x$ and $\partial p_{eyz}/\partial z$ off-diagonal electron
pressure stress components. We find that the electron anisotropy upstream of
the EDR imposes large values of $\partial p_{exy}/\partial x$ within the EDR,
and along the direction of the reconnection $X$-line this stress cancels with
the stress of a previously determined theoretical form for $\partial
p_{eyz}/\partial z$. The electron frozen-in law is instead broken by pressure
tensor gradients related to the direct heating of the electrons by the
reconnection electric field. The reconnection rate is free to adjust to the
value imposed externally by the plasma dynamics at larger scales.
###### pacs:
## I Introduction
A key issue in plasma physics with importance to space weather modeling is
related to how electrons decouple from the motion of magnetic field lines in
small scale electron diffusion regions (EDRs). The phenomenon is known as
magnetic reconnection dungey:1953 , which permits the magnetic field topology
to rearrange, often accompanied by large scale release of stored magnetic
energy. Well-known examples include reconnection associated with solar flares
masuda:1994 and magnetic storms in the Earth’s magnetosphere mcpherron:1973 .
Despite substantial efforts including laboratory, spacecraft, numerical and
theoretical investigations zweibel:2016 , a detailed understanding of the
electron dynamics within the EDR is still not fully developed. In particular,
a full mathematical description of how the electron frozen-in condition is
violated (permitting ${\bf E}+{\bf u}_{e}\times{\bf B}$ to be finite within
the EDR) has still not been reached.
Reconnection is observed in many different environments and geometries. For
the Earth’s magnetotail, symmetric and anti-parallel reconnection (where the
opposing inflow regions have similar strength magnetic fields but are
oppositely directed) is of particular importance torbert:2018 . Open questions
are concerned with the extent to which 3D effects fundamentally alter the
structure of the reconnection region. For example, there is no doubt that
reconnection current layers in the Earth’s magnetotail only have finite
lengths and therefore must be 3D in nature. However, observations from the
Magnetospheric Multiscale (MMS) mission demonstrate how the widths of
reconnection layers typically approach the small length scales associated with
the local electron orbit width. The length scales perpendicular to a
particular reconnection current layer are therefore usually much shorter than
the length scale along the current layer, and a range of observations further
suggest that the local reconnection dynamics are well captured by 2D laminar
models egedal:2018 ; egedal:2019 ; greess:2021 ; schroeder:2022 .
For laminar and near steady state reconnection the force balance of the
electron fluid is described by the generalized Ohm’s law, which similar to the
Navier-Stokes equation for a regular fluid includes pressure and inertial
effects:
${\bf E}+{\bf u}_{e}\times{\bf B}+\frac{\displaystyle 1}{\displaystyle
ne}\nabla\cdot{\bf p}_{e}+\frac{\displaystyle m_{e}}{\displaystyle e}{\bf
u}_{e}\cdot\nabla{\bf u}_{e}=0\quad.$ (1)
Using a range of kinetic simulations, in the present paper we will discuss how
the various terms of Eq. (1) become important within separate spatial areas of
the IDR and EDR. Compared to similar investigations a decade or more ago
hesse:1999 ; daughton:2006 ; drake:2008 , a main difference is that modern
computing facilities now permit routine fully kinetic simulations at the full
proton to electron mass ratio $m_{i}/m_{e}=1836$. The present analysis reveals
that for simulations where this ratio approaches its natural value, the main
assumption of whistler reconnection drake:2008 , that ${\bf E}+{\bf
u}_{e}\times{\bf B}\simeq 0$ within the IDR, is not accurate. Instead, it is
found that $|E_{\mbox{\scriptsize rec}}|\ll|\nabla\cdot{\bf p}_{e}/(ne)|$ such
that ${\bf u}_{e}\times{\bf B}\simeq-\nabla\cdot{\bf p}_{e}/(ne)$ becomes the
main driver of the Hall magnetic field perturbation. In fact, the familiar
assumption of $u_{\mbox{\scriptsize in}}=E_{\mbox{\scriptsize
rec}}/B_{\mbox{\scriptsize rec,up}}$ significantly underestimates the inflow
speed of the electron fluid into the EDR, with important implications for
previous theoretical work as well as the analysis of spacecraft data. Here,
$E_{\mbox{\scriptsize rec}}$ is the reconnection electric field and
$B_{\mbox{\scriptsize rec,up}}$ is the in-plane magnetic field just upstream
of the EDR.
The main topic of the paper, however, is the development of a theory to
explain the terms of Eq. (1) which break the electron frozen-in condition at
the very center of the EDR. The study reveals the importance of the electron
anisotropy upstream of the EDR in driving the current within the EDR and
cancelling an off-diagonal stress term identified in previous work
kuznetsova1998kinetic ; hesse:1999 . Ultimately, the off-diagonal stress of
${\bf p}_{e}$ that is responsible for breaking the frozen-in condition is
generated by $E_{\mbox{\scriptsize rec}}$ itself. Thus, the theory suggests
that the electron fluid does not represent an obstacle (or bottleneck) for
reconnection, which may then proceed at the rate imposed by dynamics external
to the EDR stanier:2015 ; liu:2017 .
The paper is organized as follows: In Section II we discuss the formation and
role of electron pressure anisotropy within the IDR inflow regions, whereas
Section III examines the length scales characterizing the EDR. A detailed
account of the electron momentum balance at the $X$-line region is provided in
Section IV, and the paper is summarized and concluded in Section V.
## II Formation of electron pressure anisotropy within the reconnection
region
### II.1 Setup of kinetic simulations
The 2D kinetic simulations applied in our study were implemented for anti-
parallel Harris sheet reconnection using the code VPIC bowers:2009 with open
boundary conditions daughton:2006 . The initial magnetic field and the density
are $B_{x}=B_{0}\tanh(z/\lambda)$ and
$n=n_{0}\cosh^{2}(z/\lambda)+n_{\infty}$, respectively, where
$\lambda=0.5d_{i0}=0.5\sqrt{\varepsilon_{0}m_{i}c^{2}/n_{0}e^{2}}$. Other
parameters are initial uniform temperatures with $T_{i\infty}/T_{e\infty}=5$,
$\omega_{pe0}/\omega_{ce0}=2$, and $\sim 400$ particles per species per cell.
The study incorporates 35 separate simulation runs implementing a matrix of 5
values for $m_{i}/m_{e}$ and 7 values for the normalized upstream electron
pressure, $\beta_{e\infty}$. To be more specific, we use mass ratios of
$m_{i}/m_{e}\in\\{100,200,400,800,1836\\}$, and the background density
$n_{\infty}$ is varied so the initial upstream electron beta is
$\beta_{e\infty}=2\mu_{0}n_{\infty}T_{e\infty}/B_{0}^{2}=2^{-k}$ with
$k\in\\{1,2,\dots,7\\}$. In VPIC “natural” units the systems are implemented
using $m_{e}=1$, $d_{e0}=1$, $c=1$, $n_{0}=1$, $B_{0}=1/2$,
$T_{e\infty}=1/48\,m_{e}c^{2}$, and
$n_{\infty}/n_{0}=(1+T_{i\infty}/T_{e\infty})2^{-k}$. For runs with
$m_{i}/m_{e}\leq 400$ the domains were $4096\times 4096$ cells, corresponding
to $50d_{i0}\times 50d_{i0}$, while runs at $m_{i}/m_{e}\geq 400$ were carried
out at system sizes of $800d_{e0}\times 800d_{e0}$. Here, again, $d_{i0}$ and
$d_{e0}$ are the ion and electron skin-depths, respectively, based on
$n_{0}=1$. Note that $d_{e0}$ is defined with respect to the central Harris
sheet density, $n_{0}$. Meanwhile, the upstream electron skin-depth becomes
central to our analysis below and can be expressed as
$d_{e\infty}=d_{0}/\sqrt{n_{\infty}/n_{0}}$. More details about the
similarities and differences between the individual simulation runs are
provided in Section III, where the length of the EDR electron jets is
examined.
### II.2 Formation of electron pressure anisotropy in reconnection inflows
Figure 1: VPIC simulations profiles of $B/B_{\infty}$, $T_{e\|}/T_{e\perp}$,
$|{\bf u}_{e}|/v_{te}$, and $B_{y}/B_{\infty}$, obtained with
$m_{i}/m_{e}=1836$ and $\beta_{e\infty}=2^{-4}=0.0625$. The arrows in b)
represent the enhanced electron flow provided by ${\bf J}_{\mbox{\scriptsize
extra}}=[({p_{e\parallel}}-{p_{e\perp}})/B]{\bf b}\times\mathbf{\kappa}$.
In the inflow regions the electrons follow the magnetic flux-tubes in their
convection toward the EDR, while the ions are unmagnetized. The ions by their
inertia decouple from the motion of the magnetic field, and they dictate a
near uniform plasma density within the region. In turn, the electrons respond
strongly to match this uniform density and maintain quasi-neutrality. As
illustrated in Fig. 1(a), the declining magnetic field strength $|{\bf B}|$
causes the widths of the magnetic flux-tubes to expand as the EDR is
approached. To avoid a reduction in the electron density, field-aligned
electric fields $E_{\|}$ develop egedal:2009pop , compressing the range of the
parallel motion for trapped electrons. This boosts the electron density such
that quasi-neutrality (i.e. $n_{e}\simeq n_{i}$) is maintained. The profiles
of $E_{\|}$ in many cases trap all thermal electrons, limiting thermal heat
conduction and yielding a regime that differs significantly from the standard
Boltzmann regime where $T_{e}$ is constant. Rather, the reduction in $B$ and
perpendicular expansion of the flux-tubes results in $T_{e\perp}$ cooling,
whereas the parallel compression by $E_{\|}$ to maintain quasi-neutrality
yields $T_{e\|}$ heating le:2009 ; le:2010grl ; egedal:2013 ; wetherton:2019 .
Generally it is found that runs with $\beta_{e\infty}^{2}m_{i}/m_{e}\gtrsim 1$
are characterized by a regime of double adiabatic electron dynamics, where at
the edge of the EDR the main electron pressure components are well described
by the CGL-limit chew:1956 where ${p_{e\parallel}}\propto n^{3}/B^{2}$ and
${p_{e\perp}}\propto nB$. For $\beta_{e\infty}^{2}m_{i}/m_{e}\lesssim 1$
electron holes and double layers yield even stronger ${p_{e\parallel}}$
heating egedal:2013 ; egedal:2015 . The effects of the anisotropic heating are
evident in the profile in Fig. 1(b), where $T_{e\|}/T_{e\perp}\simeq 4$ is
observed just upstream of the EDR, and the marginal electron firehose
condition is approached, ${p_{e\parallel}}-{p_{e\perp}}\simeq B^{2}/\mu_{0}$.
Within the EDR the adiabatic invariance of the electron magnetic moments,
$\mu=m{v_{\perp}}^{2}/(2B)$, break, leading to isotropization and pitch angle
mixing. As a result, the electron pressure is approximately isotropic in the
reconnection exhaust le:2013 ; egedal:2016b .
The physical mechanisms which underpin the CGL-scalings can be understood
through relatively simple arguments applicable to the electrons just upstream
of the EDR, where the described trapping and parallel compression dynamics are
sufficiently strong that they dominate the properties of the electron fluid.
The area of a given flux-tube scales as $1/B$, and the total number of
particles in the flux-tube section (of length $l$) therefore scales as
$N\propto nl/B$. Particle conservation then requires that $l\propto B/n$.
Next, similar to Fermi heating, the conservation of the parallel action for
each trapped electron requires that $lv_{\parallel}$ is constant such that
$v_{\parallel}\propto n/B$, yielding $T_{e\|}\propto v_{\parallel}^{2}\propto
n^{2}/B^{2}$. Furthermore, because $\mu=m{v_{\perp}}^{2}/(2B)$ is conserved it
is clear that ${v_{\perp}}^{2}\propto B$, such that for this trapped electron
population $T_{e\perp}\propto B$. It then follows that
${p_{e\parallel}}=nT_{e\|}\propto n^{3}/B^{2}$ and
${p_{e\perp}}=nT_{e\perp}\propto nB$ coinciding with the CGL-scaling laws
chew:1956 . A more accurate and rigorous model is provided in le:2009 ;
egedal:2013 taking into account that not all electrons become trapped. For
the present paper, however, it is sufficient to assume that within the inflow
regions the electron pressures approximately follow the aforementioned CGL-
scaling laws.
### II.3 The reconnection rate from the perspective of the electrons
In Section III we document how the size of the EDR normalized by $d_{e\infty}$
remains approximately constant for varying values of $m_{i}/m_{e}$. Meanwhile,
the size of the IDR scales with $d_{i\infty}=d_{e\infty}\sqrt{m_{i}/m_{e}}$,
and from the perspective of the electrons the size of the ion diffusion region
thus increases by a factor of $\sqrt{m_{i}/m_{e}}$. Likewise, relative to the
time scale of the electron motion, the increasing inertia of the ions will
slow the rate of reconnection. As a dimensionless measure of the reconnection
electric field relevant to the electron orbit dynamics we introduce
$\hat{E}_{\mbox{\scriptsize rec}}=ed_{e\infty}E_{\mbox{\scriptsize
rec}}/T_{e\infty}$. This quantity represents the temperature-normalized energy
gain an electron will acquire when traveling $d_{e\infty}$ in the direction of
$E_{\mbox{\scriptsize rec}}$. Consistent with external MHD scale constraints
for system sizes larger than $10d_{i}$ liu:2017 , in the present and previous
kinetic simulations the absolute reconnection rate obeys $E_{\mbox{\scriptsize
rec}}\simeq 0.1v_{A}B$, where the relevant Alfvén speed and $B$ are evaluated
$1d_{i}$ upstream of the EDR shay:2004 . Corresponding to the shaded region in
Fig. 2(a), it then follows that $\hat{E}_{\mbox{\scriptsize
rec}}\propto\sqrt{m_{e}/(m_{i}\beta_{e\infty}^{2})}$, where reduced values of
$m_{i}/m_{e}$ and low $\beta_{e\infty}$ impose the largest values of
$\hat{E}_{\mbox{\scriptsize rec}}$.
In Fig. 2(b) we show the profiles of $T_{e\|}/T_{e\infty}$ and
$T_{e\perp}/T_{e\infty}$ corresponding to values in the various runs recorded
just upstream of the electron diffusion region. Consistent with the discussion
in Section II.B, for simulations within the double adiabatic regime
($\beta_{e\infty}^{2}m_{i}/m_{e}\gtrsim 1$) marked by the full lines, we
observe that both of these profiles are largely independent of $m_{i}/m_{e}$.
The normalized electron pressure anisotropy
$(p_{e\|}-p_{e\perp})/(n_{\infty}T_{e\infty})$ is then also independent of
$m_{i}/m_{e}$, and given the scaling of $\hat{E}_{\mbox{\scriptsize
rec}}\propto\sqrt{m_{e}/m_{i}}$ the forces associated with the pressure
anisotropy become most significant when compared to
$e\hat{E}_{\mbox{\scriptsize rec}}$ at large value of $m_{i}/m_{e}$. In fact,
as will be demonstrated below, the thermal forces of the electron pressure
anisotropy strongly dominate the force balance of the electrons for
$m_{i}/m_{e}=1836$, rendering the electron dynamics of the IDR and EDR
significantly different when compared to reduced fluid models invoking
isotropic electron pressure.
Figure 2: Obtained from a matrix of 35 kinetic simulations, profiles as
functions of $\log_{10}(\beta_{e\infty})$ are shown for a range of parameters
as marked in the panels. As indicated in b) the line-type denotes the value of
$m_{i}/m_{e}$. In a) $\hat{E}_{\mbox{\scriptsize rec}}$ is normalized by
$T_{e\infty}/(ed_{e\infty})$, and the magenta shaded region is the expected
range based on $E_{\mbox{\scriptsize rec}}=\alpha v_{A}B_{\mbox{\scriptsize
rec}}$, with $\alpha=0.1$ and $100\leq m_{i}/m_{e}\leq 1836$ (see text). The
wider yellow shaded region correspond to $0.08\leq\alpha\leq 0.12$. In b) the
regions shaded in magenta are obtained from the CGL-limit of the Lê-2009
Equations of State le:2009 . The dashed lines mark runs in the regime
$\beta_{e\infty}<\sqrt{m_{i}/m_{e}}$ where additional $T_{e\|}$ heating
typically is observed egedal:2015 ; le:2016twostage .
### II.4 Dominant role of $p_{e\|}\gg p_{e\perp}$ for large $m_{i}/m_{e}$
Figure 3: Considering the same simulation as in Fig. 1 ($m_{i}/m_{e}=1836$ and
$\beta_{e\infty}=2^{-4}$), color contours are presented for the LHS and RHS of
Eq. (3), respectively. The EDR is outlined by the cyan rectangle. For the
extended regions where $({\bf J}\times{\bf B})_{y}/(n_{e0}E_{y0})\gtrsim 4$
the electron fluid is moving significantly faster than the magnetic field.
This motion is driven by electron pressure anisotropy such that $|{\bf
J}\times{\bf B}|\simeq|\nabla\cdot{\bf p}_{e}|\gg en_{0}E_{\mbox{\scriptsize
rec}}$.
In the region outside the EDR where the electrons are well magnetized, the
electron pressure anisotropy drives additional currents, ${\bf
J}_{\mbox{\scriptsize extra}}=[({p_{e\parallel}}-{p_{e\perp}})/B]{\bf
b}\times\mathbf{\kappa}$, which are not observed for isotropic pressure. Here
$\mathbf{\kappa}=({\bf b}\cdot\nabla){\bf b}$ is the curvature vector which
becomes large where the magnetic field lines have strong curvature. In Fig.
1(b) the arrows are proportional to $-{\bf J}_{\mbox{\scriptsize extra}}$
corresponding to a flow boosting the perpendicular drift of electrons into the
EDR.
To quantify the relative importance of the flow by ${\bf J}_{\mbox{\scriptsize
extra}}$, in Fig. 3 we provide additional profiles for the simulation of Fig.
1, here focusing on the force balance in the $y$-direction in the vicinity of
the EDR. Within the IDR we generally have $({\bf u}_{e}\times{\bf
B})_{y}\gg({\bf u}_{i}\times{\bf B})_{y}$ and we thus introduce the
approximation that $(-en_{e}{\bf u}_{e}\times{\bf B})_{y}\simeq({\bf
J}\times{\bf B})_{y}$. Furthermore, it turns out that generally $\left|({\bf
J}\times{\bf B})_{y}\right|\gg\left|(enm_{e}{\bf u}_{e}\cdot\nabla{\bf
u}_{e})_{y}\right|$, and the $y$-component of the electron inertia can for the
present analysis be neglected. With these approximation the $y$-component of
the generalized Ohm’s law in Eq. (1) then becomes
$-en_{e}E_{y}+({\bf J}\times{\bf B})_{y}=(\nabla\cdot{\bf p}_{e})_{y}\quad.$
(2)
For the time point considered in Fig. 3 the reconnection geometry and rate is
approximately steady. Furthermore, the density profile in this inner region is
approximately uniform at a density we denote as $n_{e0}$. It then follows that
$n_{e}E_{y}\simeq n_{e0}E_{y0}$, where $E_{y0}$ is the value of $E_{y}$
observed at the $X$-line. After dividing Eq. (2) by $n_{e0}E_{y0}$ we obtain:
$\frac{\displaystyle({\bf J}\times{\bf B})_{y}}{\displaystyle
n_{e0}E_{y0}}\simeq 1+\frac{\displaystyle(\nabla\cdot{\bf
p}_{e})_{y}}{\displaystyle n_{e0}E_{y0}}\quad.$ (3)
The LHS (left hand side) of Eq. (3) is shown in Fig. 3(a), while the RHS is
shown in Fig. 3(b). The fact that these profiles are near-identical validates
the approximations introduced above.
From Fig. 3(a) it is apparent that $({\bf J}\times{\bf B})_{y}$ is up to 5
times larger that $n_{e0}E_{y0}$ in a significant fraction of the cross-
section. Thus, rather than the $en_{e}E_{y}\simeq({\bf J}\times{\bf B})_{y}$
relation applied in whistler-wave reconnection models, we see that
$|en_{e}E_{y}|\ll|({\bf J}\times{\bf B})_{y}|\simeq|(\nabla\cdot{\bf p})_{y}|$
throughout the majority of the ion and electron diffusion regions. This
strongly enhanced inflow is provided by $p_{e\|}\gg p_{e\perp}$ through the
drifts associated with ${\bf J}_{\mbox{\scriptsize extra}}$. In other words,
while $E_{y}$ by Faraday’s law is fundamental for moving the magnetic flux
across the reconnection region, Fig. 3 reveals how the $(\nabla\cdot{\bf
p}_{e})_{y}$-term is responsible for driving the electrons into the EDR and
through the elongated jets of the EDR at high rates, which for the present run
are up to 5 times the speed of the magnetic field line motion.
The electric field in the local rest frame of the electron fluid is given by
${\bf E}^{\prime}={\bf E}+{\bf v}_{e}\times{\bf B}$. In resistive Hall-fluid
models with isotropic pressure the energization measure $-en_{e}{\bf
v}_{e}\cdot{\bf E}^{\prime}$ is generally positive, such that magnetic energy
is dissipated and energizes the electron fluid. This observation is altered by
the presence of the strong electron pressure anisotropy. For example, within
the regions in Fig. 3(a) where $({\bf J}\times{\bf
B})_{y}/(en_{e}E_{y0})\simeq 4$ it follows that $E_{y}^{\prime}=E_{y}+({\bf
v}_{e}\times{\bf B})_{y}\simeq-3E_{y}$. Thus, in these regions where the
electrons “run ahead” of the magnetic field lines $-en{\bf v}_{e}\cdot{\bf
E}^{\prime}$ may become negative, such that energy from the electrons is given
back to the magnetic field. Such regions, known as generator regions, have
been directly observed by the MMS mission payne:2021 . Furthermore, given the
strong electron flow generated by $\nabla\cdot{\bf p}$ the aspects ratio of
the electron diffusion region (i.e. the aspect ratio of the cyan boxes in Fig.
3) is not necessarily an accurate measure of the normalized reconnection rate,
and may be important to newly develop methods for analysis of spacecraft data
burch:2020 .
### II.5 Force balance along the length of the EDR and associated scaling
laws for the electron pressure anisotropy
a-c)
Figure 4: Profiles of $u_{ex}/v_{te}$ for the full simulation domains for
three separate runs at $\beta_{e\infty}=2^{-4}$ and $m_{i}/m_{e}=800$. The run
in a) has the “standard” size described in Section II.A, whereas the runs in
b,c) are for systems enhanced by factors $2\times$ and $4\times$,
respectively. Zoomed-in views of the EDRs are shown in d-f), whereas g-i) show
cuts of $u_{ex}$ and $u_{ey}$ for $z=0$, from which the length scale $l_{u}$
of current layer rotation is determined.
Our analysis relies strongly on previous results regarding the overall force
balance of the EDR le:2010grl ; egedal:2013 ; montag:2020 . For the
convenience of the reader we here provide a short summary of how the upstream
electron pressure anisotropy impacts the structure of the EDR. Also, in
Appendix A we provide additional mathematical considerations to demonstrate
how the upstream electron pressure anisotropy drives the current jets within
the EDR.
Consistent with Fig. 3, in Ref. le:2010grl it was demonstrated that the two
dominant terms of Eq. (1) along the EDR are ${\bf u}_{e}\times{\bf B}$ and
$(\nabla\cdot{\bf p}_{e})/ne$ le:2010grl . Expressed in terms of the Maxwell
stress tensor, ${\bf T}=({\bf B}{\bf B}-B^{2}{\bf I})/(2\mu_{0})$, we have
$-en{\bf u}_{e}\times{\bf B}\simeq{\bf J}\times{\bf B}=\nabla\cdot{\bf T}$,
and momentum balance then requires that $\nabla\cdot({\bf p}_{e}-{\bf
T})\simeq 0$. Considering the area encircled by the red square in Fig. 1(c),
it can then be shown by the divergence theorem le:2010grl ; montag:2020 that
at the edge of the EDR the electron firehose condition must be approximately
satisfied,
${p_{e\parallel}^{\mbox{\scriptsize edge}}}-{p_{e\perp}^{\mbox{\scriptsize
edge}}}\simeq B_{H}^{2}/\mu_{0}\quad,$ (4)
where $B_{H}$ represents the magnetic field strength just upstream of the EDR
(this field rotates into the Hall magnetic field explaining the subscript
$H$). For example, for the red square in Fig. 1(c) the magnetic tension acting
on the sides of the fluid element (corresponding to the integral over the
element of the $-en{\bf u}_{e}\times{\bf B}$ force) is offset by pressure
anisotropy. Again, this is discussed in more detail in Appendix A.
We can derive important scaling laws for the level of pressure anisotropy for
locations just upstream of the EDR. Using that
${p_{e\parallel}^{\mbox{\scriptsize edge}}}\gg{p_{e\perp}^{\mbox{\scriptsize
edge}}}$ with the CGL-limits the maximal pressure differential
${\Delta p_{e\|\perp}^{\mbox{\scriptsize
max}}}\equiv\mbox{max}({p_{e\parallel}}-{p_{e\perp}})={p_{e\parallel}^{\mbox{\scriptsize
edge}}}-{p_{e\perp}^{\mbox{\scriptsize edge}}}\quad,$ (5)
adheres to the asymptomatic scaling ${\Delta p_{e\|\perp}^{\mbox{\scriptsize
max}}}\simeq{p_{e\parallel}^{\mbox{\scriptsize
edge}}}\simeq(\pi\tilde{n}^{3}/(6\tilde{B}_{H}^{2}))\beta_{e\infty}B_{\infty}^{2}/(2\mu_{0})$.
The factor $\pi/6$ is part of the Lê 2009 Equations of State le:2009 . The
firehose condition in Eq. (4), representing the dominant momentum balance
condition of the EDR, may then be expressed as
$\frac{\displaystyle B_{H}}{\displaystyle
B_{\infty}}=\frac{\displaystyle\mu_{0}K}{\displaystyle
2B_{\infty}}=\left(\frac{\displaystyle\pi\tilde{n}^{3}\beta_{e\infty}}{\displaystyle
12}\right)^{1/4}\quad.$ (6)
Here $K$ is the total current integrated across the layer. $\tilde{n}\simeq 1$
is the local electron density normalized by the upstream value $n_{\infty}$
(and similar for $\tilde{B}_{H}$ used above) le:2010grl . Eq. (6) is
significant as it demonstrates how the current per unit length of the EDR,
$K$, is mainly controlled by the external electron pressure anisotropy.
The above result is in contrast to predictions of two-fluid models with
isotropic pressure where ${\bf u}_{e}\times{\bf B}$ is balanced by resistive
terms, ${\bf E}$, and/or $(m_{e}/e){\bf u}_{e}\cdot\nabla{\bf u}_{e}$. It
should also be noted that the half-width $\delta\simeq 2d_{e\infty}$ of the
EDR region current layer is set by the meandering orbit width
roytershteyn:2013 . From Ampère’s law it then follows that
$B_{H}=\mu_{0}en_{e}u_{e}\delta\simeq 2\mu_{0}en_{e}u_{e}d_{e\infty}$, such
that $u_{e}=B_{H}/(2en_{e}u_{e}d_{e\infty})\simeq
B_{H}/(2\sqrt{\mu_{0}n_{e}m_{e}})=v_{Ae}/2$. Thus, $u_{e}$ must approach the
corresponding value of $\sim v_{Ae}/2$, simply because of the narrow width of
the current layer $\delta\simeq 2d_{e\infty}$. This analysis is also
applicable to the Harris sheet type current layer and the condition
$u_{e}=v_{Ae}/2$ is therefore reflective of the underlying meandering orbit
dynamics setting the narrow width of the electron flow region
roytershteyn:2013 . The EDR current layer rotates in the $xy$-plane hesse:2008
; le:2010pop , and yields the $B_{y}$ perturbations shown in Fig. 1(d)
hesse:2008 ; le:2010grl . The above results have also been applied in deriving
scaling laws for the absolute electron heating within the reconnection inflow
and EDR le:2016twostage .
## III Length scales of the EDR
In resistive fluid models the aspect ratio of the EDR controls the rate of
magnetic reconnection parker:1957 . This follows from the condition of mass
continuity and the assumption that the electron fluid is frozen-in to the
magnetic field (i.e. ${\bf E}+{\bf v}_{e}\times{\bf B}=0$) outside the EDR.
However, as discussed above and documented in Fig. 3, in our kinetic runs at
large values of $m_{i}/m_{e}$ the electrons decouple from the magnetic field
within the IDR and “run ahead” of the magnetic field into the EDR. This simple
observation has important implications, as it follows that for the kinetic
limit with $p_{e\|}\gg p_{e\perp}$ upstream of the EDR, the rate of
reconnection is then no longer tied to the aspect ratio of the EDR.
Below we introduce the length $l_{u}$, which is a measure of the spatial scale
at which the out-of-plane EDR electron current at the $X$-line rotates into
the exhaust direction. Our analysis show that this length scale is shortened
by the enhanced inflow of ${\bf J}_{\mbox{\scriptsize extra}}$ and we observe
that $l_{u}\simeq 20\pm 7d_{e\infty}$ for all the simulation parameters
considered with $\beta_{e\infty}\leq 2^{-3}$
### III.1 Numerical evidence that $l_{u}$ does not scale with simulation
system size
In Ref. daughton:2006 , which employed relatively low mass ratio
$m_{i}/m_{e}=25$, it was observed that the length of the EDR increases with
the size of the numerical simulation domain, and evidence was presented that
the long electron current layers of the EDR could act as a bottleneck for
reconnection. These results do not to apply to our present simulations where
$m_{i}/m_{e}$ approaches its natural value of 1836. For example, in Fig. 4(a)
profiles of $u_{ex}$ are shown for a simulation at $m_{i}/m_{e}=800$ and
$\beta_{e\infty}=2^{-4}$ for the “standard” setup described in Section II.A,
whereas Figs. 4(b,c) apply domains factors of $2\times$ and $4\times$ larger,
respectively. Zoomed-in views of the EDRs for the three runs are shown in
Figs. 4(d-f), where we notice how the $2\times$ and $4\times$ runs have
similarly sized EDRs. This indicates that for simulations at large
$m_{i}/m_{e}$ implemented in large domains, the length of the EDR becomes
independent of the system size.
For the analytical analysis of the EDR the following length scale $l_{u}$
becomes essential to the presented theory for the force balance of the
electron fluid. Here
$l_{u}\equiv\left|\frac{u_{ey}}{du_{ex}/dx}\right|_{\mbox{\scriptsize
X-line}},$ (7)
represents a measure of the spatial distance that characterizes the rotation
of the EDR electron current in the $xy$-plane . Figs. 4(g-i) provide the
geometric interpretation of this quantity, which is observed to be on the
order of $20d_{e}$ for the three runs. Below we show how $l_{u}$ is
significantly shorter than the full length $l_{\mbox{\scriptsize jet}}$ of the
EDR for many runs in our study, consistent with previously observed two-scale
EDR structures karimabadi:2007 ; shay:2007 .
### III.2 Numerical results for $l_{u}$ over the full
$(\beta_{e\infty},m_{i}/m_{e})$–matrix of kinetic runs
Figure 5: Color contours of $u_{ex}$ for the matrix of numerical simulations
applied in the study, $m_{i}/m_{e}\in\\{100,200,400,800,1836\\}$ and
$\beta_{e\infty}=2^{-k}$ with $k\in\\{1,2,\dots,7\\}$. When falling within the
domains considered, the ends of the jets are marked by the red stars
corresponding to $|u_{ex}|$ dropping to 60% of its peak value. The similarly
defined width of the $u_{ey}$ out-of-plane electron drift is marked in each
panel at $x=0$. Figure 6: Recorded values of $l_{u}$ and
$l_{\mbox{\scriptsize jet}}$. In b) the dashed lines indicate runs in the non-
adiabatic regime with enhanced ${p_{e\parallel}}$-heating observed for
$\beta_{e\infty}^{2}m_{i}/m_{e}\lesssim 1$. Evidently, this regime is also
characterized by an enhancement in the length of the EDRs (while $l_{u}$
remains near constant).
Fig. 5 provides zoomed-in views of the EDRs recorded in the full set of runs
applied in our study. As is typical for kinetic simulations of reconnection
based on the Harris sheet geometry, after fast reconnection has commenced the
reconnection rate initially increases and then reduces to a near steady state.
Each of the profiles considered here correspond to times $t\simeq
70\Omega_{ci}^{-1}$ during these later intervals of mostly steady state
reconnection geometries. For $\beta_{e\infty}\geq 2^{-2}$ elongated EDRs are
mostly observed $l_{\mbox{\scriptsize jet}}>100d_{e\infty}$. Here the out-of-
plane electron flows are largely provided by the diamagnetic drifts similar to
those of the Harris sheet. Meanwhile, for $\beta_{e\infty}\leq 2^{-3}$, the
force balance condition of the EDR as expressed in Eq. (6) requires current
densities boosted beyond those provided by the diamagnetic effect, and the
EDRs acquire a shorter length. The marked ends of the jets are defined by the
locations where $|u_{ex}|$ has fallen to 60% of its respective peak value. For
the adiabatic regime ($\beta_{e\infty}^{2}m_{i}/m_{e}\gg 1$), the lengths of
the electron diffusion region are consistently observed to be
$l_{\mbox{\scriptsize jet}}\simeq 20d_{e\infty}$. For the regime
$\beta_{e\infty}^{2}m_{i}/m_{e}\ll 1$ of enhanced ${p_{e\parallel}}$
energization egedal:2015 , a sharp transition occurs where the EDRs are
characterized by longer outflow jets.
In Fig. 6 we show the values of $l_{u}$ and $l_{\mbox{\scriptsize jet}}$ for
all the runs applied in the study. The runs with $\beta_{e\infty}\leq 2^{-3}$
(most relevant to the analysis in Section IV) are all characterized by
$l_{u}\simeq 18d_{e\infty}$. In Fig. 6(b) the dashed lines represent runs in
the aforementioned regime $\beta_{e\infty}^{2}m_{i}/m_{e}>1$ of enhanced
${p_{e\parallel}}$ heating, and it is evident that this regime is
characterized by larger values of $l_{\mbox{\scriptsize jet}}$.
### III.3 How continuity of electrons in the EDR sets the $l_{u}$ length
scale
Directly related to ${\bf J}_{\mbox{\scriptsize extra}}$, we find that $l_{u}$
is shortened by the enhancement of the electron flow into the EDR. Let
$u_{ey0}$ be the flow in the $y$-direction at the $X$-line; given the
definition of $l_{u}$ the flow in the $x$-direction is then
$u_{ex}=u_{ey0}x/l_{u}$. For a uniform inflow velocity $u_{ez}$ mass
continuity requires that
$nu_{ez}x=nu_{ex}\delta=nu_{ey0}x\delta/l_{u}$
such that
$l_{u}=\frac{\displaystyle u_{ey0}\delta}{\displaystyle
u_{ez}}=\frac{\displaystyle B_{H}}{\displaystyle en\mu_{0}u_{ez}}\quad.$
Meanwhile, the strong parallel streaming of electrons in toward the EDR causes
the inflow speed of the electrons to increase. Empirically we find
$u_{ez}\simeq 4E_{y}/B_{H}\quad.$
This enhancement of the inflow speed above $E_{y}/B_{xz}$ is caused by the
inflow pressure anisotropy through the term ${{\bf
J}_{\perp}}_{\mbox{\scriptsize extra}}$, and is consistent with the profiles
in Figs. 10(a,b).
The reconnection electric field, $E_{y}$ follows the scaling laws of Ref.
shay:2004 where
$E_{y}\simeq 0.1v_{A}B_{1di}=\frac{\displaystyle
0.1B_{1di}^{2}}{\displaystyle\sqrt{\mu_{0}nm_{i}}}\quad.$
It then follows that
$l_{u}=\frac{\displaystyle B_{H}^{2}}{\displaystyle
0.4B_{1di}^{2}}\sqrt{\frac{\displaystyle m_{i}}{\displaystyle
m_{e}}}d_{e}\quad.$
Empirically, the magnetic field 1$d_{i}$ upstream of the EDR is approximately
$B_{1di}\simeq B_{\infty}/2$. Furthermore, due to non-adiabatic effects
$B_{H}$ depends on $m_{i}/m_{e}$ (see Fig. 9(b)), and the scaling is
consistent with $B_{H}^{2}\sqrt{m_{i}/m_{e}}/(0.4B_{1di}^{2})\simeq 15$ for
all the runs. In turn, this is in reasonable agreement with the observation
that $l_{u}\simeq 18d_{e}$ in Fig. 6(a).
## IV Breaking the frozen-in law at the reconnection $X$-line
The major force terms of the EDR are ${\bf J}\times{\bf B}$ and
$\nabla\cdot{\bf p}_{e}$ which largely balance. As discussed above this force
balance requirement can ultimately be expressed through the conditions in Eqs.
(4) or (6). However, as seen in Fig. 3(a), in a small region ($4d_{e}\times
2d_{e}$) centered on the $X$-line the $({\bf J}\times{\bf B})_{y}$ force
vanishes, such that the more detailed force balance right at the $X$-line then
requires that the RHS of Eq. (3) also vanishes (consistent with Fig. 3(b)).
Thus, for the considered anti-parallel and symmetric configurations and within
this limited region around the $X$-line, the force balance constraint of Eq.
(1) only involves the off-diagonal stress in the electron pressure tensor
$E_{\mbox{\scriptsize rec}}=-\frac{1}{en}\left(\frac{\displaystyle\partial
p_{exy}}{\displaystyle\partial x}+\frac{\displaystyle\partial
p_{eyz}}{\displaystyle\partial z}\right)\quad.$ (8)
Again, here $E_{\mbox{\scriptsize rec}}$ is the $E_{y}$ electric field along
the $X$-line (that is aligned with the $y$ axis and runs through
$(x,z)=(0,0)$). By Faraday’s law $E_{\mbox{\scriptsize rec}}$ then represents
the rate at which magnetic flux cross the $X$-line. Below we will derive a new
model for the terms in Eq. (8), providing a theory consistent with fast
reconnection controlled by larger scale dynamics external to the EDR.
### IV.1 Meandering orbit motion of electrons within the EDR, setting the
striated structure of the electron distribution function
To elucidate how the upstream electron pressure anisotropy drives the current
of the EDR, in Fig. 7 we consider the trajectories of electrons with field-
aligned velocities ${\bf v}=3.3v_{te}{\bf B}/B$ injected with
$z=-4d_{e\infty}$ at various values of $x<0$ just upstream of the EDR. These
trajectories are representative because the upstream distributions (like in
Figs. 8(a,d)) with ${p_{e\parallel}^{\mbox{\scriptsize
edge}}}\gg{p_{e\perp}^{\mbox{\scriptsize edge}}}$ “feed” electrons to the EDR
with $|v_{\parallel}|\gg{v_{\perp}}$ ng:2011 . In Figs. 7(a,b) the parts of
the trajectories with $z<0$ ($z>0$) are represented by the blue (red) full
lines. For each transit across $z=0$ in Fig. 7(b) additional orbits are
initialized and shown by the dashed lines. This visualizes how the average
force of $-e{\bf v}\times B_{z}{\bf e}_{z}$ causes the meandering Speiser-type
motion speiser:1965 to diverge away from the $X$-line.
The EDR current is caused in part by the $E\times B$-drift of the $E_{z}$
field displayed in Fig. 7(a). The large values of $E_{z}$ are consistent with
momentum balance with the strong gradients of $p_{\|}$ at the interface of the
EDR. The direction of the parallel thermal streaming is sensitive to the
$B_{y}$ Hall magnetic field, and from Fig. 7(b) it is evident how this sets
the angle in the $xy$-plane at which the electrons are injected into the EDR
region.
In Figs. 7(c-e) the colored and encircled triangles represent the particle
velocities corresponding to the similarly colored velocity vectors in Fig.
7(b). For example, the magenta vector at $x=0$ in Fig. 7(b) pointing mostly in
the $x$-direction, yields the velocity of an electron reaching the $X$-line
directly from the upstream region, and the corresponding encircled magenta
triangle in Fig. 7(d) has $|v_{x}|\gg|v_{y}|$. The $X$-line may also be
reached through multiple meandering motions within the layer, steadily
directing the velocity into the $-y$-direction. As shown in Figs. 7(c-e),
repeating this procedure for four more initial velocities ${\bf
v}=v_{\parallel}{\bf B}/B$ within the interval $0.1\leq
v_{\parallel}/v_{te}\leq 3.3$ yields the lines of colored triangles,
corresponding to the centers of the striated structures of the EDR electron
distributions first described in Ref. ng:2011 . In Figs. 7(c-e), the similar
points marked by green circles are the result of injecting electrons along
field lines from the opposing sides of the EDR. Also, going from
$x/d_{e\infty}=-3$ to $x/d_{e\infty}=3$, we note how the displayed structures
rotates in the $v_{x}v_{y}$-plane (and similarly in Fig. 8), corresponding to
the rotation of the EDR current layer mentioned above ng:2011 ; shuster:2015 .
Figure 7: a,b) Trajectories of electrons injected along field lines into the
EDR with $v_{\parallel}=3.3v_{te}$. Three points,
$(x,z/d_{e\infty})=(-3,0);(0,0);(3,0)$, are selected in a) indicating the
spatial locations where the velocity arrows in b) are evaluated. As indicated,
the panels c-e) also correspond to these locations. The angles indicated by
arrows in b) at which the trajectories reach the selected points, determine
the location of the encircled triangles in c), d), and e), respectively.
Similar trajectories are obtained by using a range of initial $v_{\parallel}$
values and result in the striated structures marked by the symbols in c),
characteristic of the EDR distributions ng:2011 and observed in Fig. 8.
### IV.2 Origin for the off-diagonal $\partial p_{exy}/\partial x$-stress at
the reconnection $X$-line
To understand how the upstream pressure anisotropy influences $p_{exy}$ we
consider the electron distributions in Fig. 8. At the edge of the EDR we have
$p_{eyy}\simeq p_{ezz}$, and given Eq. (5), $p_{eyy}-p_{exx}=-{\Delta
p_{e\|\perp}^{\mbox{\scriptsize max}}}$. Comparing the distributions in Figs.
8(a,b)) it is clear that only a fraction of this upstream anisotropy is
carried by the orbit motion to the $X$-line. Empirically at the $X$-line we
find that
$\left.\left(p_{eyy}-p_{exx}\right)\right|_{\mbox{\scriptsize
X-line}}=-\dfrac{3}{4}{\Delta p_{e\|\perp}^{\mbox{\scriptsize max}}}+\Delta
p_{eyy,E}\quad.$ (9)
Here the additional term, $\Delta p_{eyy,E}$, corresponds to an increase in
$p_{eyy}$ due to direct heating of electrons by the reconnection electric
field. From Fig. 7 it is clear that the “tips” of the triangular shaped
distributions are composed of electrons meandering multiple times while
streaming mostly in the $-y$-direction, energized most significantly by
$E_{\mbox{\scriptsize rec}}$. The more pronounced triangular shape in Fig.
8(e) compared to Fig. 8(b) is consistent with stronger $\Delta p_{eyy,E}$
heating, as the value of $E_{\mbox{\scriptsize rec}}$ for $m_{i}/m_{e}=400$ is
about twice as large compared to that observed in the $m_{i}/m_{e}=1836$ run.
As detailed with Figs. 7 and 8, within the EDR the $xy$-projections of the
electron distributions rotate approximately as a “solid-body” such that for
small $x/l_{u}$
$p_{exy}=\frac{x}{l_{u}}\left.\left(p_{eyy}-p_{exx}\right)\right|_{\mbox{\scriptsize
X-line}}\quad.$ (10)
Combining Eqs. (9) and (10) we then obtain
$\frac{\displaystyle\partial p_{exy}}{\displaystyle\partial
x}=\frac{1}{l_{u}}\left(-\dfrac{3}{4}{\Delta p_{e\|\perp}^{\mbox{\scriptsize
max}}}+\Delta p_{eyy,E}\right)\quad.$ (11)
This equation represents the major new result of the present paper. Below we
will find that the term proportional to ${\Delta
p_{e\|\perp}^{\mbox{\scriptsize max}}}$ cancels out the contributions to
$E_{\mbox{\scriptsize rec}}$ provided by $\partial p_{eyz}/\partial y$,
leaving only the second term in Eq. (11) for balancing $E_{\mbox{\scriptsize
rec}}$.
Figure 8: Projections, $f_{xy}(v_{x},v_{y})=\int f({\bf v})dv_{z}$ of EDR
electron distributions, recorded in simulations for $m_{i}/m_{e}=1836$ in a-c)
and $m_{i}/m_{e}=400$ in d-f). The differences between the profiles are
consistent with enhanced $\Delta p_{eyy,E}$ for $m_{i}/m_{e}=400$, caused by
$\hat{E}_{\mbox{\scriptsize rec,400}}\simeq 2\hat{E}_{\mbox{\scriptsize
rec,1836}}$.
### IV.3 Cancellation of the Hesse-$\partial p_{eyz}/\partial z$-stress
Refs. kuznetsova1998kinetic ; hesse:1999 provide approximations for the off-
diagonal pressure tensor derivatives ($\partial p_{exy}^{\mbox{\scriptsize
Hesse}}/\partial x$ and $\partial p_{eyz}^{\mbox{\scriptsize Hesse}}/\partial
z$).
$\frac{\displaystyle\partial p_{exy}^{\mbox{\scriptsize
Hesse}}}{\displaystyle\partial x}=\frac{\displaystyle\partial
p_{eyz}^{\mbox{\scriptsize Hesse}}}{\displaystyle\partial
z}=n\sqrt{\frac{\displaystyle m_{e}T_{e}}{\displaystyle
2}}\,\frac{\displaystyle\partial u_{ex}}{\displaystyle\partial x}\quad.$
These were derived through an assumption of isotropic upstream electron
pressure. As such, given that upstream of the EDR $p_{exx}-p_{eyy}=\Delta
p_{\|,\perp}$ is large, the $\partial p_{exy}^{\mbox{\scriptsize
Hesse}}/\partial x$-approximation becomes inaccurate and is not consistent
with the simulation data presented below. In contrast, the conditions required
for the accuracy of $\partial p_{eyz}^{\mbox{\scriptsize Hesse}}/\partial z$
are well satisfied, as the upstream pressure in the $yz$-plane is isotropic,
$p_{eyy}\simeq p_{ezz}\simeq{p_{e\perp}^{\mbox{\scriptsize edge}}}$. With
$\partial u_{ex}/\partial x=u_{e}/l_{u}$, and following Refs.
kuznetsova1998kinetic ; hesse:1999 that the width of the EDR is
$\rho_{e}=m{v_{\perp}}/(eB_{H})$ such that from Ampère’s law
$u_{e}=B_{H}/(\mu_{0}n\rho_{e})=B_{H}^{2}/(\mu_{0}nm{v_{\perp}})$, it then
follows that
$\frac{\displaystyle\partial p_{eyz}^{\mbox{\scriptsize
Hesse}}}{\displaystyle\partial
z}=\sqrt{\frac{m_{e}T_{e\perp}}{2}}\frac{B_{H}^{2}}{\mu_{0}m_{e}{v_{\perp}}l_{u}}=\frac{B_{H}^{2}}{2\mu_{0}l_{u}}=\frac{5}{7}\frac{{\Delta
p_{e\|\perp}^{\mbox{\scriptsize max}}}}{l_{u}}\,\,.$ (12)
As illustrated by the black contours in Figs. 1(c), for the present
configurations the momentum balance in Eq. (6) corresponds to the condition
$0.7B_{H}^{2}/\mu_{0}\simeq{\Delta p_{e\|\perp}^{\mbox{\scriptsize max}}}$,
explaining the last equality above.
It is now interesting to observe that both in Eqs. (11) and (12) the term
${\Delta p_{e\|\perp}^{\mbox{\scriptsize max}}}/l_{u}$ occurs with similar
prefactors ($5/7$ vs. $3/4$), but opposite signs. Thus, to within the accuracy
of their derivations, when calculating the sum of $\partial p_{exy}/\partial
x$ and $\partial p_{eyz}^{\mbox{\scriptsize Hesse}}/\partial z$ the stress
imposed on the layer by the rotation of the upstream pressure anisotropy
cancels the $\partial p_{eyz}^{\mbox{\scriptsize Hesse}}/\partial z$-stress
term.
### IV.4 The term breaking the electron frozen in law
Given the described cancellation of terms, the theory suggests that the
electron dynamics of the EDR do not represent a bottleneck for reconnection.
In fact, inserting Eqs. (11) and (12) into Eq. (8) only one term remains for
balancing the reconnection electric field
$E_{\mbox{\scriptsize rec}}\simeq-\frac{1}{enl_{u}}\Delta p_{eyy,E}\quad.$
(13)
Separate from the force balance constraint expressed in Eq. (13) we can obtain
a second independent estimate for $\Delta p_{eyy,E}$: In Fig. 7 we observe
that electrons in the $X$-line region typically travel a distance between $0$
and 2$l_{u}$ in the $y$-direction such that $\Delta p_{eyy,E}\simeq-
enl_{u}E_{\mbox{\scriptsize rec}}$. Applying this in Eq. (13) we obtain the
seemingly mundane expression that $E_{\mbox{\scriptsize rec}}\simeq
E_{\mbox{\scriptsize rec}}$. Nevertheless, this is an important result, as it
shows that the present geometry can accommodate any reconnection rate imposed
at larger scales onto the EDR region. Consistent with the behavior of
$\hat{E}_{\mbox{\scriptsize rec}}$ in Fig. 9(c), it confirms previous
numerical results and theoretical conjectures that, with the formation of
electron scale current layers, the electron dynamics can readily accommodate
the rates of reconnection imposed by the plasma behavior at larger scales
stanier:2015 ; liu:2017 . We also note that because
$\hat{E}_{\mbox{\scriptsize rec}}\propto\Delta p_{eyy,E}/l_{u}$ and $\Delta
p_{eyy,E}\propto l_{u}$, the reconnection rate becomes independent of $l_{u}$.
### IV.5 Numerical validation of the model
As described above, the theory is aided by fully kinetic VPIC simulations
carried out for a matrix of $\beta_{e\infty}$ and $m_{i}/m_{e}$ values. We
next validate each of the separate theoretical predictions against the
numerical results. First, the described dynamics involving trapped electrons
require that within the inflow regions $|u_{e\|}|\ll v_{te}$, which is valid
for $\beta_{e\infty}\gtrsim\sqrt{m_{e}/m_{i}}$ egedal:2015 ; le:2015 , and the
predicted profiles of $T_{e\|}/T_{e\perp}$ as well as $B_{H}$ are confirmed in
Figs. 9(a,b).
Figure 9: Obtained from a matrix of 35 kinetic simulations, profiles as
functions of $\log_{10}(\beta_{e\infty})$ are shown for a range of parameters
as marked in the panels. In a) and b) the regions shaded in magenta are
obtained from the CGL-limit of the Lê-2009 Equations of State le:2009 . In c)
$\hat{E}_{\mbox{\scriptsize rec}}$ is normalized by
$T_{e\infty}/(ed_{e\infty})$, and the magenta shaded region is the expected
range based on $E_{\mbox{\scriptsize rec}}=\alpha v_{A}B_{\mbox{\scriptsize
rec}}$, with $\alpha=0.1$ and $100\leq m_{i}/m_{e}\leq 1836$ (see text). The
wider yellow shaded region correspond to $0.08\leq\alpha\leq 0.12$. The
profiles in d–i) are various electron pressure tensor derivatives, normalized
by $-nT_{e\infty}/d_{e\infty}$, such that the resulting dimensionless values
can be compared directly to those of $\hat{E}_{\mbox{\scriptsize rec}}$ in c).
For all the numerical runs in our study, the individual contributions of
$\partial p_{exy}/\partial x$ and $\partial p_{eyz}/\partial z$ are shown in
Figs. 9(d,e), respectively. Similar to $\hat{E}_{\mbox{\scriptsize rec}}$, all
terms in Fig. 9 involving pressure derivatives are normalized by
$-nT_{e\infty}/d_{e\infty}$. With this normalization, confirming Eq. (8), the
sum of $\partial p_{exy}/\partial x+\partial p_{eyz}/\partial z$ in Fig. 9(f)
reproduces with good accuracy $\hat{E}_{\mbox{\scriptsize rec}}$ in Fig. 9(c).
Eqs. (9)-(11) represent a key insight for the theory which is confirmed
numerically, as Fig. 9(g) provides an accurate representation of Fig. 9(d).
Consistent with recent spacecraft observations egedal:2019 , $\partial
p_{exy}/\partial x$ is mostly negative for runs with $m_{i}/m_{e}\geq 400$,
and is thus dominated by the external stress imposed by
${p_{e\parallel}^{\mbox{\scriptsize edge}}}\gg{p_{e\perp}^{\mbox{\scriptsize
edge}}}$. Likewise, Eq. (12) is confirmed numerically, as the profiles of
$\frac{3}{4}{\Delta p_{e\|\perp}^{\mbox{\scriptsize max}}}/l_{u}$ in Fig.
9(h), are observed to provide a match to $\partial p_{eyz}/\partial z$ in Fig.
9(e). This is especially the case for $m_{i}/m_{e}\geq 400$, corresponding to
the adiabatic limit required for the validity of Eq. (6), and for which the
sums shown in Fig. 9(i) of the profiles of Figs. 9(g,h) provide a good match
to $\hat{E}_{\mbox{\scriptsize rec}}$ in Fig. 9(c), confirming Eq. (13).
## V Summary and discussion
The use of the VPIC code implemented in a modern super-computing facility
enabled a matrix of simulations to be carried for anti-parallel magnetic
reconnection. This matrix spans a range of the normalized electron pressures,
$\beta_{e\infty}$, as well as the ion to electron mass ratios, $m_{i}/m_{e}$.
Our study reveals a range of results, which require new interpretations of the
electron dynamics of the EDR and IDR that are significantly different from
previous models including those in Refs. hesse:1999 ; drake:2008 ;
daughton:2006 . A main difference from the numerical studies a decade or more
ago is our recent ability to carry out routine kinetic simulations at the
natural proton to electron mass ratio of $m_{i}/m_{e}=1836$. At full mass
ratio the effect of the electron pressure anisotropy that develops in the
reconnection inflow becomes the dominant force-term not only within the EDR
but also for a significant fraction of the IDR.
It has previously been determined le:2009 ; egedal:2013 that upstream of the
EDR strong electron pressure anisotropy with $p_{e\|}\gg p_{e\perp}$ is driven
by the convection of electrons into the reconnection region characterized by
low values of $B$. This upstream pressure anisotropy is responsible for
driving the strong electron currents within the EDR. In addition, with Fig. 3
we visualize (for the first time) how the electron inflow speed for locations
inside the IDR (but outside the traditional EDR) exceeds the inflow speed of
the magnetic field by a factor of 5. Thus, the main assumption of “whistler
reconnection” drake:2008 that ${\bf E}+{\bf v}_{e}\times{\bf B}=0$ within the
IDR is not supported by the simulations. Instead, the strong Hall field
$B_{y}$ perturbations are mainly a consequence of dynamics related to the
electron pressure anisotropy yielding a new and dominate force balance
constraint, namely $({\bf J}\times{\bf B})_{y}\simeq(\nabla\cdot{\bf
p}_{e})_{y}\gg en_{e}E_{\mbox{\scriptsize rec}}$. While we cannot rule out
that the condition $({\bf J}\times{\bf B})_{y}\simeq(\nabla\cdot{\bf
p}_{e})_{y}$ is compatible with dispersive waves rogers:2001 , we conclude
that for $m_{i}/m_{e}=1836$ the wave dynamics associated with the standing
$B_{y}$-Hall structure around the EDR is not governed by the common whistler
wave. Rather, the inflow into the EDR is strongly enhanced directly by ${\bf
J}_{\mbox{\scriptsize extra}}=[({p_{e\parallel}}-{p_{e\perp}})/B]{\bf
b}\times\mathbf{\kappa}$. This enhanced perpendicular flow into the EDR must
in turn be sourced by electrons flowing along the field lines on the inflow-
side of the separatrix layers. This is consistent with spacecraft observations
that the Hall currents extend many tens of $d_{i}$ away from the EDR
manapat:2006 .
Aside from the results summarized above, the main goal of this paper is to
develop a theory that can account for the electron momentum balance directly
at the $X$-line within the center of the EDR for anti-parallel reconnection.
Previously, Refs. kuznetsova1998kinetic ; hesse:1999 provided approximate
expressions for the off-diagonal stress terms $\partial p_{exy}/\partial x$
and $\partial p_{eyz}/\partial z$. We observe that their theoretical form for
$\partial p_{eyz}/\partial z$ is in excellent agreement with our numerical
results. Meanwhile, the electron pressure anisotropy that develops upstream of
the EDR turns out to have a large impact on $\partial p_{exy}/\partial x$ not
previously considered. Our updated theory for $\partial p_{exy}/\partial x$ is
cast in two separate terms. The first term is caused by the upstream electron
pressure anisotropy and cancels with high accuracy the stress of $\partial
p_{eyz}/\partial z$ predicted in Refs. kuznetsova1998kinetic ; hesse:1999 . In
hindsight, this cancellation is not surprising as the EDR current, mainly
driven by the upstream $p_{e\|}\gg p_{e\perp}$, must be in force balance
independent of the value of $E_{\mbox{\scriptsize rec}}$.
The additional second term for $\partial p_{exy}/\partial x$ in Eq. (11) is
the term that scales with $E_{\mbox{\scriptsize rec}}$ and as such, can be
considered the term that actually breaks the electron frozen-in condition at
the $X$-line. This term is related to the increase of the $p_{eyy}$ pressure
tensor element caused directly through heating by $E_{\mbox{\scriptsize
rec}}$. Because the electron flow in the $y$-direction is largely fixed by the
upstream $p_{e\|}\gg p_{e\perp}$ (see Eq. (6)), the increase in $p_{eyy}$
becomes linear in $E_{\mbox{\scriptsize rec}}$, and the rotation with length
scale $l_{u}$ of $p_{eyy}$ in the $xy$-plane then yields the stress required
to balance $E_{\mbox{\scriptsize rec}}$. Because the term is proportional to
$E_{\mbox{\scriptsize rec}}$, it facilitates reconnection at any external rate
imposed onto the EDR. From Fig. 9(c) it is clear how numerical runs with
$m_{i}/m_{e}\ll 1836$ impose (from the electrons’ perspective) rates of
reconnection which will be larger than those observed in naturally occurring
systems (with $m_{i}/m_{e}=1836)$. Even so, the identified reconnection
mechanism can still accommodate the resulting unrealistically large values of
$\hat{E}_{\mbox{\scriptsize rec}}$.
In closing, we note that our new theory predicts that $\partial
p_{exy}/\partial x$ and $\partial p_{eyz}/\partial z$ have opposite signs for
$m_{i}/m_{e}=1836$. This result has been directly confirmed in spacecraft
observations of an EDR encounter by MMS in the Earth’s magnetotail egedal:2019
. Other MMS observations also support the approximation that reconnection in
the Earth’s magnetotail occurs in regimes consistent with laminar 2D kinetic
models. One reason for the apparent success of such laminar 2D models is
perhaps that for $m_{i}/m_{e}=1836$ the force by $E_{\mbox{\scriptsize rec}}$
is small compared to the forces associated with electron pressure anisotropy,
such that $E_{\mbox{\scriptsize rec}}$ only slightly perturbs the electron
orbit dynamics within the EDR. Again, from the perspective of the electrons
the reconnection rates imposed by the inertia and dynamics of the much heavier
ions are in fact very modest and the small modifications imposed by
$E_{\mbox{\scriptsize rec}}$ on the electron motion is not sufficient to drive
strong instabilities. In order for instabilities to alter the momentum
equation in Eq. (1) they much have an inverse growth rate similar to the short
electron transit time through the EDR. For example, the Lower-Hybrid-Drift-
Instability (LHDI) may perturb the out-of-plane structure of the EDR, but this
does not necessarily cause a fundamental change in its underlying 2D dynamics
le:2017 ; greess:2021 ; schroeder:2022 . Given recent progress in the
understanding of how the larger scale ion dynamics influence the reconnection
rate stanier:2015 ; liu:2017 , a more detailed and complete picture for
reconnection now emerges consistent with local observations of reconnection
within the Earth’s magnetotail.
## DATA AVAILABILITY
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
## Appendix A: Force balance along the EDR
Figure 10: Considering the same simulation ($m_{i}/m_{e}=1836$ and
$\beta_{e\infty}=2^{-4}$) as in Fig. 1, color contours are presented for terms
related to Eq. (3). The EDR is outlined by the cyan rectangle, and a fluid-
element is selected for analysis by the black/magenta square.
In Fig. 10 we explore how the large values of $({\bf J}\times{\bf B})_{y}$
inside the EDR are related to the electron pressure anisotropy that forms
upstream of the EDR. For this, consider the small fluid element which in each
panel is marked by the black/magenta square. Naturally, we can integrate Eq.
(3) over the surface of this fluid element. For the RHS of Eq. (3) the largest
contribution is from $\int(\nabla\cdot{\bf p}_{e})_{y}dA$. In turn, by the
divergence theorem $\int(\nabla\cdot{\bf p}_{e})_{y}dA=\Delta{\bf
p}_{exy}\,\Delta z+\Delta{\bf p}_{eyz}\,\Delta x$. Here, $\Delta{\bf p}_{exy}$
is the change in ${\bf p}_{exy}$ between the two black sides of the element,
while $\Delta{\bf p}_{eyz}$ is the change in ${\bf p}_{eyz}$ between the two
magenta sides. Comparing Figs. 10(a,b) it is clear that $|\Delta{\bf
p}_{exy}|\ll|\Delta{\bf p}_{eyz}|$, such that $\int(\nabla\cdot{\bf
p}_{e})_{y}dA\simeq\Delta{\bf p}_{eyz}\,\Delta x$.
We have now demonstrated that $\int({\bf J}\times{\bf
B})_{y}dA\simeq\Delta{\bf p}_{eyz}\,\Delta x$. On the other hand, ${\bf
J}\times{\bf B}=\nabla\cdot{\bf T}$, where ${\bf T}=(B^{2}/\mu_{0})({\bf
bb}-{\bf I}/2)$ is the Maxwell stress tensor. By similar arguments to those
applied when integrating $\int(\nabla\cdot{\bf p}_{e})_{y}dA$, we then find
that $\int({\bf J}\times{\bf B})_{y}dA=\int(\nabla\cdot{\bf
T})_{y}dA\simeq\Delta{\bf T}_{yz}\,\Delta x$. Neglecting $E_{y}$, the force
balance constraint in Eq. (2) can therefore be expressed as $\Delta{\bf
T}_{yz}\simeq\Delta{\bf p}_{eyz}$, and given these elements are asymmetric
about $z=0$ this further reduces to ${\bf T}_{yz}\simeq{\bf p}_{eyz}$ along
the edge of the EDR. An identical analysis can be carried out for the forces
in the $x$-direction, which yields the similar result that ${\bf
T}_{xz}\simeq{\bf p}_{exz}$ along the edge of the EDR.
The integrals over the fluid element involved the full pressure tensor ${\bf
p}_{e}$, but by the divergence theorem the results only depend on the values
of ${\bf p}_{e}$ at the edge of the EDR; here ${\bf p}_{e}$ is well
approximated by its CGL-form ${\bf p}_{e}\simeq{\bf
p}_{eg}=({p_{e\parallel}}-p_{e\perp}){\bf bb}+p_{e\perp}{\bf I}$. To
illustrate that ${\bf p}_{e}\simeq{\bf p}_{eg}$ outside the EDR, in Figs.
10(b,c) the profiles of ${\bf p}_{eyz}$ and ${\bf p}_{eg,yz}$ can be compared
directly. Simple manipulations then show that the two force balance
constraints that ${\bf T}_{yz}\simeq{\bf p}_{eyz}$ and ${\bf T}_{xz}\simeq{\bf
p}_{exz}$, can then be expressed on a common form through the marginal
firehose condition ${p_{e\parallel}}-p_{e\perp}=B^{2}/\mu_{0}$, again,
applicable along the edge of the EDR.
To summarize this analysis, we have shown that for the present simulation at
$m_{i}/m_{e}=1836$ that $|en_{e}E_{y}|\ll|({\bf J}\times{\bf
B})_{y}|\simeq|(\nabla\cdot{\bf p}_{e})_{y}|$. By integrating over a fluid
element that spans the width of the EDR, it is then found that the strong
${\bf J}\times{\bf B}$-forces within the EDR are balanced by the force of the
pressure anisotropy at the edge of the EDR. In turn, at the edge of the EDR
the Lê-2009 equations of state le:2009 are applicable, and together with the
force balance condition ${p_{e\parallel}}-p_{e\perp}=B^{2}/\mu_{0}$, this
leads to the scaling law for the current across the EDR as expressed in Eq.
(6). It is noteworthy here how the current across the EDR is expressed in
terms of parameters evaluated far upstream of the reconnection region.
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spacing=nonfrench
# Unsafe at Any Copy: Name Collisions from Mixing Case Sensitivities
Aditya Basu$*$
<EMAIL_ADDRESS>John Sampson$*$
<EMAIL_ADDRESS>Zhiyun Qian†
<EMAIL_ADDRESS>Trent Jaeger$*$
<EMAIL_ADDRESS>$*$The Pennsylvania State University
†University of California, Riverside
###### Abstract
File name confusion attacks, such as malicious symlinks and file squatting,
have long been studied as sources of security vulnerabilities. However, a
recently emerged type, i.e., case-sensitivity-induced name collisions, has not
been scrutinized. These collisions are introduced by differences in name
resolution under case-sensitive and case-insensitive file systems or
directories. A prominent example is the recent Git vulnerability
(CVE-2021-21300) which can lead to code execution on a victim client when it
clones a maliciously crafted repository onto a case-insensitive file system.
With trends including ext4 adding support for per-directory case-insensitivity
and the broad deployment of the Windows Subsystem for Linux, the prerequisites
for such vulnerabilities are increasingly likely to exist even in a single
system.
In this paper, we make a first effort to investigate how and where the lack of
any uniform approach to handling name collisions leads to a diffusion of
responsibility and resultant vulnerabilities. Interestingly, we demonstrate
the existence of a range of novel security challenges arising from name
collisions and their inconsistent handling by low-level utilities and
applications. Specifically, our experiments show that utilities handle many
name collision scenarios unsafely, leaving the responsibility to applications
whose developers are unfortunately not yet aware of the threats. We examine
three case studies as a first step towards systematically understanding the
emerging type of name collision vulnerability.
## 1 Introduction
A fundamental file system design choice is whether it will allow file names to
be case sensitive or not, and modern file systems are diverse in their
selection. A case-sensitive file system is one that allows the definition of
multiple files whose names differ only in their case, such as Foo.c and foo.c.
In a case-insensitive file system, only one file can be defined whose names
differ only in their case. Historically, UNIX file systems are case sensitive,
whereas Windows file systems are case insensitive. Further, case-insensitive
file systems may be either case preserving (e.g. Apple File System (APFS),
NTFS, etc.) or not (FAT), where a case-preserving file system preserves the
case chosen (i.e., either Foo.c or foo.c), rather than converting all names to
one case choice (e.g., all lowercase). Importantly, while choices in case
sensitivity for a single file system may appear to be arbitrary or
aesthetically driven, the precise semantics of interactions between two file
systems with different case sensitivities can range from subtle to ill-
defined, with associated consequences.
Practitioners have long had concerns about the implications of leaving case
sensitivity as an open design choice [31] Historically, these concerns were
not considered as pressing when file systems were associated with their
respective operating systems and associated singular assumptions about case.
However, individual systems now frequently support a mixture of case-sensitive
and case-insensitive file systems, creating opportunities for files to be
moved between file systems with different cases and file identifier encodings.
More troublingly, several file systems now support allowing the choice of case
for individual directories [12], complicating file operations by having
multiple case and encoding semantics within the same file system.
Security risks related to this design choice therefore appear to be
increasing. First, the Windows Subsystem for Linux [58] (WSL) integrates Linux
and Windows platforms leading to expectations that files may be routinely
copied from Linux (i.e., case-sensitive) to Windows (i.e., case-insensitive)
file systems. Second, Linux ext4 now supports case-sensitive and case-
insensitive naming in the same partition, configurable per directory [12, 34].
Linus Torvalds expressed concerns about adding such support to ext4 [31],
stating that such features often cause "actual and very subtle security
issues."
Indeed, security issues caused by moving files from case-sensitive to case-
insensitive file systems are starting to appear. For example, the git
distributed version control system has suffered from multiple vulnerabilities
(e.g., CVE-2014-9390, CVE-2021-21300), caused by how git clones repositories
from case-sensitive file systems to case-insensitive file systems. To exploit
this, an adversary creates a repository in a case-sensitive file system with a
directory whose name will collide (i.e., only differs in case) with a symbolic
link (to another directory) added by git when the repository is cloned to a
case-insensitive file system. The name collision between the directory and the
symbolic link enables adversaries to overwrite the scripts that git executes.
Such attacks can alter both the target resource’s content and/or its metadata,
including its permission assignments.
Researchers have long been aware of hazards that may occur during file system
name resolution [3, 4], particularly that programmers must validate safe use
of symbolic links and check for "squatted" files when creating a new files.
Many defenses have been proposed [7, 8, 55, 9, 40, 41, 42, 30, 50, 51, 52].
However, to the best of our knowledge, ours is the first work studying how
case interplays cause name collisions that lead to incorrect, and in some
cases, vulnerable behaviors. We show that utilities and applications currently
do not recognize unsafe use of case-insensitive file systems, leading to these
problems. In particular, we study the behavior of six utilities used to copy
files, which is how case-sensitive and case-insensitive file systems can
interact. We find a wide variety of responses to name collisions, including
many that overwrite existing data and change file permissions silently. In
addition, we examine three case studies in using such utilities that result in
unsafe and sometimes exploitable behaviors. This paper demonstrates the
potential implications of the name collision problem, focusing on Linux and
its supported file systems, thereby motivating both more and broader (e.g.,
other OS-FS combination) investigations.
This paper makes the following contributions:
* •
We examine the security and correctness implications of name collisions, when
two distinct file system resources with two distinct names map to to a single
name, due to file system case sensitivity and/or encoding mismatches.
* •
We develop an automated method to test common Linux utilities for unsafe
reactions to name collisions, finding a wide variety of responses, many of
which are unsafe and possibly exploitable.
* •
We perform detailed case studies of the impact of name collisions on three
programs dpkg, rsync, and Apache, showing how they operate incorrectly in the
face of name collisions and how they would be exploited when deployed on case-
insensitive directories.
## 2 Background: From Cases to Collisions
Beyond traditional, i.e. operating-system-entailed, decisions made with
respect to case sensitivity, even Linux files systems now represent a
surprising diversity of case sensitivity decisions. In particular, the desire
to support some non-native applications, such as WINE and Samba from Windows
systems, has motivated Linux file systems to support the case-insensitive file
naming used in these non-native file systems.
The ability to create case-insensitive file systems has long been possible in
some Linux file systems, such as ZFS, JFS, and ciopfs. However, these options
are applied to the entire filesystem, rather than just the relevant
directories for individual applications. In 2019, Linux kernel version 5.2
added support for per-directory case-insensitivity to ext4[12, 34]. Later in
2019, similar support was added to the Flash-Friendly File System (F2FS) in
Linux kernel version 5.4 [13, 14]. For case-insensitive directories, these
file systems are case-preserving in nature.
### 2.1 Motivations for Increasing Case Diversity
Samba: Samba [45] implements the Common Internet File System (CIFS) protocol
which allows for sharing file systems over a network. Its primary use is
sharing files with Windows clients that expect a case-insensitive file system.
Hence, Samba implements user-space case-insensitive lookups even if the
underlying file system is case-sensitive. Furthermore, it allows turning
on/off case-sensitivity and case-preservation on a per-mount basis [46]. Note
that this feature only works for non-Windows clients, which means that the
actual file system can contain files differing only in case. This can lead to
unexpected behaviors where Samba will choose to show only a subset of files.
Deleting files which have collisions will now show the alternate versions,
thereby giving rise to inconsistent behavior from the end user’s perspective.
Samba’s requirement of case-insensitive matching, which is done in user-space,
incurs a huge performance overhead [37] thereby motivating the support for
case-insensitivity in the ext4 file system [34, 35, 36]. Other
programs/systems such as Wine [57], Network File System (NFS), SteamOS [49,
48] and Android [32, 59] would also benefit from in-kernel case-insensitivity
support.
ext4: For ext4, the idea is that the filesystem at large can be configured to
be "casefolding," which permits the mixing of case-sensitive and case-
insensitive directories in the same file system. When creating an ext4 file
system, the _casefold_ option is applied, e.g., mkfs -t ext4 -O casefold
/dev/sda. Setting the +F inode attribute on an empty directory makes it case-
insensitive, e.g., mkdir foo; chattr +F foo. Note that case-insensitive
directories can contain case-sensitive directories. This means that for a
given path, /foo/bar/bin/baz, any of foo, bar and bin can either be case-
sensitive or case-insensitive.
tmpfs: tmpfs recently added case-insensitivity support [33]. The use cases are
similar to that of ext4 with the addition of supporting sandboxing and
container tools such as Flatpak.
### 2.2 Name Collisions
A name collision occurs when a file system maps two distinct names of two
distinct resources to the same name. Name collisions can cause problems to
occur if the names of distinct resources _collide_ when those resources are
replicated to a target directory that does not provide a 1:1 mapping for all
replicated objects. Suppose one directory has two files with distinct names in
that file system. Should those files be copied to a second directory in which
the two file names collide (i.e., are mapped to the same name), then only one
file will be created, which may be either of the original files or an
unpredictable combination of the two files’ content and metadata. Variation in
case sensitivity between two file systems is a common origin of collisions,
but diversity in other encoding properties, such as character choice (e.g.,
FAT does not support $``$, $:$, $*$, etc. 111http://elm-
chan.org/fsw/ff/doc/filename.html) and canonicalization processes, can lead to
the same effect. For example, NTFS uses UTF-16 while APFS (macOS) and ext4
(Linux) use UTF-8 and older file systems can use other encoding schemes, such
as iso8859-1.
Modern file systems perform the canonicalization of names using a technique
called _case folding_[6]. Unlike traditional techniques, case folding uses
lookup tables to transform each character of the filename to a pre-determined
case. Furthermore, in Unicode several different characters (or code points)
can be used to represent the same abstract character. Hence, a normalization
step is needed after performing case folding to ensure that the binary
representations match.
Unfortunately, these case folding rules can differ across file systems.
Additionally, the _locale_ (or language) also influences the case folding
rules. Due to such differences, ‘temp_200K’ (where K = Kelvin Sign, U+212A in
UTF-8) and ‘temp_200k’ are considered identical on NTFS and APFS, but on ZFS
these filenames are considered different when using case-insensitive lookups.
As a result, when two files of these names are copied from a ZFS file system
to an NTFS file system, they will collide and only one filename and only one
file will be created. Similarly, consider the filenames floß, FLOSS and floss.
All can coexist on a case-sensitive file system supporting reasonable
character encodings, but, since case-folding for both floß and FLOSS is floss,
attempting to move these files to a case-insensitive system may only preserve
one of the original triple.
Modern encoding schemes such as UTF-8 have support for non-English characters
which require case folding to perform case-insensitive matching. This only
increases the number of case-insensitive matches, making the problem of name
collisions even worse. For clarity and conciseness, we will use examples of
ASCII-based, case-insensitive matching throughout the rest of the paper.
We propose a taxonomy for _name confusions_ , shown in Figure 1, that captures
the types of incorrect program behaviors that may stem from the ambiguous uses
of names for file system resources. Name collisions are a subset of this
broader class. Name confusions may be caused by three reasons: (1) because
multiple names may refer to the same resource (i.e., aliasing); (2) because an
adversary may create a resource of that name before the victim (i.e., squat);
and (3) because the multiple resources may be associated with the same name
(i.e., collisions). Of these, however, name collisions are the least explored
for their correctness and security implications. As Linux is adding more
support for case-insensitivity, it is crucial to understand the pitfalls and
problems such functionality may incur. This work aims to study these issues.
[Name Confusion (NC) [Alias [Symlink] [Hardlink] [Bind mount] ] [Squat [File]
[Other] ] [Collision [Case] [Encoding] ] ]
Figure 1: Taxonomy of name confusion vulnerabilities divided into alias (i.e.,
multiple names for a resource), collision (i.e., multiple resources for a
name), and squat (temporal ambiguities in names vs. resources) classes
## 3 From Collisions to Calamities
Name collisions can impair system functionality by modifying the content
and/or metadata of files and directories in unexpected ways. Some name
collisions have already led to security vulnerabilities [24]. In this section,
we define the conditions in which a name collision occurs, the conditions
under which such a collision may be exploitable by an adversary, and describe
a known vulnerability that is caused by a name collision.
### 3.1 Causes of Name Collisions
A process may cause a name collision under the following conditions.
* •
There exists a _source resource_ (e.g., file or directory) in a case-sensitive
file system, whose name is _source name_.
* •
The process uses a relocation operation to place the source resource in a
target directory, where the target directory is a case-insensitive or case-
preserving directory. Examples of relocation operations include copy (e.g.,
cp, rsync, or an archive operation, such as tar or unzip) or move (e.g., mv).
* •
The relocation operation produces a destination name from the source name for
the name of the source resource when placed in the target directory.
* •
There is a target resource with a target name whose name differs from the
source name, but maps to the same name as the destination name does in the
target directory (e.g., due to differences in case-folding rules between the
source and target directory).
* •
If the process is authorized to modify the target resource, the process’s
relocation operation results in a name collision between the target and source
resources.
* •
If the relocation operation proceeds despite the name collision, then the
target resource’s content and/or its metadata may be modified using the source
resource content and/or metadata.
When these conditions are met, a name collision occurs such that the target
resource in the target directory will be modified using the source resource.
In most cases, modifying a target resource using a source resource of a
different name is an unexpected result. We test how common Linux utilities
react to name collisions and examine case studies where name collisions cause
incorrect operation.
Given the above conditions, there are several clear scenarios where the
movement of files involving the following types of file systems (following the
categorization in § 2.2) could result in name collisions:
* •
Case-sensitive and case-insensitive file systems.
* •
Two distinct case-insensitive file systems with different case folding rules,
e.g. ZFS to NTFS, etc. 222ASCII encoding contains a strict subset of
characters representable under the UTF-8 encoding scheme. Hence, without loss
of generality, we only consider ASCII encoded names for the examples in this
paper.
* •
Two file system whose locales are different but they still use the same file
system format (such as ext4).
* •
A single file system that supports per-directory case-insensitivity, e.g.
ext4.
Clearly, name collisions may impact system functionality by causing collateral
damage to resources supposedly unrelated to the operation, even removing the
target resource entirely. In addition, name collisions may be used to exploit
the process performing the relocation operation in a version of a confused
deputy attack [25]. An adversary only requires write access to the source
directory to produce source names that may lead to name collisions to perform
an attack. We note that adversaries require fewer permissions to perform
attacks using name collisions than other name confusion classes, which require
write access to a directory used in name resolution of the target resource
[54]. Thus, remote attacks using file system archives, such as tarballs and
zip files, as well as file repositories, such as GitHub, can be the sources of
attacks.
In practice, to perform a successful attack using a name collision, the victim
process has to help the adversary in two ways. First, the victim process has
to use the source resource in a relocation operation planted by an adversary
as described above. In addition to archives, other activities, such as
backups, may provide opportunities for exploitation of name collisions. In
addition, ad hoc user actions copying files, e.g., from Linux to Windows in
the Windows Subsystem for Linux, may result in unexpected and exploitable
collisions. Second, the target directory of the relocation operation has to be
predictable by the adversary to enable them to produce a source name that
leads to a colliding destination name. Archives make this task much easier
because the archive itself may be crafted to provide the target resource that
is exploited by creating a collision with another archive file. A recent
vulnerability in the git distributed revision control system demonstrates
exactly this, as described below.
### 3.2 An Example Collision Vulnerability
Security vulnerabilities due to filename collisions across different file
systems have been demonstrated in the wild. Consider a recent vulnerability in
the git distributed version control system (CVE-2021-21300). This
vulnerability results in remote code execution after cloning a maliciously
crafted repository created on a case-sensitive file system to a case-
insensitive file system.
.1 repo/. .2 .git/(contents omitted). .2 A/. .3 file1. .3 file2. .3 post-
checkout(executable script). .2 a(symlink to .git/hooks/).
Figure 2: Example for Git CVE-2021-21300
Figure 2 depicts the maliciously crafted repository structure. Note that this
directory structure works correctly on a case-sensitive file system. However,
on case-insensitive file systems, the presence of the ‘a’ (small) and ‘A’
(capital) directories creates a collision that exposes a vulnerability. This
collision results in a vulnerability when using git’s out-of-order checkout
machinery. Git Large File Storage (LFS) uses out-of-order checkouts for
downloading binaries in the background. Say the repository creator (adversary)
marks ‘A/post-checkout’ for an out-of-order checkout. When a user clones this
repository to a case-insensitive file system (e.g., NTFS), git performs a
sequence of operations that: (1) replaces ‘A’ with the symbolic link ‘a’ and
(2) writes the script file ‘A/post-checkout’ to ‘.git/hooks/post-checkout’ due
to the symbolic link ‘a’. After the files are downloaded, git runs the script
‘.git/hooks/post-checkout’ that the adversary provided, which is obviously
undesirable.
In this case, a maliciously crafted git repository can be designed to provide
a target resource of the symbolic link ‘a’, which when collided by ‘A’ in
resolving the source resource ‘A/post-checkout’ redirects the operation to a
directory chosen by the adversary using the symbolic link.
### 3.3 The State of Defenses for Name Confusions
Currently, operating systems provide no innate defenses to prevent name
collisions, leaving the challenge to programmers. However, researchers have
studied problems due to other types of name confusions extensively, proposing
a variety of defenses [7, 8, 9, 40, 41, 42, 30, 50, 51, 52, 44]. However,
researchers have shown that comprehensive program defenses are expensive [55]
and that system-only defenses will always be prone to some false positives
[5]. Leveraging limited program information [53, 28] still results in some
false positives.
As a result, library commands for opening files have been extended in a
variety of ways to prevent name confusions from occurring. The open command
has been extended with flags to detect file squats (i.e., O_EXCL|O_CREAT to
detect the presence of an existing file during file creation) and prevent
unexpected use of aliases (i.e., O_NOFOLLOW to prevent following symbolic
links). However, the use of squats and aliases is desirable in some
applications, despite their risks. Thus, the openat command has been added to
enable programmers to avoid TOCTTOU attacks, by opening a file from a
validated directory (i.e., file descriptor to the directory of the desired
file). The successful use of openat requires the programmer to check for
unwanted squats or aliases themselves. An alternative is proposed by the
openat2 command, which instead controls how files may be opened, such as
requiring all file components accessed to be descendants of the directory from
which the operation originates. openat and openat2 limit the attack surface of
squat and alias attacks, but do not eliminate them entirely, depending on the
programmer’s additional actions to check for TOCTTOU attacks and configure the
commands.
At present, the above commands make no effort to help programmers address name
collisions. As a result, utilities to perform copy and move operations and
applications that may utilize file systems with multiple or mixed (e.g., ext4
and F2FS) case sensitivities or encodings may not detect and resolve name
collisions correctly. We will examine the possible defenses for name
collisions in § 8.
## 4 Overview
In this paper, we aim to explore the impact that name collisions may have on
file system security. To do this, we propose to examine three research
questions.
RQ1: How do applications invoke utilities that may allow unsafe name
collisions? In § 6, we examine Linux packages to determine the most common
options that applications employ for the utilities used to perform copies. We
examine how frequently application packages use utilities in copy operations
by scanning their scripts for such operations, as shown in Table 1.
RQ2: When do the utilities for performing copy operations allow unsafe name
collisions? Recall that § 3.1 defines the conditions under which an unsafe
name collision may occur. This research question asks whether the utilities
that applications may use to perform copy operations (e.g., cp and tar)
prevent unsafe effects when a name collisions occur. For these utilities and
the common options found in RQ1, we examine a variety of name collision
scenarios to determine whether the utilities allow name collisions and their
unsafe effects to occur as shown in Table 2.
RQ3: What correctness and security problems are caused by name collisions? In
§ 7, we examine three case studies where we show how name collisions cause
programs to behave incorrectly. In particular, we show concretely how
applications can be vulnerable to name collisions when target resources are
deployed on case-insensitive or case-preserving file systems.
Impacts: A preview of our result is that: (1) many applications rely on these
utilities to copy file system resources and repositories/archives; (2) the
utilities used to copy file system resources and repositories/archives often
allow unsafe name collisions, although the specific responses vary in ad hoc
ways; and (3) applications currently lack defenses against name collisions,
which can lead to incorrect operation and exploitable vulnerabilities.
## 5 Testing for Name Collisions
This section details an automated tool for testing the responses of common
Linux utilities used for relocation operations to name collisions. As
described in § 3.1, a name collision is caused by creating a source name that
will be converted to a destination name by the relocation operation that is
equal to a target name in the target directory of the operation. Thus, our aim
is develop a method to automate the generation of source resources with names
that will lead to name collisions when relocated to case-insensitive targets
and identify when operations allow the name collision to occur, detecting the
effects of those operations.
### 5.1 Test Case Generation
The individual test cases are generated to test file system resources of
various types, including regular files, directories, symbolic links (to files
and directories), hard links, pipes, and devices. In addition, we have found
that creating collisions in non-trivial directory structures may also lead to
incorrect behaviors. Figure 3 shows an example test case where the directories
as well as their contents result in a collision when transferred to a case-
insensitive file system. As a result, we aim to generate test cases that
result in name collisions at different depths of the directory being copied,
as evidenced by the collision between directory names at depth 2 (i.e., "dir"
and "DIR") and the impact on colliding resources of different types (i.e., a
regular file "foo" and a pipe "foo").
Input .1 src/. .2 dir/. .3 foo*. .2 DIR/. .3 foo|.
copy
Effect .1 target/. .2 dir/. .3 foo*|.
Here, ‘*’ means a regular file and ‘|’ means file type is a named pipe.
Figure 3: An Example Test Case
Since we are testing the behavior of various utilities that perform relocation
operations, we can control the source and target names in creating test cases.
As a result, the choice of names is trivial. We create source directories that
contain both the target resource (i.e., a resource copied first from the
source to the target) and the source resource (i.e., a resource copied later
by the utility to collide with the target resource (i.e., now in the target
directory). This is similar to the way name collisions would occur when
copying an archive or repository that causes a collision, as the git
vulnerability. Since different utilities may process resources in different
orders, we generate test cases with both orderings of resources that may cause
collisions.
USE[msg=10960,‘cp’.openat] 00:39|2389| /mnt/folding/dst/ROOT $\hookleftarrow$ CREATE[msg=10957,‘cp’. openat] 00:39|2389| /mnt/folding/dst/root device | inodeprogramaccessed pathsyscallauditd idoperation Figure 4: Example violation reported by name collision testing
The only decisions then are what are the resource types of the source and
target resources and where to place them in the source directory hierarchy to
cause the desired collision to be created. Symbolic links, pipes, and devices
only create interesting behaviors when used as target resources. For symbolic
links, the unsafe effect is to follow the link to another file system
resources, which only happens with the symbolic link as the target resource.
For pipes and devices, the unsafe effect is to send the source resource’s
content to the pipe or device, which also is only possible if these are target
resources.
As a result, the automated test generation produces test cases consisting of
source and target resources of all combinations of potentially unsafe resource
types and places these test cases at depth one and/or two of the file system
hierarchy. For rsync, we specifically found an issue caused by a collision at
depth two, without any collision at depth one, as detailed in Section 7.2.
### 5.2 Detecting Collision Effects
The key idea is to record the file system operations sufficiently to detect
that an unsafe name collision has occurred. Since we design the test cases to
create a name collision on a relocation operation, we want to detect when such
an operation is a successful collision. Then, we need to determine the impact
of the operation to classify the effect according to one of the ten effect
options defined in § 6.1.
We monitor file system operations using auditd to detect successful
collisions. An example of a log indicating a collision is shown in Figure 4.
In this example, a create operation creates a target resource named “root”
using openat command, but a later use operation to the same resource (i.e.,
same device-inode pair, see below) is associated with a name “ROOT”, which
differs from the name used when the resource was created. Note that although
the target resource was created on a case-insensitive file system, multiple
names may be used that are resolved to the same name.
We say that a collision is successful when we detect a use of a target
resource with a different name than that used to create the target resource.
To detect such collisions, we first identify the file system operations that
create a target resource, recording its combination of device and inode
identifiers, which form a unique resource identifier and its pathname. In
Figure 4, the name component “root” will be important to detecting the
collision. We then capture all the file system operations that use the target
resource. In general, if a file pathname in a successful use operation on a
target resource does not match the file pathname of a create for the same
resource, then we record a positive (i.e., a collision). In Figure 4, the
pathname of the use operation differs between “root” and “ROOT”, indicating a
name collision.
We also record a positive when a use operation deletes and replaces a resource
from a prior create operation, as some collisions may cause the target
resource to be deleted and the source resource to replace it. We validate that
there is a create operation for the colliding destination name to verify the
cause of the deletion is a collision.
To detect the effect of a name collision, we examine the resulting resource
that now maps to the target name. We compare the source resource and target
resource content and metadata to the resultant resource to determine whose
content and/or metadata (i.e., source, target, or neither) the resource has.
For tests on directories and hardlinks, we examine the directories and the
resultant directory entries.
## 6 Name Collisions on Linux Copy Utilities
Table 1: Prevalence of copy utilities
tar | zip | cp | cp* | rsync
---|---|---|---|---
| 10 | mc
---|---
8 | perl-modules
7 | libkf5libkleo-data
6 | pluma
6 | mc-data
| …
107 | TOTAL
| 21 | texlive-plain-generic
---|---
15 | aspell
11 | libarchive-zip-perl
7 | texlive-latex-recommended
5 | texlive-pictures
| …
69 | TOTAL
| 78 | hplip-data
---|---
32 | dkms
22 | libltdl-dev
20 | autoconf
18 | ucf
| …
538 | TOTAL
| 12 | dkms
---|---
2 | udev
2 | debian-reference-it
2 | debian-reference-es
1 | zsh-common
| …
25 | TOTAL
| 28 | mariadb-server
---|---
5 | duplicity
4 | texlive-pictures
2 | vim-runtime
1 | rsync
| …
42 | TOTAL
We calculate the number of times that each command (tar, zip, etc.) is used
inside scripts from various packages. We investigate 4752 .deb packages from
the installation disk (DVD #1) of _Debian 11.2.0_. Only the top-five packages
are shown (entries are sorted in descending order for each command).
In this section, we examine how common Linux utilities that applications use
to copy files333We focus on copy operations. The impact on move operations is
similar because (unless both target and source are on the same file system) it
simply performs a copy first and then deletes the source. from one part of the
file system to another react when the copy operation causes a name collision
in a case-insensitive directory. To quantify the ubiquity of these utilities,
we survey their use by packages on Debian 11.2.0. We retrieve all packages
from the Debian installation DVD and count the number of times the copy
utilities are used inside the packages’ scripts. The results are summarized in
Table 1. Note that the listed uses of these utilities are lower bounds because
we do not parse executable binaries. Hence, we miss uses where the utilities
are invoked via system calls such as system(...), execve(...), etc.
### 6.1 Collecting Responses to Name Collisions
Table 2: Name Collision Responses for Popular Linux Utilities
Name Collision between | | | | | |
---|---|---|---|---|---|---
Target Type | Source Type | tar | zip | cp | cp* | rsync | Dropbox
file | file | $\times$ | $A$ | $E$ | $+$$\neq$ | $+$$\neq$ | $R$
symlink (to file) | file | $\times$ | $A$ | $E$ | $+$$T$ | $+$$\neq$ | $R$
pipe/device | file | $\times$ | $-$ | $E$ | $+$ | $+$ | $-$
hardlink | file | $\times$ | $-$ | $E$ | $+$$\neq$ | $+$$\neq$ | $-$
hardlink | hardlink | $C$$\times$ | $-$ | $E$ | $C$$\times$ | $C$$+$$\neq$ | $-$
directory | directory | $+$$\neq$ | $+$$\neq$ | $E$ | $+$$\neq$ | $+$$\neq$ | $R$
symlink (to directory) | directory | $+$ | $\infty$ | $E$ | $E$ | $+$$T$ | $R$
This table shows results of copying files/directories from a case-sensitive to
a case-insensitive file system. cp* refers to cp being used with shell
completion. For e.g., ‘cp * /target’ which copies all items from the current
directory to /target directory.
$\times$
Delete existing file and create new file
$+$
Overwrite existing file. For directories, merge their contents.
$\neq$
Mismatch between content and metadata
$A$
Ask user to resolve the collision
$T$
Traverse symlink
$C$
Corrupts non-colliding files
$E$
Deny operation and report error
$\infty$
Program crashes, or hangs
$-$
Ignore unsupported file type (for hardlinks create regular file instead)
$R$
Rename colliding file/directory
The name collision test cases and the responses of copy utilities are shown in
Table 2. The ‘Target Type’ column represents the resource type of the target
resource that may be overwritten. The ‘Source Type’ represents the resource
type of the source that collides with the target. The rest of the columns
represent individual utilities and their responses to name collisions between
a source resource of the source type and a target resource of the target type.
Below is a comprehensive list of the types of responses observed. Only "Deny"
and "Rename" prevent name collisions from causing unsafe and possibly
exploitable behaviors, although both may block legitimate functionality in
some cases. "Ask the User" may result in an unsafe response if the user allows
the target resource to be overwritten. Note that more than one response is
possible for each test case.
Delete & Recreate ($\times$)
_Delete_ the target resource and _create_ a new resource based on the source
resource. The new resource’s type, as well as its data and metadata, is
determined by the source resource. The target resource is lost without any
notification.
Overwrite ($+$)
Overwrite the data and metadata of the target resource using the source
resource. Unlike Delete & Recreate, the name of the target resource is
preserved. If file foo is being overwritten with file FOO, then the final file
will be named foo but will have the contents and metadata of file FOO.
Corrupt ($C$)
Contents of a resource that is not involved in name collision (i.e., not the
target resource) is modified. For a more in-depth discussion, refer to §
6.2.5.
Metadata Mismatch ($\neq$)
After a successful copy of a given source resource, some metadata, such as its
name, UNIX permissions, user or group ID, extended attributes, or timestamp,
remain from the target resource, creating a resource with a mismatch between
the data (from the source) and the metadata (from the target).
Follow Symlink ($T$)
Follow symbolic links, even when explicitly directed not to do so.
Rename ($R$)
The source name is renamed automatically to avoid creating a name collision,
such as by appending a counter, resulting in a copy of the source resource in
the target resource’s directory with a non-colliding name.
Ask the User ($A$)
To resolve a collision, ask the user to choose from a list of actions, such as
to overwrite the target resource, skip copying the source resource, rename the
target resource, abort, etc.
Note that the user can still choose a response that results in adverse
consequences. For instance, if the user chooses to overwrite the target, the
target’s data and metadata are modified using the source.
Deny ($E$)
Deny the copy associated with a collision and report an error.
Crashes ($\infty$)
Collisions can result in the program hanging (e.g., going into an infinite
loop) or crashing.
Unsupported file type ($-$)
Does not support copying a resource if the source resource is of this file
type. Note that if hardlinks are not recognized by a utility, then it simply
creates a fresh copy of the underlying file.
The exact command-line flags used used to generate Table 2 are listed in
Appendix A. To identify these flags, we analyzed 4,752 .deb packages on Debian
11.2.0’s installation DVD. We found that the most commonly used flags enabled
the following functionality.
* •
Support recursively copying all directories.
* •
Support copying symbolic-links and hard-links as-is but do not follow them.
* •
Preserve metadata such as UNIX permissions, extended attributes (xattr),
timestamps, and owner/group IDs (uid/gid).
Before examining the responses in Table 2, we briefly note some additional
context for two of the columns.
##### cp vs. cp*
Both of these represent the same executable binary. The difference is in the
way the command-line arguments are passed to the binary. Specifically, the
format of specifying the source directories is different.
Consider that the source directory (to be copied) is foo. For cp, we will pass
it as foo/ while for cp* we will use foo. Note the trailing / is missing in
the latter case. Just this difference significantly changes the behavior of cp
as noted in Table 2.
We use the cp* method of invocation coupled with shell completion, e.g., ‘cp
src/* /target’ where the shell replaces src/* with each individual entry
present inside src sans the trailing /. When testing the cp method, we change
the command to ‘cp src/ /target’.
##### Dropbox
Strictly speaking, _Dropbox_[11] is not a copy utility but a popular file
synchronization utility. It is intended to replicate entire directories across
multiple machines and file systems.
We mention Dropbox to highlight its distinct response to handling potential
name collisions. Even when the underlying file system is case-sensitive,
Dropbox treats it as case-insensitive. It proactively renames the files and
directories to avoid name collisions that could occur if they were transferred
to a case-insensitive file system. Note, however, that its renaming strategy
is not even uniform across platforms: For example, the Dropbox application
appends “(Case Conflicts)”, “(Case Conflicts 1)”, etc. to the file/directory
names in case of a potential collision, whereas, when using their web-based
interface, they append “(1)”, “(2)”, etc. instead.
### 6.2 Unsafe Responses to Name Collisions
Several responses shown in Table 2 demonstrate that utilities often allow
unsafe responses to name collisions. In this section, we examine some of the
more concerning responses to show how utilities delegate responsibility for
security against name collisions to the applications that invoke them. For the
examples in upcoming sections, src/ and target/ are on case-sensitive and
case-insensitive file systems respectively.
#### 6.2.1 Silent data loss with tar, cp* & rsync
Name collisions involving files generally result in silent data loss. From
Table 2, we can see that tar deletes and recreates ($\times$) files when
collisions occur. Hence, when there is a name collision between foo and FOO,
only one of these files will remain in the target directory. The other file is
permanently lost without any notification.
Similar to tar, cp* and rsync also lose files silently. However, their
behavior of overwriting ($+$) files results in other problems that are
discussed later in this section.
Unlike tar, zip and cp will ask a user for next steps ($A$) or report an error
($E$) respectively. Hence, they are not prone to silently losing files.
#### 6.2.2 Merge directories with tar, zip, rsync & cp*
Name collisions involving two directories results in their contents (files,
directories, etc. inside the directory) being merged. All of tar, zip, rsync,
and cp* will silently merge directory contents without notifying the user.
Figure 5 highlights this issue using a directory listing.
.1 src/. .2 dir/. .3 subdir/. .3 file1. .3 file2. .2 DIR/. .3 file2.
— copy $\rightarrow$
.1 target/. .2 dir/. .3 subdir/. .3 file1. .3 file2.
Figure 5: Impact of merging directories
In this example in Figure 5, the data of file file2 is overwritten by the
content written last in the copy operation. For example, if src/DIR’s contents
are written last, then its content for file2 is preserved and src/dir’s is
lost.
Furthermore, when the colliding directories have different UNIX permissions, a
collision results in metadata mismatch ($\neq$). With respect to Table 2, the
UNIX permissions of the target resource are overwritten with permissions of
the source resource.
In Figure 5, consider src/dir/ with perms=700 and an adversary who creates
src/DIR/ with perms=777. After a copy (using any of the above utilities),
target/dir/ will have perms=777 effectively giving the adversary permission to
the contents of the original src/dir/.
#### 6.2.3 Stale names
Whenever utilities resort to overwriting ($+$), we end up with stale
file/directory names. For example, consider a name collision between a target
resource foo (file content: ‘bar’) and a source resource FOO (file content:
‘BAR’). After copying with rsync or cp*, we will end up with file foo whose
contents are ‘BAR’.
The problem with such name collisions is that to the end user (or other
programs), it will appear that foo was successfully copied while in reality
FOO was copied. Just using the filename is not enough to discern which files
were successfully copied. This is especially true for case-preserving file
systems where the user has the expectation of the filenames being preserved.
Hence, it is not unreasonable for the user to expect foo should contain bar.
#### 6.2.4 Symbolic link traversal at target
Name collisions between symlink (to file) and a regular file results in cp*
following the symlink ($T$) and overwriting ($+$) its target’s contents with
that of the regular source file. With regards to Table 2, if the target
resource is a symbolic link and the source resource is a file, then cp* ends
up following the symlink and writing data to the resource referenced by the
symlink.
.1 src/. .2 dat(to /foo). .2 DAT= pawn. .1 /foo= bar.
— cp* $\rightarrow$
.1 target/. .2 dat(to /foo). .1 /foo= pawn.
Figure 6: Following symlink
Figure 6 illustrates this case with an example. src/dat is a symbolic link to
/foo and /foo contains ‘bar’. Mallory (our adversary) does not have write
access to /foo but does have access to src/. She creates src/DAT which
contains ‘pawn’.
Then the administrator starts the copy using: cp -a src/* target/. At this
point, cp first creates the symlink target/dat. Then it overwrites ($+$) this
symlink with the contents of src/DAT, effectively updating the file /foo.
After the copy has completed, /foo contains ‘pawn’.
cp* has no command-line options to prevent traversal of symbolic links at the
target. Only link traversal at the source can be turned off via command-line
flags.
#### 6.2.5 The case of hardlink – hardlink name collisions
During a copy when hardlinks (whose targets are different) collide, it can
corrupt ($C$) other non-colliding files and create spurious hardlinks. Table 2
shows that this behavior is exhibited by tar, cp*, and rsync. An interesting
observation is that, regardless of whether the utility’s behavior is Delete &
Recreate ($\times$) or Overwrite ($+$), this problem affects both.
.1 src/. .2 hfoo=foo. .2 zzz=foo. .2 hbar=bar. .2 ZZZ=bar.
— rsync $\rightarrow$
.1 target/. .2 hfoo=bar. .2 zzz=bar. .2 hbar=bar.
Figure 7: hardlink – hardlink name collision
To understand this scenario, consider Figure 7 that uses rsync to perform the
copy. The same color coding represents files that are hard-linked to each
other. So src/hfoo and src/zzz are hard-linked, representing the same file.
These files contain ‘foo’. Similarly, src/hbar and src/ZZZ are hard-linked and
they contain ‘bar’.
After copying using rsync, target/ contains three files that are all hard-
linked to each other. Unlike the src/ directory, target/hfoo, target/hbar,
target/zzz are all hardlinks of each other and they contain ‘bar’.
Additionally, note that although the name collision happened between zzz and
ZZZ, the contents of hfoo were replaced. Even tar, which deletes the old file
and recreates it, exhibits this behavior.
The following order of operations undertaken by rsync result in this behavior.
1. 1.
Copy src/hbar to target/hbar. Now target/hbar contains ‘bar’.
2. 2.
Copy src/zzz to target/zzz. Now target/zzz contains ‘foo’.
3. 3.
In target/, hardlink ZZZ to hbar. Due to NC, this effectively changes zzz to
be hard-linked to hbar. Now target/zzz contains ‘bar’.
4. 4.
In target/, hardlink hfoo to zzz. Now target/foo contains ‘bar’. Additionally,
all three files inside target/ are hard-linked to each other.
The above copy is semantically different from the src/. Specifically, name
collision results in distinct sets of files getting hard-linked with each
other at the target/.
## 7 Case Studies
In this section, we examine case studies where name collisions cause unsafe
behaviors, some of which are exploitable.
### 7.1 dpkg Package Manager
dpkg is the package manager on Debian OS and its derivatives such as Ubuntu.
dpkg packages are compressed tarballs with extension .deb. When dpkg processes
a package, it tracks all files it creates during package installations in a
database. Before installing a new package, dpkg leverages this database to
ensure that any files of previously installed packages will not by overwritten
by this new package thereby preventing potentially malformed packages from
corrupting the system.
On the other hand, we have observed that dpkg will allow a package
installation to replace any file whose name is not in its database, even
privileged user files. Thus, as long as a file in a package has a filename
that does not match the filename of another package’s file, dpkg will install
the file, silently replacing any existing file.
However, regardless of the underlying file system, the above database is
matched in a _case-sensitive_ manner. This allows new packages to _replace
files_ of previously installed packages via name collisions effectively
circumventing the safeguards in dpkg.
In addition, and perhaps even more seriously, dpkg may allow an adversary to
replace a package’s customized config file with the default, reverting
important changes. deb packages can mark certain files as configuration (or
config) files. During package upgrades, if dpkg spots modifications to these
config files then it prompts the user to review the changes.
However, the config files are also matched in a case-sensitive manner. Under
name collisions, dpkg will just replace the original package’s config file
with the config file of the new package. For services, such as sshd, httpd,
etc., config files are critical to their security, so such overwrites can
potentially make the system vulnerable..
##### Reporting
We have reported these issues to the maintainers of dpkg. The maintainers of
dpkg have since updated their package documentation[10] to warn end user
communities not to use dpkg where targets may be case-insensitive (i.e.
specific directories, or entire file systems).
During our discussions, we analyzed 74,688 packages and found 12,237 filenames
from those packages would collide if a case-insensitive file system were used,
breaking multiple packages that contain these files. The name collision
problem is fundamentally entrenched into the way dpkg is implemented because
it reasons about _names_ without involving the underlying file system(s).
### 7.2 Rsync
rsync demonstrates vulnerable behavior when processing name collisions
involving _directories_. During copy, the default behavior of rsync is to
simply recreate the symbolic links present at source. However, when colliding
directories contain sub-directories and symbolic links with the same name, the
collision causes rsync to suffer from link traversal444In this case, the name
collision makes the alias exploitable, again combining name confusions..
Consider the source directory listed in Figure 8. Here, the directories
topdir/ and TOPDIR/ only differ in case. So when copying to a case-insensitive
file system, rsync will encounter a name collision.
.1 src/. .2 topdir/. .3 secret/symlink to /tmp. .2 TOPDIR/. .3 secret/. .4
confidentialregular file.
Figure 8: Case-sensitive source that rsync is copying
We use the following command to perform the copy:
rsync -a src/ dst/
where, |
---|---
-a | recursively copy directories, preserve symlinks, timestamps, DAC
src/ | is case-sensitive
dst/ | is case-insensitive
After the copy is completed, the newly created files are shown in Figure 9.
Note that the file named confidential ends up in /tmp.
.1 dst/. .2 TOPDIR/. .3 secret/symlink to /tmp. .1 /tmp/confidentiallink
traversal.
Figure 9: After copying to case-insensitive destination
rsync has created the /tmp/confidential file by following the symbolic link
dst/TOPDIR/secret.
Below, we describe how this situation can be exploited. Consider an adversary
who wants to access a confidential file in TOPDIR/ to which she lacks any
access. However, she knows that TOPDIR/ is processed by a backup operation
using rsync. If she can create a sibling directory topdir/, to which she will
have read-write access, she can direct rsync to write the confidential file
(inside TOPDIR/) to any directory of her choosing by creating a symbolic link
inside topdir/ to that directory.
##### Reporting
We reported this issue to the rsync maintainers, and they told us that user’s
should not use rsync with non-case honoring file systems. However, we have
concerns about the user community following such a recommendation in this
case, since rsync is often used by individuals.
In the course of these discussions, we learned the cause of the incorrect
behavior. rsync assumes a one-to-one mapping of directories between source and
target file systems. When a name collision results in two source directories
being mapped to a single directory in the target, rsync can be tricked into
incorrectly predicting the target file type. In the presented scenario, a
symbolic link src/topdir/secret (to a directory) is incorrectly inferred to be
a regular directory src/TOPDIR/secret.
rsync uses the O_NOFOLLOW flag with open() to prevent link traversal and uses
openat()/openat2() to contain link traversals within a directory hierarchy,
but it fails correctly use them because it assumes the symbolic link
topdir/secret is a directory.
### 7.3 Apache httpd
Security of certain applications relies on the security parameters of the
underlying file system. One such application is Apache’s httpd. It allows
access to the underlying file system via the HTTP protocol, relying on the
UNIX Discretionary Access Control (DAC) permissions555If the system supports
Mandatory Access Control (MAC), then DAC is used in conjunction with MAC. to
mediate the access. For example, files are accessible over HTTP only if: (i)
its UNIX group is www-data and has read permission for the group, or (ii) has
world-readable UNIX permissions.
Using utilities for copying directories between systems can silently alter
these DAC permissions in unintended ways, leading to serious security lapses.
We illustrate this scenario using Apache httpd and migration of its data using
tar. To study the impact of name collisions on the security parameters, we
assume that the migration happens from a case-sensitive to a case-insensitive
file system. The behavior of tar discussed below draws from the discussion of
Table 2.
To protect sensitive directories, httpd can be configured to only allow
authenticated users to access specific directories. A commonly used approach
is to configure authentication via the .htaccess file [1] which lists the
valid users/groups allowed to access a specific directory over HTTP. All sub-
directories inside the sensitive directory are also protected. We show that
the use of additional security-oriented files can be exploited under the
presence of name collisions.
##### Scenario
httpd serves the contents of www/ (of Figure 10) over HTTP. Initially, www/ is
stored on a case-senstive file system. The directory hidden/ is inaccessible
over HTTP since the others permissions are cleared. Next, protected/ is
configured to be accessible only to specific users using the .htaccess file.
.1 www/. .2 hidden/perm=700. .3 secret.txt. .2 protected/group=www-data,
perm=750. .3 .htaccess(only allow valid users). .3 user-file1.txt. .2
index.html.
Figure 10: www/ on case-sensitive file system
##### Adversary
A UNIX user called Mallory has read-write access to www/ directory. However,
DAC permissions prevent her from accessing hidden/ directory because its owner
is another user. Additionally, protected is inaccessible since Mallory does
not belong to the group www-data.
She modifies www/ as shown in Figure 11 and adds the HIDDEN/ and PROTECTED/
directories with the intent of gaining access to hidden/ and protected/ via a
name collision.
.1 www/. .2 hidden/perm=700. .3 secret.txt. .2 HIDDEN/perm=755. .2
protected/group=www-data, perm=750. .3 .htaccess(only allow valid users). .3
user-file1.txt. .2 PROTECTED/perm=755. .3 .htaccess(empty file). .2
index.html.
Figure 11: Adversary modified www/ on the case-sensitive file system
##### Vulnerability
tar is used to migrate the adversary-modified www/ directory to another system
that uses a case-insensitive file system. Figure 12 shows the state of the
file system once the tarball (archive format of tar) is extracted.
Now, the previously inaccessible hidden/ directory is now accessible over
HTTP. Additionally, since the .htaccess file is cleared, unauthenticated users
will be allowed to view protected/ over HTTP.
.1 www/. .2 hidden/perm=755. .3 secret.txt. .2 protected/perm=755. .3
.htaccess(empty file). .3 user-file1.txt. .2 index.html.
Figure 12: www/ after migrating to case-insensitive file system
##### Reporting
We have reported this scenario to the Apache maintainers, but have not yet
reached a resolution. Using Table 2, we can reason about the above problems.
Under a directory – directory collision, tar incorrectly modifies metadata.
This happens for the hidden/ – HIDDEN/ collision. Here, DAC permissions of the
latter are applied to the former resulting in the leakage of secret files.
For directory – directory collisions, tar will also merge contents of both
directories. For protected/ – PROTECTED/ collision, this merger results in the
empty .htaccess file overwriting the original one that restricts access to
authorized users. The end result is that all users are now allowed access to
the new protected/ directory.
## 8 Potential Defenses
As discussed in the context of name confusion attacks in general in § 3.3, it
can be difficult to produce defenses to prevent name collisions as well. In
this section, we discuss some options and their limitations.
Name collisions are due to differences in case folding rules among file
systems, e.g., case sensitivity and encodings, so it is difficult to ensure
that name collisions cannot happen. Suppose a system has only one file system.
Even then, an archive constructed on another file system using conflicting
case folding rules may cause name collisions to occur when copying the
archive. Since user-space programs cannot determine the case-folding rules
that may be applied to a file, user-space solutions alone will be unreliable.
In addition, they may be prone to TOCTTOU attacks [4, 3]. Thus, extending
library calls like realpath to detect name collisions will not sufficiently
solve the problem. In addition, system solutions lack knowledge of the
programmer intent that caused the collision, so as for name confusion
defenses, systems defenses will suffer from one or more of the follow
limitations: it must have side information about the programs it protects, it
must protect only a subset of all programs, it must be vulnerable to DoS
attacks, it must have false-positives, or it must fail to prevent some
exploits [5].
For example, one idea may be to write a wrapper to vet archives prior to
expansion operations (e.g., tar and zip) to validate that each file in the
archive will result in distinct file after expansion. One way to do this is to
check for name collisions due to case among all the files in the archive.
Although the notion that no files in an archive should collide with any other
files within the archive seems intuitively reasonable, there are critical
drawbacks to implementing such a defense. For example, there may be files
already in the target directory that may result in collisions, limit its
utility. More fundamentally, the case folding rules applied by such a wrapper
are not guaranteed to be the same as those of the target directory.
As a result, we envision that defenses for name collisions will evolve in a
manner similar to defenses for name confusions that utilize the open commands.
Consider that the open command has flags to check whether a file of a
corresponding name exists a creation time, only opening that file when created
anew (i.e., O_CREAT|O_EXCL). This call prevents a name collision from
overwriting an existing file, but it may be too strong a defense. Suppose one
really wants to overwrite files of the same name, but prevent name collisions
from modifying files that actually have differing names (i.e., that only match
due to case folding). In this case, a new flag is necessary, such as
O_EXCL_NAME, which prevents opening a file when the names differ (e.g., based
on strcmp), but not when such names match.
Unfortunately, even with variants of the open command and other defenses, such
as FileProvider classes in Android, programmers continue to make mistakes that
lead to errors and vulnerabilities. The challenge is for programmers to
determine the intent of their operation, understand the threats faced in such
an operation, and configure these complex, low-level commands in such a way
that they block the threats while satisfying the intent. Until file system
APIs enable this combination of requirements, errors will remain common.
## 9 Related Work
Researchers have proposed system and program defenses to thwart name confusion
attacks for alias and squat cases. To date, no defenses that are specific to
name collisions have been proposed.
##### System Defenses
Researchers have long known about name confusion attacks [3, 4] and have
proposed a variety of system defenses [7, 8, 9, 40, 41, 42, 30, 50, 51, 52,
44]. In a system defense, the operating system aims to enforce an invariant
that prevents name confusion attacks from succeeding. A challenge is that
whether an operation is a feature or a vulnerability depends on the
programmer’s intent. As a result, system defenses cannot prevent attacks
completely without introducing the possibility of false positives [5]. Hybrid
defenses have also been proposed [55, 53] where the operating introspects into
the process to leverage program state along with file system state in
enforcement. While reducing false positives, these techniques still lack
programmer intent explicitly, resulting in some false positives.
##### Program Defenses
As a result, systems provide APIs for programmers to decide how to handle name
confusion attacks. Several file system APIs include flags to avoid using
symbolic links entirely (e.g., O_NOFOLLOW flag for the open system call), but
in many cases programmers want to be able to use symbolic links. Researchers
have proposed program-specific defenses to configure APIs or program
frameworks for preventing name confusion attacks [27, 43, 47, 56]. More
advanced commands for file allow programmers to manage how files are open,
including the impact of symbolic links. For example, the openat system call
enables the user to open a directory first to validate its legitimacy before
opening the remaining path. openat2 explicitly constrains how name resolution
is performed to reduce the potential for attacks.
## 10 Conclusion
Interactions among file systems with differing encoding/case-sensitivity
design decisions can lead to name collisions when performing maliciously
crafted, or even ostensibly benign, copy operations. In this paper, we
explored the impact that these name collisions can have on file system
security. Current operating systems do not directly prevent name collision-
based attacks, delegating that responsibility to programmers. In investigating
the utilities used to copy file system resources and repositories/archives, we
demonstrate that they often allow unsafe name collisions and lack the sort of
uniformity in name-collision handling against which safer use policies could
be easily crafted. Further, we show that many applications rely on potentially
unsafe use of these utilities, opening themselves up to exploitable
vulnerabilities. We examine three case studies in greater detail and
demonstrate concrete vulnerabilities to name collisions. Finally, we suggest a
series of directions for future research to systematically defend against name
collision attacks.
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## Appendix A Command-line Flags for Utilities
### A.1 tar
tar can be used to copy files/directories across different systems. In the
first step, tar reads files and directories hierarchies at a given source and
creates an archive (also known as a tarball) that contains information about
the source contents. Subsequently, tar can recreate the source at another
target using the previously created archive. The archive is portable and can
be moved across computer systems.
To generate Table 2, we used GNU tar (v1.30) with the following command-line
flags.
-cf
create new archive and specify its name
-x
extract contents from the archive
-v
verbose mode (does not impact behavior)
### A.2 zip
zip is similar to tar. It also creates an intermediate archive to copy files
across computer systems. The following flags were used with zip (v3.0) to
produce Table 2.
-r
recursively traverse all directories (during archive creation)
–symlinks
store symbolic links in zip archive
### A.3 cp and cp*
cp is a commonly used utility to copy files from a source to a target. Both
the source and target need to be accessible on the same machine.
Unlike tar and zip, cp does not create an intermediate archive. The files and
directories are directly created at the target by reading the source.
The following flags were used with cp (GNU Coreutils v8.30) to generate Table
2.
-a
do not deference symlinks, recursively copy directories, and preserve
attributes (mode, ownership, timestamps, xattr, hardlinks and context)
cp and cp* use the same executable binary. The difference is in the way the
command-line arguments are passed to the binary. Specifically, the format of
specifying the source directories is different.
Consider that the source directory (to be copied) is foo. For cp, we will pass
it as foo/ while for cp* we will use foo. Note the trailing / is missing in
the latter case. Just this difference changes the behavior of cp as noted in
Table 2.
### A.4 rsync
rsync can be used to copy, transfer, or sync file/directories on the same
machine, or across machines (over a custom rsync:// protocol).
The following flags were used with rsync (v3.1.3) to generate Table 2.
-a
recursively copy directories, preserve symlinks, timestamps, DAC permissions,
owners, groups, device files and special files
-H
preserve hardlinks
|
# $f\left(R\right)$ gravity in the Jordan Frame as a Paradigm for the Hubble
Tension
Tiziano Schiavone,1,2 Giovanni Montani,3,4
1Department of Physics “E. Fermi", University of Pisa, Polo Fibonacci, Largo
B. Pontecorvo 3, I-56127 Pisa, Italy
2INFN, Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Polo Fibonacci,
Largo B. Pontecorvo 3, I-56127 Pisa, Italy
3ENEA, Fusion and Nuclear Safety Department, C.R. Frascati, Via E. Fermi 45,
Frascati, I-00044 Rome, Italy
4Physics Department, “Sapienza" University of Rome, P.le Aldo Moro 5, I-00185
Rome, Italy
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We analyze the $f(R)$ gravity in the so-called Jordan frame, as implemented to
the isotropic Universe dynamics. The goal of the present study is to show
that, according to recent data analyses of the supernovae Ia Pantheon sample,
it is possible to account for an effective redshift-dependence of the Hubble
constant via the dynamics of a non-minimally coupled scalar field, emerging in
the $f(R)$ gravity. We face the question both from an analytical and purely
numerical point of view, following the same technical paradigm. We arrive to
establish that the expected decay of the Hubble constant with the redshift $z$
is ensured by a form of the scalar field potential, which remains essentially
constant for $z\lesssim 0.3$, independently if this request is made a priori,
as in the analytical approach, or obtained a posteriori, when the numerical
procedure is addressed. Thus, we demonstrate that an $f(R)$ dark energy model
is able to account for an apparent variation of the Hubble constant due to the
rescaling of the Einstein constant by the $f(R)$ scalar mode.
###### keywords:
cosmology: theory – dark energy – cosmological parameters – galaxies:
distances and redshifts – supernovae: general
††pubyear: 2022††pagerange: $f\left(R\right)$ gravity in the Jordan Frame as a
Paradigm for the Hubble Tension–$f\left(R\right)$ gravity in the Jordan Frame
as a Paradigm for the Hubble Tension
## Introduction
The measurement of the Hubble constant $H_{0}$ has been the most important and
challenging effort of large-scale observations of the present Universe since
the very beginning of cosmological studies. For many decades, the value of
$H_{0}$ has been determined with a very low degree of precision. However,
since the beginning of the new century, the emergence of the so-called
’precision cosmology’ has allowed the determination of $H_{0}$ with an
appropriate degree of accuracy, and the possibility to test an increasingly
large set of cosmological parameters became concrete. Nowadays, a large number
of different and independent measurements of $H_{0}$ are available (Di
Valentino et al., 2021) and also other crucial cosmological indicators can be
found, like the position of peaks in the cosmic microwave background (CMB)
data thanks to the Planck satellite (Aghanim et al., 2020).
However, the end of the previous century has been characterized by a really
non-trivial surprise, coming from low-redshift observations of Type Ia
supernovae (SNe Ia) recovered as standard candles: the present Universe is
accelerating (Riess et al., 1998; Perlmutter et al., 1999). This experimental
issue opened a new era on the understanding of the present Universe dynamics
and physics, since either a dark energy component living in the cosmological
space (identified in a cosmological constant term in the so-called
$\Lambda$CDM model (Weinberg, 2008)), or a modified gravity theory in the late
expansion phase must be postulated to represent the emerging acceleration
phenomenology.
In this very puzzling panorama, in recent years, in particular after the
observations of the Planck satellite, an additional non-trivial observational
evidence came out, called the Hubble tension (Di Valentino et al., 2021), i.e.
a discrepancy in 4.9 $\sigma$ between the determination of $H_{0}$ via the CMB
data ($H_{0}=67.4\pm 0.5\,\textrm{km s}^{-1}\,\textrm{Mpc}^{-1}$) (Aghanim et
al., 2020) and that one coming out by using low-redshift cosmological testers,
like the Cepheid-SN Ia sample ($H_{0}=73.04\pm 1.04\,\textrm{km
s}^{-1}\,\textrm{Mpc}^{-1}$) (Riess et al., 2022). This discrepancy appears
hard to be straightforwardly interpreted.
Indeed, the emergence of different values of $H_{0}$ from several cosmological
probes can be attributed essentially to two questions only: either the
astrophysical characterization of the tester is inadequate (for instance,
because their calibration or redshift evolution are not properly fixed) or new
physics, in addition to the dark energy Universe component, must be considered
for describing the late phase of expansion. In both these cases, the resulting
observational or physical picture seems to require a deed revision. The
analysis pursued by the SNe Ia community, mainly represented by the data of
the Pantheon (Scolnic et al., 2018) and Pantheon+ (Riess et al., 2022)
samples, seem to exclude the existence of a redshift evolution of these
objects and show a reliable control of all the main sources of errors.
However, some recent studies may report a possible redshift dependence of the
marginalized absolute magnitude or the Hubble constant itself (Krishnan et
al., 2020, 2021; Kazantzidis & Perivolaropoulos, 2020; Dainotti et al., 2021,
2022; Krishnan & Mondol, 2022; Schiavone et al., 2022; Colgáin et al., 2022).
In particular, in the two analyses by Dainotti et al. (2021, 2022) it is
argued that a binned distribution of the SNe Ia with equipopulated bins
outlines a variation of $H_{0}$ as $(1+z)^{-\alpha}$ within 2 $\sigma$
confidence level, where the power exponent $\alpha$ is of the order of few
percents. Furthermore, in Dainotti et al. (2021), it has been stressed that
the extrapolation of the behavior of $H_{0}(z)$ up to the CMB redshift seems
to naturally account for the Hubble tension feature, since a larger value of
the Hubble constant today would slowly decrease to higher redshifts.
Several alternative proposals have been addressed to explain the $H_{0}$
tension. First of all, a local underdensity of the Universe could account for
the discrepancy between low redshift observations and the Planck data, simply
because the latter are referred to the (larger) average density of the
Universe, according to the original Hubble-bubble model (Zehavi et al., 1998).
However, these proposals are not quite working, since we would need an
enormous local underdensity of the actual Universe, which is not currently
observed (Kenworthy et al., 2019; Camarena et al., 2022) and would affect
critically photon paths. We can note that the Hubble tension is a serious
puzzle because the observed discrepancy is a significant effect when stated on
dynamical level. Moreover, to account for it, we can not alter other well-
established observations.
Another possibility to explain the Hubble tension is provided by modified
gravity theories, as applied to the late Universe. In particular, in Dainotti
et al. (2021), it was argued that the observed dependence of $H_{0}(z)$,
predicted by the binning analysis of the Pantheon sample, can be interpreted
as a variation of the Einstein constant, which can be naturally achieved with
the $f(R)$ gravity theory (Nojiri & Odintsov, 2007; Starobinsky, 2007;
Tsujikawa, 2008; Sotiriou & Faraoni, 2010) in the Jordan frame (Faraoni &
Capozziello, 2011) via the non-minimally coupled scalar field related to the
$f$ functional form.
However, in Dainotti et al. (2022) it was shown how one of the most reliable
dark energy models, the Hu-Sawicki proposal (Hu & Sawicki, 2007), is
inappropriate to reproduce the desired effect, i.e. the resulting luminosity
distance does not correspond to a restated constant $H_{0}$ value across the
whole Pantheon sample. This negative result suggested the necessity to
construct ad hoc a dark energy model able to account for the Universe
acceleration, and at the same time responsible for the apparent variation
$H_{0}(z)$.
The present letter is dedicated to constructing a model having the
requirements stated above, constructed following the prescriptions emerging
from the binning analysis in Dainotti et al. (2021, 2022). We start from the
evidence of a decreasing trend of $H_{0}(z)$ associated with the above
sketched power-law decay with $z$, and then we construct the profile of the
potential term of the non-minimally coupled scalar field $\phi$, which fixes
the morphology of the required $f(R)$ theory.
The analysis is divided into two parts: the first one is characterized by an
analytical approach, while the second one relies on a pure numerical study.
The analytical formulation starts with the assumption that a flat region of
the potential exists, and then the deviation from this constant value (i.e.
the global form of the scalar field potential) is calculated as the dynamics
is defrozen, i.e. in correspondence to increasingly large redshifts. Unlike
the analysis in Dainotti et al. (2021, 2022), we do not use an ansatz for
$\phi(z)$, but we obtain it from the cosmological dynamics in the Jordan
frame. In the numerical analysis, we assume again the evolution of $H_{0}(z)$,
but we relax the request of a constant potential term in a given region. It is
remarkable that both the analytical and the numerical formulations are
consistent and predict a flat potential profile in a region $0<z\lesssim 0.3$,
associated with a slow-rolling-like dynamics and a dark energy candidate.
An important consistency check for the model formulation is provided by the
determination of the $f(R)$ form in the limit of the low-redshift
approximation. We get three contributions: a cosmological constant, a linear
contribution in the Ricci scalar $R$, and eventually a quadratic correction as
in the $R^{2}$-gravity theory (Starobinsky, 1980). It is worth emphasizing
that this modified theory reduces to the standard $\Lambda$CDM model if the
function $H_{0}(z)$ is frozen to a constant value and $df/dR\equiv 1$ today.
The present study shows that the emerging $f(R)$ gravity in the Jordan frame
can account for the behavior $H_{0}(z)$ within the whole redshift range of the
Pantheon sample and, as it must be, also reproduces a reliable explanation for
the present Universe acceleration.
This work is organized as follows: in Sect. 1 we briefly introduce the $f(R)$
modified gravity in the Jordan frame within the framework of a homogeneous and
isotropic Universe; in Sect. 2 we derive the scalar field potential, inferred
from a running Hubble constant with the redshift; in Sect. 3 we show our
numerical solutions; in Sect. 4 we obtain the functional form of $f(R)$ in the
low-redshift limit; in Sect. 5 we summarize our key findings.
The metric signature adopted here is $\left(-,+,+,+\right)$, and the speed of
light is $c=1$. The Newton constant is denoted with $G$, while the Einstein
constant is defined as $\chi\equiv 8\,\pi\,G$.
## 1 $f\left(R\right)$ gravity in the Jordan frame for a homogeneous and
isotropic Universe
The $f(R)$ modified gravity theories provide a useful extension of General
Relativity (GR) up to cosmological scales. An extra scalar degree of freedom
with respect to GR occurs in this environment, since the gravitational
Lagrangian density is given by a function $f(R)$ of the Ricci scalar $R$. The
$f(R)$ metric formalism involves higher-order derivatives in field equations
up to the fourth-order with respect to the metric, but the Einstein
gravitational field equations in GR are recovered if $f\left(R\right)=R$. The
$f(R)$ proposal can be reformulated using a dynamically equivalent action in
the so-called Jordan frame:
$S_{J}=\frac{1}{2\,\chi}\,\int
d^{4}x\,\sqrt{-g}\,\left[\phi\,R-V\left(\phi\right)\right]+S_{M}\left(g_{\mu\nu},\psi\right)\,,$
(1)
where $g$ is the determinant of the metric tensor, $S_{M}$ is the matter
action with matter fields $\psi$. Note that in the Jordan frame the additional
degree of freedom is provided by the scalar field $\phi$, which is related to
$f(R)$ through the relation $\phi=f^{\prime}\left(R\right)=df/dR$, and it is
controlled by the scalar field potential
$V\left(\phi\right)=\phi\,R\left(\phi\right)-f\left(R\left(\phi\right)\right)$.
Considering a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric
(Weinberg, 2008), the field equations in the Jordan frame are the following:
$\displaystyle
H^{2}=\frac{\chi\,\rho}{3\,\phi}+\frac{V\left(\phi\right)}{6\,\phi}-H\,\frac{\dot{\phi}}{\phi}$
(2a)
$\displaystyle\frac{\ddot{a}}{a}=-\frac{\chi}{6\,\phi}\,\left(\rho+3P\right)+\frac{V\left(\phi\right)}{6\,\phi}-\frac{H}{2}\,\frac{\dot{\phi}}{\phi}-\frac{1}{2}\,\frac{\ddot{\phi}}{\phi}$
(2b) $\displaystyle
3\ddot{\phi}-2\,V\left(\phi\right)+\phi\,\frac{dV}{d\phi}+9\,H\,\dot{\phi}=\chi\,\left(\rho-3P\right),$
(2c)
where $\dot{}=\frac{d}{dt}$, being $t$ the cosmic time in the synchronous
gauge, $H(t)$ the Hubble parameter, $\rho\left(t\right)$ and $P\left(t\right)$
the energy density and pressure of a cosmological fluid, respectively. Eq.
(2a) is the generalized Friedmann equation, Eq. (2b) gives us the modified
cosmic acceleration, and Eq. (2c) is the scalar field equation.
Moreover, the divergenceless of the stress-energy tensor for a perfect fluid
gives
$\dot{\rho}+3\,H\,\left(\rho+P\right)\,.$ (3)
In particular, Eqs. (2a) and (2c) are the two independent equations regarding
the dynamics in the Jordan frame. To solve them, one must also specify the
equation of state that for a barotropic fluid is $P\left(\rho\right)=w\,\rho$,
according to the equation of state parameter of a cosmological fluid (i.e.
$w=0$ for the matter component).
The advantage of the Jordan frame is that now field equations contain
derivatives with respect to the metric or the scalar field up to second order
only, however the scalar field is non-minimally coupled to the metric. As a
consequence of the scalar field dynamics, an effective Einstein constant
$\chi/\phi$ emerges in the Jordan frame, as it can be seen in Eq. (2a).
Cosmological models based on $f(R)$ modified gravity theories have been
employed to predict deviations from the $\Lambda$CDM model and provide the
cosmic acceleration in late times without a true cosmological constant.
## 2 Approximated solution for the scalar field potential
We consider the dynamics of the gravitational field equations in the Jordan
frame of the $f(R)$ modified gravity scenario to build the profile of the
scalar field potential $V\left(\phi\right)$, assuming the evolution of $H_{0}$
with the redshift
$H_{0}\left(z\right)=\frac{\tilde{H}_{0}}{\left(1+z\right)^{\alpha}}\,,$ (4)
as a phenomenological result of the redshift binned analysis performed in
Dainotti et al. (2021, 2022). Hence, the respective Hubble parameter in a flat
almost like $\Lambda$CDM model is given by
$H\left(z\right)=\frac{\tilde{H}_{0}}{\left(1+z\right)^{\alpha}}\,\sqrt{\Omega_{m0}\left(1+z\right)^{3}+1-\Omega_{m0}}\,,$
(5)
where $\Omega_{m0}$ is the cosmological density parameter for the matter
component. Moreover, we focus on a cosmological dust (pressureless matter
$P=0$) in the late Universe, hence we neglect relativistic components, and
$\rho\propto\left(1+z\right)^{3}$ by solving the continuity equation (3).
Without loss of generality, we now set $\phi=F\left(z\right)$, so that Eq.
(2a) rewrites
$H^{2}=\frac{1}{F-\left(1+z\right)\,F^{\prime}}\frac{\chi}{3}\left(\rho+\frac{V\left(F\right)}{2\,\chi}\right)\,,$
(6)
where $F^{\prime}\equiv dF/dz$. We used the definition of redshift
$a_{0}/a=1+z$ with the standard assumption that the scale factor today is
$a_{0}=1$, and also the fact that $dz/dt=-\left(1+z\right)\,H\left(z\right)$.
In order to rewrite the generalized Friedmann equation in a form similar to
the respective one in GR and discuss the $\Lambda$CDM limit, we assume the
existence of a region in which $V\left[F\left(z\right)\right]\approx
2\chi\rho_{\Lambda}=\textrm{const}$ for $0<z\lesssim z^{*}$, where
$\rho_{\Lambda}$ is the present value of the Universe dark energy density, and
$z^{*}\sim 0.3$ is the redshift of matter-dark energy equality. Hence, we
rewrite Eq. (6) as
$H^{2}=\frac{1}{F-\left(1+z\right)\,F^{\prime}}\frac{\chi}{3}\left(\rho+\rho_{\Lambda}\right)\,,$
(7)
where you can recognize the usual terms in the Friedmann equation in the
$\Lambda$CDM scenario, unless a factor related to the scalar field $\phi$.
Then, since we look for a rescaling of the Hubble constant by a factor
$\left(1+z\right)^{-\alpha}$, comparing Eqs. (5) and (7), it is natural to
require the following condition
$F-\left(1+z\right)\,F^{\prime}=\left(1+z\right)^{2\alpha}\,,$ (8)
which admits the solution:
$\phi\left(z\right)=F\left(z\right)=\frac{\left(1+z\right)^{2\alpha}}{1-2\,\alpha}+\left(K-\frac{1}{1-2\,\alpha}\right)\,\left(1+z\right)\,.$
(9)
We fixed the initial condition $\phi\left(0\right)=K$ at $z=0$, where $K$
denotes the deviation from a pure GR scenario ($\phi=1$).
Let us now take into account the scalar field equation (2c). Using the
relation $\phi=F\left(z\right)$ and the approximation for the scalar field
potential $V\left[F\left(z\right)\right]\approx 2\chi\rho_{\Lambda}$, we
obtain:
$\displaystyle
3\,H^{2}\,\left[\left(1+z\right)^{2}\,F^{\prime\prime}-\left(1+z\right)\,F^{\prime}\right]-3\,\left(1+z\right)\,\frac{\ddot{a}}{a}\,F^{\prime}+F\,\frac{dV}{dF}=$
$\displaystyle=\chi\,\left(\rho+4\rho_{\Lambda}\right)\,.$ (10)
It should be emphasized that we do not neglect the term $dV/dF$, since we want
to check a posteriori the viability of the approximation for the scalar field
potential at low redshifts.
Furthermore, by substituting the term $\ddot{\phi}$ from Eq. (2c) in Eq. (2b),
we have
$\frac{\ddot{a}}{a}=-\frac{\chi}{3\,F}\,\left(\rho+\rho_{\Lambda}\right)+\frac{1}{6}\frac{dV\left(F\right)}{dF}-H^{2}\,\left(1+z\right)\frac{F^{\prime}}{F}\,.$
(11)
Then, we combine Eqs. (10) and (11), and we employ the solution for
$\phi\left(z\right)$ given by Eq. (9). Finally, we obtain an expression for
$\displaystyle\frac{d\tilde{V}}{dz}=\frac{6\left[1+z-2\alpha\left(1+z\right)^{2\alpha}-K\left(1+z\right)\left(1-2\alpha\right)\right]}{\Omega_{m0}\left(1+z\right)\left[1+z+2\left(1+z\right)^{2\alpha}\left(\alpha-1\right)+K\left(1+z\right)\left(2\alpha-1\right)\right]}$
$\displaystyle\times\left[2\left(2+\alpha\right)+\Omega_{m0}\left(z\left(1+2\alpha\right)\left(3+z\left(3+z\right)\right)-3\right)\right]\,,$
(12)
where we rescaled the potential as a dimensionless quantity $\tilde{V}\equiv
V/m^{2}$ with the constant
$m^{2}\equiv\chi\rho_{0}/3=\tilde{H}_{0}^{2}/\Omega_{m0}$. We denote with
$\rho_{0}$ the actual matter density, and we used the definitions of the
critical energy density of the Universe today
$\rho_{c0}=3\tilde{H}_{0}^{2}/\chi$ and also of the cosmological density
parameter $\Omega_{m0}=\rho_{0}/\rho_{c0}$.
Note that Eq. (12) cannot be solved analytically, but we need to proceed
numerically to obtain $\tilde{V}(z)$. We evaluate the numerical integral of
Eq. (12) for different redshifts within the range of the Pantheon sample, i.e.
$0<z<2.3$, and we build a grid with a step $\Delta z=10^{-3}$. If
$V\left[F\left(z\right)\right]=2\chi\rho_{\Lambda}$ at $z=0$, then it is easy
to show we need to add the following integration constant
$\tilde{V}(0)=6\left(1-\Omega_{m0}\right)/\Omega_{m0}$, when we integrate Eq.
(12). We fixed the values for $\alpha=9\cdot 10^{-3}$ (Dainotti et al., 2021),
$\Omega_{m0}=0.298$ (Scolnic et al., 2018), and $K=1-10^{-7}$ (Hu & Sawicki,
2007). After obtaining different points as a result of numerical integrals, we
use a polynomial fit up to the eighth-order in the redshift to calculate
$\tilde{V}(z)$.
The evolution of $\phi(z)$, given by Eq. (9), is plotted in Fig. 1 with a blue
line. As a final check, in Fig. 2 you can see the resulting profile of
$\tilde{V}$, which appears flat for $0<z\lesssim z^{*}$ in the dark-energy-
dominated era, according to our preliminary assumptions on the flatness of
$V(z)$ for low redshifts.
## 3 Numerical analysis of the model
We now relax the assumption on the existence of a flat region of the scalar
field potential for $0<z\lesssim z^{*}$, but we continue to consider the
evolution of $H_{0}(z)$ given by Eq. (4). Let us proceed with a complete
numerical analysis to solve the cosmological dynamics in the Jordan frame
(Sect. 1) in the presence again of a matter component, in which we have two
independent equations (2a) and (2c), and two unknown quantities, i.e.
$\phi(z)$ and $V\left[\phi(z)\right]$.
Firstly, we rewrite the generalized Friedmann equation (2a) in the variable
$z$, isolating the dimensionless scalar field potential
$\displaystyle\tilde{V}\left[\phi\left(z\right)\right]$
$\displaystyle=6\,\left[\frac{\phi(z)}{\left(1+z\right)^{2\alpha}}-\left(1+z\right)^{1-2\alpha}\,\frac{d\phi}{dz}\right]\left[\left(1+z\right)^{3}+\frac{1-\Omega_{mo}}{\Omega_{m0}}\right]$
$\displaystyle-6\left(1+z\right)^{3}\,,$ (13)
where we used Eq. (5) and the fact that $\rho\sim\left(1+z\right)^{3}$.
Secondly, we rewrite the scalar field equation (2c) likewise:
$\displaystyle\left[\left(1+z\right)^{2-2\alpha}\frac{d^{2}\phi}{dz^{2}}-\left(2+\alpha\right)\left(1+z\right)^{1-2\alpha}\frac{d\phi}{dz}\right]\left[\left(1+z\right)^{3}+\frac{1-\Omega_{m0}}{\Omega_{m0}}\right]$
$\displaystyle+\frac{3}{2}\frac{d\phi}{dz}\left(1+z\right)^{4-2\alpha}-\frac{2}{3}\tilde{V}\left[\phi(z)\right]+\frac{\phi(z)}{3}\left(\frac{d\phi}{dz}\right)^{-1}\frac{d\tilde{V}}{dz}=\left(1+z\right)^{3}\,.$
(14)
Then, by substituting $\tilde{V}\left[\phi(z)\right]$ from Eq. (13) into Eq.
(14), we obtain a second-order differential equation in $\phi(z)$. We solve
numerically this equation with the following initial conditions for $z=0$:
$\phi(0)=K$, and $d\phi/dz\,(0)=-10^{-3}$, where we consider a slowly varying
scalar field. We fixed the same values for $\alpha$, $\Omega_{m0}$, and $K$
adopted in Sect. 2.
In Fig. 1 we show the evolution of $\phi$ with $z$ using a red line, while in
Fig. 2 we plot the profile of $\tilde{V}$ in terms of $z$ and $\phi$. In all
these figures, we also compare our numerical results with the respective
profiles obtained from the approximated solution based on the assumption of a
flat potential at low redshifts in Sect. 2, noting that corresponding
solutions are very close for $z\ll 1$. It should be noted that the potential
$\tilde{V}$ exhibits a flat profile for $0<z\lesssim z^{*}$ also for the
numerical solution.
Figure 1: Behavior of the scalar field $\phi$ VS redshift $z$ in the Jordan
frame, assuming the phenomenological decreasing trend of $H_{0}\left(z\right)$
with redshift in Eq. (4). The blue line is referred to the approximated
solution developed in Sect. 2, where we started from the hypothesis of a flat
scalar field potential for low redshifts and $\phi(z)$ is given by Eq. (9),
while the red line is obtained from the numerical analysis discussed in Sect.
3, after solving Eqs. (13) and (14).
Figure 2: Profile of the scalar field potential in terms of the redshift $z$
(top panel) and the scalar field $\phi$ (bottom panel) in the Jordan frame,
inferred from the assumption of a running Hubble constant
$H_{0}\left(z\right)$, according to Eq. (4). Note that
$\tilde{V}=V(\phi)/m^{2}$ is a dimensionless potential. The blue line is
referred to the approximated solution developed in Sect. 2, while the red line
is obtained from the numerical analysis discussed in Sect. 3. The potential is
flat for $0<z\lesssim z^{*}$ for both these solutions.
## 4 The low-redshift $f(R)$ profile
We are interested in obtaining an analytical expression for the scalar field
potential and the respective $f(R)$ function, which provides both the late-
time cosmic acceleration and a running Hubble constant with the redshift,
according to Eq. (4). To this end, we expand the solution for $\phi(z)$ and
the term $d\tilde{V}/dz$ in the limit of low redshifts for $z\ll 1$ up to the
linear order in $z$.
More specifically, starting from Eq. (9) and Eq. (12), we get
$\displaystyle\phi(z)\approx K+\left(K-1\right)\,z\,,$ (15)
$\displaystyle\frac{d\tilde{V}}{dz}\approx A_{1}+2\,A_{2}\,z\,,$ (16)
respectively, where the constants $A_{1}$ and $A_{2}$ are defined as
$\displaystyle
A_{1}=6\,\frac{K-1}{K+1}\,\frac{4+2\alpha-3\Omega_{m0}}{\Omega_{m0}}\,,$ (17)
$\displaystyle
A_{2}=-6\,\frac{\alpha+2}{\Omega_{m0}}\,\frac{\left(K-1\right)^{2}+4\alpha
K}{\left(K+1\right)^{2}}\,,$ (18)
Looking at Eq. (16), we can integrate it analytically to write $\tilde{V}(z)$
in the low-redshift limit:
$\tilde{V}(z)\approx A_{0}+A_{1}\,z+A_{2}\,z^{2}\,,$ (19)
where the integration constant $A_{0}$ is set to be
$A_{0}=6\frac{1-\Omega_{m0}}{\Omega_{m0}}\,,$ (20)
assuming $V\left[F\left(0\right)\right]=2\chi\rho_{\Lambda}$. Note that
$A_{0}$, $A_{1}$, and $A_{2}$ depend on the values of $\alpha$, $K$, and
$\Omega_{m0}$.
By inverting the relation (15) and substituting $z\left(\phi\right)$ in Eq.
(19), we can easily write the scalar field potential
$\tilde{V}\left(\phi\right)\approx
A_{0}+A_{1}\,\frac{\phi-K}{K-1}+A_{2}\,\left(\frac{\phi-K}{K-1}\right)^{2}\,.$
(21)
It should be noted that $\tilde{V}\left(\phi\right)$ has constant, linear, and
quadratic contributions in $\phi$.
Once we have the expression for $\tilde{V}\left(\phi\right)$, we use the
equation $R=dV/d\phi$, which was obtained by varying the action in Eq. (1)
with respect to $\phi$. Then, we end up in a relation $R=R\left(\phi\right)$,
which admits the existence of the inverse $\phi=\phi\left(R\right)$. Finally,
using the relation in the Jordan frame
$f\left(R\right)=R\,\phi\left(R\right)-V\left[\phi\left(R\right)\right]$, we
obtain
$f\left(R\right)\approx
B_{0}+B_{1}\,\frac{R}{m^{2}}+B_{2}\,\frac{R^{2}}{m^{2}}\,,$ (22)
where we have defined the constants
$\displaystyle
B_{0}=m^{2}\,\left(\frac{A_{1}^{2}}{4A_{2}}-A_{0}\right)\qquad\quad
B_{1}=m^{2}\,\left[K-\frac{A_{1}\,\left(K-1\right)}{2\,A_{2}}\right]$ (23)
$\displaystyle B_{2}=\frac{\left(K-1\right)^{2}}{4\,A_{2}}$ (24)
It should be stressed that Eq. (22) provides an approximated solution of the
$f\left(R\right)$ function for $z\ll 1$, which contains constant, linear, and
quadratic terms in $R$. Note that if $K\rightarrow 1$ and $\alpha\rightarrow
0$, we recover the $\Lambda$CDM model.
## 5 Conclusions
We started our analysis from the results obtained by Dainotti et al. (2021,
2022), which outlined a dependence of the value of $H_{0}$ with the redshift
via a binned data analysis of the SNe Ia Pantheon sample within 2 $\sigma$.
The specific form of the decaying $H_{0}(z)$ given in Eq. (4) was the
phenomenological input of our theoretical study.
The idea proposed above consists in setting up a dark energy model that is
also able to account for a variation with $z$ of the $H_{0}$ value. More
specifically, we adopted the theoretical paradigm of a $f(R)$ extended theory
of gravity, as viewed in the Jordan frame (Sect. 1). Thus, the non-minimally
coupled scalar field dynamics was implemented to account for an effective
variation of the Einstein constant, just the required feature to account the
phenomenology predicted by the binned analysis.
Starting from the basic equations for an isotropic Universe in the proposed
theoretical framework, we assumed, without any loss of generality, that the
scalar field is a function of the redshift, and we determined such specific
relation in Eq. (9) by imposing the desired decaying of $H_{0}(z)$. Then, the
scalar field dynamics was able to provide us the corresponding form of the
scalar field potential term, which fixes the adopted $f(R)$ model.
This picture has been performed both analytically and numerically: in the
former case, in Sect. 2 we assumed the existence of flat region of the scalar
field potential, there approximated via constant value, and then we pursued
the explicit determination of the potential term derivative in Eq. (12); in
the latter case, the scheme was implemented directly on the two basic
equations (2a) and (2c) without any assumption on the potential form. It was
rather remarkable that, in both analyses, the potential term singled out a
flat region for $z\lesssim 0.3$ (Fig. 2), which is exactly when the dark
energy contribution of the Universe dominates on the matter content.
It is worth noting how the low-redshift limit of our model in Sect. 4 allowed
an analytical determination of the potential term, and hence of the underlying
$f(R)$ model. The resulting expression (22) for the modified Lagrangian of the
modified gravity contains a cosmological constant, and also linear and
quadratic contributions in the Ricci scalar. In particular, this result in the
low-redshift limit is consistent with other $f\left(R\right)$ proposals based
on quadratic gravity or the $f\left(R\right)$ power series expansion to
describe deviations from GR and the $\Lambda$CDM cosmological model in the
late Universe without introducing dark energy (Starobinsky, 1980; Sotiriou &
Faraoni, 2010; Cosmai et al., 2016; Fanizza et al., 2020).
It is very remarkable for the robustness of our proposed model that this
modified scheme approaches the $\Lambda$CDM scenario when $\alpha\rightarrow
0$ and $df/dR\rightarrow 1$. In other words, even if we reduce the function
$H_{0}(z)$ to a fixed constant value, our model can still contain a small
deviation from the $\Lambda CDM$ Universe, according to its nature of a new
proposal for explaining the dark energy phenomenon.
Thus, we can claim that our study is able to simultaneously address two key
points: on one hand, we get a modified gravity model as a suitable dark energy
candidate; on the other hand, we provided a natural interpretation for the
profile of $H_{0}(z)$ obtained in [nostri articoli], by identifying in the
proposed modified gravity approach the lacking physics in the emerging picture
from the Pantheon sample.
The present study calls attention to further investigations on the proposed
model as the redshift increases towards the CMB observations, in order to
understand if it can satisfactory solve the Hubble tension.
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# The binary and the disk: the beauty is found within NGC3132 with JWST
Raghvendra Sahai Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, CA 91109, USA Valentin Bujarrabal Observatorio
Astroómico Nacional (OAN/IGN), Ap 112, 28803 Alcalá de Henares, Spain
Guillermo Quintana-Lacaci Dept. of Molecular Astrophysics, IFF-CSIC. C/
Serrano 123, E-28006, Madrid, Spain Nicole Reindl Institut für Physik und
Astronomie, Universität Potsdam, Karl-Liebknecht-Stra$\beta$e 24/25, D-14476,
Potsdam, Germany Griet Van de Steene Royal Observatory of Belgium, Astronomy
and Astrophysics, Ringlaan 3, 1180 Brussels, Belgium Carmen Sánchez Contreras
Centro de Astrobiologia (CAB), CSIC-INTA. Postal address: ESAC, Camino Bajo
del Castillo s/n, 28692, Villanueva de la Cañada, Madrid, Spain Michael E.
Ressler Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, CA 91109, USA
###### Abstract
The planetary nebula (PN) NGC 3132 is a striking example of the dramatic but
poorly understood, mass-loss phenomena that $(1-8)$ $M_{\odot}$ stars undergo
during their death throes as they evolve into white dwarfs (WDs). From an
analysis of JWST multiwavelength ($0.9-18$ µm) imaging of NGC 3132, we report
the discovery of an asymmetrical dust cloud around the WD central star (CS) of
NGC 3132, seen most prominently in the 18 µm image, with a surface-brightness
limited radial extent of $\gtrsim 2{{}^{\prime\prime}}$. We show that the A2V
star located $1\farcs 7$ to CS’s North-East (and 0.75 kpc from Earth) is
gravitationally-bound to the latter, by the detection of relative orbital
angular motion of $0.24\arcdeg\pm 0.045\arcdeg$ between these stars over
$\sim$20 yr. Using aperture photometry of the CS extracted from the JWST
images, together with published optical photometry and an archival UV
spectrum, we have constructed the spectral-energy distribution (SED) of the CS
and its extended emission over the UV to mid-IR ($0.091-18$ µm) range. We find
that fitting the SED of the CS and the radial intensity distributions at
$7.7,12.8$ and $18$ µm with thermal emission from dust requires a cloud that
extends to a radius of $\gtrsim$1785 au, with a dust mass of $\sim 1.3\times
10^{-2}$ M⊕, and grains that are 70% silicate and 30% amorphous carbon. We
propose plausible origins of the dust cloud and an evolutionary scenario in
which a system of three stars – the CS, a close low-mass companion, and a more
distant A2V star – forms a stable hierarchical triple system on the main-
sequence but becomes dynamically unstable later, resulting in the spectacular
mass-ejections that form the current, multipolar PN.
circumstellar matter – stars: AGB and post-AGB – stars: individual (NGC 3132)
– stars: mass loss – stars: jets
cont#1 (cont.)#2#3
## 1 Introduction
In our quest to identify the signatures of extraterrestrial life, the search
for planets and planetary systems around stars other than our Sun has become
one of the most exciting areas of current astrophysical research. Planets are
found to be common around other solar-type stars, but the disks in which these
are produced dissipate as the stars reach the main-sequence. A Spitzer 24 µm
survey of main-sequence A-type stars shows that up to $\sim$50% of young
($\lesssim$30 Myr) stars have little or no 24 µm excess emission from debris
disks; and large debris-disk excesses decrease significantly at ages of
$\sim$150 Myr, and much of the dust detected in these is likely produced
(episodically) by large planetesimal collisions (Rieke et al., 2005). The dust
in these debris disks would have dissipated long before the stars evolve off
the main sequence. Thus it is remarkable that once these stars reach the end
of their life cycle, observational surveys reveal disks or equatorially-
flattened disk-like or toroidal structures (e.g., Hillen et al. (2017); Sahai
et al. (2007, 2011)) – i.e., both the birth and the death of stars represent
similar highly aspherical states that sandwich the more spherical life of
stars on the main sequence (MS).
The demise of most stars in the Universe that evolve in a Hubble time (i.e.,
in the 1–8 $M_{\odot}$ range) is believed to occur as a result of heavy mass-
loss (with rates up to $10^{-4}$ $M_{\odot}$ yr-1) on the Asymptotic Giant
Branch (AGB), when the stars are very luminous ($L\sim 5000-10,000$
$L_{\odot}$) and cool ($T_{eff}<3000$ K) (see e.g., review by Decin (2021)).
After mass-loss – which may be via a quasi-steady wind over $\sim 10^{5}$ yr
or rather sudden (if driven by a strong binary interaction, as e.g., in the
Boomerang Nebula (Sahai et al., 2017)) – has depleted most of the stellar
envelope, the stars evolve to higher temperatures through the post-AGB phase
at almost constant luminosity, fading and becoming white dwarfs (WDs) at the
ends of their lives. It is during these post-AGB and WD phases, that the
renewed presence of puzzling disk-like structures around the central stars
becomes obvious. How are such disks formed? Are there multiple mechanisms that
can make disks during this phase? Can some of these be sufficiently dense (and
long-lived), leading to a second phase of planet-formation?
Such disks can generally be divided into two broad observational classes by
their sizes – (Type 1) very small disks which are contained within the Roche
limits of the central WD stars ($\lesssim 0.01$ AU), and (Type 2) large disks
extending to radii of $\sim 10-100$ AU. The majority of the Type 2 disks are
found around the central stars of PNe (CSPNe) (e.g., Bilíková et al. (2012))
and around post-AGB stars (e.g., De Ruyter et al. (2006)), whereas most Type 1
disks have been identified around old, naked WDs (i.e. those for which the PN
shell is no longer visible). Type 1 dust disks around WDs were first detected
around G29-38 and GD 362 through their excess IR emission (confirmed
spectroscopically to be continuum), and since their discovery, it has been
commonly accepted that they originate from tidally disrupted planetesimals
(Becklin et al. (2005); Kilic et al. (2005); Jura (2003)), especially since
the chemical composition of these planetesimals is similar to rocky bodies in
the inner regions of our Solar System. The required gravitational scattering
of planetesimals towards the WD implies the presence of a planet, and recent
discoveries provide indirect and direct evidence for these. The WD
J091405.30+191412.25 is found to be accreting hydrogen, oxygen, and sulfur,
material which likely comes from the deep atmospheric layers of an icy giant
planet (Gänsicke et al., 2019). From TESS data, Vanderburg et al. (2020) find
a giant planet transiting the WD 1856+354 every 1.4 days. However, tidal
shredding of a companion during a Common Envelope phase can also make such a
disk (Kuruwita et al., 2016).
The origin(s) of Type 2 disks is(are) more difficult to understand. Using
Spitzer/MIPS imaging and IRS spectroscopy, Su et al. (2007) find evidence for
the presence of a compact dust cloud around the central star of the Helix
Nebula (WD 2226-210: $T_{eff}=110,000$ K) with a temperature of 90–130 K,
located 40–100 AU from the central star, and inferred it to be a dusty disk.
A more extensive search for 24 µm excesses around a sample of CSPNe by
(Bilíková et al. (2012); Chu et al. (2011)) revealed spectral-energy-
distributions (SEDs) and spectra (for a few objects) with a variety of IR
excess characteristics, implying the operation of a different mechanism than
planetesimal destruction for producing the dusty disk responsible for the
excess emission. This mechanism involves the presence of a binary companion –
binaries provide a source of angular momentum and free energy to form
accretion discs. Several mechanisms for making disks in binary systems with a
mass-losing companion have been identified (including Bondi-Hoyle-Eddington
accretion, Roche-lobe and wind-Roche-lobe overflow, common envelope ejection,
grazing envelope ejection). Numerical simulations of binary systems where the
companion is detached show that the latter can gravitationally capture a
fraction of the mass outflow from an AGB star into a small disk (hereafter
Type 1b, $\sim$1 AU, Mastrodemos & Morris (1998)) – too small to be
representative of Type 2 disks. Thus, the large Type 2 disks, if resulting
from binarity, likely require interaction with a close binary companion that
results in the ejection of a substantial fraction of the AGB mass outflow
being directed along directions near the orbital plane forming a disk-like
structure. But it has been observationally difficult to confirm the
association of dust disks with binary CSPNe, which requires the detection of
companions and disks around the same CSPNe. The spectral-type distribution for
companions of WDs (which should be representative of CSPNe), peaks at spectral
types M 3-4 (Farihi et al. (2005)). Spectra or sensitive photometry in the
$\sim 0.6-5$ µm range can reveal a companion – e.g., an M3V companion would
emit $\sim 210(D/0.5\,kpc)^{2}$ $\mu$Jy at 5 µm – or provide upper limits.
In this paper, we report the detection of a dust cloud around the central star
(CS) of the planetary nebula NGC 3132 using data from the JWST’s Early Release
Observations (Pontoppidan et al. (2022)). This PN, also known as the Southern
Ring, has been well-studied, with high-resolution optical images obtained with
HST (e.g., Monteiro et al. (2000)), 2D spectroscopic imaging (Monreal-Ibero &
Walsh (2020)), and mapping of molecular-line emission – 2.1 µm H2 (Storey
(1984)) and 1.3 mm CO J=2–1 (Sahai et al. (1990)). These data reveal that NGC
3132 belongs to the primary morphological class “multipolar” (Sahai et al.
(2011)) – with a bright central elliptical shell structure surrounded by
fainter structures aligned along different axes. Located at the center of the
elliptical structure, the CS of NGC 3132 is a hot white dwarf (discovered by
Kohoutek & Laustsen (1977)) with $T_{eff}\sim 105,000$ K: Monreal-Ibero &
Walsh (2020)). North-East of the CS, at a separation111all separations are as
measured on the sky-plane, unless noted otherwise of $1\farcs 696$, lies a
companion (HD 87892) that is a slightly evolved main-sequence star of spectral
type A2V (Méndez (1978)).
We focus in this study on the CS and its immediate surroundings; description
and analysis of the full PN morphology is not within the scope of this paper,
but has been reported by De Marco et al. (2022).
## 2 Observations
We retrieved the pipeline-calibrated Level3 data on NGC 3132 (Proposal 2733,
PI=K.M. Pontoppidan) from the MAST archive. NGC 3132 was imaged using filters
F090W, F187N, F212N, F356W, F405N, F470N with NIRCam (plate-scale $0\farcs
031$ for $\lambda\lesssim 2.1$ µm and $0\farcs 063$ for longer wavelengths)
(Jakobsen et al. (2022)), and F770W, F1130W, F1280W, and F1800W with MIRI
(plate-scale $0\farcs 11$) (Bouchet et al. (2015)). JWST is diffraction-
limited at wavelengths $\gtrsim$2 µm with a PSF size (FWHM) of $0\farcs 075$
at $\gtrsim$2 µm (Rigby et al. (2022)). The images have been processed and
calibrated using data processing/calibration software version nos. “2022_2a”
and “1.5.3”. Each of the Level 3 images is a combination of 8 dithered images
(hereafter “sub-images”). For NIRCam data, 6/8 sub-images in each of the
filters F090W, F187N and F202N include the CS in the FOV; for the remaining
filters, the CS is within the FOV for all 8. For MIRI data, 6/8 sub-images in
each of the filters include the CS in the FOV. We have examined each of these
sub-images to look for artifacts that may affect the CS, and found that only
one of these, for F090W, has 4 bad pixels at the location of the CS. Hence,
for this filter only, the photometry described in the next section (§ 3) was
carried out on each of the 5 good sub-images, and the reported value is a
median average of these.
## 3 Observational Results
The JWST images of NGC 3132 reveal the well-known, extended multipolar
morphology of this object, the hot central star that excites the nebular
material, and the nearby A2V star. The images (in selected filters) of the CS
and its immediate surroundings, are shown in Fig. 1. A small but clearly
extended emission region is seen around the CS in the F1800W image. In
addition, the emission is not circularly symmetric in its outer parts (i.e.,
at radial offsets $\gtrsim 0\farcs 65$) – the contours appear flattened on the
side closest to the A2V star, thus giving the emission source an overall
elongated shape (Fig. 2a,b). Furthermore, there is a small “tail” at the
Northern side of the emission source, that bends towards the A2V star. Radial
intensity cuts centered on the CS, averaged over 90° wedges pointing towards
and away from the A2V star, clearly show a significant mismatch between the
intensities at offsets $\gtrsim 0\farcs 7$ (Fig. 2c). Although there is
considerable structure in the foreground/background of the CS due to the PN
itself, it is very unlikely that the above asymmetries are the result of a
chance projection of a large extended nebular feature coinciding with the
central dust clump, distorting its shape and producing such a specific tail
shape curved towards the A2V star. Also, no hint of such an extended feature
close to the compact dust clump is seen in the large-scale 18 µm image of the
nebula (Appendix).
A comparison of the CS’s radial intensity with that of relatively-bright field
stars in the field-of-view of these images, shows that at 18 µm, the width
(FWHM) of the emitting region, $0\farcs 96$, is significantly larger than that
of the PSF ($0\farcs 66$). The observed radial extent of the dust cloud is
surface-brightness limited – e.g., on the side away from the A2V star, it
falls to an intensity equal to the standard deviation of the sky background,
$\sim$4 MJy sr-1, at $r\sim 2\farcs 5$.
The 12.8, 11.3 & 7.7 µm images also appear mildly extended, but to a
progressively lesser extent; for shorter wavelengths, the CS appears to be
point-like.
### 3.1 PSF subtraction
The CS images are affected by the presence of the bright A2V type star which
spreads its flux through its PSF. We describe our 5-step (iterative) PSF
subtraction process below:
1. 1.
Generate a simulated PSF (“sPSF”) using the PSF simulator tool
(webbpsf222https://www.stsci.edu/jwst/science-planning/proposal-planning-
toolbox/psf-simulation-tool), taking into account the spectral type of the
source (A2V) and its offset from the center of the FOV in the NIRCam and MIRI
images. The optics of the system are affected by additional effects (jitter,
deformations,…) which are equivalent to an additional gaussian smoothing but
are not accounted for in the simulator. We therefore applied a gaussian
smoothing function to “sPSF”, whose FWHM was set iteratively (see step 3
below).
2. 2.
Remove a median averaged sky-background from the “i2d” image. The value for
the box size used for estimating a median background was set iteratively for
each filter/camera.
3. 3.
Apply the gaussian smoothing function to the sPSF, and scale the sPSF peak
(central) intensity to that of the A2V star when the latter was not saturated
in the center (F1800W, F1280W, F1130W, F770W), or to the intensity in the
wings when the A2V star is saturated in the center (F090W, F187N, F212N,
F356W, F405N, F470N). The width of the gaussian was set iteratively for each
filter/camera.
4. 4.
Compare intensity slices along the vertical and horizontal axis crossing
through the center of the A2V star and the sPSF, and check the result of the
subtraction of the current sPSF with the background-subtracted image. This
allowed us to estimate the background subtraction performed in step 3, as well
as re-adjust the gaussian spread applied to the PSF. Also, the registration of
the sPSF relative to that of A2V star was re-adjusted, especially important in
the NIRCam images for which the central pixels covering the A2V star are
saturated.
5. 5.
Iterate steps 2–4 (i.e. gaussian smoothing, registration, flux scaling) to
bring the average flux in the region close to the A2V star as close to the
median error in the sky background in this region.
The background estimate and removal was performed by using the
photutils333https://photutils.readthedocs.io/en/stable/background.html package
(Bradley et al. (2015)). We estimated a 2D median background, by dividing the
field of view into boxes and substituting the pixel intensities in each box
with the median background for that box. This generated a low-resolution
background which was subtracted from the image. The box size was estimated to
remove as much background as possible but avoiding any source removal,
including the PSF sidelobes. The box size, the width of the Gausian smoothing
function, and the error in the PSF scale-factor444The actual value of the
scale factor is unimportant, since it depends on the normalisation of the PSF,
which is arbitrary are given in Table 3. The images of the CS and its
immediate surroundings with the A2V star subtracted, for the same set of
selected filters as in Fig. 1, are shown in Fig. 3. Most of the negative
residuals in these images lie relatively close to the sharp diffraction
structures and most likely are a result of inherent limitations in the PSF
simulator tool’s ability to reproduce the observed PSF.
### 3.2 Photometry
We have carried out aperture photometry of the CS in the NIRCam and MIRI
images as follows. Except for the F1800W image, we have used circular
apertures with sizes adjusted to include as much of the light from the CS, and
exclude as much light from the surrounding structured background, as possible.
For F1800W, we have used multiple methods to extract the photometry. Using the
pipeline Level 3 image, we use an elliptical aperture of size $1\farcs 1\times
1\farcs 3$ that roughly matches the observed elongated shape of the source.
Second, we extracted the average radial intensity over an angular wedge
covering the position angle (PA) range $100\arcdeg-315\arcdeg$ that best
excludes the region contaminated by the PSF of the A2V star; the background
intensity was estimated from the cut intensity at large radii and subtracted.
The resulting radial intensity was then integrated over circular apertures
with varying outer radii to estimate the flux density as a function of
aperture radius (Table 1). Radial intensity cuts have also been extracted from
the F770W, F1130W and F1280W filter images in a similar manner as for F1800W.
The background appears to be dominated by nebular emission in the F1130W,
F1280W and F1800W filters; for shorter wavelengths the PSF of the A2V star
becomes increasingly dominant while the CS emission becomes progressively
fainter. Aperture corrections have been determined using relatively bright
field stars in the images, and applied to the aperture photometry of the CS
(Table 1). In order to assess errors, for each filter, we used apertures of
different diameters, each with its own aperture correction; the results shown
in Table 1 are those obtained with largest aperture that could be feasibly
used. We have assigned conservative errors of $\pm$15% to the NIRCam
photometry, and $\pm 10$% to the MIRI photometry. Noting the 0.1 to 0.2 mag
difference between pipeline and manually-determined photometry for the HST
data at 0.44 and 0.56 µm (Table 1), we have assumed $\pm$15% errors for the
published photometry at shorter wavelengths, i.e. $\lambda<0.9$ µm. The
aperture photometry was also carried out for the PSF-subtracted images, and
yielded consistent results.
We have verified that our photometry methodology is correct by measuring the
F356W and F770W flux densities for a few field stars, and comparing these to
those measured with Spitzer, IRAC 1 and IRAC 4 (Spitzer Science Center, 2021)
– we find that our photometry is in good agreement with the latter (Table 2),
accounting for the possibility of temporal variability in these stars.
### 3.3 UV Spectrum
We downloaded a FUSE UV spectrum of NGC 3132 that covers the $910-1190$ Å
wavelength range with a spectral resolving power $R\sim 20,000$ (Moos et al.
(2000); Sahnow et al. (2000)) from the Mikulski Archive for Space Telescopes,
calibrated with the final version of the CalFUSE calibration pipeline software
package, CalFUSE 3.2.3 (Dixon et al. (2007)). The spectrum was obtained as
part of FUSE program ID P133 (PI: L. Bianchi), and an analysis of the same
(reporting the discovery of Ge III $\lambda$ 1088.46 in NGC 3132 and other
PNe) was first presented in Sterling et al. (2002). The spectra are of
relatively modest S/N at the short wavelength end, and we have re-binned it to
reduce the resolution by a factor 50. The spectrum was taken through the LWRS
aperture, which has a size of $30{{}^{\prime\prime}}\times
30{{}^{\prime\prime}}$, and therefore covers not only the CS, but also the A2V
companion and a significant fraction of the ionized region. However, the
continuum flux in the $910-1190$ Å wavelength region is dominated by the CS,
allowing us to use the FUSE spectrum to complete the SED of the CS to 0.091 µm
at the short wavelength end (see § 4.2).
## 4 Analysis
### 4.1 A Central Binary and Orbital Motion
We have investigated the relationship between the CS and the A2V star as
follows. First, there is little doubt that these two stars are a physical pair
(De Marco et al., 2022), given the close agreement in their $Gaia$-based
proper motions (A2V star: $PMra=-7.747\pm 0.027$ mas yr-1, $PMdec=-0.125\pm
0.031$ mas yr-1: CS: $PMra=-7.677\pm 0.235$ mas yr-1, $PMdec=0.197\pm 0.28$
mas yr-1) and radial velocities (A2V star: $VLSR=-24.1\pm 1.6$ km s-1, CS:
$VLSR=-25\pm 0.9$ km s-1). We have therefore inferred a distance to NGC 3132
of 0.75 kpc by inverting the parallax of the A2V star companion555no parallax
is available for the CS ($1.3198\pm 0.0344$ mas) listed in the $Gaia$ DR3
database (Gaia Collaboration, 2022). Hence, the separation between the CS and
its A2V companion and the CS is $1277\pm 34$ au.
We have found direct evidence for the CS and the A2V star to be a
gravitationally-bound binary system from the detection of orbital motion, over
a time interval of $\sim 20$ yr between the epoch of the first high angular-
resolution image of the CS-A2V pair with HST (Epoch 1), and the epoch for
$Gaia$ measurements (Epoch 2) as follows. We used the Epoch 1 F555W image
(from GO 06119, PI: H. Bond; image taken on 1995 Dec 04), obtained with the PC
camera of WFPC2 (pixel-size of 0$\farcs$05), to determine the coordinates of
the CS and the A2V star. Since the S/N is quite high ($>50$ for the CS,
$>4000$ for the A2V), the uncertainty in the location of the CS is $\sim$1
mas, and that of the A2V star much smaller. The position angle of the
separation vector of the A2V, CS pair666measured from N towards E, using the
CS as the origin, in the HST image is, $PA(HST)=47.82\arcdeg\pm 0.045\arcdeg$
(Fig. 4). The position angle of this vector during the $Gaia$ measurement
epoch (Epoch 2: $\sim$JD$\sim$2457410), estimated from the DR3 coordinates of
the CS and A2V stars, is $PA(gaia)=47.43\arcdeg$.
The JWST data could not be used to determine the current orientation of the
CS-A2V separation vector accurately enough because the locations of these
stars cannot be measured with the required accuracy – at the shortest
wavelengths, where the telescope is diffraction limited and the PSF is small
(e.g., $0\farcs 075$ at $\sim 2$ µm) the A2V star is badly saturated and the
CS is very faint and its image is partly contaminated by the wings of the A2V
star’s PSF.
We have checked for any (minute) difference between the cardinal directions in
the $Gaia$ DR3 reference frame ($Gaia$ CRF3: Gaia Collaboration et al. (2022))
and the HST image reference frame, $PA(gaia-HST)$, using the following
methodology. We identified 4 field stars in the HST image (fs1, fs4, fs6, and
fs8)777these are the 4/5 stars seen in the full FOV for which $Gaia$ DR3 data
are available, and computed the $PA$s of their separation vectors (Table 4).
Then, using the proper motions of these stars from $Gaia$ DR3, we have
computed their expected locations in Epoch 1, and from these, the expected
$PA$s of their separation vectors in Epoch 1 (Table 4). Because the separation
vectors for these stars are quite large, the errors in the $PA$s are quite
small ($\lesssim 0\farcs 01$). The average of the $PA(gaia-HST)$ values is
$0.147\arcdeg\pm 0.016$°. Applying this correction to the value of $PA(HST)$
computed above, the corrected value of the $PA$ of the CS-A2V separation
vector is, $PA(HST,c)=PA(HST)-0.147\arcdeg=47.67\arcdeg$. Thus, the separation
vector has rotated by $0.24\arcdeg\pm 0.045\arcdeg$ (clockwise) in 20.1 yr
(time interval between Epoch 1 and Epoch 2), in good agreement with the
expected 0.20° cos$(i)$/cos 45° rotation for an orbital period of $P\sim
25,500$ yr, based on the measured separation of 1277 au, assuming an
intermediate angle between the orbital and sky planes of $i=45\arcdeg$ and
masses of 0.7 $M_{\odot}$ and 2.5 $M_{\odot}$ for the CS and A2V stars.
The angular distance between the CS-A2V pair in Epoch 1 is
$\epsilon(HST)=1\farcs 693\pm 0\farcs 001$, compared to $1\farcs 696$ in Epoch
2. We have checked the “plate-scale” of the HST image, using the same 4 fields
stars and a similar methodology as above. We computed the angular distances
between the star pairs used above for Epoch 1 using their current $Gaia$ DR3
coordinates and proper motions to compute their Epoch 1 coordinates, and
compared these with the directly measured angular distances in the HST image
(Table 4): the average fractional difference for the distances between star
pairs between Epoch 1 and 2 (Table 4) is $(3.92\pm 1.35)\times 10^{-5}$. This
difference is too small to affect the value of $\epsilon(HST)$ derived above.
Therefore, there is weak evidence for an increase in the angular distance
between the CS-A2V pair of $2.7\pm 1$ mas from Epoch 1 to Epoch 2.
Although additional HST imaging is available at intermediate epochs, these
images proved to be inadequate for determining the locations of the two stars
with sufficient accuracy, because of one or more of the following factors: (i)
the pixelsize was twice as large (e.g., $0\farcs 1$ using the WFPC2 Wide-Field
cameras), (ii) only emission-line images were available providing inadequate
S/N to centroid the CS accurately, (iii) the core of the PSF of the A2V star
appeared non-circular and distorted.
### 4.2 Dust Radiative Transfer Modeling
We have used the MIRI and NIRCam photometry, together with the FUSE UV
spectrum and published optical photometry, to construct the observed SED of
the CS and the dust cloud around it over the UV to mid-IR ($0.091-18$ µm)
region (Fig. 5).
The photometry is not co-eval; we assume that there are no significant
variations in the CS’s spectrum over the $\sim$4 decades spanned by epoch of
the U-band photometry and that of the JWST observations. Although the CS is
included in the catalog of large amplitude variables from Gaia DR2 (Gaia DR2
5420219732233481472: Mowlavi et al. (2021)) is probably erroneous. Mowlavi et
al. (2021) report a variability amplitude proxy of 0.3566 mag for the CS –
this proxy was computed from the uncertainty (keyword “pho_g_mean_mag_error”)
in the G-band magnitude (keyword “phot_g_mean_mag”) published in DR2. A
comparison of the Gaia DR2 and DR3 G-band magnitudes – 15.216140 (mean
magnitude based on 125 single epoch measurements) for DR2, and 16.105730 (mean
magnitude based on 193 single epoch measurements888these include the 125
measurements used for the DR2 photometry) for DR3 – shows that the CS was
almost one order of magnitude fainter in DR3 compared to DR2. However, since
the DR3 data supersede the DR2 data, it is likely that the DR2 G-band
magnitude and error are incorrect, and possibly result from incorrect removal
of the contribution by the bright A2V star’s PSF to the measured photometry of
the CS.
The simplest explanation for the extended emission seen most clearly at 18 µm
is that it arises from thermal emission from warm dust. Although there are
strong nebular emission lines that have been detected in NGC 3132 in the
wavelength bands covered by the filters listed above, it is very unlikely that
such gas is present in the immediate vicinity of the CS. We have made a rough
estimate of the width of the intrinsic radial intensity distribution at 18 µm
(FWHMint) as follows. We approximated the observed radial intensity
distribution as well as the PSF at 18 µm, using Gaussians of widths
FWHM${}_{obs}\sim 1\farcs 0$ and FWHM${}_{psf}\sim 0\farcs 7$, and determined
FWHM${}_{int}\sim 0\farcs 7$ using ${\rm FWHM}_{obs}^{2}={\rm
FWHM}_{int}^{2}+{\rm FWHM}_{psf}^{2}$. This intrinsic width implies a radial
extent of $\sim$260 au at NGC 3132’s distance of 0.75 kpc. Any gas this close
to the star would be quite hot, $>10^{4}$ K, and at the sound speed for such
gas, 10 km s-1, would expand to radii $>10,000-20,000$ au in (say) a typical
post-AGB age of $>5,000-10,000$ yr. Furthermore, the SED of NGC 3132’s CS dust
emission is similar to that of the Helix, for which a spectrum of the CS shows
no obvious line emission.
We have used the DUSTY dust radiative transfer code (Ivezić et al., 2012) to
model the CS and its dust emission. Although it is plausible, depending on its
origin, that the dust cloud is flattened or disk-like (see § 5), since we have
no direct information about this aspect, we have chosen to use 1D modeling.
This is not a limitation, because (as shown below), the optical depth of this
cloud is $\ll 1$, even at relatively short wavelengths ($\lesssim 0.001$ at
$\lambda\gtrsim 0.05$ µm), hence the results are not sensitive to the specific
geometry (i.e., sphere or disk) of the cloud. Even if the dust cloud had a
disk configuration, the radial optical depth near and in the equatorial plane
would remain well below unity.
The main input parameters are (i) the dust temperature at the inner shell
boundary ($T_{\mathrm{d}}$), (ii) the total radial optical depth at
0.55$\micron$ ($\tau_{V}$), (iii) the shell density (iv) the grain-size
distribution for a choice of grain composition, (v) the relative shell
thickness ($Y$ = ratio of the shell’s outer radius, $R_{ou}$, to its inner
radius, $R_{in}$), (vi) the spectrum of the central star. The shell density
was assumed to be a single power-law exponent, $n$, of the power-law
($\rho_{d}(r)\propto r^{-n}$), or a combination of two power-laws (described
below). We have computed the stellar spectrum using the Tübingen NLTE Model
Atmosphere Package (TMAP) (Rauch & Deetjen (2003); Werner et al. (2003, 2012))
for $T_{eff}=105,000$ K – as derived by Monreal-Ibero & Walsh (2020) using
photoionization modeling. We have accounted for nebular extinction by using a
visual extinction of $A_{V}=0.31$ – derived by Kohoutek & Laustsen (1977)) –
to attenuate and redden the model SED.
We require the grains to be relatively cool ($\sim 100$ K at $r\sim
1{{}^{\prime\prime}}$) in order for the average slope of the model spectrum
from 7.7 µm to 18 µm to match the observed one (Fig. 5). The shape of the
model SED longwards of $\sim 5$ µm, where dust emission is the dominant
component, is most sensitive to the dust grain properties (composition, size
distribution). We have investigated models with “cold” (“Sil-Oc”), “warm”
silicate grains (“Sil-Ow”), and amorphous C (amC) grains in DUSTY999DUSTY
provides 6 in-built choices for grain composition. The SED has a very
distinctive shape in the wavelength region dominated by dust emission, i.e.,
longwards of $\sim 5$ µm – there is a steep rise from $\sim 7$ µm to $\sim$11
µm, followed by a flat region between 11 and 13 µm, and then a rise towards
longer wavelengths. While models with silicate grains produce the observed
roughly flat shape of the SED in the 11-13 µm region, pure amC-grain models
produce a smooth rise with increasing wavelength, discrepant from the observed
SED. Warm silicates produce modestly smaller ratios flux F(18 µm)/F(12.8 µm),
F(18 µm)/F(11.3 µm) and F(18 µm)/F(7.7 µm) compared to cold silicates. We used
a modified version of the Mathis, Rumpl, Nordsieck (MRN) distribution function
for grain radius $a$, $n(a)\propto$ a-q for $a_{\mathrm{min}}$ $\leq a\leq$
$a_{\mathrm{max}}$ (Mathis et al. (1977)), with q=1.101010The standard MRN
parameters are: $q=3.5$, $a_{\mathrm{min}}$ = 0.005$\micron$ and
$a_{\mathrm{max}}$ = 0.25$\micron$. Models with a substantial fraction of
large grains e.g., $a_{\mathrm{m}ax}\gtrsim 1$µm produced radial intensity
profiles that are significantly more compact than observed; in addition, the
detailed shaped of the SED between 7.7 and 18 µm (described above) cannot be
reproduced – large grains produce a dust emission spectrum in which there is a
smooth rise with increasing wavelength. De Marco et al. (2022) present a
simple large-grain (size $\sim$100 µm) dust shell model, which suffers from
these deficiencies. Keeping the grain size distribution the same, but allowing
the grains to be cooler by decreasing $T_{\mathrm{d}}$ results in a decrease
in the 7.7 µm to 18 µm flux ratio, and simultaneously makes the model emission
more extended due to an increase in $R_{in}$. Keeping the value of
$T_{\mathrm{d}}$ fixed, an increase in the proportion of larger grains by
increasing a(max) or decreasing $q$, makes the dust emission more compact. The
model radial intensity distribution, especially at the longer wavelengths
($\sim 11-18$ µm) is sensitive to (i) $T_{d}$, and (ii) the radial density
gradient of the dust shell. Lower values of $T_{d}$ lead to smaller values of
$R_{in}$, and therefore a more compact radial intensity distribution of the
dust emission. Higher values of $n$ also result in more compact dust emission.
The above competing constraints allow us to provide reasonable constraints on
the model parameters.
The DUSTY code generates a model SED, normalized to the bolometric flux,
$F_{bol}$. We find that $F_{bol}=(6.5\pm 0.5)\times 10^{-9}$ erg s-1 cm-2
(implying a CS luminosity of 114 $L_{\odot}$) by scaling the (reddened) model
SED to match the observed SED (Fig. 5a). The value of $F_{bol}$ is well-
constrained (for a given choice of $T_{eff}$), independently of the specific
dust model, because the model SED in the optical to near-IR wavelength range
(i.e., $\sim 0.1-3$ µm) (Fig. 5d) is only affected by nebular extinction
(which is modest), and not by the dust cloud which has a very low optical
depth. While determining the best-fit model, we have ignored (i) photometry
from the narrow band filters F187N, F212N, F405N, and F470N, as the data from
these are most affected by the presence of nebular atomic or molecular
hydrogen lines, and (ii) $Gaia$ DR3 photometry due to potential imperfect
removal of the PSF of the A2V star. We did not convolve the SED with the
filter responses, but we have checked that the difference between convolved
and non-convolved model photometry is relatively small and well below
difference between the best-fit model and the data.
In order to quantitatively distinguish between models, we have defined a
‘goodness-of-fit’ measure for the SED modeling, $G(sed)$, as follows: $G(sed)$
=$\sum\left(\frac{O_{\rm j}-M_{\rm j}}{\sigma_{\rm j}}\right)^{2}$, where, Oj
($\sigma_{\rm j}$) is the observed flux (error), Mj is the model flux, and the
index j refers to different wavelengths. The summation was carried out only
for $\lambda\geq 3.56$ µm because at shorter wavelengths, the SED is dominated
by the central star, and including the data for these would reduce the
sensitivity of $G(sed)$ to the contribution of the dust emission. The
‘goodness-of-fit’ for the radial intensity disributions, $G(Fx)$, where $Fx$
is F770W, F1280W, or F1800W, is: $G(Fx)$ =$\sum[r_{\rm j}\,\delta\,r_{\rm
j}\,(Oint_{\rm j}-Mint_{\rm j})/\sigma_{\rm j}]^{2}/\sum[r_{\rm
j}\,\delta\,r_{\rm j}]^{2}$, where $Oint_{\rm j}$ ($\sigma_{\rm j}$) is the
observed normalized intensity (error), $Mint_{\rm j}$ is the model normalized
intensity, $r_{\rm j}$ ($\delta\,r_{\rm j}$) is the radial offset (width of
annulus) at grid-point j, and the index j refers to different radial offsets.
The inclusion of $r_{\rm j}\,\delta\,r_{\rm j}$ in the formula for $G(Fx)$
means that we have used the difference of between (normalized) observed and
model flux at each radial grid point for determining the goodness-of-fit.
Better-fitting models have lower values of $G(sed)$ and $G(Fx)$.
The above “1-shell” models provide a reasonably good fit to the SED (Fig. 5a)
as well as the radial intensity in the F1800W, F1280W and F770W images (Fig.
6a,b,c). The grain composition needed to fit the specific shape of the SED,
requires a mixture of 70% cold silicate (Sil-cW) and 30% amorphous carbon
(amC) grains. However, there are small but noticeable discrepancies – (i) the
F1800W model radial intensity lies a little below the observed one for radial
offsets $r\lesssim 1{"}$, and (ii) the F1280W model radial intensity lies a
little above the observed one for radial offsets $r\lesssim 0\farcs 65$.
Since the radial dust emission intensity is sensitive to the dust radial
density distribution $\rho_{d}(r)$, we explore the possibility that a more
complex density structure than the ones used for the 1-shell models can reduce
the above discrepancy. We have investigated models in which $\rho_{d}(r)$ is
described by a broken power-law, such that $\rho_{d}(r)\propto r^{-n1}$ for
$1\leq Y\leq Y1$, and $\rho_{d}(r)\propto r^{-n2}$ for $Y1\leq Y\leq Y$
(hereafter “2-shell” models).
We find that a 2-shell model with $n1=0.2$, $n2=-0.4$, and $Y1=5$ provides the
best fit (Fig. 7a,b,c). For this model, $G(sed)$ is a factor 0.75 times and
$G(Fx)$ is a factor 1.2, 0.96, and 0.41 times (for $Fx$ equal to F770W,
F1280W, and F1800W, respectively), of that for the 1-shell model. While
comparing the $G(Fx)$ values for these models, we give the most weight to
$G(F1800W)$ and the least wight to $G(F770W)$ since the dust distribution is
most (least) resolved at $18\,\micron$ ($7.7\,\micron$). Thus the 2-shell
model provides an overall better fit to both the SED and the radial instensity
distribution.
The density structure in the 2-shell model suggests that the dust cloud around
the CS has two shells with a low density region in between. The presence of
double shells is not uncommon for the central stars of PNe that show dust
emission – Bilíková et al. (2012) find the presence of double shells in
$\gtrsim$50% of the CSs with IR-excesses due to dust.
We have therefore also investigated models with a different variant of the
2-shell density structure – in this class of models (hereafter “gap” models),
we have a geometrically thin shell close to the star, separated by a density
gap from the extended shell. In these models, we varied $T_{\mathrm{d}}$, the
ratio of the density in the inner shell (assumed constant) to the density of
the outer shell at the outer edge of the gap, the radius and width of the gap,
and the power-law density exponent for the outer shell. We find that although
the best “gap” model can fit the SED and the F1800W and F1280W radial
intensities as well as the 1-shell and 2-shell models, the F770W radial
intensity distribution is signifiantly narrower than observed. Hence, we do
not discuss these models further.
We have derived the mass of the dust shell as follows, since DUSTY does not
provide a direct measure of the shell dust mass. For objects obeying a
$r^{-n}$ density distribution, the dust mass in the circumstellar component is
given by (for $n\neq 1,3$):
$M_{d}=4\pi\,[(n-1)/(3-n)]\,y(Y)\,R^{2}_{in}(\tau_{18}/\kappa_{18})$ (1)
where $y(Y)=(Y^{3-n}-1)/(1-Y^{1-n})$, and $\tau_{18}$ and $\kappa_{18}$ are,
respectively, the radial optical depth of the shell and the dust mass
absorption coefficient, at 18 $\micron$. We assume $\kappa_{18}=10^{3}$
cm2g-1, based on the dust properties for silicate dust tabulated by Ossenkopf
et al. (1992). For the ‘2-shell” models, we apply Eqn. 1 piecewise to the
inner and outer shell. The input parameters and output properties for the
best-fit models are given in Table 5. The total dust mass is about $1.45\times
10^{-2}$ M⊕. In the 2-shell model, the mass in the inner shell is a small
percentage (0.5%) of the total. For comparison, De Marco et al. (2022) derive
a dust mass of $\sim 5\times 10^{-2}$ M⊕ from their simple model.
For both the 1-shell and 2-shell model classes, models covering an
appropriately large input parameter space were computed, allowing us to
estimate rough uncertainties in the derived output parameters for a fixed
value of $T_{eff}$. The most accurately determined parameter is $T_{d}$
(uncertainty $\lesssim 5$%), followed by $R_{in}$ (uncertainty $\lesssim
10$%). The uncertainty in the value of q is about 50%; the maximum grain
radius may be as high as $\sim 0.5$ µm. The optical depth has an uncertainty
of $\sim$20%. The dust masses have an uncertainty of $\sim$50%.
The temperature of the CS also affects the model results. At the relatively
high effective temperature estimated for the CS, the flux at $\lambda\gtrsim
0.4$ µm follows the Rayleigh-Jeans approximation, i.e.,
$F(\nu,T)\propto(2\,k\,T_{eff}/\lambda^{2})$, and
$F_{bol}=F(\nu,T)\,\lambda^{2}\,(\sigma\,T_{eff}^{3})/(2\,\pi\,k)$ where $k$
and $\sigma$ are the Boltzmann and Stefan-Boltzmann constants. Hence the slope
of the observed SED at short wavelengths (UV to near-IR), where the stellar
emission dominates, does not provide a constraint on $T_{eff}$. Since
$F_{bol}$ depends on $T_{eff}^{3}$ for a given value of the flux $F(\nu,T)$
and the latter is constrained by the observed photometry, variations in
$T_{eff}$ lead to corresponding variations in the CS’s inferred luminosity,
$L=114\,\mbox{$L_{\odot}$}\,(T_{eff}/105,000K)^{3}$. Since a larger value of
$L$ increases $R_{in}$ for a given value of $T_{d}$, making the model dust
emission more extended than observed, models with larger $L$ require higher
values of $T_{d}$. For example, if $T_{eff}=110,000$ K, then good 1-shell
model fits (i.e., similar to the one shown in Fig. 6) are obtained with
$F_{bol}=(110/105)^{3}\,(6.5\times 10^{-9})=7.5\times 10^{-9}$ erg s-1 cm-2,
and $T_{d}\sim 255-260$ K, $\tau_{V}=8.6\times 10^{-4}$, and mass
$\sim(0.78-0.72)\times 10^{-2}$ M⊕.
## 5 Discussion
The dust cloud around the central star of NGC 3132 is probably not the result
of current mass loss, since such phenomena are not expected in a WD surrounded
by a relatively old planetary nebula. It is more likely a stable disk in
Keplerian or quasi-Keplerian rotation. As mentioned earlier (§ 1), disks have
been found to be associated with quite evolved WDs, often too old to show
detectable PNe around them (e.g.Tokunaga et al. (1990); Manser et al. (2020)).
These disks are very small, occupying just a few au, and have been proposed to
be the result of disruption of planets or planetoids from a former planetary
system. Central stars of relatively young PNe tend to show disks with smaller
radii, inferred to typically $\sim$ 50 au (Bilíková et al. (2012); Su et al.
(2007)). Many post-AGB stars (thus less evolved than old WDs) also show disks
(e.g. De Ruyter et al. (2006); Bujarrabal et al. (2013, 2016)), but in this
case their extents are much larger,
$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$ 1000 au. In this case, the
rotating disks are systematically associated with the presence of tight binary
systems and probably consist of material ejected during the previous AGB
phase, which gains angular momentum from interaction with the system and forms
rotating circumbinary disks.
We now discuss possible origins of the dust cloud around NGC 3132’s CS.
### 5.1 An Oort Cloud
The large radial extent ($>1500$ au) of the dust cloud in NGC 3132, makes it
unlikely that it is formed from material extracted from a planetary system, or
a Kuiper Belt/debris-disk analog – the typical radius of the latter is about
$\sim 50-150$ au. However, an Oort cloud analog might explain the origin of
this dust cloud. The Oort cloud in our Solar system is thought to occupy a
radius between 2,000 and 5,000 au and may extend as far as 50,000 au from the
Sun.
The dust cloud around the CS of NGC 7293 may also be extended – Su et al.
(2007) find that the 24 µm emission from this cloud, after background
subtraction, appeared slightly resolved with a FWHM of $\sim
9{{}^{\prime\prime}}$, 1.5 times that of a true point source – although, as
stated by Su et al. (2007), this could be due to an imperfect subtraction of
the background, it may also be real (as in the case of NGC 3132.)
We now estimate the survival probability of Oort’s rocky bodies after
collisions with the gas and dust ejected by the CS while it was on the AGB or
RGB. For a cometary nucleus with a typical radius of $\sim$1 km, located at a
distance of $\gtrsim 2000$ au, only about $0.56\times 10^{10}$ g of
circumstellar ejecta will hit the nucleus assuming a total $\sim$1 $M_{\odot}$
ejected via mass-loss, much smaller than the typical mass of the comet nucleus
($>10^{16}$ g). So the cometary nuclei in the Oort cloud should easily survive
the impact of the CS ejecta. Assuming that the impacting gas is expanding at
15 km s-1, and all of its linear momentum is transmitted to the cometary
nucleus, the latter will gain a velocity of 0.85 cm s-1, significantly smaller
than the escape velocity at 2000 au ($\sim 1$ km s-1), implying that an Oort
cloud will easily survive the AGB mass-loss phase. Thus, the Oort cloud, with
an estimated mass of $\sim$1.9 M⊕(Weissman (1983)), can easily supply the mass
of dust observed around the CS.
However, in this scenario, we require a mechanism to transport the dust
inwards, since the inner radius of the dust cloud in our model is $\sim 70$
au. We think it is possible that the A2V companion provides sufficient
perturbation to the orbits of the cometary bodies in the Oort cloud, causing
them to collide and/or increase their eccentricity sufficiently so as to reach
the inner regions of the dust cloud.
### 5.2 Interaction with a Binary Companion
As mentioned earlier, the central star of NGC 3132 has a detached A2V sp. type
companion (§ 4.1). A closer companion has not been detected but cannot be
discarded (we discuss this possibility in more detail below) – the highest
mass unresolved main-sequence companion that can be present and remain
undetected is of lower mass than that of a M-dwarf with spectral type M6V
(with L$\sim 7\times 10^{-4}$ $L_{\odot}$ and $T_{eff}\sim 2500$ K, e.g.,
Cifuentes et al. (2020)), i.e., $\lesssim 0.1$ $M_{\odot}$. Including the
theoretical spectrum of such a companion – BT-NextGen (AGSS2009) model with
$T_{eff}=2500$ K, log $g$ (cm s-2)=5, extracted from http://svo2.cab.inta-
csic.es/theory/newov2/index.php, Allard et al. (2011) – increases the model
SED flux in the F356W filter by 30%, well above the observed value (see inset
of Fig. 5d). De Marco et al. (2022) propose that the CS has at least 2, and
maybe even 3 close companions, in order to explain the origin of the dust
cloud and the morphological structure of the extended PN.
From angular momentum balance considerations alone, the dust cloud may be a
result of interaction with either such a close companion or the detached A2V
companion. The angular momentum of the cloud, assuming it have a disk geometry
and in Keplerian rotation around a $\sim 0.65$ $M_{\odot}$ CS, is low
($\lesssim 0.5\times 10^{4}$ $M_{\odot}$ km2 s-1), orders of magnitude less
than that of a compact binary system even if the secondary is just a big
planet – e.g., the angular momentum of a Jupiter-mass planet in a 1 au orbit
around the CS is $3.54\times 10^{6}$ $M_{\odot}$ km2 s-1. The A2V star’s
angular momentum, $\sim 1.5\times 10^{11}$ $M_{\odot}$ km2 s-1, is also much
larger than that of the CS’s dust cloud, assuming the latter has a disk
geometry.
From the point of view of the momentum transfer mechanism, however, the
situation is very different for the above two scenarios. The currently adopted
wind Roche-Lobe Overflow (wRLOF) mechanism of hydrodynamical interaction
between a companion and a stellar wind in the red giant phase predicts strong
effects when the separation of both stars is small, typically under 50–100 au
(Mohamed & Podsiadlowski (2012); Chen et al. (2017); Kim et al. (2019)). The
circumstellar gas at such distances is still being accelerated and shows a
relatively low velocity, which is a basic ingredient for allowing strong
interaction effects, including the formation of rotating circumbinary disks
and symbiotic phenomena (e.g., Sánchez Contreras et al. (2022)). Rotation in
inner suborbital regions can also be induced in those cases of strong
interaction (e.g. Bermúdez-Bustamante et al. (2020)). However, when the
distance between the two stars is significantly larger, the circumstellar
expansion velocity is expected to be large and models predict that the gas
passes by the companion with a minor interaction; the main effect is then the
formation of spiral arcs due to the oscillation of the mass-ejecting primary.
Formation of rotating disks is therefore not expected in detached binary
systems. Hence, if the dust cloud is a disk resulting from binary interaction,
we require that the CS has (or had in the past) another companion that is
(was) much closer than the A2V star, and that this close companion (hereafter
“Compc”) underwent a strong gravitational interaction with the CS.
The A2V star may have played an active role in inducing the strong
gravitational interaction of Compc with the CS. In this scenario, three stars
– the CS, Compc (say at a separation of $\lesssim 5$ au), and the A2V star (in
a more distant orbit, say at $\sim$400 au) – formed a stable hierarchical
triple system while these stars were on the main-sequence. Such systems can
become dynamically active on much longer time-scales due to the “Eccentric
Kozai-Lidov” (EKL) mechanism causing the inner binary to undergo large-
amplitude eccentricity and inclination oscillations (Kozai (1962); Lidov
(1962); Naoz (2016)). The oscillations tend to drive the inner binary to have
very small pericenter distances and even to merge (e.g., Prodan et al. (2015);
Stephan et al. (2018)). Salas et al. (2019) have simulated such a triple
system with a 2.2$M_{\odot}$ primary, a close companion (with a range of
masses $\lesssim$0.9 $M_{\odot}$) and a tertiary star with a range of masses
M$\lesssim$0.9 $M_{\odot}$) – they find that in 37% of their simulations, the
tight binary merges. Although in the case of NGC 3132, the tertiary star is
more massive ($>$2.5 $M_{\odot}$), Salas et al. (2019)’s results indicate that
there is a significant possibility of the inner binary to merge. Such a merger
would lead to a common envelope ejection of most of the stellar envelope of
the primary star – the multipolar morphology observed in NGC 3132 would then
be a result of interaction of multipolar collimated outflows with the ejecta,
as appears to be the case in the Boomerang Nebula (Sahai et al. (2017)). The
Boomerang is a pre-planetary nebula which has most likely resulted from CEE
while the central star was still on the RGB and therefore in a much earlier
post-AGB phase than NGC 3132. Since the primary must be initially more massive
than the A2V star, say $\sim 2.8$ $M_{\odot}$, simple conservation of angular
momentum indicates that the tertiary’s orbit, if initially equal to $\sim$400
au, would expand to a semi-major axis of $\sim$1200 au after the merger,
bringing it to its current observed location.
### 5.3 The A2V Companion’s Effects on the Dust Cloud around the CS
The JWST images show that the radial extent of the dust cloud is almost
identical to the orbital radius of the CS-A2V detached binary system. In
addition, there is a flattening of the 18 µm intensity contours defining the
shape of the dust cloud on the side facing the A2V star. These two features
suggest a physical interaction between the dust cloud formation and the A2V
companion; we now discuss several physical mechanisms for such an interaction.
The first is that the A2V star preferentially illuminates the dust closest to
it, making it relatively hotter, and therefore produce an extra brightening of
the dust emission, relative to the diametrically-opposed side of the dust
cloud, contrary to what is observed.
Next, we consider the effect of radiation pressure and a possible wind from
the A2V star on the dust cloud around NGC 3132’s CS. The ratio of the stellar
wind force ($F_{sw}$) to the gravitational force of star ($F_{gr}$) on a dust
grain, is given by:
$\beta_{sw}=3\mbox{$\dot{M}$}_{sw}V_{sw}C_{D}/(32\,\pi GM_{*}\rho_{g}a_{g}),$
(2)
where $M_{*}$, $\mbox{$\dot{M}$}_{sw}$, and $V_{sw}$, and are the mass, mass-
loss rate and outflow velocity for the A2V star, $G$ is the gravitational
constant, $\rho_{g}$ and $a_{g}$ are the dust grain material density and
radius, $r$ is the radial distance from the A2V star, $C_{D}$ is a coefficient
$\sim 2$ (e.g., Eqn. 28 in Augereau & Beust (2006)). Taking
$\mbox{$\dot{M}$}_{sw}=10^{-10}$ $M_{\odot}$ yr-1 (e.g., Lanz & Catala
(1992)), $V_{sw}=300$ km s-1, $\rho_{g}=3$ g cm-3, $a_{g}=0.25$ µm,
$M_{*}=2.5$ $M_{\odot}$, we get $\beta_{sw}=0.45$. The ratio of the radiation
pressure force ($F_{rd}$) to the gravitational force of star ($F_{gr}$) on a
dust grain, $\beta_{rd}$, has been computed by Lamy & Perrin (1997) for stars
of various spectral types and luminosity classes, two of which bracket the A2V
star – these are $\alpha$Aql (A5 IV-V) and $\alpha$CMi (F6 IV-V). This study
shows that, for silicate grains of radii in the range 0.1–1 µm,
$\beta_{rd}\sim 1-3$, i.e., significantly greater than the value of
$\beta_{sw}$ derived above. Thus, radiation pressure is much more effective in
pushing the grains away from the A2V star than the A2V star’s wind. Adopting
an intermediate value, $\beta_{rd}=2$, the acceleration due to radiation
pressure at a radial offset that is (say) halfway between the A2V star and the
CS, i.e. at $r=0\farcs 85$, is $\sim 7.2\times 10^{-6}$ cm s-2 (and varies as
$r^{-2}$). Thus, within about 1000 yr, radiation pressure will move the grains
that are located (say) halfway between the A2V star and the CS, roughly 100 au
or $0\farcs 13$ towards the CS. This pressure would push the outer regions of
the dust cloud on the side facing the A2V star towards the CS, providing a
plausible explanation for the observed flattening of the 18 µm intensity
contours defining the shape of the dust cloud on the side facing the A2V star.
But this process would also lead to an increase in the column density near
this edge, which would result in a brightening of the dust emission compared
to the outer regions of the dust cloud that are located on the side facing
away from the A2V star, contrary to what is observed.
A plausible explanation of why the dust cloud closer to the A2V star does not
show enhanced emission, is that the wind from the A2V star also destroys some
fraction of dust grains via sputtering. For example, e.g., Gray & Edmunds
(2004) have studied sputtering as a function of impact energy of hydrogen
nuclei, and their Fig. 2 shows a substantial sputtering yield for impact
energies of $\sim$1 keV (H and He nuclei moving at 300 km s-1 have energies
corresponding to 0.47 and 1.9 keV).
Given the clockwise rotation that we find for the A2V star around the CS, the
grains in the dust cloud must also be rotating clockwise (set by the sign of
the global angular momentum of the primordial dense core in which these stars
were formed). Hence, the dust grains approach the A2 star from the south-east
as a result of their orbital motion. When they are far from the separation
vector, the dominating force is the gravitational attraction to the CS, but
that force becomes progressively less as the grains get closer to the A2V
star. The “tail” seen in the 18 µm image in the north/north-west direction is
thus a signature of grains trapped by the A2V star (as they pass from a
dynamical regime dominated by gravitation attraction toward the CS to one that
is dominated by attraction toward the A2V star) or perhaps escaping the CS-A2V
system (because they pass a region with a very weak net gravitational force).
### 5.4 The Origin of the Density Discontinuity in the Dust Cloud
The discontinuity in the 2-shell dust model suggests the presence of a radial
gap in the dust cloud density, at a radius of $\sim$350 au. A plausible origin
of this gap is the presence of a giant planet or brown dwarf with an initial
orbital radius of $\sim 10$ au; following the mass-loss from the central star
as described above (§ 5.2), the orbital radius (as in the case of the A2V
companion) would increase to $\sim$350 au. This giant planet/ brown dwarf
could then open up a fairly wide gap in the disk – e.g., Crida et al. (2006)
have computed gap density profiles using semi-analytic calculations and
numerical simulations for a Jupiter mass planet in a disk, and find that the
widths are comparable or larger than the orbital radius for low viscosities.
The migration of the giant planet/ brown dwarf radially outwards could also
contribute to further broadening the gap. Multiple planets would produce
larger gaps but with shallower depths (e.g., see Fig. 1 of Duffell & Dong
(2015)).
## 6 Conclusions
We have analysed new imaging data of the central star (CS) of NGC 3132
obtained using its NIRCam and MIRI instruments onboard JWST, through a set of
filters spanning the $0.9-18$ µm range. Our main findings are as follows:
1. 1.
The CS is located at an angular distance of $1\farcs 696$ from a bright A2V
star to its north-east. We find that these stars form a wide gravitationally-
bound system, separated by 1277 au and located at a distance of 0.75 kpc from
Earth. The proper motions and radial velocities of these stars are consistent
within uncertainties. In addition, we detect relative orbital motion between
the two stars over a $20$ yr period, which is consistent with the expected
value from the 25,500 yr period of the binary system estimated from the
current separation and assuming an intermediate inclination angle of 45°
between the orbital and sky-planes.
2. 2.
The A2V star outshines the CS at all but the longest wavelength of 18 µm.
Using PSF subtraction of the A2V star, we find that CS is clearly seen in the
JWST images even at the shortest wavelength 0.9 µm. The CS is surrounded by
extended emission, seen directly at $18$ µm. Radial intensity cuts show that
the emission is extended in the $7.7-12.8$ µm range as well. This emission,
which is surface-brightness limited and is somewhat asymmetrical in its outer
regions, extends to a radius $\gtrsim$1600 au and most likely results from
thermal dust emission.
3. 3.
We have carried out aperture photometry of the CS in the JWST images. Using
these data, together with an archival UV spectrum and published optical
photometry, we have constructed the spectral-energy distribution (SED) of the
CS and its extended emission over the UV to mid-IR ($0.091-18$ µm) range. The
SED has a very distinctive shape in the $\sim 5-18$ µm region.
4. 4.
Using dust radiative modeling, we have fitted the SED of the CS and the radial
intensity distributions at $7.7,12.8$ and $18$ µm with a dust cloud that
extends to a radius of $\gtrsim$1785 au. Models with a dust composition of 70%
silicate and 30% amorphous carbon, and a modified MRN grain-size distribution
that increases the proportion of larger grains, i.e., one in which the number
of grains with radius $a$, varies as $n(a)\propto$ a-q for $a=0.005-0.25$ µm,
with q=1, provide the best fit to the specific shape of the SED in the $5-18$
µm range. The radial dust optical depth of this cloud is $\sim 10^{-4}$ at
0.55 µm; and the dust temperature decreases from about $232$ to $73$ K from
the inner to the outer radius. Our best-fit models give a total dust mass of
$(1.3\pm 0.15)\times 10^{-2}$ M⊕ within a radius of $1785$ au; the dust mass
estimate has a conservative uncertainty of about $\pm 25$%.
5. 5.
The material in the dust cloud may have come from a pre-existing Oort cloud
analog, or it may lie in a disk-like structure produced as a result of binary
interaction.
6. 6.
The radial extent of the dust cloud is almost identical to the orbital radius
of the CS-A2V binary stellar system; the cloud appears flattened on the side
facing the A2V star. These features suggest a physical interaction between the
disk and the A2V companion, due to a combination of radiation pressure (due to
the A2V star’s radiation), together with partial destruction of the dust
grains by sputtering (due to a tenuous wind from the A2V star).
7. 7.
A plausible evolutionary scenario that explains the spectacular mass-ejection
that has resulted in the current, multipolar planetary nebula, is one in which
three stars – the CS, a close low-mass companion (with, say, $a\lesssim 5$
au), and a much more distant A2V star (with, say, $a\sim$400 au) – formed a
stable hierarchical triple system on the main-sequence, but which then became
dynamically active much later due to the Eccentric Kozai-Lidov mechanism
causing a strong binary interaction between the inner pair and leading to a
loss of most of the primary’s envelope. The resulting severe reduction in the
CS mass caused the A2V star’s orbit to expand, resulting in its current
separation from the CS.
8. 8.
We set an upper limit of $\sim$0.1 $M_{\odot}$ on the mass of any main-
sequence star, i.e., spectral-type M6V (L$\sim 7\times 10^{-4}$ $L_{\odot}$,
$T_{eff}\sim 2500$ K) or later, that may be located close enough to the CS to
be unresolved and faint enough to be undetectable. Such a star could be the
close binary companion in the above evolutionary scenario, provided the binary
interaction did not lead to its merger with the CS.
## 7 acknowledgements
We thank an (anonymous) referee for his/her timely and thorough review which
has helped us improve our paper.
The Early Release Observations and associated materials were developed,
executed, and compiled by the ERO production team: Hannah Braun, Claire Blome,
Matthew Brown, Margaret Carruthers, Dan Coe, Joseph DePasquale, Nestor
Espinoza, Macarena Garcia Marin, Karl Gordon, Alaina Henry, Leah Hustak, Andi
James, Ann Jenkins, Anton Koekemoer, Stephanie LaMassa, David Law, Alexandra
Lockwood, Amaya Moro-Martin, Susan Mullally, Alyssa Pagan, Dani Player, Klaus
Pontoppidan, Charles Proffitt, Christine Pulliam, Leah Ramsay, Swara
Ravindranath, Neill Reid, Massimo Robberto, Elena Sabbi, Leonardo Ubeda. The
EROs were also made possible by the foundational efforts and support from the
JWST instruments, STScI planning and scheduling, and Data Management teams.
Most of the data presented in this paper were obtained from the Mikulski
Archive for Space Telescopes (MAST) at the Space Telescope Science Institute.
The specific observations analyzed can be accessed via
http://dx.doi.org/10.17909/6wd1-0170 (catalog
http://dx.doi.org/10.17909/6wd1-0170). This work is also based in part on
observations made with the Spitzer Space Telescope, which is operated by the
Jet Propulsion Laboratory, California Institute of Technology under a contract
with NASA, that can be accessed via https://doi.org/10.26131/irsa3 (catalog
https://doi.org/10.26131/irsa3).
RS’s contribution to the research described here was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with NASA. VB acknowledges support from the EVENTs/NEBULAE WEB research
program, Spanish AEI grant PID2019-105203GB-C21. CSC’s work is part of the
I+D+i project PID2019-105203GB-C22 funded by the Spanish
MCIN/AEI/10.13039/501100011033. The TMAW tool (http://astro.uni-tuebingen.de/
TMAW) used for this paper was constructed as part of the activities of the
German Astrophysical Virtual Observatory.
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Table 1: Photometry of the Central Star of NGC 3132 Filter | Wavelength | Flux | ErroraaPercentage Error in Flux in previous column | Apert. | Apert. | Phot
---|---|---|---|---|---|---
| (µm) | (mJy) | (%) | Rad.(′′) | Corr. | Ref.bbReferences for photometry: (1) Kohoutek & Laustsen 1977, (2) photometry on UVIS/WFC3 images (HST proposal 11699) from Monreal-Ibero & Walsh (2020), (3) Hubble Source Catalog V.3 (Whitmore et al. 2016), (4) $Gaia$ DR3, (5) this work; when pohotometry reference provides magnitudes, these are listed here
U | 0.36 | 2.15 | 15 | … | … | 14.8ccVega Magnitude (1)
F438W | 0.438 | 1.80 | 15 | … | … | 15.76ccVega Magnitude (3)
F438W | 0.438 | 1.62 | 15 | … | … | 15.876ddAB Magnitude (2)
F555W | 0.555 | 1.45 | 15 | … | … | 16.0ccVega Magnitude (3)
F555W | 0.555 | 1.19 | 15 | … | … | 16.212ddAB Magnitude (2)
G-band | 0.639 | 1.18 | 15 | … | … | (4)
F814W | 0.814 | 0.56 | 15 | … | … | 17.03ddAB Magnitude (3)
F090W | 0.90 | 0.38 | 15 | $0\farcs 10$ | 0.76 | (5)
F187N | 1.87 | 0.11 | 15 | $0\farcs 10$ | 0.75 | (5)
F212N | 2.12 | 0.106 | 15 | $0\farcs 10$ | 0.74 | (5)
F356W | 3.56 | 0.037 | 15 | $0\farcs 15$ | 0.71 | (5)
F405N | 4.05 | 0.020 | 15 | $0\farcs 15$ | 0.73 | (5)
F470N | 4.70 | 0.018 | 15 | $0\farcs 15$ | 0.72 | (5)
F770W | 7.7 | 0.070 | 10 | $0\farcs 35$ | 0.76 | (5)
F1130W | 11.3 | 1.16 | 10 | $0\farcs 75$ | 0.97 | (5)
F1280W | 12.8 | 1.2 | 10 | $0\farcs 75$ | 0.93 | (5)
F1800W | 18.0 | 9.9 | 10 | $1\farcs 1\times 1\farcs 3$ | … | (5)
F1800W | 18.0 | 10.3eeFlux derived from integration of radial intensity to outer radius in Col. (3) | 10 | $1\farcs 6$ | … | (5)
F1800W | 18.0 | 11.1eeFlux derived from integration of radial intensity to outer radius in Col. (3) | 10 | $1\farcs 8$ | … | (5)
F1800W | 18.0 | 12.1eeFlux derived from integration of radial intensity to outer radius in Col. (3) | 10 | $2\farcs 0$ | … | (5)
Table 2: Photometry of Field Stars Star | WavelengthaaWavelength of JWST filter-passband | FluxbbMeasured photometry from this study | ErrorccPercentage Error in Flux in previous column | WavelengthddWavelength of filter-passband for published IRAC photometry (Fazio et al. (2004)) | FluxeePublished photometry | ErrorccPercentage Error in Flux in previous column | Instr.ffInstrument/ Detector for published photometry | Publ.Phot.
---|---|---|---|---|---|---|---|---
| (µm) | (mJy) | (%) | (µm) | (mJy) | (%) | | Ref.ggReference for published photometry: (1) The Spitzer (SEIP) source list (SSTSL2) (Spitzer Science Center, SSC)
fs3 | 3.56 | 0.38 | 5 | 3.56 | 0.359 | 0.4 | IRAC 1 | 1
fs3 | 7.7 | 0.085 | 10 | 7.91 | 0.075 | 7 | IRAC 4 | 1
fs4 | 7.7 | 0.22 | 5 | 7.91 | 0.27 | 2.5 | IRAC 4 | 1
fs5 | 7.7 | 3.9 | 5 | 7.91 | 4.6 | 0.2 | IRAC 4 | 1
Table 3: Instr. | Filter | BoxaaBox size used for estimating a median background | % ErrorbbPercentage error in the scale-factor applied to the PSF for subtraction from A2V star image | G. smoothccFWHM of gaussian smoothing function applied to PSF before PSF-subtraction
---|---|---|---|---
| | (px/′′) | in PSF scale-factor | (′′)
NIRCam | F090W | 50/1.54 | 0.16$\times 10^{-2}$ | 0.05
NIRCam | F187N | 50/1.54 | 0.20$\times 10^{-2}$ | 0.03
NIRCam | F212N | 200/6.17 | 0.13$\times 10^{-2}$ | 0.02
NIRCam | F356W | 50/3.15 | 0.47$\times 10^{-2}$ | 0.05
NIRCam | F405N | 250/15.75 | 0.15$\times 10^{-1}$ | 0.05
NIRCam | F470N | 50/3.15 | 0.17$\times 10^{-1}$ | 0.045
MIRI | F770W | 200/22.18 | 0.26$\times 10^{-1}$ | 0.045
MIRI | F1130W | 200/22.18 | 0.97$\times 10^{-1}$ | 0.07
MIRI | F1280W | 169/18.74 | 0.15 | 0.1
MIRI | F1800W | 135/14.97 | 0.49 | 0.1
Table 4: Position Angles and Separations of Field Star Pairs StarPair | $PA(HST)$aaPA of separation vector between star pair in Epoch 1, using HST image. PA is measured from N towards E, with the second star in the pair as the origin | $PA($Gaia$,Epoch1)$bbPA of separation vector between star pair in Epoch 1, estimated from $Gaia$ DR3 data | Ang.Dist.(HST)ccAngular distance between star pair in Epoch 1, using HST image | Err(HST)ddError in angular distance between star pair in Epoch 1, using HST image | Frac.Diff.eeAngular distance between star pair in Epoch 1 (using HST image) minus the angular distance between star pair in Epoch 1 (estimated from $Gaia$ DR3 data), divided by the average angular distance between star pair
---|---|---|---|---|---
| ° | ° | ′′ | ′′ | $10^{-5}$
fs6, fs1 | 22.016 | 21.850 | 98.1236 | 0.00055 | -2.58
fs6, fs8 | 21.219 | 21.071 | 162.7896 | 0.0012 | -4.25
fs1, fs8 | 20.031 | 19.912 | 64.6894 | 0.0011 | -6.36
fs4, fs8 | 13.816 | 13.674 | 141.6898 | 0.0017 | -3.55
fs4, fs1 | 8.647 | 8.486 | 77.6963 | 0.0013 | -2.85
Table 5: Models of the Dust Emission towards the Central Star of NGC 3132 Td(in)aaThe (input) dust temperature at shell inner radius | $R_{in}$bbThe (inferred) inner radius of the dust shell | $R_{out}$ccThe (inferred) outer radius of the dust shell | nddThe (input) exponent of the density power law ($\rho_{d}(r)\propto r^{-n}$) in the dust shell | $\tau_{V}$eeThe (input) dust shell’s optical depth at $0.55\micron$ | $F_{\rm bol}$ffBolometric Flux | MdggThe (inferred) circumstellar dust mass
---|---|---|---|---|---|---
(K) | (arcsec, au) | (arcsec, au) | | | (erg s-1 cm-2) | (M⊕)
1-shell model
232 | $0\farcs 095$, 71 | $\gtrsim 2\farcs 37$, 1785 | -0.05 | $9.9\times 10^{-4}$ | $6.5\times 10^{-9}$ | $1.15\times 10^{-2}$
2-shell model
232 | $0\farcs 095$, 71 | $0\farcs 47$, 357 | 0.2 | $1.4\times 10^{-4}$ | $6.5\times 10^{-9}$ | $7.4\times 10^{-5}$
125 | $0\farcs 47$, 357 | $\gtrsim 2\farcs 37$, 1785 | -0.4 | $9.6\times 10^{-4}$ | … | $1.45\times 10^{-2}$
Figure 1: Images of the central region of NGC 3132, taken with MIRI (a)
F1800W, (b) F1280W, (c), F1130W, (d) F770W), and NIRCam (e) F187N (f) F212N
(g) F356W (h) F470N, shown using a logarithmic strech for the intensity
(MJy/sr). Black (white) dashed circles of diameter $1\farcs 0$ ($0\farcs 5$)
locate the central white-dwarf star (CS) in the MIRI (NIRC) images; the A2V
companion is located 1$\farcs$7 to the NE of the CS. Figure 2: Images of the
dust cloud around the CS, taken with F1800W (a) pipeline image (logarithmic
stretch), (b) PSF-sbtracted image (square-root strech). Contour levels are at
0.88, 0.50, 0.33, 0.25, 0.20, 0.167, 0.125, 0.10, 0.067, 0.05, 0.04, 0.033,
0.025 times the background-subtracted peak intensity (308 My/sr) in panel $a$;
the median background level is 135 MJy/sr. Dashed box in panel $b$ demarcates
the region of the “tail” feature. (c) Radial intensity cuts centered on the
CS, averaged over 90° wedges pointing away (solid red curve) and towards
(dashed red curve) from the A2V star. Brown curve shows the PSF extracted
using a field star. Figure 3: As in Fig. 1, but with the A2V star subtracted.
Figure 4: HST 0.55 µm (F555W) image of the CS and the A2V stars at the center
of NGC 3132. Dashed yellow line shows the vector separating the two stars,
with length $1\farcs 693$ and $PA=47.82\arcdeg$. North is up and East is to
the left. Figure 5: Observed photometry (black symbols) and model spectral-
energy-distributions (green curves) for the CS – panels $a$ and $b$
respectively show the 1-shell and 2-shell models. Panels $c$ and $d$ show the
model SED at short wavelengths where it is dominated by direct starlight
(reddened and attenuated by nebular extinction) and is unaffected by the low-
optical depth dust cloud, together with observed spectrum (panel $c$) or
photometry (panel $d$). The inset in panel $d$ shows the model in panel $a$ or
$b$ (green curve), an M6V dwarf companion (red curve), and the addition of the
two (black curve). Error bars ($\pm 15$%) on the observed photometry are
conservative estimates. Figure 6: Observed (azimuthally-averaged) and model
radial intensity distributions of the CS (+dust) of NGC 3132 as seen in the
(a) F1800W, (b) F1280 and (c) F770W filters, together with (d) the radial
distribution of density, visual optical depth, and dust temperature for the
best-fit 1-shell model. All distributions have been normalized to their peak
values. This 1-shell model uses a grain mixture consisting of 70% warm-
silicates and 30% amorphous Carbon (amC). The model intensities (blue) have
been convolved with the PSFs at 18, 12.8 and 7.7 µm, to generate the convolved
models (green), for direct comparion with the observations (red). The sharp
intensity peaks in the intrinsic model correspond to the inner radius of the
dust shell. The F1280W and F770W intensities are shown over a smaller range of
offsets compared to F1800W, because the observed radial intensity
distributions for the former are much more narrow than for the latter. Figure
7: As in Fig. 6, but for a 2-shell model.
We show here the large-scale structure of the PN, NGC 3132, imaged through the
F1800W (18 µm) filter (Fig. A.1A). The image does not show the presence of any
extended nebular feature (background or foreground) close to the compact dust
clump around the central star (CS).
Figure A.1A: The PN NGC 3132 imaged with MIRI, through the F1800W (18 µm)
filter, displayed using a square-root stretch. The green dashed circle
(diameter $5{{}^{\prime\prime}}$) encloses the dust cloud around the CS, and
the A2V star to its North-East. The panel size is $80{{}^{\prime\prime}}\times
80{{}^{\prime\prime}}$.
|
aainstitutetext: School of Science, Huzhou University, Huzhou, Zhejiang
313000, Chinabbinstitutetext: National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012, Chinaccinstitutetext: School of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, Chinaddinstitutetext: International Centre for
Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
# Unitarity bounds on extensions of Higgs sector
Bo-Qiang Lu b,c,d,2 Da Huang***Corresponding Author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
It is widely believed that extensions of the minimal Higgs sector is one of
the promising directions for resolving many puzzles beyond the Standard Model
(SM). In this work, we study the unitarity bounds on the models by extending
the two-Higgs-doublet model with an additional real or complex Higgs triplet
scalar. By noting that the SM gauge symmetries $SU(2)_{L}\times U(1)_{Y}$ are
recovered at high energies, we can classify the two-body scattering states by
decomposing the direct product of two scalar multiplets into their direct sum
of irreducible representations of electroweak gauge groups. In such state
bases, the s-wave amplitudes of two-body scalar scatterings can be written in
the form of block-diagonalized scattering matrices. Then the application of
the perturbative unitarity conditions on the eigenvalues of scattering
matrices leads to the analytic constraints on the model parameters. Finally,
we numerically investigated several specific models, finding that the
perturbative unitarity places stringent bounds on the extra scalar mass
spectrum, especially when there exist significant mixings among scalars in the
extended Higgs sectors.
## 1 Introduction
Despite the overwhelming success of the Standard Model (SM) by discovering the
125 GeV Higgs boson at the Large Hadron Collider (LHC) in 2012 ATLAS:2012yve ;
CMS:2012qbp , it has been widely believed that new physics is required to
explain various phenomena beyond the SM, such as tiny neutrino masses
Formaggio:2021nfz , the nature of dark matter Planck:2015fie , and the origin
of matter-antimatter asymmetry in the Universe Kuzmin:1985mm . One of the
promising directions for resolving these puzzles is to extend the minimal SM
Higgs sector by including additional scalars. Note that the shape of the SM
Higgs potential is fully determined by the vacuum expectation value (VEV),
$v$, and the quartic self-coupling, $\lambda_{H}$. However, in the non-minimal
extension of the Higgs section, there will be inevitable deviations of the
Higgs self-couplings with respect to the SM predictions. Therefore, the
precise measurement of the Higgs self-couplings can help us to probe the new
physics and to understand the electroweak symmetry breaking mechanism. Until
now, the determination of the trilinear Higgs coupling has been performed at
the LHC Run 2 and will be further searched for at the Run 3, by directly
detecting the single and double Higgs boson productions Degrassi:2021uik and
other indirect probes McCullough:2013rea ; Cao:2015oxx ; Bizon:2016wgr ;
deBlas:2016ojx . In addition to the above experimental endeavors, there has
already been considerable theoretical explorations in order to constrain the
Higgs sector, such as the perturbative unitarity Gell-Mann:1969cuq ;
Weinberg:1971fb ; Lee:1977yc ; Lee:1977eg , vacuum stability and triviality
Cabibbo:1979ay ; Lindner:1985uk .
In this work, we shall focus on the systematic derivation of the perturbative
unitarity bounds on the non-minimal Higgs sector with two or three Higgs
multiplets. As early as 1977, Lee, Quigg, and Thacker Lee:1977yc ; Lee:1977eg
made use of the perturbative unitarity and found the Higgs boson mass upper
bound $m_{h}<870$ GeV in the minimal SM. The perturbative unitarity has
recently been calculated in various extensions of the Higgs sector and been
identified as a significant constraint on the new physics. One of the most
popular extensions is the two-Higgs-doublet model (2HDM) (see Refs.
Branco:2011iw ; Wang:2022yhm for recent reviews) whose perturbative unitarity
was firstly calculated in Refs. Huffel:1980sk ; Maalampi:1991fb ;
Akeroyd:2000wc ; Arhrib:2000is with the assumptions of a softly broken
$Z_{2}$ symmetry and $CP$ conservation. The perturbative unitarity for the
most general 2HDM was given in Ref. Ginzburg:2005dt and the associated
numerical investigationwas carried out in detail in Ref. Kanemura:2015ska .
For other Beyond-SM theories, the unitarity bounds have been explored for the
Georgi-Machacek model Georgi:1985nv in Ref. Aoki:2007ah , for the Type-II
seesaw model Konetschny:1977bn ; Cheng:1980qt ; Magg:1980ut ; Schechter:1980gr
; Lazarides:1980nt ; Mohapatra:1980yp in Ref. Arhrib:2011uy , for extended
scalar sector with a real triplet scalar in Ref. Khan:2016sxm , and for a
complex triplet extension of the 2HDM with $CP$ conservation and a softly
broken $Z_{2}$ symmetry in Ref. Ouazghour:2018mld , respectively.
In this paper, we systematically study the unitarity bounds in extensions of
the 2HDM by including an additional real Higgs triplet $\Sigma$ with
hypercharge $Y_{\Sigma}=0$ or a complex Higgs triplet scalar $\Delta$ with
$Y_{\Delta}=1$ in the most general setup, in order to ensure the validity of
perturbation theory. Here we only concentrate on the high-energy limit where
the SM gauge symmetry broken effects can be ignored. Thus, we can classify the
two-scalar-particle states according to their conserved isospin and
hypercharge quantum numbers, and construct the associated 2-to-2 scattering
amplitude matrices in terms of the bases of $SU(2)_{L}\times U(1)_{Y}$
irreducible representations. Then we will consider the unitarity bounds in a
few special cases, including the extensions of the SM or 2HDM by one
additional real or complex scalar triplet, with or without a softly broken
$Z_{2}$ symmetry. Finally, we will numerically apply our derived unitarity
bounds to various concrete models, and show the corresponding constraints on
the extended scalar potentials.
This work is organized as follows. In Sec. 2, we provide the two-particle
state eigenbasis according to the irreducible representations of given
hypercharges and isospins. In Secs. 3 and 4, we present the scattering
amplitude matrices in extensions of the 2HDM by including an additional real
triplet and complex triplet, respectively. In Sec. 5, we show the unitarity
bounds on the model parameters in the 2HDM with $CP$ conservation and a softly
broken $Z_{2}$ symmetry, the complex 2HDM, and a complex triplet scalar
extension of 2HDM, respectively. The conclusions are summarized in Sec. 6. We
also include several appendix. In Appendix A, we provide the analytical
solutions for eigenvalues of a 5-dimensional scattering matrix appearing in
the real or complex triplet augment of the 2HDM with a softly broken $Z_{2}$
symmetry. In Appendix B, we provide elements of the mass matrices in the
complex triplet extension of 2HDM. The relations between parameters in the
generic scalar basis and the Higgs basis in this model are provided in
Appendix C. Finally, the trilinear couplings of a neutral scalar with two
charged Higgs particles are summarized in Appendix D.
## 2 Two-particle eigenstates and unitarity bounds on scattering matrices
The calculation of the unitarity bounds in the minimal SM was firstly
investigated in Refs. Lee:1977yc ; Lee:1977eg and has been applied to various
extensions of the SM. It requires that the eigenvalues of this scattering
matrix should be less than the unitarity limit Gell-Mann:1969cuq ;
Weinberg:1971fb ; Ginzburg:2005dt , otherwise the perturbative calculation of
scattering amplitudes at tree level is no more reliable. From another
perspective, one can make a partial wave expansion of the scattering
amplitudes for the interaction channels and put the unitarity bounds on the
partial wave amplitudes. Concretely, the cross section of scalar scattering
processes $s_{1}s_{2}\to s_{3}s_{4}$ can be expressed in terms of the partial
wave decomposition as
$\sigma=\frac{16\pi}{s}\sum_{l=1}^{\infty}(2l+1)\left|a_{l}(s)\right|^{2},$
(1)
where $s$ is the Mandelstam variable and $a_{l}$ is the partial wave
coefficients with the specific angular momenta $l$. Together with the optical
theorem one finds the following bound of unitarity:
$\left|\text{Re}\left(a_{l}\right)\right|<\frac{1}{2}\,,\quad\text{for
all}~{}l.$ (2)
In the high energy limit, it is found that the $s$-wave amplitude $a_{0}(s)$
is dominated by the point vertex processes since the $s$-, $t$-, $u$-channel
processes are suppressed by the scattering energy. Furthermore, the
equivalence theorem Cornwall:1973tb ; Cornwall:1974km ; Yao:1988aj ;
Veltman:1989ud ; He:1992nga declares that at very high energy, the amplitudes
of scattering processes involving longitudinal gauge bosons in the initial and
final states are equivalent to those in which gauge bosons are replaced by the
corresponding Nambu-Goldstone bosons. Thus, in the high energy limit
$a_{0}(s)$ is fully determined by the quartic couplings of the scalar
potential.
Using the equivalence theorem, we can write down the two-particle state bases
in terms of the components of the Higgs multiplets. Once given the scalar
potential, we can determine the amplitudes for the $2\to 2$ scattering
processes with the bases. This largely simplifies the calculations for
scattering amplitudes. In Refs. Arhrib:2000is ; Aoki:2007ah ; Kanemura:2015ska
; Ginzburg:2005dt , the bases are further classified according to their
electroweak (EW) charges, i.e., total hypercharge $Y$ and total isospin $I$,
since the EW $SU(2)_{L}\times U(1)_{Y}$ gauge symmetries are recovered at high
energies so that their associated quantum numbers becomes conserved again. In
this approach, we decompose the direct product of two Higgs multiplets into
the direct sums of irreducible representations under EW gauge symmetries.
Field | $\Phi$ | $\tilde{\Phi}$ | $\Sigma$ | $\Delta$ | $\tilde{\Delta}$
---|---|---|---|---|---
$SU(2)_{L}$ isospin | 2 | 2 | 3 | 3 | 3
Hypercharge | 1 | $-1$ | 0 | 2 | $-2$
Table 1: A summary of the quantum numbers of the Higgs multiplets. $\Phi$,
$\Sigma$, and $\Delta$ denotes the $SU(2)_{L}$ doublet, real triplet, and
complex triplet, respectively. We define $\tilde{\Phi}=i\tau_{2}\Phi^{*}$ and
$\tilde{\Delta}_{ab}=(i\tau_{2})_{ac}(i\tau_{2})_{bd}(\Delta^{\dagger})^{cd}$,
which have negative hypercharge.
In this work, we adopt an intermediate route for the classification of the
bases. Firstly, we classify the direct product of the two Higgs multiplets
according to their total hypercharge, where the isospins and hypercharges of
the Higgs multiplets considered in the present work are summarized in Table 1.
Then we decompose the direct product into direct sums of the irreducible
representations of the EW $SU(2)_{L}$ symmetry. There are three types of
direct products of Higgs multiplets we are concerned about, which are given as
follows:
$2\otimes 2=1\oplus 3,~{}2\otimes 3=2\oplus 4,~{}{\rm and}~{}3\otimes
3=1\oplus 3\oplus 5.$ (3)
In this way, we classify the two-particle bases according to their total
isospins and hypercharges of the two Higgs multiplets. Furthermore, we express
the bases of the irreducible representation in terms of components in the
multiplets. The results are summarized in Tables 2-7, in which the eigenstates
are rescaled so that they are normalized. Moreover, due to the symmetry
property when exchanging two identical bosons, some representations of the
two-particle eigenstates vanish, e.g., the $(Y,I)=(2,0)$ state in the
$\Phi_{i}\times\Phi_{j}$ when $i=j$ in Table 2, the $(Y,I)=(0,1)$ state from
$\Sigma\times\Sigma$ in Table 5, and the $(Y,I)=(4,0)$ state from
$\Delta\times\Delta$ in Table 6.
| $I=0$ | $I=1$
---|---|---
$Y=0$ | $\frac{1}{\sqrt{2}}\left(w_{i}^{+}w_{j}^{-}+H_{i}^{0}H_{j}^{0*}\right)$ | $w_{i}^{+}H_{j}^{0*}$ $\frac{1}{\sqrt{2}}\left(-w_{i}^{+}w_{j}^{-}+H_{i}^{0}H_{j}^{0*}\right)$ $-H_{i}^{0}w_{j}^{-}$
$Y=2$ | $\frac{1}{\sqrt{2}}\left(-w_{i}^{+}H_{j}^{0}+H_{i}^{0}w_{j}^{+}\right)$ | $w_{i}^{+}w_{j}^{+}~{}\left(\times\frac{1}{\sqrt{2}}~{}{\rm for}~{}i=j\right)$ $\frac{1}{\sqrt{2}}\left(w_{i}^{+}H_{j}^{0}+H_{i}^{0}w_{j}^{+}\right)$ $H_{i}^{0}H_{j}^{0}~{}\left(\times\frac{1}{\sqrt{2}}~{}{\rm for}~{}i=j\right)$
Table 2: The bases of the irreducible representation for the two Higgs doublets direct product. The bases in the first and second row are corresponding to the direct product $\Phi_{i}\times\tilde{\Phi}_{j}$ $(Y=0)$ and $\Phi_{i}\times\Phi_{j}$ $(Y=2)$, respectively. Note that $i$ and $j$ indicate the Higgs doublet. We observe that the bases with $(Y=2,~{}I=2)$ vanish when the two Higgs doublets are identical, i.e., $i=j$. | $I=\frac{1}{2}$ | $I=\frac{3}{2}$
---|---|---
$Y=1$ | $\sqrt{\frac{2}{3}}\left(-\frac{i}{\sqrt{2}}w^{+}\sigma^{0}+H^{0}\sigma^{+}\right)$ $\sqrt{\frac{2}{3}}\left(-w^{+}\sigma^{-}+\frac{i}{\sqrt{2}}H^{0}\sigma^{0}\right)$ | $w^{+}\sigma^{+}$ $\frac{1}{\sqrt{3}}\left(i\sqrt{2}w^{+}\sigma^{0}+H^{0}\sigma^{+}\right)$ $\frac{1}{\sqrt{3}}\left(w^{+}\sigma^{-}+i\sqrt{2}H^{0}\sigma^{0}\right)$ $H^{0}\sigma^{-}$
Table 3: The bases of the irreducible representation for the direct product of a Higgs doublet and a real Higgs triplet scalar, $\Phi\times\Sigma$. | $I=\frac{1}{2}$ | $I=\frac{3}{2}$
---|---|---
$Y=1$ | $\sqrt{\frac{2}{3}}\left(-\frac{1}{\sqrt{2}}H^{0*}\delta^{+}+w^{-}\delta^{++}\right)$ $\sqrt{\frac{2}{3}}\left(-H^{0*}\delta^{0}-\frac{1}{\sqrt{2}}w^{-}\delta^{+}\right)$ | $-H^{0*}\delta^{++}$ $\frac{1}{\sqrt{3}}\left(\sqrt{2}H^{0*}\delta^{+}+w^{-}\delta^{++}\right)$ $\frac{1}{\sqrt{3}}\left(H^{0*}\delta^{0}-\sqrt{2}w^{-}\delta^{+}\right)$ $-w^{-}\delta^{0}$
$Y=3$ | $\sqrt{\frac{2}{3}}\left(-\frac{1}{\sqrt{2}}w^{+}\delta^{+}-H^{0}\delta^{++}\right)$ $\sqrt{\frac{2}{3}}\left(-w^{+}\delta^{0}+\frac{1}{\sqrt{2}}H^{0}\delta^{+}\right)$ | $-w^{+}\delta^{++}$ $\frac{1}{\sqrt{3}}\left(\sqrt{2}w^{+}\delta^{+}-H^{0}\delta^{++}\right)$ $\frac{1}{\sqrt{3}}\left(w^{+}\delta^{0}+\sqrt{2}H^{0}\delta^{+}\right)$ $H^{0}\delta^{0}$
Table 4: The bases of the irreducible representation for the direct product of a Higgs doublet and a complex Higgs triplet scalar. The bases in the first and second row are corresponding to the direct product $\tilde{\Phi}\times\Delta$ $(Y=1)$ and $\Phi\times\Delta$ $(Y=3)$, respectively. | $I=0$ | $I=1$ | $I=2$
---|---|---|---
$Y=0$ | $\sqrt{\frac{2}{3}}\left(\sigma^{+}\sigma^{-}+\frac{1}{2}\sigma^{0}\sigma^{0}\right)$ | 0 | $\frac{1}{\sqrt{2}}\sigma^{+}\sigma^{+}$ $i\sigma^{+}\sigma^{0}$ $\frac{1}{\sqrt{3}}\left(\sigma^{+}\sigma^{-}-\sigma^{0}\sigma^{0}\right)$ $i\sigma^{-}\sigma^{0}$ $\frac{1}{\sqrt{2}}\sigma^{-}\sigma^{-}$
Table 5: The bases of the irreducible representation for the two real Higgs triplets direct product, $\Sigma\times\Sigma$. We only consider the case of the direct product of two identical triplet scalar (i.e., $i=j$). We find the bases with $I=1$ vanish in this case. | $I=0$ | $I=1$ | $I=2$
---|---|---|---
$Y=0$ | $\frac{1}{\sqrt{3}}\left(\delta^{++}\delta^{--}+\delta^{+}\delta^{-}+\delta^{0}\delta^{0*}\right)$ | $\frac{1}{\sqrt{2}}\left(-\delta^{++}\delta^{-}+\delta^{+}\delta^{0*}\right)$ $-\frac{1}{\sqrt{2}}\left(\delta^{++}\delta^{--}-\delta^{0}\delta^{0*}\right)$ $-\frac{1}{\sqrt{2}}\left(-\delta^{+}\delta^{--}+\delta^{-}\delta^{0}\right)$ | $-\delta^{++}\delta^{0*}$ $\frac{1}{\sqrt{2}}\left(\delta^{++}\delta^{-}+\delta^{+}\delta^{0*}\right)$ $\frac{1}{\sqrt{6}}\left(-2\delta^{+}\delta^{-}+\delta^{++}\delta^{--}+\delta^{0}\delta^{0*}\right)$ $\frac{1}{\sqrt{2}}\left(-\delta^{+}\delta^{--}-\delta^{0}\delta^{-}\right)$ $-\delta^{--}\delta^{0}$
$Y=4$ | $\sqrt{\frac{2}{3}}\left(-\delta^{++}\delta^{0}-\frac{1}{2}\delta^{+}\delta^{+}\right)$ | 0 | $\frac{1}{\sqrt{2}}\delta^{++}\delta^{++}$ $-\delta^{++}\delta^{+}$ $\frac{1}{\sqrt{3}}\left(\delta^{+}\delta^{+}-\delta^{++}\delta^{0}\right)$ $\delta^{0}\delta^{+}$ $\frac{1}{\sqrt{2}}\delta^{0}\delta^{0}$
Table 6: The bases of the irreducible representation for the direct product of two complex Higgs triplet scalars. The bases in the first and second row are corresponding to the direct product $\Delta\times\tilde{\Delta}$ $(Y=0)$ and $\Delta\times\Delta$ $(Y=4)$, respectively. We observe again that the bases with $(Y=4,~{}I=1)$ vanish because the two triplets are identical. | $I=0$ | $I=1$ | $I=2$
---|---|---|---
$Y=2$ | $\frac{1}{\sqrt{3}}\left(\sigma^{+}\delta^{0}-i\sigma^{0}\delta^{+}-\sigma^{-}\delta^{++}\right)$ | $-\frac{1}{\sqrt{2}}\left(\sigma^{+}\delta^{+}+i\sigma^{0}\delta^{++}\right)$ $-\frac{1}{\sqrt{2}}\left(\sigma^{+}\delta^{0}+\sigma^{-}\delta^{++}\right)$ $-\frac{1}{\sqrt{2}}\left(i\sigma^{0}\delta^{0}-\sigma^{-}\delta^{+}\right)$ | $\sigma^{-}\delta^{++}$ $\frac{1}{\sqrt{2}}\left(\sigma^{+}\delta^{+}-i\sigma^{0}\delta^{++}\right)$ $\frac{1}{\sqrt{6}}\left(\sigma^{+}\delta^{0}+i2\sigma^{0}\delta^{+}-\sigma^{-}\delta^{++}\right)$ $\frac{1}{\sqrt{2}}\left(i\sigma^{0}\delta^{0}+\sigma^{-}\delta^{+}\right)$ $\sigma^{-}\sigma^{0}$
Table 7: The bases of the irreducible representation for the direct product of
a real Higgs triplet and a complex Higgs triplet, $\Sigma\times\Delta$.
Based on the above two-particle basis, we can determine the $2\to 2$
scattering amplitudes as follows Ginzburg:2005dt
$S_{(Y,I)}=\left\langle(\phi\phi)_{Y,I}^{f}|\hat{S}|(\phi\phi)_{Y,I}^{i}\right\rangle,$
(4)
in the sector with definite EW charges $(Y,I)$ with $Y$ and $I$ as the total
hypercharge and isospin, respectively. We do not distinguish states by the
third components of isospin $I_{3}$, since the states with same $(Y,I)$ but
different $I_{3}$ would lead to exactly the same scattering matrix. In the
tree-level approximation, the elements of the scattering matrix among scalars
are determined by the quartic couplings in the scalar potential. Here we do
not decompose a complex scalar field into its real and imaginary parts either
in the external state basis or in the scalar potential due to the recovered
$SU(2)_{L}\times U(1)_{Y}$ symmetry at high energies.
As argued before, the unitarity of scattering amplitudes requires that the
$s$-wave amplitude $a_{0}(s)$ in the partial-wave expansion should fulfill the
bound in Eq. (2). Note that the amplitude of two scalar scatterings is
dominated by the $s$-wave one at the tree level, so that the unitarity bounds
can be transformed into the following condition on the eigenvalues
$\Lambda_{(Y,I)}$ of the scattering matrices $16\pi S_{(Y,I)}$ as follows
$|\Lambda_{(Y,I)}|\leq 8\pi\,.$ (5)
## 3 Two-Higgs-doublet model plus a real triplet
In this section, we will focus on the model that contains two Higgs doublets
and a real Higgs triplet scalar. The scattering matrix for the scalar
potential is provided in sec. 3.2. Using these results, we consider the
perturbative unitarity constraints on two simplified cases: the model with a
softly broken $Z_{2}$ symmetry and the $\Sigma$SM model FileviezPerez:2008bj .
### 3.1 The scalar potential
The scalar potential for the extension of the 2HDM with an additional real
Higgs triplet field $\Sigma$ is given by
$V_{r}=V\left(\Phi_{1},\Phi_{2}\right)+V(\Sigma)+V\left(\Phi_{1},\Phi_{2},\Sigma\right),$
(6)
where the Higgs doublet and real triplet scalar are
$\Phi_{i}=\begin{pmatrix}w_{i}^{+}\\\ H_{i}^{0}\end{pmatrix},~{}{\rm
and}~{}\Sigma=\left(\begin{array}[]{cc}\sigma^{0}/\sqrt{2}&\sigma^{+}\\\
\sigma^{-}&-\sigma^{0}/\sqrt{2}\end{array}\right),$ (7)
where we can further expand the
$H_{i}^{0}=\frac{1}{\sqrt{2}}(\varphi_{i}+iz_{i})$. The most general
renormalizable scalar potential for the 2HDM in the generic basis
$\\{\Phi_{1},\Phi_{2}\\}$ is commonly written as Davidson:2005cw
$\displaystyle V\left(\Phi_{1},\Phi_{2}\right)=$ $\displaystyle
m_{1}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{2}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left(m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\text{H.c.}\right)+\frac{1}{2}\lambda_{1}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}$
(8)
$\displaystyle+\frac{1}{2}\lambda_{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}+\lambda_{3}\Phi_{1}^{\dagger}\Phi_{1}\Phi_{2}^{\dagger}\Phi_{2}+\lambda_{4}\Phi_{1}^{\dagger}\Phi_{2}\Phi_{2}^{\dagger}\Phi_{1}+\left[\frac{1}{2}\lambda_{5}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}\right.$
$\displaystyle\left.+\lambda_{6}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)+\lambda_{7}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)+\text{H.c.}\right].$
The parameters $m_{12}^{2}$, $\lambda_{5}$, $\lambda_{6}$, and $\lambda_{7}$
should be real if we impose the CP conservation on the potential. If the
$Z_{2}$ symmetry with $\Phi_{1}\to\Phi_{1}$ and $\Phi_{2}\to-\Phi_{2}$ is only
softly broken by the term proportional to $m_{12}^{2}$, we should require
$\lambda_{6}=\lambda_{7}=0$. The potential for the real Higgs triplet scalar
is given by
$V(\Sigma)=\frac{1}{2}m_{\Sigma}^{2}\operatorname{Tr}\Sigma^{2}+\frac{1}{4}\lambda_{\Sigma}\operatorname{Tr}\Sigma^{4}\,.$
(9)
while the interactions between the Higgs doublets and the real triplet read as
follows
$\displaystyle V(\Phi_{1},\Phi_{2},\Sigma)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[a_{1}\Phi_{1}^{\dagger}\Sigma\Phi_{1}+a_{2}\Phi_{2}^{\dagger}\Sigma\Phi_{2}+\left(a_{12}\Phi_{1}^{\dagger}\Sigma\Phi_{2}+\text{H.c.}\right)\right]$
(10) $\displaystyle+$
$\displaystyle\frac{1}{2}\operatorname{Tr}\Sigma^{2}\left[\lambda_{8}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{9}\Phi_{2}^{\dagger}\Phi_{2}+\left(\lambda_{10}\Phi_{1}^{\dagger}\Phi_{2}+\text{H.c.}\right)\right]\,.$
For a real triplet, the possible terms
$\operatorname{Tr}\left(\Sigma^{4}\right)$ and $\Phi^{\dagger}\Sigma^{2}\Phi$
are not independent since they can be expressed as the combination of
$\left[\operatorname{Tr}\left(\Sigma^{2}\right)\right]^{2}$ and
$\operatorname{Tr}\left(\Sigma^{2}\right)\Phi^{\dagger}\Phi$. Also, the
potential cubic terms in the first line of Eq. (10) break the $Z_{2}^{\Sigma}$
symmetry: $\Sigma\to-\Sigma$, and are negligible for the $2\to 2$ scalar
scattering in the high energy limit. Therefore, these terms play no roles in
deriving the perturbative unitarity bounds. Furthermore, $\lambda_{10}$ can be
a complex parameter and should vanish when the $Z_{2}$ symmetry involving the
two Higgs doublets, softly-broken or not, is imposed.
### 3.2 Scattering matrix
Based on the two-particle bases given in Tables 2, 3, and 5 classified
according to the conserved quantum numbers $(Y,I)$, we can expand the general
potential in Eq. (6) with the scalar components defined in Eq. (7) and obtain
the following scattering matrices of given $(Y,I)$:
$16\pi
S_{(0,0)}=\left(\begin{array}[]{ccccc}3\lambda_{1}&2\lambda_{3}+\lambda_{4}&3\lambda_{6}&3\lambda_{6}^{*}&\sqrt{3}\lambda_{8}\\\
2\lambda_{3}+\lambda_{4}&3\lambda_{2}&3\lambda_{7}&3\lambda_{7}^{*}&\sqrt{3}\lambda_{9}\\\
3\lambda_{6}^{*}&3\lambda_{7}^{*}&\lambda_{3}+2\lambda_{4}&3\lambda_{5}^{*}&\sqrt{3}\lambda_{10}^{*}\\\
3\lambda_{6}&3\lambda_{7}&3\lambda_{5}&\lambda_{3}+2\lambda_{4}&\sqrt{3}\lambda_{10}\\\
\sqrt{3}\lambda_{8}&\sqrt{3}\lambda_{9}&\sqrt{3}\lambda_{10}&\sqrt{3}\lambda_{10}^{*}&5\lambda_{\Sigma}\end{array}\right)$
(11) $16\pi
S_{(0,1)}=\left(\begin{array}[]{cccc}\lambda_{1}&\lambda_{4}&\lambda_{6}&\lambda_{6}^{*}\\\
\lambda_{4}&\lambda_{2}&\lambda_{7}&\lambda_{7}^{*}\\\
\lambda_{6}^{*}&\lambda_{7}^{*}&\lambda_{3}&\lambda_{5}^{*}\\\
\lambda_{6}&\lambda_{7}&\lambda_{5}&\lambda_{3}\end{array}\right)$ (12) $16\pi
S_{(0,2)}=2\lambda_{\Sigma}$ (13) $16\pi S_{(1,\frac{1}{2})}=16\pi
S_{(1,\frac{3}{2})}=\left(\begin{array}[]{cc}\lambda_{8}&\lambda_{10}^{*}\\\
\lambda_{10}&\lambda_{9}\end{array}\right)$ (14) $16\pi
S_{(2,0)}=\lambda_{3}-\lambda_{4}$ (15) $16\pi
S_{(2,1)}=\left(\begin{array}[]{ccc}\lambda_{1}&\lambda_{5}^{*}&\sqrt{2}\lambda_{6}^{*}\\\
\lambda_{5}&\lambda_{2}&\sqrt{2}\lambda_{7}\\\
\sqrt{2}\lambda_{6}&\sqrt{2}\lambda_{7}^{*}&\lambda_{3}+\lambda_{4}\end{array}\right)$
(16)
Comparing with the 2HDM results in Ref. Ginzburg:2005dt , the scattering
matrix $16\pi S_{(0,0)}$ now becomes 5-dimensional, since there is an
additional state with $(Y,I)=(0,0)$ composed solely by components in the
triplet $\Sigma$. Furthermore, the scattering processes in the sectors with
$(Y,I)=(0,2),~{}(1,\frac{1}{2})$, and $(1,\frac{3}{2})$ take place only
between two scalar triplets.
### 3.3 The case with a $Z_{2}$ symmetry
Now we simplify our discussion by imposing the softly broken $Z_{2}$ symmetry
with $\Phi_{1}\to\Phi_{1}$ and $\Phi_{2}\to-\Phi_{2}$ on the scalar potential
(6), so that we have $\lambda_{6}=\lambda_{7}=\lambda_{10}=0$ but leaving a
nonzero $m_{12}^{2}$. Such a model is phenomenologically important because it
protects the theory from flavor changing neutral currents at tree level. Using
the results given in Appendix A, the matrix (11) can be block diagonalized
into a $2\times 2$ matrix $16\pi S_{(0,0)}^{(2)}$ and a $3\times 3$ one $16\pi
S_{(0,0)}^{(2)}$ as follows
$16\pi
S_{(0,0)}^{(2)}=\left(\begin{array}[]{cc}\lambda_{3}+2\lambda_{4}&3\lambda_{5}^{*}\\\
3\lambda_{5}&\lambda_{3}+2\lambda_{4}\end{array}\right)\,,\,\,16\pi
S_{(0,0)}^{(3)}=\left(\begin{array}[]{ccc}3\lambda_{1}&2\lambda_{3}+\lambda_{4}&\sqrt{3}\lambda_{8}\\\
2\lambda_{3}+\lambda_{4}&3\lambda_{2}&\sqrt{3}\lambda_{9}\\\
\sqrt{3}\lambda_{8}&\sqrt{3}\lambda_{9}&5\lambda_{\Sigma}\end{array}\right)\,.$
(17)
The eigenvalues for $16\pi S_{(0,0)}^{(3)}$ can be found numerically or
analytically by applying Eq. (76). Furthermore, the matrix $16\pi S_{(0,1)}$
in Eq. (12) can also be decomposed into the following two matrices,
$\displaystyle 16\pi
S_{(0,1)}^{\text{u}}=\left(\begin{array}[]{cc}\lambda_{1}&\lambda_{4}\\\
\lambda_{4}&\lambda_{1}\end{array}\right)\,,\quad 16\pi
S_{(0,1)}^{\text{d}}=\left(\begin{array}[]{cc}\lambda_{3}&\lambda_{5}^{*}\\\
\lambda_{5}&\lambda_{3}\end{array}\right)\,.$ (22)
Apart from $16\pi S_{(0,0)}^{(3)}$, the eigenvalues for the scattering
matrices are summarized as follows:
$\displaystyle\Lambda_{(0,0)}^{(2)\pm}=\lambda_{3}+2\lambda_{4}\pm
3\left|\lambda_{5}\right|,$ (23)
$\displaystyle\Lambda_{(0,1)}^{\text{u}\pm}=\frac{1}{2}\left(\lambda_{1}+\lambda_{2}\pm\sqrt{\left(\lambda_{1}-\lambda_{2}\right)^{2}+4\lambda_{4}^{2}}\right),$
$\displaystyle\Lambda_{(0,1)}^{\text{d}\pm}=\lambda_{3}\pm\left|\lambda_{5}\right|,$
$\displaystyle\Lambda_{(2,0)}=\lambda_{3}-\lambda_{4},~{}~{}\Lambda_{(2,1)}=\lambda_{3}+\lambda_{4},$
$\displaystyle\Lambda_{(2,1)}^{\pm}=\frac{1}{2}\left(\lambda_{1}+\lambda_{2}\pm\sqrt{\left(\lambda_{1}-\lambda_{2}\right)^{2}+4\left|\lambda_{5}\right|^{2}}\right),$
$\displaystyle\Lambda_{(0,2)}=2\lambda_{\Sigma},~{}~{}\Lambda_{(1,\frac{1}{2})}^{1}=\Lambda_{(1,\frac{3}{2})}^{1}=\lambda_{8},~{}~{}\Lambda_{(1,\frac{1}{2})}^{2}=\Lambda_{(1,\frac{3}{2})}^{2}=\lambda_{9},$
where $\Lambda_{(0,0)}^{2\pm}$, $\Lambda_{(0,1)}^{\text{u}\pm}$, and
$\Lambda_{(0,1)}^{\text{d}\pm}$ are the eigenvalues for $16\pi
S_{(0,0)}^{(2)}$, $16\pi S_{(0,1)}^{\text{u}}$, and $16\pi
S_{(0,1)}^{\text{d}}$. By further assuming
$\lambda_{8}=\lambda_{9}=\lambda_{\Sigma}=0$ in the matrix $S_{(0,0)}^{(3)}$,
we can reproduce the eigenvalues $\Lambda_{00\pm}^{\text{even }}$ in Eq. (10)
of Ref. Ginzburg:2005dt . The last line of Eq. (LABEL:eq:2HDMEV) gives the
eigenvalues for the scattering matrices involving only components of the real
triplet. Together with numerical eigenvalues of $16\pi S_{(0,0)}^{(3)}$, we
have provided all eigenvalues for the model (6) with the softly broken $Z_{2}$
symmetry.
### 3.4 The $\Sigma$SM model
The $\Sigma$SM model is a simple extension of the SM by a real triplet scalar,
many aspects of which has been extensively investigated in the literature,
such as the dark matter phenomenology FileviezPerez:2008bj ;
YaserAyazi:2014jby ; Chiang:2020rcv , the LHC searches Wang:2013jba ;
Bandyopadhyay:2014vma , and the strongly first-order electroweak phase
transition Niemi:2018asa ; Bell:2020gug . Now we only have one Higgs doublet
$\Phi$ which should be identified as the $\Phi_{1}$ in the 2HDM case. Thus,
$\Sigma$SM can be reproduced by setting all the couplings in Eq. (6) involving
the second Higgs doublet $\Phi_{2}$ to vanish, with the potential given by
$V_{r}=V(\Phi)+V(\Sigma)+V(\Phi,\Sigma)\,,$ (24)
where $V(\Sigma)$ is already given in Eq. (9) and
$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle\mu^{2}\Phi^{\dagger}\Phi+\lambda_{\Phi}\left(\Phi^{\dagger}\Phi\right)^{2}\,,$
$\displaystyle V(\Phi,\Sigma)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}a_{1}\Phi^{\dagger}\Sigma\Phi+\frac{\lambda_{8}}{2}\left({\rm
Tr}\Sigma^{2}\right)\Phi^{\dagger}\Phi\,.$ (25)
By using Eqs. (17) and (LABEL:eq:2HDMEV), the unitarity bounds on the
$\Sigma$SM are then found to be
$\displaystyle|\lambda_{\Phi}|\leq 4\pi,~{}|\lambda_{\Sigma}|\leq
4\pi,~{}|\lambda_{8}|\leq 8\pi,$ (26)
$\displaystyle|6\lambda_{\Phi}+5\lambda_{\Sigma}\pm\sqrt{\left(6\lambda_{\Phi}-5\lambda_{\Sigma}\right)^{2}+12\lambda_{8}}|\leq
16\pi\,,$
which confirm the unitarity bounds provided in Ref. Khan:2016sxm .
## 4 Two-Higgs-doublet model plus a complex triplet
In this section, we will consider the extension of 2HDM by a complex Higgs
triplet $\Delta$ with $Y=2$ Chen:2021jok . By using the state bases provided
in Sec. 2, we shall calculate the scattering matrix for the most general case
of the model. We shall then impose the perturbative unitarity constraints on
the eigenvalues of the scattering matrix for several simplified models, such
as the one with a softly broken $Z_{2}$ symmetry and the Type-II seesaw model
Konetschny:1977bn ; Cheng:1980qt ; Magg:1980ut ; Schechter:1980gr ;
Lazarides:1980nt ; Mohapatra:1980yp ; Mohapatra:1999zr .
### 4.1 The general scalar potential
The general scalar potential for the model with two Higgs doublets and a
complex Higgs triplet scalar $\Delta$ is given by
$V_{c}=V\left(\Phi_{1},\Phi_{2}\right)+V(\Delta)+V\left(\Phi_{1},\Phi_{2},\Delta\right),$
(27)
where the complex Higgs triplet is written as
$\Delta=\left(\begin{array}[]{cc}\delta^{+}/\sqrt{2}&\delta^{++}\\\
\delta^{0}&-\delta^{+}/\sqrt{2}\end{array}\right)\,.$ (28)
Note that the neutral component $\delta^{0}$ is a complex scalar. The 2HDM
potential $V\left(\Phi_{1},\Phi_{2}\right)$ has been provided in Eq. (8),
while the part related to the self-interactions of the complex Higgs triplet
is given by
$V(\Delta)=m_{\Delta}^{2}\operatorname{Tr}\Delta^{\dagger}\Delta+\lambda_{\Delta
1}\left(\operatorname{Tr}\Delta^{\dagger}\Delta\right)^{2}+\lambda_{\Delta
2}\operatorname{Tr}\left(\Delta^{\dagger}\Delta\right)^{2}.$ (29)
The third part in Eq. (27) gives the interactions among the Higgs doublets and
the triplet Chen:2021jok
$\displaystyle V\left(\Phi_{1},\Phi_{2},\Delta\right)=$
$\displaystyle\left(\mu_{1}\Phi_{1}^{T}i\tau_{2}\Delta^{\dagger}\Phi_{1}+\mu_{2}\Phi_{2}^{T}i\tau_{2}\Delta^{\dagger}\Phi_{2}+\mu_{3}\Phi_{1}^{T}i\tau_{2}\Delta^{\dagger}\Phi_{2}+\text{H.c.}\right)$
(30)
$\displaystyle+\left[\lambda_{8}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{9}\Phi_{2}^{\dagger}\Phi_{2}+\left(\lambda_{10}\Phi_{1}^{\dagger}\Phi_{2}+\text{H.c.}\right)\right]\operatorname{Tr}\Delta^{\dagger}\Delta$
$\displaystyle+\lambda_{11}\Phi_{1}^{\dagger}\Delta\Delta^{\dagger}\Phi_{1}+\lambda_{12}\Phi_{2}^{\dagger}\Delta\Delta^{\dagger}\Phi_{2}+\left(\lambda_{13}\Phi_{1}^{\dagger}\Delta\Delta^{\dagger}\Phi_{2}+\text{H.c.}\right).$
The $Z_{2}^{\Delta}$ symmetry of the transformation $\Delta\to-\Delta$ is only
softly broken by the cubic terms in the first line of Eq. (30), which are
negligible for the $2\to 2$ scalar scatterings in the high energy limit, so
that they cannot be constrained by the unitarity bounds. Note that there are
possibly additional cubic interactions like
$\Delta^{a}_{b}\Delta^{b}_{c}\Delta^{c}_{a}$. But we ignore them since they do
not contribute to the unitarity bounds. The parameters $\lambda_{10}$ and
$\lambda_{13}$ can be complex, and the associated interactions explicitly
break the $Z_{2}$ symmetry involved in the two Higgs doublets.
### 4.2 Scattering matrix
We expand the scalar potential (27) in terms of components in the two doublets
and the triplet. With the two-particle eigenstates given in Tables. 2, 4, and
6, we can determine the scattering matrices for different conserved quantum
numbers $(Y,I)$, which are summarized as follows:
$16\pi
S_{(0,0)}=\left(\begin{array}[]{ccccc}3\lambda_{1}&2\lambda_{3}+\lambda_{4}&3\lambda_{6}&3\lambda_{6}^{*}&\lambda_{a}\\\
2\lambda_{3}+\lambda_{4}&3\lambda_{2}&3\lambda_{7}&3\lambda_{7}^{*}&\lambda_{b}\\\
3\lambda_{6}^{*}&3\lambda_{7}^{*}&\lambda_{3}+2\lambda_{4}&3\lambda_{5}^{*}&\lambda_{c}^{*}\\\
3\lambda_{6}&3\lambda_{7}&3\lambda_{5}&\lambda_{3}+2\lambda_{4}&\lambda_{c}\\\
\lambda_{a}&\lambda_{b}&\lambda_{c}&\lambda_{c}^{*}&\lambda_{\Delta}\end{array}\right)~{}{\rm
with}~{}\left\\{\begin{matrix}\lambda_{a}&=&\sqrt{\frac{3}{2}}(2\lambda_{8}+\lambda_{11})\\\
\lambda_{b}&=&\sqrt{\frac{3}{2}}(2\lambda_{9}+\lambda_{12})\\\
\lambda_{c}&=&\sqrt{\frac{3}{2}}(2\lambda_{10}+\lambda_{13})\\\
\lambda_{\Delta}&=&2(4\lambda_{\Delta 1}+3\lambda_{\Delta
2})\end{matrix}\right.$ (31) $16\pi
S_{(0,1)}=\left(\begin{array}[]{ccccc}\lambda_{1}&\lambda_{4}&\lambda_{6}&\lambda_{6}^{*}&\lambda_{11}\\\
\lambda_{4}&\lambda_{2}&\lambda_{7}&\lambda_{7}^{*}&\lambda_{12}\\\
\lambda_{6}^{*}&\lambda_{7}^{*}&\lambda_{3}&\lambda_{5}^{*}&\lambda_{13}^{*}\\\
\lambda_{6}&\lambda_{7}&\lambda_{5}&\lambda_{3}&\lambda_{13}\\\
\lambda_{11}&\lambda_{12}&\lambda_{13}&\lambda_{13}^{*}&2\lambda_{\Delta
1}+4\lambda_{\Delta 2}\end{array}\right)$ (32) $16\pi
S_{(0,2)}=2\lambda_{\Delta 1}$ (33) $16\pi
S_{(1,\frac{1}{2})}=\left(\begin{array}[]{cc}\lambda_{8}+3\lambda_{11}/2&\lambda_{10}+3\lambda_{13}/2\\\
\lambda_{10}^{*}+3\lambda_{13}^{*}/2&\lambda_{9}+3\lambda_{12}/2\end{array}\right)$
(34) $16\pi
S_{(1,\frac{3}{2})}=\left(\begin{array}[]{cc}\lambda_{8}&\lambda_{10}\\\
\lambda_{10}^{*}&\lambda_{9}\end{array}\right)$ (35) $16\pi
S_{(2,0)}=\lambda_{3}-\lambda_{4}$ (36) $16\pi
S_{2,1}=\left(\begin{array}[]{ccc}\lambda_{1}&\lambda_{5}^{*}&\sqrt{2}\lambda_{6}\\\
\lambda_{5}&\lambda_{2}&\sqrt{2}\lambda_{7}^{*}\\\
\sqrt{2}\lambda_{6}^{*}&\sqrt{2}\lambda_{7}&\lambda_{3}+\lambda_{4}\end{array}\right)$
(37) $16\pi
S_{(3,\frac{1}{2})}=\left(\begin{array}[]{cc}\lambda_{8}-\lambda_{11}/2&\lambda_{10}^{*}-\lambda_{13}^{*}/2\\\
\lambda_{10}-\lambda_{13}/2&\lambda_{9}-\lambda_{12}/2\end{array}\right)$ (38)
$16\pi
S_{(3,\frac{3}{2})}=\left(\begin{array}[]{cc}\lambda_{8}+\lambda_{11}&\lambda_{10}^{*}+\lambda_{13}^{*}\\\
\lambda_{10}+\lambda_{13}&\lambda_{9}+\lambda_{12}\end{array}\right)$ (39)
$16\pi S_{(4,0)}=2\lambda_{\Delta 1}-\lambda_{\Delta 2}$ (40) $16\pi
S_{(4,2)}=2(\lambda_{\Delta 1}+\lambda_{\Delta 2})$ (41)
We observe that the scattering matrices $16\pi S_{(0,0)}$ and $16\pi
S_{(0,1)}$ are now five-dimensional, which is compared with the four-
dimensional matrices in the 2HDM. This is caused by the fact that irreducible
representations of the product of two Higgs triplets contain the states with
$(Y,I)=(0,0)$ and $(0,1)$, which can scatter into two components of Higgs
doublets with the same quantum numbers. On the other hand, the scattering
processes with $(Y,I)=(0,2),~{}(1,\frac{1}{2}),~{}(1,\frac{3}{2})$,
$(3,\frac{1}{2}),~{}(3,\frac{3}{2})$, $(4,0)$, and $(4,2)$ take place only
among the complex Higgs triplets.
### 4.3 Special case with a softly broken $Z_{2}$ symmetry
Now we consider some simplified models in the above complex triplet extension
of the 2HDM, which may allow us to obtain the eigenvalues of scattering
matrices analytically. The first example is to impose a softly broken $Z_{2}$
symmetry on the potential of the two Higgs doublets in Eq. (27), so that the
$m_{12}^{2}$, we have the condition
$\lambda_{6}=\lambda_{7}=\lambda_{10}=\lambda_{13}=0$ for the potential. In
this case, the $5\times 5$ scattering matrix (31) and (32) can be decomposed
into a 2-dimensional (which is correspondingly denoted as $16\pi
S_{(0,0)}^{(2)}$ and $16\pi S_{(0,1)}^{(2)}$) and a 3-dimensional matrices.
The corresponding 2-dimensional and 3-dimensional matrices are given by
$16\pi
S_{(0,0)}^{(2)}=\left(\begin{array}[]{cc}\lambda_{3}+2\lambda_{4}&3\lambda_{5}^{*}\\\
3\lambda_{5}&\lambda_{3}+2\lambda_{4}\end{array}\right),~{}16\pi
S_{(0,1)}^{(2)}=\left(\begin{array}[]{cc}\lambda_{3}&\lambda_{5}^{*}\\\
\lambda_{5}&\lambda_{3}\end{array}\right).$ (42) $16\pi
S_{(0,0)}^{(3)}=\left(\begin{array}[]{ccc}3\lambda_{1}&2\lambda_{3}+\lambda_{4}&\lambda_{a}\\\
2\lambda_{3}+\lambda_{4}&3\lambda_{2}&\lambda_{b}\\\
\lambda_{a}&\lambda_{b}&\lambda_{\Delta}\end{array}\right),~{}16\pi
S_{(0,1)}^{(3)}=\left(\begin{array}[]{ccc}\lambda_{1}&\lambda_{4}&\lambda_{11}\\\
\lambda_{4}&\lambda_{2}&\lambda_{12}\\\
\lambda_{11}&\lambda_{12}&2\lambda_{\Delta 1}+4\lambda_{\Delta
2}\end{array}\right).$ (43)
It is convenient to find the eigenvalues for $16\pi S_{(0,0)}^{(3)}$ and
$16\pi S_{(0,1)}^{(3)}$ numerically, we also provide the analytical solutions
in Eq. (76). We collect the eigenvalues for the remaining scattering matrices
as follows:
$\displaystyle\Lambda_{(0,0)}^{2\pm}=\lambda_{3}+2\lambda_{4}\pm
3\left|\lambda_{5}\right|,$ (44)
$\displaystyle\Lambda_{(0,1)}^{2\pm}=\lambda_{3}\pm\left|\lambda_{5}\right|,$
$\displaystyle\Lambda_{(0,2)}=2\lambda_{\Delta 1},$
$\displaystyle\Lambda_{(1,\frac{1}{2})}^{1}=\lambda_{8}+3\lambda_{11}/2,~{}~{}\Lambda_{(1,\frac{1}{2})}^{2}=\lambda_{9}+3\lambda_{12}/2,$
$\displaystyle\Lambda_{(1,\frac{3}{2})}^{1}=\lambda_{8},~{}~{}\Lambda_{(1,\frac{3}{2})}^{2}=\lambda_{9},$
$\displaystyle\Lambda_{(2,0)}=\lambda_{3}-\lambda_{4},~{}~{}\Lambda_{(2,1)}^{1}=\lambda_{3}+\lambda_{4},$
$\displaystyle\Lambda_{(2,1)}^{2\pm}=\frac{1}{2}\left(\lambda_{1}+\lambda_{2}\pm\sqrt{\left(\lambda_{1}-\lambda_{2}\right)^{2}+4\left|\lambda_{5}\right|^{2}}\right),$
$\displaystyle\Lambda_{(3,\frac{1}{2})}^{1}=\lambda_{8}-\lambda_{11}/2,~{}~{}\Lambda_{(3,\frac{1}{2})}^{2}=\lambda_{9}-\lambda_{12}/2,$
$\displaystyle\Lambda_{(3,\frac{3}{2})}^{1}=\lambda_{8}+\lambda_{11},~{}~{}\Lambda_{(3,\frac{3}{2})}^{2}=\lambda_{9}+\lambda_{12},$
$\displaystyle\Lambda_{(4,0)}=2\lambda_{\Delta 1}-\lambda_{\Delta
2},~{}~{}\Lambda_{(4,2)}=2(\lambda_{\Delta 1}+\lambda_{\Delta 2}),$
where $\Lambda_{(0,0)}^{2\pm}$ and $\Lambda_{(0,1)}^{2\pm}$ are the
eigenvalues for $16\pi S_{(0,0)}^{(2)}$ and $16\pi S_{(0,1)}^{(2)}$,
respectively. The remaining results in Eq. (44) represent the eigenvalues for
the matrices (33)-(41). Combining with numerical eigenvalues of the matrices
$16\pi S_{(0,0)}^{(3)}$ and $16\pi S_{(0,1)}^{(3)}$, we have provided all
eigenvalues for the scattering matrices in the model (27) with a softly broken
$Z_{2}$ symmetry.
### 4.4 The $\Delta$SM
Another simple example belonging the present class is to consider a model with
only one Higgs doublet and one complex triplet, the so-called $\Delta$SM,
which has been widely employed to explain the tiny neutrino masses by the
type-II seesaw mechanism Konetschny:1977bn ; Cheng:1980qt ; Magg:1980ut ;
Schechter:1980gr ; Lazarides:1980nt ; Mohapatra:1980yp ; Mohapatra:1999zr ;
Gu:2006wj ; Chao:2007mz ; FileviezPerez:2008jbu . The investigations of this
model are extended to the searches at colliders Melfo:2011nx ; Chen:2013dh ;
Han:2015sca ; Dev:2018sel ; Du:2018eaw ; Cheng:2022jyi , dark matter
Ding:2017jdr and electroweak phase transition (EWPT) phenomena Arhrib:2011uy
; Primulando:2019evb ; Zhou:2022mlz . From potential (27), we can reproduce
the Higgs sector of $\Delta$SM by keeping the quartic terms with couplings
$\lambda_{1}$, $\lambda_{8}$, $\lambda_{11}$, $\lambda_{\Delta 1}$, and
$\lambda_{\Delta 2}$ while setting all other quartic couplings to zero. Again,
we take $\lambda_{1}=2\lambda_{\Phi}$ to reproduce the commonly seen SM Higgs
potential in Eq. (3.4). By using the matrices in Eq. (43) and the eigenvalues
in Eq. (44), the unitarity bounds for the $\Delta$SM are given by
$\displaystyle|\lambda_{\Phi}|\leq 4\pi,~{}|\lambda_{\Delta 1}|\leq
4\pi,~{}|\lambda_{8}|\leq 8\pi$ (45) $\displaystyle|2\lambda_{\Delta
1}-\lambda_{\Delta 2}|\leq 8\pi,~{}|\lambda_{\Delta 1}+\lambda_{\Delta 2}|\leq
4\pi,$ $\displaystyle|\lambda_{8}+3\lambda_{11}/2|\leq
8\pi,~{}|\lambda_{8}-\lambda_{11}/2|\leq
8\pi,~{}|\lambda_{8}+\lambda_{11}|\leq 8\pi,$
$\displaystyle|\lambda_{\Phi}+\lambda_{\Delta 1}+2\lambda_{\Delta
2}\pm\sqrt{(\lambda_{\Phi}-\lambda_{\Delta 1}-2\lambda_{\Delta
2})+\lambda_{11}^{2}}|\leq 8\pi,$
$\displaystyle|6\lambda_{\Phi}+8\lambda_{\Delta 1}+6\lambda_{\Delta
2}\pm\sqrt{(6\lambda_{\Phi}-8\lambda_{\Delta 1}-6\lambda_{\Delta
2})^{2}+6(2\lambda_{8}+\lambda_{11})^{2}}|\leq 16\pi\,,$
which are in agreement with the results given in Ref. Arhrib:2011uy .
## 5 Applications
The unitarity constraint on the quartic couplings can be translated into the
upper bounds on the Higgs boson masses if $\sqrt{\lambda_{i}}v$ dominates the
contributions to the Higgs boson masses. In this section, we will apply the
perturbative unitarity to three specific models and show the quantitative
constraints on the model parameters.
### 5.1 $CP$-conserving 2HDM with a $Z_{2}$ symmetry
Let us firstly focus on the $CP$-conserving 2HDM with a softly broken $Z_{2}$
symmetry, in which the potential is given by Eq. (8) . In this simplified
case, we have $\lambda_{6}=\lambda_{7}=0$ and all parameters in the scalar
potential are real. We can rotate the generic scalar basis
$\\{\Phi_{1},\Phi_{2}\\}$ into the Higgs basis $\\{H_{1},H_{2}\\}$ via the
transformation
$\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos\beta&\sin\beta\\\
-\sin\beta&\cos\beta\end{array}\right)\left(\begin{array}[]{c}\Phi_{1}\\\
\Phi_{2}\end{array}\right)$ (46)
so that only $H_{1}$ has a non-vanishing VEV
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}\simeq 246$ GeV. Here the quantity $\tan\beta$ is
defined by the ratio of two Higgs field VEVs, i.e., $\tan\beta=v_{2}/v_{1}$,
and therefore, depends on the choices of the doublet field basis. In the most
general 2HDM, $\tan\beta$ is of no particular importance since there is no
preferred basis choice in this model Davidson:2005cw ; Haber:2006ue . On the
other hand, in the case with a discrete $Z_{2}$ symmetry:
$\Phi_{1}\to\Phi_{1}$ and $\Phi_{2}\to-\Phi_{2}$, $\tan\beta$ is a meaningful
parameter that connects the $Z_{2}$ charge basis with the Higgs basis. In the
following, we will use shorthanded notations $t_{\theta}\equiv\tan\theta$,
$s_{\theta}\equiv\sin\theta$, and $c_{\theta}\equiv\cos\theta$, in which
$\theta$ is an arbitrary angle. In the Higgs basis, the two doublet scalars
can be expressed in components as follows,
$H_{1}=\left(\begin{array}[]{c}H_{1}^{+}\\\
\frac{1}{\sqrt{2}}\left(v+h_{1}^{0}+iA_{1}\right)\end{array}\right),~{}~{}{\rm
and}~{}~{}H_{2}=\left(\begin{array}[]{c}H^{+}\\\
\frac{1}{\sqrt{2}}\left(h_{2}^{0}+iA_{2}\right),\end{array}\right)$ (47)
where $h_{1}^{0}$ and $h_{2}^{0}$ are CP-even neutral Higgs bosons, $A_{2}$
and $H^{+}$ are the physical neutral pseudoscalar and the charged scalar,
respectively, while $H_{1}^{\pm}$and $A_{1}$ are the Goldstone bosons
associated with the $W^{\pm}$ and $Z$ gauge bosons. Since $A_{2}$ here is a
mass eigenstate, we will denote it as $A\equiv A_{2}$ for latter convenience.
The scalar potential (8) in the Higgs basis $\\{H_{1},H_{2}\\}$ takes the same
form as that in the generic scalar basis $\\{\Phi_{1},\Phi_{2}\\}$, with the
coefficients in Eq. (8) being replaced by
$m_{11}^{2}\to\tilde{m}_{11}^{2},~{}m_{22}^{2}\to\tilde{m}_{22}^{2},~{}m_{12}^{2}\to\tilde{m}_{12}^{2},~{}{\rm
and}~{}\lambda_{i}\to\tilde{\lambda}_{i}~{}(i=1,2,...,7).$ (48)
The explicit relations between two sets of parameters can be found in Appendix
C. With Eq. (47), the scalar potential can be expanded in terms of component
fields. By using the global minimum conditions for the 2HDM potential in the
Higgs basis, we obtain
$\tilde{m}_{11}^{2}=-\frac{1}{2}\tilde{\lambda}_{1}v^{2}~{}~{}{\rm
and}~{}~{}\tilde{m}_{12}^{2}=\frac{1}{2}\tilde{\lambda}_{6}v^{2}.$ (49)
With the second derivatives of the potential with respect to the component
fields as in Appendix B, the masses of the pseudoscalar and singly-charged
Higgs boson can be obtained as follows
$\displaystyle m_{A}^{2}$ $\displaystyle=$
$\displaystyle\tilde{m}_{22}^{2}+\frac{v^{2}}{2}\left(\tilde{\lambda}_{3}+\tilde{\lambda}_{4}-\tilde{\lambda}_{5}\right),$
(50) $\displaystyle m_{H^{\pm}}^{2}$ $\displaystyle=$
$\displaystyle\tilde{m}_{22}^{2}+\frac{v^{2}}{2}\tilde{\lambda}_{3},$ (51)
and the mass matrix for the CP-even Higgs scalars $h_{1}^{0}$ and $h_{2}^{0}$
is given by
$\mathcal{M}_{h_{1}^{0}h_{2}^{0}}^{2}=\left(\begin{array}[]{cc}\tilde{\lambda}_{1}v^{2}&\tilde{\lambda}_{6}v^{2}\\\
\tilde{\lambda}_{6}v^{2}&m_{A_{2}}^{2}+\tilde{\lambda}_{5}v^{2}\end{array}\right).$
(52)
The mass eigenstates $h$ and $H$ of the CP-even Higgs scalars $h_{1}^{0}$ and
$h_{2}^{0}$ can be obtained by the rotation Haber:2006ue
$\left(\begin{array}[]{c}h\\\
H\end{array}\right)=\left(\begin{array}[]{cc}s_{\beta-\alpha}&c_{\beta-\alpha}\\\
c_{\beta-\alpha}&-s_{\beta-\alpha}\end{array}\right)\left(\begin{array}[]{l}h_{1}^{0}\\\
h_{2}^{0}\end{array}\right),$ (53)
where the mixing angle $\beta-\alpha$ is determined by
$s_{2(\beta-\alpha)}=-\frac{2\tilde{\lambda}_{6}v^{2}}{m_{H}^{2}-m_{h}^{2}}\,.$
(54)
The masses of $h$ and $H$ are given by
$m_{H,h}^{2}=\frac{1}{2}\left\\{m_{A}^{2}+v^{2}\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{5}\right)\pm\sqrt{\left[m_{A}^{2}+v^{2}\left(\tilde{\lambda}_{5}-\tilde{\lambda}_{1}\right)\right]^{2}+4v^{4}\tilde{\lambda}_{6}^{2}}\right\\}\,,$
(55)
where $m_{h}=125$ GeV is the mass of the SM-like Higgs boson $h$. In the
present work, we assume that the mass of $H$ is larger than that of the SM-
like Higgs $h$. Combining Eq. (54) and Eq. (55) gives us the following
relation
$\tilde{\lambda}_{6}=\frac{t_{2(\beta-\alpha)}}{2v^{2}}\left[m_{A}^{2}+v^{2}\left(\tilde{\lambda}_{5}-\tilde{\lambda}_{1}\right)\right].$
(56)
It is seen from Eq. (53) that the angle $\beta-\alpha$ represents the mixing
between the two $CP$-even neutral Higgs states. In the alignment or decoupling
limit, $s_{\beta-\alpha}$ or $\tilde{\lambda}_{6}$ approaches zero so that $h$
becomes more SM-like.
Figure 1: Unitarity bound on the scalar masses. Left: scalar mass
distributions as a function of $c_{\beta-\alpha}$. Right: scalar mass
distributions as a function of $t_{\beta}$. The red, blue, and green scatter
points denote the mass distributions for $H$, $A$, and $H^{\pm}$,
respectively. Figure 2: Unitarity bound on the scalar mass distributions.
It is well-known Ginzburg:2005dt ; Kanemura:2015ska that The perturbative
unitarity can set strong limits on scalar masses in new-physics models. In
order to achieve this, we make a random scan over model parameter space to
numerically study the unitarity bounds. There are two distinctive sets of
parameters: one set is those directly appearing in the scalar potential,
including quartic couplings $\lambda_{i}$ while the other set contains the
scalar masses and mixing angles, which are derived but more directly related
to observations. In Figs. 1 and 2, the scan is performed for quartic couplings
in the following ranges:
$0<\lambda_{1},\lambda_{2}<8\pi,~{}{\rm
and}~{}-8\pi<\lambda_{3},\lambda_{4},\lambda_{5}<8\pi.$ (57)
Note that the bound $\lambda_{1},\lambda_{2}>0$ is the requirement from the
positivity of the scalar potential. In the left plots of Figs. 1 and 2, we
take $t_{\beta}=5$ and scan over the mixing angle $c_{\beta-\alpha}$ in the
range $10^{-3}-10^{-1}$. The observations from collider searches have strongly
constrained the mixing between the SM Higgs $h$ and the heavy one $H$. The
recent global fit Herrero-Garcia:2019mcy ; Eberhardt:2020dat ; Athron:2021auq
to the Higgs signal strengths and flavor measurements has restricted
$c_{\beta-\alpha}\lesssim 10^{-1}$. Therefore, we only consider the case with
a small mixing. On the other hand, in the right plot of Fig. 1, we fix
$c_{\beta-\alpha}=0.02$ and scan over $t_{\beta}$ in the range of
$10^{-1}-10$. The masses of scalars can be obtained with Eqs. (50)- (55) once
the values of quartic couplings and mixing angles are given.
In the left plot of Fig. 1, we show the scanning results on the scalar masses
and $c_{\beta-\alpha}$ plane. The green, blue, and red points represent the
masses for $H^{\pm}$, $A$, and $H$, respectively. As clearly shown by this
figure, the unitarity puts upper bounds on scalar masses for a given
$c_{\beta-\alpha}$. Specifically, when the mixing angle
$c_{\beta-\alpha}\simeq 0.1$, the scalar masses have been restricted to be
$\lesssim 1.4$ TeV. The bounds become weak with the decrease of
$c_{\beta-\alpha}$. When the mixing angle $c_{\beta-\alpha}\simeq 10^{-3}$,
the unitarity allowed upper bound climbs to around $\sim 12$ TeV. This feature
can be easily understood as folows. From Eqs. (50) and (51) we observe that
the masses for $H^{\pm}$, $A$ and $H$ can all be expressed by the form
$\tilde{m}_{22}^{2}+\lambda_{i}v^{2}$. For small values of
$\tilde{m}_{22}^{2}$, $\lambda_{i}v^{2}$ can dominate the contribution to
scalar masses with large values of $\lambda_{i}$’s and $c_{\beta-\alpha}$
which are, however, strongly constrained by unitarity. Thus, the mass
contributions from the term $\sqrt{\lambda_{i}}v$ should be smaller than
$\sim$ TeV. On the other hand, for $m_{H,A,H^{\pm}}^{2}\gg v^{2}$, the scalar
masses mainly come from the parameter $\tilde{m}_{22}$, which is not
constrained by the perturbative unitarity. In this case, the scalar masses can
be arbitrarily large. From Fig. 2 we can directly observe that the ranges of
$m_{H}$, $m_{A}$, and $m_{H^{\pm}}$ become overlapped with each other when
their values are much above several TeV. In the right plot of Fig. 1, we show
the scan results on the scalar masses as a function of $t_{\beta}$. The upper
limit on the scalar masses has its maximum value at $t_{\beta}\simeq 0.5$ and
3. When $t_{\beta}$ deviates from these two values, the unitarity bounds get
stronger, so that the allowed mass ranges shrink.
### 5.2 Complex 2HDM
Next we consider the so-called complex 2HDM (C2HDM) model WahabElKaffas:2007xd
, in which the model admits a softly-broken $Z_{2}$ symmetry and the
$CP$-violation by the complex $m_{12}$ and $\lambda_{5}$. Therefore, there are
two $CP$-violating phases, which are defined by
$m_{12}^{2}=\left|m_{12}^{2}\right|e^{i\phi\left(m_{12}^{2}\right)},\quad\lambda_{5}=\left|\lambda_{5}\right|e^{i\phi\left(\lambda_{5}\right)}\,.$
(58)
However, these two phases are not independent. They are related to each other
by one of the minimum conditions
$2\operatorname{Im}\left(m_{12}^{2}\right)=v_{1}v_{2}\operatorname{Im}\left(\lambda_{5}\right).$
(59)
As in the CP-conserving case, we can first rotate the scalar basis into the
Higgs basis by Eq. (46). In the presence of the explicit $CP$ violation,
$A_{2}$ is not yet a mass eigenstate, but would further mix with the scalars
$h_{1}^{0}$ and $h_{2}^{0}$. This can be observed directly from the following
off-diagonal elements in the neutral scalar mass matrix Fontes:2017zfn
$m_{h_{1}^{0}A_{2}}^{2}=-\frac{1}{2}{\rm Im}\lambda_{5}s_{\beta}\,,\quad
m_{h_{2}^{0}A_{2}}^{2}=-\frac{1}{2}{\rm Im}\lambda_{5}c_{\beta},$ (60)
where the minimum conditions for the potential have been taken into account.
We define an orthogonal rotation matrix $R$ to transform from the basis of the
neutral components $\\{h_{1}^{0},h_{2}^{0},A_{2}\\}$ into the mass basis
$\\{h,H,A\\}$, where the associated mass matrix is diagonalized as
$R\mathcal{M}^{2}R^{\mathrm{T}}=\operatorname{diag}\left(m_{h}^{2},m_{H}^{2},m_{A}^{2}\right)$
with the rotation matrix $R$ given by WahabElKaffas:2007xd ; Fontes:2017zfn
$R=\left(\begin{array}[]{ccc}c_{1}c_{2}&s_{1}c_{2}&s_{2}\\\
-\left(c_{1}s_{2}s_{3}+s_{1}c_{3}\right)&c_{1}c_{3}-s_{1}s_{2}s_{3}&c_{2}s_{3}\\\
-c_{1}s_{2}c_{3}+s_{1}s_{3}&-\left(c_{1}s_{3}+s_{1}s_{2}c_{3}\right)&c_{2}c_{3}\end{array}\right),$
(61)
where $s_{i}=\sin\alpha_{i},~{}c_{i}=\cos\alpha_{i}(i=1,2,3)$ with the
rotation angles defined in the following range
$-\pi/2<\alpha_{1\,,2\,,3}\leq\pi/2\,.$ (62)
The angles $\alpha_{2}$ and $\alpha_{3}$ parametrize the mixing between the
pseudoscalar $A_{2}$ with two scalars $h_{1,2}^{0}$, which signals the $CP$
violation Khater:2003wq ; Inoue:2014nva . The five quartic couplings in the
potential as functions of scalar masses and mixing angles have been provided
in previous literature ElKaffas:2007rq . The ranges of other input parameters
are given as follows: $m_{H}\in[126~{}{\rm GeV},15~{}{\rm TeV}]$,
$m_{H^{\pm}}\in[10~{}{\rm GeV},15~{}{\rm TeV}]$, $t_{\beta}\in[10^{-1},10]$,
and $\text{Re}m_{12}^{2}/(v^{2}s_{2\beta})\in[-10,10]$, where $m_{H}$ and
$m_{H^{\pm}}$ stand for the masses of bosons $H$ and $H^{\pm}$, and
$t_{\beta}$ for the VEV ratio between two Higgs doublet as usual. Here we have
assumed the neutral state $h$ to be the SM-like Higgs with its mass to be
$m_{h}=125$ GeV, while the mass squared of the third component $A$ is given by
Fontes:2017zfn
$m_{A}^{2}=\frac{m_{h}^{2}R_{13}\left(R_{12}t_{\beta}-R_{11}\right)+m_{H}^{2}R_{23}\left(R_{22}t_{\beta}-R_{21}\right)}{R_{33}\left(R_{31}-R_{32}t_{\beta}\right)},$
(63)
where $R_{ij}$ are elements of the rotation matrix in Eq. (61).
Figure 3: The perturbative unitarity bounds on the scalar masses. The
colorbars in the left and right two plots represent the values of mixing
angles $\alpha_{1}$ and $\alpha_{3}$, respectively.
In Fig. 3, we have shown the results by imposing the perturbative unitarity
bounds on the scalar masses in the C2HDM.It is shown that the upper limits on
the masses of the scalar $H$, $A$, and $H^{\pm}$ from the unitarity are
estimated as
$m_{H}\lesssim 800~{}{\rm GeV},~{}~{}m_{A}\lesssim 2~{}{\rm TeV},~{}~{}{\rm
and}~{}~{}m_{H^{\pm}}\lesssim 1~{}{\rm TeV}\,.$ (64)
Comparing with the $CP$-conserving case, the unitarity bounds become stronger
in the C2HDM. This is because both the neutral component $h_{2}^{0}$ and
$A_{2}$ mix with $h_{1}^{0}$, the SM-like component of the Higgs boson. In
most of parameter space, terms proportional to $\lambda_{i}v^{2}$ dominate the
contribution to neutral Higgs masses squared, so that the constraints on
quratic couplings from unitarity can significantly restrict the masses of
neutral scalars. Note that the mass bounds in Eq. (64) becomes ineffective in
the decoupling and $CP$-conserving limits, where the mixings among scalars
vanish. The colorbars in the two left plots of Fig. 3 denotes the
distributions of the mixing angle $\alpha_{1}$, which show that the model with
a large $m_{A}$ tends to have a big value of $\alpha_{1}$. In the right two
plots of Fig. 3, the colorbars represent the size of the mixing angle
$\alpha_{3}$. We find that large values of $|\alpha_{3}|$ mainly concentrate
around the upper boundary of the distribution of $m_{A}$, while small values
of $|\alpha_{3}|$ mainly distribute around $m_{A}\sim 100$ GeV.
### 5.3 Complex triplet extension of the 2HDM
The muon anomalous magnetic dipole moment (denoted by $(g-2)_{\mu}$) is one of
the long-standing anomalies in the particle physics. The recent muon $g-2$
measurement performed by the Muon experiment at Fermilab Muong-2:2021ojo has
further confirmed this discrepancy. Possible solutions to the muon $g-2$
anomaly has been widely discussed in the 2HDM content in Refs. Cheung:2001hz ;
Cheung:2003pw ; Zhou:2001ew ; Aoki:2009ha ; Han:2022juu . At one-loop level,
both the charged and neutral Higgs bosons in the 2HDM contribute to the muon
$g-2$, but it is found that these corrections are too small to explain the
observed deviation. On the other hand, the two-loop Barr-Zee diagrams can give
rise to the dominant contribution to the muon $g-2$ in some parameter space.
However, it has been shown that the explanation of the muon $g-2$ anomaly with
the Barr-Zee mechanism requires a light pseudo-scalar mass with $m_{A}\lesssim
100$ GeV and $t_{\beta}\sim 50$ when various constraints are imposed
Cheung:2003pw ; Ferreira:2021gke ; Kim:2022xuo . Note that one class of the
strictest constraints is provided by the unitarity bounds in the theory. In
particular, Ref. Ferreira:2021gke has shown that most of the parameter space
with $m_{A}\gtrsim 100$ GeV in the typical 2HDM is already excluded by the
unitarity alone.
Figure 4: The Barr-Zee type Feynman diagram for the muon $g-2$, with charged
scalar $\delta^{\pm}$ and $\delta^{\pm\pm}$ running in the loop.
Figure 5: Upper: unitarity bound on the mass difference of
$m_{\delta^{\pm}}-m_{\delta^{\pm\pm}}$. Lower: unitarity bound on the
trilinear couplings $\lambda_{H\delta^{\pm}\delta^{\pm}}$ and
$\lambda_{H\delta^{\pm\pm}\delta^{\pm\pm}}$. The colorbar represents the
values of $\Delta a_{\mu}(\times 10^{-9})$.
The recent work in Ref. Chen:2021jok shows that if a Higgs triplet with
hypercharge $Y=2$ is added to the 2HDM, the charged components of the Higgs
triplet can induce new Barr-Zee-type contributions illustrated in Fig. 4,
which may explain the muon $g-2$ while easily evading other experimental
constraints. From these Feynman diagrams, it is clear that the new
contribution to muon $g-2$ is proportional to the trilinear scalar couplings
$\lambda_{i}v$ which might be well constrained by the perturbative unitarity.
However, Ref. Chen:2021jok has not appropriately taken into account the
unitarity issue, which is the main topic in this subsection.
Following Ref. Chen:2021jok , we will consider the decoupling limit of the
model. By using the minimization conditions for the scalar potential in Eq.
(27) and the mass matrices provided in Appendix B, we find that the decoupling
between components in Higgs doublets and those in the triplet can be achieved
when $v_{\Delta},~{}\tilde{\mu}_{3}\ll 1$ GeV. In the two-Higgs-doublet
sector, we consider the case with a softly broken $Z_{2}$ symmetry and $CP$
conservation, in which all parameters in the scalar potential are real Also,
following Ref. Chen:2021jok , we shall consider the aligned limit of the two
Higgs doublets, i.e., $c_{\beta-\alpha}\approx 0$. In this case, the mass
eigenstates $h$ and $H$ are almost $h_{1}^{0}$ and $h_{2}^{0}$ in Eq. (47), so
that the trilinear scalar couplings are given by Eq. (D) in Appendix D.
To calculate the Barr-Zee diagram shown in Fig. 4, we need to know the
coupling between muons and $H$. As in Ref. Chen:2021jok , we shall consider
the aligned two-Higgs-doublet model (A2HDM) case Pich:2009sp (see Ref.
Eberhardt:2020dat for the recent global fit of A2HDM), in which the lepton
Yukawa couplings with $H$ are given by
$-{\cal L}_{Y}=\sum_{f}y_{f}^{H}\frac{M_{f}}{v}\bar{f}_{L}f_{R}H+{\rm H.c.},$
(65)
where $M_{f}$ is the mass of the lepton flavor $f$ and
$y_{f}^{H}=\left(s_{\beta-\alpha}\zeta_{f}-c_{\beta-\alpha}\right)\,.$ (66)
Here $\zeta_{f}$ is a parameter in the A2HDM, whose benchmark value is taken
to be $\zeta_{f}=-100$ following Ref. Chen:2021jok .
The contribution to the muon $(g-2)$ from the Barr-Zee diagrams is given by
Ilisie:2015tra
$\Delta a_{\mu}=\sum_{\phi_{i}}\frac{\alpha
m_{\mu}^{2}}{8\pi^{3}m_{H}^{2}}\operatorname{Re}\left(y_{f}^{H}\right)\lambda_{H\phi_{i}\phi_{i}^{*}}\mathcal{F}\left(\frac{m_{\phi_{i}}^{2}}{m_{H}^{2}}\right),$
(67)
where $\phi_{i}=\delta^{\pm},~{}\delta^{\pm\pm}$, the trilinear couplings
$\lambda_{H\phi_{i}\phi_{i}^{*}}$ are given in Eq. (D), and the loop function
is given by
$\mathcal{F}(\omega)=\frac{1}{2}\int_{0}^{1}dx\frac{x(x-1)}{\omega-x(1-x)}\ln\left(\frac{\omega}{x(1-x)}\right).$
(68)
Since the Barr-Zee diagrams in Fig. 4 dominate the anomalous muon $g-2$, we
can ignore other one- or two-loop $(g-2)_{\mu}$ contributions in our following
numerical calculations.
In order to search for the parameter space allowed by the perturbative
unitarity, we scan over the quartic couplings $\lambda_{8}$, $\lambda_{9}$,
$\lambda_{11}$, and $\lambda_{12}$ in the range of $(-8\pi-8\pi)$ and the
doubly-charged scalar mass $m_{\delta^{\pm\pm}}$ in the range of $(10-1000)$
GeV. The mass of $\delta^{\pm}$ is determined by equations given in Appendix
B. For the 2HDM sector, we take $\lambda_{1,2,...,5}=0.2$, $t_{\beta}=5$, and
$m_{H}=300$ GeV for conservative estimations.
We show the scan results in Fig. 5. From the upper two plots, we observe that
the mass squared difference between the singly-charged and doubly-charged
scalars in the triplet should be
$|m^{2}_{\delta^{\pm}}-m^{2}_{\delta^{\pm\pm}}|/v^{2}\lesssim 6$, which is
restricted by unitarity bound on the quartic coupling $\tilde{\lambda}_{11}$
as seen in Eqs. (B) and (82). Furthermore, for $m_{\delta^{\pm\pm}}\lesssim
200$ GeV we have $m_{\delta^{\pm}}-m_{\delta^{\pm\pm}}>0$. The colorbar of
this figure represents the value distribution of $\Delta a_{\mu}$. The lower
two plots of Fig. 5 show that large values of $|\Delta a_{\mu}|$ prefer large
values of trilinear scalar couplings, $|\lambda_{H\delta^{\pm}\delta^{\pm}}|$
and $|\lambda_{H\delta^{\pm\pm}\delta^{\pm\pm}}|$, as well as small values of
$m_{\delta^{\pm\pm}}$. Note that if $\Delta a_{\mu}$ is positive as required
by experiments, it picks the parameter space with negative values of
$\lambda_{H\delta^{\pm}\delta^{\pm}}$ and/or
$\lambda_{H\delta^{\pm\pm}\delta^{\pm\pm}}$, which are well constrained by the
unitarity consideration with $|\lambda_{H\delta^{\pm}\delta^{\pm}}|\lesssim
2.6$ and $|\lambda_{H\delta^{\pm\pm}\delta^{\pm\pm}}|\lesssim 5.0$. From the
Barr-Zee Feynman diagrams and their expressions in Eq. (67), the trilinear
couplings are directly related to the dominant contribution to the muon $g-2$,
and perturbative unitarity can thus put very useful constraints on this model.
Finally, we note that the unitarity bounds given in this section are rather
conservative since the doublet-triplet mixings are ignored due to the nearly
vanishing triplet VEV. In the case with $v_{\Delta}\sim 1$ GeV, the mixings
between Higgs doublet and triplet components can become significant, which
would further enhance the unitarity bounds on the scalar masses.
## 6 Conclusions
The perturbative unitarity is one of the most significant theoretical
constraints on the Higgs sector, beyond which the perturbation calculation in
the theory breaks down. It has proven to be successful in predicting the upper
limit on the Higgs boson mass in the minimal SM, and in constraining many new
physics models such the 2HDM. In this work, we focus on deriving the
perturbative unitarity bounds on several interesting extensions of the 2HDM
with an additional real or complex Higgs triplet scalar. Since the total
hypercharge and isospin are conserved in the high-energy limit of scatterings,
we explicitly give the two-particle state basis according to their
$SU(2)_{L}\times U(1)_{Y}$ charges by decomposing the direct product of two
Higgs multiplets into direct sums of irreducible representations under the
electroweak gauge groups. The classification of the two-particle state basis
is summarized in Tables 2-7, in which the states are expressed in terms of
component fields. With these two-particle bases, the $2\to 2$ scattering
amplitudes among scalars can be simplified into the block-diagonal forms,
which are easily determined by expanding the quartic scalar terms in the
potential. We then impose the unitarity bounds on the eigenvalues of the
scattering matrices. The associated analytical results are summarized in Secs.
3.2 and 4.2.
We then numerically apply our derived unitarity bounds to several models of
phenomenological interest, including the softly broken $Z_{2}$ symmetric 2HDM
with and without $CP$-conservation, as well as the extension of 2HDM with a
complex Higgs triplet. We have shown that the perturbative unitarity can put
useful and stringent constraints on the Higgs scalar masses and mixings. The
bounds become stronger when the mixings among Higgs scalars are significant.
It is found that the neutral scalar masses in the C2HDM have been restricted
to be below about $1-2$ TeV. Also, in the complex triplet extension of the
2HDM, the unitarity bounds on the couplings can also constrain new solutions
to the long-standing muon $g-2$ anomaly. In the near future, together with the
experimental measurements of the Higgs trilinear coupling and the Higgs signal
strengths of different channels at the LHC Run 3, we hope that the unitarity
bounds would help us to understand the structure of Higgs sector more deeply.
## Acknowledgments
BQL is supported in part by the Huzhou University under startup Grant No.
RK21094 and National Natural Science Foundation of China (NSFC) under Grant
No. 12147219, No. 12175066, and No. 11975009. DH is supported in part by the
National Natural Science Foundation of China (NSFC) under Grant No. 12005254,
the National Key Research and Development Program of China under Grant No.
2021YFC2203003, and the Key Research Program of Chinese Academy of Sciences
under grant No. XDPB15
## Appendix A Eigenvalues of the $(Y,I)=(0,0)$ scattering matrix in the
triplet extension of the 2HDM with a $Z_{2}$ symmetry
In this section we analytically solve the eigenvalues for the 5-dimensional
scattering matrix, which appears in the $(Y,I)=(0,0)$ sector of the extension
of 2HDM with a softly-broken $Z_{2}$ symmetry. By imposing the $Z_{2}$
symmetry, the 5-dimensional scattering matrix in Eq. (31) can be simplified
into the following form
$X=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&0&0&c_{5}\\\
a_{3}&a_{4}&0&0&c_{6}\\\ 0&0&b_{1}&b_{2}&0\\\ 0&0&b_{3}&b_{4}&0\\\
c_{5}&c_{6}&0&0&c_{7}\end{array}\right)\,,$ (69)
which can be further decomposed into a $2\times 2$ matrix and a $3\times 3$
one as follows
$X^{(2)}=\left(\begin{array}[]{cc}b_{1}&b_{2}\\\ b_{3}&b_{4}\\\
\end{array}\right),~{}~{}X^{(3)}=\left(\begin{array}[]{ccc}a_{1}&a_{2}&c_{5}\\\
a_{3}&a_{4}&c_{6}\\\ c_{5}&c_{6}&c_{7}\end{array}\right).$ (70)
The eigenvalues for $X^{(2)}$ and $X^{(3)}$ are the same as that directly
obtained from the 5-dimensional matrix $X$. Note that, for a general matrix
$A$, the eigenvalue $f$ can be obtained by solving the equation
$|fI-A|=0\,.$ (71)
For $X^{(2)}$ and $X^{(3)}$, the eigenvalue equation can be transformed into
the following equations
$(f-b_{1})(f-b_{4})-b_{2}b_{3}=0$ (72)
$(f-a_{1})(f-a_{4})(f-c_{7})-(f-a_{1})c_{6}^{2}-a_{2}a_{3}(f-c_{7})-a_{2}c_{5}c_{6}-a_{3}c_{5}c_{6}-(f-a_{4})c_{5}^{2}=0\,,$
(73)
respectively. The solutions for Eq. (72) can be easily solved by
$f_{1,2}=\frac{1}{2}\left(b_{1}+b_{4}\pm\sqrt{(b_{1}-b_{4})^{2}+4b_{2}b_{3}}\right).$
(74)
For the case with $b_{1}=b_{4}$ and $b_{2}=b_{3}^{*}$, we have
$f_{1,2}=b_{1}\pm|b_{2}|.$ (75)
Note that the scattering matrix should be Hermitian, which means
$a_{2}=a_{3}^{*}$ and $b_{2}=b_{3}^{*}$. Furthermore, the eigenvalues for the
Hermitian matrix are always real. Note that, for a general cubic equation
$f^{3}+bf^{2}+cf+d=0$, one representations of the three roots is given by
$\displaystyle f_{3}=-\frac{b}{3}+2\sqrt[3]{r}\cos\theta\,,$ (76)
$\displaystyle
f_{4}=-\frac{b}{3}+2\sqrt[3]{r}\cos\left(\theta+\frac{2}{3}\pi\right)\,,$
$\displaystyle
f_{5}=-\frac{b}{3}+2\sqrt[3]{r}\cos\left(\theta+\frac{4}{3}\pi\right)\,,$
where
$r=\sqrt{-\left(\frac{p}{3}\right)^{3}},~{}\theta=\frac{1}{3}\arccos\left(-\frac{q}{2r}\right),~{}{\rm
with}~{}p=\frac{3c-b^{2}}{3},~{}q=\frac{27d-9bc+2b^{3}}{27}\,.$ (77)
By comparing Eq. (73) with the general cubic equation, it is found that
$\displaystyle b=-(a_{1}+a_{4}+c_{7})\,,$ (78) $\displaystyle
c=a_{1}a_{4}+a_{1}c_{7}+a_{4}c_{7}-a_{2}a_{3}-c_{5}^{2}-c_{6}^{2}\,,$
$\displaystyle
d=a_{1}c_{6}^{2}+a_{2}a_{3}c_{7}+a_{4}c_{5}^{2}-a_{2}c_{5}c_{6}-a_{3}c_{5}c_{6}\,.$
In this way, we give the analytic solutions to the eigenvalues for the three-
rank scattering matrix $X^{(3)}$.
## Appendix B Mass matrices in the complex triplet extension of 2HDM
Here we provide some of the mass matrices elements for the extension of 2HDM
with an additional complex Higgs triplet. In this appendix we express the
neutral scalar in the triplet (28) as
$\delta^{0}=\frac{1}{\sqrt{2}}(d^{0}+i\eta^{0})$.
The mass matrix elements for the $CP$-even components of the neutral scalars
in the model are given by
$\displaystyle m_{h_{1}^{0}h_{1}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{3\tilde{\lambda}_{1}v^{2}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{8}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{11}}{2}-\sqrt{2}v_{\Delta}\tilde{\mu}_{1}+\tilde{m}_{1}^{2},$
$\displaystyle m_{h_{2}^{0}h_{2}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}_{3}v^{2}}{2}+\frac{\tilde{\lambda}_{4}v^{2}}{2}+\frac{1}{4}v^{2}{\rm
Re}\tilde{\lambda}_{5}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{9}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{12}}{2}-\sqrt{2}v_{\Delta}\tilde{\mu}_{2}+\tilde{m}_{2}^{2},$
$\displaystyle m_{d^{0}d^{0}}^{2}$ $\displaystyle=$ $\displaystyle
3v_{\Delta}^{2}\tilde{\lambda}_{\Delta
1}+3v_{\Delta}^{2}\tilde{\lambda}_{\Delta
2}+\tilde{m}_{\Delta}^{2}+\frac{\tilde{\lambda}_{8}v^{2}}{2}+\frac{\tilde{\lambda}_{11}v^{2}}{2},$
$\displaystyle m_{h_{1}^{0}h_{2}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{4}v^{2}{\rm
Re}\tilde{\lambda}_{6}+\frac{1}{4}v_{\Delta}^{2}{\rm
Re}\tilde{\lambda}_{10}+\frac{1}{4}v_{\Delta}^{2}{\rm
Re}\tilde{\lambda}_{13}-\frac{v_{\Delta}{\rm
Re}\tilde{\mu}_{3}}{2\sqrt{2}}-\frac{{\rm Re}\tilde{m}_{12}^{2}}{2},$
$\displaystyle m_{h_{1}^{0}d^{0}}^{2}$ $\displaystyle=$ $\displaystyle
v_{\Delta}\tilde{\lambda}_{8}v+v_{\Delta}\tilde{\lambda}_{11}v-\sqrt{2}\tilde{\mu}_{1}v,$
$\displaystyle m_{h_{2}^{0}d^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}v_{\Delta}v{\rm
Re}\tilde{\lambda}_{10}+\frac{1}{2}v_{\Delta}v{\rm
Re}\tilde{\lambda}_{13}-\frac{v{\rm Re}\tilde{\mu}_{3}}{2\sqrt{2}}\,,$ (79)
while the mass matrix elements for the $CP$-odd components are shown as
follows
$\displaystyle m_{A_{1}^{0}A_{1}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}_{1}v^{2}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{8}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{11}}{2}+\sqrt{2}v_{\Delta}\tilde{\mu}_{1}+\tilde{m}_{1}^{2},$
$\displaystyle m_{A_{2}^{0}A_{2}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}_{3}v^{2}}{2}+\frac{\tilde{\lambda}_{4}v^{2}}{2}-\frac{1}{4}v^{2}{\rm
Re}\tilde{\lambda}_{5}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{9}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{12}}{2}+\sqrt{2}v_{\Delta}\tilde{\mu}_{2}+\tilde{m}_{2}^{2},$
$\displaystyle m_{\eta^{0}\eta^{0}}^{2}$ $\displaystyle=$ $\displaystyle
v_{\Delta}^{2}\tilde{\lambda}_{\Delta 1}+v_{\Delta}^{2}\tilde{\lambda}_{\Delta
2}+\tilde{m}_{\Delta}^{2}+\frac{\tilde{\lambda}_{8}v^{2}}{2}+\frac{\tilde{\lambda}_{11}v^{2}}{2},$
$\displaystyle m_{A_{1}^{0}A_{2}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{\Delta}^{2}{\rm
Re}\tilde{\lambda}_{10}}{4}+\frac{v_{\Delta}^{2}{\rm
Re}\tilde{\lambda}_{13}}{4}+\frac{v_{\Delta}{\rm
Re}\tilde{\mu}_{3}}{2\sqrt{2}}-\frac{{\rm Re}\tilde{m}_{12}^{2}}{2}+\frac{{\rm
Re}\tilde{\lambda}_{6}v^{2}}{4},$ $\displaystyle m_{A_{1}^{0}\eta^{0}}^{2}$
$\displaystyle=$ $\displaystyle-\sqrt{2}\tilde{\mu}_{1}v,$ $\displaystyle
m_{A_{2}^{0}\eta^{0}}^{2}$ $\displaystyle=$ $\displaystyle-\frac{{\rm
Re}\tilde{\mu}_{3}v}{2\sqrt{2}}.$ (80)
The mass matrix elements for the singly-charged scalars are
$\displaystyle m_{H_{1}^{+}H_{1}^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}_{1}v^{2}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{8}}{2}+\tilde{m}_{1}^{2},$
$\displaystyle m_{H_{2}^{+}H_{2}^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}_{3}v^{2}}{2}+\frac{v_{\Delta}^{2}\tilde{\lambda}_{9}}{2}+\tilde{m}_{2}^{2},$
$\displaystyle m_{\delta^{+}\delta^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}_{8}v^{2}}{2}+\frac{\tilde{\lambda}_{11}v^{2}}{4}+v_{\Delta}^{2}\tilde{\lambda}_{\Delta
1}+v_{\Delta}^{2}\tilde{\lambda}_{\Delta 2}+\tilde{m}_{\Delta}^{2},$
$\displaystyle m_{H_{1}^{+}H_{2}^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}v^{2}\tilde{\lambda}_{6}^{*}+\frac{1}{2}v_{\Delta}^{2}\tilde{\lambda}_{10}^{*}-\tilde{m}_{12}^{2*}=m_{H_{1}^{-}H_{2}^{+}}^{2*},$
$\displaystyle m_{H_{1}^{+}\delta^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{\Delta}\tilde{\lambda}_{11}v}{2\sqrt{2}}-\tilde{\mu}_{1}v=m_{H_{1}^{-}\delta^{+}}^{2*},$
$\displaystyle m_{H_{2}^{+}\delta^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{\Delta}\tilde{\lambda}_{13}v}{2\sqrt{2}}-\frac{\tilde{\mu}_{3}v}{2}=m_{H_{2}^{-}\delta^{+}}^{2*}.$
(81)
Since there is only one doubly-charged scalar in the model, so there is not
any mixing and its mass is simply given by
$m_{\delta^{++}\delta^{--}}^{2}=v_{\Delta}^{2}\tilde{\lambda}_{\Delta
1}+\tilde{m}_{\Delta}^{2}+\frac{\tilde{\lambda}_{8}v^{2}}{2}$ (82)
The parameters with tilde denote those in the Higgs basis, with the
transformation relation to the parameters in the generic basis given in
Appendix C. Due to $CP$-violating effects, there may exist mixings between the
$CP$-even and $CP$-odd components in the neutral scalars. Since we do not use
them in our work, we do not provide their explicit formulae here.
## Appendix C Parameters in the Higgs basis for the complex triplet extension
of 2HDM
The electroweak gauge symmetry is spontaneously broken when the neutral
components of the Higgs multiplets obtain VEVs. The VEVs can be complex and
there may be a relative phase between them. Here we use $\xi$ to denote the
phase between the VEVs of doublets $\Phi_{1}$ and $\Phi_{2}$ in the triplet
extension of the 2HDM. Concretely, one assumes real $v_{1}$ and complex
$v_{2}e^{i\xi}$. Such a phase can be absorbed by the following phase
redefinitions of the complex parameters:
$\lambda_{5}\rightarrow e^{2i\xi}\lambda_{5}\text{ and
}m_{12}^{2},\lambda_{6},\lambda_{7},\lambda_{10},\lambda_{13}\rightarrow
e^{i\xi}\left\\{m_{12}^{2},\lambda_{6},\lambda_{7},\lambda_{10},\lambda_{13}\right\\}.$
(83)
so that the form of the potential keeps unchanged. Thus, we can start with
real VEVs for scalars. In the following, we summarize the parameters of the
potential in the Higgs basis as functions of those defined in the generic
basis. For the parameters with mass dimensions, we have
$\displaystyle\tilde{m}_{11}^{2}$ $\displaystyle=$ $\displaystyle
c_{\beta}^{2}m_{11}^{2}+s_{\beta}^{2}m_{2}^{2}-c_{\beta}s_{\beta}\left(m_{12}^{2}+m_{12}^{2*}\right),$
$\displaystyle\tilde{m}_{22}^{2}$ $\displaystyle=$ $\displaystyle
s_{\beta}^{2}m_{11}^{2}+c_{\beta}^{2}m_{2}^{2}+c_{\beta}s_{\beta}\left(m_{12}^{2}+m_{12}^{2*}\right),$
$\displaystyle\tilde{m}_{12}^{2}$ $\displaystyle=$ $\displaystyle
c_{\beta}s_{\beta}\left(m_{11}^{2}-m_{22}^{2}\right)+\left(c_{\beta}^{2}m_{12}^{2}-s_{\beta}^{2}m_{12}^{2*}\right),$
$\displaystyle\tilde{m}_{\Delta}^{2}$ $\displaystyle=$ $\displaystyle
m_{\Delta}^{2},$ $\displaystyle\tilde{\mu}_{1}$ $\displaystyle=$
$\displaystyle\mu_{1}c_{\beta}^{2}+\mu_{3}c_{\beta}s_{\beta}+\mu_{2}s_{\beta}^{2},$
$\displaystyle\tilde{\mu}_{2}$ $\displaystyle=$
$\displaystyle\mu_{2}c_{\beta}^{2}-\mu_{3}c_{\beta}s_{\beta}+\mu_{1}s_{\beta}^{2},$
$\displaystyle\tilde{\mu}_{3}$ $\displaystyle=$
$\displaystyle\mu_{3}\left(c_{\beta}^{2}-s_{\beta}^{2}\right)+2\left(\mu_{2}-\mu_{1}\right)c_{\beta}s_{\beta}.$
(84)
The quartic couplings that relate only to the two Higgs doublets are given by
$\displaystyle\tilde{\lambda}_{1}$ $\displaystyle=$
$\displaystyle\lambda_{1}c_{\beta}^{4}+\lambda_{2}s_{\beta}^{4}+2\left(\lambda_{3}+\lambda_{4}+{\rm
Re}\lambda_{5}\right)c_{\beta}^{2}s_{\beta}^{2}+4{\rm
Re}\lambda_{6}c_{\beta}^{3}s_{\beta}+4{\rm
Re}\lambda_{7}c_{\beta}s_{\beta}^{3},$ $\displaystyle\tilde{\lambda}_{2}$
$\displaystyle=$
$\displaystyle\lambda_{1}s_{\beta}^{4}+\lambda_{2}c_{\beta}^{4}+2\left(\lambda_{3}+\lambda_{4}+{\rm
Re}\lambda_{5}\right)c_{\beta}^{2}s_{\beta}^{2}-4{\rm
Re}\lambda_{6}c_{\beta}s_{\beta}^{3}-4{\rm
Re}\lambda_{7}c_{\beta}^{3}s_{\beta},$ $\displaystyle\tilde{\lambda}_{3}$
$\displaystyle=$
$\displaystyle\frac{1}{4}s_{2\beta}^{2}\left[\lambda_{1}+\lambda_{2}-2\left(\lambda_{3}+\lambda_{4}+{\rm
Re}\lambda_{5}\right)\right]+\lambda_{3}-\left({\rm Re}\lambda_{6}-{\rm
Re}\lambda_{7}\right)c_{2\beta}s_{2\beta},$ $\displaystyle\tilde{\lambda}_{4}$
$\displaystyle=$
$\displaystyle\frac{1}{4}s_{2\beta}^{2}\left[\lambda_{1}+\lambda_{2}-2\left(\lambda_{3}+\lambda_{4}+{\rm
Re}\lambda_{5}\right)\right]+\lambda_{4}-\left({\rm Re}\lambda_{6}-{\rm
Re}\lambda_{7}\right)c_{2\beta}s_{2\beta},$ $\displaystyle\tilde{\lambda}_{5}$
$\displaystyle=$
$\displaystyle\left[\lambda_{1}+\lambda_{2}-2\left(\lambda_{3}+\lambda_{4}\right)\right]c_{\beta}^{2}s_{\beta}^{2}+\lambda_{5}c_{\beta}^{4}+\lambda_{5}^{*}s_{\beta}^{4}-2\left(\lambda_{6}-\lambda_{7}\right)c_{\beta}^{3}s_{\beta}$
$\displaystyle+2\left(\lambda_{6}^{*}-\lambda_{7}^{*}\right)c_{\beta}s_{\beta}^{3},$
$\displaystyle\tilde{\lambda}_{6}$ $\displaystyle=$
$\displaystyle\left(-\lambda_{1}+\lambda_{3}+\lambda_{4}+\lambda_{5}^{*}\right)c_{\beta}^{3}s_{\beta}+\left(\lambda_{2}-\lambda_{3}-\lambda_{4}-\lambda_{5}\right)c_{\beta}s_{\beta}^{3}+\lambda_{6}^{*}c_{\beta}^{4}-\lambda_{7}s_{\beta}^{4}$
$\displaystyle-\left(\lambda_{6}^{*}-2\lambda_{7}^{*}+2\lambda_{6}-\lambda_{7}\right)c_{\beta}^{2}s_{\beta}^{2},$
$\displaystyle\tilde{\lambda}_{7}$ $\displaystyle=$
$\displaystyle\left(-\lambda_{1}+\lambda_{3}+\lambda_{4}+\lambda_{5}\right)c_{\beta}s_{\beta}^{3}-\left(-\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}^{*}\right)c_{\beta}^{3}s_{\beta}-\lambda_{6}s_{\beta}^{4}+\lambda_{7}^{*}c_{\beta}^{4}$
(85)
$\displaystyle+\left(2\lambda_{6}^{*}-\lambda_{7}^{*}+\lambda_{6}-2\lambda_{7}\right)c_{\beta}^{2}s_{\beta}^{2}.$
Finally, the quartic couplings involving the complex Higgs triplet are give by
$\displaystyle\tilde{\lambda}_{8}$ $\displaystyle=$
$\displaystyle\lambda_{8}c_{\beta}^{2}+\lambda_{9}s_{\beta}^{2}+\lambda_{10}^{*}c_{\beta}s_{\beta},$
$\displaystyle\tilde{\lambda}_{9}$ $\displaystyle=$
$\displaystyle\lambda_{8}s_{\beta}^{2}+\lambda_{9}c_{\beta}^{2}-\lambda_{10}^{*}c_{\beta}s_{\beta},$
$\displaystyle\tilde{\lambda}_{10}$ $\displaystyle=$
$\displaystyle-\left(\lambda_{8}-\lambda_{9}\right)c_{\beta}s_{\beta}-\lambda_{10}^{*}s_{\beta}^{2},$
$\displaystyle\tilde{\lambda}_{11}$ $\displaystyle=$
$\displaystyle\lambda_{11}c_{\beta}^{2}+\lambda_{12}s_{\beta}^{2}+{\rm
Re\lambda_{13}}s_{2\beta},$ $\displaystyle\tilde{\lambda}_{12}$
$\displaystyle=$
$\displaystyle\lambda_{11}s_{\beta}^{2}+\lambda_{12}c_{\beta}^{2}-{\rm
Re\lambda_{13}}s_{2\beta},$ $\displaystyle\tilde{\lambda}_{13}$
$\displaystyle=$
$\displaystyle\left(-\lambda_{11}+\lambda_{12}\right)c_{\beta}s_{\beta}+\lambda_{13}c_{\beta}^{2}-\lambda_{13}^{*}s_{\beta}^{2},$
$\displaystyle\tilde{\lambda}_{\Delta 1}$ $\displaystyle=$
$\displaystyle\lambda_{\Delta 1},~{}~{}\tilde{\lambda}_{\Delta
2}=\lambda_{\Delta 2}.$ (86)
Note that under the $U(1)$ transformation $H_{1}\rightarrow e^{i\chi}H_{1}$
and $H_{2}\rightarrow e^{-i\chi}H_{2}$, the scalar potential remains unchanged
if the complex parameters of the scalar potential in the Higgs basis are
transformed by the corresponding phase rotation Davidson:2005cw :
$\tilde{\lambda}_{5}\rightarrow e^{4i\chi}\tilde{\lambda}_{5}\text{ and
}\tilde{m}_{12}^{2},\tilde{\lambda}_{6},\tilde{\lambda}_{7},\tilde{\lambda}_{10},\tilde{\lambda}_{13}\rightarrow
e^{2i\chi}\left\\{\tilde{m}_{12}^{2},\tilde{\lambda}_{6},\tilde{\lambda}_{7},\tilde{\lambda}_{10},\tilde{\lambda}_{13}\right\\}.$
(87)
Therefore, beginning with the Eqs. (C)-(C) in the phase $\\{\xi=0,\chi=0\\}$,
we can obtain the relations between the two sets of parameters with arbitrary
phases $\\{\xi,\chi\\}$ by applying the replacements (83) and (87) to the Eqs.
(C)-(C). Finally, The inversion of Eqs. (C)-(C) can be obtained by making the
replacements $\tilde{m}_{12}^{2}\to m_{12}^{2}$,
$\tilde{\lambda}_{i}\to\lambda_{i}$, and $\beta\to-\beta$.
## Appendix D Trilinear couplings in the complex triplet extension of 2HDM
The trilinear couplings between a neutral scalar and a pair of charged scalars
in the triplet extension of the 2HDM are summarized as follows:
$\displaystyle\lambda_{h_{1}^{0}H_{2}^{+}H_{2}^{-}}=\tilde{\lambda}_{3}$ ,
$\displaystyle~{}~{}\lambda_{h_{2}^{0}H_{2}^{+}H_{2}^{-}}={\rm
Re}\tilde{\lambda}_{7},$
$\displaystyle\lambda_{h_{1}^{0}\delta^{+}\delta^{-}}=\tilde{\lambda}_{8}+\frac{1}{2}\tilde{\lambda}_{11}$
, $\displaystyle~{}~{}\lambda_{h_{2}^{0}\delta^{+}\delta^{-}}={\rm
Re}\tilde{\lambda}_{10}+\frac{1}{2}{\rm Re}\tilde{\lambda}_{13},$
$\displaystyle\lambda_{h_{1}^{0}\delta^{++}\delta^{--}}=\tilde{\lambda}_{8}$ ,
$\displaystyle~{}~{}\lambda_{h_{2}^{0}\delta^{++}\delta^{--}}={\rm
Re}\tilde{\lambda}_{10}.$ (88)
Note that in the aligned and decoupling limits, we have $h_{1}^{0}\equiv h$
and $h_{2}^{0}\equiv H$.
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# Spin-polarized antichiral exciton-polariton edge states
Ruiqi Bao S. Mandal<EMAIL_ADDRESS>Huawen Xu Xingran Xu R.
Banerjee Timothy C. H. Liew<EMAIL_ADDRESS>Division of Physics and
Applied Physics, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore
###### Abstract
We consider theoretically a system of exciton-polariton micropillars arranged
in a honeycomb lattice. The naturally present TE-TM splitting and an
alternating Zeeman splitting, where the different sublattices experience
opposite Zeeman splitting, shifts the Dirac points in energy, giving rise to
antichiral behavior. In a strip geometry having zigzag edges, two pairs of
edge states exist and propagate in the same direction (including the states at
the opposite edges). The edge modes localized at the opposite edges have
opposite spins (circular polarizations), which leads to co-propagating
$``\pm"$ spin channels. The antichiral edge states are protected by non-zero
winding numbers and can propagate around a 60 degree bend without being
reflected. We further compare the transport properties of these edge states
with chiral edge modes and propose a scheme to realize them experimentally.
## I Introduction
The quantum Hall effect Klitzing _et al._ (1980), where free electron gas
subjected to a perpendicular magnetic field shows quantized Hall conductance,
has laid the platform for topological phases. A few years later, the Haldane
model was proposed Haldane (1988), where the quantum Hall effect was shown
without any net magnetic field. The Haldane model has gone onto become the
backbone of many noble topological systems such as anomalous Chern insulators
Jotzu _et al._ (2014); Zhao _et al._ (2020), quantum spin-Hall insulators
Kane and Mele (2005), etc.
Recently, a new type of topological phase, which shows antichiral edge states,
has been introduced based on a modified-Haldane model Colomés and Franz
(2018). In contrast to previous topological phases, systems with antichiral
edge states do not host a bulk band gap. Instead, in a strip geometry both the
edge states propagate in the same direction and counter-propagating states
within the same energy window lie in the bulk. Although, the antichiral edge
states reside along with the bulk modes, they were shown to be reasonably
robust against disorder Colomés and Franz (2018). Because of their peculiar
properties, the antichiral edge states have been under intense investigation
in various system including in exciton-polaritonics Mandal _et al._ (2019a),
photonics Chen _et al._ (2020); Zhou _et al._ (2020); Chen and Li (2022),
acoustics Yu _et al._ (2021); Wu _et al._ (2020), electric circuits Yang
_et al._ (2021) and others Bhowmick and Sengupta (2020); Mannaï and Haddad
(2020); Breitkreiz (2020); Mizoguchi and Koma (2021); Cheng _et al._ (2021).
Figure 1: Schematic of a honeycomb lattice consisting exciton-polariton
micropillars. “A” and “B” correspond to the different sublattices of the
system, which experience opposite Zeeman splittings. In presence of the
sublattice dependent Zeeman splitting and the TE-TM splitting, spin-polarized
antichiral edge states appear, where opposite edge attain opposite spins and
propagate in the same direction.
Although, the antichiral edge states have been explored over a variety of
systems, the spin degree of freedom has not been associated with them till
now. In this work, we propose theoretically spin-polarized antichiral edge
states in an exciton-polariton system. Exciton-polaritons are quasi particles
that arise due to the strong coupling between quantum well excitons and
microcavity photons Deng _et al._ (2010); Carusotto and Ciuti (2013); Byrnes
_et al._ (2014). Due to their excitonic component, polaritons exhibit strong
nonlinearity, which has lead to a variety of optical devices Sanvitto and
Kéna-Cohen (2016); Feng _et al._ (2021), optical computation Kyriienko _et
al._ (2019); Xu _et al._ (2020); Banerjee and Liew (2020); Harrison _et al._
(2022), and others Xu _et al._ (2021a); Zvyagintseva _et al._ (2022). The
finite lifetime of the photons gives rise to interesting non-Hermitian
physics. Circularly polarized photons provide a spin degree of freedom to the
polaritons, which show many rich effects. For example, the naturally present
energy splitting between the transverse electric (TE) and transverse magnetic
(TM) modes (also known as the TE-TM splitting) acts as an effective magnetic
field Kavokin _et al._ (2005), which leads to the realization of the optical
spin-Hall effect both in the linear Leyder _et al._ (2007) and nonlinear
Kammann _et al._ (2012) regimes. The excitonic component along with the spin
degree of freedom leads to the Zeeman splitting in presence of perpendicular
magnetic field Larionov _et al._ (2010). These fascinating properties of
polaritons make them ideal for studying topological phases in polariton
lattices.
Polariton Chern insulators Karzig _et al._ (2015); Bardyn _et al._ (2015);
Nalitov _et al._ (2015); Banerjee _et al._ (2018); Li _et al._ (2018); Sun
_et al._ (2019) and related phases Solnyshkov _et al._ (2018); Mandal _et
al._ (2019b) have been explored theoretically and later realized
experimentally Klembt _et al._ (2018). Strong nonlinear polariton-polariton
interaction gives rise to different kinds of nonlinear topological behavior
including topological solitons Kartashov and Skryabin (2016); Gulevich _et
al._ (2017); Pernet _et al._ (2022); Solnyshkov _et al._ (2017), interaction
induced chiral edge states Bleu _et al._ (2017), bistable topological
insulators Kartashov and Skryabin (2017), nonlinear higher-order topological
insulators Banerjee _et al._ (2020), etc. In some cases, nonlinearity can
alone induce topological phase transition Bardyn _et al._ (2016); Sigurdsson
_et al._ (2017); Banerjee _et al._ (2021). The inherent non-Hermiticity of
the polaritons has also lead to a variety of non-Hermitian topological
behavior such as exceptional points Gao _et al._ (2015a, 2018); Khurgin
(2020); Comaron _et al._ (2020); Su _et al._ (2021); Liao _et al._ (2021);
Krol _et al._ (2022), non-reciprocal transport Mandal _et al._ (2020); Xu
_et al._ (2021b), skin effect Mandal _et al._ (2022), double-sided skin
effect Xu _et al._ (2021c), non-Hermitian topological edge-modes Pickup _et
al._ (2020); Comaron _et al._ (2020), etc. However, none of the above works
can show topologically protected spin transport and the search for a system
exhibiting robust spin transport is still an open question.
Here, we consider exciton-polaritons in a honeycomb lattice in the presence of
the TE-TM splitting and subjected to an alternating Zeeman splitting, where
all the “A” (“B”) sublattices are considered to have positive (negative)
Zeeman splitting. The same system with a uniform Zeeman splitting is known to
give rise to a Chern insulating phase, where in a strip geometry, the bulk
band gap hosts two pairs of counter-propagating edge states Nalitov _et al._
(2015); Bardyn _et al._ (2015); Klembt _et al._ (2018). Unlike the Chern
insulator case, our present system does not show a bulk bandgap. Instead, the
Dirac points split in energy. The non-trivial topology of the system connects
the Dirac points with two pairs of zigzag edge modes with each pair residing
on the opposite edges of a finite strip. Consequently, the edge modes
including the opposite edges obtain the same sign of the group velocity.
Unlike the Chern insulator case, here the system becomes antichiral, where all
the edge modes propagate in the same direction. We further find that the edge
states residing at the opposite edges of the strips have opposite circular
polarization, which enables robust spin transport along the edges of the
system. However, due to the fact that the edge modes reside with the counter-
propagating bulk modes, the system is less robust compared to their Chern
insulator counterpart. Our simulation shows that the system shows fair
robustness against lattices defects and the efficiency of pulse propagation
remains lower than but close to the Chern insulators.
Figure 2: (a)Honeycomb lattice consisting of two types of sublattices, “A” and
“B”. $a_{1}$ and $a_{2}$ are the primitive lattice vectors. (b) Lines
connecting the high symmetry points of the 2D Brillouin zone, along which the
band structures are calculated. (c) Band structure for a trivial Honeycomb
lattice ($\Delta_{T}=\Delta_{A}=\Delta_{B}=0$). (d) Effect of Zeeman splitting
on the honeycomb lattice band structure
($\Delta_{T}=0,~{}\Delta_{A}=\Delta_{B}=0.3J$). (e) Effect of the TE-TM
splitting on the band structure
($\Delta_{T}=0.3J,~{}\Delta_{A}=\Delta_{B}=0$). (f) Chern insulating band
structure with a non-trivial bulk bandgap
($\Delta_{T}=\Delta_{A}=\Delta_{B}=0.3J$). (g) Antichiral band structure with
the two valleys shifted in energy ($\Delta_{T}=\Delta_{A}=-\Delta_{B}=0.3J$).
(h) Winding numbers as a function of $k_{1}$ for the antichiral bands, which
become nontrivial for $k_{1}>|k_{D}|$, predicting the existence of two pairs
of antichiral edge states in that momentum window.
## II The model
We start by considering two coupled micropillars “A” and “B” in presence of
the Zeeman splitting and the TE-TM splitting. The corresponding Hamiltonian
using the bispinor basis
$\Psi=[\Psi_{A}^{+},\Psi_{A}^{-},\Psi_{B}^{+},\Psi_{B}^{-}]^{T}$ can be
expressed in the tight-binding limit as
$\displaystyle\mathcal{H}_{AB}=\begin{bmatrix}\Delta_{A}&0&J&\Delta_{T}e^{-2i\theta}\\\
0&-\Delta_{A}&\Delta_{T}e^{2i\theta}&J\\\
J&\Delta_{T}e^{-2i\theta}&\Delta_{B}&0\\\
\Delta_{T}e^{2i\theta}&J&0&-\Delta_{B}\end{bmatrix}.$ (1)
Here $J$ is the coupling strength between the pillars, $\Delta_{T}$ is the TE-
TM splitting, $\theta$ is the orientation angle connecting the two pillars
with respect to a reference axis, and $\Delta_{A(B)}$ is the Zeeman splitting
corresponding to the pillar “A”(“B”). We stress that while staggered
potentials were considered in previous work predicting a valley-Hall effect
Bleu _et al._ (2018), here it is the Zeeman splitting that is staggered. We
discuss explicit ways that it can be realized in section VI. The difference
between staggered potentials and staggered Zeeman splitting is that in the
case that a uniform magnetic field is applied together with a staggered
potential, we would generally obtain the chiral edge states of a Chern
insulator rather than antichiral ones. However, by properly tunning the
parameters it is possible to completely gap out one of the valleys, leaving
the other one gapped. Although, the system no longer remains a Chern
insulator, it can act as a robust spin filter, where only one of the two spins
are allowed to propagate along one of the edges. Such a regime is discussed in
details in Ref. Mandal _et al._ (2019b).
Next, we arrange such pillar pairs in a honeycomb lattice structure and write
down its corresponding Hamiltonian in the reciprocal space by taking periodic
boundaries in both the directions. This can be achieved by writing
$\Psi^{\pm}_{A,B,n}=\Psi^{\pm}_{A,B}e^{i\mathbf{k}.{\mathbf{x}}_{n}}$ where
${\mathbf{x}}_{n}$ is the position of each site and reducing the basis back to
that of $[\Psi_{A}^{+},\Psi_{A}^{-},\Psi_{B}^{+},\Psi_{B}^{-}]^{T}$.
$\displaystyle\mathcal{H}_{\mathbf{k}}=\begin{bmatrix}\Delta_{A}&0&-g_{\mathbf{k}}J&-g^{+}_{\mathbf{k}}\Delta_{T}\\\
0&-\Delta_{A}&-g^{-}_{\mathbf{k}}\Delta_{T}&-g_{\mathbf{k}}J\\\
-g^{*}_{\mathbf{k}}J&-{g^{-}_{\mathbf{k}}}^{*}\Delta_{T}&\Delta_{B}&0\\\
-{g^{+}_{\mathbf{k}}}^{*}\Delta_{T}&-g^{*}_{\mathbf{k}}J&0&-\Delta_{B}\end{bmatrix},$
(2)
where
$g_{\mathbf{k}}=\sum_{n=1}^{3}\exp\left({-i\mathbf{k}.{\mathbf{r}}_{n}}\right)$
and
$g^{\pm}_{\mathbf{k}}=\sum_{n=1}^{3}\exp\left({-i\left[\mathbf{k}.{\mathbf{r}}_{n}\mp
2\theta_{n}\right]}\right).$
Here $\mathbf{r}_{n}$ represent the vectors connecting the three nearest “B”
sites from a single “A” site (Fig. 2(a)) and $\theta_{n}=2\pi(n-1)/3$ are the
angles of those vectors with respect to one of them (say $\mathbf{r}_{1}$).
To investigate the effect of $\Delta_{T}$, $\Delta_{A}$, and $\Delta_{B}$, we
plot the band structure of the system along a line connecting the high
symmetry points in the 2D Brillouin zone. Such a line is shown in Fig. 2(b).
Before proceeding, we define the degree of circular polarization corresponding
to a state as
$\displaystyle
S_{z}=\frac{\left|\Psi^{+}_{A}\right|^{2}+\left|\Psi^{+}_{B}\right|^{2}-\left|\Psi^{-}_{A}\right|^{2}-\left|\Psi^{-}_{B}\right|^{2}}{\left|\Psi^{+}_{A}\right|^{2}+\left|\Psi^{+}_{B}\right|^{2}+\left|\Psi^{-}_{A}\right|^{2}+\left|\Psi^{-}_{B}\right|^{2}}.$
(3)
The Hamiltonian of the system for $\Delta_{T}=\Delta_{A}=\Delta_{B}=0$
corresponds to trivial Graphene. Consequently, we obtain the Graphene band
structure by diagonalizing Eq. (2), where the lower and upper band touch each
other at the $K$ and $K^{\prime}$ valleys forming the Dirac cones (see Fig.
2(c)). In this case, the bands corresponding to the “$\pm$” spins are
degenerate or equivalently they are linearly polarized with $S_{z}=0$. This
degeneracy can be lifted by making $\Delta_{A}=\Delta_{B}=\Delta$, which
breaks the time-reversal symmetry (TR) and shifts the bands corresponding to
the “$\pm$” spins in energy by $2\Delta$ (see Fig. 2(d)). Next, we make
$\Delta_{T}\neq 0$ keeping $\Delta_{A}=\Delta_{B}=0$. In this case, the TR
symmetry is restored and the bands become linearly polarized. One significant
difference from the previous two cases is that the band structure is no longer
linear near the $K$ and $K^{\prime}$ valleys (see Fig. 2(e)). Making
$\Delta_{T}\neq 0$ and $\Delta_{A}=\Delta_{B}\neq 0$ gives rise to the non-
trivial topological band structure with the system transiting to a Chern
insulating phase having Chern number 2 Nalitov _et al._ (2015); Bardyn _et
al._ (2015); Klembt _et al._ (2018). A nontrivial bulk band gap opens up due
to the band inversion, which can be identified by the opposite values of
$S_{z}$ of the second and third bands near the valleys (see Fig. 2(f)).
Finally, we introduce the antichiral band structure in Fig. 2(g) by choosing
$\Delta_{T}\neq 0$ and $\Delta_{A}=-\Delta_{B}\neq 0$. Unlike the Chern
insulator band structure, the bulk bands are not gapped here. Instead the
states near the $K$ and $K^{\prime}$ prime valleys shift in energy. The
introduction of $\Delta_{A(B)}$ breaks the TR symmetry. However, unlike Figs.
2(d,f) the bulk bands do not have any preferred spins and stay linearly
polarized.
The antichiral system does not have a bulk band gap. Consequently, it is not
possible to assign a Chern number to characterize the topology of the system.
It is possible to calculate a relevant winding number. However, unlike all the
previous antichiral systems, instead of one, two bands are involved here.
Consequently, one needs to define the non-abelian Berry connection in order to
calculate the winding number (see appendix). This is defined as Qi _et al._
(2008)
$\displaystyle\mathcal{F}^{m,n}_{k_{1}}=i\langle
u^{m}_{k_{1}}(k_{2})\left|\frac{\partial}{\partial
k_{2}}\right|u^{n}_{k_{1}}(k_{2})\rangle,$ (4)
where $m,n=1,2$ corresponds to the lowest two bands as indicated in Fig. 2(g).
$u^{n}_{k_{1}}(k_{2})$ is the Bloch wave function. $k_{1}$ and $k_{2}$ are the
wave vectors along the $a_{1}$ and $a_{2}$ directions, respectively (see Fig.
2(a)). The winding number is given as
$\displaystyle\nu_{k_{1}}=\nu^{+}_{k_{1}}+\nu^{-}_{k_{1}}=\frac{1}{\pi}\oint_{\text{BZ}}dk_{2}\text{Tr}[\mathcal{F}_{k_{1}}],$
(5)
where $\nu_{k_{1}}$ is the total winding number of the system,
$\nu^{\pm}_{k_{1}}$ correspond to the contribution from the two bands, and the
integration is taken over the Brillouin zone along the ${k_{2}}$ direction for
each value of $k_{1}$. In Fig. 2(h) the winding numbers are plotted as a
function of $k_{1}$, which shows
$\displaystyle\nu^{+}_{k_{1}}=-\nu^{-}_{k_{1}}\approx\left\\{\begin{matrix}0,~{}\text{for}\left|k_{1}\right|<k_{D}\\\
1,~{}\text{for}\left|k_{1}\right|>k_{D}\end{matrix}\right.$ (6)
where $k_{D}$ is the position of the Dirac point. The total winding number of
the system is always 0. However, $\nu^{\pm}_{k_{1}}$ deviates from 0 and
becomes close to $\pm 1$ as shown in the shaded region in Fig. 2(h). Non-zero
winding corresponds to the non-trivial topology and predicts the existence of
antichiral edge states Colomés and Franz (2018). Two non-trivial branches in
the winding number predicts two pairs of antichiral edge modes. This is
different from previously studied systems, where only a pair of antichiral
edge modes are obtained.
Figure 3: Topological band structures for antichiral (a-b) and Chern insulator
(c-d) cases. The states are color coded with respect to the contribution from
the edges in (a,c) and with respect to the contribution from the spins in
(b,d). Antichiral edge states are co-propagating having almost pure spins in a
certain energy window. The chiral edge states in the Chern insulator case are
counter-propagating and have mixed spins. Here $a$ is the periodicity along
the $x$ direction, $\langle y\rangle$ represents of the mean position of the
states along the $y$ axis with $-L$ and $+L$ being the boundaries. $S_{z}$ is
defined in Eq. (3). All the parameters are kept the same as those in Fig. 2.
## III Topological band structure
In this section, we provide topological band structures by considering the
system periodic along the $x$ direction and truncated along the $y$ direction
with the zigzag boundaries. In Figs. 3(a-b) the numerically calculated band
structure for the antichiral system $(\Delta_{T}=\Delta_{A}=-\Delta_{B})$ is
shown. As predicted from the non-zero winding numbers, indeed two pairs of
antichiral edge modes connecting the Dirac points appear and co-exist with the
bulk modes. Generally, the edge modes localized at the opposite edges are
degenerate. However, for better visualization, we have lifted the degeneracy
of the edge modes by adding a inversion symmetry breaking term
$m\sigma_{z}\otimes\sigma_{o}$ to the Hamiltonian in Eq. (2). Here
$\sigma_{z}$ is the $z$ component of the Pauli matrix, $\sigma_{o}$ is a
$2\times 2$ identity matrix, and $\otimes$ represents the tensor product. We
have chosen a small $m=10^{-2}$ for Fig. 3(a-b), which does not alter any
property of the system. This is necessary for visualizing the properties of
the edge modes from the band structure, otherwise the opposite edge modes in
each pair would overlap with each other making it difficult to distinguish
them.
To study the properties of the edge modes, we color code the band structure as
a contribution of localization in Fig. 3(a) and as a contribution of $S_{z}$
in Fig. 3(b). All the edge modes have the same group velocity, which makes
them propagate in the same direction. Such one directional propagation at the
edges is balanced by the counter propagating bulk modes. Although, the bulk
remains linearly polarized, the antichiral edge modes at the two opposite
edges have opposite values of $S_{z}$ and in a particular energy window
$S_{z}\rightarrow\pm 1$. As we show in the next section, this leads to
different spin channels in the same sample, where opposite spins propagate
along the different edges of a finite sample in the same direction.
In Figs. 3(c-d), we provide the Chern insulating band structure
$(\Delta_{T}=\Delta_{A}=\Delta_{B})$ for comparison. Unlike the antichiral
system, here the bulk bands are gapped and two pairs of edge modes appear.
Since the edge states at the different edges have opposite group velocities,
they propagate in the opposite directions. Here the bulk bands are not
linearly polarized and have some preferred value of $S_{z}$. Near the Dirac
points, $S_{z}\rightarrow\pm 1$. However, the edge modes deviate from the pure
spins and acquire some mixed spins.
## IV Spin propagation
Figure 4: (a-c) Polariton propagation under the effect of two pulses. The
arrows in (b) represents the positions of two defects deliberately introduced
by removing a site at each edge. Polaritons can propagate around the defect
without significant backscattering. (d) Polariton propagation under continuous
coherent pumps. Polaritons with “$-$” spin propagate along the upper edge,
while the polaritons with “$+$” spin propagate along the lower edge. The
yellow stars correspond to the positions of linearly polarized coherent pulses
(a-c) and pumps (d). Parameters: $F_{0}=J$, $\Gamma=0.002J$, $\sigma=3$,
$\tau_{0}/\hbar=50/J$, $\hbar\omega_{p}=-0.207J$, $t_{u}=\hbar/J$ and
$ak_{p}=3.42$. All other parameters are kept the same as those in Figs.
3(a-b). Figure 5: Schematic diagram of antichiral system arranged in a
triangle (a) and a parallelogram shape (d). The triangle shaped system has
only one type of edge. (b-c) Only spin “+” polaritons can propagate around the
bending without being backscattered. (e-f) The parallelogram consists of two
types of zigzag edges and “$\pm$” spins propagate along those edges,
separately. (g) At the top left corner, where both edges meet, polaritons
scatter into the bulk. (h) For the Chern insulator system, polaritons can
propagate along the edges and do not scatter into the bulk. Each arm consists
of 15 sites in (a-c) and 25 sites in (d-h). The yellow stars represent
linearly polarized coherent pumps. Parameters: (a-g) All other parameters are
kept the same as those in Fig. 4. (h) $\Delta_{T}=\Delta_{A}=\Delta_{B}=0.3J$,
$\hbar\omega_{p}=0.07J$, and $ak_{x}=2.89$.
In this section, we investigate the polariton propagation in the antichiral
system and find out their robustness by removing one site from each edge. We
consider a stripe lattice formed by 12 unit cells along the $y$ direction and
20 unit cells along the $x$ direction. The upper (lower) edge of the system is
“A” (“B”) type zigzag edge (see Fig. 1). The evolution of the polaritons under
the coherent excitation is governed by the following time dependent
Schrödinger equation
$i\hbar\frac{\partial\Psi_{\pm}}{\partial
t}=H\Psi_{\pm}-i\Gamma\Psi_{\pm}+F_{\pm}(x,y,t)\exp{\left[i\left(k_{p}x-\omega_{p}t\right)\right]},$
(7)
where
$\displaystyle
F_{\pm}=F_{0}\left\\{\begin{matrix}\exp\left[{-\frac{[(x-x_{0})^{2}+(y-y_{0})^{2}]}{2\sigma^{2}}-\frac{t^{2}}{2\tau_{0}^{2}}}\right],~{}\text{for
pulse,}\\\
\exp\left[{-\frac{[(x-x_{0})^{2}+(y-y_{0})^{2}]}{2\sigma^{2}}}\right],~{}\text{for
pump}\end{matrix}\right..$ (8)
Here $H$ is the Hamiltonian representing the system which can be written by
repeating the Hamiltonian $H_{AB}$, $F_{\pm}$ represents the circularly
polarized components of Gaussian shaped coherent pulses or pumps having
amplitude $F_{0}$, energy $\hbar\omega_{p}$, and momentum $k_{p}$; and
$\Gamma$ is the decay rate of polaritons. The expression in Eq. (8) represents
the profiles of $F_{\pm}$ in space and in time. $(x_{0},y_{0})$ represent the
position of $F_{\pm}$ having widths $\sigma$ in space and $\tau_{0}$ in time.
In Fig. 4 the polariton propagation under two linearly polarized
($F_{+}=F_{-}$) pulses positioned at the two edges (represented by the yellow
stars) are shown. As it can be seen from the band structure in Fig. 3(a-b) the
edge modes have negative group velocity. Consequently, polaritons propagate
from right to left along both the edges. To check the robustness, we have
removed one site at each edge as indicated by the arrows in 4(b). As it can be
seen in Figs. 4(a-c) the polaritons can propagate around the defect without
significant backscattering. However, the propagation around the defect is not
perfect and in the next section, we have shown that the backscattering is
around 10%. Because of the presence of the decay term $\Gamma$, the intensity
created by the pulse decreases exponentially. Consequently, for visulization
purpose the intensity is renormalized at each time step in Figs. 4(a-c). Next,
to show that the edges are circularly polarized, we use continuous pump
instead of pulses. It should be noted that we use linearly polarized pumps to
inject both $\pm$ polaritons. However, the system only allows $``+"$
polaritons to propagate along the lower edge and $``-"$ polaritons to
propagate along the upper edge, creating two spin channels.
The dynamics of the polaritons become more interesting if instead of a strip
geometry, triangular and parallelogram shaped geometries are considered. In
Fig. 5(a), (d) schematics of triangular and parallelogram shaped geometries
are shown, respectively. For triangular geometry, one of the edges of the
system can be considered as periodic while the other edge does not appear in
the structure. As a result, under a linearly polarized coherent pump, spin “+”
polaritons can propagate along the edges without being backscattered into the
bulk (see Figs. 5(b-c)). This scenario changes for parallelogram shaped
geometry as it consists of both type of edges. Consequently, $``\pm"$
polaritons propagate along different edges (see Figs. 5(e-f)). Due to the
presence of $\Gamma$ the intensity of the polaritons decreases as they
propagate away from the excitation spot. It should be noted that in both the
structures, polaritons can go around the 60 degree bends perfectly. At 120
degree bends the two types of edge states meet (see the top left corner in
Fig. 5(d)). Since both the edge states propagate in the same direction,
polaritons can not continue propagating along the edges. Instead they scatter
into the bulk. Such a situation is shown in Fig. 5(g), where the pump is
positioned at the top left corner, where the opposite edges meet. For
comparison, in Fig. 5(h) we have considered the case for the Chern insulator,
which understandably does not scatter into the bulk and continues propagating
along the edges.
## V Efficiency comparison
To estimate the backscattering while going around the defect, we consider the
similar geometry as shown in Figs. 4(a-c), where the edge contains a defect.
Next, we position a linearly polarized coherent pulse at the right end of the
strip and define the efficiency of the system as
$\eta=\frac{I_{l}-I_{r}}{I_{l}+I_{r}}$ (9)
where $I_{l}$ is the total intensity of the sites situated at the left of the
defect (from both edge and bulk) and $I_{r}$ is the total intensity of sites
situated at the right of the defect (from both edge and bulk). The intensities
are considered at longer times when the polaritons pass the defect. In Fig. 6
the efficiency of the antichiral system is plotted in blue as a function of
different system size $N$ as defined in the inset. For larger $N$, the
efficiency stays near the 90%, but never reaches 100%, which is an indication
that there is always around 10% backscattering when the antichiral edge states
go around a defect. This is understandable, as the antichiral edge modes
reside together with the bulk modes in the same energy window, a defect can
couple the edge and bulk modes. However, Since the edge modes are spatially
separated from the bulk modes, the system still shows fair robustness and the
efficiency remains around 90 %. The efficiency decreases rapidly as $N$
decreases.
For comparison with the Chern insulator edge states, we also perform the same
steps and obtain the intensity as shown in red. As expected, the Chiral edge
modes of the Chern insulators show near perfect transmission around a defect
and the efficiency remains around 100 %. Similar to antichiral system the
efficiency decreases rapidly for lower $N$. The low $\eta$ for lower $N$ is
understandable from the fact that, being topological in nature, the robustness
of the edge modes in both the systems is associated with the bulk properties.
However, if sufficient bulk is not provided to the system, the system can not
show the expected robustness. We should notice that the nature of
backscattering is different for the antichiral and Chern insulator case. For
the antichiral system, the drop in efficiency arises due to the coupling of
co-propagating antichiral edge states with the counter propagating bulk
states. However, for the Chern insulator case, when $N$ is small, a defect
couples the two counter-propagating edge states, reducing the efficiency.
Figure 6: Transmission efficiency comparison between anti-chiral edge states
(in blue) and the Chern insulator edge states (in red) with different number
of layers $N$, as defined in the inset. The Chern insulator can reach almost
perfect transmission without backscattering when number of layers is larger
than 10. Understandably, the efficiency of antichiral edge states remains
lower than that of the Chern insulator, but still reaches around $90\%$. Both
types of system has a site missing at the edge.
## VI Experimental proposal
Figure 7: (a) Spatial profile of $P_{+}$ with peaks occurring in only one type
of sublattices. The green circles represent the micropillars. (b-e) Band
structure of the system. All four of them corresponds to the same system. For
presentation purpose, upper edge states are omitted in (b-c) and lower edge
states are omitted in (d-e). The bulk is linearly polarized, while the
opposite edge states have opposite spins.
In this section, we discuss and propose a scheme to realize the spin-polarized
antichiral edge states in the experiments. The key elements in our scheme are
1) honeycomb lattice potential for the polaritons, 2) TE-TM splitting, and 3)
sublattice dependent Zeeman splitting. Arranging exciton-polariton
micropillars in a variety of periodic lattices is a routine task in
experiments. The TE-TM splitting is known to occur naturally in microcavities
due to the polarization dependent reflection from the cavity mirrors Panzarini
_et al._ (1999). A Chern insulator was realized based on these ingredients,
when a honeycomb lattice was subjected to a strong perpendicular magnetic
field, which induced a uniform Zeeman splitting Klembt _et al._ (2018). In
order to realize the antichiral system, instead of a uniform Zeeman splitting,
the sign of the Zeeman splitting needs to be opposite for different
sublattices. For this reason, the set up with real magnetic field used for the
realization of Chern insulators seems challenging and requires an alternative
way. In Ref. Sun _et al._ (2019), an additional layer of ferromagnetic
material was proposed to induced the Zeeman splitting. Instead of one, two
ferromagnetic material layers with opposite predefined magnetic momentums may
be used and etched to make the Zeeman splitting opposite in different
sublattices.
However, using the inherent nonlinearity of the polaritons seems to be the
most straight-forward way. In a honeycomb lattice under the nonpolarized
incoherent pump, it is possible to form polariton condensate in an
antiferromagnetic configuration, where different types of sublattices attain
opposite circular polarization Sigurdsson _et al._ (2019). In such a
scenario, due to the nonlinear interactions, polaritons with opposite spins
will experience blueshifts in different sublattices, giving rise to the
necessary sublattice dependent Zeeman splitting in our scheme.
Another approach could be using circularly polarized incoherent pumps. It was
recently verified experimentally that in micropillar structures, a circularly
polarized incoherent pump can give rise to optically induced Zeeman splitting
up to 0.2 meV Real _et al._ (2021). In what follows, we implement this recent
experimental finding in our scheme and realize the spin-polarized antichiral
edge states. Polaritons under the effect of external incoherent pump can be
described using the following Gross-Pitaevskii equation (GPE) Carusotto and
Ciuti (2013)
$\displaystyle i\hbar\frac{\partial\psi_{\pm}}{\partial t}=$
$\displaystyle\left[-\frac{\hbar^{2}\nabla^{2}}{2m_{o}}+V(x,y)+gP_{\pm}(x,y)\right]\psi_{\pm}$
$\displaystyle+\left(\alpha_{1}|\psi_{\pm}|^{2}+\alpha_{2}|\psi_{\mp}|^{2}-i\alpha_{NL}|\psi_{\pm}|^{2}\right)\psi_{\pm}$
$\displaystyle+i\left[P_{\pm}(x,y)-\Gamma\right]\psi_{\pm}+\Delta_{T}\left(i\frac{\partial}{\partial
x}\pm\frac{\partial}{\partial y}\right)^{2}\psi_{\mp}.$ (10)
Here $\psi_{\pm}$ is the polariton wave function corresponding to the “$\pm$”
spins, $m_{o}$ is the polartion effective mass, and $V(x,y)$ is the underlying
honeycomb lattice potential. $gP_{\pm}$ is the spin dependent potential
experienced by the polaritons due to interaction with the excitonic reservoir
created by the circularly polarized incoherent pumps $P_{\pm}$ with $g$ being
a dimensionless parameter. $\Delta_{T}$ is the TE-TM splitting.
$\alpha_{1(2)}$ is the polariton-polariton interaction coefficient between
same (opposite) spins. $\alpha_{NL}$ is the nonlinear loss and $\Gamma$ is the
linear decay.
The incoherent pump can be patterned into a lattice shape. One way is to use a
spatial light-modulator, which has been used to make lattice potentials in a
variety of works Ohadi _et al._ (2017, 2018); Töpfer _et al._ (2021); Opala
_et al._ (2022). An alternative technique is to interference plane waves with
different in-plane wavevector components which has been demonstrated
experimentally in Luo _et al._ (2020). Here in this work, we use
superposition of four waves to prepare the incoherent pumps, such that the
resultant wave has maxima at one type of sublattice, while minima at the other
type of sublattices. Here we emphasize that the four waves we are using here
come from the same laser by using additional external optics devices. In this
case, $P_{+}$ can be expressed as
$\displaystyle P_{+}(x,y)=P_{0}f(x,y),$ (11)
where
$\displaystyle f(x,y)=$ $\displaystyle\cos\left(K_{1}\cdot
R\right)+\cos\left(K_{2}\cdot R\right)+\cos\left(K_{3}\cdot R\right)$
$\displaystyle+\cos\left(K_{4}\cdot R\right)$ (12)
with $K_{1}=4\pi(0,1/\sqrt{3})/a$, $K_{2,3}=2\pi(1,\pm 1/\sqrt{3})/a$,
$K_{4}=(0,0)$. In the above $a$ is the periodicity of the lattice along the
$x$ direction, $P_{0}$ controls the amplitude of the pump, $f$ is the
superposition of four waves that creates the maxima at a particular type of
sublattices, and the last term is a wave which has zero in-plane wave vector
that fixes the minima of $P_{+}$ to zeros. In Fig. 7(a) the spatial profile of
$P_{+}$ is plotted, which shows that indeed the maxima of $P_{+}$ coincide
with only one type of sublattices. $P_{-}$ also has a similar spatial form,
however, one needs to shift $f$ spatially such that the maxima now coincide
with the other type of sublattice.
We choose the pillar diameter to be $2~{}\mu$m, the lattice periodicity around
$a=2.95~{}\mu$m Whittaker _et al._ (2018); Su _et al._ (2020); Jacqmin _et
al._ (2014), effective mass $m_{0}=5\times 10^{-5}m_{e}$, where $m_{e}$ is the
free electron mass, and the potential depth of the micropillars to be around 4
meV with 0 meV being the minimum. We also fix $\Delta_{T}=0.12$ meV.$\mu$m2
Klembt _et al._ (2018); Dufferwiel _et al._ (2015); Sedov _et al._ (2019)
and $\Gamma=5.9~{}\mu$eV Nelsen _et al._ (2013), which corresponds to a
polariton lifetime around 55 ps. By fixing $P_{0}=2.3~{}\mu$eV we stay near
the condensation threshold, such that $|\psi_{\pm}|^{2}\approx$0\. This allows
us to be in the linear limit and calculate the band structure of the system
using the Bloch theorem. We also set $g=2.3$, which induces optical Zeeman
shift of the lower energy modes around 0.2 meV, which is consistent with
experiment Real _et al._ (2021).
Now that we have all the ingredients, by applying the Bloch theorem to the
linear Hamiltonian corresponding to the Eq. (VI), we calculate the band
structure of the system corresponding to a strip geometry with zigzag edges as
shown in Figs. 7(b-e). Similar to the tight binding model, our continuous
model also predicts the antichiral band structure, where the Dirac points
shifts in energy and two pairs of edge states appear. As already discussed,
the opposite edge states in our system are degenerate. Consequently, to
clearly show that opposite edge states have opposite spins, we plot the same
band structure four times. In Figs. 7(b-c) we have omitted the edge states
located at the upper edge of the strip ($\langle y\rangle=+L$), while in Figs.
7(d-e) we have omitted the edge states located at the lower edge of the strip
($\langle y\rangle=-L$). Indeed, the opposite edge states attain opposite
spins, while the bulk remains linearly polarized. Although, the topological
properties of the continuous model band structure is the same with those from
the tight-binding band structure, there are a few differences. For example,
the upper of pair edge states does not have similar dispersion as the lower
pair. This may arise due to the next nearest neighbour hopping, which is
naturally present in the continuous model as well as due to the continuous
nature of the effective Zeeman and TE-TM splittings.
The parameters that we have used are flexible. However, the TE-TM splitting
and effective sublattice dependent Zeeman splitting are the crucial
parameters. For the present parameters, the energy window, where the
antichiral edge states resides is around 0.12 meV. This is similar to the
topological bandgap obtained experimentally for the polariton Chern insulator
Klembt _et al._ (2018). Being topological in nature, a slight change in the
parameters will not hamper the antichiral modes. However, the case where the
Zeeman splitting decreases will result in a decrease in the energy window of
interest.
For simplicity, we have only considered only the necessary terms in the GPE in
Eq. (VI) to obtain the desired effect. For example, we have ignored the spin
relaxation of the excitonic reservoir and assumed it to preserve their
circular polarization perfectly. Although this is not entirely true,
experiment has shown that under circularly polarized incoherent pump the
reservoir can have up to 17% preferred $S_{z}$ C. Zambon _et al._ (2019),
which should be enough as long as the blueshift difference between opposite
circularly polarized polariton modes in a pillar is around 0.2 meV.
## VII Discussion and conclusion
The spin degree of freedom of exciton-polaritons makes them suitable for
optical spintronic devices. Devices based on polariton spins have been
realized experimentally, which includes spin switches Amo _et al._ (2010),
logic gates Gao _et al._ (2015b), etc. However, for the realization of a
complete optical network, it is important to be able to communicate among
different elements without losing the spin information. Unfortunately, the
Chiral polaritonic edge modes do not preserve spins while propagating. Our
proposed antichiral edge states could be useful in this purpose. Besides all
the edge modes in our system propagate in one direction, which may be helpful
in transferring oneway information through both the edges, unlike in the Chern
insulator where only one edge can be used.
We note that antichiral edge states for exciton-polaritons were proposed in a
previous work Mandal _et al._ (2019a). The present work is significantly
different from the previous one in many ways. First, in the previous work the
antichiral edge modes were not coherent polaritonic states. Instead, those
were fluctuations on top of a steady state. Second, in Ref. Mandal _et al._
(2019a) the antichiral edge states located at opposite edges did not have
opposite spins. Last, in the previous work, the antichiral edge states were
shown to propagate only in a straight strip, while their propagation around
different types of bendings was not discussed.
To conclude, we have presented a theoretical scheme to obtain spin-polarized
antichiral edge states for the first time. We use the naturally present TE-TM
splitting and sublattice depend Zeeman splitting in a honeycomb lattice of
exciton-polariton micropillar, where the Dirac points shift in energy and two
pairs of antichiral edge states appear that propagate in the same directions.
While the bulk of the system is linearly polarized, states located at the
opposite edges have opposite spins. Although the antichiral edge states reside
together with the bulk modes, we found that they show fair robustness and can
go around 60 degree bend without being reflected. Our work may be useful in
transferring spins and connecting different spin dependent polariton logic
elements.
## VIII Acknowledgements
The work was supported by the Ministry of Education, Singapore (Grant No.
MOE2019-T2-1-004).
## IX Appendix
Here in the appendix we discuss how to calculate the winding number in detail.
Different from the model considered in Colomés and Franz (2018), which has
only one band below the Dirac points, our present model involves two bands.
This makes the Hamiltonian complex and restricts us from calculating the
winding number analytically. Additionally, because of the presence of two
bands the Berry connections becomes non-Abelian (a matrix instead of a scalar)
and one must rely on numerical techniques to calculate the winding number.
To make the integral Gauge invariance, we use the Wilson loop approach
Benalcazar _et al._ (2017). We first transform the hexagonal BZ to a rhombic
shape with its axes aligned with the $k_{1}$ and $k_{2}$ direction as shown in
Fig. 8.We notice that such a choice of the BZ is not unique, and the
directions $(k_{1},k_{2})$ used in Colomés and Franz (2018) is related to ours
by a rotation of $\pi/12$. However, we choose such a BZ as the Dirac points
are situated at the central of the BZ when projected on any of the two
directions. The Wilson loop operator along $k_{2}$ for a fixed value of
$k_{1}$ can be defined as:
Figure 8: A rhombic shape Brillouin zone with its axis aligned with the
$k_{1}$ and $k_{2}$ direction.
$\displaystyle W(k_{1})=$ $\displaystyle F(k_{1},k_{2}+(N-1)\Delta
k)F(k_{1},k_{2}+(N-2)\Delta k)...$ $\displaystyle F(k_{1},k_{2}+\Delta
k)F(k_{1},k_{2}),$ (13)
where $F$ is a matrix whose elements are given by
$\left[F(k_{1},k_{2})\right]^{mn}=\left\langle u^{m}(k_{1},k_{2}+\Delta
k)|u^{n}(k_{1},k_{2})\right\rangle.$ (14)
Here $\Delta k$ is the grid spacing in the reciprocal space, the integral is
over the unit cell, $u$ is the Bloch state and $m,n=1,2$ corresponds to the
lowest two bands. Next we define a Wannier Hamiltonian $H_{W(k_{1})}$ as
$W(k_{1})=e^{iH_{W(k_{1})}}.$ (15)
The eigenvalues of $H_{W(k_{1})}$ gives the winding numbers for each $k_{1}$
as shown in Fig. 2(h) in the main text.
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|
A Pseudo-Value Regression Approach for Differential Network Analysis of Co-
Expression Data–A Pseudo-Value Regression Approach for Differential Network
Analysis of Co-Expression Data October
# A Pseudo-Value Regression Approach for Differential Network Analysis of Co-
Expression Data
Seungjun Ahn1 Tyler Grimes2 and Somnath Datta1
1Department of Biostatistics University of Florida Gainesville Florida
U.S.A.
2Department of Mathematics and Statistics University of North Florida
Jacksonville Florida U.S.A
(2022)
###### Abstract
The differential network (DN) analysis detects changes in measures of
association among genes under two or more conditions. In this study, we
introduce a Pseudo-value Regression Approach for Network Analysis (PRANA).
This is a novel method of DN analysis that also adjusts for additional
clinical covariates. We start from mutual information (MI) criteria, followed
by pseudo-value calculations, which are then entered into a robust regression
model. Performance in terms of precision, recall, and F1 score of
differentially connected (DC) genes is assessed in both univariable and
multivariable settings through variety of simulations. By and large, PRANA
outperformed dnapath and DINGO, neither of which is equipped to adjust for
available covariates such as patient-age. Lastly, we employ PRANA in a real
data application from the Gene Expression Omnibus (GEO) database to identify
DC genes that are associated with chronic obstructive pulmonary disease (COPD)
to demonstrate its utility. This is the first attempt of utilizing a
regression modeling for DN analysis by collective gene expression levels
between two or more groups with the inclusion of additional covariates.
###### keywords:
Differential network analysis; Gene regulatory network; Pseudo-value;
Regression method; RNA-Seq data.
## 1 Introduction
The rapid advancement of RNA-sequencing (RNA-Seq) data from high-throughput
sequencing technologies has provided clear advantages in gene expression
studies. It has broadened our understanding of genetics and pathogenesis of
human diseases (Reuter et al., 2015; Gao et al., 2017). Compared to
microarrays, RNA-Seq has a wider dynamic range, the ability to detect novel
transcripts, and often results in higher sensitivity and specificity of
detection of differential gene expression (Wang et al., 2009; Zhao et al.,
2014). With the increase of gene expression studies, the statistical methods
to analyze these gene expression data have also accordingly adapted and
progressed. The regression modelling for RNA-Seq differential expression (DE)
analysis has been established to compare the number of DE genes under
different biological or clinical states, including linear model based limma
(Ritchie et al., 2015), negative binomial model based edgeR (Robinson et al.,
2010), or Poisson log-linear model approach (Li et al., 2012). The analysis of
DE has a major limitation in that it looks at one gene at a time, even though
a set of genes are often involved in the same biological process (Ideker and
Krogan, 2012; Kim et al., 2018). In contrast, the differential network (DN)
analysis complements the DE analysis (de la Fuente, 2010) by looking at genes
collectively.
The DN analysis identifies changes in measures of association (i.e. network
properties or topologies) of the networks across biological conditions, which
makes it distinct from a single network analysis. Several groups have proposed
statistical methods for DN analysis (Ha et al., 2015; McKenzie et al., 2016;
Gill et al., 2010; Grimes et al., 2019). In particular, DINGO (Ha et al.,
2015) and dnapath (Grimes et al., 2019) have developed methods for RNA-Seq
data, to find differentially connected (DC) genes in subnetworks corresponding
to different pathways, between two groups of patients; e.g. ‘high-risk’ vs.
‘low-risk’ or ‘long-term survivors’ vs. ‘short-term survivors.’ While these
methods are convenient to use and applicable, they do not consider other
observed covariates that may be associated with gene expression.
For instance, a previous study has shown that the expression levels of
oxidative stress-associated genes were differentially expressed with smokers
with chronic obstructive pulmonary disease (COPD) through gene set enrichment
analysis using microarray data (Marballi et al., 2016). Let us suppose we want
to carry out DN analysis on expression data that includes oxidative stress-
associated genes and smoking status, which would be used as a grouping
variable. In practice, clinicians would also want to include additional
covariates such as patient history of cardiac arrhythmia (Bruha et al., 2012)
and lung carcinoma (Herreros-Villanueva et al., 2013) to garner more
information for better prognosis. However, there is no available direct
regression modeling for DN analysis regressing gene expression level between
the smoking statuses with the inclusion of additional clinical covariates
described above.
The pseudo-value approach was first developed from the leave-one-out jackknife
subsampling procedure, applied to a marginal quantity representing some aspect
of a marginal distribution of the response variable. It was originally
introduced by Andersen and his colleagues (Andersen et al., 2003; Andersen and
Klein, 2007) for multi-state survival models. Several studies (Dutta et al.,
2017; Graw et al., 2009) purported that the pseudo-value regression has
advantages that the pseudo-values derived from an asymptotically linear and
unbiased estimator are approximately independent and identically distributed
with the same conditional expectation. Ahn and Logan (Ahn and Logan, 2016) and
Ahn and Mendolia (Ahn and Mendolia, 2014) showed that their pseudo-value
approaches controlled the type I error while maintaining high power with
clustered survival data. With these benefits, we propose a regression modeling
method that regresses the jackknife pseudo-values (Efron and Tibshirani, 1993)
derived from a measure of connectivity of genes in a network to estimate the
effects of predictors. Note that the grouping variable itself could also be
included in the regression model along with additional clinical covariates
while regressing the pseudo-values. We loosely refer to this as a
“multivariate setting”, whereas in “univariate settings” only the group
variable is utilized in a DN analysis.
Thus, in this paper, we introduce a Pseudo-value Regression Approach for
Network Analysis (PRANA). This is a novel method of DN analysis that can
adjust for additional covariates. We start from mutual information (MI)
criteria, followed by pseudo-value calculations, which are then entered into a
robust regression model. This article assesses the model performances of our
pseudo-value approach in a multivariable setting, followed by a comparison to
dnapath and DINGO in the univariable setting through simulations. Lastly, we
employ our method in a real data application (Wang et al., 2021) from the Gene
Expression Omnibus (GEO) database (Barrett et al., 2013) to identify DC genes
that are associated with COPD. All statistical analyses are performed in R
version 4.0.2 (R Foundation for Statistical Computing, Vienna, Austria).
## 2 Methods
### 2.1 Mutual Information and ARACNE
Mutual information (MI) determines whether and how two genes interact. That
is, it is a measure of their relatedness and calculated from their joint
expression profiles. MI is zero if and only if the joint distribution between
the expression level of gene $j$ and gene $k$ satisfies
$P(g_{j},g_{k})=P(g_{j})P(g_{k})$ for $j\neq k$, or if $j=k$; in other words,
they are statistically independent. Algorithm for the Reconstruction of
Accurate Cellular Networks (ARACNE) which are proposed by Margolin et al.
(2006) and Zoppoli et al. (2010) estimates MI using a computationally
efficient Gaussian kernel estimator. The estimate of MI is used to quantify
the connectivity of each pair of genes in a network. Given a set of two genes
measurements, $\overrightarrow{u_{i}}\equiv(g_{ij},g_{ik})$, $i=1,\ldots,n$,
the joint probability distribution is approximated as
$f(\overrightarrow{u})=1/n\sum_{i}h^{-2}\Phi(h^{-1}(\overrightarrow{u}-\overrightarrow{u_{i}}))$,
where $\Phi$ is the bivariate standard normal density and $h$ is the position-
dependent kernel width. Then the MI can be expressed as (Margolin et al.,
2006):
$\hat{I}_{jk}=\frac{1}{n}\sum_{i}\log\frac{f(g_{ij},g_{ik})}{f(g_{ij})f(g_{ik})},$
where $f(g_{j})$ and $f(g_{k})$ are the marginals of $f(\overrightarrow{u})$.
The matrix containing entries $\hat{I}_{jk}$ is defined as the association
matrix. The ARACNE algorithm copula-transform the profiles for MI estimation
because MI is reparameterization invariant; thus, the range of these
transformed variables is between 0 and 1 (Basso et al., 2005).
### 2.2 Pseudo-Value Approach
Let $\hat{I}_{jk}$ be the MI estimate for a pair of genes
$j,k\in\\{1,\dots,p\\}$ of an estimated network from $n$ individuals. For each
gene $k$, we sum the edges (MI estimates) around the gene by taking the column
sum of the association matrix to obtain the total connectivity (which can be
deemed as a continuous version of degree centrality) of gene $k$:
$\hat{\theta}_{k}=\sum_{j=1}^{p}\hat{I}_{jk},$
where $k=1,\dots,p$.
The jackknife pseudo-values (Efron and Tibshirani, 1993) for the
$i^{\text{th}}$ individual and $k^{\text{th}}$ gene are defined by:
$\tilde{\theta}_{ik}=n\hat{\theta}_{k}-(n-1)\hat{\theta}_{k(i)},$ (1)
where $\hat{\theta}_{k(i)}$ is the column sum of a gene calculated from the
re-estimated association matrix using the RNA-Seq data without the
$i^{\text{th}}$ subject. For each gene $k$, the re-estimation process requires
$n$ such calculations with the data size of $n-1$.
Let $Z$ a binary group indicator. Let $\mathcal{G}_{1}=\\{i:Z_{i}=1\\}$,
$\mathcal{G}_{2}=\\{i:Z_{i}=2\\}$, and $n_{z}=|\mathcal{G}_{z}|$ is the sample
size for the two groups $z=1,2$ and $n=\sum n_{z}$. The jackknife pseudo-
values are separately obtained within groups. Following the general formula
above, for gene $k$ and group $z$, we similarly define $\hat{\theta}_{k}^{z}$
and $\hat{\theta}_{k(i)}^{z}$, where $i=1,\ldots,n_{z}$. Then for each
$i\in\mathcal{G}_{z}$, the $k^{\text{th}}$ gene jackknife pseudo-values are
calculated by
$\tilde{\theta}_{ik}=n_{z}\hat{\theta}_{k}^{z}-(n_{z}-1)\hat{\theta}_{k(i)}^{z}$.
Next, a robust regression is applied to regress the pseudo-values on a set of
covariates, including $Z$ and X, where $Z$ is the group indicator and
$\textbf{X}=(X_{1},\dots,X_{q}$) are the potential confounders, such as age
and gender. For the $i^{\text{th}}$ individual and $k^{\text{th}}$ gene, we
posit the model
$E[\tilde{\theta}_{ik}|Z_{i},\textbf{X}_{i}]=\alpha_{k}+\beta_{k}Z_{i}+\gamma_{k1}X_{i1}+\dots+\gamma_{kq}X_{iq},$
(2)
where $\alpha_{k}$ is the intercept, $\beta_{k}$ is the regression coefficient
for $Z$, and $\gamma_{k1},\dots,\gamma_{kq}$ is the set of regression
coefficients to be estimated for $X$. The main parameter of interest is
$\beta_{k}$ to test for the change in total connectivity (or degree
centrality) of the $k^{\text{th}}$ gene between groups. Least trimmed squares
(LTS), also known as least trimmed sum of squares (Rousseeuw, 1984), is then
implemented to perform a robust regression. The LTS estimator is defined by
$\min_{\alpha_{k},\beta_{k},\gamma_{k1},\ldots,\gamma_{kq}}\sum_{i=1}^{h}r_{(i)}(\alpha_{k},\beta_{k},\gamma_{k1},\ldots,\gamma_{kq})^{2},$
where $r_{(i)}$ is the set of ordered absolute values of the residuals (in
increasing order of absolute value) and $h$ may depend on some pre-defined
trimming proportion $c$, for instance by means of $h=[n(1-c)]+1$. In general,
$c$ is chosen between 0.5 and 1 (Pison et al., 2002).
### 2.3 Hypothesis Testing
To test whether the true difference in total connectivity of $k^{\text{th}}$
gene differs between groups, we test the null hypothesis of
$H_{0}:\beta_{k}=0$ against the research hypothesis $H_{1}:\beta_{k}\neq 0$.
The t-statistic is computed by $\hat{\beta}_{k}/SE(\hat{\beta}_{k})$, where
$SE(\hat{\beta}_{k})$ standard error of $\hat{\beta}_{k}$, obtained using
large sample theory, and which is the least-squares estimator of $\beta_{k}$
for $k=1,\dots,p$ from the robust regression described in equation (2).
P-values are calculated using a t-distribution as in robustbase R package
(Todorov and Filzmoser, 2009).
It is important to control the false discovery rate (FDR), since multiple
hypothesis tests are conducted in the DN analysis. The FDR measures the
proportion of false discoveries among a set of genes which are significantly
DC between groups. The empirical Bayes screening (EBS) approach (Datta and
Datta, 2005) has been applied to control the FDR, which is an extension of
Westfall and Young step-down adjusted p-values (Westfall and Young, 1993). The
EBS procedure is robust against model mis-specification, as it utilizes
nonparametric function estimation techniques for the estimation of the
marginal density of the transformed p-values.
## 3 Materials
This section details step-by-step procedures how the simulation is performed.
The performance of our proposed method is assessed by an extensive simulation
study. Data are simulated with different number of genes $p$ and sample size
$n$. In this simulation, the regression model includes two covariates $Z$ and
$X$, where $Z$ is the group indicator and $X\sim N(55,10)$ is a continuous
covariate such as the age of a patient. Three different simulation scenarios
are considered.
### 3.1 Data Generation
Simulate weighted networks and RNA-Seq data with a dependence structure that
depends on $Z$ and/or $X$ using the SeqNet R package (Grimes and Datta, 2021).
In this setting, there are total of six networks for the combination of two
groups and three age categories (younger than 50, 50-60, and older than 60).
We consider three different scenarios incorporating group information only
(scenario I), age and group information (scenario II), and age and group
information with unequal sampling proportions with different distributions of
the age in the two groups (scenario III) (see Figures 1–3 for visual
demonstrations).
#### Scenarios I (a–b) and II (a–c)
* a.
Generate the first random network with $p$ nodes for $z=1$. The $p\times p$
adjacency matrix, where the diagonal elements are 0 and non-diagonal elements
are in $\\{0,1\\}$, is extracted from this first graph. It is a symmetric
matrix indicating whether a pair of nodes are connected by an edge. Take the
column sum of the adjacency matrix to see the total number of connected edges
to the node. Record the indices of this vector with column sum for the use of
effect size adjustment in later step.
* b.
Perturb the first network to generate the second network for $z=2$ by removing
the edges around nodes using the indices obtained in previous step. To assess
the effect size of group, the top $5\%$, $10\%$, and $20\%$ of total nodes
with the most number of edges in a network lose their edges (e.g. 2 nodes with
the most number of edges for a network with $p=20$ for the effect size of
$10\%$, Figure 1). This is the end of scenario I.
* c.
For scenario II, further perturb remaining networks by removing edges of one
additional node with the next most number of edges, coming after Step (b)
above. This is to simulate networks with a covariate dependence structure on
both age and group (see Figure 2).
#### Scenario III
We created a scenario where age is acting like a confounder. In other words,
for a given each category that two networks are the same, but the
distributions of the age of the patients are different in the two groups.
Therefore, there will be an observed difference in network connectivity, which
is explained through age.
* a.
Generate the first random network with $p$ nodes for younger than 50 category.
The $p\times p$ adjacency matrix, where the diagonal elements are 0 and non-
diagonal elements are either $\\{0,1\\}$, is extracted from this first graph.
Record the indices of connected edges for the perturbation of network in later
steps below.
* b.
Perturb the first network to generate the second network for age 50-60
category by removing the edges of the two nodes with the most number of edges
in a network lose all of their edges. In other words, we refer to the indices,
recorded in the adjacency matrix from the earlier step, and remove all the
connected edges around the two nodes.
* c.
Next, we repeat the same to perturb the second network to obtain the third
network for older than 60 category (see Figure 3).
Figure 1: Network plots visualizing the gene network (p = 20) without a
covariate dependence structure that depends on binary group only (scenario I).
The row represents group whereas the column represents age categories. The
three networks in each row are identical, since there is no effect of age on
the structure of network. The edges of the hub nodes are removed based on the
effect size of the binary group.
Figure 2: Network plots visualizing the gene network (p = 20) with a covariate
dependence structure that depends on age and group information (scenario II).
The row represents group whereas the column represents age categories. All six
networks have unique structure of the network. The edges of the hub nodes are
firstly removed based on the effect size of the group, as shown in Figure 1
above. For this scenario II, additional edges of nodes with greater number of
connected edges are removed for each age category.
Figure 3: Network plots visualizing the gene network (p = 20) with a covariate
dependence structure that depends on age and group information with unequal
sampling proportions with respect to different distribution of the age in the
two groups (scenario III). The row represents group whereas the column
represents age categories. All six networks have unique structure of the
network. The edges of the two hub nodes are removed for each age category. To
employ the effect of group, $10\%/10\%/80\%$ of the subjects in $z=1$ will
have a network structure to each of the first, second, and third networks in
the first row. In contrast, $80\%/10\%/10\%$ of the subjects in $z=2$ will
have a network structure to each of the first, second, and third networks in
the second row.
For all three scenarios, we assign a partial correlation to edges to obtain
weighted networks (Grimes and Datta, 2021). Note that adjacency matrices of
these weighted networks are used for the the true connection per gene.
Generate RNA-Seq samples based on weighted networks with equal sampling
proportions for scenarios I and II. However, specifically for scenario III, a
sampling proportion differs across age categories and groups. That is,
$10\%/10\%/80\%$ for $z=1$ and $80\%/10\%/10\%$ for $z=2$. The data generation
involves with two major steps. Firstly, we generate gene expressions (Gaussian
values) from a group-specific weighted network for each gene, denoted as
$\tilde{x}_{i}\sim N(0,1)$. These Gaussian values are then mapped into RNA-Seq
data column-wise by using the inverse CDF of empirical distribution of the
reference data using expression data with accession number GSE158699 (Wang et
al., 2021) from the Gene Expression Omnibus (GEO) database (Barrett et al.,
2013). We will have $n_{z}\times p$ matrices for each group $z=1,2$.
### 3.2 Algorithm
[enumerate] Obtain an association matrix with ARACNE from the data generated
in steps from the “Data Generation” section to fit an estimated network using
minet (Margolin et al., 2006; Meyer et al., 2008) for each group. For each
gene $k$, calculate the column sums of association matrix for each group $z$
separately, denoted by $\hat{\theta}_{k}^{z}$. For each gene $k$ and
individual $i\in\mathcal{G}_{z}$, compute $\hat{\theta}_{k(i)}^{z}$ from the
association matrix that is re-estimated based on RNA-Seq data without the
$i^{\text{th}}$ subject of $n_{z}\times p$ data from the “Data Generation”
section for each group $z$ separately, where $i=1,\dots,n_{z}$.
Calculate $\tilde{\theta}_{ik}$ using equation (1) based on Step 2 and 3. For
each gene $k$, fit a multivariable robust regression with binary group
variable and continuous age variable to obtain the p-values of the group
variable, computed from the t-test. These p-values are used to compute the
performance measures of simulation study. More details on the performance
measures are stated next.
### 3.3 Performance Measures
To evaluate the performance of our proposed method, precision, recall, and the
F1 score are calculated. Let $\Omega^{z}\in\mathbf{R}^{p\times p}$ be the
adjacency matrix for group $z$, where
$\Omega_{jk}^{z}=\begin{cases}1&\text{if }j^{\text{th}}\text{ gene and
}k^{\text{th}}\text{ gene are connected}\\\ 0&\text{otherwise},\end{cases}$
for $z=1,2$. Then, for each gene $k$, we calculate
$\eta_{k}=I\bigg{(}\sum_{j=1}^{p}|\Omega_{jk}^{1}-\Omega_{jk}^{2}|\geq
1\bigg{)},$
where $I(\cdot)$ is an indicator function to determine whether gene $k$ has
differential connectivity. The gene $k$ is truly DC if $\eta_{k}=1$, and is
not DC if $\eta_{k}=0$ for the true gene network. Similarly, for the covariate
dependence structure, the following quantities are obtained
$\eta_{k}=I\bigg{(}\frac{1}{c}\sum_{c}\sum_{j=1}^{p}|\Omega_{jk}^{1,c}-\Omega_{jk}^{2,c}|\geq
1\bigg{)},$
where $\Omega^{z,c}\in\mathbf{R}^{p\times p}$ be the adjacency matrix for
group $z$ and age category $c=1,2,3$. Denote that $S$ is the total number of
Monte Carlo simulation replicates. Let $q_{ks}$ be adjusted p-value as in
following procedure (Datta and Datta, 2005) of $k^{\text{th}}$ gene at the
$s^{\text{th}}$ simulation replicate. $\alpha$ represents the magnitude of
error control, and 0.05 was used throughout the simulation.
* -
Precision is the proportion of genes that are inferred to be significantly DC
from the test which have true connection between two comparing groups:
$\text{Precision}=\frac{\sum_{k=1}^{p}\eta_{k}\,I(q_{ks}<\alpha)}{\sum_{k=1}^{p}I(q_{ks}<\alpha)}.$
* -
Recall is the proportion of genes that have true connection which are
correctly inferred to be significantly DC between two comparing groups from
the test:
$\text{Recall}=\frac{\sum_{k=1}^{p}\eta_{k}\,I(q_{ks}<\alpha)}{\sum_{k=1}^{p}\eta_{k}}.$
* -
F1 is calculated based on the harmonic mean of precision and recall obtained
from the simulation. A higher F1 score suggests lower false negative and false
positive rate:
$\text{F1}=2\cdot\frac{\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}}.$
### 3.4 COPDGene Data
A recent genome-wide association study (Sakornsakolpat et al., 2019)
identified 35 new COPD-related genes from the UK Biobank and International
COPD Genetics Consortium data. Among these 35 COPD-related genes, 28 genes are
available in the data from the COPDGene study for the analysis using PRANA and
other DN analysis methods including dnapath and DINGO. The 28 COPD-related
genes are the following: CITED2, TESK2, COL15A1, AMZ1, RASEF, DDX1, DMWD,
MED13L, ZBTB38, CCDC69, EML4, HSPA4, ITGB8, TEPP, TNPO1, ARNTL, DTWD1,
ADAMTSL3, RREB1, THRA, SLMAP, DENND2D, STN1, SYN3, ASAP2, IER3, MFHAS1, and
VGLL4.
Among 2,561 samples from the initial phenotype data, we have used 406 samples
that were provided as the validation set in the analysis of the original
study. For the analysis with PRANA, binary current smoking status variable is
used as the grouping variable, and smoking pack years, age, gender, race, and
FEV1 are included as additional covariates in a multivariable model. The
binary current smoking status variable is used as the grouping variable for
dnapath and DINGO.
## 4 Results
### 4.1 Simulation Study
More details of the simulation setup are available in the “Materials” section
above. We select $p=20,50,100$ genes to test our pseudo-value approach in
smaller to larger gene networks. For each gene network, five different sample
sizes $n=40,100,200,500,1000$ are considered and in each setting, 1,000 Monte
Carlo replicates. We draw 1,000 random samples first, then take the subsamples
from this pool for a simulation with a smaller sample size to reduce
computational burden. A random network is generated at each simulation
replicate in which a layer of randomness is imposed to account for biological
variability of the network structure. For additional details on the generation
of simulated RNA-Seq data, see the Materials section. Simulations are repeated
to show the performance of our method by altering the effect size from 5$\%$,
10$\%$, to 20$\%$ for simulation scenarios I and II.
Results are compared with the true parent network in order to compute the
performance measures described in the “Performance Measures” section. In the
true network setting, a gene is considered truly DC between groups if it has
at least one DC edge connected to other genes. Tables 1 and 2 summarize
simulation results in the multivariable setting for scenarios I and II,
respectively, when the continuous variable is added as a covariate with the
binary group variable in the regression. Table 2 incorporates the effect of
covariate when generating random networks, whereas Table 1 does not. For both
scenarios, results show that the pseudo-value regression method generally
yields a high precision and recall across all specifications of network size,
sample size, and effect size. The pseudo-value regression method maintains a
high precision while having an acceptable recall, especially, when a smaller
sample size is considered.
Table 1: Scenario I simulation results of binary group variable in the multivariable robust regression model (continuous age and binary group) using pseudo-value approach with 1,000 replicates. Random network is generated at each simulation replicate. | | Effect size
---|---|---
| | 5$\%$ | 10$\%$ | 20$\%$
$p$ | $n$ | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1
20 | 40 | 0.76 | 0.81 | 0.77 | 0.90 | 0.79 | 0.83 | 0.98 | 0.79 | 0.87
| 100 | 0.75 | 0.91 | 0.82 | 0.90 | 0.91 | 0.90 | 0.98 | 0.91 | 0.94
| 200 | 0.74 | 0.94 | 0.82 | 0.89 | 0.94 | 0.91 | 0.97 | 0.94 | 0.96
| 500 | 0.73 | 0.97 | 0.83 | 0.88 | 0.97 | 0.92 | 0.97 | 0.97 | 0.97
| 1,000 | 0.73 | 0.98 | 0.83 | 0.88 | 0.98 | 0.92 | 0.97 | 0.98 | 0.97
50 | 40 | 0.95 | 0.65 | 0.77 | 0.98 | 0.65 | 0.78 | 1.00 | 0.64 | 0.77
| 100 | 0.95 | 0.77 | 0.85 | 0.98 | 0.77 | 0.86 | 1.00 | 0.77 | 0.86
| 200 | 0.96 | 0.85 | 0.90 | 0.99 | 0.85 | 0.91 | 1.00 | 0.85 | 0.91
| 500 | 0.95 | 0.93 | 0.94 | 0.99 | 0.92 | 0.95 | 1.00 | 0.92 | 0.96
| 1,000 | 0.95 | 0.96 | 0.95 | 0.98 | 0.96 | 0.97 | 1.00 | 0.96 | 0.98
100 | 40 | 0.96 | 0.57 | 0.71 | 0.98 | 0.57 | 0.72 | 1.00 | 0.57 | 0.72
| 100 | 0.96 | 0.67 | 0.79 | 0.99 | 0.68 | 0.80 | 1.00 | 0.67 | 0.80
| 200 | 0.96 | 0.74 | 0.83 | 0.99 | 0.74 | 0.84 | 1.00 | 0.74 | 0.85
| 500 | 0.97 | 0.82 | 0.89 | 0.99 | 0.81 | 0.89 | 1.00 | 0.81 | 0.89
| 1,000 | 0.97 | 0.90 | 0.93 | 0.99 | 0.89 | 0.94 | 1.00 | 0.89 | 0.94
Table 2: Scenario II simulation results of binary group variable in the multivariable robust regression model (continuous age and binary group) using pseudo-value approach with 1,000 replicates. Random network is generated at each simulation replicate. | | Effect size
---|---|---
| | 5$\%$ | 10$\%$ | 20$\%$
$p$ | $n$ | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1
20 | 40 | 0.75 | 0.59 | 0.64 | 0.90 | 0.69 | 0.70 | 0.98 | 0.61 | 0.73
| 100 | 0.78 | 0.70 | 0.71 | 0.91 | 0.72 | 0.79 | 0.98 | 0.74 | 0.83
| 200 | 0.78 | 0.81 | 0.78 | 0.91 | 0.83 | 0.86 | 0.98 | 0.85 | 0.91
| 500 | 0.76 | 0.90 | 0.82 | 0.90 | 0.92 | 0.91 | 0.98 | 0.94 | 0.95
| 1,000 | 0.75 | 0.94 | 0.83 | 0.89 | 0.95 | 0.92 | 0.97 | 0.96 | 0.97
50 | 40 | 0.95 | 0.56 | 0.70 | 0.98 | 0.56 | 0.71 | 1.00 | 0.57 | 0.72
| 100 | 0.95 | 0.65 | 0.77 | 0.98 | 0.66 | 0.78 | 1.00 | 0.68 | 0.80
| 200 | 0.96 | 0.75 | 0.84 | 0.99 | 0.75 | 0.85 | 1.00 | 0.77 | 0.87
| 500 | 0.96 | 0.88 | 0.92 | 0.99 | 0.87 | 0.92 | 1.00 | 0.89 | 0.94
| 1,000 | 0.96 | 0.93 | 0.94 | 0.99 | 0.93 | 0.96 | 1.00 | 0.94 | 0.97
100 | 40 | 0.96 | 0.55 | 0.69 | 0.99 | 0.55 | 0.70 | 0.99 | 0.55 | 0.70
| 100 | 0.95 | 0.63 | 0.75 | 0.98 | 0.63 | 0.76 | 0.98 | 0.63 | 0.76
| 200 | 0.96 | 0.68 | 0.79 | 0.99 | 0.68 | 0.81 | 0.99 | 0.68 | 0.80
| 500 | 0.97 | 0.77 | 0.86 | 0.99 | 0.77 | 0.86 | 0.99 | 0.77 | 0.86
| 1,000 | 0.97 | 0.87 | 0.92 | 0.99 | 0.86 | 0.92 | 0.99 | 0.86 | 0.92
Tables 3 and 4 summarize simulation results for scenarios I and II,
respectively, when only the binary group variable is included in the model for
pseudo-value calculation. Thus, age dependent networks are simulated for Table
4 but not for Table 3. Two competing univariable methods, dnapath and DINGO
are included for these simulations.
Overall, a similar pattern is observed in the univariable setting; i.e., PRANA
consistently reaches a high precision and recall. The performance improves as
n increases, as to be expected. It is noteworthy that PRANA outperforms
dnapath in simulation when the sample size is relatively small regardless of
the network size. Our method also shows a better recall value and F1-score
than dnapath with small sample sizes ($\textit{n}=40,100$). As DINGO requires
substantially large computational time, it was considered for the gene network
with smaller sample sizes only. To be more specific, simulations with larger
sample sizes ($n=500,1000$) are stopped after 20 days for DINGO from the
University of Florida Research Computing Linux server, HiPerGator 3.0 with
10CPU cores and 10GB of RAM per node. See Table S1 in Additional File 1 for
more details.
Table 3: Scenario I simulation results of binary group variable in the univariable robust regression model using pseudo-value approach with 1,000 replicates. The network structure does not depend on age covariate. Random network is generated at each simulation replicate. Sample size n = (500, 1000) for gene size p = 100 were not included for DINGO due to heavy computational time. The best results are highlighted in boldface. | | Precision | Recall | F1
---|---|---|---|---
$p$ | $n$ | PRANA | dnapath | DINGO | PRANA | dnapath | DINGO | PRANA | dnapath | DINGO
20 | 40 | 0.90 | 0.95 | 0.87 | 0.81 | 0.64 | 0.78 | 0.84 | 0.76 | 0.82
| 100 | 0.90 | 0.93 | 0.87 | 0.90 | 0.88 | 0.79 | 0.84 | 0.90 | 0.82
| 200 | 0.89 | 0.91 | 0.87 | 0.94 | 0.94 | 0.76 | 0.91 | 0.92 | 0.81
| 500 | 0.88 | 0.89 | - | 0.97 | 0.98 | - | 0.92 | 0.93 | -
| 1,000 | 0.88 | 0.89 | - | 0.98 | 0.99 | - | 0.92 | 0.93 | -
50 | 40 | 0.98 | 1.00 | 0.99 | 0.65 | 0.39 | 0.70 | 0.78 | 0.56 | 0.81
| 100 | 0.98 | 1.00 | 0.98 | 0.77 | 0.61 | 0.84 | 0.86 | 0.75 | 0.90
| 200 | 0.99 | 1.00 | 0.98 | 0.85 | 0.85 | 0.85 | 0.91 | 0.91 | 0.91
| 500 | 0.99 | 0.99 | - | 0.92 | 0.95 | - | 0.95 | 0.97 | -
| 1,000 | 0.98 | 0.99 | - | 0.96 | 0.98 | - | 0.97 | 0.98 | -
100 | 40 | 0.98 | 1.00 | 0.98 | 0.57 | 0.27 | 0.69 | 0.72 | 0.42 | 0.85
| 100 | 0.98 | 1.00 | 0.99 | 0.68 | 0.32 | 0.75 | 0.80 | 0.48 | 0.85
| 200 | 0.99 | 1.00 | 0.98 | 0.74 | 0.62 | 0.82 | 0.84 | 0.77 | 0.89
| 500 | 0.99 | 1.00 | - | 0.81 | 0.88 | - | 0.89 | 0.93 | -
| 1,000 | 0.99 | 0.99 | - | 0.89 | 0.94 | - | 0.94 | 0.97 | -
Table 4: Scenario II simulation results of binary group variable in the univariable robust regression model using pseudo-value approach with 1,000 replicates. The network structure depends on age covariate. Random network is generated at each simulation replicate. Sample size n = (500, 1000) or gene size p = 100 were not included for DINGO due to heavy computational time. The best results are highlighted in boldface. | | Precision | Recall | F1
---|---|---|---|---
$p$ | $n$ | PRANA | dnapath | DINGO | PRANA | dnapath | DINGO | PRANA | dnapath | DINGO
20 | 40 | 0.90 | 0.97 | 0.89 | 0.59 | 0.38 | 0.63 | 0.69 | 0.53 | 0.72
| 100 | 0.91 | 0.97 | 0.88 | 0.72 | 0.66 | 0.75 | 0.79 | 0.77 | 0.80
| 200 | 0.91 | 0.96 | 0.89 | 0.83 | 0.83 | 0.73 | 0.86 | 0.88 | 0.79
| 500 | 0.90 | 0.93 | - | 0.93 | 0.94 | - | 0.91 | 0.93 | -
| 1,000 | 0.89 | 0.91 | - | 0.95 | 0.97 | - | 0.92 | 0.94 | -
50 | 40 | 0.98 | 1.00 | 0.99 | 0.56 | 0.28 | 0.67 | 0.71 | 0.43 | 0.80
| 100 | 0.98 | 1.00 | 0.98 | 0.65 | 0.45 | 0.71 | 0.78 | 0.61 | 0.82
| 200 | 0.99 | 1.00 | 0.98 | 0.75 | 0.68 | 0.80 | 0.85 | 0.80 | 0.88
| 500 | 0.99 | 0.99 | - | 0.87 | 0.91 | - | 0.93 | 0.95 | -
| 1,000 | 0.99 | 0.99 | - | 0.94 | 0.97 | - | 0.96 | 0.98 | -
100 | 40 | 0.99 | 1.00 | 0.55 | 0.55 | 0.21 | 0.77 | 0.70 | 0.34 | 0.63
| 100 | 0.98 | 1.00 | 0.55 | 0.63 | 0.27 | 0.75 | 0.76 | 0.42 | 0.63
| 200 | 0.99 | 1.00 | 0.55 | 0.68 | 0.46 | 0.77 | 0.80 | 0.62 | 0.63
| 500 | 0.99 | 1.00 | - | 0.77 | 0.82 | - | 0.86 | 0.90 | -
| 1,000 | 0.99 | 1.00 | - | 0.86 | 0.92 | - | 0.92 | 0.96 | -
Table 5 presents results of scenario III, where age acts as a confounder. That
is, an observed difference in connectivity may be due to a difference in the
distribution of age in the two groups. Higher precision values from the
multivariable pseudo-value regression indicate that PRANA correctly identifies
the DC genes, compared with dnapath and DINGO, neither of which accounts for
the effects of age. By and large, PRANA has higher precision than DINGO and
higher recall than dnapath.
Table 5: Scenario III simulation results of binary group variable in the multivariable and univariable robust regression model using pseudo-value approach with 1,000 replicates. The network structure depends on age covariate by unequal sampling proportion depending on age categories. Random network is generated at each simulation replicate. Sample size n = (500, 1000) or gene size p = 100 were not included for DINGO due to heavy computational costs. The best results are highlighted in boldface. | | Precision | Recall | F1
---|---|---|---|---
$p$ | $n$ | PRANA (Mult) | PRANA (Univ) | dnapath | DINGO | PRANA (Mult) | PRANA (Univ) | dnapath | DINGO | PRANA (Mult) | PRANA (Univ) | dnapath | DINGO
20 | 40 | 0.67 | 0.60 | 0.64 | 0.58 | 0.57 | 0.74 | 0.50 | 0.75 | 0.59 | 0.65 | 0.55 | 0.64
| 100 | 0.67 | 0.58 | 0.61 | 0.58 | 0.65 | 0.85 | 0.73 | 0.79 | 0.64 | 0.68 | 0.65 | 0.66
| 200 | 0.66 | 0.58 | 0.59 | 0.58 | 0.76 | 0.91 | 0.85 | 0.80 | 0.69 | 0.70 | 0.69 | 0.67
| 500 | 0.64 | 0.57 | 0.58 | - | 0.87 | 0.95 | 0.95 | - | 0.73 | 0.71 | 0.71 | -
| 1,000 | 0.64 | 0.57 | 0.58 | - | 0.92 | 0.97 | 0.97 | - | 0.75 | 0.71 | 0.72 | -
50 | 40 | 0.57 | 0.50 | 0.54 | 0.49 | 0.47 | 0.62 | 0.33 | 0.67 | 0.50 | 0.54 | 0.40 | 0.55
| 100 | 0.58 | 0.49 | 0.52 | 0.49 | 0.50 | 0.71 | 0.51 | 0.79 | 0.52 | 0.58 | 0.51 | 0.60
| 200 | 0.58 | 0.49 | 0.51 | 0.49 | 0.52 | 0.76 | 0.60 | 0.83 | 0.53 | 0.59 | 0.54 | 0.61
| 500 | 0.55 | 0.48 | 0.48 | - | 0.67 | 0.88 | 0.83 | - | 0.59 | 0.61 | 0.60 | -
| 1,000 | 0.54 | 0.48 | 0.48 | - | 0.81 | 0.93 | 0.91 | - | 0.64 | 0.62 | 0.62 | -
100 | 40 | 0.57 | 0.49 | 0.53 | 0.48 | 0.44 | 0.57 | 0.22 | 0.75 | 0.49 | 0.52 | 0.30 | 0.58
| 100 | 0.58 | 0.49 | 0.53 | 0.48 | 0.47 | 0.64 | 0.35 | 0.76 | 0.51 | 0.55 | 0.41 | 0.58
| 200 | 0.58 | 0.49 | 0.53 | 0.48 | 0.46 | 0.65 | 0.40 | 0.80 | 0.50 | 0.55 | 0.45 | 0.60
| 500 | 0.56 | 0.47 | 0.49 | - | 0.44 | 0.74 | 0.60 | - | 0.48 | 0.57 | 0.53 | -
| 1,000 | 0.53 | 0.47 | 0.47 | - | 0.63 | 0.85 | 0.82 | - | 0.57 | 0.60 | 0.59 | -
Table S1: Comparison of computational time of PRANA with that of dnapath and DINGO. Scenario I is depicted for the illustrative purposes. Random network is generated at each simulation replicate. Large sample sizes n = (500, 1000) are stopped after 20 days for DINGO from the high-performance Linux cluster using machines with 10CPU cores and 10GB of RAM per node. | | Time (in hours)
---|---|---
$p$ | $n$ | PRANA | dnapath | DINGO
20 | 40 | 0.12 | 0.13 | 29.25
| 100 | 0.22 | 0.2 | 86.58
| 200 | 0.23 | 0.17 | 206.18
| 500 | 0.42 | 0.27 | $>$ 480
| 1,000 | 0.68 | 0.45 | $>$ 480
50 | 40 | 0.2 | 0.22 | 26.57
| 100 | 0.35 | 0.32 | 31.48
| 200 | 0.67 | 0.45 | 137.58
| 500 | 1.52 | 0.98 | $>$ 480
| 1,000 | 2.48 | 1.08 | $>$ 480
100 | 40 | 0.63 | 0.52 | 68.73
| 100 | 0.82 | 0.7 | 61.78
| 200 | 1.1 | 1.23 | 147.57
| 500 | 3.58 | 2.37 | $>$ 480
| 1,000 | 4.68 | 3.57 | $>$ 480
### 4.2 Analysis of COPDGene Data
23 out of 28 COPD-related genes are predicted to be DC between current and
non-current smokers with PRANA while accounting for smoking pack years, age,
gender, race, and FEV1. A complete list of DC genes found from the pseudo-
value approach are CITED2, TESK2, AMZ1, DDX1, DMWD, MED13L, ZBTB38, EML4,
HSPA4, ITGB8, TEPP, TNPO1, ARNTL, DTWD1, ADAMTSL3, THRA, SLMAP, DENND2D, STN1,
SYN3, ASAP2, IER3, and MFHAS1.
We compared results of PRANA with dnapath (Grimes et al., 2019) and DINGO (Ha
et al., 2015). With DINGO, a total of 19 out of 28 COPD-related genes were
selected as DC genes between current and non-current smokers. A complete list
of DC genes found in DINGO are the following: ARNTL, DDX1, HSPA4, ITGB8,
SLMAP, SYN3, ASAP2, IER3, MFHAS1, VGLL4, CITED2, TESK2, CCDC69, EML4,
ADAMTSL3, DENND2D, AMZ1, RASEF, and ZBTB38. Lastly, 3 genes were found DC
between current smoking groups with dnapath, namely DTWD1, EML4, and TEPP.
Of the 23 DC genes from PRANA, 5 are found exclusive to PRANA (DMWD, MED13L,
TNPO1, THRA, and STN1). Notably, DMWD is linked to myotonic dystrophy, a rare
genetic muscular disorder (Westerlaken et al., 2003). Thyroid hormone receptor
alpha (THRA) is related to congenital hypothyroidism (Tylki-Szymańska et al.,
2015). These findings about additional genes will facilitate harnessing of the
possible mechanisms at work in COPD exacerbation.
Heat shock protein family A (Hsp70) member 4 (HSPA4) is associated with
gastric ulcer (Sakurai et al., 2015). Multifunctional ROCO family signaling
regulator 1 (MFHAS1) is linked to soft tissue tumor and cell cycle (Stelzer et
al., 2016). HSPA4 and MFHAS1 are DC genes identified in both PRANA and DINGO.
Echinoderm microtubule-associated protein-like 4 (EML4) is found in all three
methods. It has been studied for its association with lung cancer (Stelzer et
al., 2016; Adib et al., 2019). A Venn diagram is provided to show the overlap
between and among three methods (Figure 4). In addition, a diagram is included
to summarize the findings of this application study (Figure 5).
Figure 4: A Venn diagram displaying the number of overlapping DC genes between
and among univariable analysis such as DINGO and dnapath versus multivariable
robust regression with pseudo-value approach using COPDGene study data from
GEO database.
Figure 5: A diagram summarizing results using each methods analyzing the
COPDGene study data from GEO database. A full list of DCGs (differentially
connected genes) are provided in each box.
## 5 Discussion
Simulations and real-data analysis have elucidated that PRANA is superior to
existing alternatives and a practical tool, which includes covariates in the
model. To the best of our knowledge, this is the first attempt to develop a
regression modeling in DN analysis. Our working objective is to propose a
statistical method that determines whether a gene is significantly DC between
groups with the covariate included in the model. In this paper, we have shown
through simulations that PRANA reaches a consistently high degree of precision
and recall to identify DC genes with varying simulation parameters such as
network size, sample size, and effect size. We also analyzed a COPD-related
gene expression data from the GEO database. When comparing results from our
method to dnapath and DINGO, five COPD-related genes are additionally found DC
between current versus non-current smokers: DMWD, MED13L, TNPO1, THRA, and
STN1.
There are a number of limitations to be highlighted in this study. We have
used the absolute value of the differences between the two adjacency matrices
as a proxy to determine the true DC genes. Certainly, this is a practical way
to detect differences in the number of edges for each genes in a network. The
comparison of maximum values between adjacency matrices was also considered.
However, we concluded that they are more useful describing the global
characteristic of a network, which deviates from our objective, namely, a
gene-specific characteristic of a network.
Another limitation is the inability to perturb simulated networks in a
continuous way. Right now, we have discretized the effect of a covariate into
three groups. Perhaps, there are other models where a truly continuous
covariate could be incorporated.
Lastly, the Pearson correlation, partial correlation, and degree-weighted
LASSO were also examined as alternatives to the ARACNE as a measure of
association or connectedness, albeit not reported in the paper, due to
relatively poor performance and heavy computational costs. It remains an
interesting task for future studies to extend our work to other measures of
association of a network which better assess different structural changes in
the network. We conclude by foregrounding the future direction of the pseudo-
value regression approach for the DN analysis, which are potentially
extensible to other data types, such as the microbiome data.
## 6 Conclusion
The adjustment of covariate is an important step in differential network
analysis. In this paper, we presented PRANA, a novel pseudo-value regression
approach for the DN analysis, which can incorporate additional clinical
covariates in the model. This is a direct regression modeling, and it is
therefore computationally amenable for the most users.
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# Approximate minimum cuts and their enumeration111University of Illinois,
Urbana-Champaign. Email: {calvinb2, karthe<EMAIL_ADDRESS>Supported
in part by NSF grants CCF-1814613 and CCF-1907937.
Calvin Beideman Karthekeyan Chandrasekaran Weihang Wang
###### Abstract
We show that every $\alpha$-approximate minimum cut in a connected graph is
the _unique_ minimum $(S,T)$-terminal cut for some subsets $S$ and $T$ of
vertices each of size at most $\lfloor 2\alpha\rfloor+1$. This leads to an
alternative proof that the number of $\alpha$-approximate minimum cuts in a
$n$-vertex connected graph is $n^{O(\alpha)}$ and they can all be enumerated
in deterministic polynomial time for constant $\alpha$.
## 1 Introduction
Let $G=(V,E)$ be a graph with positive edge costs
$c:E\rightarrow\mathbb{R}_{+}$. A partitioning of the vertex set into $2$ non-
empty parts is known as a _cut_. For a non-empty proper subset $U$ of
vertices, we will use $\overline{U}$ to denote $V\setminus U$,
$(U,\overline{U})$ to denote the cut associated with $U$, $\delta(U)$ to
denote the set of edges crossing the cut $(U,\overline{U})$, and
$d(U):=\sum_{e\in\delta(U)}c(e)$ to denote the cut value of $U$. We will use
$\lambda$ to denote the value of a minimum cut in $G$, i.e.,
$\lambda:=\min\\{d(U):\emptyset\neq U\subsetneq V\\}.$
For $\alpha\geq 1$, a cut $(U,\overline{U})$ is an _$\alpha$ -approximate
minimum cut_ if $d(U)\leq\alpha\lambda$. A fundamental graph structural result
states that the number of $\alpha$-approximate minimum cuts in a $n$-vertex
connected graph is $O(n^{2\alpha})$. This structural result forms the backbone
of several algorithmic and representation results in graphs—e.g., fast
randomized construction of skeletons [12], existence of cut sparsifiers which
in turn finds applications in streaming and sketching [1, 2, 3, 14],
approximation algorithms for TSP [17, 4, 6, 5], computing reliability of
probabilistic networks [11], and polynomial time algorithms for connectivity
augmentation [16]. The structural result along with a randomized polynomial-
time algorithm to enumerate all constant-approximate minimum cuts was first
shown via Karger’s random contraction technique [13]. Subsequently, the
splitting-off technique and the tree packing technique have been used to show
the structural result along with a _deterministic_ polynomial-time algorithm
to enumerate all constant-approximate minimum cuts [15, 8].
In this work, we give a fourth technique to bound the number of
$\alpha$-approximate minimum cuts by $n^{O(\alpha)}$ along with a
deterministic polynomial-time algorithm to enumerate all constant-approximate
minimum cuts. Let $S$, $T$ be disjoint non-empty subsets of vertices. A
$2$-partition $(U,\overline{U})$ is an $(S,T)$-terminal cut if $S\subseteq
U\subseteq V\setminus T$. Here, the set $U$ is known as the _source set_ and
the set $\overline{U}$ is known as the _sink set_. An $(S,T)$-terminal cut
with minimum cut value will be denoted as a _minimum $(S,T)$-terminal cut_.
The following is our main result.
###### Theorem 1.1.
Let $G=(V,E)$ be a connected graph with positive edge costs and
$(U,\overline{U})$ be an $\alpha$-approximate minimum cut for some $\alpha\geq
1$. Then, there exist subsets $S,T\subseteq V$ with $|S|,|T|\leq\lfloor
2\alpha\rfloor+1$ such that $(U,\overline{U})$ is the unique minimum
$(S,T)$-terminal cut.
In other words, every $\alpha$-approximate minimum cut $(U,\overline{U})$ in a
connected graph can be recovered as the _unique_ minimum $(S,T)$-terminal cut
for some subsets $S$ and $T$ of sizes at most $\lfloor 2\alpha\rfloor+1$. A
few remarks are in order.
###### Remark 1.1.
An immediate consequence of Theorem 1.1 is that the number of distinct
$\alpha$-approximate minimum cuts in a $n$-vertex connected graph is at most
$n^{4\alpha+2}$ and they can all be enumerated using $n^{4\alpha+2}$ minimum
$(S,T)$-terminal cut computations. We recall that minimum $(S,T)$-terminal cut
in a given graph with given subsets $S$ and $T$ of vertices can be computed in
deterministic polynomial time.
###### Remark 1.2.
The size bound for subsets $S$ and $T$ in Theorem 1.1 is tight up to an
additive factor of one: Consider the $n$-vertex cycle graph $G$ with $\alpha$
being a positive integer where $n\geq 4\alpha$ and with all edge costs being
$1$. Let $(U,\overline{U})$ be a cut in $G$ with $d(U)=2\alpha$ and moreover,
each connected component in $G-\delta(U)$ has at least $2$ vertices (see
Figure 1 for such an example where $\alpha=4$). For such a cut
$(U,\overline{U})$, the only choice of $S$ and $T$ for which
$(U,\overline{U})$ is the _unique_ minimum $(S,T)$-terminal cut is given by
$\displaystyle S$ $\displaystyle:=\\{v\in U:v\text{ is an end vertex of an
edge in }\delta(U)\\}\text{ and}$ $\displaystyle T$
$\displaystyle:=\\{v\in\overline{U}:v\text{ is an end vertex of an edge in
}\delta(U)\\}.$
Hence, this example shows that both $S$ and $T$ need to have at least
$2\alpha$ vertices in order to recover $(U,\overline{U})$ as the _unique_
minimum $(S,T)$-terminal cut.
Figure 1: An example showing the tightness of Theorem 1.1: The set $U$
consists of vertices within all gray boxes and the set $\overline{U}$ consists
of all vertices outside gray boxes; the dotted edges correspond to the edges
in the cut set $\delta(U)$. The only choice of $S$ and $T$ for which
$(U,\overline{U})$ is the unique minimum $(S,T)$-terminal cut is as follows:
$S$ consists of all vertices that are shaded black and $T$ consists of all
vertices that are shaded gray.
Theorem 1.1 is inspired by a result of Goemans and Ramakrishnan [9]. They
showed the following structural theorem for $(4/3-\epsilon)$-approximate
minimum cuts: fix an arbitrary vertex $t$ in a connected graph $G=(V,E)$ and
let $(U,\overline{U})$ be a $(4/3-\epsilon)$-approximate minimum cut with
$t\in\overline{U}$ for some $\epsilon>0$. Then, there exists a subset $S$ of
size at most $2$ such that $(U,\overline{U})$ is the unique minimum
$(S,\\{t\\})$-terminal cut. Their result leads to the following natural
question: for every $\alpha\geq 1$, can every $\alpha$-approximate minimum cut
$(U,\overline{U})$ be obtained as a minimum $(S,T)$-terminal cut for some
subsets $S$ and $T$ of vertices each of size at most some function of
$\alpha$? Our Theorem 1.1 answers this question affirmatively. However, our
proof of Theorem 1.1 deviates significantly from the proof approach of
Goemans-Ramakrishnan’s structural result for $(4/3-\epsilon)$-approximate
minimum cuts. Their proof proceeds via contradiction and relies on the
_submodular triple inequality_ which holds for the graph cut function. The
submodular triple inequality shows that $3$ non-empty sets satisfying certain
crossing properties can be uncrossed to obtain $4$ disjoint non-empty sets
without increasing the sum of their cut values; thus, if all $3$ sets have cut
value less than $(4/3)\lambda$, then one of the $4$ sets should have cut value
strictly smaller than $\lambda$, a contradiction to the definition of
$\lambda$. However, there does not seem to be a generalization of the
submodular triple inequality to larger number of sets. Our proof of Theorem
1.1 circumvents the submodular triple inequality but achieves its intended
purpose in this context—we show that a sufficiently large number of
$\alpha$-approximate minimum cuts with certain crossing properties can be
uncrossed to obtain a cut of value cheaper than $\lambda$, thus leading to a
contradiction again.
### 1.1 Preliminaries
Let $G=(V,E)$ be a graph with positive edge costs
$c:E\rightarrow\mathbb{R}_{+}$. For disjoint non-empty subsets $S,T\subseteq
V$, there can be multiple minimum $(S,T)$-terminal cuts. We will be interested
in _source minimal_ minimum $(S,T)$-terminal cuts. A minimum $(S,T)$-terminal
cut $(U,\overline{U})$ is a source minimal minimum $(S,T)$-terminal cut if for
all minimum $(S,T)$-terminal cut $(X,\overline{X})$, we have $U\subseteq X$.
For every pair of disjoint non-empty subsets $S$ and $T$ of vertices, there
exists a unique source minimal minimum $(S,T)$-terminal cut (this is due to
submodularity—e.g., see [9]).
We recall that the graph cut function $d:2^{V}\rightarrow\mathbb{R}$ is
symmetric and submodular, i.e., $d(A)=d(V\setminus A)$ and $d(A)+d(B)\geq
d(A\cap B)+d(A\cup B)$ for all $A,B\subseteq V$. We will need an uncrossing
result that relies on more careful counting of edges than simply employing the
submodularity inequality. We begin with some notation that will help in such
careful counting. Let $(Y_{1},\ldots,Y_{p},W,Z)$ be a partition of $V$ into
$p+2$ non-empty parts. We define
$\displaystyle\sigma(Y_{1},\ldots,Y_{p},W,Z)$
$\displaystyle:=2\left(\sum_{uv\in E:\ u\in Y_{i},\ v\in Y_{j}\text{ for
distinct }i,\ j\in[p]}c(uv)+\sum_{uv\in E:\ u\in W,\ v\in Z}c(uv)\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{uv\in E:\
u\in\cup_{i\in[p]}Y_{i},\ v\in W\cup Z}c(uv)$
We note that
$\displaystyle\sum_{i\in[p]}d(Y_{i})$
$\displaystyle\leq\sigma(Y_{1},\ldots,Y_{p},W,Z).$ (1)
The following lemma uncrosses a collection of $p$ sets to obtain a partition
of the vertex set into $p+2$ non-empty parts with small $\sigma$-value. The
lemma was shown by Kamidoi, Yoshida, and Nagamochi in the context of minimum
$k$-cuts [10]. It was also extended to hypergraphs by Chandrasekaran and
Chekuri [7]. See Figure 2 for an illustration of the sets that appear in the
statement of the lemma.
###### Lemma 1.1.
[10, 7] Let $G=(V,E)$ be a graph with positive edge costs and $\emptyset\neq
R\subsetneq U\subsetneq V$. Let $S=\\{u_{1},\ldots,u_{p}\\}\subseteq
U\setminus R$ for $p\geq 2$. For each $i\in[p]$, let
$(\overline{A_{i}},A_{i})$ be a minimum $((S\cup
R)\setminus\\{u_{i}\\},\overline{U})$-terminal cut. Suppose that $u_{i}\in
A_{i}\setminus(\cup_{j\in[p]\setminus\\{i\\}}A_{j})$ for every $i\in[p]$. Let
$Z:=\cap_{i=1}^{p}\overline{A_{i}},\ W:=\cup_{1\leq i<j\leq p}(A_{i}\cap
A_{j}),\ \text{and}\ Y_{i}:=A_{i}-W\ \forall i\in[p].$
Then, $(Y_{1},\ldots,Y_{p},W,Z)$ is a partition of $V$ into $p+2$ non-empty
parts with
$\sigma(Y_{1},\ldots,Y_{p},W,Z)\leq\min\\{d(A_{i})+d(A_{j}):i,j\in[p],i\neq
j\\}.$
Figure 2: Illustration of the sets that appear in the statement of Lemma 1.1.
We briefly remark on the lemma: known proofs of the lemma are by induction on
$p$ with an uncrossing argument that involves careful edge counting. The use
of $\sigma(Y_{1},\ldots,Y_{p},W,Z)$ instead of $\sum_{i\in[p]}d(Y_{i})$ on the
LHS is crucial for the induction argument—this is why we prefer to state the
lemma as giving an upper bound on $\sigma(Y_{1},\ldots,Y_{p},W,Z)$ although we
will use it only to conclude an upper bound on $\sum_{i\in[p]}d(Y_{i})$ via
inequality (1).
## 2 Proof of Theorem 1.1
We will derive Theorem 1.1 from the following theorem.
###### Theorem 2.1.
Let $G=(V,E)$ be a connected graph with positive edge costs and
$(U,\overline{U})$ be an $\alpha$-approximate minimum cut for some $\alpha\geq
1$. Then, there exists a subset $S\subseteq U$ with $|S|\leq\lfloor
2\alpha\rfloor+1$ such that $(U,\overline{U})$ is the unique minimum
$(S,\overline{U})$-terminal cut.
###### Proof.
Consider the collection $\mathcal{C}:=\\{Q\subsetneq V:\overline{U}\subsetneq
Q,d(Q)\leq d(U)\\}$. Suppose $\mathcal{C}$ is empty. Let $S=\\{x\\}$ for some
arbitrary $x\in U$ and let $(X,\overline{X})$ be a minimum
$(S,\overline{U})$-terminal cut. Then,
$\overline{U}\subseteq\overline{X}\subsetneq V$ and $d(\overline{X})\leq
d(U)$. Since $\overline{X}\not\in\mathcal{C}$ (because $\mathcal{C}$ is
empty), we must have $\overline{X}=\overline{U}$. Therefore,
$(U,\overline{U})$ is the unique minimum $(S,\overline{U})$-terminal cut.
Next, suppose $\mathcal{C}\neq\emptyset$. Let $S\subseteq U$ be a minimal
transversal of the collection $\mathcal{C}$, i.e., $S\subseteq U$ is a minimal
set with $S\cap Q\neq\emptyset$ for all $Q\in\mathcal{C}$. Proposition 2.1 and
Lemma 2.1 complete the proof of Theorem 2.1. ∎
###### Proposition 2.1.
$(U,\overline{U})$ is the unique minimum $(S,\overline{U})$-terminal cut.
###### Proof.
Suppose $(X,\overline{X})$ be the source minimal minimum
$(S,\overline{U})$-terminal cut with $X\neq U$. Since $(U,\overline{U})$ is a
$(S,\overline{U})$-terminal cut, we have $d(\overline{X})=d(X)\leq d(U)$.
Moreover, we have $\emptyset\neq X\subsetneq U$, which implies that
$\overline{U}\subsetneq\overline{X}\subsetneq V$. Hence, we have
$\overline{X}\in\mathcal{C}$, and thus $S\cap\overline{X}\neq\emptyset$. This
contradicts with the assumption that $(X,\overline{X})$ is a
$(S,\overline{U})$-terminal cut. Therefore, we must have $X=U$. ∎
###### Lemma 2.1.
$|S|\leq\lfloor 2\alpha\rfloor+1$.
###### Proof.
For the sake of contradiction, suppose that $|S|\geq\lfloor 2\alpha\rfloor+2$.
Let $\lambda$ denote the value of a minimum cut in $G$. Our proof strategy is
to arrive at a cut with value cheaper than $\lambda$, thus contradicting the
definition of $\lambda$. For convenience, we will denote $p:=|S|$ and write
$S=\\{u_{1},\ldots,u_{p}\\}$. For each $i\in[p]$, let
$(\overline{A_{i}},A_{i})$ be the source minimal minimum
$(S-u_{i},\overline{U})$-terminal cut.
###### Claim 2.1.
For every $i\in[p]$, we have $d(A_{i})\leq d(U)$ and $u_{i}\in A_{i}$.
###### Proof.
Let $i\in[p]$. Since $S$ is a minimal transversal of the collection
$\mathcal{C}$, there exists a set $B_{i}\in\mathcal{C}$ such that $B_{i}\cap
S=\\{u_{i}\\}$. Hence, $(\overline{B_{i}},B_{i})$ is a
$(S-u_{i},\overline{U})$-terminal cut. Therefore,
$d(A_{i})\leq d(B_{i})\leq d(U).$
We will show that $A_{i}$ is in the collection $\mathcal{C}$. By definition,
$A_{i}\subseteq V\setminus(S-u_{i})\subsetneq V$ and $\overline{U}\subseteq
A_{i}$. If $A_{i}=\overline{U}$, then the above inequalities are equations
(since $d(A_{i})=d(\overline{U})=d(U)$) implying that
$(\overline{B_{i}},B_{i})$ is a minimum $(S-u_{i},\overline{U})$-terminal cut,
and consequently, $(\overline{B_{i}},B_{i})$ contradicts source minimality of
the minimum $(S-u_{i},\overline{U})$-terminal cut $(\overline{A_{i}},A_{i})$.
Therefore, $\overline{U}\subsetneq A_{i}$. Hence, $A_{i}$ is in the collection
$\mathcal{C}$.
We recall that the set $S$ is a transversal for the collection $\mathcal{C}$
and moreover, none of the elements of $S-u_{i}$ are in $A_{i}$ by definition
of $A_{i}$. Therefore, the vertex $u_{i}$ must be in $A_{i}$. ∎
We recall our assumption that $p\geq\lfloor 2\alpha\rfloor+2$. We note that
$p-1>2\alpha\geq 2$. Using Claim 2.1, we observe that the sets $U$,
$R:=\\{u_{p}\\}$, $S=\\{u_{1},\ldots,u_{p-1}\\}$, and partitions
$(\overline{A_{i}},A_{i})$ for $i\in[p-1]$ satisfy the conditions of Lemma
1.1. Therefore, applying Lemma 1.1 shows the existence of a partition
$(Y_{1},\ldots,Y_{p-1},W,Z)$ of $V$ into $p+1$ non-empty parts such that
$\displaystyle\sigma(Y_{1},\ldots,Y_{p-1},W,Z)$
$\displaystyle\leq\min\\{d(A_{i})+d(A_{j}):i,j\in[p-1],i\neq j\\}\leq
2d(U)\leq 2\alpha\lambda.$
By inequality (1), this implies that
$\displaystyle\sum_{i\in[p-1]}d(Y_{i})\leq 2\alpha\lambda.$
Therefore, there exists $i\in[p-1]$ such that
$\displaystyle d(Y_{i})\leq\frac{2\alpha\lambda}{p-1}<\lambda.$
The last inequality is because $p-1>2\alpha$ and $\lambda>0$ since the graph
is connected. Thus, we have a cut $(Y_{i},\overline{Y_{i}})$ with cut value
smaller than $\lambda$, a contradiction to the definition of $\lambda$. ∎
Applying Theorem 2.1 to $(\overline{U},U)$ yields the following corollary.
###### Corollary 2.1.
Let $G=(V,E)$ be a connected graph with positive edge costs and
$(U,\overline{U})$ be an $\alpha$-approximate minimum cut for some $\alpha\geq
1$. Then, there exists a subset $T\subseteq\overline{U}$ with $|T|\leq\lfloor
2\alpha\rfloor+1$ such that $(U,\overline{U})$ is the unique minimum
$(U,T)$-terminal cut.
We now restate Theorem 1.1 and prove it using Theorem 2.1 and Corollary 2.1.
See 1.1
###### Proof.
By Theorem 2.1, there exists a subset $S\subseteq U$ with $|S|\leq\lfloor
2\alpha\rfloor+1$ such that $(U,\overline{U})$ is the unique minimum
$(S,\overline{U})$-terminal cut. By Corollary 2.1, there exists a subset
$T\subseteq\overline{U}$ with $|T|\leq\lfloor 2\alpha\rfloor+1$ such that
$(U,\overline{U})$ is the unique minimum $(U,T)$-terminal cut. For these
choices of subsets $S$ and $T$, we now show that $(U,\overline{U})$ is the
unique minimum $(S,T)$-terminal cut.
Let $(Y,\overline{Y})$ be an arbitrary minimum $(S,T)$-terminal cut. It
suffices to show that $Y=U$. Since $(U,\overline{U})$ is a $(S,T)$-terminal
cut, we have that
$d(Y)\leq d(U).$
Since $(Y\cap U,\overline{Y\cap U})$ is a $(S,\overline{U})$-terminal cut and
$(U,\overline{U})$ is a minimum $(S,\overline{U})$-terminal cut, we have that
$d(U)\leq d(Y\cap U).$
Since $(Y\cup U,\overline{Y\cup U})$ is a $(U,T)$-terminal cut and
$(U,\overline{U})$ is a minimum $(U,T)$-terminal cut, we have that
$d(U)\leq d(Y\cup U).$
Using the above inequalities in conjunction with the submodularity of the cut
function, we obtain that
$2d(U)\leq d(U\cap Y)+d(U\cup Y)\leq d(U)+d(Y)\leq 2d(U).$
Hence, all of the above inequalities should be equations. Consequently,
$(Y\cap U,\overline{Y\cap U})$ is a minimum $(S,\overline{U})$-terminal cut,
and $(Y\cup U,\overline{Y\cup U})$ is a minimum $(U,T)$-terminal cut. This
implies that $Y\cap U=U$ since $(U,\overline{U})$ is the unique minimum
$(S,\overline{U})$-terminal cut and $Y\cup U=U$ since $(U,\overline{U})$ is
the unique minimum $(U,T)$-terminal cut. Thus, we have that $Y=U$. ∎
## 3 An Open Problem
The fundamental structural result that the number of $\alpha$-approximate
minimum cuts in a $n$-vertex connected graph is at most $n^{O(\alpha)}$ has
been extended to $r$-rank hypergraphs (the _rank_ of a hypergraph is the size
of the largest hyperedge): Kogan and Krauthgamer [14] used the random
contraction technique to show that the number of $\alpha$-approximate minimum
cuts in a $r$-rank $n$-vertex connected hypergraph is $O(2^{\alpha
r}n^{2\alpha})$ and they can all be enumerated in randomized polynomial time
for constant $\alpha$ and $r$. It would be interesting to give a
_deterministic_ proof of this bound.
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|
# Extreme nature of four blue-excess dust-obscured galaxies revealed by
optical spectroscopy
Akatoki Noboriguchi School of General Education, Shinshu University, 3-1-1
Asahi, Matsumoto, Nagano 390-8621, Japan Frontier Research Institute for
Interdisciplinary Sciences, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai,
Miyagi 980-8578, Japan Graduate School of Science and Engineering, Ehime
University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan Tohru Nagao
Research Center for Space and Cosmic Evolution, Ehime University, 2-5 Bunkyo-
cho, Matsuyama, Ehime 790-8577, Japan Yoshiki Toba National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Department
of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Kyoto 606-8502,
Japan Academia Sinica Institute of Astronomy and Astrophysics, 11F of
Astronomy-Mathematics Building, AS/NTU, No.1, Section 4, Roosevelt Road,
Taipei 10617, Taiwan Research Center for Space and Cosmic Evolution, Ehime
University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan Kohei Ichikawa
Frontier Research Institute for Interdisciplinary Sciences, Tohoku University,
6-3 Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan Masaru Kajisawa Graduate
School of Science and Engineering, Ehime University, 2-5 Bunkyo-cho,
Matsuyama, Ehime 790-8577, Japan Nanako Kato Graduate School of Science and
Engineering, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577,
Japan Toshihiro Kawaguchi Department of Economics, Management and Information
Science Onomichi City University, Hisayamada 1600-2, Onomichi, Hiroshima
722-8506, Japan Hideo Matsuhara Institute of Space and Astronautical
Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku,
Sagamihara, Kanagawa 252-5210, Japan Department of Space and Astronautical
Science, The Graduate University for Advanced Studies, SOKENDAI, 3-1-1
Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan Yoshiki Matsuoka
Research Center for Space and Cosmic Evolution, Ehime University, 2-5 Bunkyo-
cho, Matsuyama, Ehime 790-8577, Japan Kyoko Onishi Department of Space, Earth
and Environment, Chalmers University of Technology, Onsala Observatory, SE-439
92 Onsala, Sweden Masafusa Onoue Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Kavli Institute for Astronomy and
Astrophysics, Peking University, Beijing 100871, China Max-Planck-Institut
für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany Nozomu Tamada
Graduate School of Science and Engineering, Ehime University, 2-5 Bunkyo-cho,
Matsuyama, Ehime 790-8577, Japan Koki Terao Subaru Telescope, National
Astronomical Observatory of Japan, 650 North A’ohoku Place, Hilo, HI 96720,
USA Astronomical Institute, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai,
Miyagi 980-8578, Japan Yuichi Terashima Graduate School of Science and
Engineering, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577,
Japan Research Center for Space and Cosmic Evolution, Ehime University, 2-5
Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan Yoshihiro Ueda Department of
Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Kyoto 606-8502, Japan
Takuji Yamashita National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan
###### Abstract
We report optical spectroscopic observations of four blue-excess dust-obscured
galaxies (BluDOGs) identified by Subaru Hyper Suprime-Cam. BluDOGs are a sub-
class of dust-obscured galaxies (DOGs, defined with the extremely red color
$(i-[22])_{\rm AB}\geq 7.0$; Toba et al. 2015), showing a significant flux
excess in the optical $g$\- and $r$-bands over the power-law fits to the
fluxes at the longer wavelengths. Noboriguchi et al. (2019) has suggested that
BluDOGs may correspond to the blowing-out phase involved in a gas-rich major
merger scenario. However the detailed properties of BluDOGs are not understood
because of the lack of spectroscopic information. In this work, we carry out
deep optical spectroscopic observations of four BluDOGs using Subaru/FOCAS and
VLT/FORS2. The obtained spectra show broad emission lines with extremely large
equivalent widths, and a blue wing in the C iv line profile. The redshifts are
between 2.2 and 3.3. The averaged rest-frame equivalent widths of the C iv
lines are $160\pm 33$ Å, $\sim$7 times higher than the average of a typical
type-1 quasar. The FWHMs of their velocity profiles are between 1990 and 4470
${\rm km\ s^{-1}}$, and their asymmetric parameters are 0.05 and 0.25. Such
strong C iv lines significantly affect the broad-band magnitudes, which is
partly the origin of the blue excess seen in the spectral energy distribution
of BluDOGs. Their estimated supermassive black hole masses are $1.1\times
10^{8}<M_{\rm BH}/M_{\odot}<5.5\times 10^{8}$. The inferred Eddington ratios
of the BluDOGs are higher than 1 ($1.1<\lambda_{\rm Edd}<3.8$), suggesting
that the BluDOGs are in a rapidly evolving phase of supermassive black holes.
galaxies: active — galaxies: evolution — infrared: galaxies — quasars: general
— techniques: spectroscopic
††software: Astropy, Astro-SCRAPPY, X-CIGALE, Numpy,
Scipy.optimaize.curve_fit, Recipe flexible execution workbench
(Reflex)††facilities: Subaru (FOCAS, HSC), VLT (FORS2), VISTA, WISE, and
Herschel (PACS, SPIRE)
## 1 Introduction
In the last two decades, observations of low-redshift galaxies have revealed
tight correlations between the mass of supermassive black holes (SMBHs) and
the host galaxy properties such as bulge mass (e.g., Magorrian et al. 1998;
Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Marconi
& Hunt 2003; Kormendy & Ho 2013; Ding et al. 2020). Such scaling relations
suggest the so-called co-evolution between galaxies and SMBHs. It has been
argued that a major merger of gas-rich galaxies triggers active star-forming
activity and subsequent mass accretion onto SMBHs (e.g., Sanders et al. 1988;
Hopkins et al. 2008; Treister et al. 2012; Goulding et al. 2018). In this
scenario, the merging two galaxies first evolve into a dusty star-forming (SF)
galaxy. Then it evolves into a dusty active galactic nucleus (AGN) as gas
accretion to the nuclear region triggers the activity of SMBHs. Finally, a
dusty AGN evolves into an optically-thin quasar after the surrounding dust is
blown out by the powerful AGN outflow. The most active period of such SF and
AGN activity is generally heavily obscured by dust, which prevents us from
investigating these phases observationally.
By combining optical, near-infrared (NIR), and mid-infrared (MIR) catalogs
obtained from the Subaru Hyper Suprime-Cam (HSC; Miyazaki et al. 2018)-Subaru
Strategic Program (SSP; Aihara et al. 2018), the VISTA Kilo-degree Infrared
Galaxy survey (VIKING; Arnaboldi et al. 2007), and the Wide-field Infrared
Survey Explorer (WISE; Wright et al. 2010) all-sky survey (ALLWISE; Cutri
2014), Toba et al. (2015, 2017b) and Noboriguchi et al. (2019) selected dusty
SF galaxies and/or powerful AGNs as dust-obscured galaxies (DOGs; Dey et al.
2008; Fiore et al. 2008; Bussmann et al. 2009; Desai et al. 2009; Bussmann et
al. 2011). DOGs are defined with a very red optical-MIR color ($(i-[22])_{\rm
AB}\geq 7.0$; Toba et al. 2015). DOGs represent a transition phase from a gas-
rich major merger to an optically-thin quasar in the gas-rich major merger
scenario (Dey et al. 2008), suggesting that some DOGs are expected to have
buried AGNs. Recently, eight blue-excess DOGs (BluDOGs; Noboriguchi et al.
2019) were discovered from the HSC-selected DOGs based on their optical
spectral slopes (i.e., $\alpha_{\rm opt}<0.4$, where $\alpha_{\rm opt}$ is the
observed-frame optical spectral index for the HSC $g$-, $r$-, $i$-, $z$-, and
$y$-bands in the power-law fit, $f_{\nu}\propto\lambda^{\alpha_{\rm opt}}$),
and are a very rare population (eight BluDOGs out of 571 HSC-selected DOGs).
Noboriguchi et al. (2019) suggested that the BluDOGs with such blue excess may
be in the blowing-out phase involved in the gas-rich major merger scenario.
However, the detailed properties of BluDOGs are not well understood because of
the lack of spectroscopic information. Spectroscopic observations will give us
accurate redshifts, and thus reliable AGN luminosities as a measure of the
accretion rates, as well as the SMBH masses.
Another interesting population that may represent the transition phase between
optically-thick AGNs and optically-thin quasars is extremely red quasars
(ERQs; e.g., Ross et al. 2015; Hamann et al. 2017; Perrotta et al. 2019;
Villar Martín et al. 2020). ERQs were identified by combining the optical
photometric data of Sloan Digital Sky Survey (SDSS; York et al. 2000), the
optical spectroscopic data from SDSS-III (Eisenstein et al. 2011) Baryon
Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013), and the MIR
photometric data of WISE catalog. They are also defined with very red optical
to MIR colors ($F_{\nu}(24{\rm\mu m})/F_{\nu}(R)\geq 1000$), and their spectra
show broad emission lines with extremely large equivalent widths (Ross et al.
2015; Hamann et al. 2017). Hamann et al. (2017) refined the definition of ERQs
as $(i-[12])_{\rm AB}>4.6$111All of the BluDOGs also satisfy the criterion of
the ERQ (see Table 1)., and reported notable blue-wing features in their C iv
profiles, which suggests the presence of powerful outflow. However, the ERQ
sample is limited to optically bright objects since their selection requires
SDSS spectra. Detailed studies of optically-faint populations in the
transition phase between the optically-thick and optically-thin stages are
required to understand the whole scenario of the merger-driven evolution of
SMBHs. Therefore, it is important to execute the spectroscopic observations
for BluDOGs and to research their spectroscopic properties.
In this work, we present the results of spectroscopic observations and
subsequent analyses of four BluDOGs. This paper is organized as follows. We
describe sample selection of our targets and observations in Section 2. In
Section 3, we present properties of the detected emission lines, the estimated
dust extinctions, bolometric luminosities of an AGN ($L^{\rm AGN}_{\rm bol}$),
and SMBH masses ($M_{\rm BH}$). The discussion on the large equivalent widths
of the C iv emission, their SMBH mass, and Eddington ratios is given in
Section 4. Then we give a brief summary in Section 5. Throughout this paper,
the adopted cosmology is a flat universe with $H_{0}=70\ {\rm km\ s^{-1}\
Mpc^{-1}}$, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$. Unless otherwise
noted, all magnitudes refer to the AB system.
Table 1: Photometric data of BluDOGs Name | HSC $r$-band | HSC $i$-band | WISE $W3$-band | WISE $W4$-band
---|---|---|---|---
| [AB mag] | [AB mag] | [AB mag] | [AB mag]
HSC J090705.64$+$020955.8 (HSC J0907) | 22.56$\pm$0.01 | 22.59$\pm$0.01 | 16.06$\pm$0.13 | 14.89$\pm$0.34
HSC J120200.84$-$011846.4 (HSC J1202) | 20.92$\pm$0.00 | 20.87$\pm$0.00 | 14.47$\pm$0.04 | 13.46$\pm$0.10
HSC J120728.71$-$005808.4 (HSC J1207) | 22.12$\pm$0.01 | 22.31$\pm$0.01 | 16.28$\pm$0.16 | 15.01$\pm$0.36
HSC J141435.21$+$003547.4 | 23.32$\pm$0.02 | 23.11$\pm$0.02 | 17.24aaThe magnitude is a 95% confidence upper limit.
https://wise2.ipac.caltech.edu/docs/release/allwise/expsup/sec2_1a.html | 15.33$\pm$0.33
HSC J143727.40$-$011726.5 | 23.17$\pm$0.02 | 23.10$\pm$0.01 | 16.94$\pm$0.23 | 15.37$\pm$0.31
HSC J144333.84$-$000830.3 (HSC J1443) | 22.34$\pm$0.01 | 22.24$\pm$0.01 | 16.14$\pm$0.10 | 15.04$\pm$0.23
HSC J144813.65$+$002244.3 | 23.55$\pm$0.02 | 23.43$\pm$0.02 | 16.83$\pm$0.16 | 15.37$\pm$0.34
HSC J144900.84$+$002350.2 | 23.95$\pm$0.03 | 23.74$\pm$0.02 | 17.15$\pm$0.22 | 15.47$\pm$0.36
Table 2: Observation log Name | Exp. time [s] | Date | Standard star | Instrument
---|---|---|---|---
HSC J0907 | 900$\times$2 | 2019 October 8 | G191-B2B | FOCAS (Subaru)
| 600$\times$1 | | |
HSC J1202 | 900$\times$6 | 2019 February 27 | LTT 6248 | FORS2 (VLT)
HSC J1207 | 900$\times$12 | 2019 March 1, 2, 6 | LTT 4816 | FORS2 (VLT)
HSC J1443 | 900$\times$12 | 2019 March 7, 8 | LTT 4816, EG 274 | FORS2 (VLT)
## 2 Sample and the data
### 2.1 Sample selection
In Noboriguchi et al. (2019), 571 DOGs were selected by combining $\sim$105
deg2 imaging data obtained from the survey of HSC-SSP222 We utilize the
photometric data of S16A HSC-SSP, which was released internally within the HSC
survey team and is based on data obtained from 2014 March to 2016 April. ($g$,
$r$, $i$, $z$, and $y$), VIKING ($Z$, $Y$, $J$, $H$, and $Ks$), and ALLWISE
($W1$, $W2$, $W3$, and $W4$). The eight BluDOGs were defined among the DOG
sample with the smallest observed-frame optical slope ($\alpha_{\rm
opt}\textless 0.4$, where $\alpha_{\rm opt}$ is the observed-frame optical
spectral index of the power-law fits to the HSC $g$, $r$, $i$, $z$, and
$y$-band fluxes, $f_{\nu}\propto\lambda^{\alpha_{\rm opt}}$). We selected the
four brightest BluDOGs ($r_{\rm AB}<23$: see Table 1) as the targets of our
spectroscopic observations presented in this paper.
### 2.2 Spectroscopic observations and data reductions
We executed the observations by using Faint Object Camera and Spectrograph
(FOCAS; Kashikawa et al. 2002) installed on the Subaru Telescope of National
Astronomical Observatory of Japan, and FORS2 (Appenzeller et al. 1998)
installed on Very Large Telescope (VLT-UT1) of European Southern Observatory
(ESO). We present the observation log in Table 2.
#### 2.2.1 Subaru FOCAS
By using FOCAS, we observed HSC J090705.64$+$020955.8 (hereafter J0907) on
October 8th in 2019, with airmass $\sim$1.76 and seeing $\sim$0.5 arcsec. We
used the 300B grism and the SY47 filter to cover $\lambda_{\rm
obs}\sim$4700–9200 Å, with the resultant spectral resolution of $R\sim$800 for
the used 0″.8-width slit. To reduce the obtained data, we performed bias
correction, flat fielding with dome flat, removal of cosmic-rays, spectral
extraction, sky subtraction, wavelength calibration, and flux calibration with
a standard star (G191-B2B) using the Python packages of `Astropy` and `Numpy`.
For removing cosmic-rays, we utilized `Astro-SCRAPPY` (McCully & Tewes 2019).
`Astro-SCRAPPY` is based on the algorithm of `L.A.Cosmic`, which removes
cosmic-rays based on a variation of Laplacian edge detection (van Dokkum
2001). The final spectrum is an inverse-variance weighted mean of the
individual shots, corrected for the Galactic extinction (Schlegel et al.
1998).
#### 2.2.2 VLT FORS2
By using FORS2, we observed HSC J120200.84$-$011846.4, HSC
J120728.71$-$005808.4, and HSC J144333.84$-$000830.3 (hereafter J1202, J1207,
and J1443, respectively) between February 27th and March 8th, 2019. We used
the GRISM_600RI$+19$ and the GG435 filter to cover $\lambda_{\rm
obs}\sim$5200–8000 Å, which results in the spectral resolution of $R\sim$1500
with 0″.7-width slit. The typical airmasses of the observations for J1202,
J1207, and J1443 were 1.17, 1.24, and 1.12, and the typical seeing sizes were
$\sim$1.0, 0.5, and 0.5 arcsec, respectively. For the data reduction, we
utilized the `Recipe flexible execution workbench` (`Reflex`; Freudling et al.
2013) software. `Reflex` performed bias correction, flat fielding with dome
flat, sky subtraction, removing cosmic-rays, spectral extraction, wavelength
calibration, and flux calibration with a standard star (LTT 6248, LTT 4816,
and EG 274). The final spectrum of each target is the inverse-variance
weighted mean of the individual shots, corrected for the Galactic extinction.
Figure 1: The reduced spectra of the BluDOGs. The spectra are for J0907,
J1202, J1207 and J1443 from the top to bottom. Detected lines are marked by
arrows and labels.
#### 2.2.3 Spectrophotometric re-calibration
We re-calibrated the reduced spectra to match the HSC photometry, in order to
correct for the effects of the slit loss of the flux, systematic errors in the
photometric and spectroscopic calibrations, and any other possible systematic
errors. In our observations, the spectra cover the wavelength range of the HSC
$r$-band. We calculate the calibration factor, $f_{\rm photo\\_calib}=F_{{\rm
photo}\\_r}/F_{{\rm spec}\\_r}$, where $F_{{\rm photo}\\_r}$ and $F_{{\rm
spec}\\_r}$ are the photometric and spectroscopic fluxes in the HSC $r$-band.
The derived calibration factors of J0907, J1202, J1207, and J1443 are 0.97,
1.50, 1.40, and 1.36, respectively. We multiply the spectra with the derived
calibration factors.
## 3 Results
Figure 2: Spectral fits to the C iv emission lines of the BluDOGs. The top
left, top right, bottom left, and bottom right show the C iv emission lines of
J0907, J1202, J1207, and J1443, respectively. The green, magenta, red, blue,
and black lines represent the observed spectrum, linear fit to the continuum
emission, two Gaussians for the red and blue components, and best-fit model,
respectively. The orange line on the J1202 and J1443 panels represents the C
iv doublet absorption line. The horizontal black bars denote the wavelength
range used to fit the continuum emission. In each panel, the lower part
presents the residual of the best-fit, with the same flux scale as in the
upper part.
### 3.1 Emission-line measurements
Figure 1 shows the reduced spectra of the four BluDOGs. In order to measure
the emission-line properties, we divide emission lines into six groups as
follows; (1) Ly$\alpha$1216, N v$\lambda$1240, and Si ii$\lambda$1263, (2) Si
iv$\lambda$1397 and O iv]$\lambda$1402, (3) He ii$\lambda$1640 and O
iii]$\lambda$1663, and (4) Al iii$\lambda$1857, Si iii]$\lambda$1892, and C
iii]$\lambda$1909, (5) C iv 1549, and (6) Mg ii. We fit the emission lines in
each group simultaneously, with a linear continuum model subtracted from the
observed spectrum. We adopt a single-Gaussian profile for Ly ${\alpha}$, N v,
Si ii, Si iv, O iv], O iii], Al iii, Si iii and Mg ii. The C iii] of J1202 is
fitted with a single-Gaussian profile, while those of J0907 and J1207 are
fitted with a double-Gaussian profile. For the fit around the Si iv and O iv]
of J1202, we add an additional Gaussian profile to reproduce the observed
broad component. We fit C iv and He ii with double-Gaussian profiles, and
denote the blue and red components with the suffixes of “_B” and “_R”,
respectively. Additionally, we fit the doublet absorption lines observed
around the C iv emission lines of J1202 and J1443. The C iv absorption lines
observed at $\lambda_{\rm obs}$ = 5916.8 Å and 5926.6 Å for J1202 and those at
$\lambda_{\rm obs}$ = 6678.8 Å and 6689.9 Å for J1443 are fitted using the
Voigt profile, respectively. The doublet absorption line ratio is fixed as 2:1
(Feibelman, 1983). The best values and standard deviations for emission and
absorption lines parameters are estimated by using
scipy.optimize.curve_fit333https://docs.scipy.org/doc/scipy/reference/, while
we calculate full width at half maximum (FWHM) of emission lines with double
Gaussian by using a Monte Carlo method. For this Monte Carlo simulation, we
created 10,000 mock spectra using the noise arrays of the observed spectra,
and calculate the mean and standard deviation of the line properties. The
results for emission lines are listed in Tables 3–6. For absorption lines, the
observed-frame equivalent widths and redshifts of the doublet absorption lines
on the J1202 C iv emission line are 47.7$\pm$19.6 Å, 23.6$\pm$9.7 Å, and
2.822, respectively, while the observed-frame equivalent widths and redshift
of the doublet absorption lines on J1443 C iv emission line are 14.0$\pm$6.3
Å, 6.94$\pm$3.14 Å, and 3.314, respectively. Therefore, the co-moving distance
between J1202 and its C iv absorber is 8.73 Mpc, while that between J1443 and
its C iv absorber is 2.79 Mpc.
The flux ratios of N v/Ly$\alpha$ and N v/C iv for J1443 are 3.9 and 1.8,
respectively, whereas the values for the typical quasar (Vanden Berk et al.,
2001) are 0.02 and 0.10. One possible reason of these unusual flux ratios in
J1443 is the presence of absorption lines, which absorb most of the Ly$\alpha$
and the C iv fluxes around the peak. The unusual flux ratios cannot be
explained by the dust reddening, given too small wavelength separations among
emission lines of Ly$\alpha$, N v, and C iv.
Figure 2 showcases the best-fit models to the C iv emission lines in the four
BluDOGs. We adopt the C iv redshift taking C iv_R $+$ C iv_B into account as
the systemic redshift of the targets. The determined systemic redshifts of
J0907, J1202, J1207, and J1443 are 2.258$\pm$0.002, 2.830$\pm$0.002,
2.511$\pm$0.001, and 3.317$\pm$0.006, respectively.
### 3.2 Emission-line contributions to the HSC $g$\- and $r$-band magnitudes
Figure 1 suggests the very large equivalent width (EW) of the emission lines.
The average rest-frame EW (REW) of the C iv line of the four BluDOGs is
$160\pm 33$ Å, $\sim$7 times higher than the average of SDSS type-1 quasars
($23.8\pm 0.1$ Å; Vanden Berk et al. 2001). Here we investigate the effect of
the large REWs on the HSC $g$\- and $r$-band magnitudes.
First, we calculate the expected magnitudes at the $g$\- and $r$-bands from an
extrapolation of the power-law fit to the longer wavelength bands ($i$, $z$,
$y$, $Z$, $Y$, $J$, $H$, $Ks$, $W1$, $W2$, $W3$, and $W4$). Figure 3 clearly
shows that the observed $g$\- and $r$-band magnitudes exceed the extrapolation
of the power-law fit. The excesses of the $g$-band magnitudes for J0907,
J1202, J1207, and J1443 are 1.27, 1.13, 1.48, and 0.88 mag, and those for the
$r$-band excesses are 0.47, 0.44, 0.65, and 0.40 mag, respectively.
Furthermore, we estimate the effect of the strong emission lines, based on
their observed-frame EWs and the band widths (BW) of the HSC $g$ and
$r$-bands. The BWs of the HSC $g$ and $r$-bands are 1468 and 1508 Å
(Kawanomoto et al., 2018), respectively. By taking all of the emission lines
(Tables 3 – 6) covered by the HSC $g$-band (4000–5500Å) and $r$-band
(5500–7000Å) into account, the total observed-frame EWs for J0907, J1202,
J1207, and J1443 in the $g$-band are 604, 258, 608, and 1380 Å respectively,
while those in the $r$-band are 129, 836, 329, and 721 Å (Table 7). Note that
the total observed-frame EWs in the $g$-band are lower limits, because our
optical spectra do not cover the entire wavelength range of the band (Section
2.2) and thus some emission lines are not taken into account in the derived
total observed-frame EWs. Especially Ly$\alpha$, the strongest emission line
in the rest-frame UV spectrum of typical AGNs, is not covered in our spectra
of J0907, J1202, and J1207, thus the total observed-frame EWs for these 3
objects are largely underestimated444The Ly$\alpha$ line of J0907 is at the
shorter edge of the HSC $g$-band coverage but the flux contribution to the
$g$-band magnitude is likely to be significant owing to its broad nature..
Since the magnitude excess by the emission lines is given by $\Delta{\rm
mag}=2.5\log{(1+EW/BW)}$, the estimated effects in the $g$-band for J0907,
J1202, J1207, and J1443 are 0.37, 0.18, 0.38, and 0.72 mag, respectively.
Similarly, the estimated effects of emission lines to the $r$-band magnitudes
are 0.09, 0.48, 0.21, and 0.42 mag, respectively (see Table 7 for a summary).
We will discuss the implication from these estimates in Section 4.2.
Table 3: The detected lines of J0907
Line name | $\lambda_{\rm rest}$ [Å] | $z_{\rm line}$ | $FWHM_{\rm rest}$ [Å] | $F_{\rm line}$ [${\rm erg\ s^{-1}\ cm^{-2}}$] | $EW_{\rm rest}$ [Å] | $v_{\rm width}$ [${\rm km\ s^{-1}}$]
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
C iv_R | 1549.5 | 2.258$\pm$0.001 | 12.8$\pm$0.7 | (1.10$\pm$0.07)E$-$15 | 118$\pm$10 | 2470$\pm$130
C iv_B | 1549.5 | 2.227$\pm$0.004 | 13.8$\pm$2.5 | (2.87$\pm$0.60)E$-$16 | 31.5$\pm$6.7 | 2670$\pm$490
C iv_R + C iv_B | 1549.5 | 2.258$\pm$0.002 | 15.2$\pm$0.8 | (1.39$\pm$0.09)E$-$15 | 148$\pm$12 | 2940$\pm$150
He ii_R | 1640.4 | 2.260$\pm$0.002 | 3.63$\pm$2.32 | (3.50$\pm$2.36)E$-$17 | 4.43$\pm$3.01 | 663$\pm$425
He ii_B | 1640.4 | 2.235$\pm$0.005 | 38.1$\pm$5.9 | (2.13$\pm$0.48)E$-$16 | 26.8$\pm$6.3 | 6960$\pm$1080
He ii_R + He ii_B | 1640.4 | 2.260$\pm$0.002 | 5.54$\pm$1.29 | (2.48$\pm$0.53)E$-$16 | 31.4$\pm$7.0 | 1010$\pm$240
O iii] | 1663.5 | 2.264$\pm$0.001 | 3.29$\pm$1.69 | (4.49$\pm$1.48)E$-$17 | 5.84$\pm$1.98 | 593$\pm$305
Si iii | 1892.0 | 2.258$\pm$0.002 | 4.03$\pm$4.16 | (2.97$\pm$2.55)E$-$17 | 3.56$\pm$3.06 | 639$\pm$659
C iii]_R | 1908.7 | 2.261$\pm$0.005 | 30.3$\pm$5.3 | (2.34$\pm$0.66)E$-$16 | 28.2$\pm$8.1 | 4760$\pm$830
C iii]_R | 1908.7 | 2.258$\pm$0.002 | 6.19$\pm$3.04 | (6.50$\pm$3.21)E$-$17 | 7.83$\pm$3.89 | 971$\pm$477
C iii]_R + C iii]_B | 1908.7 | 2.258$\pm$0.003 | 11.6$\pm$1.7 | (2.99$\pm$0.73)E$-$16 | 36.0$\pm$9.0 | 1830$\pm$260
Mg ii | 2799.1 | 2.259$\pm$0.001 | 17.6$\pm$1.6 | (2.44$\pm$0.29)E$-$16 | 49.0$\pm$7.3 | 1890$\pm$170
Note. — Column (1): Line name, (2): Rest-frame wavelength of the line, (3):
Line redshift, (4): Rest-frame FWHM, (5): Line flux, (6): Rest-frame EW, (7):
Velocity width after the correction for the instrumental broadening.
Table 4: The detected lines of J1202
Line name | $\lambda_{\rm rest}$ [Å] | $z_{\rm line}$ | $FWHM_{\rm rest}$ [Å] | $F_{\rm line}$ [${\rm erg\ s^{-1}\ cm^{-2}}$] | $EW_{\rm rest}$ [Å] | $v_{\rm width}$ [${\rm km\ s^{-1}}$]
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
Si iv | 1393.8 | 2.831$\pm$0.001 | 4.27$\pm$0.73 | (1.56$\pm$0.33)E$-$16 | 5.55$\pm$1.17 | 919$\pm$157
Broad componenta | — | — | — | (1.23$\pm$0.13)E$-$15 | — | 4970$\pm$170
O iv] | 1399.9 | 2.842$\pm$0.001 | 9.90$\pm$0.87 | (5.13$\pm$0.68)E$-$16 | 18.1$\pm$2.4 | 2120$\pm$190
C iv_R | 1549.5 | 2.831$\pm$0.001 | 15.0$\pm$0.4 | (5.46$\pm$0.26)E$-$15 | 177$\pm$9 | 2900$\pm$70
C iv_B | 1549.5 | 2.793$\pm$0.002 | 13.0$\pm$1.0 | (7.98$\pm$1.03)E$-$16 | 27.7$\pm$3.6 | 2510$\pm$190
C iv_R + C iv_B | 1549.5 | 2.830$\pm$0.002 | 16.0$\pm$0.5 | (6.26$\pm$0.28)E$-$15 | 203$\pm$10 | 3100$\pm$90
He ii_R | 1640.4 | 2.838$\pm$0.002 | 11.0$\pm$1.7 | (1.68$\pm$0.47)E$-$16 | 5.06$\pm$1.43 | 2010$\pm$310
He ii_B | 1640.4 | 2.806$\pm$0.005 | 21.1$\pm$3.4 | (3.16$\pm$0.55)E$-$16 | 9.31$\pm$1.64 | 3870$\pm$620
He ii_R + He ii_B | 1640.4 | 2.834$\pm$0.004 | 25.5$\pm$2.5 | (4.84$\pm$0.73)E$-$16 | 14.5$\pm$2.2 | 4650$\pm$450
O iii] | 1663.5 | 2.852$\pm$0.003 | 7.57$\pm$2.94 | (2.06$\pm$1.07)E$-$17 | 0.661$\pm$0.343 | 1360$\pm$530
Al iii | 1858.8 | 2.839$\pm$0.002 | 21.5$\pm$2.4 | (3.05$\pm$0.41)E$-$16 | 10.5$\pm$1.4 | 3470$\pm$390
Si iii | 1892.0 | 2.819$\pm$0.004 | 13.6$\pm$5.5 | (9.40$\pm$4.81)E$-$17 | 3.19$\pm$1.64 | 2160$\pm$870
C iii] | 1908.7 | 2.835$\pm$0.002 | 30.6$\pm$1.9 | (1.01$\pm$0.07)E$-$15 | 33.5$\pm$2.4 | 4810$\pm$300
Note. — See Table 3 for the description of each column.
aThe observed-frame wavelength, continuum flux, and FWHM of the broad
component are 5349.5$\pm$2.0 Å, (7.36$\pm$0.12)E$-$18 ${\rm erg\ s^{-1}\
cm^{-2}}$ Å-1, and 88.9$\pm$2.9 Å, respectively. See Section 3.1 for details.
Table 5: The detected lines of J1207
Line name | $\lambda_{\rm rest}$ [Å] | $z_{\rm line}$ | $FWHM_{\rm rest}$ [Å] | $F_{\rm line}$ [${\rm erg\ s^{-1}\ cm^{-2}}$] | $EW_{\rm rest}$ [Å] | $v_{\rm width}$ [${\rm km\ s^{-1}}$]
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
C iv_R | 1549.5 | 2.512$\pm$0.001 | 7.87$\pm$0.43 | (1.02$\pm$0.08)E$-$15 | 74.2$\pm$6.0 | 1520$\pm$80
C iv_B | 1549.5 | 2.500$\pm$0.001 | 20.2$\pm$0.5 | (1.36$\pm$0.09)E$-$15 | 99.3$\pm$6.9 | 3900$\pm$90
C iv_R + C iv_B | 1549.5 | 2.511$\pm$0.001 | 10.3$\pm$0.4 | (2.39$\pm$0.12)E$-$15 | 173$\pm$9 | 1990$\pm$70
He ii_R | 1640.4 | 2.511$\pm$0.001 | 6.53$\pm$1.12 | (6.61$\pm$1.50)E$-$17 | 5.32$\pm$1.21 | 1190$\pm$210
He ii_B | 1640.4 | 2.499$\pm$0.002 | 17.7$\pm$1.3 | (1.47$\pm$0.24)E$-$16 | 11.7$\pm$1.9 | 3240$\pm$240
He ii_R + He ii_B | 1640.4 | 2.509$\pm$0.002 | 11.6$\pm$1.2 | (2.13$\pm$0.28)E$-$16 | 17.2$\pm$2.3 | 2120$\pm$220
O iii] | 1663.5 | 2.516$\pm$0.002 | 12.2$\pm$2.4 | (3.70$\pm$0.95)E$-$17 | 3.18$\pm$0.82 | 2200$\pm$430
Al iii | 1858.8 | 2.508$\pm$0.002 | 19.7$\pm$1.7 | (9.09$\pm$1.02)E$-$17 | 9.23$\pm$1.04 | 3180$\pm$280
Si iii | 1892.0 | 2.511$\pm$0.003 | 10.7$\pm$4.9 | (2.64$\pm$1.60)E$-$17 | 2.79$\pm$1.69 | 1690$\pm$770
C iii]_R | 1908.7 | 2.510$\pm$0.001 | 12.9$\pm$1.4 | (1.69$\pm$0.25)E$-$16 | 18.2$\pm$2.7 | 2030$\pm$220
C iii]_B | 1908.7 | 2.503$\pm$0.002 | 42.0$\pm$2.9 | (4.02$\pm$0.65)E$-$16 | 43.1$\pm$7.0 | 6600$\pm$450
C iii]_R + C iii]_B | 1908.7 | 2.509$\pm$0.002 | 19.6$\pm$1.2 | (5.71$\pm$0.70)E$-$16 | 61.4$\pm$7.5 | 3080$\pm$180
Note. — See Table 3 for the description of each column.
Table 6: The detected lines of J1443
Line name | $\lambda_{\rm rest}$ [Å] | $z_{\rm line}$ | $FWHM_{\rm rest}$ [Å] | $F_{\rm line}$ [${\rm erg\ s^{-1}\ cm^{-2}}$] | $EW_{\rm rest}$ [Å] | $v_{\rm width}$ [${\rm km\ s^{-1}}$]
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
Ly ${\alpha}$ | 1215.7 | 3.341$\pm$0.003 | 10.7$\pm$2.6 | (5.28$\pm$1.31)E$-$16 | 64.3$\pm$17.3 | 2650$\pm$640
N v | 1240.8 | 3.312$\pm$0.001 | 16.8$\pm$0.6 | (2.06$\pm$0.09)E$-$15 | 242$\pm$23 | 4070$\pm$160
Si ii | 1262.6 | 3.326$\pm$0.003 | 16.0$\pm$1.9 | (1.24$\pm$0.17)E$-$16 | 13.6$\pm$2.0 | 3790$\pm$450
Si iv | 1393.8 | 3.324$\pm$0.010 | 16.4$\pm$3.2 | (3.18$\pm$1.01)E$-$16 | 33.8$\pm$10.7 | 3520$\pm$700
O iv] | 1399.9 | 3.339$\pm$0.008 | 12.9$\pm$2.5 | (1.74$\pm$1.11)E$-$16 | 18.6$\pm$11.8 | 2760$\pm$540
C iv_R | 1549.5 | 3.328$\pm$0.005 | 14.2$\pm$1.6 | (6.15$\pm$1.74)E$-$16 | 60.0$\pm$17.0 | 2740$\pm$310
C iv_B | 1549.5 | 3.296$\pm$0.007 | 15.7$\pm$2.3 | (5.47$\pm$1.67)E$-$16 | 54.9$\pm$16.8 | 3030$\pm$440
C iv_R + C iv_B | 1549.5 | 3.317$\pm$0.006 | 23.1$\pm$1.8 | (1.16$\pm$0.24)E$-$15 | 114$\pm$24 | 4470$\pm$350
He ii_R | 1640.4 | 3.337$\pm$0.001 | 6.48$\pm$1.56 | (3.23$\pm$1.01)E$-$17 | 3.24$\pm$1.02 | 1180$\pm$290
He ii_B | 1640.4 | 3.307$\pm$0.004 | 20.3$\pm$2.9 | (9.15$\pm$1.53)E$-$17 | 9.25$\pm$1.56 | 3710$\pm$530
He ii_R + He ii_B | 1640.4 | 3.335$\pm$0.003 | 20.1$\pm$2.5 | (1.24$\pm$0.18)E$-$16 | 12.4$\pm$1.9 | 3680$\pm$460
Note. — See Table 3 for the description of each column.
Table 7: Emission-line contribution and excess magnitude to the power-law fit
in the $g$\- and $r$-band
| $g$-band | $r$-band
---|---|---
| Total EWs | $\Delta$mag | Excess mag | Total EWs | $\Delta$mag | Excess mag
| [Å] | [AB mag] | [AB mag] | [Å] | [AB mag] | [AB mag]
HSC J0907 | $>$604 | $>$0.37 | 1.27 | 129 | 0.09 | 0.47
HSC J1202 | $>$258 | $>$0.18 | 1.13 | 836 | 0.48 | 0.44
HSC J1207 | $>$608 | $>$0.38 | 1.48 | 329 | 0.21 | 0.65
HSC J1443 | 1380 | 0.72 | 0.88 | 721 | 0.43 | 0.40
Note. — $\Delta$mag: $2.5\log{(1+EW/BW)}$, Excess mag: The excesses of the
$g$\- and $r$\- band magnitudes between the observed magnitudes and the
expected magnitudes from an extrapolation of the power-law fit to the longer
wavelength bands.
Figure 3: The SED of J0907 (top), J1202 (middle upper), J1207 (middle lower),
and J1443 (bottom). The red dots denote the $g$\- and $r$-band magnitudes,
while the blue dots denote the longer-wavelength optical and near-infrared
magnitudes that are used for the power-law fit (black line). The green lines
represent the observed spectra. Figure 4: J1207 images in the WISE $W1$-band
(left) and H-ATLAS 250 $\mu$m-band (right). The orange stars, green crosses,
and red cross denote source detections in the HSC-SSP, ALLWISE, and H-ATLAS
catalogs, respectively. The size of each image is 60″$\times$ 60″, centered at
the HSC position of J1207.
### 3.3 Estimating the dust extinction
We need to estimate dust extinction, $E(B-V)$ of AGN radiation, and $L^{\rm
AGN}_{\rm bol}$ to calculate the SMBH mass and Eddington ratio. Since Balmer
decrement or other spectral measures of $E(B-V)$ is not available, we perform
the SED fitting to the broad-band photometry to estimate the $E(B-V)$ and
$L^{\rm AGN}_{\rm bol}$. In this work, we utilize the new version of Code
Investigating GAlaxy Emission (CIGALE; Burgarella et al. 2005; Noll et al.
2009; Boquien et al. 2019) called X-CIGALE (Yang et al., 2020), to perform the
SED fit in a self-consistent framework by considering an energy balance
between the UV/optical absorption and IR emission. X-CIGALE generates the
best-fit model including the stellar, AGN, and SF components that fits the
photometric data in the rest-frame UV to far-infrared (FIR) bands. We utilize
the Herschel Space Observatory (Pilbratt et al. 2010) Astrophysical Terahertz
Large Area Survey (H-ATLAS; Eales et al. 2010; Valiante et al. 2016; Bourne et
al. 2016) data observed with Photodetector Array Camera and Spectrometer
(PACS; Poglitsch et al. 2010) at 100 and 160 $\mu$m and with Spectral and
Photometric Imaging REceiver (SPIRE; Griffin et al. 2010) at 250, 350, and 500
$\mu$m in the FIR, in addition to optical, NIR, and MIR data obtained by
Subaru HSC, VISTA, and WISE. The 1$\sigma$ limiting fluxes at 100, 160, 250,
350, and 500 $\mu$m are 44, 49, 7.4, 9.4, and 10.2 mJy, respectively (Valiante
et al. 2016).
To search for the H-ATLAS counterpart of the four BluDOGs, we adopt a search
radius of 10 arcsec by following Toba et al. (2019) (and Toba et al. 2022).
Accordingly we found the counterparts of two BluDOGs (J1202 and J1207). The
separation between the HSC position and the H-ATLAS counterpart position is
0.94 arcsec for J1202, and 9.7 arcsec for J1207. The relatively large
separation in the latter case suggests the counterpart being a coincidental
detection. There are two WISE sources around J1207 (Figure 4); one probably
corresponds to J1207 itself (the angular separation between the HSC and WISE
potions is 0.65 arcsec) and another is located at 19 arcsec away to the north-
east direction. The H-ATLAS source is located between these two WISE sources,
and thus the FIR fluxes given in the H-ATLAS catalog are possibly attributed
to the two WISE sources. Therefore we regard the H-ATLAS fluxes of J1207 as
the upper limit. For the remaining two BluDOGs (J0907 and J1443), we adopt the
5$\sigma$ upper limit fluxes.
As for the optical–MIR photometric data, we utilize $g$, $r$, $i$, $z$, $y$
(HSC-SSP), $Z$, $Y$, $J$, $H$, $Ks$ (VIKING DR2), $W1$, $W2$, $W3$, and $W4$
(ALLWISE) bands (see Noboriguchi et al. 2019). Note that the SNRs in these
bands are more than 5, except for the $W4$-band with SNR more than 3 because
we adopted such SNR cut in the selection of DOGs (Noboriguchi et al. 2019).
Since the $g$\- and $r$-band photometry are significantly affected by the
strong emission lines (see Figure 1 and Tables 3–6) which cannot be treated
properly in X-CIGALE, we corrected for their contribution by referring to the
estimates given in Table 7.
Table 8: Parameters adopted in the X-CIGALE fit Parameter | Value
---|---
Delayed SFH (Ciesla et al. 2015)
${\tau}_{\rm main}$ [Myr] | 100, 250, 500
${\tau}_{\rm burst}$ [Myr] | 10, 50
$f_{\rm burst}$ | 0.0, 0.5, 0.99
Agemain [Myr] | 500, 800, 1000
Ageburst [Myr] | 1, 5, 10
Single stellar population (Bruzual & Charlot 2003)
IMF | Chabrier (2003)
Metallicity | 0.02
Ageseparation [Myr] | 10
Nebular emission (Inoue 2011)
$\log{U}$ | $-$2.0
$f_{\rm esc}$ | 0.0
$f_{\rm dust}$ | 0.0
Lines width [${\rm km\ s^{-1}}$] | 300.0
Dust attenuation (Calzetti et al. 2000)
$E(B-V)_{\rm line}$ | 3, 4, 5, 6, 7, 8, 9, 10
$f_{E(B-V)}$ | 0.44
$\lambda_{\rm UV,bump}$ [nm] | 217.5
FWHMUV,bump [nm] | 35.0
$A_{\rm UV,bump}$ | 0.0
$\delta$ | 0.0
Extinction law of emission lines | the Milky Way
$R_{V}$ | 3.1
Dust emission (Dale et al. 2014)
AGN fraction | 0.0
$\alpha_{\rm IR,AGN}$ | 0.0625, 0.2500, 2.0000
AGN model (Stalevski et al. 2016)
$\tau_{9.7}$ | 3, 7
$p$ | 1.0
$q$ | 1.0
$oa$ [deg] | 10, 20, 30, 40, 50, 60, 70, 80
$R_{\rm ratio}$ | 20
$M_{\rm cl}$ | 0.97
$i$ [deg] | 0, 10, 20, 30, 40,
| 50, 60, 70, 80, 90
$f_{\rm AGN}$ | 0.1, 0.3, 0.5, 0.7, 0.9
Extinction law of polar dust | Calzetti et al. 2000
$E(B-V)^{\rm AGN}_{\rm polar\ dust}$ | 0.1, 0.2, 0.3, 0.4, 0.5
$T^{\rm AGN}_{\rm polar\ dust}$ [K] | 600, 700, 800, 900,
| 1000, 1100, 1200, 1300, 1400
Emissivity of polar dust | 1.6
The models and parameters of X-CIGALE adopted in this work are summarized in
Table 8. We assume a delayed star formation history (SFH; Ciesla et al. 2015)
with the e-folding times of the main stellar population ($\tau_{\rm main}$)
and late starburst population ($\tau_{\rm burst}$), mass fraction of the late
burst population ($f_{\rm burst}$), and age of the main stellar population
(Agemain) and the late burst (Ageburst). As the stellar population, we assume
the initial mass function of Chabrier (2003), solar metallicity, and 10-Gyr
separation between young and old stellar population (Ageseparation). The
nebular emission model (Inoue 2011) is characterized by the ionization
parameter ($U$), fractions of Lyman continuum photons escaping the galaxy
($f_{\rm esc}$) and absorbed by dust ($f_{\rm dust}$), and line width. We
utilize a modified dust attenuation model presented by Boquien et al. (2019).
The dust attenuation model for the continuum is taken from Calzetti et al.
(2000) with the extension taken from Leitherer et al. (2002) between the Lyman
break and 1500 Å. The emission lines are attenuated with a Milky Way
extinction with $R_{V}=3.1$ (Cardelli et al., 1989). We assumed $E(B-V)_{\rm
continuum}=0.44E(B-V)_{\rm line}$, following Calzetti et al. (2000). The
$E(B-V)_{\rm line}$ is varied between 3 and 10. We utilize the SKIRTOR model
as the AGN emission model, which takes geometric parameters of the AGN into
account and also allows us to incorporate the effect of extinction by the
polar dust. The parameters of the AGN model are the average edge-on optical
depth at 9.7 $\mu$m ($\tau_{9.7}$), the torus density parameters ($p$ and $q$;
Stalevski et al. 2016), the angle between the equatorial plane and the edge of
the torus ($oa$), the ratio of the maximum to minimum radii of the dust torus
($R_{\rm ratio}$), the fraction of total dust mass inside clumps ($M_{\rm
cl}$), the inclination ($i$), the AGN fraction ($f_{\rm AGN}$), the extinction
law, color excess ($E(B-V)^{\rm AGN}_{\rm polar\ dust}$), dust temperature
($T^{\rm AGN}_{\rm polar\ dust}$), and emissivity index of the polar dust.
The best-fit SED models are shown in Figure 5. The reduced $\chi^{2}$ of the
fits are 1.38, 3.19, 0.93, and 1.74 for J0907, J1202, J1207, and J1443,
respectively. The best-fit values and associated errors for $E(B-V)^{\rm
AGN}_{\rm polar\ dust}$ and $L^{\rm AGN}_{\rm bol}$ are estimated with a
Bayesian-like strategy presented in Noll et al. (2009), and are reported in
Table 9. On the other hands, we cannot quantitatively constrain the parameters
of the host galaxies because the $E(B-V)$ values are too large and the optical
parts in their SEDs are dominated by their AGN emission (see Figure 5).
Figure 5: The results of the SED fitting for the four BluDOGs. The upper left,
upper right, lower left and lower right panels show the results of J0907,
J1202, J1207, and J1443, respectively. The black, blue, green, red, and orange
lines represent the best-fit model, stellar component (with dust attenuation),
AGN component, SF component (FIR re-emission from the dust heated by SF), and
nebular component, respectively. The magenta plots represent the photometric
data. The arrows denote 5$\sigma$ upper limit flux.
### 3.4 Measurement of the SMBH mass
Table 9: Physical properties of the four BluDOGs | HSC J0907 | HSC J1202 | HSC J1207 | HSC J1443
---|---|---|---|---
Redshift | $2.258\pm 0.002$ | $2.830\pm 0.002$ | $2.511\pm 0.001$ | $3.317\pm 0.006$
$E(B-V)^{\rm AGN}_{\rm polar\ dust}$ | $0.26\pm 0.05$ | $0.33\pm 0.04$ | $0.30\pm 0.02$ | $0.20\pm 0.01$
$L^{\rm AGN}_{\rm bol}$/$L_{\odot}$ | $(6.11\pm 0.95)\times 10^{12}$ | $(6.11\pm 1.18)\times 10^{13}$ | $(7.95\pm 1.47)\times 10^{12}$ | $(2.52\pm 0.13)\times 10^{13}$
$M_{\rm BH}$ (C iv)/$M_{\odot}$ | $(1.69\pm 0.17)\times 10^{8}$ | $(4.95\pm 0.30)\times 10^{8}$ | $(1.11\pm 0.08)\times 10^{8}$ | $(5.48\pm 0.86)\times 10^{8}$
$M_{\rm BH}$ (Mg ii)/$M_{\odot}$ | $(9.85\pm 1.80)\times 10^{7}$ | — | — | —
$\lambda_{\rm Edd}$ (C iv) | $1.10\pm 0.20$ | $3.75\pm 0.76$ | $2.19\pm 0.43$ | $1.40\pm 0.23$
We have detected the C iv emission line for all the four BluDOGs and Mg ii
emission line for J0907, both of which are widely used to calculate the SMBH
mass of type-1 AGNs. Note that the systematic uncertainty is larger in the C
iv-based SMBH mass than in the Mg ii-based SMBH mass, due to a powerful
outflow sometimes seen in the C iv velocity profile (e.g., Baskin & Laor,
2005; Netzer, 2015; Coatman et al., 2017). We calculate the single-epoch mass
of SMBHs with the C iv and Mg ii emission lines, following the calibrations
given in Vestergaard & Peterson (2006) and Vestergaard & Osmer (2009)
respectively:
$\displaystyle M_{\rm BH}$ $\displaystyle=$ $\displaystyle
10^{6.66}\Big{(}\frac{{\rm FWHM(C~{}{\sc IV})}}{10^{3}\ {\rm km\
s^{-1}}}\Big{)}^{2}\Big{(}\frac{\lambda L_{\lambda}(1350{\rm\AA})}{10^{44}\
{\rm erg\ s^{-1}}}\Big{)}^{0.53}M_{\odot},$
and
$\displaystyle M_{\rm BH}$ $\displaystyle=$ $\displaystyle
10^{6.86}\Big{(}\frac{{\rm FWHM(Mg~{}{\sc II})}}{10^{3}\ {\rm km\
s^{-1}}}\Big{)}^{2}\Big{(}\frac{\lambda L_{\lambda}(3000{\rm\AA})}{10^{44}\
{\rm erg\ s^{-1}}}\Big{)}^{0.5}M_{\odot},$
where FWHM(C iv), FWHM(Mg ii), $\lambda L_{\lambda}(1350{\rm\AA})$ and
$\lambda L_{\lambda}(3000{\rm\AA})$ are the FWHM of the C iv and Mg ii
velocity profile, and the monochromatic luminosity at 1350 Å and 3000 Å,
respectively. Note that we use the FWHM of C iv_R + C iv_B as the FWHM of the
C iv. We cannot eliminate the possiblility that the estimated SMBH masses are
overestimated because the C iv profiles are affected by nucleus outflows
(Section 4.1). For estimating the reddening-corrected monochromatic
luminosity, we use the optical spectra presented in Section 3.1. We converted
the spectra to the rest-frame, de-reddened them with $E(B-V)^{\rm AGN}_{\rm
polar\ dust}$ derived in the SED fit, and masked out emission and absorption
lines as well as pixels with negative values. Then, we fit a power-law
continuum model to the spectra and estimate the monochromatic luminosities
from the best fits. The estimated $\lambda L_{\lambda}$(1350) of J0907, J1202,
J1207, and J1443 are $(1.54\pm 0.05)\times 10^{45}$, $(9.64\pm 0.27)\times
10^{45}$, $(3.06\pm 0.06)\times 10^{45}$, and $(2.93\pm 0.03)\times 10^{45}$
${\rm erg\ s^{-1}}$, respectively. The $\lambda L_{\lambda}$(3000) of J0907 is
estimated to be $(1.45\pm 0.04)\times 10^{45}$ ${\rm erg\ s^{-1}}$.
The resultant SMBH masses are summarized in Table 9. It should be noted that
the C iv-based $M_{\rm BH}$ and Mg ii-based $M_{\rm BH}$ of J0907 is not
consistent within the statistical error. This is probably attributed to a
systematic error especially in the C iv-based $M_{\rm BH}$, known to be
accompanied with a large systematic error ($\sim$0.5 dex; see, e.g., Shen
2013). Hereafter we use only the C iv-based $M_{\rm BH}$, since it is measured
in all the four BluDOGs.
## 4 Discussion
### 4.1 Spectral features and nuclear outflows
We found that the redshifts of the four BluDOGs are in the range of
$2.2\lesssim z_{\rm sp}\lesssim 3.3$. They are systematically higher than the
typical redshifts of DOGs ($z_{\rm sp}=1.99\pm 0.45$; Dey et al. 2008; Pope et
al. 2008). One possible reason for this systematically high redshift is a
selection effect related to the blue-excess criterion. When we select BluDOGs
from the parent DOG sample, the $g$\- and $r$-band magnitudes show an excess
of the expected magnitudes estimated by the power-law extrapolation from
$i$-band to $W4$-band. Thus we may select DOGs in a preferred redshift range
where strong emission lines such as Ly$\alpha$ and C iv shifts into the two
bands (see Section 4.2 for more quantitative assessments). The reason for the
underestimated photometric redshift ($\sim 1$; Noboriguchi et al. 2019) is the
unusual emission lines with the large REW.
The detected emission lines have large velocity widths, $\gtrsim 2000\ {\rm
km\ s^{-1}}$ in most cases. This suggests that the broad-line region (BLR) of
the BluDOGs is not completely obscured; in other words, the observed BluDOGs
are classified as type-1 AGNs. This is an unexpected result, because their
very red color between optical and mid-IR suggests the heavily obscured
nature. One possible interpretation is that we are looking at a phase where
the surrounding dust is just blown away by the nuclear activity (outflow,
radiation pressure, or both), as discussed more in Section 4.3. It should be
noted that the type-1 nature is seen not only in the presented BluDOGs but
also in some other DOGs (e.g., Toba & Nagao, 2016; Toba et al., 2017a; Zou et
al., 2020). Systematic spectroscopic observations for the whole populations of
DOGs are required to study the nature of obscuration occurring in various
populations of DOGs.
As shown in Figure 2, the velocity profile of the observed C iv lines show a
notable excess feature in the blue wing. Such an excess in the C iv velocity
profile has been observed in other type-1 AGNs, and interpreted as a result of
powerful nuclear outflows (e.g., Baskin & Laor, 2005; Netzer, 2015; Coatman et
al., 2017). To evaluate quantitatively how the nuclear outflow in BluDOGs is
strong compared to ordinary AGNs, we examine the “asymmetry parameter
($\alpha_{\beta}$)” defined by De Robertis (1985) as
$\displaystyle\alpha_{\beta}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{c}(3/4)-\lambda_{c}(1/4)}{\Delta\lambda(1/2)},$
(3)
where $\lambda_{c}(h)$ and $\Delta\lambda(1/2)$ are the central wavelength at
which the flux falls to a $h$ time the peak flux and FWHM of the broad
profile, respectively. The positive and negative values of $\alpha_{\beta}$
express the blue and red excesses, respectively. The derived values of
$\alpha_{\beta}$ for J0907, J1202, J1207, and J1443 are 0.216, 0.102, 0.246,
and 0.051, respectively. As a reference, the C iv velocity profile in the
composite spectrum of SDSS type-1 quasars given by Vanden Berk et al. (2001)
shows $\alpha_{\beta}=0.110$. Thus J0907, and J1207 may possess a significant
nuclear outflow that is more powerful than typical quasars.
In order to compare $\alpha_{\beta}$ of the BluDOG with that of another dusty
AGN population, we fitted the C iv profile of 97 “core” ERQs (ERQs with REW(C
iv) $>100$ Å) in Hamann et al. (2017) and measured $\alpha_{\beta}$ by
adopting a single or double Gaussian profile. The core ERQ sample consists of
80 objects without BAL and 17 objects with BAL, and we investigate the
statistics of $\alpha_{\beta}$ for the two subsamples separately because the
BAL feature can affect the C iv line profile. Here we exclude J1443 from the
BluDOG sample when comparing the $\alpha_{\beta}$ index because its velocity
profile is largely affected by narrow absorption lines (hereafter the limited-
BluDOG sample to infer the 3 BluDOGs; i.e., J0907, J1202, and J1207). Figure 6
shows the cumulative fraction of $\alpha_{\beta}$ for the limited-BluDOGs,
core ERQs without BAL, and core ERQs with BAL. The averaged values of the
limited-BluDOGs, core ERQs without BAL, and core ERQs with BAL are
0.15$\pm$0.08, 0.02$\pm$0.13, and 0.01$\pm$0.09, respectively. We performed
the Kolmogorov-Smirnov test (KS-test) to examine the statistical significance
of the difference in $\alpha_{\beta}$ among the samples. The p-values of the
limited-BluDOGs-core ERQs without BAL, and limited-BluDOG-core ERQs with BAL
are 0.0178 and 0.0175, respectively. Thus we conclude that the distributions
of $\alpha_{\beta}$ of the limited-BluDOGs and core ERQs with/without BAL are
marginally different with $>2$ sigma significance. This suggests that the
BluDOGs show nuclear outflow that is possibly more powerful than the nuclear
outflow in core ERQs with/without BAL.
Figure 6: Cumulative distribution of the $\alpha_{\beta}$ indices for the
limited-BluDOGs (see the main text; blue line), core ERQs without BAL (green
line), and core ERQs with BAL (orange line). The red dashed line denotes the
$\alpha_{\beta}$ index measured for the composite spectrum of SDSS type-1
quasars. Figure 7: Cumulative distribution of the $kt_{80}$ indices for the
limited-BluDOGs (blue line), core ERQs without BAL (green line), and core ERQs
with BAL (orange line). The red dashed line denotes the $kt_{80}$ index for
the single Gaussian profile.
We also focus on the kurtosis index ($kt_{80}$) defined as follows (see Hamann
et al. 2017 for detailes): $kt_{80}=\Delta v$(80%) $/\Delta v$(20%), where
$\Delta v$($x$%) is the velocity width at $x$% of the peak flux height. In
addition to $\alpha_{\beta}$, this $kt_{80}$ index is useful to characterize
the C iv wing (a more prominent blue wing results in smaller $kt_{80}$). By
using the best-fit double Gaussian profile of the BluDOGs, $kt_{80}$ of J0907,
J1202, J1207, and J1443 are 0.276, 0.313, 0.252, and 0.440, respectively.
Again we exclude J1443 from the BluDOG sample when comparing the $kt_{80}$
index as the discussion of the $\alpha_{\beta}$ index. For comparison,
$kt_{80}$ of a single Gaussian is $\sqrt{1-\frac{2\ln(2)}{\ln(5)}}$ ($\sim
0.37$), whereas most quasars have $kt_{80}\sim 0.15$–$0.30$ (see Figure 7 in
Hamann et al. 2017). The limited-BluDOGs, core ERQs without BAL, and core ERQs
with BAL show $kt_{80}=0.28\pm 0.03$, $0.33\pm 0.06$, and $0.34\pm 0.05$,
respectively (see also Figure 7). Note that the C iv profile of 41 core ERQs
without BAL and 7 core ERQs with BAL is fitted by a single Gaussian, which is
the reason why many objects have $kt_{80}\sim 0.37$ as shown in Figure 7. C iv
velocity profiles of core ERQs with/without BAL are roughly consistent with
the Gaussian without a blue wing. However, the $kt_{80}$ index of the limited-
BluDOGs is less than $\sqrt{1-\frac{2\ln(2)}{\ln(5)}}$, suggesting that their
C iv line profile has a wing. We performed the KS-test to examine the
statistical significance of the difference in $kt_{80}$ among the samples. The
p-values of the limited-BluDOGs-core ERQs without BAL, and limited-BluDOGs-
core ERQs with BAL are 0.0637 and 0.0175, respectively. Therefore, we conclude
that the distributions of $kt_{80}$ between the samples of the limited-BluDOGs
and core ERQs with/without BAL feature are marginally different with $>2$
sigma significance.
It has been reported that AGNs with a high Eddington ratio tend to show a
Lorentzian-line velocity profile in BLR lines (e.g., Moran et al. 1996; Véron-
Cetty et al. 2001; Collin et al. 2006; Zamfir et al. 2010). Therefore, the
small $kt_{80}$ value of BluDOGs can be caused by the contribution of extended
Lorentzian wings instead of the asymmetric blue wing. For a symmetric
Lorentzian profile, $kt_{80}\sim 1/16$ (much smaller than a Gaussian profile,
$kt_{80}\sim 0.37$) and $\alpha_{\beta}=0$ are expected. However, the BluDOGs
are inconsistent with this expectation (Figure 8). This Figure 8 also shows
that the BluDOGs follow the trend made by core ERQs with/without BAL in the
$kt_{80}-\alpha_{\beta}$ plane, while a systematic deviation of BluDOGs toward
($\alpha_{\beta}$, $kt_{80}$) = (0, 0) is expected if a Lorentzian component
significantly contributes to the C iv line of BluDOGs. Thus, we conclude that
extended Lorentzian wings do not affect the C iv line profile of BluDOGs, but
the small $kt_{80}$ of BluDOGs is caused by the asymmetric blue excess due to
the stronger nuclear outflow than that of ERQs.
Figure 8: The $kt_{80}$ vs. $\alpha_{\beta}$ plot for the limited-BluDOGs
(blue dots), core ERQs without BAL (green dots), and core ERQs with BAL
(orange dots). The black solid line and dashed line denote the $kt_{80}$
values of single Gaussian and single Lorentzian, respectively. Note that 41
core ERQs without BAL and 7 core ERQs with BAL are fitted with a single
Gaussian, and they are plotted at ($\alpha_{\beta}$, $kt_{80}$) = (0,
$\sqrt{1-\frac{2\ln(2)}{\ln(5)}}$).
### 4.2 Large equivalent widths of the CIV emission
As we summarized in Table 7, the blue excess in J1443 can be almost explained
by the contribution of the strong emission lines. This is also the case for
J1202 by taking into account of the additional contribution of unobserved
Ly$\alpha$ to $g$-band. On the other hand, the blue excess of the remaining
two BluDOGs cannot be explained only by the contribution of BLR emission
lines. Figure 3 strongly suggests that a part of the excess flux comes from
the continuum emission, which deviates at $\lesssim$7000Å from the
extrapolation of the power-law fit. These results demonstrate the complexity
and diversity of BluDOGs; systematic exploration of a larger sample is
required to statistically understand the origin of the blue excess.
Figure 9: Rest-frame EW ratio vs. rest-frame wavelength. REW ratios are
defined as REWs of objects over REWs from the composite spectrum of SDSS
type-1 quasar measured by Vanden Berk et al. (2001). The red, blue, green, and
orange plots show the REW ratios of J0907, J1202, J1207, and J1443,
respectively. Figure 10: Rest-frame EW of the C iv vs. the monochromatic
luminosity at 1350 Å. Blue and orange dots represent the four BluDOGs and
WISSH quasars (Vietri et al. 2018). The grey 2D histogram represents the
number density of the SDSS quasars (Shen et al., 2011). The green line
represents the linear fit to the distribution of the SDSS quasars. The red
plots show the mean and standard deviation in luminosity bins with 0.5 dex
width. The numbers of the SDSS quasars in the individual bins are shown at the
bottom of the panel.
Not only REW(C iv), but the REW of other BLR emission lines are also
systematically larger than observed in typical type-1 quasars (see Tables 3–6,
Figure 9, and also Table 2 in Vanden Berk et al. 2001). Such a trend may be
explained if the observed BluDOGs have lower UV luminosity than typical
quasars owing to the Baldwin effect (Baldwin 1977; Kinney et al. 1990; Baskin
& Laor 2004), i.e., the negative correlation between the REWs and the
continuum luminosities of quasars. Figure 10 shows the four BluDOGs on the C
iv REW vs. $\lambda L_{\lambda}$(1350Å) diagram. Note that the REW of J1443
($114\pm 24$ Å) is somewhat smaller than that of the remaining three BluDOGs
($148\pm 12$, $203\pm 10$ and $173\pm 9$ Å for J0907, J1202 and J1207,
respectively; see Tables 3–6). This is partly because of an underestimation of
the C iv flux caused by the absorption features. The figure also shows SDSS
type-1 quasars with reliable measurement of C iv REW (`EWCIV/e_EWCIV > 5`) and
without broad absorption lines (`BAL < 1`) taken from Shen et al. (2011).
Since the Baldwin effect does not significantly depend on redshift (e.g.,
Croom et al. 2002; Dietrich et al. 2002; Niida et al. 2016), we do not adopt
any redshift criterion to select the SDSS quasars so that a wide luminosity
range is covered. We also use another comparison sample taken from the
WISE/SDSS selected hyper-luminous quasar sample (WISSH; Bischetti et al. 2017;
Vietri et al. 2018)555The C iv REW and C iv line luminosity of WISSH quasars
are given by Vietri et al. (2018). To calculate $L_{\lambda}$(1350Å) of WISSH
quasars, we assume that the continuum spectrum of WISSH quasars is a power-law
and adopt the following formula: $\displaystyle L_{\lambda}({\rm 1350\AA})$
$\displaystyle=$ $\displaystyle\frac{L^{\rm line}({\rm C~{}{\sc
IV}})}{REW({\rm C~{}{\sc
IV}})}\times\Big{(}\frac{1350}{1549}\Big{)}^{\alpha_{\lambda}},$ (4) where
$L_{\lambda}({\rm 1350\AA})$, $L^{\rm line}({\rm C~{}{\sc IV}})$, and
$\alpha_{\lambda}$ are the monochromatic luminosity at 1350 Å, the line
luminosity of C iv, and power-law index, respectively. Here we adopt
$\alpha_{\lambda}=-1.7$ (Vanden Berk et al., 2001) as the power-law index. ,
in order to add objects at the high-luminosity end.
Figure 10 clearly shows that the C iv REWs of BluDOGs are larger than the
comparison samples at a given UV luminosity. The excess REW over the average
relation of the Baldwin effect (shown with a green solid line in Figure 10)
for J0907, J1202, J1207, and J1443 are 0.29, 0.66, 0.44, and 0.26 dex,
respectively. This excess is larger than the scatter of the comparison samples
(see red plots in Figure 10). Therefore the large REW seen in the BluDOGs are
not due to the Baldwin effect.
The averages and standard deviations of REW(C iv) for core ERQs and ERQ-like
objects are $178\pm 74$ and $86\pm 45$ Å, respectively (Hamann et al., 2017).
The distributions of REW(C iv) and $(i-W3)_{\rm AB}$ color for BluDOGs are
consistent with these of core ERQs although the most of core ERQs and ERQ-like
objects do not show a blue-wing profile in C iv (Section 4.1). Hamann et al.
(2017) proposed a scenario that the large REW of ERQs are possibly due to the
spatially extended geometry of BLRs caused by the powerful nuclear outflow. If
the obscuration is heavier for the accretion disk than for the BLRs which have
extended geometry, the continuum emission is more heavily extinct than the BLR
emission lines and thus the observed-frame EW becomes larger. Such a scenario
may also apply to BluDOGs. Unfortunately it is not observationally feasible to
confirm this idea by resolving the spatial structure of BLRs in ERQs or
BluDOGs due to the required angular resolution, even with the JWST or
exisiting ground-based interferometers. Without spatially resolving them, a
possible approach is the velocity-resolved reverberation mapping of the
geometry and kinematics of BLR clouds (e.g., Horne et al., 2004; Denney et
al., 2009; Li et al., 2013; Kollatschny et al., 2014; Pancoast et al., 2014).
### 4.3 Possible extreme accretion and the nature of BluDOGs
To understand the nature of BluDOGs especially in the context of the major-
merger scenario for the quasar evolution, we compare the SMBH accretion of the
four BluDOGs with other AGN populations. Figure 11 is a diagram of $L_{\rm
bol}$ vs. the SMBH mass. As in Section 4.2, SDSS quasars (Shen et al., 2011)
and WISSH quasars (Vietri et al., 2018) are used as comparison samples. For
the SDSS quasars, we select only non-BAL quasars (`BAL < 1`) with the
uncertainty of $L_{\rm bol}$ and $M_{\rm BH}$ less than 0.5 dex (`e_logBHCV <
0.5 & e_logLbol < 0.5`), and adopt the C iv-based SMBH mass for a fair
comparison with those of the BluDOGs. We also plot samples of 28 ERQs
(Perrotta et al., 2019), 5 Hot DOGs (Wu et al., 2018), 2 power-law DOGs
(Melbourne et al., 2011), and 1 Compton-thick (CT) DOG (Toba et al., 2020).
Hot DOGs are DOGs with a special color of WISE (very faint in the 3.4 $\mu$m
and 4.5 $\mu$m bands, but bright in the longer bands; Eisenhardt et al. 2012;
Wu et al. 2012), while power-law DOGs are DOGs with a featureless power-law
SED from optical to mid-IR (e.g., Dey et al., 2008; Bussmann et al., 2012;
Toba et al., 2015; Noboriguchi et al., 2019). The CT DOG was identified by
Nuclear Spectroscopic Telescope Array (Harrison et al., 2013) from SDSS-WISE
DOGs sample. All but the CT DOG have spectroscopic redshifts. The SMBH masses
of the ERQs and the WISSH quasars are estimated from H$\beta$, while those of
the Hot DOGs and the DOGs are estimated from H$\alpha$. The SMBH mass of the
CT DOG was estimated by Toba et al. (2020) from the stellar mass by using an
empirical relation between the stellar mass and SMBH mass (Kormendy & Ho,
2013). Since Perrotta et al. (2019) and Melbourne et al. (2011) did not
correct the absorption of dust, the $M_{\rm BH}$ of ERQs and DOGs are lower
limits.
Figure 11: The SMBH mass vs. the bolometric luminosity diagram. The filled-
blue stars and gray contour denote the BluDOGs and SDSS quasars, while the
filled hexagons with green, orange, purple, cyan, and light green colors
denote ERQs (Perrotta et al., 2019), WISSH quasars (Vietri et al., 2018), Hot
DOGs (Wu et al., 2018), DOGs (Melbourne et al., 2011, 2012), and a CT DOG
(Toba et al., 2020), respectively. The red dashed lines represent a constant
Eddington ratio of $\lambda_{\rm Edd}=$ 0.01, 0.1, 1.0, and 10.0.
Figure 11 shows that the four BluDOGs are more luminous than the other AGN
populations at a given SMBH mass, or equivalently, they have lower-mass SMBHs
than the other AGN populations at a given bolometric luminosity. This suggests
that the SMBH growth in the BluDOGs is more rapid than AGNs in comparison
samples. Indeed, the Eddington ratios ($\lambda_{\rm Edd}$) of J0907, J1202,
J1207, and J1443 are $1.10\pm 0.20$, $3.89\pm 0.78$, $2.19\pm 0.44$, and
$1.40\pm 0.23$, respectively (Table 9), with the average value of 2.26. In
other words, the SMBHs in the BluDOGs are now in the stage of the Eddington-
limit or super-Eddington accretion. Even if the intrinsic SMBH masses are
lower than those estimated (Section 3.4), the conclusion of this study remains
qualitatively unchanged. The higher Eddington ratios compared to other
populations suggest that the SMBHs in BluDOGs are in the most rapidly evolving
phase during the whole evolutionary history of SMBHs. In the gas-rich major
merger scenario of Hopkins et al. (2008), the peak of the AGN activity (i.e.,
the mass growth of SMBHs) corresponds to the transition phase from the
optically thick to optically thin quasars, where the surrounding dust is blown
out by the powerful AGN activity. Note that optically thick quasars in the
major merger scenario should be recognized as type-2 quasars in optical (the
BLR cannot be observed due to the heavy dust reddening). Since optically-thick
quasars in the final stage of the evolution can be recognized as both type-1
and type-2 due to the orientation effect toward the dusty torus, the observed
type-1 nature suggests the object is not in the early (optically-thick) stage
in the major merger evolutionary scenario. Preferentially in AGNs with high
$\lambda_{\rm Edd}$, a blue-wing feature tends to be observed (e.g., Aoki et
al. 2005; Komossa et al. 2008). The observed characteristics of the BluDOGs
such as the type-1 nature and the blue-wing feature of the C iv velocity
profile are consistent with the picture that BluDOGs are in such a peak stage
of the SMBH evolution.
To discuss the evolutionary relation among populations of dusty galaxies
(BluDOGs, core ERQs, and hot DOGs), we focus on $E(B-V)$ and $kt_{80}$.
$E(B-V)$ for BluDOGs, core ERQs (Hamann et al., 2017), and hot DOGs (Wu et
al., 2018) are $0.273\pm 0.049$, $0.242\pm 0.127$, and $4.781\pm 1.986$,
respectively. The $E(B-V)$ of the hot DOGs is significantly larger than that
of the BluDOG and core ERQ samples, suggesting the hot DOGs are thought to be
in a heavily obscured phase. Since the $kt_{80}$ of BluDOGs is smaller than
that of core ERQs (Section 4.1), and the $kt_{80}$ of Mid-IR detected quasars
is close to that of BluDOGs (Figure 1 of Monadi & Bird 2022), the BluDOG phase
is thought to be close to the optically-thin quasar phase. Therefore, it is
suggested that the evolutionary path of various AGN populations is “Hot DOGs –
core ERQs – BluDOGs – optically-thin quasars”.
For AGNs in general, the mass accretion efficiency ($\eta$) is defined as
following:
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{L_{\rm
bol}}{\dot{M}c^{2}},$ (5)
where $\dot{M}$ is the mass accretion rate. By multiplying the $\dot{M}$ and
lifetime of BluDOGs ($t_{\rm life}$), we can roughly estimate the accreted
mass ($M_{\rm Acc}$) in the BluDOGs phase. Bian & Zhao (2003) estimated
$\log\eta=-1.61$ of Seyfert 1 galaxies and Palomar-Green quasars by assuming
that the geometrically-thin and optically-thick standard $\alpha$-prescription
accretion disk model (Shakura & Sunyaev 1973). By assuming $\log\eta=-1.61$
and $t_{\rm life}=1$ Myr (Noboriguchi et al., 2019), the estimated $M_{\rm
Acc}$ of J0907, J1202, J1207, and J1443 are about $1.68\times 10^{7}$,
$1.68\times 10^{8}$, $2.19\times 10^{7}$, and $6.93\times 10^{7}\ M_{\odot}$.
The SMBH masses reached when the SMBH masses of the BluDOGs are increased by
the observed mass accretion rate during the typical lifetime of BluDOGs
($M^{+}_{\rm BH}=M_{\rm BH}+M_{\rm Acc}$) of J0907, J1202, J1207, and J1443
are $1.86\times 10^{8}$, $6.63\times 10^{8}$, $1.32\times 10^{8}$, and
$6.17\times 10^{8}\ M_{\odot}$, respectively. Therefore the SMBH mass of
BluDOGs increases by $\sim$20% during the short BluDOG phase, suggesting that
BluDOGs are actually in a rapidly glowing phase.
Figure 12 shows the Eddington ratios of various populations of AGNs as a
function of redshift. The excess of $\lambda_{\rm Edd}$ of the four BluDOGs is
more significant than the scatter of the $\lambda_{\rm Edd}$ distribution,
suggesting that BluDOGs are a special class of AGNs that harbor SMBHs in the
most actively evolving phase. Then, why such a class of AGNs is found only in
a limited redshift range, $2.2<z_{\rm sp}<3.3$? A possible reason comes from
their selection criteria, as briefly mentioned in Sec 4.1. Since the BluDOGs
are selected by the blue excesses which are largely caused by the contribution
of strong BLR emission lines, the resultant redshift distribution would be
biased such that the blue bands contain strong emission lines. It is also not
clear whether the whole population of DOGs have systematically larger
$\lambda_{\rm Edd}$ than ordinary type-1 quasars, due to the paucity of the
spectroscopic data. In order to reveal the total picture of the dust-
enshrouded evolution of SMBHs, more systematic spectroscopic observations for
various populations of BluDOGs and DOGs are needed.
Figure 12: The redshift vs. the Eddington ratio diagram. The filled-blue stars
and hexagons are the same as in Figure 11. The gray 2D histogram represents
the number density of SDSS quasars (Shen et al., 2011). The red plots show the
mean and standard deviation of $\lambda_{\rm Edd}$ in redshift bins with the
width of $\Delta z=0.5$. The numbers shown at the upper part denote the
numbers of SDSS quasars in the redshift bins.
## 5 CONCLUSION
We carried out spectroscopic observations of the four BluDOGs selected by
Noboriguchi et al. (2019) using Subaru/FOCAS and VLT/FORS2. The analysis of
the obtained spectroscopic data revealed the following spectroscopic
properties of the BluDOGs:
1. 1.
The rest-frame UV spectra of the BluDOGs show broad ($\gtrsim$2000 ${\rm km\
s^{-1}}$) emission lines. This suggests that the BLRs of the BluDOGs are not
completely obscured, albeit the very dusty nature inferred from their optical-
IR SED.
2. 2.
The C iv lines of the BluDOGs show a significant blue wing, which is more
prominent than in ordinary SDSS type-1 quasars. This suggests a presence of
powerful nuclear outflow at the spatial scale of the BLR in the BluDOGs.
3. 3.
The REWs of their BLR lines are very large, REW(C iv)$\sim$160 Å, $\sim$7
times larger than the average of SDSS type-1 quasars. Such strong lines cause
the flux excess of the two BluDOGs in the HSC $g$\- and $r$-bands, while blue
continuum emission also contributes the blue excess in the remaining two
objects. The large REWs are not explained by the Baldwin effect. A possible
origin is a powerful nuclear outflow in the BluDOGs causing a selective
obscuration of the nuclear region, as suggested for ERQs.
4. 4.
The Eddington ratios of the BluDOGs are higher than 1.0 and are systematically
larger than other AGN populations. The mass accretion onto the SMBH in BluDOGs
is in the mode of the Eddington-limit or super-Eddington accretion.
All of the above results support the scenario that BluDOGs represent a
population of AGNs in the transition phase from optically thick to optically
thin quasars, i.e., in the blowing-out phase of the major-merger scenario for
the SMBH evolution. The spectroscopic properties of the BluDOGs are similar to
those of ERQs. For further understandings of the complete picture, more
systematic spectroscopic observations are crucial, not only of BluDOGs but
also of the whole population of DOGs.
The authors gratefully acknowledge the anonymous referee for a careful reading
of the manuscript and very helpful comments. This study is based on data
collected at Subaru Telescope, which is operated by the National Astronomical
Observatory of Japan, and observations collected at the European Southern
Observatory under a ESO programme 0102.B-0614(A). We are honored and grateful
for the opportunity of observing the Universe from Maunakea, which has the
cultural, historical and natural significance in Hawaii. The Hyper Suprime-Cam
(HSC) collaboration includes the astronomical communities of Japan and Taiwan,
and Princeton University. The HSC instrumentation and software were developed
by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute
for the Physics and Mathematics of the Universe (Kavli IPMU), the University
of Tokyo, the High Energy Accelerator Research Organization (KEK), the
Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA),
and Princeton University. Funding was contributed by the FIRST program from
Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science
and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS),
Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ,
Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of
software developed for the Large Synoptic Survey Telescope. We thank the LSST
Project for making their code available as free software at
http://dm.lsstcorp.org. The Pan-STARRS1 Surveys (PS1) have been made possible
through contributions of the Institute for Astronomy, the University of
Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its
participating institutes, the Max Planck Institute for Astronomy, Heidelberg
and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns
Hopkins University, Durham University, the University of Edinburgh, Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation under Grant No. AST-1238877, the
University of Maryland, and Eotvos Lorand University (ELTE) This publication
has made use of data from the VIKING survey from VISTA at the ESO Paranal
Observatory, program ID 179.A-2004. Data processing has been contributed by
the VISTA Data Flow System at CASU, Cambridge and WFAU, Edinburgh. This
publication makes use of data products from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration. Herschel is an
ESA space observatory with science instruments provided by European-led
Principal Investigator consortia and with important participation from NASA.
This research made use of Astropy,666http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013, 2018). We would like to thank Enago (www.enago.jp) for the English
language review and Kohei Iwashita for helpful discussion. This study was
financially supported by the Japan Society for the Promotion of Science (JSPS)
KAKENHI grant Nos. 19J10458, 21H01126 (A.N.), 16H03958, 17H01114, 19H00697,
20H01949 (T.N.), 18J01050, 19K14759, 22H01266 (Y. Toba), 20H01939 (K.I.), and
20K04014 (Y. Terashima).
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|
# DiProber: Using Dual Probing to Estimate Tor Relay Capacities in Underloaded
Networks
Hussein Darir University of Illinois at
Urbana-Champaign
<EMAIL_ADDRESS>Nikita Borisov University of Illinois at
Urbana-Champaign
<EMAIL_ADDRESS>Geir Dullerud University of Illinois at
Urbana-Champaign
<EMAIL_ADDRESS>
###### Abstract
Tor is the most popular anonymous communication network. It has millions of
daily users seeking privacy while browsing the internet. It has thousands of
relays to route and anonymize the source and destinations of the users
packets. To create a path, Tor authorities generate a probability distribution
over relays based on the estimates of the capacities of the relays. An
incoming user will then sample this probability distribution and choose three
relays for their paths. The estimates are based on the bandwidths of
observation probes the authority assigns to each relay in the network. Thus,
in order to achieve better load balancing between users, accurate estimates
are necessary. Unfortunately, the currently implemented estimation algorithm
generate inaccurate estimates causing the network to be under utilized and its
capacities unfairly distributed between the users paths. We propose
$\mathit{DiProber}$, a new relay capacity estimation algorithm. The algorithm
proposes a new measurement scheme in Tor consisting of two probes per relay
and uses maximum likelihood to estimate their capacities. We show that the new
technique works better in the case of under-utilized networks where users tend
to have very low demand on the Tor network.
## I Introduction
Tor [7] is the most popular anonymous communication network, with several
million estimated daily users [17]. It offers users a way to communicate
online while preserving their identity and relationship to third parties. Tor
operates by using a network of volunteer _relays_ to forward an encrypted
version of users’ traffic in order to obscure the source and/or the
destination of network traffic. To ensure consistent performance, users’
traffic needs to be load-balanced across the relays. The network capacities of
the relays are quite heterogeneous, spanning many orders of magnitude; an
accurate estimate of these capacities is an important input to the load-
balancing process.
Initially Tor relied on relays’ own measurements of their own capacities to
generate estimates ; this, however, created the possibility of low-resource
attacks on the Tor network [2]. This motivated the development of
$\mathit{TorFlow}$, a bandwidth monitoring system [12]. $\mathit{TorFlow}$
uses external probes to monitor the performance of individual relays and uses
this value to adjust the bandwidth value reported by the relay itself. The
capacity estimates produced by $\mathit{TorFlow}$, however, vary considerably
over time and between different $\mathit{TorFlow}$ instance, impacting the
traffic allocation algorithm’s ability to properly balance load across relays.
A new estimation algorithm based on maximum likelihood estimation was also
developed, $\mathit{MLEFlow}$ [5]. While the estimation accuracy of this new
algorithm showed a lot of promises, the probabilistic model used to derive the
estimator had many simplifying assumptions that are not necessarily true in
practise. Notably, the model used assumed that users utilize all the bandwidth
allocated to them while in practise users’ demand fluctuate and is generally
lower than the available bandwidth. Moreover, while the estimation algorithm
used had significantly lower estimation error for exit relays, it still led to
relatively high error for guard and middle relays.
We propose a new maximum likelihood estimator based on the work of
$\mathit{MLEFlow}$, where we relax the assumption aforementioned about clients
usage of the available bandwidth. The developed algorithm proposes a new
measurement scheme of two probes per relay and uses the results of both
measurements to identify whether a relay is bottlenecked or not during each
epoch. Then, depending on the case, the algorithm uses a distinct
probabilistic model relating measurements to actual capacity and performs
maximum likelihood estimation. We derive analytical bounds of convergence for
the estimates. We then validate the results of our analysis in flow-based
Python simulations, where we simulate both loaded and under-loaded networks
and show the benefits of using the new measurements scheme.
## II Path allocation in Tor
The current Tor network consists of around $6000$ relays [16] that are used to
forward user traffic. To create a connection, a user chooses a _path_ of three
different relays to construct a circuit that forwards traffic in both
directions. Only the user knows the entire path; the relays know only their
predecessor and successor, obscuring the relationship between clients and
destinations. The traffic is also encrypted / decrypted at each node to hide
the correspondence between incoming and outgoing traffic from a network
observer.
Figure 1: Tor relay selection and capacity estimation. Users select three
relays from collections $G$, $G\cup M$, and $E$ using respective weight
vectors $w_{t}^{g},w_{t}^{m}$, and $w_{t}^{e}$, to form a _path_ (black
arrows). The bandwidth authority collects measurements $m_{t}$ of each relay
(red arrows), and updates the capacity estimates $C_{t+1}$ in the next
consensus document, which are then used to generate new weight vectors
$w^{x}_{t+1}$.
Relays in Tor have heterogeneous capabilities and have network capacity111By
“capacity” we refer to the smaller of upload and download bandwidth limit on
the relay. This may be imposed by the ISP, the network configuration, or
manually configured by the relay operator. In some cases, there may exist
other bottlenecks on the path between two relays but a per-node bandwidth
limit is a common and useful model of network capacity constraints. sizes that
differ by orders of magnitude (see Figure 5). Relays also have different
capabilities and can be divided into three classes: _exits_ , which can be
used in the last position of the path, _guards_ , which can be used in the
first or second position, and _middles_ which can only be used in the second
position [15]. We denote the corresponding sets of relays by $E$, $G$, and
$M$, respectively. To create a path, nodes are sampled from these sets with a
probability proportional to their estimated capacity. For example, if we
define $C[j]$ to be the estimated capacity of relay $j$, then the probability
of choosing relay $j\in E$ as the last node in a path is
$w^{e}[j]=C[j]/\left(\sum_{j^{\prime}\in E}C[j^{\prime}]\right)$; likewise for
guard nodes being chosen in the first position. The middle position can be
chosen from both guard and middle nodes; to balance bandwidth among classes,
guard node capacity is adjusted by a multiplier $W_{mg}$; i.e., a guard node
$j\in G$ is chosen for the middle position with probability:
$w^{m}[j]=\frac{W_{mg}C[j]}{\sum_{j^{\prime}\in
G}W_{mg}C[j^{\prime}]+\sum_{j^{\prime}\in M}C[j^{\prime}]}$
The multiplier is computed as:
$W_{mg}=\frac{\sum_{j^{\prime}\in G}C[j^{\prime}]-\sum_{j^{\prime}\in
M}C[j]}{2\sum_{j^{\prime}\in G}C[j^{\prime}]}$
This is a somewhat simplified presentation that describes the scenario where
exit bandwidth is scarce and there is more guard bandwidth than middle
bandwidth, as is the case in the actual Tor network. (See the Tor Directory
Specification for more details on how other cases would be handled [15].) See
Figure 1 for a description of this process.
It is easy to see that, in this scenario, if the estimated capacities are
equal to the true relay capacities, which we will call $C^{*}[j]$, the
expected number of paths using each exit relay will be proportional to its
bandwidth; likewise, the expected number of paths using each guard and middle
node will be proportional to their bandwidth. Using $X[j]$ to denote the
number of paths on relay $j$, we have:
$\displaystyle E[X[j]]/C^{*}[j]=E[X[j^{\prime}]]/C^{*}[j^{\prime}]\quad$ for
$j,j^{\prime}\in E$ $\displaystyle
E[X[j]]/C^{*}[j]=E[X[j^{\prime}]]/C^{*}[j^{\prime}]\quad$ for $j,j^{\prime}\in
G\cup M$
Thus, in expectation, each path would have the same
bandwidth—$C^{*}[j]/E[X[j]]$ for $j\in E$. Our goal is therefore to estimate
these capacities as accurately as possible.222Note that some research suggests
allocation other than proportional to bandwidth results in better performance
[14, 8]; nevertheless, an accurate capacity estimate is still needed for these
alternative path allocation strategies.
### II-A Capacity Estimation
Each relay estimates its own network capacity by computing the maximum
sustained download and upload bandwidth over a 5-second period over the last 5
days. It reports this value (called the _observed bandwidth_) to directory
authorities, who then compile it across all relays and distribute the
information to the clients in a _consensus_ document, published every hour. We
will use $b_{t}[j]$ to refer to the observed bandwidth of relay $j$ in the
consensus document published at time $t$.
Using the observed bandwidth directly for load-balancing creates the
opportunity for a low-resource attack on the Tor network [2]. In particular, a
relay can publish a high observed bandwidth for itself, which will cause more
clients to choose it, and create more chances for it to break users anonymity.
This motivated the design of $\mathit{TorFlow}$ [12], which used observations
of actual relay performance to estimate capacities, rather than simply
trusting the value reported by the relay itself. In $\mathit{TorFlow}$, a
_bandwidth authority_ creates probe circuits through each relay and downloads
a file of a certain size, measuring the realized bandwidth.333Since Tor does
not allow one-hop circuits, these circuits use two relays: the relay under
measurement and a high-bandwidth relay. We will call this the _measured
bandwidth_ , $m_{t}[j]$. Note that in a perfectly load-balanced network, all
of these observations should be equal, regardless of the chosen relay. The
design of $\mathit{TorFlow}$ uses a PID controller to attempt to bring these
observations into balance.
More specifically, let $C^{\mathit{TF}}_{t}[j]$ be the capacity of relay $j$,
as estimated by $\mathit{TorFlow}$, at time $t$. $\mathit{TorFlow}$ computes
an error term, $e_{t}[j]$ as the difference of the measured bandwidth and the
average measured bandwidth, normalized by the average bandwidth:
$e_{t}[j]=(m_{t}[j]-\bar{m_{t}})/\bar{m_{t}}$
where
$\bar{m_{t}}=\sum_{j\in G\cup M\cup E}m_{t}[j]/\left(|G|+|M|+|E|\right)$
It then computes the new estimate as [11, §3.1]:
$C^{\mathit{TF}}_{t+1}[j]=C^{\mathit{TF}}_{t}[j]\left(1+K_{p}e_{t}[j]+K_{i}\int_{0}^{t}e_{t^{\prime}}[j]dt^{\prime}+K_{d}\frac{de_{t}[j]}{dt}\right)$
The constants $K_{p},K_{i},$ and $K_{d}$ control the proportional, integral,
and derivative components of the PID controller. In the default configuration
of $\mathit{TorFlow}$, $K_{i}=K_{d}=0$ and $K_{p}=1$, so we will call this
version of $\mathit{TorFlow}$ $\mathit{TorFlow\text{-}P}$. In this case the
update equation can be simplified as:
$C^{\mathit{TF}}_{t+1}[j]=C^{\mathit{TF}}_{t}[j]m_{t}[j]/\bar{m_{t}}$
Since the update equation does not have a normalization step, when
$\mathit{TorFlow\text{-}P}$ was enabled in late 2011 in the actual Tor
network, the absolute values of estimated bandwidth grew without bound444As
can be seen on this graph:
https://metrics.torproject.org/totalcw.html?start=2011-06-01&end=2011-12-31.
This caused the Tor network to turn off $\mathit{TorFlow\text{-}P}$ and switch
to using a version of $\mathit{TorFlow}$ that uses adjusted observed bandwidth
instead, called $\mathit{sbws}$. We will call estimates produced by this
version $C^{A}$, with:
$C^{A}_{t+1}[j]=b_{t}[j]m_{t}[j]/\bar{m_{t}}$
This version uses the observed bandwidth published by the relay itself, but
adjusts it down if the observed bandwidth is below average or up if it is
above. It has been in use in Tor since 2012; however, it has a number of
disadvantages. A relay that is not sufficiently loaded may underestimate its
observed bandwidth; this leads to a well-documented ramp-up period of new
relays, where their low observed bandwidth leads to a small estimated capacity
and low load, which in turn leads to low observed bandwidth [6]. But even
established relays see their observed bandwidth change. Figure 2 shows the
observed bandwidth of 10 randomly selected relays over the month of May 2020,
demonstrating that the observed values vary significantly over time.
Figure 2: Variation in observed bandwidth in Tor relays over the month of May
2020: plot of 10 randomly selected relays.
A new method for estimating the capacity of relays based on maximum likelihood
estimation was proposed with $\mathit{MLEFlow}$ [5]. To develop
$\mathit{MLEFlow}$, a probabilistic model relating the actual relay capacities
$C^{*}[j]$’s and the bandwidth measurements $m[j]$’s was derived. In order to
derive this relationship, a number of assumptions was made in [5]:
1. 1.
Relays fall into a single category and each user path goes through only a
single relay.
2. 2.
A synchronized model where time is divided into epochs and user connections
all terminate at the end of each epoch. At the end of the epoch the weight
vector is updated and the new vector is used by all users in the next epoch.
3. 3.
Users arrive to the network randomly following a Poisson process with rate
$\lambda_{s}$, denoted by $\text{Pois}(\lambda_{s})$.
4. 4.
Clients circuits use all the bandwidth allocated to them and are only
bottlenecked by the relays.
The estimates produced by this algorithm, denoted $C^{H}$ are then computed
using:
$C_{t+1}^{H}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
f(\kappa,m_{[t]}[j],w_{[t]}[j]),\text{ where}$
$f(\kappa,m_{[t]}[j],w_{[t]}[j])=\prod_{i=0}^{t}\frac{e^{-\lambda_{s}w_{i}[j]}}{\big{(}\frac{\kappa}{m_{i}[j]}-1\big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m_{i}[j]}-1}$
While the algorithm derived showed a lot of promises, the model used to derive
the estimator assumed that users can utilize arbitrary amounts of bandwidth
and are only bottlenecked on the Tor network. In practice, at times the client
demand on Tor is lower than the overall available bandwidth. When simulated in
a low-load scenario, $\mathit{MLEFlow}$ tended to misestimate relays
capacities. Also, due to this assumption, the probabilistic model in [5] is
expected to produce useful results for exit relays since those relays are
generally scarce in the Tor network and are expected to be paths bottleneckes.
This is not true for guard and middle relays. As shown in both Python and
Shadow simulations, guard and middle relays had larger estimation errors when
using $\mathit{MLEFlow}$.
## III $\mathit{DiProber}$: Maximum Likelihood Estimation of Relays
Capacities using dual probing
We propose a new method $\mathit{DiProber}$ for estimating the capacity of
relays based on maximum likelihood estimation (MLE). We use the same
assumptions used to derive $\mathit{MLEFlow}$ while relaxing the last
assumption made. The paths in our model consist of single relays. We assume
that time is divided into epochs and connections terminate at the end of each
epoch. We denote the number of paths passing through the $j^{\mathit{th}}$
relay in the $i^{\mathit{th}}$ epoch by $x_{i}[j]$. Hence, given $w_{i}$, the
number of paths using the $j^{\mathit{th}}$ relay during the $i^{\mathit{th}}$
epoch is a random variable $X_{i}[j]$ with distribution
$\text{Pois}(\lambda_{s}w_{i}[j])$. We design an updated measurement mechanism
based on two measurement probes assigned by the authorities to each relay
instead of just one. The result of the two measurements is used to group the
relays in two different groups, then derive a probabilistic model for each
case and apply MLE. In this section, we will present the new measurement
scheme proposed as well as the probabilistic models derived to relate the
actual capacity of a relay to the measurements obtained. We then derive
analytical guarantees to the new estimation algorithm proposed.
### III-A Measurement mechanism
Currently, Tor authorities assign a measurement probe to each relay in the
network, download a file of a certain size and measure the realized bandwidth
of the probe. Instead of having only one measurement probe for each relay, we
propose having two measurement probes added sequentially to each relay. The
authorities start by activating the first probe and measure its realized
bandwidth, denoted $m^{1}_{i}[j]$. Then, while the first probe is still
active, the authority starts the second probe circuit and measure its
bandwidth, denoted $m^{2}_{i}[j]$.
Using both _non-noisy_ measurements of a relay $j$ at a given epoch $t$, we
can divide the relays in two groups and derive a probabilistic model for each
one. First, we let $C_{client,i}[j]$ be the total bandwidth used by clients
using relay $j$ during epoch $i$.
Case 1: Relay is not bottlenecked by clients and when adding the two probes.
When the clients using relay $j$, during the $i^{th}$ epoch, are not using all
the available bandwidth of the relay we have $C_{client,i}[j]<C^{*}[j]$. After
adding the first probe, in the case where the capacity of the relay was
equally divided between the users and the probe, each path using relay $j$
will have a bandwidth of $\frac{C^{*}[j]}{x_{i}[j]+1}$. Hence all the clients
using this relay will have a total bandwidth of
$x_{i}[j]\frac{C^{*}[j]}{x_{i}[j]+1}$. If
$C_{client,i}[j]<x_{i}[j]\frac{C^{*}[j]}{x_{i}[j]+1}$, then client utilization
is not affected by adding the first probe and the probe will use all the
remaining _unused capacity_ of the relay, notably
$m^{1}_{i}[j]=C^{*}[j]-C_{client,i}[j]$. The same logic is true when adding
the second probe. If $C_{client,i}[j]<x_{i}[j]\frac{C^{*}[j]}{x_{i}[j]+2}$
then client utilization is not affected by adding the second probe. The
remaining _unused capacity_ will then be equally divided between the two
probes, notably
$m^{2}_{i}[j]=\frac{C^{*}[j]-C_{client,i}[j]}{2}=\frac{m^{1}_{i}[j]}{2}$.
Figure 3 illustrate the idea aforementioned.
Figure 3: (Case 1) Relay not bottlenecked by users.
Case 2: Relay bottlenecked by clients or when adding any of the two probes.
1. (a)
If the clients are using all the available capacity of the relay there will be
no remaining unused capacity. Hence, as in [5], the capacity of the relay will
be divided equally between all the paths going through it, notably
$m^{1}_{i}[j]=\frac{C^{*}[j]}{x_{i}[j]+1}$ and
$m^{2}_{i}[j]=\frac{C^{*}[j]}{x_{i}[j]+2}$. Figure 4 illustrates this case.
2. (b)
In this case, the clients are not using all the available capacity of relay.
However, when adding the first probe, the client utilization can be affected
if $C_{client,i}[j]>x_{i}[j]\frac{C^{*}[j]}{x_{i}[j]+1}$. In other words,
clients will then be using all the capacity available to them after the first
probe was added and there will be no remaining unused capacity. Thus as case
(a), the capacity of the relay will be divided equally between all the paths
going through it, notably $m^{1}_{i}[j]=\frac{C^{*}[j]}{x_{i}[j]+1}$ and
$m^{2}_{i}[j]=\frac{C^{*}[j]}{x_{i}[j]+2}$.
3. (c)
In this case, clients are not using all the available capacity of the relay,
nor adding the first probe will affect their utilization. Thus, the first
probe will be first assigned all the remaining _unused capacity_ ,
$m^{1}_{i}[j]=C^{*}[j]-C_{client,i}[j]$. However, when adding the second
probe, the clients utilization is affected if
$C_{client,i}[j]>x_{i}[j]\frac{C^{*}[j]}{x_{i}[j]+2}$ and there will be no
remaining capacity. Thus, $m^{2}_{i}[j]=\frac{C^{*}[j]}{x_{i}[j]+2}$.
Figure 4: (Case 2) Relay bottlenecked by users.
Thus, if the relay falls under case 1, the second observation of the relay
will be equal to half the first observation obtained and the relationship
between the actual capacity of the relay $C^{*}[j]$ and the observation is
$M^{2}_{i}[j]=\frac{C^{*}[j]-C_{client,i}[j]}{2}$. While for relays that fall
under the second case, we can’t make the same conclusion about $m^{1}_{i}[j]$
and $m^{2}_{i}[j]$. However, for all subcases of case 2, we have a
relationship between $C^{*}[j]$ and $M^{2}_{i}[j]$ with
$M^{2}_{i}[j]=\frac{C^{*}[j]}{X_{i}[j]+2}$.
### III-B MLE capacities estimation
In this section, we will show the method used to compute the maximum
likelihood estimation of relays capacities using the sequence of pairs of non-
noisy measurements and the weights published by the Tor authority. More
specifically, for any relay $j\in[n]$, the MLE estimate of its actual capacity
$C^{*}[j]$ is the maximizer in $\mathcal{C}\subset\mathbb{R}^{n}_{\geq 0}$ of
the probability of observing the full history of measurements
$(m^{1}_{[t]}[j],m^{2}_{[t]}[j])$, given the published weights $w_{[t]}[j]$
over the first $t+1$ periods. First at each epoch $i\in[t]$, and in order to
use the correct probabilistic model, the algorithm compares the values of
$m^{1}_{i}[j]$ and $m^{2}_{i}[j]$ and determine which case of the two cases
discussed in III-A is true for the relay $j$ at the $i^{th}$ epoch.
* •
If $m^{1}_{i}[j]=2m^{2}_{i}[j]$, then
$M^{2}_{i}[j]=\frac{C^{*}[j]-C_{client,i}[j]}{2}$.
* •
If $m^{1}_{i}[j]\neq 2m^{2}_{i}[j]$, then
$M^{2}_{i}[j]=\frac{C^{*}[j]}{X_{i}[j]+2}$
Assuming that we know total client utilization of each relay during each epoch
is not a practical assumption. In this paper, we assume that the average
utilization of a client on the Tor network is known and denoted
$C^{avg}_{client}$. Thus the total utilization of a relay $j$ during the
$i^{th}$ epoch will be $X_{i}[j]C^{avg}_{client}$
We add the superscript $D$ to the capacity estimate to denote that the result
of two probes is considered.
###### Theorem 1 (MLE estimates using dual probing).
For any $j\in[n]$ and $t\in\mathbb{N}$, the MLE estimate of $C^{*}[j]$ given
the weight and observation pairs vectors $w_{[t]}[j]$ and
$(m^{1}_{[t]}[j],m^{2}_{[t]}[j])$ is
$\displaystyle
C_{t+1}^{D}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\prod_{i=0}^{t}f(\kappa,m_{i}[j],w_{i}[j]),\text{ where}$ (1)
$f(\kappa,m_{i}[j],w_{i}[j])=$
$\left\\{\begin{array}[]{ll}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(\frac{\kappa-
X_{i}[j]C^{avg}_{client}}{2}=m^{2}_{i}[j]\right)\text{if}\hskip
2.84544ptm^{1}_{i}[j]=2m^{2}_{i}[j]\\\
\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(\frac{\kappa}{X_{i}[j]+2}=m^{2}_{i}[j]\right)\text{if}\hskip
2.84544ptm^{1}_{i}[j]\neq 2m^{2}_{i}[j]\end{array}\right.$
We now write the probability in equation (1) in terms of the known quantities:
$\lambda_{s}$, $w_{[t]}$, and $m_{[t]}$. For any $j\in[n]$ and
$t\in\mathbb{N}$ the function $f$ of equation (1) can be written as follows:
$f(\kappa,m_{i}[j],w_{i}[j])=$
(5)
###### Definition 1 ($\mathit{DiProber\text{-}WH}$).
For any relay $j\in[n]$ and $t\in\mathbb{N}$, $\mathit{DiProber\text{-}WH}$
updates the weight vector by:
$\displaystyle
w_{t+1}^{\mathit{WH}}[j]=\frac{C_{t}^{D}[j]}{\sum_{k=0}^{n}C_{t}^{D}[k]}.$ (6)
where we use the supersctipt $\mathit{WH}$ in $w_{t}^{\mathit{WH}}$ to
identify $\mathit{DiProber\text{-}WH}$.
Equation (1) does not account for non-modeled noise in the measurements. Once
noise affects $m^{1}_{i}[j]$ and $m^{2}_{i}[j]$, we will not be able to
accurately discern between the two cases.
### III-C One step maximum likelihood
In this section, we look at the special case of only considering the last
relay measurement at each epoch instead of the whole history of measurements.
We add the superscript $D1$ to the capacity estimate to denote that only the
last result of the two probes is considered. Equation 1 can now be written as,
$\displaystyle
C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
f(\kappa,m_{t}[j],w_{t}[j]),\text{ where}$ (7)
$f(\kappa,m_{t}[j],w_{t}[j])$ is defined in Equation 5.
To approximate the maximizer of equation (1), we find $\kappa$ that sets the
derivative of $f$ to zero. The result is given in the following theorem.
###### Theorem 2 (One step MLE closed form).
For any $j\in[n]$ and $t\in\mathbb{N}$, the MLE estimate of $C^{*}[j]$ given
the last weight and observations pair, $w_{t}[j]$ and
($m^{1}_{t}[j],m^{2}_{t}[j]$ is
$\displaystyle C_{t+1}^{D1}[j]$
$\displaystyle=\left\\{\begin{array}[]{ll}\lambda_{s}w_{t}[j]C^{avg}_{client}+2m_{t}^{2}[j],\text{if}\hskip
2.84544ptm^{1}_{i}[j]=2m^{2}_{i}[j]\\\ \\\
m_{t}^{2}[j]\left(\lambda_{s}w_{t}[j]+2\right),\text{if}\hskip
2.84544ptm^{1}_{i}[j]\neq 2m^{2}_{i}[j]\end{array}\right.$ (11)
From Theorem 2, in the case where the relay is not bottlenecked, i.e.
$m^{1}_{i}[j]=2m^{2}_{i}[j]$, the estimated capacity is equal to the sum of
the unused capacity, $2m^{2}_{t}[j]$, and the expected capacity used by
clients, $\lambda_{s}w_{t}[j]C^{avg}_{client}$.
$C_{t+1}^{D1}[j]=\underbrace{\lambda_{s}w_{t}[j]C^{avg}_{client}}_{\text{expected
capacity used by clients}}+\underbrace{m_{t}^{1}[j]}_{\text{capacity unused by
clients}}$
While in the case where the relay is bottlenecked, the estimated capacity is
the result of the product of the observation, $m^{2}_{t}[j]$, and the expected
number of users, $\lambda_{s}w_{t}[j]$. The added two in the denominator
refers to the two probes added.
$C_{t+1}^{D1}[j]=m_{t}^{2}[j]\left(\underbrace{\lambda_{s}w_{t}[j]}_{\text{expected
number of users}}+2\right)$
###### Definition 2 ($\mathit{DiProber\text{-}O}$).
For any relay $j\in[n]$ and $t\in\mathbb{N}$, $\mathit{DiProber\text{-}O}$
updates the weight vector by:
$\displaystyle
w_{t+1}^{\mathit{O}}[j]=\frac{C_{t}^{D1}[j]}{\sum_{k=0}^{n}C_{t}^{D1}[k]}.$
(12)
where we use the superscript $\mathit{O}$ in $w_{t}^{\mathit{O}}$ to identify
$\mathit{DiProber\text{-}O}$.
### III-D Convergence of $\mathit{DiProber\text{-}WH}$ and
$\mathit{DiProber\text{-}O}$ estimates
In this section we show that, starting with any initial weight vector, the
mean of the dual probing algorithm estimates for any relay capacity, whether
considering the full history in every update as in
$\mathit{DiProber\text{-}WH}$ (Definition 1) or only the most recent
measurement in every update as in $\mathit{DiProber\text{-}O}$ (Definition 2),
converges to the actual relay capacity. We also show that the variance of the
estimates when using $\mathit{DiProber\text{-}WH}$ estimation goes to zero as
the number of epochs increases.
###### Theorem 3 (Estimates of both methods converge).
For any $j\in[n]$, $t\in\mathbb{N}$, and a method
$y\in\\{\mathit{TorFlow\text{-}P},\mathit{MLEFlow\text{-}CF}\,\mathit{DiProber\text{-}WH}\\}$,
$\displaystyle\mathbb{E}[C_{t}^{y}[j]]\leq C^{*}[j].$ (13) Moreover, as
$t\rightarrow\infty$, $\displaystyle\mathbb{E}[C_{t}^{y}[j]]\geq
C^{*}[j]\left(1-\frac{1}{\lambda_{s}w^{*}[j]}\right).$ (14)
###### Corollary 4 (More users paths leads to a better convergence).
As the rate of users arrival $\lambda_{s}\rightarrow\infty$, for any
$j\in[n]$, $t\in\mathbb{N}$, and method
$y\in\\{\mathit{DiProber\text{-}O},\mathit{DiProber\text{-}WH}\\}$,
$\mathbb{E}[C_{t}^{y}[j]]\rightarrow C^{*}[j]$.
Furthermore, we show that the variance of the estimates of
$\mathit{DiProber\text{-}WH}$ converge to zero. This shows that
$\mathit{DiProber\text{-}WH}$ provides stable and consistent estimates.
###### Theorem 5 (Convergence of variance of $\mathit{DiProber\text{-}WH}$).
As $t\rightarrow\infty$, $\mathit{Var}[C_{t}^{\mathit{WH}}[j]]\rightarrow 0$.
Hence the variances of $\mathit{DiProber}$ estimates when considering the full
history of weights and measurements are close to the actual capacities.
We will show this experimentally in the next section.
### III-E $\mathit{DiProber}$: Implementation
In order to solve the maximization of Theorem 1, we discretize the bounded
capacity set $\mathcal{C}$ and iteratively find the maximizer of (1). Note
that quantization requires knowing a lower and upper bound on the relay
capacity, which can be estimated based on past observations.
Consider a partition $\bar{\mathcal{C}}$ of $\mathcal{C}$ into bins. The set
$\bar{\mathcal{C}}$ contains the centers of the bins of $\mathcal{C}$. For any
$j\in[n]$, $t\in\mathbb{N}$, and $\kappa\in\bar{\mathcal{C}}$, we define
${L_{t}}(j,\kappa)$ to be:
(18)
the $t^{\mathit{th}}$ term of the sum when taking the $\log$ of (5).
For $\mathit{DiProber\text{-}O}$, since we are only considering the last
measurement, the estimate of the capacity of relay $j$ is computed by
iteratively searching for the maximizer $\kappa$ over the discretized capacity
set $\bar{\mathcal{C}}$ in the following equation:
$\displaystyle
C_{t+1}^{\mathit{O}}[j]:=\max_{\kappa\in\bar{\mathcal{C}}}{L_{t}}(j,\kappa).$
(19)
For $\mathit{DiProber\text{-}WH}$, the sum of ${L_{t}}(j,\kappa)$ over
measurement periods is stored in a variable $S_{t+1}(j,\kappa)$:
${S_{t+1}}(j,\kappa)={S}_{t}(j,\kappa)+{L_{t}}(j,\kappa)$,
with ${S}_{0}(j,\kappa)=0$. Then, the maximum likelihood estimate is computed
by iteratively searching for the maximizer $\kappa$ over the discretized
capacity set $\bar{\mathcal{C}}$ in the following equation:
$\displaystyle
C_{t+1}^{\mathit{WH}}[j]:=\max_{\kappa\in\bar{\mathcal{C}}}{S_{t+1}}(j,\kappa).$
(20)
## IV Analytical Model Simulations: Low Fidelity Experiments
To better understand the properties and performance of
$\mathit{TorFlow\text{-}P}$, $\mathit{MLEFlow}$, $\mathit{sbws}$ and
$\mathit{DiProber}$, we evaluated them using simulations of our analytical
model of the Tor network. These simulations are implemented in Python; we will
make our implementation publicly available at publication time.
We evaluate the performance of the different methods using two metrics: (a)
the accuracy of the relay capacity estimates in a network where clients use
all the bandwidth allocated to them, (b) the accuracy of the relay capacity
estimates in an underloaded network and (c) the amount of bandwidth allocated
to the user paths resulting from the weight vectors generated using the
capacity estimates.
The simulation algorithm we have used is shown in Algorithm 1. The algorithm
takes as input: the Poisson arrival rate of users $\lambda_{s}$, the total
number of measurement periods $T$ to be simulated, a method
$\in\\{\mathit{Actual},\mathit{TorFlow\text{-}P},$
$\mathit{MLEFlow},\mathit{DiProber\text{-}O},\mathit{DiProber\text{-}WH}\\}$
to compute the capacities of the relays from measurements. The algorithm
outputs the bandwidths allocated for users paths and the weight vectors
published over all periods between $0$ and $T$.
The simulation algorithm iterates over measurement periods. In each period
$i$, it generates the total number of users paths $N_{i}$ that will join the
network by sampling a Poisson distribution with rate $\lambda_{s}$ in line 3.
Then, it uses the weight vector $w_{i}$ computed in the previous period as a
probability distribution for the users to choose the relays of their paths
from in line 4. In line 5, it adds the first probe to each relay and uses the
max-min fairness bandwidth allocation algorithm to get the bandwidth allocated
for each path, and thus generate the observation vector $m^{1}_{i}$. In line
6, it adds the second probe and generates $m^{2}_{i}$. After that, it computes
$w_{i+1}$ using the given method in line 7. Finally, it deletes all the paths
for a fresh start of the next period.
Algorithm 1 Low fidelity simulation
1:input: $\lambda_{s},T$, $\mathit{method}\in\\{\mathit{Actual}$,
$\mathit{TorFlow\text{-}P},\mathit{MLEFlow},\mathit{DiProber\text{-}WH},\mathit{DiProber\text{-}O}\\}$.
2:for $i\in[0,...,T]$ do
3: Pick the number of users $N_{i}\sim Poi(\lambda_{s})$.
4: Construct users paths of $3$ relays using $w_{i}$.
5: Compute ${m^{1}_{i}}$ using max-min bandwidth allocation.
6: Compute ${m^{2}_{i}}$ using max-min bandwidth allocation.
7: Compute $w_{i+1}$ based on $m_{i}$ and $w_{i}$ using $\mathit{method}$.
8: Delete all paths in the network.
9:return: $m_{0:T},w_{0:T}$
We consider a network analogous to the current Tor network with 6037 relays as
of June $23^{\mathit{rd}}$ 2020\. The relays are distributed as follows:
$2351$ are guard relays, $2576$ are middle relays, and $1110$ are exit relays
(this includes any relays that have both the Exit and Guard flags set).
Lacking a ground truth, we used the measured capacity in the Tor consensus
document as the actual capacity of the Tor relays in our simulation. The
maximum capacity of all relays was $169\,000$ kb/s, while the total capacity
of the guard, middle and exit relays in the network are around $42.6\times
10^{6}$, $6.7\times 10^{6}$ and $17.7\times 10^{6}$ kb/s respectively. Hence,
the capacity set is $\mathcal{C}=[0,169000]$. The capacity distributions of
the guard, middle, and exit relays are shown in Figure 5.
The max-min bandwidth allocation algorithm [4, 3], in Python assumes that
clients will use the full bandwidth allocated to them. Thus we simulated two
types of networks: (1) a full utilization network using the max-min bandwidth
allocation algorithm where each client path consists of three relays and (2)
an under loaded network scenario taking into account client side bottlenecks.
To simulate this idea, we adjusted the simulation to add a bandwidth cap to
each client flow. This bandwidth cap is enforced by a fourth relay added to
each client flow, with a capacity selected uniformly at random from the
interval [8,18] KB/s. This interval was chosen based on the full utilization
scenario bandwidth distribution. Since the average bandwidth of flows in the
full utilization scenario was approximately 22 KB/s, the cap means that the
flows can utilize at most about 60% of the Tor network capacity.
To use $\mathit{DiProber}$, we need to partition $\mathcal{C}$ into bins. From
Figure 5, we use the same technique used in [5]. We choose the bins to be
intervals of the form $[a^{b-1},a^{b}]$ where $a$ is a strictly positive real
number and $b\in[1,...,b_{max}]$ where
$b_{max}=\lceil\frac{\log(\max(C[j]))}{\log(a)}\rceil$ for $j\in[n]$.
Figure 5: relays capacity distribution.
### IV-A Low-fidelity sims results and analysis
In all of the simulation runs, we chose the number of periods $t$ to be $50$,
the initial weight vector $w_{0}$ to be uniform, the rate of arrival
$\lambda_{s}$ to be $10^{6}$.
#### $\mathit{DiProber\text{-}WH}$ performs better than
$\mathit{DiProber\text{-}O}$
While both algorithms of $\mathit{DiProber}$ outperformed the other
algorithms, $\mathit{DiProber\text{-}WH}$ had a lower average error than
$\mathit{DiProber\text{-}O}$ for all classes of relays. The results are shown
in Figure 6.
Figure 6: Estimation error distribution after the $50^{th}$ measurement period
in a fully utilized network for both $\mathit{DiProber}$ algorithms.
#### $\mathit{DiProber}$ leads to better guard and middle relays estimates in
fully loaded networks
We tested the different estimation algorithms on three-relays paths networks.
Both $\mathit{DiProber}$ algorithms and $\mathit{MLEFlow}$ had lower average
error than $\mathit{TorFlow\text{-}P}$ and $\mathit{sbws}$. The results are
shown in Figure 7. While the errors for exit relays when using
$\mathit{DiProber}$ and $\mathit{MLEFlow}$ were close, $\mathit{DiProber}$
leads to lower average error for guard and middle relays. The average
estimation errors for $\mathit{DiProber}$ and $\mathit{MLEFlow}$ stayed below
$10\%$ for exit relays, while it was higher for $\mathit{TorFlow\text{-}P}$
with $72\%$. For guard and middle relays, the error was lowest for
$\mathit{DiProber\text{-}WH}$ at $18\%$ and $14\%$ respectively. It was higher
for $\mathit{MLEFlow}$ at around $25\%$ error and even higher for
$\mathit{TorFlow\text{-}P}$ at $75\%$.
Figure 7: Estimation error distribution after the $50^{th}$ measurement period
in a fully utilized network.
#### The dual probing algorithm advantage extend to the under-loaded network
scenario:
We simulate the case of under-loaded networks as was done in [5] and as
described above by adding a forth relay to each path created.
$\mathit{DiProber}$ outperforms all other algorithms. The error for all
classes of relays was below $25\%$ for $\mathit{DiProber}$. While the error
for exit relays remained relatively low when using $\mathit{MLEFlow}$, other
classes estimates all had an average error above $45\%$. The results are shown
in Figure 8.
Figure 8: Estimation error distribution after the $50^{th}$ measurement period
in an under-loaded network.
#### $\mathit{DiProber\text{-}WH}$ and $\mathit{DiProber\text{-}O}$ give
better and fairer bandwidth allocation than than the other algorithms
The means of the bandwidths allocated for paths using $\mathit{MLEFlow}$ and
$\mathit{DiProber}$ are equal to that of the Actual scenario, while that of
$\mathit{TorFlow\text{-}P}$ is slightly smaller. The advantage of both those
methods is the stability of their estimates. The standard deviation of
$\mathit{DiProber}$ is a maximum of 1.4, while $\mathit{MLEFlow}$ was 1.7,
when that of Actual is around 0.6 and $\mathit{TorFlow\text{-}P}$ is around
165, orders of magnitude larger. Moreover, the maximum and minimum bandwidths
allocated of our methods are similar to that of Actual while
$\mathit{TorFlow\text{-}P}$ had orders of magnitude larger maximum. That means
that our methods distribute bandwidths more fairly than
$\mathit{TorFlow\text{-}P}$.
## V Related Work
Improving the performance of the Tor network has been the subject of much
research; we refer the reader to the survey by AlSabah and Goldberg for an
overview [1]. Here we summarize related work specifically focusing on relay
capacity estimation.
Snader and Borisov proposed using _opportunistic measurements_ , where each
relay measures the bandwidth of each other relay it communicates with as part
of normal operation, and designed EigenSpeed [13], which combines these
measurements using principal component analysis to derive a single relay
capacity. EigenSpeed was designed to avoid certain types of collusion and
misreporting attacks; however, Johnson et al. [10] discovered that it is
subject to a number of other attacks that allow colluding adversaries to
inflate their bandwidth. They also designed PeerFlow, which is a more robust
mechanism to combine opportunistic measurements from relays with provable
limits on inflation attacks. These bounds, however, depend on having a
fraction of bandwidth being on trusted nodes, and it has slow convergence
properties due to its limitations on changing bandwidth values.
FlashFlow [18] is a new proposal to replace TorFlow. FlashFlow uses several
servers that measure a relay simultaneously, generating a large network load
intended to max out its capacity. FlashFlow has a guaranteed inflation bound
of only 33% but it is based on the assumption that a relay capacity is based
on a hard limit that cannot be exceeded, as TorFlow uses traffic that is
explicitly labeled for for bandwidth probing. In practice, it is often easier
and cheaper to obtain high peak bandwidth capability than sustaining the same
bandwidth continuously.
At the same time, TorFlow probes, though not explicitly labeled, can also be
identified by their distinct characteristics, and can be used to
preferentially forward probe traffic to inflate bandwidth estimates, or to
perform sophisticated denial-of-service attacks [9]. A more stealthy approach
for bandwidth measurement probing remains an open research question. The
simplest and most effective attack on TorFlow, however, is to inflate the
observed bandwidth published by the relay [10]; this attack does not apply to
$\mathit{DiProber}$ as it does not use the observed bandwidth.
## VI Conclusion
We have developed a new method for estimating the relay capacities in the Tor
network, $\mathit{DiProber}$. We show that $\mathit{MLEFlow}$ fails to
accurately estimate relays capacities in under-loaded networks. Our detailed
mathematical analysis showed that $\mathit{DiProber}$ capacity estimates
converge to their true value, while the estimate variance converges to 0, as
the number of observations grows. We validated the performance of
$\mathit{MLEFlow}$ with extensive simulations using a our custom flow-based
simulator. Our results show that $\mathit{DiProber}$ produces much more
accurate estimates of relay capacities, which in turn results in much better
load balancing of user traffic across the network, as compared with current
methods.
## References
* [1] M. AlSabah and I. Goldberg, “Performance and security improvements for Tor: A survey,” _ACM Computing Surveys (CSUR)_ , vol. 49, no. 2, p. 32, 2016.
* [2] K. Bauer, D. McCoy, D. Grunwald, T. Kohno, and D. Sicker, “Low-resource routing attacks against Tor,” in _Proceedings of the 2007 ACM Workshop on Privacy in Electronic Society (WPES)_ , 2007, pp. 11–20.
* [3] H. Darir, “Privacy-preserving network congestion control,” 2019.
* [4] H. Darir, H. Sibai, N. Borisov, G. Dullerud, and S. Mitra, “Tightrope: Towards optimal load-balancing of paths in anonymous networks,” in _Proceedings of the 2018 Workshop on Privacy in the Electronic Society_ , 2018, pp. 76–85.
* [5] H. Darir, H. Sibai, C.-Y. Cheng, N. Borisov, G. Dullerud, and S. Mitra, “Mleflow: Learning from history to improve load balancing in tor,” _Proceedings on Privacy Enhancing Technologies_ , vol. 2022, no. 1, pp. 75–104, 2022. [Online]. Available: https://doi.org/10.2478/popets-2022-0005
* [6] R. Dingledine, “The lifecycle of a new relay,” The Tor Project Blog, https://blog.torproject.org/lifecycle-new-relay, Sep. 2013.
* [7] R. Dingledine, N. Mathewson, and P. F. Syverson, “Tor: The second-generation onion router,” in _USENIX Security Symposium_. USENIX, 2004, pp. 303–320.
* [8] S. Herbert, S. J. Murdoch, and E. Punskaya, “Optimising node selection probabilities in multi-hop m/d/1 queuing networks to reduce latency of tor,” _Electronics letters_ , vol. 50, no. 17, pp. 1205–1207, 2014.
* [9] R. Jansen, T. Vaidya, and M. Sherr, “Point break: a study of bandwidth denial-of-service attacks against Tor,” in _28th USENIX Security Symposium_ , 2019, pp. 1823–1840.
* [10] A. Johnson, R. Jansen, N. Hopper, A. Segal, and P. Syverson, “PeerFlow: Secure load balancing in Tor,” _Proceedings on Privacy Enhancing Technologies_ , vol. 2017, no. 2, pp. 74–94, 2017.
* [11] K. Loesing, M. Perry, and A. Gibson, “Bandwidth scanner specification,” https://gitweb.torproject.org/torflow.git/tree/NetworkScanners/BwAuthority/README.spec.txt, 2011\.
* [12] M. Perry, “TorFlow: Tor network analysis,” in _Proceedings of the 2nd Workshop on Hot Topics in Privacy Enhancing Technologies (HotPETs)_ , 2009, pp. 1–14.
* [13] R. Snader and N. Borisov, “EigenSpeed: Secure peer-to-peer bandwidth evaluation,” in _8th International Workshop on Peer-To-Peer Systems_ , R. Rodrigues and K. Ross, Eds. Berkeley, CA, USA: USENIX Association, Apr. 2009.
* [14] ——, “Improving security and performance in the Tor network through tunable path selection,” _IEEE Transactions on Dependable and Secure Computing_ , vol. 8, no. 5, pp. 728–741, 2011.
* [15] The Tor Project, “Tor directory protocol, version 3,” https://gitweb.torproject.org/torspec.git/tree/dir-spec.txt, 2020.
* [16] ——, “Tor metrics: Servers,” https://metrics.torproject.org/networksize.html, 2020.
* [17] ——, “Tor metrics: Users,” https://metrics.torproject.org/userstats-relay-country.html, 2020.
* [18] M. Traudt, R. Jansen, and A. Johnson, “Flashflow: A secure speed test for tor,” 2020.
### -A Proofs
See 1
###### Proof.
As discussed, at each epoch, a relay can fall into the two cases discussed in
section III-A. We start with Case 1, where the second measurement is equal to
double the first measurement of a relay. When evaluating the objective
function of the MLE, the observation random variable of the $j^{\mathit{th}}$
relay $M_{i}[j]$ can be written as a function of $\kappa$, as if we are
assuming $\kappa=C^{*}[j]$, and the random variable $X_{i}[j]$ for $i\in[t]$:
$M^{2}_{i}[j]=\frac{\kappa-X_{i}[j]C_{Client}^{avg}}{2}.$ (21)
Recall that we assume that the random variable $X_{i}[j]$ follows a Poisson
distribution with parameter $\lambda_{s}w_{i}[j]$ and all users leave at the
end of each epoch. Hence, given $w_{i}[j]$ for $j\in[n]$, the $M_{i}[j]$’s at
different iterations are independent random variables. Thus eq. 21 can be
written as the product of the probability of the independent random variables
$[M_{1}[j],...,M_{t}[j]]$:
$\begin{multlined}C_{t+1}^{H}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\Pr_{X_{[t]}[j]\sim\text{Pois}(\lambda_{s}W_{[t]}[j])}(M^{2}_{[t]}[j]=m^{2}_{[t]}[j]\hskip
2.84544pt|W_{[t]}[j]=w_{[t]}[j])\\\
C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}\Pr(M^{2}_{i}[j]=m^{2}_{i}[j]\hskip 2.84544pt|\hskip
2.84544ptW_{i}[j]=w_{i}[j]).\\\
\end{multlined}C_{t+1}^{H}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\Pr_{X_{[t]}[j]\sim\text{Pois}(\lambda_{s}W_{[t]}[j])}(M^{2}_{[t]}[j]=m^{2}_{[t]}[j]\hskip
2.84544pt|W_{[t]}[j]=w_{[t]}[j])\\\
C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}\Pr(M^{2}_{i}[j]=m^{2}_{i}[j]\hskip 2.84544pt|\hskip
2.84544ptW_{i}[j]=w_{i}[j]).\\\ $
Rearranging eq. 21 results in:
$X_{i}[j]=\frac{\kappa-2M^{2}_{i}[j]}{C_{Client}^{avg}}.$ (22)
When the measurement is made and the observation is fixed, i.e.
$M^{2}_{i}[j]=m^{2}_{i}[j]$, the probability in LABEL:eq:Mal3 can be expressed
in terms of the random variable $X_{i}[j]$:
$x_{i}[j]=\frac{\kappa-2m^{2}_{i}[j]}{C_{Client}^{avg}}$.
$C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}\Pr(X_{i}[j]=x_{i}[j]|\hskip
2.84544ptW_{i}[j]=w_{i}[j])$ (23)
Using the Poisson distribution probability mass function, we can write:
$\begin{multlined}C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{(x_{i}[j])!}(\lambda_{s}w_{i}[j])^{x_{i}[j]}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\big{(}\frac{\kappa-2m^{2}_{i}[j]}{C_{Client}^{avg}}\big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa-2m^{2}_{i}[j]}{C_{Client}^{avg}}}\\\
\end{multlined}C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{(x_{i}[j])!}(\lambda_{s}w_{i}[j])^{x_{i}[j]}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\big{(}\frac{\kappa-2m^{2}_{i}[j]}{C_{Client}^{avg}}\big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa-2m^{2}_{i}[j]}{C_{Client}^{avg}}}\\\
$ (24)
Considering Case 2, the objective function of the MLE, the observation random
variable of the $j^{\mathit{th}}$ relay $M_{i}[j]$ can be written as a
function of $\kappa$, as if we are assuming $\kappa=C^{*}[j]$, and the random
variable $X_{i}[j]$ for $i\in[t]$:
$M^{2}_{i}[j]=\frac{\kappa}{X_{i}[j]+2}.$ (25)
Recall that we assume that the random variable $X_{i}[j]$ follows a Poisson
distribution with parameter $\lambda_{s}w_{i}[j]$ and all users leave at the
end of each epoch. Hence, given $w_{i}[j]$ for $j\in[n]$, the $M^{2}_{i}[j]$’s
at different iterations are independent random variables. Thus eq. 25 can be
written as the product of the probability of the independent random variables
$[M^{2}_{1}[j],...,M^{2}_{t}[j]]$:
$\begin{multlined}C_{t+1}^{H}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\Pr_{X_{[t]}[j]\sim\text{Pois}(\lambda_{s}W_{[t]}[j])}(M_{[t]}[j]=m_{[t]}[j]\hskip
2.84544pt|W_{[t]}[j]=w_{[t]}[j])\\\
C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}\Pr(M_{i}[j]=m_{i}[j]\hskip 2.84544pt|\hskip
2.84544ptW_{i}[j]=w_{i}[j]).\\\
\end{multlined}C_{t+1}^{H}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\Pr_{X_{[t]}[j]\sim\text{Pois}(\lambda_{s}W_{[t]}[j])}(M_{[t]}[j]=m_{[t]}[j]\hskip
2.84544pt|W_{[t]}[j]=w_{[t]}[j])\\\
C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}\Pr(M_{i}[j]=m_{i}[j]\hskip 2.84544pt|\hskip
2.84544ptW_{i}[j]=w_{i}[j]).\\\ $
Rearranging eq. 25 results in:
$X_{i}[j]=\frac{\kappa}{M^{2}_{i}[j]}-2.$ (26)
When the measurement is made and the observation is fixed, i.e.
$M^{2}_{i}[j]=m_{i}[j]$, the probability in LABEL:eq:Maln3 can be expressed in
terms of the random variable $X_{i}[j]$:
$X_{i}[j]=\frac{\kappa}{m^{2}_{i}[j]}-2$.
$C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}\Pr(X_{i}[j]=x_{i}[j]|\hskip
2.84544ptW_{i}[j]=w_{i}[j])$ (27)
Using the Poisson distribution probability mass function, we can write:
$\begin{multlined}C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{(x_{i}[j])!}(\lambda_{s}w_{i}[j])^{x_{i}[j]}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\big{(}\frac{\kappa}{m^{2}_{i}[j]}-2\big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m^{2}_{i}[j]}-2}\\\
\end{multlined}C^{H}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{(x_{i}[j])!}(\lambda_{s}w_{i}[j])^{x_{i}[j]}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\big{(}\frac{\kappa}{m^{2}_{i}[j]}-2\big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m^{2}_{i}[j]}-2}\\\
$ (28)
∎
See 2
###### Proof.
Case 1: As discussed previously, we know that we are in this case if
$m_{i}^{1}[j]=2m_{i}^{2}[j]$. We denote $C_{avg,i}$ the average bandwidth used
by each client in the network during the $i^{th}$ epoch. Hence since the relay
is not bottlenecked, we can say that
$m_{i}^{1}[j]=C^{*}[j]-x_{i}[j]C_{avg,i}$; hence, given $w_{i}$, the
measurement of the $j^{\mathit{th}}$ relay at the $i^{\mathit{th}}$ iteration
is a random variable $M_{i}^{1}[j]=C^{*}[j]-X_{i}[j]C_{avg,i}$.
W will use superscript $D1$ to indicate that we are only using the most recent
value of $m_{t}[j]$ and $w_{t}[j]$. Using maximum likelihood estimation, we
have
$\displaystyle
C_{t+1}^{R}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
f(\kappa,m_{t}[j],w_{t}[j]),\text{ where}$ (29) $\displaystyle
f(\kappa,m_{t}[j],w_{t}[j])=\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(\kappa-
X_{t}[j]C_{avg,t}=m_{i}^{1}[j]\right),$ (30)
$\begin{multlined}C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(\kappa-
X_{t}[j]C_{avg,t}=m_{t}^{1}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\exp\left(-\lambda_{s}w_{t}[j]\right)\frac{1}{\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)!}\left(\lambda_{s}w_{t}[j]\right)^{\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\log\left(\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)!\right)+\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\
\end{multlined}C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(\kappa-
X_{t}[j]C_{avg,t}=m_{t}^{1}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\exp\left(-\lambda_{s}w_{t}[j]\right)\frac{1}{\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)!}\left(\lambda_{s}w_{t}[j]\right)^{\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\log\left(\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)!\right)+\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\ $
Using Stirling approximation for the second term, we get
$\begin{multlined}C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\\\ +\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)+\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\
\end{multlined}C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\\\ +\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)+\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\ $
In order to find maximum, we differentiate the right side with respect to
$\kappa$ and equate it to zero:
$\begin{multlined}-\frac{1}{C_{avg,t}}\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)-\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\frac{\frac{1}{C_{avg,t}}}{\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)}+\frac{1}{C_{avg,t}}+\frac{1}{C_{avg,t}}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\frac{1}{C_{avg,t}}\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)-\frac{1}{C_{avg,t}}+\frac{1}{C_{avg,t}}+\frac{1}{C_{avg,t}}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)+\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
\log\left(\frac{C_{avg,t}\lambda_{s}w_{t}[j]}{\kappa-m_{t}^{1}[j]}\right)=0\\\
\frac{C_{avg,t}\lambda_{s}w_{t}[j]}{\kappa-m_{t}^{1}[j]}=1\\\
\end{multlined}-\frac{1}{C_{avg,t}}\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)-\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)\frac{\frac{1}{C_{avg,t}}}{\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)}+\frac{1}{C_{avg,t}}+\frac{1}{C_{avg,t}}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\frac{1}{C_{avg,t}}\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)-\frac{1}{C_{avg,t}}+\frac{1}{C_{avg,t}}+\frac{1}{C_{avg,t}}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\log\left(\frac{\kappa-
m_{t}^{1}[j]}{C_{avg,t}}\right)+\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
\log\left(\frac{C_{avg,t}\lambda_{s}w_{t}[j]}{\kappa-m_{t}^{1}[j]}\right)=0\\\
\frac{C_{avg,t}\lambda_{s}w_{t}[j]}{\kappa-m_{t}^{1}[j]}=1\\\ $
Thus $\kappa=\underbrace{\lambda_{s}w_{t}[j]C_{avg,t}}_{\text{expected
bandwidth used by clients}}+\underbrace{m_{t}^{1}[j]}_{\text{bandwidth left
unused}}$
Case 2: We know that we are in this case if $m_{i}^{1}[j]\neq 2m_{i}^{2}[j]$.
Since the relay is bottlenecked when we add the second probe in all cases
described previously, we can say that
$m_{i}^{2}[j]=\frac{C^{*}[j]}{x_{i}[j]+2}$; hence, given $w_{i}$, the
measurement of the $j^{\mathit{th}}$ relay at the $i^{\mathit{th}}$ iteration
is a random variable $M_{i}^{2}[j]=\frac{C^{*}[j]}{X_{i}[j]+2}$.
Using maximum likelihood estimation, we have
$\displaystyle
C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
f(\kappa,m_{t}[j],w_{t}[j]),\text{ where}$ (31) $\displaystyle
f(\kappa,m_{t}[j],w_{t}[j])=\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(\frac{\kappa}{X_{i}[j]+2}=m_{i}^{2}[j]\right),$
(32)
$\begin{multlined}C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(\frac{\kappa}{X_{i}[j]+2}=m_{t}^{2}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa}{m_{t}^{2}[j]}-2\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa}{m_{t}^{2}[j]}-2\right)\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\exp\left(-\lambda_{s}w_{t}[j]\right)\frac{1}{\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)!}\left(\lambda_{s}w_{t}[j]\right)^{\frac{\kappa}{m_{t}^{2}[j]}-2}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\log\left(\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)!\right)+\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\
\end{multlined}C_{t+1}^{D1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(\frac{\kappa}{X_{i}[j]+2}=m_{t}^{2}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa}{m_{t}^{2}[j]}-2\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\Pr_{X_{t}[j]\sim\text{Pois}(\lambda_{s}w_{t}[j])}\left(X_{t}[j]=\frac{\kappa}{m_{t}^{2}[j]}-2\right)\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\exp\left(-\lambda_{s}w_{t}[j]\right)\frac{1}{\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)!}\left(\lambda_{s}w_{t}[j]\right)^{\frac{\kappa}{m_{t}^{2}[j]}-2}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\log\left(\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)!\right)+\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\
$
Using Stirling approximation for the second term, we get
$\begin{multlined}C_{t+1}^{R}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)+\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)+\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\
\end{multlined}C_{t+1}^{R}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
-\lambda_{s}w_{t}[j]-\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)+\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)+\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\log\left(\lambda_{s}w_{t}[j]\right)\\\
$
In order to find the optimum, we differentiate the right side with respect to
$\kappa$ and equate it to zero:
$\begin{multlined}-\frac{1}{m_{t}^{2}[j]}\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)-\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\frac{\frac{1}{m^{2}_{t}[j]}}{\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)}+\frac{1}{m_{t}^{2}[j]}+\frac{1}{m_{t}^{2}[j]}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\frac{1}{m_{t}^{2}[j]}\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)-\frac{1}{m_{t}^{2}[j]}+\frac{1}{m_{t}^{2}[j]}+\frac{1}{m_{t}^{2}[j]}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)+\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
\log\left(\frac{\lambda_{s}w_{t}[j]m_{t}^{2}[j]}{\kappa-2m_{t}^{2}[j]}\right)=0\\\
\frac{\lambda_{s}w_{t}[j]m_{t}^{2}[j]}{\kappa-2m_{t}^{2}[j]}=1\\\
\end{multlined}-\frac{1}{m_{t}^{2}[j]}\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)-\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)\frac{\frac{1}{m^{2}_{t}[j]}}{\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)}+\frac{1}{m_{t}^{2}[j]}+\frac{1}{m_{t}^{2}[j]}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\frac{1}{m_{t}^{2}[j]}\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)-\frac{1}{m_{t}^{2}[j]}+\frac{1}{m_{t}^{2}[j]}+\frac{1}{m_{t}^{2}[j]}\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
-\log\left(\frac{\kappa}{m_{t}^{2}[j]}-2\right)+\log\left(\lambda_{s}w_{t}[j]\right)=0\\\
\log\left(\frac{\lambda_{s}w_{t}[j]m_{t}^{2}[j]}{\kappa-2m_{t}^{2}[j]}\right)=0\\\
\frac{\lambda_{s}w_{t}[j]m_{t}^{2}[j]}{\kappa-2m_{t}^{2}[j]}=1\\\ $
Thus
$\kappa=m_{t}^{2}[j]\left(\underbrace{\lambda_{s}w_{t}[j]}_{\text{expected
number of users}}+2\right)$
∎
See 3
###### Proof.
We start by considering a relay that falls into case 1 for its whole operation
and hence is never bottlenecked. From Theorem 1, we can write:
$\begin{multlined}C_{t+1}^{D}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\prod_{i=0}^{t}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(\kappa-
X_{i}[j]C_{avg}=m_{i}^{1}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\prod_{i=0}^{t}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(X_{i}[j]=\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\prod_{i=0}^{t}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(X_{i}[j]=\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\prod_{i=0}^{t}\exp\left(-\lambda_{s}w_{i}[j]\right)\frac{1}{\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)!}\left(\lambda_{s}w_{i}[j]\right)^{\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]-\log\left(\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)!\right)+\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\log\left(\lambda_{s}w_{i}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]-\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)+\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)+\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\log\left(\lambda_{s}w_{i}[j]\right)\\\
\end{multlined}C_{t+1}^{D}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\prod_{i=0}^{t}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(\kappa-
X_{i}[j]C_{avg}=m_{i}^{1}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\prod_{i=0}^{t}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(X_{i}[j]=\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\prod_{i=0}^{t}\Pr_{X_{i}[j]\sim\text{Pois}(\lambda_{s}w_{i}[j])}\left(X_{i}[j]=\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\
\log\left(\prod_{i=0}^{t}\exp\left(-\lambda_{s}w_{i}[j]\right)\frac{1}{\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)!}\left(\lambda_{s}w_{i}[j]\right)^{\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}}\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]-\log\left(\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)!\right)+\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\log\left(\lambda_{s}w_{i}[j]\right)\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]-\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)+\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)+\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\log\left(\lambda_{s}w_{i}[j]\right)\\\ $
We differentiate the right side with respect to $\kappa$ and equate it to
zero:
$\begin{multlined}\sum_{i=0}^{t}-\frac{1}{C_{avg}}\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)-\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\frac{\frac{1}{C_{avg}}}{\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)}+\frac{1}{C_{avg}}+\frac{1}{C_{avg}}\log\left(\lambda_{s}w_{i}[j]\right)=0\\\
\sum_{i=0}^{t}-\frac{1}{C_{avg}}\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)-\frac{1}{C_{avg}}+\frac{1}{C_{avg}}+\frac{1}{C_{avg}}\log\left(\lambda_{s}w_{i}[j]\right)=0\\\
\sum_{i=0}^{t}-\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)+\log\left(\lambda_{s}w_{i}[j]\right)=0\\\
\end{multlined}\sum_{i=0}^{t}-\frac{1}{C_{avg}}\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)-\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)\frac{\frac{1}{C_{avg}}}{\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)}+\frac{1}{C_{avg}}+\frac{1}{C_{avg}}\log\left(\lambda_{s}w_{i}[j]\right)=0\\\
\sum_{i=0}^{t}-\frac{1}{C_{avg}}\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)-\frac{1}{C_{avg}}+\frac{1}{C_{avg}}+\frac{1}{C_{avg}}\log\left(\lambda_{s}w_{i}[j]\right)=0\\\
\sum_{i=0}^{t}-\log\left(\frac{\kappa-
m_{i}^{1}[j]}{C_{avg}}\right)+\log\left(\lambda_{s}w_{i}[j]\right)=0\\\ $
Using linearization to solve for the optimum $\kappa$, we have that
$\begin{multlined}f^{\prime}_{t}(\kappa)=\sum_{i=0}^{t}-\log\left(\kappa-
m_{i}^{1}[j]\right)+\sum_{i=0}^{t}\log\left(\lambda_{s}w_{i}[j]C_{avg}\right)\\\
=f_{t-1}^{\prime}(\kappa)+\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa-
m_{t}^{1}[j]\right)\\\
\end{multlined}f^{\prime}_{t}(\kappa)=\sum_{i=0}^{t}-\log\left(\kappa-
m_{i}^{1}[j]\right)+\sum_{i=0}^{t}\log\left(\lambda_{s}w_{i}[j]C_{avg}\right)\\\
=f_{t-1}^{\prime}(\kappa)+\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa-
m_{t}^{1}[j]\right)\\\ $
We know that $f^{\prime}_{t}(\kappa_{t})=0$ and
$f^{\prime}_{t-1}(\kappa_{t-1})=0$ thus
$\begin{multlined}f^{\prime}_{t}(\kappa_{t-1})=f_{t-1}^{\prime}(\kappa_{t-1})+\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa_{t-1}-m_{t}^{1}[j]\right)\\\
=\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa_{t-1}-m_{t}^{1}[j]\right)\\\
\end{multlined}f^{\prime}_{t}(\kappa_{t-1})=f_{t-1}^{\prime}(\kappa_{t-1})+\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa_{t-1}-m_{t}^{1}[j]\right)\\\
=\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa_{t-1}-m_{t}^{1}[j]\right)\\\
$
We also have $f^{\prime\prime}_{t}(\kappa)=-\sum_{i=0}^{t}\frac{1}{\kappa-
m_{i}^{1}[j]}=f^{\prime\prime}_{t-1}(\kappa)-\frac{1}{\kappa-m_{t}^{1}[j]}$.
By linearization we have
$f^{\prime}_{t}(\kappa_{t})=f^{\prime}_{t}(\kappa_{t-1})+f^{\prime\prime}_{t}(\kappa_{t-1})(\kappa_{t}-\kappa_{t-1})$
and thus,
$\begin{multlined}\kappa_{t}=\frac{f^{\prime}_{t}(\kappa_{t})-f^{\prime}_{t}(\kappa_{t-1})}{f^{\prime\prime}_{t}(\kappa_{t-1})}+\kappa_{t-1}\end{multlined}\kappa_{t}=\frac{f^{\prime}_{t}(\kappa_{t})-f^{\prime}_{t}(\kappa_{t-1})}{f^{\prime\prime}_{t}(\kappa_{t-1})}+\kappa_{t-1}$
Thus we can find an iterative solution of the optimization,
$\begin{multlined}\kappa_{0}=m_{0}^{1}[j]+\lambda_{s}w_{0}[j]C_{avg}\hskip
8.53581pt\text{and}\\\
\kappa_{t}=\frac{\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa_{t-1}-m_{t}^{1}[j]\right)}{\sum_{i=0}^{t}\frac{1}{\kappa_{t-1}-m_{i}^{1}[j]}}+\kappa_{t-1}\end{multlined}\kappa_{0}=m_{0}^{1}[j]+\lambda_{s}w_{0}[j]C_{avg}\hskip
8.53581pt\text{and}\\\
\kappa_{t}=\frac{\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa_{t-1}-m_{t}^{1}[j]\right)}{\sum_{i=0}^{t}\frac{1}{\kappa_{t-1}-m_{i}^{1}[j]}}+\kappa_{t-1}$
Finding the steady state convergence of the above iterative formulation:
$\begin{multlined}\kappa=\frac{\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa-
m_{t}^{1}[j]\right)}{\sum_{i=0}^{t}\frac{1}{\kappa-m_{i}^{1}[j]}}+\kappa\\\
\log(\lambda_{s}w_{t}[j]C_{avg})=\ log(\kappa-m_{t}^{1}[j])\\\
\lambda_{s}w_{t}[j]C_{avg}=\kappa-m_{t}^{1}[j]\\\
\end{multlined}\kappa=\frac{\log\left(\lambda_{s}w_{t}[j]C_{avg}\right)-\log\left(\kappa-
m_{t}^{1}[j]\right)}{\sum_{i=0}^{t}\frac{1}{\kappa-m_{i}^{1}[j]}}+\kappa\\\
\log(\lambda_{s}w_{t}[j]C_{avg})=\ log(\kappa-m_{t}^{1}[j])\\\
\lambda_{s}w_{t}[j]C_{avg}=\kappa-m_{t}^{1}[j]\\\ $ (33)
Hence the expected value of the estimate $E(\kappa_{t})=C^{*}[j]$ since the
right hand side of Equation 33 is equal to the actual capacity of the relay if
it was never bottlenecked on average.
Now we consider case 2, where the relay is always bottlenecked for its whole
measurement history. In this proof we will drop the upperscript $2$ from the
measurement since we are only dealing with the second measurement of a
relay.As we derived in eq. 28, we know that for any $j\in[n]$, the weight at
iteration $(t+1)$ should satisfy the following equation:
$C^{D}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\Big{(}\frac{\kappa}{m_{i}[j]}-2\Big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m_{i}[j]}-2}$
(34)
Since the logarithm function is a strictly increasing function, the maximum
likelihood estimate of the capacity of a relay $j\in[n]$ using full history
can be found:
$\begin{multlined}C^{D}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\Big{(}\frac{\kappa}{m_{i}[j]}-2\Big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m_{i}[j]}-2}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{(\frac{\kappa}{m_{i}[j]})^{2}}{\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m_{i}[j]}}(\lambda_{s}w_{i}[j])^{-2}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]+2\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}-\log\bigg{(}\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}!\bigg{)}+\frac{\kappa}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])-2\log(\lambda_{s}w_{i}[j])\end{multlined}C^{D}_{t+1}[j]=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{1}{\Big{(}\frac{\kappa}{m_{i}[j]}-2\Big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m_{i}[j]}-2}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\prod_{i=0}^{t}e^{-\lambda_{s}w_{i}[j]}\frac{(\frac{\kappa}{m_{i}[j]})^{2}}{\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}!}(\lambda_{s}w_{i}[j])^{\frac{\kappa}{m_{i}[j]}}(\lambda_{s}w_{i}[j])^{-2}\\\
=\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]+2\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}-\log\bigg{(}\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}!\bigg{)}+\frac{\kappa}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])-2\log(\lambda_{s}w_{i}[j])$
(35)
Using Stirling’s approximation, we have $\log(x!)\approx x\log(x)-x$. Thus
substituting in eq. 35:
$\begin{multlined}C^{H}_{t+1}[j]=\\\
\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]+2\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}-\frac{\kappa}{m_{i}[j]}\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}+\frac{\kappa}{m_{i}[j]}+\frac{\kappa}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])-2\log(\lambda_{s}w_{i}[j])\end{multlined}C^{H}_{t+1}[j]=\\\
\underset{\kappa\in\mathcal{C}}{\operatorname{argmax}}\hskip
2.84544pt\sum_{i=0}^{t}-\lambda_{s}w_{i}[j]+2\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}-\frac{\kappa}{m_{i}[j]}\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}+\frac{\kappa}{m_{i}[j]}+\frac{\kappa}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])-2\log(\lambda_{s}w_{i}[j])$
(36)
Hence in order to find $C^{H}_{t+1}[j]$, we differentiate the right hand side
of eq. 36 with respect to $\kappa$, and find the value of $C^{H}_{t+1}[j]$ for
which the derivative is zero.
$\begin{multlined}\sum_{i=0}^{t}\frac{2}{m_{i}[j]}\frac{1}{\frac{\kappa}{m_{i}[j]}}-\frac{1}{m_{i}[j]}\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}-\frac{1}{m_{i}[j]}+\frac{1}{m_{i}[j]}+\frac{1}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])=0\\\
\sum_{i=0}^{t}\frac{2}{\kappa}-\frac{1}{m_{i}[j]}\log(\kappa)+\frac{1}{m_{i}[j]}\log(m_{i}[j])+\frac{1}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])=0\\\
\sum_{i=0}^{t}\frac{2}{\kappa}-\frac{1}{m_{i}[j]}\log(\kappa)+\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])=0\\\
\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\kappa)=\sum_{i=0}^{t}\frac{2}{\kappa}+\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])\\\
\log(\kappa)\Big{(}\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\Big{)}=\frac{2(t+1)}{\kappa}+\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])\\\
\log(\kappa)=\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}+\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\\\
\log(\kappa)-\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\\\
\kappa
e^{-\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}}=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}\kappa
e^{-\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
\end{multlined}\sum_{i=0}^{t}\frac{2}{m_{i}[j]}\frac{1}{\frac{\kappa}{m_{i}[j]}}-\frac{1}{m_{i}[j]}\log\Big{(}\frac{\kappa}{m_{i}[j]}\Big{)}-\frac{1}{m_{i}[j]}+\frac{1}{m_{i}[j]}+\frac{1}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])=0\\\
\sum_{i=0}^{t}\frac{2}{\kappa}-\frac{1}{m_{i}[j]}\log(\kappa)+\frac{1}{m_{i}[j]}\log(m_{i}[j])+\frac{1}{m_{i}[j]}\log(\lambda_{s}w_{i}[j])=0\\\
\sum_{i=0}^{t}\frac{2}{\kappa}-\frac{1}{m_{i}[j]}\log(\kappa)+\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])=0\\\
\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\kappa)=\sum_{i=0}^{t}\frac{2}{\kappa}+\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])\\\
\log(\kappa)\Big{(}\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\Big{)}=\frac{2(t+1)}{\kappa}+\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])\\\
\log(\kappa)=\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}+\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\\\
\log(\kappa)-\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\\\
\kappa
e^{-\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}}=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}\kappa
e^{-\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
$
(37)
Letting $z=\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}$ in eq. 37,
we have:
$\begin{multlined}\frac{1}{z}e^{-z}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
\frac{1}{ze^{z}}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
ze^{z}=\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
\end{multlined}\frac{1}{z}e^{-z}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
\frac{1}{ze^{z}}=\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{2(t+1)}e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
ze^{z}=\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
$ (38)
We know that the inverse image of the function $ze^{z}$ is the Lambert W
function which has real solutions along its principal branch for
$z>-\frac{1}{e}$, denoted $W_{0}$. Thus we can solve for $z$:
$\begin{multlined}\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}=z=W_{0}\Bigg{(}\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\Bigg{)}\\\
\end{multlined}\frac{2(t+1)}{\kappa(\sum_{i=0}^{t}\frac{1}{m_{i}[j]})}=z=W_{0}\Bigg{(}\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\Bigg{)}\\\
$
(39)
And hence solving for $\kappa$:
$\begin{multlined}C^{D}_{t+1}[j]=\kappa=\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\frac{1}{W_{0}\Bigg{(}\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\Bigg{)}}\\\
\end{multlined}C^{D}_{t+1}[j]=\kappa=\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\frac{1}{W_{0}\Bigg{(}\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\Bigg{)}}\\\
$
(40)
$\begin{multlined}C^{D}_{t+1}[j]=\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\frac{1}{W_{0}\Bigg{(}\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\Bigg{)}},\\\
\end{multlined}C^{D}_{t+1}[j]=\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}\frac{1}{W_{0}\Bigg{(}\frac{2(t+1)}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}e^{-\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\Bigg{)}},\\\
$
(41)
where $W_{0}$ is the Lambert $W$ function along the principal branch. The
Lambert $W$ function is the multi-valued complex function $(ze^{z})^{-1}$ and
$W_{0}$ is the unique-valued real function that takes the unique real value of
$W$ when $z>\frac{-1}{e}$. Implementations of Lambert function exist in
multiple software libraries 555https://kite.com/python/docs/mpmath.lambertw.
$W_{0}$ has the following Taylor series expansion for $z$ in the neighborhood
of 0: $W_{0}(z)=z+o(z^{2})$. Moreover, the argument of $W_{0}$ is small if the
rate of users arrival to the network $\lambda_{s}$ is large enough. Hence, the
Taylor expansion around zero is valid and therefore:
$\begin{multlined}C^{D}_{t+1}[j]\approx
e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]C^{*}[j]}{C^{*}[j]})}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]}{C^{*}[j]})+\frac{1}{m_{i}[j]}\log(C^{*}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]}{C^{*}[j]})}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}e^{\frac{\log(C^{*}[j])\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=C^{*}[j]e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]}{C^{*}[j]})}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}.\end{multlined}C^{D}_{t+1}[j]\approx
e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(m_{i}[j]\lambda_{s}w_{i}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]C^{*}[j]}{C^{*}[j]})}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]}{C^{*}[j]})+\frac{1}{m_{i}[j]}\log(C^{*}[j])}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]}{C^{*}[j]})}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}e^{\frac{\log(C^{*}[j])\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}\\\
=C^{*}[j]e^{\frac{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}\log(\frac{m_{i}[j]\lambda_{s}w_{i}[j]}{C^{*}[j]})}{\sum_{i=0}^{t}\frac{1}{m_{i}[j]}}}.$
(42)
We refer the reader to the Appendix of [5] for a proof of the expected value
of the closed form found above since this form exactly matches the form
derived for $\mathit{MLEFlow\text{-}CF}$.
∎
See 5
###### Proof.
We refer the reader to the Appendix of [5] for a complete proof of the
theorem. ∎
|
# Nuclear Volterra composition operators between Bloch and weighted type
spaces
Aakriti Sharma Department of Mathematics, Central University of Jammu,
Central University of Jammu, (Bagla) Rahya-Suchani, Samba-181143, J&K, India.
<EMAIL_ADDRESS>and Ajay K. Sharma Department of Mathematics,
Central University of Jammu, Central University of Jammu, (Bagla) Raya-
Suchani, Samba-181143, J&K, India<EMAIL_ADDRESS>
###### Abstract.
In this paper, we completely characterize nuclear Volterra composition
operators
$T^{\phi}_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
and
$S^{\phi}_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
acting between weighted type spaces in terms of the symbols $g$ and $\phi$ of
$T^{\phi}_{g}$ and $S^{\phi}_{g}$ and weights $\nu$ and $\mu$, when the
weights $\nu$ and $\mu$ are normal weights in the sense of Shields and
Williams. Moreover, nuclear Volterra composition operators acting between
little weighted type spaces and Bloch spaces of order $\beta$ are also
characterized.
###### Key words and phrases:
Nuclear operator; Volterra composition operator, weighted Banach space,
Bergman space, weighted Bloch space, little weighted Banach space, little
weighted Bloch space.
###### 2010 Mathematics Subject Classification:
47B33, 47B38, 46E10, 32A37.
The first author is thankful to CSIR(India), Grant Number (File no.
09/1231(0001)/2019-EMR-I).
The second author is thankful to DST(SERB) for the project grant under MATRICS
scheme. Grant No. MTR/2018/000479.
## 1\. Introduction
Denote by ${H}(\mathbb{D})$, the class of all analytic functions on
$\mathbb{D}$, where $\mathbb{D}$ is the open unit disk in the complex plane
$\mathbb{C}$. For $g\in H(\mathbb{D}),$ Ch. Pommerenke [28] introduced an
integral operator $T_{g}:H(\mathbb{D})\to H(\mathbb{D})$ which maps every
$f\in H(\mathbb{D})$ to a function vanishing at $0,$ defined as
$T_{g}f(z)=\int_{0}^{z}f(\zeta)dg(\zeta)=\int_{0}^{z}f(\zeta)g^{\prime}(z)d\zeta,\;\;\;z\in\mathbb{D}.$
Another integral operator, generally known as a companion operator of $T_{g}$
was defined by Yoneda [31] as
$S_{g}f(z)=\int_{0}^{z}f^{\prime}(\zeta)g(\zeta)d\zeta,\;\;\;z\in\mathbb{D}.$
The operators $T_{g}$ and $S_{g}$ can be viewed as cousins of Hankel and
Toeplitz operators and they appeared in the literature under various names
such as Volterra operators, Cesaro operators, Riemann Stieltjes operators and
integration operators. For any $g\in{H}(\mathbb{D})$ and $\phi$ an analytic
self-map of $\mathbb{D}$, the Volterra composition operators $T^{\phi}_{g}$
and $S^{\phi}_{g}$ are defined, respectively on ${H}(\mathbb{D})$ by
$(T^{\phi}_{g}f)(z)=\int_{0}^{z}f(\phi(\zeta))g^{\prime}(\zeta)d\zeta,\;\;\;z\in\mathbb{D}$
and
$(S^{\phi}_{g}f)(z)=\int_{0}^{z}f^{\prime}(\phi(\zeta))g(\zeta)d\zeta,\;\;\;z\in\mathbb{D}.$
Recently, these type of operators are considered on spaces of analytic
functions by several authors, see [1]-[6], [10],[11],[16],[20]-[22] and
references therein. Motivated by results in [8] and [17], in this paper, we
characterize nuclear Volterra composition operators between Bloch and weighted
type spaces of analytic functions, when the weights are normal. Recall that a
radial weight $\nu$ is a non-increasing, non-negative continuous function
defined on $\mathbb{D}$ such that $\nu(z)=\nu(|z|)$ for all $z\in\mathbb{D}.$
A radial weight $\nu$ is normal in the sense of Shields and Williams if it
satisfies following two conditions:
1. ($\mathbf{I}$).
There exists $\beta>0$ such that the function
$r\rightarrow\frac{\nu(r)}{(1-r^{2})^{\beta}}$ is almost increasing, or
equivalently, $\inf_{n}\frac{\nu(1-2^{-(n+1)})}{\nu(1-2^{-n})}>0$.
2. ($\mathbf{II}$).
There exists $\gamma>0$ such that the function
$r\rightarrow\frac{\nu(r)}{(1-r^{2})^{\gamma}}$ is almost decreasing, or
equivalently, these is some $k\in\mathbb{N}$ such that
$\limsup_{n}\frac{\nu(1-2^{-(n+k)})}{\nu(1-2^{-n})}<1$.
It is clear that the standard weights
$\nu_{\alpha}(z)=(1-|z|^{2})^{\alpha},(\alpha>0)$ are normal weights.
Throughout this paper, we denote the set of all normal weights on $\mathbb{D}$
by $N_{W}(\mathbb{D}).$ For more about normal weights, we refer to [29],[6]
and [15]. The weighted Banach spaces $H^{\infty}_{\nu}$ and $H_{\nu}^{0}$ are
defined as follows:
$\mathcal{H}_{\nu}^{\infty}=\bigg{\\{}{f\in{H}(\mathbb{D}):\|f\|_{\mathcal{H}_{\nu}^{\infty}}=\sup_{z\in\mathbb{D}}\nu(z)|f(z)|<\infty\bigg{\\}}}$
and
$H_{\nu}^{0}=\bigg{\\{}{f\in{H}(\mathbb{D}):\lim_{|z|\rightarrow
1^{-}}\nu(z)|f(z)|=0\bigg{\\}}}.$
The weighted Bloch space $\mathcal{B}_{\nu}$ is the space of all functions $f$
in ${H}(\mathbb{D})$ such that
$\|f\|_{\nu}=\sup_{z\in\mathbb{D}}\nu(z)|f^{\prime}(z)|<{\infty}.$ Note that
$\|\cdot\|_{\nu}$ is a semi-norm on $\mathcal{B}_{\nu}$ and a norm can be
defined on $\mathcal{B}_{\nu}$ as
$\|f\|_{\mathcal{B}_{\nu}}=|f(0)|+\sup_{z\in\mathbb{D}}\nu(z)|f^{\prime}(z)|.$
Endowed with the norm $\|\cdot\|_{\mathcal{B}_{\nu}}$, $\mathcal{B}_{\nu}$
becomes a Banach space. Similarly, the little weighted Bloch space
$\mathcal{B}^{0}_{\nu}$ is defined as $\lim_{|z|\rightarrow
1^{-}}\nu(z)|f^{\prime}(z)|=0.$ If $\lim_{|z|\rightarrow 1^{-}}\nu(z)=0$, then
the weight $\nu$ is called typical and for typical weights $H_{\nu}^{\infty}$
is the bidual space of $H_{\nu}^{0}$. If the weight $\nu$ satisfies the
condition ($\mathbf{I}$), then $\nu$ satisfies condition ($\mathbf{II}$) if
and only if $H_{\nu}^{0}=\mathcal{B}^{0}_{\nu(z)(1-|z|^{2})}$, see [25]. In
particular, if $\nu\in N_{W}(\mathbb{D})$, then
$H_{\nu}^{0}=\mathcal{B}_{\nu(z)(1-|z|^{2})}^{0}$ and by duality,
$H_{\nu}^{\infty}=\mathcal{B}_{\nu(z)(1-|z|^{2})}$, see [23]. Moreover, if
$\nu\in N_{W}(\mathbb{D})$, then the topological isomorphisms
$H_{\nu}^{0}\simeq c_{0}$ and $H_{\nu}^{\infty}\simeq l_{\infty}$ hold, see
[8],[30] and [23]-[26]. Next, let us recall a duality pairing between
$\mathcal{H}^{\infty}_{\nu}$ and a weighted Bergman space, which was
established by Shields and Williams in [29]. Following [29], let $\nu$ be a
typical weight and $\omega$ a radial, positive and continuous function which
satisfies $\int_{0}^{1}\omega(r)dr<\infty.$ Let $A^{1}_{\omega}$ be the
subspace of $H(\mathbb{D})$ consists of functions $f$ such that
$\|f\|_{A^{1}_{\omega}}=\int_{\mathbb{D}}|f(z)|\omega(z)dA(z)<\infty,$ where
$dA$ is the normalized Lebesgue measure on $\mathbb{D}.$ For a normal weight
$\nu$ and $\omega$ a function defined as above, the pair $\\{\nu,\omega\\}$ is
called a normal pair if
$\nu(r)\omega(r)=(1-r^{2})^{\alpha},\;\;\;0\leq r<1$ (1.1)
for some $\alpha>\beta-1$, where $\beta$ is a positive real number defined as
in $(\mathbf{I}).$ For a normal pair $\\{\nu,\omega\\}$, the following pairing
between $\mathcal{H}^{\infty}_{\nu}$ and $A^{1}_{\omega}$ is defined in [29],
see p. 292.
$\displaystyle(f,g)$
$\displaystyle=\int_{\mathbb{D}}f(z)g(\bar{z})(1-|z|^{2})^{\alpha}dA(z)=\int_{\mathbb{D}}f(\bar{z})g({z})(1-|z|^{2})^{\alpha}dA(z).$
(1.2)
The pairing $(\cdot,\cdot)$ define a duality relation
$(\mathcal{H}^{0}_{\nu})^{\prime}=A^{1}_{\omega},$ see [29], Theorem 2, p.
296. Thus we also have
$(\mathcal{H}^{0}_{\nu})^{\prime\prime}=\mathcal{H}^{\infty}_{\nu}.$ Moreover,
if $K_{\zeta}(z)=(\alpha+1)/(1-{\zeta}z)^{\alpha+2},$ $\alpha>-1$ is the
kernel function and $\\{\nu,\omega\\}$ a normal pair, then
$K_{\zeta}\in\mathcal{H}^{0}_{\nu}\cap A^{1}_{\omega},$
$g(\zeta)=(K_{\zeta},g)$ for all $g\in A^{1}_{\omega}$ and
$f(\zeta)=(f,K_{\zeta})$ for all $f\in\mathcal{H}^{\infty}_{\nu},$ see Lemma
10 in [29].
Recall that a linear operator $T$ between two Banach space $X$ and $Y$ is
1. (a)
Nuclear if there is a sequence $(x^{\prime}_{k})\subset X^{\prime}$ and a
sequence $(y_{k})\subset Y$ such that
$\sum_{k}\|x^{\prime}_{k}\|\|y_{k}\|<{\infty}\mbox{ and
}T=\sum_{k=1}^{\infty}x^{\prime}_{k}\otimes y_{k},$
where $x^{\prime}_{k}\otimes y_{k}:X\rightarrow Y$ is the mapping defined as
$x\rightarrow x^{\prime}_{k}(x)y_{k}$.
2. (b)
Absolutely summing if there is $C>0$ such that for all finite sequences
$(x_{i})_{i=1}^{n}\subset X$ we have
$\sum_{i=1}^{n}\|Tx_{i}\|\leq C\sup_{\|x^{\prime}\|\leq
1}\sum_{i=1}^{n}|x^{\prime}(x_{i})|=C\sup_{|\eta_{i}|=1}\|\sum_{i=1}^{n}\eta_{i}x_{i}\|.$
Moreover, it is well known that any nuclear operator is compact and absolutely
summing. Also if $X=c_{0}$ or $l_{\infty}$, then a linear operator
$T:X\rightarrow Y$ is absolutely summing if and only if it is nuclear, see
[17].
Throughout this paper, $C$ denotes a positive constant, the exact value of
which may be not be same at each occurrence. The expression $A\lesssim B$
means that there is a positive constant $C$ such that $A\leq CB$ and the
expression $A\asymp B$ means that there are positive constants $C$ and $D$
such that $CB\leq A\leq DB$.
## 2\. Nuclear Volterra composition operators
In this section, we characterize nuclear Volterra composition operator between
Bloch and weighted spaces of analytic functions.
Next we present characterizations of bounded Volterra composition operators
between weighted spaces of analytic functions. The arguments are standard and
may possibly appear in literature in a more general sense, so we give an
outline proof for completeness.
###### Theorem 1.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $g\in H(\mathbb{D})$ and $\phi$ a self-map
of $\mathbb{D}$. Then
1. ($i$)
The following are equivalent.
1. ($a$)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.
2. ($b$)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.
3. ($c$)
$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:
$M_{1}=\sup_{z\in\mathbb{D}}(1-|z|^{2})\frac{\mu(z)}{\nu(\phi(z))}|g^{\prime}(z)|<\infty.$
(2.1)
Moreover,
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded if and only if $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$ and (2.1)
holds.
2. ($ii$)
The following are equivalent.
1. ($d$)
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.
2. ($e$)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded.
3. ($f$)
$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:
$\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})}{(1-|\phi(z)|^{2})}\frac{\mu(z)}{\nu(\phi(z))}|g(z)|<\infty.$
(2.2)
Moreover,
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded if and only if $g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$ and (2.2)
holds.
###### Proof.
$(i)$ Since $\mu$ is a normal weight, so
$\mathcal{H}_{\mu(z)}=\mathcal{B}_{(1-|z|^{2})\mu(z)}.$ Thus boundedness of
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to the boundedness of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}.$
By Proposition 2 in [15], every normal weight is essential, that is there is a
constant $C>0$ such that $\nu(z)\leq\tilde{\nu}(z)\leq C\nu(z)$ for each
$z\in\mathbb{D}$.
$(a)\Leftrightarrow(c)$ Then by Proposition 3.1 in [9], we have that
$(a)\Leftrightarrow(c).$
It is obvious that $(a)\Rightarrow(b).$ Thus to complete the proof, we need to
show that $(b)\Rightarrow(c).$
$(b)\Rightarrow(c).$ Suppose that
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$
is bounded. For a fixed $\zeta\in\mathbb{D},$ consider the function
$f_{\zeta}$ defined on $\mathbb{D}$ as
$f_{\zeta}(z)=\frac{(1-|\phi(\zeta)|^{2})^{\beta+1}}{\nu(\phi(\zeta))(1-\overline{\phi(\zeta)}z)^{\beta+1}},$
where $\beta>0$ is as in condition $(\mathbf{I})$ of normal weight. Then it is
easy to see that $f_{\zeta}\in\mathcal{H}^{0}_{\nu}$ and
$\sup_{\zeta\in\mathbb{D}}\|f_{\zeta}\|_{\mathcal{H}^{\infty}_{\nu}}\lesssim
1.$ Moreover, $f_{\zeta}(\phi(\zeta))=1/\nu(\phi(\zeta)).$ Thus by the
boundedness of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$,
we have that
$(1-|\zeta|^{2})\mu(\zeta)|g^{\prime}(\zeta)||f_{\zeta}(\phi(\zeta))|\leq\|M_{g^{\prime}}C_{\phi}f_{\zeta}\|_{\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}}\lesssim\|f_{\zeta}\|_{\mathcal{H}^{\infty}_{\nu}}\lesssim
1.$
Taking supermum over $\zeta\in\mathbb{D},$ we get (2.1).
By Proposition 3.2 in [9], we have that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded if and only if $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$ and (2.1)
holds. This completes the proof of $(i)$.
$(ii)$ Thus boundedness of
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to the boundedness of
$M_{g}C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)},$
where $D$ is the differentiation operator. We write an outline proof of
$(d)\Leftrightarrow(f)$ and $(e)\Leftrightarrow(f)$ The details of all other
cases are omitted.
$(d)\Leftrightarrow(f)$ By Theorem 7 in [27], we have that
$(d)\Leftrightarrow(f).$
$(e)\Rightarrow(f).$ Suppose that
$M_{g}C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$
is bounded. For a fixed $\zeta\in\mathbb{D},$ consider the function
$f_{\zeta}$ defined as in case $(i).$ Then proceeding as in case $(i)$, we
have that
$f^{\prime}_{\zeta}(\phi(\zeta))=\overline{\phi(\zeta)}/(1-|\phi(\zeta)|^{2})\nu(\phi(\zeta)).$
Thus by the boundedness of
$M_{g}C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}$,
we have that
$(1-|\zeta|^{2})\mu(\zeta)|g(\zeta)||f^{\prime}_{\zeta}(\phi(\zeta))|\leq\|M_{g}C_{\phi}Df_{\zeta}\|_{\mathcal{H}^{\infty}_{(1-|z|^{2})\mu(z)}}\lesssim\|f_{\zeta}\|_{\mathcal{H}^{\infty}_{\nu}}\lesssim
1.$
Taking supermum over $\zeta\in\mathbb{D},$ we get
$\sup_{z\in\mathbb{D}}\frac{(1-|\zeta|^{2})\mu(\zeta)}{(1-|\phi(\zeta)|^{2})\nu(\phi(\zeta))}|\phi(\zeta)||g(\zeta)|<\infty.$
Thus
$\sup_{|\phi(\zeta)|>1/2}\frac{(1-|\zeta|^{2})\mu(\zeta)}{(1-|\phi(\zeta)|^{2})\nu(\phi(\zeta))}|g(\zeta)|<\infty.$
(2.3)
The above condition easily yields (2.2) by bifurcating the supermum over
$z\in\mathbb{D}$ in (2.2) as sum of supermum over $|\phi(\zeta)|\leq 1/2$ and
supermum over $|\phi(\zeta)|>1/2$. Thus the proof can be finished, using the
fact that the first term in this sum is definitely finite and the second term
in this sum is finite by (2.3).∎
###### Theorem 2.
Let $\nu,\mu\in N_{W}(\mathbb{D}).$ If $g\in H(\mathbb{D})$ and $\phi$ a self-
map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then
1. ($i$)
The following are equivalent.
1. ($a$)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is compact.
2. ($b$)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is weakly compact.
3. ($c$)
The bi-transpose $(T_{g}^{\phi})^{\prime\prime}$ of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
satisfies $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$.
2. ($ii$)
If $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$, then
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded and the bi-transpose $(T^{\phi}_{g})^{\prime\prime}$ of $T^{\phi}_{g}$
satisfies $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$. Moreover,
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
compact if and only if
$T^{\phi}_{g}(H^{\infty}_{\nu})\subset\mathcal{H}^{0}_{\mu}.$
###### Proof.
$(i)$ $(a)\Leftrightarrow(b)$ Since
$\mathcal{H}^{\infty}_{\mu}=\mathcal{B}_{(1-|z|^{2})\mu(z)}$, so
$T_{g}^{\phi}f\in H^{\infty}_{\mu}$ if and only if $M_{g^{\prime}}C_{\phi}f\in
H^{\infty}_{(1-|z|^{2})\mu}.$ Moreover, $T_{g}^{\phi}f$ and
$M_{g^{\prime}}C_{\phi}f$ have comparable norm. Thus compactness (and/or weak
compactness) of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to compactness (and/or weak compactness) of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}.$
By Theorem 1 in [7], $M_{g^{\prime}}C_{\phi}f$ is compact if and only if it is
weakly compact. Therefore, it follows that $(a)$ and $(b)$ are equivalent.
$(a)\Rightarrow(c)$ Assume that $(a)$ holds. Then by Gantmacher-Nakamura’s
theorem, see Theorem 5.5 in [12], the bi-transpose
$(T_{g}^{\phi})^{\prime\prime}$ of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
acts continuously from $\mathcal{H}^{\infty}_{\nu}$ with $w^{*}$-topology to
$\mathcal{H}^{\infty}_{\mu}$ with $w^{*}$-topology. Since pointwise topology
$\tau_{p}$ is weaker than $w^{*}$-topology on $\mathcal{H}^{\infty}_{\nu}$ and
$\mathcal{H}^{\infty}_{\mu}$, so $(T_{g}^{\phi})^{\prime\prime}$ and
$T_{g}^{\phi}$ are both $w^{*}$ and $\tau_{p}$ continuous and they agree in
$\mathcal{H}^{0}_{\nu}$ which is $w^{*}$ dense in
$\mathcal{H}^{\infty}_{\nu}.$ Therefore, $(c)$ holds.
$(c)\Rightarrow(b)$ Assume that the bi-transpose
$(T_{g}^{\phi})^{\prime\prime}$ of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
satisfies $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$. Then it holds that
$(T^{\phi}_{g})^{\prime\prime}(\mathcal{H}^{\infty}_{\nu})\subset\mathcal{H}^{\infty}_{\mu}.$
Using the fact that a bounded linear operator $T$ acting from a Banach space
$X$ to a Banach space $Y$ is weakly compact if and only if
$T^{\prime\prime}(X^{\prime\prime})\subset Y,$ see p. 482 in [14], we have
that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is weakly compact.
$(ii)$ If $g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$, then by Theorem 1,
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded. Moreover, by replacing $T_{g}$ in Lemma 1 in [6] by $T^{\phi}_{g}$,
we can easily prove that $(T^{\phi}_{g})^{\prime\prime}=T^{\phi}_{g}$.
Finally, by Gantmacher-Nakamura’s theorem it holds that the compactness of
$T^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
equivalent to $T^{\phi}_{g}(H^{\infty}_{\nu})\subset\mathcal{H}^{0}_{\mu}.$
This completes the proof. ∎
###### Theorem 3.
Let $\nu,\mu\in N_{W}(\mathbb{D}).$ If $g\in H(\mathbb{D})$ and $\phi$ a self-
map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then
1. ($i$)
The following are equivalent.
1. ($a$)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is compact.
2. ($b$)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is weakly compact.
3. ($c$)
The bi-transpose $(S_{g}^{\phi})^{\prime\prime}$ of
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
satisfies $(S^{\phi}_{g})^{\prime\prime}=S^{\phi}_{g}$.
2. ($ii$)
If $g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$, then
$S^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded and the bi-transpose $(S^{\phi}_{g})^{\prime\prime}$ of $S^{\phi}_{g}$
satisfies $(S^{\phi}_{g})^{\prime\prime}=S^{\phi}_{g}$. Moreover,
$S^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
compact if and only if
$S^{\phi}_{g}(H^{\infty}_{\nu})\subset\mathcal{H}^{0}_{\mu}.$
###### Proof.
(i) Proceeding as in Theorem 2, $S_{g}^{\phi}f\in H^{\infty}_{\mu}$ if and
only if $M_{g^{\prime}}C_{\phi}Df\in H^{\infty}_{(1-|z|^{2})\mu(z)}$ with
comparable norm. Thus compactness (and/or weak compactness) of
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is equivalent to compactness (and/or weak compactness) of
$M_{g^{\prime}}C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
which is further equivalent to is compactness (and/or weak compactness) of
$M_{g^{\prime}}C_{\phi}:\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}\longrightarrow\mathcal{H}^{\infty}_{\mu}.$
Thus the first part of theorem can now be settles as the proof of first part
of Theorem 2.
(ii) If $g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$, then by Theorem 1,
$S^{\phi}_{g}:\mathcal{H}_{\nu}^{0}\rightarrow\mathcal{H}_{\mu}^{0}$ is
bounded. Moreover, by replacing $S_{g}$ in Lemma 2.12 in [21], we can easily
prove that $(S^{\phi}_{g})^{\prime\prime}=S^{\phi}_{g}$. The last part can be
settle by Gantmacher-Nakamura’s theorem as in the second part of Theorem 2.∎
Finally, we recall Piestsch’s theorem which plays an important role in the
proof of main results of this paper. For more about the Piestsch’s theorem, we
refer the readers to Theorem 2.12 of [13].
###### Theorem 4.
(Pietsch’s Theorem) Let $E$ and $F$ be Banach spaces and $1\leq p<+\infty$. A
bounded linear operator $T:E\longrightarrow F$ is absolutely p-summing if and
only if there is a constant $C$ and a regular Borel probability measure
$\varrho$ on closed unit ball $B_{E^{*}}$ of $E^{*}$ with the weak∗-topology
$\sigma(E^{*},E)$ such that for every $x\in E$,
$\|T(x)\|\leq
C\left(\int_{B_{E^{*}}}|\phi(x)|^{p}d\varrho(\phi)\right)^{1/p}.$
###### Theorem 5.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $g\in
H(\mathbb{D})$ and $\phi$ a self-map of $\mathbb{D}$ be such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.
1. (a)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.
2. (b)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.
3. (c)
$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:
$M_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Proof.
$(c)\Rightarrow(a)$ Since $\mathcal{H}^{0}_{\nu}\simeq c_{0}$, so
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear if and only if
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing. Thus to complete the proof of the implication
$(c)\Rightarrow(a)$, we need to prove that
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing. First note that if $\mu$ is normal,
$\mathcal{H}^{\infty}_{\mu(z)}=\mathcal{B}_{\mu(z)(1-|z|^{2})}$,
$\|T_{g}^{\phi}(f)\|_{\mathcal{H}^{\infty}_{\mu}}\asymp\|T_{g}^{\phi}(f)\|_{\mathcal{B}_{\mu(z)(1-|z|^{2})}}$
for any $f\in H^{\infty}_{\nu}$ and $T_{g}^{\phi}(f)$ vanishes at origin for
any $f\in H(\mathbb{D})$. Now polynomials are dense in
$\mathcal{H}^{0}_{\mu}$, so it is sufficient to consider polynomials
$f_{1},f_{2},\cdots,f_{N}$ in the definition of absolutely summing. Now for
any $C>1,$ we can select $z_{i}$, $i=1,2,\cdots N$ such that
$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{\mathcal{H}^{\infty}_{\mu}}$
$\displaystyle\asymp\displaystyle\sum_{i=1}^{N}\left(\big{|}(T_{g}^{\phi}f_{i})(0)\big{|}+\sup_{z\in\mathbb{D}}(1-|z|^{2})\mu(z)\big{|}(T_{g}^{\phi}f_{i})^{\prime}(z)\big{|}\right)$
$\displaystyle=\sum_{i=1}^{N}\displaystyle\sup_{z\in\mathbb{D}}(1-|z|^{2})\mu(z)\big{|}f_{i}(\phi(z))g^{\prime}(z)\big{|}$
$\displaystyle\leq
C\displaystyle\sum_{i=1}^{N}(1-|z_{i}|^{2})\mu(z_{i})\big{|}f_{i}(\phi(z_{i}))g^{\prime}(z_{i})\big{|}$
(2.4)
Using the duality pairing in (1.2), we have that
$\displaystyle\big{|}f_{i}(\phi(z_{i}))\big{|}=\big{|}(f_{i},K_{\phi(z_{i})})\big{|}=\bigg{|}(\alpha+1)\int_{\mathbb{D}}f_{i}(\zeta)\frac{(1-|\zeta|^{2})^{\alpha}}{|1-\bar{\zeta}\phi(z_{i})|^{\alpha+2}}dA(\zeta)\bigg{|}.$
Thus from (2.4), we have that
$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{{}_{\mathcal{H}^{\infty}_{\mu}}}$
$\displaystyle\leq
C\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}$
$\displaystyle\quad\times\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)\big{|}g^{\prime}(z)\big{|}}{\big{|}1-\bar{\zeta}\phi(z)\big{|}^{\alpha+2}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle=C\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}M_{\alpha}$
(2.5)
Using the equality
$\sup_{w\in\mathbb{D}}\sum_{i=1}^{N}|f_{i}(w)|\nu(w)=\sup_{|\eta_{i}|=1}\sup_{w\in\mathbb{D}}\bigg{|}\displaystyle\sum_{i=1}^{N}\eta_{i}f_{i}(w)\bigg{|}\nu(w)=\sup_{|\eta_{i}|=1}\bigg{\|}\displaystyle\sum_{i=1}^{N}\eta_{i}f_{i}\bigg{\|}_{\mathcal{H}^{\infty}_{\mu}}$
in (2.5), we have that
$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{{}_{\mathcal{H}^{\infty}_{\mu}}}$
$\displaystyle\leq
C\sup_{|\eta_{i}|=1}\bigg{\|}\displaystyle\sum_{i=1}^{N}\eta_{i}f_{i}\bigg{\|}_{\mathcal{H}^{\infty}_{\mu}}.$
Since $C>0$ is arbitrary, so
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing.
$(a)\Rightarrow(c)$ Since every nuclear operator is absolutely summing,
therefore,
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing. By Theorem 4, there is a probability Borel measure
$\varrho$ on $\sigma(A^{1}_{\omega},\mathcal{H}^{0}_{\mu})$-compact unit ball
$\mathbb{B}_{1}$ of $A^{1}_{\omega}$, where
$\sigma(A^{1}_{\omega},\mathcal{H}^{0}_{\mu})$ is the weak∗-topology on
$(\mathcal{H}^{0}_{\mu})^{\prime}=A^{1}_{\omega},$ and some $\xi$ in
$A^{1}_{\omega}$ such that
$\|T_{g}^{\phi}f\|_{\mathcal{H}_{\mu}^{\infty}}\leq
C\int_{\mathbb{B}_{1}}|\xi(f)|d\varrho(\xi)$ (2.6)
for every $f\in\mathcal{H}^{0}_{\mu}$ and some constant $C>0$ independent of
$f$. For every $\zeta\in\mathbb{D}$, we have that
$K_{{\zeta}}\in\mathcal{H}^{0}_{\mu}$. Therefore,
$\displaystyle(\alpha+1)\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-{\zeta}\phi(z)|^{\alpha+2}}\leq\|T_{g}^{\phi}K_{{\zeta}}\|_{\mathcal{B}_{\mu(z)(1-|z|^{2})}}\leq
C\|T_{g}^{\phi}K_{{\zeta}}\|_{\mathcal{H}^{\infty}_{\mu(z)}}.$ (2.7)
Replacing $f$ in (2.6) by $K_{{\zeta}}$, and then using (2.7), we have that
$\displaystyle\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-{\zeta}\phi(z)|^{\alpha+2}}\leq\frac{C}{\alpha+1}\int_{\mathbb{B}_{1}}|\xi(K_{w})|d\varrho(\xi).$
Integrating over $\mathbb{D}$ with respect to $\omega dA$, using (1.1) and
then applying Fubini’s theorem, we get
$\displaystyle\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}$
$\displaystyle\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\quad\quad\leq\frac{C}{\alpha+1}\int_{\mathbb{B}_{1}}\int_{\mathbb{D}}|\xi(K_{\zeta})|\omega(z)dA(\zeta)d\varrho(\xi).$
Now $K_{\zeta}\in\mathcal{H}^{0}_{\mu}$ and $\xi$ in $A^{1}_{\omega}$, so
$\xi(K_{\zeta})=\langle K_{\zeta},\xi\rangle=\xi(w).$ Thus using the fact that
$\varrho$ is a probability Borel measure, we get
$\displaystyle\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}$
$\displaystyle\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\quad\quad\leq\frac{C}{\alpha+1}\sup_{\xi\in\mathbb{B}_{1}}\int_{\mathbb{D}}|\xi(\zeta)|\omega(\zeta)dA(\zeta)$
$\displaystyle\quad\quad=\frac{C}{\alpha+1}\sup_{\xi\in\mathbb{B}_{1}}\|\xi\|_{A^{1}_{\omega}}=\frac{C}{\alpha+1}.$
This completes the proof of the implication $(a)\Rightarrow(c)$.
$(a)\Rightarrow(b)$ Since transpose of a nuclear operator is nuclear, see
3.1.8 in [15], so by an application of Theorem 2(1), it follows that
$(a)\Rightarrow(b)$.
$(b)\Rightarrow(a)$ Recall that class of nuclear operators forms an operator
ideal, that is, nuclear operators are a class of operators which do not fixes
a copy of $l^{\infty},$ (see [7] and [8]), and $\mathcal{H}^{0}_{\nu}$ is a
closed subspace of $\mathcal{H}^{\infty}_{\mu}$, so it is obvious that
$(b)\Rightarrow(a)$. This completes the proof. ∎
###### Theorem 6.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{(1-|z|^{2})\mu(z)}$ and $\phi$ a self-map of
$\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.
1. (a)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.
2. (b)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.
3. (c)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
nuclear.
4. (d)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$
is nuclear.
5. (e)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$M_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Proof.
By Theorem 4, we have that $(a)\Leftrightarrow(b)\Leftrightarrow(e).$ To
complete the proof it is sufficient to prove that
$(a)\Rightarrow(c)\Rightarrow(d)\Rightarrow(a).$
$(a)\Rightarrow(c)$ Suppose that $(a)$ holds. Then by Theorem 2(2)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
bounded. Moreover,
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is absolutely summing if and only if
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}.$ Thus
the implication $(a)\Rightarrow(d)$ follows.
$(c)\Rightarrow(d)$ Using Theorem 2(ii) and the fact that transpose of a
nuclear operator is nuclear the implication follows.
$(d)\Rightarrow(a).$ The proof follows as the proof of implication
$(b)\Rightarrow(a)$ in Theorem 4. This completes the proof.∎
###### Corollary 1.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $g\in
H(\mathbb{D})$ and $\phi$ a self-map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded,
then the following statements are equivalent.
1. (a)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
2. (b)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 2.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu}$ and $\phi$ a self-map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.
1. (a)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
2. (b)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
$T_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
4. (d)
$T_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$
is nuclear.
5. (e)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
We can easily obtain the following corollaries.
###### Corollary 3.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $\phi$ is a
self-map of $\mathbb{D}$ such that
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.
1. (a)
$C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
$\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 4.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$\phi\in\mathcal{B}^{0}_{\mu(z)}$ and
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.
1. (a)
$C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
$C_{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
4. (d)
$C_{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
5. (e)
$\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 5.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in{H}(\mathbb{D})$ be such that
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded, then the
following statements are equivalent.
1. (a)
$T_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
3. (c)
For $g$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 6.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu(z)}$ be such that
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is bounded,
then the following statements are equivalent.
1. (a)
$T_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
3. (c)
$T_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is nuclear.
4. (d)
$T_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
5. (e)
For $g$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+2}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Theorem 7.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $g\in
H(\mathbb{D})$ and $\phi$ a self-map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.
1. (a)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
nuclear.
2. (b)
$S_{g}^{\phi}:\mathcal{H}_{\nu}\longrightarrow\mathcal{H}_{\mu}$ is nuclear.
3. (c)
$g$, $\nu,$ $\mu$ and $\phi$ satisfy the following condition:
$N_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Proof.
That $(a)\Leftrightarrow(b)$ can be proved, proceeding as in the proof of
Theorem 5. Thus to complete the proof, we need to prove that
$(a)\Leftrightarrow(c)$. $(c)\Rightarrow(a)$ Proceeding as in the proof of
$(c)\Rightarrow(a)$ of Theorem 5, for any $C>1,$ we can select $z_{i}$,
$i=1,2,\cdots N$ such that
$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{\mathcal{H}_{\mu}^{\infty}}\leq
C\displaystyle\sum_{i=1}^{N}(1-|z_{i}|^{2})\mu(z_{i})\big{|}f_{i}(\phi(z_{i}))g(z_{i})\big{|}$
(2.8)
and
$\displaystyle f_{i}(w)=\langle
f_{i},K_{w}\rangle=(\alpha+1)\int_{\mathbb{D}}f_{i}(\zeta)\frac{(1-|\zeta|^{2})^{\alpha}}{(1-\bar{\zeta}w)^{\alpha+2}}dA(\zeta).$
Differentiating with respect to $w,$ we have that
$\displaystyle
f^{\prime}_{i}(w)=(\alpha+1)(\alpha+2)\int_{\mathbb{D}}f_{i}(\zeta)\frac{\bar{\zeta}(1-|\zeta|^{2})^{\alpha}}{(1-\bar{\zeta}w)^{\alpha+3}}dA(\zeta).$
Thus
$\displaystyle\big{|}f^{\prime}_{i}(\phi(z_{i}))\big{|}\leq(\alpha+1)(\alpha+2)\int_{\mathbb{D}}|f_{i}(\zeta)|\frac{\bar{\zeta}(1-|\zeta|^{2})^{\alpha}}{|1-\bar{\zeta}\phi(z_{i})|^{\alpha+3}}dA(\zeta).$
(2.9)
Using (2.8) and (2.9 ), we have that
$\displaystyle\sum_{i=1}^{N}\|T_{g}^{\phi}f_{i}\|_{{}_{\mathcal{H}_{\mu}^{\infty}}}$
$\displaystyle\leq
C(\alpha+1)(\alpha+2)\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}$
$\displaystyle\quad\times\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)\big{|}g^{\prime}(z)\big{|}}{\big{|}1-\bar{\zeta}\phi(z)\big{|}^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle=C(\alpha+1)(\alpha+2)\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}f_{i}(w)\big{|}\nu(w)\bigg{)}N_{\alpha}$
(2.10)
The proof can now be completed proceeding as in the Theorem 5. We omit the
details.
$(a)\Rightarrow(c)$ Once again, proceeding as in the proof of Theorem 5, there
exists a probability Borel measure $\varrho$ on
$\sigma(A^{1}_{\omega},\mathcal{H}^{0}_{\mu})$-compact unit ball
$\mathbb{B}_{1}$ of $A^{1}_{\omega}$ and $\xi$ in $A^{1}_{\omega}$ such that
$\|S_{g}^{\phi}f\|_{\mathcal{H}_{\mu}^{\infty}}\leq
C\int_{\mathbb{B}_{1}}|\xi(f)|d\varrho(\xi)$ (2.11)
for every $f\in\mathcal{H}^{0}_{\mu}$ and some constant $C>0$ independent of
$f$. For every $\zeta\in\mathbb{D}$, we have that
$K_{\zeta}\in\mathcal{H}^{0}_{\mu}$.
$\displaystyle(\alpha+1)(\alpha+2)\sup_{z\in\mathbb{D}}\frac{|\zeta|(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\leq\|S_{g}^{\phi}K_{\zeta}\|_{\mathcal{B}_{\mu(z)(1-|z|^{2})}}\leq
C\|S_{g}^{\phi}K_{\zeta}\|_{\mathcal{H}_{\mu(z)}^{\infty}}.$ (2.12)
Replacing $f$ in (2.11) by $K_{\zeta}$, and then using (2.12), we have that
$\displaystyle\sup_{z\in\mathbb{D}}\frac{|\zeta|(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\leq\frac{C}{(\alpha+1)(\alpha+2)}\int_{\mathbb{B}_{1}}|\xi(K_{\zeta})|d\varrho(\xi).$
(2.13)
Integrating (2.13) over $\mathbb{D}$ with respect to $\omega dA$ and
proceeding as in the Theorem 5, we get
$\displaystyle\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{|\zeta|(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)\leq\frac{C}{(\alpha+1)(\alpha+2)}.$
(2.14)
Now if $|\zeta|>1/2,$ then by (2.14), we have that
$\displaystyle\int_{|\zeta|>1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)\leq\frac{2C}{(\alpha+1)(\alpha+2)}.$
(2.15)
Further, if $|\zeta|\leq 1/2,$ then we have that
$\displaystyle\int_{|\zeta|\leq
1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)\leq\frac{2^{\alpha+1}\|g\|_{\mathcal{H}_{\mu}^{\infty}}}{\nu(1/2)}.$
(2.16)
Thus
$\displaystyle N_{\alpha}$ $\displaystyle\leq\int_{|\zeta|\leq
1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\quad+\int_{|\zeta|>1/2}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)$
$\displaystyle\leq\frac{2^{\alpha+1}\|g\|_{\mathcal{H}_{\mu}^{\infty}}}{\nu(1/2)}+\frac{2C}{(\alpha+1)(\alpha+2)}.$
This completes the proof. ∎
###### Corollary 7.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{H}^{0}_{(1-|z|^{2})\mu(z)}$ and $\phi$ an analytic self-map of
$\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is bounded, then the following statements are equivalent.
1. (a)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.
2. (b)
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{\infty}_{\mu}$
is nuclear.
3. (c)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$ is
nuclear.
4. (d)
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{H}^{0}_{\mu}$
is nuclear.
5. (e)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$N_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 8.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in{H}(\mathbb{D})and$ $\phi$ an analytic self-map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded,
then the following statements are equivalent.
1. (a)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
2. (b)
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 9.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu(z)}$ and $\phi$ an analytic self-map of $\mathbb{D}$
such that
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.
1. (a)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
2. (b)
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
$S_{g}^{\phi}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
4. (d)
$S_{g}^{\phi}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$
is nuclear.
5. (e)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
We can also obtain the following corollaries.
###### Corollary 10.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $\phi$ is an
analytic self-map of $\mathbb{D}$ such that
$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded, then
the following statements are equivalent.
1. (a)
$C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
$\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 11.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If $\phi$ an
analytic self-map of $\mathbb{D}$ such that $\phi\in\mathcal{H}^{0}_{\mu(z)}$
and $C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
bounded, then the following statements are equivalent.
1. (a)
$C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is
nuclear.
3. (c)
$C_{\phi}D:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
4. (d)
$C_{\phi}D:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
5. (e)
$\phi$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|\phi^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 12.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If Let
$g\in{H}(\mathbb{D})$ be such that
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow B_{\mu}$ is bounded, then the
following statements are equivalent.
1. (a)
$S_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
3. (c)
For $g$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
###### Corollary 13.
Let $\nu,\mu\in N_{W}(\mathbb{D}),$ $\alpha>-1$ and $\omega$ a weight function
such that $\\{\nu,\omega\\}$ is a normal pair and (1.1) holds. If
$g\in\mathcal{B}^{0}_{\mu(z)}$ be such that
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is bounded,
then the following statements are equivalent.
1. (a)
$S_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
2. (b)
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}_{\mu}$ is nuclear.
3. (c)
$S_{g}:\mathcal{H}^{0}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is nuclear.
4. (d)
$S_{g}:\mathcal{H}^{\infty}_{\nu}\longrightarrow\mathcal{B}^{0}_{\mu}$ is
nuclear.
5. (e)
$g$ and $\nu$ satisfy the following condition:
$\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{\mu(z)|g^{\prime}(z)|}{|1-\bar{\zeta}z|^{\alpha+3}}\bigg{]}\frac{(1-|\zeta|^{2})^{\alpha}}{\nu(\zeta)}dA(\zeta)<\infty.$
## 3\. Nuclear $T_{g}^{\phi}$ and $S_{g}^{\phi}$ between Bloch spaces of
order $\beta$
Finally, we apply our results to $T_{g}^{\phi}$ and $S_{g}^{\phi}$ acting
between weighted Banach spaces and Bloch spaces of order $\beta,$ where
$\beta>0.$ Recall that if $\nu_{\beta}$ is a classical weight, that is
$\nu_{\beta}(z)=(1-|z|^{2})^{\beta}$, then the spaces
$\mathcal{H}^{\infty}_{\nu}$, $\mathcal{H}^{0}_{\nu}$, $\mathcal{B}_{\nu}$ and
$\mathcal{B}^{0}_{\mu}$ reduces respectively to weighted and Bloch spaces of
order $\beta$, denoted by $\mathcal{H}^{\infty}_{\beta}$,
$\mathcal{H}^{0}_{\beta}$, $\mathcal{B}_{\beta}$ and
$\mathcal{B}^{0}_{\beta}$. Moreover, for $\beta>1$, we have that
$\mathcal{H}^{0}_{\beta-1}=\mathcal{B}^{0}_{\beta}$ and
$\mathcal{H}^{\infty}_{\beta-1}=\mathcal{B}^{\infty}_{\beta}$ with equivalent
norm. Thus we have the following corollary.
###### Corollary 14.
Let $\beta>1$, $\gamma>0$, $g\in{H}(\mathbb{D})$ and $\phi$ an analytic self-
map of $\mathbb{D}$ such that
$T_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
bounded. Then the following statements are equivalent.
1. (a)
$T_{g}^{\phi}:\mathcal{B}^{0}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.
2. (b)
$T_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.
3. (c)
$g$, $\beta,$ $\gamma$ and $\phi$ satisfy the following condition:
$P_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})^{\gamma}|g^{\prime}(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+2}}\bigg{]}{(1-|\zeta|^{2})^{\alpha-\beta+1}}dA(\zeta)<\infty.$
The case $\beta=1$ is settled in the next theorem.
###### Theorem 8.
Let $g\in{H}(\mathbb{D})$ and $\phi$ an analytic self-map of $\mathbb{D}$ such
that $T_{g}^{\phi}:\mathcal{B}\longrightarrow\mathcal{B}$ is bounded. Then the
following statements are equivalent.
1. (a)
$T_{g}^{\phi}:\mathcal{B}_{0}\longrightarrow\mathcal{B}$ is nuclear.
2. (b)
$g$ and $\phi$ satisfy the following condition:
$M=\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|\phi^{\prime}(z)|}{|1-\bar{w}\phi(z)|^{3}}|g(z)|dA(w)<\infty.$
###### Proof.
$(b)\Rightarrow(a)$ Since $c_{0}\simeq\mathcal{B}_{0}$ and polynomials are
dense in $\mathcal{B}_{0}$. Moreover, by Corollary $1.5$ in [18], we have that
$f\in\mathcal{B},$ then we have that
$f^{\prime}(z)=2\int_{\mathbb{D}}\frac{1-|w|^{2}}{(1-z\bar{w})^{3}}f^{\prime}(w)dA(w).$
(3.1)
Consider polynomials $p_{1},p_{2},\cdots,p_{N}$ in the definition of
absolutely summing and then using (3.1), we have
$\sum_{i=1}^{N}\|T_{g}^{\phi}p_{i}\|_{\mathcal{B}}\leq
C\bigg{(}\sup_{w\in\mathbb{D}}\displaystyle\sum_{i=1}^{N}\big{|}p^{\prime}_{i}(w)\big{|}(1-|w|^{2})\bigg{)}\int_{\mathbb{D}}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})\big{|}\phi^{\prime}(z)\big{|}}{\big{|}1-\bar{w}\phi(z)\big{|}^{3}}\big{|}g(z)\big{|}dA(w).$
The proof can be settled proceeding as in the proof of Theorem 5.
$(a)\Rightarrow(b)$ By Theorem 1, there exists a probability Borel measure
$\varrho$ on $\sigma((\mathcal{B}_{0})^{*},\mathcal{B})$-compact unit ball
$B^{\prime}$ of $(\mathcal{B}_{0})^{*}=A^{1}$ such that
$\|T_{g}^{\phi}f\|_{\mathcal{B}}\leq
C\int_{B^{{}^{\prime}}}|\xi(f)|d\varrho(\xi)$
for every $f\in\mathcal{B}_{0}$ and some constant $C>0$ independent of $f$.
Next for every $w\in\mathbb{D}$, we consider $f_{w}(z)={1}/{(1-\bar{w}z)^{2}}$
which lies in $\mathcal{B}_{0}\cap H^{\infty}$. By duality and reproducing
property of $f_{w}$ in the Bergman space $(\mathcal{B}_{0})^{*}=A^{1}$, we
have
$\xi(f_{w})=\langle
f_{w},h\rangle=\int_{\mathbb{D}}h(z)\overline{f_{w}(z)}dA(z)=\int_{\mathbb{D}}\dfrac{h(z)}{(1-\bar{z}w)^{2}}dA(z)=h(w).$
Also,
$\sup_{z\in\mathbb{D}}\dfrac{2|w|(1-|z|^{2})|\phi^{\prime}(z)||g(z)|}{|1-\bar{w}\phi(z)|^{3}}=\|T_{g}^{\phi}f_{w}\|_{1}\leq\|T_{g}^{\phi}f_{w}\|_{\mathcal{B}}$
Now proceeding as in the proof of Theorem 7, we can easily see that $(b)$
holds. This completes the proof.∎
Similarly, we have the following corollary.
###### Corollary 15.
Let $\beta>1$, $\gamma>0$, $g\in{H}(\mathbb{D})$ and $\phi$ an analytic self-
map of $\mathbb{D}$ such that
$S_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
bounded. Then the following statements are equivalent.
1. (a)
$S_{g}^{\phi}:\mathcal{B}^{0}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.
2. (b)
$S_{g}^{\phi}:\mathcal{B}_{\beta}\longrightarrow\mathcal{B}_{\gamma}$ is
nuclear.
3. (c)
For $g$, $\phi$ and $\nu$ satisfy the following condition:
$Q_{\alpha}=\int_{\mathbb{D}}\bigg{[}\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})^{\gamma}|g(z)|}{|1-\bar{\zeta}\phi(z)|^{\alpha+3}}\bigg{]}{(1-|\zeta|^{2})^{\alpha-\beta+1}}dA(\zeta)<\infty.$
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|
# Split-PU: Hardness-aware Training Strategy for Positive-Unlabeled Learning
Chengming Xu School of Data Science, Fudan University China
<EMAIL_ADDRESS>, Chen Liu Department of Mathematics Hong Kong
University of Science and Technology China<EMAIL_ADDRESS>, Siqian
Yang , Yabiao Wang Tencent Youtu Lab China<EMAIL_ADDRESS><EMAIL_ADDRESS>, Shijie Zhang School of Data Science, Fudan
University China<EMAIL_ADDRESS>, Lijie Jia Shanghai Jiaotong
University China<EMAIL_ADDRESS>and Yanwei Fu School of Data Science,
Fudan University China<EMAIL_ADDRESS>
(2022)
###### Abstract.
Positive-Unlabeled (PU) learning aims to learn a model with rare positive
samples and abundant unlabeled samples. Compared with classical binary
classification, the task of PU learning is much more challenging due to the
existence of many incompletely-annotated data instances. Since only part of
the most confident positive samples are available and evidence is not enough
to categorize the rest samples, many of these unlabeled data may also be the
positive samples. Research on this topic is particularly useful and essential
to many real-world tasks which demand very expensive labelling cost. For
example, the recognition tasks in disease diagnosis, recommendation system and
satellite image recognition may only have few positive samples that can be
annotated by the experts. While this problem is receiving increasing
attention, most of the efforts have been dedicated to the design of
trustworthy risk estimators such as uPU (Du Plessis et al., 2014) and nnPU
(Kiryo et al., 2017) and direct knowledge distillation, e.g., Self-PU (Chen et
al., 2020a). These methods mainly omit the intrinsic hardness of some
unlabeled data, which can result in sub-optimal performance as a consequence
of fitting the easy noisy data and not sufficiently utilizing the hard data.
In this paper, we focus on improving the commonly-used nnPU (Kiryo et al.,
2017) with a novel training pipeline. We highlight the intrinsic difference of
hardness of samples in the dataset and the proper learning strategies for easy
and hard data. By considering this fact, we propose first splitting the
unlabeled dataset with an early-stop strategy. The samples that have
inconsistent predictions between the temporary and base model are considered
as hard samples. Then the model utilizes a noise-tolerant Jensen-Shannon
divergence loss for easy data; and a dual-source consistency regularization
for hard data which includes a cross-consistency between student and base
model for low-level features and self-consistency for high-level features and
predictions, respectively. Our method achieves much better results compared
with existing methods on CIFAR10 and two medical datasets of liver cancer
survival time prediction, and low blood pressure diagnosis of pregnant,
individually. The experimental results validates the efficacy of our proposed
method. Codes and models are available at
https://github.com/loadder/SplitPU_MM2022.
Positive-Unlabeled learning; Consistency Regularization
††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
30th ACM International Conference on Multimedia; October 10–14, 2022; Lisboa,
Portugal††booktitle: Proceedings of the 30th ACM International Conference on
Multimedia (MM ’22), October 10–14, 2022, Lisboa, Portugal††price: 15.00††doi:
10.1145/3503161.3548045††isbn: 978-1-4503-9203-7/22/10††ccs: Computing
methodologies††ccs: Computing methodologies Artificial intelligence††ccs:
Computing methodologies Computer vision††ccs: Computing methodologies Computer
vision problems
## 1\. Introduction
The deep models of classical recognition task, especially binary
classification, has been supported by plenty of positive and negative samples,
which are comprehensively annotated in general. On the other hand, the
collection of training data could not always be perfect. In many Multimedia
applications, only part of the positive samples can be annotated, while the
other samples including both positive and negative ones can only be left
without labels. These cases may happen quite often. For example, the annotator
source is severely limited for tasks, such as military targets recognition
from satellite images (Fan et al., 2020), since expertise and confidential
knowledge is required. In medical image analysis such as liver cancer
diagnosis, the annotators can only provide confident judgement to few positive
samples, while the other images without obvious symptom could also be positive
but cannot be explicitly annotated. Essentially, the Positive-Unlabeled (PU)
learning is studied to handle these tasks. With a well-trained PU learning
model, many real-world applications of different modalities with only positive
and unlabeled data can thus be effectively solved, e.g. medical diagnosis,
recommendation system, satellite image recognition, etc.
Figure 1. The binary classification versus PU learning. Left: The most
commonly-seen visual data with distinguished categories. The data can be
easily identified as an animal or not animal. Right: Data for low blood
pressure diagnosis. Without sufficient expertise annotators, only part of the
most confident positive samples can be recognized. Therefore the data in such
an area is often taken as PU data.
Due to the special structure of training set, PU learning is generally more
challenging than standard supervised binary classification. Directly applying
existing methods in both fully-supervised and semi-supervised learning would
lead to great performance degradation. The classical PU learning methods can
be traced back to confident negative sample selection(Liu et al., 2002, 2003)
in the early stages, and focus on designing unbiased risk estimator (Du
Plessis et al., 2014; Kiryo et al., 2017) recently. While some previous works
like Self-PU (Chen et al., 2020a) and PUUPL (Dorigatti et al., 2022) have
explored combining together the well-designed objective function and sample
selection for better performance, these methods neglect an important fact in
the PU data: Some samples are relatively much harder to learn by deep neural
networks than most data.
Such hardness is not similar to the definition of ’easiness’ depicted in Self-
PU which is measured by the confidence of predictions. It is the intrinsic
property of image data that we focus on, which could be resulted from many
different reasons. For example, for commonly-used image datasets, those images
with extremely small, distorted or camouflaged objects can be relatively
harder to learn than other normal one. Similarly, in medical image analysis,
some samples may have severe artifacts, affecting training and inference of
models. These artifacts may be caused by the errors of medical instruments,
which can hardly be avoided in all cases.
By considering this fact, we advocate that the unlabeled data should be
further divided into easy and hard set. The easy data dominates the unlabeled
set, being easy to fit but vulnerable to noise, while the hard data is on the
opposite. Specifically, inspired by Self-PU, we can annotate these unlabeled
samples with pseudo-labels generated by a base model, e.g. a nnPU model. Then
both sets contain wrongly-labeled (noisy) data and correctly-labeled (clean)
data. We highlight two important points which imply that these different types
of data serve as different roles in training. (1) Easy data is not totally
reliable. Especially, since the easy noisy samples can be fitted by the model
facilely, it is more likely for these samples than the hard noisy ones to hurt
the generalization ability of the learned PU models. Thus potentially, it may
be questionable to directly utilize all pseudo-labels from the easy set in a
vanilla objective functions like cross entropy losses. (2) Hard data is more
difficult to deal with and requires objective functions that exclude the
pseudo-labels. As the name of hard data suggests, the model can hardly learn
information from the pseudo-labels of hard data no matter whether they are
clean or not. Nevertheless, the hard clean samples are essentially valuable
for estimating the decision boundary. For example, as a common rule of the
thumb, the hard and ambiguous training instances may also be taken as support
vectors to compute the classifier margins in support vector machine.
Therefore, it is a problem to make use of the hard data in a more effective
way. Given these two points, the naïve selection-distillation pipeline as in
Self-PU, i.e. gradually selecting confident samples and training with their
pseudo-labels, may lead to sub-optimal performance, as all unlabeled samples
are treated with the same form of objective functions.
To this end, we propose a novel Split-PU method that employs a novel hardness-
aware training strategy for positive-unlabeled learning. Our Split-PU method
takes apart the easy and hard data from all PU data to deal with,
respectively. Specifically, the whole method contains four steps. (1) We first
train a base teacher model of nnPU (Kiryo et al., 2017) as objective function.
The base model is applied to generate the initial pseudo-labels for all
unlabeled data. (2) Then a temporary student model is trained via knowledge
distillation from the teacher model, with recording training accuracy of each
epoch. When the temporary model reaches the specified performance, we split
out those training samples as hard data that have the inconsistent predictions
between the temporary model and the base one, with the rest as easy data. (3)
Further, different learning strategies are employed to handle the easy and
hard data. For easy data, we find out that those easy noisy samples cannot be
detected as effectively as the hard noisy samples by using existing noise
detection methods. As an alternative, we thus for the first time introduce the
Jensen-Shannon divergence loss (Englesson and Azizpour, 2021) to optimize the
PU learning model for noisy-robust PU learning. For hard data, we develop a
dual-source consistency regularization which covers low-level and high-level
features together with predictions, thus fully utilizing the supervision from
both the base and the student model. We have to regularize such consistency
here: a cross-consistency between the low-level features of base model and
student model, which is aimed to guide the student model with well-trained
low-level features like edges and colors from base model; and a self-
consistency on the high-level features and predictions of student model
between different views of each image, which is used to improve the stability
of training for better visual representations. With the above training
strategy, we can better handle the unlabeled data with the pseudo-labels
predicted by the base model, leading to a student model with better
performance. (4) Finally, we replace the base model with the student model to
iterate the training algorithm for further improvement.
To validate the effectiveness of our proposed method, we conduct extensive
experiments. First, one of the most widely-used visual benchmarks CIFAR10 is
adopted, following the existing methods. Additionally, we adopt a MRI dataset
on revival time prediction for liver cancer and a myometrioum image dataset of
low blood pressure diagnosis for the pregnant to further testify the
capability of our model on real-life applications. The results on these three
datasets illustrate that our model can significantly improve the base model
trained with nnPU and surpass the state-of-the-art methods. Moreover, when
trained with fewer positive labeled data, our model still performs better than
competitors which use more labeled data, which supports the argument that our
model can solve the data scarcity problem to some extent.
In summary, our paper has the following contributions:
* •
We analyze the essence of the PU data and find that those hard data could make
the existing pseudo-labeling methods less effective.
* •
To take advantage of the analysis, we propose to use an early-stop splitting
strategy to excavate the easy and hard data from the unlabeled set.
* •
We leverage the Jensen-Shannon divergence loss to learn the easy data, and
propose a novel dual-source consistency regularization to efficiently utilize
the hard unlabeled data.
* •
With the help of our proposed techniques, we consistently improve the base
model among different datasets when using various amount of labeled samples.
In the following context, we first briefly introduce the related works in Sec.
2, which includes PU learning, learning with noisy labels and consistency in
deep learning. Then in Sec. 3 we give the problem formulation of PU learning
and our proposed method. Experiment results among multiple datasets are listed
in Sec. A.
## 2\. Related Works
Positive-unlabeled Learning Positive-unlabeled learning is a surging research
topic that aims to learn patterns on a training set with only few labeled
positive data and sufficient unlabeled data. The research on PU learning can
be dated back to (Liu et al., 2002, 2003; Yu et al., 2002) in which reliable
negative samples are selected and used to train a set of classifiers. The
recent effort on PU learning with deep deep learning can be generally
categorized into two buckets. One is about the design of objective function
that can unbiasedly estimate the risk. uPU (Du Plessis et al., 2014) shows
that a cost-sensitive classifier can be used by reform the risk in original
classification, with known class prior. nnPU (Kiryo et al., 2017) points out
the tendency of overfitting complex models when using uPU. As a solution they
impose a non-negative constraint on the objective function of uPU, which leads
to better generalization. Some other papers follow these two works to find
powerful objective functions in different settings such as imbalanced data (Su
et al., 2021), biased negative data (Hsieh et al., 2019), data selection bias
(Kato et al., 2018), etc. Some other works focus on sample selection from the
unlabeled samples based on the former methods. For example, (Xu et al., 2019)
analyze the reasonability of only selecting positive data, Self-PU (Chen et
al., 2020a) leverages self-paced learning to gradually update the base model
with newly-learned knowledge, PUUPL (Dorigatti et al., 2022) strengthens the
calibration of base model so that the pseudo-labels are all uncertainty-aware.
Note that the concept of hardness has also been mentioned in Self-PU. However,
they refer to the this concept as the reliability of pseudo-labels, which is
reflected by the confidence of model predictions. Our method split the data
into easy and hard set based on the intrinsic property of data, which is
different with the existing ideas. Moreover, we point out that directly using
cross entropy on pseudo-labeled data and PU loss on unlabeled data is
inappropriate. Instead, we propose to conquer the easy and hard unlabeled data
based on noise-tolerant learning and dual-source consistency regularization
respectively. Additionally, PU learning is different with semi-supervised
learning which also uses the unlabeled data as the regularizer. The unlabeled
data in PU learning plays a more important role of estimating the data
distribution since no labeled negative samples are available.
Learning with noisy labels Since the pseudo-labeling strategy in PU learning
inevitably generates wrongly-labeled samples, i.e. noisy samples, such methods
are correlated with the task of learning with noisy labels, which is aimed to
train a more robust model from the noisy dataset. Methods in learning with
noisy labels mainly belong to two groups: noise robust algorithm and noise
detection. Noise robust algorithms target designing network architectures
(Chen and Gupta, 2015; Goldberger and Ben-Reuven, 2016) or objective function
(Zhang and Sabuncu, 2018; Ghosh et al., 2017; Englesson and Azizpour, 2021)
that are robust against side effect of noisy samples. Noise detection methods
involve a pipeline for selecting and processing the noisy data from the
training set. Typical selection basis includes large error (Shen and Sanghavi,
2019), incoherent gradient (Chatterjee, 2020), inconsistency among different
networks or different views of the same input (Yu et al., 2019; Zhou et al.,
2020) and so on. As for the strategies for handling noisy samples, label-
correction modules are proposed in many works (Li et al., 2017; Tanaka et al.,
2018; Vahdat, 2017). We in this paper leverage the technique in this topic to
solve PU learning, which has never been studied in the previous works.
Moreover, while directly applying these advantaged noise detection methods may
also help dealing with the noisy pseudo-labels in PU learning, we will show in
the experiments that these methods could easily fail when facing with some of
the training data.
Consistency Regularization Recently consistency regularization has been widely
used in deep learning as extra supervision besides the annotations.
Specifically, contrastive learning methods (He et al., 2020; Chen et al.,
2020b; Caron et al., 2020; Chen and He, 2021a) take as objective function the
similarity among features of different views of the same image, which can be
seen as the feature level consistency. Such objectives are utilized to enhance
the instance discriminative ability for networks. Moreover, for semi-
supervised learning, many works (Sohn et al., 2020; Verma et al., 2019; Jeong
et al., 2019) adopt prediction level consistency to increase the stability of
training. Compared with these methods, our proposed dual-source consistency
regularization takes into account the various properties of features and
predictions, which leads to consistency with more capacity from multiple
sources.
## 3\. Methodology
Problem Formulation Suppose we have a data distribution $\mathcal{T}$ which
can be represented as a joint distribution of image distribution $\mathcal{I}$
and label distribution $\mathcal{Y}$, i.e.,
$\mathcal{T}=\mathcal{I}\times\mathcal{Y}$. In PU learning, the training set
$\mathcal{T}^{train}$ is composed of two parts including $\mathcal{T}^{p}$
which only contains few labeled positive samples and $\mathcal{T}^{u}$ which
contains abundant unlabeled samples. The labeled positive
$\mathcal{T}^{p}=\\{(I_{i}^{p},y_{i}^{p})\\}_{i=1}^{n_{p}}$, where
$I_{i}^{p}\sim\mathcal{I},y_{i}^{p}=1$. The unlabeled
$\mathcal{T}^{u}=\\{I_{i}^{u}\\}_{i=1}^{n_{u}}$, where
$I_{i}^{u}\sim\mathcal{I}$. Our goal is to learn a network $\phi$, which can
well generalize to the test set
$\mathcal{T}^{test}=\\{(I_{i},y_{i})\\}_{i=1}^{n_{t}}$.
Figure 2. Overview of our model. We first train a base teacher model using
nnPU as objective function. Then the unlabeled data is split into easy and
hard set via a temporary model which is trained with knowledge distillation
given the base model. Then we iterate a new training process applying
different learning strategies to the easy and hard data to train the student
model.
Overview. We propose a hardness-based splitting strategy to deal with the PU
learning, which is schematically shown in Fig. 2. Concretely, we first train a
base model $\phi_{base}$ with nnPU (Kiryo et al., 2017) (Sec. 3.1), from which
all of the following supervision is generated. After attaining this base
model, we split the unlabeled set $\mathcal{T}^{u}$ into easy set
$\mathcal{T}^{e}$ and hard set $\mathcal{T}^{h}$ by the training accuracy of a
temporary student network $\phi_{tmp}$ which is trained with naive knowledge
distillation from $\phi_{base}$ (Sec. 3.2). Then we re-train a network $\phi$
with different learning strategies on $\mathcal{T}^{easy}$ and
$\mathcal{T}^{hard}$. For the easy set $\mathcal{T}^{easy}$, we utilize a
noise-tolerant objective function to make use of the pseudo-labels from
$\phi_{base}$ (Sec. 3.3). For the hard set $\mathcal{T}^{hard}$, we adopt a
dual-source consistency regularization to take advantage of the helpful
knowledge from $\phi_{base}$ without using the pseudo-labels (Sec. 3.4). Such
a learning strategy can be repeated by replacing the teacher model
$\phi_{base}$ with the newly-learned $\phi$ to further improve the performance
not only for commonly-used visual datasets, but also for PU learning settings
in real scenario.
### 3.1. Preliminary: Base Model Training
We first briefly introduce the nnPU (Kiryo et al., 2017) as our model. The
nnPU is a non-negative version of an unbiased risk estimator as following:
(1) $\displaystyle L_{nnPU}$
$\displaystyle=\frac{\pi}{n_{p}}\sum_{i=1}^{n_{p}}L(\phi(I_{i}^{p}),1)+$ (2)
$\displaystyle
max\\{0,\frac{1}{n_{u}}\sum_{i=1}^{n_{u}}L(\phi(I_{i}^{u}),-1)-\frac{\pi}{n_{u}}\sum_{i=1}^{n_{u}}L(\phi(I_{i}^{p}),-1)\\}$
where $\pi$ is the class prior of positive samples, i.e.
$\pi=P(I_{i}|y_{i}=1)$, which is supposed to be known as (Kiryo et al., 2017).
By training with $L_{nnPU}$ on $\mathcal{T}^{train}$, we can get our base
model $\phi_{base}$ and correspondingly a pseudo-labeled extension of
$\mathcal{T}^{u}$:
(3) $\displaystyle\hat{\mathcal{T}}^{u}$
$\displaystyle=\\{(I_{i}^{u},\hat{y_{i}}^{u})\\}_{i=1}^{n_{u}}$ (4)
$\displaystyle\hat{y_{i}}^{u}$ $\displaystyle=\phi_{base}(I_{i}^{u})$
Note that $\hat{y_{i}}^{u}$ can be either soft-label or hard-label. In this
way, $\hat{\mathcal{T}}^{u}$ is composed of correctly-labeled data (i.e. clean
data), and wrongly-labeled data (i.e., noisy data). A straightforward way,
like in Self-PU (Chen et al., 2020a), to make use of $\hat{\mathcal{T}}^{u}$
is to train a supervised learning model based on these pseudo-labels and then
iteratively rectify the noisy or wrongly pseudo-labeled samples. However, as
we will show in the experiments (see Fig. 8), some training samples are
intrinsically hard to learn. This implies that the models can barely receive
knowledge from these samples no matter they are affiliated with correct
pseudo-labels or not. Thus such a learning strategy will lead to sub-optimal
performance. As a solution, we propose to use the following method to separate
these hard samples from $\mathcal{T}^{u}$ and then learn the easy and hard
data respectively.
### 3.2. Early-stop Splitting Strategy
Given the base model, we consider splitting $\mathcal{T}^{u}$ apart into easy
and hard data, and then conquer them respectively. Exactly, we learn a new
student model $\phi_{tmp}$ based on the pseudo-labels generated by
$\phi_{base}$ with soft-label cross entropy as objective function:
(5) $L_{tmp}=\sum_{i=1}^{n_{u}}-\hat{y}_{i}^{u}log(\phi_{tmp}(I_{i}^{u}))$
We set a small learning rate for this optimization process to track the
variety of $\phi_{tmp}$ in detail. After each epoch, we record the training
accuracy. Once the accuracy is over a threshold $\tau$, we stop the training
and split the unlabeled set $\hat{\mathcal{T}}^{u}$ based on the prediction
coherence between $\phi_{tmp}$ and $\phi_{base}$. The samples that are still
not learned by $\phi_{tmp}$ are categorized as hard samples, which compose the
hard set $\mathcal{T}^{hard}$. The rest are called easy set, denoted as
$\mathcal{T}^{easy}$. Since the noisy samples generally have different
patterns from clean samples as in existing works (Chatterjee, 2020; Arpit et
al., 2017), most of them would be learned more slowly than clean ones. As a
consequence, $\mathcal{T}^{hard}$ can have a much larger proportion of noisy
samples than $\mathcal{T}^{u}$, and the noise rate of $\mathcal{T}^{easy}$ is
much smaller. Meanwhile, the roles of these two subsets are totally different:
(1) The easy clean samples take up the most part of $\mathcal{T}^{u}$, from
which the model learns its patterns. (2) The easy noisy samples, despite with
a small sample size, can be easily fitted by the model and hurt the
generalization ability. (3) The hard samples, while can hardly be learned by
$\phi_{tmp}$ and have high noise rate, can regularize the model for better
performance. Especially, the hard clean samples can provide data knowledge
which is not contained in the easy set. Therefore, it is essential to utilize
a learning strategy robust to noise on $\mathcal{T}^{easy}$ and a new
supervision other than pseudo-labels on $\mathcal{T}^{hard}$. To this end, we
propose the following noise-tolerant learning on easy samples and dual-source
consistency regularization on hard samples.
### 3.3. Noise-tolerant Learning on Easy Samples
As mentioned above, the easy set $\mathcal{T}^{easy}$ has two noticeable
properties: (1) The noise rate is much smaller, thus most pseudo-labels are
reliable. (2) The negative effect of easy noisy samples on generalization
ability cannot be neglected. Directly learning the easy samples with plain
objective like cross entropy will make the model be trapped by the noisy
samples. A straightforward thought is if we can use noise detection methods to
further cull the noisy easy samples. However, due to the fact that the slowly-
learned hard noisy samples have already been taken out from
$\mathcal{T}^{easy}$, most methods based on the difference of fitting speed or
fitting performance between noisy and clean data could fail in our case, as we
will show in experiments. As an alternative solution, we adopt a noise-robust
objective function to process the easy data, which was proposed in (Englesson
and Azizpour, 2021). Concretely, for each easy sample
$I_{i}^{easy}\in\mathcal{T}^{easy}$ and its corresponding pseudo-label
$\hat{y}_{i}$, the following objective function is used:
(6) $L_{easy}=\frac{(\rho
D_{KL}(\phi(I_{i}^{easy})\|m)+(1-\rho)D_{KL}(\hat{y}_{i}\|m))}{-(1-\rho)log(1-\rho)}$
where $m=\rho\phi(I_{i}^{easy})+(1-\rho)\hat{y}_{i}$ which is an interpolation
between the pseudo-label and prediction from $\phi$, $D_{KL}$ denotes the
Kullback-Leibler divergence, $\rho$ is a hyper-parameter. Accordingly, Eq. 6
can take into account both robustness to noise and the convergence speed. As
we will show in our experiments, this term is more effective than plain
knowledge distillation using pseudo-labels.
### 3.4. Dual-source Consistency Regularization on Hard Samples
The hard set $\mathcal{T}^{hard}$ has a totally different characteristic
compared with $\mathcal{T}^{easy}$ and $\hat{\mathcal{T}}^{u}$.
$\mathcal{T}^{hard}$ contains much less samples than the other two sets, while
its proportion of noisy samples are much larger as we have explained above.
Besides, these samples can be hardly learned with the pseudo-labels, which is
reflected by the low training accuracy in the splitting phase. Therefore, it
is not proper to adopt the same training strategy as for the easy samples. To
this end, we explore to use these data via a dual-source consistency
regularization without using the pseudo-labels. Specifically, we use the
following objective function to learn the hard samples:
(7) $L_{hard}=L_{low}+L_{high}$
where $L_{low}$ is a low-level cross-consistency between $\phi_{base}$ and
$\phi$, and $L_{high}$ is a high-level self-consistency of $\phi$, which will
be introduced in the following.
Low-level cross-consistency. While the pseudo-label generated by $\phi_{base}$
is not usable due to the high noisy rate of $\mathcal{T}^{hard}$, the teacher
model can still provide some guidance of visual knowledge embedded in the low-
level features. According to the existing study (Lee et al., 2016), compared
with high-level features that is correlated with the semantic category of each
sample, these low-level features contain more basic and class-agnostic visual
patterns, such as edges and colors. Such patterns are still reusable no matter
if it leads to noisy pseudo-prediction in the $\mathcal{T}^{hard}$. Therefore,
we regularize $\phi$ with the consistency between low-level features of $\phi$
and $\phi_{base}$ on the same sample. Specifically, given a hard sample
$I_{i}^{hard}\in\mathcal{T}^{hard}$, we simultaneously extract the
corresponding features from $\phi$ and $\phi_{base}$ and compute their
similarity as the objective function:
(8) $L_{cross}=\|\phi^{1}(I_{i}^{hard})-\phi_{base}^{1}(I_{i}^{hard})\|_{2}$
where $\phi^{1}$ denotes the first convolutional layer of $\phi$, and the same
for $\phi_{base}^{1}$.
High-level self-consistency. The guidance from $\phi_{base}$ is sub-optimal
for the high-level features and the predictions. Therefore, inspired by recent
research on self-supervised learning (Chen and He, 2021b) and semi-supervised
learning (Sohn et al., 2020), we propose to utilize self-consistency on these
high-level information. Specifically, for a hard sample $I_{i}^{hard}$, we
first process it via both weak and strong augmentation pipelines
$A^{weak},A^{strong}$, which leads to two different views of this image
$\hat{I}_{i}^{hard}=A^{weak}(I_{i}^{hard}),\tilde{I}_{i}^{hard}=A^{strong}(I_{i}^{hard})$.
Then the high-level features are generated from the last convolutional layer
of $\phi$, together with the classification prediction from which the
consistency can be calculated:
(9) $\displaystyle\hat{X}_{i}$
$\displaystyle=\phi^{l}(\hat{I}_{i}^{hard}),\hat{y}_{i}=\phi(\hat{I}_{i}^{hard})$
(10) $\displaystyle\tilde{X}_{i}$
$\displaystyle=\phi^{l}(\tilde{I}_{i}^{hard}),\tilde{y}_{i}=\phi(\tilde{I}_{i}^{hard})$
(11) $\displaystyle L_{pred}$
$\displaystyle=D_{KL}(\hat{y}_{i}\|\tilde{y}_{i})$
After that a new predictor $f$ is introduced to transform the high-level
features into another embedding space. Based on that we can regularize $\phi$
with high-level feature consistency:
(12)
$L_{feat}=\frac{1}{2}D(\hat{X}_{i},f(\tilde{X}_{i}))+\frac{1}{2}D(\tilde{X}_{i},f(\hat{X}_{i}))$
where $D$ denotes the normalized negative cosine similarity. The high-level
self-consistency can then be summarized as the combination of $L_{pred}$ and
$L_{feat}$ with tunable weights $\alpha,\beta$:
(13) $L_{high}=\alpha L_{pred}+\beta L_{feat}$
This objective function can help the model with better instance discrimination
and to be more stable against disturbance on the samples. Compared with the
self-supervised consistency used in Self-PU (Chen et al., 2020a), our method
considers the intrinsically high noise-rate of hard data, thus making better
use of teacher model and leading to more effective regularization. Meanwhile,
our proposed consistency does not require training multiple networks at one
time so that we can reduce the computation cost.
Remark: why not use nnPU on $\mathcal{T}^{hard}$. One would ask if a simple
solution works for the hard samples, i.e., still taking them as unlabeled
samples and learning them by nnPU loss together with $\mathcal{T}^{p}$ as in
Eq. 1. We here provide an intuitive explanation about this problem. There are
two key components that makes $L_{nnPU}$ works: (1) The $\mathcal{T}^{u}$ can
well represent the data distribution. (2) We can have an accurate estimation
of class prior $\pi$. However, since the the sample size of
$\mathcal{T}^{hard}$ is much smaller than that of $\mathcal{T}^{u}$, there is
little chance that it can provide as much information to estimate the data
distribution as $\mathcal{T}^{u}$ can. On the other hand, if we think of
$\mathcal{T}^{p}$ and $\mathcal{T}^{hard}$ as another training set in PU
setting, then using $L_{nnPU}$ requires a new class prior $\pi^{\prime}$,
which cannot be easily estimated. Therefore it is inappropriate to use
$L_{nnPU}$ on $\mathcal{T}^{hard}$, which will be empirically verified in Sec.
A.
### 3.5. Iterative Training
With the above pipeline, we can get a $\phi$ that not only inherits the useful
knowledge from $\phi_{base}$, but also eliminate the side-effect of noisy
samples. Moreover, $\phi$ can be used as the teacher model again to train
another new student model for further improvement.
## 4\. Experiments
Figure 3. Image examples in the liver cancer dataset and myometrium dataset.
Dataset To validate the efficacy of our model, we first follow Self-PU (Chen
et al., 2020a) to conduct extensive experiments on CIFAR10 (Krizhevsky et al.,
2009). CIFAR10 is a basic visual dataset containing 50000 training images and
10000 testing images from 10 categories. We follow the setting in Self-PU to
take 4 categories of vehicles (i.e., ’airplane’, ’automobile’, ’ship’,
’truck’) to compose the positive class and the other animal classes (i.e.,
’bird’, ’cat’, ’deer’, ’dog’, ’frog’, ’horse’) for the negative class. In each
experiment, 500/1000/3000 positive samples are randomly chosen and the rest
are set as unlabeled samples. Moreover, as shown in Fig. 3, we further adopt
two new benchmarks of liver cancer data and myometrium data, which would be
publicly released. The liver cancer dataset contains MRI data of 259 patients
and their corresponding survival time. We split the dataset by setting samples
with survival time less than 90 as positive. This leads to totally 145
positive samples, from which we randomly choose 40 as labeled samples, and 114
negative samples. For test set, we have 65 negative samples and 35 positive
samples. The Myometrium data is used to diagnose the low blood pressure of the
pregnant, with 327 positive images and 233 negative images in total. We split
the dataset into 395 training images, 53 validation images and 112 testing
images, and then randomly sample 40 positive images as labeled set.
Implementation Details For CIFAR10 we use the same network architecture as in
Self-PU, i.e. a 13-layer CNN for CIFAR10. About Liver, we adopt a multi-branch
ResNet18, with the detail in supplementary. For Myometrium we finetune the
whole layers of an ImageNet-pretrained ResNet18.
For all datasets, we first train a base nnPU model for 50 epoches, using Adam
(Kingma and Ba, 2014) with learning rate of $10^{-4}$. Batch size is set to
512 for CIFAR10 and 64 for other datasets. After training the base model, we
fix the splitting of labeled and unlabeled samples and then further train
$\phi_{tmp}$ using SGD as optimizer with learning rate of $10^{-3}$ to avoid
converging to quickly. $\phi$ is trained for 100 epoches using Adam with
learning rate of $0.00005$. As for the hyper-parameters,
$\tau,\rho,\alpha,\beta$ are set as 92%, 0.7, 0.3, 0.1, respectively. We use
random cropping and flipping as weak augmentation. Color jittering is adopted
as strong augmentation for CIFAR10 and CutOut is adopted for the other two
medical datasets.
Metrics For all results, we report the mean accuracies and standard deviations
among 5 repetitions of each experiment.
Competitors We choose several former methods in PU learning as our competitors
including uPU (Du Plessis et al., 2014), nnPU (Kiryo et al., 2017), DAN (Liu
et al., 2019), Self-PU (Chen et al., 2020a), P3MIX-C (Li et al., 2021), PUUPL
(Dorigatti et al., 2022).
Model | CIFAR10
---|---
500 | 1000 | 3000
uPU (Du Plessis et al., 2014) | — | 88.00$\pm$0.62 | —
nnPU (Kiryo et al., 2017) | — | 88.60$\pm$0.40 | —
nnPU∗ | 87.22$\pm$0.46 | 89.02$\pm$0.39 | 90.53$\pm$0.36
Self-PU (Chen et al., 2020a) | — | 89.68$\pm$0.22 | 90.77$\pm$0.21
PUbN (Hsieh et al., 2019) | — | 89.30$\pm$0.57 | —
P3MIX-C (Li et al., 2021) | — | 87.90$\pm$0.50 | —
VPU (Chen et al., 2020c) | — | 85.06$\pm$0.55 | 87.50$\pm$1.05
DAN (Liu et al., 2019) | — | — | 89.70$\pm$0.40
PUUPL (Dorigatti et al., 2022) | — | 89.84$\pm$0.13 | 91.37$\pm$0.05
Ours | 89.18$\pm$0.12 | 90.51$\pm$0.10 | 92.51$\pm$0.10
Table 1. Test accuracies and standard deviation on CIFAR10 with three positive size, * denotes results reproduced by us. Model | Myometrium | Liver
---|---|---
uPU (Du Plessis et al., 2014) | 60.18$\pm$2.33 | 40.40$\pm$5.50
nnPU (Kiryo et al., 2017) | 62.32$\pm$2.71 | 56.60$\pm$5.90
Self-PU (Chen et al., 2020a) | 59.11$\pm$1.16 | 55.60$\pm$8.17
Ours | 65.89$\pm$2.13 | 63.80$\pm$1.64
Supervised | 72.32 | 68.00
Table 2. Test accuracies and standard deviation on Myometrium and Liver. All
results of competitors are reproduced by us.
### 4.1. Main Results
We compare our model with the competitors in Tab. 1 and Tab. 2. Note that for
nnPU on CIFAR10 we both report the results in the original paper and the one
reproduced by us.
CIFAR10 result. For CIFAR10, we train our model among three different positive
sample size. When $n_{p}=500$, the test accuracy of our model is 1.96% better
than that of nnPU, and comparable with nnPU model trained with 1000 labeled
positive samples. This means that support nnPU with our proposed method is
able to solve the problem of data scarcity for positive samples. When
$n_{p}=1000$, which is the most common setting, our proposed model attains
1.49% accuracy increasement against nnPU and performs 0.67% better than the
best counterpart PUUPL. When $n_{p}=3000$, the improvement on nnPU of our
model reaches 1.6%, leading to a much better result than all of the
competitors. Such an improvement is consistently significant among these three
settings, which validates the effectiveness of our proposed method and shows
the potential that our model can be used as a fixed suppplement for nnPU. As
for the standard deviation, our model is on par with the competitors, which
shows our method is stable enough to be safely utilized.
Myometrium and Liver results. We report results of different methods and one
oracle model trained with whole labeled training set together on the two
medical datasets in Tab. 2. For Myometrioum, our model is better than nnPU by
3.75% with comparable standard deviation. For Liver, our model improves nnPU
by 7.2% with remarkably more stability. It is noticeable that Self-PU is worse
than nnPU on these datasets. We suspect this is due to the poor performance of
nnPU leads to unreliable pseudo-labels, thus increasing the size of clean
noisy samples. In this way, since Self-PU cannot well handle such data, the
performance could be damaged.
### 4.2. Ablation Study
To further verify the efficacy of our contributions, we conduct several
ablation studies on CIFAR10, including each part of our model and the choice
of hyper-parameters.
Figure 4. (a) Size of noisy samples in easy and hard set with different $\tau$. (b) Proportion of noisy samples in easy and hard set with different $\tau$. (c) Test accuracy when trained with the easy and hard set that are generated with different $\tau$. (d) Number of clean and noisy samples that are selected from the clean set by SPR in each epoch. Early stop | Acc.
---|---
500 | 1000 | 3000
w/o | 88.57 | 89.86 | 91.73
w | 89.28 | 90.51 | 92.51
Table 3. Test accuracies on CIFAR10 of models trained with and without early stop splitting. Loss | label type | Acc.
---|---|---
500 | 1000 | 3000
CE | hard | 87.51 | 89.02 | 91.55
soft | 88.15 | 89.48 | 92.06
DJS | hard | 88.61 | 89.58 | 92.11
soft | 89.28 | 90.51 | 92.51
Table 4. Test accuracies on CIFAR10 of our model trained with soft cross entropy, hard cross entropy, soft Jensen-Shannon Divergence and hard Jensen-Shannon Divergence. Loss | Acc.
---|---
500 | 1000 | 3000
no | 88.52 | 89.71 | 91.46
nnPU | 87.91 | 89.43 | 90.66
self-consistency | 89.03 | 90.09 | 92.11
cross consistency | 89.17 | 90.42 | 92.27
dual-source consistency | 89.28 | 90.51 | 92.51
Table 5. Test accuracies on CIFAR10 of our model trained with different
losses on hard data.
What can early-stop splitting do? First of all, we would like to show the
effectiveness of the early-stop splitting strategy. To do so, we conduct an
oracle experiment where all training set labels are available for indicating
which samples are wrongly labeled by $\phi_{base}$. Then we gradually increase
the accuracy threshold $\tau$ and visualize the sample size and proportion of
noisy samples that are categorized as easy and hard ones respectively. The
results are shown in Fig. 4. We can find that (1) As $\tau$ increases, less
samples can be filtered out from $\mathcal{T}^{u}$, but the noise rate of
$\mathcal{T}^{hard}$ gets larger. (2) When $\tau$ is relatively small as 70%,
the proportion of noisy sample in $\mathcal{T}^{hard}$ is twice as larger than
that of $\mathcal{T}^{u}$, which coincides with the common sense that noisy
samples are generally learned more slowly than clean samples. Also, such a
strategy can effectively reduce the noise rate of $\mathcal{T}^{easy}$. (3)
There is a balance between the threshold and the test accuracy. The benefit
from increasing $\tau$ saturates at about 92%, then the test accuracy
gradually decreases. This is reasonable since larger $\tau$ means smaller hard
set, which leads to a less representative hard set. (4) As shown in 3,
performance is worse when early-stop strategy is not used , i.e., $L_{easy}$
is applied to all unlabeled data, which indicates the efficacy of this
strategy.
Can noise detection methods fit in our case? As analyzed in Sec. 3.3, noise
detection methods may not be proper for further detecting noisy samples in
$\mathcal{T}^{easy}$. To verify this, we adopt two state-of-the-art methods in
noisy label learning, SPR (Wang et al., 2022) and CGH (Chatterjee, 2020). SPR
is based on penalized regression to build a regularization path, which is used
to detect the most confident noisy samples. Fig. 4(d) shows the result using
SPR. With about 9% noisy samples in $\mathcal{T}^{easy}$, SPR can only filter
out a set with noise rate of 10% in each epoch, which is even worse than
uniform sampling. The result of CGH which is provided in the supplementary is
also unsatisfactory, failing to detect enough noisy samples. Considering these
results, we speculate that the easy set is too challenging for these noise
detection methods to work.
Choices of objective functions for easy data. In Tab. 4 we compare the noise-
tolerant DJS loss $L_{easy}$ with cross entropy, using both hard and soft
labels. The results demonstrate that: (1) DJS loss is more effective than
cross entropy. When using soft label, DJS loss is better than cross entropy by
1.13%, 1.03% and 0.45% when using 500, 1000 and 3000 positive samples. (2)
Using soft label is better than hard label no matter which objective function
is utilized in training. We think this is due to the base model is calibrated
enough. Therefore transferring low-confidence soft labels to hard labels could
hurt the model training.
#Iter | Acc.
---|---
500 | 1000 | 3000
1 | 88.86 | 90.07 | 92.02
2 | 89.28 | 90.51 | 92.51
3 | 89.19 | 90.55 | 92.64
4 | 89.12 | 90.49 | 92.93
Table 6. Test accuracies on CIFAR10 of our model trained with different
number of iterations.
Choices of objective functions for hard data. In Tab. 5 we compare several
choices to optimize hard data, including (1) not using hard data. (2) nnPU
loss on hard data and positive labeled data. (3) cross consistency as in Eq.
8. (4) self-consistency as in Eq. 13. (5) dual-source consistency as in Eq. 7.
The results reflect several important points: (1) Either forms of consistency
can improve the model. Specifically, the model using self-consistency is
better than the model without using hard data by 0.51%, 0.32% and 0.65% among
three settings, and the model using cross consistency is better by 0.65%,
0.71% and 0.81%. (2) Using $L_{nnPU}$ worsens the performance by 0.61%, 0.32%
and 0.80%, which is compatible with our analysis in Sec. 3.4. (3) Cross
consistency is more effective than self-consistency, which is reasonable
because cross consistency can guide the model to learn better low-level
feature. Based on such features, the model can then learn to build useful
high-level features easily.
Relationship between test accuracy and number of iterations. Since our method
utilizes an iterative training strategy, we investigate how the number of
iterations could affect the final result. Tab. 6 shows that while training for
two iterations is much better than single iteration, i.e. 0.44% improvement,
repeating the algorithm more times does not significantly boost the
performance when using 500 or 1000 positive samples. Under the setting of 3000
positive samples, training 4 iterations leads to the best result. The drawback
of using more iterations is the longer training time, which could make the
algorithm less practical and not comparable with the existing methods. Also,
since the variation caused by the randomness can stack during each iteration,
the final result with high iteration number would be more unstable.
Usage | Acc.
---|---
500 | 1000 | 3000
all data | 88.96 | 90.49 | 92.25
hard data | 89.28 | 90.51 | 92.51
Table 7. Test accuracies on CIFAR10 of our model trained with dual-source
consistency applied to all unlabeled data or only hard data.
Usage of consistency regularization As a straightforward thought, our proposed
dual-source consistency regularization can also be applied to easy data. We
testify this idea with results in Tab. 7. By applying dual-source consistency
on all unlabeled data, the model gets accuracy of 88.96%, 90.49% and 92.25%,
receiving no marginal advancement. We think the reason is that since most
pseudo-labels of easy data is reliable and easy to fit, the supervision with
regard to semantic labels would be stronger than other forms of guidance.
Therefore, using DJS loss is enough for the easy data.
What kind of data is selected by early-stop splitting? We visualize some
positive and negative data from easy and hard set of three datasets in Fig. 8.
We can find that the easy data is discriminative enough, with obvious patterns
and symptoms. On the other side, the hard samples are generally much harder to
distinguish. For example the hard negative sample of CIFAR10 is actually an
aeroplane but also looks like a bird, and the hard positive sample in
Myometrium does not have the same symptoms like apparent blood vessels with
dark red color.
## 5\. Conclusion
In this paper we explore the task of PU learning. The problem of intrinsic
hardness of some unlabeled samples is highlighted which may affect the
existing methods’ performance. We analyze the different role of easy and hard
data in training and propose a novel training strategy as a solution to this
problem. Our model includes an early-stop splitting strategy, a noise-tolerant
learning strategy for easy data and a dual-source consistency regularization
for hard data. We achieve remarkable results on CIFAR10 and two medical
datasets, which shows the strength of our method as a promising choice for
solving PU learning both in scientific research and in real-life applications.
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## Appendix A Experimental Detail
### A.1. Detailed Information for Liver dataset
Each sample in the Liver dataset denotes a patient with liver cancer. The data
consists of several modalities of MRI images from all patient, including T1WI,
T1CE1, T1IP and T2WI. We simply adopt T1CE1 as the input modality of our
experiment and leave the full usage of all modalities as future works. the
model reads all of the slices of each sample with Pydicom and resize them to
$256\times 256$ without additional normalization. The labels are transformed
from the survival time of each patient, which ranges from 0 to 120.
Figure 5. Detailed architecture of our network for Liver dataset. N, C, H, W
denotes number of slices, number of channels, image height and image width
respectively.
### A.2. Detailed Architecture for Liver dataset
As we have mentioned in our main context, since each liver cancer sample is
composed of several slices, we adopt a special architecture to deal with such
data. Essentially, as shown in Fig. 5, the basic ResNet18 is used to extract
features for all slices. The network weights are shared among different
slices. Then for slices from one sample, a multi-head attention module is used
to aggregate these features. Then a pooling layer and a classifier are used,
as the same in original ResNet18.
## Appendix B Additional Experiment Results
Figure 6. Training accuracy of clean and noisy data respectively in easy set.
Figure 7. Gradient visualization of both clean and noisy samples on CIFAR10.
### B.1. Results on noise detection methods
To support our claim in the main context that noise detection methods may not
be proper for further detecting noisy samples in $\mathcal{T}^{easy}$, we
first plot the training accuracy of noisy and clean samples in the easy set.
As depicted in Fig. 6, while the training accuracy of clean samples is higher
than that of noisy ones for each epoch, the actual size of wrongly-predicted
clean samples is also much higher than noisy set. The proportion of wrongly-
predicted noisy samples among all samples that are wrongly predicted is much
smaller than that in the original unlabeled data as. This means these easy
noisy samples do not enjoy the slow-learning property to the same extend as
the hard noisy ones. Moreover, the result of CGH (Chatterjee, 2020) is shown
in Fig. 7, in which we visualize the gradient of the first weight in the first
convolutional layer of the whole network with regard to both easy noisy
samples and easy clean samples. We can find that both the scale and
distribution are similar between these two kinds of data, which means the
coherent gradient hypothesis also does not hold so well on the easy data.
$\rho$ | Acc.
---|---
500 | 1000 | 3000
0.3 | 88.26 | 89.47 | 91.75
0.4 | 88.68 | 89.51 | 92.03
0.5 | 88.55 | 90.02 | 92.06
0.6 | 88.92 | 90.24 | 92.03
0.7 | 89.28 | 90.51 | 92.51
0.8 | 89.18 | 90.31 | 92.13
0.9 | 89.16 | 90.23 | 92.17
Table 8. Test accuracies on CIFAR10 of our model trained with different
$\rho$ in Eq. (6) in the main context.
### B.2. Choices of $\rho$
We compare different $\rho$ in Eq. (6) in the main context, with the results
in Tab. 8. As mentioned in the original paper (Englesson and Azizpour, 2021),
smaller $\rho$ leads to faster convergence and less robustness, which is
roughly consistent with our results. When $\rho=0.3$, the model gets 88.26%,
89.47% and 91.75%, which is the worst among all $\rho$ and almost similar to
the results when using soft cross entropy. As $\rho$ gets larger, the
performance gets better and saturates at $\rho=0.7$.
Model | Acc.
---|---
500 | 1000 | 3000
uPU | 85.98$\pm$1.09 | 88.22$\pm$0.28 | 89.83$\pm$0.42
uPU+Split-PU | 87.44$\pm$0.18 | 90.57$\pm$0.26 | 91.51$\pm$0.13
Table 9. Test accuracies on CIFAR10 of models trained with and without early
stop splitting.
### B.3. Choices of $\alpha$ and $\beta$
We conduct the comparison on $\alpha$ and $\beta$ with the same setting as in
the main paper, i.e. 500/1000/3000 positive labeled samples on CIFAR10. As
shown in Tab. 10 and Tab. 11, the performance of our model among different
hyper-parameter settings is relatively robust. Meanwhile, the trend of
performance is generally consistent with few exceptions. For example,
$\alpha=0.5$ or $\beta=0.2$ lead to the highest accuracy when using 3000
positive labeled samples. We advocate that these two hyper-parameters control
the balance among different loss terms adopted by our method. In particular,
larger $\alpha$ and $\beta$ lead to the scale of $L_{pred}$ and $L_{feat}$, as
depicted in Eq. (11) and Eq. (12), larger than that of $L_{easy}$. The
strength of annotations is thus weakened in this way. Consequently, the model
becomes poorer at learning label information from the dataset, which leads to
worse performance. On the other hand, smaller $\alpha$ makes the effect of
prediction consistency negligible, hence resulting in lower accuracy. We will
add these results to our paper.
$\alpha$ | Acc.
---|---
500 | 1000 | 3000
0.1 | 87.68 | 89.87 | 90.84
0.2 | 88.05 | 90.08 | 92.05
0.3 | 89.18 | 90.51 | 92.51
0.4 | 88.61 | 90.03 | 92.27
0.5 | 88.52 | 90.08 | 92.55
0.6 | 88.48 | 89.93 | 92.41
0.7 | 88.60 | 90.09 | 92.13
0.8 | 88.68 | 89.56 | 91.17
0.9 | 88.30 | 89.30 | 90.53
Table 10. Test accuracies on CIFAR10 of our model trained with different $\alpha$ in Eq. (13) in the main context. $\beta$ | Acc.
---|---
500 | 1000 | 3000
0.1 | 89.18 | 90.51 | 92.51
0.2 | 88.96 | 90.31 | 92.63
0.3 | 89.02 | 90.39 | 92.37
0.4 | 88.67 | 90.11 | 92.43
0.5 | 88.84 | 89.87 | 92.14
0.6 | 88.81 | 90.09 | 92.37
0.7 | 89.07 | 90.08 | 91.71
0.8 | 88.48 | 89.14 | 91.24
0.9 | 88.26 | 89.49 | 91.54
Table 11. Test accuracies on CIFAR10 of our model trained with different
$\beta$ in Eq. (13) in the main context.
### B.4. Application of Split-PU on other base models
To further verify the efficacy of our proposed method, we use another popular
objective function uPU (Du Plessis et al., 2014) as our base model, i.e.
replacing $L_{nnPU}$ with the following $L_{uPU}$ when training $\phi_{base}$:
(14) $\displaystyle L_{uPU}$
$\displaystyle=\frac{\pi}{n_{p}}\sum_{i=1}^{n_{p}}L(\phi(I_{i}^{p}),1)+$ (15)
$\displaystyle\frac{1}{n_{u}}\sum_{i=1}^{n_{u}}L(\phi(I_{i}^{u}),-1)-\frac{\pi}{n_{u}}\sum_{i=1}^{n_{u}}L(\phi(I_{i}^{p}),-1)$
The results on CIFAR10 are shown in Tab. 9, which indicates that the
improvement of our Split-PU on both nnPU and uPU is consistent. It is
noticeable that when using 1000 positive labeled samples, using uPU with
Split-PU is even better than using nnPU with Split-PU, receiving accuracy of
90.57% with comparable standard deviation.
Figure 8. Visualization of the easy and hard sets.
|
# Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO
Scheme
Alexander Vidal
Dept. of Applied Mathematics and Statistics
Colorado School of Mines
&Samy Wu Fung
Dept. of Applied Mathematics and Statistics
Dept. of Computer Science
Colorado School of Mines
&Luis Tenorio
Dept. of Applied Mathematics and Statistics
Colorado School of Mines
&Stanley Osher
Dept. of Mathematics
University of California, Los Angeles
&Levon Nurbekyan
Dept. of Mathematics
University of California, Los Angeles
<EMAIL_ADDRESS>
###### Abstract
A normalizing flow (NF) is a mapping that transforms a chosen probability
distribution to a normal distribution. Such flows are a common technique used
for data generation and density estimation in machine learning and data
science. The density estimate obtained with a NF requires a change of
variables formula that involves the computation of the Jacobian determinant of
the NF transformation. In order to tractably compute this determinant,
continuous normalizing flows (CNF) estimate the mapping and its Jacobian
determinant using a neural ODE. Optimal transport (OT) theory has been
successfully used to assist in finding CNFs by formulating them as OT problems
with a soft penalty for enforcing the standard normal distribution as a target
measure. A drawback of OT-based CNFs is the addition of a hyperparameter,
$\alpha$, that controls the strength of the soft penalty and requires
significant tuning. We present JKO-Flow, an algorithm to solve OT-based CNF
without the need of tuning $\alpha$. This is achieved by integrating the OT
CNF framework into a Wasserstein gradient flow framework, also known as the
JKO scheme. Instead of tuning $\alpha$, we repeatedly solve the optimization
problem for a fixed $\alpha$ effectively performing a JKO update with a time-
step $\alpha$. Hence we obtain a "divide and conquer" algorithm by repeatedly
solving simpler problems instead of solving a potentially harder problem with
large $\alpha$.
## 1 Introduction
A normalizing flow (NF) is a type of generative modeling technique that has
shown great promise in applications arising in physics noe2019boltzmann ;
brehmer2020madminer ; carleo2019machine as a general framework to construct
probability densities for continuous random variables in high-dimensional
spaces papamakarios2019normalizing ; kobyzev2019normalizing ; rezende2015 . An
NF provides a $\mathcal{C}^{1}$-diffeomorphism $f$ (i.e., a normalizing
transformation) that transforms the density $\rho_{0}$ of an initial
distribution $P_{0}$ to the density $\rho_{1}$ of the standard multivariate
normal distribution $P_{1}$ – hence the term "normalizing." Given such mapping
$f$, the density $\rho_{0}$ can be recovered from the Gaussian density via the
change of variables formula,
$\log\rho_{0}(x)=\log\rho_{1}\left(f(x)\right)+\log|\det J_{f}(x)|,$ (1)
where $J_{f}\in\mathbb{R}^{d\times d}$ is the Jacobian of $f$. Moreover, one
can obtain samples with density $\rho_{0}$ by pushing forward Gaussian samples
via $f^{-1}$.
###### Remark 1.
Throughout the paper we slightly abuse the notation, using the same notation
for probability distributions and their density functions. Additionally, given
a probability distribution $P_{0}$ on $\mathbb{R}^{d}$ and a measurable
mapping $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$, we define the pushforward
distribution of $P_{0}$ through $f$ as $(f\sharp P_{0})(B)=P_{0}(f^{-1}(B))$
for all Borel measurable $B\subseteq\mathbb{R}^{d}$ villani2003topics .
There are two classes of normalizing flows: finite and continuous. A finite
flow is defined as a composition of a finite number of
$\mathcal{C}^{1}$-diffeomorphisms: $f=f_{1}\circ f_{2}\circ\cdots\circ f_{n}$.
To make finite flows computationally tractable, each $f_{i}$ is chosen to have
some regularity properties such as a Jacobian with a tractable determinant;
for example, $J_{f_{i}}$ may have a triangular structure
papamakarios2017masked ; baptista2021learning ; zech2022sparse .
On the other hand, continuous normalizing flows (CNFs) estimate $f$ using a
neural ODE of the form chen2017continuous :
$\partial_{t}z(x,t)=v_{\theta}(z(x,t),t),\qquad z(x,0)=x,\qquad 0\leq t\leq
T,$ (2)
where $\theta$ are the parameters of the neural ODE. In this case, $f$ is
defined as $f(x)=z(x,T)$ (for simplicity, we remove the dependence of $z$ on
$\theta$). One of the main advantages of CNFs is that we can tractably
estimate the log-determinant of the Jacobian using Jacobi’s identity, which is
commonly used in fluid mechanics (see, e.g., (villani2003topics, , p.114)):
$\begin{split}\partial_{t}\log|\det\nabla_{x}z(x,t)|&=\nabla_{z}\cdot
v_{\theta}(z(x,t),t)=\operatorname{trace}\left(\nabla_{z}v_{\theta}(z(x,t),t)\right).\end{split}$
(3)
This is computationally appealing as one can replace the expensive determinant
calculation by a more tractable trace computation of
$\nabla_{z}v_{\theta}(z(x,t),t)$. Importantly, no restrictions on
$\nabla_{z}v_{\theta}(z(x,t),t)$ (e.g., diagonal or triangular structure) are
needed; thus, these Jacobians are also referred to as “free-form Jacobians”
grathwohl2019ffjord .
The goal in training a CNF is to find parameters, $\theta$, such that
$f=z(\,\cdot,T)$ leads to a good approximation of $\rho_{1}$ or, assuming $f$
is invertible, the pushforward of $\rho_{1}$ through $f^{-1}$ is a good
approximation of $\rho_{0}$ rezende2015 ; papamakarios2017masked ;
papamakarios2019normalizing ; grathwohl2019ffjord . Indeed, let
$\widehat{\rho}_{0}$ be this pushforward density obtained with a CNF $f$; that
is, $\widehat{\rho}_{0}=f^{-1}\sharp\rho_{1}$. We then minimize the Kullback-
Leibler (KL) divergence from $\widehat{\rho}_{0}$ to $\rho_{0}$ given by
$\min_{\theta}~{}\mathbb{E}_{x\sim\rho_{0}}\log(\rho_{0}(x)/\widehat{\rho}_{0}(x))=\min_{\theta}~{}\mathbb{E}_{x\sim\rho_{0}}\left[\log\rho_{0}(x)-\log\rho_{1}(z(x,T))-\ell(x,T)\right],$
where $\ell(x,T)=\log|\det\nabla z(x,T)|$. Dropping the $\theta$-independent
term $\log\rho_{0}$ and using (2) and (3), this previous optimization problem
reduces to the minimization problem
$\min_{\theta}\;\;\mathbb{E}_{x\sim\rho_{0}}\;C(x,T),\quad
C(x,T):=-\log\rho_{1}(z(x,T))-\ell(x,T)$ (4)
subject to ODE constraints
$\partial_{t}\begin{bmatrix}z(x,t)\\\
\ell(x,t)\end{bmatrix}=\begin{bmatrix}v_{\theta}(z(x,t),t)\\\
\operatorname{trace}\left(\nabla_{z}v_{\theta}(z(x,t),t)\right)\end{bmatrix},\qquad\begin{bmatrix}z(x,0)\\\
\ell(x,0)\end{bmatrix}=\begin{bmatrix}x\\\ 0\end{bmatrix}.$ (5)
The ODE (5) might be stiff for certain values of $\theta$, leading to
extremely long computation times. Indeed, the dependence of $v$ on $\theta$ is
highly nonlinear and might generate vector fields that lead to highly
oscillatory trajectories with complex geometry.
Some recent work leverages optimal transport theory to find the CNF
finlay2020train ; onken2021ot . In particular, a kinetic energy regularization
term (among others) is added to the loss to “encourage straight trajectories”
$z(x,t)$. That is, the flow is trained by minimizing the following instead of
(4):
$\begin{split}\min_{\theta}\;\;\mathbb{E}_{x\sim\rho_{0}}\;\int_{0}^{T}\frac{1}{2}\|v_{\theta}(z(x,t),t)\|^{2}dt+\alpha
C(x,T)\end{split}$ (6)
subject to (5). The key insight in finlay2020train ; onken2021ot is that (4)
is an example of a degenerate OT problem with a soft terminal penalty and
without a transportation cost. The first objective term in (6) given by the
time integral is the transportation cost term, whereas $\alpha$ is a
hyperparameter that balances the soft penalty and the transportation cost.
Including this cost makes the problem well-posed by forcing the solution to be
unique villani2008optimal . Additionally, it enforces straight trajectories so
that (5) is not stiff. Indeed finlay2020train ; onken2021ot empirically
demonstrate that including optimal transport theory leads to faster and more
stable training of CNFs. Intuitively, we minimize the KL divergence _and_ the
arclength of the trajectories.
Although including optimal transport theory into CNFs has been very successful
onken2021ot ; finlay2020train ; yang2019 ; zhang2018monge , there are two key
challenges that render them difficult to train. First, estimating the log-
determinant in (4) via the trace in (5) is still computationally taxing and
commonly used methods rely on stochastic approximations grathwohl2019ffjord ;
finlay2020train , which adds extra error. Second, including the kinetic energy
regularization requires tuning of the hyperparameter $\alpha$. Indeed, if
$\alpha$ is chosen too small in (6), then the kinetic regularization term
dominates the training process, and the optimal solution consists of not
moving, i.e., $f(x)=x$. On the other hand, if $\alpha$ is chosen too large, we
return to the original setting where the problem is ill-posed, i.e., there are
infinitely many solutions. Finally, finding an "optimal" $\alpha$ is problem
dependent and requires tuning on a cases-by-case basis.
### 1.1 Our Contribution
We present JKO-Flow, an optimal transport-based algorithm for training CNFs
without the need to tune the hyperparameter $\alpha$ in (6). Our approach also
leverages fast numerical methods for exact trace estimation from the recently
developed optimal transport flow (OT-Flow) onken2021ot ; ruthotto2020machine .
The key idea is to integrate the OT-Flow approach into a Wasserstein gradient
flow framework, also known as the Jordan, Kinderlehrer, and Otto (JKO) scheme
jordan1998variational . Rather than tuning the hyperparameter $\alpha$
(commonly done using a grid search), the idea is to simply pick any $\alpha$
and solve a sequence of "easier" OT problems that gradually approach the
target distribution. Each solve is precisely a gradient descent in the space
of distributions, a Wasserstein gradient descent, and the scheme provably
converges to the desired distribution for all $\alpha>0$ salim20Wasserstein .
Our experiments show that our proposed approach is effective in generating
higher quality samples (and density estimates) and also allows us to reduce
the number of parameters required to estimate the desired flow.
Our strategy is reminiscent of debiasing techniques commonly used in inverse
problems. Indeed, the transportation cost that serves as a regularizer in (6)
introduces a bias – the smaller $\alpha$ the more bias is introduced (see,
e.g., tenIP ), so good choices of $\alpha$ tend to be larger. One way of
removing the bias and the necessity of tuning the regularization strength is
to perform a sequence of Bregman iterations osher05iterative ;
burger05nonlinear also known as nonlinear proximal steps. Hence our approach
reduces to debiasing via Bregman or proximal steps in the Wasserstein space.
In the context of CNF training, Bregman iterations are advantageous due to the
flexibility of the choice for $\alpha$. Indeed, the resulting loss function is
non-convex and its optimization tends to get harder for large $\alpha$. Thus,
instead of solving one harder problem we solve several “easier” problems.
## 2 Optimal Transport Background and Connections to CNFs
Denote by $\mathcal{P}_{2}(\mathbb{R}^{d})$ the space of Borel probability
measures on $\mathbb{R}^{d}$ with finite second-order moments, and let
$\rho_{0},\rho_{1}\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The quadratic optimal
transportation (OT) problem (which also defines the Wasserstain metric
$W_{2}$) is then formulated as
$W_{2}^{2}(\rho_{0},\rho_{1})=\inf_{\pi\in\Gamma(\rho_{0},\rho_{1})}\int_{\mathbb{R}^{2d}}\|x-y\|^{2}d\pi(x,y),$
(7)
where $\Gamma(\rho_{0},\rho_{1})$ is the set of probability measures
$\pi\in\mathcal{P}(\mathbb{R}^{2d})$ with fixed $x$ and $y$-marginals
densities $\rho_{0}$ and $\rho_{1}$, respectively. Hence the cost of
transporting a unit mass from $x$ to $y$ is $\|x-y\|^{2}$, and one attempts to
transport $\rho_{0}$ to $\rho_{1}$ as cheaply as possible. In (7), $\pi$
represents a transportation plan, and $\pi(x,y)$ is the mass being transported
from $x$ to $y$. One can prove that $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$
is a complete separable metric space villani2003topics . OT has recently
become a very active research area in PDE, geometry, functional inequalities,
economics, data science and elsewhere partly due to equipping the space of
probability measures with a (Riemannian) metric villani2003topics ;
villani2008optimal ; peyre2018computational ; santambrogio2015optimal .
As observed in prior works, there are many similarities between OT and NFs
onken2021ot ; finlay2020train ; yang2020potential ; zhang2018monge . This
connection becomes more transparent when considering the dynamic formulation
of (7). More precisely, the Benamou-Brenier formulation of the OT problem is
given by benamou2000computational :
$\begin{split}\frac{T}{2}W_{2}^{2}(\rho_{0},\rho_{1})=\inf_{v,\rho}\;\;&\int_{0}^{T}\int_{\mathbb{R}^{d}}\frac{1}{2}\|v(x,t)\|_{2}^{2}\rho(x,t)dxdt\\\
\mbox{ s.t. }\;\;&\partial_{t}\rho(x,t)+\nabla\cdot(\rho(x,t)v(x,t))=0\\\
&\rho(x,0)=\rho_{0}(x),\;\;\rho(x,T)=\rho_{1}(x).\end{split}$ (8)
Hence, the OT problem can be formulated as a problem of flowing $\rho_{0}$ to
$\rho_{1}$ with a velocity field $v$ that achieves minimal kinetic energy. The
optimal velocity field $v$ has several appealing properties. First, particles
induced by the optimal flow $v$ travel in straight lines. Second, particles
travel with constant speed. Moreover, under suitable conditions on $\rho_{0}$
and $\rho_{1}$, the optimal velocity field is unique villani2003topics .
Given a velocity field $v$, denote by $z(x,t)$ the solution of the ODE
$\partial_{t}z(x,t)=v(z(x,t),t),\qquad z(x,0)=x,\qquad 0\leq t\leq T.$
Then, under suitable regularity conditions, we have that the solution of the
continuity equation is given by $\rho(\cdot,t)=z(\cdot,t)\sharp\rho_{0}$. Thus
the optimization problem in (8) can be written as
$\begin{split}\inf_{v}\;\;&\int_{0}^{T}\int_{\mathbb{R}^{d}}\frac{1}{2}\|v(z(x,t),t)\|_{2}^{2}\rho_{0}(x)dxdt\\\
\mbox{ s.t.
}\;\;&\partial_{t}z(x,t)=v(z(x,t),t),~{}z(x,0)=x,~{}z(\cdot,T)\sharp\rho_{0}=\rho_{1}.\end{split}$
(9)
This previous problem is very similar to (4) with the following differences:
* •
the objective function in (4) does not have the kinetic energy of
trajectories,
* •
the terminal constraint is imposed as a soft constraint in (4) and as a hard
constraint in (9), and
* •
$v$ in (4) is $\theta$-dependent, whereas the formulation in (9) is in the
non-parametric regime.
So the NF (4) can be thought of as an approximation to a degenerate
transportation problem that lacks transportation cost. Based on this insight
one can regularize (4) by adding the transportation cost and arrive at (6) or
some closely related version of it onken2021ot ; finlay2020train ;
yang2020potential ; zhang2018monge . It has been observed that the
transportation cost (kinetic energy) regularization significantly improves the
training of NFs.
## 3 JKO-Flow: Wasserstein Gradient Flows for CNFs
While the OT-based formulation of CNFs in (6) has been found successful in
some applications onken2021ot ; finlay2020train ; yang2020potential ;
zhang2018monge , a key difficulty arises in choosing how to balance the
kinetic energy term and the KL-divergence, i.e., on selecting $\alpha$. This
difficulty is typical in problems where the constraints are imposed in a soft
fashion. Standard training of CNFs typically involves tuning for a “large but
hopefully stable enough” step size $\alpha$ so that the KL divergence term is
sufficiently small after training. To this end, we propose an approach that
avoids the need to tune $\alpha$ by using the fact that the solution to (6) is
an approximation to a backward Euler (or proximal point) algorithm when
discretizing the Wasserstein gradient flow using the Jordan-Kinderlehrer-Otto
(JKO) scheme jordan1998variational . The seminal work in jordan1998variational
provides a gradient flow structure of the Fokker-Planck equation using an
implicit time discretization. That is, given $\alpha>0$, density at
$k^{\text{th}}$ iteration, $\rho^{(k)}$, and terminal density $\rho_{1}$, one
finds
$\begin{split}\rho^{(k+1)}=&\operatorname*{arg\,min}_{\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})}\;\frac{1}{2\alpha}W_{2}^{2}(\rho,\rho^{(k)})+KL(\rho||\rho_{1})\\\
=&\operatorname*{arg\,min}_{v}\;\frac{1}{\alpha}\int_{0}^{1}\int_{\mathbb{R}^{d}}\frac{1}{2}\|v(z(x,t),t)\|_{2}^{2}\rho_{0}(x)dxdt+KL(z(\cdot,1)\sharp\rho^{(k)}||\rho_{1})\\\
\mbox{ s.t.
}\;\;&\partial_{t}z(x,t)=v(z(x,t),t),~{}z(x,0)=x,~{}z(\cdot,T)\sharp\rho_{0}=\rho_{1},\end{split}$
(10)
for $k=0,1,\ldots$, and $\rho^{(0)}=\rho_{0}$. Here, $\alpha$ takes the role
of a step size when applying a proximal point method to the KL divergence
using the Wasserstein-2 metric, and $\\{\rho^{(k)}\\}$ provably converges to
$\rho_{1}$ jordan1998variational ; salim20Wasserstein . Hence, repeatedly
solving (9) with the KL penalty acting as a soft constraint yields an
arbitrarily accurate approximation of $\rho_{1}$. In the parametric regime
each iteration takes the form
$\operatorname*{arg\,min}_{\theta}\;\;\mathbb{E}_{x\sim\rho^{(k)}}\;\int_{0}^{T}\frac{1}{2}\|v_{\theta}(x,t)\|^{2}dt+\alpha
C(x,T)\quad\text{subject to}\quad\eqref{eq: min_likelihood_constraints}.$ (11)
Thus we solve a sequence of problems (6), where the initial density of the
current subproblem is given by the pushforward of the density generated in the
previous subproblem.
Algorithm 1 Proposed Algorithm
1: Input: Samples from $\rho_{0}$, step size $\alpha>0$, number of steps $K$
2: Initialize $\theta_{1}$ at random
3: for $k=1,\ldots,K$ do
4: Solve for $\theta_{k}$ using samples by solving (11)
5: Update distribution of samples using $v_{\theta_{k}}$
6: end for
7: Output: saved weights $\theta_{1},\ldots,\theta_{K}$
Importantly, our proposed approach _does not require tuning $\alpha$_.
Instead, we solve a sequence of subproblems that is guaranteed to converge to
$\rho_{1}$ jordan1998variational prior to the neural network
parameterization; see Alg. 1. While our proposed methodology can be used in
tandem with any algorithm used to solve (11), an important numerical aspect in
our approach is to leverage fast computational methods that use _exact_ trace
estimation in (5); this approach is called OT-Flow onken2021ot . Consequently,
we avoid the use of stochastic approximation methods for the trace, e.g.,
Hutchinson’s estimator avron2011 ; hutchinson1990stochastic ; tenIP , as is
typically done in CNF methods grathwohl2019ffjord ; finlay2020train . A
surprising result of our proposed method is that it empirically shows improved
performance even with fewer number of parameters (see Fig. 3).
## 4 Related Works
Density Estimation. One of the main advantages of NFs over other generative
models is that they provide density estimates of probability distributions
using (1). That is, we do not need to apply a separate density estimation
technique after generating samples from a distribution, e.g., as in GANs
goodfellow2020generative . Multivariate density estimation is a fundamental
problem in statistics silverman1986density ; scott2015multivariate , High
Energy Physics (HEP) cranmer2001kernel and in other fields of science dealing
with multivariate data. For instance, particle physicists in HEP study
possible distributions from a set of high energy data. Another application of
density estimation is in confidence level calculations of particles in Higgs
searches at Large Electron Positron Colliders (LEP) opal2000search and
discriminant methods used in the search for new particles cranmer2001kernel .
Finite Flows. Finite normalizing flows tabak2013family ; rezende2015 ;
papamakarios2019normalizing ; kobyzev2019normalizing use a composition of
discrete transformations, where specific architectures are chosen to allow for
efficient inverse and Jacobian determinant computations. NICE dinh2014nice ,
RealNVP dinh2016density , IAF kingma2016improved , and MAF
papamakarios2017masked use either autoregressive or coupling flows where the
Jacobian is triangular, so the Jacobian determinant can be tractably computed.
GLOW kingma2018glow expands upon RealNVP by introducing an additional
invertible convolution step. These flows are based on either coupling layers
or autoregressive transformations, whose tractable invertibility allows for
density evaluation and generative sampling. Neural Spline Flows
durkan2019neural use splines instead of the coupling layers used in GLOW and
RealNVP. Using monotonic neural networks, NAF huang2018neural require
positivity of the weights, which UMNN wehenkel2019unconstrained circumvent
this requirement by parameterizing the Jacobian and then integrating
numerically.
Continuous and Optimal Transport-based Flows. Modeling flows with differential
equations is a natural and commonly used method suykens1998 ;
welling2011bayesian ; neal2011mcmc ; salimans2015markov ;
ruthotto2021introduction ; huang2020convex . In particular, CNFs model their
flow via a neural ordinary differential equation chen2017continuous ;
chen2018neural ; grathwohl2019ffjord . Among the most well-known CNFs are
FFJORD grathwohl2019ffjord , which estimates the determinant of the Jacobian
by accumulating its trace along the trajectories, and the trace is estimated
using Hutchinson’s estimator avron2011 ; hutchinson1990stochastic ; tenIP . To
promote straight trajectories, RNODE finlay2020train regularizes FFJORD with
a transport cost $L(\boldsymbol{x},T)$. RNODE also includes the Frobenius norm
of the Jacobian $\|\nabla\mathbf{v}\|_{F}^{2}$ to stabilize training. The
trace and the Frobenius norm are estimated using a stochastic estimator and
report speedup by a factor of 2.8.
Monge-Ampère Flows zhang2018monge and Potential Flow Generators yang2019
similarly draw from OT theory but parameterize a potential function instead of
the dynamics directly. OT is also used in other generative models sanjabi2018
; salimans2018improving ; lei2018geometric ; lin2019fluid ;
avraham2019parallel ; tanaka2019discriminator . OT-Flow onken2021ot is based
on a discretize-then-optimize approach onken2020do that also parameterizes
the potential function. To evaluate the KL divergence, OT-Flow estimates the
density using an _exact_ trace computation following the work of
ruthotto2020machine .
Wasserstein Gradient Flows. Our proposed method is most closely related to
fan2021variational , which also employs a JKO-based scheme to perform
generative modeling. But a key difference is that fan2021variational
reformulates the KL-divergence as an optimization over difference of
expectations (See (fan2021variational, , Prop. 3.1)); this makes their
approach akin to GANs, where the density cannot be obtained without using a
separate density estimation technique. Our proposed method is also closely
related to methods that use input-convex CNNs mokrov2021large ;
bunne2022proximal ; alvarez2021optimizing . mokrov2021large focuses on the
special case with KL divergence as objective function. alvarez2021optimizing
solve a sequence of subproblems different from the fluid flow formulation
presented in (11). They also require an end-to-end training scheme that
backpropagates to the initial distribution; this can become a computational
burden when the number of time discretizations is large. bunne2022proximal
utilizes a JKO-based scheme to approximate a population dynamics given an
observed trajectory and focus on applications in computational biology. Other
related works include natural gradient methods nurbekyan2022efficient and
implicit schemes based on the Wasserstein-1 distance heaton2020wasserstein .
## 5 Numerical Experiments
We demonstrate the effectiveness of our proposed JKO-Flow on a series of
synthetic and real-world datasets. As previously mentioned, we compute each
update in (10) by solving (6) using the OT-Flow solver onken2021ot , which
leverages fast and exact trace computations. We also use the same architecture
provided in onken2021ot . Henceforth, we shall also call the traditional CNF
approach the “single-shot” approach.
$\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
3.58e-2 | 3.56e-3 | 1.42e-3 | 1.26e-3
| | |
MMD2 Values, JKO-Flow (five iterations)
4.90e-4 | 5.70e-4 | 6.40e-4 | 9.00e-4
| | |
Figure 1: Checkerboard dataset: Generated samples of $\widehat{\rho}_{0}$
using the standard one-shot approach (top row). Generated using our proposed
JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10,
50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_.
Data
$x$ | | | | | | |
---|---|---|---|---|---|---|---
Estimate
$\rho_{0}$ | | | | | | |
Generation
$f^{-1}(y)$ | | | | | | |
Figure 2: Density estimation on 2D toy problems using five JKO-Flow
iterations. Top: samples from the unknown distribution $\rho_{0}$. Middle:
density estimate for $\rho_{0}$ computed by inverting the flow through the
five iterations of JKO-Flow from $\rho_{1}$ via (2). Bottom: samples generated
by inverse JKO-Flow through five iterations where $y$ has density $\rho_{1}$.
Maximum Mean Discrepancy Metric (MMD). Our density estimation problem requires
approximating a density $\rho_{0}$ by finding a transformation $f$ such that
$f^{-1}\sharp\rho_{1}$ has density $\widehat{\rho}_{0}$ close to $\rho_{0}$
where $\rho_{1}$ is the standard Gaussian. However, $\rho_{0}$ is not known in
real-world density estimation scenarios, such as in physics applications, all
we have are samples $X=\\{x_{i}\\}_{i=1}^{n}$ from the unknown distribution.
Consequently, we use the observed samples $X$ and samples
$\widehat{X}=\\{\widehat{x}_{j}\\}_{j=1}^{m}$,
$\widehat{x}_{j}=f^{-1}(q_{j})$, generated by the CNF and samples
$Q=\\{q_{j}\\}_{j=1}^{m}$ from $\rho_{1}$ to determine if their corresponding
distributions are close in some sense. To measure the discrepancy we use a
particular integral probability metric zolotarev1976metric ; rachev2013methods
; muller1997integral known as maximum mean discrepancy (MMD) defined as
follows gretton2012mmd : Let $x$ and $y$ be random vectors in $\mathbb{R}^{d}$
with distributions $\mu_{x}$ and $\mu_{y}$, respectively, and let
$\mathcal{H}$ be a reproducing kernel Hilbert space of functions on
$\mathbb{R}^{d}$ with Gaussian kernel (see paulsen2016introduction for an
introduction)
$k(x_{i},x_{j})=\exp{\left(-\frac{1}{2}\|x_{i}-x_{j}\|^{2}\right)}.$ (12)
Then the MMD of $\mu_{x}$ and $\mu_{y}$ is given by
$\mathrm{MMD}_{\mathcal{H}}(\mu_{x},\mu_{y})=\sup_{\|f\|_{\mathcal{H}}\leq
1}\,|\,\mathbb{E}\,f(x)-\mathbb{E}\,f(y)\,|.$
It can be shown that $\mathrm{MMD}_{\mathcal{H}}$ defines a metric on the
class of probability measures on $\mathbb{R}^{d}$ gretton2012mmd ;
fukumizu2007kernel . The squared-MMD can be written in terms of the kernel as
follows:
$\mathrm{MMD}^{2}_{\mathcal{H}}(\mu_{x},\mu_{y})=\mathbb{E}\,k(x,x^{\prime})+\mathbb{E}\,k(y,y^{\prime})-2\,\mathbb{E}\,k(x,y),$
where $x,x^{\prime}$ are iid $\mu_{x}$ independent of $y,y^{\prime}$ which are
iid $\mu_{y}$. An unbiased estimate of the squared-MMD based on the samples
$X$ and $\widehat{X}$ defined above is given by gretton2012mmd :
$\mathrm{MMD}^{2}_{\mathcal{H}}(X,\widehat{X})=\frac{1}{n(n-1)}\sum_{i\neq
j}k(x_{i},x_{j})+\frac{1}{m(m-1)}\sum_{k\neq\ell}k(\widehat{x}_{k},\widehat{x}_{\ell})-\frac{2}{nm}\sum_{i,\ell}k(x_{i},\widehat{x}_{\ell}).$
Note that the MMD is not used for algorithmic training of the CNF, it is only
used to compare the densities $\rho_{0}$ and $\widehat{\rho}_{0}$ based on the
samples $X$ and $\widehat{X}$.
Synthetic 2D Data Set. We begin by testing our method on seven two-dimensional
(2D) benchmark datasets for density estimation algorithms commonly used in
machine learning grathwohl2019ffjord ; wehenkel2019unconstrained ; see Fig. 2.
We generate results with JKO-Flow for different values of $\alpha$ and for
different number of iterations. We use $\alpha=1,\,5,\,10,\,$ and $50$, and
for each $\alpha$ we use the single shot approach $k=1$ and JKO-Flow with
$k=5$ iterations from (10). Note that in CNFs, we are interested in estimating
the density (and generating samples) from $\rho_{0}$; consequently, once we
have the optimal weights $\theta^{(1)},\theta^{(2)},\ldots,\theta^{(5)}$, we
must “flow backwards” starting with samples from the normal distribution
$\rho_{1}$. Fig. 1 shows that JKO-Flow outperforms the single shot approach
for different values of $\alpha$. In particular, the performance for the
single shot approach varies drastically for different values of $\alpha$, with
$\alpha=1$ being an order of magnitude higher in MMD than $\alpha=5$. On the
other hand, JKO-Flow is _consistent regardless of the value of $\alpha$_. As
previously mentioned, this is expected as JKO-Flow is a proximal point
algorithm that converges regardless of the step size $\alpha$. In this case,
five JKO-Flow iterations are enough to obtain this consistency. Additional
plots and hyperparameter setups for different benchmark datasets with similar
performance results are shown in the Appendix. Tab. 1 summarizes the
comparison between the single shot and JKO-Flow on all synthetic 2D datasets
for different values of $\alpha$. We also show an illustration of all the
datasets, estimated densities, and generated samples with JKO-Flow in Fig. 2.
Varying Network Size. In addition to obtaining consistent results for
different values of $\alpha$, we also _empirically_ observe that JKO-Flow
outperforms the single shot approach for different numbers of network
parameters, i.e., network size. We illustrate this in Fig. 3. This is also
intuitive as we reformulate the problem of finding a single “difficult”
optimal transportation problem as a sequence of “smaller and easier” OT
problems. In this setup, we vary the width of a two-layer ResNet he2016deep .
In particular, we choose the widths to be $m=3,4,5,8$, and $16$. These
correspond to $40,53,68,125$, and $365$ parameters. The hyperparameter
$\alpha$ is chosen to be the best performing value for each synthetic dataset.
All datasets vary $m$ for fixed $\alpha=5$, except the 2 Spiral dataset, which
uses $\alpha=50$; we chose these $\alpha$ values as they performed the best in
the fixed $m$ experiments. Similar results are also shown for the remaining
synthetic datasets in the appendix. Tab. 2 summarizes the comparison between
the single shot and JKO-Flow on all synthetic 2D datasets.
$m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
1.1e-2 | 5.6e-3 | 2.46e-3 | 3.03e-3 | 2.7e-3
| | | |
MMD2 Values, JKO-Flow (five iterations)
5.6e-3 | 1.07e-3 | 2.7e-4 | 2.32e-4 | 4.16e-4
| | | |
Figure 3: Checkerboard dataset: Generated samples of $\widehat{\rho}_{0}$
using the standard single shot approach (top row). Generated samples using our
proposed JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=5$
and vary the network width $m=3,4,5,8$, and $16$. JKO-Flow performs
competitively even with fewer parameters.
Density Estimation on a Physics Dataset We train JKO-Flow on the
43-dimensional Miniboone dataset which is a high-dimensional, real-world
physics dataset used as benchmark for high-dimensional density estimation
algorithms in physics misc_miniboone_particle_identification_199 . For this
physics problem, our method is trained for $\alpha=0.5,\,1,\,5,\,10,\,50$ and
using $10$ JKO-Flow iterations. Fig 4 shows generated samples with JKO-Flow
and the standard single-shot approach for $\alpha=5$. Since Miniboone is a
high-dimensional dataset, we follow onken2021ot and plot two-dimensional
slices. JKO-Flow generates better quality samples. Similar experiments for
$\alpha=1,10$, and $50$ are shown in Figs 17, 18, and 19 in the Appendix. Tab.
3 summarizes the results for all values of $\alpha$. Note that we compute MMD
values for all the dimensions as well as 2D slices; this is because we only
have limited data ( 3000 testing samples) and the 2D slice MMD give a better
indication on the improvement of the generated samples. Results show that _the
MMD is consistent across all $\alpha$ values for JKO-Flow_. We also show the
convergence (in MMD2) of the miniboone dataset across each 2D slice in Fig. 5.
As expected, smaller step size $\alpha$ values converge slower (see
$\alpha=0.5)$, but all converge to similar accuracy (unlike the single-shot).
Samples | Single Shot | JKO-Flow
---|---|---
$x\sim\rho_{0}(x)$ | $f(x)$ | $f(x)$
| |
$y\sim\rho_{1}(y)$ | $f^{-1}(y)$ | $f^{-1}(y)$
| |
(a) Miniboone dimension 16 vs 17
Samples | Single Shot | JKO-Flow
---|---|---
$x\sim\rho_{0}(x)$ | $f(x)$ | $f(x)$
| |
$y\sim\rho_{1}(y)$ | $f^{-1}(y)$ | $f^{-1}(y)$
| |
(b) Miniboone dimension 28 vs 29
Figure 4: Generated samples for the 43-dimensional Miniboone dataset using the single shot approach and JKO-Flow with 10 iterations. To visualize the dataset, we show 2-dimensional slices. We show the forward flow $f(x)$ where $x\sim\rho_{0}$ and the genereated samples $f^{-1}(y)$ where $y\sim\rho_{1}$. Dimension 16 vs. 17 | Dimension 28 vs. 29
---|---
|
Figure 5: MMD2 per iteration when using JKO-Flow after training. MMD2 values
vs. JKO-Flow iteration for 2-dimensional slice (dimensions 16-17 on the left
and dimensions 28-29 on the right) of the Miniboone dataset. JKO-Flow achieves
same accuracy _regardless_ of the value of $\alpha$.
## 6 Conclusion
We propose a new approach we call JKO-Flow to train OT-regularized CNFs
without having to tune the regularization parameter $\alpha$. The key idea is
to embed an underlying OT-based CNF solver into a Wasserstein gradient flow
framework, also known as the JKO scheme; this approach makes the
regularization parameter act as a “time” variable. Thus, instead of tuning
$\alpha$, we repeatedly solve proximal updates for a fixed (time variable)
$\alpha$. In our setting, we choose OT-Flow onken2021ot , which leverages
exact trace estimation for fast CNF training. Our numerical experiments show
that JKO-Flow leads to improved performance over the traditional approach.
Moreover, _JKO-Flow achieves similar results regardless of the choice of
$\alpha$_. We also empirically observe improved performance when varying the
size of the neural network. Future work will investigate JKO-Flow on similar
problems such as deep learning-based methods for optimal control
fleming06controlled ; onken2022neural ; onken2021neural and mean field games
lin2021alternating ; ruthotto2020machine ; agrawal2022random .
## Acknowledgments
LN and SO were partially funded by AFOSR MURI FA9550-18-502, ONR
N00014-18-1-2527, N00014-18-20-1-2093 and N00014-20-1-2787.
Table 1: Synthetic 2D Data: JKO-Flow performance for different values of
$\alpha$. JKO-Flow returns consistent performance for different $\alpha$.
$\alpha$ | | 1 | 5 | 10 | 50
---|---|---|---|---|---
Dataset | Approach | MMD 2
Checkerboard | single Shot | 3.58e-2 | 3.56e-3 | 1.42e-3 | 1.26e-3
JKO-Flow (5 iters) | 4.9e-4 | 5.67e-4 | 6.40e-4 | 9.00e-4
2 Spirals | Single Shot | 7.21e-2 | 2.30e-2 | 1.84e-2 | 7.73e-4
JKO-Flow (5 iters) | 2.10e-2 | 4.62e-4 | 1.02e-4 | 5.37e-5
Swiss Roll | Single Shot | 4.74e-3 | 7.33e-4 | 2.86e-4 | 7.03e-4
JKO-Flow (5 iters) | 5.16e-4 | 8.3e-5 | 3.27e-5 | 6.07e-4
8 Gaussians | Single Shot | 9.18e-3 | 2.69e-4 | 3.94e-4 | 7.10e-4
JKO-Flow (5 iters) | 1.07e-4 | 4.13e-5 | 2.67e-4 | 7.27e-6
Circles | Single Shot | 9.84e-3 | 2.24e-4 | 6.51e-4 | 1.04e-4
JKO-Flow (5 iters) | 9.49e-4 | 9.97e-6 | 2.38e-5 | 9.28e-5
Pinwheel | Single Shot | 1.18e-2 | 1.8e-3 | 1.37e-3 | 2.2e-5
JKO-Flow (5 iters) | 4.7e-4 | 2.63e-4 | 3.84e-4 | 4.30e-4
Moons | Single Shot | 1.45e-3 | 2.05e-3 | 2.49e-4 | 2.42e-4
JKO-Flow (5 iters) | 1.92e-4 | 4.65e-5 | 4.3e-5 | 1.08e-4
Table 2: Synthetic 2D Data: Network width comparison for 1 and 5 iterations
given a fixed, best performing $\alpha$. JKO-Flow performs better than the
single shot approach for different network sizes.
[2mm] $m$ 3 4 5 8 16 Dataset Approach MMD 2 Checkerboard Single Shot 1.10e-2
5.60e-3 2.46e-3 3.03e-3 2.70e-3 JKO-Flow (5 iters) 5.60e-3 1.07e-3 2.7e-4
2.32e-4 4.16e-4 2 Spirals Single Shot 5.98e-3 4.54e-3 5.47e-3 1.19e-3 3.96e-3
JKO-Flow (5 iters) 1.42e-3 1.49e-5 6.11e-4 3.93e-5 2.19e-3 Swiss Roll Single
Shot 8.89e-3 7.71e-3 1.41e-3 1.37e-3 1.52e-3 JKO-Flow (5 iters) 1.49e-3
2.90e-4 6.13e-4 2.29e-4 8.40e-5 8 Gaussians Single Shot 2.20e-3 1.05e-3
1.04e-3 2.3e-4 5.05e-4 JKO-Flow (5 iters) 1.33e-4 9.85e-4 2.40e-5 3.96e-4
1.07e-4 Circles Single Shot 2.06e-3 1.72e-3 1.37e-3 1.69e-3 1.34e-3 JKO-Flow
(5 iters) 1.94e-3 3.24e-4 7.71e-4 5.9e-5 1.01e-4 Pinwheel Single Shot 1.10e-2
4.03e-3 2.27e-3 3.80e-3 5.43e-4 JKO-Flow (5 iters) 1.20e-3 8.23e-4 1.60e-3
7.00e-5 2.69e-4 Moons Single Shot 4.98e-3 4.54e-3 5.47e-3 1.2e-3 3.96e-3 JKO-
Flow (5 iters) 1.42e-3 1.5e-5 6.11e-4 3.90e-5 2.19e-3
Table 3: Miniboone: Comparison of Single Shot and JKO-Flow for different
values of $\alpha$.
[2mm] $\alpha$ 0.5 1 5 10 50 Dimensions Approach MMD 2 2d: 16 vs. 17 Single
Shot 4.62e-3 4.50e-3 5.88e-3 4.89e-3 5.2e-3 JKO-Flow (10 iters) 3.42e-4
2.94e-4 1.27e-4 1.43e-4 1.71e-4 2d: 28 vs. 29 Single Shot 4.33e-2 4.60e-2
5.02e-2 4.97e-2 4.74e-2 JKO-Flow (10 iters) 8.02e-5 3.31e-5 4.43e-5 6.33e-5
9.97e-5 Full Single Shot 4.7e-2 4.51e-3 4.75e-3 4.21e-3 4.27e-3 JKO-Flow (10
iters) 4.72e-4 4.72e-4 4.71e-4 4.72e-4 4.72e-4
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## Appendix A Appendix
$\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
7.21e-2 | 2.30e-2 | 1.84e-2 | 7.73e-4
| | |
MMD2 Values, JKO-Flow (five iterations)
2.10e-2 | 4.62e-4 | 1.02e-4 | 5.37e-5
| | |
Figure 6: 2 Spirals dataset: Generated samples of $\hat{\rho}_{0}$ using the standard one-shot approach (top row). Generated using our proposed JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10, 50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_. $\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
4.73e-3 | 7.33e-4 | 2.86e-4 | 7.03e-4
| | |
MMD2 Values, JKO-Flow (five iterations)
5.16e-4 | 8.3e-5 | 3.27e-5 | 6.07e-4
| | |
Figure 7: Swiss Roll dataset: Generated samples of $\hat{\rho}_{0}$ using the standard one-shot approach (top row). Generated using our proposed JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10, 50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_. $\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
9.18e-3 | 2.69e-4 | 3.94e-4 | 7.10e-4
| | |
MMD2 Values, JKO-Flow (five iterations)
1.07e-4 | 4.13e-4 | 2.67e-4 | 7.27e-6
| | |
Figure 8: 8 Gaussians dataset: Generated samples of $\hat{\rho}_{0}$ using the standard one-shot approach (top row). Generated using our proposed JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10, 50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_. $\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
9.84e-3 | 2.24e-4 | 6.51e-4 | 1.04e-4
| | |
MMD2 Values, JKO-Flow (five iterations)
9.49e-4 | 1.0e-5 | 2.4e-5 | 9.3e-5
| | |
Figure 9: Circles dataset: Generated samples of $\hat{\rho}_{0}$ using the standard one-shot approach (top row). Generated using our proposed JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10, 50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_. $\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
1.18e-2 | 1.8e-3 | 1.37e-4 | 2.2e-5
| | |
MMD2 Values, JKO-Flow (five iterations)
4.78e-4 | 2.63e-4 | 3.84e-4 | 4.30e-4
| | |
Table 4: Pinwheel dataset: Generated samples of $\hat{\rho}_{0}$ using the standard one-shot approach (top row). Generated using our proposed JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10, 50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_. $\alpha=1$ | $\alpha=5$ | $\alpha=10$ | $\alpha=50$
---|---|---|---
MMD2 Values, Single Shot
1.45e-3 | 2.05e-3 | 2.49e-4 | 2.42e-4
| | |
MMD2 Values, JKO-Flow (five iterations)
1.9e-5 | 5.0e-5 | 4.3e-5 | 1.08e-4
| | |
Figure 10: Moons dataset: Generated samples of $\hat{\rho}_{0}$ using the standard one-shot approach (top row). Generated using our proposed JKO-Flow using five iterations (bottom row). Here, we use $\alpha=$ 1, 5, 10, 50. JKO-Flow returns consistent results _regardless of the value of $\alpha$_. $m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
5.98e-3 | 4.54e-3 | 5.47e-3 | 1.19e-3 | 3.96e-3
| | | |
MMD2 Values, JKO-Flow (five iterations)
1.42e-3 | 1.49e-5 | 6.11e-4 | 3.93e-5 | 2.19e-3
| | | |
Figure 11: 2 Spirals dataset: Generated samples of $\hat{\rho}_{0}$ using the standard single shot approach (top row). Generated samples using our proposed JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=50$ and vary the network width $m=3,4,5,8$, and $16$. JKO-Flow performs competitively even with lower number of parameters. $m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
8.89e-3 | 7.71e-3 | 1.41e-3 | 1.37e-3 | 1.52e-3
| | | |
MMD2 Values, JKO-Flow (five iterations)
1.49e-3 | 2.90e-4 | 6.13e-4 | 2.29e-4 | 8.4e-5
| | | |
Figure 12: Swiss Roll dataset: Generated samples of $\hat{\rho}_{0}$ using the standard single shot approach (top row). Generated samples using our proposed JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=5$ and vary the network width $m=3,4,5,8$, and $16$. JKO-Flow performs competitively even with lower number of parameters. $m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
2.2e-3 | 1.05e-3 | 1.04e-3 | 2.3e-4 | 5.05e-4
| | | |
MMD2 Values, JKO-Flow (five iterations)
1.33e-4 | 9.85e-4 | 2.4e-5 | 3.96e-4 | 1.07e-4
| | | |
Figure 13: 8 Gaussians dataset: Generated samples of $\hat{\rho}_{0}$ using the standard single shot approach (top row). Generated samples using our proposed JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=5$ and vary the network width $m=3,4,5,8$, and $16$. JKO-Flow performs competitively even with lower number of parameters. $m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
2.06e-3 | 1.72e-3 | 1.37e-3 | 1.69e-3 | 1.34e-3
| | | |
MMD2 Values, JKO-Flow (five iterations)
1.94e-3 | 3.24e-4 | 7.71e-4 | 5.9e-5 | 1.01e-4
| | | |
Figure 14: Circles dataset: Generated samples of $\hat{\rho}_{0}$ using the standard single shot approach (top row). Generated samples using our proposed JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=5$ and vary the network width $m=3,4,5,8$, and $16$. JKO-Flow performs competitively even with lower number of parameters. $m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
1.10e-2 | 4.03e-3 | 2.27e-3 | 3.8e-3 | 5.43e-4
| | | |
MMD2 Values, JKO-Flow (five iterations)
1.20e-3 | 8.23e-4 | 1.60e-3 | 7e-5 | 2.69e-4
| | | |
Figure 15: Pinwheel dataset: Generated samples of $\hat{\rho}_{0}$ using the standard single shot approach (top row). Generated samples using our proposed JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=5$ and vary the network width $m=3,4,5,8$, and $16$. JKO-Flow performs competitively even with lower number of parameters. $m=3$ | $m=4$ | $m=5$ | $m=8$ | $m=16$
---|---|---|---|---
MMD2 Values, Single Shot
5.98e-3 | 4.54e-3 | 5.47e-3 | 1.2e-3 | 3.96e-3
| | | |
MMD2 Values, JKO-Flow (five iterations)
1.42e-3 | 1.5e-5 | 6.11e-4 | 3.9e-5 | 2.19e-3
| | | |
Figure 16: Moons dataset: Generated samples of $\hat{\rho}_{0}$ using the
standard single shot approach (top row). Generated samples using our proposed
JKO-Flow using five iterations (bottom row). Here, we fix $\alpha=5$ and vary
the network width $m=3,4,5,8$, and $16$. JKO-Flow performs competitively even
with lower number of parameters.
Samples | Single Shot | JKO Flow
---|---|---
$\boldsymbol{x}\sim\rho_{0}(\boldsymbol{x})$ | $f(\boldsymbol{x})$ | $f(\boldsymbol{x})$
| |
$\boldsymbol{y}\sim\rho_{1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$
| |
(a) Miniboone dimension 16 vs 17
Samples | Single Shot | JKO Flow
---|---|---
$\boldsymbol{x}\sim\rho_{0}(\boldsymbol{x})$ | $f(\boldsymbol{x})$ | $f(\boldsymbol{x})$
| |
$\boldsymbol{y}\sim\rho_{1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$
| |
(b) Miniboone dimension 28 vs 29
Figure 17: Generated samples for the 43-dimensional Miniboone dataset using
the single shot approach and JKO-Flow with 10 iterations for $\alpha=1$. To
visualize the dataset, we show 2-dimensional slices. We show the forward flow
$f(x)$ where $x\sim\rho_{0}$ and the genereated samples $f^{-1}(y)$ where
$y\sim\rho_{1}$.
Samples | Single Shot | JKO Flow
---|---|---
$\boldsymbol{x}\sim\rho_{0}(\boldsymbol{x})$ | $f(\boldsymbol{x})$ | $f(\boldsymbol{x})$
| |
$\boldsymbol{y}\sim\rho_{1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$
| |
(a) Miniboone dimension 16 vs 17
Samples | Single Shot | JKO Flow
---|---|---
$\boldsymbol{x}\sim\rho_{0}(\boldsymbol{x})$ | $f(\boldsymbol{x})$ | $f(\boldsymbol{x})$
| |
$\boldsymbol{y}\sim\rho_{1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$
| |
(b) Miniboone dimension 28 vs 29
Figure 18: Generated samples for the 43-dimensional Miniboone dataset using
the single shot approach and JKO-Flow with 10 iterations for $\alpha=10$. To
visualize the dataset, we show 2-dimensional slices. We show the forward flow
$f(x)$ where $x\sim\rho_{0}$ and the genereated samples $f^{-1}(y)$ where
$y\sim\rho_{1}$.
Samples | Single Shot | JKO Flow
---|---|---
$\boldsymbol{x}\sim\rho_{0}(\boldsymbol{x})$ | $f(\boldsymbol{x})$ | $f(\boldsymbol{x})$
| |
$\boldsymbol{y}\sim\rho_{1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$
| |
(a) Miniboone dimension 16 vs 17
Samples | Single Shot | JKO Flow
---|---|---
$\boldsymbol{x}\sim\rho_{0}(\boldsymbol{x})$ | $f(\boldsymbol{x})$ | $f(\boldsymbol{x})$
| |
$\boldsymbol{y}\sim\rho_{1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$ | $f^{-1}(\boldsymbol{y})$
| |
(b) Miniboone dimension 28 vs 29
Figure 19: Generated samples for the 43-dimensional Miniboone dataset using
the single shot approach and JKO-Flow with 10 iterations for $\alpha=50$. To
visualize the dataset, we show 2-dimensional slices. We show the forward flow
$f(x)$ where $x\sim\rho_{0}$ and the genereated samples $f^{-1}(y)$ where
$y\sim\rho_{1}$.
|
# Extracting maximal entanglement from linear cluster states
J. de Jong1, F. Hahn1, N. Tcholtchev2, M. Hauswirth1,2, and A. Pappa1,2 1
Electrical Engineering and Computer Science Department, Technische Universität
Berlin, 10587 Berlin, Germany 2 Fraunhofer Institute for Open Communication
Systems - FOKUS, 10589 Berlin, Germany
(August 28, 2024)
###### Abstract
Most quantum information processing architectures only allow for nearest-
neighbour entanglement creation. In many cases, this prevents the direct
generation of maximally entangled states, which are commonly used for many
communication and computation tasks. Here we show how to obtain maximally
entangled $\mathrm{GHZ}$ states between vertices initially connected by a
minimum number of connections, which specifically allows them to share linear
cluster states. We prove that the largest $\mathrm{GHZ}$ state that a linear
cluster state on $n$ qubits can be transformed into by means of local Clifford
unitaries, local Pauli measurements and classical corrections, is of size
$\lfloor(n+3)/2\rfloor$. We demonstrate exactly which qubit selection patterns
are possible below this threshold and which are not, and implement the
transformation on the IBMQ Montreal quantum device for linear cluster states
of up to $n=19$ qubits.
††preprint: APS/123-QED
## I Introduction
Recent years have seen exciting developments in quantum computation and
communication, both in theory and experiment. Building upon the year-long
research on bipartite settings, focus has now also turned towards multipartite
settings, where multiple vertices in a network share quantum resources between
them. While the correlations of maximally entangled states, in particular
Greenberger-Horne-Zeilinger ($\mathrm{GHZ}$) states [1] have naturally been
the first to explore, other types of graph states have also been extensively
examined [2, 3, 4, 5]. The possible transformations between quantum states is
a topic that is heavily studied [6, 7, 8, 9, 10], and while several hardness
results have emerged [11, 12], a lot of practical questions remain unanswered.
But why should we be interested in transforming one quantum state to another
in the first place? For one, the underlying network architecture might not
allow for the direct distribution of large maximally entangled states. Here we
show that there is an indirect remedy for this deficiency using suitable
transformations of the distributed quantum states. In particular, we focus on
the transformation of linear cluster states [13] to $\mathrm{GHZ}$ states,
i.e. on transforming to maximally entangled states the graph states that are
naturally distributable in linear quantum networks.
Such transformations require the removal of some of the qubits from the state
by measuring them, such that only a selected subset of the qubits of the
resource linear cluster state form the final $\mathrm{GHZ}$ state; these
transformations are therefore also called $\mathrm{GHZ}$ extractions. A
previous study [14] by some of the authors showed how to extract three- and
four-partite $\mathrm{GHZ}$ states from linear cluster states. Here, we
conclude this study by providing a complete characterisation of which
$\mathrm{GHZ}$ extractions are possible and which are not. Very importantly,
we provide a tight upper bound to the size of the largest $\mathrm{GHZ}$ state
that can be extracted, equal to $\lfloor(n+3)/2\rfloor$; interestingly this is
slightly higher than the one of $n/2$ conjectured in Ref. [13]. In addition to
our theoretical analysis, we perform experimental implementations of the
$\mathrm{GHZ}$ extractions from linear cluster states with
$n\in\\{5,7,\dots,19\\}$ qubits on the IBMQ Montreal device.
Our manuscript is organized as follows: The notation, technical terminology
and main definitions are introduced in Section II. Section III contains the
main theoretical results. In Section IV the experiments are introduced,
discussed, and their results presented. Finally, Section V discusses the
results presented and opportunities for future research. Various technical
details are diverted to topical appendices: Appendix A contains the proof of
an important lemma stated in the theoretical section, Appendix B contains
technical details regarding the post-processing steps during the extractions,
and Appendix C contains technical details regarding the data analysis of the
experimental section.
Figure 1: Example of extracting GHZ states from a linear cluster state with
seven qubits: The only $5$-partite GHZ state that can be extracted from this
resource is on the qubits corresponding to $1,2,4,6,7$ and is highlighted in
green. For $4$-partite $\mathrm{GHZ}$ states, we also highlight all $15$
possible extraction patterns in green, while the patterns in brown are
impossible due to Lemma 1 and the patterns shown in violet are impossible due
to both Corollary 1 and Lemma 1. Note that due to Theorem 1 it is impossible
to extract $\mathrm{GHZ}$ states with six or more qubits from this resource –
it is however trivially possible to extract all combinations of three-partite
$\mathrm{GHZ}$ states.
## II Notation and terminology
In this work, two quantum graph states play a central role: we define linear
cluster states $\ket{\mathrm{L}}$ and GHZ states $\ket{\mathrm{GHZ}}$ as
$\begin{split}\ket{\mathrm{L}}_{\\{1,\dots,n\\}}&\mathrel{\mathop{:}}=\frac{1}{2^{n}}\bigotimes_{i=1}^{n}\left(\ket{0}+\ket{1}\sigma_{z}^{i+1}\right)\\\
\ket{\mathrm{GHZ}}_{\\{1,\dots,m\\}}&\mathrel{\mathop{:}}=\frac{1}{\sqrt{2}}\left(\bigotimes_{i=1}^{m}\ket{0}+\bigotimes_{i=1}^{m}\ket{1}\right)\end{split}$
(1)
and $\ket{\mathrm{L}}_{V}$, $\ket{\mathrm{GHZ}}_{V}$ as corresponding to the
vertex set $V$. When context permits, with e.g. $\ket{\mathrm{L}}_{n}$ we
denote the linear cluster state of size $n$.
Our resource state is the $n$-partite linear cluster state
$\ket{\mathrm{L}}_{V_{L}}$. As a graph state it corresponds to a line graph on
the vertices $V_{L}{}\mathrel{\mathop{:}}=\\{1,2,\ldots,n\\}$. Here, each
vertex $i$ corresponds to the $i$-th qubit of $\ket{\mathrm{L}}_{V_{L}}$ and
the edges of the graph correspond to nearest-neighbour entangling controlled
phase gates. This structure allows us to use the terms left and right
neighbours of $i$ to indicate any vertices $h$, $j$ with $h<i$, $i<j$,
respectively; e.g. the direct left and right neighbours of $i$ are $i\pm 1$.
Let $V_{G}{}\subset V_{L}{}$ be a set of vertices for which we can extract a
$\mathrm{GHZ}$ state from the linear cluster resource state. Performing Pauli
measurements on the qubits corresponding to
$V_{M}\mathrel{\mathop{:}}=V_{L}\setminus V_{G}{}$, we obtain a post-
measurement state which is local-Clifford equivalent to the
$\ket{\mathrm{GHZ}}_{V_{G}{}}$ state. By performing local operations based on
the measurement outcomes, the state can then be locally transformed into this
$\mathrm{GHZ}$ state.
This construction allows for $V_{G}$ to inherit the neighbour structure from
the linear network $V_{L}{}$: For a vertex $j\in V_{G}{}$, we use $j_{-}$ and
$j_{+}$ to indicate the left and right neighbour of $j$ in $V_{G}$,
respectively. We refer to the smallest and largest element of $V_{G}$ as the
boundaries of the $\mathrm{GHZ}$ state. We finally define as a $k$-island any
selection of consecutive vertices $i,i+1,\dots,i+k\in V_{G}{}$.
## III Main results
We now examine what are the different types of GHZ states one can obtain from
a given linear cluster state. We first provide an upper bound for the size
$\absolutevalue{V_{G}{}}$ of the extracted GHZ state and we then show how to
saturate it. In order to achieve this, we use Lemma 1, which provides an
impossibility result for $2$-islands (the proof can be found in App. A).
###### Lemma 1.
No $2$-island can have both a left and a right neighbour in $V_{G}$. If two
vertices $i,i+1$ are in $V_{G}{}$, then there is either no vertex to the left
of $i$ or no vertex to the right of $i+1$.
Lemma 1 implies that all vertices $i$ in the target $\mathrm{GHZ}$ state must
be ‘isolated’ in the linear cluster state; $i-1$ and $i+1$ cannot be in
$V_{G}{}$ (with the exception of the boundaries). A corollary for $3$-islands
follows directly:
###### Corollary 1.
If $V_{G}$ contains a $3$-island, then $\absolutevalue{V_{G}{}}{}=3$.
###### Proof.
Let $i,i+1,i+2$ be a $3$-island in $V_{G}{}$ and assume that $|V_{G}{}|\geq
4$, i.e. that we have $h<i$ or $j>i+2$ in $V_{G}$. Now, either $i,i+1$ form a
$2$-island with both left-neighbour $h$ and right-neighbour $i+2$ or $i+1,i+2$
form a $2$-island with both left-neighbour $i$ and right-neighbour $j$. Both
are in direct contradiction to Lemma 1. ∎
By the same argument, $k$-islands with $k\geq 4$ are impossible; ultimately,
each such $k$-island would contain several $3$-islands in contradiction to
Corollary 1.
This allows us to calculate the upper bound to $\absolutevalue{V_{G}{}}{}$.
###### Theorem 1.
The size of a GHZ state extractable from an $n$-partite linear cluster state
via local Clifford operations, local Pauli measurements, and local unitary
corrections, is upper-bounded as
$\absolutevalue{V_{G}{}}\leq\left\lfloor\frac{n+3}{2}\right\rfloor$.
###### Proof.
As there are at most two $2$-islands, for every other $i$ in $V_{G}{}$ both
$i\pm 1$ were measured. Thus, to maximize $\absolutevalue{V_{G}{}}$, we may
have $1,2,n-1,n$ in $V_{G}{}$, and $V_{M}{}$ containing every other vertex in
between: For $n$ odd, $V_{M}{}=\\{3,5,\dots,n-2\\}$; for $n$ even
$V_{M}{}=\\{3,5,\dots,n-5,n-3,n-2\\}$ (see 111 In the case of $n$ being even,
there is more than one such pattern. While we have chosen here to measure the
two consecutive vertices, $n-3$ and $n-2$, other possibilities would have been
to measure consecutive vertices further to the left and measure only the even
vertices to the right. Another option would have been to measure not two
consecutive vertices, but a vertex of one of the $2$ islands, i.e. either
$1$,$2$,$n-1$ or $n$. It is important to note that all resulting sets
$V_{M}{}$ have the same size. ). In the even case, $n-2$ must be measured due
to Corollary 1. In both cases $\absolutevalue{V_{G}{}}{}=n-|V_{M}{}|$ is upper
bounded by $\left\lfloor\frac{n+3}{2}\right\rfloor$. ∎
We now show that there is a set of measurements that saturates the bound of
Theorem 1 by explicitly giving such a measurement pattern. For $n\leq 5$ this
pattern was shown in Ref. [14]; the special case $n=7$ is visualized in Figure
1. For the general case, let us consider a case distinction with respect to
the parity of $n$:
For odd $n$, we can choose $V_{M}=\\{2i+1\\}_{i=1}^{\frac{n-3}{2}}$ and every
corresponding qubit to be measured in the $\sigma_{x}$-basis; we refer to this
measurement pattern as the maximal pattern.
The linear cluster state is a stabilizer state, i.e. it is an element of the
shared $+1$ eigenspace of the operators
$\\{l_{i}=\sigma_{z}^{i-1}\sigma_{x}^{i}\sigma_{z}^{i+1}\\}_{i\in V_{L}}$,
where $\sigma_{z}^{0}$ and $\sigma_{z}^{n+1}$ are set equal to the identity.
This set of operators forms the set of canonical generators of an Abelian
subgroup of the $n$-qubit Pauli group known as the stabilizer of the linear
cluster states. For an overview of the stabilizer formalism and stabilizer
measurements in particular see [16], [17].
Consider the generator transformation
$l_{2}\rightarrow l_{2}^{{}^{\prime}}=l_{2}l_{4}\dots
l_{n-4}l_{n-2}\mathrel{\mathop{:}}=\sigma_{z}^{1}\sigma_{z}^{n}\prod_{2i\in
V_{L}}\sigma_{x}^{2i},$ (2)
which ensures that $l^{{}^{\prime}}_{2}$ and all odd-indexed generators
commute with all measurement operators $\\{\sigma_{x}^{j_{0}}\\}_{j_{0}\in
V_{M}{}}$. The post-measurement state is determined by replacing the other
$\absolutevalue{V_{M}{}}$ generators $\\{l_{2i}\\}_{i=2}^{\frac{n-1}{2}}$ with
the measurement operators –together with a multiplicative phase depending on
the respective measurement outcome. Then (after removing the support on the
measured qubits and applying a Hadamard transformation to $1$ and $n$) the
post measurement state on $V_{G}$ is characterized by the generators
$\sigma_{x}^{V_{G}{}}$ and
$\\{m_{j_{0}}\sigma_{z}^{j_{0}}\sigma_{z}^{j_{0+}}\\}_{j_{0}\in
V_{G}{}\setminus\\{n\\}}$, where the $m_{j_{0}}=\pm 1$ are phases due to the
measurement outcomes. These phases can be accounted for by applying
$\sigma_{x}$-operations to a selection of the nodes, recovering the generators
of the $\ket{\mathrm{GHZ}}_{V_{G}{}}$ state. The number of measurements
implies
$\absolutevalue{V_{G}{}}{}=n-\absolutevalue{\\{3,5,\dots,n-2\\}}=\frac{n+3}{2}$
which saturates the bound for odd $n$.
For even $n$, it suffices to observe that a $\sigma_{z}$-measurement on the
qubit corresponding to $n$ yields a linear cluster state
$\ket{\mathrm{L}}_{\\{1,2,\dots,n-1\\}}$ up to a randomized
$\sigma_{z}^{n-1}$-correction depending on the measurement outcome. In analogy
to the odd parity case we then obtain $V_{M}=\\{3,5,\dots,n-3,n\\}$ such that
$\absolutevalue{V_{G}{}}{}=n-\absolutevalue{V_{M}{}}{}=\frac{n+2}{2}=\left\lfloor\frac{n+3}{2}\right\rfloor$
for even $n$.
Note that the even-case-analysis above also applies for measuring an
‘internal’ node in the $\sigma_{y}$-basis, rather than the first or last; this
does introduce a Clifford rotation on the two neighbours of the node which
needs to be accounted for [18]. The resulting state is then also LOCC
equivalent to an ($n-1$)-partite linear cluster state on the remaining nodes,
from which in turn a $\ket{\mathrm{GHZ}}_{\frac{n+2}{2}}$ state can be
extracted through the maximal pattern. This approach can be extended to more
measurements, where additional ‘inside’ nodes are measured in the
$\sigma_{y}$-basis, and ‘outside’ nodes are measured in the
$\sigma_{z}$-basis. It is straightforward to see that any choice $V_{G}$
allowed by Lemma 1 can be seen as arising from such a setting.
Finally, note that while Lemma 1 does allow $2$-islands on the boundaries of
the extracted $\mathrm{GHZ}$ states, they do not necessarily have to be
contained in them. For example, $\ket{\mathrm{GHZ}}_{\\{1,3,5,7\\}}$ can be
extracted from $\ket{\mathrm{L}}_{\\{1,\ldots,7\\}}$ as shown in Figure 1.
Rigorously stated, this pattern does not arise from one of the maximal
patterns defined above, but can instead be considered as a maximal pattern
$\ket{\mathrm{GHZ}}_{\\{0,1,3,5,7,8\\}}$ extracted from
$\ket{\mathrm{L}}_{\\{0,\ldots,8\\}}$. Here, the additional qubits
corresponding to $0$ and $8$ are just “virtual” and not really there; they
simply help visualize all possible patterns:
$\ket{\mathrm{GHZ}}_{\\{1,3,5,7\\}}$ can be extracted from the “virtual” state
$\ket{\mathrm{GHZ}}_{0,1,3,5,7,8}$ by measuring qubits $0$ and $8$ in the
$\sigma_{x}$-basis. The measurements on the other qubits are unaffected by
this; the physical measurements of $2,4,6$ to obtain
$\ket{\mathrm{GHZ}}_{\\{1,3,5,7\\}}$ from
$\ket{\mathrm{L}}_{\\{1,\ldots,7\\}}$ are exactly the same as the ones that
would be required to obtain $\ket{\mathrm{GHZ}}_{\\{0,1,3,5,7,8\\}}$ from the
“virtual” $\ket{\mathrm{L}}_{\\{0,\ldots,8\\}}$. In this sense, all possible
selections of $V_{G}$ can be seen as subsets of the maximal measurement
patterns defined above.
## IV Experimental implementation
We used the IBMQ Montreal device to experimentally demonstrate our protocol
for the maximal extraction of $\mathrm{GHZ}$ states from resource linear
cluster states. For odd $n\in\\{5,7,\dots,19\\}$ we prepared the state
$\ket{\psi}_{n}=\bigotimes_{i\text{
odd}}H^{i}\ket{\mathrm{L}}_{\\{1,\ldots,n\\}},$ (3)
i.e. the linear cluster state with every odd qubit rotated to the
$\sigma_{x}$-basis. We then extract $\mathrm{GHZ}$ states for
$m\in\\{\frac{n+3}{2}\\}_{n}=\\{4,5,\dots,11\\}$ using the maximal pattern
described in the previous section.
Implementing $\ket{\psi}_{n}$ instead of $\ket{\mathrm{L}}_{\\{1,\ldots,n\\}}$
allows us to reduce the circuit depth of the preparation circuit by one, when
compiling for the gateset of the IBMQ Montreal device (Pauli-basis rotations,
$CNOT$; see Figure 2). When considering the $\mathrm{GHZ}$ extraction, this
approach has further benefits: The necessary Hadamard transformations on the
first and last qubit have, in essence, been applied ‘in advance’, and the
$\sigma_{x}$-measurements prescribed by the maximal pattern become
$\sigma_{z}$-measurements, which are native to the device. The Pauli-based
flips due to the measurement outcomes that are necessary to obtain the
$\mathrm{GHZ}$ state can be performed in post-processing, as all the
subsequent measurements on the $\mathrm{GHZ}$ state itself are in the Pauli
basis.
Figure 2: The circuit on the left and on the right are equal; the circuit on
the left is the standard preparation of the $\ket{\mathrm{L}}_{3}$ state. The
generalization to $\ket{\mathrm{L}}_{n}$ for higher (odd) $n$ follows
naturally. The circuit on the right is the compiled version for the IBMQ
Montreal chip. By not implementing the single-qubit gates in the grey box, the
circuit depth is reduced by $25\%$. This results in a rotated linear cluster
state $\ket{\psi}_{n}$ as defined in Eq.(3). This state carries the same
entanglement properties as the $\ket{\mathrm{L}}_{n}$ state, and most notably
can still be used to extract $\ket{\mathrm{GHZ}}$ states by adapting the
measurement bases. Figure 3: A lower bound of the fidelity of the rotated
linear cluster and $\mathrm{GHZ}$ states prepared on the IBMQ Montreal device.
We prepared rotated (see Eq. 3) linear cluster states $\ket{\psi}_{n}$ for
$n\in\\{5,7,\dots,19\\}$ and extracted $\ket{\mathrm{GHZ}}_{m}$ states for
$m\in\\{4,5,\dots,11\\}$using the maximal pattern introduced in Sec. III. For
states with a higher number of qubits, the lower bound on the linear cluster
state is increasingly worse due to a technical aspect of the estimation method
(see App. C for details). The results are ordered such that each linear
cluster state $\ket{\mathrm{L}}_{n}$ is paired with the
$\ket{\mathrm{GHZ}}_{m}$ state extracted from it.
To benchmark the results, we compute an estimate for the lower bound of the
fidelity for both the linear cluster states and the GHZ states extracted from
them. For the linear cluster states we use methods adapted from [19] using
insights originally presented in [20]; two measurement settings suffice to
estimate the lower bound –one in which all qubits are measured in the
$\sigma_{z}$-basis, and one in which all qubits are measured in the
$\sigma_{x}$-basis. For the $\mathrm{GHZ}$ states we derive a similar
technique. Again two measurement settings suffice –one where all the qubits of
the $\mathrm{GHZ}$ state are measured in the $\sigma_{z}$-basis, and one where
all the qubits are measured in the $\sigma_{x}$-basis. These measurements are
performed in parallel to the $\sigma_{z}$-measurements of the qubits not
included in the $\mathrm{GHZ}$ state that are required for the extraction. All
measurements are repeated $32000$ times to obtain estimates for the
expectation values.
Figure 3 shows the lower bounds on the fidelity for all linear cluster states
that we generated with the IBMQ Montreal device –as well as for the
$\mathrm{GHZ}$ states we extracted from them. Note that our estimation method
imposes a relative penalty for linear cluster state fidelity estimations
compared to $\mathrm{GHZ}$ state fidelity estimations. However, this does not
mean that the fidelity of the $\mathrm{GHZ}$ states is truly higher than that
of the linear cluster states from which they were extracted: It simply means
that we have used a method of bounding the fidelity from below, which works
comparatively better for $\mathrm{GHZ}$ states than it does for linear cluster
states. For the details of the estimation method we refer to App. C.
## V Discussion
$\mathrm{GHZ}$ states are an indispensable resource in quantum communication
settings. Our results show that their extraction in linear architectures is
costly, since about half of the qubits in a linear cluster state must be
measured to obtain a $\mathrm{GHZ}$ state on the remaining qubits. We give an
exhaustive characterization of all possible $\mathrm{GHZ}$ states that can be
extracted and provide a constructive method to obtain them, including
calculations for the necessary local rotations on the remaining qubits.
Finally, extending our methods to other simple graph states does not seem to
be straightforward and requires more research. We note however that deriving a
similar characterization for ring graph states, i.e. , those where the
leftmost and rightmost qubits of the linear cluster state are connected, can
be easily achieved using our methods. In this case, only a single $2$-island
is possible, so the upper bound for $\absolutevalue{V_{G}{}}{}$ becomes
$\lfloor\frac{n+1}{2}\rfloor$.
## VI Acknowledgements
A.P., J.d.J. and F.H. acknowledge support from the German Research Foundation
(DFG, Emmy Noether Grant No. 418294583). Upon completion of this work, we
became aware of work similarly motivated [21].
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* Hein _et al._ [2004] M. Hein, J. Eisert, and H. J. Briegel, Multiparty entanglement in graph states, Physical Review A 69, 062311 (2004).
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* Note [1] In the case of $n$ being even, there is more than one such pattern. While we have chosen here to measure the two consecutive vertices, $n-3$ and $n-2$, other possibilities would have been to measure consecutive vertices further to the left and measure only the even vertices to the right. Another option would have been to measure not two consecutive vertices, but a vertex of one of the $2$ islands, i.e. either $1$,$2$,$n-1$ or $n$. It is important to note that all resulting sets $V_{M}{}$ have the same size.
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## Appendix A Proof of Lemma 1
In this section we proof Lemma 1. We prove the theorem by contradiction. Fix a
set $V_{G}$ such that $\\{i,i+1\\}\subset V_{G}{}$ and let the post-
measurement state $\ket{\psi}_{V_{G}{}}$ be locally equivalent to
$\ket{\mathrm{GHZ}}_{V_{G}{}}$. Assume now that there are both $j<i$ and
$k>i+1$ for which both $j,k\in V_{G}{}$. W.l.o.g. assume that $j$ and $k$ are
the direct left- and right neighbour of $i$ and $i+1$, respectively.
Recall that a set of generators for the linear cluster state is
$\\{l_{i_{0}}=\sigma_{z}^{i_{-}}\sigma_{x}^{i_{0}}\sigma_{z}^{i_{+}}\\}_{i_{0}\in
V_{L}}$. If the post-measurement state is locally equivalent to the
$\mathrm{GHZ}$ state then there must exist a (reversible) generator
transformation such that their support on $i$ and $i+1$ coincides exactly with
(the generators of) the $\mathrm{GHZ}$ state - up to local Clifford rotations.
We will now show that, from a reversible transformation of the
$\\{l_{i_{0}}\\}$, it is impossible to obtain such a set of generators when
$j,i,i+1,k\in V_{G}{}$. This directly implies that a measurement pattern such
that the $\mathrm{GHZ}$ state can be obtained is not possible.
(A set of) generators for the $\mathrm{GHZ}$ state are,
$\\{\sigma_{x}^{V_{G}{}}\\}\cup\\{\sigma_{z}^{i_{0}}\sigma_{z}^{i_{+}}\\}_{i_{0}\in
V_{G}{}}$, where it is implied that $\sigma_{z}^{i_{+}}=1$ when $i_{+}\not\in
V_{G}{}$. Focusing on $i$ and $i+1$, the only generators with non-trivial
support are
$\\{\alpha,\beta,\gamma,\delta\\}=\\{\sigma_{a_{i}}^{i},\sigma_{a_{i}}^{i}\sigma_{a_{i+1}}^{i+1},\sigma_{a_{i+1}}^{i+1},\sigma_{b_{i}}^{i}\sigma_{b_{i+1}}^{i+1}\\}$,
where $a_{i},a_{i+1},b_{i},b_{i+1}\in\\{x,y,z\\}$ reflect the fact that the
state is locally equivalent to the $\mathrm{GHZ}$ state. This implies that
$a_{i}\not=b_{i}$ and $a_{i+1}\not=b_{i+1}$.
Similarly, only the generators $l_{i-1},l_{i},l_{i+1}$ and $l_{i+2}$ of the
linear cluster state (i.e. those with support on $i$ or $i+1$) can have a non-
trivial contribution to the generator transformation on the vertices in
question. Therefore, w.l.o.g., we can focus on just these four generators and
restrict our attention to vertices $i$ and $i+1$. If we show that there is no
reversible transformation of $\\{l_{k}\\}_{k=\\{i-1,i,i+1,i+2\\}}$ to obtain
$\\{\alpha,\beta,\gamma,\delta\\}$ when only considering these nodes, the
lemma follows. We show there is no such transformation by exhaustive
contradiction.
There are three different ways of creating generator $\alpha$: i)
$\alpha\propto l_{i-1}=\sigma_{z}^{i}$, ii) $\alpha\propto l_{i}\circ
l_{i+2}=\sigma_{x}^{i}$, iii) $\alpha\propto l_{i-1}\circ l_{i}\circ
l_{i+2}=\sigma_{y}^{i}$, where ‘$\alpha\propto l_{i-1}$’ should be read as
‘$l_{i-1}$ takes the role of $\alpha$’, and $\circ$ denotes the (qubit-wise)
product (e.g. $l_{i}\circ
l_{i+1}=\sigma_{x}^{i}\sigma_{z}^{i+1}\circ\sigma_{z}^{i}\sigma_{x}^{i+1}\hat{=}\sigma_{y}^{i}\sigma_{y}^{i+1}$,
where the last equality is up to an irrelevant global phase). Similarly, there
are three different ways of creating generator $\gamma$: j) $\gamma\propto
l_{i+2}=\sigma+{z}^{i+2}$, jj) $\gamma\propto l_{i-1}\circ
l_{i+1}=\sigma_{x}^{i+2}$, jjj $\gamma\propto l_{i-1}\circ l_{i+1}\circ
l_{i+2}=\sigma_{y}^{i+2}$. Picking e.g. i) and j) one sees that $\beta$ is
fixed at $\propto\sigma_{z}^{i}\sigma_{z}^{i+1}$. But this is $l_{i-1}\circ
l_{i+2}\propto\alpha\circ\gamma$, which would not be a reversible
transformation of the generators $l_{i-1},l_{i},l_{i+1}$ and $l_{i+2}$. Any
other pair from $\\{\textbf{i)},\textbf{ii)},\textbf{iii)}\\}$ and
$\\{\textbf{j)},\textbf{jj)},\textbf{jjj)}\\}$ would also necessitate such a
non-reversible transformation.
In essence, when viewing the generators as vectors over $\mathbf{F}^{2n}_{2}$
through the binary representation [22], the argument follows from the
observation that (the vector associated with) $\beta$ lies in the subspace
spanned by (the vectors associated with) $\alpha$ and $\gamma$. As such there
can never be a reversible (i.e. basis-preserving) operation on (the vectors
associated with) $l_{i-1},l_{i},l_{i+1}$ and $l_{i+2}$ that obtains
$\alpha,\beta$ and $\gamma$.
## Appendix B Local-Clifford corrections to obtain GHZ states.
We provide a jupyter notebook for determining the required correction
operations under [18].
## Appendix C Estimation of lower bound for fidelity in experimental
implementation.
We here provide details for the method of estimation of the lower bound of the
fidelity of both the linear cluster state and $\mathrm{GHZ}$ state, that has
been used in the experimental implementation. The method is presented in and
adapted from [19] using insights originally presented in [20]. The state that
is prepared is
$\ket{\psi}_{n}=\bigotimes_{i\text{ odd}}H^{i}\ket{\mathrm{L}}_{n},$ (4)
which is a linear cluster state rotated by Clifford operations and thus a
stabilizer state. Note that the generators $G^{L}$ for the stabilizer of
$\ket{\psi}_{n}$ can be grouped into ‘odd’ generators
$G^{L}_{o}=\\{\sigma_{z}^{i-1}\sigma_{z}^{i}\sigma_{z}^{i+1}\\}_{i\text{
odd}}$ and ‘even’ generators
$G^{L}_{e}=\\{\sigma_{x}^{i-1}\sigma_{x}^{i}\sigma_{x}^{i+1}\\}_{i\text{
even}}$, where again $\sigma_{z}^{0}=\sigma_{z}^{n+1}=1$. The fidelity of the
prepared state $\rho$ with the rotated linear cluster state is
$F(\rho,\ket{\psi}_{n})=\tr\left[\rho\outerproduct{\psi}{\psi}_{n}\right]$.
Writing $G_{o(e)}=\prod_{g\in G^{L}_{o(e)}}\frac{\mathbb{I}+g}{2}$, and using
$\outerproduct{\psi}{\psi}_{n}=\prod_{g\in
G}\frac{\mathbb{I}+g}{2}=G_{o}G_{e}$, we can write
$\displaystyle F(\rho,\ket{\psi}_{n})$
$\displaystyle=\tr\left[G_{o}G_{e}\rho\right]$ (5)
$\displaystyle=\tr\left[G_{o}\rho\right]+\tr\left[G_{e}\rho\right]-\tr\left[\mathbb{I}\rho\right]+\tr\left[K\rho\right],$
(6)
where $K=\left(\mathbb{I}-G_{o}\right)\left(\mathbb{I}-G_{e}\right)$. $K$ is
positive semidefinite and thus we can discard the last term to obtain a lower
bound for the fidelity:
$F(\rho,\ket{\psi_{n}})\geq\mathbb{E}\left[G_{o}\right]+\mathbb{E}\left[G_{e}\right]-1,$
(7)
where
$\mathbb{E}\left[G_{o(e)}\right]=\frac{1}{2^{\absolutevalue{\mathcal{S}_{o(e)}}}}\sum_{\sigma\in\mathcal{S}_{o(e)}}\tr\left[\rho\sigma\right]$
with $\mathcal{S}_{o(e)}=\langle G^{L}_{o(e)}\rangle\subset\mathcal{S}$ the
subgroup generated by the ‘odd’ (‘even’) generators of $\ket{\psi}_{n}$.
Notably, all terms $\tr\left[\rho\sigma\right]$ comprise of only
$\sigma_{z}$-basis ($\sigma\in\mathcal{S}_{o}$) or $\sigma_{x}$-basis
($\sigma\in\mathcal{S}_{e}$) measurements. This means that just two
measurement settings suffice to estimate the lower bound: measuring all
vertices in the $\sigma_{z}$-basis, and measuring all vertices in the
$\sigma_{x}$-basis. By repeating these measurements $32000$ times and
obtaining the outcome statistics, we estimate all terms
$\tr\left[\rho\sigma\right]$ by selecting the outcomes associated with the
$+1$ and $-1$ eigenspaces of all different observables.
For the $\mathrm{GHZ}$ state we use a similar method, where we now group the
generators $G^{G}$ of the $\mathrm{GHZ}$ state into
$G_{o}^{G}=\\{\sigma_{x}^{V_{G}{}}\\}$ and
$G_{e}^{G}=\\{\sigma_{z}^{j}\sigma_{z}^{j_{+}}\\}_{j\in V_{G}{}}$, which again
allows for an estimate of the lower bound with just two measurement settings.
A caveat is that now there is only one ‘odd’ generator and thus
$\mathbb{E}\left[G^{G}_{o}\right]=\frac{1}{2}\tr\left[\rho\mathbb{I}\right]+\frac{1}{2}\tr\left[\rho\sigma_{x}^{V_{G}{}}\right]$.
By definition $\tr\left[\rho\mathbb{I}\right]=1$ and therefore the expectation
value is more skewed towards $1$ than for the linear cluster state estimation.
In other words it gives a higher bound on the fidelity when compared to the
linear cluster state, since $G^{L}_{o}=O(2^{n})$ and as such the identity does
not have such a strong impact on the estimate, especially for larger linear
cluster states. To give another comparison between the two states, Figure 4
contains the same results as Figure 3 from the main text, but with the
identity-term omitted. This gives a lower but more equal estimate for both
classes of states.
Figure 4: Lower bound on the fidelity using an adapted estimate method. In
comparison with Figure 3, positive terms that favour the $\mathrm{GHZ}$ states
are dropped, which renders a lower but more equal estimate on the fidelities
for all states.
|
# Ferromagnetic Crossover within the Ferromagnetic Order of U7Te12
Petr Opletal Hironori Sakai Yoshinori Haga Yoshifumi Tokiwa Etsuji
Yamamoto Shinsaku Kambe Yo Tokunaga Advanced Science Research Center, Japan
Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
###### Abstract
We investigate the physical properties of a single crystal of uranium
telluride U7Te12. We have confirmed that U7Te12 crystallizes in the hexagonal
structure with three nonequivalent crystallographic uranium sites. The
paramagnetic moments are estimated to be approximately 1 $\mu_{\rm B}$ per the
uranium site, assuming a uniform moment on all the sites. A ferromagnetic
phase transition occurs at $T_{\rm C}=48$ K, where the in-plane magnetization
increases sharply, whereas the out-of-plane component does not increase
significantly. With decreasing temperature further below $T_{\rm C}$ under
field-cooling conditions, the out-of-plane component increases rapidly around
$T^{\star}=26$ K. In contrast, the in-plane component hardly changes at
$T^{\star}$. Specific heat measurement indicates no $\lambda$-type anomaly
around $T^{\star}$, so this is a cross-over suggesting a reorientation of the
ordering moments or successive magnetic ordering on the part of the multiple
uranium sites.
††preprint: APS/123-QED
## I Introduction
Uranium 5$f$ electrons play an important role in bringing out various solid-
state properties. In comparison to 4$f$ electron states in lanthanide
compounds, which are strongly localized, 5$f$ electron states extend into
space [1]. This leads to inter-site interaction with direct overlap by 5$f$
electron states of the nearest uranium atoms, and/or hybridization with
conduction electron states of ligand atoms. Meanwhile, the intrasite coupling
with 6$d$ electron states is also important for microscopic understanding. To
investigate the role of 5$f$ electrons in physical properties, it is important
to systematically study a group of compounds constructed from the same
elements. In this sense, uranium chalcogenides are promising research targets,
which exhibit various physical properties, such as, piezomagnetism in UO2 [2],
metal-insulator crossover induced by magnetic field in $\beta$-US2 [3, 4], and
spin-triplet superconductivity recently discovered in UTe2 [5].
Especially, there has been a growing interest in the adjacent phase of UTe2,
which is not a line phase, including nonstoichiometry caused by U deficiency
[6]. In addition, U7Te12 can be grown with stoichiometric UTe2 by the molten
salt flux method [7]. U7Te12 is reported to crystallize in non-centrosymmetric
hexagonal Cr12P7-like structure (space group $P\overline{6}$, No. 174) [8, 9].
This structure is characterized by three nonequivalent uranium sites – U1 with
multiplicity 1 and U2 and U3 with multiplicity 3. Only reported data for
U7Te12 were obtained on polycrystalline samples showing ferromagnetic
transition at 52 K [10, 9]. Although anisotropic magnetism is expected from
the crystal structure, the details of this magnetism are unknown. Moreover, it
was reported as a semimetallic compound [9]. However, no data on temperature-
dependent resistivity has been presented until now.
In this study, we prepared single crystals of U7Te12 by chemical vapor
transport (CVT) and molten salt flux methods. Single crystals enable us to
examine the anisotropic physical properties of this compound. Two different
magnetic anomalies are observed through magnetic, electrical transport, and
heat capacity measurements: one is a ferromagnetic transition at $T_{\rm
C}=48$ K, and another is a ferromagnetic crossover at $T^{\star}=26$ K. The
former ferromagnetic ordering is characterized by an increase of the $a$-axis
magnetization, while the latter crossover is characterized by an increase of
$c$-axis magnetization. We have also revealed that the material exhibits
semimetallic conductivity, of which resistivity slightly decreases below
$T_{\rm C}$ and conversely increases below $T^{\star}$. It may be a half-
gapped semimetallic state induced by the ferromagnetic crossover. We discuss
the possible nature of these magnetic anomalies and their connection to
nonequal uranium sites.
## II Experimental methods
Single crystals of U7Te12 were prepared by the CVT method with NH4Cl as a
transport agent. Precursor elements of etched U metal and Te (Te 6N purity
produced by Rare Metallic Co. Ltd.) in a 1:1.5 ratio were used and the quartz
tube for growth was etched and smoothed inside by hydrofluoric acid. Precursor
elements were sealed inside the quartz tube under high vacuum (10-5 mbar) with
approx. 0.04 mg/cm3 of NH4Cl. During the evacuation, the whole quartz tube was
heated up to 100∘C to desorb any water in NH4Cl. The sealed tube was placed in
a two-zone horizontal electric tube furnace (ARF2-370-50KC, Asahi-rika Co.,
Ltd.). The entire tube was heated up to 950∘C for 24 hours to allow the
reaction of contents. An opposite gradient was applied to clean the growth
zone for three days. Following that, the gradient was flipped with the lower
temperature - 850∘C at the growth zone (etched by hydrofluoric acid) and
higher temperature - 950∘C at the charge zone and was left for 10 days. We
were also able to prepare a single crystal by the molten-salt method as
described in [7].
Single crystals grown by the molten-salt method tend to be prism-like with a
hexagonal base growing along the c-axis with a thickness of 0.5 to 1 mm and a
length of 2 to 6 mm. Single crystals obtained by the CVT method tend to have a
block shape with dimensions ranging from 2 to 4 mm. Samples obtained by both
methods have dark metallic color.
The crystal structure was determined by single crystal diffraction at room
temperature using graphite monochromated Mo $K_{\alpha}$ radiation. The
scattered X-ray beam was recorded on an image plate detector (R-AXIS RAPID,
Rigaku Corp.). Absorption correction with the empirical method was applied
prior to the structural solution. Structural solution and crystallographic
parameters fitting were performed by the SHELX program. The absence of any
different phases was checked by electron-probe microanalysis using wavelength-
dispersive spectrometers installed in a scanning electron microscope
(JXA-8900, JEOL Ltd.).
To measure the physical properties, samples were cut from larger single
crystals by an electric spark saw and polished. Magnetic properties were
measured by a commercial superconducting quantum interference device
magnetometer (MPMS, Quantum Design) in the temperature range of 4.2 K to 300 K
and a maximum magnetic field of 7 T. The obtained data were not corrected for
the demagnetization effect. We estimate the demagnetization field to be below
0.05 T for the highest magnetic moment. The heat capacity was measured by
relaxation method by two $\tau$ models in Physical Properties Measurement
System (PPMS, Quantum Design) from 1.8 K up to 300 K. Electrical resistivity
was measured by the four-probe method in PPMS.
## III Results
### III.1 Crystal structure
Figure 1: Crystal structure of U7Te12 with coordination polyhedra for (a) U1 in tricapped trigonal prismatic and (b) U2 and (c) U3 in bicapped trigonal prismatic geometry. (d) Crystal structure only with uranium (for better clarity) highlighting triangular prism of U2 and U3. Distances between uranium atoms are shown. Table 1: Wyckoff notation, local symmetry, atomic coordinates, equivalent isotropic displacement parameter, and site occupancy for each atomic position in U7Te12 which are obtained from the single crystal X-ray refinement. atom | site | local symm. | $x$ | $y$ | $z$ | $B_{\mathrm{e}q}$ (Å2) | occupancy
---|---|---|---|---|---|---|---
U1 | $1a$ | $\overline{6}..$ | 0 | 0 | 0 | 0.53(3) | 1
U2 | $3k$ | $m..$ | 0.17296(11) | 0.46524(10) | 1/2 | 0.58(3) | 0.983(6)
U3 | $3j$ | $m..$ | 0.43295(10) | 0.26651(11) | 0 | 0.56(2) | 1
Te1 | $3k$ | $m..$ | 0.21589(19) | 0.2081(2) | 1/2 | 0.57(4) | 1
Te2 | $3k$ | $m..$ | 0.52583(19) | 0.1345(2) | 1/2 | 0.59(3) | 1
Te3 | $3j$ | $m..$ | 0.3765(2) | 0.4949(2) | 0 | 0.71(4) | 1
Te4 | $3j$ | $m..$ | 0.0173(2) | 0.2649(2) | 0 | 0.72(5) | 0.955(10)
We have confirmed U7Te12 crystallizes in Cr12P7-like structure (space group
$P\overline{6}$, No. 174) with crystal lattice parameters $a=12.277(3)$ Å and
$c=4.2646(5)$ Å. The resulting coordinates of the atoms are presented in Table
1. The structure was determined with final agreement factor $R_{1}$ = 3.05%.
If the atomic occupancies on the U2 and Te4 sites were fixed as 1, the
obtained equivalent displacement coefficients ($B_{\rm eq}$) were too large.
Alternatively, if these occupancies were treated as fitting parameters, the
occupancies were slightly reduced. However the $B_{\rm eq}$ values become
reasonable, and then the $R_{1}$-value became slightly reduced. The resulting
displacement factor of U2 sites remains large and qualitatively consistent
with the result by Tougait et al. [9] As shown in Fig. 1(b), U2 is located in
the polyhedron formed by Te3 and Te4, where U2 is much closer to Te4 leaving
more space in the opposite direction. The atomic displacement would likely be
larger in this direction, resulting in a large displacement factor.
As shown in Fig. 1, two different coordination polyhedra of uranium and
tellurium are seen in U7Te12. The coordination polyhedra for U1 sites is
tricapped trigonal prismatic (Fig. 1(c)), while those for U2 and U3 sites are
bicapped trigonal prismatic geometry (Figs. 1(d) and 1(e)). Therefore, two
types (at least) of crystalline electric field (CEF) effects are expected to
exist for different polyhedral U sites.
At first glance, the polyhedral arrays of U2 and U3 appear similar, but their
polyhedral sequences are quite different as follows. In the U7Te12 structure,
the shortest distance between uranium sites is the U2-U2 distance of 3.917 Å
which is longer than Hill’s limit [11]. This shortest U2-U2 corresponds to a
side of the U2 triangle in the basal plane (Fig. 1). Since each U2 site is
aligned in a straight line in the $c$-axis direction, every $c=4.2646$ Å of
the lattice constant, a geometrical frustration effect may exist in the U2
prism if antiferromagnetic interactions were dominant. Meanwhile, the shortest
distance in the U3 prisms corresponds to that along the $c$-axis, i.e., it is
equal to the lattice parameter of $c$, whereas the side in the U3 triangle is
equal to 4.45 Å. The shortest distance in U1 prism is the same as in U3, but
for U1 no triangle exists. In between different uranium sites, the shortest
one is found between U1-U3 equal to 4.67 $\mathrm{\AA}$. The distances between
U1-U2 and between U2-U3 are more than 5 $\mathrm{\AA}$, as a result of the
stacking sequence of U2 being shifted by $c/2$ as shown in Fig. 1(a).
### III.2 Magnetic properties
Figure 2: Temperature dependence of magnetic susceptibility defined as $M/H$
for U7Te12 with an application of $\mu_{0}H=0.01$ T along the $a$-axis and
$c$-axis. The data were taken under field-cooled (FC) conditions. Closed and
open squares show data obtained from the field applied along the $a$-axis and
$c$-axis, respectively. The inset shows detailed behavior close to
ferromagnetic transition at 48 K for the field applied along the $c$-axis.
Figure 3: Left figure shows magnetic curves at different temperatures for the
field applied along the $a$-axis. Right figure shows magnetic curves at
different temperatures for the field applied along the $c$-axis with an inset
showing detail of magnetic curves around the zero field region. Figure 4:
Temperature dependence of inverse magnetic susceptibility for magnetic field
applied along the $a$-axis and $c$-axis. The fit to data by Curie-Weiss law
and modified Curie-Weiss law are shown as dashed lines (for more information
see text). The inset shows the low-temperature data of $M/H$ for different
magnetic fields applied along the $a$-axis. Data at 1 T and 7 T are scaled and
shifted for clarity.
Figure 2 shows the temperature dependence of magnetic susceptibility $\chi$
which is defined as the magnetization ($M$) divided by the applied magnetic
field ($H$) along the $a$\- and $c$-axes for U7Te12. The data shown in Fig. 2
were taken in the field cooled (FC) condition. As seen in Fig. 2, it appears
as if U7Te12 underwent two ferromagnetic transitions along the $a$-axis and in
the $c$-axis, respectively. For the magnetic susceptibility defined as $M/H$
along the $a$-axis, a ferromagnetic transition is observed at $T_{\mathrm{C}}$
= 48 K with a steep increase in magnetization. Below $T_{\mathrm{C}}$, a local
maximum is observed in susceptibility at $T^{\star}\approx$ 26 K. A large
increase on $c$-axis magnetic susceptibility is seen below $T_{\mathrm{C}}$
and an inflection point is seen at $T^{\star}$. This increase is observed even
in higher magnetic fields indicating a ferromagnetic-like origin. By
increasing $H$ along the $a$-axis, the local maximum around $T^{\star}$ in
magnetic susceptibility is suppressed (inset of Fig. 4). The polarization of
ferromagnetically ordered magnetic moments in high magnetic fields can result
in an additional contribution to magnetic susceptibility which may appear as
an additional shift of the local maximum at $T^{\star}$.
U7Te12 is a soft ferromagnet with a small magnetic coercivity for a magnetic
field applied along the $a$-axis. Magnetization curves for a magnetic field
applied along the $a$-axis (Fig. 3) show almost no hysteresis down to 4.2 K,
and quickly become nearly saturated, with a magnetization of 1.41
$\mu_{\mathrm{B}}$/U at 7 T. Spontaneous magnetization is observed below
$T_{\mathrm{C}}$, but the magnetization curve at 50 K shows an ‘S’-shape
characteristic just above $T_{\rm C}$. Although the $M$-$H$ data near $T_{\rm
C}$ were analyzed by the the generalized Arrott–Noakes relationship
$\left(H/M\right)^{1/\gamma}=(T-T_{\rm C})/T_{1}+(M/M_{1})^{1/\beta}$ with
free parameters of $T_{1}$, and $M_{1}$. The fitting could not yield
reasonable critical exponents in the temperature region. It may be due to the
geometrically frustrated magnetism or competing interactions between in-plane
and out-of-plane magnetism.
In the case of $H\parallel c$, the magnetization curve (Fig. 3) exhibits a
rectangular shape below $T^{\star}$. The saturation magnetization at 4.2 K is
observed to be 0.89 $\mu_{\mathrm{B}}$/U at 7 T, which is nearly half the
value for the $a$-axis. Similarly, the magnetization curve at 30 K above
$T^{\star}$ shows an ‘S’-like shape again (Fig. 3(b)), except for the low-
field region near zero field. In the field region above $T^{\star}$ but below
$T_{\rm C}$, the $M$-$H$ curve exhibits a tiny remanent magnetization and a
small coercive field (inset of the left figure in Fig. 3), which completely
disappears above $T_{\rm C}$. The small remanent magnetization suggests a
minor contribution from the $a$-axis due to misalignment.
Paramagnetic susceptibility in the high-temperature region above 150 K for the
$a$-axis obeys the Curie-Weiss law, as shown in Fig. 4. The Curie-Weiss
fitting yields an effective moment $\mu_{\rm eff}$ of 1.23
$\mu_{\mathrm{B}}$/U assuming a uniform moment and Weiss temperature
$\Theta_{a}$ of 63 K. In the case of $H\parallel c$, since the inverse
magnetic susceptibility is not linear, the modified Curie-Weiss law is
required, which yields $\chi_{0}=3.7\times 10^{-8}$ m3/mol,
$\mu_{\mathrm{eff}}$ = 0.92 $\mu_{\mathrm{B}}$/U, and $\Theta_{c}$ = 36 K.
Magnetic susceptibility for the $c$-axis follows the modified Curie-Weiss law
down to $\approx$ 60 K. For both directions, the effective magnetic moment is
much smaller than that for free U3+/U4+ ion. Positive $\Theta$ means that
ferromagnetic interactions are dominant in the system, and
$\Theta_{a}>\Theta_{c}$ suggests the in-plane interactions are stronger.
Suski [10] previously discussed the possibility of different magnetic moments
by assuming 3H4 term for U4+ ($5f^{2}$) for the uranium but different
splitting based on the two types of polyhedral U sites, shown in Fig. 1.
However, because U7Te12 can be formally considered as
U${}^{4+}_{3}$U${}^{3+}_{4}$Te${}_{12}^{2-}$ from the charge neutral
principle, valence conditions are probably needed, e.g. U3+ state on the U1
site, and U3+ or U4+ state assigned on the U2 and U3 sites, respectively.
Additional experiments are desirable to determine the CEF and valence states.
The magnetic anisotropy is evidently expected to be site-dependent in such a
situation, but based on the ratio of magnetic moments $M_{a}/M_{c}$(2 K, 7 T)
$\approx$ 4/3 and multiplicity of uranium sites we expect uraniums on U1 and
U2 or U3 to be ordering in-plane at $T_{\mathrm{C}}$ while uraniums on U2 or
U3 site are responsible for the $c$-axis magnetization increase at
$T^{\star}$.
### III.3 Heat capacity
Figure 5: Heat capacity of U7Te12 in different magnetic applied along the
$c$-axis and $a$-axis. Data for $H\parallel a$ (empty symbols) are shifted
downward by 40 J/mol$\cdot$K for clarity. Figure 6: Plot of $C/T$ vs. $T$ of
U7Te12 in the magnetic field applied along the $c$-axis. The inset shows the
low-temperature part of $C/T$ vs. $T^{2}$ to determine the Sommerfeld
coefficient.
Specific heat measurements were performed to examine the magnetic anomalies at
$T_{\rm C}$ and $T^{\star}$. The results for various magnetic fields applied
in both directions are shown in Figs. 5 and 6. Only one phase transition is
observed at 48.6 K corresponding to $T_{\mathrm{C}}$. Concurrently, no
$\lambda$-type anomaly is observed around $T^{\star}$, indicating that the
anomaly at $T^{\star}$ is a magnetic crossover.
As shown in Fig. 5, in the case of $H\parallel a$, the $\lambda$-type anomaly
at $T_{\rm C}$ is easily blurred by a tiny external field of 0.1 T. Meanwhile,
in the case of $H\parallel c$, the anomaly remains at 0.1 T, despite being
blurred by an $a$-projected field due to a field misalignment within several
degrees. The Sommerfeld coefficient well below $T_{\rm C}$ is extrapolated to
be $\gamma=48$ mJ/mol$\cdot\mathrm{K}^{2}$ from data below 10 K, which is
similar to the $\gamma$ values well below $T_{\rm C}$ in other ferromagnetic
uranium compounds with itinerant $5f$ electrons (e.g., UCoAl $\gamma=48$
mJ/mol$\cdot\mathrm{K}^{2}$ [12], UCu2Ge2 $\gamma=30$
mJ/mol$\cdot\mathrm{K}^{2}$ [13], UGe2 $\gamma=35$ mJ/mol$\cdot\mathrm{K}^{2}$
[14], UIr $\gamma=50$ mJ/mol$\cdot\mathrm{K}^{2}$ [15], U5Sb4 $\gamma=37$
mJ/mol$\cdot\mathrm{K}^{2}$ [16]).
In the $C/T$ vs. $T$ plot as shown in Fig. 6, instead of a $\lambda$-type
anomaly, there is a sign of hump anomaly that spreads widely around
$T^{\star}=26$ K, which does not seem to be affected by external fields along
the $c$ axis and no change was observed for field applied along the $a$-axis.
For a crossover from in-plane ferromagnetism to $c$-axis ferromagnetism of
uniform moments, applying an external magnetic field will change $C/T$ vs. $T$
at the crossover, but it is not the case in U7Te12. Therefore, the crossover
at $T^{\star}$ is most likely a crossover from a paramagnetic state to a
ferromagnetic state with an out-of-plane component, in the background of in-
plane ferromagnetism. Namely, it suggests the presence of partially disordered
(paramagnetic) uranium sites in the intermediate region of $T^{\star}<T<T_{\rm
C}$. It should be also noted that the $T^{\star}$ is reduced by the fields
applied along the $a$-axis in electrical resistivity and magnetization. As
shown in the inset of Fig. 6, the $\gamma$ value is reduced by an external
field along the $c$ axis, but it is almost independent of $H$ along the
$a$-axis. This indicates that the low-temperature state is only sensitive to
the magnetic field in the $c$ direction.
### III.4 Electrical resistivity
Figure 7: (a) Temperature dependence of electrical resistivity $\rho_{c}$ and
$\rho_{a}$ for U7Te12 measured with the electrical current ($I$) applied along
the $c$\- and $a$-axes, respectively. (b) $\rho_{c}(T)$ with applying fields
along the $a$-axis. (c) $\rho_{a}(T)$ with applying fields along the $c$-axis.
The inset shows a magnified $\rho$-$T$ plot around $T_{\mathrm{C}}$.
$T_{\mathrm{C}}$ and $T^{\star}$ are highlighted by red and blue lines,
respectively.
As shown in Fig. 7(a), U7Te12 exhibits a semimetallic behavior. The electrical
resistivity $\rho(T)$ is much lower than the reported value of $\rho({\rm 300\
K})\simeq$3 m$\Omega\cdot$cm [9], indicating the high-quality of our crystal.
The $\rho_{c}(T)$ in the $c$-axis direction is smaller than $\rho_{a}(T)$ in
the $a$-direction, confirming the high anisotropy of U7Te12. Both
$\rho_{a}(T)$ and $\rho_{c}(T)$ show a gradual increase with decreasing
temperature from 300 K. At $T_{\rm C}=48$ K, $\rho_{a}(T)$ shows a local
maximum and $\rho_{c}(T)$ shows a kink anomaly. Notably, around $\sim$60 K
above $T_{\rm C}$, a broad maximum is seen. Below 48 K, $\rho_{a}(T)$ keeps
increasing without any other anomalies. For $\rho_{c}(T)$, a sharp decrease is
observed below $T_{\rm C}$, but increases sharply below $T^{\star}=26$ K. The
$\rho_{c}(T)$ below $T^{\star}$ does not follow the Arrhenius-type
$T$-dependence, i.e., it is not a semiconducting behavior with a fixed energy
gap. It may be a half-gapped semimetallic state induced by the ferromagnetic
crossover.
Figures 7(b) and 7(c) show the magnetic field dependence of $\rho_{c}(T)$ and
$\rho_{a}(T)$ measured with applying magnetic fields along the perpendicular
directions to the current ($I$) directions. For $\rho_{c}$, as shown in Fig.
7(b), when the magnetic field is applied along the $a$-axis, the ferromagnetic
transition at $T_{\rm C}(0)=48$ K moves to higher temperatures, and it is
easily smeared out as magnetic fluctuations are suppressed. In other words,
the negative magnetoresistance effect of $\rho_{c}$ above $T_{\rm C}$ is
significant in the in-plane field direction. The $T^{\star}$ slightly moves to
lower temperatures with an increasing magnetic field, which is similar to the
behavior observed in magnetic susceptibility in $H\parallel a$. The
$\rho_{c}(T)$ below $T^{\star}$ does not show any significant
magnetoresistance effect by applying fields along the $a$ axis. Alternatively,
in $H\parallel c$ shown in Fig. 7(c), the $T_{\rm C}$ does not shift
significantly below $\sim$3 T. There is no significant magnetoresistance
effect in this field direction. The sample was easily moved from the fixed
position in the higher fields along the hard magnetic axis due to the strong
magnetic torque. Therefore, the data above 3 T were unable to be collected
under these conditions.
## IV Discussions
To begin with the discussion, the main experimental results are summarized as
follows. Two anomalies at $T_{\rm C}=48$ K and around $T^{\star}=26$ K are
observed in U7Te12. A ferromagnetic ordering occurs at $T_{\rm C}$ where the
ferromagnetic component emerges in the basal plane. In the ferromagnetic state
below $T_{\rm C}$, an additional ferromagnetic crossover along the $c$-axis
occurs around $T^{\star}$. In the specific heat measurement, a clear
$\lambda$-type anomaly is observed at $T_{\rm C}$, whereas a broad hump is
barely visible around $T^{\star}$. Such an anisotropic ferromagnetic response
along the $a$\- and $c$-axes can be due to the multiple uranium sites of U1,
U2, and U3. As shown in Sec. IIIB, the in-plane ferromagnetism below $T_{\rm
C}$ can be ascribed to the four out of the seven uranium moments, while the
rest uranium moments would be responsible for the ferromagnetic crossover
along the $c$ axis below $T^{\star}$. Based on these observations, the nature
and origin of the anomaly at $T^{\star}$ would be discussed.
In general, a crossover between paramagnetic and ferromagnetic states is often
induced by external magnetic fields. This occurs in U7Te12 even at zero
magnetic field. The ferromagnetic transition at $T_{\rm C}$ is a second-order
transition, breaking the time-reversal symmetry.
Considering the in-plane ferromagnetic ordering at the $T_{\mathrm{C}}$, the
symmetry of U1, U2, and U3 positions denoted in Table 1 seems to be reduced to
$m^{\prime}..$ by time-reversal symmetry breaking. If this were the case, the
appearance of ferromagnetic moments along the $c$-axis below $T^{\star}$ broke
the $m^{\prime}$ symmetry to the lowest symmetry 1, which should induce a
phase transition at $T^{\star}$. This contradicts the observed crossover
behavior below $T^{\star}$, indicating that the symmetry at the uranium sites
is already lowered to 1 at $T_{\mathrm{C}}$. At the present, the origin of
this symmetry lowering to 1 at $T_{\mathrm{C}}$ is unknown.
Let us consider the in-plane ferromagnetic structure below $T_{\rm C}$ again.
Considering the in-plane arrangement of uranium atoms, the U1 moments can form
a collinear ferromagnetic structure without the cost of anisotropy energy.
However for U2 sites, because the local easy axis is not parallel between the
neighboring sites, a collinear ferromagnetic structure costs the anisotropy
energy. Thus, non-colinear ferromagnetic states, such as 2-in-1-out
(1-in-2-out), are considered more favorable.
The same situation might be realized for the U3 site. In reality, the
interaction between the out-of-plane site and the uranium moments of the other
sites must be considered further. As a result, the overall ferromagnetic
structure seems to be more complex. To identify the magnetic structure, it is
necessary to perform neutron scattering experiments.
## V Conclusion
We have succeeded in growing the single crystals of U7Te12 and characterized
the physical properties. The ferromagnetic transition is observed at $T_{\rm
C}=48$ K in different measurements with magnetic moment parallel to the basal
plane. The crossover is observed around $T^{\star}=26$ K and is characterized
by an increase of magnetic moment along the $c$-axis and no dependence on the
magnetic field. We explain the low-temperature state below $T^{\star}$ as the
ferromagnetic state with magnetic moment along the $c$-axis or canted from the
$c$-axis. We propose that ordered magnetic moments originate from uranium at
different crystallographic positions giving rise to ordering at $T_{\rm C}$
and $T^{\star}$.
## VI Acknowledgement
We wish to thank C. Tabata, K. Kaneko, and M.-T. Suzuki, for the helpful
discussion. A part of this work was supported by JSPS KAKENHI Grant Nos.
JP20KK0061, and JP20K20905. This work was also partly supported by the JAEA
Fund for Exploratory Researches (Houga fund) and REIMEI Research Program.
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|
# Improving Cross-Modal Retrieval with Set of Diverse Embeddings
Dongwon Kim1 Namyup Kim1 Suha Kwak1,2
1Dept. of CSE, POSTECH 2 Graduate School of AI, POSTECH
{kdwon, namyup<EMAIL_ADDRESS>
###### Abstract
Cross-modal retrieval across image and text modalities is a challenging task
due to its inherent ambiguity: An image often exhibits various situations, and
a caption can be coupled with diverse images. Set-based embedding has been
studied as a solution to this problem. It seeks to encode a sample into a set
of different embedding vectors that capture different semantics of the sample.
In this paper, we present a novel set-based embedding method, which is
distinct from previous work in two aspects. First, we present a new similarity
function called smooth-Chamfer similarity, which is designed to alleviate the
side effects of existing similarity functions for set-based embedding. Second,
we propose a novel set prediction module to produce a set of embedding vectors
that effectively captures diverse semantics of input by the slot attention
mechanism. Our method is evaluated on the COCO and Flickr30K datasets across
different visual backbones, where it outperforms existing methods including
ones that demand substantially larger computation at inference.
## 1 Introduction
Cross-modal retrieval is the task of searching for data relevant to a query
from a database when the query and database have different modalities. While
it has been studied for various pairs of modalities such as video-text [19, 3,
7] and audio-text [5, 16], the most representative setting for the task is the
retrieval across image and text modalities [30, 47, 10, 38]. A naïve solution
to cross-modal retrieval is a straightforward extension of the conventional
unimodal retrieval framework [18, 17, 27], _i.e._ , learning a joint embedding
space of the different modalities with known ranking losses (_e.g._ ,
contrastive loss [9] and triplet loss [45]). In this framework, each sample is
represented as a single embedding vector and the task reduces to neighbor
search on the joint embedding space.
However, this naïve approach has trouble in handling the inherent ambiguity of
the cross-modal retrieval across image and text modalities [30, 47, 10]. A
cause of the ambiguity is the fact that even a single image often contains
various situations and contexts. Consider an image in Figure 1, which
illustrates a group of children in a skate park. One of the captions coupled
with it could be about children carrying up a bike, while another may describe
the child riding a skateboard. Indeed, different local features of the image
are matched to different captions. Similarly, visual manifestations of a
caption could vary significantly as text descriptions are highly abstract.
This ambiguity issue suggests that a sample should be embedded while
reflecting its varying semantics in cross-modal retrieval. Embedding models
dedicated to the uni-modal retrieval do not meet this requirement since they
represent a sample as a single embedding vector.
Figure 1: An example of the ambiguity problem introduced in the cross-modal
retrieval task; an image region and a word corresponding to each other are
highlighted in the same color. This example demonstrates that a single image
can be coupled with multiple heterogeneous captions.
Various methods have been studied to mitigate the ambiguity issue of cross-
modal retrieval. Most of them adopt cross-attention networks that directly
predict the similarity of an input image-caption pair [31, 38, 51, 12, 25, 39,
50, 54, 14]. These models successfully address the ambiguity since they
explicitly infer relations across the modalities by drawing attentions on both
modalities at once. However, they inevitably impose a large computation burden
on retrieval systems since they demand both image and caption to be processed
together for computing their similarity; all data in a database thus have to
be reprocessed whenever a query arrives. In contrast, methods using separate
textual and visual encoders [30, 17, 21, 24] seek to find samples relevant to
query through nearest-neighbor search on pre-computed embedding vectors, which
are more suitable to nowadays retrieval systems working on huge databases.
However, most of the previous arts in this direction cannot address the
ambiguity issue since their encoders return a single embedding vector for a
given input.
Recent studies [47, 10] have advanced embedding model architectures to tackle
the ambiguity issue even with separate encoders for the two modalities. To be
specific, their models compute a set of heterogeneous embedding vectors for a
given sample using self-attention layers on top of visual or textual features;
such a set of embedding vectors is called _embedding set_ in the remainder of
this paper. Then the ambiguity issue is addressed through elements of the
embedding set that encode diverse semantics of input.
Although the set-based embedding models enable a retrieval system to be
powerful yet efficient, however, similarity functions used for their training
do not consider the ambiguity of the data. Hence, training the models with the
similarity functions often causes the following two side effects: (1) _Sparse
supervision_ –An embedding set most of whose elements remain untrained, or (2)
_Set collapsing_ –An embedding set with a small variance where elements do not
encode sufficient ambiguity. Further, self-attention modules used for set
prediction in the previous work do not explicitly consider disentanglement
between set elements. These limitations lead to an embedding set whose
elements encode redundant semantics of input, which also causes the _set
collapsing_ and degrades the capability of learned embedding models.
To address the aforementioned limitations of previous work, we propose a novel
set-based embedding method for cross-modal retrieval. The proposed method is
distinct from previous work in mainly two aspects. First, we design a novel
similarity function for sets, called smooth-Chamfer similarity, that is
employed for both training and evaluation of our model. In particular, our
loss based on the smooth-Chamfer similarity addresses both limitations of the
existing similarity functions, _i.e._ , sparse supervision and set collapsing.
Second, we propose a model with a novel set prediction module motivated by
slot attention [34]. In the proposed module, learnable embeddings called
element slots compete with each other for aggregating input data while being
transformed into an embedding set by progressive update. Therefore, our model
captures the diverse semantic ambiguity of input successfully, with little
redundancy between elements of the embedding set.
The proposed method is evaluated and compared with previous work on two
realistic cross-modal retrieval benchmarks, COCO [33] and Flickr30K [43],
where it outperforms the previous state of the art in most settings. In
summary, our contribution is three-fold as follows:
* •
We address issues on previous set-based embedding methods by proposing a novel
similarity function for sets, named smooth-Chamfer similarity.
* •
We introduce a slot attention based set prediction module where elements of
embedding set iteratively compete with each other for aggregating input data,
which can capture semantic ambiguity of input without redundancy.
* •
Our model achieved state-of-the-art performance on the COCO and Flickr30K
datasets, two standard benchmarks for cross-modal retrieval.
## 2 Related Work
Cross-modal retrieval: Previous work can be categorized into two classes,
methods using two separate encoders and those using a cross-attention network.
Models of the first class consist of disjoint visual and textual encoders and
learn an embedding space where embedding vectors of matching pairs are located
nearby. Retrieval is then performed by finding nearest neighbors of query on
the embedding space, which is efficient since their embedding vectors are pre-
computed. To improve the quality of the embedding space, loss functions [17,
10], model architectures [24, 32, 28], and embedding methods [47, 10] have
been proposed. Though the separate encoders enable simple and fast retrieval,
they often failed to handle the inherent ambiguity of cross-modal retrieval,
leading to inferior retrieval performance. On the other hand, models of the
other class adopt a cross-attention network that directly predicts the
similarity score between an image-caption pair [31, 51, 50, 54, 14, 53].
Though these methods make their models explicitly consider relations across
modalities, they are not suitable for real-world retrieval scenarios due to
the large computation burden imposed during evaluation. We thus focus on
improving the separate encoders in this paper.
Figure 2: An overview of our model. (a) The overall framework of our model.
The model consists of three parts: visual feature extractor, textual feature
extractor, and set-prediction modules $f^{\mathcal{V}}$ and $f^{\mathcal{T}}$.
First, the feature extractors of each modality extract local and global
features from input samples. Then, the features are fed to the set prediction
modules to produce embedding sets $\mathbf{S}^{\mathcal{V}}$ and
$\mathbf{S}^{\mathcal{T}}$. The model is trained with the loss using our
smooth-Chamfer similarity. (b) Details of our set prediction module and
attention maps that slots of each iteration produce. A set prediction module
consists of multiple aggregation blocks that share their weights. Note that
$f^{\mathcal{V}}$ and $f^{\mathcal{T}}$ have the same model architecture.
Embedding beyond vector representation: Most of existing work projects each
sample into a single embedding vector [17, 21, 24]. However, it has been
empirically proven that such single vector representation is not sufficient to
deal with inherent ambiguity of data [49, 46]. To mitigate this issue for the
cross-modal retrieval task, PVSE [47] and PCME [10] introduce novel embedding
methods. To be specific, PVSE represents each sample as a set of embedding
vectors and PCME introduces probabilistic embedding where each sample is
represented as a set of vectors sampled from a normal distribution.
Slot attention: Recently, slot attention [34] is proposed to learn object-
centric representation, which is especially beneficial for tasks that require
perception of objectness, such as object discovery and set property
prediction. Slots, which are embedding vectors sampled from a random
distribution, compete with each other for explaining the input in an iterative
manner. This attention mechanism enables the final outputs to encode
heterogeneous semantics that appear in the input without any explicit
supervision. To produce informative embedding sets with sufficient within-set
variance, our set prediction module employs the slot attention mechanism.
However, the ability of slot attention to discover individual objects is only
verified on synthetic datasets [29] and is known to fail on real-world images
[4]. We make three architectural modifications to resolve this issue: (1)
using learnable embeddings for initial element slots instead of random
vectors, (2) replacing GRU [8] with a residual sum, and (3) adding a global
feature to the final element slots. We will detail these differences in
Section 3.2. Without them, we observed that the loss does not converge, and
thus training fails.
## 3 Proposed Method
This section first describes the overall model architecture and elaborates on
the proposed set prediction module. Then we introduce our smooth-Chamfer
similarity and illustrate how it differs from existing set similarity
functions. Finally, details of the training and inference of our model are
provided.
### 3.1 Overall Model Architecture
The overall framework of our method is illustrated in Figure 2. Our model
architecture consists of a visual feature extractor, a textual feature
extractor, and a set prediction module of each modality: $f^{\mathcal{V}}$ and
$f^{\mathcal{T}}$. The feature extractors have two branches that compute local
features and global features from the input sample, respectively. The
extracted features are given as input to the set prediction modules, each of
which fuses local and global features to encode an embedding set. For the
visual and textual feature extractors, we followed the conventional setting of
the previous work [47, 10, 28].
Visual feature extractor: We consider two different types of visual feature
extractors for fair comparisons with previous work. One employs a flattened
convolutional feature map as local features and their average pooled feature
as the global feature. The other uses ROI features of a pretrained object
detector as local features and their max pooled feature as the global feature.
In any of these cases, the local and global features are denoted by
$\psi^{\mathcal{V}}(\mathbf{x})\in\mathbb{R}^{N\times D}$ and
$\phi^{\mathcal{V}}(\mathbf{x})\in\mathbb{R}^{D}$, respectively, where
$\mathbf{x}$ is input image.
Textual feature extractor: For the textual feature, we also employ two
different types of extractors: bi-GRU [8] and BERT [13]. When using bi-GRU,
for $L$-words input caption $\mathbf{y}$, we take GloVe [42] word embedding of
each word as a local feature
$\psi^{\mathcal{T}}(\mathbf{y})\in\mathbb{R}^{L\times 300}$. Then we apply bi-
GRU with $D$ dimensional hidden states on the top of the
$\psi^{\mathcal{T}}(\mathbf{y})$. The last hidden state is used as the global
feature $\phi^{\mathcal{T}}(\mathbf{y})\in\mathbb{R}^{D}$. Similarly, when
using BERT, output hidden states of BERT and their max pooled feature are used
as a $\psi^{\mathcal{T}}(\mathbf{y})\in\mathbb{R}^{L\times D}$ and
$\phi^{\mathcal{T}}(\mathbf{y})\in\mathbb{R}^{D}$, respectively.
### 3.2 Set Prediction Module
Elements of an embedding set should encode heterogeneous semantics that appear
in the input data. Otherwise, it will degenerate into a set with small
variations that cannot handle the ambiguity of the input.
Inspired by slot attention [34], we introduce an aggregation block, where
element slots compete with each other for aggregating input data. The slots
are progressively updated to capture various semantics of the input through
multiple iterations of the aggregation block, and then the slots are fused
with a global feature to be used as elements of the output embedding set. In
this manner, the proposed module produces an embedding set whose elements
encode substantially different semantics, meanwhile preserving the global
context of the input data. In this section, we only present how the visual set
prediction module $f^{\mathcal{V}}$ works; $f^{\mathcal{T}}$ has the same
model architecture and works in the same manner with $f^{\mathcal{V}}$.
As shown in Figure 2(b), the proposed set prediction module consists of
multiple aggregation blocks, which share their weights. We define the initial
element slots as learnable embedding vectors
$\mathbf{E}^{0}\in\mathbb{R}^{K\times D}$, where $K$ is the cardinality of the
embedding set. Then, the element slots of the $t$-th iteration
$\mathbf{E}^{t}$ are computed by
$\displaystyle\mathbf{E}^{t}=\mathrm{AggBlock}(\psi^{\mathcal{V}}(\mathbf{x});\mathbf{E}^{t-1})\in\mathbb{R}^{K\times
D}.$ (1)
The aggregation block first layer-normalizes [2] inputs and then linearly
projects input local feature $\psi^{\mathcal{V}}(\mathbf{x})$ to
$\mathbf{k}\in\mathbb{R}^{N\times D_{h}}$ and
$\mathbf{v}\in\mathbb{R}^{N\times D_{h}}$, and $\mathbf{E}^{t-1}$ to
$\mathbf{q}\in\mathbb{R}^{K\times D_{h}}$. 111When using convolutional
features for visual feature extraction, we add sinusoidal positional encoding
before projecting $\psi^{\mathcal{V}}(\mathbf{x})$ to $\mathbf{k}$. Then, the
attention map $A$ between $\psi^{\mathcal{V}}(\mathbf{x})$ and
$\mathbf{E}^{t-1}$ is obtained via
$\displaystyle A_{n,k}=\frac{e^{M_{n,k}}}{\sum_{k=1}^{K}e^{M_{n,k}}},\text{
where }M=\frac{\mathbf{k}\mathbf{q}^{\top}}{\sqrt{D_{h}}}.$ (2)
It is worth noting that the attention map is normalized over slots, not keys
as in the transformer [48]. Since this way of normalization lets slots compete
with each other, each slot attends to nearly disjoint sets of local features,
and these sets will correspond to the distinctive semantics of the input.
Using the obtained attention map, we update the element slot with the weighted
mean of local features and then feed it to a multi-layer perceptron (MLP) with
layer normalization, residual connection, and GELU [23] activation:
$\displaystyle\hat{A}_{n,k}=\frac{A_{n,k}}{\sum_{n=1}^{N}A_{n,k}},\text{
}\overline{\mathbf{E}^{t}}=\hat{A}^{\top}\mathbf{v}W_{o}+\mathbf{E}^{t-1},$
(3)
$\displaystyle\mathbf{E}^{t}=\mathrm{AggBlock}(\psi^{\mathcal{V}}(\mathbf{x});\mathbf{E}^{t-1})=\mathrm{MLP}(\overline{\mathbf{E}^{t}})+\overline{\mathbf{E}^{t}},$
(4)
where $W_{0}\in\mathbb{R}^{D_{h}\times D}$ is a learnable linear projection.
The output of the aggregation block is used as an element slot for the $t$-th
iteration. As shown in Figure 2, attention maps that slots of early stage
produce are sparse and noisy, but refined to aggregate local features with
distinctive semantics as proceeds.
Finally, at $T$-th iteration, the model predicts an embedding set $\mathbf{S}$
by adding the global feature $\phi^{\mathcal{V}}(\mathbf{x})$ to each element
slot $\mathbf{E}^{T}$:
$\displaystyle\mathbf{S}=\mathrm{LN}(\mathbf{E}^{T})+\big{[}\mathrm{LN}(\phi^{\mathcal{V}}(\mathbf{x}))\stackrel{{\scriptstyle\scriptscriptstyle\times
K}}{{\cdots}}\big{]}),$ (5)
where $\mathrm{LN}$ is a layer normalization, and
$\big{[}\phi^{\mathcal{V}}(\mathbf{x})\stackrel{{\scriptstyle\scriptscriptstyle\times
K}}{{\cdots}}\big{]}\in\mathbb{R}^{K\times D}$ is $K$ repetitions of
$\phi^{\mathcal{V}}(\mathbf{x})$. The module benefits from the global feature
when treating samples with little ambiguity since the global feature allows
them to be represented as an embedding set of small within-set variance.
### 3.3 Smooth-Chamfer Similarity
Figure 3: Comparison between our similarity function and existing ones used
for embedding set. Illustrations of embedding space before and after
optimizing each similarity function are presented. Presented two sets are
matching pairs, and thus optimized to maximize their similarity. Lines
indicate the association that similarity functions consider, where their
intensities represent the weights given to each association.
Our smooth-Chamfer similarity is proposed to resolve the drawbacks of the
existing set-based embedding approaches. Before presenting smooth-Chamfer
similarity, we first review the similarity functions used in PVSE [47] and
PCME [10]. Figure 3 illustrates embedding spaces trained with these set
similarity functions. Let $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ be the
embedding sets, which are the set of vectors. $c(x,y)$ and $d(x,y)$ denote the
cosine similarity and $\ell_{2}$-normalized Euclidean distance between vectors
$x$ and $y$, respectively.
PVSE adopts the multiple instance learning (MIL) framework [15] during
training and inference. Its similarity function, which we call MIL similarity,
is given by
$s_{\mathrm{MIL}}(\mathbf{S}_{1},\mathbf{S}_{2})=\max_{x\in\mathbf{S}_{1},y\in\mathbf{S}_{2}}c(x,y)$.
MIL similarity only takes account of the closest pair of elements, as shown in
Figure 3(a). While this behavior simplifies similarity measurement, MIL
similarity suffers from the _sparse supervision_ problem. In other words, the
majority of elements are not sampled as the closest pair and thus remain
untrained.
On the other hand, PCME uses the match probability (MP) as a similarity
function, which is defined as
$s_{\mathrm{MP}}(\mathbf{S}_{1},\mathbf{S}_{2})=\sum_{x\in\mathbf{S}_{1},y\in\mathbf{S}_{2}}\sigma(\alpha
d(x,y)+\beta),$ where $\alpha$ and $\beta$ are learnable parameters and
$\sigma$ is the sigmoid function. MP takes average on every pairwise distance
between elements. Though MP resolves the sparse supervision, it introduces the
_set collapsing_ problem. As presented in Figure 3(b), directly optimizing MP
leads to a collapsed embedding set since elements of embedding sets pull each
other without the consideration of their relative proximity. To mitigate this
issue, MP requires a specialized loss function [10, 40], which makes MP
incompatible with other losses or frameworks.
The aforementioned problems stem from the way how similarity functions
associate set elements. Associating only the nearest pair of elements like MIL
leads to insufficient supervisory signals, while equally taking account of
every possible association like MP results in a collapsed set. Unlike these
similarity functions, our smooth-Chamfer similarity associates every possible
pair with different degrees of weight, as illustrated in Figure 3(c). Smooth-
Chamfer similarity is formulated as
$\begin{split}s_{\mathrm{SC}}(\mathbf{S}_{1},\mathbf{S}_{2})=\frac{1}{2\alpha|\mathbf{S}_{1}|}\sum_{x\in\mathbf{S}_{1}}\log\bigg{(}\sum_{y\in\mathbf{S}_{2}}e^{\alpha
c(x,y)}\bigg{)}\\\
+\frac{1}{2\alpha|\mathbf{S}_{2}|}\sum_{y\in\mathbf{S}_{2}}\log\bigg{(}\sum_{x\in\mathbf{S}_{1}}e^{\alpha
c(x,y)}\bigg{)},\end{split}$ (6)
where $|\cdot|$ is the set cardinality and $\alpha>0$ is a scaling parameter.
Using Log-Sum-Exp, it can be reformulated in a more interpretable form as
shown below:
$\small\begin{split}s_{\mathrm{SC}}(\mathbf{S}_{1},\mathbf{S}_{2})=\frac{1}{2\alpha|\mathbf{S}_{1}|}\sum_{x\in\mathbf{S}_{1}}\underset{y\in\mathbf{S}_{2}}{\mathrm{LSE}}\bigg{(}\alpha
c(x,y)\bigg{)}\\\
+\frac{1}{2\alpha|\mathbf{S}_{2}|}\sum_{y\in\mathbf{S}_{2}}\underset{x\in\mathbf{S}_{1}}{\mathrm{LSE}}\bigg{(}\alpha
c(x,y)\bigg{)}.\end{split}$ (7)
LSE indicates Log-Sum-Exp, which is the smooth approximation of the max
function. Thanks to the property of LSE, smooth-Chamfer similarity softly
assigns elements in one set to those in the other set, where the weight for a
pair of elements is determined by their relative proximity. Hence, smooth-
Chamfer similarity can provide dense supervision like MP but without the
collapsing issue.
By replacing LSE with the max function, we can consider a non-smooth version
of smooth-Chamfer similarity, which we refer to as Chamfer similarity. Chamfer
similarity clears up the problem of set collapsing by assigning individual
elements of $\mathbf{S}_{1}$ to their closest neighbor in $\mathbf{S}_{2}$,
and vice-versa. However, unlike smooth-Chamfer similarity, it does not provide
supervision that is dense as MP since most associations are not considered.
The behavior of smooth-Chamfer similarity can be clearly demonstrated by its
gradient with respect to $c(x,y)$, which is given by
$\begin{split}\frac{\partial
s_{\mathrm{SC}}(\mathbf{S}_{1},\mathbf{S}_{2})}{\partial
c(x^{\prime},y^{\prime})}=\frac{1}{2|\mathbf{S}_{1}|}\frac{e^{\alpha
c(x^{\prime},y^{\prime})}}{\sum_{y\in\mathbf{S}_{2}}e^{\alpha
c(x^{\prime},y)}}\\\ +\frac{1}{2|\mathbf{S}_{2}|}\frac{e^{\alpha
c(x^{\prime},y^{\prime})}}{\sum_{x\in\mathbf{S}_{1}}e^{\alpha
c(x,y^{\prime})}},\end{split}$ (8)
where $x^{\prime}$ and $y^{\prime}$ are elements of $\mathbf{S}_{1}$ and
$\mathbf{S}_{2}$, respectively. One can see that the gradient is the sum of
two relative similarity scores, suggesting that the gradient for
$c(x^{\prime},y^{\prime})$ is emphasized when $x^{\prime}$ and $y^{\prime}$
are close to each other. Using the weighting scheme based on the relative
proximity, we can give denser supervision during training, while preserving
sufficient within-set variance.
### 3.4 Training and Inference
Training: Our model is trained with the objective functions presented in [47],
which consists of the triplet loss, diversity regularizer, and Maximum Mean
Discrepancy (MMD) [20] regularizer. Following previous work [17, 47, 51], we
adopt the triplet loss with hard negative mining. Let
$B=\\{(S^{\mathcal{V}}_{i},S^{\mathcal{T}}_{i})\\}^{N}_{i=1}$ be a batch of
embedding sets. Then the triplet loss is given by
$\displaystyle\begin{split}\mathcal{L}_{\mathrm{tri}}(B)=\sum_{i=1}^{N}\max_{j}[\delta+s_{\mathrm{SC}}(\mathbf{S}^{\mathcal{V}}_{i},\mathbf{S}^{\mathcal{T}}_{j})-s_{\mathrm{SC}}(\mathbf{S}^{\mathcal{V}}_{i},\mathbf{S}^{\mathcal{T}}_{i})]_{+}\\\
+\sum_{i=1}^{N}\max_{j}[\delta+s_{\mathrm{SC}}(\mathbf{S}^{\mathcal{T}}_{i},\mathbf{S}^{\mathcal{V}}_{j})-s_{\mathrm{SC}}(\mathbf{S}^{\mathcal{T}}_{i},\mathbf{S}^{\mathcal{V}}_{i})]_{+},\end{split}$
(9)
where $[\cdot]_{+}$ indicates the hinge function and $\delta>0$ is a margin.
The MMD regularizer minimizes MMD between embedding sets of image and text. It
prevents embeddings of different modalities from diverging at the early stage
of training. The diversity regularizer penalizes similar element slots, which
helps the model produce a set of diverse embeddings. The two regularizers are
formulated as
$\mathcal{L}_{\mathrm{mmd}}=\mathrm{MMD}(B^{\mathcal{V}},B^{\mathcal{T}})$ and
$\mathcal{L}_{\mathrm{div}}=\sum_{x,x^{\prime}\in\mathbf{E}^{T}}e^{-2||x-x^{\prime}||^{2}_{2}}$,
where $B^{\mathcal{V}}$ and $B^{\mathcal{T}}$ denote subsets of batch $B$
consisting of each modality embeddings.
Inference: We pre-compute the embedding set of size $K$ for every sample in
the database. Then, the sample most relevant to a query is retrieved via
nearest-neighbor search on embedding sets, using smooth-Chamfer similarity.
## 4 Experiments
Method | CA | 1K Test Images | 5K Test Images
---|---|---|---
Image-to-Text | Text-to-Image | RSUM | Image-to-Text | Text-to-Image | RSUM
R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10
ResNet-152 + Bi-GRU
VSE++ [17] | ✗ | 64.6 | 90.0 | 95.7 | 52.0 | 84.3 | 92.0 | 478.6 | 41.3 | 71.1 | 81.2 | 30.3 | 59.4 | 72.4 | 355.7
PVSE [47] | ✗ | 69.2 | 91.6 | 96.6 | 55.2 | 86.5 | 93.7 | 492.8 | 45.2 | 74.3 | 84.5 | 32.4 | 63.0 | 75.0 | 374.4
PCME [10] | ✗ | 68.8 | - | - | 54.6 | - | - | - | 44.2 | - | - | 31.9 | - | - | -
Ours | ✗ | 70.3 | 91.5 | 96.3 | 56.0 | 85.8 | 93.3 | 493.2 | 47.2 | 74.8 | 84.1 | 33.8 | 63.1 | 74.7 | 377.7
Faster R-CNN + Bi-GRU
SCAN† [31] | ✓ | 72.7 | 94.8 | 98.4 | 58.8 | 88.4 | 94.8 | 507.9 | 50.4 | 82.2 | 90.0 | 38.6 | 69.3 | 80.4 | 410.9
VSRN† [32] | ✗ | 76.2 | 94.8 | 98.2 | 62.8 | 89.7 | 95.1 | 516.8 | 53.0 | 81.1 | 89.4 | 40.5 | 70.6 | 81.1 | 415.7
CAAN [54] | ✓ | 75.5 | 95.4 | 98.5 | 61.3 | 89.7 | 95.2 | 515.6 | 52.5 | 83.3 | 90.9 | 41.2 | 70.3 | 82.9 | 421.1
IMRAM† [6] | ✓ | 76.7 | 95.6 | 98.5 | 61.7 | 89.1 | 95.0 | 516.6 | 53.7 | 83.2 | 91.0 | 39.7 | 69.1 | 79.8 | 416.5
SGRAF† [14] | ✓ | 79.6 | 96.2 | 98.5 | 63.2 | 90.7 | 96.1 | 524.3 | 57.8 | - | 91.6 | 41.9 | - | 81.3 | -
VSE∞ [28] | ✗ | 78.5 | 96.0 | 98.7 | 61.7 | 90.3 | 95.6 | 520.8 | 56.6 | 83.6 | 91.4 | 39.3 | 69.9 | 81.1 | 421.9
NAAF† [53] | ✓ | 80.5 | 96.5 | 98.8 | 64.1 | 90.7 | 96.5 | 527.2 | 58.9 | 85.2 | 92.0 | 42.5 | 70.9 | 81.4 | 430.9
Ours | ✗ | 79.8 | 96.2 | 98.6 | 63.6 | 90.7 | 95.7 | 524.6 | 58.8 | 84.9 | 91.5 | 41.1 | 72.0 | 82.4 | 430.7
Ours† | ✗ | 80.6 | 96.3 | 98.8 | 64.7 | 91.4 | 96.2 | 528.0 | 60.4 | 86.2 | 92.4 | 42.6 | 73.1 | 83.1 | 437.8
ResNeXt-101 + BERT
VSE∞ [28] | ✗ | 84.5 | 98.1 | 99.4 | 72.0 | 93.9 | 97.5 | 545.4 | 66.4 | 89.3 | 94.6 | 51.6 | 79.3 | 87.6 | 468.9
VSE∞† [28] | ✗ | 85.6 | 98.0 | 99.4 | 73.1 | 94.3 | 97.7 | 548.1 | 68.1 | 90.2 | 95.2 | 52.7 | 80.2 | 88.3 | 474.8
Ours | ✗ | 86.3 | 97.8 | 99.4 | 72.4 | 94.0 | 97.6 | 547.5 | 69.1 | 90.7 | 95.6 | 52.1 | 79.6 | 87.8 | 474.9
Ours† | ✗ | 86.6 | 98.2 | 99.4 | 73.4 | 94.5 | 97.8 | 549.9 | 71.0 | 91.8 | 96.3 | 53.4 | 80.9 | 88.6 | 482.0
Table 1: Recall@K (%) and RSUM on the COCO dataset. Evaluation results on
both 1K test setting (average of 5-fold test dataset) and 5K test setting are
presented. The best RSUM scores are marked in bold. CA and $\dagger$ indicate
models using cross-attention and ensemble models of two hypotheses,
respectively.
Method | CA | Image-to-text | Text-to-image | RSUM
---|---|---|---|---
R@1 | R@5 | R@10 | R@1 | R@5 | R@10
ResNet-152 + Bi-GRU
VSE++ | ✗ | 52.9 | 80.5 | 87.2 | 39.6 | 70.1 | 79.5 | 409.8
PVSE∗ | ✗ | 59.1 | 84.5 | 91.0 | 43.4 | 73.1 | 81.5 | 432.6
PCME∗ | ✗ | 58.5 | 81.4 | 89.3 | 44.3 | 72.7 | 81.9 | 428.1
Ours | ✗ | 61.8 | 85.5 | 91.1 | 46.1 | 74.8 | 83.3 | 442.6
Faster R-CNN + Bi-GRU
SCAN† | ✓ | 67.4 | 90.3 | 95.8 | 48.6 | 77.7 | 85.2 | 465.0
VSRN† | ✗ | 71.3 | 90.6 | 96.0 | 54.7 | 81.8 | 88.2 | 482.6
CAAN | ✓ | 70.1 | 91.6 | 97.2 | 52.8 | 79.0 | 87.9 | 478.6
IMRAM† | ✓ | 74.1 | 93.0 | 96.6 | 53.9 | 79.4 | 87.2 | 484.2
SGRAF† | ✓ | 77.8 | 94.1 | 97.4 | 58.5 | 83.0 | 88.8 | 499.6
VSE∞ | ✗ | 76.5 | 94.2 | 97.7 | 56.4 | 83.4 | 89.9 | 498.1
NAAF† | ✓ | 81.9 | 96.1 | 98.3 | 61.0 | 85.3 | 90.6 | 513.2
Ours | ✗ | 77.8 | 94.0 | 97.5 | 57.5 | 84.0 | 90.0 | 500.8
Ours† | ✗ | 80.9 | 94.7 | 97.6 | 59.4 | 85.6 | 91.1 | 509.3
ResNeXt-101 + BERT
VSE∞ | ✗ | 88.4 | 98.3 | 99.5 | 74.2 | 93.7 | 96.8 | 550.9
VSE∞† | ✗ | 88.7 | 98.9 | 99.8 | 76.1 | 94.5 | 97.1 | 555.1
Ours | ✗ | 88.8 | 98.5 | 99.6 | 74.3 | 94.0 | 96.7 | 551.9
Ours† | ✗ | 90.6 | 99.0 | 99.6 | 75.9 | 94.7 | 97.3 | 557.1
Table 2: Recall@K(%) and RSUM on the Flickr30K dataset. CA, $\dagger$, and *
indicate models using cross-attention, ensemble models of two hypotheses, and
models we reproduce, respectively.
### 4.1 Datasets and Evaluation Metric
We validate the effectiveness of our method on COCO [33] and Flickr30K [43]
datasets. In both datasets, we follow the split proposed by [30]. COCO dataset
consists of a train split of 113,287 images, a validation split of 5,000
images, and a test split of 5,000 images. Retrieval results of our method on
1K test setting and 5K test setting are both reported, following [30]. In the
1K test setting, the average retrieval performance on the 5-fold test split is
reported. For the Flickr30K dataset, we use 28,000 images for training, 1,000
images for validation, and 1,000 images for testing. We report the retrieval
results on the Flick30K using a test split of 1,000 images. In both datasets,
each image is given with five matching captions. For evaluation, we use the
Recall@$K$, which is the percentage of the queries that have matching samples
among top-$K$ retrieval results. Following [28], We also report the RSUM,
which is the sum of the Recall@$K$ at $K\in\\{1,5,10\\}$ in the image-to-text
and text-to-image retrieval settings.
### 4.2 Implementation Details
Feature extractor: For the visual feature extractor, convolutional visual
features are obtained by applying $1\times 1$ convolution to the last feature
map of CNN. We obtain ROI visual features by feeding the pre-extracted
features [1] from a Faster R-CNN [44] to the 2-layer MLP with residual
connection, following [28]. In every model, we set $D$ to 1024 and $K$ to 4.
Figure 4: For each element of the image embedding set, we present its
attention map and the caption nearest to the element in the embedding space.
Matching captions are colored in green. Entities corresponding to the
attention maps are underlined.
Set prediction module: In every experiment, we set $T$ to 4. When using
convolutional visual features, $D_{h}$ is set to 1024 and otherwise set to
2048.
Similarity and loss function: For smooth-Chamfer similarity, we set scaling
parameter $\alpha$ to 16. In $\mathcal{L}_{\mathrm{tri}}$, we use margin
$\delta$ of 0.1 for convolutional visual feature and 0.2 for ROI visual
feature. We multiply the factor of $0.1$ and $0.01$ to
$\mathcal{L}_{\mathrm{div}}$ and $\mathcal{L}_{\mathrm{mmd}}$, respectively.
Training: The model is trained with the AdamW optimizer [36]. We construct the
input batch with 200 images and all of their captions. When using ROI visual
features, the model is trained for 80 epochs with the initial learning rate of
1e-3, which is scheduled with cosine annealing [35]. For the models employing
convolutional visual features, we follow the training setting of previous work
[47, 28], which will be discussed in detail in the supplementary material.
### 4.3 Comparisons with Other Methods
Results on COCO and Flickr30K are summarized in Table 1 and Table 2,
respectively. For fair comparisons with previous work, our method is evaluated
under three different visual extractors: ResNet-152 [22] pretrained on
ImageNet [11], ROI features [1] pre-extracted by Faster R-CNN [44], and
ResNeXt-101 [52] pretrained on the Instagram [37] dataset. For ResNet-152 and
ResNeXt-101, input image resolution is set to respectively $224\times 224$ and
$512\times 512$, following previous work [47, 28]. For the textual feature
extractor, we use either bi-GRU or BERT. The ensemble results are obtained by
averaging the similarity scores of two models trained with different random
seeds.
Our method outperforms previous methods in terms of RSUM in every setting
except for Flickr30K with ROI visual features. Even in this setting, ours
achieves the second-best. Note that NAAF, currently the best in this setting,
requires two orders of magnitude heavier computations than ours at inference
since it relies on cross-attention, as will be shown in the next section. The
ensemble of ours with ResNeXt-101 and BERT clearly outperforms all existing
records, improving the previous best RSUM by $7.2\%$p and $2.0\%$p on COCO 5K
and Flickr30K. Comparisons with PVSE and PCME demonstrate the superiority of
our set-based embedding framework. Our model outperforms both of them in every
case. Specifically, our model improves RSUM by $3.3\%$p on COCO 5K and
$10.0\%$p on Flickr30K, compared to PVSE. Also, our method shows better
results than models involving cross-attention networks [31, 6, 14], which
impose substantial computation as noted. We finally emphasize that ours using
a single model often outperforms previous work using ensemble [31, 32, 6, 14].
### 4.4 Computation Cost Analysis
To demonstrate the efficiency of our model during evaluation time, we measure
FLOPs required for computing a similarity score between representations of an
image and a caption. Ours demands $16.4$K FLOPs, while SCAN [31], a
representative cross-attention method, requires $1.24$M FLOPs. This result
shows that our method demands about $80$ times less floating-point operations
than SCAN since the cross-attention requires re-processing of the image or
caption representations for each query. Also, it is worth noting that our
method outperforms SCAN in terms of RSUM, as shown in Table 1 and 2.
### 4.5 Analysis of Attention
In Figure 4, attentions from the last aggregation block in the
$f^{\mathcal{V}}$ are visualized. Each attention is used to encode individual
elements of the image embedding set. For each element, its nearest caption in
the embedding space is presented together. Thanks to slot attention, the set
prediction module produces heterogeneous attention maps that focus on
individual objects or contexts, which enable retrieving diverse captions. In
particular, on the top-right, elements are matched with the captions that
describe different semantics (helmet, children, park, and bicycle).
Interestingly, we observed that one of the element slots attends to regions
left out after other slots capture distinctive semantics, which wholly charges
for retrieving related but highly abstract captions.
Similarity | Arch. | RSUM
---|---|---
MIL | Ours | 491.7
MP | Ours | 490.5
Ours (Chamfer) | Ours | 499.6
Ours (S-Chamfer) | PIE-Net | 483.3
Ours (S-Chamfer) | Ours | 500.8
Setting | $\log$(Var.) | RSUM
---|---|---
PIE-Net | -7.35 | 483.3
Ours \w MP | -5.27 | 490.5
Transformer | -2.27 | 496.1
Ours | -2.13 | 500.8
Table 3: Ablation studies of the proposed similarity function and set
prediction module.
### 4.6 Ablation Study
We perform ablation studies to investigate the contributions of the proposed
similarity function and set prediction module. For all ablation experiments,
we use the models employing ROI visual features.
Importance of set similarity function: In the left of Table 3, we ablate the
proposed similarity function and replace it with the MP, MIL, and Chamfer
similarity. The results show that the models trained with smooth-Chamfer
similarity surpass those trained with MIL or MP similarity under the same set
prediction architecture. Though Chamfer similarity improves the performance by
resolving set collapsing, smooth-Chamfer similarity further improves
performance by providing dense supervision during training.
Importance of set prediction module: We also ablate the proposed set
prediction module and substitute it with the PIE-Net, which is the baseline
set prediction architecture of the previous set-based embedding methods, PVSE
and PCME. The results in the left of Table 3 suggest that the impact of our
set prediction module is significant.
Verification of set collapsing: In the right of Table 3, we report the average
circular variance of embedding set. Formally, the circular variance of
embedding set $\mathbf{S}$ is denoted as
$1-\|\sum_{x\in\mathbf{S}}x/|\mathbf{S}|\|_{2}$. As discussed earlier, results
show that employing PIE-Net or MP leads to set collapsing. Moreover, we report
the result when aggregation block is replaced to transformer. This variant
results in a 13$\%$ lower variance on a linear scale and inferior performance
since it does not guarantee disentanglement between elements.
$K$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
RSUM | 492.6 | 495.5 | 497.4 | 500.8 | 498.4 | 499.3
$T$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
RSUM | 492.6 | 497.8 | 499.3 | 500.8 | 499.4 | 499.1
$\alpha$ | 1 | 2 | 4 | 8 | 16 | 32 | 64
---|---|---|---|---|---|---|---
RSUM | 495.2 | 497.2 | 497.7 | 498.9 | 500.8 | 499.3 | 499.0
Table 4: Impact of hyperparameters for the proposed similarity function and
set prediction module.
### 4.7 Impact of Hyperparameters
In Table 4, we report RSUM of the model while varying the cardinality of
embedding sets $K$, the total number of iterations $T$, and the scaling
parameter $\alpha$. Results demonstrate that the model shows consistently high
RSUM when $K>3$, $T>2$, and $\alpha>4$. Specifically, we notice that when
$K=1$, which is identical to using a single embedding vector, the model shows
significantly lower accuracy compared to others. It suggests that embedding
sets enable a more accurate retrieval by addressing the semantic ambiguity.
When $T=1$, we observe substantial degradation of accuracy, which underpins
that progressive refinement done by multiple aggregation blocks helps element
slots aggregate semantic entities, as shown in Figure 2.
### 4.8 Analysis of Embedding Set Elements
For further analysis on the embedding set elements, we use only one of the
elements during the evaluation of the model. Let $\mathbf{S}(i)$ be an element
of $\mathbf{S}$ produced by the $i$-th element slot. Table 5 summarizes the
accuracy of the model in terms of RSUM when elements of
$\mathbf{S}^{\mathcal{V}}$ and $\mathbf{S}^{\mathcal{T}}$ are ablated during
evaluation. Results present that using only one of the elements degrades
accuracy. Specifically, using only $\mathbf{S}^{\mathcal{V}}(2)$ during
evaluation leads to far lower accuracy compared to using the other elements.
To check whether $\mathbf{S}^{\mathcal{V}}(2)$ is just a noisy element that
hinders final performance, we also report RSUM when only
$\mathbf{S}^{\mathcal{V}}(2)$ is ablated. However, interestingly, we still
observed the degraded accuracy compared to using a complete embedding set.
Though one of the elements often retrieves non-matching samples, the results
demonstrate that using them together during evaluation helps the model find
accurate matching samples.
We hypothesize that $\mathbf{S}^{\mathcal{V}}(2)$ is trained to capture highly
ambiguous semantics. Therefore, using them alone during evaluation leads to
less accurate retrievals, while it improves performance when used together by
successfully representing ambiguous situations and context. A similar tendency
could be observed in Figure 4, where one of the slots often encodes an element
that is located nearby to a related but highly ambiguous caption in the
embedding space.
Evaluation |
---|---
$\mathbf{S}^{\mathcal{V}}(1)$ | $\mathbf{S}^{\mathcal{V}}(2)$ | $\mathbf{S}^{\mathcal{V}}(3)$ | $\mathbf{S}^{\mathcal{V}}(4)$ | RSUM
✓ | ✓ | ✓ | ✓ | 500.8
✓ | | | | 491.1
| ✓ | | | 309.6
| | ✓ | | 484.9
| | | ✓ | 486.0
✓ | | ✓ | ✓ | 500.2
Evaluation |
---|---
$\mathbf{S}^{\mathcal{T}}(1)$ | $\mathbf{S}^{\mathcal{T}}(2)$ | $\mathbf{S}^{\mathcal{T}}(3)$ | $\mathbf{S}^{\mathcal{T}}(4)$ | RSUM
✓ | ✓ | ✓ | ✓ | 500.8
✓ | | | | 481.9
| ✓ | | | 483.0
| | ✓ | | 481.7
| | | ✓ | 497.2
Table 5: RSUM on the Flickr30K dataset when elements of the
$\mathbf{S}^{\mathcal{V}}$(left) and $\mathbf{S}^{\mathcal{T}}$(right) are
ablated during evaluation.
## 5 Conclusion
We propose the novel set embedding framework for cross-modal retrieval,
consisting of the set prediction module and smooth-Chamfer similarity. The
proposed set prediction module outputs an embedding set that successfully
captures ambiguity, while smooth-Chamfer similarity successfully resolves
undesirable effects of the existing similarity functions. As a result, our
model surpasses most of the previous methods, including methods involving
higher computation costs, on COCO and Flickr30K datasets. We will extend our
model for the different modalities and tasks in the future.
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Appendix
## Appendix A Implementation details
Our model is implemented with PyTorch [41] v 1.8.1. Automatic mixed precision
is used for faster and more efficient training. Additional to implementation
details provided in the main paper, settings for training differ by which
feature extractors are used.
ResNet-152 + bi-GRU: In this setting, the model is trained for 120 epochs, and
the learning rate for the set prediction module is scaled with $0.1$ on both
datasets. The learning rate decays by a multiplicative factor of $0.1$ for
every 10 epochs. Following [47], the entire model is trained in an end-to-end
manner, but CNN is fixed for the first 50 epochs. Training is done with a
single RTX 3090 GPU.
Faster-RCNN + bi-GRU: The learning rate for the set prediction module is
scaled with $0.1$ and $0.05$ on COCO and Flickr30K, respectively. Following
[28], size augmentation is used, which drops 20% of ROI features and words
during training. Training is done with a single RTX 3090 GPU.
ResNeXt-101 + BERT: In this setting, we construct a batch with 128 images and
their entire matching captions. The learning rate for the set prediction
module is scaled with $0.1$. The model is trained for 50 epochs, where the
learning rate decays by a multiplicative factor of $0.1$ for every 20 epochs.
Following [28], the learning rates for CNN and BERT are scaled with $0.01$ and
$0.1$, respectively. The statistics of the batch normalization [26] layer are
fixed during training. The entire model is trained in an end-to-end manner
except for CNN in the first epoch. In the first epoch, triplet loss without
mining is used, whereas the hardest negative mining is used for later epochs.
Training is done with two A100 PCIe GPU.
## Appendix B Additional qualitative results
In Figure A1 and Figure A2, we present additional visualization of attention
map from the visual set prediction module $f^{\mathcal{V}}$. Visualizations of
attention maps, including ones presented in the main paper, are obtained from
the model using ResNeXt + BERT feature extractors. Attention maps from each
iteration are presented together, where $t=4$ is the last iteration. For each
attention map from the last iteration, its corresponding element of embedding
set and nearest caption are provided together. Results show that the
aggregation block produces heterogeneous attention maps capturing various
semantics, such as different objects (1st row of Figure A1) and action (2nd
row of Figure A2). Moreover, in every case, we can observe that element slots
are progressively updated to capture distinctive semantics, starting from
sparse and noisy attention maps.
Specifically, in Figure A2, we present the examples where the nearest captions
of multiple elements are the same. For instance, in the 2nd row of Figure A2,
each element attends to individual entities (sky, larger giraffe, grassy area,
and baby giraffe), but their nearest captions, which describe the entire
scene, are the same. Results imply that by fusing element slots with the
global feature, elements of the embedding set can preserve the global context
while focusing on distinctive semantics. These characteristics help model when
samples with little ambiguity are given, such as a caption describing the
entire scene (2nd row of Figure A2) or an image containing a single iconic
entity (3rd row of Figure A2).
Figure A1: For each element of the image embedding set, we present its
attention map and the caption nearest to the element in the embedding space.
Matching captions are colored in green. Entities corresponding to the
attention maps are underlined.
Figure A2: For each element of the image embedding set, we present its
attention map and the caption nearest to the element in the embedding space.
Matching captions are colored in green. Entities corresponding to the
attention maps are underlined.
|
# GeoUDF: Surface Reconstruction from 3D Point Clouds via
Geometry-guided Distance Representation
Siyu Ren1,2 Junhui Hou1 Xiaodong Chen2 Ying He3 Wenping Wang4
1City University of Hong Kong 2Tianjin University
3 Nanyang Technological University 4Texas A&M University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author. This work was
supported by the Hong Kong Research Grants Council under Grants 11202320 and
11219422.
###### Abstract
The recent neural implicit representation-based methods have greatly advanced
the state of the art for solving the long-standing and challenging problem of
reconstructing a discrete surface from a sparse point cloud. These methods
generally learn either a binary occupancy or signed/unsigned distance field
(SDF/UDF) as surface representation. However, existing SDF/UDF-based methods
usually use neural networks to implicitly regress the distance in a purely
data-driven manner, thus limiting the accuracy and generalizability to some
extent. In contrast, we propose a geometry-guided method for UDF and its
gradient estimation that explicitly formulates the unsigned distance of a
query point as the learnable affine averaging of its distances to the tangent
planes of neighbouring points. Besides, we model the local geometric structure
of the input point clouds by explicitly learning a quadratic polynomial for
each point. This not only facilitates upsampling the input sparse point cloud
but also naturally induces unoriented normal, which further augments UDF
estimation. Finally, to extract triangle meshes from the predicted UDF we
propose a customized edge-based marching cube module. We conduct extensive
experiments and ablation studies to demonstrate the significant advantages of
our method over state-of-the-art methods in terms of reconstruction accuracy,
efficiency, and generalizability. The source code is publicly available at
https://github.com/rsy6318/GeoUDF.
## 1 Introduction
Surface reconstruction from point clouds is a fundamental and challenging
problem in 3D vision, graphics and robotics [17, 26, 9]. Traditional
approaches, such as Poisson surface reconstruction [18], compute an occupancy
or signed distance field (SDF) by solving Poisson’s equation, resulting in a
sparse line system. Then they utilize the Marching Cubes (MC) [22] algorithm
to extract iso-surfaces as the reconstructed meshes. These methods are
computationally efficient, scalable, resilient to noisy data and tolerant to
registration artifacts. However, they work only for oriented point samples.
Recent deep learning methods can learn implicit fields, such as binary
occupancy fields (BOFs) and SDFs, directly from raw point clouds, making them
an ideal tool for real-world scans, of which orientation is often unavailable.
In spite of many advantages, BOF and SDF-based approaches can only deal with
watertight and manifold surfaces, lacking the ability to represent more
general ones, such as those with open boundaries or interior structures. In
contrast, the unsigned distance field (UDF) is more powerful in terms of
representation ability, and can overcome the above-mentioned limitations of
BOFs and SDFs. The existing deep UDF-based methods, such as [11, 38, 16],
employ neural networks to regress UDFs from input point clouds. Due to their
reliance on the training data, these methods have limited generalizability and
cannot deal with unseen objects well. Another technical challenge diminishing
UDFs’ practical usage is that one cannot use the conventional MC to extract
iso-surfaces directly because UDFs do not have zero-crossings and thus do not
provide the required sign-change information of inside or outside. Some
methods [16, 40] locally convert a UDF to an SDF in a cubic cell and then
apply MC to extract the surface. However, they often fail at corners or edges.
Recently, GIFS [38] modifies MC by predicting whether the surface interacts
with each cube edge via a neural network.
In this paper, we propose GeoUDF, a new learning-based framework
reconstructing the underlying surface of a sparse point cloud. We particularly
aim at addressing two fundamental issues of UDFs, namely, (1) how to predict
an accurate UDF and its gradients for arbitrary 3D inputs? and (2) how to
extract triangle meshes, given a UDF? More importantly, the constructed
methods should be well generalized to unseen data. To solve the first issue,
we propose a geometry-guided learning module, adaptively approximating the UDF
and its gradient of a point cloud by leveraging its local geometry in a
decoupled manner. As for the second issue, we propose an edge-based MC
algorithm, extracting triangle mesh from any UDF according to the intersection
relationship between the surface and the connections of any two vertices of
the cubes. In addition, we propose to enhance input point cloud quality by
explicitly and concisely modeling its local geometry structure, which is
beneficial for surface reconstruction. The elegant formulations distinguish
our GeoUDF from existing methods, making it more lightweight, efficient, and
accurate and better generalizability. Besides, the proposed modules contained
in GeoUDF can be independently used as a general method in its own right.
Extensive experiments on various datasets show the significant superiority of
our method over state-of-the-art methods.
In summary, the main contributions of this paper are three-fold:
* •
novel accurate, compact, efficient, and explainable learning-based UDF and its
gradient estimation methods that are decoupled;
* •
a concise yet effective learning-based point cloud upsampling and
representation method; and
* •
a general method for extracting triangle meshes from any unsigned distance
field.
## 2 Related Work
Figure 1: The pipeline of GeoUDF, a lightweight, explainable, and efficient
learning-based surface reconstruction framework that produces 3D meshes with
high accuracy and strong generalizability.
Learning Implicit Surface Representation. With the development of deep
learning, many advances have been made for the implicit representation of 3D
shapes in recent years. For example, Binary Occupancy Field (BOF) casts the
problem into binary classification, i.e., any point in the 3D space can be
classified as inside or outside of a closed shape. ONet [25] uses a latent
vector to represent the whole input, then uses a decoder to get the occupancy
of a query point. Subsequently, IF-NET [10] and CONet [29] adopt convolutions
on voxelized features, which increase the reconstruction accuracy. Referring
to SPSR [18], SAP [28] introduces a differentiable formulation of SPSR and
then incorporates it into a neural network to reconstruct the surfaces from
point clouds. Recently, POCO [5] designs an attention-based convolution and
further improves the reconstruction quality. SDFs, distinguishing interior and
exterior by assigning a sign to the distance, are a more accurate
representation. DeepSDF [27] uses an auto-decoder to optimize the latent
vector in order to refine the SDF. Based on DeepSDF, DeepLS [7] uses a grid to
store the independent latent codes, each only responsible for a local area.
Neural-Pull [1] and OSP [23] concentrate on the differential property and use
the gradient of the SDF to pull the point onto the surface, where they
estimate and refine the SDF. Some traditional methods can also be adapted
through neural networks, such as DeepIMLS [21] and DOG [34], which incorporate
implicit moving least-squares (IMLS) [19] into neural networks to estimate the
SDF of a point cloud.
BOF and SDF can only deal with the watertight shapes because they assume a
surfacee that partition the whole space into inside and outside, making it
impossible to represent general shapes, such as those with open surface with
boundaries or non-manifold surfaces. UDF can represent general shapes since it
represents the (unsigned) distance from a query point to the surface. NDF [11]
and GIFS [38] use 3D convolution to regress the UDF from the voxelized point
cloud. CAP-UDF [40] employs a field consistency constraint to get consistency-
aware UDF. To our best knowledge, there are no methods learning a UDF from
geometric information explicitly.
Surface Extraction from Implicit Fields. After learning implicit fields, such
as BOFs or SDFs, the MC algorithm [22] is commonly used to obtain a triangle
mesh. However, MC cannot be applied to UDFs because there is no inside/outside
information. NDF [11] uses the gradient of the UDF to generate a much denser
point cloud and then uses ball-pivoting algorithm [3] to extract the triangle
mesh, which inefficient with poor surface quality. GIFS [38] not only predicts
the UDF, but also determines whether any two vertices of the cubes are
separated by the surface, which may be regarded as a binary classification.
Then it modifies the MC to extract a triangle mesh from the UDF. Note that the
Marching Cubes method modified by GIFS cannot be applied to any UDF because it
needs an extra network to achieve do the binary classification on each edge of
the cube. MeshUDF [16] uses one vertex of the cube as a reference and computes
the inner product of the gradients of UDF with the other seven points, thus
converting UDF to SDF in this cube according to the sign of the inner product.
But such a criterion is not robust enough and often predicts wrong faces at
the edges or corners.
Learning-based Point Cloud Upsampling. Point cloud upsampling (PU) aims to
densify sparse point clouds to recover more geometry details, which will be
beneficial to downstream surface reconstruction. As the first learning-based
method for point cloud upsampling, PU-Net [39] employs PointNet++ [30] to
extract features and then expands the features through multi-branch MLPs to
achieve the upsampling. Furthermore, the generative adversarial network-based
PU-GAN [20] achieves impressive results on non-uniform data, but the network
is difficult to train. PU-GCN [31], a graph convolutional network-based
approach, introduces the NodeShuffle module into the existing upsampling
pipeline. The first geometry-centric method PUGeo-Net [32] links deep learning
with differentiable geometry to learn the local geometry structure of point
clouds. MPU [36] appends a 1D code $\\{-1,~{}1\\}$ to separate replicated the
point-wise features. Built upon the linear approximation theorem, MAFU [33]
adaptively learns the interpolation weights in a flexible manner, as well as
high-order approximation errors. Alternatively, PU-Flow [24] conducts
learnable interpolation in feature space regularized by normalizing flows.
Recently, NP [15] adopts a neural network to represent the local geometric
shape in a continuous manner, embedding more shape information.
## 3 Proposed Method
Overview. As shown in Fig. 1, our GeoUDF consists of three modules, i.e.,
local geometry representation (LGR), geometry-guided UDF estimation (GUE), and
edge-based marching cube (E-MC). Specifically, given a sparse 3D point cloud,
we first model its local geometry through LGR, producing a dense point cloud
associated with un-oriented normal vectors. Then, we predict the unsigned
distance field of the resulting dense point cloud via GUE from which we
customize an E-MC module to extract the triangle mesh for the zero level set.
Note that our GeoUDF works for both watertight and open surfaces with interior
structures. Besides, each of the three modules can be independently used as a
general method in its own right. In what follows, we will detail each module.
### 3.1 Local Geometry Representation
Problem Formulation. Let
$\mathcal{P}=\\{\mathbf{p}_{i}|\mathbf{p}_{i}\in\mathbb{R}^{3}\\}_{i=1}^{N}$
be an input sparse point cloud of $N$ points sampled from a surface
$\mathcal{S}$ to be reconstructed. Based on differential geometry theory of
surfaces [14], the local neighborhood of a regular point of a surface is
completely determined by the first and second fundamental forms. Therefore we
use a quadratic polynomial surface to approximate the local surface centered
at each point:
$f_{i}(\mathbf{u})=\mathbf{p}_{i}+\mathbf{A}_{i}\texttt{E}(\mathbf{u}),\vspace{-0.15cm}$
(1)
where $\mathbf{u}=[u_{1},u_{2}]^{\mathsf{T}}\in\mathbb{R}^{2}$ is the
coordinate in the 2D local parameter domain, $\texttt{E}(\mathbf{u}):=[1\
u_{1}\ u_{2}\ u_{1}^{2}\ u_{1}u_{2}\
u_{2}^{2}]^{\mathsf{T}}\in\mathbb{R}^{6}$, and
$\mathbf{A}_{i}\in\mathbb{R}^{3\times 6}$ is the coefficient matrix.
With this representation, we can perform the following two operations, which
will bring benefits to the subsequent module. (1) Densifying $\mathcal{P}$.
For each point $\mathbf{p}_{i}\in\mathcal{P}$, we can uniformly sample $M$ 2D
coordinates from a pre-defined local paramterization
$\mathcal{D}=[-\delta,\delta]^{2}\subset\mathbb{R}^{2}$, which are then
substituted into Eq. (1) to generate additional $M$ 3D points around
$\mathbf{p}_{i}$, producing a dense point cloud
$\mathcal{P}_{M}=\\{\mathbf{p}_{j}|\mathbf{p}_{j}\in\mathbb{R}^{3}\\}_{j=1}^{NM}$.
(2) Inducing un-oriented normal vectors of $\mathcal{P}_{M}$. From the
explicit parameterization function in Eq. (1), we can calculate the un-
oriented normal vector for each point of $\mathcal{P}_{M}$, denoted as
$\mathcal{N}_{M}=\\{\mathbf{n}_{j}|\mathbf{n}_{j}\in\mathbb{R}^{3},\|\mathbf{n}_{j}\|=1\\}$,
based on the differential geometry property [2]. We refer the readers to the
supplementary material for details.
Learning-based Implementation. As depicted in Fig. 2, we design a sub-network
to realize this representation process in a data-driven manner, which is
boiled down to predicting the coefficient matrix
$\\{\mathbf{A}_{i}\\}_{i=1}^{N}$. Specifically, we adopt 3-layer EdgeConvs
[35], which can capture structural information of point cloud data, to embed
$\mathcal{P}$ into a high-dimensional feature space, generating point-wise
features $\\{\mathbf{c}^{(l)}_{i}\in\mathbb{R}^{d_{1}}\\}_{i=1}^{N}$ at the
$l$-th layer. Then, we concatenate the features of three layers and feed them
into an MLP to predict 18-dimension vectors, which are further reshaped into
$\\{\mathbf{A}_{i}\\}_{i=1}^{N}$. Besides, in Sec. 4.3, we experimentally
demonstrate the advantages of this concise operation over state-of-the-art
point cloud upsampling methods.
Figure 2: Flowchart of the LGR module. Note that the 2D coordinates are
randomly sampled from $\mathcal{D}$ via PDS [6] at each iteration. We refer
readers to the supplementary material for the detailed network architecture.
### 3.2 Geometry-guided UDF Estimation
This module aims to estimate the unsigned distance field of $\mathcal{P}_{M}$,
where the iso-surface with the value of zero indicates the surface. As
aforementioned, existing learning-based UDF estimation methods, such as NDF
[11] and GIFS [38], utilize a neural network to implicitly regress the UDFs
from point clouds, thus limiting their accuracy. Besides, the trained models
often yield poor results on unseen data. In sharp contrast to the regression-
based methods, GUE leverages the inherent geometric property of input point
clouds, leading to a more accurate UDF estimation method with better
generalizability.
(a)
(b)
(c)
(d)
Figure 3: Illustration of the UDF and its gradient estimation processes. (a)
Un-oriented normal vectors. (b) Aligned normal vectors. (c) UDF estimation.
(d) UDF gradient estimation.
Formulation of UDF and its Gradient Estimation. Given a query point
$\mathbf{q}\in\mathbb{R}^{3}$, we can find its $K$ nearest points from
$\mathcal{P}_{M}$ in a Euclidean distance sense, denoted as
$\Omega(\mathbf{q})=\\{\mathbf{p}_{k}|\mathbf{p}_{k}\in\mathcal{P}_{R}\\}_{k=1}^{K}$.
Denote by $\mathbf{n}_{k}$ the un-oriented normal of $\mathbf{p}_{k}$. Before
calculating the UDF, as shown in Fig. 3(b), we first align
$\\{\mathbf{n}_{k}\\}_{k=1}^{K}$ with reference to $\mathbf{q}$ via
$\mathbf{\tilde{n}}_{k}(\mathbf{q})=\texttt{sgn}\left(\langle\mathbf{n}_{k},\mathbf{\tilde{p}}_{k}\rangle\right)\mathbf{n}_{k},\vspace{-0.15cm}$
(2)
where $\mathbf{\tilde{p}}_{k}:=\mathbf{q}-\mathbf{p}_{k}$,
$\langle\cdot,\cdot\rangle$ computes the inner product of two vectors, and
$\texttt{sgn}(\cdot)$ extracts the sign of the input. Eq. (2) makes
$\mathbf{\tilde{n}}_{k}$ align with $\mathbf{\tilde{p}}_{k}$, i.e.,
$\langle\mathbf{\tilde{n}}_{k},\mathbf{\tilde{p}}_{k}\rangle>0$.
Denote by $\mathbf{p}^{*}$ the point on surface $\mathcal{S}$ closest to
$\mathbf{q}$, and $\mathbf{\tilde{n}}^{*}$ its aligned normal vector. Assume
the points of $\Omega(\mathbf{q})$ are exactly distributed on $\mathcal{S}$,
and $\\{\tilde{\mathbf{n}}_{k}\\}_{k=1}^{K}$ are accurate, i.e, they are
perpendicular to $\mathcal{S}$. As shown in Fig. 3(c), as $\Omega(\mathbf{q})$
generally covers a small region/area, $\mathbf{p}^{*}$ should be close to the
tangent plane of any $\mathbf{p}_{k}$. Based on this observation, we propose
to approximate the UDF of $\mathbf{q}$, denoted as $\texttt{U}(\mathbf{q})$,
using the weighted sum of the distances from $\mathbf{q}$ to the tangent
planes:
$\texttt{U}(\mathbf{q})\approx\Phi(\mathbf{q}):=\sum_{\mathbf{p}_{k}\in\Omega(\mathbf{q})}w_{1}(\mathbf{q},\mathbf{p}_{k})\cdot\langle\mathbf{\tilde{n}}_{k},\mathbf{\tilde{p}}_{k}\rangle,\vspace{-0.2cm}$
(3)
where the weights $\\{w_{1}(\mathbf{q},\mathbf{p}_{k})\\}_{k=1}^{K}$ are non-
negative and satisfy partition of unity
$\sum_{k=1}^{K}w_{1}(\mathbf{q},\mathbf{p}_{k})=1$.
Figure 4: Flowchart of the GUE module. We refer readers to the supplementary
material for the detailed network architecture.
Similarly, we propose to approximate the gradient of $\texttt{U}(\mathbf{q})$,
denoted as $\nabla\texttt{U}(\mathbf{q})$, using the continuity property of
the normal vectors of a surface and the fact
$\nabla\texttt{U}(\mathbf{q})=\mathbf{\tilde{n}}^{*}$. See Fig. 3(d). We use
$\mathbf{\tilde{n}}_{k}$, each of which is around $\mathbf{\tilde{n}}^{*}$, to
approximate $\nabla\texttt{U}(\mathbf{q})$ as
$\nabla\texttt{U}(\mathbf{q})\approx\Theta(\mathbf{q}):=\frac{\sum_{\mathbf{p}_{k}\in\Omega(\mathbf{q})}w_{2}(\mathbf{q},\mathbf{p}_{k})\cdot\tilde{\mathbf{n}}_{k}}{\|\sum_{\mathbf{p}_{k}\in\Omega(\mathbf{q})}w_{2}(\mathbf{q},\mathbf{p}_{k})\cdot\tilde{\mathbf{n}}_{k}\|},\vspace{-0.15cm}$
(4)
where the weights $\\{w_{2}(\mathbf{q},\mathbf{p}_{k})\\}_{k=1}^{K}$ are non-
negative and satisfy $\sum_{k=1}^{K}w_{2}(\mathbf{q},\mathbf{p}_{k})=1$. We
refer readers to the supplementary material for the detailed derivation
process.
Learning the Weights. Based on the above formulation, the problem of UDF
estimation is boiled down to obtaining
$\\{w_{1}(\mathbf{q},\mathbf{p}_{k}),~{}w_{2}(\mathbf{q},\mathbf{p}_{k})\\}_{k=1}^{K}$.
As illustrated in Fig. 4, we construct a sub-network to learn them adaptively.
Intuitively, the values of the weights should be relevant to the relative
position between $\mathbf{q}$ and $\mathbf{p}_{k}$ and the overall shape of
$\Omega(\mathbf{q})$. Thus, we embed $\mathbf{\tilde{p}}_{k}$ and
$\mathbf{\tilde{n}}_{k}$ via an MLP to obtain point-wise features for all
neighbors, $\mathbf{F}_{k}(\mathbf{q})$, which are then maxpooled, leading to
a global feature, $\mathbf{F}(\mathbf{q})$, to encode the overall shape of
$\Omega(\mathbf{q})$. Finally, we concatenate the above-mentioned features and
feed them into two separated MLPs followed with the Softmax layer to regress
the two sets of weights.
Remark. Although $\Theta(\mathbf{q})$ could be calculated through the back-
propagation (BP) of the neural network (the NDF method [11] adopts this
manner), such a process is time-consuming. Compared with BP-based UDF gradient
estimation, our method is more efficient. Besides, owing to the separate
estimation processes, the UDF error would not be transferred to its gradient,
making it more accurate. To the best of our knowledge, this is the first to
decouple the estimation processes of UDF and its gradient. See the
experimental comparison in Sec. 4.3.
### 3.3 Edge-based Marching Cube
Inspired by the classic marching cube algorithm [22] that extracts triangles
in a cube according to the occupancy of its eight vertices, we propose edge-
based marching cube (E-MC) to extract triangle meshes from the predicted
unsigned distance field in the preceding module. Generally, we first detect
whether the connection between any pair of vertices of a cube intersects with
the surface, then find the most matched condition with the detection results
from the lookup table of Marching Cube.
(a)
(b)
(c)
(d)
Figure 5: The position relationship between the surface and edge. (a) The
surface interacts with the edge. (b)(c) The surface does not interact with the
edge. (d) One vertex of the edge is extremely close to the surface. See the
supplementary material for comprehensive examination of different cases.
Edge Intersection Detection. As illustrated in Fig. 5, if the surface
interacts with the connection between vertices $\mathbf{q}_{1}$ and
$\mathbf{q}_{2}$, denoted as $\mathbf{q}_{1}\mathbf{q}_{2}$, the following
constraints must be satisfied:
$\displaystyle\langle\Theta(\mathbf{q}_{1}),\Theta(\mathbf{q}_{2})\rangle<0,$
$\displaystyle\langle\Theta(\mathbf{q}_{1}),\overrightarrow{\mathbf{o}\mathbf{q}_{2}}\rangle>0,~{}~{}\langle\Theta(\mathbf{q}_{2}),\overrightarrow{\mathbf{o}\mathbf{q}_{1}}\rangle>0,\vspace{-0.3cm}$
(5)
where $\mathbf{o}$ is the midpoint between $\mathbf{q}_{1}$ and
$\mathbf{q}_{2}$. Specifically, the first constraint indicates the directions
of the UDF gradients of $\mathbf{q}_{1}$ and $\mathbf{q}_{2}$ must be
opposite, i.e., the angle between them is larger than $90^{\circ}$. The last
two constraints ensure that $\mathbf{q}_{1}$ and $\mathbf{q}_{2}$ are located
at different sides of an identical part of a surface to eliminate the case
shown in Fig. 5(c). Once the above three constraints are satisfied, the
intersection of the surface with $\mathbf{q}_{1}\mathbf{q}_{2}$, denoted as
$\mathbf{v}(\mathbf{q}_{1},\mathbf{q}_{2})$, can be calculate via
$\mathbf{v}(\mathbf{q}_{1},\mathbf{q}_{2})=\frac{\mathbf{q}_{2}\Phi(\mathbf{q}_{1})+\mathbf{q}_{1}\Phi(\mathbf{q}_{2})}{\Phi(\mathbf{q}_{1})+\Phi(\mathbf{q}_{2})}.\vspace{-0.15cm}$
(6)
In addition, if at least one of $\Phi(\mathbf{q}_{1})$ and
$\Phi(\mathbf{q}_{2})$ is less than a small threshold $\tau$, their connection
is determined to interact with the surface, and the vertex with the smaller
UDF is the intersection.
Triangle Extraction. After utilizing the edge intersection detection on the
connections between any two vertices of a cube, we find the most similar
condition to the detection results in the lookup table of Marching Cube to
extract the triangles. We refer the readers to the supplementary material for
more details.
Remark. With reference to a typical vertex of the 3D cube, MeshUDF [16]
converts UDFs to SDFs in a small cube by using only the first constraint of
Eq. (3.3). GIFS [38] utilizes a sub-network to determine whether
$\mathbf{q}_{1}\mathbf{q}_{2}$ intersects the surface, which is not as general
as ours. In Sec. 4.3, we experimentally demonstrate the advantage of our E-MC
over MeshUDF.
### 3.4 Loss Function
We design a joint loss function to train GeoUDF in an end-to-end manner. (1)
Loss for upsampling $\mathcal{L}_{\text{PU}}$. We compute the Chamfer Distance
(CD) between the upsampled point cloud $\mathcal{P}_{M}$ and its ground-truth
dense point cloud to drive the learning of the LGR module. Note that we do not
supervise the normal vectors. (2) Losses for UDF and its gradient learning
$\mathcal{L}_{\rm UDF}$ and $\mathcal{L}_{\rm Grad}$. We use the mean absolute
error between predicted and ground-truth UDFs and the cosine distance between
predicted and ground-truth gradients to supervise the learning of the GUE
module. The overall loss function is finally written as
$\mathcal{L}=\lambda_{1}\mathcal{L}_{\rm PU}+\lambda_{2}\mathcal{L}_{\rm
UDF}+\lambda_{3}\mathcal{L}_{\rm Grad},\vspace{-0.15cm}$ (7)
where $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are hyper-parameters for
balancing the three terms.
## 4 Experiments
Implementation Details. During training, we set the upsampling factor $M=16$.
For each shape, we sampled 2048 query points near the surface in one training
iteration. We set the size of neighbourhood $K=10$. We trained our framework
in two stages: first, we only trained LGR, i.e., $\lambda_{1}=100$ and
$\lambda_{2}=\lambda_{3}=0$, with the learning rate $10^{-3}$ for 100 epochs;
second, we trained the whole network with $\lambda_{1}=100$, $\lambda_{2}=1$,
and $\lambda_{3}=0.1$ for 300 epochs with the learning rate $10^{-4}$. We
conducted all experiments on an NVIDIA RTX 3090 GPU with Intel(R) Xeon(R) CPU.
Method | Clean | Noisy (0.005)
---|---|---
CD $(\times 10^{-2})$ $\downarrow$ | F-Score $\uparrow$ | CD $(\times 10^{-2})$ $\downarrow$ | F-Score $\uparrow$
Mean | Median | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$ | Mean | Median | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$
ONet [25] | $0.894$ | $0.716$ | $0.535$ | $0.756$ | $1.175$ | $1.005$ | $0.256$ | $0.624$
CONet [29] | $0.549$ | $0.489$ | $0.624$ | $0.910$ | $0.454$ | $0.398$ | $0.766$ | $0.936$
SAP [28] | $0.411$ | $0.370$ | $0.841$ | $0.953$ | $0.393$ | $0.358$ | $0.821$ | $0.959$
POCO [5] | 0.321 | 0.283 | 0.913 | 0.971 | 0.369 | 0.337 | 0.863 | 0.965
DOG [34] | $0.429$ | $0.353$ | $0.792$ | $0.956$ | $0.381$ | $0.337$ | $0.814$ | $0.967$
NDF [11] | $0.341$ | $0.320$ | $0.840$ | $0.976$ | $0.431$ | $0.419$ | $0.685$ | $0.961$
GIFS [38] | $0.328$ | $0.276$ | $0.860$ | $0.974$ | $0.418$ | $0.358$ | $0.731$ | $0.958$
Ours | $\boldsymbol{0.234}$ | $\boldsymbol{0.226}$ | $\boldsymbol{0.938}$ | $\boldsymbol{0.992}$ | $\boldsymbol{0.289}$ | $\boldsymbol{0.278}$ | $\boldsymbol{0.893}$ | $\boldsymbol{0.987}$
Table 1: Quantitative comparison of different methods on the 13 classes of the
ShapeNet dataset. (a) Input(b) GT (c) ONet [25] (d) CONet [29] (e) SAP [28]
(f) POCO [5] (g) DOG [34] (h) GIFS [38] (i) Ours Figure 6: Visual comparisons
on the ShapeNet dataset [8]. We refer readers to the supplementary material
for more visual results.
### 4.1 Watertight Surface Reconstruction
Dataset and Metrics. Following previous works [25, 29, 28, 5], we chose 13
classes of the ShapeNet dataset [8] and followed DISN [37] to remove the non-
manifold structures of the original shapes. We split the data into the
train/val/test sets according to 3D-R2N2 [12]. For each shape, we randomly
sampled 3000 points from the surface as the input sparse point cloud. Besides,
we also added the Gaussian noise of standard deviation 0.005 to generate noisy
inputs. To measure reconstruction quality, we followed previous works [25, 29,
28, 5, 38] to randomly sample $10^{5}$ points from each reconstructed surface
to compute CD and F-Score with the threshold of 0.5% and 1% with respect to
ground-truths.
Comparisons. We compared our GeoUDF with state-of-the-art methods, including
ONet [25], CONet [29], SAP [28], POCO [5], DOG [34], NDF [11], and GIFS [38].
For fair comparisons, we trained all methods with the same dataset until the
minimum validation error was reached. We quantitatively compared those methods
in Table 1, where it can be seen that our method outperforms the other methods
in terms of all metrics. Besides, as illustrated in Fig. 6, our GeoUDF can
reconstruct surfaces with sharp edges and correct structures that are closer
to GTs. However, the methods, i.e., ONet, CONet, SAP, and POCO, which divide
the whole space into inside and outside, fail to recover the thin structures,
such as the back of a chair. Although GIFS can reconstruct such thin
structures, its resulting surfaces are not smooth.
(d) Ours (c) GIFS [38] (b) GT (a) Input Figure 7: Visual comparisons on the
MGN dataset [4].
### 4.2 Unseen Non-watertight Surface Reconstruction
Dataset and Metrics. Following GIFS [38], we evaluated our method on the MGN
[4] and the raw ShapeNet car111These raw data were not used in the training of
the experiment in Sec. 4.1. [8] datasets, containing the shapes with
boundaries and interior structures. Besides, we also evaluated our method on
the ScanNet dataset [13], in which the point clouds were collected through an
RGB-D camera from the real scenes. For the MGN dataset, we randomly sampled
3000 points from the surface as the input. As for the shapes and scenes in the
ShapeNet car and ScanNet dataset, we randomly sampled 6000 points from the
surface or the dense point cloud as the input.
Comparisons. To verify the generalizability, for all methods, we used the
trained models on the ShapeNet dataset (watertight shapes) to perform
inference directly. As shown in Tables 2 and 3 and Figs. 7 and 8, our GeoUDF
exceeds GIFS [38] both quantitatively and visually on the MGN and ShapeNet car
datasets. Besides, Fig. 9 visualizes the results of different methods on the
ScanNet dataset, where it can be seen that CONet [29] and POCO [5] fail to
reconstruct the whole scene; GIFS can reconstruct the objects in the scene,
but the details are not well preserved. By contrast, the surfaces by our
GeoUDF contain more details and are closer to GT ones.
(a) Input (b) GT (c) GIFS [38] (d) Ours Figure 8: Visual comparisons on the
ShapeNet cars.
Method | CD ($\times 10^{-2}$) $\downarrow$ | F-Score $\uparrow$
---|---|---
Mean | Media | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$
GIFS [38] | $0.248$ | $0.234$ | $0.951$ | $0.991$
Ours | $\boldsymbol{0.194}$ | $\boldsymbol{0.190}$ | $\boldsymbol{0.975}$ | $\boldsymbol{0.996}$
Table 2: Quantitative comparisons on the MGN dataset.
Method | CD ($\times 10^{-2}$) $\downarrow$ | F-Score $\uparrow$
---|---|---
Mean | Media | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$
GIFS [38] | $0.408$ | $0.346$ | $0.758$ | $0.954$
Ours | $\boldsymbol{0.310}$ | $\boldsymbol{0.271}$ | $\boldsymbol{0.870}$ | $\boldsymbol{0.986}$
Table 3: Quantitative comparisons on the ShapeNet Car dataset. (f) Ours (e)
GIFS [38] (d) POCO [5] (c) CONet [29] (b) GT (a) Input Figure 9: Visual
comparisons of different surface reconstruction methods on the ScanNet dataset
[13].
### 4.3 Ablation Study
Upsampling process. We evaluated the upsampling function of our LGR module on
the ShapeNet [8] dataset with $M=16$ and compared with recent representative
PU methods, i.e., PUGeo-Net [32], MAFU [33], and NP [15]. For a fair
comparison, we retrained the official codes of the compared methods with the
same training data as ours. As listed in Table 4, it can be seen that our
method with the fewest number of parameters achieves the best performance. NP
is much worse than others because its modeling manner makes it hard to deal
with very sparse point clouds. Besides, Fig. 10 visually compares the results
of different methods, further demonstrating the advantage of our method.
Method | # Param | Normal Supervision | CD ($\downarrow$) ($\times 10^{-2}$) | P2F ($\downarrow$) ($\times 10^{-3}$)
---|---|---|---|---
PUGeo [32] | 1.287M | Yes | $0.287$ | $1.309$
MAFU [33] | 1.390M | No | $0.289$ | $1.220$
NP [15] | 0.637M | Yes | $0.493$ | $4.572$
Ours | 0.522M | No | $\boldsymbol{0.245}$ | $\boldsymbol{0.512}$
Table 4: Comparisons of different PU methods ($M=16$). P2F: point-to-face
distance.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 10: Visual comparison of the results by different upsampling methods on
the ShapeNet dataset [8] .
Effectiveness of LGR. We removed LGR from our GeoUDF to reconstruct surfaces
from input sparse point clouds directly. Under this scenario, we estimated the
normal vectors by using the principal component analysis (PCA). Besides, we
also equipped GIFS [38] with our LGR, i.e., input point clouds were upsampled
by our LGR before voxelization. From Table 5, it can be seen that LRG can
boost the reconstruction accuracy of both GIFS and ours, validating its
effectiveness. Besides, as visualized in Fig. 11, without LGR, the
reconstructed surfaces are much worse, i.e., there are many holes caused by
the sparsity of the input point cloud.
Method | L1-CD $(\times 10^{-2})$ $\downarrow$ | F-Score $\uparrow$
---|---|---
Mean | Median | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$
GIFS | 0.328 | 0.276 | 0.860 | 0.974
GIFS w/ LGR | 0.316 | 0.292 | 0.894 | 0.981
Ours w/o LGR | 0.284 | 0.264 | 0.882 | 0.967
Ours | 0.234 | 0.226 | 0.938 | 0.992
Table 5: The reconstruction accuracy of GIFS [38] and our method with and
without LGR.
(a)
(b)
(c)
(d)
Figure 11: Visual results of our method w/ and w/o LGR.
Accuracy of UDF and its gradient estimation. The accuracy of reconstructed
surfaces cannot exactly reflect that of UDFs and corresponding gradients due
to the discrete cubes of MC. Thus, we directly compared the accuracy of UDFs
and gradients estimated by different methods. In addition to NDF [11] and GIFS
[38] for comparison, we also set another baseline (named Regress) by replacing
Eqs. (3) and (4) of our GUE with an MLP to regress UDFs and gradients. As
listed in Table 6, our GUE produces the most accurate UDFs and gradients.
Besides, BP can obtain gradients that are slightly worse than those of our
GUE, but it is much slower. Finally, we validated that the shape encoding (SE)
of $\Omega(\mathbf{q})$ is essential to our GUE, i.e., GUE w/o SE produces
UDFs and gradients with much larger error than GUE.
UDF Estimation | UDF Error $\downarrow$ | Time $\downarrow$ ($\upmu$s/point)
---|---|---
UDF $(\times 10^{-3})$ | Grad $(^{\circ})$
NDF [11] | 2.536 | 13.930 | 1.173
GIFS [38] | 2.439 | - | 4.228
Regress | 0.994 | 8.371 | 1.762
BP | 0.615 | 7.877 | 12.125
GUE w/o SE | 1.453 | 10.707 | 1.271
GUE | 0.615 | 7.237 | 3.405
Table 6: UDF accuracy of different methods. Note that GIFS does not rely on
the gradients of UDFs. “BP” means in our GUE, the gradients of UDFs are
directly obtained by the back-propagation of the UDF estimation network.
Size of $\Omega(\mathbf{q})$. From Table 7, we can conclude that our GeoUDF is
robust to the size of neighborhood used in Eqs. (3) and (4). This is credited
to the manner of learning adaptive weights, i.e., very smaller weights would
be predicted for not important neighbours.
$K$ | CD ($\times 10^{-2}$) $\downarrow$ | F-Score $\uparrow$
---|---|---
Mean | Median | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$
5 | $0.235$ | $0.227$ | $0.937$ | $0.992$
10 | $0.234$ | $0.226$ | $0.938$ | $0.992$
20 | $0.232$ | $0.224$ | $0.939$ | $0.993$
Table 7: The reconstruction accuracy of different $K$-NN sizes.
Superiority of E-MC. We also compared our E-MC with MeshUDF [16], the SOTA
method for extracting triangle meshes from unsigned distance fields. For a
fair comparison, we fed the two methods with identical unsigned distance
fields estimated by our GUE. From Table 8, it can be seen that our E-MC can
extract triangle meshes with higher quality than MeshUDF. Besides, we also
studied how the resolution of the 3D grid affects reconstruction accuracy.
From Table 9, we can see that the reconstruction accuracy gradually improves
with the resolution increasing, which is consistent with the visual results in
Fig. 12, but more times are consumed. Such an observation is fundamentally
credited to the highly-accurate UDFs by our method.
Method | CD ($\times 10^{-2}$) $\downarrow$ | F-Score $\uparrow$
---|---|---
Mean | Median | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$
GUE+MeshUDF [16] | $0.266$ | $0.251$ | $0.916$ | $0.978$
GUE+E-MC | $\boldsymbol{0.234}$ | $\boldsymbol{0.226}$ | $\boldsymbol{0.938}$ | $\boldsymbol{0.992}$
Table 8: Quantitative comparison of our E-MC with MeshUDF [16].
(a)
(b)
(c)
(d)
(e)
(f)
Figure 12: Visual illustration of the effect of the resolution of E-MC on the
reconstructed surface.
Res. | CD ($\times 10^{-2}$) $\downarrow$ | F-Score $\uparrow$ | Time (s) $\downarrow$
---|---|---|---
Mean | Median | $\text{F1}^{0.5\%}$ | $\text{F1}^{1\%}$ | UDF | E-MC
32 | $0.584$ | $0.554$ | $0.646$ | $0.824$ | 0.051 | 3.295
64 | $0.312$ | $0.285$ | $0.860$ | $0.958$ | 0.138 | 4.744
128 | $0.234$ | $0.226$ | $0.938$ | $0.992$ | 0.759 | 15.282
192 | $0.223$ | $0.218$ | $0.949$ | $0.995$ | 2.264 | 32.052
Table 9: The reconstruction accuracy under various 3D grid resolutions used in
E-MC.
### 4.4 Efficiency Analysis
We also evaluated the efficiency of our GeoUDF on the ShapeNet dataset. As
listed in Table 10, our method has the fewest number of parameters while
achieving the highest reconstruction accuracy among all methods, which is
credited to our explicit and elegant formulations to this problem. We also
evaluated the time consumption of our method, demonstrating that all modules
before surface extraction are very efficient. As for the surface extraction
process, due to the global optimization in our E-MC, our method is slower than
ONet, CONet and SAP, but faster than GIFS.
Method | # Param | Time (s) $\downarrow$
---|---|---
Inference | Surface Extraction
ONet [25] | 10.373M | 0.653 | 0.219
CONet [29] | 1.978M | 0.941 | 0.589
SAP [28] | 1.085M | 0.247 | 0.384
POCO [5] | 12.790M | 4.652 | 8.560
DOG [34] | 2.182M | 0.302 | 0.228
GIFS [38] | 3.682M | 27.771 | 26.915
Ours | 0.775M | 0.759 | 15.282
Table 10: Efficiency comparisons. The time of “Inference” refers to the
overall time minus the time for iso-surface extraction.
## 5 Conclusion
We introduced GeoUDF, a new learning-based framework for reconstructing
general surfaces from sparse 3D point clouds. GeoUDF is featured with
lightweight, efficient, accurate, explainable, and generalizability
properties, which are validated by extensive experiments and ablation studies.
The various advantages of our framework are fundamentally credited to the
explicit and elegant formulations of the problems of UDF and its gradient
estimation and local structure representation from the geometric perspective,
as well as the edge-based marching cube.
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# Inverse molecular design and parameter optimization with Hückel theory using
automatic differentiation
Rodrigo A. Vargas–Hernández Chemical Physics Theory Group, Department of
Chemistry, University of Toronto, Toronto, Ontario, M5S 3H6, Canada. Vector
Institute for Artificial Intelligence, 661 University Ave. Suite 710, Toronto,
Ontario M5G 1M1, Canada. current affiliation: Department of Chemistry and
Chemical Biology, McMaster University, 1280 Main Street West, Hamilton,
Ontario, L8S 4M1 Canada<EMAIL_ADDRESS>Kjell Jorner Chemical Physics
Theory Group, Department of Chemistry, University of Toronto, Toronto,
Ontario, M5S 3H6, Canada. Department of Computer Science, University of
Toronto, 40 St. George St, Ontario M5S 2E4, Canada. Department of Chemistry
and Chemical Engineering, Chalmers University of Technology, Kemigården 4,
SE-41258, Gothenburg, Sweden. Robert Pollice Chemical Physics Theory Group,
Department of Chemistry, University of Toronto, Toronto, Ontario, M5S 3H6,
Canada. Department of Computer Science, University of Toronto, 40 St. George
St, Ontario M5S 2E4, Canada. current affiliation: Stratingh Institute for
Chemistry, University of Groningen, Nijenborgh 4, Groningen, 9747 AG, The
Netherlands. Alán Aspuru–Guzik Chemical Physics Theory Group, Department of
Chemistry, University of Toronto, Toronto, Ontario, M5S 3H6, Canada.
Department of Computer Science, University of Toronto, 40 St. George St,
Ontario M5S 2E4, Canada. Department of Chemical Engineering & Applied
Chemistry, 200 College St., University of Toronto, Ontario M5S 3E5, Canada.
Department of Materials Science & Engineering, 184 College St., University of
Toronto, Ontario M5S 3E4, Canada. Vector Institute for Artificial
Intelligence, 661 University Ave. Suite 710, Toronto, Ontario M5G 1M1, Canada.
Lebovic Fellow, Canadian Institute for Advanced Research (CIFAR), 661
University Ave., Toronto, Ontario M5G 1M1, Canada.
###### Abstract
Semi-empirical quantum chemistry has recently seen a renaissance with
applications in high-throughput virtual screening and machine learning. The
simplest semi-empirical model still in widespread use in chemistry is Hückel’s
$\pi$-electron molecular orbital theory. In this work, we implemented a Hückel
program using differentiable programming with the JAX framework, based on
limited modifications of a pre-existing NumPy version. The auto-differentiable
Hückel code enabled efficient gradient-based optimization of model parameters
tuned for excitation energies and molecular polarizabilities, respectively,
based on as few as 100 data points from density functional theory simulations.
In particular, the facile computation of the polarizability, a second-order
derivative, via auto-differentiation shows the potential of differentiable
programming to bypass the need for numeric differentiation or derivation of
analytical expressions. Finally, we employ gradient-based optimization of atom
identity for inverse design of organic electronic materials with targeted
orbital energy gaps and polarizabilities. Optimized structures are obtained
after as little as 15 iterations, using standard gradient-based optimization
algorithms.
## 1 Introduction
Mathematical models that are both predictive and provide insight are a
cornerstone of the physical sciences. However, accurate models for complicated
processes often have no analytical solution and require large computational
resources to solve numerically. At the same time, they also tend to be hard to
interpret, as highlighted by Mulliken’s famous quote ”the more accurate the
calculations became, the more the concepts tended to vanish into thin air”
[1]. Approximate models with problem-specific parameters are therefore used in
practice, but finding optimal values for these parameters can be non-trivial.
Parameter optimization normally requires considerable amounts of reference
data and is done either manually or with algorithms that do not take advantage
of first or higher order derivatives as the corresponding analytical
expressions are often unavailable.
In chemistry, the Schrödinger equation is an archetype of such a mathematical
model that describes the interactions between nuclei and electrons in both
atoms and molecules. However, (near) exact solutions are too computationally
expensive for most molecules of interest. Quantum chemistry is an entire
research field dedicated to finding computationally efficient solutions to the
Schrödinger equation by introducing prudent approximations or reformulations
[2]. One approach that was extremely successful in the early days of quantum
chemistry is the use of so-called semiempirical (SE) approximations [3]. The
central idea is the use of problem-specific parameters to simplify the
mathematical form of the Schrödinger equation. One of the earliest SE models
was Hückel’s method to treat the $\pi$-electrons in organic molecules [4, 5,
6, 7]. Traditionally, the parameters in the Hückel method were derived
manually by human scientists with the aim to reproduce properties for well-
known reference molecules [8, 9], or they were derived from more accurate
calculations [10]. Over the years, the Hückel method has been used for
pedagogical purposes and for obtaining physical insight into problems in
organic chemistry [11] and photochemistry and photophysics [12, 13]. However,
it can also be used as a fast method for the prediction of molecular
properties [14], and for inverse design of molecules with desired target
properties [15, 16].
The recent upsurge in machine learning (ML), and specifically deep neural
networks, created a need for robust and efficient algorithms to co-optimize a
very large number of model parameters for various architectures. This problem
is now solved by automatic differentiation (AD), a technique to evaluate the
derivatives of mathematical expressions via the chain rule [17]. Importantly,
AD removes the need to determine analytic expressions for derivatives and
makes complicated mathematical models amenable to gradient-based optimization,
allowing them to be applied in the same way as general supervised machine
learning models. Regular machine learning approaches like deep neural networks
are meant to be very general mathematical models with a large number of
parameters. Through learning, they can adapt to essentially any problem
provided sufficient training data is available. In contrast, physics-based
mathematical models have expressions that are specific to a certain type of
problem to be solved and feature a much smaller number of parameters.
Implementing physical models such as quantum chemistry within AD frameworks
enables the use of default learning algorithms for parameter optimization with
a potentially much smaller training data requirement. Along these lines,
autodifferentiable versions of Hartree-Fock [18, 19], density functional
theory (DFT) [20, 21, 22, 19], excited state mean-field theory [23]. For semi-
empirical methods [24, 25, 26, 27], and other applications [28, 20, 29, 30,
30, 31, 32, 33, 34, 35, 36, 37], AD has been used to accelerate the
calculation of gradients physical methods and to blend with ML algorithms.
In this work, we developed an auto-differentiable implementation of the Hückel
method, by minimal adaptation of an initially developed NumPy [38] version
into the JAX [39] AD framework. We use this model to demonstrate the ease and
efficiency of parameter fitting based on computational reference data sets for
both excitation energies and molecular polarizabilities, a property calculated
via a second order derivative. Additionally, we demonstrate that our AD model
allows for gradient-based inverse design by regarding the atomic composition
of a molecular system as an adjustable parameter to find molecules with
targeted properties.[16] The corresponding code is made publicly available,
allowing it to be applied to a large variety of chemical problems. As the
Hückel calculations are extremely fast, our workflow allows for facile
development of property-specific models that can be readily used in molecular
generative models that require on the order of 105–106 property evaluations.
The paper is structured as follows: we first present a short introduction on
automatic differentiation and the Hückel model (Sections 2.1 and 2.2).
Following that, we execute inverse design of molecules as a fully
differentiable procedure (Section 3.1) and perform optimization of the Hückel
model parameters using modern gradient-based methods (Section 3.2).
## 2 Methods
### 2.1 Automatic differentiation
Gradients and high-order derivatives are at the core of physical simulations.
For physical models, common approaches to evaluate derivatives of any order
are closed-form solutions, symbolic differentiation, and numerical
differentiation, i.e., finite differences [40, 17]. For any function
represented as a computer program, AD [17] is an alternative way to compute
gradients and higher order derivatives. AD makes use of the chain rule for
differentiation to create a program that computes the gradients during
evaluation. There are two main modes in AD, forward and reverse mode. For
scalar functions, reverse mode is more efficient as differentiation requires a
single evaluation of the function to fully compute the Jacobian. An example of
reverse mode differentiation is the backpropagation algorithm that is used for
training neural networks. For more details about AD, we refer the reader to
Ref. [17].
The optimization of ML models is mostly done with methods that require the
gradient of the loss or error function (${\cal L}$) with respect to the model
parameters ($\boldsymbol{\theta}$), $\nabla_{\boldsymbol{\theta}}{\cal
L}(\boldsymbol{\theta})$. All contemporary ML libraries, e.g., Tensorflow
[41], PyTorch [42] and JAX [39], are built on top of an AD engine which
computes $\nabla_{\boldsymbol{\theta}}{\cal L}(\boldsymbol{\theta})$ for any
ML model. Given the robustness of AD libraries, differentiating physical
models [40] could be done similarly to modern ML algorithms.
### 2.2 Hückel model
The Hückel model, a well-known semi-empirical quantum chemistry model [4, 5,
6, 7], was first proposed to describe the interactions of $\pi$-electrons in
conjugated unsaturated hydrocarbons. In the Hückel model, this interaction is
restricted to electrons centered at nearest neighbour atoms. Generally, the
Hückel model is considered a tight-binding type Hamiltonian (Eq. 1) where the
on-site and hopping parameters are commonly denoted in the literature as
$\alpha_{\ell}$ and $\beta_{\ell,k}$, respectively. The matrix elements of the
Hückel Hamiltonian are given by
$H_{\ell,k}=\begin{cases}\alpha_{\ell},&\ell=k\\\ \beta_{\ell,k},&\ell\text{
and }k\text{ are adjacent }\\\ 0,&\text{ otherwise, }\end{cases}$ (1)
where the $\alpha_{\ell}$ parameters roughly represent the energy of an
electron in a 2p orbital, and the $\beta_{\ell,k}$ parameters describe the
energy of an electron in the bond $\ell-k$. Extensions of the Hückel
Hamiltonian are possible and can, for instance, incorporate distance-
dependence via $\beta_{\ell,k}=\beta^{0}_{\ell,k}\;g(\mathbf{R}_{\ell,k})$
(cf. Section 3.2). For more details, we refer the reader to standard quantum
chemistry textbooks [43, 44].
Notably, any molecular property computed with the Hückel method depends
directly on the $\alpha_{\ell}$ and $\beta_{\ell,k}$ parameters. Therefore, by
tuning their values, one can either construct a more accurate Hückel model for
a given molecule and property (cf. Section 3.2), or search for atomic
compositions that optimize target properties given a preset connectivity (cf.
Section 3.1). In the following sections, we demonstrate how AD can be used to
facilitate both these types of problems.
## 3 Results and Discussion
### 3.1 Inverse molecular design
Inverse molecular design can be carried out via gradient-based optimization
methods, as shown in Ref. [16]. The Hückel model can be extended to search for
the molecular structure with a desired property. Both the diagonal and off-
diagonal elements of the Hückel Hamiltonian matrix can be described by a
weighted average of atom types at each site,
$H_{\ell,k}=\begin{cases}\sum_{i}^{M}b_{\ell}^{i}\alpha^{i}_{\ell},&\ell=k\\\
\sum_{i}^{M}\sum_{j}^{M}b_{\ell}^{i}b_{k}^{j}\beta^{ij}_{\ell,k},&\ell\text{
and }k\text{ are adjacent }\\\ 0,&\text{ otherwise },\end{cases}$ (2)
where $b_{\ell}^{i}$ is the weight of the atom of type $i$ for site $\ell$.
For a meaningful description, the weights of each site must be normalized,
i.e., $\sum_{i}^{M}b_{\ell}^{i}=1$. $M$ is the total number of atom types
considered in the search. As a proof of concept, we consider eight different
molecular frameworks [16], which are displayed in Fig. 1. The $x$-symbol
indicates the sites with variable atom types to be optimized. We only
considered carbon ($C$), nitrogen ($N$) and phosphorus ($P$), i.e., $M=3$, as
these atom types each contribute one electron, assuming that the remaining
valences of carbon will be satisfied with a bond to an implicit hydrogen atom,
and can be incorporated interchangeably at all sites with two neighbors in the
$\pi$-framework (Fig. 1). Therefore we defined the following vector of atom
type weight parameters:
$\mathbf{b}_{\ell}=[b_{\ell}^{C},b_{\ell}^{N},b_{\ell}^{P}]$. For clarity,
$\mathbf{b}$ jointly describes the $\mathbf{b}_{\ell}$ parameters for all
search sites in a molecule, i.e., $\mathbf{b}=\\{\mathbf{b}_{\ell}\\}^{N}$.
For all results presented, the values of the $\alpha_{\ell}$ and
$\beta^{0}_{\ell,k}$ parameters were previously optimized with respect to the
desired property (cf. Section 3.2).
For the set of eight molecules considered (cf. Fig. 1), we search for the type
of atom $\mathbf{b}$ at each site that gives the lowest HOMO-LUMO gap (Eq. 3),
denoted as $\epsilon_{HL}$, and the maximum polarizability denoted as
$\langle\tilde{\alpha}\rangle$, (Eq, 4). $\epsilon_{HL}$ is defined as,
$\epsilon_{HL}=\epsilon_{LUMO}-\epsilon_{HOMO},$ (3)
where $\epsilon_{HOMO}$ and $\epsilon_{LUMO}$ are the eigenvalues of the
highest occupied molecular orbital (HOMO), and lowest unoccupied molecular
orbital (LUMO), respectively. The polarizability function is defined as
$\langle\tilde{\alpha}\rangle=\frac{1}{3}\left(\tilde{\alpha}_{xx}+\tilde{\alpha}_{yy}+\tilde{\alpha}_{zz}\right),$
(4)
where the $\tilde{\alpha}_{ij}$ elements are the polarizability components
defined as
$\tilde{\alpha}_{ij}=-\frac{\partial^{2}E}{\partial F_{i}\partial F_{j}}.$ (5)
The $F_{i}$ terms are the components of the electric field,
$\vec{\boldsymbol{F}}=[F_{x},F_{y},F_{z}]$, and $E$ is the electronic energy
of the system. The elements of the polarizability tensor are usually computed
using a finite-difference (FD) approach [16],
$\tilde{\alpha}_{ii}=\frac{2E(0)-\left[E(-F_{i})+E(+F_{i})\right]}{F_{i}^{2}},$
(6)
where the electronic energy is evaluated several times, typically three times
for each diagonal element, and four times for each cross term. Notably, if the
parameters $\mathbf{b}$ are to be optimized using a gradient-based method, the
Jacobians $\nabla_{\mathbf{b}}\epsilon_{HL}$ and
$\nabla_{\mathbf{b}}\langle\tilde{\alpha}\rangle$ will also be constructed
using an FD approach. However, this will increase the number of energy
evaluations needed, especially for $\langle\tilde{\alpha}\rangle$ as the
elements of $\nabla_{\mathbf{b}}\langle\tilde{\alpha}\rangle$ are third-order
derivatives:
$\frac{\partial}{\partial\mathbf{b}^{i}}\frac{\partial^{2}\tilde{\alpha}_{kk}}{\partial
F^{2}_{k}}$. Thus, for a single element of
$\nabla_{\mathbf{b}}\langle\tilde{\alpha}\rangle$, using FD will require 18
energy calculations, ${\cal O}(18\times\|\mathbf{b}\|)$, where
$\|\mathbf{b}\|$ is the total number of parameters in $\mathbf{b}$. For
$\epsilon_{HL}$, using FD, we only require ${\cal O}(2\times\|\mathbf{b}\|)$,
as $\nabla_{\mathbf{b}}\epsilon_{HL}$ is a first-order derivative.
In contrast, using modern AD frameworks, we can efficiently compute, with a
single energy calculation (i.e., one forward pass), the Jacobian of
$\epsilon_{HL}$ with respect to $\mathbf{b}$. For
$\langle\tilde{\alpha}\rangle$, the number of total energy evaluations depends
on the dimension of the external field to construct the diagonal elements of
the Hessian (Eq. 5), which we also computed using AD. The Jacobian of
$\langle\tilde{\alpha}\rangle$ with respect to $\mathbf{b}$, a third order
derivative, can be constructed from only three energy evaluations using AD
[17], a drastic reduction from the 18 required for FD.
After implementation of the Hückel model using the JAX ecosystem [39], we
could fully differentiate both observables, $\epsilon_{HL}$ and
$\langle\tilde{\alpha}\rangle$. Importantly, using JAX allowed us to convert
our existing Python-based Hückel code very easily by replacing calls to NumPy
with almost equivalent calls to the JAX.Numpy package. For optimization, we
used the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm via the JaxOpt
library [45]. Instead of using a constrained optimization scheme to satisfy
the site-normalization restriction for $\mathbf{b}$, we used the softmax
function,
$b^{i}=\frac{\exp(\underline{b}^{i})}{\sum_{i}^{M}\exp(\underline{b}^{i})},$
(7)
where $\underline{b}$ represents the unnormalized $b$ parameters.
Given the flexibility of the JAX ecosystem, we were able to test other
gradient optimization algorithms such as Adam [46] and canonical gradient-
descent, but found the BFGS to be most efficient as it required, on average,
fifteen or less total iterations to reach convergence (cf. Fig. 2). The Adam
and gradient-descent algorithms, with an exponential learning rate decay, each
needed more than thirty iterations to minimize $\epsilon_{HL}$ or
$\langle\tilde{\alpha}\rangle$. Notably, we initialized the values for all
$\underline{b}^{i}$ parameters by sampling a uniform distribution,
$\underline{b}^{i}\sim{\cal U}(-1,1)$. Instead of using literature Hückel
parameters, we used our optimized parameters for each target observable where
5,000 training molecules were used. More details are described in Section 3.2.
As seen in Figs. 3 and 4, we found that our random initialization of
$\mathbf{\underline{b}}$ allows us to sample a wide range of molecules with a
broad range of values for both objectives, $\epsilon_{HL}^{\mathrm{initial}}$
and $\langle\tilde{\alpha}\rangle_{\mathrm{initial}}$.
Because of the statistical description of the molecules by the $\mathbf{b}$
parameters, the optimal parameters ($\mathbf{b}^{*}$) found by optimization
are not one-hot vectors that correspond to only one atom type per site in the
molecule but rather a linear combination of multiple atom types. We define the
observable value for this unphysical molecule as $y_{\mathrm{virtual}}$ and
that for the real molecule as $y_{\mathrm{feasible}}$ (i.e., the most probable
atom is picked for each site to define the real molecule). An example of this
is displayed for framework 3 in Fig. 5 where we show the change in
$\mathbf{b}$ throughout the optimization for both objectives ($\epsilon_{HL}$
and $\langle\tilde{\alpha}\rangle$). As we can observe, the change in
$\epsilon_{HL}$ from the initial random molecule to the optimal one is close
to 1 eV. The optimizations in Fig. 5 were done using Adam only to properly
illustrate the change in $\mathbf{b}$ as the change between iterations is
smoother and more readily discernable. Importantly, this shows that the
generative model can shift the distribution of properties towards the target
property with little dependence on the random initialization of $\mathbf{b}$
(Figs. 3 and 4). We also observe that, for the majority of the optimized
molecules, $y_{virtual}$ and $y_{feasible}$ are linearly correlated,
indicating that the optimization converged essentially to feasible molecules.
The property distributions of $y_{\mathrm{virtual}}$ and
$y_{\mathrm{feasible}}$ are reasonably close, even in the few cases when the
correlations are poor.
Fig. 6 displays the molecules with the lowest $\epsilon_{HL}$ and maximum
$\langle\tilde{\alpha}\rangle$ from the ensemble of different optimizations.
First, we notice that there is a higher amount of phosphorus atoms in the
molecules when $\langle\tilde{\alpha}\rangle$ was the target property. This is
not unexpected as molecular polarizabilities, while not simply a sum of the
atomic polarizabilities, are strongly influenced by the atomic
polarizabilities of the constituent atoms [47, 48]. As phosphorus is a third-
row element in the same group with nitrogen and atomic polarizability
increases significantly when increasing the row number, its atomic
polarizability, both in free atoms [49] and in molecules [50], is
significantly larger than both nitrogen and carbon. Therefore, incorporating a
large number of phosphorus atoms is expected to be a viable strategy to
maximize the molecular polarizability in all of the molecular frameworks
considered. For the molecules with the lowest $\epsilon_{HL}$, we see
extensive incorporation of both nitrogen and phosphorous atoms. This can be
understood in terms of the effect of heteroatom substitution on
$\epsilon_{HL}$ within the HMO framework [51]. For alternant compounds such as
1–8, $\epsilon_{HL}$ is unaffected by changing $\alpha_{\ell}$. The main
effect comes from changing $\beta_{\ell,k}$, and is expected to be largest for
bonds that feature a bonding interaction in the HOMO and an anti-bonding
interaction in the LUMO. A lowered $\beta_{\ell,k}$ leads to decreased bonding
interactions in the HOMO and consequently, a higher HOMO energy. For the LUMO,
decreasing the antibonding interactions by a lowered $\beta_{\ell,k}$ leads to
a lowering of the energy. The net effect by raising the HOMO and lowering the
LUMO is a decrease in the $\epsilon_{HL}$ gap. We can therefore expect
optimization to favor atom pairs with a low $\beta_{\ell,k}$ for bonds that
feature a bonding interaction in the HOMO and an antibonding interaction in
the LUMO. Inspection of the molecular orbitals of the optimized frameworks
(cf. Figure S1) indeed reveals that these bonds are dominantly between two N
atoms, which feature the by far lowest $\beta_{\ell,k}$ at 0.159 (the next
lowest is for P–P at 0.539). Control optimizations with the original parameter
set by Van-Catledge [10] instead gives molecules with P–P at those bonds (cf.
Figure S2), consistent with the fact that $\beta_{P,P}$ = 0.63 is the lowest
for this parameter set.
For this proof of principle work, we picked two distinct molecular target
properties. However, based on the framework employed, a significant number of
alternative properties could also be predicted and, thus, used for inverse
design via gradient-based optimization. Additionally, any combined objective
that is derived from multiple target properties can equally be optimized for
via the same types of algorithms out of the box. This is particularly
interesting for properties where Hückel models are known to provide reasonable
prediction accuracies such as HOMO-LUMO gaps. The use of gradient-based
optimization algorithms enables fast convergence towards the closest local
optimum solution reducing the number of evaluations and leading to
significantly increased computation time. This is particularly important as
one of the main bottlenecks in current approaches to inverse molecular design
is the number of property evaluations needed to find an optimal structure
[52]. Going beyond single-objective optimization, one possible extension of
our presented approach is targeting multiple objectives via genuine gradient-
based multi-objective optimization, for example, both $\epsilon_{HL}$ and
$\langle\tilde{\alpha}\rangle$. The standard approach to perform multi-
objective optimization is via via property concatenation into a single
function to use standard single-objective algorithms, where algorithms like
Bayesian optimization are used [53, 54, 55, 56, 57]. However, gradient-based
multi-objective optimization algorithms [58] have been developed and they,
together with automatic differentiation, could be employed for both parameter
optimization and inverse molecular design in order to explore the
corresponding Pareto fronts in a systematic manner.
From a conceptual point of view, representing chemical structure subspaces in
a parameterized form can greatly facilitate inverse design [16] as it allows
the use of well-established approaches for parameter optimization to be used
for the design of molecules. This is particularly effective when used in
combination with AD due to its numerical stability and computational
efficiency compared to alternative means to compute gradients. Consequently,
this also makes the molecular size that can still be feasibly treated in such
an approach larger and thus, essentially, expands the chemical subspace the
generative model can explore. However, one of the main downsides of the
approach implemented in this work is the reliance on fixed molecular
frameworks, which is common for alchemical formulations [59] strongly limiting
the structural space considered in the optimization. Simple extensions would
be i) the combination of methods to change the molecular framework without
relying on gradients with the method presented here to modify the atom
identities within the respective framework, or ii) differentiable supermatrix
structure where atom vacancies are allowed [60]. Ideally, future extensions
should aim to find prudent ways allowing for framework modifications based on
gradients as this potentially can lead to a dramatic reduction in the number
of structure optimization steps and, thus, the number of property evaluations
necessary. The extended Hückel model is also compatible with the proposed
methodology, even with ML learned parameters [25], by considering a
description of the overlap integrals between different atoms types, similar to
Eq. 2.
Figure 1: The eight molecular frameworks considered for inverse molecular
design in this work [16]. The $x$-symbol represents atomic sites whose
identity is optimized.
Figure 2: Average learning curve for $\epsilon_{HL}$ (left panel) and
$\langle\tilde{\alpha}\rangle$ (right panel) for 250 random initial molecules
based on the eight different molecular frameworks. We use the BFGS algorithm
to optimize both observables. More details about the random initialization are
provided in the text.
Figure 3: Histograms of the optimized HOMO-LUMO gap, Eq. 3, for 250 random
initial molecules ($\epsilon_{HL}^{\mathrm{initial}}$). The inset panels
compare the similarity between $\epsilon_{HL}$ computed with the value of
$\mathbf{b}$ at the end of the optimization protocol
($\epsilon_{HL}^{\mathrm{virtual}}$), and the values of $\epsilon_{HL}$
selecting the most probable atoms given $\mathbf{b}^{*}$
($\epsilon_{HL}^{\mathrm{feasible}}$). Curves represent the derived histograms
using kernel density estimation, (solid) $\epsilon_{HL}^{\mathrm{feasible}}$,
(dashed) $\epsilon_{HL}^{\mathrm{virtual}}$, and (dotdash)
$\epsilon_{HL}^{\mathrm{initial}}$. All molecules were optimized using the
BFGS algorithm. Molecular frameworks are displayed in Fig. 1.
Figure 4: Histograms of the optimized $\langle\tilde{\alpha}\rangle$, Eq. 4,
for 250 random initial molecules
($\langle\tilde{\alpha}\rangle^{\mathrm{initial}}$). The inset panels compare
the similarity between $\langle\tilde{\alpha}\rangle$ computed with the value
of $\mathbf{b}$ at the end of the optimization protocol
($\langle\tilde{\alpha}\rangle^{\mathrm{virtual}}$), and the values of
$\langle\tilde{\alpha}\rangle$ selecting the most probable atoms given
$\mathbf{b}^{*}$ ($\langle\tilde{\alpha}\rangle^{\mathrm{feasible}}$). Curves
represent the derived histograms using kernel density estimation, (solid)
$\langle\tilde{\alpha}\rangle^{\mathrm{feasible}}$, (dashed)
$\langle\tilde{\alpha}\rangle^{\mathrm{virtual}}$, and (dotdash)
$\langle\tilde{\alpha}\rangle^{\mathrm{initial}}$. All molecules were
optimized using the BFGS algorithm. Molecular frameworks are displayed in Fig.
1.
Figure 5: Change in the parameters $\mathbf{b}$ for $\epsilon_{HL}$ (left
panel) and $\langle\tilde{\alpha}\rangle$ (right panel) during the
optimization of a single random initial virtual molecule based on framework 3.
For the initial, one intermediate and the final $\mathbf{b}$, we plot the
values of $\mathbf{b}$ for molecular framework 3. For both panels, the filled
markers represent the values of $\epsilon_{HL}$ and
$\langle\tilde{\alpha}\rangle$ computed with $\mathbf{b}$ at each iteration
(virtual), and the empty markers represent the observable values computed only
with the most probable atoms given $\mathbf{b}$ at each iteration (feasible).
For each search site we only considered three different atom types, namely
$C$, $N$ and $P$;
$\mathbf{b}_{\ell}=[b_{\ell}^{C},b_{\ell}^{N},b_{\ell}^{P}]$. The optimization
of $\epsilon_{HL}$ and $\langle\tilde{\alpha}\rangle$ w.r.t. $\mathbf{b}$ was
carried out with Adam using a learning rate of 0.2. More details are provided
in the main text.
Figure 6: The molecular structures with the lowest HOMO-LUMO gap and maximum
polarizability for each of the eighth molecules considered (Fig. 1).
### 3.2 Parameter optimization
Another important task that might sometimes be underappreciated in
computational chemistry is model parameter optimization. Here, we leverage the
flexibility of AD and optimize all free parameters of the Hückel model in the
same way as it is done for modern ML algorithms. Originally, the Hückel model
is solely based on electronic interactions between nearest-neighbour atoms,
which is typically also referred to as the tight-binding approximation (cf.
$\beta_{\ell,k}$ parameters in Eq. 1). Beyond the standard Hückel model, one
can introduce atomic distance-dependence of the corresponding interactions via
$\beta_{\ell,k}=\beta^{0}_{\ell,k}g(\mathbf{R}_{\ell,k})$. For example, based
on previous work by Longuett-Higgins and Salem [61], $\beta_{\ell,k}$ has an
exponential dependence on $\mathbf{R}_{\ell,k}$,
$\beta_{\ell,k}^{exp}=-\beta^{0}_{\ell,k}\exp^{-\frac{\Delta
R_{\ell,k}}{y_{\ell,k}}}.$ (8)
A second functional form, which is based on the work of Su, Schrieffer and
Heege [62, 63] on conducting polymers, uses a linear distance-dependence of
the interactions,
$\beta_{\ell,k}^{lr}=-\beta^{0}_{\ell,k}\left(1-y^{-1}_{\ell,k}\Delta
R_{\ell,k}\right).$ (9)
For both expressions (see Eqs. 8–9), $\Delta R_{\ell,k}$ is the difference
with respect to the reference bond length distance $R^{0}_{\ell,k}$, and
$y_{\ell,k}$ is a length scale parameter. By including $R^{0}_{\ell,k}$ and
$y_{\ell,k}$ in the set of parameters for the Hückel model, the complete set
of parameters becomes
$\boldsymbol{\theta}=[\alpha_{\ell},\beta^{0}_{\ell,k},y_{\ell,k},R^{0}_{\ell,k}]$.
For this work, all initial $\alpha_{\ell}$ and $\beta^{0}_{\ell,k}$ parameters
were taken from Van-Catledge [10], and the initial $R^{0}_{\ell,k}$ parameters
were approximated from tables of standard bond lengths [64]. The length scale
parameters ($y_{\ell,k}$) were initially set to 0.3 Å, which corresponds to
the value that has been used for C–C in the literature [65, 66].
We used a subset of the GDB-13 data [67] set that only consists of molecules
with $\pi$-systems for fitting our model parameters (see Supplementary
Material for details on how the dataset was generated). Note that some
molecules in the dataset could have n-$\pi$* transitions as their lowest
excited state. We used a pool of 60,000 molecules and randomly sampled 100,
1,000, and 5,000 molecules from this set, and used 1,000 additional molecules
as validation set to monitor the optimization procedure. To optimize
$\boldsymbol{\theta}$, we used the mean squared error as a loss function,
${\cal
L}(\boldsymbol{\theta})=\frac{1}{2}\sum_{i}^{N}\left(\hat{\epsilon}_{HL}({\cal
M}_{i})-\epsilon_{HL}(\boldsymbol{\theta};{\cal M}_{i})\right)^{2},$ (10)
where ${\cal M}_{i}$ is a single molecule of the training set, and
$\hat{\epsilon}_{HL}$ is the vertical excitation energy between the ground
state and the first excited singlet state computed at the TDA-
SCS-$\omega$PBEPP86/def2-SVP level of theory [68, 69]. At the Hückel level of
theory, this excitation energy simply corresponds to the HOMO-LUMO gap
($\epsilon_{HL}$) due to the disregard for electron correlation. To compare
the prediction of $\epsilon_{HL}$ with the DFT reference values properly, we
linearly transformed the results of the Hückel model using two additional
parameters, $w_{0}$ and $w_{1}$ (Eq. 11),
$\epsilon_{HL}(\boldsymbol{\theta};{\cal
M}_{i})=w_{1}\times\epsilon_{HL}(\alpha_{\ell},\beta^{0}_{\ell,k},y_{\ell,k},R^{0}_{\ell,k};{\cal
M}_{i})+w_{0},$ (11)
where $\boldsymbol{\theta}$ jointly represents all parameters of the model,
i.e.,
$\boldsymbol{\theta}=[\alpha_{\ell},\beta^{0}_{\ell,k},y_{\ell,k},R^{0}_{\ell,k},w_{0},w_{1}]$.
For the optimization of all free parameters, we used the AdamW optimization
algorithm [46], as implemented in the Optax library [70], with a learning rate
of $0.02$, and a weight decay of $10^{-4}$. Notably, we considered various
training scenarios that included different values for the weight decay, and
the regularization of different sets of the Hückel parameters. However, we
found no impact on the accuracy of the model. The initial model parameters
were gathered from Refs. [10, 64].
We optimized the parameters of three different Hückel models, i) the original
one where $\beta_{\ell,k}$ is distance-independent
($\beta_{\ell,k}=\beta^{0}_{\ell,k}$), and both ii) the exponential (Eq. 8),
and iii) the linear (Eq. 9) distance-dependence functional forms. We want to
emphasize that any other analytic form for $\beta_{\ell,k}$ could be
considered as well as AD makes any of these expressions fully differentiable.
Following the convention in the literature, we scaled the parameters
$\beta^{0}_{\ell,k}$ and $\alpha_{\ell}$ with respect to the carbon atom
parameters according to $\alpha_{\ell}=\alpha_{\ell}-\alpha_{C}$, and
$\beta^{0}_{\ell,k}=\beta^{0}_{\ell,k}/\beta^{0}_{C,C}$. Notably, at least for
our results in this work, we found that including a regularization term in the
loss function did not impact the accuracy of the model. Finally, we found 20
epochs to be enough to minimize the loss function when the parameters are
initialized with values from Refs. [10, 64].
In Figures 7–10 we display the optimized values of the parameters for the
three different Hückel models considered. From the optimized parameters, we
observe that $\alpha_{O}$ (i.e., the 2p orbital energy parameter for oxygen),
for all three models, is the one that differs the most from the literature
[10, 64]. While there is no good reference data for $y_{\ell,k}$ to compare
to, we observe, nevertheless, that the C–C parameter value changes
considerably from the initial value of 0.3. For the $\beta^{0}_{\ell,k}$
parameters, only the values for N–C resemble the literature values.
Furthermore, the optimal values of $R^{0}_{\ell,k}$ change the least from the
values in Ref. [64].
Using these optimized parameters, we predicted $\epsilon_{HL}$ for 40,000
additional test set molecules and compare the results with DFT reference data.
The direct comparison is depicted in Figures 11–13. By optimizing all
parameters there is a significant improvement in the prediction of
$\epsilon_{HL}$ using our semi-empirical model. Notably, we also found that
considering a larger training data set does not impact the accuracy of the
Hückel model which suggests that either the corresponding molecules do not
provide any additional information with respect to the relevant interaction
parameters or that the model already is close to its best expected performance
and cannot be improved further. Another important observation in that regard
is that the analytical form of $g(\mathbf{R}_{\ell,k})$ in $\beta_{\ell,k}$
does not impact the accuracy of the model when optimized parameters are used.
Next, we also optimized the parameters with respect to the polarizability for
the distance-independent Hückel model. Here, the gradients needed for training
are of third-order, e.g.,
$\frac{\partial}{\partial\alpha_{\ell}}\frac{\partial^{2}\tilde{\alpha}_{kk}}{\partial
F^{2}_{k}}$ or
$\frac{\partial}{\partial\beta_{k,\ell}}\frac{\partial^{2}\tilde{\alpha}_{kk}}{\partial
F^{2}_{k}}$, and can be computed more efficiently via AD, illustrating the
potential of this approach. The molecular polarizabilities that were used as
reference data were computed using dftd4 (version 3.4.0) [71, 72, 73] via the
default methodology summing atomic polarizabilities. Even though we observe a
higher prediction accuracy when the optimized parameters are used compared to
the model before parameter refinement, as depicted in Fig. 14, the accuracy of
the model still remains relatively low and does not improve anymore when more
training data is used. We suspect that this prediction task is particularly
challenging for the Huc̈kel model as the simulated polarizability only
corresponds to the contribution from $\pi$-electrons, while that of the
reference data accounts for all the electrons in the molecules. Even though we
expect a significant portion of the molecular polarizabilities to stem from
the $\pi$-electrons, the contributions of the $\sigma$-electrons cannot be
neglected and can dominate this property. Nevertheless, this proof-of-concept
application example demonstrates the operational ease of conducting parameter
refinement of a given physics-based prediction model based on reference data,
even when derivative properties are targeted.
Figure 7: Optimized $\alpha_{\ell}$ parameters for different Hückel models, a)
$\beta^{0}_{\ell,k}$, b) $\beta^{lr}_{\ell,k}$ (Eq. 9), and c)
$\beta^{exp}_{\ell,k}$ (Eq. 8). All parameters reported were averaged over ten
different training data sets, and different number of training molecules.
Colored symbols and bars represent the mean and standard deviation of the
optimized parameters averaged over ten different data sets. The reference
parameters ($\blacksquare$-symbol) were taken from Refs. [10], and used as the
initial parameters for all optimizations. We refer the reader to the main text
for the optimization details.
Figure 8: Optimized $\beta^{0}_{\ell,k}$ parameters for different Hückel
models, a) $\beta^{0}_{\ell,k}$, b) $\beta^{lr}_{\ell,k}$ (Eq. 9), and c)
$\beta^{exp}_{\ell,k}$ (Eq. 8). All parameters reported were averaged over ten
different training data sets, and different number of training molecules.
Colored symbols and bars represent the mean and standard deviation of the
optimized parameters averaged over ten different data sets. The reference
parameters ($\blacksquare$-symbol) were taken from Refs. [10], and used as the
initial parameters for all optimizations. We refer the reader to the main text
for the optimization details.
Figure 9: Optimized $y_{\ell,k}$ parameters for different Hückel models, a)
$\beta^{lr}_{\ell,k}$ (Eq. 9), and b) $\beta^{exp}_{\ell,k}$ (Eq. 8). All
parameters reported were averaged over ten different training data sets, and
different number of training molecules. Colored symbols and bars represent the
mean and standard deviation of the optimized parameters averaged over ten
different data sets.The reference parameters ($\blacksquare$-symbol) were set
to 0.3 Å, and used as the initial parameters for all optimizations. We refer
the reader to the main text for the optimization details.
Figure 10: Optimized $R^{0}_{\ell,k}$ parameters for different Hückel models,
a) $\beta^{lr}_{\ell,k}$ (Eq. 9), and b) $\beta^{exp}_{\ell,k}$ (Eq. 8). All
parameters reported were averaged over ten different training data sets, and
different number of training molecules. Colored symbols and bars represent the
mean and standard deviation of the optimized parameters averaged over ten
different data sets. The reference parameters ($\blacksquare$-symbol) were
taken from Refs. [64], and used as the initial parameters for all
optimizations. We refer the reader to the main text for the optimization
details.
Figure 11: $\epsilon_{HL}$ predicted with the Hückel models and with DFT level
for 40K molecules not considered during training. The atom-atom interaction of
the Hückel model is described by a distance-independent parameter,
$\beta^{0}_{\ell,k}$. Results computed with a Hückel with parameters taken
from the literature (a), and parameters optimized with N=100 (b) and N=5,000
(c) data points. We refer the reader to the main text for the optimization
details.
Figure 12: $\epsilon_{HL}$ predicted with the Hückel models and with DFT level
for 40K molecules not considered during training. The atom-atom interaction of
the Hückel model is described by a distance-dependent parameter,
$\beta^{lr}_{\ell,k}$ (Eq. 9). Results computed with a Hückel with parameters
taken from the literature (a), and parameters optimized with N=100 (b) and
N=5,000 (c) data points. We refer the reader to the main text for the
optimization details.
Figure 13: $\epsilon_{HL}$ predicted with the Hückel models and with DFT level
for 40K molecules not considered during training. The atom-atom interaction of
the Hückel model is described by a distance-dependent parameter,
$\beta^{exp}_{\ell,k}$ (Eq. 8). Results computed with a Hückel model with
parameters taken from the literature (a), and parameters optimized with N=100
(b) and N=5,000 (c) data points. We refer the reader to the main text for the
optimization details.
Figure 14: $\langle\tilde{\alpha}\rangle$ predicted with the Hückel models and
with DFT level for 40K molecules not considered during training. The atom-atom
interaction of the Hückel model is described by a distance-independent
parameter, $\beta^{0}_{\ell,k}$. Results computed with a Hückel model with
parameters taken from the literature (a), and (b) parameters optimized with
N=100 data points. We refer the reader to the main text for the optimization
details.
## 4 Conclusions
In this work, we demonstrate the power of automatic differentiation to enable
the efficient use of physics-inspired models for gradient-based optimization
problems in the realm of molecular chemistry via semi-empirical Hückel models.
In particular, we showcase inverse molecular design via an alchemical problem
formulation using fixed molecular frameworks. This allows us to perform
structure optimization requiring only a very small number of intermediate
structures to find local minima with respect to the properties of interest
utilizing gradients with respect to atom identities at specific sites. While
our approach is currently limited to a fixed molecular framework, performing
optimizations over the molecular composition space alone is far from trivial.
Compared to various alternative approaches, our implementation shows a
remarkably high molecular sampling efficiency due to efficient utilization of
gradient information in combination with powerful gradient-based optimization
algorithms. Additionally, we showcase the ease of generating calibrated
physics-based property prediction models using high quality reference training
data of relatively modest size, again allowing for quick convergence of model
parameters. This is particularly important as most physical models that rely
on empirical parameters such as semi-empirical quantum chemistry models and
density functional approximations are still largely optimized by hand, making
the corresponding procedures tedious. Thus, we believe that our work will
serve as an inspiration for the field of computational chemistry in order to
adopt the readily available AD capabilities of mature ML programming
frameworks allowing to accelerate the construction of ever more accurate
physics-based property simulation models.
## Acknowledgments
R.P. acknowledges funding through a Postdoc.Mobility fellowship by the Swiss
National Science Foundation (SNSF, Project No. 191127). A.A.-G. thanks Anders
G. Frøseth for his generous support. A.A.-G. acknowledges the generous support
of Natural Resources Canada and the Canada 150 Research Chairs program. We
also thank the SciNet HPC Consortium for support regarding the use of the
Niagara supercomputer. SciNet is funded by the Canada Foundation for
Innovation, the Government of Ontario, Ontario Research Fund - Research
Excellence, and the University of Toronto.
## Data Availability
The data that support the findings of this study are available within the
article and its supplementary material and in GitHub (inverse molecular
design) https://github.com/RodrigoAVargasHdz/huxel_molecule_desing and
(parameter optimization) https://github.com/RodrigoAVargasHdz/huxel.
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|
# Electron-phonon interactions in the Andreev Bound States of aluminum
nanobridge Josephson junctions
James T. Farmer<EMAIL_ADDRESS>Azarin Zarassi Sadman Shanto Darian
Hartsell Eli M. Levenson-Falk<EMAIL_ADDRESS>Department of Physics,
University of Southern California Center for Quantum Information Science and
Technology, University of Southern California
###### Abstract
We report continuous measurements of quasiparticles trapping and clearing from
Andreev Bound States in aluminum nanobridge Josephson junctions integrated
into a superconducting-qubit-like device. We find that trapping is well
modeled by independent spontaneous emission events. Above 80 mK the clearing
process is well described by absorption of thermal phonons, but other
temperature-independent mechanisms dominate at low temperature. We find
complex structure in the dependence of the low-temperature clearing rate on
the Andreev Bound State energy. Our results shed light on quasiparticle
behavior in qubit-like circuits.
††preprint: APS/123-QED
Non-equilibrium quasiparticles (QPs) in superconducting quantum circuits can
hinder device operation, limiting coherence in most qubit architectures [1, 2]
and inducing correlated, difficult-to-correct errors across multiple qubits on
the same chip [3, 4]. The QPs are generated by non-thermal mechanisms such as
pair-breaking infrared photons [5] or energy dissipation from local
radioactivity and cosmic rays [2]. Significant non-equilibrium QP populations
with fractional densities $x_{qp}\sim 10^{-9}-10^{-5}$ are ubiquitously
observed [5, 6, 7, 8] and have proven difficult to eliminate. Mitigation
strategies such as improved light-tight shielding [5], input/output filtering
with infrared absorbers [9, 10, 7], and device engineering [11, 12, 13, 14,
15, 16, 17] have reduced QP densities over the last decade. Many works have
probed QP populations by detecting single charge tunneling across Josephson
junctions [18, 19, 20, 21, 22, 1, 7, 15] or observing QPs trapped inside the
Andreev Bound States (ABS) of a junction [23, 24, 25, 26, 27]. These ABS
provide a complementary measurement of QP behavior, and can be used as qubit
modes themselves [28, 29]. In many implementations the ABS qubit relies on a
non-equilibrium QP trapping in order to initialize the state; such qubits are
vulnerable to additional trapping events and to accidental clearing of the QP
from the ABS. There is thus a great need to better understand the behavior of
QPs in ABS and the mechanisms for QPs transitions between ABS and bulk
continuum states.
In this letter we investigate the electron-phonon interactions involved in
trapping a quasiparticle into / clearing a quasiparticle from an Andreev Bound
State. We show continuous, real-time measurements of ABS trapping dynamics as
a function of ABS energy and device temperature in a superconducting-qubit-
like device. We find that QP trapping is consistent with independent
spontaneous emission events from a bulk QP population that is a combination of
a temperature-independent non-equilibrium background and a thermal equilibrium
density. We further find that QP clearing from an ABS is consistent with a
process dominated by absorption of a thermal phonon at temperatures above 80
mK. At low temperatures, we find evidence that absorption of microwave photons
by trapped QPs is the dominant clearing mechanism, even at low drive powers.
We analyze the mean QP occupancy of our ABS device and find independent
confirmation of our trapping and clearing models. Our results shed light on
quasiparticle behavior in ABS and in qubit-like circuits in general.
To study QP trapping, we require a circuit element which is sensitive to the
occupation of single electron states with tunability below the superconducting
gap. We find such an element in the aluminum nanobridge Josephson junction, an
all-superconducting junction which was shown [30, 31, 32] to follow the KO-1
current-phase relation [33] while providing several hundred conduction
channels. Each conduction channel hosts a pair of ABS with energies
$E_{A}(\delta)=\pm\Delta\sqrt{1-\tau\sin{\frac{\delta}{2}}}$ (1)
measured from the Fermi energy. The transparency $\tau$ is the probability
that an incident Cooper pair is transmitted across the junction and $\delta$
is the phase bias across the junction; $\Delta$ is the superconducting gap.
For short ($\lesssim 100$ nm) aluminum nanobridges, $\tau$ approximately
follows a Dorokhov distribution with a strong preference to be 0 or 1 [31,
34]. When occupied, each ABS in a given channel carries equal and opposite
contributions to the supercurrent. The negative state is usually occupied
while the positive state is unoccupied. However, the positive ABS dips below
the gap $\Delta$ when both $\delta$ and $\tau$ are nonzero, making it
energetically favorable for a quasiparticle above the gap (i.e. in the bulk
continuum) to relax into the ABS and become trapped. When this occurs the
supercurrent contribution of the given channel is cancelled and the channel is
“poisoned". This is the mechanism of our detection: the Josephson inductance
becomes a function of the number of trapped quasiparticles.
By embedding a DC SQUID with symmetric aluminum nanobridge junctions in a
$\lambda/4$ coplanar waveguide resonator, we are able to measure the trapping
of a QP as a resonant frequency shift of the resonator. This allows for a high
bandwidth, continuous measurement of the ABS occupation in a qubit-like
circuit using a standard dispersive measurement setup [27]. A constant flux
bias on the SQUID introduces a constant, symmetric phase bias to the junctions
$\delta=\pi\phi$ (where $\phi$ is the applied flux in units of flux quanta),
tuning the ABS energies. The fundamental mode $f_{0}(\phi)$ of our resonator
is flux tunable from 4.301 GHz to 4̃.25 GHz with a linewidth
$\kappa=2\pi\times 250$ kHz and the shift due to trapping a single
quasiparticle $\chi(\phi)/2\pi$ ranges from 100 kHz to 400 kHz.
We perform continuous microwave reflection measurements on our device which is
mounted in a dilution refrigerator with a base temperature of 30 mK. The
reflected signal is homodyne demodulated with an IQ mixer and the two
quadratures of signal are recorded as a gapless voltage record in 3 s segments
by an Alazar 9371 digitizer operating at 300 MHz sample rate. This is down
sampled to 1 MHz sample rate before fitting each record to a Hidden Markov
Model (HMM) [35, 36] with further down sampling if required to maintain signal
to noise ratio greater than 3. The HMM analysis models the tranisition rates
between each trapped QP number and is used to extract a time series of the
number of trapped QPs from the continuous voltage record. Data was collected
and processed in this way over a range of parameters: the dilution
refrigerator temperature, the ABS energy, and the applied microwave power. For
brevity, we restrict ourselves in this analysis to a constant power of -133
dBm ($\sim$ 25 photons) at the device. More details are given in the
Supplement [37].
We present three quantities of interest: $\Gamma_{trap}$ is the rate of QPs
relaxing from the bulk into available ABS of the junction, $\Gamma_{release}$
is the rate of clearing QPs from ABS to the bulk, and $\bar{n}$ which we call
the mean occupation is the time average of the number of trapped QPs.
$\Gamma_{trap}$ and $\Gamma_{release}$ are found from the off-diagonal
elements of the HMM transition matrix—that is, they are parameters of the
model used to extract the ABS occupation time series—while $\bar{n}$ is found
from averaging the extracted occupation over the full 3 second record.
We begin our modeling with the trap rate. Assuming trapping events are
independent of each other and spontaneous emission dominates the QP relaxation
into the ABS, each QP in the bulk has a temperature-independent trapping rate.
This implies the overall trap rate is separable:
$\Gamma_{trap}(\Delta_{A},T)=f(\Delta_{A})x(T)$, where $x(T)$ is the
fractional quasiparticle density and $\Delta_{A}\equiv\Delta-E_{A}$ is the
trap depth. We take the limit $\tau\rightarrow 1$ as the Dorkhov distribution
$\rho(\tau)$ is sharply peaked at 0 and 1, and channels with 0 transmittivity
do not contribute to the transport. The fractional quasiparticle density
should be the sum of a non-equilibrium background $x_{ne}$ and a thermal
population:
$x(T)=x_{ne}+\sqrt{\frac{2\pi
k_{B}T}{\Delta}}\exp{\left(\frac{-\Delta}{k_{B}T}\right)}.$ (2)
We expect that most bulk QPs are near the gap energy, so for spontaneous
emission we take $f(\Delta_{A})\propto\Delta_{A}^{3}$. Putting this together,
we obtain a model for the trap rate
$\Gamma_{trap}=\beta\Delta_{A}^{3}\left(x_{ne}+\sqrt{\frac{2\pi
k_{B}T}{\Delta}}\exp\left(\frac{-\Delta}{k_{B}T}\right)\right)$ (3)
where $\beta$, $\Delta$, and $x_{ne}$ are the free parameters. To improve the
quality of our fit, we take advantage of the low temperature saturation of
trap rate $\Gamma_{trap}^{0}(\Delta_{A})\approx\beta\Delta_{A}^{3}x_{ne}$ for
$T\leq 120$ mK. We first subtract $\Gamma_{trap}^{0}(\Delta_{A})$ from Eq. 3
and fit the resulting quantity to find the gap $\Delta$ and scaling factor
$\beta$. Next we divide Eq. 3 by $\Gamma_{trap}^{0}(\Delta_{A})$ and fit this
normalized rate with the fractional non-equilibrium density $x_{ne}$ as the
only free parameter. This fitting procedure is covered in detail in the
supplement [37]. In Figure 1, we show the full model (Eq. 3) using the
combined results of this fitting procedure.
Figure 1: Measured trap rate (circles) and model (solid lines). The dependence
on the trap depth $\Delta_{A}$ is shown on the left, while temperature
dependence is shown on the right. We note the peak in 30 mK data around 9 GHz
on the left was observed as a period of significantly larger than normal mean
occupation which lasted approximately 1 hour in laboratory time. The source of
this peak has not been found and it is not reproducible.
We find $\beta=8.73\pm 0.68\times 10^{15}$ MHz/eV3, $x_{ne}=8.50\pm 0.10\times
10^{-7}$, and $\Delta=185.0\pm 1.5\mu\text{eV}$. We note that the fractional
non-equilibrium density $x_{ne}$ is quite high compared to recent works [7, 2,
8] which show a fractional density on the order of $10^{-9}$. Our setup uses
light-tight radiation shields on all stages of the fridge, with Berkeley Black
infrared-absorbing coating [38] on the interior of the 100 mK and mixing
chamber shields. In addition, the sample package is mounted inside of an
Amumetal 4K shield with a tin-plated copper can nested inside, also with a
Berkeley Black interior coating. We use custom-made Eccosorb filters as well
as K&L 12 Ghz low-pass filters on all inputs and outputs. A full diagram is
available in Supplementary Figure 1 of the supplement [37]. We suspect that
our device geometry may contribute to the higher-than-expected density, as
large areas of superconducting aluminum are galvanically coupled to the SQUID.
The left panel of Figure 1 shows a peak in the 30 mK data near 9 GHz. This
anomaly was present in the trap rate and mean occupation, while the release
rate was marginally increased. We attribute this to a temporary increase in
the bulk QP density, as repeated measurement under nearly identical conditions
did not show this effect. The period of increased trapping lasted for
approximately one hour with no change in fridge conditions and no obvious
environment factors to blame. We note the duration of the effect is too long
to be caused by adhesive strain [39] or a strong cosmic ray [40].
We now turn our attention to $\Gamma_{release}$. To promote a trapped QP from
ABS to the continuum of states above the gap, sufficient energy (at least
$\Delta_{A}$) must be absorbed. In a well shielded dilution refrigerator, we
expect this energy to come from the absorption of phonons. The clearing rate
due to electron-phonon interactions should be linear in the phonon density,
$\Gamma_{phonon}(\Delta_{A},T)\propto\rho_{\epsilon\geq\Delta_{A}}(T),$ (4)
where $\rho_{\epsilon\geq\Delta_{A}}(T)$ is the density of phonons with energy
exceeding the trap depth. In the supplement [37], we integrate the Debye
density of states and Bose-Einstien distribution over energies exceeding the
trap depth to obtain the model for QP clearing due to phonons:
$\Gamma_{phonon}(\Delta_{A},T)=\alpha
T^{3}\left[-\left(\frac{\Delta_{A}}{k_{B}T}\right)^{2}\ln{\left(1-e^{\frac{-\Delta_{A}}{k_{B}T}}\right)}+\frac{2\Delta_{A}}{k_{B}T}\text{Li}_{2}\left(e^{\frac{-\Delta_{A}}{k_{B}T}}\right)+2\text{Li}_{3}\left(e^{\frac{-\Delta_{A}}{k_{B}T}}\right)\right].$
(5)
In the above, $\text{Li}_{n}(x)$ is the polylogarithm function of order $n$
and $\alpha=C_{ABS\rightarrow bulk}k_{B}^{3}/2\pi^{2}\hbar^{3}\nu^{3}$ is an
overall scaling factor; $\nu$ is the speed of sound in our sample and
$C_{ABS\rightarrow bulk}$ relates the ABS clearing rate to the phonon density.
The formal foundation for $C_{ABS\rightarrow bulk}$ is a matter worthy of
study as it represents the coupling between ABS and an incoherent bath.
In our measurements, we observe that the release rate saturates at $T\leq$ 60
mK to a value which depends on the power of our microwave readout tone,
suggesting that low-temperature clearing is dominated by driven electron-
photon interactions. This is surprising because a single readout photon
($\approx$ 4.27 GHz) has insufficient energy to clear the ABS trap
($\Delta_{A}(\phi)>5$ GHz $\forall$ measured $\phi$). Accounting for this
readout-dominated electron-photon clearing, we can model the total release
rate as
$\Gamma_{release}(\Delta_{A},T)=\Gamma_{RO}(\Delta_{A})+\Gamma_{phonon}(\Delta_{A},T),$
(6)
where the electron-photon clearing rate $\Gamma_{RO}$ is the subject of future
work. For now, we take advantage of the low temperature saturation
$\Gamma_{release}^{0}\approx\Gamma_{RO}$ to eliminate this photon contribution
and maintain focus on the electron-phonon clearing rate. Our model is
$\Gamma_{release}(\Delta_{A},T)-\Gamma_{release}^{0}(\Delta_{A})\approx\Gamma_{phonon}(\Delta_{A},T)$
(7)
which is equivalent to the right hand side of Eq. 5.
Figure 2: (top) The measured release rate vs trap depth and temperature. The
top left panel shows structure in the trap depth dependence which is
attributed to the driven electron-photon interactions which dominate at low
temperature. In the top right panel, the low temperature saturation is
visible. The grey dashed line indicates the cutoff temperature (90 mK) for the
fit. (bottom) The measured release rate minus the low temperature saturation
is shown as circles, while the phonon clearing model (Eq. 7) is shown as solid
curves.
We keep $\Delta=185\mu$eV and fit Eq. 7 with $\alpha$ as the only free
parameter as shown in Figure 2. We obtain $\alpha=38.51\pm 0.36\text{
MHz/K}^{3}$. Clearly the high temperature release rate is dominated by a
thermal distribution of phonons, but this result shows that non-thermal
sources may dominate at typical qubit operating temperatures. We point out
that the 240 mK and 260 mK data in the top left panel show some clipping of
the release rate data to the 1 MHz sample rate – A limitation of our
measurement rather than a physical effect.
Our last feature of interest is the mean occupation $\bar{n}$, which is taken
directly from the extracted time series of ABS occupations, not from HMM
parameters. We start with a simple sum over weighted probabilities:
$\bar{n}=\sum_{i}iP(i),$ (8)
where $P(i)$ is the probability of having $i$ trapped QPs. In this analysis,
we are only distinguishing between 1 trapped QP and 0 trapped QPs, as the
incidence of 2 or more trapped QPs is quite rare. We can therefore assume a
stationary distribution to obtain
$P(0)\Gamma_{trap}=P(1)\Gamma_{release}.$ (9)
Plugging (9) into (8), we obtain the model for the mean occupation:
$\bar{n}(\Delta_{A},T)=P(0)\frac{\Gamma_{trap}(\Delta_{A},T)}{\Gamma_{release}(\Delta_{A},T)}.$
(10)
Unfortunately, we are unable to eliminate the driven electron-photon
contribution as we did in Eq. (7) so we simply leave $\Gamma_{RO}(\Delta_{A})$
as a free parameter and fit each line cut along temperature separately. We
normalize by dividing out the low-temperature saturation ($T\leq 60$ mK) to
obtain the model
$\|\bar{n}_{\Delta_{A}}(T)\|=\frac{1+\frac{1}{x_{ne}}\sqrt{\frac{2\pi
k_{B}T}{\Delta}}e^{\frac{-\Delta}{k_{B}T}}}{1+\alpha_{M}T^{3}\left[-\left(\frac{\Delta_{A}}{k_{B}T}\right)^{2}\ln{\left(1-e^{\frac{-\Delta_{A}}{k_{B}T}}\right)}+\frac{2\Delta_{A}}{k_{B}T}\text{Li}_{2}\left(e^{\frac{-\Delta_{A}}{k_{B}T}}\right)+2\text{Li}_{3}\left(e^{\frac{-\Delta_{A}}{k_{B}T}}\right)\right]}.$
(11)
We fit this independently for each trap depth, while holding $x_{ne}=8.5\times
10^{-7}$ and $\Delta=185\mu\text{eV}$ fixed. The only fit parameter is
$\alpha_{M}\equiv\alpha/\Gamma_{RO}(\Delta_{A})$. The results are shown in
Figure 3.
Figure 3: (top) The measured mean occupation (circles) and the corresponding
fit (solid) are shown against temperature. Note that a different fit is
performed at each value of $\Delta_{A}$. (bottom) The fit parameter
$\alpha_{M}$ vs trap depth. Stars indicate the value of $\alpha_{M}$ for the
three curves of the same color displayed in the top panel.
We note the characteristic dip in mean occupation for $T\in[80,150]$ mK arises
from an increased phonon population leading to faster clearing of ABS, while
the rise for $T\geq 150$ mK is due to large population of thermal QPs.
We may check for self-consistency in our description by examining the
relationship between $\alpha_{M}(\Delta_{A})$ and the driven electron-photon
clearing rate $\Gamma_{RO}(\Delta_{A})$. We directly measure
$\Gamma_{RO}(\Delta_{A})$ as the low-temperature saturation of the release
rate and compare this with the estimate obtained from $\alpha/\alpha_{M}$, as
shown in Figure 4. Note that the former quantity comes entirely from the HMM
parameters, while the latter quantity comes from direct analysis of the ABS
occupation time series. These quantities agree very closely, indicating that
our analysis is robust.
Figure 4: Two sources of estimate for the rate of readout photons clearing QPs
from the ABS traps. The measured low temperature release rate (blue) and the
fit parameter from the mean occupation, shown as $\alpha/\alpha_{M}$ (orange),
where $\alpha=38.51$ is found from fitting the phonon contribution to the
release rate as shown in Figure 2. We point out that these agree in shape and
magnitude despite coming from different sources.
The driven electron-photon clearing rate has significant structure in its
dependence on $\Delta_{A}$ which is repeatable. There is additional structure
when one looks at the dependence on the microwave power, which is the focus of
our future work with this system.
By utilizing the many ABS of aluminum nanobridge Josephson junctions, we are
able to measure and explain the behavior of quasiparticle trapping in qubit-
like circuits over a range of trap depth and temperature. We show that QPs
relax into traps primarily by spontaneous emission of a phonon. The close
agreement between our data and our model suggests that most QPs entering the
trap are originally at or near the superconducting gap $\Delta$. This
indicates that any “hot” non-equilibrium quasiparticles are first relaxing to
the gap in an independent process before trapping or that the majority of non-
equilibrium quasiparticles exist at the gap edge, in agreement with past
results [41]. We do not see any evidence of “photon-assisted trapping” (in
analogy to the photon-assisted tunneling observed in tunnel junctions) where
an infrared photon breaks a Cooper pair, promoting a QP directly into an ABS.
This process may occur at lower rates, and is the subject of future work. We
also show that clearing of QPs from ABS traps at temperatures above 90 mK
occurs primarily through absorption of phonons which are distributed according
to the Debye model. Other sources, such as microwave photons, are the dominant
source of ABS-clearing energy at qubit operating temperatures. Our results
further elucidate the behavior of equilibrium and non-equilibrium
quasiparticles in superconducting circuits.
###### Acknowledgements.
We thank Leonid Glazman for useful discussions and the MIT Lincoln Laboratory
for providing the TWPA used in these measurements. This work was funded by the
AFOSR under FA9550-19-1-0060 and the NSF under DMR-1900135. JF, AZ, DH
fabricated the device. JF, AZ, SS performed measurements. JF and AZ performed
the analysis. JF wrote the manuscript with input from all authors. ELF was the
PI supervising all aspects of the work.
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# Arithmetic autocorrelation distribution of binary $m$-sequences
Xiaoyan Jing, Aixian Zhang and Keqin Feng Xiaoyan Jing is with Research Center
for Number Theory and Its Applications, Northwest University, Xi’an 710127,
China (e-mail: jxymg@126.com). Aixian Zhang is with Department of Mathematical
Sciences, Xi’an University of Technology, Xi’an 710054, China (e-mail:
zhangaixian1008@126.com). Keqin Feng is with Department of Mathematical
Sciences, Tsinghua University, Beijing 100084, China (e-mail:
fengkq@tsinghua.edu.cn).This work is supported by the National Natural Science
Foundation of China under Grant No. 12031011 and the Natural Science Basic
Research Plan in Shaanxi Province of China under Grant No. 2022JM-017.
###### Abstract
Binary $m$-sequences are ones with the largest period $n=2^{m}-1$ among the
binary sequences produced by linear shift registers with length $m$. They have
a wide range of applications in communication since they have several
desirable pseudorandomness such as balance, uniform pattern distribution and
ideal (classical) autocorrelation. In his reseach on arithmetic codes,
Mandelbaum [9] introduces a 2-adic version of classical autocorrelation of
binary sequences, called arithmetic autocorrelation. Later, Goresky and
Klapper [3, 4, 5, 6] generalize this notion to nonbinary case and develop
several properties of arithmetic autocorrelation related to linear shift
registers with carry. Recently, Z. Chen et al. [1] show an upper bound on
arithmetic autocorrelation of binary $m$-sequences and raise a conjecture on
absolute value distribution on arithmetic autocorrelation of binary
$m$-sequences.
In this paper we present a general formula for computing arithmetic
autocorrelations, from which we totally determine the arithmetic
autocorrelation distribution of arbitrary binary $m$-sequences. Particularly,
the conjecture raised in [1] is verified.
###### Index Terms:
arithmetic autocorrelation, binary $m$-sequences, 2-adic expression
## I Introduction
Let $q=2^{m}\ (m\geq 2)$, $\mathbb{F}_{q}$ be the finite field with $q$
elements, $T:\mathbb{F}_{q}\rightarrow\mathbb{F}_{2}$ be the trace mapping
defined by
$T(a)=a+a^{2}+a^{2^{2}}+\cdots+a^{2^{m-1}}\quad(a\in\mathbb{F}_{q}).$
Let $\pi$ be a primitive element of $\mathbb{F}_{q}$,
$\mathbb{F}^{\ast}_{q}=\mathbb{F}_{q}\setminus\\{0\\}=\langle\pi\rangle.$ The
binary $m$-sequence with period $n=2^{m}-1$ is defined by
$s=(s_{\lambda}=T(\pi^{\lambda}))^{n-1}_{\lambda=0}$
and shift sequences $s^{(\tau)}=(s^{(\tau)}_{\lambda})^{n-1}_{\lambda=0}\
(1\leq\tau\leq n-1)$ are defined by
$s_{\lambda}^{(\tau)}=s_{\lambda+\tau}=T(\pi^{\lambda+\tau})\
(s_{\lambda}^{(0)}=s_{\lambda})$.
Binary $m$-sequences have a wide range of applications in the field of
communications such as error-correcting coding, spread spectrum
communications, code divison multiple access systems and cryptography since
such sequences have the longest period $n=2^{m}-1$ among all binary sequences
generated by linear shift registers with length $m$ and have good
pseudorandomness. We list several basic properties of binary $m$-sequences
which we need in this paper.
###### Lemma 1.
Let $s=(s_{\lambda})^{n-1}_{\lambda=0}$ be a binary $m$-sequence with period
$n=2^{m}-1$. Then
(1). (linearity) All shift sequences $s^{(\tau)}\ (0\leq\tau\leq n-1)$ plus
zero sequence form a $\mathbb{F}_{2}$-vector space of dimension $m$.
(2). (uniform pattern distribution) For $1\leq l\leq m$,
$a=(a_{1},\ldots,a_{l})\in\mathbb{F}_{2}^{l}$, let
$N(a)=\sharp\\{0\leq i\leq
n-1\mid(s_{i},s_{i+1},\ldots,s_{i+l-1})=a\\}\quad(\text{where}\
s_{n+\lambda}=s_{\lambda}).$
Then
$\displaystyle N(a)=\left\\{\begin{array}[]{ll}2^{m-l}-1,&\mbox{ if}\
a_{1}=a_{2}=\cdots=a_{l}=0\\\ 2^{m-l},&\mbox{otherwise}\end{array}\right.$
(3). (ideal classical autocorrelation)
$\sum\limits_{\lambda=0}^{n-1}(-1)^{s_{\lambda}+s_{\lambda+\tau}}=-1$ for all
$\tau$, $1\leq\tau\leq n-1.$
In 1967, Mandelbaum [9] introduced a new version of autocorrelation on binary
(periodical) sequences, called arithmetic autocorrelation, in his research on
coding theory (arithmetic codes). Later, such notion is generalized to non-
binary case and investigated by Goresky and Klapper [3, 4, 5, 6] related to
their research on sequences generated by linear shift registers with carry.
Now we state the definition of the arithmetic autocorrelation of (periodical)
binary sequences. For any integer $a\geq 1$, we have the 2-adic expression of
$a$ as
$a=a_{0}+a_{1}2+a_{2}2^{2}+\cdots+a_{r}2^{r}\quad(a_{i}\in\\{0,1\\},\ r\geq
0).$
The 2-adic weight of $a$ is defined by
$w_{2}(a)=\sum_{\lambda=0}^{r}a_{\lambda}\ (=\text{the number of}\
a_{\lambda}=1).$
It is obvious that $1\leq w_{2}(a)\leq r$ and $w_{2}(2^{l}a)=w_{2}(a)$ for all
$l\geq 1$.
Let $a=(a_{\lambda})_{\lambda=0}^{n-1}$ be a binary sequence with period
$n\geq 2$, $a_{\lambda}\in\\{0,1\\}$, $a_{n+\lambda}=a_{\lambda}$. For any
$\tau$, $0\leq\tau\leq n-1$, the shift sequence
$a^{(\tau)}=(a^{(\tau)}_{\lambda})_{\lambda=0}^{n-1}$ of $a$ is defined by
$a^{(\tau)}=a_{\lambda+\tau}$. Then $a^{(0)}=a$ and $a^{(\tau)}\
(1\leq\tau\leq n-1)$ are distinct sequences. Let
$\sigma(a)=\sum_{\lambda=0}^{n-1}a_{\lambda}2^{\lambda}\in\mathbb{Z}.$
By assumption $n\geq 2$, we know that $(a_{0},\
\ldots,a_{n-1})\neq(0,\ldots,0),\ (1,\ldots,1)$ and $1\leq\sigma(a)\leq
2^{n}-2$. For $1\leq\tau\leq n-1$, let
$\sigma(a,\tau)=\sigma(a)-\sigma(a^{(\tau)})=\sum_{\lambda=0}^{n-1}(a_{\lambda}-a_{\lambda+\tau})2^{\lambda}\in\mathbb{Z}.$
From $1\leq\sigma(a),\sigma(a^{\tau})\leq 2^{n}-2$ and
$\sigma(a)\neq\sigma(a^{\tau})$ we get
$1\leq\arrowvert\sigma(a)-\sigma(a^{\tau})\arrowvert\leq 2^{n}-3$. Let
$\sigma(a)-\sigma(a^{\tau})=\varepsilon\cdot\sum_{\lambda=0}^{n-1}c_{\lambda}2^{\lambda}\quad(c_{\lambda}\in\\{0,1\\},\
\varepsilon\in{\\{\pm 1\\}}).$
Then
$1\leq\sum\limits_{\lambda=0}^{n-1}c_{\lambda}2^{\lambda}=\arrowvert\sigma(a)-\sigma(a^{\tau})\arrowvert\leq
2^{n}-3$ and $1\leq w_{2}(\arrowvert\sigma(a)-\sigma(a^{\tau})\arrowvert)\leq
n-1.$
###### Definition 1.
Let $a=(a_{\lambda})_{\lambda=0}^{n-1}$ be a binary sequence with period
$n\geq 2$. For $1\leq\tau\leq n-1$, let
$\sigma(a,\tau)=\sigma(a)-\sigma(a^{(\tau)})=\varepsilon\cdot\sum_{\lambda=0}^{n-1}c_{\lambda}2^{\lambda}\quad(c_{\lambda}\in\\{0,1\\},\
\varepsilon\in{\\{\pm 1\\}}).$
The arithmetic autocorrelation $A_{a}(\tau)$ is defined by
(1). For $\varepsilon=1\ (\sigma(a)>\sigma(a^{(\tau)})),$
$\displaystyle A_{a}(\tau)$ $\displaystyle=\text{``number of
$c_{\lambda}=1$"-``number of $c_{\lambda}=0$"}$
$\displaystyle=2w_{2}(\sum_{\lambda=0}^{n-1}c_{\lambda}2^{\lambda})-n=2w_{2}(\arrowvert\sigma(a,\tau)\arrowvert)-n.$
(2). For $\varepsilon=-1\ (\sigma(a)<\sigma(a^{(\tau)})),$
$\displaystyle A_{a}(\tau)$ $\displaystyle=\text{``number of
$c_{\lambda}=0$"-``number of $c_{\lambda}=1$"}$
$\displaystyle=n-2w_{2}(\sum_{\lambda=0}^{n-1}c_{\lambda}2^{\lambda})=n-2w_{2}(\arrowvert\sigma(a,\tau)\arrowvert).$
From $1\leq w_{2}(\sum_{\lambda=0}^{n-1}c_{\lambda}2^{\lambda})\leq n-1$ we
know that $\arrowvert A_{a}(\tau)\arrowvert\leq n-2$ for all $\tau$,
$1\leq\tau\leq n-1$.
The arithmetic autocorrelation distribution of $a$ is the following multiset
$D(a)=\\{A_{a}(\tau)\arrowvert 1\leq\tau\leq n-1\\}.$
Comparing with the classical autocorrelation, there exist few results
currently on arithmetic auto (and cross) correlation on binary sequences ([1,
2, 3, 4, 5, 6, 7, 8, 9, 10]). It is desirable that the absolute values
$\arrowvert A_{a}(\tau)\arrowvert\ (1\leq\tau\leq n-1)$ are as small as
possible. Hofer, Merai and Winterhof [8] proved that for a truly random binary
sequence with period $n$, $\arrowvert
A_{a}(\tau)\arrowvert=O(n^{\frac{3}{4}}(\log_{2}n)^{\frac{1}{2}}$. There exist
a family of binary sequences, called $l$-sequences, having the ideal
arithmetic autocorrelation which means that $A_{a}(\tau)=0\ (1\leq\tau\leq
n-1)$ [4]. Hofer and Winterhof [7] present nontrivial bounds of arithmetic
autocorrelations on Legendre sequences. Recently Chen et al. proved that
$\arrowvert A_{s}(\tau)\arrowvert\leq 2^{m-1}-1$ for binary $m$-sequences with
period $n=2^{m}-1$ ([1], Theorem 1). Based on examples in cases $m$=3 and 4,
they raise the following conjecture.
###### Conjecture.
([1]. Conjecture 1) Let $s$ be a binary $m$-sequence with period $n=2^{m}-1,\
m\geq 2$. Then
1\. the arithmetic autocorrelation of $s$ satisfies
$A_{s}(\tau)\in\\{\pm(2^{k}-1):\ 1\leq k\leq m-1\\}\quad\text{for}\
1\leq\tau\leq n-1.$
2\. for each $k,\ 1\leq k\leq m-1$, the total number of $\tau\ (1\leq\tau\leq
n-1)$ with $\arrowvert A_{s}(\tau)\arrowvert=2^{k}-1$ is $2^{n-k}.$
In this paper we prove this conjecture. Firstly we show several preliminary
results on $A_{a}(\tau)$ and present a formula of $A_{a}(\tau)$ for general
(periodic) binary sequences in Section 2 (Theorem 1). Then we totally
determine the arithmetic autocorrelation distribution of all binary
$m$-sequences (Theorem 2) in Section 3. As a direct consequence, the above
conjecture is verified.
## II A general formula of arithmetic autocorrelation
In this section we present a formula on arithmetic autocorrelation
$A_{a}(\tau)\ (1\leq\tau\leq n-1)$ for arbitrary binary sequence
$a=(a_{\lambda})_{\lambda=0}^{n-1}$ with period $n$. Firstly we need some
observations.
Let $a=(a_{\lambda})_{\lambda=0}^{n-1}$ and
$b=(b_{\lambda})_{\lambda=0}^{n-1}$ be two distinct binary sequences with
period $n\geq 2$. We have $2\times n\ \\{0,1\\}$-matrix
$M=\left(\begin{array}[]{cc}a_{0}\ a_{1}\ \ldots\ a_{n-1}\\\ b_{0}\ b_{1}\
\ldots\ b_{n-1}\end{array}\right)\ \ \quad(a_{\lambda},\
b_{\lambda}\in\\{0,1\\})$
and (cyclic) shift matrices of $M$
$M^{(t)}=\left(\begin{array}[]{cc}a_{t}\ a_{t+1}\ \ldots\ a_{t+n-1}\\\ b_{t}\
b_{t+1}\ \ldots\ b_{t+n-1}\end{array}\right)\ \ \quad(0\leq t\leq n-1)$
where $a_{n+l}=a_{l}$, $b_{n+l}=b_{l}$ and $M^{(0)}=M$. Let
$\sigma(a^{(t)})=\sum_{\lambda=0}^{n-1}a_{\lambda+t}2^{\lambda},\
\sigma(b^{(t)})=\sum_{\lambda=0}^{n-1}b_{\lambda+t}2^{\lambda}\in\mathbb{Z}.$
From $a\neq b$ we know that $\sigma(a^{(t)})\neq\sigma(b^{(t)}).$ Let
$\displaystyle
A(M^{(t)})=\left\\{\begin{array}[]{ll}n-2w_{2}(\arrowvert\sigma(a^{(t)})-\sigma(b^{(t)})\arrowvert),&\mbox{
if}\ \sigma(a^{(t)})>\sigma(b^{(t)})\\\
2w_{2}(\arrowvert\sigma(a^{(t)})-\sigma(b^{(t)})\arrowvert)-n,&\mbox{ if}\
\sigma(a^{(t)})<\sigma(b^{(t)}).\end{array}\right.$ (3)
###### Lemma 2.
With above assumptions and notations, we have
$A(M)=A(M^{(t)})\quad\text{for all}\ 1\leq t\leq n-1.$
###### Proof.
We just need to show that $A(M)=A(M^{(1)})$ since the general case can be done
from $M^{(t+1)}=(M^{(t)})^{(1)}$. We have
$M=\left(\begin{array}[]{cc}a_{0}\ a_{1}\ \ldots\ a_{n-1}\\\ b_{0}\ b_{1}\
\ldots\ b_{n-1}\end{array}\right),\ \ \quad
M^{(1)}=\left(\begin{array}[]{cc}a_{1}\ \ldots\ a_{n-1}\ a_{0}\\\ b_{1}\
\ldots\ b_{n-1}\ b_{0}\end{array}\right)$
and
$\displaystyle\left.\begin{array}[]{ll}~{}~{}~{}~{}~{}~{}\sigma(a)-\sigma(b)&=\sum\limits_{\lambda=0}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}=(a_{0}-b_{0})+\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}\\\
&=(a_{0}-b_{0})+2\cdot\sum\limits_{\lambda=0}^{n-2}(a_{\lambda+1}-b_{\lambda+1})2^{\lambda}\\\
\sigma(a^{(1)})-\sigma(b^{(1)})&=\sum\limits_{\lambda=0}^{n-1}(a_{\lambda+1}-b_{\lambda+1})2^{\lambda}=\sum\limits_{\lambda=0}^{n-2}(a_{\lambda+1}-b_{\lambda+1})2^{\lambda}+(a_{0}-b_{0})2^{n-1}\end{array}\right\\}$
(7)
$(\@slowromancap i@)$ Case $a_{0}=b_{0}.$ By (7) we have
$\sigma(a)-\sigma(b)=2\cdot\sum\limits_{\lambda=0}^{n-2}(a_{\lambda+1}-b_{\lambda+1})2^{\lambda}=2(\sigma(a^{(1)})-\sigma(b^{(1)})).$
Therefore $\sigma(a)-\sigma(b)$ and $\sigma(a^{(1)})-\sigma(b^{(1)})$ have the
same sign and
$w_{2}(\arrowvert\sigma(a)-\sigma(b)\arrowvert)=w_{2}(2\arrowvert\sigma(a)^{(1)}-\sigma(b)^{(1)}\arrowvert)=w_{2}(\arrowvert\sigma(a)^{(1)}-\sigma(b)^{(1)}\arrowvert).$
From (3) we get $A(M)=A(M^{(1)})$.
$(\@slowromancap ii@)$ Case $(a_{0},b_{0})=(1,0).$ From (7) we have
$\displaystyle\begin{array}[]{ll}\sigma(a^{(1)})-\sigma(b^{(1)})=\sum\limits_{\lambda=0}^{n-2}a_{\lambda+1}2^{\lambda}-\sum\limits_{\lambda=0}^{n-2}b_{\lambda+1}2^{\lambda}+2^{n-1}>0,\\\
\sigma(a)-\sigma(b)=1+\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}.\end{array}$
$(\@slowromancap ii@.1)$ If
$\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}\geq 0$,
then $\sigma(a)-\sigma(b)>0$. From (7) we have
$\displaystyle
w_{2}(\sigma(a)-\sigma(b))=w_{2}(1+\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda})=1+w_{2}(\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}),$
$\displaystyle
w_{2}(\sigma(a^{(1)})-\sigma(b^{(1)}))=w_{2}(\sum\limits_{\lambda=0}^{n-2}(a_{\lambda+1}-b_{\lambda+1})2^{\lambda}+2^{n-1})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=w_{2}(\sum\limits_{\lambda=0}^{n-2}(a_{\lambda+1}-b_{\lambda+1})2^{\lambda})+1=w_{2}(\sigma(a)-\sigma(b))$
From (3) we get $A(M)=A(M^{(1)})$.
$(\@slowromancap ii@.2)$ If
$\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}<0$, then
$\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}=2\sum\limits_{\lambda=1}^{n-1}(a_{\lambda}-b_{\lambda})2^{\lambda}\leq-2$
and $\sigma(a)-\sigma(b)\leq 1-2<0$. From (7) we have
$\displaystyle
w_{2}(\arrowvert\sigma(a)-\sigma(b)\arrowvert)=w_{2}(-1+\sum\limits_{\lambda=1}^{n-1}(b_{\lambda}-a_{\lambda})2^{\lambda})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=n-w_{2}(2^{n}-1-(-1+\sum\limits_{\lambda=1}^{n-1}(b_{\lambda}-a_{\lambda})2^{\lambda}))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=n-w_{2}(2^{n}-(\sum\limits_{\lambda=1}^{n-1}(b_{\lambda}-a_{\lambda})2^{\lambda}))$
$\displaystyle
w_{2}(\arrowvert\sigma(a^{(1)})-\sigma(b^{(1)})\arrowvert)=w_{2}(2^{n-1}-\sum\limits_{\lambda=0}^{n-2}(b_{\lambda+1}-a_{\lambda+1})2^{\lambda})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=w_{2}(2^{n}-\sum\limits_{\lambda=1}^{n-1}(b_{\lambda}-a_{\lambda})2^{\lambda})=n-w_{2}(\arrowvert\sigma(a)-\sigma(b)\arrowvert).$
From (3) we get
$\displaystyle A(M^{(1)})$
$\displaystyle=n-2w_{2}(\arrowvert\sigma(a^{(1)})-\sigma(b^{(1)})\arrowvert)=n-2(n-w_{2}(\arrowvert\sigma(a)-\sigma(b)\arrowvert))$
$\displaystyle=2w_{2}(\arrowvert\sigma(a)-\sigma(b)\arrowvert)-n=A(M).$
$(\@slowromancap iii@)$ Case $(a_{0},b_{0})=(0,1).$ Let
$\widetilde{M}=\left(\begin{array}[]{cc}b_{0}\ \ldots\ b_{n-1}\\\ a_{0}\
\ldots\ a_{n-1}\end{array}\right),\ \
\quad\widetilde{M}^{(1)}=\left(\begin{array}[]{cc}b_{1}\ \ldots\ b_{n-1}\
b_{0}\\\ a_{1}\ \ldots\ a_{n-1}\ a_{0}\end{array}\right).$
From the formula (3) we know that $A(M)=-A(\widetilde{M})$,
$A(M^{(1)})=-A(\widetilde{M}^{(1)})$. By Case $(\@slowromancap ii@)$ we get
that $A(\widetilde{M})=A(\widetilde{M}^{(1)})$. Therefore $A(M)=A(M^{(1)})$.
This completes the proof of Lemma 2. ∎
As a direct consequence of Lemma 2, we obtain the following result which means
that for fixed $\tau$, the value $A_{a}(\tau)$ is a shift-invariant of binary
sequence $a$.
###### Lemma 3.
Let $a=(a_{\lambda})_{\lambda=0}^{n-1}$ be a binary sequence with period
$n\geq 2$. Then for any $t$, $1\leq t\leq n-1$,
$A_{a}(\tau)=A_{a^{(t)}}(\tau)$.
###### Proof.
Take $b=a^{(\tau)}$ in Lemma 2. It is easy to see that $A(M)=A_{a}(\tau)$ and
$A(M^{(t)})=A_{a^{(t)}}(\tau)$. Then from $A(M)=A(M^{(t)})$ (Lemma 2) we get
$A_{a}(\tau)=A_{a^{(t)}}(\tau)$. ∎
We continue to compute the value of $A(M)$ where
$M=\left(\begin{array}[]{cc}a_{0}\ \ldots\ a_{n-1}\\\ b_{0}\ \ldots\
b_{n-1}\end{array}\right)$
and $a=(a_{\lambda})_{\lambda=0}^{n-1}$, $b=(b_{\lambda})_{\lambda=0}^{n-1}$
are distinct binary sequences with period $n\geq 2$. Now we put a weak
assumption.
$(*)\ \text{There\ exists}\ t,\ 0\leq t\leq n-1\ \text{such\ that}\
(a_{t},b_{t})=(1,0).$
Namely, the matrix $M$ have at least one column ${a_{t}\choose
b_{t}}={1\choose 0}$. Then the shift matrix $W=M^{(t+1)}$ is
$W=\left(\begin{array}[]{cc}a_{t+1}\ \ldots\ a_{t-1}\ a_{t}\\\ b_{t+1}\
\ldots\ b_{t-1}\ b_{t}\end{array}\right)=\left(\begin{array}[]{cc}a_{t+1}\
\ldots\ a_{t-1}\ 1\\\ b_{t+1}\ \ldots\ b_{t-1}\ 0\end{array}\right).$
In this case,
$\sigma=\sum\limits_{\lambda=0}^{n-1}a_{\lambda+t+1}2^{\lambda}=\sum\limits_{\lambda=0}^{n-2}a_{\lambda+t+1}2^{\lambda}+2^{n-1}$
is bigger than
$\sigma^{\prime}=\sum\limits_{\lambda=0}^{n-1}b_{\lambda+t+1}2^{\lambda}=\sum\limits_{\lambda=0}^{n-2}b_{\lambda+t+1}2^{\lambda}$,
we get $A(W)=n-2w_{2}(\sigma-\sigma^{\prime})$ and $A(M)=A(M^{(t+1)})=A(W)$ by
Lemma 2.
In order to give a closed formula for $A(W)$, we introduce some notations.
###### Definition 2.
For $\alpha,\ \beta,\ \gamma_{1},\ \ldots,\ \gamma_{l}\in\\{0,1\\}$ and $l\geq
0$, we call the matrix
$\left(\begin{array}[]{cc}\alpha\ \ \ \gamma_{1}\ \ldots\ \gamma_{l}\ \ \ \
\beta\\\ 1-\alpha\ \gamma_{1}\ \ldots\ \gamma_{l}\ 1-\beta\end{array}\right)$
as one with type $[\alpha,\ \beta;l]$.
###### Theorem 4.
Let $a=(a_{\lambda})_{\lambda=0}^{n-1}$ and
$b=(b_{\lambda})_{\lambda=0}^{n-1}$ be distinct binary sequences with period
$n\geq 2$, and
$M=\left(\begin{array}[]{cc}a_{0}\ \ldots\ a_{n-2}\ a_{n-1}\\\ b_{0}\ \ldots\
b_{n-2}\ b_{n-1}\end{array}\right).$
Suppose that there exists $\lambda$, $0\leq\lambda\leq n-1$ such that
$(a_{\lambda},b_{\lambda})=(1,0)$. Then
$A(M)=n-2g(M)$
where
$g(M)=\sum\limits_{l=0}^{n-1}l(N(0,\ 0;l)+N(0,\
1;l))+\sum\limits_{l=0}^{n-1}(N(1,\ 0;l)+N(1,\ 1;l))$
and for $\alpha,\ \beta\in\\{0,1\\}$, $N(\alpha,\ \beta;l)$ is the number of
matrices $\left(\begin{array}[]{cc}a_{\lambda}\ a_{\lambda+1}\ \ldots\
a_{\lambda+l+1}\\\ b_{\lambda}\ b_{\lambda+1}\ \ldots\
b_{\lambda+l+1}\end{array}\right)$ $(0\leq\lambda\leq n-1)$ having type
$[\alpha,\ \beta;l]$ $($namely, $\left(\begin{array}[]{cc}a_{\lambda}\
a_{\lambda+1}\ \ldots\ a_{\lambda+l+1}\\\ b_{\lambda}\ b_{\lambda+1}\ \ldots\
b_{\lambda+l+1}\end{array}\right)=\left(\begin{array}[]{cc}\alpha\ \ \
\gamma_{1}\ \ldots\ \gamma_{l}\ \ \ \ \beta\\\ 1-\alpha\ \gamma_{1}\ \ldots\
\gamma_{l}\ 1-\beta\end{array}\right)$ for some $\gamma_{1},\ \ldots,\
\gamma_{l}\in\\{0,1\\})$.
###### Proof.
By assumption $(a_{\lambda},b_{\lambda})=(1,0)$ for some $\lambda$, we have a
(cyclic) shift matrix $W$ of $M$
$W=\left(\begin{array}[]{ll}\alpha_{0}\ldots\ \alpha_{n-1}\\\ \varepsilon_{0}\
\ldots\ \varepsilon_{n-1}\end{array}\right)$
such that $\left(\begin{array}[]{cc}\alpha_{n-1}\\\
\varepsilon_{n-1}\end{array}\right)=\left(\begin{array}[]{cc}a_{\lambda}\\\
b_{\lambda}\end{array}\right)=\left(\begin{array}[]{cc}1\\\
0\end{array}\right)$. Let
$\sigma(\alpha)=\sum\limits_{\lambda=0}^{n-1}\alpha_{\lambda}2^{\lambda}$,
$\sigma(\varepsilon)=\sum\limits_{\lambda=0}^{n-1}\varepsilon_{\lambda}2^{\lambda}$.
From $\alpha_{n-1}=1,\ \varepsilon_{n-1}=0$ we know that $\sigma(\alpha)\geq
2^{n-1}>\sigma(\varepsilon).$ Since $W$ is shift matrix of $M$, we get
$A(M)=A(W)=n-2w_{2}(\sigma(\alpha)-\sigma(\varepsilon)).$
We compute $\sigma(\alpha)-\sigma(\varepsilon)$ by 2-adic subtraction on the
digits of $\sigma(\alpha)$ and $\sigma(\varepsilon)$ as shown in Figure 1.
Figure 1 2-adic subtraction $\sigma(\alpha)-\sigma(\varepsilon)$
$\sigma(\alpha)$ | $\alpha_{0}$ | $\alpha_{1}$ | $\ldots$ | $\alpha_{n-1}$ | $(\alpha_{n-1}=1)$
---|---|---|---|---|---
$\sigma(\varepsilon)$ | $\varepsilon_{0}$ | $\varepsilon_{1}$ | $\ldots$ | $\varepsilon_{n-1}$ | $(\varepsilon_{n-1}=0)$
$\sigma(\alpha)-\sigma(\varepsilon)$ | $\delta_{0}$ | $\delta_{1}$ | $\ldots$ | $\delta_{n-1}$ | $(\delta_{\lambda}\in\\{0,1\\})$
Then
$\sigma(\alpha)-\sigma(\varepsilon)=\sum\limits_{\lambda=0}^{n-1}\delta_{\lambda}2^{\lambda}$
and $w_{2}(\sigma(\alpha)-\sigma(\varepsilon))$ is the number of
$\delta_{\lambda}=1\ (0\leq\lambda\leq n-1).$
Now we write the matrix $W$ in the following block form
$W=\left(W_{1}\ W_{2}\ \ldots\ W_{m}\right)\quad m\geq 1.$
Each bloack $W_{i}$ has form
$W_{i}=\left(\begin{array}[]{ll}\gamma_{1}^{(i)}\ \ldots\
\gamma_{l_{i}}^{(i)}\ \ \ \ \beta_{i}\\\ \gamma_{1}^{(i)}\ \ldots\
\gamma_{l_{i}}^{(i)}\ 1-\beta_{i}\end{array}\right)\quad(l_{i}\geq 0)$
where $\gamma_{1}^{(i)},\ldots,\gamma_{l_{i}}^{(i)}\in\\{0,1\\}$ and the last
block is $W_{m}=\left(\begin{array}[]{cc}\gamma_{1}^{(m)}\ \ldots\
\gamma_{l_{m}}^{(m)}\ 1\\\ \gamma_{1}^{(m)}\ \ldots\ \gamma_{l_{m}}^{(m)}\
0\end{array}\right)\ (\beta_{m}=\alpha_{n-1}=1).$ Put the last column of
$W_{i-1}$ to the left-hand side of $W_{i}$, we get
$\widetilde{W_{i}}=\left(\begin{array}[]{cc}\beta_{i-1}\ \ \ \gamma_{1}^{(i)}\
\ldots\ \gamma_{l_{i}}^{(i)}\ \ \ \ \beta_{i}\\\ 1-\beta_{i-1}\
\gamma_{1}^{(i)}\ \ldots\ \gamma_{l_{i}}^{(i)}\
1-\beta_{i}\end{array}\right)=\left(\begin{array}[]{cc}\beta_{i-1}\\\
1-\beta_{i-1}\end{array}W_{i}\right)$
which has type $[\beta_{i-1},\ \beta_{i};l_{i}]$. For the first block $W_{1}$,
we assume $\widetilde{W_{1}}=\left(\begin{array}[]{cc}\beta_{m}\\\
1-\beta_{m}\end{array}W\right)=\left(\begin{array}[]{ll}1\ \gamma_{1}^{(1)}\
\ldots\ \gamma_{l_{1}}^{(1)}\ \ \ \ \beta_{1}\\\ 0\ \gamma_{1}^{(1)}\ \ldots\
\gamma_{l_{1}}^{(1)}\ 1-\beta_{1}\end{array}\right)$. Then we can see the
2-adic subtraction given in Figure 1 locally. For each $\widetilde{W_{i}}$
with type $[\beta_{i-1},\ \beta_{i};l_{i}]$, the digits $\delta_{\lambda}$ (in
Figure 1) under the columns of $W_{i}$ can be determined as shown in Figure 2.
Figure 2 2-adic subtraction at $W_{i}$
$(\beta_{i-1},\beta_{i})=(1,0)$
(1) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 0 | $\ldots$
---|---|---|---|---|---
(0) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 1 | $\ldots$
$\ldots$ | 0 | $\ldots$ | 0 | 1 | $\ldots$
$(\beta_{i-1},\beta_{i})=(0,0)$
(0) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 0 | $\ldots$
---|---|---|---|---|---
(1) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 1 | $\ldots$
$\ldots$ | 1 | $\ldots$ | 1 | 0 | $\ldots$
$(\beta_{i-1},\beta_{i})=(1,1)$
(1) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 1 | $\ldots$
---|---|---|---|---|---
(0) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 0 | $\ldots$
$\ldots$ | 0 | $\ldots$ | 0 | 1 | $\ldots$
$(\beta_{i-1},\beta_{i})=(0,1)$
(0) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 1 | $\ldots$
---|---|---|---|---|---
(1) | $\gamma_{1}^{(i)}$ | $\ldots$ | $\gamma_{l_{i}}^{(i)}$ | 0 | $\ldots$
$\ldots$ | 1 | $\ldots$ | 1 | 0 | $\ldots$
From Figure 2 we know that the number $\delta_{\lambda}=1$ given by $W_{i}$ is
1 if the type of $\widetilde{W_{i}}$ is $[\beta_{i-1},\ \beta_{i};l_{i}]=[1,\
0;l_{i}]$ or $[1,\ 1;l_{i}]$; or $l_{i}$ if the type of $\widetilde{W_{i}}$ is
$[\beta_{i-1},\ \beta_{i};l_{i}]=[0,\ 0;l_{i}]$ or $[0,\ 1;l_{i}]$. Put all
local data together, we get
$\displaystyle
w_{2}(\sigma(\alpha)-\sigma(\varepsilon))=\sum_{\begin{subarray}{c}i=1\\\
\text{``type of $\widetilde{W_{i}}$"=$[1,\ \beta_{i};l_{i}]$}\\\
\beta_{i}\in\\{0,1\\}\end{subarray}}^{m}1+\sum_{\begin{subarray}{c}i=1\\\
\text{``type of $\widetilde{W_{i}}$"=$[0,\ \beta_{i};l_{i}]$}\\\
\beta_{i}\in\\{0,1\\}\end{subarray}}^{m}l_{i}$ $\displaystyle=$
$\displaystyle\sum_{l\geq 0}l(N_{W}(0,\ 0;l)+N_{W}(0,\ 1;l))+\sum_{l\geq
0}(N_{W}(1,\ 0;l)+N_{W}(1,\ 1;l)),$
where for $l\geq 0$, $\alpha,\ \beta\in\\{0,1\\}$, $N_{W}(\alpha,\ \beta;l)$
is the number of $\left(\begin{array}[]{cc}\alpha_{\lambda}\
\alpha_{\lambda+1}\ \ldots\ \alpha_{\lambda+l+1}\\\ \varepsilon_{\lambda}\
\varepsilon_{\lambda+1}\ \ldots\ \varepsilon_{\lambda+l+1}\end{array}\right)\
(0\leq\lambda\leq n-1)$ having type $[\alpha,\ \beta;l]$. Since $W$ is a shift
of $M$, the number $N_{W}(\alpha,\ \beta;l)$ is the same as the number
$N(\alpha,\ \beta;l)=N_{M}(\alpha,\ \beta;l)$. Therefore
$w_{2}(\sigma(\alpha)-\sigma(\varepsilon))=\sum_{l\geq 0}l(N(0,\ 0;l)+N(0,\
1;l))+\sum_{l\geq 0}(N(1,\ 0;l)+N(1,\ 1;l)).$
At last, if $l\geq n$ and there exists a matrix
$\left(\begin{array}[]{cc}a_{\lambda}\ a_{\lambda+1}\ \ldots\ a_{\lambda+l}\
a_{\lambda+l+1}\\\ b_{\lambda}\ b_{\lambda+1}\ \ldots\ b_{\lambda+l}\
b_{\lambda+l+1}\end{array}\right)$ of type $[\alpha,\ \beta;l]$ for some
$\lambda$, then $a_{\lambda+i}=b_{\lambda+i}\ (i=1,\ldots,n)$ which means
$a=b$ since the period of both sequences $a$ and $b$ is $n$. This contradicts
to the assumption $a\neq b$. Therefore $N(\alpha,\ \beta;l)=0$ for any
$\alpha,\ \beta\in\\{0,1\\}$ and $l\geq n$. We get the final formula
$\displaystyle\sum_{l\geq 0}l(N(0,\ 0;l)+N(0,\ 1;l))+\sum_{l\geq 0}(N(1,\
0;l)+N(1,\ 1;l))$ $\displaystyle=$ $\displaystyle\sum_{l=0}^{n-1}l(N(0,\
0;l)+N(0,\ 1;l))+\sum_{l=0}^{n-1}(N(1,\ 0;l)+N(1,\ 1;l))=g(M)$
and $A(M)=n-2g(M)$. This completes the proof of Theorem 1. ∎
By Theorem 4, the value of $A(M)$ is expressed by the numbers $N(1,\
0;l)+N(1,\ 1;l)$ and $N(0,\ 0;l)+N(0,\ 1;l)$ ($0\leq l\leq n-1,\ N(\alpha,\
\beta;l)=N_{M}(\alpha,\ \beta;l)$) for $M=\left(\begin{array}[]{cc}a_{0}\
\ldots\ a_{n-1}\\\ b_{0}\ \ldots\ b_{n-1}\end{array}\right)$. To determine
these numbers is not easy for arbitrary binary sequences
$a=(a_{\lambda})_{\lambda=0}^{n-1}$ and $b=(b_{\lambda})_{\lambda=0}^{n-1}$
with period $n\geq 2$ in general case. Fortunately, if $a$ is a binary
$m$-sequence and $b$ is a shift sequence of $a$, these numbers can be computed
and then the arithmetic autocorrelation distribution of all binary
$m$-sequences can be determined. We do this in next section.
## III Arithmetic autocorrelation distribution of binary $m$-sequences
Let $q=2^{m}\ (m\geq 2)$, $\pi$ be a primitive element of the finite field
$\mathbb{F}_{q}$, which means that
$\mathbb{F}^{\ast}_{q}=\mathbb{F}_{q}\setminus\\{0\\}=\langle\pi\rangle$,
$\pi$ is the generator of the cyclic multiplicative group
$\mathbb{F}^{\ast}_{q}$. The binary $m$-sequence
$s=(s_{\lambda})^{n-1}_{\lambda=0}$ with period $n=2^{m}-1$ is defined by
$s_{\lambda}=T(\pi^{\lambda})\in\mathbb{F}_{2}\ (0\leq\lambda\leq n)$, where
$T:\ \mathbb{F}_{q}\rightarrow\mathbb{F}_{2},\
T(\alpha)=\alpha+\alpha^{2}+\alpha^{2^{2}}+\cdots+\alpha^{2^{m-1}}$
is the trace mapping.
In Theorem 4, we take $a=s$,
$b=s^{(\tau)}=(s^{(\tau)}_{\lambda})_{\lambda=0}^{n-1}$, the shift sequence of
$s$ where $s^{(\tau)}_{\lambda}=s_{\lambda+\tau}$. Then
$M=\left(\begin{array}[]{ll}s_{0}\ \ \ s_{1}\ \ \ \ldots\ s_{n-1}\\\
s^{(\tau)}_{0}\ s^{(\tau)}_{1}\ \ldots\
s^{(\tau)}_{n-1}\end{array}\right)=\left(\begin{array}[]{ll}s_{0}\ \ \ s_{1}\
\ \ldots\ s_{n-1}\\\ s_{\tau}\ s_{\tau+1}\ \ldots\
s_{n-1+\tau}\end{array}\right)\ (s_{n+t}=s_{t})$ and the arithmetic
autocorrelation $A_{s}(\tau)$ is $A(M)$ for $1\leq\tau\leq n-1$. From Theorem
4 we know that $A_{s}(\tau)=n-2g(\tau)$ where
$\displaystyle g(\tau)=\sum\limits_{l=0}^{n-1}l(N^{(\tau)}(0,\
0;l)+N^{(\tau)}(0,\ 1;l))+\sum\limits_{l=0}^{n-1}(N^{(\tau)}(1,\
0;l)+N^{(\tau)}(1,\ 1;l))$ (8)
and for $\alpha,\ \beta\in\mathbb{F}_{2}=\\{0,1\\}$
$\displaystyle N^{(\tau)}(\alpha,\ \beta;l)$ $\displaystyle=$
$\displaystyle\sharp\left\\{0\leq t\leq
n-1\Bigg{\arrowvert}\begin{array}[]{ll}\left(\begin{array}[]{ll}s_{t}\ \ \
s_{t+1}\ \ldots\ s_{t+l+1}\\\ s^{(\tau)}_{t}\ s^{(\tau)}_{t+1}\ \ldots\
s^{(\tau)}_{t+l+1}\end{array}\right)=\left(\begin{array}[]{cc}\alpha\ \ \
\gamma_{1}\ \ldots\ \gamma_{l}\ \ \ \ \beta\\\ 1-\alpha\ \gamma_{1}\ \ldots\
\gamma_{l}\ 1-\beta\end{array}\right),\
\gamma_{1},\ldots,\gamma_{l}\in\mathbb{F}_{2}\end{array}\right\\}$
$\displaystyle=$ $\displaystyle\sharp\left\\{0\leq t\leq
n-1\Bigg{\arrowvert}\begin{array}[]{ll}T(\pi^{t})=\alpha,\
T(\pi^{t+\tau})=1+\alpha,\ T(\pi^{t+l+1})=\beta,\
T(\pi^{t+l+1+\tau})=1+\beta,\\\ T(\pi^{t+\lambda})=T(\pi^{t+\lambda+\tau})\
\text{for}\ 1\leq\lambda\leq l\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\sharp\left\\{x\in\mathbb{F}^{\ast}_{q}\Bigg{\arrowvert}\begin{array}[]{ll}T(x)=\alpha,\
T(\pi^{\tau}x)=1+\alpha,\ T(\pi^{l+1}x)=\beta,\ T(\pi^{l+1+\tau}x)=1+\beta,\\\
T(\pi^{\lambda}x)=T(\pi^{\lambda+\tau}x)\ \text{for}\ 1\leq\lambda\leq
l\end{array}\right\\}.$
In order to determine $A_{s}(\tau)$ by formula (8), we need to compute
$\displaystyle N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\ 1;l)$ $\displaystyle=$
$\displaystyle\sharp\left\\{x\in\mathbb{F}^{\ast}_{q}\Bigg{\arrowvert}\begin{array}[]{ll}T(x)=0,\
T(\pi^{\tau}x)=1,\ T(\pi^{l+1}x)+T(\pi^{l+1+\tau}x)=1,\\\
T(\pi^{\lambda}x)+T(\pi^{\lambda+\tau}x)=0\ \text{for}\ 1\leq\lambda\leq
l\end{array}\right\\}$ (11)
and
$\displaystyle N^{(\tau)}(1,\ 0;l)+N^{(\tau)}(1,\ 1;l)$ $\displaystyle=$
$\displaystyle\sharp\left\\{x\in\mathbb{F}^{\ast}_{q}\Bigg{\arrowvert}\begin{array}[]{ll}T(x)=1,\
T(\pi^{\tau}x)=0,\ T(\pi^{l+1}x)+T(\pi^{l+1+\tau}x)=1,\\\
T(\pi^{\lambda}x)+T(\pi^{\lambda+\tau}x)=0\ \text{for}\ 1\leq\lambda\leq
l\end{array}\right\\}.$ (14)
###### Lemma 5.
Let $1\leq\tau\leq n-1.$ Then
$(\@slowromancap i@).$ If $l\geq m$, then $N^{(\tau)}(\alpha,\ \beta;l)=0$ for
any $\alpha,\ \beta\in\mathbb{F}_{2}.$
$(\@slowromancap
ii@).\begin{array}[]{ll}\sum\limits_{l=0}^{m-1}(N^{(\tau)}(1,\
0;l)+N^{(\tau)}(1,\ 1;l))=\sum\limits_{l=0}^{m-1}(N^{(\tau)}(0,\
0;l)+N^{(\tau)}(0,\ 1;l))=2^{m-2}.\end{array}$
$(\@slowromancap iii@).$
$g(\tau)=2^{m-2}+\sum\limits_{l=1}^{m-1}l(N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\
1;l)).$
###### Proof.
$(\@slowromancap i@)$ Suppose that $l\geq m$ and $N^{(\tau)}(\alpha,\
\beta;l)\geq 1$. Then there exists $t$ such that $s_{t+i}=s_{t+i}^{(\tau)}$
for $1\leq i\leq l$. From $1\leq\tau\leq n-1$ we know that $s\neq s^{(\tau)}$
and $s+s^{\tau}=(s_{\lambda}+s_{\lambda}^{(\tau)})_{\lambda=0}^{n-1}$ is a
shift sequence of $s$ (Lemma 1. (1)). But there is no pattern
$(s_{t+1}+s_{t+1}^{(\tau)},\ldots,s_{t+l}+s_{t+l}^{(\tau)})=(0,\ldots,0)$ in
sequence $s+s^{\tau}$ for $l\geq m$ (Lemma 1. (2)). This contradiction shows
that $N(\alpha,\ \beta;l)=0$ for $l\geq m$ and any $\alpha,\
\beta\in\mathbb{F}_{2}.$
$(\@slowromancap ii@)$ Since the $m$-sequence $s$ has no pattern
$(0,\ldots,0)\in\mathbb{F}_{2}^{m}$, from $(\@slowromancap i@)$ and formula
(III) we can see that
$\displaystyle\sum_{l=0}^{m-1}N^{(\tau)}(1,\ 0;l)+N^{(\tau)}(1,\ 1;l)$
$\displaystyle=$
$\displaystyle\sharp\left\\{x\in\mathbb{F}^{\ast}_{q}\Bigg{\arrowvert}T(x)=1,\
T(\pi^{\tau}x)=0\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{x\in\mathbb{F}_{q}^{\ast}}\big{(}1-(-1)^{T(x)}\big{)}\big{(}1+(-1)^{T(\pi^{\tau}x)}\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{x\in\mathbb{F}_{q}}\big{(}1-(-1)^{T(x)}\big{)}\big{(}1+(-1)^{T(\pi^{\tau}x)}\big{)}\quad(\text{since}\
1-(-1)^{T(0)}=1-1=0)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(q+\sum_{x\in\mathbb{F}_{q}}\left[(-1)^{T(\pi^{\tau}x)}-(-1)^{T(x)}-(-1)^{T((1+\pi^{\tau})x)}\right]\right)$
(15)
It is well-known that $\sum\limits_{x\in\mathbb{F}_{q}}(-1)^{T(\alpha
x)}=\sum\limits_{x\in\mathbb{F}_{q}}(-1)^{T(x)}=0$ if
$\alpha\in\mathbb{F}_{q}^{\ast}$. Then by $1+\pi^{\tau}\neq 0$ (for
$1\leq\tau\leq n-1$) we know that the summation at right-hand side of (III) is
zero. Therefore $\sum\limits_{l=0}^{m-1}(N^{(\tau)}(1,\ 0;l)+N^{(\tau)}(1,\
1;l))=\frac{q}{4}=2^{m-2}$.
We can show $\sum\limits_{l=0}^{m-1}(N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\
1;l))=2^{m-2}$ in similar way.
$(\@slowromancap iii@)$ Directly from $(\@slowromancap i@)$, $(\@slowromancap
ii@)$ and formula (8). ∎
Now we come to place of computing $N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\ 1;l)$
for each $l$, $1\leq l\leq m-1$ and $1\leq\tau\leq n-1$. For $1\leq\tau\leq
n-1$, $1+\pi^{\tau}\neq 0,\ 1$, because the order of $\pi$ is $n=q-1$. Since
$\\{1,\pi,\ldots,\pi^{m-1}\\}$ is an $\mathbb{F}_{2}$-basis of
$\mathbb{F}_{q}$, $(1+\pi^{\tau})^{-1}(\neq 0,\ 1)$ has the following unique
expression
$(1+\pi^{\tau})^{-1}=b_{0}+b_{1}\pi+\cdots+b_{s-1}\pi^{e-1}+\pi^{e}$
where $b_{0},\cdots,b_{s-1}\in\mathbb{F}_{2}$, $1\leq e\leq m-1$. Another
thing we need is that the minimal polynomial of $\pi$ over $\mathbb{F}_{2}$ is
an irreducible (primitive) polynomial
$f(x)=1+a_{1}x+\cdots+a_{m-1}x^{m-1}+x^{m}\
(a_{1},\ldots,a_{m-1}\in\mathbb{F}_{2})$ and $f(\pi)=0$. Therefore
$\pi^{m}=1+a_{1}\pi+\cdots+a_{m-1}\pi^{m-1}$.
###### Lemma 6.
Let $q=2^{m}\ (m\geq 2)$, $\pi$ be a primitive element of $\mathbb{F}_{q}$,
$s=(s_{\lambda})^{n-1}_{\lambda=0}$ is the binary $m$-sequence with period
$n=q-1$ defined by $s_{\lambda}=T(\pi^{\lambda})$, where $T:\
\mathbb{F}_{q}\rightarrow\mathbb{F}_{2}$ is the trace mapping. Let
$1\leq\tau\leq n-1$ and
$(1+\pi^{e})^{-1}=b_{0}+b_{1}\pi+\cdots+b_{s-1}\pi^{e-1}+\pi^{e}\quad(b_{0},\ldots,b_{s-1}\in\mathbb{F}_{2},\
1\leq e\leq m-1)$
Then
$(\@slowromancap i@).$ For $1\leq l\leq m-2$,
$\displaystyle N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\
1;l)=\left\\{\begin{array}[]{ll}2^{m-l-3},&\text{if}\ l\leq e-2\\\
2^{m-l-3}(1-(-1)^{b_{0}}),&\text{if}\ l=e-1(\geq 1)\\\
2^{m-l-3}(1+(-1)^{b_{0}}),&\text{if}\ l\geq e.\end{array}\right.$
$(\@slowromancap ii@).$ For $l=m-1$,
$N^{(\tau)}(0,\ 0;m-1)+N^{(\tau)}(0,\ 1;m-1)=\frac{1}{2}(1+(-1)^{b_{0}}).$
###### Proof.
From formula (III) we know that
$\displaystyle N^{(\tau)}(0,\ 0;$ $\displaystyle l)+N^{(\tau)}(0,\ 1;l)$
$\displaystyle=\frac{1}{2^{l+3}}\sum_{x\in\mathbb{F}_{q}^{\ast}}$
$\displaystyle\big{(}(1+(-1)^{T(x)})(1-(-1)^{T(\pi^{\tau}x)})(1-(-1)^{T(\pi^{l+1}x+\pi^{l+1+\tau}x)})\prod\limits_{\lambda=1}^{l}(1+(-1)^{T(\pi^{\lambda}x+\pi^{\lambda+\tau}x)})\big{)}$
$\displaystyle=\frac{1}{2^{l+3}}\sum_{x\in\mathbb{F}_{q}}$
$\displaystyle\big{(}(1+(-1)^{T(x)})(1-(-1)^{T(\pi^{\tau}x)})(1-(-1)^{T(\pi^{l+1}(1+\pi^{\tau})x)})\prod\limits_{\lambda=1}^{l}(1+(-1)^{T(\pi^{\lambda}(1+\pi^{\tau})x)})\big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad(\text{for}\
x=0,\ 1-(-1)^{T(\pi^{\tau}x)}=0)$
$\displaystyle=\frac{1}{2^{l+3}}\sum_{x\in\mathbb{F}_{q}}$
$\displaystyle\sum_{c^{\prime},c_{0},\ldots,c_{l+1}=0}^{1}(-1)^{T[x(c^{\prime}+c_{0}\pi^{\tau}+(1+\pi^{\tau})(c_{1}\pi+c_{2}\pi^{2}+\cdots+c_{l+1}\pi^{l+1}))]}(-1)^{c_{0}+c_{l+1}}$
$\displaystyle=\frac{1}{2^{l+3}}\sum_{x\in\mathbb{F}_{q}}$
$\displaystyle\sum_{c,c_{0},\ldots,c_{l+1}=0}^{1}(-1)^{T[x(c+(1+\pi^{\tau})(c_{0}+c_{1}\pi+c_{2}\pi^{2}+\cdots+c_{l+1}\pi^{l+1}))]}(-1)^{c_{0}+c_{l+1}}\quad(\text{let}\
c=c^{\prime}+c_{0})$
If
$\delta=c+(1+\pi^{\tau})(c_{0}+c_{1}\pi+c_{2}\pi^{2}+\cdots+c_{l+1}\pi^{l+1})\neq
0$, then $\sum_{x\in\mathbb{F}_{q}}(-1)^{T(\delta x)}=0$. Therefore
$\displaystyle N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\
1;l)=\frac{q}{2^{l+3}}\sum_{\begin{subarray}{c}c,c_{0},\ldots,c_{l+1}=0\\\
c_{0}+c_{1}\pi+\cdots+c_{l+1}\pi^{l+1}=c(1+\pi^{\tau})^{-1}\end{subarray}}^{1}(-1)^{c_{0}+c_{l+1}}$
(16)
where the summation in the right-hand side is under the condition
$\displaystyle
c_{0}+c_{1}\pi+\cdots+c_{l+1}\pi^{l+1}=c(1+\pi^{\tau})^{-1}=c(b_{0}+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e}).$
(17)
From now on we write $N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\ 1;l)$ by
$N^{(\tau)}(l)$ briefly.
$(\@slowromancap i@)$ For $1\leq l\leq m-2$, $l+1\leq m-1$ and $e\leq m-1$.
Since $\\{1,\pi,\ldots,\pi^{m-1}\\}$ is a basis of $\mathbb{F}_{q}$ over
$\mathbb{F}_{2}$, the coefficients of $\pi^{\lambda}$ in both sides of formula
(17) should be equal for each $\lambda$, $0\leq\lambda\leq m-1$.
$(\@slowromancap i@.1)$ If $l\leq e-2$, then $e\geq l+2$. From the
coefficients of $\pi^{e}$ we get $0=c$. Then $c_{\lambda}=cb_{\lambda}=0$ for
all $\lambda$, $0\leq\lambda\leq l+1\ (\leq e-1)$. By formula (16) we get
$N^{(\tau)}(l)=\frac{q}{2^{l+3}}\cdot 1=2^{m-l-3}$.
$(\@slowromancap i@.2)$ For $l=e-1$, formula (17) becomes
$c_{0}+c_{1}\pi+\cdots+c_{l+1}\pi^{l+1}=c(b_{0}+b_{1}\pi+\cdots+b_{l}\pi^{l}+\pi^{l+1})$.
Therefore $c_{l+1}=c$ and $c_{\lambda}=cb_{\lambda}\ (0\leq\lambda\leq l)$.
For $c=0$, we have $c_{0}=c_{1}=\cdots=c_{l+1}=0$ and $c_{0}+c_{l+1}=0$. For
$c=1$, we have $c_{l+1}=1$, $c_{\lambda}=b_{\lambda}\ (0\leq\lambda\leq l)$
and $c_{0}+c_{l+1}=b_{0}+1$. We obtain
$N^{(\tau)}(e-1)=2^{m-l-3}(1+(-1)^{b_{0}+1})=2^{m-l-3}(1-(-1)^{b_{0}})$.
$(\@slowromancap i@.3)$ For $l\geq e$, then $c_{\lambda}=cb_{\lambda}\
(0\leq\lambda\leq e-1)$, $c_{e}=c$, $c_{\lambda}=0\ (e+1\leq\lambda\leq l+1)$
and $c_{0}+c_{l+1}=cb_{0}$. From formula (16) we get
$\begin{array}[]{ll}N^{(\tau)}(l)=2^{m-l-3}(&~{}~{}~{}~{}~{}1~{}~{}~{}~{}~{}+~{}~{}(-1)^{b_{0}}~{}~{}).\\\
&\small{\text{(for\ c=0)}~{}~{}~{}~{}~{}~{}~{}~{}\text{(for\
c=1)}}\end{array}$
$(\@slowromancap ii@)$ For $l=m-1$, formula (17) becomes
$c(b_{0}+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e})=c_{0}+c_{1}\pi+\cdots+c_{m-1}\pi^{m-1}+c_{m}\pi^{m}.$
Let $f(x)=1+a_{1}x+\cdots+a_{m-1}x^{m-1}+x^{m}\ (a_{i}\in\mathbb{F}_{2})$ be
the minimal polynomial of $\pi$ over $\mathbb{F}_{2}$. Then
$\pi^{m}=1+a_{1}\pi+\cdots+a_{m-1}\pi^{m-1}$ and
$\begin{array}[]{ll}c(b_{0}+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e})=c_{0}+c_{1}\pi+\cdots+c_{m-1}\pi^{m-1}+c_{m}(1+a_{1}\pi+\cdots+a_{m-1}\pi^{m-1})\end{array}$
which imply that
$\begin{array}[]{ll}cb_{0}=c_{0}+c_{m}=c_{0}+c_{l+1},\
cb_{\lambda}=c_{\lambda}+c_{l+1}a_{\lambda}\ (1\leq\lambda\leq e-1),\
c=c_{e}+c_{l+1}a_{e},\\\ 0=c_{\lambda}+c_{l+1}a_{\lambda}\ (e+1\leq\lambda\leq
m-1=l).\end{array}$
For $c=0$ we have $c_{0}=c_{l+1}$, $c_{\lambda}=c_{l+1}a_{\lambda}\
(1\leq\lambda\leq l)$. For $c=1$ we have $c_{0}=b_{0}+c_{l+1}$,
$c_{\lambda}=b_{\lambda}+c_{l+1}a_{\lambda}\ (1\leq\lambda\leq e-1)$,
$c_{e}=1+c_{l+1}a_{e}$, $c_{\lambda}=c_{l+1}a_{\lambda}\ (e+1\leq\lambda\leq
l)$. By formula (16) we get
$\displaystyle\begin{array}[]{ll}N^{(\tau)}(m-1)=2^{m-l-3}(\sum\limits_{c_{l+1}=0}^{1}(-1)^{c_{l+1}+c_{l+1}}+\sum\limits_{c_{l+1}=0}^{1}(-1)^{b_{0}+c_{l+1}+c_{l+1}})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\small{\text{(for\
c=0)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{(for\
c=1)}}\end{array}=2^{m-l-2}(1+(-1)^{b_{0}})=\frac{1}{2}(1+(-1)^{b_{0}}).$
This completes the proof of Lemma 6. ∎
Now we show our main result on arithmetic autocorrelation distribution
$\\{A_{s}(\tau)\arrowvert 1\leq\tau\leq n\\}$ of any binary $m$-sequence with
period $n=2^{m}-1$.
###### Theorem 7.
Let $m\geq 2$, $n=2^{m}-1$,
$s=(s_{\lambda}=T(\pi^{\lambda}))^{n-1}_{\lambda=0}$ be the binary
$m$-sequence with period $n$ where $\pi$ is a primitive element of
$\mathbb{F}_{q}\ (q=2^{m})$ and $T:\ \mathbb{F}_{q}\rightarrow\mathbb{F}_{2}$
is the trace mapping. For $1\leq\tau\leq n-1$, let $A(\tau)=A_{s}(\tau)$ be
the arithmetic autocorrelation of $s$ and
$(1+\pi^{\tau})^{-1}=b_{0}+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e}\quad(1\leq
e\leq m-1,\ b_{0},b_{1},\ldots,b_{e-1}\in\mathbb{F}_{2}).$
Then
$(\@slowromancap i@)$. For $N^{(\tau)}(l)=N^{(\tau)}(0,\ 0;l)+N^{(\tau)}(0,\
1;l)$,
$\displaystyle\sum_{l=0}^{m-1}lN^{(\tau)}(l)=\left\\{\begin{array}[]{ll}2^{m-2}+2^{m-e-1}-1,&\text{if}\
b_{0}=0\\\ 2^{m-2}-2^{m-e-1},&\text{if}\ b_{0}=1.\\\ \end{array}\right.$
$(\@slowromancap ii@)$. (arithmetic autocorrelation distribution) For all
$\tau$, $1\leq\tau\leq n-1$, $A_{s}(\tau)\in\\{\pm(2^{k}-1)\arrowvert 1\leq
k\leq m-1\\}$. Moreover, for each $\varepsilon\in\\{0,1\\}$ and $0\leq k\leq
n-1$, the number of $\tau\ (1\leq\tau\leq n-1)$ such that
$A_{s}(\tau)=\varepsilon(2^{k}-1)$ is $2^{m-k-1}$.
###### Proof.
For $1\leq\tau\leq n-1$ and
$(1+\pi^{\tau})^{-1}=b_{0}+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e}\quad(1\leq
e\leq m-1,\ b_{0},\ldots,b_{e-1}\in\mathbb{F}_{2})$, by Lemma 6 we know that
$\displaystyle\sum_{l=1}^{m-1}lN^{(\tau)}(l)=$ $\displaystyle\sum_{1\leq l\leq
e-2}l\cdot 2^{m-l-3}+(e-1)(1-(-1)^{b_{0}})2^{m-(e-1)-3}$
$\displaystyle+\sum_{e\leq l\leq m-2}l\cdot
2^{m-l-3}(1+(-1)^{b_{0}})+\frac{1}{2}(m-1)(1+(-1)^{b_{0}})$ $\displaystyle=$
$\displaystyle 2^{m-3}[\sum_{1\leq l\leq e-2}l\cdot
2^{-l}+(e-1)(1-(-1)^{b_{0}})2^{-e+1}$ $\displaystyle+\sum_{e\leq l\leq
m-1}(1+(-1)^{b_{0}})l\cdot 2^{-l}]+\frac{1}{4}(m-1)(1+(-1)^{b_{0}})$
$(\@slowromancap i@.1)$ If $b_{0}=0$, then
$\displaystyle\sum_{l=1}^{m-1}lN^{(\tau)}(l)$ $\displaystyle=$ $\displaystyle
2^{m-3}[\sum_{1\leq l\leq e-2}l\cdot 2^{-l}+\sum_{e\leq l\leq m-1}l\cdot
2^{-(l-1)}]+\frac{1}{2}(m-1)$ $\displaystyle=$ $\displaystyle
2^{m-3}[\sum_{1\leq l\leq e-2}l\cdot 2^{-l}+\sum_{e-1\leq l\leq m-2}(l+1)\cdot
2^{-l}]+\frac{1}{2}(m-1)$ $\displaystyle=$ $\displaystyle
2^{m-3}[\sum_{l=1}^{m-2}l\cdot
2^{-l}+\sum_{l=e-1}^{m-2}2^{-l}]+\frac{1}{2}(m-1).$
Now we use the identity $\sum\limits_{l=1}^{d}l\cdot
2^{-l}=2-\frac{d+2}{2^{d}}\ (d\geq 1)$ which can be proved by induction. We
obtain
$\sum_{l=1}^{m-1}lN^{(\tau)}(l)=2^{m-3}[2-\frac{m}{2^{m}-2}+\frac{1}{2^{e-2}}-\frac{1}{2^{m-2}}]+\frac{1}{2}(m-1)=2^{m-2}+2^{m-e-1}-1.$
$(\@slowromancap i@.2)$ If $b_{0}=1$, then
$\sum_{l=1}^{m-1}lN^{(\tau)}(l)=2^{m-3}[\sum_{1\leq l\leq e-2}l\cdot
2^{-l}+(e-1)2^{-e+2}].$
For $e\geq 3$,
$\sum_{l=1}^{m-1}lN^{(\tau)}(l)=2^{m-3}[2-\frac{e}{2^{e-2}}+\frac{e-1}{2^{e-2}}]=2^{m-3}[2-\frac{1}{2^{e-2}}]=2^{m-2}+2^{m-e-1}.$
For $e=2$, there is no $l$ satisfing $1\leq l\leq e-2=0$,
$\sum\limits_{l=1}^{m-1}lN^{(\tau)}(l)=2^{m-3}[0+2^{0}]=2^{m-3}=2^{m-2}-2^{m-e-1}\
(\text{for}\ e=2).$ For $e=1$,
$\sum\limits_{l=1}^{m-1}lN^{(\tau)}(l)=2^{m-3}[0+0]=0=2^{m-2}-2^{m-e-1}\
(\text{for}\ e=1).$
$(\@slowromancap ii@)$ By Lemma 5, for $1\leq\tau\leq n-1$,
$A_{s}(\tau)=n-2g(\tau)=n-2(2^{m-2}+\sum_{l=1}^{m-1}lN^{(\tau)}(l)).$
For $b_{0}=0$,
$A_{s}(\tau)=2^{m}-1-2(2^{m-2}+2^{m-2}+2^{m-e-1}-1)=-(2^{m-e}-1).$ For
$b_{0}=1$, $A_{s}(\tau)=2^{m}-1-2(2^{m-2}+2^{m-2}-2^{m-e-1})=2^{m-e}-1.$
Therefore
$A_{s}(\tau)\in\\{\pm(2^{m-e}-1)\arrowvert 1\leq e\leq
m-1\\}=\\{\pm(2^{k}-1)\arrowvert 1\leq k\leq m-1\\}.$
Moreover,
$\displaystyle\\{(1+\pi^{\tau})^{-1}\arrowvert 1\leq\tau\leq
n-1\\}=\mathbb{F}_{2^{m}}\setminus\\{0,1\\}$ $\displaystyle=$
$\displaystyle\\{b_{0}+b_{1}\pi+\cdots+b_{m-1}\pi^{m-1}\neq 0,\ 1\arrowvert
b_{0},\ldots,b_{m-1}\in\mathbb{F}_{2}\\}$ $\displaystyle=$
$\displaystyle\\{b_{0}+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e}\arrowvert
b_{0},\ldots,b_{e-1}\in\mathbb{F}_{2},\ 1\leq e\leq m-1\\}.$
From part $(\@slowromancap i@)$ we know that for $1\leq\tau\leq n-1$ and
$1\leq k\leq m-1$,
$A_{s}(\tau)=2^{k}-1\Leftrightarrow(1+\pi^{\tau})^{-1}=1+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e},\
e=m-k.$
Therefore
$\displaystyle\sharp\\{1\leq\tau\leq n-1\arrowvert A_{s}(\tau)=2^{k}-1\\}$
$\displaystyle=$ $\displaystyle\sharp\\{(b_{1},\ldots,b_{e-1})\arrowvert
b_{1},\ldots,b_{e-1}\in\mathbb{F}_{2}\\}$ $\displaystyle=$ $\displaystyle
2^{e-1}=2^{m-k-1}.$
Similarly,
$A_{s}(\tau)=-(2^{k}-1)\Leftrightarrow(1+\pi^{\tau})^{-1}=0+b_{1}\pi+\cdots+b_{e-1}\pi^{e-1}+\pi^{e}$
and
$\sharp\\{1\leq\tau\leq n-1\arrowvert
A_{s}(\tau)=-(2^{k}-1)\\}=2^{e-1}=2^{m-k-1}.$
This completes Theorem 7. ∎
###### Remark.
From Theorem 7 we know that the conjecture, stated in Section I, is true. The
conjecture is concern on the distribution on absolute values $\lvert
A_{s}(\tau)\rvert\ (1\leq\tau\leq n-1)$. In fact, from Theorem 7 and the proof
we get more informations as following.
$(\@slowromancap i@)$. $\lvert A_{s}(\tau)\rvert=2^{k}-1$ if and only if
$(1+\pi^{\tau})^{-1}=b_{0}+b_{1}\pi+\cdots+b_{m-k-1}\pi^{m-k-1}+\pi^{m-k}\
(1\leq k\leq m-1).$
$(\@slowromancap ii@)$. $A_{s}(\tau)=2^{k}-1\Leftrightarrow b_{0}=1$ and
$A_{s}(\tau)=-(2^{k}-1)\Leftrightarrow b_{0}=0$.
$(\@slowromancap iii@)$. Both of multiplicity of $\tau\ (1\leq\tau\leq n-1)$
satisfying $A_{s}(\tau)=2^{k}-1$ and $A_{s}(\tau)=-(2^{k}-1)$ are the same
number $2^{m-k-1}$.
## IV Conflicts of Interest:
The authors declare that they have no conflicts of interest to report
regarding the present study.
## References
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* [3] M. Goresky, A.Klapper, Arithmetic crosscorrelations of feedback with carry shift register sequences, IEEE Trans. on Inform. Theory, 43(4) (1997), 1342-1345.
* [4] M. Goresky, A.Klapper, Some results on the arithmetic correlation of sequences. In $\langle$Sequences and Their Applications-SETA 2008$\rangle$, Lecture Notes in Computer Science, vol.5203, (2008), 71-80.
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* [6] M. Goresky, A.Klapper, $\langle$Algebraic Shift Register Sequences$\rangle$, Combiridege University Press, 2012.
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|
# NMR evidence for a Peierls transition in the layered square-net compound
LaAgSb2
Seung-Ho Baek Department of Physics, Changwon National University, Changwon
51139, Korea Department of Materials Convergence and System Engineering,
Changwon National University, Changwon 51139, Korea Sergey L. Bud’ko Ames
Laboratory, U.S. Department of Energy and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA Paul C. Canfield Ames
Laboratory, U.S. Department of Energy and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA F. Borsa Ames Laboratory, U.S.
Department of Energy and Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA Byoung Jin Suh<EMAIL_ADDRESS>Department of Physics,The Catholic University of Korea, Bucheon 14662, Korea
###### Abstract
We measured the central ($1/2\leftrightarrow-1/2$) and first satellite ($\pm
3/2\leftrightarrow\pm 1/2$) lines of the 139La NMR spectra as a function of
temperature in LaAgSb2, in order to elucidate the origin and nature of the
charge-density-wave (CDW) transitions at $T_{\text{CDW1}}=207$ K and
$T_{\text{CDW2}}=186$ K. In the normal state, the Knight shift $\mathcal{K}$
reveals a fairly linear relationship with decreasing temperature, which is
ascribed to a pseudogap in the spin excitation spectrum, pointing towards the
material being an unconventional metal. Upon further cooling, $\mathcal{K}$
decreases more steeply below $T_{\text{CDW1}}$, indicative of the partial
Fermi surface gap opening on top of the pseudogap. The most remarkable finding
in our study is a clear splitting of the satellite lines at $T_{\text{CDW1}}$
observed for $H\parallel c$, whose temperature dependence behaves as the BCS
order parameter in the weak-coupling limit, evidencing that the CDW transition
induces the periodic lattice distortion. Our NMR findings therefore
demonstrate that the CDW transition in LaAgSb2 is of Peierls type, being
driven by the electronic instability in the vicinity of the Fermi level.
## I Introduction
The charge-density-wave (CDW) phenomenon has been observed in numerous two-
dimensional (2D) layered metallic compounds such as transition-metal
dichalcogenides (TMDs) and high-$T_{c}$ cuprates, as well as quasi-one-
dimensional (1D) metals [1, 2, 3, 4, 5]. However, whereas CDW in 1D is well
understood by the conventional Peierls’ model [6] in which the Fermi surface
nesting (FSN) drives a periodic lattice distortion (PLD), the origin of CDW in
higher dimensions is much more complicated than in 1D [7] and strongly
material dependent [8]. As examples, rather than FSN, strong electron
correlations appear to be a crucial ingredient for charge ordering in
high-$T_{c}$ cuprates [5], or local electron-phonon coupling (EPC) may be a
main driving force for the CDW in TMDs [9, 10].
LaAgSb2 is a unique member of the layered Sb square-net family $R$AgSb2
($R$=rare earth) in space group $P4/nmm$ [11, 12], isostructural to the Fe-
oxypnictide LaFeAsO [13]. The square-net is featured by a shorter in-plane
atomic distance ($a/\sqrt{2}$) than the lattice constant $a$, resulting in a
twice larger unit cell in real space [see Fig. 1(a)] or a folding of the
Brillouin zone in reciprocal space [14]. The resultant linear band crossings
may occur near the Fermi level, allowing to host Dirac fermions [15], as
indeed observed in the Bi square-net, SrMnBi2 [16]. Interestingly, unlike
other members of $R$AgSb2, most of which have a magnetically ordered ground
state, LaAgSb2 is nonmagnetic and undergoes two successive CDW transitions
(CDW1 at 207 K and CDW2 at 186 K) [17] while exhibiting unusual large linear
magnetoresistance (MR) [11], which may be attributed to the quantum limit of
the Dirac fermions [18]. LaAgSb2 reveals not only a Dirac-cone-like band
structure near the Fermi level [18, 19], but also well-nested segments of the
Fermi surface directly linked to the Dirac cone, suggesting a close
relationship between the Dirac fermions and the CDW ordering [19].
In this work we have performed 139La NMR to investigate the microscopic origin
of the CDW transitions in LaAgSb2. As 139La has a large quadrupole moment, it
could probe changes of both the local spin susceptibility and charge
environments, via the Knight shift $\mathcal{K}$ and the quadrupole frequency
$\nu_{Q}$. Our NMR data indicate that the CDW1 transition opens up a Fermi
surface gap and causes the periodic lattice distortion, whose order parameter
is well described by BCS mean-field theory, at the same time. Together with
the observation of phonon softening at the CDW wave vector [20, 21], we
conclude that LaAgSb2 is a rare quasi-2D material that undergoes a nesting-
driven weak-coupling CDW transition.
## II Experimental details
High-quality LaAgSb2 single crystals were flux grown from a typical Sb rich
self-flux, La0.045Ag0.091Sb0.864, as described in Ref. [11]. The size of the
crystal used in this NMR measurement is roughly $1\times 2\times 4$ mm3.
139La (nuclear spin $I=7/2$) NMR measurements were carried out at the external
field of 4.7 T in the range of temperature 100–300 K. The signal intensity
turned out very weak in our experiments, which is ascribed largely to the low
filling factor owing to the plate-like single crystal within the NMR coil, and
partly to small rf penetration in a metal that is limited by the skin depth,
$\delta=\sqrt{2\rho/\omega\mu}$ where $\rho$ is the resistivity, $\omega$ is
the angular frequency of the rf, and $\mu$ is the magnetic permeability [22].
Due to the weak signal intensity, we focused solely on measuring the
temperature dependence of the NMR spectra, without attempting to measure the
spin-lattice relaxation rate, $T_{1}^{-1}$. Further, among the three pairs of
satellites expected for $I=7/2$, we have measured the first satellite pair
only, because it yields sufficient information regarding the electric field
gradient and its temperature evolution at the 139La in an axial symmetry. The
139La NMR spectra were obtained by a conventional Hahn spin-echo technique
with a typical $\pi/2$ pulse length of 3 $\mu$s.
## III Results
### III.1 Quadrupole-perturbed 139La NMR spectra
For a nuclear spin $I>1/2$, there are central
($\frac{1}{2}\leftrightarrow-\frac{1}{2}$) and satellite transitions between
the $m$th and $(m-1)$th levels ($m=-I,-I+1,\cdots,+I$). In an axial symmetric
surrounding (asymmetry parameter $\eta=0$), the allowed transitions to first
order are given by [22]
$\begin{split}\nu(m\leftrightarrow m-1)=\;&\nu_{0}(1+\mathcal{K})\\\
&+\frac{1}{2}\nu_{Q}(3\cos^{2}\theta-1)\left(m-\frac{1}{2}\right),\end{split}$
(1)
where $\mathcal{K}$ is the Knight shift, $\nu_{0}$ is the unshifted Larmor
frequency, $\nu_{Q}$ is the nuclear quadrupole frequency, and $\theta$ is the
angle between the principal axis $z$ of the electric field gradient (EFG) and
an external field $\mathbf{H}$.
Figure 1(b) shows the central ($1/2\leftrightarrow-1/2$) and first satellite
($\pm 3/2\leftrightarrow\pm 1/2$) lines of the 139La NMR spectra measured at
$H=4.7$ T for $H\perp c$ and $H\parallel c$, respectively, at 292 K. The
observed NMR spectra are excellently described by Eq. (1), in which the
distance between the first satellite lines is given by $\nu_{Q}$ and
$2\nu_{Q}$ for $H\perp c$ and $H\parallel c$, respectively, indicating that
the local symmetry at 139La is indeed axial with respect to the principal axis
of the EFG which lies along the $c$ axis. Interestingly, the line shape of
each satellites appears to be asymmetric while the satellite pairs are
symmetric about the central transition, similarly for both field orientations
(see Fig. 2). It should be noted that the FWHM of the NMR lines are very
narrow, less than 10 kHz for the satellites which corresponds to only 1% of
$\nu_{Q}$ (see Fig. 3), evidencing the very high homogeneity of our sample.
Therefore, the asymmetric line shape of the satellites seems to reflect a
feature of the satellite transitions ($\pm 1/2\leftrightarrow\pm 3/2$).
Figure 1: (a) Crystal structure of LaAgSb2 viewed along the $b$ (upper) and
$c$ (lower) axes. The La atoms above and below the Ag-Sb2 plane are staggered,
and the Ag and Sb1 atoms form an identical square-net, resulting in the
$P4/nmm$ structure. The dashed lines denote the unit cell. (b) 139La central
and first satellite transitions for $H\perp c$ (upper) and $H\parallel c$
(lower) measured at 292 K and an external field of 4.7 T. The separations of
the satellites for both directions are well described in terms of the single
quadrupole frequency $\nu_{Q}=1.339$ MHz, as expected for an axial symmetry
about the $c$ axis.
### III.2 Knight shift and Fermi surface gap
The temperature dependence of the NMR lines is presented in Fig. 2. To begin
with, we discuss the temperature dependence of the 139La central line, shown
in the middle panels of Figs. 2(a) and 2(b) for $H\parallel c$ and $H\perp c$,
respectively. While the FWHM of the central line remains very narrow down to
100 K, the line clearly shifts to lower frequency with decreasing temperature,
involving a weak line broadening. One can note that a small shoulder peak,
which persists up to room temperature, is detected at the high-frequency side
for both field directions. The small peak suggests that a small portion of the
139La sites experiences slightly different hyperfine fields from the majority.
Regardless of its origin and nature, the peak is so small that it could hardly
affect the satellites, and it will be ignored for our purpose.
Figure 2: Temperature dependence of 139La NMR spectra for the central and
first satellite transitions measured at $H=4.7$ T applied (a) parallel and (b)
perpendicular to the $c$ axis. The left axis refers to the temperature at
which each spectrum was obtained, whereas the amplitude of the spectra is
plotted in arbitrary units. The central lines for both field directions
similarly shift to lower frequency, i.e., the Knight shift is decreased, with
decreasing temperature. Whereas the satellite lines for $H\parallel c$
abruptly split below $T_{\text{CDW1}}=207$ K (marked by the red arrows), those
for $H\perp c$ significantly broaden below $T_{\text{CDW2}}=186$ K (marked by
the cyan arrows).
The Knight shift $\mathcal{K}\equiv(\nu-\nu_{0})/\nu_{0}\times 100$ % as a
function of temperature is plotted in Fig. 3(a). For $H\perp c$
($\theta=90^{\circ}$), the second order quadrupole shift of the central line
given by $15\nu_{Q}^{2}/16\nu_{0}(1-\cos^{2}\theta)(1-9\cos^{2}\theta)$ for
$I=7/2$ [22], was subtracted from the total shift. In the normal state, we
find that $\mathcal{K}$ decreases in an almost linear fashion with
temperature, albeit very small, for both field orientations. For a simple
Pauli metal, $\mathcal{K}$ or the local spin susceptibility is proportional to
the density of states (DOS) at the Fermi level, $n(\epsilon_{F})$, so it would
have been expected to be independent of temperature in the normal state.
Therefore, the unusual decrease of $\mathcal{K}$ may be ascribed to a
pseudogap, which is frequently observed in many layered strange metals such as
high-$T_{c}$ cuprates [23], Fe-based superconductors [24, 25], and TMDs [26,
27]. In fact, LaAgSb2 shows a rather atypical linear temperature dependence of
resistivity in the normal state [11, 17], and is believed to be a topological
semimetal that hosts Dirac fermions [18, 19], being likely responsible for the
pseudogap-like behavior.
Upon further lowering temperature, we find that $\mathcal{K}$ decreases more
rapidly just below $T_{\text{CDW1}}$, deviating from the linear temperature
dependence in the normal state. The small but clear additional reduction of
$\mathcal{K}$ at $T_{\text{CDW1}}$ observed for the two field directions
suggests that $n(\epsilon_{F})$ is further suppressed by the CDW1 transition,
greatly supporting that the CDW1 transition opens up a gap at the Fermi
surface [20]. Interestingly, no additional notable change of $\mathcal{K}$ was
observed at $T_{\text{CDW2}}$. This may suggest either that the CDW2
transition is too weak to yield a sizable gap, or that the nature of the CDW2
is different from that of the CDW1, as conjectured by the very different CDW
wave vectors [17].
### III.3 Periodic lattice distortion in the CDW state
Now we evaluate how the charge environment surrounding the 139La nuclei
evolves through the CDW transitions, via analysis on the temperature
dependence of the satellites, giving rise to the information of the EFG and
its spatial distribution.
Most remarkably, we observed a clear splitting of the satellites for
$H\parallel c$ just below $T_{\text{CDW1}}$, as shown in Fig. 2(a). The
$\nu_{Q}$ values extracted from the distance between the central and lower
satellite lines are plotted in Fig. 3(b), which reveals the sharp splitting of
the satellite lines. This means that the CDW1 transition induces two
inequivalent 139La sites which experience different EFG on average. Moreover,
we confirmed that the difference of the $\nu_{Q}$ values between the split
satellites, $\Delta\nu_{Q}$, serves as an order parameter for the CDW1 phase,
which is consistent with the BCS mean field theory in the weak coupling limit,
i.e., $\Delta\nu_{Q}\propto(1-T/T_{\text{CDW1}})^{1/2}$ [see the inset of Fig.
3(b)]. This result bears strong resemblance to the behavior of the CDW
modulation intensity below $T_{\text{CDW1}}$ observed by x-ray scattering
measurements [17], indicating that $\Delta\nu_{Q}$ is directly related to the
lattice modulation associated with CDW order. That is, the local periodic
lattice distortion emerges as a direct consequence of the charge modulation.
Nevertheless, the presence of the two distinct 139La sites is surprising,
because the extremely long wavelength of the CDW modulation ($\lambda_{1}\sim
39a$) would result in a spatial distribution of the EFG at the 139La sites,
leading to an inhomogeneous quadrupole broadening rather than the split lines.
This suggests that the PLD/CDW is formed in a way that discriminates the
staggered La atoms above and below Ag-Sb2 plane in the unit cell [see Fig.
1(a)].
Figure 3: (a) Temperature dependence of the Knight shift $\mathcal{K}$. For
both field directions, $\mathcal{K}$ decreases linearly with lowering
temperature. Below $T_{\text{CDW1}}$ the slope becomes larger suggesting a
partial gap opening at the Fermi surface. (b) The quadrupole frequency
$\nu_{Q}$ as a function of temperature. $\nu_{Q}$ sharply splits into two at
$T_{\text{CDW1}}$. Inset shows an order parameter-like behavior of
$\Delta\nu_{Q}$. The solid line is given by the weak coupling BCS theory. (c)
and (d) FWHM for the central and first satellite lines as a function of
temperature. Whereas the FWHM of all the NMR lines increases weakly for both
field directions at $T_{\text{CDW1}}$, a significant line broadening was
observed below $T_{\text{CDW2}}$ only for $H\perp c$.
In contrast, the satellites for $H\perp c$ are nearly unchanged through
$T_{\text{CDW1}}$. The anisotropic changes of the spectra in the CDW1 phase
may be understood by the strong dependence of the quadrupole effect on the
angle between the direction of the EFG and $H$. Since the lattice modulation
for the CDW1 is developed along the $a$ axis [17], $H$ is perpendicular to the
EFG modulation which generates the two 139La sites. In other words, $H$ has
the same effects on the two 139La sites, revealing the difference between
them. However, the situation is quite different for $H\perp c$. Namely, the
otherwise differentiated 139La sites may feel a similar quadrupole-perturbed
Zeeman field for $H\perp c$ because $H$ is applied along an arbitrary
direction in the $ab$ plane in our experimental setup. Note that the similar
weak broadening observed for all the NMR lines below $T_{\text{CDW1}}$, as
shown in Figs. 3(c) and 3(d), is what would be expected from a spatial
distribution of the EFG.
Another prominent feature found in Fig. 2 is a significant line broadening of
the NMR lines below the CDW2 transition temperature $T_{\text{CDW2}}$ that
occurs only for $H\perp c$ [see Figs. 3(c) and 3(d)]. Further, we find that
the satellite lines for $H\perp c$ rapidly spread out in the CDW2 phase,
becoming vanishingly small below $\sim 170$ K. While the external magnetic
field anisotropy is naturally ascribed to the EFG modulation induced by the
CDW2 modulation that is along the $c$ axis [17], the considerable broadening
of the satellite lines for $H\perp c$ suggests that the EFG distribution is
much wider for the CDW2, probably due to the much shorter modulation
wavelength of the CDW2 ($\lambda_{2}\sim 6.3c$) than the CDW1
($\lambda_{1}\sim 39a$) [17].
## IV Discussion
We have established that the CDW transition at 207 K causes a (partial) gap
opening at the Fermi level and generates two inequivalent 139La sites, the
difference in quadrupole frequency $\Delta\nu_{Q}$ of which is well described
by the BCS mean field theory. These NMR findings microscopically prove that
the CDW1 transition in LaAgSb2 is governed by the conventional (Peierls type)
weak-coupling mechanism in which the Fermi surface nesting instability drives
the periodic lattice distortion as well as the Fermi surface gap opening via
electron-phonon interactions. This is consistent with the observation of
acoustic phonon softening and the corresponding Kohn anomalies at the CDW wave
vector [20, 21].
It is worthwhile to compare the aforementioned with the recent NMR results
obtained in the transition metal dichalcogenide, $2H$-TaSe2, within which a
strong-coupling mechanism driven by local electron-phonon coupling appears to
underlie the CDW transition [28]. The seemingly different CDW mechanism in the
two layered 2D materials may be related to the structural motif of the TMDs
and the square-net materials. Namely, unlike TMDs whose structural symmetry is
lower than tetragonal, the square-net has an inherent instability against a
Peierls transition, similar to 1D metals which are always unstable toward a
lattice distortion [6]. Nevertheless, the square-net could be stabilized,
e.g., by the overlap of $d$-orbitals from the transition-metal elements with
the $s$ and $p$ orbitals in the square-net [14].
As to the CDW2 transition at 186 K, the anomalous anisotropic line broadening
of the 139La NMR spectra below 186 K reveals that the EFG modulation is formed
along the $c$ axis due to the CDW2 ordering. However, neither a Fermi surface
gap opening, although it is suggested by the resistivity anomaly for
$H\parallel c$ [17], nor the order parameter associated with the CDW2 was
observed in our work, thus leaving the origin and nature of the CDW2 still an
open question.
## V Summary
We carried out the 139La NMR investigation in the Sb square-net LaAgSb2. In
the normal state, we found an unusual pseudogap behavior, reflecting the
unconventional metallic state of the material. At the CDW transition at 207 K,
the 139La NMR spectra undergo sharp changes, establishing that a CDW gap opens
up at the Fermi surface and two inequivalent 139La sites are generated as a
result of the periodic lattice distortion. These provide microscopic evidence
that LaAgSb2 undergoes a Peierls transition, which is quite rare in non-1D
materials.
Our NMR studies strongly suggest that the square-net not only induces the band
inversions being a good structural motif for topological semimetals that host
2D Dirac fermions, it could be also an excellent platform to study the CDW
phenomenon in quasi-two dimensions and its relationship with Dirac fermions.
###### Acknowledgements.
We thank Changyong Song for useful discussions. This work was supported by a
National Research Foundation of Korea (NRF) grant funded by the Korea
Government(MSIT) (Grant No. NRF-2020R1A2C1003817). Work at the Ames Laboratory
was supported by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, Materials Sciences and Engineering Division. The Ames
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DE-AC02-07CH11358.
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|
# Non-stationary difference equation,
affine Laumon space and quantization of
discrete Painlevé equation
Hidetoshi Awata , Koji Hasegawa , Hiroaki Kanno , Ryo Ohkawa , Shamil
Shakirov , Jun’ichi Shiraishi and Yasuhiko Yamada H.Awata: Graduate
School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
<EMAIL_ADDRESS>K.Hasegawa: Mathematical Institute, Tohoku
University, Sendai 980-8578, Japan<EMAIL_ADDRESS>H.Kanno: Graduate
School of Mathematics and Kobayashi-Maskawa Institute, Nagoya University,
Nagoya 464-8602, Japan<EMAIL_ADDRESS>R.Ohkawa: Osaka Central
Advanced Mathematical Institute, Osaka Metropolitan University, Osaka
558-8585, Japan; Research Institute for Mathematical Sciences, Kyoto
University, Kyoto 606-8502, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>Sh.Shakirov: University of Geneva, Switzerland;
Institute for Information Transmission Problems, Moscow, Russia
<EMAIL_ADDRESS>J.Shiraishi: Graduate School of Mathematical
Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan
<EMAIL_ADDRESS>Y.Yamada: Department of Mathematics, Kobe
University, Rokko, Kobe 657-8501, Japan<EMAIL_ADDRESS>
###### Abstract.
We show the relation of the non-stationary difference equation proposed by one
of the authors and the quantized discrete Painlevé VI equation. The five
dimensional Seiberg-Witten curve associated with the difference equation has a
consistent four dimensional limit. We also show that the original equation can
be factorized as a coupled system for a pair of functions
$(\mathcal{F}^{(1)},\mathcal{F}^{(2)})$, which is a consequence of the
identification of the Hamiltonian as a translation element in the extended
affine Weyl group. We conjecture that the instanton partition function coming
from the affine Laumon space provides a solution to the coupled system.
Dedicated to the memories of our friends, Omar Foda and Yaroslav Pugai
###### Contents
1. 1 Introduction
2. 2 Non-stationary difference equation
3. 3 Quantized discrete Painlevé VI equation
4. 4 Quantum Seiberg-Witten curve
5. 5 Relation to affine Laumon space
6. A Difference analogue of Painlevé VI equation
7. B Evidences for the Conjecture
8. C Four dimensional limit in a factorized form
## 1\. Introduction
The conformal blocks, more precisely, the matrix elements or the traces of the
intertwiners between the Verma modules of the Virasoro algebra, or the chiral
algebra in general define special functions which are ubiquitous in
mathematics and physics. As special functions originated from the
representation theory of the symmetry, the hypergeometric series and Nekrasov
function [36] to mention a few, the conformal blocks should satisfy
sufficiently simple but significant equations.The Belavin-Polyakov-
Zamolodchikov (BPZ) equation for the Virasoro conformal block with degenerate
field insertion is a celebrated example [7].
For the deformed Virasoro algebra [46] such an equation is expected to be some
(non-stationary) difference equation. Though the conformal blocks allow
integral representations of Dotsenko-Fateev type or the deformed matrix model
type, the desired equations were not known for a long time, more than a
quarter century after the discovery of the algebra. While most attempts to
work out explicit form of the expected difference equation were not
successful, the non-stationary Ruijsenaar function has been proposed [45] (see
also [32]). In an appropriate limit, we can see that it satisfies the non-
stationary affine Toda equation which is a difference equation that involves
the $q$-exponent of the Laplace operator (the $q$-Borel transformation
$\mathcal{B}$, see Definition 2.2). Recently a remarkable progress has been
made by one of the authors [44] based on AGT correspondence [2]. Namely a non-
stationary difference equation for the Nekrasov partition function of the five
dimensional gauge theory with a surface defect was discovered. The AGT
correspondence tells that if the theory has four matter hypermultiplets in the
fundamental representation, the partition function agrees with the genus zero
five point function with one degenerate field coming from the defect
[1],[47],[33]. In [44] the five dimensional lift of AGT correspondence [5] was
applied, where the surface defect is realized by imposing the Higgsing
condition on the $SU(2)\times SU(2)$ quiver gauge theory. Hence the non-
stationary difference equation proposed in [44] is regarded as a $q$-deformed
version of BPZ equation. In the decoupling limit of the matter multiplets it
reproduces the non-stationary affine Toda equation for the non-stationary
Ruijsenaar function. A distinguished feature of these equations is the
appearance of the $q$-Borel transformation $\mathcal{B}$.
The quantization of the Painlevé equations (or isomonodromic deformations more
generally) has been studied for many years. One of the motivation of such
studies was its relation to the conformal field theories. This relation plays
a key role in the recent studies of AGT correspondence (see [40] and
references therein). Also, there has been a large progress in the study of
discrete (or difference) analog of Painlevé equations in the last decades. The
discrete Painlevé equations are classified into additive, multiplicative (i.e.
$q$-), and elliptic-difference equations [43], and each class corresponds to
gauge theories in four, five and six dimensions;
${\begin{array}[]{lcccccccccccccccccc}{\rm elliptic}&E_{8}^{(1)}\\\
&&&&&&&&&&&&&&&&A_{1}^{(1)}\\\\[-8.53581pt] &&&&&&&&&&&&&&&\nearrow\\\ {\rm
multiplicative}&E_{8}^{(1)}&\rightarrow&E_{7}^{(1)}&\rightarrow&E_{6}^{(1)}&\rightarrow&{\color[rgb]{1,0,0}D_{5}^{(1)}}&\rightarrow&A_{4}^{(1)}&&\rightarrow&A_{2+1}^{(1)}&\rightarrow&A_{1+1}^{(1)}&\rightarrow&A_{1}^{(1)}&\rightarrow&A_{0}^{(1)}\\\\[14.22636pt]
{\rm
additive}&E_{8}^{(1)}&\rightarrow&E_{7}^{(1)}&\rightarrow&E_{6}^{(1)}&&\rightarrow&&D_{4}^{(1)}&&\rightarrow&A_{3}^{(1)}&\rightarrow&A_{1+1}^{(1)}&\rightarrow&A_{1}^{(1)}&\rightarrow&A_{0}^{(1)}\\\
&&&&&&&&&&&&&\searrow&&\searrow\\\
&&&&&&&&&&&&&&A_{2}^{(1)}&\rightarrow&A_{1}^{(1)}&\rightarrow&A_{0}^{(1)}\\\
\end{array}}$
Such a correspondence can be easily seen at classical level, however, the
understanding at quantum level is incomplete so far. Fortunately, for the
$q$-difference Painlevé VI equation [26] relevant for this paper, a natural
quantization was worked out in [23] (see also [24], [31]) based on the
extended affine Weyl group symmetry of type $D_{5}^{(1)}$. Recall that the
Bäcklund transformations for the discrete Painlevé equations are generated by
the affine Weyl group and the automorphisms of the Dynkin diagram, which act
on the dynamical variables as birational transformations. In this paper, the
discrete Painlevé VI equation always (except for Appendix A) means the
quantized one in the sense of [23], where the dynamical variables $(F,G)$ are
non-commutative and the time evolution is defined by the adjoint action of the
Hamiltonian. Since the prefix $q$\- is already used for classical
$q$-difference analogue of Painlevé VI equation, we will call the quantization
of the equation $qq$-Painlevé VI for short, namely the double ‘$q$’ for
$q$-difference and quantization. We warn the reader that the use of $qq$\-
does not mean any direct connection to the $qq$-character introduced in [37].
But there is a similarity in the sense that the full $\Omega$-background
parameters $(q,t)$ [36] are turned on in both cases. The fact that one of the
$\Omega$-background parameters, say $q$, plays the role of the quantization
parameter of integrable systems is the same as the Nekrasov-Shatashvili limit
[39] , which corresponds to the autonomous limit of the Painlevé equations.
The recent paper [44] gives us valuable lessons on the problem of the
quantization of the discrete Painlevé equation. The problem is also discussed
from the viewpoint of the cluster integrable system [8]. The relevant cluster
algebras are associated with the BPS quiver of five dimensional superconformal
field theories [10].
As we have mentioned above, in [23] by constructing a representation of the
extended Weyl group $\widetilde{W}(D_{5}^{(1)})$ on the non-commutative
variables $(F,G)$, explicit forms of the $qq$-Painlevé equation were derived
both in the Heisenberg and the Schrödinger forms;
###### Definition 1.1 (Heisenberg form of $qq$-Painlevé VI).
$\displaystyle\overline{F}F=qb_{7}b_{8}{G+b_{5}\over G+b_{7}}{G+b_{6}\over
G+b_{8}},\qquad G\underline{G}=qb_{3}b_{4}{F+b_{1}\over F+b_{3}}{F+b_{2}\over
F+b_{4}},$
where $\overline{F}=T\cdot F,~{}\underline{G}=T^{-1}\cdot G$ and $T$ is a
translation element in $\widetilde{W}(D_{5}^{(1)})$.
$\mathbf{b}=(b_{1},\cdots,b_{8})$ are the standard parameters for the Painlevé
VI equation (See Appendix A).
###### Definition 1.2 (Schrödinger form of $qq$-Painlevé VI).
$\displaystyle
H_{\mathrm{VI}}\cdot\mathsf{u}(b,G,Q|q,t^{-1})=\mathsf{u}(b,G,Q|q,t^{-1}),$
(1.1)
with the Hamiltonian given by
$\displaystyle
H_{\mathrm{VI}}:=\frac{1}{\varphi(-qb_{5}G^{-1})\varphi(-qb_{6}G^{-1})\varphi(-b_{7}^{-1}G)\varphi(-b_{8}^{-1}G)}\theta(F^{-1}G;q)^{-1}$
$\displaystyle\times\frac{1}{\varphi(qt^{-1/2}b_{1}G^{-1})\varphi(qt^{-1/2}b_{2}G^{-1})\varphi(t^{-1/2}b_{3}^{-1}G)\varphi(t^{-1/2}b_{4}^{-1}G)}\theta(F^{-1}G;q)^{-1}T_{t^{1/4},\mathbf{b}},$
where $\varphi(x):=(x;q)_{\infty}$ and
$\theta(X;q)=(X;q)_{\infty}(q/X;q)_{\infty}$. $T_{t^{1/4},\mathbf{b}}$ is a
shift operator of the parameters $\mathbf{b}$ and $Q$ appears as a parameter
of wave functions.
In this paper we first show that the non-stationary equation proposed in [44]
is successfully identified with the $qq$-Painlevé VI equation. Namely we prove
###### Proposition 1.3.
By an appropriate gauge transformation from $\mathsf{U}(\Lambda,x)$ to
$\mathsf{u}(b,G,Q|q,t^{-1})$, the non-stationary difference equation in [44]
$\displaystyle\mathsf{U}(t\Lambda,x)=\mathcal{A}_{1}(\Lambda,x)\cdot\mathcal{B}\cdot\mathcal{A}_{2}(\Lambda,x)\cdot\mathcal{B}\cdot\mathcal{A}_{3}(\Lambda,x)\mathsf{U}(\Lambda,{x\over
tqQ})$ (1.2)
is transformed to the $qq$-Painlevé VI equation (1.1), where $\mathcal{B}$ is
the $q$-Borel transformation and $\mathcal{A}_{i}(\Lambda,x)$ are
multiplications of $\varphi(x)$ and $\Phi(x):=(x;q,t)_{\infty}$ (see section
2).
In contrast to the original form of the $qq$-Painlevé equation the double
infinite product $\Phi(x):=(x;q,t)_{\infty}$ arises as a consequence of the
gauge transformation. In (1.2) the parameter $x$ is related to the dynamical
variable $G$ and $\Lambda$ plays the role of the time variable. (See
subsection 2.2 for the dictionary of variables between Painlevé side and the
gauge theory side). $\mathsf{U}(\Lambda,x)$ is regarded as a formal power
series in $x$ and $\Lambda/x$, which is motivated by the following conjecture
that the Nekrasov partition function solves the above equation. Hence the
virtue of the gauge transformation in Proposition 1.3 is due to the conjecture
that the Nekrasov partition functions, which allow a combinatorial
description, provide solutions to (1.2).
###### Conjecture 1.4 ([44]; Conjecture 2.4).
Let $Z(\Lambda,x)$ be the the Nekrasov partition function of five dimensional
$SU(2)\times SU(2)$ gauge theory (See Definition 2.3). If we define a function
$\Psi(\Lambda,x)$ by imposing the Higgsing condition on $Z(\Lambda,x)$, then
it gives a solution $\mathsf{U}(\Lambda,x)=\Psi(\Lambda,x)$ to the equation
(1.2).
The time evolution of the discrete Painlevé VI equation is given by a
translation by the element in $D_{5}^{(1)}$ root lattice, which is orthogonal
to the symmetry $D_{4}^{(1)}$ of discrete $P_{\rm VI}$. If we write the
translation element in terms of the generators of the extended affine Weyl
groups, it is a product of two factors which are exchanged by the automorphism
$\tau$ of $D_{5}^{(1)}$ Dynkin graph. The factorization of the original
difference equation as the coupled system reflects this fact. The
decomposition of the discrete time evolution (or the Hamiltonian) by the
Bäcklund transformations implies that the original equation, which is the
second order in the $q$-exponent of the Laplace operator $\mathcal{B}$, can be
rewritten as a coupled system of the first order difference equations in
$\mathcal{B}$, as was already suggested in the original paper [44].
###### Proposition 1.5.
The following coupled system is gauge equivalent to the non-stationary
difference equation of [44] and hence, to the $qq$-Painlevé VI equation;
$\displaystyle\mathsf{V}^{(1)}$
$\displaystyle={\Phi(qt^{-1}b_{2}/b_{4})\Phi(b_{1}/b_{3})\over\Phi(tb_{6}/b_{8})\Phi(qb_{5}/b_{7})}{1\over\varphi(-qb_{6}/G)\varphi(-G/b_{8})}$
$\displaystyle\qquad\cdot(\widetilde{\mathcal{B}}\cdot\widetilde{T}_{\mathsf{p},b}){1\over\varphi(-\mathsf{p}^{-1}qb_{2}/G)\varphi(-\mathsf{p}^{-1}G/b_{4})}\widetilde{T}_{\mathsf{p},b}\cdot\mathsf{V}^{(2)},$
$\displaystyle\widetilde{T}_{\mathsf{p},b}\cdot\mathsf{V}^{(2)}$
$\displaystyle={\Phi(\mathsf{p}^{-2}tb_{6}/b_{8})\Phi(\mathsf{p}^{-2}qb_{5}/b_{7})\over\Phi(\mathsf{p}^{-2}qb_{2}/b_{4})\Phi(\mathsf{p}^{-2}tb_{1}/b_{3})}{1\over\varphi(-\mathsf{p}^{-1}qb_{1}/G)\varphi(-\mathsf{p}^{-1}G/b_{3})}$
$\displaystyle\cdot(\widetilde{\mathcal{B}}\cdot\widetilde{T}_{\mathsf{p},b})\cdot{1\over\varphi(-qb_{5}/G)\varphi(-G/b_{7})}\mathsf{V}^{(1)}.$
To motivate an analogous conjecture to Conjecture 1.4, let us recall that the
instanton counting with a surface defect allows another description in terms
of the affine Laumon spaces [18],[3],[4]. In this method the partition
functions are identified with the conformal blocks of the affine Kac-Moody
algebra (the current algebra) without degenerate fields [30]. For example, in
the present case the parameter $x$ which originally comes from the insertion
point of the degenerate field is replaced by the $SU(2)$ spin variable of
$\widehat{\mathfrak{sl}}_{2}$. The existence of the surface defect is taken
into account by introducing the orbifold action [19],[27]. The relation of two
methods for incorporating a surface defect is discussed in [20] from the
viewpoint of integrable systems. In fact the role of the affine Kac-Moody
algebra was already revealed in [12],[13], where the conjecture that the
prepotential of the Seiberg-Witten theory is obtained from the leading term of
the Nekrasov partition function was proved. We conjecture that the solutions
to the coupled system are provided by the $K$-theoretic instanton partition
function derived from the equivariant character of the affine Laumon space
[18].
###### Conjecture 1.6 (Conjecture 5.4).
The partition function (5.17) gives a solution to the coupled system in
Proposition 1.5 by the following specialization of parameters;
$\displaystyle\mathcal{F}^{(1)}=f\left(\left.\left.\begin{array}[]{ll}q^{1/2}b_{4}/b_{8},&q^{1/2}b_{6}/b_{2}\\\
(\mathsf{p}^{2}Q)^{-1/2},&(\mathsf{p}^{2}Q)^{1/2}\\\
q^{-1/2}b_{2}/b_{5},&q^{-1/2}b_{7}/b_{4}\end{array}\right|q^{1/2}\mathsf{p}^{-1}\mathsf{t}G^{-1},q^{-1/2}\mathsf{p}^{-1}\mathsf{t}G\right|q,t^{-1/2}\right),$
$\displaystyle\mathcal{F}^{(2)}=f\left(\left.\left.\begin{array}[]{ll}q^{1/2}b_{4}/b_{8},&q^{1/2}b_{6}/b_{2}\\\
(\mathsf{p}^{2}Q)^{-1/2},&(\mathsf{p}^{2}Q)^{1/2}\\\
q^{-1/2}b_{1}/b_{6},&q^{-1/2}b_{8}/b_{3}\end{array}\right|-q^{1/2}\mathsf{t}G^{-1},-q^{-1/2}\mathsf{t}G\right|q,t^{-1/2}\right).$
The point here is that due to the symmetry of the translation element $T$
which defines the discrete time evolution of the $qq$-Painlevé VI equation,
the pair of solutions $(\mathcal{F}^{(1)},\mathcal{F}^{(2)})$ comes from the
common instanton partition function of the affine Laumon space with different
specialization of parameters. An action of the quantum toroidal algebra of
$A_{r}$ type on the equivariant cohomology group and the equivariant $K$ group
of the affine Laumon space can be defined geometrically [35]. In four
dimensional case (cohomological version) it has been shown that the instanton
partition function satisfies the Knizhnik-Zamolodchikov (KZ) equation for the
affine Kac-Moody algebra [38],[40]. Hence, the non-stationary difference
equation should be derived as a KZ type equation for the quantum affine
algebra $U_{q}(\widehat{\mathfrak{sl}}_{2})$ or more likely
$U_{q}(\widehat{\mathfrak{gl}}_{2})$. Moreover, recalling that the affine
Laumon space has elliptic cohomology, we believe it is an interesting problem
to show our non-stationary difference equation can be generalized to elliptic
case.
The paper is organized as follows; In the next section we first summarize the
non-stationary difference equation proposed in [44]. We make a gauge
transformation to rewrite it in a form which is natural from the viewpoint of
the $qq$-Painlevé VI equation. We also propose a dictionary between the
variables on the gauge theory side and those on the Painlevé side. In section
3, following [23], we describe the quantization of the discrete Painlevé VI
equation and explain the relation of the non-stationary difference equation
and the $qq$-Painlevé VI equation. In section 4, we introduce the five
dimensional quantum Seiberg-Witten curve. We show the quantum Seiberg-Witten
allows a four dimensional limit and it is consistent with our previous result
[4]. This is regarded as a good support for the conjecture in [44]. In section
5, we propose a coupled system which is gauge equivalent to the $qq$-Painlevé
VI equation. Finally we conjecture the instanton partition functions of the
affine Laumon space are solutions $(\mathcal{F}^{(1)},\mathcal{F}^{(2)})$ to
the coupled system. As a consequence of the Dynkin automorphism symmetry of
the factorization, $\mathcal{F}^{(1)}$ and $\mathcal{F}^{(2)}$ are obtained
from a common instanton partition function with two kinds of specialization of
parameters, which are related by the automorphism $\tau$. A summary of the
discrete Painlevé VI equation is provided in Appendix A. Some of notations and
conventions for the discrete Painlevé VI equation are fixed there. A few
examples for supporting our conjecture in section 5 are presented in Appendix
B. The four dimensinal limit for a factorized form of the Hamiltonian is
discussed in Appendix C.
## 2\. Non-stationary difference equation
###### Definition 2.1.
Let $T_{a,\Lambda}$ and $T_{b,x}$ be the shift operators acting on the
variables $\Lambda$ and $x$ by
$T_{a,\Lambda}f(\Lambda,x)\mathbin{:=}f(a\Lambda,x)$,
$T_{b,x}f(\Lambda,x)\mathbin{:=}f(\Lambda,bx)$. Let
$\vartheta_{x}\mathbin{:=}x\partial_{x}$ be the Euler operator in $x$. We have
$\vartheta_{x}x=x(\vartheta_{x}+1)$, indicating that
$p\mathbin{:=}q^{\vartheta_{x}}$ acts as the $q$-shift operator
$q^{\vartheta_{x}}=T_{q,x}$.
###### Definition 2.2.
Set $\mathcal{B}\mathbin{:=}q^{\vartheta_{x}(\vartheta_{x}+1)/2}$. We define
the action of $\mathcal{B}$ on a formal Laurent series in $x$ as the $q$-Borel
transformation:
$\mathcal{B}(\sum_{n}c_{n}x^{n})=\sum_{n}q^{n(n+1)/2}c_{n}x^{n}.$ (2.1)
The fundamental relations among $x,~{}p=q^{\vartheta_{x}}$ and
$\mathcal{B}=q^{\vartheta_{x}(\vartheta_{x}+1)/2}$ are
$px=qxp,\qquad\mathcal{B}p=p\mathcal{B},\qquad\mathcal{B}x=px\mathcal{B}.$
(2.2)
One can see the last relation by looking at the action on $x^{n}$. In fact,
both sides give the same result; $q^{\frac{1}{2}(n+1)(n+2)}x^{n+1}$. The
$q$-Borel transformation $\mathcal{B}$ (see [21] section 2 and references
therein) plays a significant role in the non-stationary difference equation in
[44].
It is convenient to introduce the notations $\varphi(x)\in\mathbb{Q}(q)[[x]]$
and $\Phi(x)\in\mathbb{Q}(q,t)[[x]]$ for the infinite products;
$\displaystyle\varphi(x):=$
$\displaystyle(x;q)_{\infty}=\prod_{n=0}^{\infty}(1-q^{n}x)=\exp\left(-\sum_{n=1}^{\infty}{1\over
n}{1\over 1-q^{n}}x^{n}\right),$ (2.3) $\displaystyle\Phi(x):=$
$\displaystyle(x;q,t)_{\infty}=\prod_{n,m=0}^{\infty}(1-q^{n}t^{m}x)=\exp\left(-\sum_{n=1}^{\infty}{1\over
n}{1\over(1-q^{n})(1-t^{n})}x^{n}\right).$ (2.4)
They satisfy
$\frac{\Phi(x)}{\Phi(tx)}=\varphi(x),\qquad\Phi(ta\Lambda)^{-1}T_{t,\Lambda}^{-1}\Phi(ta\Lambda)=\varphi(a\Lambda)T_{t,\Lambda}^{-1}.$
(2.5)
We also use the standard notation for the $q$-shifted factorial
$\displaystyle(u;q)_{n}=\prod_{i=0}^{n-1}(1-uq^{i})\qquad(n\in\mathbb{Z}_{\geq
0}).$
See [22] for useful formulas for $(u;q)_{n}$.
### 2.1. Five point function with a degenerate field
The correlation functions of the chiral primary fields $\Phi_{\Delta}(z)$ are
the most fundamental objects in two dimensional conformal field theory with
the energy-momentum tensor $T(z)$ (the generating currents of the Virasoro
algebra). The BPZ equation describes the response of the correlation functions
under the insertion of the descendant fields created by the action of the
Virasoro algebra. The BPZ equation for the five point function on
$\mathbb{P}^{1}$111The instanton partition function we are going to discuss is
expanded in $x$ and $\Lambda/x$. Hence here we assume the radial ordering with
$|x|<1$ and $|\Lambda/x|<1$.
$\Psi_{\mathrm{CFT}}(\Lambda,x):=\langle\Phi_{\Delta_{4}}(\infty)\Phi_{\Delta_{3}}(1)\varphi(x)\Phi_{\Delta_{2}}(\Lambda)\Phi_{\Delta_{1}}(0)\rangle_{\mathbb{P}^{1}}$
(2.6)
with a level two degenerate field $\varphi(x)$ at $x$ is the linear
differential equation of the form
$\left(\partial_{x}^{2}+a(\Lambda,x)\partial_{x}+b(\Lambda,x)+c(\Lambda,x)\partial_{\Lambda}\right)\Psi_{\mathrm{CFT}}(\Lambda,x)=0,$
(2.7)
which has regular singularities at $\\{0,\Lambda,x,1\\}$ and hence, is
identified with the non-stationary Heun equation. We will reserve $t$ for one
of the equivariant parameters ($\Omega$ background) of the torus action on
$\mathbb{C}^{2}$ and $\Lambda$ plays the role of “time” variable in (2.7). In
the non-stationary case, the constant part of the Heun operator involves the
time derivative $\partial_{\Lambda}$. Up to the gauge transformation with the
factor
$x^{\alpha}(x-1)^{\beta}(x-\Lambda)^{\gamma}$ (2.8)
the equation (2.7) agrees with the quantization of the Painlevé VI equation.
The BPZ equation is also obtained from the deformed Seiberg-Witten curve of
four dimensional supersymmetric gauge theory in the Nekrasov-Shatashvili limit
as the non-stationary Schrödinger equation [14],[42]. This is also regarded as
the quantization of (continuous, additive) Painlevé VI equation [4]. What we
are going to discuss in this paper is an uplift of these stories to the
triality of the deformed Virasoro algebra, discrete Painlevé equation and five
dimensional supersymmetric gauge theories.
Recall that in the AGT correspondence $(r+3)$ point conformal blocks on the
genus zero curve are identified with the instanton partition functions of the
linear quiver gauge theory of type $A_{r}$. Let us consider the five
dimensional uplift of the AGT correspondence. . The instanton partition
function is expressed in terms of the $K$-theoretic Nekrasov factor
${\mathsf{N}}_{\lambda,\mu}(u)={\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)$ defined
by
$\displaystyle{\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)$
$\displaystyle=\prod_{(i,j)\in\lambda}(1-uq^{\lambda_{i}-j}\kappa^{-\mu^{\prime}_{j}+i-1})\cdot\prod_{(k,l)\in\mu}(1-uq^{-\mu_{k}+l-1}\kappa^{\lambda^{\prime}_{l}-k}),$
(2.9)
or equivalently
$\displaystyle{\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)=\prod_{j\geq i\geq
1}(uq^{-\mu_{i}+\lambda_{j+1}}\kappa^{-i+j};q)_{\lambda_{j}-\lambda_{j+1}}\cdot\prod_{\beta\geq\alpha\geq
1}(uq^{\lambda_{\alpha}-\mu_{\beta}}\kappa^{\alpha-\beta-1};q)_{\mu_{\beta}-\mu_{\beta+1}}.$
Here $q$ and $\kappa$ are regarded independent indeterminates. The Nekrasov
factor ${\mathsf{N}}_{\lambda,\mu}(u)$ depends on a pair of partitions
$(\lambda,\mu)$, namely $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ is non-
increasing non-negative integers with finitely many positive parts.
$\lambda^{\prime}$ denotes the conjugate of $\lambda$. In [44] the instanton
partition function of five dimensional $SU(2)\times SU(2)$ theory with four
fundamental matter multiplets and one bi-fundamental matter multiplet is
considered. On the deformed conformal block side this corresponds to the five
point function on $\mathbb{P}^{1}$.
###### Definition 2.3.
$\displaystyle\mathcal{Z}(\Lambda,x)$
$\displaystyle:=\sum_{\nu_{1},\nu_{2},\mu_{1},\mu_{2}\in\mathsf{P}}\mathfrak{p}_{1}^{|\nu_{1}|+|\nu_{2}|}\mathfrak{p}_{2}^{|\mu_{1}|+|\mu_{2}|}$
(2.10) $\displaystyle\cdot\prod_{1\leq a,b\leq
2}{{\mathsf{N}}_{\emptyset,\nu_{b}}(v\mathfrak{f}_{a}^{+}/\mathfrak{n}_{b}|q,t^{-1}){\mathsf{N}}_{\nu_{a},\mu_{b}}(w\mathfrak{n}_{a}/\mathfrak{m}_{b}|q,t^{-1}){\mathsf{N}}_{\mu_{a},\emptyset}(v\mathfrak{m}_{a}/\mathfrak{f}^{-}_{b}|q,t^{-1})\over{\mathsf{N}}_{\nu_{a},\nu_{b}}(\mathfrak{n}_{a}/\mathfrak{n}_{b}|q,t^{-1}){\mathsf{N}}_{\mu_{a},\mu_{b}}(\mathfrak{m}_{a}/\mathfrak{m}_{b}|q,t^{-1})},$
where ${\mathsf{P}}$ denotes the set of all partitions and $\emptyset$ is the
empty partition.
The following parametrization was used in [44].
$\displaystyle v=q^{1/2}t^{-1/2},\qquad
w=v\phi_{1},\qquad\mathfrak{p}_{1}=v^{-2}T_{2}\phi_{2}x,\qquad\mathfrak{p}_{2}=v^{-2}{T_{4}\Lambda\over\phi_{1}x},$
$\displaystyle\mathfrak{n}_{1}=1,\qquad\mathfrak{n}_{2}=Q,\qquad\mathfrak{m}_{1}=1,\qquad\mathfrak{m}_{2}=\phi_{1}\phi_{2}Q,$
(2.11)
$\displaystyle\mathfrak{f}_{1}^{+}=T_{1}Q,\qquad\mathfrak{f}_{2}^{+}=T_{2}^{-1},\qquad\mathfrak{f}_{1}^{-}=T_{3}^{-1},\qquad\mathfrak{f}_{2}^{-}=T_{4}\phi_{1}\phi_{2}Q.$
The coefficients of the expansion depend on parameters
$(Q,\phi_{1},\phi_{2},T_{1},\cdots,T_{4})$ and the equivariant parameters
$(q,t)$ of the torus action on $\mathbb{C}^{2}$. The parameters
$(Q,\phi_{1},\phi_{2})$ correspond to the equivariant parameters of the Cartan
subalgebra $U(1)\times U(1)\subset SU(2)\times SU(2)$ of the gauge group and
the mass of the bi-fundamental matter222If we extend the gauge group to
$U(2)\times U(2)$, the (exponentiated) mass parameter of the bi-fundamental
matter may be identified with the equivariant parameter of the relative $U(1)$
factor of the gauge group $U(2)\times U(2)$. Note that the diagonal $U(1)$
factor decouples.. On the other hand the mass parameters $M_{i}$ of the
fundamental hypermultiplets are related to $\log T_{i}$ up to the appropriate
shifts of $\log q=\epsilon_{1},~{}\log t=-\epsilon_{2},~{}\log Q=-2a$. The
instanton partition function $\mathcal{Z}(\Lambda,x)$ is a formal power series
in $(x,\Lambda/x)$, where they are related to the insertion points of the
intertwiners up the $SL(2,\mathbb{R})$ transformation.
Let us consider the degenerate conformal block with the insertion of a level
two degenerate field. Then one of the external Liouville momentum has a
special value and the degenerate fusion rule tells that there are two allowed
values for the intermediate momentum. According to the AGT correspondence this
imposes two conditions on parameters $(Q,\phi_{1},\phi_{2})$ on the quiver
gauge theory side, which is often referred to the Higgsing condition. In the
present case the conditions are explicitly
$\phi_{1}=q^{-1/2}t^{3/2},\phi_{2}=q^{1/2}t^{-1/2}$ (or
$\phi_{1}=q^{-1/2}t^{1/2},\phi_{2}=q^{3/2}t^{-1/2}$). Recall that there are
two possibilities for the intermediate momentum. As a consequence, one of the
equivariant parameters, say $t$, is transmuted to the Higgsing mass parameter.
Later we will see $t^{1/4}=\mathsf{p}$ becomes a basic shift parameter, or the
non-autonomous parameter ($t\to 1$ is the autonomous limit) on the Painlevé
side. In the five brane web realization of the surface defect, the parameter
$t$ is identified with the volume of $S^{3}$ connecting $\mathrm{NS}5$ brane
and $\mathrm{D}5$ brane which are non-intersecting. Note that $\mathrm{D}3$
brane can wrap $S^{3}$ in type IIB string theory.
After imposing the Higgsing condition the instanton partition function
$\Psi(\Lambda,x)=\mathcal{Z}(\Lambda,x)|_{\phi_{1}=q^{-1/2}t^{3/2},\phi_{2}=q^{1/2}t^{-1/2}}$
(2.12)
has parameters $(Q;T_{1},\cdots,T_{4};\Lambda,x;q,t)$. Physically this is the
instanton partition function of the $SU(2)$ gauge theory with a surface
defect333In general the gauge theory allows several types of the surface
defect according to the breaking patters of the total gauge group $SU(N)$
which are labelled by partitions of $N$. But for $SU(2)$ the breaking pattern
is unique $SU(2)\to U(1)$. Mathematically the breaking pattern defines a
parabolic structure along a defect (or a divisor).. The parameter $x$ is the
insertion point of the degenerate field. Then, the conjecture in [44] says
###### Conjecture 2.4 ([44]).
$\displaystyle\Psi(t\Lambda,x)=\mathcal{A}_{1}(\Lambda,x)\cdot\mathcal{B}\cdot\mathcal{A}_{2}(\Lambda,x)\cdot\mathcal{B}\cdot\mathcal{A}_{3}(\Lambda,x)\Psi(\Lambda,{x\over
tqQ}),$ (2.13)
where
$\displaystyle\mathcal{A}_{1}(\Lambda,x)={1\over\varphi(T_{1}tvx)}{\Phi(T_{3}t^{2}v\Lambda
x^{-1})\over\Phi(T_{3}qv\Lambda x^{-1})}{\Phi(T_{4}t^{2}v\Lambda
x^{-1})\over\Phi(T_{4}t^{2}v^{-1}\Lambda x^{-1})},$
$\displaystyle\mathcal{A}_{2}(\Lambda,x)={\varphi(qT_{2}T_{3}\Lambda)\varphi(tT_{1}T_{4}\Lambda)\over\varphi(-T_{1}T_{2}x)\varphi(-Q^{-1}x)\varphi(-T_{3}T_{4}Qqt\Lambda
x^{-1})\varphi(-q\Lambda x^{-1})},$
$\displaystyle\mathcal{A}_{3}(\Lambda,x)={1\over\varphi(T_{2}Q^{-1}q^{-1}vx)}{\Phi(T_{3}Qq^{2}v\Lambda
x^{-1})\over\Phi(T_{3}Qq^{2}v^{-1}\Lambda x^{-1})}{\Phi(T_{4}Qt^{3}v\Lambda
x^{-1})\over\Phi(T_{4}Qq^{2}v^{-1}\Lambda x^{-1})},$
with $v:=q^{1/2}t^{-1/2}$ and $\mathcal{B}$ is the $q$-Borel transformation.
In [44] several evidences for the conjecture have been provided. For example
it was proved for a special choice of mass parameters (external Liouville
momenta) $(T_{1},T_{2},T_{3},T_{4})=(vt^{-1},v^{-1},v^{-1},vt^{-1})$444Recall
that the Higgsing condition is $(\phi_{1},\phi_{2})=(tv^{-1},v)$., where the
solution is expressed in terms of the Macdonald polynomials.
In this paper we will investigate the structure of the difference equation
(2.13), which is independent of the validity of Conjecture 2.4. Hence, let us
replace $\Psi(\Lambda,x)$, which was defined by the Nekrasov partition
function, with a generic function $\mathsf{U}(\Lambda,x)$ and write the
equation as follows;
$\displaystyle\mathsf{U}(t\Lambda,x)=\mathcal{A}_{1}(\Lambda,x)\cdot\mathcal{B}\cdot\mathcal{A}_{2}(\Lambda,x)\cdot\mathcal{B}\cdot\mathcal{A}_{3}(\Lambda,x)\mathsf{U}(\Lambda,{x\over
tqQ}).$ (2.14)
The difference equation (2.14) is called non-stationary, since there appears
the shift $\Lambda\to t\Lambda$ on the left hand side. It was pointed out [44]
that in the mass decoupling limit $T_{1},T_{2},T_{3},T_{4}\to 0$, the
difference equation (2.14) reduces to the non-stationary relativistic affine
Toda equation [45].
###### Remark 2.5.
In the mass decoupling limit the difference equation (2.14)degenerates to
$\mathsf{U}_{\mathrm{Toda}}(t\Lambda,x)=\widehat{H}\mathsf{U}_{\mathrm{Toda}}(\Lambda,x/tqQ),\qquad\widehat{H}=\mathcal{B}\cdot\frac{1}{\varphi(-Q^{-1}x)\varphi(-q\Lambda
x^{-1})}\cdot\mathcal{B}.$ (2.15)
The name ‘Toda’ comes from the following fact.
###### Proposition 2.6.
The operator $\widehat{H}$ commutes with the quantum Hamiltonian of the two-
particle relativistic affine Toda system;
$\hat{H}_{\mathrm{Toda}}=T_{q,x}+tQT_{q,x}^{-1}+tx+\Lambda x^{-1}.$ (2.16)
One can confirm the existence and uniqueness of the solution to the non-
stationary difference equation (2.14):
###### Proposition 2.7.
The equation (2.14) has a formal series solution of the form
$\mathsf{U}(\Lambda,x)=\sum_{i,j=0}^{\infty}c_{i,j}x^{i}(\Lambda/x)^{j},$
(2.17)
and it is unique up to a normalization.
###### Proof.
It is easy to see that the operator
$T_{t,\Lambda}-{\mathcal{A}}_{1}\cdot\mathcal{B}\cdot{\mathcal{A}}_{2}\cdot\mathcal{B}\cdot{\mathcal{A}}_{3}T_{tqQ,x}^{-1}$
is “triangular” in the sense that it sends the monomials
$x^{i}(\Lambda/x)^{j}$ to linear combinations of $x^{i+m}(\Lambda/x)^{j+n}$
$(m,n\geq 0)$. Moreover the leading coefficient with $(m,n)=(0,0)$ is
$t^{j}-q^{(i-j)(i-j+1)}(tqQ)^{j-i}$, which is non vanishing for generic
parameters $t,q$ and $Q$. Hence the coefficients $c_{i,j}$ are uniquely solved
order by order with respect to $i+j$, once the initial value $c_{0,0}$ is
fixed. ∎
We can eliminate $\Phi$-factors (the double infinite products) completely from
the non-stationary difference equation (2.14), by the the following gauge
transformation;
$\mathsf{u}(\Lambda,x)=\Phi(qtT_{2}T_{3}\Lambda)\Phi(t^{2}T_{1}T_{4}\Lambda)\mathcal{A}_{3}(t\Lambda,tqQx)\mathsf{U}(t\Lambda,x).$
(2.18)
Using the relations
$\displaystyle\mathcal{A}_{3}(t\Lambda,tqQx)\mathcal{A}_{1}(\Lambda,x)={1\over\varphi(T_{1}vtx)\varphi(T_{2}vtx)\varphi(T_{3}vt\Lambda
x^{-1})\varphi(T_{4}vt\Lambda x^{-1})},$
$\displaystyle{\Phi(qT_{2}T_{3}\Lambda)\Phi(tT_{1}T_{4}\Lambda)\over\Phi(qtT_{2}T_{3}\Lambda)\Phi(t^{2}T_{1}T_{4}\Lambda)}=\varphi(qT_{2}T_{3}\Lambda)\varphi(tT_{1}T_{4}\Lambda),$
which follow from (2.5), we obtain
###### Proposition 2.8.
The difference equation (2.14) is gauge equivalent to
$\displaystyle H\mathsf{u}(\Lambda,x)=\mathsf{u}(\Lambda,x),$ (2.19)
with the Hamiltonian
$\displaystyle H=$
$\displaystyle{1\over\varphi(T_{1}vtx)\varphi(T_{2}vtx)\varphi(T_{3}vt\Lambda
x^{-1})\varphi(T_{4}vt\Lambda x^{-1})}\cdot\mathcal{B}$
$\displaystyle\cdot{1\over\varphi(-T_{1}T_{2}x)\varphi(-Q^{-1}x)\varphi(-T_{3}T_{4}Qqt\Lambda
x^{-1})\varphi(-q\Lambda x^{-1})}\cdot\mathcal{B}\cdot
T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1}.$ (2.20)
### 2.2. Dictionary of five dimensional gauge/Painlevé correspondence
In this paper we will employ the following relations of the parameters in [44]
and the root variables for the discrete Painlevé VI equation (see Appendix A);
$\displaystyle a_{0}=(T_{4}/T_{3})^{1/4},\quad a_{1}=(T_{1}/T_{2})^{1/4},\quad
a_{2}=(t\Lambda T_{1}T_{4})^{-1/4},$ $\displaystyle a_{3}=(t^{2}\Lambda
T_{3}T_{4})^{1/4},\quad a_{4}=(QT_{1}T_{2})^{1/4},\quad
a_{5}=(tQT_{3}T_{4})^{-1/4}.$ (2.21)
Namely
$\sqrt{Q}T_{1}={a_{1}^{2}a_{4}^{2}},\quad\sqrt{Q}T_{2}={a_{4}^{2}\over
a_{1}^{2}},\quad\sqrt{tQ}T_{3}={1\over
a_{0}^{2}a_{5}^{2}},\quad\sqrt{tQ}T_{4}={a_{0}^{2}\over a_{5}^{2}},$ (2.22)
and
$\displaystyle\mathsf{p}:=e^{\delta}=a_{0}a_{1}a_{2}^{2}a_{3}^{2}a_{4}a_{5}=t^{1/4},\quad\mathsf{t}:=a_{3}^{2}a_{4}a_{5}=t^{1/2}\Lambda(T_{1}T_{2}T_{3}T_{4})^{1/4},$
$\displaystyle{t\Lambda\over Q}=a_{3}^{4}a_{5}^{4}.$ (2.23)
We assume that the parameter $Q$ is invariant under the action of the extended
affine Weyl group. In [23], $\mathsf{t}$ is identified with the “time”
variable. The parameter $\mathsf{p}$ defines the shift parameter of the
discrete time evolution. On the gauge theory side $\Lambda$ plays the role of
the corresponding time variable. The dictionary (2.21) tells that the exchange
of $T_{1}$ and $T_{2}$ corresponds to the Weyl reflection
$r_{1}:(a_{1},a_{2})\to(a_{1}^{-1},a_{1}a_{2})$. Similarly the exchange of
$T_{3}$ and $T_{4}$ gives the action of
$r_{0}:(a_{0},a_{2})\to(a_{0}^{-1},a_{0}a_{2})$ on the Painlevé side. We have
to accept that (2.21) is not invertible. This is due to the fact that the root
variables $a_{i}$ and the parameters $(T_{i},\Lambda,Q)$ on the gauge theory
side are two kinds of variables with constraints in the ambient ten
dimensional Picard lattice of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with eight
points blow-ups (or $\mathbb{P}^{2}$ with nine points blow-ups).
The parameter $x$ is related to a pair of $q$-commuting dynamical variables
$(F,G)$ as
$\displaystyle G=-\xi\qquad\mbox{\rm({\it i.e.} as multiplication by $-\xi$)}$
(2.24) $\displaystyle F=(qt^{1/2}Q)^{-1/2}\xi q^{-\vartheta_{\xi}},$ (2.25)
where
$\xi=\left({q^{2}\Lambda^{2}T_{3}T_{4}\over T_{1}T_{2}}\right)^{1/4}x^{-1}$
(2.26)
and $\vartheta_{\xi}=\xi\partial/\partial\xi$. Note that the dynamical
variables $(F,G)$ satisfy
$FG=q^{-1}GF.$ (2.27)
Parameters | Higgsed quiver theory [44] | Painlevé VI
---|---|---
$T_{i}$ | (dressed) mass parameters | root variables of the outer nodes
$Q$ | $SU(2)$ Coulomb modulus | parameter of solutions
$\Lambda$ | instanton expansion parameter | root variable of the inner node
$x$ | position of degenerate field insertion | dynamical variable (coordinate)
$q$ | $\Omega$ background along the defect | quantization parameter
$t$ | $\Omega$ background orthogonal to the defect | non-autonomous parameter
Table 1. Dictionary between the quiver gauge theory and Painlevé VI equation
By the above dictionary the Hamiltonian (2.8) in terms of the variables of
Painlevé VI equation takes the following formula (for the definition of the
variables $b_{i}$ see Definition A.5 in Appendix A);
$\displaystyle H_{\mathrm{VI}}=$
$\displaystyle{1\over\varphi(-qb_{5}G^{-1})\varphi(-qb_{6}G^{-1})\varphi(-b_{7}^{-1}G)\varphi(-b_{8}^{-1}G)}\cdot\widetilde{\mathcal{B}}$
$\displaystyle\cdot{1\over\varphi(q\mathsf{p}^{-2}b_{1}G^{-1})\varphi(q\mathsf{p}^{-2}b_{2}G^{-1})\varphi(\mathsf{p}^{-2}b_{3}^{-1}G)\varphi(\mathsf{p}^{-2}b_{4}^{-1}G)}\cdot\widetilde{\mathcal{B}}\cdot
T_{t^{1/2},x}^{-1}T_{t,\Lambda}^{-1}.$ (2.28)
For later convenience, by writing
$T_{qtQ,x}^{-1}=(T_{qt^{1/2}Q,x}^{-1/2})^{2}T_{t^{1/2},x}^{-1}$, we have
distributed $T_{qt^{1/2}Q}^{-1/2}$ and combined it with $\mathcal{B}$ to
define $\widetilde{\mathcal{B}}=\mathcal{B}\cdot
T_{(qt^{1/2}Q)^{-1/2},x}=T_{(qt^{1/2}Q)^{-1/2},x}\cdot\mathcal{B}$. Since
$G\sim-x^{-1}$, we have the $(qt^{1/2}Q)^{\pm 1/2}$-shift of the arguments of
$\varphi$ involving $b_{1},\cdots,b_{4}$ after commuting them with
$T_{qt^{1/2}Q,x}^{-1/2}$ in (2.2), compared with (2.8).
In terms of the variables $b_{i}$, the symmetry of the Hamiltonian (2.2),
exchanging $(b_{1},\cdots,b_{4})$ and $(b_{5},\cdots,b_{8})$, becomes more
manifest555The additional factor $\mathsf{p}^{-2}=t^{-1/2}$ for
$b_{1},\cdots,b_{4}$ should be related to the redefinition of $\mathsf{t}$ and
$\mathsf{p}$ to be discussed in the next section (see (3.1)).. The symmetry is
related to the diagram automorphism $\tau$ of $D_{5}^{(1)}$, see Fig.1 in
Appendix A. Geometrically this is the exchange of $\mathbb{P}^{1}$ in the
product $\mathbb{P}^{1}\times\mathbb{P}^{1}$ that appears in the description
of the space of initial conditions, see Fig.2 in Appendix A. We will get back
to this point in section 5.
## 3\. Quantized discrete Painlevé VI equation
### 3.1. Heisenberg form of discrete $P_{\mathrm{VI}}$
We recapitulate a formulation of the $qq$-Painlevé VI equation. Consider a
linear operator ${\bf H}$ of the form
${\bf H}=A(x)\cdot\mathcal{B}\cdot B(x)\cdot\mathcal{B},$ (3.1)
where $A(x),B(x)$ are some formal Laurent series in $x$.
###### Proposition 3.1.
Putting $a(x)\mathbin{:=}A(qx)A(x)^{-1}$, $b(x)\mathbin{:=}B(qx)B(x)^{-1}$, we
have
${\bf H}^{-1}x^{-1}{\bf H}=b(p^{-1}x)x^{-1}p^{2},\quad{\bf H}x^{-1}p{\bf
H}^{-1}=x^{-1}p^{-1}a(x).$ (3.2)
###### Proof.
Since $\mathcal{B}^{-1}x^{-1}\mathcal{B}=x^{-1}p$,
$\mathcal{B}^{-1}x\mathcal{B}=p^{-1}x$, we have
$\displaystyle{\bf H}^{-1}x^{-1}{\bf H}$ $\displaystyle=$
$\displaystyle\mathcal{B}^{-1}B(x)^{-1}\mathcal{B}^{-1}A(x)^{-1}x^{-1}A(x)\mathcal{B}B(x)\mathcal{B}$
$\displaystyle=$
$\displaystyle\mathcal{B}^{-1}B(x)^{-1}\mathcal{B}^{-1}x^{-1}\mathcal{B}B(x)\mathcal{B}$
$\displaystyle=$
$\displaystyle\mathcal{B}^{-1}B(x)^{-1}x^{-1}pB(x)\mathcal{B}=\mathcal{B}^{-1}B(x)^{-1}B(qx)x^{-1}\mathcal{B}p$
$\displaystyle=$
$\displaystyle\mathcal{B}^{-1}b(x)x^{-1}\mathcal{B}p=b(p^{-1}x)x^{-1}p^{2}.$
Similarly, using $\mathcal{B}x^{-1}p\mathcal{B}^{-1}=x^{-1}$.
$\mathcal{B}x^{-1}\mathcal{B}^{-1}=x^{-1}p^{-1}$, we have
$\displaystyle{\bf H}x^{-1}p{\bf H}^{-1}$ $\displaystyle=$ $\displaystyle
A(x)\mathcal{B}B(x)\mathcal{B}x^{-1}p\mathcal{B}^{-1}B(x)^{-1}\mathcal{B}^{-1}A(x)^{-1}$
$\displaystyle=$ $\displaystyle
A(x)\mathcal{B}B(x)x^{-1}B(x)^{-1}\mathcal{B}^{-1}A(x)^{-1}$ $\displaystyle=$
$\displaystyle
A(x)\mathcal{B}x^{-1}\mathcal{B}^{-1}A(x)^{-1}=A(x)x^{-1}p^{-1}A(x)^{-1}$
$\displaystyle=$ $\displaystyle x^{-1}p^{-1}A(qx)A(x)^{-1}=x^{-1}p^{-1}a(x).$
Hence the desired equations are proved. ∎
We specify the functions $A(x),B(x)$ as
$A(x)=\frac{1}{\varphi(\mathsf{a}_{1}x)\varphi(\mathsf{a}_{2}x)\varphi(\frac{q}{\mathsf{a}_{3}x})\varphi(\frac{q}{\mathsf{a}_{4}x})},\quad
B(x)=\frac{1}{\varphi(-\mathsf{b}_{1}x)\varphi(-\mathsf{b}_{2}x)\varphi(-\frac{q}{\mathsf{b}_{3}x})\varphi(-\frac{q}{\mathsf{b}_{4}x})},$
(3.3)
where the infinite products are considered as formal Laurent series in $x$ by
the expansion
$\frac{1}{\varphi(z)}=\frac{1}{(z;q)_{\infty}}=\sum_{n=0}^{\infty}\frac{z^{n}}{(q)_{n}}.$
(3.4)
We note that the gauge transformed Hamiltonian (2.8) is reproduced by the
choice of parameters;
$\displaystyle\mathsf{a}_{1}=q^{1/2}t^{1/2}T_{1},\qquad\mathsf{a}_{2}=q^{1/2}t^{1/2}T_{2},$
$\displaystyle\mathsf{a}_{3}=q^{1/2}t^{-1/2}T_{3}^{-1}\Lambda^{-1},\qquad\mathsf{a}_{4}=q^{1/2}t^{-1/2}T_{4}^{-1}\Lambda^{-1},$
(3.5) $\displaystyle\mathsf{b}_{1}=T_{1}T_{2},\qquad\mathsf{b}_{2}=Q^{-1},$
$\displaystyle\mathsf{b}_{3}=t^{-1}Q^{-1}(T_{3}T_{4})^{-1}\Lambda^{-1},\qquad\mathsf{b}_{4}=\Lambda^{-1}.$
(3.6)
###### Corollary 3.2.
We put
$\tilde{F}\mathbin{:=}x^{-1}p,\quad\tilde{G}\mathbin{:=}-x^{-1}\quad{\rm
i.e.}\quad x=-\tilde{G}^{-1},\quad
p=\tilde{G}^{-1}\tilde{F},\quad\tilde{G}\tilde{F}=q\tilde{F}\tilde{G},$ (3.7)
then we have
${\bf H}\tilde{F}{\bf
H}^{-1}=\frac{\mathsf{a}_{3}\mathsf{a}_{4}}{q\tilde{F}}{\displaystyle\frac{(\tilde{G}+\mathsf{a}_{1})(\tilde{G}+\mathsf{a}_{2})}{(\tilde{G}+\mathsf{a}_{3})(\tilde{G}+\mathsf{a}_{4})}},\quad{\bf
H}^{-1}\tilde{G}{\bf
H}={\displaystyle\frac{(\tilde{F}+\mathsf{b}_{1})(\tilde{F}+\mathsf{b}_{2})}{(\tilde{F}+\mathsf{b}_{3})(\tilde{F}+\mathsf{b}_{4})}}\frac{\mathsf{b}_{3}\mathsf{b}_{4}}{q\tilde{G}}.$
(3.8)
###### Proof.
For the function $A(x),B(x)$ in (3.3), we see
$a(x)={\displaystyle\frac{(1-\mathsf{a}_{1}x)(1-\mathsf{a}_{2}x)}{(1-\mathsf{a}_{3}x)(1-\mathsf{a}_{4}x)}}\mathsf{a}_{3}\mathsf{a}_{4}x^{2},\quad
b(x)={\displaystyle\frac{(1+\mathsf{b}_{1}x)(1+\mathsf{b}_{2}x)}{(1+\mathsf{b}_{3}x)(1+\mathsf{b}_{4}x)}}\mathsf{b}_{3}\mathsf{b}_{4}x^{2}.$
(3.9)
Then from eq. (3.2), we have
$\displaystyle{\bf H}\tilde{F}{\bf H}^{-1}$ $\displaystyle=$ $\displaystyle
x^{-1}p^{-1}a(x)=\frac{\mathsf{a}_{3}\mathsf{a}_{4}}{q\tilde{F}}{\displaystyle\frac{(1+\mathsf{a}_{1}/\tilde{G})(1+\mathsf{a}_{2}/\tilde{G})}{(1+\mathsf{a}_{3}/\tilde{G})(1+\mathsf{a}_{4}/\tilde{G)}}},$
(3.10) $\displaystyle{\bf H}^{-1}\tilde{G}{\bf H}$ $\displaystyle=$
$\displaystyle
b(p^{-1}x)x^{-1}p^{2}={\displaystyle\frac{(1-\mathsf{b}_{1}/\tilde{F})(1-\mathsf{b}_{2}/\tilde{F})}{(1-\mathsf{b}_{3}/\tilde{F})(1-\mathsf{b}_{4}/\tilde{F})}}\frac{\mathsf{b}_{3}\mathsf{b}_{4}}{q\tilde{G}},$
(3.11)
as desired. ∎
The equation (3.8) can be viewed as the Heisenberg form of the $qq$-Painlevé
VI equation.
Now let us check by an appropriate rescaling
$\displaystyle x^{-1}p=\alpha^{-1}F,\quad x^{-1}=-\beta^{-1}G,$ (3.12)
the equation (3.8) agrees with the Heisenberg equations first presented in
[23]. Combining the dictionary (3.1)– (3.6) with (3.9), we can rewrite (3.2)
as follows;
$\displaystyle{\bf H}^{-1}x^{-1}{\bf H}=$
$\displaystyle{(1+T_{1}T_{2}p^{-1}x)(1+Q^{-1}p^{-1}x)\over(1+T_{3}T_{4}Qt\Lambda
x^{-1}p)(1+\Lambda x^{-1}p)}x^{-1}p^{2}$ $\displaystyle=$ $\displaystyle
q^{-1}x\mathsf{b}_{3}\mathsf{b}_{4}{(x^{-1}p+q^{-1}\mathsf{b}_{1})(x^{-1}p+q^{-1}\mathsf{b}_{2})\over(x^{-1}p+q^{-1}\mathsf{b}_{3})(x^{-1}p+q^{-1}\mathsf{b}_{4})},$
$\displaystyle{\bf H}x^{-1}p{\bf H}^{-1}=$ $\displaystyle
x^{-1}p^{-1}{(1-q^{1/2}t^{1/2}T_{1}x)(1-q^{1/2}t^{1/2}T_{2}x)\over(1-q^{-1/2}t^{1/2}T_{3}\Lambda
x^{-1})(1-q^{-1/2}t^{1/2}T_{4}\Lambda x^{-1})}$ $\displaystyle=$
$\displaystyle
q^{-1}\mathsf{a}_{3}\mathsf{a}_{4}\frac{(x^{-1}-q^{-1}\mathsf{a}_{1})(x^{-1}-q^{-1}\mathsf{a}_{2})}{(x^{-1}-q^{-1}\mathsf{a}_{3})(x^{-1}-q^{-1}\mathsf{a}_{4})}p^{-1}x.$
We see an agreement with (3.8), which confirms the consistency of the
dictionary (3.1)– (3.6). Under the scaling (3.12), we have
$\displaystyle{\bf H}F{\bf H}^{-1}$ $\displaystyle={\alpha^{2}\over
T_{3}T_{4}\Lambda^{2}t}{(G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{2})(G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2})\over(G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{2})(G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2})}F^{-1},$
(3.13) $\displaystyle{\bf H}^{-1}G{\bf H}=$ $\displaystyle{\beta^{2}\over
T_{3}T_{4}Qqt\Lambda^{2}}G^{-1}{(F+\mathsf{t}a_{4}^{2})(F+\mathsf{t}a_{4}^{-2})\over(F+\mathsf{t}^{-1}a_{5}^{2})(F+\mathsf{t}^{-1}a_{5}^{-2})},$
(3.14)
where we have defined
$\displaystyle\mathsf{t}\mathsf{p}^{-1}a_{1}^{2}=\beta
q^{-1/2}t^{1/2}T_{1},\quad\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2}=\beta
q^{-1/2}t^{1/2}T_{2},$
$\displaystyle\mathsf{t}^{-1}\mathsf{p}a_{0}^{2}={\beta\over
q^{1/2}t^{1/2}T_{3}\Lambda},\quad\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2}={\beta\over
q^{1/2}t^{1/2}T_{4}\Lambda},$ (3.15)
and
$\mathsf{t}a_{4}^{2}=\alpha q^{-1}T_{1}T_{2},\quad\mathsf{t}a_{4}^{-2}=\alpha
q^{-1}Q^{-1},\quad\mathsf{t}^{-1}a_{5}^{2}={\alpha\over
T_{3}T_{4}Qqt\Lambda},\quad\mathsf{t}^{-1}a_{5}^{-2}={\alpha\over q\Lambda}.$
(3.16)
From (3.1) and (3.16) we find
$\displaystyle\beta^{4}{T_{1}T_{2}\over
T_{3}T_{4}\Lambda^{2}q^{2}}=1,\quad\alpha^{4}{T_{1}T_{2}\over
T_{3}T_{4}Q^{2}q^{4}t\Lambda^{2}}=1,$
which implies
$\alpha^{2}=(qt^{1/2}Q)\cdot\beta^{2}.$ (3.17)
Hence, we can eliminate $\alpha$ and $\beta$ from (3.13) and (3.14) as
follows;
$\displaystyle{\bf H}F{\bf H}^{-1}=(qt^{1/2}Q)\times
q\mathsf{p}^{2}\mathsf{t}^{-2}{(G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{2})(G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2})\over(G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{2})(G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2})}F^{-1},$
$\displaystyle{\bf H}^{-1}G{\bf H}=(qt^{1/2}Q)^{-1}\times
q\mathsf{t}^{-2}G^{-1}{(F+\mathsf{t}a_{4}^{2})(F+\mathsf{t}a_{4}^{-2})\over(F+\mathsf{t}^{-1}a_{5}^{2})(F+\mathsf{t}^{-1}a_{5}^{-2})}.$
The final step to a complete matching with [23],[24] requires a further
redefinition
$\displaystyle\widetilde{F}=(qt^{1/2}Q)^{-1}F,\qquad\widetilde{G}=G,$
$\displaystyle\widetilde{\mathsf{t}}=t^{1/2}\mathsf{t},\qquad\widetilde{\mathsf{p}}=t^{1/2}\mathsf{p},\qquad\widetilde{\mathsf{p}}^{4}=t.$
(3.18)
Then we have
###### Proposition 3.3.
$\displaystyle{\bf H}\widetilde{F}{\bf
H}^{-1}=q\widetilde{\mathsf{p}}^{2}\widetilde{\mathsf{t}}^{-2}{(\widetilde{G}+\widetilde{\mathsf{t}}\widetilde{\mathsf{p}}^{-1}a_{1}^{2})(\widetilde{G}+\widetilde{\mathsf{t}}\widetilde{\mathsf{p}}^{-1}a_{1}^{-2})\over(\widetilde{G}+\widetilde{\mathsf{t}}^{-1}\widetilde{\mathsf{p}}a_{0}^{2})(\widetilde{G}+\widetilde{\mathsf{t}}^{-1}\widetilde{\mathsf{p}}a_{0}^{-2})}\widetilde{F}^{-1},$
(3.19) $\displaystyle{\bf H}^{-1}\widetilde{G}{\bf
H}=q\widetilde{\mathsf{t}}^{-2}\widetilde{G}^{-1}{(\widetilde{F}+\widetilde{\mathsf{t}}a_{4}^{2})(\widetilde{F}+\widetilde{\mathsf{t}}a_{4}^{-2})\over(\widetilde{F}+\widetilde{\mathsf{t}}^{-1}a_{5}^{2})(\widetilde{F}+\widetilde{\mathsf{t}}^{-1}a_{5}^{-2})}.$
(3.20)
Summarizing all the above definitions and rescalings, we arrive at the
dictionary presented in section 2.2.
### 3.2. Coxeter relations for quantum variables
To formulate the quantization of the discrete $P_{\rm VI}$ Equation reviewed
in Appendix A, we first quantize the commutative canonical pair of variables
$f,g$. Let $F$ and $G$ be non commutative variables satisfying the
$q$-commutation relation666$F$ and $G$ are exponentiated variables of $f$ and
$g$.
$\displaystyle FG=q^{-1}GF.$ (3.21)
Recall the notation $\mathbb{K}=\mathbb{C}(a)=\mathbb{C}(a_{0},\ldots,a_{5})$
for the rational function field in the root variables $a_{i}$. Let
$\mathbb{K}\langle F,G\rangle$ be the $\mathbb{K}$-algebra generated by $F,G$
with the relation (3.21). It is known that $\mathbb{K}\langle F,G\rangle$ is
an Ore domain (see [31] section 2 and references therein). Denote by
$\mathbb{F}=\mathbb{K}(F,G)$ the quotient skew field of $\mathbb{K}\langle
F,G\rangle$. Note that $\mathbb{F}$ is generated by $a_{0},\ldots,a_{5},F$ and
$G$. For any formal power series $h(z)$ in $z$, we use the formula;
${\rm Ad}(F)\cdot h(G)=h(q^{-1}G),\qquad{\rm Ad}(G)\cdot h(F)=h(qF),$ (3.22)
in our computations.
As is summarized in Appendix A, the time evolution of the discrete Painlevé VI
equation is derived from the translation element $T$ in the extended affine
Weyl group $\widetilde{W}$. Hence, we need an action of $\widetilde{W}$ on the
quantum pair of dynamical variables $(F,G)$.
###### Definition 3.4.
Define the actions of
$r_{0},\ldots,r_{5},\sigma_{01},\sigma_{45},\tau\in\widetilde{W}$ on the
generators of $\mathbb{F}$ by the rules:
$\displaystyle r_{0}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0}^{-1},a_{1},a_{0}a_{2},a_{3},a_{4},a_{5},F,G),$
$\displaystyle r_{1}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0},a_{1}^{-1},a_{1}a_{2},a_{3},a_{4},a_{5},F,G),$
$\displaystyle r_{2}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0}a_{2},a_{1}a_{2},a_{2}^{-1},a_{2}a_{3},a_{4},a_{5},F{a_{0}a_{1}^{-1}G+a_{2}^{2}\over
a_{0}a_{1}^{-1}a_{2}^{2}G+1},G),$ $\displaystyle r_{3}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0},a_{1},a_{2}a_{3},a_{3}^{-1},a_{3}a_{4},a_{3}a_{5},F,{a_{3}^{2}a_{4}a_{5}^{-1}F+1\over
a_{4}a_{5}^{-1}F+a_{3}^{2}}G),$ $\displaystyle r_{4}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0},a_{1},a_{2},a_{3}a_{4},a_{4}^{-1},a_{5},F,G),$
$\displaystyle r_{5}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0},a_{1},a_{2},a_{3}a_{5},a_{4},a_{5}^{-1},F,G),$
$\displaystyle\sigma_{01}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{1}^{-1},a_{0}^{-1},a_{2}^{-1},a_{3}^{-1},a_{4}^{-1},a_{5}^{-1},qF^{-1},G),$
$\displaystyle\sigma_{45}$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{0}^{-1},a_{1}^{-1},a_{2}^{-1},a_{3}^{-1},a_{5}^{-1},a_{4}^{-1},F,qG^{-1}),$
$\displaystyle\tau$
$\displaystyle:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\mapsto(a_{5}^{-1},a_{4}^{-1},a_{3}^{-1},a_{2}^{-1},a_{1}^{-1},a_{0}^{-1},G,F).$
Then extend them to the actions on $\mathbb{F}$ by the requirements: (1)
$r_{0},\ldots,r_{5}$ are the ring homomorphisms, (2)
$\sigma_{01},\sigma_{45},\tau$ are the ring anti-homomorphisms.
Note that we have defined $\sigma_{01},\sigma_{45},\tau$ as ring anti-
homomorphisms. Compare it with Definition A.1 in Appendix A, where they are
ring homomorphisms.
###### Proposition 3.5.
These actions on $\mathbb{F}$ are compatible with the group structure of the
extended affine Weyl group $\widetilde{W}$. Namely they satisfy the Coxeter
relations given in Proposition A.2.
One can check the Coxeter relations by straightforward calculations. We only
present two nontrivial cases.
###### Lemma 3.6.
We have
$\displaystyle r_{3}(F)=F,\qquad r_{2}r_{3}r_{2}(F)=r_{3}r_{2}(F),$
$\displaystyle r_{2}(G)=G,\qquad r_{3}r_{2}r_{3}(G)=r_{2}r_{3}(G).$
Hence we have the Coxeter relations
$\displaystyle r_{2}r_{3}r_{2}(F)=r_{3}r_{2}r_{3}(F),\qquad
r_{2}r_{3}r_{2}(G)=r_{3}r_{2}r_{3}(G).$
###### Proof.
We have777The variables $b_{i}$ are defined in Appendix A (see Definition
A.5).
$\displaystyle r_{2}(F)=\sqrt{b_{5}\over b_{7}}F{G+b_{7}\over G+b_{5}},$
$\displaystyle r_{3}r_{2}(F)=\sqrt{b_{3}b_{5}\over b_{1}b_{7}}F{{F+b_{3}\over
F+b_{1}}G+b_{7}\over{F+b_{3}\over F+b_{1}}G+b_{1}^{-1}b_{3}b_{5}},$
$\displaystyle r_{2}r_{3}r_{2}(F)=\sqrt{b_{3}b_{5}\over
b_{1}b_{7}}F{G+b_{7}\over G+b_{5}}{A\over B},$
where
$\displaystyle A={F{G+b_{7}\over G+b_{5}}+b_{3}\over F{G+b_{7}\over
G+b_{5}}+b_{1}b_{5}^{-1}b_{7}}G+b_{5},\qquad B={F{G+b_{7}\over
G+b_{5}}+b_{3}\over F{G+b_{7}\over
G+b_{5}}+b_{1}b_{5}^{-1}b_{7}}G+b_{1}^{-1}b_{3}b_{5}.$
Note that we have $AB=BA$.
Using
$\displaystyle{F+b_{3}\over F+b_{1}}G+b_{7}={1\over
F+b_{1}}(F(G+b_{7})+b_{3}(G+b_{1}b_{3}^{-1}b_{7})),$
$\displaystyle{F+b_{3}\over F+b_{1}}G+b_{1}^{-1}b_{3}b_{5}={1\over
F+b_{1}}(F(G+b_{1}^{-1}b_{3}b_{5})+b_{3}(G+b_{5})),$
we obtain
$\displaystyle r_{3}r_{2}(F)=\sqrt{b_{3}b_{5}\over b_{1}b_{7}}F{1\over
F(G+b_{1}^{-1}b_{3}b_{5})+b_{3}(G+b_{5})}(F(G+b_{7})+b_{3}(G+b_{1}b_{3}^{-1}b_{7})).$
On the other hand, from
$\displaystyle A=C^{-1}(F(G+b_{7})+b_{3}(G+b_{1}b_{3}^{-1}b_{7})),$
$\displaystyle B=C^{-1}(F(G+b_{1}^{-1}b_{3}b_{5})+b_{3}(G+b_{5})){G+b_{7}\over
G+b_{5}},$ $\displaystyle C=F{G+b_{7}\over G+b_{5}}+b_{1}b_{5}^{-1}b_{7},$
we have
$\displaystyle r_{2}r_{3}r_{2}(F)=\sqrt{b_{3}b_{5}\over b_{1}b_{7}}F{1\over
F(G+b_{1}^{-1}b_{3}b_{5})+b_{3}(G+b_{5})}(F(G+b_{7})+b_{3}(G+b_{1}b_{3}^{-1}b_{7})),$
indicating that we have $r_{2}r_{3}r_{2}(F)=r_{3}r_{2}(F)$. ∎
###### Lemma 3.7.
We have
$\displaystyle\sigma_{01}r_{2}(F,G)=\sigma_{01}(F{a_{0}a_{1}^{-1}G+a_{2}^{2}\over
a_{0}a_{1}^{-1}a_{2}^{2}G+1},G)=({a_{0}a_{1}^{-1}G+a_{2}^{-2}\over
a_{0}a_{1}^{-1}a_{2}^{-2}G+1}qF^{-1},G),$ $\displaystyle
r_{2}\sigma_{01}(F,G)=r_{2}(qF^{-1},G)=(q{a_{0}a_{1}^{-1}a_{2}^{2}G+1\over
a_{0}a_{1}^{-1}G+a_{2}^{2}}F^{-1},G),$
and
$\displaystyle\sigma_{01}r_{3}(F,G)=\sigma_{01}(F,{a_{3}^{2}a_{4}a_{5}^{-1}F+1\over
a_{4}a_{5}^{-1}F+a_{3}^{2}}G)=(qF^{-1},G{a_{3}^{-2}a_{4}^{-1}a_{5}qF^{-1}+1\over
a_{4}^{-1}a_{5}qF^{-1}+a_{3}^{-2}}),$ $\displaystyle
r_{3}\sigma_{01}(F,G)=r_{3}(qF^{-1},G)=(qF^{-1},{a_{3}^{2}a_{4}a_{5}^{-1}F+1\over
a_{4}a_{5}^{-1}F+a_{3}^{2}}G).$
indicating that we have
$\sigma_{01}r_{2}(F,G)=r_{2}\sigma_{01}(F,G),\sigma_{01}r_{3}(F,G)=r_{3}\sigma_{01}(F,G)$.
###### Proposition 3.8.
Let $T=r_{2}r_{1}r_{0}r_{2}\sigma_{01}r_{3}r_{4}r_{5}r_{3}\sigma_{45}$ be the
translation element in $\widetilde{W}$. Writing $\overline{F}=T\cdot
F,~{}\underline{G}=T^{-1}\cdot G$ for short, we have the $qq$-Painlevé VI
equation;
$\displaystyle\overline{F}F$
$\displaystyle=q\mathsf{p}^{2}\mathsf{t}^{-2}{G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{2}\over
G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{2}}{G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2}\over
G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2}},$ (3.23) $\displaystyle G\underline{G}$
$\displaystyle=q\mathsf{t}^{-2}{F+\mathsf{t}a_{4}^{2}\over
F+\mathsf{t}^{-1}a_{5}^{2}}{F+\mathsf{t}a_{4}^{-2}\over
F+\mathsf{t}^{-1}a_{5}^{-2}},$ (3.24)
where the discrete shift of the time variable is
$T(\mathsf{t})=\mathsf{p}^{-2}\mathsf{t}$.
Note that compared with the classical version, there appears the factor $q$ on
the right hand side.
###### Proof.
We can compute the action of the reflections and the diagram automorphisms as
follows;
$\displaystyle
F\mathop{\longmapsto}^{\sigma_{45}}F\mathop{\longmapsto}^{r_{3}}F\mathop{\longmapsto}^{r_{5}}F\mathop{\longmapsto}^{r_{4}}F\mathop{\longmapsto}^{r_{3}}F$
$\displaystyle\mathop{\longmapsto}^{\sigma_{01}}qF^{-1}$
$\displaystyle\mathop{\longmapsto}^{r_{2}}q{a_{0}a_{1}^{-1}a_{2}^{2}G+1\over
a_{0}a_{1}^{-1}G+a_{2}^{2}}F^{-1}$
$\displaystyle\mathop{\longmapsto}^{r_{0}}q{a_{0}a_{1}^{-1}a_{2}^{2}G+1\over
a_{0}^{-1}a_{1}^{-1}G+a_{0}^{2}a_{2}^{2}}F^{-1}$
$\displaystyle\mathop{\longmapsto}^{r_{1}}q{a_{0}a_{1}^{3}a_{2}^{2}G+1\over
a_{0}^{-1}a_{1}G+a_{0}^{2}a_{1}^{2}a_{2}^{2}}F^{-1}$
$\displaystyle\mathop{\longmapsto}^{r_{2}}q{a_{0}a_{1}^{3}a_{2}^{2}G+1\over
a_{0}^{-1}a_{1}G+a_{0}^{2}a_{1}^{2}a_{2}^{2}}{a_{0}a_{1}^{-1}a_{2}^{2}G+1\over
a_{0}a_{1}^{-1}G+a_{2}^{2}}F^{-1}$
$\displaystyle=q\mathsf{p}^{2}\mathsf{t}^{-2}{G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{2}\over
G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{2}}{G+\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2}\over
G+\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2}}F^{-1}.$
Similarly for $G$;
$\displaystyle
G\mathop{\longmapsto}^{r_{2}}G\mathop{\longmapsto}^{r_{1}}G\mathop{\longmapsto}^{r_{0}}G\mathop{\longmapsto}^{r_{2}}G\mathop{\longmapsto}^{\sigma_{01}}G$
$\displaystyle\mathop{\longmapsto}^{r_{3}}{a_{3}^{2}a_{4}a_{5}^{-1}F+1\over
a_{4}a_{5}^{-1}F+a_{3}^{2}}G$
$\displaystyle\mathop{\longmapsto}^{r_{4}}{a_{3}^{2}a_{4}a_{5}^{-1}F+1\over
a_{4}^{-1}a_{5}^{-1}F+a_{3}^{2}a_{4}^{2}}G$
$\displaystyle\mathop{\longmapsto}^{r_{5}}{a_{3}^{2}a_{4}a_{5}^{3}F+1\over
a_{4}^{-1}a_{5}F+a_{3}^{2}a_{4}^{2}a_{5}^{2}}G$
$\displaystyle\mathop{\longmapsto}^{r_{3}}{a_{3}^{2}a_{4}a_{5}^{3}F+1\over
a_{4}^{-1}a_{5}F+a_{3}^{2}a_{4}^{2}a_{5}^{2}}{a_{3}^{2}a_{4}a_{5}^{-1}F+1\over
a_{4}a_{5}^{-1}F+a_{3}^{2}}G$
$\displaystyle\mathop{\longmapsto}^{\sigma_{45}}qG^{-1}{a_{3}^{-2}a_{4}a_{5}^{-1}F+1\over
a_{4}a_{5}^{-1}F+a_{3}^{-2}}{a_{3}^{-2}a_{4}^{-3}a_{5}^{-1}F+1\over
a_{4}^{-1}a_{5}F+a_{3}^{-2}a_{4}^{-2}a_{5}^{-2}}$
$\displaystyle=qG^{-1}\mathsf{t}^{-2}{F+\mathsf{t}a_{4}^{2}\over
F+\mathsf{t}^{-1}a_{5}^{2}}{F+\mathsf{t}a_{4}^{-2}\over
F+\mathsf{t}^{-1}a_{5}^{-2}}.$
∎
In terms of the variables $b_{i}$ defined by Definition A.5, we can write the
equations (3.23) and (3.24) as follows;
$\displaystyle\overline{F}F=qb_{7}b_{8}{G+b_{5}\over G+b_{7}}{G+b_{6}\over
G+b_{8}},\qquad G\underline{G}=qb_{3}b_{4}{F+b_{1}\over F+b_{3}}{F+b_{2}\over
F+b_{4}}.$
By taking the conjugation (the adjoint action) by $F$ or $G^{-1}$ (see
(3.22)), we can also write the equations in the following manner;
$\displaystyle
F\overline{F}=q^{-1}\tilde{b}_{7}\tilde{b}_{8}{G+\tilde{b}_{5}\over
G+\tilde{b}_{7}}{G+\tilde{b}_{6}\over
G+\tilde{b}_{8}},\qquad\underline{G}G=q^{-1}\tilde{b}_{3}\tilde{b}_{4}{F+\tilde{b}_{1}\over
F+\tilde{b}_{3}}{F+\tilde{b}_{2}\over F+\tilde{b}_{4}},$
where $\tilde{b}_{i}=qb_{i}$.
### 3.3. Adjoint action and Yang-Baxter relation
In the Heisenberg form of the discrete Painlevé equations the Hamiltonian acts
on the dynamical variables $(F,G)$ by the adjoint action. Hence, we have to
work out the adjoint action of the affine Weyl group generators including the
diagram automorphism $\sigma=\sigma_{01}\sigma_{45}$ in the translation (See
Appendix A). The fundamental part is to realize the birational transformation
of the non-commutative variables $(F,G)$ by the Weyl reflections $r_{2}$ and
$r_{3}$ as the adjoint action. We can achieve it by using the following
function [23];
###### Definition 3.9.
For $X\in\mathbb{F}$, and $z\in\mathbb{K}$, i.e. when we have $zX=Xz$, set
$\displaystyle\theta(X;q)=(X;q)_{\infty}(q/X;q)_{\infty},$ (3.25)
$\displaystyle
R(z,X)={(-X;q)_{\infty}(-qX^{-1};q)_{\infty}\over(-z^{-1}X;q)_{\infty}(-z^{-1}qX^{-1};q)_{\infty}}={\theta(-X;q)\over(-z^{-1}X;q)_{\infty}(-z^{-1}qX^{-1};q)_{\infty}}.$
(3.26)
Recall that $\mathbb{F}=\mathbb{K}(F,G)$ is the quotient skew field of
$\mathbb{K}\langle F,G\rangle$. The function $R(z,X)$ satisfies the following
formulas;
###### Lemma 3.10.
$\displaystyle R(z,X)=R(z,qX^{-1}),$ (3.27) $\displaystyle
R(z,X)R(z^{-1},X)={\theta(-X;q)\theta(-X;q)\over\theta(-zX;q)\theta(-z^{-1}X;q)}=R(z,qX)R(z^{-1},qX).$
(3.28)
We will need the following Yang-Baxter relation to check the Coxeter relations
among the adjoint actions of $R_{i}$ to be defined shortly (see Prop 3.17)
[23].
###### Proposition 3.11 ([17]).
We have the Yang-Baxter equation
$\displaystyle R(x,F)R(xy,G)R(y,F)=R(y,G)R(xy,F)R(x,G).$ (3.29)
The proof given in [16] was based on the Ramanujan’s summation formula, which
implies the expansion formula of $R(z,X)$;
###### Remark 3.12.
Ramanujan’s summation formula for the bilateral basic hypergeometric series
$\displaystyle{}_{1}\psi_{1}(a;b;q;z)=\sum_{n=-\infty}^{\infty}{(a;q)_{n}\over(b;q)_{n}}z^{n}$
$\displaystyle={(q;q)_{\infty}(b/a;q)_{\infty}(az;q)_{\infty}(q/az;q)_{\infty}\over(b;q)_{\infty}(q/a;q)_{\infty}(z;q)_{\infty}(b/az;q)_{\infty}}\qquad(|b/a|<z<1),$
(3.30)
gives
$\displaystyle{(q;q)_{\infty}(q/z^{2};q)_{\infty}\over(q/z;q)_{\infty}(q/z;q)_{\infty}}R(z,X)=\sum_{n=-\infty}^{\infty}{(z;q)_{n}\over(q/z;q)_{n}}(-z^{-1}X)^{n}.$
(3.31)
Essentially the same relation as (3.29) is proved in [34], where an elementary
proof by the Heine’s formula is provided. It is also worth mentioning that the
Yang-Baxter relation (3.29) is closely related to the quantum dilogarithmic
identities;
###### Proposition 3.13 ([29]).
We have the five term identity
$\displaystyle(-F;q)_{\infty}(-G;q)_{\infty}=(-G;q)_{\infty}(-FG;q)_{\infty}(-F;q)_{\infty}.$
(3.32)
Let us begin with the action of the affine Weyl group. The action on the root
variables is easily obtained by introducing the dual letters.
###### Definition 3.14.
Let $\partial_{0},\ldots,\partial_{5}$ be the dual letters associated with the
simple roots satisfying $[\partial_{j},\alpha_{k}]=a_{jk}$. Set
$\displaystyle\rho_{i}=e^{{\pi\over 2}\sqrt{-1}\alpha_{i}\partial_{i}}.$
(3.33)
Later we also use the dual letters associated with the fundamental weights
satisfying $[\partial_{j}^{\prime},\alpha_{k}]=\delta_{jk}$ for a realization
of the adjoint action of the diagram automorphism.
###### Lemma 3.15.
The action of the affine Weyl group on $\mathbb{K}=\mathbb{C}(a)$ is realized
by the adjoint action
$\displaystyle r_{i}\cdot a_{j}=\rho_{i}a_{j}\rho_{i}^{-1}.$ (3.34)
Among the Weyl reflections $r_{i}$, only $r_{2}$ and $r_{3}$ act on $(F,G)$
non-trivially. We can show they are realized by the adjoint action of
$R(z,X)$.
###### Lemma 3.16.
$\displaystyle R\left(a_{2}^{2},{a_{0}\over
a_{1}}G\right)F\,R\left(a_{2}^{2},{a_{0}\over
a_{1}}G\right)^{-1}=F{{a_{0}\over a_{1}}G+a_{2}^{2}\over{a_{0}\over
a_{1}}a_{2}^{2}G+1}=r_{2}\cdot F,$ (3.35) $\displaystyle
R\left(a_{3}^{2},{a_{5}\over a_{4}}F\right)G\,R\left(a_{3}^{2},{a_{5}\over
a_{4}}F\right)^{-1}={{a_{5}\over a_{4}}a_{3}^{2}F+1\over{a_{5}\over
a_{4}}F+a_{3}^{2}}G=r_{3}\cdot G.$ (3.36)
###### Proof.
For any parameter $z$, the computation goes as follows;
$\displaystyle R(z,G)FR(z,G)^{-1}=FR(z,qG)R(z,G)^{-1}=F{1+z^{-1}G\over
1+z^{-1}G^{-1}}G^{-1}=F{G+z\over zG+1},$ $\displaystyle
R(z,F)GR(z,F)^{-1}=R(z,F)R(z,qF)^{-1}G={1+z^{-1}F^{-1}\over
1+z^{-1}F}FG={zF+1\over F+z}G.$
∎
Combining these two lemmas, we obtain the following result.
###### Proposition 3.17.
If we define
$\displaystyle R_{i}=\rho_{i}\qquad(i=0,1,4,5),$ (3.37) $\displaystyle
R_{2}=R\left(a_{2}^{2},{a_{0}\over a_{1}}G\right)\rho_{2},\qquad
R_{3}=R\left(a_{3}^{2},{a_{4}\over a_{5}}F\right)\rho_{3},$ (3.38)
the action of the affine Weyl group on the skew field $\mathbb{F}$ is realized
by the adjoint action
$\displaystyle r_{i}\cdot u={\rm Ad}(R_{i})u=R_{i}uR_{i}^{-1}\qquad(0\leq
i\leq 5,u\in\mathbb{F}).$ (3.39)
As we have mentioned before, one can check ${\rm Ad}(R_{i})$ satisfy the
Coxeter relation by using the Yang-Baxter relation (3.29).
Next let us consider the adjoint representation of the diagram automorphism
$\sigma:=\sigma_{01}\sigma_{45}$888Though $\sigma_{01}$ and $\sigma_{45}$ are
anti-homomorphisms, the composition $\sigma$ is a ring homomorphism.. Since
$\sigma:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\longmapsto(a_{1},a_{0},a_{2},a_{3},a_{5},a_{4},qF^{-1},qG^{-1}),$
(3.40)
the action on the root variables $a_{i}$ is realized by
${\rm
Ad}(\rho_{2}\rho_{1}\rho_{0}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{3}):(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},F,G)\longmapsto(a_{1},a_{0},\mathsf{p}a_{2},\mathsf{p}^{-1}a_{3},a_{5},a_{4},F,G).$
(3.41)
up to the scaling of $(a_{2},a_{3})$, which is nothing but the action of the
translation element $T$. On the other hand, the adjoint representation of
$\sigma$ on the dynamical variables $(F,G)$ is given by suitable combinations
of the theta functions with non-commutative variables in $\mathbb{F}$;
###### Lemma 3.18.
We have
${\rm Ad}\Bigl{(}\theta(F^{-1}G;q)^{-1}\Bigr{)}F=-G,\qquad{\rm
Ad}\Bigl{(}\theta(F^{-1}G;q)^{-1}\Bigr{)}G=-q^{-1}F^{-1}G^{2}.$ (3.42)
###### Proof.
Similarly to the proof of Lemma 3.16, we can compute;
$\displaystyle\theta(F^{-1}G;q)^{-1}F$
$\displaystyle=F\frac{1}{(qF^{-1}G;q)_{\infty}(G^{-1}F;q)_{\infty}}$
$\displaystyle=F\frac{1-F^{-1}G}{1-G^{-1}F}\theta(F^{-1}G;q)^{-1}=-G\cdot\theta(F^{-1}G;q)^{-1},$
$\displaystyle\theta(F^{-1}G;q)^{-1}G$
$\displaystyle=G\frac{1}{(qF^{-1}G;q)_{\infty}(G^{-1}F;q)_{\infty}}$
$\displaystyle=-GF^{-1}G\theta(F^{-1}G;q)^{-1}=-q^{-1}F^{-1}G^{2}\cdot\theta(F^{-1}G;q)^{-1}.$
∎
###### Lemma 3.19.
For any $z\in\mathbb{K}$, i.e. when $zF=Fz$ and $zG=Gz$, we have
$\displaystyle{\rm
Ad}\Bigl{(}\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}\Bigr{)}F=FG^{2},\quad{\rm
Ad}\Bigl{(}\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}\Bigr{)}F^{-1}=G^{-2}F^{-1},$
$\displaystyle{\rm
Ad}\Bigl{(}\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}\Bigr{)}G=G.$
Note that the right hand sides of the above relations are independent of $z$.
###### Proof.
The second equation follows from the first. The third one is trivial. We can
check the first equation as follows;
$\displaystyle\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}F$
$\displaystyle=F\frac{(1-zG)(1-z^{-1}G)}{(1-z^{-1}G^{-1})(1-zG^{-1})}\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}$
$\displaystyle=FG^{2}\cdot\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}.$
∎
Combining these lemmas we find
$\displaystyle{\rm
Ad}\left(\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}\theta(F^{-1}G;q)^{-1}\right)F=-G,$
(3.43) $\displaystyle{\rm
Ad}\left(\left(\theta(zG;q)\theta(z^{-1}G;q)\right)^{-1}\theta(F^{-1}G;q)^{-1}\right)G=-qF^{-1}.$
(3.44)
Hence, the square of this adjoint action999We can specialize the parameter $z$
of the first and the second actions at our disposal. agrees with the action of
$\sigma$ on $(F,G)\mapsto(qF^{-1},qG^{-1})$.
On the other hand, if we define
$S:=e^{\frac{\pi}{2}\sqrt{-1}(\alpha_{0}-\alpha_{1})(\partial^{\prime}_{0}-\partial^{\prime}_{1})}e^{\frac{\pi}{2}\sqrt{-1}(\alpha_{4}-\alpha_{5})(\partial^{\prime}_{4}-\partial^{\prime}_{4})},$
(3.45)
the adjoint action $\mathrm{Ad}(S)$ gives the same action as
$(\rho_{2}\rho_{1}\rho_{0}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{3})^{-1}T_{\mathsf{p},a_{2}}T_{\mathsf{p},a_{3}}^{-1}:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5})\mapsto(a_{1},a_{0},a_{2},a_{3},a_{5},a_{4})$
on the root variables. Hence, the desired realization of $\sigma$ by the
adjoint action is
###### Proposition 3.20.
If we define
$\displaystyle\Sigma:=\left(\theta\left(-{a_{0}\over
a_{1}}G;q\right)\theta\left(-{a_{1}\over
a_{0}}G;q\right)\right)^{-1}\theta(F^{-1}G;q)^{-1}$
$\displaystyle\qquad\cdot\left(\theta\left({a_{4}\over
a_{5}}G;q\right)\theta\left({a_{5}\over
a_{4}}G;q\right)\right)^{-1}\theta(F^{-1}G;q)^{-1}\cdot S,$
the action of the element $\sigma=\sigma_{01}\sigma_{45}$ on the skew field
$\mathbb{F}$ is realized by the adjoint action
$\displaystyle\sigma\cdot u={\rm Ad}(\Sigma)u\qquad(u\in\mathbb{F}).$
### 3.4. Hasegawa operator
###### Proposition 3.21.
Define the Hasegawa operator $H$ for the $qq$-Painlevé VI equation as
$\displaystyle
H_{\mathrm{VI}}=R_{2}R_{1}R_{0}R_{2}R_{3}R_{4}R_{5}R_{3}\Sigma,$ (3.46)
then
$\displaystyle
H_{\mathrm{VI}}=\frac{1}{\varphi(-qb_{5}G^{-1})\varphi(-qb_{6}G^{-1})\varphi(-b_{7}^{-1}G)\varphi(-b_{8}^{-1}G)}\theta(F^{-1}G;q)^{-1}$
$\displaystyle\times\frac{1}{\varphi(q\mathsf{p}^{-2}b_{1}G^{-1})\varphi(q\mathsf{p}^{-2}b_{2}G^{-1})\varphi(\mathsf{p}^{-2}b_{3}^{-1}G)\varphi(\mathsf{p}^{-2}b_{4}^{-1}G)}\theta(F^{-1}G;q)^{-1}T_{\mathsf{p},a_{2}}T_{\mathsf{p},a_{3}}^{-1}$
(3.47)
and the non-stationary difference equation (2.14) is equivalent to
$\displaystyle
H_{\mathrm{VI}}\cdot\mathsf{u}(b,G,Q|q,t^{-1})=\mathsf{u}(b,G,Q|q,t^{-1}).$
###### Proof.
We proceed as follows;
$\displaystyle R_{2}R_{1}R_{0}R_{2}R_{3}R_{4}R_{5}R_{3}\Sigma$
$\displaystyle=$ $\displaystyle R\left(a_{2}^{2},{a_{0}\over
a_{1}}G\right)\rho_{2}\rho_{1}\rho_{0}R\left(a_{2}^{2},{a_{0}\over
a_{1}}G\right)\rho_{2}R\left(a_{3}^{2},{a_{4}\over
a_{5}}F\right)\rho_{3}\rho_{4}\rho_{5}R\left(a_{3}^{2},{a_{4}\over
a_{5}}F\right)\rho_{3}\Sigma$ $\displaystyle=$ $\displaystyle
R\left(a_{2}^{2},{a_{0}\over
a_{1}}G\right)R\left((a_{0}a_{1}a_{2})^{2},{a_{1}\over
a_{0}}G\right)R\left(\mathsf{p}^{2}(a_{3}a_{4}a_{5})^{-2},{a_{4}\over
a_{5}}F\right)R\left(\mathsf{p}^{2}a_{3}^{-2},{a_{5}\over a_{4}}F\right)$
$\displaystyle\cdot\left(\theta\left(-{a_{0}\over
a_{1}}G;q\right)\theta\left(-{a_{1}\over
a_{0}}G;q\right)\right)^{-1}\theta(F^{-1}G;q)^{-1}\left(\theta\left({a_{4}\over
a_{5}}G;q\right)\theta\left({a_{5}\over a_{4}}G;q\right)\right)^{-1}$
$\displaystyle\theta(F^{-1}G;q)^{-1}\cdot
T_{\mathsf{p},a_{2}}T_{\mathsf{p},a_{3}}^{-1}$ $\displaystyle=$ $\displaystyle
R\left(a_{2}^{2},{a_{0}\over
a_{1}}G\right)R\left((a_{0}a_{1}a_{2})^{2},{a_{1}\over
a_{0}}G\right)\left(\theta\left(-{a_{0}\over
a_{1}}G;q\right)\theta\left(-{a_{1}\over
a_{0}}G;q\right)\right)^{-1}\theta(F^{-1}G;q)^{-1}$ $\displaystyle\cdot
R\left(\mathsf{p}^{2}(a_{3}a_{4}a_{5})^{-2},-{a_{4}\over
a_{5}}G\right)R\left(\mathsf{p}^{2}a_{3}^{-2},-{a_{5}\over
a_{4}}G\right)\left(\theta\left({a_{4}\over
a_{5}}G;q\right)\theta\left({a_{5}\over a_{4}}G;q\right)\right)^{-1}$
$\displaystyle\theta(F^{-1}G;q)^{-1}\cdot
T_{\mathsf{p},a_{2}}T_{\mathsf{p},a_{3}}^{-1}$ $\displaystyle=$
$\displaystyle{1\over(-{a_{0}\over a_{1}a_{2}^{2}}G;q)_{\infty}(-{a_{1}\over
a_{0}a_{2}^{2}}qG^{-1};q)_{\infty}}{1\over(-{1\over
a_{0}^{3}a_{1}a_{2}^{2}}G;q)_{\infty}(-{1\over
a_{0}a_{1}^{3}a_{2}^{2}}qG^{-1};q)_{\infty}}\theta(F^{-1}G;q)^{-1}$
$\displaystyle\cdot{1\over({a_{3}^{2}a_{4}a_{5}^{3}\over\mathsf{p}^{2}}G;q)_{\infty}({a_{3}^{2}a_{4}^{3}a_{5}\over\mathsf{p}^{2}}qG^{-1};q)_{\infty}}{1\over({a_{3}^{2}a_{4}\over\mathsf{p}^{2}a_{5}}G;q)_{\infty}({a_{3}^{2}a_{5}\over\mathsf{p}^{2}a_{4}}qG^{-1};q)_{\infty}}.\theta(F^{-1}G;q)^{-1}T_{\mathsf{p},a_{2}}T_{\mathsf{p},a_{3}}^{-1}.$
∎
We note that the combination of the shift operators
$T_{\mathsf{p},a_{2}}T_{\mathsf{p},a_{3}}^{-1}$ keeps the constraints
$a_{0}a_{1}a_{2}^{2}a_{3}^{2}a_{4}a_{5}=t^{1/4}$ intact. Hence, the desired
identification is completed, if we can show the adjoint action of
$\theta(F^{-1}G;q)^{-1}$ and
$\widetilde{\mathcal{B}}=T_{(qt^{1/2}Q)^{-1/2},x}\cdot\mathcal{B}$ are the
same. The adjoint action of $\theta(F^{-1}G;q)^{-1}$ is already given in Lemma
3.18. On the other hand the fundamental commutation relation (2.2) for
$\mathcal{B}$ implies
${\rm Ad}\mathcal{B}(\tilde{F})=-\tilde{G},\qquad{\rm
Ad}\mathcal{B}(\tilde{F})=-q^{-1}\tilde{F}^{-1}\tilde{G}^{2},$ (3.48)
for $\tilde{F}=x^{-1}p,~{}\tilde{G}=-x^{-1}$. Recall that we have rescaled
$\tilde{F}$ and $\tilde{G}$. This is the reason why we have to combine
$\mathcal{B}$ with the shift operator $T_{(qt^{1/2}Q)^{-1/2},x}$ whose adjoint
action produces the necessary multiplication factor for $\tilde{F}$ and
$\tilde{G}$.
As is well-known, the Painlevé equations are derived from the isomonodromic
deformation of linear system. It is an interesting problem to find the Lax
operators which give rise to the $qq$-Painlevé VI equation. Note that the
monodromy problem is naturally associated with the Yang-Baxter relation of the
universal $R$ matrix. In fact in [24] the universal $R$ matrix of
$U_{q}(\widehat{\mathfrak{sl}}_{2})$ was used to define local Lax matrices for
the $qq$-Painlevé VI equation, which is a good starting point to work out the
problem completely.
## 4\. Quantum Seiberg-Witten curve
### 4.1. Gauge transformation and $U(1)$ factor
By the following gauge transformation;
$\displaystyle{1\over\Phi(t^{2}T_{1}T_{3}\Lambda)\Phi(qtT_{2}T_{4}\Lambda)}\varphi(q^{1/2}t^{1/2}T_{2}x)\varphi(q^{1/2}t^{1/2}T_{4}\Lambda/x)\mathsf{U}(\Lambda,x)=\mathsf{v}^{(1)}(\Lambda,x),$
(4.1)
the $qq$-Painlevé VI equation is recast to
$\widetilde{H}\mathsf{v}^{(1)}(\Lambda,x)=\mathsf{v}^{(1)}(\Lambda,x)$ with
$\displaystyle\widetilde{H}={1\over\varphi(T_{1}q^{1/2}t^{1/2}x)\varphi(T_{3}q^{1/2}t^{1/2}\Lambda
x^{-1})}\cdot\mathcal{B}$
$\displaystyle\qquad\cdot{\varphi(tT_{1}T_{3}\Lambda)\varphi(qT_{2}T_{4}\Lambda)\over\varphi(-T_{1}T_{2}x)\varphi(-Q^{-1}x)\varphi(-T_{3}T_{4}Qqt\Lambda
x^{-1})\varphi(-q\Lambda x^{-1})}\cdot\mathcal{B}$ $\displaystyle\qquad\cdot
T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1}\cdot{1\over\varphi(T_{2}q^{1/2}t^{1/2}x)\varphi(T_{4}q^{1/2}t^{1/2}\Lambda
x^{-1})}.$ (4.2)
Recall that we have made the gauge transformation twice;
$\mathsf{U}(\Lambda,x)\to\mathsf{u}(\Lambda,x)\to\mathsf{v}^{(1)}(\Lambda,x)$.
We can see the total gauge factor from the original system of Higgsed quiver
gauge theory is simply
$\frac{\Phi(qtT_{2}T_{3}\Lambda)\Phi(t^{2}T_{1}T_{4}\Lambda)}{\Phi(t^{2}T_{1}T_{3}\Lambda)\Phi(qtT_{2}T_{4}\Lambda)}\frac{\Phi(T_{3}q^{3/2}t^{-1/2}\Lambda/x)\Phi(T_{4}q^{-1/2}t^{5/2}\Lambda/x)}{\Phi(T_{3}q^{1/2}t^{1/2}\Lambda/x)\Phi(T_{4}q^{1/2}t^{3/2}\Lambda/x)}.$
(4.3)
Note that in the original definition of $\mathcal{A}_{3}$ in [44] the four
dimensional limit is not well-defined. But after the above gauge
transformation we can take the four dimensional limit. In fact let us look at
the gauge transformation factor in the plethystic form;
$\displaystyle\frac{\Phi(qtT_{2}T_{3}\Lambda)\Phi(t^{2}T_{1}T_{4}\Lambda)}{\Phi(t^{2}T_{1}T_{3}\Lambda)\Phi(qtT_{2}T_{4}\Lambda)}$
$\displaystyle=\exp\left(-\sum_{n=1}^{\infty}\frac{(q^{n}T_{2}^{n}-t^{n}T_{1}^{n})(T_{3}^{n}-T_{4}^{n})}{n(1-q^{n})(1-t^{n})}(t\Lambda)^{n}\right)$
$\displaystyle\to(1-\Lambda)^{\frac{(\mu_{3}-\mu_{4})(\mu_{2}-\mu_{1}-\epsilon_{1}-\epsilon_{2})}{\epsilon_{1}\epsilon_{2}}},\qquad\hbar\to
0,$ (4.4)
$\displaystyle\frac{\Phi(T_{3}q^{3/2}t^{-1/2}\Lambda/x)\Phi(T_{4}q^{-1/2}t^{5/2}\Lambda/x)}{\Phi(T_{3}q^{1/2}t^{1/2}\Lambda/x)\Phi(T_{4}q^{1/2}t^{3/2}\Lambda/x)}$
$\displaystyle=\exp\left(-\sum_{n=1}^{\infty}\frac{(1-q^{n}t^{-n})(T_{4}^{n}q^{-n/2}t^{5n/2}-T_{3}^{n}q^{n/2}t^{n/2})}{n(1-q^{n})(1-t^{n})}(\Lambda/x)^{n}\right)$
$\displaystyle\to(1-\Lambda/x)^{\frac{(\epsilon_{1}+\epsilon_{2})(\mu_{4}-\mu_{3}+\epsilon_{1}+2\epsilon_{2})}{\epsilon_{1}\epsilon_{2}}},\qquad\hbar\to
0,$ (4.5)
where we set $q=e^{\hbar\epsilon_{1}},t^{-\hbar\epsilon_{2}}$ and
$T_{i}=e^{-\hbar\mu_{i}}$. This four dimensional limit completely matches with
what we have found for the four dimensional instanton partition functions.
This gauge transformation is also regarded as what is called the $U(1)$ factor
in the AGT correspondence [2]. Thus the gauge transformation (4.3) is a five
dimensional uplift of the $U(1)$ factor. The five dimensional $U(1)$ factor
for the Nekrasov partition function has been proposed in [25],[6], where it is
associated with pairs of parallel external lines in the fivebrane web diagram.
However, this $U(1)$ factor does not recover that of [2] in the four
dimensional limit. In this sense the relation of the $U(1)$ factor in [25],[6]
and our $U(1)$ factor derived above is not clear at the moment.
To simplify the computation with the quantum Seiberg-Witten curve, let us make
the change of parameters and variables;
$\displaystyle\Lambda^{\prime}$ $\displaystyle=tT_{1}T_{3}\Lambda,\qquad
x^{\prime}=T_{1}q^{-1/2}t^{1/2}x,$ (4.6) $\displaystyle
d_{1}=T_{1}^{-1}q^{1/2}t^{-1/2}Q^{-1},\quad d_{2}$
$\displaystyle=T_{2}q^{1/2}t^{-1/2},\quad
d_{3}=T_{3}^{-1}q^{1/2}t^{-1/2},\quad d_{4}=T_{4}q^{1/2}t^{1/2}Q,$ (4.7)
so that $\tilde{H}$ can be written as $\tilde{H}=\SS
T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1}$ with
$\displaystyle\SS=$
$\displaystyle\frac{1}{\varphi(qx)\varphi(\Lambda/x)}\cdot\mathcal{B}\cdot\frac{\varphi(\Lambda)\varphi(q^{-1}d_{1}d_{2}d_{3}d_{4}\Lambda)}{\varphi(-d_{1}x)\varphi(-d_{2}x)\varphi(-d_{3}\Lambda/x)\varphi(-d_{4}\Lambda/x)}$
$\displaystyle~{}~{}\cdot\mathcal{B}\cdot\frac{1}{\varphi(q^{-1}d_{1}d_{2}x)\varphi(d_{3}d_{4}\Lambda/x)}.$
(4.8)
where we delete ′ of $x^{\prime}$ and $\Lambda^{\prime}$. Note that by the
change of variables (4.6) and (4.7), we have eliminated the parameter $t$ from
(4.1). In fact we can transform $\SS$ to ${\bf H}$ defined by (3.1) by the
multiplication of
$\varphi(\Lambda)\varphi(q^{-1}d_{1}d_{2}d_{3}d_{4}\Lambda)$, which commutes
with $\mathcal{B}$ and the adjoint action of
$\varphi(q^{-1}d_{1}d_{2}x)\varphi(d_{3}d_{4}\Lambda/x)$. It is this form of
$\tilde{H}$ that naturally leads to the solutions by the instanton counting of
the affine Laumon space with the following property. The instanton partition
function is a formal double power series in $x$ and $\Lambda/x$;
$\mathcal{Z}(\Lambda,x)=\sum_{m,n\geq 0}c_{m,n}x^{m}(\Lambda/x)^{n},$ (4.9)
where the coefficients $c_{m,n}$ are functions of $Q,T_{i},q$ and $t$. One of
the characterestic features is that the “boundary” coefficients $c_{0,n}$ and
$c_{m,0}$ factorize, which is a consequence of the fact that there is a unique
fixed point of the torus action on the instanton moduli space, which
corresponds to the topological number $n=0$ or $m=0$ This means $\SS$ should
be regarded as the Hamiltonian on the gauge theory side.
### 4.2. Five dimensional quantum Seiberg-Witten curve
When $t=1$, the $qq$-Painlevé VI equation becomes an autonomous system which
admits a conserved curve. The curve is identified with a quantization of the
Seiberg-Witten curve for the corresponding gauge theory. Here we work out such
a quantum curve based on the commutativity with the operator $\SS$.
###### Proposition 4.1.
Let $\mathcal{D}=\mathcal{D}(x,p)$ be an operator of the form
$\mathcal{D}=\sum_{i,j}c_{i,j}x^{i}p^{j},$ (4.10)
with nonzero coefficients
$\left[\begin{array}[]{ccc}c_{-1,1}&c_{0,1}&c_{1,1}\\\
c_{-1,0}&c_{0,0}&c_{1,0}\\\ c_{-1,-1}&c_{0,-1}&c_{1,-1}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}\Lambda\mu&-\frac{\mu
q^{2}+\Lambda d_{1}d_{2}}{q}&d_{1}d_{2}\\\
-\Lambda\mu\left(d_{3}+d_{4}\right)&u&-d_{1}-d_{2}\\\ \Lambda\mu
d_{3}d_{4}&-\Lambda\mu d_{3}d_{4}-1&1\\\ \end{array}\right].$ (4.11)
Then it satisfies the relation
$\SS^{-1}\mathcal{D}(x,p)\SS=\mathcal{D}(\mu x,p),$ (4.12)
where $u,\mu\in\mathbb{C}$ are free parameters.
###### Proof.
We compute the successive transformations of the operators
$\mathcal{D}\mapsto\mathcal{D}_{1}\mapsto\cdots\mapsto\mathcal{D}_{6},$ (4.13)
under the adjoint actions ${\rm Ad}(X):\mathcal{D}\mapsto X\mathcal{D}X^{-1}$
for factors $X$ in $\SS^{-1}$. In the following we will display the operators
$\mathcal{D}_{i}$ by their coefficient matrices $(c_{i,j})$101010 Initialy the
range (support) of the indices $(i,j)$ is $-1\leq i,j\leq 1$. However, by the
adjoint action the range of the power of $x$ is extended to $-2\leq i\leq 2$.
See also the Nowton polygons displayed below..
First, under the gauge transformation ${\rm
Ad}\big{(}{\varphi(qx)\varphi(\frac{\Lambda}{x})}\big{)}:p\mapsto{\displaystyle\frac{1-qx}{1-\frac{\Lambda}{qx}}}p,$
we have
$\mathcal{D}\mapsto\mathcal{D}_{1}=\left[\begin{array}[]{ccccc}0&0&-q\mu&\mu
q^{2}+d_{1}d_{2}&-qd_{1}d_{2}\\\
0&-\Lambda\mu\left(d_{3}+d_{4}\right)&u&-d_{1}-d_{2}&0\\\ -\Lambda^{2}\mu
d_{3}d_{4}&\Lambda\left(\mu d_{3}d_{4}+1\right)&-1&0&0\end{array}\right].$
(4.14)
Next, by the action of ${\rm Ad}(\mathcal{B}^{-1}):x^{i}\mapsto
q^{-i(i+1)/2}x^{i}p^{-i}$, we obtain
$\mathcal{D}_{1}\mapsto\mathcal{D}_{2}=\left[\begin{array}[]{ccccc}-\frac{\Lambda^{2}\mu
d_{3}d_{4}}{q}&-\Lambda\mu\left(d_{3}+d_{4}\right)&-q\mu&0&0\\\
0&\Lambda\left(\mu d_{3}d_{4}+1\right)&u&\frac{\mu q^{2}+d_{1}d_{2}}{q}&0\\\
0&0&-1&-\frac{d_{1}+d_{2}}{q}&-\frac{d_{1}d_{2}}{q^{2}}\end{array}\right].$
(4.15)
Then by the gauge transformation ${\rm
Ad}\big{(}\varphi(-d_{1}x)\varphi(-d_{3}\frac{\Lambda}{x})\big{)}:p\mapsto{\displaystyle\frac{(1+d_{1}x)}{(1+d_{3}\frac{\Lambda}{qx})}}p,$
we have
$\mathcal{D}_{2}\mapsto\mathcal{D}_{3}=\left[\begin{array}[]{ccc}-\Lambda\mu
d_{4}&-\mu\left(q+\Lambda d_{1}d_{4}\right)&-q\mu d_{1}\\\ \Lambda\left(\mu
d_{3}d_{4}+1\right)&u&\frac{\mu q^{2}+d_{1}d_{2}}{q}\\\ -\Lambda
d_{3}&-\frac{q+\Lambda d_{2}d_{3}}{q}&-\frac{d_{2}}{q}\end{array}\right].$
(4.16)
And by further gauge transformation ${\rm
Ad}\big{(}\varphi(-d_{2}x)\varphi(-d_{4}\frac{\Lambda}{x})\big{)}:p\mapsto{\displaystyle\frac{(1+d_{2}x)}{(1+d_{4}\frac{\Lambda}{qx})}}p,$
we have
$\mathcal{D}_{3}\mapsto\mathcal{D}_{4}=\left[\begin{array}[]{ccccc}0&0&-q\mu&-q\mu\left(d_{1}+d_{2}\right)&-q\mu
d_{1}d_{2}\\\ 0&\Lambda\left(\mu d_{3}d_{4}+1\right)&u&\frac{\mu
q^{2}+d_{1}d_{2}}{q}&0\\\
-\Lambda^{2}d_{3}d_{4}&-\Lambda\left(d_{3}+d_{4}\right)&-1&0&0\\\
\end{array}\right].$ (4.17)
Then the action of ${\rm Ad}(\mathcal{B}^{-1})$ gives
$\mathcal{D}_{4}\mapsto\mathcal{D}_{5}=\left[\begin{array}[]{ccccc}-\frac{\Lambda^{2}d_{3}d_{4}}{q}&\Lambda\left(\mu
d_{3}d_{4}+1\right)&-q\mu&0&0\\\
0&-\Lambda\left(d_{3}+d_{4}\right)&u&-\mu\left(d_{1}+d_{2}\right)&0\\\
0&0&-1&\frac{\mu q^{2}+d_{1}d_{2}}{q^{2}}&-\frac{\mu
d_{1}d_{2}}{q^{2}}\end{array}\right].$ (4.18)
Finally, by the gauge transformation ${\rm
Ad}\big{(}\varphi(\frac{d_{1}d_{2}x}{q})\varphi(d_{3}d_{4}\frac{\Lambda}{x})\big{)}:p\mapsto{\displaystyle\frac{1-\frac{d_{1}d_{2}}{q}x}{1-\frac{\Lambda
d_{3}d_{4}}{qx}}}p,$ we have
$\mathcal{D}_{5}\mapsto\mathcal{D}_{6}=\left[\begin{array}[]{ccc}\Lambda&-\frac{\mu
q^{2}+\Lambda d_{1}d_{2}}{q}&\mu d_{1}d_{2}\\\
-\Lambda\left(d_{3}+d_{4}\right)&u&-\mu\left(d_{1}+d_{2}\right)\\\ \Lambda
d_{3}d_{4}&-\Lambda\mu d_{3}d_{4}-1&\mu\end{array}\right].$ (4.19)
In total, we obtained the relation
$\SS^{-1}\mathcal{D}\SS=\mathcal{D}_{6}=\mathcal{D}|_{x\to\mu x},$ (4.20)
as desired. ∎
###### Corollary 4.2.
If we set $\mu=(tqQ)^{-1}$, then we have
${\tilde{H}}^{-1}\mathcal{D}{\tilde{H}}=T_{t,\Lambda}(\mathcal{D}),$ (4.21)
where ${\tilde{H}}=\SS T_{tqQ,x}^{-1}T_{t,\Lambda}^{-1}$. Namely, the operator
$\mathcal{D}$ with $\mu=(tqQ)^{-1}$ is conserved under the evolution by
$\tilde{H}$ in autonomous case: $t=1$.
###### Remark 4.3.
The converse of the Proposition 4.1 is also true. Namely, for a general
operator $\mathcal{D}$ of the form (4.10) with the initial support
$i,j\in\\{-1,0,1\\}$, the condition (4.12) with fixed $\mu$ determines the
coefficients $c_{i,j}$ up to two free parameters. In fact, in order to keep
the form of the operator $\mathcal{D}$ under the successive transformations as
above, one has six linear constraints on the nine coefficients $c_{i,j}$. Then
the condition (4.12) gives one more constraint. Hence two coefficients (the
constant term $u=c_{0,0}$ and overall normalization) remain free.
###### Remark 4.4.
The position of the nonzero coefficients $c_{i,j}$ shows the Newton polygon of
the operators $\mathcal{D}_{k}$. Their transitions are as follows:
$\mathcal{D}$$\mapsto$$\mathcal{D}_{1}$$\mapsto$${\rm
Ad}\mathcal{B}^{-1}$$\mathcal{D}_{2}$$\mapsto$$\mathcal{D}_{3}$$\mapsto$$\mathcal{D}_{4}$$\mapsto$${\rm
Ad}\mathcal{B}^{-1}$$\mathcal{D}_{5}$$\mapsto$$\mathcal{D}_{6}$.
We see the Newton polygon of $\mathcal{D}_{3}$ has also a rectangular shape.
More precisely we have
$\mathcal{D}_{3}=\mathcal{D}\Big{|}_{\\{d_{2}\to\frac{\mu q^{2}}{d_{2}},\
d_{4}\to\frac{1}{d_{4}\mu},\ \Lambda\to\frac{d_{2}d_{4}\Lambda}{q},\
x\to-\frac{d_{2}x}{q}\\}}.$ (4.22)
This corresponds to the factorization property Proposition 5.1 which will be
discussed in the next section.
Since the parameter $u$ is stable under the adjoint action, we can regard it
as a trivial free parameter. Setting $u=0$ the quantum Seiberg-Witten curve is
explicitly
$\displaystyle\mathcal{D}_{\rm SW}(\Lambda,x)=(d_{1}d_{2}x-\mu
q)(1-\frac{\Lambda}{qx})p-(d_{1}+d_{2})x-\frac{(d_{3}+d_{4})\Lambda\mu}{x}+(x-1)(1-\frac{d_{3}d_{4}\Lambda\mu}{x})\frac{1}{p}$
$\displaystyle~{}~{}~{}=x\frac{(1-d_{1}p)(1-d_{2}p)}{p}-(1+d_{3}d_{4}\Lambda\mu)\frac{1}{p}-(\mu
q+\frac{d_{1}d_{2}\Lambda}{q})p+\frac{\Lambda\mu}{x}\frac{(p-d_{3})(p-d_{4})}{p}.$
(4.23)
This is a non-commutative Laurent polynomial in $(x,p)$: $px=qxp$. Note that
the highest and lowest terms in $x$ and $p$ are all factorized. When $(x,p)$
are commutative, the curve (4.2) reduces to the $M$-theoretic curve, which is
obtained from the toric diagram, or the five-brane web [11],[15],[28],[9].
### 4.3. Four dimensional limit
We study the $q\to 1$ limit of the operator (4.1). Let us use the short hand
notation such as
$F^{a,b,b}_{c}=T_{q,a}T_{q,b}^{2}T_{q,c}^{-1}F=F(qa,q^{2}b,q^{-1}c,\ldots),$
(4.24)
to represent the $q$-shifts of a function $F=F(a,b,c,\ldots)$.
###### Proposition 4.5.
We have
$\begin{array}[]{ll}(1-\Lambda)(\SS)^{\Lambda}_{d_{1}}=(1-x)(\SS)_{x}+d_{2}(x-\Lambda)\SS
p.&\qquad(d_{1}\leftrightarrow d_{2}),\\\
(1-\Lambda)(\SS)^{\Lambda}_{d_{3}}=(1-{\displaystyle\frac{\Lambda}{x}})\SS+d_{4}({\displaystyle\frac{\Lambda}{x}}-\Lambda)p^{-1}\SS,&\qquad(d_{3}\leftrightarrow
d_{4}),\end{array}$ (4.25)
and
$\SS x=q^{2}x(\SS)^{d_{1},d_{2}}_{d_{3},d_{4}}p^{2}.$ (4.26)
###### Proof.
We write $\SS$ as $\SS=A_{1}\mathcal{B}A_{2}\mathcal{B}A_{3}$, where
$\displaystyle A_{1}$
$\displaystyle=\frac{1}{\varphi(qx)\varphi(\Lambda/x)}\quad
A_{2}=\frac{\varphi(\Lambda)\varphi(q^{-1}d_{1}d_{2}d_{3}d_{4}\Lambda)}{\varphi(-d_{1}x)\varphi(-d_{2}x)\varphi(-d_{3}\Lambda/x)\varphi(-d_{4}\Lambda/x)}$
$\displaystyle A_{3}$
$\displaystyle=\frac{1}{\varphi(q^{-1}d_{1}d_{2}x)\varphi(d_{3}d_{4}\Lambda/x)}.$
(4.27)
Then we have
${A_{1}}^{\Lambda,x}_{d_{1}}=(1-qx)A_{1},\quad{A_{2}}^{\Lambda,x}_{d_{1}}={\displaystyle\frac{1+d_{2}x}{1-\Lambda}}A_{2},\quad{A_{3}}^{\Lambda,x}_{d_{1}}=A_{3}.$
(4.28)
Using these relations, we have
$\displaystyle(\SS)^{\Lambda,x}_{d_{1}}\SS^{-1}$
$\displaystyle={A_{1}}^{\Lambda,x}_{d_{1}}\mathcal{B}{A_{2}}^{\Lambda,x}_{d_{1}}{A_{2}}^{-1}\mathcal{B}^{-1}A_{1}^{-1}={\displaystyle\frac{1-qx}{1-\Lambda}}A_{1}(1+d_{2}px)A_{1}^{-1}$
$\displaystyle={\displaystyle\frac{1-qx}{1-\Lambda}}\Bigl{(}1+qd_{2}x{\displaystyle\frac{1-{\displaystyle\frac{\Lambda}{qx}}}{1-qx}}p\Bigr{)},$
hence
$(1-\Lambda)(\SS)^{\Lambda,x}_{d_{1}}=(1-qx)\SS+qd_{2}x(1-{\displaystyle\frac{\Lambda}{qx}})(\SS)^{x}p.$
(4.29)
Putting $x\to x/q$, we obtain the first relation of (4.25). Similarly, the
second relation of (4.25) follows from
${A_{1}}^{\Lambda}_{d_{3}}=(1-{\displaystyle\frac{\Lambda}{x}})A_{1},\quad{A_{2}}^{\Lambda}_{d_{3}}={\displaystyle\frac{1+d_{4}{\displaystyle\frac{\Lambda}{x}}}{1-\Lambda}}A_{2},\quad{A_{3}}^{\Lambda}_{d_{3}}=A_{3}.$
(4.30)
The relation (4.26) follows easily as
$\begin{array}[]{l}\SS
x=A_{1}\mathcal{B}A_{2}\mathcal{B}A_{3}x=A_{1}\mathcal{B}A_{2}px\mathcal{B}A_{3}=qA_{1}\mathcal{B}xA_{2}\mathcal{B}{A_{3}}^{x}p\\\
\quad=qA_{1}px\mathcal{B}A_{2}\mathcal{B}{A_{3}}^{x}p=q^{2}x{A_{1}}\mathcal{B}{A_{2}}^{x}\mathcal{B}{A_{3}}^{x,x}p^{2}\\\
\quad=q^{2}x(\SS)^{d_{1},d_{2}}_{d_{3},d_{4}}p^{2}.\end{array}$ (4.31)
∎
###### Theorem 4.6.
We put $q=e^{h}$, $d_{i}=e^{hm_{i}}$, then we have
$\displaystyle\SS$ $\displaystyle\to 1+hH_{\rm 4d}+O(h^{2}),\quad(h\to 0)$
$\displaystyle H_{\rm 4d}$
$\displaystyle=\vartheta_{x}(\vartheta_{x}+1)+{\displaystyle\frac{\Lambda-x}{1-\Lambda}}(\vartheta_{x}+m_{1})(\vartheta_{x}+m_{2})+{\displaystyle\frac{\Lambda}{x}}{\displaystyle\frac{x-1}{1-\Lambda}}(\vartheta_{x}-m_{3})(\vartheta_{x}-m_{4}).$
(4.32)
###### Proof.
First we consider the expansion
$v\mathbin{:=}\SS.1=v_{0}+hv_{1}+O(h^{2})$ (4.33)
Obviously $v_{0}=1$. For the first order term $v_{1}$, we have from (4.25)
$\begin{array}[]{ll}(1-\Lambda)\\{({v_{1}}|_{m_{1}\to
m_{1}-1})-v_{1}\\}=m_{2}(x-\Lambda)v_{0},&\qquad(m_{1}\leftrightarrow
m_{2}),\\\ (1-\Lambda)\\{({v_{1}}|_{m_{3}\to
m_{3}-1})-v_{1}\\}=\frac{\Lambda}{x}m_{4}(1-x)v_{0},&\qquad(m_{3}\leftrightarrow
m_{4}),\end{array}$ (4.34)
hence we have
$v_{1}={\displaystyle\frac{\Lambda-x}{1-\Lambda}}m_{1}m_{2}+{\displaystyle\frac{\Lambda}{x}}{\displaystyle\frac{x-1}{1-\Lambda}}m_{3}m_{4}.$
(4.35)
Then from this and iterative use of (4.26), we obtain
$\begin{array}[]{l}\SS
x^{n}=q^{n(n+1)}x^{n}\Big{\\{}(T_{q,d_{1}}T_{q,d_{2}}T^{-1}_{q,d_{3}}T^{-1}_{q,d_{4}})^{n}\SS\Big{\\}}p^{n}.1=q^{n(n+1)}x^{n}(T_{q,d_{1}}T_{q,d_{2}}T^{-1}_{q,d_{3}}T^{-1}_{q,d_{4}})^{n}v\\\
=1+hn(n+1)+h\left({\displaystyle\frac{\Lambda-x}{1-\Lambda}}(n+m_{1})(n+m_{2})+{\displaystyle\frac{\Lambda}{x}}{\displaystyle\frac{x-1}{1-\Lambda}}(n-m_{3})(n-m_{4})\right)+O(h^{2}).\end{array}$
(4.36)
This is the desired result. ∎
In a similar way, we can compute higher order corrections for $v=\SS.1$ as
$v=\exp\left(\frac{h}{1-\Lambda}C_{1}+\frac{h^{2}}{(1-\Lambda)^{2}}C_{2}+\frac{h^{3}}{(1-\Lambda)^{3}}C_{3}+O(h^{4})\right),\\\
$ (4.37)
where
$\begin{array}[]{l}C_{1}=m_{1}m_{2}(\Lambda-x)+\frac{\Lambda}{x}m_{3}m_{4}(x-1),\\\\[11.38109pt]
C_{2}=\frac{(1-x)}{2x}\Big{[}\frac{\Lambda}{x}m_{3}m_{4}\left(x+\Lambda-(m_{3}+m_{4})(x-\Lambda)\right)-m_{1}m_{2}x\left(x+\Lambda+(m_{1}+m_{2})(x-\Lambda)\right)\Big{]},\\\\[11.38109pt]
C_{3}=(x-1)(x-\Lambda)\Big{[}m_{1}m_{2}\left(-\frac{\Lambda^{2}+\Lambda+8x^{2}+3\Lambda
x-x}{12(x-\Lambda)}+\frac{(m_{1}+m_{2})(\Lambda^{2}+\Lambda+\Lambda
x+x-4x^{2})}{4(x-\Lambda)}\right.\\\
\left.+\frac{(m_{1}+m_{2}){}^{2}(1+\Lambda-2x)}{6}-\frac{m_{1}m_{2}(1+\Lambda+4x)}{12}\right)+m_{3}m_{4}\left(\frac{\Lambda(\Lambda
x^{2}+x^{2}+8\Lambda^{2}+3\Lambda
x-\Lambda^{2}x)}{12x^{3}(x-\Lambda)}\right.\\\
\left.-\frac{\Lambda(m_{3}+m_{4})(\Lambda x^{2}+x^{2}+\Lambda^{2}x+\Lambda
x-4\Lambda^{2})}{4x^{3}(x-\Lambda)}+\frac{\Lambda(m_{3}+m_{4}){}^{2}(1+\Lambda-2\frac{\Lambda}{x})}{6x^{4}}-\frac{\Lambda
m_{3}m_{4}(1+\Lambda+4\frac{\Lambda}{x})}{12x^{4}}\right)+\frac{\Lambda
m_{1}m_{2}m_{3}m_{4}}{x}\Big{]}.\end{array}$ (4.38)
###### Remark 4.7.
The quantum Seiberg-Witten curve for four dimensional $\mathcal{N}=2$ $SU(2)$
gauge theory with $N_{f}=4$ is equivalent to the quantum $P_{\rm VI}$ equation
$\mathcal{D}^{4d}_{\rm SW}\Psi(\Lambda,x)=0$, where $\mathcal{D}^{4d}_{\rm
SW}$ can be written as [4]
$\mathcal{D}^{4d}_{\rm SW}=(1-\Lambda)\Lambda\partial_{\Lambda}+D_{\rm Heun},$
(4.39)
and the Heun operator is
$\displaystyle D_{\rm Heun}$
$\displaystyle=(1-\Lambda)(v-a_{1})(v-a_{2})-\frac{\Lambda}{z}(1-z)(v-\mu_{1})(v-\mu_{2})-(z-\Lambda)(v-\mu_{3})(v-\mu_{4}),$
where $v=bz\partial_{z}$ and $b^{2}=\epsilon_{1}/\epsilon_{2}$. The result of
Theorem 4.6 is consistent with the correspondence of the quantum $P_{\rm VI}$
equation and $SU(2)$ Seiberg-Witten theory with $N_{f}=4$ in four dimensions.
In five dimensions there are two mutually commuting Hamiltonians, one of which
requires the infinite product to generate the discrete time evolution, the
other is related to the conserved quantities and takes a simple expression
(like the relativistic affine Toda Hamiltonian). We have explicitly seen this
is the case in the decoupling limit of the hypermultiplets (the pure Yang-
Mills case). In four dimensional limit these two Hamiltonians degenerate to a
single Hamiltonian, which is obtained from the Seiberg-Witten curve.
## 5\. Relation to affine Laumon space
### 5.1. Factorization as a coupled system
We may use the same gauge transformation as (4.1) but exchanging $T_{1}$ and
$T_{2}$ in the gauge factor. By the dictionary in section 2.2, we can see this
is nothing but the action of the Weyl reflection $r_{1}$ (See Appendix A). By
this $r_{1}$-reflected gauge transformation, we obtain the following
Hamiltonian from (4.1) by exchanging $T_{1}$ and $T_{2}$.
$\displaystyle\widetilde{\mathcal{H}}:={1\over\varphi(T_{2}q^{1/2}t^{1/2}x)\varphi(T_{3}q^{1/2}t^{1/2}\Lambda
x^{-1})}\cdot\mathcal{B}$
$\displaystyle\qquad\cdot{\varphi(tT_{2}T_{3}\Lambda)\varphi(qT_{1}T_{4}\Lambda)\over\varphi(-T_{1}T_{2}x)\varphi(-Q^{-1}x)\varphi(-T_{3}T_{4}Qqt\Lambda
x^{-1})\varphi(-q\Lambda x^{-1})}\cdot\mathcal{B}$ $\displaystyle\qquad\cdot
T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1}\cdot{1\over\varphi(T_{1}q^{1/2}t^{1/2}x)\varphi(T_{4}q^{1/2}t^{1/2}\Lambda
x^{-1})}.$ (5.1)
It turns out that this exchange of $T_{1}$ and $T_{2}$ is better for the
purpose of factorizing the original non-stationary difference
equation111111The virtue of the gauge transformation (4.1) is that it cancels
$\varphi(q^{1/2}t^{1/2}T_{2}x)$ appearing in (2.18) so that the total gauge
factor is written in terms the double infinite product $\Phi(x)$ only, as we
have seen in the beginning of Section 4..
Our main point is that one can transform the non-stationary difference
equation $\widetilde{\mathcal{H}}\mathsf{V}^{(1)}=\mathsf{V}^{(1)}$ to the
following coupled system;
$\displaystyle\mathsf{V}^{(1)}={\Phi(qt^{-1}b_{2}/b_{4})\Phi(b_{1}/b_{3})\over\Phi(tb_{6}/b_{8})\Phi(qb_{5}/b_{7})}{1\over\varphi(-qb_{6}/G)\varphi(-G/b_{8})}\cdot\widetilde{\mathcal{B}}$
$\displaystyle\qquad\cdot{1\over\varphi(\mathsf{p}^{-2}qb_{2}/G)\varphi(\mathsf{p}^{-2}G/b_{4})}(T_{t^{1/2},x}^{-1}T_{t,\Lambda}^{-1})\mathsf{V}^{(2)},$
(5.2)
$\displaystyle\mathsf{V}^{(2)}={\Phi(tb_{6}/b_{8})\Phi(qb_{5}/b_{7})\over\Phi(qb_{2}/b_{4})\Phi(tb_{1}/b_{3})}{1\over\varphi(qb_{1}/G)\varphi(G/b_{3})}\cdot\widetilde{\mathcal{B}}$
$\displaystyle\qquad\cdot{1\over\varphi(-qb_{5}/G)\varphi(-G/b_{7})}\mathsf{V}^{(1)}.$
(5.3)
where $\widetilde{\mathcal{B}}:=T_{(qt^{1/2}Q)^{1/2},x}^{-1}\mathcal{B}$. Note
that by making use of the equality
$\displaystyle{\Phi(qt^{-1}b_{2}/b_{4})\Phi(b_{1}/b_{3})\over\Phi(tb_{6}/b_{8})\Phi(qb_{5}/b_{7})}T_{t,\Lambda}^{-1}{\Phi(tb_{6}/b_{8})\Phi(qb_{5}/b_{7})\over\Phi(qb_{2}/b_{4})\Phi(tb_{1}/b_{3})}$
$\displaystyle=\varphi(b_{6}/b_{8})\varphi(qt^{-1}b_{5}/b_{7})T_{t,\Lambda}^{-1}=\varphi(tT_{2}T_{3}\Lambda)\varphi(qT_{1}T_{4}\Lambda)T_{t,\Lambda}^{-1}.$
(5.4)
we have called back the double infinite product $\Phi(z)$, which we once
eliminated to reveal the relation to the $qq$-Painlevé VI equation, to
factorize the original equation as a coupled system. The possibility of such a
factorization was already suggested in the proof of Conjecture 2.4 in the
special case of the “Macdonald” limit [44]. We believe this factorization is a
significant step towards a general proof of Conjecture 2.4. By the dictionary
in section 2.2 and Definition A.5 of the variables $b_{i}$, it is
straightforward to check the matching of parameters
$qt^{-1}b_{2}/b_{4},b_{1}/b_{3},tb_{6}/b_{8}$ and $qb_{5}/b_{7}$. Note that
the parameters $b_{i}$ involve neither $x$ nor $\Lambda$. On the other hand,
$G$ is a monomial in $x$ and $\Lambda$. But it is also easy to see that
matcing for $-qb_{6}/G,-G/b_{8},-qb_{5}/G$ and $-G/b_{7}$. For remaining
parameters we have to take the commutation with the shift operators into
account. Namely the parameters $\mathsf{p}^{-2}qb_{2}/G$ and
$\mathsf{p}^{-2}G/b_{4}$ are affected by $T_{(qt^{1/2}Q)^{1/2},x}^{-1}$ and
for $qb_{1}/G$ and $G/b_{3}$ the action of
$T_{t^{1/2},x}^{-1}T_{t,\Lambda}^{-1}$ is also involved.
To rewrite the coupled system (5.1) and (5.1) to more symmetric one in
$\mathsf{V}^{(1)}$ and $\mathsf{V}^{(2)}$, let us introduce the following
shift operator $\widetilde{T}_{\mathsf{p},b}$ which commutes with
$\widetilde{\mathcal{B}}$.
$\widetilde{T}_{\mathsf{p},b}:=T_{-t^{-1/4},x}T_{t^{-1/2},\Lambda}:b\mapsto(-\mathsf{p}^{-1}b_{1},-\mathsf{p}^{-1}b_{2},-\mathsf{p}b_{3},-\mathsf{p}b_{4},-\mathsf{p}^{-1}b_{5},-\mathsf{p}^{-1}b_{6},-\mathsf{p}b_{7},-\mathsf{p}b_{8}),$
(5.5)
where $\mathsf{p}=e^{\delta}=a_{0}a_{1}a_{2}^{2}a_{3}^{2}a_{4}a_{5}=t^{1/4}$.
Note that $G$ is invariant under $\widetilde{T}_{\mathsf{p},b}$. For the
transformation (5.5) of the parameters $b_{i}$, the shift operator
$T_{t^{-1/2},\Lambda}$ is enough, but to make $G$ invariant up to sign we need
the combination with $T_{-t^{-1/4},x}$. We also note that
$(\widetilde{T}_{\mathsf{p},b})^{2}=T~{}(\hbox{the discrete time evolution})$
as far as the $b$ variables are concerned (see Lemma A.8). Then we have
###### Proposition 5.1.
We can rewrite (5.1) and (5.1) as follows;
$\displaystyle\mathsf{V}^{(1)}$
$\displaystyle={\Phi(qt^{-1}b_{2}/b_{4})\Phi(b_{1}/b_{3})\over\Phi(tb_{6}/b_{8})\Phi(qb_{5}/b_{7})}{1\over\varphi(-qb_{6}/G)\varphi(-G/b_{8})}$
$\displaystyle\qquad\cdot(\widetilde{\mathcal{B}}\cdot\widetilde{T}_{\mathsf{p},b}){1\over\varphi(-\mathsf{p}^{-1}qb_{2}/G)\varphi(-\mathsf{p}^{-1}G/b_{4})}\widetilde{T}_{\mathsf{p},b}\cdot\mathsf{V}^{(2)},$
(5.6) $\displaystyle\widetilde{T}_{\mathsf{p},b}\cdot\mathsf{V}^{(2)}$
$\displaystyle={\Phi(\mathsf{p}^{-2}tb_{6}/b_{8})\Phi(\mathsf{p}^{-2}qb_{5}/b_{7})\over\Phi(\mathsf{p}^{-2}qb_{2}/b_{4})\Phi(\mathsf{p}^{-2}tb_{1}/b_{3})}{1\over\varphi(-\mathsf{p}^{-1}qb_{1}/G)\varphi(-\mathsf{p}^{-1}G/b_{3})}$
$\displaystyle\cdot(\widetilde{\mathcal{B}}\cdot\widetilde{T}_{\mathsf{p},b})\cdot{1\over\varphi(-qb_{5}/G)\varphi(-G/b_{7})}\mathsf{V}^{(1)}.$
(5.7)
The coupled system is gauge equivalent to the non-stationary difference
equation (2.14).
Recall that the discrete time evolution $T$ of Painlevé VI equation is a
translation element in the extended affine Weyl group of $D_{5}^{(1)}$. It is
remarkable that $T$ allows a square root;
$T:=r_{2}r_{1}r_{0}r_{2}\sigma_{01}r_{3}r_{4}r_{5}r_{3}\sigma_{45}=(r_{2}r_{1}r_{0}r_{2}\sigma_{01}\tau)(r_{2}r_{1}r_{0}r_{2}\sigma_{01}\tau).$
(5.8)
In fact the factorization into (5.1) and (5.1) is not unrelated to the
existence of the square root $T^{1/2}:=(r_{2}r_{1}r_{0}r_{2}\sigma_{01}\tau)$,
which acts on the $b$ variables as follows;
$T^{1/2}=(r_{2}r_{1}r_{0}r_{2}\sigma_{01}\tau):b\mapsto(b_{6},b_{5},b_{8},b_{7},\mathsf{p}^{-2}b_{2},\mathsf{p}^{-2}b_{1},\mathsf{p}^{2}b_{4},\mathsf{p}^{2}b_{3}).$
(5.9)
We define an operator
$X:b\mapsto(\mathsf{p}b_{6},\mathsf{p}b_{5},\mathsf{p}^{-1}b_{8},\mathsf{p}^{-1}b_{7},\mathsf{p}^{-1}b_{2},\mathsf{p}^{-1}b_{1},\mathsf{p}b_{4},\mathsf{p}b_{3}),$
(5.10)
and assume that $X$ does not act on $(F,G)$. Then we have
###### Lemma 5.2.
The action (5.9) on the parameters $b_{i}$ is represented by
$T^{1/2}=(-1)\cdot X\cdot\widetilde{T}_{\mathsf{p},b},$ (5.11)
where $(-1)$ is the overall sign flip of $b_{i}$. Two functions
$\mathsf{V}^{(1)}$ and $\mathsf{V}^{(2)}$ are related by
$X\mathsf{V}^{(1)}=\widetilde{T}_{\mathsf{p},b}\cdot\mathsf{V}^{(2)}.$ (5.12)
Hence, the coupled system in the form of Prop.5.1 is quite natural from the
viewpoint of the $qq$-Painlevé VI equations, because the second equation (5.1)
is obtained by applying $X$ to (5.1).
### 5.2. Instanton counting with a surface defect
Now we want to point out that solutions to the coupled system (5.1) and (5.1)
are given by the instanton partition function of the affine Laumon space. In
fact we have already mentioned that the gauge transformation introduced in the
beginning of Section 4 is a five dimensional uplift of the gauge
transformation from the Higgsed quiver gauge theory to the gauge theory with a
surface defect. A torus action on the affine Laumon space is induced by the
standard torus action on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. The fixed
points of the torus action on the affine Laumon space of type $A_{r}^{(1)}$
are labelled by $(r+1)$-tuples of partitions [18].
###### Definition 5.3.
Set
$\displaystyle[u;q]_{n}=u^{-n/2}q^{-n(n-1)/4}(u;q)_{n}$
$\displaystyle=(u^{-1/2}-u^{1/2})(q^{-1/2}u^{-1/2}-q^{1/2}u^{1/2})\cdots(q^{-(n-1)/2}u^{-1/2}-q^{(n-1)/2}u^{1/2}).$
For a pair $(\lambda,\mu)$ of partitions, the $\mathbb{Z}_{N}$ orbifolded
Nekrasov factor with color $k$ is121212We associate a $\sinh$ factor with each
monomial term in the equivariant character.
$\displaystyle{\mathsf{N}}^{(k|N)}_{\lambda,\mu}(u|q,\kappa)={\mathsf{N}}^{(k)}_{\lambda,\mu}(u|q,\kappa)$
(5.13) $\displaystyle=$ $\displaystyle\prod_{j\geq i\geq 1\atop j-i\equiv
k\,\,({\rm
mod}\,N)}[uq^{-\mu_{i}+\lambda_{j+1}}\kappa^{-i+j};q]_{\lambda_{j}-\lambda_{j+1}}\cdot\prod_{\beta\geq\alpha\geq
1\atop\beta-\alpha\equiv-k-1\,\,({\rm
mod}\,N)}[uq^{\lambda_{\alpha}-\mu_{\beta}}\kappa^{\alpha-\beta-1};q]_{\mu_{\beta}-\mu_{\beta+1}}.$
Note that the equivariant parameters of the torus action on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ are not $(q,t)$, but $(q,\kappa)$. We
will substitute $\kappa=t^{-\frac{1}{2}}$ later131313The square root comes
from the $\mathbb{Z}_{2}$ orbifolding..
From the equivariant character evaluated at each fixed point of the affine
Laumon space of type $A_{1}^{(1)}$ [18], we obtain
$\displaystyle
f(u_{1},u_{2};v_{1},v_{2};w_{1},w_{2}|x_{1},x_{2}|q,\kappa)=f\left(\left.\left.\begin{array}[]{c}u_{1},u_{2}\\\
v_{1},v_{2}\\\ w_{1},w_{2}\end{array}\right|x_{1},x_{2}\right|q,\kappa\right)$
(5.17)
$\displaystyle=\sum_{\lambda^{(1)},\lambda^{(2)}\in\mathsf{P}}\prod_{i,j=1}^{2}{\mathsf{N}^{(j-i|2)}_{\emptyset,\lambda^{(j)}}(u_{i}/v_{j}|q,\kappa)\mathsf{N}^{(j-i|2)}_{\lambda^{(i)},\emptyset}(v_{i}/w_{j}|q,\kappa)\over\mathsf{N}^{(j-i|2)}_{\lambda^{(i)},\lambda^{(j)}}(v_{i}/v_{j}|q,\kappa)}\cdot
x_{1}^{|\lambda^{(1)}|_{o}+|\lambda^{(2)}|_{e}}x_{2}^{|\lambda^{(1)}|_{e}+|\lambda^{(2)}|_{o}},$
(5.18)
where $\emptyset$ denotes the empty partition and for a partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots)$, we set
$|\lambda|_{o}\mathbin{:=}\sum_{k\geq
1}\lambda_{2k-1},\qquad|\lambda|_{e}\mathbin{:=}\sum_{k\geq 1}\lambda_{2k}.$
(5.19)
Note that the function
$f(u_{1},u_{2};v_{1},v_{2};w_{1},w_{2}|x_{1},x_{2}|q,\kappa)$ is invariant
under the overall scaling of the equivariant parameters
$(u_{1},u_{2};v_{1},v_{2};w_{1},w_{2})$. The partition function (5.17) is a
five dimensional uplift of the instanton partition function which is given,
for example, in [4]. The parameters $(v_{1},v_{2})$ are the Coulomb moduli of
$U(2)$ gauge theory, or the equivariant parameters of the Cartan subgroup
$U(1)\times U(1)\subset U(2)$. The parameters $(u_{1},u_{2})$ and
$(w_{1},w_{2})$ are exponentiated mass parameters of the hypermultiplets in
the fundamental and the anti-fundamental representations. They are also
regarded as equivariant parameters for the flavor symmetry. The expansion
parameters are parametrized as $x_{1}=x,x_{2}=\Lambda/x$, where $x$ counts the
monopole number (the first Chern number of the $U(1)$ connection on the
defect), while $\Lambda$ counts the instanton number (the second Chern
number). When $\Lambda=0$, the terms with
$|\lambda^{(1)}|_{e}+|\lambda^{(2)}|_{o}\neq 0$ do not contribute to the
partition function. This means the sum in (5.17) is restricted to
$\lambda^{(1)}=(m)$ (a partition with a single row) and
$\lambda^{(2)}=\emptyset$. Physically this corresponds to the topological
sector with instanton number zero. As we see in Appendix B, the partition
function is given by the Heine’s $q$-hypergeometric series. The six parameters
$(u_{1},u_{2}),(w_{1},w_{2})$ and $(x_{1},x_{2})$ are “external” spectral
parameters which correspond to the (independent) dynamical variables on
Painlevé side. On the other hand the parameters $(v_{1},v_{2})$ are “internal”
parameters or the loop parameters.
The function (5.17) should be compared with the non-stationary Ruijsenaars
function [45];
$\displaystyle
f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)=\sum_{\lambda^{(1)},\ldots,\lambda^{(N)}\in{\mathsf{P}}}\prod_{i,j=1}^{N}{{\mathsf{N}}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(ts_{j}/s_{i}|q,\kappa)\over{\mathsf{N}}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(s_{j}/s_{i}|q,\kappa)}\cdot\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}(px_{\alpha+\beta}/x_{\alpha+\beta-1})^{\lambda^{(\beta)}_{\alpha}},$ (5.20)
for $N=2$. Both functions come from the affine Laumon space of type
$A_{1}^{(1)}$. The non-stationary Ruijsenaars function (5.20) corresponds to
the theory with an adjoint matter, or the tangent bundle overt the affine
Laumon space, while the partition function (5.17) is for the theory with four
matter hypermultiplets in the (anti-)fundamental representation, or the
tautological bundle. In the AGT correspondence, the former is identified with
the conformal block on a punctured torus and the latter on $\mathbb{P}^{1}$
with four punctures. In the mass decoupling limit, both functions are
conjectured to give solutions to the non-stationary affine Toda equation [45].
###### Conjecture 5.4.
The partition function (5.17) gives a solution to the coupled system (5.1) and
(5.1) by the following specialization of parameters;
$\displaystyle\mathcal{F}^{(1)}=f\left(\left.\left.\begin{array}[]{ll}q^{1/2}b_{4}/b_{8},&q^{1/2}b_{6}/b_{2}\\\
(\mathsf{p}^{2}Q)^{-1/2},&(\mathsf{p}^{2}Q)^{1/2}\\\
q^{-1/2}b_{2}/b_{5},&q^{-1/2}b_{7}/b_{4}\end{array}\right|q^{1/2}\mathsf{p}^{-1}\mathsf{t}G^{-1},q^{-1/2}\mathsf{p}^{-1}\mathsf{t}G\right|q,t^{-1/2}\right),$
(5.24)
$\displaystyle\mathcal{F}^{(2)}=f\left(\left.\left.\begin{array}[]{ll}q^{1/2}b_{4}/b_{8},&q^{1/2}b_{6}/b_{2}\\\
(\mathsf{p}^{2}Q)^{-1/2},&(\mathsf{p}^{2}Q)^{1/2}\\\
q^{-1/2}b_{1}/b_{6},&q^{-1/2}b_{8}/b_{3}\end{array}\right|-q^{1/2}\mathsf{t}G^{-1},-q^{-1/2}\mathsf{t}G\right|q,t^{-1/2}\right).$
(5.28)
It is remarkable that the only difference between $\mathcal{F}^{(1)}$ and
$\mathcal{F}^{(2)}$ is the exchanges of
$(b_{1},b_{2}),(b_{3},b_{4}),(b_{5},b_{6})$ and $(b_{7},b_{8})$ (see Figure 2)
in the specialization of $w_{i}$ and the scaling $-\mathsf{p}$ of
$\mathsf{t}$. Note that these exchanges of four pairs of $b_{i}$ are nothing
but the action of the Weyl reflections $r_{4},r_{5},r_{1}$ and $r_{0}$,
respectively. We can see that the specializations (5.24) and (5.28) are
consistent with the relation (5.12). In fact under the action of $X$
$\displaystyle\frac{b_{4}}{b_{8}}\mapsto\mathsf{p}^{-2}\frac{b_{7}}{b_{3}}=\frac{b_{4}}{b_{8}},\qquad\frac{b_{2}}{b_{6}}\mapsto\mathsf{p}^{-2}\frac{b_{1}}{b_{5}}=\frac{b_{2}}{b_{6}},$
$\displaystyle\frac{b_{2}}{b_{5}}\mapsto\mathsf{p}^{2}\frac{b_{5}}{b_{2}}=\frac{b_{1}}{b_{6}},\qquad\frac{b_{7}}{b_{4}}\mapsto\mathsf{p}^{2}\frac{b_{4}}{b_{7}}=\frac{b_{8}}{b_{3}},$
where we have used the constraints (A.1). On the other hand, these ratios are
invariant by the action of $\widetilde{T}_{\mathsf{p},b}$. Finally
$\widetilde{T}_{\mathsf{p},b}$ generates the square root of the discrete time
shift $\mathsf{t}\mapsto-\mathsf{p}^{-1}\mathsf{t}$.
In terms of the variables on the gauge theory side the specialization of
parameters is given as follows;
$\displaystyle\mathcal{F}^{(1)}=f\left(\left.\left.\begin{array}[]{ll}q^{1/2}t^{1/2}Q^{1/2}T_{3},&q^{1/2}Q^{-1/2}T_{1}^{-1}\\\
Q^{-1/2}&t^{1/2}Q^{1/2}\\\
q^{-1/2}t^{1/2}Q^{1/2}T_{2},&q^{-1/2}Q^{-1/2}T_{4}^{-1}\end{array}\right|-t^{1/2}T_{1}^{1/2}T_{2}^{1/2}x,-t^{1/2}T_{3}^{1/2}T_{4}^{1/2}\Lambda/x\right|q,t^{-1/2}\right),$
(5.32)
$\displaystyle\mathcal{F}^{(2)}=f\left(\left.\left.\begin{array}[]{ll}q^{1/2}t^{1/2}Q^{1/2}T_{3},&q^{1/2}Q^{-1/2}T_{1}^{-1}\\\
Q^{-1/2},&t^{1/2}Q^{1/2}\\\
q^{-1/2}t^{1/2}Q^{-1/2}T_{2}^{-1},&q^{-1/2}tQ^{1/2}T_{4}\end{array}\right|t^{3/4}T_{1}^{1/2}T_{2}^{1/2}x,t^{3/4}T_{3}^{1/2}T_{4}^{1/2}\Lambda/x\right|q,t^{-1/2}\right),$
(5.36)
where by making use of the scaling symmetry of the partition function (5.17),
we have made the overall scaling of parameters by $\mathsf{p}$. One can check
that $\mathcal{F}^{(2)}$ is invariant under the exchange of $T_{1}$ and
$T_{2}$. We have examined our conjecture in several cases. The results are
summarized in Appendix B.
Thus we can formulate Conjecture 2.4 in terms of the instanton partition
functions from the affine Laumon space. According to the (four dimensional)
AGT correspondence, the partition functions coming from the affine Laumon
space should be identified with the conformal blocks of the current algebra.
In the present case it is the affine algebra
$A_{1}^{(1)}=\widehat{\mathfrak{sl}}_{2}$. In the original formulation in
[44], the five dimensional Nekrasov partition function are regarded as a
conformal block of the deformed Virasoro algebra. In our formulation it is
natural to expect that the coupled system of non-stationary difference
equations defines a conformal block of the quantum deformation of
$A_{1}^{(1)}=\widehat{\mathfrak{sl}}_{2}$, namely the quantum affine algebra
$U_{q}(\widehat{\mathfrak{sl}}_{2})$. The advantage of
$U_{q}(\widehat{\mathfrak{sl}}_{2})$ to the deformed Virasoro algebra is that
it is a quantum group (in particular we have a coproduct), while the latter is
not.
###### Acknowledgements.
We would like to thank H.Hayashi, A.N.Kirillov, G.Kuroki and H.Nakajima for
useful discussions. Our work is supported in part by Grants-in-Aid for
Scientific Research (Kakenhi); 18K03274 (H.K.), 21K03180 (R.O.), 19K03512
(J.S.), 19K03530 (J.S.) and 22H01116 (Y.Y.). The work of R.O. was partly
supported by Osaka Central Advanced Mathematical Institute: MEXT Joint
Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849,
and the Research Institute for Mathematical Sciences, an International Joint
Usage/Research Center located in Kyoto University.
## Appendix A Difference analogue of Painlevé VI equation
### A.1. Affine Weyl group and Bäcklund transformation
Let $A=A(D^{(1)}_{5})=(a_{ij})_{i,j=0}^{5}$ be the generalized Cartan matrix
of type $D^{(1)}_{5}$ associated with the Dynkin diagram in Fig. 1.
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\scriptstyle
0$$\scriptstyle 1$$\scriptstyle 2$$\scriptstyle 3$$\scriptstyle
5$$\scriptstyle
4$$\sigma_{01}\,\,\,\updownarrow$$\updownarrow\,\,\,\sigma_{45}$$\longleftrightarrow$$\tau$
Figure 1. Dynkin diagram of $D^{(1)}_{5}$ and its automorphism
Fix a realization $(\mathfrak{h},\Pi,\Pi^{\vee})$ of $A$, where
$\Pi=\\{\alpha_{0},\ldots,\alpha_{5}\\}\subset\mathfrak{h}^{*}$ denotes the
set of simple roots,
$\Pi^{\vee}=\\{\alpha_{0}^{\vee},\ldots,\alpha_{5}^{\vee}\\}\subset\mathfrak{h}$
the set of simple coroots, with $\Pi$ and $\Pi^{\vee}$ being linearly
independent. We have $\langle\alpha_{i}^{\vee},\alpha_{j}\rangle=a_{ij}$
$(i,j=0,\ldots,5)$. Let $Q=\sum_{i=0}^{5}\mathbb{Z}\alpha_{i}$ be the root
lattice. Denote by $\Delta,\Delta_{+}$ and $\Delta_{-}$ the sets of all roots,
positive and negative roots respectively. We have
$\Delta=\Delta_{+}\cup\Delta_{-}$ (a disjoint union). The center of
$\mathfrak{g}=\mathfrak{g}(D^{(1)}_{5})$ is 1-dimensional and is spanned by
the canonical central element
$K=\alpha_{0}^{\vee}+\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+2\alpha_{3}^{\vee}+\alpha_{4}^{\vee}+\alpha_{5}^{\vee}.$
Denote by $\delta\in Q$ the null root
$\delta=\alpha_{0}+\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4}+\alpha_{5}$.
Let $d\in\mathfrak{h}$ be the scaling element (defined up to a summand
proportional to $K$) satisfying $\langle\alpha_{i},d\rangle=0$ for
$i=1,\ldots,5$ and $\langle\alpha_{0},d\rangle=1$. The elements
$\alpha_{0}^{\vee},\ldots,\alpha_{5}^{\vee},d$ form a basis of $\mathfrak{h}$.
We have $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]+\mathbb{C}d$,
$\mathfrak{h}=\sum_{i=0}^{5}\mathbb{C}\alpha_{i}^{\vee}+\mathbb{C}d$. Define
the element $\Lambda_{0}\in\mathfrak{h}^{*}$ by
$\langle\Lambda_{0},\alpha_{i}^{\vee}\rangle=\delta_{0i}$ for $i=0,\ldots,5$
and $\langle\Lambda_{0},d\rangle=0$. The elements
$\alpha_{0},\ldots,\alpha_{5},\Lambda_{0}$ form a basis of $\mathfrak{h}^{*}$
and we have
$\mathfrak{h}^{*}=\sum_{i=0}^{5}\mathbb{C}\alpha_{i}+\mathbb{C}\Lambda_{0}$.
The affine Weyl group $W=W(D^{(1)}_{5})$ is defined to be the group generated
by the fundamental reflections $r_{0},r_{1},\ldots,r_{5}$ which act on
$\mathfrak{h}^{*}$ by
$\displaystyle
r_{i}(\lambda)=\lambda-\langle\lambda,\alpha_{i}^{\vee}\rangle\alpha_{i}\qquad(\lambda\in\mathfrak{h}^{*}).$
Let $a_{i}$ $(i=0,\ldots,5)$ be the formal exponentials
$a_{i}=e^{\alpha_{i}}$. Write $a=(a_{0},\ldots,a_{5})$ for short. Denote by
$\mathbb{K}=\mathbb{C}(a)$ the field of rational functions in $a$. Define the
actions of the generators $r_{i}\in W$ on $\mathbb{K}$ by setting the rules
$\displaystyle r_{i}\cdot
a_{j}=a_{j}a_{i}^{-a_{ij}}=e^{\alpha_{j}-a_{ij}\alpha_{i}}\qquad(0\leq i,j\leq
5),$
and extending them as ring homomorphisms. It is clear that these actions are
compatible with the group structure, namely they satisfy the Coxeter
relations: $r_{i}^{2}={\rm id}$ $(a_{ij}=2)$,
$r_{i}r_{j}r_{i}=r_{j}r_{i}r_{j}$ $(a_{ij}=-1)$, and $r_{i}r_{j}=r_{j}r_{i}$
$(a_{ij}=0)$. We regard $W$ as a group of birational isomorphisms of
$\mathbb{K}$.
Let $\sigma_{01},\sigma_{45}$ and $\tau$ be the automorphisms shown in Fig. 1,
namely
$\displaystyle\sigma_{01}:(0,1,2,3,4,5)\mapsto(1,0,2,3,4,5),$
$\displaystyle\sigma_{45}:(0,1,2,3,4,5)\mapsto(0,1,2,3,5,4),$
$\displaystyle\tau:(0,1,2,3,4,5)\mapsto(5,4,3,2,1,0).$
Let $\widetilde{W}$ denotes the extended affine Weyl group generated by $W$
together with $\sigma_{01},\sigma_{45}$ and $\tau$. Let $f$ and $g$ be
independent indeterminates, and consider the rational function field
$\mathbb{K}(f,g)=\mathbb{C}(a)(f,g)$.
###### Definition A.1.
Define the actions of the generators
$r_{0},\ldots,r_{5},\sigma_{01},\sigma_{45},\tau\in\widetilde{W}$ on
$\mathbb{K}(f,g)$ by setting the rules
$\displaystyle r_{i}\cdot a_{j}=a_{j}a_{i}^{-a_{ij}}\qquad(0\leq i,j\leq 5),$
$\displaystyle\sigma_{01}\cdot
a_{i}=(a_{\sigma_{01}(i)})^{-1},\quad\sigma_{45}\cdot
a_{i}=(a_{\sigma_{45}(i)})^{-1},\quad\tau\cdot
a_{i}=(a_{\tau(i)})^{-1}\qquad(0\leq i\leq 5),$ $\displaystyle r_{i}\cdot
f=f,\qquad r_{i}\cdot g=g\qquad(i\neq 2,3),$ $\displaystyle r_{2}\cdot
f=f{a_{0}a_{1}^{-1}g+a_{2}^{2}\over a_{0}a_{1}^{-1}a_{2}^{2}g+1},\qquad
r_{2}\cdot g=g,\qquad r_{3}\cdot f=f,\qquad r_{3}\cdot
g={a_{3}^{2}a_{4}a_{5}^{-1}f+1\over a_{4}a_{5}^{-1}f+a_{3}^{2}}g,$
$\displaystyle\sigma_{01}\cdot f=f^{-1},\qquad\sigma_{01}\cdot
g=g,\qquad\sigma_{45}\cdot f=f,\qquad\sigma_{45}\cdot g=g^{-1},$
$\displaystyle\tau\cdot f=g,\qquad\tau\cdot g=f,$
and extending them as ring homomorphisms.
###### Proposition A.2.
The actions are compatible with the group structure of the extended affine
Weyl group $\widetilde{W}$. Namely they satisfy the Coxeter relations:
$\displaystyle r_{i}^{2}={\rm id}\qquad(\mbox{if }a_{ij}=2),$ $\displaystyle
r_{i}r_{j}r_{i}=r_{j}r_{i}r_{j}\qquad(\mbox{if }a_{ij}=-1),\qquad
r_{i}r_{j}=r_{j}r_{i}\qquad(\mbox{if }a_{ij}=0),$
$\displaystyle\sigma_{01}^{2}=\sigma_{45}^{2}=\tau^{2}={\rm
id},\qquad\sigma_{01}\sigma_{45}=\sigma_{45}\sigma_{01},\qquad\sigma_{01}\tau=\tau\sigma_{45},$
$\displaystyle\sigma_{01}r_{0}=r_{1}\sigma_{01},\qquad\sigma_{01}r_{i}=r_{i}\sigma_{01}\qquad(i\neq
0,1),$
$\displaystyle\sigma_{45}r_{4}=r_{5}\sigma_{45},\qquad\sigma_{45}r_{i}=r_{i}\sigma_{45}\qquad(i\neq
4,5),$ $\displaystyle\tau r_{i}=r_{5-i}\tau.$
### A.2. Difference analogue of Painlevé VI equation
Set for simplicity of display that
$\displaystyle\mathsf{p}=a_{0}a_{1}a_{2}^{2}a_{3}^{2}a_{4}a_{5}=e^{\delta},\qquad\mathsf{t}=a_{3}^{2}a_{4}a_{5}.$
Recall that the element
$\displaystyle
T=r_{2}r_{1}r_{0}r_{2}\sigma_{01}r_{3}r_{4}r_{5}r_{3}\sigma_{45}\in\widetilde{W},$
acts on $\mathfrak{h}^{*}$ as a translation. Namely, we have
$\displaystyle
T:(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5})\mapsto(a_{0},a_{1},\mathsf{p}a_{2},\mathsf{p}^{-1}a_{3},a_{4},a_{5}).$
Write $\overline{f}=T\cdot f$ and $\underline{g}=T^{-1}\cdot g$ for short.
###### Proposition A.3.
We have the difference analogue of the Painlevé VI equation;
$\displaystyle f\overline{f}$
$\displaystyle=\mathsf{p}^{2}\mathsf{t}^{-2}{g+\mathsf{t}\mathsf{p}^{-1}a_{1}^{2}\over
g+\mathsf{t}^{-1}\mathsf{p}a_{0}^{2}}{g+\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2}\over
g+\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2}},$ $\displaystyle g\underline{g}$
$\displaystyle=\mathsf{t}^{-2}{f+\mathsf{t}a_{4}^{2}\over
f+\mathsf{t}^{-1}a_{5}^{2}}{f+\mathsf{t}a_{4}^{-2}\over
f+\mathsf{t}^{-1}a_{5}^{-2}}.$
###### Proposition A.4.
The translation $T$ admits the symmetry of the type $D^{(1)}_{4}$ in the
following sense. (1) The translation $T$ commutes with the action of the
subgroup $\langle
r_{0},r_{1},r_{2}r_{3}r_{2}=r_{3}r_{2}r_{3},r_{4},r_{5}\rangle\simeq
W(D^{(1)}_{4})$ of $W(D^{(1)}_{5})$, (2) We have $\sigma_{01}T=T\sigma_{01}$,
$\sigma_{45}T=T\sigma_{45}$ and $\tau T=T^{-1}\tau$.
$\mathbb{P}^{1}\times\\{0\\}$$\mathbb{P}^{1}\times\\{\infty\\}$$\\{0\\}\times\mathbb{P}^{1}$$\\{\infty\\}\times\mathbb{P}^{1}$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$-b_{1}$$-b_{2}$$-b_{3}$$-b_{4}$$-b_{5}$$-b_{6}$$-b_{7}$$-b_{8}$
Figure 2. Space of initial data for Painlevé VI (8 points blow up of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$)
###### Definition A.5.
Introduce the following variables:
$\displaystyle b_{1}=\mathsf{t}a_{4}^{-2}={a_{3}^{2}a_{5}\over a_{4}},\quad
b_{2}=\mathsf{t}a_{4}^{2}=a_{3}^{2}a_{4}^{3}a_{5},\quad
b_{3}=\mathsf{t}^{-1}a_{5}^{2}={a_{5}\over a_{3}^{2}a_{4}},\quad
b_{4}=\mathsf{t}^{-1}a_{5}^{-2}={1\over a_{3}^{2}a_{4}a_{5}^{3}},$
$\displaystyle b_{5}=\mathsf{t}\mathsf{p}^{-1}a_{1}^{2}={a_{1}\over
a_{0}a_{2}^{2}},\quad b_{6}=\mathsf{t}\mathsf{p}^{-1}a_{1}^{-2}={1\over
a_{0}a_{1}^{3}a_{2}^{2}},\quad
b_{7}=\mathsf{t}^{-1}\mathsf{p}a_{0}^{-2}={a_{1}a_{2}^{2}\over a_{0}},\quad
b_{8}=\mathsf{t}^{-1}\mathsf{p}a_{0}^{2}=a_{0}^{3}a_{1}a_{2}^{2}.$
The parameters $b_{i}$ are not independent and they satisfy the constraints;
$b_{1}b_{2}b_{3}b_{4}=1,\qquad b_{5}b_{6}b_{7}b_{8}=1,\qquad
b_{1}b_{2}b_{7}b_{8}=\mathsf{p}^{2},\qquad
b_{3}b_{4}b_{5}b_{6}=\mathsf{p}^{-2}.$ (A.1)
Regarding $\mathsf{p}$ as a new parameter, we have six independent parameters.
The root variables are expressed as the ratios of $b_{i}$ (see Figure 1);
$\displaystyle a_{0}^{4}=\frac{b_{8}}{b_{7}},\qquad
a_{1}^{4}=\frac{b_{5}}{b_{6}},\qquad a_{2}^{4}=\frac{b_{7}}{b_{5}},\qquad
a_{3}^{4}=\frac{b_{1}}{b_{3}},\qquad a_{4}^{4}=\frac{b_{2}}{b_{1}},\qquad
a_{5}^{4}=\frac{b_{3}}{b_{4}}.$
###### Corollary A.6.
We have
$\displaystyle f\overline{f}=b_{7}b_{8}{g+b_{5}\over g+b_{7}}{g+b_{6}\over
g+b_{8}},\qquad g\underline{g}=b_{3}b_{4}{f+b_{1}\over f+b_{3}}{f+b_{2}\over
f+b_{4}}.$
### A.3. Tables of the action of $\widetilde{W}$ on $\mathbb{K}(f,g)$
###### Lemma A.7.
Write $a=(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5})$ for short. We have
$\displaystyle r_{0}$
$\displaystyle:(a,f,g)\mapsto(a_{0}^{-1},a_{1},a_{0}a_{2},a_{3},a_{4},a_{5},f,g),$
$\displaystyle r_{1}$
$\displaystyle:(a,f,g)\mapsto(a_{0},a_{1}^{-1},a_{1}a_{2},a_{3},a_{4},a_{5},f,g),$
$\displaystyle r_{2}$
$\displaystyle:(a,f,g)\mapsto(a_{0}a_{2},a_{1}a_{2},a_{2}^{-1},a_{2}a_{3},a_{4},a_{5},f{a_{0}a_{1}^{-1}g+a_{2}^{2}\over
a_{0}a_{1}^{-1}a_{2}^{2}g+1},g),$ $\displaystyle r_{3}$
$\displaystyle:(a,f,g)\mapsto(a_{0},a_{1},a_{2}a_{3},a_{3}^{-1},a_{3}a_{4},a_{3}a_{5},f,{a_{3}^{2}a_{4}a_{5}^{-1}f+1\over
a_{4}a_{5}^{-1}f+a_{3}^{2}}g),$ $\displaystyle r_{4}$
$\displaystyle:(a,f,g)\mapsto(a_{0},a_{1},a_{2},a_{3}a_{4},a_{4}^{-1},a_{5},f,g),$
$\displaystyle r_{5}$
$\displaystyle:(a,f,g)\mapsto(a_{0},a_{1},a_{2},a_{3}a_{5},a_{4},a_{5}^{-1},f,g),$
$\displaystyle\sigma_{01}$
$\displaystyle:(a,f,g)\mapsto(a_{1}^{-1},a_{0}^{-1},a_{2}^{-1},a_{3}^{-1},a_{4}^{-1},a_{5}^{-1},f^{-1},g),$
$\displaystyle\sigma_{45}$
$\displaystyle:(a,f,g)\mapsto(a_{0}^{-1},a_{1}^{-1},a_{2}^{-1},a_{3}^{-1},a_{5}^{-1},a_{4}^{-1},f,g^{-1}),$
$\displaystyle\tau$
$\displaystyle:(a,f,g)\mapsto(a_{5}^{-1},a_{4}^{-1},a_{3}^{-1},a_{2}^{-1},a_{1}^{-1},a_{0}^{-1},g,f),$
$\displaystyle T$
$\displaystyle:a\mapsto(a_{0},a_{1},\mathsf{p}a_{2},\mathsf{p}^{-1}a_{3},a_{4},a_{5}).$
###### Lemma A.8.
Write $b=(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7},b_{8})$ for short. We have
$\displaystyle r_{0}$
$\displaystyle:(b,f,g)\mapsto(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{8},b_{7},f,g),$
$\displaystyle r_{1}$
$\displaystyle:(b,f,g)\mapsto(b_{1},b_{2},b_{3},b_{4},b_{6},b_{5},b_{7},b_{8},f,g),$
$\displaystyle r_{2}$ $\displaystyle:(b,f,g)\mapsto(b_{1}\sqrt{b_{7}\over
b_{5}},b_{2}\sqrt{b_{7}\over b_{5}},b_{3}\sqrt{b_{5}\over
b_{7}},b_{4}\sqrt{b_{5}\over b_{7}},b_{7},b_{6},b_{5},b_{8},f\sqrt{b_{5}\over
b_{7}}\frac{g+b_{7}}{g+b_{5}},g),$ $\displaystyle r_{3}$
$\displaystyle:(b,f,g)\mapsto(b_{3},b_{2},b_{1},b_{4},b_{5}\sqrt{b_{3}\over
b_{1}},b_{6}\sqrt{b_{3}\over b_{1}},b_{7}\sqrt{b_{1}\over
b_{3}},b_{8}\sqrt{b_{1}\over b_{3}},f,\sqrt{b_{1}\over
b_{3}}\frac{f+b_{3}}{f+b_{1}}g),$ $\displaystyle r_{4}$
$\displaystyle:(b,f,g)\mapsto(b_{2},b_{1},b_{3},b_{4},b_{5},b_{6},b_{7},b_{8},f,g),$
$\displaystyle r_{5}$
$\displaystyle:(b,f,g)\mapsto(b_{1},b_{2},b_{4},b_{3},b_{5},b_{6},b_{7},b_{8},f,g),$
$\displaystyle\sigma_{01}$
$\displaystyle:(b,f,g)\mapsto(b_{1}^{-1},b_{2}^{-1},b_{3}^{-1},b_{4}^{-1},b_{7},b_{8},b_{5},b_{6},f^{-1},g),$
$\displaystyle\sigma_{45}$
$\displaystyle:(b,f,g)\mapsto(b_{3},b_{4},b_{1},b_{2},b_{5}^{-1},b_{6}^{-1},b_{7}^{-1},b_{8}^{-1},f,g^{-1}),$
$\displaystyle\tau$
$\displaystyle:(b,f,g)\mapsto(b_{5},b_{6},b_{7},b_{8},b_{1},b_{2},b_{3},b_{4},g,f),$
$\displaystyle T$
$\displaystyle:b\mapsto(\mathsf{p}^{-2}b_{1},\mathsf{p}^{-2}b_{2},\mathsf{p}^{2}b_{3},\mathsf{p}^{2}b_{4},\mathsf{p}^{-2}b_{5},\mathsf{p}^{-2}b_{6},\mathsf{p}^{2}b_{7},\mathsf{p}^{2}b_{8}).$
The action of $T$ on $(f,g)$ is read from the Painlevé VI equation.
## Appendix B Evidences for the Conjecture
### B.1. Heine limit (the sector with vanishing instanton number)
The power of the parameter $\Lambda$ counts the instanton number. In the limit
$\Lambda\to 0$ only the fixed points with instanton number zero survive and
the partition function degenerates to the $q$-hypergeometric series. Note that
in this limit the non-stationary shift $T_{t,\Lambda}$ becomes trivial.
###### Lemma B.1.
The equation
$\displaystyle Sy(x)=y(x),\quad
S:=\frac{1}{\varphi(x)}\cdot\mathcal{B}\cdot\frac{1}{\varphi(-\frac{a}{q}x)\varphi(-\frac{b}{q}x)}\cdot\mathcal{B}\cdot\frac{1}{\varphi(\frac{ab}{q^{2}}x)}T_{\frac{c}{q^{2}},x}$
(B.1)
has a solution given by the Heine’s $q$-hypergeometric series:
$y(x)={}_{2}\phi_{1}(x)={}_{2}\phi_{1}\left[\left.{a,b\atop
c}\right|q;x\right]:=\sum_{n=0}^{\infty}\frac{(a;q)_{n}(b;q)_{n}}{(q,q)_{n}(c;q)_{n}}x^{n}.$
(B.2)
###### Proof.
Recall that the Heine’s series ${}_{2}\varphi_{1}(x)$ satisfies the
$q$-difference equation
$\displaystyle D_{\rm Heine}$ $\displaystyle\ {}_{2}\phi_{1}(x)=0,$
$\displaystyle D_{\rm Heine}$
$\displaystyle:=x(1-ap)(1-bp)p^{-1}-(1-p)(q-cp)p^{-1},\quad p=T_{q,x}.$ (B.3)
In a similar way to the proof of Prop. 4.1, we see the adjoint actions on
$D_{\rm Heine}$ as
$\mapsto$${\rm Ad}\varphi(x)$$\mapsto$${\rm Ad}\mathcal{B}^{-1}$$\mapsto$${\rm
Ad}\varphi(-\frac{a}{q}x)\varphi(-\frac{b}{q}x)$$\mapsto$${\rm
Ad}\mathcal{B}^{-1}$$\mapsto$${\rm Ad}\varphi(\frac{ab}{q^{2}}x)$,
and we have
$S^{-1}D_{\rm Heine}\ S=D_{\rm Heine}.$ (B.4)
Hence we obtain
$D_{\rm Heine}\,S\,{}_{2}\phi_{1}(x)=SD_{\rm Heine}\,{}_{2}\phi_{1}(x)=0.$
(B.5)
Since the power series solution of the equation $D_{\rm Heine}y(x)=0$ of the
form $y(x)=1+O(x)$ is unique, we have the conclusion
$S\,{}_{2}\phi_{1}(x)={}_{2}\phi_{1}(x)$. ∎
###### Remark B.2.
Lemma B.1 shows the factorization of the coefficients $c_{m,0}$ of the
“boundary” terms of the solution (4.9). The other cases $c_{0,n}$ are similar.
### B.2. Macdonald limit
#### B.2.1. Macdonald function of types $A_{1}$ and $A_{2}$ [41]
The asymptotically free Macdonald function $f^{\mathfrak{gl}_{2}}(x|s|q,t)$ of
type $A_{1}$ is written as
$\displaystyle f^{\mathfrak{gl}_{2}}(x|s|q,t)=\sum_{\theta\geq
0}{(t;q)_{\theta}(ts_{2}/s_{1};q)_{\theta}\over(q;q)_{\theta}(qs_{2}/s_{1};q)_{\theta}}(qx_{2}/tx_{1})^{\theta}.$
(B.6)
We have the eigenvalue equation
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{2}}(s|q,t)=s_{1}{1-x_{2}/tx_{1}\over
1-x_{2}/x_{1}}T_{q,x_{1}}+s_{2}{1-tx_{2}/x_{1}\over
1-x_{2}/x_{1}}T_{q,x_{2}},$
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{2}}(s|q,t)f^{\mathfrak{gl}_{2}}(x|s|q,t)=(s_{1}+s_{2})f^{\mathfrak{gl}_{2}}(x|s|q,t).$
The asymptotically free Macdonald function $f^{\mathfrak{gl}_{3}}(x|s|q,t)$ of
type $A_{2}$ is written as
$\displaystyle f^{\mathfrak{gl}_{3}}(x|s|q,t)$
$\displaystyle=\sum_{\theta_{12},\theta_{13},\theta_{23}\geq
0}{(t;q)_{\theta_{12}}(q^{-\theta_{23}+\theta_{13}}ts_{2}/s_{1};q)_{\theta_{12}}\over(q;q)_{\theta_{12}}(q^{-\theta_{23}+\theta_{13}}qs_{2}/s_{1};q)_{\theta_{12}}}(qx_{2}/tx_{1})^{\theta_{12}}$
$\displaystyle\cdot{(t;q)_{\theta_{13}}(ts_{3}/s_{1};q)_{\theta_{13}}\over(q;q)_{\theta_{13}}(qs_{3}/s_{1};q)_{\theta_{13}}}{(ts_{2}/s_{1};q)_{\theta_{13}}(q^{\theta_{23}-\theta_{13}}ts_{1}/s_{2};q)_{\theta_{13}}\over(qs_{2}/s_{1};q)_{\theta_{13}}(q^{\theta_{23}-\theta_{13}}qs_{1}/s_{2};q)_{\theta_{13}}}(q^{2}x_{3}/t^{2}x_{1})^{\theta_{13}}$
$\displaystyle\cdot{(t;q)_{\theta_{23}}(ts_{3}/s_{2};q)_{\theta_{23}}\over(q;q)_{\theta_{23}}(qs_{3}/s_{2};q)_{\theta_{23}}}(qx_{3}/tx_{2})^{\theta_{23}}.$
(B.7)
The eigenvalue equation is
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{3}}(s|q,t)=s_{1}{1-x_{2}/tx_{1}\over
1-x_{2}/x_{1}}{1-x_{3}/tx_{1}\over 1-x_{3}/x_{1}}T_{q,x_{1}}$
$\displaystyle\qquad\qquad\quad+s_{2}{1-tx_{2}/x_{1}\over
1-x_{2}/x_{1}}{1-x_{3}/tx_{2}\over
1-x_{3}/x_{2}}T_{q,x_{2}}+s_{3}{1-tx_{3}/x_{1}\over
1-x_{3}/x_{1}}{1-tx_{3}/x_{2}\over 1-x_{3}/x_{2}}T_{q,x_{3}},$
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{3}}(s|q,t)f^{\mathfrak{gl}_{3}}(x|s|q,t)=(s_{1}+s_{2}+s_{3})f^{\mathfrak{gl}_{3}}(x|s|q,t).$
#### B.2.2. Macdonald limit by tuning mass parameters
Consider the particular choice of the mass parameters;
$\displaystyle T_{1}=v^{-1},\qquad T_{2}=v,\qquad T_{3}=v^{-1},\qquad
T_{4}=vt^{-1}.$ (B.8)
###### Proposition B.3.
Let (B.8) be satisfied. Then the Nekrasov partition function
$\Psi(t\Lambda,x)$ degenerates to the asymptotically free Macdonald function
of type $\mathfrak{gl}_{3}$ with $(x_{1},x_{2},x_{3})=(1/\Lambda,1/x,1)$ and
$(s_{1},s_{2},s_{3})=(1/Q,1,0)$;
$\displaystyle\Psi(t\Lambda,x)=f^{\mathfrak{gl}_{3}}(1/\Lambda,1/x,1|1/Q,1,0|q,t).$
(B.9)
Therefore, we have the eigenvalue equation
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{3}}(1/Q,1,0|q,q/t)\Psi(t\Lambda,x)=(1/Q+1)\Psi(t\Lambda,x),$
where
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{3}}(1/Q,1,0|q,q/t)=Q^{-1}{1-t\Lambda/qx\over
1-\Lambda/x}{1-t\Lambda/q\over
1-\Lambda}T_{q,\Lambda}^{-1}+{1-q\Lambda/tx\over 1-\Lambda/x}{1-tx/q\over
1-x}T_{q,x}^{-1}.$
We need the following lemma which is easily confirmed by the explicit formula
of the Nekrasov factor.
###### Lemma B.4.
(I) $\mathsf{N}_{\lambda,\mu}(1|q,\kappa)\neq 0$ if and only if
$\lambda_{i}\leq\mu_{i}$ $(i\geq 1)$. (II)
$\mathsf{N}_{\lambda,\mu}(\kappa^{-1}|q,\kappa)\neq 0$ if and only if
$\lambda_{i+1}\leq\mu_{i}$ $(i\geq 1)$.
Then the proof of Proposition B.3 is given as follows;
###### Proof.
Under the condition (B.8), we have (see (2.11))
$\displaystyle v\mathfrak{f}^{+}_{2}/\mathfrak{n}_{1}=1,\qquad
v\mathfrak{f}^{+}_{1}/\mathfrak{n}_{2}=1,$ $\displaystyle
w\mathfrak{n}_{1}/\mathfrak{m}_{1}=t,\qquad
w\mathfrak{n}_{2}/\mathfrak{m}_{2}=1,$ $\displaystyle
v\mathfrak{m}_{1}/\mathfrak{f}^{-}_{1}=1,\qquad
v\mathfrak{m}_{2}/\mathfrak{f}^{-}_{2}=t.$
Hence, in view of the Lemma above, we have the following parametrization of
the quadruples of partitions which give rise to the non vanishing
contributions in the partition function (2.10):
$\displaystyle\nu_{1}=(\theta_{23}),\qquad\nu_{2}=(\theta_{13}),$
$\displaystyle\mu_{1}=\emptyset,\qquad\mu_{2}=(\theta_{12}+\theta_{13}),$
where $\theta_{12},\theta_{13},\theta_{23}\geq 0$. Then by making
straightforward calculation and comparison with (B.7) under the identification
$(x_{1},x_{2},x_{3})=(1/\Lambda,1/x,1)$, and $(s_{1},s_{2},s_{3})=(1/Q,1,0)$,
we obtain (B.9).
∎
###### Proposition B.5.
Let (B.8) be satisfied. Then we have
$\displaystyle\mathcal{F}^{(1)}=\sum_{m\geq
0}{(q/t;q)_{m}(qQ/t;q)_{m}\over(q;q)_{m}(qQ;q)_{m}}t^{m}(\Lambda/x)^{m}=f^{\mathfrak{gl}_{2}}(1/\Lambda,1/x|1/Q,1|q,t),$
(B.10) $\displaystyle\mathcal{F}^{(2)}=\sum_{m,n\geq
0}{(t;q)_{m}(q^{n}q/t;q)_{m}\over(q;q)_{m}(q^{n}qQ;q)_{m}}\left(-(qt^{1/2}Q)^{1/2}\right)^{m}(\Lambda/x)^{m}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{2n}\Lambda^{n}.$
(B.11)
Hence with the identification $(x_{1},x_{2})=(1/\Lambda,1/x)$, we have the
eigenvalue equation
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{2}}(1/Q,1|q,q/t)\mathcal{F}^{(1)}=(1/Q+1)\mathcal{F}^{(1)},$
where
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{2}}(1/Q,1|q,q/t)=Q^{-1}{1-t\Lambda/qx\over
1-\Lambda/x}T_{q,\Lambda}^{-1}+{1-q\Lambda/tx\over 1-\Lambda/x}T_{q,x}^{-1}.$
###### Proof.
Under the condition (B.8), we have
$\displaystyle\mathcal{F}^{(1)}=f\left(\left.\left.\begin{array}[]{ll}tQ^{1/2},&qt^{-1/2}Q^{-1/2}\\\
Q^{-1/2},&t^{1/2}Q^{1/2}\\\
Q^{1/2},&q^{-1}t^{3/2}Q^{-1/2}\end{array}\right|-t^{1/2}x,-\Lambda/x\right|q,t^{-1/2}\right),$
and
$\displaystyle\mathcal{F}^{(2)}=f\left(\left.\left.\begin{array}[]{ll}tQ^{1/2},&qt^{-1/2}Q^{-1/2}\\\
Q^{-1/2},&t^{1/2}Q^{1/2}\\\
q^{-1}tQ^{-1/2},&t^{-1/2}Q^{1/2}\end{array}\right|t^{3/4}x,t^{1/4}\Lambda/x\right|q,t^{-1/2}\right).$
One finds that $\mathcal{F}^{(1)}$ has the factor
$\displaystyle\mathsf{N}^{(1|2)}_{\lambda^{(2)},\emptyset}(t^{1/2}|q,t^{-1/2}),$
which is non vanishing, if and only if $\ell(\lambda^{(2)})\leq 1$. The
$\mathcal{F}^{(1)}$ also has the factor
$\displaystyle\mathsf{N}^{(1|2)}_{\emptyset,\lambda^{(1)}}(qt^{-1/2}|q,t^{-1/2}),$
which is non vanishing, if and only if $\lambda^{(1)}=\emptyset$. Hence the
non vanishing contributions for $\mathcal{F}^{(1)}$ arise if and only if we
have $(\lambda^{(1)},\lambda^{(2)})=(\emptyset,(m))$ $(m\geq 0)$. Making
explicit calculation, we have (B.10).
The $\mathcal{F}^{(2)}$ has the factor
$\displaystyle\mathsf{N}^{(0|2)}_{\lambda^{(2)},\emptyset}(t|q,t^{-1/2}),$
which is non vanishing, if and only if $\ell(\lambda^{(2)})\leq 2$. The
$\mathcal{F}^{(2)}$ also has the factor
$\displaystyle\mathsf{N}^{(1|2)}_{\emptyset,\lambda^{(1)}}(qt^{-1/2}|q,t^{-1/2}),$
which is non vanishing, if and only if $\lambda^{(1)}=\emptyset$. Hence the
non vanishing contributions for $\mathcal{F}^{(2)}$ arise if and only if we
have $(\lambda^{(1)},\lambda^{(2)})=(\emptyset,(n+m,n))$ $(m,n\geq 0)$. Making
explicit calculation, we have (B.11).
∎
We have seen how both partition functions $\Psi(t\Lambda,x)$ and
$\mathcal{F}^{(1)}$ are simplified to the asymptotically free Macdonald
functions in the limit (B.8). If Conjectures 2.4 and 5.4 are true, they are
related by the following gauge transformation.
###### Conjecture B.6.
$\displaystyle\mathcal{F}^{(1)}$
$\displaystyle={1\over\Phi(t^{2}T_{2}T_{3}\Lambda)\Phi(qtT_{1}T_{4}\Lambda)}\varphi(q^{1/2}t^{1/2}T_{1}x)\varphi(q^{1/2}t^{1/2}T_{4}\Lambda/x)\psi(\Lambda,x)$
$\displaystyle={\Phi(qtT_{2}T_{3}\Lambda)\Phi(t^{2}T_{1}T_{4}\Lambda)\over\Phi(t^{2}T_{2}T_{3}\Lambda)\Phi(qtT_{1}T_{4}\Lambda)}\varphi(q^{1/2}t^{1/2}T_{1}x)\varphi(q^{1/2}t^{1/2}T_{4}\Lambda/x)\mathcal{A}_{3}(t\Lambda,tqQx)\Psi(t\Lambda,x).$
We can prove this is indeed the case.
###### Lemma B.7.
We have
$\displaystyle
f^{\mathfrak{gl}_{2}}(1/\Lambda,1/x|1/Q,1|q,t)={(t\Lambda;q)_{\infty}\over(q\Lambda;q)_{\infty}}{(tx;q)_{\infty}\over(qx;q)_{\infty}}f^{\mathfrak{gl}_{3}}(1/\Lambda,1/x,1|1/Q,1,0|q,t).$
(B.12)
###### Proof.
We find that the Macdonald operators
$\mathcal{D}^{\mathfrak{gl}_{3}}(1/Q,1,0|q,q/t)$ and
$\mathcal{D}^{\mathfrak{gl}_{2}}(1/Q,1|q,q/t)$ are gauge equivalent
$\displaystyle{(t\Lambda;q)_{\infty}\over(q\Lambda;q)_{\infty}}{(tx;q)_{\infty}\over(qx;q)_{\infty}}\mathcal{D}^{\mathfrak{gl}_{3}}(1/Q,1,0|q,q/t){(q\Lambda;q)_{\infty}\over(t\Lambda;q)_{\infty}}{(qx;q)_{\infty}\over(tx;q)_{\infty}}=\mathcal{D}^{\mathfrak{gl}_{2}}(1/Q,1|q,q/t).$
Then the equality (B.12) follows from the uniqueness of the normalized
asymptotic eigenfunctions of $\mathcal{D}^{\mathfrak{gl}_{3}}(1/Q,1,0|q,q/t)$
and $\mathcal{D}^{\mathfrak{gl}_{2}}(1/Q,1|q,q/t)$. ∎
###### Proposition B.8.
Conjecture B.6 holds in the Macdonald limit (B.8).
###### Proof.
Under the condition (B.8), we have
$\displaystyle{\Phi(qtT_{2}T_{3}\Lambda)\Phi(t^{2}T_{1}T_{4}\Lambda)\over\Phi(t^{2}T_{2}T_{3}\Lambda)\Phi(qtT_{1}T_{4}\Lambda)}\varphi(q^{1/2}t^{1/2}T_{1}x)\varphi(q^{1/2}t^{1/2}T_{4}\Lambda/x)\mathcal{A}_{3}(t\Lambda,tqQx)={(t\Lambda;q)_{\infty}\over(q\Lambda;q)_{\infty}}{(tx;q)_{\infty}\over(qx;q)_{\infty}}.$
∎
Now we are ready to prove Conjecture 5.4 in the Macdonald limit. We need the
following formulas for the proof.
###### Lemma B.9 ([44]).
For $n\in\mathbb{Z}$, we have
$\displaystyle\mathcal{B}\cdot{1\over\varphi(\alpha
x)\varphi(\beta\Lambda/x)}x^{n}={\varphi(-q^{1+n}\alpha
x)\varphi(-q^{-n}\beta\Lambda/x)\over\varphi(\alpha\beta\Lambda)}q^{n(n+1)/2}x^{n},$
(B.13) $\displaystyle\mathcal{B}^{-1}\cdot{\varphi(\alpha
x)\varphi(\beta\Lambda/x)}x^{n}={\varphi(q^{-1}\alpha\beta\Lambda)\over\varphi(-q^{-1-n}\alpha
x)\varphi(-q^{n}\beta\Lambda/x)}q^{-n(n+1)/2}x^{n}.$ (B.14)
###### Proposition B.10 ($q$-Chu-Vandermonde sums [22]).
For $n\in\mathbb{Z}_{\geq 0}$, we have
$\displaystyle{}_{2}\phi_{1}\left[\left.{a,q^{-n}\atop
c}\right|q;q\right]={(c/a;q)_{n}\over(c;q)_{n}}a^{n},$ (B.15)
$\displaystyle{}_{2}\phi_{1}\left[\left.{a,q^{-n}\atop c}\right|q;{cq^{n}\over
a}\right]={(c/a;q)_{n}\over(c;q)_{n}}.$ (B.16)
###### Proposition B.11.
Conjecture 5.4 holds in the Macdonald limit (B.8).
$\displaystyle
T_{(qt^{1/2}Q)^{1/2},x}\mathcal{F}^{(2)}=\varphi(q\Lambda){1\over\varphi(-qtx)\varphi(-\Lambda/x)}\cdot\mathcal{B}\cdot{1\over\varphi(tx)\varphi(q\Lambda/tx)}\mathcal{F}^{(1)},$
(B.17)
$\displaystyle\mathcal{F}^{(1)}=\varphi(t\Lambda){1\over\varphi(qx)\varphi(t\Lambda/x)}\cdot\mathcal{B}\cdot{1\over\varphi(-x)\varphi(-q\Lambda/x)}(T_{t^{1/2},x}^{-1}T_{t,\Lambda}^{-1})T_{(qt^{1/2}Q)^{-1/2},x}\mathcal{F}^{(2)}.$
(B.18)
###### Proof.
First, we show (B.17). By using (B.13) and the $q$-binomial formula, we have
$\displaystyle\varphi(q\Lambda){1\over\varphi(-qtx)\varphi(-\Lambda/x)}\cdot\mathcal{B}\cdot{1\over\varphi(tx)\varphi(q\Lambda/tx)}\mathcal{F}^{(1)}$
$\displaystyle={}$
$\displaystyle\varphi(q\Lambda){1\over\varphi(-qtx)\varphi(-\Lambda/x)}\cdot\mathcal{B}\cdot{1\over\varphi(tx)\varphi(q\Lambda/tx)}\sum_{n\geq
0}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{n}(\Lambda/x)^{n}$
$\displaystyle={}$ $\displaystyle\sum_{n\geq
0}{\varphi(-q^{1-n}tx)\varphi(-q^{1+n}\Lambda/tx)\over\varphi(-qtx)\varphi(-\Lambda/x)}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{n}q^{n(n-1)/2}(\Lambda/x)^{n}$
$\displaystyle={}$ $\displaystyle\sum_{k,l,n\geq
0}{(q^{-n};q)_{k}\over(q;q)_{k}}(-qtx)^{k}{(q^{n+1}/t;q)_{l}\over(q;q)_{l}}(-\Lambda/x)^{l}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{n}q^{n(n-1)/2}(\Lambda/x)^{n}.$
Note that we have the truncation of the summation as $\sum_{k,l,n\geq
0}=\sum_{k,l\geq 0}\sum_{n=k}^{\infty}$ because of the factor
$(q^{-n};q)_{k}$. Shifting the running index $n$ by $k$ as $n\rightarrow n+k$
(hence changing the summation range as $\sum_{k,l\geq
0}\sum_{n=k}^{\infty}\rightarrow\sum_{k,l\geq 0}\sum_{n=0}^{\infty}$
accordingly), we have
$\displaystyle={}$ $\displaystyle\sum_{k,l,n\geq
0}{(q^{-k-n};q)_{k}\over(q;q)_{k}}(-qt)^{k}{(q^{k+n+1}/t;q)_{l}\over(q;q)_{l}}(-1)^{l}$
$\displaystyle\cdot{(q/t;q)_{k+n}(qQ/t;q)_{k+n}\over(q;q)_{k+n}(qQ;q)_{k+n}}t^{k+n}q^{(k+n)(k+n-1)/2}\cdot(\Lambda/x)^{l+n}\Lambda^{k}.$
Then we change the running index $l$ to $m$ (where $m=l+n$) as $\sum_{l,n\geq
0}=\sum_{m\geq 0}\sum_{n=0}^{m}$. Simplifying the factors, we have
$\displaystyle={}$ $\displaystyle\sum_{k\geq 0}\sum_{m\geq
0}{(q^{k+1}/t;q)_{m}\over(q;q)_{m}}(-\Lambda/x)^{m}{(q/t;q)_{k}\over(q;q)_{k}}{(qQ/t;q)_{k}\over(qQ;q)_{k}}(t^{2}\Lambda)^{k}\sum_{n=0}^{m}{(q^{-m};q)_{n}\over(q;q)_{n}}{(q^{k+1}Q/t;q)_{n}\over(q^{k+1}Q;q)_{n}}q^{mn}t^{n}$
$\displaystyle={}$ $\displaystyle\sum_{k\geq 0}\sum_{m\geq
0}{(q^{k+1}/t;q)_{m}\over(q;q)_{m}}{(t;q)_{m}\over(q^{k+1}Q;q)_{m}}(-\Lambda/x)^{m}{(q/t;q)_{k}\over(q;q)_{k}}{(qQ/t;q)_{k}\over(qQ;q)_{k}}(t^{2}\Lambda)^{k}$
$\displaystyle={}$ $\displaystyle T_{(qt^{1/2}Q)^{1/2},x}\mathcal{F}^{(2)}.$
Here, we have used the $q$-Chu-Vandermonde summation formula (B.16) in the
last step.
The second equation (B.18) can be shown in exactly the same manner as above.
Note that (B.18) is equivalent to
$\displaystyle(T_{t^{1/2},x}^{-1}T_{t,\Lambda}^{-1})T_{(qt^{1/2}Q)^{-1/2},x}\mathcal{F}^{(2)}=$
$\displaystyle{}\,{1\over\varphi(t\Lambda)}{\varphi(-x)\varphi(-q\Lambda/x)}\cdot\mathcal{B}^{-1}\cdot{\varphi(qx)\varphi(t\Lambda/x)}\mathcal{F}^{(1)}.$
By using (B.14) and the $q$-binomial formula, we have
$\displaystyle{1\over\varphi(t\Lambda)}{\varphi(-x)\varphi(-q\Lambda/x)}\cdot\mathcal{B}^{-1}\cdot{\varphi(qx)\varphi(t\Lambda/x)}\mathcal{F}^{(1)}$
$\displaystyle={}$
$\displaystyle{1\over\varphi(t\Lambda)}{\varphi(-x)\varphi(-q\Lambda/x)}\cdot\mathcal{B}^{-1}\cdot{\varphi(qx)\varphi(t\Lambda/x)}\sum_{n\geq
0}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{n}(\Lambda/x)^{n}$
$\displaystyle={}$ $\displaystyle\sum_{n\geq
0}{\varphi(-x)\varphi(-q\Lambda/x)\over\varphi(-q^{n}x)\varphi(-q^{-n}t\Lambda/x)}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{n}q^{-n(n-1)/2}(\Lambda/x)^{n}$
$\displaystyle={}$ $\displaystyle\sum_{k,l,n\geq
0}{(q^{-n};q)_{k}\over(q;q)_{k}}(-q^{n}x)^{k}{(q^{n+1}/t;q)_{l}\over(q;q)_{l}}(-q^{-n}t\Lambda/x)^{l}{(q/t;q)_{n}(qQ/t;q)_{n}\over(q;q)_{n}(qQ;q)_{n}}t^{n}q^{-n(n-1)/2}(\Lambda/x)^{n}.$
Note that we have $\sum_{k,l,n\geq 0}=\sum_{k,l\geq 0}\sum_{n=k}^{\infty}$
because of the factor $(q^{-n};q)_{k}$. Shifting the running index $n$ by $k$
as $n\rightarrow n+k$ (and changing the summation range as $\sum_{k,l\geq
0}\sum_{n=k}^{\infty}\rightarrow\sum_{k,l\geq 0}\sum_{n=0}^{\infty}$
accordingly), we have
$\displaystyle={}$ $\displaystyle\sum_{k,l,n\geq
0}{(q^{-k-n};q)_{k}\over(q;q)_{k}}(-q^{n})^{k}{(q^{k+n+1}/t;q)_{l}\over(q;q)_{l}}(-q^{-n}t)^{l}$
$\displaystyle\cdot{(q/t;q)_{k+n}(qQ/t;q)_{k+n}\over(q;q)_{k+n}(qQ;q)_{k+n}}t^{k+n}q^{-(k+n)(k+n-1)/2}\cdot(\Lambda/x)^{l+n}\Lambda^{k}.$
Changing the running index $l$ to $m$ (where $m=l+n$) as $\sum_{l,n\geq
0}=\sum_{m\geq 0}\sum_{n=0}^{m}$, and simplifying the factors, we have
$\displaystyle={}$ $\displaystyle\sum_{k\geq 0}\sum_{m\geq
0}{(q^{k+1}/t;q)_{m}\over(q;q)_{m}}(-\Lambda/x)^{m}{(q/t;q)_{k}\over(q;q)_{k}}{(qQ/t;q)_{k}\over(qQ;q)_{k}}\Lambda^{k}q^{-km}t^{k+m}\sum_{n=0}^{m}{(q^{-m};q)_{n}\over(q;q)_{n}}{(q^{k+1}Q/t;q)_{n}\over(q^{k+1}Q;q)_{n}}q^{n}$
$\displaystyle={}$ $\displaystyle\sum_{k\geq 0}\sum_{m\geq
0}{(q^{k+1}/t;q)_{m}\over(q;q)_{m}}{(t;q)_{m}\over(q^{k+1}Q;q)_{m}}(-qQ\Lambda/x)^{m}{(q/t;q)_{k}\over(q;q)_{k}}{(qQ/t;q)_{k}\over(qQ;q)_{k}}(t\Lambda)^{k}$
$\displaystyle={}$
$\displaystyle(T_{t^{1/2},x}^{-1}T_{t,\Lambda}^{-1})T_{(qt^{1/2}Q)^{-1/2},x}\mathcal{F}^{(2)}.$
Here, we have used the $q$-Chu-Vandermonde summation formula (B.15) in the
last step. ∎
#### B.2.3. Macdonald limit in [44]
For the sake of readers’ convenience, we recollect the facts concerning the
choice of the parameters investigated in [44];
$\displaystyle T_{1}=vt^{-1},\qquad T_{2}=v^{-1},\qquad T_{3}=v^{-1},\qquad
T_{4}=vt^{-1}.$ (B.19)
Note that in the limit (B.19) we have (See Conjecture 2.4);
$\displaystyle\mathcal{A}_{1}(\Lambda,x)={1\over\varphi(qt^{-1}x)\varphi(t\Lambda/x)},$
$\displaystyle\mathcal{A}_{2}(\Lambda,x)={\varphi(t\Lambda)\varphi(qt^{-2}\Lambda)\over\varphi(-t^{-1}x)\varphi(-Q^{-1}x)\varphi(-qQ\Lambda/x)\varphi(-q\Lambda/x)},$
$\displaystyle\mathcal{A}_{3}(\Lambda,x)={1\over\varphi(q^{-1}Q^{-1}x)\varphi(q^{2}t^{-1}Q\Lambda/x)}.$
Set
$\displaystyle U(\Lambda,x)=\sum_{k,l\geq
0}{(q/t;q)_{k}(q^{l}qQ/t;q)_{k}\over(q;q)_{k}(q^{l}qQ;q)_{k}}(\Lambda/x)^{k}{(t;q)_{l}(tQ;q)_{l}\over(q;q)_{l}(qQ;q)_{l}}(q\Lambda/t^{2})^{l},$
$\displaystyle V(\Lambda,x)=\sum_{k,l\geq
0}{(q/t;q)_{k}(t;q)_{k}\over(q;q)_{k}(qQ;q)_{k}}(-qQ\Lambda/x)^{k}{(q^{k}t^{2};q)_{l}(tQ;q)_{l}\over(q;q)_{l}(q^{k}qQ;q)_{l}}(q\Lambda/t^{2})^{l}.$
###### Proposition B.12 ([44]).
When (B.19) is satisfied, we have
$\displaystyle\Psi(\Lambda,x)=U(\Lambda,x),$
and
$\displaystyle
V(\Lambda,x)={\varphi(t\Lambda)\over\varphi(-Q^{-1}x)\varphi(-qQ\Lambda/x)}\cdot\mathcal{B}\cdot{1\over\varphi(q^{-1}Q^{-1}x)\varphi(q^{2}t^{-1}Q\Lambda/x)}U\left(\Lambda,{x\over
tqQ}\right),$ $\displaystyle
U(t\Lambda,x)={\varphi(qt^{-2}\Lambda)\over\varphi(qt^{-1}x)\varphi(t\Lambda/x)}\cdot\mathcal{B}\cdot{1\over\varphi(-t^{-1}x)\varphi(-q\Lambda/x)}V\left(\Lambda,x\right).$
###### Proposition B.13.
In the limit (B.19), we have
$\displaystyle{\varphi(t\Lambda/x)\over\varphi(q\Lambda/tx)}\Psi(t\Lambda,x)$
$\displaystyle=\sum_{k,l\geq
0}{(t;q)_{k}(q^{l}tQ;q)_{k}\over(q;q)_{k}(q^{l}qQ;q)_{k}}(q\Lambda/tx)^{k}{(t;q)_{l}(tQ;q)_{l}\over(q;q)_{l}(qQ;q)_{l}}(q\Lambda/t)^{l}$
$\displaystyle=f^{\mathfrak{gl}_{3}}(1/\Lambda,1/x,1|tQ^{-1},t,1|q,t),$
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{3}}(tQ^{-1},t,1|q,t){\varphi(t\Lambda/x)\over\varphi(q\Lambda/tx)}\Psi(t\Lambda,x)=(tQ^{-1}+t+1){\varphi(t\Lambda/x)\over\varphi(q\Lambda/tx)}\Psi(t\Lambda,x),$
where
$\displaystyle\mathcal{D}^{\mathfrak{gl}_{3}}(tQ^{-1},t,1|q,t)$
$\displaystyle=tQ^{-1}{1-\Lambda/tx\over 1-\Lambda/x}{1-\Lambda/t\over
1-\Lambda}T_{q,\Lambda}^{-1}$ $\displaystyle+t{1-t\Lambda/x\over
1-\Lambda/x}{1-x/t\over 1-x}T_{q,x}^{-1}+{1-t\Lambda/x\over
1-\Lambda/x}{1-tx\over 1-x}T_{q,\Lambda}T_{q,x}.$
###### Remark B.14.
In the limit (B.19), the set of partitions giving rise to the non vanishing
contributions for $\mathcal{F}^{(1)}$ remains rather big. This seems to be
consistent with the appearance of double infinite products in the gauge
transformation;
$\displaystyle{\Phi(q^{-1}t^{3}\Lambda)\Phi(q^{2}t^{-2}\Lambda)\over\Phi(qt^{-1}\Lambda)\Phi(t^{2}\Lambda)}{\varphi(tx)\over\varphi(qt^{-1}x)}\mathcal{F}^{(1)}=\Psi(t\Lambda,x).$
In four dimensional theory, the set of partitions giving rise to the non
vanishing contributions is the same. The four dimensional limit of the ratio
of double infinite products is
$(1-\Lambda)^{-\frac{2(\epsilon_{1}+\epsilon_{2})^{2}}{\epsilon_{1}\epsilon_{2}}}$.
On the other hand the corresponding set of partitions for $\mathcal{F}^{(2)}$
becomes small and gives us
$T_{(qt^{1/2}Q)^{1/2},x}\mathcal{F}^{(2)}=\sum_{m,n\geq
0}{(t;1)_{m}(q^{n}q/t;1)_{m}\over(q;q)_{m}(q^{n}qQ;q)_{m}}(-\Lambda/x)^{m}{(q/t;1)_{n}(qQ/t;1)_{n}\over(q;q)_{n}(qQ;q)_{n}}(t^{2}\Lambda)^{n}.$
## Appendix C Four dimensional limit in a factorized form
In section 4, we computed the four dimensional limit of the operator $\SS$.
Here we will consider the limit of each factor
$\begin{array}[]{l}\displaystyle
K_{1}=\frac{1}{\varphi(\Lambda)}\varphi(-d_{1}x)\varphi(-d_{3}\frac{\Lambda}{x})\mathcal{B}^{-1}\varphi(qx)\varphi(\frac{\Lambda}{x}),\\\
\displaystyle
K_{2}=\frac{1}{\varphi(q^{-1}d_{2}d_{4}\Lambda)}\varphi(q^{-1}d_{1}d_{2}x)\varphi(d_{3}d_{4}\frac{\Lambda}{x})\mathcal{B}^{-1}\varphi(-d_{2}x)\varphi(-d_{4}\frac{\Lambda}{x}),\\\
\displaystyle
N=\frac{\varphi(q^{-1}d_{2}d_{4}\Lambda)}{\varphi(q^{-1}d_{1}d_{2}d_{3}d_{4}\Lambda)},\end{array}$
(C.1)
in the decomposition
$\SS^{-1}=NK_{2}K_{1},$ (C.2)
separately. We show that each factor $K_{i}$ has also a well defined four
dimensional limit, though the result is not usual differencial operator any
more. To describe the results in a simple form, it is convenient to use the
normal ordering $:\ :$ which is a $\mathbb{C}$-linear map from a commutative
ring $\mathbb{C}(x,\vartheta_{x})_{/\vartheta_{x}x=x\vartheta_{x}}$ to the
corresponding non-commutative ring
$\mathbb{C}(x,\vartheta_{x})_{/\vartheta_{x}x=x(\vartheta_{x}+1)}$ defined by
$:x^{n}F(x,\vartheta_{x}):\
=x^{n}:F(x,\vartheta_{x}):,\quad:F(x,\vartheta_{x})\vartheta_{x}^{n}:\ =\
:F(x,\vartheta_{x}):{\vartheta_{x}}^{n},\quad:1:=1.$ (C.3)
###### Theorem C.1.
For $q=e^{h},d_{i}=q^{m_{i}}$, we have
$\displaystyle K_{1}$
$\displaystyle=~{}:(1+x)^{-m_{1}-\vartheta_{x}}(1+\frac{\Lambda}{x})^{-m_{3}+\vartheta_{x}}\left(1+\frac{h}{2}A_{1}\right)+{\mathcal{O}}(h^{2}):,$
$\displaystyle K_{2}$
$\displaystyle=~{}:(1-x)^{-m_{1}-\vartheta_{x}}(1-\frac{\Lambda}{x})^{-m_{3}+\vartheta_{x}}\left(1+\frac{h}{2}A_{2}\right)+{\mathcal{O}}(h^{2}):,$
$\displaystyle N$
$\displaystyle=(1-\Lambda)^{-(m_{1}+m_{3})}\left(1+\frac{h}{2}\frac{(m_{1}+m_{3})(m_{1}+2m_{2}+m_{3}+2m_{4}-3)}{(1-\Lambda)}\right)+{\mathcal{O}}(h^{2}),$
where
$\displaystyle A_{1}$
$\displaystyle=\frac{x(\vartheta_{x}-m_{1}+1)(\vartheta_{x}+m_{1})}{1+x}+\frac{\frac{\Lambda}{x}(\vartheta_{x}+m_{3}-1)(\vartheta_{x}-m_{3})}{1+\frac{\Lambda}{x}}-\vartheta_{x}(\vartheta_{x}+1),$
$\displaystyle A_{2}$
$\displaystyle=-\frac{x(\vartheta_{x}-m_{1}-2m_{2}+3)(\vartheta_{x}+m_{1})}{1-x}-\frac{\frac{\Lambda}{x}(\vartheta_{x}+m_{3}+2m_{4}-1)(\vartheta_{x}-m_{3})}{1-\frac{\Lambda}{x}}-\vartheta_{x}(\vartheta_{x}+1).$
###### Proof.
From (B.14) in Appendix B, we have
$\mathcal{B}^{-1}\varphi(qx)\varphi(\frac{\Lambda}{x})x^{n}=\frac{\varphi(\Lambda)}{\varphi(-q^{-n}x)\varphi(-q^{n}\frac{\Lambda}{x})}q^{-n(n+1)}x^{n},$
(C.4)
and hence
$K_{1}x^{n}=\frac{\varphi(-d_{1}x)\varphi(-d_{3}\frac{\Lambda}{x})}{\varphi(-q^{-n}x)\varphi(-q^{n}\frac{\Lambda}{x})}q^{-n(n+1)}x^{n}.$
(C.5)
Then, using the limiting formula of the $q$-binomial theorem
$\frac{\varphi(q^{j}x)}{\varphi(x)}=\sum_{k=1}^{\infty}\frac{(q^{j})_{k}}{(q)_{k}}x^{k}=(1-x)^{-j}\big{\\{}1+\frac{hj(j-1)}{2}\frac{x}{1-x}\big{\\}}+{\mathcal{O}}(h^{2}),$
(C.6)
we obtain
$\begin{array}[]{l}\displaystyle
K_{1}x^{n}=(1+x)^{-m_{1}-n}(1+\frac{\Lambda}{x})^{-m_{3}+n}(1+\frac{h}{2}A_{1})+{\mathcal{O}}(h^{2})\\\
\displaystyle\qquad
A_{1}=\frac{x(n-m_{1}+1)(n+m_{1})}{1+x}+\frac{\frac{\Lambda}{x}(n+m_{3}-1)(n-m_{3})}{1+\frac{\Lambda}{x}}-n(n+1),\end{array}$
(C.7)
as desired.
The expression for $K_{2}$ follows from that for $K_{1}$ by the substitution
$x\to-q^{-1}d_{2}x,\quad\Lambda\to q^{-1}d_{2}d_{4}\Lambda.$ (C.8)
The limit of $N$ follows directly as
$\begin{array}[]{l}\displaystyle
N=\exp\sum_{n=1}^{\infty}\frac{(q^{-1}d_{2}d_{4}\Lambda)^{n}-(q^{-1}d_{1}d_{2}d_{3}d_{4}\Lambda)^{n}}{n(1-q^{n})}=\exp\sum_{n=1}^{\infty}\frac{(q^{m_{2}+m_{4}-1}\Lambda)^{n}(1-q^{m_{1}+m_{3}+n})}{n(1-q^{n})}\\\
\displaystyle=\exp\\{\sum_{n=1}^{\infty}\frac{1}{n}(1+nh(m_{2}+m_{4}-1))\Lambda^{n}(m_{1}+m_{3})(1+\frac{nh}{2}(m_{1}+m_{3}-1))\\}+{\mathcal{O}}(h^{2}),\\\
\displaystyle=\exp\\{(m_{1}+m_{3})\sum_{n=1}^{\infty}\frac{\Lambda^{n}}{n}(1+\frac{nh(m_{1}+2m_{2}+m_{3}+2m_{4}-3)}{2})\\}+{\mathcal{O}}(h^{2}),\\\
\displaystyle=(1+\frac{h(m_{1}+2m_{2}+m_{3}+2m_{4}-3)}{2}\vartheta_{\Lambda})\exp\\{(m_{1}+m_{3})\sum_{n=1}^{\infty}\frac{\Lambda^{n}}{n}\\}+{\mathcal{O}}(h^{2}),\\\
\displaystyle=(1-\Lambda)^{-(m_{1}+m_{3})}(1+\frac{h}{2}\frac{(m_{1}+m_{3})(m_{1}+2m_{2}+m_{3}+2m_{4}-3)}{(1-\Lambda)})+{\mathcal{O}}(h^{2}).\end{array}$
(C.9)
∎
###### Remark C.2.
The result of Theorem C.1 is consistent with the result in section 4. For
instance, the consistency in the leading order is given by the identity
$:(1-x)^{-m_{1}-\vartheta_{x}}(1-\frac{\Lambda}{x})^{-m_{3}+\vartheta_{x}}::(1+x)^{-m_{1}-\vartheta_{x}}(1+\frac{\Lambda}{x})^{-m_{3}+\vartheta_{x}}:=(1-\Lambda)^{m_{1}+m_{3}},$
(C.10)
which follows from formal computations such as
$:(1+x)^{-m_{1}-\vartheta_{x}}(1+\frac{\Lambda}{x})^{-m_{3}+\vartheta_{x}}:f(x)=(1+x)^{-m_{1}}(1+\frac{\Lambda}{x})^{-m_{3}}f(x\frac{1+\frac{\Lambda}{x}}{1+x}).$
(C.11)
In relation to (C.11), the formula
$:(1-x)^{-b-\vartheta_{x}}:F(a,b,c;-x)=(1-x)^{-b}F(a,b,c,\frac{x}{x-1})=F(c-a,b,c;x).$
(C.12)
for the Gauss Hypergeometric series $F(a,b,c;x)$ will be instructive.
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|
# Stellar s-process neutron capture cross sections of 69,71Ga
M. Tessler<EMAIL_ADDRESS>Soreq Nuclear Research Center,
Yavne, Israel 81800 Racah Institute of Physics, Hebrew University, Jerusalem,
Israel 91904 M. Paul Racah Institute of Physics, Hebrew University,
Jerusalem, Israel 91904 S. Halfon Soreq Nuclear Research Center, Yavne,
Israel 81800 Y. Kashiv University of Notre Dame, Notre Dame, IN 46556, USA
D. Kijel Soreq Nuclear Research Center, Yavne, Israel 81800 A. Kreisel
Soreq Nuclear Research Center, Yavne, Israel 81800 A. Shor Soreq Nuclear
Research Center, Yavne, Israel 81800 L. Weissman Soreq Nuclear Research
Center, Yavne, Israel 81800
###### Abstract
The stable gallium isotopes, 69,71Ga, are mostly produced by the weak slow
($s$) process in massive stars. We report here on measurements of
astrophysically-relevant neutron capture cross sections of the
69,71Ga$(n,\gamma)$ reactions. The experiments were performed by the
activation technique using a high-intensity ($3-5\times 10^{10}$ n/s), quasi-
Maxwellian neutron beam that closely mimics conditions of stellar $s$-process
nucleosynthesis at $kT\approx$ 40 keV. The neutron field was produced by a mA
proton beam at $E_{p}=1925$ keV (beam power of 2-3 kW) from the Soreq Applied
Research Accelerator Facility (SARAF), bombarding the Liquid-Lithium Target
(LiLiT). A 473 mg sample of Ga2O3 of natural isotopic composition was
activated in the LiLiT neutron field and the activities of 70,72Ga were
measured by decay counting via $\gamma$-spectrometry with a high-purity
germanium detector. The Maxwellian-averaged cross sections at $kT$ = 30 keV of
69Ga and 71Ga determined in this work are 136(8) mb and 105(7) mb,
respectively, in good agreement with previous experimental values.
Astrophysical implications of the measurements are discussed.
## I Introduction
The majority of elements beyond iron are produced by neutron capture reactions
in stars. The slow neutron capture process ($s$ process) is composed of the
weak and main components [1]. The main component takes place during recurrent
thermal pulses in the He shell of low-mass AGB stars ($M\lesssim 4M_{\odot}$).
The weak component is produced during core He and shell C burning in massive
stars ($M\gtrsim 8M_{\odot}$). Neutron densities during the $s$ process are
between $10^{6}$ and $10^{12}$ $n/cm^{3}$ [2]. The weak $s$ process produces
most of the $s$-process isotopes between iron and strontium ($60<A<90$).
Figure 1 shows the weak $s$-process flow in the Ga region.
Figure 1: The path of the $s$ process between zinc and arsenic. The numerical
values are the terrestrial isotopic abundances in % (half-life) for stable
(unstable) nuclides. When 69Ga captures a neutron, the product 70Ga either
decays to 70Ge (99.59%) or to 70Zn (0.41%) with a half-life of 21.14 minutes.
Following the neutron capture reaction on 71Ga, the product 72Ga decays to
72Ge with a half-life of 14.1 hours.
Recent simulations show that gallium is the most abundant $s$-element at the
end of shell carbon burning in a model of $25M_{\odot}$ star [3].
Until recently, there was only a Time-of-Flight (TOF) measurement of the
69Ga$(n,\gamma)$ cross section, with a sample of natural gallium [4] (see also
[5]). Other nuclei measured within the same experimental campaign (74Ge, 75As,
81Br) show large deviations from more recent results [5]. For 71Ga, two
previous results from activation measurements are in marginal agreement [6,
7]. Very recently, values were published for the neutron capture cross
sections of 69,71Ga [8] which are stated to disagree with available evaluated
data from KADoNiS v0.3 [5]. We report here on new measurements of the
69,71Ga$(n,\gamma)$ cross sections, which were initiated prior to the
publication of the latter results. This work took advantage of the Liquid-
Lithium Target (LiLiT) at the Soreq Applied Research Accelerator Facility
(SARAF) intense quasi-Maxwellian neutron source ([9], see Sec. II), especially
valuable for the activation measurement of short-lived nuclides like 70Ga
($t_{1/2}$= 21.1 min). Section II describes the SARAF-LiLiT neutron source,
Sec. III describes the samples used and the irradiation details, and Sec. IV
describes the activity measurements. In Sec. V we report the activation
results, in Sec. VI we calculate the experimental cross sections, Sec. VII
describes how the MACS were calculated, and in Sec. VIII we discuss the
results.
## II SARAF-LiLiT
Figure 2: Diagram of the Liquid-Lithium Target (LiLiT) and activation target
assembly. The 1-2 mA ($\approx$9 mm full width) proton beam (dashed red arrow)
impinges directly on the windowless liquid-lithium film. The light blue shows
the liquid-lithium circulating flow (see [10] for details). The activation
samples are mounted at the center of a ring target holder made of Al and
positioned in the outgoing neutron cone (green dashed lines) at a distance of
$\approx$6 mm from the liquid-lithium film surface. The targets are in a
vacuum chamber separated from the LiLiT chamber by a 0.5 mm stainless steel
concave vacuum wall.
An intense 7Li$(p,n)^{7}$Be neutron source, in the form of a Liquid-Lithium
Target (LiLiT) [10, 11], bombarded by a mA proton beam from the Soreq Applied
Research Accelerator Facility (SARAF), was developed and is used for
Maxwellian Average Cross Section (MACS) measurements. The SARAF accelerator is
based on a continuous wave (CW), proton/deuteron RF superconducting linear
accelerator. SARAF Phase I, presently undergoing a major upgrade to its Phase
II, consisted of a 20 keV/u Electron Cyclotron Resonance (ECR) ion source
injector, a Low-Energy Beam Transport section (LEBT), a four-rod Radio
Frequency Quadrupole (RFQ, 1.5 MeV/u), a Medium Energy Beam Transport section
(MEBT), a Prototype Superconducting Module (PSM) housing six half-wave
resonators and three superconducting solenoids, and a diagnostic plate
(D-plate). The beamline downstream of the accelerator transports the high
intensity beam to the target. SARAF Phase I delivered for experiments currents
of up to 2 mA of protons or deuterons, with energies of up to $\approx$ 4 and
5 MeV, respectively [12, 13, 14].
The SARAF high-intensity beam requires a Li target that can withstand its
power, which is incompatible with solid Li and Li compounds. The Liquid-
Lithium Target (LiLiT) [10, 11] consists of a liquid-lithium film (temperature
$\approx$200 °C, above the lithium melting temperature of 180.5 °C) circulated
at high velocity (3-7 $m/s$) onto a thin convex stainless-steel support wall.
The target is bombarded with a high-intensity proton beam impinging directly
on the Li-vacuum (windowless) interface at an energy above and close to the
7Li$(p,n)$ reaction threshold, $E_{th}=1.88$ MeV (Fig. 2). A rectangular-
shaped nozzle just upstream of the curved support wall determines the film
width and thickness to be 18 mm and 1.5 mm, respectively (see [10] for
details). The first few microns at the surface of the liquid-lithium film
serve as the neutron-producing thick target. The deeper Li film layers act as
a beam dump, from which the power is transported by the flow to a heat
exchanger. A spherical cap made of stainless steel foil, 0.5 mm thick and 19
cm in diameter, is located $\approx$1 mm beyond the nozzle and seals the LiLiT
vacuum chamber neutron exit port (Fig. 2). The vacuum wall curvature (convex
toward the Li flow with a curvature radius of 300 mm) allows us to locate a
secondary activation target very close (see below) to the neutron source at
the Li-vacuum interface.
## III Sample characteristics and irradiation details
In the experiment described here, a 13 mm diameter natGa2O3 (99.99+% purity,
[15]) pellet target was activated. It was sandwiched between two Au foils
(Table 1) which were used as monitors of the neutron fluence. The distance of
the 13 mm diameter Ga target from the Li neutron source was $6\pm 1$ mm,
intercepting $>75\%$ of the outgoing neutrons. The characteristics of the
samples used are summarized in Table 1.
The Ga and Au targets were inserted into the LiLiT activation chamber (Fig. 2)
and held in place by the target holder. The 9 mm full width proton beam
impinged on the free-surface lithium film, resulting in an outgoing neutron
cone due to the 7Li$(p,n)$ reaction, which irradiated the target. The setup is
shown in Fig. 2 and explained in the caption.
The number of stable nuclei $A$ per cm2, $n_{t}(A)$, for a target element with
an atomic or molecular mass $M_{A}$, target area $S$ and mass $m$, is given by
Eq. (1).
$n_{t}(A)=s_{A}\cdot a(A)\frac{m\cdot N_{A}}{S\cdot M_{A}}.$ (1)
The symbol $N_{A}$ denotes Avogadro’s number and $a(A)$ the isotopic abundance
of $A$ (Table 2). The stoichiometry of element $A$ in the target is denoted by
$s_{A}$ ($s_{A}$=2 or 1 for Ga or Au, respectively).
Table 1: Characteristics of the samples used in this work. They are listed in the order that they were placed downstream from the Li target. The first gold foil (Au #33) was necessary to monitor the beam offset (see below). The Ga oxide targets #2 (#1) were used for neutron irradiations above (below) the Li neutron emission threshold. Sample | Diam. | Mass | Nucleus | $n_{t}$
---|---|---|---|---
| (mm) | (mg) | | ($10^{19}$cm-2)
Au #33 | 25 | 109.8(1) | 197Au | 6.839(4)
Au #14 | 13 | 31.6(1) | 197Au | 7.28(1)
natGa2O3 #2 | 13 | 472.5(1) | 69Ga | 137.5(1)
| | | 71Ga | 91.25(1)
Au #15 | 13 | 32.5(1) | 197Au | 7.49(1)
natGa2O3 #1 | 13 | 443.0(1) | 69Ga | 128.9(1)
| | | 71Ga | 85.55(1)
The proton beam energy was measured by a TOF pick-up and Rutherford
backscattering off a Au target located in the diagnostic D-plate. The beam
energy was 1925 keV, with an energy spread of $\approx$15 keV. The energy
spread was estimated from beam dynamics calculations and was verified
experimentally under similar conditions [16].
To determine the position of the proton beam relative to the activation
targets, the 25 mm diameter gold foil (Au #33) was auto-radiographically
scanned after the samples irradiation (Fig. 3). An offset of 2.5 mm in the
vertical direction was found and accounted for in our detailed simulations.
Figure 3: An auto-radiographic scan of the 25 mm diameter gold foil (Au #33).
Red indicates the area with the highest neutron irradiation followed by
yellow, green and blue. A $\approx$2.5 mm offset was observed in the neutron
irradiation and is attributed to the vertical steering of the proton beam.
This offset was taken into account in the neutron irradiation simulations.
Throughout the irradiation, the neutron yield was continuously monitored and
recorded with a fission-product ionization chamber detector (PFC16A,
Centronics Ltd.), counting neutron-induced fission events from a thin 235U
internal foil (1 mg/cm2, 12.5 cm2 active area). The fission chamber was
located at 0$\degree$ to the incident proton beam, at a distance of
$\approx$80 cm downstream from the target. The fission chamber was covered
with a 1 mm thick Cd sheet to absorb scattered thermal neutrons. The count
rate of the fission chamber was calibrated to the beam current at low
intensity (10% duty cycle, using a slow chopper), with the Faraday cup located
$\approx$1 m upstream of the Li target. After the SARAF was tuned, the beam
duty cycle was ramped up to 99%. Normally, the ramp up is performed rather
slowly while monitoring the temperature and radiation along the beamline and
LiLiT. If necessary, fine-tuning of the beamline ion-optical magnetic elements
(bending magnets, steerers) is performed based on temperature reading of
sensors located on the lithium nozzle (see [10, 11] for details). It is
important to know the time dependence of the neutron yield in the case of a
short-lived activation product (e.g., 70Ga, $t_{1/2}$= 21.1 min). In such a
case, one needs to account for fluctuations of the neutron yield when
evaluating the fraction of the reaction product that decayed during the
irradiation. The time record of the proton beam current is presented in figure
4. The total integrated current in the irradiation was $\sim$0.96 mA$\times$h.
Figure 4: Time record of the fission chamber count rate (left axis) and
corresponding intensity of the proton beam current (right axis) during the
irradiation. The two low-intensity intervals and gaps at beginning and end of
the irradiation correspond to the calibration of the fission count rate
against beam current measured at low intensity with a Faraday cup (see text).
All intensity variations are taken into account by the $1/f_{b}$ correction
factor (Eq. (2)). The total integrated current was 0.96 mA$\times$h.
## IV Activity measurement
After the irradiation, the induced activities were measured separately for
each sample with a shielded HPGe detector (ORTEC GMX 25-83). The distance of
the sample to the HPGe detector was 20 cm (2 cm) for the measurement above
(below) the neutron threshold. The detector efficiency was determined by
standard calibrated radioactive sources: 22Na, 60Co, 88Y, 133Ba, 137Cs, 241Am,
152Eu and 155Eu. The measured efficiency curve at 20 cm is presented in Fig.
5.
Figure 5: Efficiency of the HPGe detector used at a distance of 20 cm (black
dots with 1$\sigma$ uncertainties) determined by standard calibrated
radioactive sources: 22Na, 60Co, 88Y, 133Ba, 137Cs, 241Am, 152Eu and 155Eu.
The red curve is a fit to the data with the expression: $\epsilon=a\times
E_{\gamma}^{-b}$ and the dotted lines are the uncertainty (1$\sigma$). The
main source of the efficiency uncertainty is the radioactive source activity;
these uncertainties are given by the source manufacturers. The small value of
the reduced chi square suggests that these uncertainties are overestimated.
The $\gamma$-ray spectrum for the neutron activated Ga sample (natGa2O3 #2) is
presented in Fig. 6, where the full-energy peaks of 70Ga and 72Ga are labeled.
Figure 6: $\gamma$-ray spectrum of the activated natGa2O3 #2 sample. The
spectrum was accumulated for 3000 s, starting 2464 s after the end of the
neutron activation. The sample was located 20 cm from the HPGe detector. The
main Ga isotope full-energy peaks, with energies in keV, are labeled.
Figure 7 presents the decay curves of the 176.3 and 1039.5 keV $\gamma$ lines
of 70Ga, showing an excellent agreement with the adopted 70Ga half-life of
21.14(5) min. [17].
Figure 7: Decay curves for 70Ga. The uncertainties of each data point in this
plot are only the statistical uncertainties of the activity measurement.
$N_{\gamma}=N_{act}{\cdot}I_{\gamma}$ (see Eq. 2) is the photon intensity
($\gamma$s per second) of the $\gamma$ transitions at the 176.3 and 1039.5 keV
lines of the 70Ga decay, determined by $\gamma$ spectrometry. Excellent
agreement is observed with the adopted half-life of 70Ga from the literature
(21.14(5) min. [17]), and between the number of activated 70Ga nuclei (see
Tables 3 and 4) derived independently from each transition, using the
respective adopted $\gamma$ intensities [17]; see text.
The numbers of activated 70Ga nuclei derived from each of the two transitions,
using the adopted $\gamma$ intensities [17], are in excellent agreement as
well. The ratio of $\gamma$ intensities of the 1039 and 176 keV lines
determined here is 2.53(6) in agreement and better precision than the ratio of
adopted intensities 2.24(19) [17]. The reported $\gamma$ intensity ratio in
[18] is 2.30(6). The precise counting of activated 70Ga nuclei, in spite of
its short half-life relative to the irradiation time and the very low
intensity of the $\gamma$ transitions (Table 2), is credited to the high
intensity of the LiLiT neutron source.
Figure 8: Secondary $\gamma$-ray spectrum of the natGa2O3 #1 sample after
irradiation with primary $\gamma$-rays. The spectrum was accumulated for 636
s, starting 1275 s after the end of the irradiation. The proton energy was
below the neutron production threshold, resulting in the production of only
$\gamma$ rays. The sample was located 2 cm from the HPGe detector. No Ga lines
are observed.
The SARAF-LiLiT setup was shown to produce high-energy $\gamma$-rays
($\approx$108 14.6 and 17.6 MeV $\gamma$-rays/(mA$\times$s)) via the
7Li$(p,\gamma)^{8}$Be reaction [19, 9]. In cases where there are a stable A-1
isotope, an unstable A isotope and a stable A+1 isotope, the nuclide A can be
produced through the A-1$(n,\gamma)$ or A+1$(\gamma,n)$ reactions. To test the
possible contribution of the 71Ga$(\gamma,n)$ reaction to the production of
70Ga, an irradiation of protons on LiLiT was conducted below the neutron
production threshold. This resulted in the irradiation of the Ga sample
(natGa2O3 #1) with $\gamma$-rays, but without neutrons. The proton energy was
measured to be 1800 keV. The total integrated current in this irradiation was
$\approx$0.46 mA$\times$h. The $\gamma$-ray spectrum for the activated Ga
sample (natGa2O3 #1) is presented in Fig. 8. As can be seen, no Ga isotopes
were created through $(\gamma,n)$ reactions.
## V Activation results
Table 2: Properties of the relevant target and product nuclei studied in this work. The half-life data were taken from [20, 17, 21]. Target | Isotopic | Product | Half-life,
---|---|---|---
nucleus | abundance | nucleus | $t_{\frac{1}{2}}$
197Au | 1 | 198Au | $2.6947(3)$ d
69Ga | 0.60108(9) | 70Ga | $21.14(5)$ m
71Ga | 0.39892(9) | 72Ga | $14.10(1)$ h
The number of activated nuclei created during the irradiation, $N_{act}$, was
obtained from the $\gamma$-ray spectra using Eq. (2),
$N_{act}=\frac{C}{\epsilon_{\gamma}I_{\gamma}K_{\gamma}}\frac{e^{\lambda
t_{cool}}}{1-e^{-\lambda t_{real}}}\frac{t_{real}}{t_{live}}\frac{1}{f_{b}},$
(2)
where $C$ is the number of counts in a full-energy peak, $\epsilon_{\gamma}$
is the detector energy-dependent full-energy efficiency for the relevant
target-detector geometry and $I_{\gamma}$ is the $\gamma$-intensity per decay.
The $I_{\gamma}$ used in this work was taken from [20, 17, 21]. The correction
due to $\gamma$-ray self absorption in the sample is $K_{\gamma}$. In the case
of a disk sample of thickness $x$, $K_{\gamma}\approx\frac{1-e^{-\mu x}}{\mu
x}$, where $\mu$ is the $\gamma$-ray absorption coefficient. The $\gamma$-ray
absorption coefficients, $\mu$, were taken from [22]. The decay constant of
the activated nucleus is $\lambda=\frac{ln(2)}{t_{1/2}}$. The cooling time
between the end of the irradiation and the start of activity measurement is
$t_{cool}$, and $t_{real}$ ($t_{live}$) is the real (live) measurement time.
The decay of activated nuclei during the irradiation is accounted for in
$f_{b}$. It is calculated using the time dependence of the neutron yield
$\Phi(t)$, obtained from the fission chamber (see Fig. 4), by
$f_{b}=\frac{\intop_{0}^{t_{a}}\Phi(t)e^{-\lambda(t_{a}-t)}dt}{\intop_{0}^{t_{a}}\Phi(t)dt}$.
$t_{a}$ is the time of the end of irradiation. The decay parameters and
correction factors used in this analysis are listed in Tables 2 and 3.
Table 3: Analysis of one of the $\gamma$-ray spectra from the natGa2O3 #2 sample, measured 5180 s after the end of the neutron irradiation, with 656.8 s real time and 600 s live time; see Eq. (2) for the definition of notations. The data for $E_{\gamma}$ and $I_{\gamma}$ were taken from [17, 21]. Nucleus | $E_{\gamma}$ (keV) | counts | $I_{\gamma}$ (%) | $\epsilon_{\gamma}$ ($10^{-4}$) | $K_{\gamma}$ | $f_{b}$ | $N_{act}$ ($10^{9}$)
---|---|---|---|---|---|---|---
70Ga | 176.115(13) | 1264(93) | 0.29(1) | 30.4(8) | 0.970(3) | 0.550 | 16.4(14)
| 1039.513(10) | 602(43) | 0.65(5) | 6.50(8) | 0.990(3) | 0.550 | 16.1(17)
72Ga | 600.912(15) | 3578(99) | 5.822(19) | 10.5(1) | 0.987(3) | 0.984 | 7.96(24)
| 629.967(19) | 15484(157) | 26.13(4) | 10.0(1) | 0.987(3) | 0.984 | 8.00(12)
| 834.13(4) | 45243(224) | 95.45(8) | 7.87(9) | 0.989(3) | 0.984 | 8.15(10)
| 894.327(18) | 4419(80) | 10.136(15) | 7.41(9) | 0.989(3) | 0.984 | 7.96(17)
| 1050.794(17) | 2643(64) | 6.991(11) | 6.44(8) | 0.990(3) | 0.984 | 7.93(22)
| 1861.996(18) | 1248(46) | 5.41(3) | 3.92(8) | 0.992(3) | 0.984 | 7.93(17)
| 2201.586(17) | 5479(79) | 26.87(12) | 3.39(7) | 0.993(3) | 0.984 | 8.10(21)
| 2491.026(17) | 1437(42) | 7.73(3) | 3.05(6) | 0.993(3) | 0.984 | 8.22(29)
| 2507.718(17) | 2454(52) | 13.33(6) | 3.03(6) | 0.993(3) | 0.984 | 8.19(25)
The numbers of activated nuclei at the end of the irradiation, $N_{act}$,
calculated with Eq. (2), are summarized in table 4.
Table 4: The number of activated nuclei at the end of the irradiation, $N_{act}$, and comparison with simulated $N_{act}$. Sample | Nucleus | $N_{act}$ ($10^{9}$) | simulated $N_{act}$ ($10^{9}$)
---|---|---|---
Au-14 | 198Au | 4.07(8) | 4.09
natGa2O3 #2 | 70Ga | 16.3(7) |
| 72Ga | 8.1(1) |
Au-15 | 198Au | 3.65(6) | 3.63
## VI Experimental cross section
Since the sample cross section is measured relative to the Au cross section
(which is considered known), the cross section of the sample, averaged over
the experimental neutron spectrum, can be expressed as
$\sigma_{exp}(i)=\sigma_{ENDF}(\textrm{Au})\frac{N_{act}(i)}{N_{act}(\textrm{Au})}\frac{n_{t}(\textrm{Au})}{n_{t}(i)},$
(3)
where $i$ denotes the Ga stable isotope (69 or 71) and
$\sigma_{ENDF}(\textrm{Au})$ is the reference Au cross section from the
ENDF/B-VIII.0 library [23] averaged over the experimental neutron spectrum. It
is defined as
$\sigma_{ENDF}(\textrm{Au})=\frac{\int\sigma_{ENDF}(E_{n};\textrm{Au})\frac{dn}{dE_{n}}dE_{n}}{\int\frac{dn}{dE_{n}}dE_{n}}.$
(4)
The energy-dependent 197Au$(n,\gamma)^{198}$Au cross section
$\sigma_{ENDF}(E_{n};\textrm{Au})$ was taken from ENDF/B-VIII.0 [23], in
agreement with high-precision experimental data [24, 25]. The neutron
spectrum, $\frac{dn}{dE_{n}}$, is obtained from our simulation code (Fig. 9),
developed and benchmarked by experiment [26, 27, 19, 9]. The simulated neutron
spectrum impinging on the Ga target, along with a $kT$ = 41.8 keV fit to a
Maxwellian neutron flux ($\propto E\cdot e^{-\frac{E}{kT}}$), is presented in
Fig. 9. This spectrum is generated by a GEANT4 [28] simulation, using the
SimLiT code [27] output as the neutron source. The SimLiT calculation uses
7Li$(p,n)$ differential cross sections taken from [29] and takes into account
the proton mean beam energy and energy spread, proton energy loss in the
liquid Li using differential $dE/dx$ values taken from SRIM [30] and a
Gaussian proton beam profile consistent with the Au monitor auto-radiography
(Fig. 3). The detailed GEANT4 simulation takes into account the LiLiT geometry
setup including the off-center position of the neutron beam relative to the
Au-Ga-Au target and the surrounding materials. The simulation explicitly
calculates (see [19, 9] for details) the number of activated 198Au nuclei,
based on the $\sigma_{ENDF}(E_{n};Au)$ cross sections and the measured proton
charge during irradiation and reproduces the experimental 198Au activity of
the Au monitors within 0.5% (Table 4).
Figure 9: The simulated neutron spectrum impinging on the Ga target (black).
Also shown is a fit (between 0 and 90 keV) to a Maxwell-Boltzmann (MB) flux
(red), where the best fit is for $kT$ = 41.8 keV.
The results of the experimental cross sections, $\sigma_{exp}$, of 69Ga and
71Ga (Eq. 3) and of the $\sigma_{ENDF}(\textrm{Au})$ values (Eq. 4) are
presented in Table 5.
Table 5: The experimental cross sections, $\sigma_{exp}$ (Eq. 3), measured in this work for the 69,71Ga$(n,\gamma)^{70,72}$Ga reactions, together with the $\sigma_{ENDF}$ cross section averaged over the neutron energy distribution (Fig. 9). See Table 7 for explanation of $\sigma_{ENDF}(^{197}$Au) uncertainty. Isotope | $\sigma_{exp}$ (mb) | $\sigma_{ENDF}$ (mb)
---|---|---
197Au | | $524(10)$
69Ga | $119(5)$ | 103
71Ga | $89(2)$ | 107
## VII Maxwellian Averaged Cross Section
The Maxwellian Averaged Cross Section (MACS) at a given thermal energy, $kT$,
is defined as:
$\Braket{\sigma}_{kT}=\frac{\Braket{\sigma
v}}{v_{T}}=\frac{2}{\sqrt{\pi}}\frac{\intop_{0}^{\infty}\sigma(E_{n})E_{n}e^{-\frac{E_{n}}{kT}}dE_{n}}{\intop_{0}^{\infty}E_{n}e^{-\frac{E_{n}}{kT}}dE_{n}},$
(5)
where $\sigma(E_{n})$ is the differential $(n,\gamma)$ reaction cross section
at neutron energy $E_{n}$. In this work, the MACS at a given thermal energy
$kT$ is calculated with the procedure developed in [19, 31, 9] and using Eq.
(6),
$\textrm{MACS}_{exp}(kT)=\frac{2}{\sqrt{\pi}}\cdot
C_{lib}(kT)\cdot\sigma_{exp},$ (6)
where the correction factor $C_{lib}(kT)$ is given by Eq. (7):
$C_{lib}(kT)=\frac{\frac{\int_{0}^{\infty}\sigma_{lib}(E_{n})E_{n}e^{-\frac{E_{n}}{kT}}dE_{n}}{\int_{0}^{\infty}E_{n}e^{-\frac{E_{n}}{kT}}dE_{n}}}{\frac{\int_{0}^{\infty}\sigma_{lib}(E_{n})\frac{dn}{dE_{n}}dE_{n}}{\int_{0}^{\infty}\frac{dn}{dE_{n}}dE_{n}}}.$
(7)
In Eq. (7), $\frac{dn}{dE_{n}}$ is the simulated experimental neutron spectrum
(Fig. 9) and $\sigma_{lib}(E_{n})$ is the energy-dependent neutron capture
cross section taken from an evaluation library.
In Table 6 we present the correction factors, $C_{lib}$, and the MACSexp at
$kT=30$ keV, calculated with Eq. (6), for the Ga isotopes. The MACSexp at
$kT=30$ keV derived in this work are obtained by using in Eq. (6) the average
$C_{lib}$ calculated using the various cross section libraries: ENDF/B-VIII.0
[23], JENDL-4.0 [32], JEFF-3.3 [33], CENDL-3.2 [34] and TENDL-2019 [35] (see
Table 6).
Table 6: Comparison of correction factors, $C_{lib}$, and MACS calculated for 69Ga and 71Ga at 30 keV using the different libraries [23, 32, 33, 34, 35]. See text for explanation. | $C_{lib}$ | | MACSexp (mb)
---|---|---|---
Library | 69Ga | 71Ga | | 69Ga | 71Ga
ENDF/B-VIII.0 [23] | 1.03 | 1.02 | | 138 | 103
JENDL-4.0 [32] | 1.03 | 1.01 | | 138 | 102
JEFF-3.3 [33] | 1.0 | 0.97 | | 134 | 98
CENDL-3.2 [34] | 0.99 | 1.08 | | 133 | 108
TENDL-2019 [35] | 1.04 | 1.15 | | 139 | 115
average | 1.018 | 1.046 | | 136.4 | 105.0
standard deviation | 0.02 | 0.07 | | 2.8 | 6.7
The experimental uncertainties, as discussed in detail in [19, 9], are
summarized in Table 7. The uncertainties for the average $C_{lib}$ were
obtained by taking the standard deviation of the $C_{lib}$ calculated for the
various cross section libraries [23, 32, 34, 33, 35] (see Table 6).
Table 7: Random (rand) and systematic (sys) uncertainties in the results
presented in this work.
| Uncertainty (%)
---|---
| 69Ga | | 71Ga
Source of uncertainty | rand | sys | | rand | sys
target thickness | 0.5 | | | 0.5 |
activity measurement | 4.0 | | | 0.5 |
full-energy eff. rel. to Au | | 0.5a | | | 0.5a
intensity per decay | | 3.4 | | | 0.1
$\sigma_{ENDF}$(Au) | | 1.9b | | | 1.9b
$C_{lib}$ | | 2.0 | | | 6.4
Total random uncertainty | 4.0 | | | 0.7 |
Total systematic uncertainty | | 4.4 | | | 6.7
Total uncertainty | 6.0 | | 6.7
* a
This contribution to the uncertainty of the 69,71Ga MACS includes only the
ratio of the full-energy efficiencies of the 69,71Ga $\gamma$ lines to that of
the 198Au $\gamma$ line. In Tables 3 and 4 the overall uncertainty (including
the systematic uncertainty of the calibration sources) is quoted.
* b
This value includes the uncertainty of beam parameters (proton beam energy,
energy spread, and distance of sample from Li) of 0.6%, the uncertainty of the
simulations of 1.5% and the uncertainty of the ENDF cross section for Au of
1.0%, $\sqrt{0.6^{2}+1.5^{2}+1.0^{2}}=1.9\%$. See [9] for more details of the
uncertainties.
Table 8: Comparison of the results of this work with previous experiments and
compilations. The MACS listed in the Table correspond to $kT$= 30 keV. The
previously measured MACS were renormalized as specified in [36] using the
recent measured 197Au$(n,\gamma)^{198}$Au cross section data [25, 24, 37] used
as standard. The KADoNiS v0.3 [5] recommended MACS (30 keV) is the average of
the experimental data renormalized to the 197Au$(n,\gamma)^{198}$Au MACS data
measured in [38]. The KADoNiS v1.0 [36] recommended MACS(30 keV) values are an
average from evaluated libraries ENDF/B-VII.1 [39], JENDL-4.0 [32] and
TENDL-2015 [40].
| KADoNiS | KADoNiS | Walter, | Anand | Walter | Göbel | This work
---|---|---|---|---|---|---|---
| v0.3 [5] | v1.0 [36] | 1984 [4] | et al., 1979 [6] | et al., 1986 [7] | et al., 2021 [8] |
MACS(69Ga) (mb) | 139(6) | 123(9) | 149(6) | | | | 136(8)
$\sigma_{exp}$(69Ga)/$\sigma_{exp}$(197Au) 1 | | | | | | 0.286(19) | 0.227(12)
MACS(71Ga) (mb) | 123(8) | 103(14) | | 79(23) | 130(8) | | 105(7)
$\sigma_{exp}$(71Ga)/$\sigma_{exp}$(197Au) 1 | | | | | | 0.173(11) | 0.170(5)
* 1
Note that Göbel et al. [8] denote $\sigma_{exp}$ as defined in Eq. (3) by SACS
(spectrum averaged cross section) for their experimental neutron spectrum.
Figure 10: Comparison of this work’s results (red) with previously measured
MACS at $kT=30$ keV (black) for 69Ga [4] (left) and 71Ga [6, 7] (right). The
previously measured MACS were renormalized as specified in the KADoNiS v1.0
website [36]. The TOF measurement is displayed as a circle and the activation
measurements as squares.
## VIII Discussion
This work’s results are compared in Table 8 and Fig. 10 with previously
measured MACS at $kT$ = 30 keV for 69Ga [4] and 71Ga [6, 7], and with
compilations of experimental values (KADoNiS v0.3, [5]) and of evaluated
values (KADoNiS v1.0, [36]). We compare also the ratios
$\sigma_{exp}$(AGa)/$\sigma_{exp}$(197Au) with the corresponding values from
Göbel et al. [8], denoted there as SACS(AGa)/SACS(197Au), calculated for the
relevant experimental spectrum.
Our result for the 69Ga MACS (30 keV) are in good agreement within
uncertainties with the previous experimental value of [4] and those
recommended by Bao et al. [41] and KADoNiS v0.3 [5]). Our MACS value for 71Ga
is larger than the experimental value of [6], but smaller than the
experimental value of [7] recommended by Bao et al. [41] and KADoNiS v0.3
[5]), though marginally consistent within the quoted uncertainties. This
smaller value is significant in view of the fact that the previous value of
KADoNiS v0.3 [5] was used in extensive network calculations [3]; see below.
Both 69,71Ga MACS values extracted in this work are in reasonable agreement
with the evaluated values of KADoNiS v1.0 [36]. We note also the good
agreement of our $\sigma_{exp}$(71Ga)/$\sigma_{exp}$(197Au) value with that of
Göbel at al. [8], but a striking discrepancy for
$\sigma_{exp}$(69Ga)/$\sigma_{exp}$(197Au), whose origin is not understood.
Gallium, like the other elements between iron and strontium (26 $<$ Z $<$ 38,
60 $\lesssim$ A $\lesssim$ 90), is produced primarily by the weak component of
the $s$ process in massive stars (M${}_{initial}>$ 8M⊙) ([42] and references
therein). Pignatari et al. [3] studied nucleosynthesis, including the weak $s$
process, in a model of a Population I (solar metallicity) 25M⊙ star. For the
69,71Ga$(n,\gamma)$ MACS, the authors used the values recommended by Bao et
al. [41] (KADoNiS 0.3 [5]). As shown in Table 8 and discussed above, the new
69Ga MACS measurement presented here supports the Bao et al. [41] value, but
the 71Ga MACS is 15% lower. This means that less 71Ga is consumed by the
$(n,\gamma)$ reaction and hence its isotopic fraction is expected to increase
compared to that calculated by Pignatari et al. [3]. In addition, since the
$s$ process flow goes from Ga to Ge,
69Ga$(n,\gamma)^{70}$Ga$(\beta^{-})^{70}$Ge and
71Ga$(n,\gamma)^{72}$Ga$(\beta^{-})^{72}$Ge (see Fig. 1), the lower 71Ga MACS
is expected to result in a lower production of 72,73,74Ge and reduce their
isotopic fraction. The lower 71Ga MACS (105(7) mb) may have as well a wider
effect, as Pignatari et al. [3] concluded that the effects of MACS $\lesssim
150$ mb tend to propagate to heavier isotopes. However, we should caution that
the above potential effects are tentative. The only way to check for the
effect of the new 71Ga MACS is to incorporate it in a network calculation like
the one performed by Pignatari et al. [3].
Such nucleosynthesis models need to be tested against isotopic compositions in
the relevant star types. This is done for many elements by isotopic analysis
of chemical elements in presolar grains. However, to date, there are no
measurements of Ga in presolar grains ([43, 44, 45]). The most studied family
of presolar grains are the carbides, mainly SiC and graphite. At the same
time, Ga is thought to usually not form carbides [46], so it is unlikely that
there is enough Ga in presolar carbides for isotopic analysis with the current
capabilities of the experimental techniques used. On the other hand, Lodders
[47] calculated that Ga condensed in the early Solar system as a trace element
into the mineral feldspar, by substituting for the major element Al in the
crystal lattice, and into Fe metal to a lesser extent. While presolar feldspar
has not been found to date, most of the presolar oxides and silicates studied
to date contain Al as a major element. This makes them more likely hosts for
stellar Ga to be analyzed.
## IX Summary
The neutron capture cross sections of 69,71Ga were measured by the activation
technique in the intense $kT\approx$ 40 keV quasi-Maxwellian neutron field of
the SARAF-LiLiT facility. The reaction products were measured by $\gamma$
spectrometry with a HPGe detector. The experimental cross sections were
converted to Maxwell-averaged cross sections at $kT$ = 30 keV using the energy
dependence of various neutron cross section library data. The MACS values
obtained in this work are MACS (69Ga) = 136(8) mb and MACS (69Ga) = 105(7) mb.
The 69Ga MACS value is in good agreement with previous experimental values and
their recommended values while that of 71Ga is smaller and with reduced
uncertainty than the experimental recommended value. This smaller value may
have implications in network calculations such as those of Pignatari et al.
[3] which used so far the recommended values. Potential natural samples to
measure stellar Ga composition are presolar silicates and oxides, in which Ga
may replace Al in the grains’ crystal lattice. We note a significant and not
understood discrepancy between our experimental cross-section value of 69Ga
with that recently published by Göbel et al. [8], beyond that expected from
the different experimental conditions.
###### Acknowledgements.
We would like to thank the SARAF and LiLiT (Soreq NRC) staffs for their
dedicated help during the experiments. This work was supported in part by
Israel Science Foundation (Grant Nr. 1387/15) and the Pazy Foundation
(Israel). M.P. acknowledges support by the European Union (ChETEC-INFRA,
project no. 101008324).
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|
# Reliable Quantum Certification of Bosonic Code Preparations
Ya-Dong Wu QICI Quantum Information and Computation Initiative, Department of
Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
###### Abstract
Bosonic fault tolerant quantum computing requires preparations of Bosonic code
states like cat states and GKP states with high fidelity and reliable quantum
certification of these states. Although many proposals on preparing these
states have been developed, few investigation has been done on how to reliably
certify these experimental states. In this paper, we develop approaches to
certify whether a continuous-variable state falls inside a certain Bosonic
qubit code space by detecting a witness using Gaussian measurements. Our
results can be applied to certification of various cat codes including two-
component cat states, four-component cat states, and squeezed two-component
cat states as well as Gottesman-Kitaev-Preskill codes. Then we further apply
our approach to certify resource states in continuous-variable universal fault
tolerant measurement-based quantum computing and quantum outputs of
instantaneous quantum polynomial circuits, which can be used to show quantum
supremacy. The sample complexity of our approach is efficient in terms of
number of modes, so significantly reducing overhead compared to quantum state
tomography in certification of many-mode quantum states.
## I Introduction
A scalable fault tolerant universal quantum computer demands quantum error
correction codes to protect qubit information from decoherence noises Shor
(1995); Nielsen and Chuang (2010). One promising type of quantum error
correcting codes is Bosonic quantum codes that encode a logical qubit into a
quantum harmonic oscillator, such that a two-dimensional code space lies in a
corner of an infinite-dimensional Hilbert space. This type of quantum error
correcting codes include cat codes Cochrane _et al._ (1999); Ralph _et al._
(2003); Leghtas _et al._ (2013); Mirrahimi _et al._ (2014); Bergmann and van
Loock (2016), Gottesman-Kiteav-Preskill (GKP) codes Gottesman _et al._
(2001), and some other rotational symmetric codes Michael _et al._ (2016);
Albert _et al._ (2018); Grimsmo _et al._ (2020).
Significant experimental progress have been made on encoding qubit information
into cat states Ofek _et al._ (2016) and GKP states Flühmann _et al._
(2019); Campagne-Ibarcq _et al._ (2020) on superconducting cavities as well
as trapped ions, along with many theoretical proposals on state preparation
and engineering Terhal and Weigand (2016); Puri _et al._ (2017); Weigand and
Terhal (2018); Shi _et al._ (2019). Then a natural question is how to
experimentally characterize these prepared states in harmonic oscillators. To
implement fault tolerant quantum computing with Bosonic quantum error
correcting codes, efficient and reliable quantum certification of scalable
Bosonic code state preparations is significant.
The most common approach to characterize a quantum state in experiments is
quantum state tomography Lvovsky and Raymer (2009); D’Ariano _et al._ (2003),
which, however, requires a huge number of measurements and long classical
postprocessing time. Quantum state tomography cannot handle multi-mode
entangled nonGaussian states. Quantum tomography with neural networks Tiunov
_et al._ (2020); Ahmed _et al._ (2021a, b); Zhu _et al._ (2022); Wu _et
al._ (2022) evidently reduces the required number of measurements and shorten
the classical postprocessing time, but still cannot deal with multi-mode
quantum states. In practical applications, physicists have the prior knowledge
about the classical description of a target quantum state. Hence, rather than
fully characterizing an experimental quantum state, in these scenarios,
physicists can apply quantum certification Eisert _et al._ (2020), to
efficiently determine whether a prepared quantum state is close enough to
certain target state. L. Aolita et al. investigated quantum certification of
photonic states Aolita _et al._ (2015), including Gaussian states, and those
states prepared in Boson sampling Aaronson and Arkhipov (2013) and
Knill–Laflamme–Milburn (KLM) schemes Knill _et al._ (2001). Later U. Chabaud
developed another verification protocol of quantum outputs in Boson sampling
Chabaud _et al._ (2021). Up to now, how to certify cat states and GKP states,
without performing quantum state tomography or fidelity estimation da Silva
_et al._ (2011); Chabaud _et al._ (2020a), is still open. On the other hand,
although lots of work have been done on efficient verification of many-qubit
states, different nature between CV states and qubit states makes
certification approaches of many-qubit states Pallister _et al._ (2018)
cannot be directly applied in CV quantum systems Liu _et al._ (2021).
In this paper, we generalize the concept of fidelity witness Aolita _et al._
(2015); Gluza _et al._ (2018) to witness of Bosonic code space, and propose
protocols to certify both two-component cat code space, four-component cat
code space, and a realistic GKP code space with finite truncation in phase
space. The measurements in all the protocols are experimentally friendly,
being either homodyne detections or heterodyne detections. We propose
certification protocols for both the resource states of CV fault-tolerant
measurement-based quantum computing Menicucci (2014) and the quantum outputs
of CV instantaneous quantum polynomial-time (IQP) circuits Douce _et al._
(2017) by estimating fidelity witnesses.
## II Certification of Cat State Code Space
Let $\mathcal{H}$ denote the infinite-dimensional Hilbert space of a quantum
harmonic oscillator. A quantum device outputs $n$ copies of quantum states
$\rho^{\otimes n}$ on $\mathcal{H}^{\otimes n}$. Given a Bosonic code logical
subspace $\bar{\mathcal{H}}\subset\mathcal{H}$, an experimenter, who has no
knowledge of $\rho$, wants to determine whether
$\operatorname{tr}_{\bar{\mathcal{H}}}\rho\geq 1-\epsilon$, by applying local
measurement at each copy of $\rho$. To certify whether a single-mode state
$\rho$ falls on $\bar{\mathcal{H}}$, we measure an observable $W$, which
satisfies the following conditions: (i) if $\rho$ falls on
$\bar{\mathcal{H}}$, i.e. $\operatorname{tr}_{\bar{\mathcal{H}}}\rho=1$, then
$\braket{W}_{\rho}=1$; (ii) otherwise, $\braket{W}_{\rho}<1$. We call the
observable $W$ a witness of the code space $\bar{\mathcal{H}}$. By estimating
$\braket{W}_{\rho}$, we can determine whether to accept $\rho$ as a reliable
Bosonic code state or not. In this subsection, we introduce code witnesses for
different cat codes and explain how to estimate their mean values by Gaussian
measurements.
A Schrodinger cat state is a superposition of coherent states with opposite
amplitudes Bužek and Knight (1995) and can be used to encode qubit information
Cochrane _et al._ (1999). We call this code two-component cat code to
differentiate it from the cat code with a superposition of four coherent
states. A two-component cat code space $\bar{\mathcal{H}}$ is spanned by
$\displaystyle\ket{\bar{0}}=\frac{1}{\sqrt{2(1+e^{-2|\alpha|^{2}})}}\left(\ket{\alpha}+\ket{-\alpha}\right)$
$\displaystyle\ket{\bar{1}}=\frac{1}{\sqrt{2(1+e^{-2|\alpha|^{2}})}}\left(\ket{\alpha}-\ket{-\alpha}\right).$
A two-component cat code witness is
$W=\mathds{1}-\frac{\left(\hat{a}^{\dagger
2}-\alpha^{*\,2}\right)\left(\hat{a}^{2}-\alpha^{2}\right)}{2}.$ (1)
The two-component cat code space is the degenerate ground space of the
Hamiltonian
$\hat{H}_{tCat}=\left(\hat{a}^{\dagger
2}-\alpha^{*\,2}\right)\left(\hat{a}^{2}-\alpha^{2}\right)$ (2)
with eigenvalue zero. $\hat{H}$ can be written as
$\hat{H}_{tCat}=0\left(\ket{\bar{0}}\bra{\bar{0}}_{tCat}+\ket{\bar{1}}\bra{\bar{1}}_{tCat}\right)+\lambda_{1}\ket{\psi}\bra{\psi}+\cdots$
(3)
where $\ket{\psi}$ is the first excited state of $\hat{H}_{tCat}$ and
$\lambda_{1}$ is the first excited energy. Then
$\mathds{1}-\frac{\hat{H}_{tCat}}{2}=\ket{\bar{0}}\bra{\bar{0}}_{tCat}+\ket{\bar{1}}\bra{\bar{1}}_{tCat}+\left(1-\frac{\lambda_{1}}{2}\right)\ket{\psi}\bra{\psi}-\cdots$
(4)
As $\lambda_{1}\geq 2$, we get
$\mathds{1}-\frac{\hat{H}_{tCat}}{2}\leq\ket{\bar{0}}\bra{\bar{0}}_{tCat}+\ket{\bar{1}}\bra{\bar{1}}_{tCat}.$
(5)
Furthermore, any state $\rho$ satisfying
$\operatorname{tr}\left[\rho\left(\mathds{1}-\frac{\hat{H}_{tCat}}{2}\right)\right]=1$
if and only if $\rho$ falls on the two-component cat code space. The witness
of squeezed two-component cat code space can be obtained by using the fact
that squeezing operation is a Gaussian operation inducing a linear
transformation on phase space.
To clarify how to measure this witness, for simplicity, we rewrite it in terms
of quadrature operators when $\alpha\in\mathbb{R}$,
$\displaystyle W_{tCat}$
$\displaystyle=\frac{3-2\alpha^{4}}{4}\mathds{1}-\frac{1}{16}(\hat{x}^{4}+\hat{p}^{4})-\frac{1}{12}\Bigg{[}\left(\frac{\hat{x}+\hat{p}}{\sqrt{2}}\right)^{4}+$
$\displaystyle\left(\frac{\hat{x}-\hat{p}}{\sqrt{2}}\right)^{4}\Bigg{]}+\frac{1}{2}\left(1+\alpha^{2}\right)\hat{x}^{2}+\frac{1}{2}\left(1-\alpha^{2}\right)\hat{p}^{2},$
(6)
where $\hat{x}=\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}}$ and
$\hat{p}=\frac{\hat{a}-\hat{a}^{\dagger}}{\sqrt{2}\text{i}}$ are the position
and momentum operators. The expectation value of such a witness can be
estimated by homodyne detections on at most four different quadrature bases,
i.e. $\hat{x}$, $\hat{p}$, $\frac{\hat{x}+\hat{p}}{\sqrt{2}}$ and
$\frac{\hat{x}-\hat{p}}{\sqrt{2}}$. The witness (1) can also be rewritten as
$W_{tCat}=-\frac{1}{2}\hat{a}^{2}\hat{a}^{\dagger\,2}+2\hat{a}\hat{a}^{\dagger}+\frac{1}{2}(\alpha^{*2}\hat{a}^{2}+\alpha^{2}\hat{a}^{\dagger\,2})-\frac{1}{2}|\alpha|^{4}.$
(7)
Each term in the above expression is a product of annihilation operators and
creation operators in anti-normal order. Using optical equivalence principle
Scully _et al._ (1997), i.e.,
$\braket{\hat{a}^{m}\hat{a}^{\dagger\,n}}=\int\text{d}^{2}\alpha
Q_{\rho}(\alpha)\alpha^{m}\alpha^{*n},\quad m,n\in\mathbb{N}$ (8)
$\braket{W_{tCat}}_{\rho}$ can be estimated by sampling Humusi-Q function
$Q_{\rho}(\alpha):=\braket{\alpha}{\rho}{\alpha}$ using heterodyne detections.
Four-component cat code, which is superposition of four coherent states, is
introduced to protect qubit information from photon loss errors Mirrahimi _et
al._ (2014). A four-component cat code space $\bar{\mathcal{H}}$ with even
parity is spanned by
$\displaystyle\bar{\ket{0}}=\frac{1}{\sqrt{2(1+e^{-2|\alpha|^{2}})}}\left(\ket{\alpha}+\ket{-\alpha}\right),$
$\displaystyle\bar{\ket{1}}=\frac{1}{\sqrt{2(1+e^{-2|\alpha|^{2}})}}\left(\ket{\text{i}\alpha}+\ket{-\text{i}\alpha}\right).$
To certify whether a single-mode state $\rho$ falls on $\bar{\mathcal{H}}$, we
measure the witness
$W_{fCat}=\frac{\mathds{1}+(-1)^{\hat{n}}}{2}-\frac{\hat{H}_{fCat}}{24},$ (9)
where $\hat{n}=\hat{a}^{\dagger}\hat{a}$ is photon number operator, and
$\hat{H}_{fCat}:=\left(\hat{a}^{\dagger
2}+\alpha^{*2}\right)\left(\hat{a}^{\dagger
2}-\alpha^{*2}\right)\left(\hat{a}^{2}-\alpha^{2}\right)\left(\hat{a}^{2}+\alpha^{2}\right)$.
The first term is the parity of photon number, equal to the value of Wigner
function at origin, and can be measured by a parity measurement. The second
term can be written as
$\displaystyle\frac{\hat{H}_{fCat}}{24}=$
$\displaystyle\hat{a}^{4}\hat{a}^{\dagger\,4}/24-2/3\hat{a}^{3}\hat{a}^{\dagger\,3}+3\hat{a}^{2}\hat{a}^{\dagger\,2}-4\hat{a}\hat{a}^{\dagger}-\alpha^{4}/24\hat{a}^{\dagger\,4}$
$\displaystyle-\alpha^{*\,4}/24\hat{a}^{4}+(|\alpha|^{8}/24+1)\mathds{1}.$
(10)
Again the expectation value of each term can be estimated by heterodyne
detections.
A squeezed two-component cat code Schlegel _et al._ (2022) consists of
superpositions of two squeezed coherent states
$\displaystyle\ket{\bar{0}}\propto\left(D(-\alpha)+D(\alpha)\right)S(r)\ket{0}$
$\displaystyle\ket{\bar{1}}\propto\left(D(-\alpha)-D(\alpha)\right)S(r)\ket{0},$
where $S(r):=\text{e}^{1/2(r^{*}\hat{a}^{2}-r\hat{a}^{\dagger\,2})}$ is a
squeezing operation, $r\in\mathbb{C}$ is a squeezing parameter, and
$\text{arg}(r)=\text{arg}(\alpha)/2$ such that the squeezed quadrature is in
the same direction as that of the displacement operation.
For simplicity, we suppose $\alpha\in\mathbb{R}$. From the fact that
$D(\alpha)S(r)=S(r)D(\alpha\text{e}^{r})$, we obtain a witness of the squeezed
two-component cat code space
$W_{stCat}=\mathds{1}-\frac{S(r)\left(\hat{a}^{\dagger
2}-\alpha^{2}\text{e}^{2r}\right)\left(\hat{a}^{2}-\alpha^{2}\text{e}^{2r}\right)S(r)^{\dagger}}{2}.$
(11)
Using the fact that $S(r)\hat{x}S(r)^{\dagger}=\text{e}^{-r}\hat{x}$ and
$S(r)\hat{p}S(r)^{\dagger}=\text{e}^{r}\hat{p}$, the witness (11) can be
rewritten in terms of quadrature operators
$\displaystyle W_{stCat}=$
$\displaystyle\frac{3-2\alpha^{4}\text{e}^{4r}}{4}\mathds{1}-\frac{1}{16}\left(\text{e}^{-4r}\hat{x}^{4}+\text{e}^{4r}\hat{p}^{4}\right)$
$\displaystyle-\frac{1}{48}\left[\left(\text{e}^{-r}\hat{x}+\text{e}^{r}\hat{p}\right)^{4}+\left(\text{e}^{-r}\hat{x}-\text{e}^{r}\hat{p}\right)^{4}\right]$
$\displaystyle+\frac{1}{2}\left(\text{e}^{-2r}+\alpha^{2}\right)\hat{x}^{2}+\frac{1}{2}\left(1-\alpha^{2}\text{e}^{2r}\right)\text{e}^{2r}\hat{p}^{2},$
which can be estimated by applying homodyne detections at four quadratures.
## III Certification of a realistic GKP state
GKP states are defined to be simultaneous eigenstates of two commutative
displacement operators $S_{q}=e^{2\text{i}\sqrt{\pi}\hat{x}}$ and
$S_{p}=e^{-2\text{i}\sqrt{\pi}\hat{p}}$, with eigenvalue one Gottesman _et
al._ (2001). A qubit is encoded into a GKP state in the following way
$\displaystyle\ket{\bar{0}}_{iGKP}$
$\displaystyle:=\sum_{k=-\infty}^{\infty}\delta(x-2k\sqrt{\pi})\ket{x},$ (12)
$\displaystyle\ket{\bar{1}}_{iGKP}$
$\displaystyle:=\sum_{k=-\infty}^{\infty}\delta\left(x-(2k+1)\sqrt{\pi}\right)\ket{x}.$
(13)
Ideal GKP states are coherent superpositions of position eigenstates,
demanding infinite squeezing, which is not physically realistic. From now on,
we consider realistic GKP states with finite squeezing.
A realistic GKP state replaces position eigenstates by finitely squeezed
states, i.e.,
$\displaystyle\ket{\bar{0}}_{rGKP}$
$\displaystyle\propto\sum_{k=-\infty}^{\infty}e^{-4k^{2}\sigma^{2}\pi}D(2k\sqrt{\pi})\ket{\psi_{x}},$
(14) $\displaystyle\ket{\bar{1}}_{rGKP}$
$\displaystyle\propto\sum_{k=-\infty}^{\infty}e^{-(2k+1)^{2}\sigma^{2}\pi}D((2k+1)\sqrt{\pi})\ket{\psi_{x}},$
(15)
where
$D(\alpha):=\text{e}^{\text{i}\sqrt{2}(-\text{Re}(\alpha)\hat{p}+\text{Im}(\alpha)\hat{x})}$
is a displacement operator,
$\ket{\psi_{x}}=\frac{1}{\sqrt{\sigma}\pi^{1/4}}\int\text{d}x\text{e}^{-\frac{x^{2}}{2\sigma^{2}}}\ket{x}$
is a position-squeezed vacuum state, and $0<\sigma<1$ is the variance of a
Gaussian distribution. When $\sigma$ is small, a GKP state that is a
superposition of Gaussian peaks with width $\sigma$, separated by
$2\sqrt{\pi}$, with Gaussian envelop of width $1/\sigma$ in position basis is
a superposition of Gaussian peaks with width $\sigma$, separated by
$\sqrt{\pi}$, with Gaussian envelop of width $\frac{1}{\sigma}$ in momentum
basis.
A straightforward way to certify a GKP state is to detect the eigenvalue of
two stabilizer operators $S_{q}$ and $S_{p}$ to check whether both eigenvalues
are one. However, a realistic GKP state with finite squeezing can never pass
this certification test. To circumvent this obstacle, one solution is to set a
threshold on the eigenvalue in the certification test to pass those realistic
GKP states which are close enough to the ideal GKP states. Nevertheless, this
approach does not exclusively certify a realistic GKP state or a GKP code
space with a certain target degree of squeezing. In this paper, we aim to
certify a realistic GKP state with a certain target degree of squeezing.
To certify a realistic GKP state, we must truncate the phase space to consider
only finite number of superpositions of squeezed coherent states. Suppose we
truncate at $x=\pm\sqrt{\pi}m$ in position quadrature and $p=\pm\sqrt{\pi}m$
in momentum quadrature, respectively, where $m\in\mathbb{N}^{+}$. Then we
obtain a GKP code space witness
$\displaystyle W_{rGKP}=$
$\displaystyle\mathds{1}-\frac{1}{(2m+1)!}\prod_{-m\leq k\leq m}\left(\cosh
r\hat{a}^{\dagger}-\sinh
r\hat{a}-\frac{\sqrt{\pi}k}{\sigma}\right)\cdot\text{h.c.}$
$\displaystyle-\frac{1}{(2m+1)!}\prod_{-m\leq k\leq m}\left(\cosh
r\hat{a}^{\dagger}+\sinh
r\hat{a}+\text{i}\frac{\sqrt{\pi}k}{\sigma}\right)\cdot\text{h.c.},$ (16)
where $r=\frac{1}{2}\ln\frac{1}{\sigma}$ and h.c. denotes Hermitian conjugate.
As GKP states are applied in fault-tolerant CV quantum computing, we use this
witness to certify output state in fault-tolerant CV quantum computing.
## IV Application in verification of fault-tolerant quantum computing
In this subsection, we apply the results in the last subsection to certify two
important types of many-mode quantum states. The first are the resource states
in universal fault-tolerant CV measurement-based quantum computing Menicucci
(2014). These states are CV cluster states Menicucci _et al._ (2006); Gu _et
al._ (2009); Larsen _et al._ (2019); Hastrup _et al._ (2021), attached with
GKP states. The second are the output states of CV IQP circuits Douce _et
al._ (2017). These states are prepared by applying unitary gates diagonal in
position quadrature on the combination of momentum-squeezed states and GKP
states. The preparations of both two types of target states are plotted in
Fig. 1.
We first slightly revise the code witness in Eq. (16) to obtain a fidelity
witness of the GKP state
$\ket{\bar{+}}_{rGKP}\propto\mathcal{N}_{0}\sum_{k=-\infty}^{\infty}e^{-\sigma^{2}\pi
k^{2}}D(\sqrt{\pi}k)\ket{\psi_{x}}.$ (17)
As $\ket{\bar{+}}_{rGKP}$ is a grid of Gaussian peaks separated by
$\sqrt{\pi}$ in position quadrature and separated by $2\sqrt{\pi}$ in momentum
quadrature. We obtain a fidelity witness
$\displaystyle W_{\ket{\bar{+}}_{rGKP}}=\mathds{1}-$
$\displaystyle\frac{1}{(2m+1)!}\prod_{-m\leq k\leq m}\left(\cosh
r\hat{a}^{\dagger}-\sinh
r\hat{a}-\frac{\sqrt{\pi}k}{\sigma}\right)\cdot\text{h.c.}-$
$\displaystyle\frac{1}{(m+1)!}\prod_{-m/2\leq k\leq m/2}\left(\cosh
r\hat{a}^{\dagger}+\sinh
r\hat{a}+2\text{i}\frac{\sqrt{\pi}k}{\sigma}\right)\cdot\text{h.c.}$ (18)
A CV cluster state can be considered as more generally a CV graph state, where
each vertex represents a quantum mode and each edge connecting vertices $i$
and $j$ represents a quantum gate
$CZ:=\text{e}^{\text{i}\hat{x}_{i}\otimes\hat{x}_{j}}$. Suppose a target
cluster state has $N_{s}+N_{GKP}$ modes in total, where $N_{s}$ modes are
initially momentum-squeezed vacuum states
$\ket{\psi_{p}}:=\frac{1}{\sqrt{\sigma}\pi^{1/4}}\int\text{d}p\text{e}^{-\frac{p^{2}}{2\sigma^{2}}}\ket{p}$
and the other $N_{GKP}$ modes are initially $\ket{\bar{+}}_{rGKP}$. The target
CV cluster state is obtained by applying a CZ gate at each pair of adjacent
modes $i$ and $j$ in the graph.
A fidelity witness of this resource state is
$\displaystyle W=$
$\displaystyle\left(1+\frac{N_{s}}{2}\right)\mathds{1}-\sum_{i=1}^{N_{GKP}}\Bigg{[}\frac{1}{(2m+1)!}\prod_{-m\leq
k\leq
m}\left(\frac{\text{e}^{-r}}{\sqrt{2}}\hat{x}_{i}-\frac{\text{e}^{r}}{\sqrt{2}}\text{i}\hat{\tilde{p}}_{i}-\frac{\sqrt{\pi}k}{\sigma}\right)\cdot
h.c.$ $\displaystyle+\frac{1}{(m+1)!}\prod_{-m/2\leq k\leq
m/2}\left(\frac{\text{e}^{r}}{\sqrt{2}}\hat{x}_{i}-\frac{\text{e}^{-r}}{\sqrt{2}}\text{i}\hat{\tilde{p}}_{i}+2\text{i}\frac{\sqrt{\pi}k}{\sigma}\right)\cdot
h.c.\Bigg{]}-\frac{1}{2}\sum_{i=1}^{N_{s}}\left(e^{2r}\hat{x}_{i}^{2}+e^{-2r}\hat{\tilde{p}}_{i}^{2}\right),$
(19)
where $\hat{\tilde{p}}_{i}=\hat{p}_{i}-\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}$
and $\mathcal{N}(i)$ denotes the set of modes adjacent to mode $i$ in the
graph. The fidelity witness in Eq. (19) contains at most $(n+2)^{4m+2}$
products of quadrature operators with maximal order $4m+2$, where $n$ is the
maximal number of neighborhood modes and is as most four in a square-lattice
cluster state. To calculate the required sample complexity to estimate the
mean value of fidelity witness, we denote $\sigma_{k}$ to be the uniform upper
bound of the mean square of all products of $k$ quadrature operators and
$\sigma_{\leq k}$ to be the maximum value of all $\sigma_{j}$ with $1\leq
j\leq k$. Suppose $m$ is a constant, that is we always truncate the phase
space of each mode at a certain fixed value in both position and momentum
quadratures. If we use the approach of importance sampling Flammia and Liu
(2011) to estimate $\braket{W}$, by Hoeffding’s inequality, we find that the
minimum number of required copies of states to obtain an estimate $\omega$
such that
$\text{Pr}\left(|\omega-\braket{W}_{\rho}|\geq\epsilon\right)\leq\delta$, is
upper bounded by
$O\left[\frac{\ln
1/\delta}{\epsilon^{2}}\left(N_{GKP}^{2}\text{e}^{r(4m+2)}\sigma_{\leq
4m+2}+N_{s}^{2}\text{e}^{2r}\sigma_{2}\right)\right].$ (20)
This sample complexity scales polynomially in both $N_{s}$ and $N_{GKP}$.
A CV IQP circuit Douce _et al._ (2017) is a uniformly random combination of
three quantum gates in the set
$\left\\{Z:=e^{\text{i}\hat{x}\sqrt{\pi}},CZ,T:=e^{\text{i}\frac{\pi}{4}[2(\hat{x}/\sqrt{\pi})^{3}+(\hat{x}/\sqrt{\pi})^{2}-2\hat{x}/\sqrt{\pi}]}\right\\},$
with the input of combination of $N_{s}$ copies of $\ket{\psi_{p}}$ and
$N_{GKP}$ copies of $\ket{\bar{+}}_{rGKP}$. Denote $n_{Z}^{i}$, and
$n_{T}^{i}$ as the number of Z gates and T gates applied at $i$th mode,
respectively. Then the fidelity witness of the output state of a CV IQP
circuit is given in Eq. (19) with
$\hat{\tilde{p}}_{i}=\hat{p}_{i}-n_{T}^{i}(3\hat{x}_{i}^{2}/(2\sqrt{\pi})+\hat{x}_{i}/2-\sqrt{\pi}/2)-n_{Z}^{i}\sqrt{\pi}-\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}$
There are at most $N_{GKP}(3+n_{CZ})^{4m+2}$ terms of products of quadrature
operators with order at most $4m+2$, and the absolute value of coefficient is
no larger than $(n_{T}\text{e}^{r})^{4m+2}$. Thus the sample complexity is
bounded by
$\displaystyle O$ $\displaystyle\Bigg{[}\frac{\ln
1/\delta}{\epsilon^{2}}\Big{(}N_{GKP}^{2}(n_{T}\text{e}^{r})^{4m+2}(n_{CZ}+3)^{8m+4}\sigma_{\leq
4m+2}+$ $\displaystyle N_{s}^{2}n_{T}^{2}(n_{CZ}+3)^{4}\sigma_{\leq
4}\Big{)}\Bigg{]}.$
The sample complexity scales polynomially with respect to both the number of
modes and the number of gates.
Figure 1: Diagrams of quantum circuits to generate the two types of target
states.
## V Methods
The four-component cat code space is the degenerate ground space of
Hamiltonian
$\displaystyle\hat{H}=$ $\displaystyle\left(\hat{a}^{\dagger
2}+\alpha^{*2}\right)\left(\hat{a}^{\dagger
2}-\alpha^{*2}\right)\left(\hat{a}^{2}-\alpha^{2}\right)\left(\hat{a}^{2}+\alpha^{2}\right)$
$\displaystyle=$ $\displaystyle\hat{a}^{\dagger
4}\hat{a}^{4}-(\alpha^{4}\hat{a}^{\dagger
4}+\alpha^{*4}\hat{a}^{4})+|\alpha^{2}|^{4}$ (21)
with even photon parity. Following idea similar to above two-component cat
code, and using the fact that $\frac{\mathds{1}+(-1)^{\hat{n}}}{2}$ is the
projection onto the even parity subspace, we obtain the witness of four-
component cat code space as shown in Eq. (9).
A realistic GKP state is a superposition of squeezed coherent states. Each
component $D(\sqrt{\pi}k)\ket{\psi_{x}}$ has the nullifier
$S(r)\left(\hat{a}^{\dagger}-\frac{\sqrt{\pi}k}{\sigma}\right)\left(\hat{a}-\frac{\sqrt{\pi}k}{\sigma}\right)S(r)^{\dagger}$.
Then the superposition of $D(\sqrt{\pi}k)\ket{\psi_{x}}$ for $k\in\mathbb{Z}$
has the nullifier
$S(r)\prod_{k\in\mathbb{Z}}\left(\hat{a}^{\dagger}-\frac{\sqrt{\pi}k}{\sigma}\right)\prod_{l\in\mathbb{Z}}\left(\hat{a}-\frac{\sqrt{\pi}l}{\sigma}\right)S(r)^{\dagger}.$
(22)
Similarly, any state $D(2\sqrt{\pi}k\text{i})\ket{\psi_{p}}$ has the nullifier
$S(\text{i}r)\left(\hat{a}^{\dagger}+\text{i}\frac{2\sqrt{\pi}k}{\sigma}\right)\left(\hat{a}-\text{i}\frac{2\sqrt{\pi}k}{\sigma}\right)S(\text{i}r)^{\dagger}.$
(23)
Thus, a realistic GKP state has another nullifier
$S(\text{i}r)\prod_{k\in\mathbb{Z}}\left(\hat{a}^{\dagger}+\text{i}\frac{2\sqrt{\pi}k}{\sigma}\right)\prod_{l\in\mathbb{Z}}\left(\hat{a}-\text{i}\frac{2\sqrt{\pi}l}{\sigma}\right)S(\text{i}r)^{\dagger}.$
(24)
Combining these two nullifiers together, we obtain the fidelity witness of a
realistic GKP state as shown in Eq. (18).
To obtain the fidelity witness of a cluster state attached with GKP states, we
also need a witness of CV cluster state. A cluster state is constructed from
tensor product of $n$ $\hat{p}$-squeezed vacuum states by applying CV CZ
gates. A nullifier of each mode $1\leq i\leq n$ in a $n$-mode CV cluster state
is ,
$\frac{e^{2r}\left(\hat{x}_{i}^{2}-\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}^{2}\right)+e^{-2r}\hat{p}_{i}^{2}-1}{2}.$
(25)
Combining all the nullifiers, we obtain a fidelity witness in Eq. (19). The
fidelity witness of output states of IQP circuits can be obtained by noticing
that
$T\hat{p}T^{\dagger}=\hat{p}-3\hat{x}^{2}/(2\sqrt{\pi})-\hat{x}/2+\sqrt{\pi}/2,$
and following a similar strategy.
The sample complexities for both the certification protocols of CV cluster
states attached with GKP states and output states of IQP circuits can be
calculated using importance sampling and Hoeffding’s inequality. Suppose a
random variable $\bm{F}=\sum_{i=1}^{m}\lambda_{i}f_{i}$, where each $f_{i}$ is
a polynomial function of quadrature operators. Then by importance sampling
approach, Hoeffindg’s inequality implies
$\text{Pr}(|F^{*}-\bm{F}|>\epsilon)\leq
8\text{e}^{-\frac{N\epsilon^{2}}{33\braket{\bm{F}^{2}}}}$, where
$\braket{\bm{F}^{2}}\leq
m^{2}\max_{i}|\lambda_{i}|\max_{i}\braket{f_{i}^{2}}$. As calculated in Ref.
Liu _et al._ (2019); Farias and Aolita (2021), to make the failure
probability of estimation less than $\delta$, the required sample complexity
is upper bounded by
$O\left(\frac{\ln(1/\delta)}{\epsilon^{2}}m^{2}\max_{i}|\lambda_{i}|\max_{i}\braket{f_{i}^{2}}\right)$.
## VI Discussion
Bosonic quantum error correcting codes is a promising way to realize universal
fault tolerant quantum computing is a qubit-into-qumode manner alternative to
the KLM scheme. Preparation of bosonic code quantum states with high fidelity
is an key issue for the implementation of Bosonic quantum error correction.
Here we propose realistic protocols to certify the preparations of
experimental Bosonic code sates using Gaussian measurements. Different from
state tomography, this protocol is extended to certification of many-mode
quantum states efficiently with respect to the number of modes. Most previous
work on certification or verification of nonGaussian states are about
nonGaussian states with finite stellar rank Chabaud _et al._ (2020b). In
contrast to quantum states in Boson sampling and KLM schemes, cat states and
GKP states have infinite stellar rank, indicating higher nonGaussianity.
In this paper, we first develop an approach to certify whether a CV state
falls inside a certain two-dimensional Bosonic code space. If a CV state
passes this certification protocol, then we can ignore the CV nature of this
state and consider only the logical code space. By combining our certification
approach, together with quantum characterization approaches on many-qubit
systems, including direct fidelity estimation Flammia and Liu (2011), qubit-
state verification Pallister _et al._ (2018) and classical shadow estimation
Huang _et al._ (2020), one can certify preparations of many-qubit Bsonic code
states.
Although here we assume independent and identical copies of prepared quantum
states, we can extend this work into non-i.i.d scenario by using the technique
developed in Ref. Wu _et al._ (2021). Hence, the protocol can be applied to
verifiable blind fault-tolerant quantum computing Hayashi and Morimae (2015);
Gheorghiu _et al._ (2019). A server prepares quantum states, which are
claimed to be resource states for CV fault-tolerant quantum computing, and
sends them to a client. After receiving these states, the client randomly
chooses some of them to perform dimension test and fidelity test. If both
tests are passed, then the remaining states can be used for measurement-based
fault tolerant quantum computing with homodyne detections.
Besides verification of resource states for universal quantum computing, we
also propose protocols to certify output states of IQP circuits. It has been
shown that the statstics of practical homodyne measurements on position
quadratures of the quantum output of IQP circuits cannot be efficiently
simulated by a classical computer Douce _et al._ (2017). Hence, a certified
quantum output state of IQP circuits can be used to show quantum supremacy.
## VII Acknowleadgement
This work was supported by funding from the Hong Kong Research Grant Council
through grants no. 17300918 and no. 17307520.
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## VIII Appendix
### VIII.1 Calculation of two-component cat code witness in Eq. (6)
By using
$\displaystyle\hat{a}^{\dagger\,2}\hat{a}^{2}=$
$\displaystyle\hat{n}^{2}-\hat{n}$ (26) $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\hat{x}^{4}+\hat{p}^{4}+\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2}\right)-\hat{x}^{2}-\hat{p}^{2}+\frac{3}{4}$
(27) $\displaystyle\alpha^{*2}\hat{a}^{2}+\alpha^{2}\hat{a}^{\dagger\,2}=$
$\displaystyle\operatorname{Re}(\alpha^{2})(\hat{x}^{2}-\hat{p}^{2})+\operatorname{Im}(\alpha^{2})(\hat{x}\hat{p}+\hat{p}\hat{x}),$
(28)
we obtain
$\displaystyle W=$ $\displaystyle\mathds{1}-\frac{\hat{H}_{tCat}}{2}$ (29)
$\displaystyle=$
$\displaystyle\mathds{1}-\frac{1}{8}\left(\hat{x}^{4}+\hat{p}^{4}+\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2}\right)+\frac{1}{2}(\hat{x}^{2}+\hat{p}^{2})-\frac{3}{8}+\frac{1}{2}\operatorname{Re}(\alpha^{2})(\hat{x}^{2}-\hat{p}^{2})$
(30)
$\displaystyle+\frac{1}{2}\operatorname{Im}(\alpha^{2})(\hat{x}\hat{p}+\hat{p}\hat{x})-\frac{|\alpha|^{4}}{2}$
(31)
Note that
$\hat{x}\hat{p}+\hat{p}\hat{x}=(\hat{x}+\hat{p})^{2}-\hat{x}^{2}-\hat{p}^{2}$
(32)
and
$\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2}=\frac{1}{6}\left[\left(\hat{x}+\hat{p}\right)^{4}+\left(\hat{x}-\hat{p}\right)^{4}\right]-\frac{1}{3}(\hat{x}^{4}+\hat{p}^{4})-1.$
(33)
The second equality is because
$\displaystyle(\hat{x}+\hat{p})^{4}+(\hat{x}-\hat{p})^{4}=$ $\displaystyle
2(\hat{x}^{4}+\hat{p}^{4}+\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2})+2(\hat{x}\hat{p}\hat{x}\hat{p}+\hat{x}\hat{p}^{2}\hat{x}+\hat{p}\hat{x}\hat{p}\hat{x}+\hat{p}\hat{x}^{2}\hat{p})$
$\displaystyle=$ $\displaystyle
2(\hat{x}^{4}+\hat{p}^{4}+\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2})+4(\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2})+6.$
So the fidelity witness can be rewritten as
$\displaystyle W=$
$\displaystyle\mathds{1}-\frac{1}{8}(\hat{x}^{4}+\hat{p}^{4})-\frac{1}{48}\left[\left(\hat{x}+\hat{p}\right)^{4}+\left(\hat{x}-\hat{p}\right)^{4}\right]+\frac{1}{24}(\hat{x}^{4}+\hat{p}^{4})$
(34)
$\displaystyle+\frac{1}{8}+\frac{1}{2}(\hat{x}^{2}+\hat{p}^{2})-\frac{3}{8}+\frac{1}{2}\operatorname{Re}(\alpha^{2})(\hat{x}^{2}-\hat{p}^{2})+\frac{1}{2}\operatorname{Im}(\alpha^{2})\left(\hat{x}+\hat{p}\right)^{2}$
(35)
$\displaystyle-\frac{1}{2}\operatorname{Im}(\alpha^{2})\hat{x}^{2}-\frac{1}{2}\operatorname{Im}(\alpha^{2})\hat{p}^{2}-\frac{|\alpha|^{4}}{2}$
(36) $\displaystyle=$
$\displaystyle\mathds{1}-\frac{1}{16}(\hat{x}^{4}+\hat{p}^{4})-\frac{1}{48}\left[\left(\hat{x}+\hat{p}\right)^{4}+\left(\hat{x}-\hat{p}\right)^{4}\right]$
(37)
$\displaystyle+\frac{1}{2}\operatorname{Im}(\alpha^{2})\left(\hat{x}+\hat{p}\right)^{2}+\frac{1}{2}\left[1+\operatorname{Re}(\alpha^{2})-\operatorname{Im}(\alpha^{2})\right]\hat{x}^{2}$
(38)
$\displaystyle+\frac{1}{2}\left[1-\operatorname{Re}(\alpha^{2})-\operatorname{Im}(\alpha^{2})\right]\hat{p}^{2}-\frac{1+2|\alpha|^{4}}{4}.$
(39)
In the case that $\alpha^{2}\in\mathbb{R}$, we have
$W=\mathds{1}-\frac{1}{16}(\hat{x}^{4}+\hat{p}^{4})-\frac{1}{48}\left[\left(\hat{x}+\hat{p}\right)^{4}+\left(\hat{x}-\hat{p}\right)^{4}\right]+\frac{1}{2}(1+\alpha^{2})\hat{x}^{2}+\frac{1}{2}(1-\alpha^{2})\hat{p}^{2}-\frac{1+2\alpha^{4}}{4}.$
(40)
### VIII.2 Calculation of four-component cat code witness in Eq. (10)
Using the fact that for any $k\in\mathbb{N}^{+}$,
$\hat{a}^{\dagger k}\hat{a}^{k}=\hat{n}(\hat{n}-1)\cdots(\hat{n}-k+1)$ (41)
and
$\displaystyle\hat{n}=\hat{a}\hat{a}^{\dagger}-1$
$\displaystyle\hat{n}^{2}=\hat{a}^{2}\hat{a}^{\dagger\,2}-3\hat{a}\hat{a}^{\dagger}+1$
$\displaystyle\hat{n}^{3}=\hat{a}^{3}\hat{a}^{\dagger\,3}-6\hat{a}^{2}\hat{a}^{\dagger\,2}+7\hat{a}\hat{a}^{\dagger}-1$
$\displaystyle\hat{n}^{4}=\hat{a}^{4}\hat{a}^{\dagger\,4}-10\hat{a}^{3}\hat{a}^{\dagger\,3}+25\hat{a}^{2}\hat{a}^{\dagger\,2}-15\hat{a}\hat{a}^{\dagger}+1$
we get
$\displaystyle\hat{a}^{\dagger
4}\hat{a}^{4}=\hat{a}^{4}\hat{a}^{\dagger\,4}-16\hat{a}^{3}\hat{a}^{\dagger\,3}+72\hat{a}^{2}\hat{a}^{\dagger\,2}-96\hat{a}\hat{a}^{\dagger}+24,$
$\displaystyle\hat{a}^{\dagger
3}\hat{a}^{3}=\hat{a}^{3}\hat{a}^{\dagger\,3}-9\hat{a}^{2}\hat{a}^{\dagger\,2}+18\hat{a}\hat{a}^{\dagger}-6,$
$\displaystyle\hat{a}^{\dagger
2}\hat{a}^{2}=\hat{a}^{2}\hat{a}^{\dagger\,2}-4\hat{a}\hat{a}^{\dagger}+2,$
$\displaystyle\hat{a}^{\dagger}\hat{a}=\hat{a}\hat{a}^{\dagger}-1.$
Combing all the above equations, we obtain Eq. (10).
### VIII.3 Calculation of state witness in fault-tolerant quantum computing
Using the fact that
$CZ\hat{a}_{i}CZ^{\dagger}=\hat{a}_{i}-\frac{\text{i}}{\sqrt{2}}\hat{x}_{j}$,
we have
$\displaystyle\prod_{j\in\mathcal{N}(i)}CZ_{ij}\left(\cosh
r\hat{a}_{i}^{\dagger}-\sinh
r\hat{a}_{i}-\frac{\sqrt{\pi}k}{\sigma}\right)\prod_{j\in\mathcal{N}(i)}CZ_{ij}^{\dagger}$
$\displaystyle=$ $\displaystyle\cosh
r\hat{a}_{i}^{\dagger}+\frac{\text{i}}{\sqrt{2}}\cosh
r\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}-\sinh
r\hat{a}_{i}+\frac{\text{i}}{\sqrt{2}}\sinh
r\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}-\frac{\sqrt{\pi}k}{\sigma}$
$\displaystyle=$
$\displaystyle\frac{\text{e}^{-r}}{\sqrt{2}}\hat{x}_{i}-\frac{\text{e}^{r}}{\sqrt{2}}\text{i}\hat{p}_{i}+\frac{\text{i}}{\sqrt{2}}\text{e}^{r}\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}-\frac{\sqrt{\pi}k}{\sigma}$
and
$\displaystyle\prod_{j\in\mathcal{N}(i)}CZ_{ij}\left(\cosh
r\hat{a}_{i}^{\dagger}+\sinh
r\hat{a}_{i}+2\text{i}\frac{\sqrt{\pi}k}{\sigma}\right)\prod_{j\in\mathcal{N}(i)}CZ_{ij}^{\dagger}$
$\displaystyle\rightarrow$ $\displaystyle\cosh
r\hat{a}_{i}^{\dagger}+\frac{\text{i}}{\sqrt{2}}\cosh
r\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}+\sinh
r\hat{a}_{i}-\frac{\text{i}}{\sqrt{2}}\sinh
r\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}+2\text{i}\frac{\sqrt{\pi}k}{\sigma}$
$\displaystyle=$
$\displaystyle\frac{\text{e}^{r}}{\sqrt{2}}\hat{x}_{i}-\frac{\text{e}^{-r}}{\sqrt{2}}\text{i}\hat{p}_{i}+\frac{\text{i}}{\sqrt{2}}\text{e}^{-r}\sum_{j\in\mathcal{N}(i)}\hat{x}_{j}+2\text{i}\frac{\sqrt{\pi}k}{\sigma}.$
Plugging them into Eq. (18), we obtain the witness in Eq. (19).
Since
$\displaystyle
Z\hat{x}Z^{\dagger}=CZ\hat{x}CZ^{\dagger}=T\hat{x}T^{\dagger}=\hat{x}$
$\displaystyle Z\hat{p}Z^{\dagger}=\hat{p}-\sqrt{\pi}$ $\displaystyle
CZ\hat{p}_{1}CZ^{\dagger}=\hat{p}_{1}-\hat{x}_{2}$ $\displaystyle
T\hat{p}T^{\dagger}=\hat{p}+\text{i}\pi/4(2/\sqrt{\pi^{3}}[\hat{x}^{3},\hat{p}]+1/\pi[\hat{x}^{2},\hat{p}]-2\text{i}/\sqrt{\pi})=\hat{p}-3\hat{x}^{2}/(2\sqrt{\pi})-\hat{x}/2+\sqrt{\pi}/2,$
for each mode in the QIP circuit, the annihilation operator is transformed to
$\hat{a}\rightarrow\hat{a}-\text{i}\sqrt{\frac{\pi}{2}}n_{Z}-\frac{\text{i}}{\sqrt{2}}\sum_{j=1}^{n_{CZ}}\hat{x}_{j}-3\text{i}n_{T}\hat{x}^{2}/(2\sqrt{2}\pi)-\text{i}n_{T}\hat{x}/2\sqrt{2}+\sqrt{\pi}n_{T}\text{i}/2\sqrt{2}.$
(42)
Hence,
$\displaystyle\cosh r\hat{a}^{\dagger}-\sinh r\hat{a}$
$\displaystyle\rightarrow$ $\displaystyle\cosh
r(\hat{a}^{\dagger}+\text{i}\sqrt{\frac{\pi}{2}}n_{Z}+\frac{\text{i}}{\sqrt{2}}\sum_{j=1}^{n_{CZ}}\hat{x}_{j}+3\text{i}n_{T}\hat{x}^{2}/(2\sqrt{2}\pi)+\text{i}n_{T}\hat{x}/2\sqrt{2}-\sqrt{\pi}n_{T}\text{i}/2\sqrt{2})$
$\displaystyle-\sinh
r(\hat{a}-\text{i}\sqrt{\frac{\pi}{2}}n_{Z}-\frac{\text{i}}{\sqrt{2}}\sum_{j=1}^{n_{CZ}}\hat{x}_{j}-3\text{i}n_{T}\hat{x}^{2}/(2\sqrt{2}\pi)-\text{i}n_{T}\hat{x}/2\sqrt{2}+\sqrt{\pi}n_{T}\text{i}/2\sqrt{2})$
$\displaystyle=$
$\displaystyle(\frac{\text{e}^{-r}}{\sqrt{2}}+\frac{n_{T}\text{i}\text{e}^{r}}{2\sqrt{2}})\hat{x}_{i}-\frac{\text{e}^{r}}{\sqrt{2}}\text{i}\hat{p}_{i}+\frac{\text{i}}{\sqrt{2}}\text{e}^{r}\sum_{j=1}^{n_{CZ}}\hat{x}_{j}+\frac{3n_{T}\text{i}\text{e}^{r}}{2\sqrt{2}\pi}\hat{x}_{i}^{2}+\text{i}\sqrt{\frac{\pi}{2}}(n_{Z}-n_{T}/2)\text{e}^{r}.$
Plugging it into Eq. (18), we obtain the witness of quantum output of IQP
circuit.
|
# Continual Learning with Optimal Transport based Mixture Model
Quyen Tran
VinAI Research
<EMAIL_ADDRESS>Hoang Phan
VinAI Research
<EMAIL_ADDRESS>Khoat Than
Hanoi University of Science and Technology
<EMAIL_ADDRESS>Dinh Phung
Monash University
<EMAIL_ADDRESS>Trung Le
Monash University
<EMAIL_ADDRESS>
###### Abstract
Online Class Incremental learning (CIL) is a challenging setting in Continual
Learning (CL), wherein data of new tasks arrive in incoming streams and online
learning models need to handle incoming data streams without revisiting
previous ones. Existing works used a single centroid adapted with incoming
data streams to characterize a class. This approach possibly exposes
limitations when the incoming data stream of a class is naturally multimodal.
To address this issue, in this work, we first propose an online mixture model
learning approach based on nice properties of the mature optimal transport
theory (OT-MM). Specifically, the centroids and covariance matrices of the
mixture model are adapted incrementally according to incoming data streams.
The advantages are two-fold: (i) we can characterize more accurately complex
data streams and (ii) by using centroids for each class produced by OT-MM, we
can estimate the similarity of an unseen example to each class more reasonably
when doing inference. Moreover, to combat the catastrophic forgetting in the
CIL scenario, we further propose Dynamic Preservation. Particularly, after
performing the dynamic preservation technique across data streams, the latent
representations of the classes in the old and new tasks become more condensed
themselves and more separate from each other. Together with a contraction
feature extractor, this technique facilitates the model in mitigating the
catastrophic forgetting. The experimental results on real-world datasets show
that our proposed method can significantly outperform the current state-of-
the-art baselines.
## 1 Introduction
Figure 1: The intuitions and motivations of our CLOT. Dynamic preservation
inspires the classes in the old and new tasks to be more condensed themselves
via minimizing $L_{2}(\boldsymbol{\theta})$ and more separate to each other
via minimizing $L_{1}(\boldsymbol{\theta},\boldsymbol{w})$. This encourages
the feature extractor $f_{\boldsymbol{\theta}}$ to have a contraction
behaviour, which helps to reduce the discrepancy in the latent representations
of the replay memory and the entire old-task data. Subsequently, we employ our
OT-MM to characterize each class with a mixture model and multiple centroids.
Artificial neural networks have created a revolution in solving many real-
world problems, especially in the field of computer vision [48, 13, 30, 10].
Along with the development of advanced artificial neural networks, many
intelligent systems have been widely applied to reality and in return posed
many more challenging problems. Additionally, many real-world problems,
typically autonomous vehicles [21, 46], sensory robot data [23, 26], and video
streaming [39, 43] require the intelligent systems to continuously interact,
learn, and adapt to the dynamic changes in learning environments. To further
address this desideratum, Continual Learning has been introduced and
investigated to tackle the dynamic changes in learning environments.
In continual learning (CL), our learning system sequentially receives incoming
data streams and need to adapt continuously to the dynamic changes in the
learning environments. Currently, most of works in CL focuses on three main
scenarios: (i) TIL where the task boundary is known during learning and task
id is available during inference [33, 16, 38, 12], (ii) DIL where there is
only a change in the distribution of tasks in the data stream [47, 2, 24], and
(iii) CIL where the system receives infinite incoming data streams only once
without revisiting previous ones [22, 44, 14]. Among these three, CIL is
possibly the most realistic scenario because it does not require the task id
and and allows not only data distribution but also labels being dynamically
changed. Therefore, in this work, we propose a novel method to tackle the CIL
setting.
One of the crucial problems in continual learning is the catastrophic
forgetting (CF), indicating the phenomenon where neural networks tend to
forget how to perform old tasks when learning new tasks. This even becomes
more serious in the CIL scenario. The reason is we use only a single
classifier to predict classes seen so far and the classifier tends to bias the
classes in the recent and new tasks, known as the bias phenomenon.
To address the catastrophic forgetting, most of the previous works [7] only
focused on avoiding forgetting for the backbone and ignored the important
classification head, hence compromising the predictive performance.
Additionally, some recent works [14, 44, 52] have investigated bias
correction, but this is a way to solve the situation without looking at the
big picture in combination with features extractor. Some others [20, 45] used
tricks on the task id of the test set when predicting, which seems to violate
the rule of CIL regarding no task id available. Another recent work [8]
proposed CoPE to learn a single centroid for each class and used the class
centroids for prediction. Although achieving impressive performance, using a
single centroid for a class might not be sufficient to characterize the
complexity of the incoming data stream of this class, since practical data
often exhibit multimodality. Moreover, there always exists a gap between train
and test latent representations, hence single class centroid tailored to the
train set might highly mismatch the test set.
In this work, to tackle the complexity of the data stream of a class in the
OCL scenario, we propose the novel optimal transport based mixture model (OT-
MM) for modelling incoming streaming data of each class. This is realized by
leveraging the rich theoretical body of optimal transport or Wasserstein (WS)
distance [40, 34] and mixture models [31] in the context of OCL.
Interestingly, by using the entropic dual form of optimal transport [9] and
Gumbel softmax distribution [15], we reach an appealing formulation in the
expectation form for the WS distance of interest, making it ready for the OCL
setting where the update is performed on the basis of incoming data batches.
As a result, we can learn the multiple centroids and covariance matrices
incrementally for each class. As shown in Figure 1, by using multiple
centroids characterizing each class to do inference, we can mitigate the
negative effect to the predictive performance caused by the shift between
train and test latent representations for further improving the predictive
performance.
Another complementary component of our proposed approach is the dynamic
preservation, performed dynamically across incoming data streams. This
component is designed to make latent representations of data in the classes of
both old and new tasks more condensed themselves and more separate to increase
the class discrimination ability. More interestingly, this makes the feature
extractor a contraction behavior, which helps reduce the discrepancy in the
latent representations of the replay memory and the entire old data (cf.
Figure 1) for mitigating the catastrophic forgetting.
Finally, we name our proposed approach Class Increamental Learning with
Optimal Transport based Mixture Model (CLOT) and conduct comprehensive
experiments on the real-world datasets to compare our CLOT with state-of-the-
art baselines. The experimental results show that CLOT significantly
outperforms state-of-the-art baselines.
To sum up, our work has the following contributions:
* •
This is the first work that leverages with the rich theoretical body of
optimal transport to propose a novel OT-MM for tacking the multimodality of
incoming data streams in the OCL scenario.
* •
Our proposed Dynamic Preservation component that strengthens the class
discrimination ability and reduces the gap between train and test latent
representations for relieving the catastrophic forgetting. This component is
complimentary to OT-MM for devising our CLOT.
* •
We conduct extensive experiments on the real-world datasets to show that our
proposed method performs significantly better than other existing methods.
## 2 Related work
Figure 2: $t$-SNE visualization on MNIST: Motivation of OT-MM. Left: the test
latent representation of CoPE [8] with one centroid (i.e., visualized by
digits) per class. Right: the test latent representation of our CLOT with four
centroids per class (i.e., visualized by digits). We observe that there exists
a shift between the test and train representations. Therefore, centroids
learned on the training set might mismatch the testing set. However, using
more centroids per class as in our CLOT can relieve this mismatch.
Previous works attempt to tackle the problem of catastrophic forgetting by
efficiently designing the training process in a number of ways. Memory-based
approaches utilize episodic memory to store past data [32, 4, 5, 33] or employ
deep generative models [37, 19] to produce pseudo samples from the previous
history. Architecture-based methods dynamically allocate a separate subnetwork
for each task [27, 42] to maintain the knowledge of old tasks. Regularization-
based approaches encourage important parameters of old tasks to lie in their
close vicinity [17, 51, 49] by penalizing their changes. Among these three
main lines of work, our method falls into the memory-based approach category,
which utilizes the advantages of buffer memory to preserve the discriminative
characteristics of data.
Current CL algorithms are often assessed in an incremental classification
scenario, where tasks or classes arrive at different timestamps. We here
consider the online class-incremental (OCL) setting, which is more realistic
but also more challenging. In particular, in comparison to task-incremental
learning, the data stream is no longer divided into tasks with clear
boundaries. We also have to deal with catastrophic forgetting without any
information about the task id that the data come from at both training and
inference time. The recent work in OCL usually focuses on: either (i) how to
choose meaningful, diverse samples to store [6, 35], also how to select old
samples to replay [29], or (ii) how to learn representation effectively [11,
3] , mostly inspired by contrastive learning, (iii) how to solve bias problem
for single categorical head [14, 44, 52]. In this work, we present a memory-
based learning strategy, also inspired by contrastive learning. However, not
only that, we bring a breeze when introducing the theory of optimal transport
into continuous learning for the first time. Specifically, an optimal
transport-based mixing model will be applied in the online scenario to learn
the multiple centroids that characterize each class. Then these centroids will
be used to make more accurate predictions and improve episodic memory.
## 3 Preliminaries
In what follows, we present the background of Wasserstein (WS) distance and
optimal transport theory that is necessary for the development of our proposed
method.
### 3.1 Optimal transport and Wasserstein distance
Consider two distributions $\mathbb{P}$ and $\mathbb{Q}$ which operate on the
domain $\Omega\subseteq\mathbb{R}^{d}$, let
$d\left(\boldsymbol{x},\boldsymbol{y}\right)$ be a non-negative and continuous
cost function or metric. Wasserstein distance [34, 40] between $\mathbb{P}$
and $\mathbb{Q}$ w.r.t the metric $d$ is defined as
$\mathcal{W}_{d}\left(\mathbb{Q},\mathbb{P}\right):=\min_{\gamma\in\Gamma\left(\mathbb{Q},\mathbb{P}\right)}\mathbb{E}_{\left(\boldsymbol{x},\boldsymbol{y}\right)\sim\gamma}\left[d\left(\boldsymbol{x},\boldsymbol{y}\right)\right],$
(1)
where $\gamma$ is a coupling that admits $\mathbb{Q},\mathbb{P}$ as its
marginals.
### 3.2 Entropic dual-form for OT and WS distance
To enable the application of optimal transport in machine learning and deep
learning, [9] developed an entropic regularized dual form. First, they
proposed to add an entropic regularization term to primal form (1)
$\displaystyle\mathcal{W}_{d}^{\varepsilon}\left(\mathbb{Q},\mathbb{P}\right)$
$\displaystyle:=$
$\displaystyle\min_{\gamma\in\Gamma\left(\mathbb{Q},\mathbb{P}\right)}$
$\displaystyle\left\\{\mathbb{E}_{\left(\boldsymbol{x},\boldsymbol{y}\right)\sim\gamma}\left[d\left(\boldsymbol{x},\boldsymbol{y}\right)\right]+\varepsilon
D_{KL}\left(\gamma\|\mathbb{Q}\otimes\mathbb{P}\right)\right\\},$ (2)
where $\varepsilon$ is the regularization rate,
$D_{KL}\left(\cdot\|\cdot\right)$ is the Kullback-Leibler (KL) divergence, and
$\mathbb{Q}\otimes\mathbb{P}$ represents the specific coupling in which
$\mathbb{Q}$ and $\mathbb{P}$ are independent.
Using Fenchel-Rockafellar theorem, they obtained the following _entropic
regularized dual form_ of (2)
$\displaystyle\mathcal{W}_{d}^{\varepsilon}\left(\mathbb{Q},\mathbb{P}\right)$
$\displaystyle=\max_{\phi}\left\\{\int\tilde{\phi}\left(\boldsymbol{x}\right)\mathrm{d}\mathbb{Q}\left(\boldsymbol{x}\right)+\int\phi\left(\boldsymbol{y}\right)\mathrm{d}\mathbb{P}\left(\boldsymbol{y}\right)\right\\}$
$\displaystyle=\max_{\phi}\left\\{\mathbb{E}_{\mathbb{Q}}\left[\tilde{\phi}\left(\boldsymbol{x}\right)\right]+\mathbb{E}_{\mathbb{P}}\left[\phi\left(\boldsymbol{y}\right)\right]\right\\},$
(3)
where
$\tilde{\phi}\left(\boldsymbol{x}\right):=-\varepsilon\log\left(\mathbb{E}_{\mathbb{P}}\left[\exp\left\\{\frac{-d\left(\boldsymbol{x},\boldsymbol{y}\right)+\phi\left(\boldsymbol{y}\right)}{\varepsilon}\right\\}\right]\right)$.
## 4 Our Proposed Method
In this section, we present the details of our proposed method. We start with
the general framework and our motivations, followed by the technical details
of online mixture model based on optimal transport theory and the simple yet
but effective dynamic preservation technique that assists the model in
maintaining the performance on the old tasks.
### 4.1 General Framework and Motivations
In the context of the online class incremental learning, at a time step, our
system receives a batch of data examples $X=[X^{c}]_{c\in\mathcal{C}_{new}}$
from a new task, where $\mathcal{C}_{new}$ represents the classes in the new
task and $X^{c}$ is the batch data for the class $c$. To mitigate catastrophic
forgetting, we also maintain a replay memory $\mathcal{M}$ of the old tasks.
We sample a batch of data examples from the replay memory
$\bar{X}=[\bar{X}^{c}]_{c\in\mathcal{C}_{old}}$, where $\mathcal{C}_{old}$
represents the classes in the old tasks and $\bar{X}^{c}$ is the batch data
for the class $c$ in the replay memory.
We feed $X$ and $\bar{X}$ to the feature extractor $f_{\boldsymbol{\theta}}$
to obtain the batches of latent representations $Z=f_{\boldsymbol{\theta}}(X)$
and $\bar{Z}=f_{\boldsymbol{\theta}}(\bar{X})$ for new and old tasks
respectively. For the batches of $X$ and $\bar{X}$, we first perform dynamic
preservation to make the classes in the old and new tasks to become more
condensed themselves and more separate to each other. By doing this, we hope
to strengthen the class discrimination ability for both old and new classes,
hence maximally preserving the performance on the old tasks. Subsequently,
given the separation of the classes in the old and new tasks, we perform OT-MM
which incrementally estimates a mixture model, using optimal transport, for
each class to tackle the complexity of incoming data streams.
Algorithm 1 summarizes the main steps of our method which is Class
Increamental Learning with Optimal Transport based Mixture Model (CLOT).
Additionally, the key motivations of our approach are presented in Figure 1.
In what follows, we present and discuss the technicality of dynamic
preservation and OT-MM.
Algorithm 1 CLOT
Input: The batches $X=[X^{c}]_{c\in\mathcal{C}_{new}}$ and
$\bar{X}=[\bar{X}^{c}]_{c\in\mathcal{C}_{old}}$.
Output: Feature extractor $f_{\boldsymbol{\theta}}$, the centroids/ covariance
matrices $[\boldsymbol{\mu}^{c}_{k},\Sigma^{c}_{k}]_{k=1}^{K}$ for each class
$c$
1: Initialize replay memory $\mathcal{M}$
2: for each batch $(X,\bar{X})$ do
3: Step 1. Perform dynamic preservation
4: Step 2. Perform OT-MM
5: Step 3. Update the replay memory $\mathcal{M}$
6: end for
### 4.2 Dynamic Preservation
For each class $c\in\mathcal{C}=\mathcal{C}_{new}\cup\mathcal{C}_{old}$, we
maintain an aligned prototype $\boldsymbol{w}^{c}$ to provide the logit to the
class $c$ for $\boldsymbol{z}=f_{\boldsymbol{\theta}}(\boldsymbol{x})$ as
$\langle\boldsymbol{w}^{c},\boldsymbol{z}\rangle$. To strengthen the
separation of the classes in the old and new tasks, we update the feature
extractor $f_{\boldsymbol{\theta}}$ and aligned prototypes
$\boldsymbol{w}^{c}$ by minimizing the following cross-entropy loss over
$X=[X^{c}]_{c\in\mathcal{C}_{new}}$ and
$\bar{X}=[\bar{X}^{c}]_{c\in\mathcal{C}_{old}}$:
$\small{}L_{1}(\boldsymbol{\theta},\boldsymbol{w})=\sum_{c\in\mathcal{C}}\mathbb{E}_{\boldsymbol{x}^{c}\sim
X^{c}\text{
or}\,\bar{X}^{c}}\left[-\log\frac{\exp\left\langle\boldsymbol{w}^{c},f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)\right\rangle}{\sum_{c^{\prime}}\exp\left\langle\boldsymbol{w}^{c^{\prime}},f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)\right\rangle}\right].$
(4)
After solving the above optimization problem (4), the representation of the
classes become more separating. We then compute the mean prototypes
$\boldsymbol{p}^{c}$ for each class $c$ as the average of the latent
representations, i.e.,
$\boldsymbol{p}^{c}=\begin{cases}\frac{1}{|X^{c}|}\sum_{\boldsymbol{x}^{c}\in
X^{c}}f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)&c\in\mathcal{C}_{new}\\\
\frac{1}{|\bar{X}^{c}|}\sum_{\boldsymbol{x}^{c}\in\bar{X}^{c}}f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)&c\in\mathcal{C}_{old}\end{cases},$
(5)
where $|\cdot|$ returns the cardinality of a set.
To encourage the representations of the class $c$ to be pulling around its
mean prototype and the representations of other classes $c^{\prime}\neq c$ to
be pushing away the mean prototype $\boldsymbol{p}^{c}$, we propose to
minimize the following loss function:
$\small{}L_{2}(\boldsymbol{\theta})=\sum_{c\in\mathcal{C}}\mathbb{E}_{\boldsymbol{x}^{c}\sim
X^{c}\text{
or}\,\bar{X}^{c}}\left[-\log\frac{\exp\left\\{-\|f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)-\boldsymbol{p}^{c}\|_{2}^{2}\right\\}}{\sum_{c^{\prime}}\exp\left\\{-\|f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)-\boldsymbol{p}^{c^{\prime}}\|_{2}^{2}\right\\}}\right]$
(6)
After performing the dynamic preservation, the latent representations of the
classes in the old and new tasks become more condensed themselves and more
separate from each other. Resultantly, we increase the class discrimination
ability for those in the old and new tasks. Additionally, the feature
extractor $f_{\boldsymbol{\theta}}$ has a contraction behavior which helps
reduce the shift in the latent representations of the data examples in the old
tasks and those in the replay memory. Summarily, we expect that the increasing
class discrimination ability and the contracting feature extractor aid CLOT to
mitigate the catastrophic forgetting (cf. Figure 1). The key steps of the
dynamic preservation is summarized in Algorithm 2.
Algorithm 2 Dynamic preservation
Input:The batches $X=[X^{c}]_{c\in\mathcal{C}_{new}}$ and
$\bar{X}=[\bar{X}^{c}]_{c\in\mathcal{C}_{old}}$
Output:The feature extractor $f_{\boldsymbol{\theta}}$
1: Update $\boldsymbol{w}^{c\in\mathcal{C}},\boldsymbol{\theta}$ by minimizing
(4).
2: Compute the mean prototypes using (5).
3: Update $\boldsymbol{\theta}$ by minimizing (6).
We observe that though becoming more condensed and separate, the latent
representations of data streams in the new task and replay memory are still
sufficiently complex with multi-modality. Therefore, we propose an online OT-
based mixture model (OT-MM) approach to incrementally learn the mixture model
for each class.
Mixture models enable our OT-MM to use multiple centroids to characterize a
class in the online learning manner. Figure 2 demonstrates the advantages of
using multiple centroids to characterize a class. Specifically, we use t-SNE
to visualize the test latent representations of CoPE [8] using one centroid
per class and our CLOT using four centroids per class. We observe that there
exists a shift between test and train representations. Hence, the centroids
tailored to the train set might mismatch the test set. However, it can be seen
that using more centroids per class as in our CLOT can mitigate this mismatch.
Together with the possibly complex distribution of incoming data streams, this
is another motivation for our OT-MM.
Figure 3: Average Accuracy by different number of centroids per class
(CIFAR10). Figure 4: Average accuracy through tasks.
### 4.3 Optimal transport based mixture model
#### 4.3.1 The Derivation of OT-MM
OT-MM is one important building-block of our framework. Given a class
$c\in\mathcal{C}$ of an old or a new task, we aim to use the mature
theoretical body of OT to develop the online OT-MM for data in this class in
the sense that the centroids and covariance matrices of the mixture model are
incrementally updated according to incoming data streams.
Given the latent representations or feature vectors of the class $c$:
$D_{c}=\left\\{\boldsymbol{z}_{1}^{c},...,\boldsymbol{z}^{c}_{N_{c}}\right\\}$,
wherein each
$\boldsymbol{z}_{i}^{c}=f_{\boldsymbol{\theta}}(\boldsymbol{x}_{i}^{c})$ is
the latent representation or feature vector of data example
$\boldsymbol{x}_{i}^{c}$ via the feature extractor $f_{\boldsymbol{\theta}}$.
We denote
$\mathbb{P}_{c}:=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}\delta_{\boldsymbol{z}_{i}^{c}}$
as the empirical data distribution over all data of the class $c$ appearing in
all data streams on the latent space. When data are arriving in streams, we
need to learn a Gaussian mixture model (GMM) that approximates the data
distribution $\mathbb{P}_{c}$. Consider the following GMM
$\mathbb{Q}_{c}:=\sum_{k=1}^{K}\pi_{k,c}\mathcal{N}\left(\boldsymbol{\mu}_{k,c},\text{diag}\left(\mathbf{\sigma}_{k,c}^{2}\right)\right),$
where $\boldsymbol{\pi}^{c}=\left[\pi_{k,c}\right]_{k=1}^{K}$ represents
mixing proportions and
$\boldsymbol{\mu}_{k,c},\text{diag}\left(\mathbf{\sigma}_{k,c}^{2}\right)$ are
the mean vector and covariance matrix of the $k$-th Gaussian. To learn this
GMM, we propose minimizing a WS distance between $\mathbb{P}_{c}$ and
$\mathbb{Q}_{c}$ as
$\min_{\boldsymbol{\pi}^{c},\boldsymbol{\mu}^{c},\Sigma^{c}}\mathcal{W}_{d}\left(\mathbb{P}_{c},\sum_{k=1}^{K}\pi_{k,c}\mathcal{N}\left(\boldsymbol{\mu}_{k,c},\text{diag}\left(\mathbf{\sigma}_{k,c}^{2}\right)\right)\right),$
where $\boldsymbol{\pi}^{c}=\left[\pi_{k,c}\right]_{k=1}^{K}$,
$\boldsymbol{\mu}^{c}=\left[\boldsymbol{\mu}_{k,c}\right]_{k=1}^{K}$,
$\Sigma^{c}=\left[\text{diag}\left(\mathbf{\sigma}_{k,c}\right)\right]_{k=1}^{K}$,
and $d$ is a distance on the latent space.
To handle the above WS distance, we need the parameterized samples
$\tilde{\boldsymbol{z}}^{c}$ from the GMM or
$\tilde{\boldsymbol{z}}^{c}\sim\mathbb{Q}_{c}=\sum_{k=1}^{K}\pi_{k,c}\mathcal{N}\left(\boldsymbol{\mu}_{k,c},\text{diag}\left(\mathbf{\sigma}_{k,c}^{2}\right)\right)$.
To this end, we first sample
$\tilde{\boldsymbol{z}}_{k}^{c}\sim\mathcal{N}\left(\boldsymbol{\mu}_{k,c},\text{diag}\left(\mathbf{\sigma}_{k,c}^{2}\right)\right)$
by using the reparameterization trick
$\tilde{\boldsymbol{z}}_{k}^{c}=\boldsymbol{\mu}_{k,c}+\boldsymbol{\epsilon}_{k}\text{diag}\left(\mathbf{\sigma}_{k,c}\right),$
where the source of randomness
$\boldsymbol{\epsilon}_{k}\sim\mathcal{N}\left(\mathbf{0},\mathbb{I}\right)$.
We then sample the one-hot vector
$\boldsymbol{y}\sim\text{Cat}\left(\boldsymbol{\pi}^{c}\right)$ and compute
$\tilde{\boldsymbol{z}}^{c}=\sum_{k=1}^{K}y_{k}\tilde{\boldsymbol{z}}_{k}^{c}=\sum_{k=1}^{K}y_{k}\left(\boldsymbol{\mu}_{k,c}+\boldsymbol{\epsilon}_{k}\text{diag}\left(\mathbf{\sigma}_{k,c}\right)\right).$
To do a continuous relaxation [15] of $\boldsymbol{y}$ for enabling learning
$\boldsymbol{\pi}^{c}$, we parameterize
$\boldsymbol{\pi}^{c}=\text{softmax}\left(\boldsymbol{\alpha}^{c}\right)$ and
compute
$\displaystyle y_{k}$
$\displaystyle=\frac{\exp\left\\{\left(\log\pi_{k,c}+G_{k}\right)/\tau\right\\}}{\sum_{j=1}^{K}\exp\left\\{\left(\log\pi_{j,c}+G_{j}\right)/\tau\right\\}},k=1,...,K$
$\displaystyle\tilde{\boldsymbol{z}}^{c}$
$\displaystyle=\sum_{k=1}^{K}y_{k}\tilde{\boldsymbol{z}}_{k}^{c}=\sum_{k=1}^{K}y_{k}\left(\boldsymbol{\mu}_{k,c}+\boldsymbol{\epsilon}_{k}\text{diag}\left(\mathbf{\sigma}_{k,c}\right)\right),$
where $\tau>0$ is a temperature parameter and random noises $G_{k}$ are i.i.d.
sampled from Gumbel distribution (i.e., $G_{k}=-\log\left(-\log u_{k}\right)$
for $u_{k}\sim\text{Uniform}(0;1)$).
To train the WS distance of interest, we use the $\varepsilon$-entropic dual-
form [9] (cf. Section 3.2) as
$\displaystyle\mathcal{W}^{\varepsilon}_{d}\left(\mathbb{P}_{c},\sum_{k=1}^{K}\pi_{k,c}\mathcal{N}\left(\boldsymbol{\mu}_{k,c},\text{diag}\left(\mathbf{\sigma}_{k,c}^{2}\right)\right)\right)$
$\displaystyle=\max_{\phi}\left\\{\mathbb{E}_{\mathbb{P}_{c}}\left[\phi\left(\boldsymbol{z}^{c}\right)\right]+\mathbb{E}_{\mathbb{Q}_{c}}\left[\tilde{\phi}\left(\tilde{\boldsymbol{z}}^{c}\right)\right]\right\\},$
(7)
where $\varepsilon>0$ is a small number, $\phi$ is the Kantorovich network,
$\tilde{\boldsymbol{z}}^{c}=\sum_{k=1}^{K}y_{k}\left(\boldsymbol{\mu}_{k,c}+\boldsymbol{\epsilon}_{k}\text{diag}\left(\mathbf{\sigma}_{k,c}\right)\right)$,
and
$\tilde{\phi}\left(\tilde{\boldsymbol{z}}^{c}\right)=-\varepsilon\log\left(\mathbb{E}_{\mathbb{P}_{c}}\left[\exp\left\\{\frac{-d\left(\boldsymbol{z}^{c},\tilde{\boldsymbol{z}}^{c}\right)+\phi\left(\boldsymbol{z}^{c}\right)}{\varepsilon}\right\\}\right]\right).$
#### 4.3.2 OT-MM in Online Continual learning Scenario
We now present how to perform our OT-MM in the online continual learning
scenario when we need to update the set of centroids and covariance matrices
for a class $c\in\mathcal{C}$ based on the batch $X^{c}$ or $\bar{X}^{c}$.
Looking into Eq. (7), this objective function is in the form of expectation,
hence perfectly fitting for online learning. Specifically, we use the current
batch $X^{c}$ or $\bar{X}^{c}$ for each class $c$ to solve
$\displaystyle\min_{\boldsymbol{\pi}^{c},\boldsymbol{\mu}^{c},\Sigma^{c}}\max_{\phi}\left\\{\mathbb{E}_{X^{c}\,or\,\bar{X}^{c}}\left[\phi\left(f_{\boldsymbol{\theta}}(\boldsymbol{x}^{c})\right)\right]+\mathbb{E}_{\mathbb{Q}_{c}}\left[\tilde{\phi}\left(\tilde{\boldsymbol{z}}^{c}\right)\right]\right\\}.$
(8)
To solve the optimization problem (8), we update the Kantorovich network
$\phi$ several times and then update the mixing propotions
$\boldsymbol{\pi}^{c}$, the set of centroids $\boldsymbol{\mu}^{c}$, and the
set of covariance matrices $\Sigma^{c}$ for class $c$ in several times.
The key steps of OT-MM is summarized in Algorithm 3. Eventually, we harvest
the centroids $[\boldsymbol{\pi}^{c}]_{c\in\mathcal{C}}$ and covariance
matrices $[\Sigma^{c}]_{c\in\mathcal{C}}$ to select diverge replay memory for
a task and do inference in the testing phase (cf. Section 4.4).
Algorithm 3 OT-GMM in the OCL scenario
Input:The batches $X=[X^{c}]_{c\in\mathcal{C}_{new}}$ and
$\bar{X}=[\bar{X}^{c}]_{c\in\mathcal{C}_{old}}$
Output:$[\boldsymbol{\pi}^{c},\boldsymbol{\mu}^{c},\Sigma^{c}]_{c\in\mathcal{C}}$
1: for each $c\in\mathcal{C}$ do
2: Update $\phi$ according to (8).
3: Update $[\boldsymbol{\pi}^{c},\boldsymbol{\mu}^{c},\Sigma^{c}]$ according
to (8).
4: end for
Method | M = 1k | M = 3k | M = 5k | M = 0.2k | M = 0.5k | M = 1k | M = 0.1k | M = 0.5k | M = 1.5k
---|---|---|---|---|---|---|---|---|---
iid-offline | $50.28\pm 0.66$ | $50.28\pm 0.66$ | $50.28\pm 0.66$ | $83.02\pm 0.60$ | $83.02\pm 0.60$ | $83.02\pm 0.60$ | $98.44\pm 0.02$ | $98.44\pm 0.02$ | $98.44\pm 0.02$
iid-online | $20.10\pm 0.90$ | $20.10\pm 0.90$ | $20.1\pm 0.90$ | $62.31\pm 1.67$ | $62.31\pm 1.67$ | $62.31\pm 1.67$ | $96.57\pm 0.14$ | $96.57\pm 0.14$ | $96.57\pm 0.14$
DVC [11] | $\mathbf{10.93\pm 0.49}$ | $14.49\pm 0.90$ | $20.83\pm 0.89$ | $28.56\pm 2.10$ | $34.92\pm 2.27$ | $45.02\pm 2.72$ | _ | _ | _
OCD [18] | $3.71\pm 0.5$ | $9.2\pm 0.21$ | $11.56\pm 0.32$ | $15.83\pm 2.30$ | $21.02\pm 5.24$ | $26.45\pm 2.40$ | _ | _ | _
MIR [1] | $8.20\pm 0.72$ | $14.9\pm 0.40$ | $20.00\pm 0.57$ | $24.3\pm 2.01$ | $35.6\pm 2.56$ | $42.80\pm 2.22$ | $73.6\pm 0.50$ | $86.5\pm 0.22$ | $91.2\pm 0.36$
OCS [50] | $8.50\pm 0.35$ | $13.76\pm 0.70$ | $19.98\pm 0.65$ | _ | _ | _ | _ | _ | _
ASER [36] | $9.60\pm 0.88$ | $13.08\pm 0.87$ | $15.00\pm 0.64$ | $22.38\pm 2.94$ | $26.42\pm 4.80$ | $30.67\pm 3.10$ | $58.02\pm 0.07$ | $75.47\pm 0.04$ | $85.68\pm 0.03$
Reservoir [41] | $8.60\pm 1.58$ | $13.5\pm 2.61$ | $19.60\pm 0.53$ | $25.96\pm 1.41$ | $33.85\pm 1.76$ | $40.531\pm 5.30$ | $69.22\pm 1.15$ | $86.50\pm 1.04$ | $91.27\pm 0.96$
GDUMB [28] | $8.48\pm 0.26$ | $11.24\pm 0.56$ | $15.63\pm 0.60$ | $24.60\pm 1.50$ | $28.70\pm 0.63$ | $33.24\pm 2.10$ | $73.5\pm 0.12$ | $86.16\pm 0.01$ | $92.07\pm 0.01$
CoPE [8] | $9.22\pm 0.55$ | $15.15\pm 0.50$ | $19.56\pm 1.22$ | $37.20\pm 1.80$ | $42.22\pm 0.54$ | $47.98\pm 2.62$ | $72.57\pm 2.10$ | $88.92\pm 0.99$ | $92.50\pm 0.33$
CLOT (Ours) | $10.62\pm 0.45$ | $\mathbf{15.80\pm 0.55}$ | $\mathbf{22.35\pm 0.40}$ | $\mathbf{42.28\pm 0.50}$ | $\mathbf{50.08\pm 1.00}$ | $\mathbf{54.45\pm 1.20}$ | $\mathbf{77.40\pm 0.21}$ | $\mathbf{90.98\pm 0.35}$ | $\mathbf{93.71\pm 0.05}$
| a) CIFAR-100 | b) CIFAR-10 | c) MNIST
Table 1: Average Accuracy (higher is better), M denotes the memory buffer
size. All numbers are the average of 5 runs. The data in the table represents
Average Accuracy ± standard deviation.
### 4.4 Replay memory selection and inference process
We now present how to utilize the output of the OT-MM for selecting the replay
memory of the new task and how to do inference.
##### Replay memory selection:
After processing each batch $(X,\bar{X})$, as shown in Step 3 (Line 5) in
Algorithm 1, we select some more data points to supplement the replay memory
of the new task as follows:
* •
For a centroid of a class, we choose some closest data points of this class in
the current batch to add to the replay memory.
* •
If the replay memory is full, we randomly pick some data points in the current
replay memory to replace by the fresh-new ones.
##### Doing inference:
Given an unseen data point $\boldsymbol{x}$, we compute the distance of
$\boldsymbol{x}$ to each class $c$ and classify $\boldsymbol{x}$ to the
closest class as follows:
$\hat{y}=\arg\min_{c\in\mathcal{C}}d\left(\boldsymbol{x},c\right)\text{ where
}d\left(\boldsymbol{x},c\right)=\min_{k\leq
K}d(f_{\boldsymbol{\theta}}\left(\boldsymbol{x}\right),\boldsymbol{\mu}_{k,c})$
## 5 Experiments
In this section, we review the experimental settings including datasets,
baselines, and metrics used to compare our method against other state-of-the-
art models. We then report and analyse the results to validate the
effectiveness of our approach.
### 5.1 Experimental setup
Datasets: We use three benchmark datasets including:
* •
Split-MNIST is constructed by splitting the MNIST dataset (with 60k training
samples) into 5 disjoint subsets, corresponding to 5 tasks, each of which
consists of 2 classes.
* •
Split-CIFAR10 is constructed by splitting CIFAR10 dataset into 5 disjoint
tasks, hence each task has 2 different classes.
* •
Split-CIFAR100 is a variant CL dataset constructed from CIFAR100 where 50k
training samples of 100 classes is divided into 20 tasks of 5 classes.
Architectures: For the experiments on Split-MNIST dataset, we use a simple MLP
neural network with 2 hidden layers of 400 units. While a slim-version of
ResNet-18 will be used to evaluate on Split-CIFAR10 and Split-CIFAR100.
Baselines: We use several state-of-the-art methods in Continual learning,
specifically in Online Continual Learning (OCL), to compare with our proposed
method.
* •
iid-online and iid-offline: give the performance when ignoring the challenges
of non-stationary data stream in OCL scenarios. Both of them train a model
from i.i.d. sampled mini-batches (i.e. there is no change of the distribution
of data in the learning process). While iid-online only allows learning once
over data, iid-offline allows a model to be trained from multiple epochs.
* •
Regarding the recent methods for OCL, the chosen baselines can divided into 2
main groups. The first group focuses on improving the quality of replay
memory: Reservoir, OCS (ICRL2022), MIR (NeurIPS2019) are well-known techniques
to build memory buffer, GDumb (ECCV2020) is interested in the way to retrieve
data more effectively, while ASER (AAAI2021) consider both updating and
retrieving data from memory. The second group includes the methods that help
improve a model mainly through more effective representation learning: DVC
(CVPR2022), OCD (IJICAI2022), CoPE (ICCV2021).
### 5.2 Performance comparison
Figure 5: Features on latent space of our method (a and b) and CoPE (c). It
can be observed that 4 centroids is better than 1 centroid. CLOT 1-centroid is
better than CoPE 1-centroid due to the effect of the dynamic preservation for
learning better latent representations.
Table 1 summarizes the experimental results on three datasets with various
memory sizes. On MNIST, we compare our method with five baselines and achieve
better results. Especially when memory sizes are small (0.1k), we exceed the
strongest baseline (MIR) by roughly 4%. On CIFAR10, CLOT again outperforms the
SOTA baselines. The largest gap between CLOT and the best baseline is around
8%. On CIFAR100, the memory size for each class and the batch size in each
learning step both are very small, making this the most challenging dataset to
learn. The results show that with a sufficient large memory size (3k, 5k), our
CLOT surpasses the baselines, while with a too-small memory size, our CLOT is
the runner-up method.
Among three datasets, the superiority of CLOT is most apparent in CIFAR10.
This because that MNIST is a fairy easy dataset, especially when the memory is
large enough to get the representational diversity of the old samples. Thus,
for the results on MNIST with memory size 1.5K, the methods do not show clear
performance differences. As for CIFAR100, we can see that our method needs a
sufficient amount of data to show its strengths in learning representation.
Because with too small memory size and batch size, some samples from all
classes are not always available.
Figure 4 shows the visual results about the average accuracy on observed tasks
of the different methods with the large enough memory size on each dataset. It
is easy to see that our method outperforms the other baselines, especially as
the number of tasks increases. This is most evident in CIFAR10, consistent
with the reasons discussed earlier.
### 5.3 Ablation study
In what follows, we present additional experimental results to analyse the
effectiveness of components in CLOT.
The role of using multiple centroids in general: For our OT-MM, we examine the
influence of number of centroids per class. Figure 3 depicts the curves of
average accuracies on CIFAR10 when Num-centroids varies. We observe that in
all cases, the performance improves when we increase the number of centroids
per class from 1 to a certain threshold. The bigger the memory size is, the
bigger the threshold is. Moreover, if these thresholds are exceeded, the model
performance depends on the support level of replay memory: the smaller memory
size leads to the lower the representation learning quality, and higher
prediction error. When memory size = 0.2K, if the number of centroids is
bigger than 3, the model performance degrades most. On the other hand, with a
larger memory size (e.g., memory size = 1K), the ideal number of centroids is
4, and when increasing the number of centroids to 5, the model quality does
not degrade significantly. Therefore, it can be seen that our prediction
method using many centroids would achieve the best results when there is a
harmonious combination with the representation learning quality.
The role of centroids in improving replay buffer: In our framework, we
leverage centroids learned from OT-MM to improve the quality of the memory
buffer. Table 2 is an ablation study on the effect of using or not using
centroids to select samples for the replay buffer. It can be observed that in
all cases, using centroids to select samples consistently offers better
results than not using centroids. Regardless that the sample selection based
on centroids is quite simple, the results support our intuition that centroids
help effectively characterize data and improve the diversity in the episode
memory.
| Num of centroids
---|---
| 1 | 2 | 3 | 4 | 5 | 8
Use centroids | 50.1 | 52.5 | 53.8 | 54.5 | 54.1 | 54
w/o centroids | 50.1 | 52.11 | 53.2 | 54.0 | 53.6 | 53.3
Table 2: Effect of using centroids on replay memory selection. We compare the
performance when using centroids to select samples (Use centroids) and when
selecting samples at random for the replay memory (w/o centroids).
The role of centroids in prediction: To verify the role of using multiple
centroids when making decisions, we present $t$-SNE visualization (Figure 5)
of the feature space of a setting on CIFAR10. Figures 5a and 5b illustrate our
approach when using multiple centroids and only 1 centroid when predicting. We
can see that, in the real scenario, the features of the classes are usually
distributed with multi-modality. Therefore, using more centroids helps to
characterize more accurately multi-modal distributions, thus generally giving
better prediction than using only 1 centroid.
Furthermore, Figure 5c illustrates the corresponding results on latent space
using CoPE where the features of different classes are not well separated and
each class has only one centroid for prediction. When comparing 5c with 5a and
5b, we again emphasize the effectiveness of our framework in improving the
quality of representation learning via dynamic preservation, along with making
decisions based on multiple centroids produced by OT-MM.
## 6 Conclusion
In this work, we have proposed a novel method for Online CIL scenarios, which
is for the first time that OT theory has been leveraged with Continual
learning. We have introduced a new perspective that helps the network to make
predictions based on many centroids for each class of data to tackle more
efficiently their multi-modality. In addition, we have further proposed a
strategy of remembering old knowledge flexibly, called Dynamic Preservation,
which complements to the prediction mechanism above. Furthermore, the
extensive experiments on three commonly used benchmark datasets demonstrate
the effectiveness of our method.
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## 7 Supplementary Material
### 7.1 Implementation Detail
We implement our proposed method and baselines on the same code base, based on
the learner-evaluator framework proposed in [8]. Our code is available at
https://github.com/tranquyenbk173/Streaming_WSD_GGM
#### 7.1.1 Hyperparameters configuration
We shared the same replay memory size (total samples of all classes) per
dataset for each method, and set the batch size for all methods to 10, the
size of the retrieved batch is the same as the batch size from data stream.
The optimizers are all SGD, the learning rate and other hyperparameters were
trying to adjust to get the best results for each method. In addition to that,
for a fair comparison, the classes in each task and the order of tasks are
fixed in all experiments of the same kind. Table 3 presents the setting.
Hyper-params | Split | Split | Split
---|---|---|---
| CIFAR100 | CIFAR10 | MNIST
batch_size | 10 | 10 | 10
memory size | [ 1k, 3k, 5k ] | [ 0.2k, 0.5k, 1k ] | [ 0.1k, 0.5k, 1.5k ]
Table 3: Shared Hyperparameter configurations among our method and baselines
for three datasets.
#### 7.1.2 Evaluation and metrics
In all experiments, we use a single-head architecture, where the final
classifier layer is shared across all the tasks (except for CLOT and CoPE,
only features extractor is available), and the task identity is not provided
during inference.
Following the CL literature [4, 25], we used the following metrics to
evaluate:
* •
Average accuracy ($\mathcal{A}_{T}$): Averaged test accuracy of all tasks
after completing learning T task.
$\mathcal{A}_{t}=\frac{1}{T}\sum_{j=1}^{T}a_{T,j},$
where $a_{i,j}$ is the performance of the model on the held-out testing set of
task $j$ after the model is trained from task $1$ to $i$.
* •
Average forgetting ($\mathcal{F}_{T}$): The averaged gap between the highest
recorded and final task accuracy at the end of continual learning on $T$
tasks.
$\mathcal{F}_{T}=\frac{1}{T-1}\sum_{j=1}^{T-1}f_{T,j},$
where $f_{i,j}=\max_{l\in\\{1,...,i-1\\}}a_{l,j}-a_{i,j}$
### 7.2 Additional Experiments
#### 7.2.1 Average forgetting
Table 4 shows the average forgetting at the end of data stream of methods on 3
datasets. Our CLOT is the method with the lowest average forgetting in most
cases. The most enormous difference is shown on the MNIST dataset, with a gap
of about 6 % from the most robust baseline (MIR). On the CIFAR10, CLOT avoids
forgetting better than the best method by more than 2.6%. With the most
challenging dataset - CIFAR100, our method also shows an advantage over other
methods.
On the other hand, Figure 6 depicts the average accuracy of methods through
tasks. We can see that, in general, CLOT is the method with the lowest
forgetting rate (on the MNIST and CIFAR100 datasets) / or in the group with
the least forgetting rate (on the CIFAR10 set) during the continual learning
process. At the end of the data stream, our proposed method shows a clear
difference in avoiding catastrophic forgetting compared to the baselines on
the two datasets: MNIST and CIFAR10.
These evidences once again confirm the effectiveness of our framework in
online continual learning scenario.
Figure 6: Average forgetting through tasks. Lower is better.
Method | M = 1k | M = 3k | M = 5k | M = 0.2k | M = 0.5k | M = 1k | M = 0.1k | M = 0.5k | M = 1.5k
---|---|---|---|---|---|---|---|---|---
DVC [11] | $\mathbf{52.25\pm 0.41}$ | $51.69\pm 0.34$ | $50.36\pm 0.46$ | $60.15\pm 5.5$ | $55.2\pm 5.99$ | $44.37\pm 1.21$ | _ | _ | _
OCD [18] | $68.55\pm 1.5$ | $65.33\pm 0.78$ | $64.1\pm 0.57$ | $68.48\pm 4.41$ | $62.34\pm 7.59$ | $51.09\pm 1.62$ | _ | _ | _
MiR [1] | $58.862\pm 0.97$ | $56.56\pm 0.81$ | $54.55\pm 0.72$ | $58.23\pm 4.3$ | $56.65\pm 1.6$ | $46.34\pm 3.2$ | $33.2\pm 2.59$ | $16.01\pm 1.51$ | $8.87\pm 1.8$
OCS [50] | $60.01\pm 1.81$ | $57.22\pm 0.55$ | $56.88\pm 0.6$ | _ | _ | _ | _ | _ | _
ASER [36] | $58.71\pm 0.84$ | $59.44\pm 0.92$ | $59.67\pm 0.79$ | $62.05\pm 3.2$ | $60.01\pm 1.73$ | $56.67\pm 3.27$ | $40.55\pm 3.12$ | $20.53\pm 2.52$ | $25.4\pm 0.25$
Reservoir [41] | $58.91\pm 1.7$ | $56.70\pm 0.5$ | $55.88\pm 0.87$ | $57.62\pm 2.45$ | $52.25\pm 1.41$ | $48.53\pm 2.3$ | $35.58\pm 2.5$ | $15.69\pm 2.1$ | $9.48\pm 1.13$
GDUMB [28] | $59.11\pm 1.17$ | $57.83\pm 0.82$ | $56.55\pm 0.66$ | $57.88\pm 1.89$ | $57.15\pm 2.1$ | $55.16\pm 1.55$ | $33.5\pm 2.89$ | $12.66\pm 0.2$ | $8.12\pm 0.93$
CoPE [8] | $55.17\pm 0.3$ | $54.65\pm 0.45$ | $48.47\pm 0.99$ | $52.35\pm 2.15$ | $49.76\pm 2.39$ | $42.99\pm 2.87$ | $34.49\pm 3.62$ | $12.915\pm 1.8$ | $7.53\pm 1.2$
CLOT (Ours) | $53.21\pm 0.51$ | $\mathbf{51.01\pm 0.55}$ | $\mathbf{47.59\pm 0.57}$ | $\mathbf{51.32\pm 2.3}$ | $\mathbf{47.08\pm 3.1}$ | $\mathbf{40.25\pm 1.1}$ | $\mathbf{27.46\pm 3.17}$ | $\mathbf{10.74\pm 0.21}$ | $\mathbf{6.27\pm 0.28}$
| a) CIFAR-100 | b) CIFAR-10 | c) MNIST
Table 4: Average Forgetting (lower is better), M denotes the memory buffer
size. All numbers are the average of 5 runs. The data in the table represents
Average Forgetting ± standard deviation.
#### 7.2.2 Additional technicalities
Clip grad proportion of prototypes
| Clip grad proportion
---|---
| 0.0 | 0.001 | 0.05 | 0.08 | 0.1 | 0.15 | 0.5
MNIST | 93.22 | 93.46 | 93.50 | 93.58 | 93.71 | 93.20 | 92.80
CIFAR10 | 53.23 | 53.55 | 54.41 | 54.45 | 52.40 | 49.12 | 46.55
Table 5: Average Accuracy by different clip grad proportion
In the Dynamic Preservation phrase, when solving the problem (4):
$\small{}L_{1}(\boldsymbol{\theta},\boldsymbol{w})=\sum_{c\in\mathcal{C}}\mathbb{E}_{\boldsymbol{x}^{c}\sim
X^{c}\text{
or}\,\bar{X}^{c}}\left[-\log\frac{\exp\left\langle\boldsymbol{w}^{c},f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)\right\rangle}{\sum_{c^{\prime}}\exp\left\langle\boldsymbol{w}^{c^{\prime}},f_{\boldsymbol{\theta}}\left(\boldsymbol{x}^{c}\right)\right\rangle}\right],$
(4)
we use a secondary technique to control the updating of the prototypes as
follows:
* •
Compute gradient of $L_{1}$ w.r.t prototypes:
$g_{\boldsymbol{p}}=\nabla_{\boldsymbol{p}}L_{1}$
* •
Adjust the gradient for prototypes: new prototypes are allowed to learn more
than old prototypes. With clip grad proportion $\alpha\in\left(0,1\right)$, we
update:
$\boldsymbol{p^{c}}=\left\\{\begin{matrix}\boldsymbol{p}^{c}-\eta*\alpha*g_{\boldsymbol{p}}&c\in\mathcal{C}_{old},\\\
\boldsymbol{p}^{c}-\eta*g_{\boldsymbol{p}}&c\in\mathcal{C}_{new}\end{matrix}\right.$
Clip grad proportion $\alpha$ is a hyperparameter that tells us how much
portability we allow for prototypes. Table 5 shows the results when changing
the value of clip grad proportion. The experiments on MNIST give the best
results with $alpha=0.001$, and the ideal value is $0.08$ on CIFAR10.
In continual learning, the representation of features often shift too much,
especially at transition times between tasks. If we learn prototypes
arbitrarily without control (especially when the old data reviewed are very
limited), they then move to unfavorable locations, making the features of
different classes to be unseparated well, on both the training and testing
sets. The table also shows that, on all 3 datasets, if the clip grad
proportion is too high, the average accuracy is highly reduced. While, if it
is too small, the model performance is greatly stable. That is, the features
of the old classes are fixedly distributed in some places that is only good
for the old tasks, and the new tasks does not have much space to place the new
features.
Prototype adjustment
Figure 7: $t$-SNE visualization of learned features and prototypes. The
features of different classes are assigned different colors. The prototypes
are located in the position of the red ”X” signs.
In this section, we conduct additional experiments and explanations regarding
why we need to adjust the aligned prototypes to the mean prototypes (a.k.a.
the prototype adjustment) when compressing the feature vectors in the dynamic
preservation.
Our practical observations show that the aligned prototypes (used to learn
feature extractor by some objective functions like Cross Entropy loss) do not
often locate at the centre of each class and also very close to each other
(Figure 7). This seems to be detrimental. Additionally, in the next update
step (6), we immobilize the prototypes to learn the feature extractor so that
the features will move to the corresponding prototypes. Therefore, if we use
the aligned prototypes directly for (6), the clusters of features (of
different classes) will be overlapped. Consequently, the predictive efficiency
may be degraded. Moreover, after performing (4), the features of different
classes are quite separable. Hinted by this observation, we compute mean
prototypes in (5) and use them to solve the optimization problem in (6) for
compressing the feature vectors around the corresponding mean prototypes.
Figure 8: Performance of CLOT when adjusting prototypes (Adjust proto) and not
(Not adjust proto), according to (5)
Figure 8 shows the effect of prototype adjustment in (5). We can see this
change helps to solve the issue mentioned above and improve the performance.
On all 3 datasets, when adjusting the prototypes according to (5), we obtain
higher average accuracies and lower average forgetting scores.
|
# Dissipative reactions with intermediate-energy beams – a novel approach to
populate complex-structure states in rare isotopes
A. Gade Facility for Rare Isotope Beams, Michigan State University, East
Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan
State University, East Lansing, Michigan 48824, USA B. A. Brown Facility for
Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824,
USA Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824, USA D. Weisshaar Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA D. Bazin
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA Department of Physics and Astronomy, Michigan State
University, East Lansing, Michigan 48824, USA K. W. Brown Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
Department of Chemistry, Michigan State University, East Lansing, Michigan
48824, USA R. J. Charity Department of Chemistry, Washington University, St.
Louis, Missouri 63130, USA P. Farris Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA Department of
Physics and Astronomy, Michigan State University, East Lansing, Michigan
48824, USA A. M. Hill Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA J. Li
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA B. Longfellow Present address: Lawrence Livermore
National Laboratory, Livermore, California 94550, USA Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA D. Rhodes Present address: TRIUMF, 4004 Wesbrook Mall,
Vancouver, BC V6T 2A3, Canada Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA W.
Reviol Physics Division, Argonne National Laboratory, Argonne, Illinois
60439, USA J. A. Tostevin Department of Physics, Faculty of Engineering and
Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United
Kingdom
###### Abstract
A novel pathway for the formation of multi-particle-multi-hole ($np-mh$)
excited states in rare isotopes is reported from highly energy- and momentum-
dissipative inelastic-scattering events measured in reactions of an
intermediate-energy beam of 38Ca on a Be target. The negative-parity, complex-
structure final states in 38Ca were observed following the in-beam
$\gamma$-ray spectroscopy of events in the
9Be($\hbox{${}^{38}$Ca},\hbox{${}^{38}$Ca}+\gamma$)X reaction in which the
scattered projectile lost longitudinal momentum of order $\Delta p_{||}=700$
MeV/c. The characteristics of the observed final states are discussed and
found to be consistent with the formation of excited states involving the
rearrangement of multiple nucleons in a single, highly-energetic projectile-
target collision. Unlike the far-less dissipative, surface-grazing reactions
usually exploited for the in-beam $\gamma$-ray spectroscopy of rare isotopes,
these more energetic collisions appear to offer a practical pathway to
nuclear-structure studies of more complex multi-particle configurations in
rare isotopes – final states conventionally thought to be out of reach with
high-luminosity fast-beam-induced reactions.
###### pacs:
Beyond the proof of existence of a rare isotope and the determination of its
ground-state half-life, the energies of excited states are typically the first
observables that become accessible in laboratory experiments. For excited
bound states, depending on their lifetime, prompt or delayed $\gamma$-ray
spectroscopy is frequently used to obtain precise excitation energies from the
measured transition energies [1]. In short-lived rare isotopes, excited states
can be populated efficiently in (direct) nuclear reactions [2] or $\beta$
decay [3], for example, most often exploiting the unique selectivity inherent
to each of these different population pathways. The selectivity of one- and
two-nucleon transfer and knockout reactions, or inelastic scattering [2, 4, 5,
6, 7, 8], often enhances the population of excited states at moderate spin
associated with the single-particle or collective degree of freedom. Here, we
report the novel, complementary in-beam $\gamma$-ray spectroscopy of higher-
spin, negative-parity states in 38Ca, observed to be populated in the
9Be($\hbox{${}^{38}$Ca},\hbox{${}^{38}$Ca}+\gamma$)X inelastic scattering at
high momentum loss. From the peculiar final states observed, we argue that
these complex-structure, projectile excited states are formed by the
rearrangement of multiple nucleons in a single, highly-energetic projectile-
target collision, giving access to multi-particle configurations not expected
to be in reach of high-luminosity fast-beam reactions.
The reaction channel analyzed here is populated in the same experiment as
reported on in Ref. [9] where the focus was on 40Sc produced in the $pn$
pickup reaction onto the 38Ca projectile. Here, we briefly summarize the
experimental scheme below and refer the reader to Ref. [9, 16] for more
details. The 38Ca rare-isotope beam was produced by fragmentation of a stable
40Ca beam, accelerated to 140 MeV/nucleon by the Coupled Cyclotron Facility at
NSCL [10]. The momentum width transported to the experiment was restricted to
$\Delta p/p=0.25$%, resulting in 160,000 38Ca/s impinging upon a
188-mg/cm2-thick 9Be foil located at the target position of the S800
spectrograph [11]. The setting subject of this publication ran for less than
40 hours. The constituents of the incoming beam and the projectile-like
reaction products were identified on an event-by-event basis using the S800
analysis beam line and focal plane with the standard detector systems [12]. As
the magnetic rigidity of the S800 spectrograph was tuned for 36Ca, only part
of the outermost (exponential) low-momentum tail of the reacted 38Ca
distribution was transmitted to the focal plane. Specifically, the S800
momentum acceptance at this setting is $p_{0}\pm 330$ MeV/c, with
$p_{0}=11.222$ GeV/c.
When compared to the parallel momentum distribution of the unreacted 38Ca
passing through the target, having suffered only in-target energy losses
($p_{0}=11.932$ GeV/c), the low-momentum, reacted 38Ca events detected in the
reaction setting have undergone an additional longitudinal momentum loss of
about 700 MeV/c (see Fig. 1). That is, approximately 18 MeV/c per nucleon in
momentum or 5.4 MeV/nucleon in energy. The cross section for finding 38Ca with
such large momentum loss was extracted to be $\sigma(p_{0}\pm
330~{}\mathrm{MeV/c})=3.8(4)$ mb, making these inelastic large-momentum-loss
events rather rare.
Figure 1: Longitudinal momentum distributions of 38Ca passing through the
target and only suffering energy loss (magenta peak) and, on log scale, for
the dissipative setting (inset (a)). Insets (b) and (c) confront the
$\gamma$-ray spectra in coincidence with less than 100,500 38Ca at high
momentum loss (black) and from nearly 179,000 38Ca in the direct setting
(magenta), highlighting a stark difference in excitation probability.
The mid-target energy of 38Ca in the 9Be reaction target was 60.9 MeV/nucleon.
The target was surrounded by GRETINA [13, 14], an array of 48 36-fold
segmented high-purity germanium crystals assembled into modules of four
crystals each, used for prompt $\gamma$-ray detection to tag the final states
of the reaction residues. Signal decomposition was employed to provide the
$\gamma$-ray interaction points. Of these, the location of the interaction
with the largest energy deposition was selected as the first hit entering the
event-by-event Doppler reconstruction of the $\gamma$ rays emitted from the
reaction residues in-flight at about 33% of the speed of light [14].
The event-by-event Doppler reconstructed $\gamma$-ray spectrum obtained in
coincidence with the 38Ca reaction residues detected in the S800 focal plane
at large momentum loss is shown in Fig. 2. Nearest neighbor addback, as
detailed in [14], was used. Of the seven $\gamma$-ray transitions compiled in
[15], those at 2213(5), 1489(5), 489(4) and 3684(8) keV are observed here,
while the transitions at 214(4), 1048(6), 2417(7), 2537(6), 2688(7), and
2758(7) keV are reported for the first time in the present work. This letter
discusses the strongly-populated states. The reader is referred to the
companion paper for details on some other weakly-populated states [16].
Figure 2: Doppler-reconstructed addback $\gamma$-ray spectrum as detected in
coincidence with the scattered 38Ca nuclei that underwent a large momentum
loss. All $\gamma$-ray transitions are labeled by their energy. The inset
magnifies the high-energy region of the spectrum.
To construct the level scheme, $\gamma\gamma$ coincidences are used. Figure 3
shows the coincidence analysis of the low-energy part of the 38Ca spectrum.
From Fig. 3, it is clear that the 1489-keV $\gamma$ ray feeds the 2213-keV
line, the 489-keV transition feeds the level depopulated by the 1489 keV, and
the 214-keV transition lies on top of the level depopulated by the 489-keV
transition. There is evidence for a weak 1048-keV transition being in
coincidence with the 2213 and 1489 keV $\gamma$ rays.
Figure 3: Left: Doppler-corrected $\gamma\gamma$ coincidence spectra obtained
from cuts on the labeled prominent transitions in the $\gamma\gamma$
coincidence matrix. Background was subtracted via a cut of equal width at
slightly higher energy. Coincidence relationships are evident in the panels.
Right: Resulting level scheme. The width of the arrows is proportional to the
$\gamma$-ray intensity of the corresponding transition. The proton separation
energy of $S_{p}=4.54727(22)$ MeV [19] places the second $3^{-}$ state above
the proton separation energy. The $0^{+}_{2}$ state is shown but was not
populated in the present work.
Figure 3 also shows the partial level scheme with the intensities of the
$\gamma$-ray transitions indicated by the arrow widths. These relative
$\gamma$-ray intensities were deduced from the efficiency-corrected peak areas
from the spectrum displayed in Fig. 2. Remarkably, the fourth strongest
$\gamma$ ray, at 214 keV, has not been reported previously. The relative
intensities and a more complete level scheme, including all $\gamma$-ray
transitions observed, is provided in Ref. [16].
The 1489-keV transition in coincidence with the $2^{+}_{1}\rightarrow
0^{+}_{1}$ decay is consistent with the previously reported ($3^{-}$) state at
3702 keV. The 489-keV transition in coincidence with the 3702-keV ($3^{-}$)
state suggests a level at 4191 keV, which is consistent with a previously
reported state at 4194 keV. However, the $J^{\pi}$ assignment proposed in the
literature of ($5^{-})$ [15] is unlikely as the 489-keV transition in our work
is prompt, on the level of a few ps or faster as evident from the good
resolution and absence of a low-energy tail, which – if of $E2$ character –
would indicate a $B(E2;5^{-}\rightarrow 3^{-})$ strength exceeding the
recommended upper limit of 100 W.u. [17]. From comparison with the mirror
nucleus, 38Ar, which has a 4480-keV $4^{-}$ level with a sole transition of
670 keV connecting to the first $3^{-}$ state, resembling the situation
described here, we propose $J^{\pi}=(4^{-})$ for the 4191-keV state in 38Ca.
The new 214-keV transition feeding the ($4^{-}$) level establishes a state at
4405 keV which appears to correspond to the 4586-keV $5^{-}$ level in the 38Ar
mirror whose far-dominant decay is a 106-keV transition to the $4^{-}$. Based
on mirror symmetry, a ($5^{-}$) assignment is proposed here for the 4405-keV
level in 38Ca. This establishes
$(5^{-})\rightarrow(4^{-})\rightarrow(3^{-}_{1})\rightarrow 2^{+}_{1}$ as the
most intense cascade seen following the 38Ca inelastic scattering populated at
large momentum loss.
The next strongest populated level is the $2^{+}_{2}$ state at 3684 keV for
which only the transition to the ground state is observed here. A $0^{+}$
state at 4748(5) keV is claimed in 38Ca from the $(\hbox{${}^{3}$He},n)$
transfer reaction, however, with the suspicion of a doublet [15]. Due to the
transition to the $(3^{-}$) state, a $0^{+}$ assignment is excluded and the
level established here is tentatively assigned ($3^{-}_{2}$), consistent with
the 4877-keV $3^{-}_{2}$ level in the 38Ar mirror, which also decays
predominantly to the $2^{+}_{1}$ and $3^{-}_{1}$ states [15].
It is interesting to explore which low-lying levels have not been observed in
the present experiment. This is, most prominently, the $0^{+}_{2}$ state
reported at 3084 keV which would decay to the first $2^{+}$ state with a
871-keV transition [15]. There is no evidence for an appreciable presence of
that transition in Figs. 2 and 3 (the 871-keV transition would be 13 keV above
the background feature originating from neutron-induced background as
indicated in Fig. 2).
In the following, we discuss the configurations of the states observed. Many
properties of 40Ca and the surrounding nuclei can be interpreted relative to a
doubly-closed shell structure for the ground state of 40Ca with the $sd$ shell
filled and the $pf$ shell empty. The first excited state of 40Ca has
$J^{\pi}=0^{+}$ and is qualitatively associated with a four-particle four-hole
(4p-4h) state relative to the 40Ca closed-shell ground state [18]. We will use
$\Delta$, the number nucleons moved from $sd$ to $fp$ orbitals, to
characterize the structure of the states. In this notation, the 4p-4h states
in 40Ca have $\Delta=4$. (To remove spurious states, the $\Delta$ basis
includes all components associated with the $\Delta\hbar\omega$ basis
constructed in the $0s$-$0p$-$0d1s$-$0f1p$ model space).
In Ref. [20], a Hamiltonian was developed for these pure $\Delta$
configurations. This Hamiltonian served as the starting point for the new
Florida State University (FSU) Hamiltonian for pure $\Delta$ states [21, 22].
The $A=38$, FSU results are compared to experiment in Fig. 4, the overall
agreement with experiment being good. The calculated configurations can be
divided into those with $\Delta=0$ with positive parity (green), those with
$\Delta=1$ with negative parity (blue) and those with $\Delta=2$ with positive
parity (red).
Figure 4: Comparison of the energies of the low-lying states of 38Ca, with the
states observed here labeled, with shell-model calculations using the FSU
$spsdfp$ interaction, and states in 38Ar [15]. In these plots, the length of
the levels indicates the $J$ value and the color positive parity, $\Delta=2$
(red), negative parity, $\Delta=1$ (blue), and $sd$-shell origin, $\Delta=0$
(green).
In the present 38Ca level scheme, the strongest $\gamma$ rays come from the
$2^{+}_{1}$ state, which is predicted to be of $sd$-shell origin, and from
states with $\Delta$=1, including the highest $J^{\pi}=5^{-}$ level possible
for this $\Delta$. The $\gamma$-ray decay of the 2${}^{+}_{2}$ state is also
observed. In the 36Ar$(\hbox{${}^{3}$He},n)$ reaction in [23] this state is
found to have a strong $(f_{7/2})^{2}$ form factor which would come from
$\Delta=2$ configurations in the FSU spectrum. However, the 0${}^{+}_{2}$
state, which also has $\Delta=2$, was not populated.
In the following, we propose a view that puts the populated states within the
context of the observed high-momentum-loss reaction events. From the
approximately 200 MeV of energy loss in the reaction, and given that the
detected 38Ca are largely within laboratory scattering angles of 3-4∘, about
150 MeV must be dissipated in the 9Be nuclei, with a total binding of 58 MeV.
Thus, there must be disintegration of the target nucleus into a number of
energetic fragments. The emerging picture is then one of multiple nucleons
interacting in a single collision with the formation of complex multi-particle
multi-hole configurations, in contrast to the situation in far-less-
dissipative, surface-grazing collisions. We exclude scenarios where a 38Ca
projectile undergoes multiple collisions within the target as an explanation
for the observed cross sections. High-momentum loss events creating $mp$-$nh$
excitations in such a scenario would require a sequence of knockout and/or
pickup processes and such pickup mechanism cross sections are small – with a
typical upper limit of 2 mb at these beam energies [24].
Connecting to the shell-model picture, excitations within the FSU model space
are described by many-body transition densities. In the simplest scenario,
excitation of the $\Delta=1$ negative-parity states involve the $\Delta=0$ to
$\Delta=1$ one-body transition densities (OBTD). The OBTD to those states
observed are all large. The $\Delta=2$, 2${}^{+}_{2}$ state involves the
$\Delta=0$ to $\Delta=2$ two-body transition density (TBTD). The TBTD
connecting the $\Delta=0$ and $\Delta=2$ 0+ wave functions are the same ones
that enter into the Hamiltonian matrix for mixing these two states. We expect
that the microscopic, two-nucleon excitation mechanism should involve an
operator similar to that of the two-body mixing Hamiltonian (e.g. dominated by
pairing). This would explain why excitation of the 0${}^{+}_{2}$ is not
observed – the mixed 0${}^{+}_{1}$ and 0${}^{+}_{2}$ eigenfunctions are
orthogonal with respect to the two-nucleon excitation operator. We note that
in 40Ca($p,t$) [25] the $0^{+}_{2}$ state is only very weakly populated
compared to the $2^{+}_{2}$ state (see Fig. 1 in Ref. [25]).
The events at momentum losses of 600-700 MeV/c, studied here, are also
reminiscent of observations in the work of Podolyak et al. [30]. There, in the
two-neutron knockout from 56Fe to 54Fe at 500 MeV/nucleon, the population of a
10+ isomer of complex structure was observed in the low-momentum tail of the
parallel momentum distribution at about the same absolute momentum loss. The
authors attributed this population to the excitation of the $\Delta$(1232)
resonance at their relativistic beam energies. This mechanism is not available
to our intermediate-energy beams of tens of MeV/nucleon. One may speculate
that the population of the complex-structure state in the two-neutron knockout
from 56Fe is rather due to a simultaneous multi-nucleon rearrangement as
hypothesized here, without evoking quark degrees of freedom and consistent
with the reduction of multi-step processes at their relativistic energies. For
example, population of the $10^{+}$ state could be due to the $\Delta J=6$
excitation of a $(f_{7/2})^{2}$ $6^{+}$ configuration in 56Fe combined with
the removal of two neutrons from the $1p_{3/2}$ and $0f_{7/2}$ orbitals having
$\Delta J\geq 4$.
In Ref. [16], from the high-spin spectroscopy of states up to $J=15/2$ in
39Ca, we argue that such simultaneous multi-nucleon rearrangement is also at
play in intermediate-energy nucleon transfer reactions, such as
9Be($\hbox{${}^{38}$Ca}^{*},\hbox{${}^{39}$Ca}+\gamma$)X. Once again, these
excitations are seen in events in the tail of the longitudinal momentum
distribution at a momentum loss of 600-700 MeV/c.
In the present work, the specific reaction dynamics at play in the observed
large momentum loss collisions are unclear and remain a challenge for future,
more complete and exclusive measurements. Specifically, it would be critical
to detect the dissociation of the 9Be target nuclei in the large-momentum-loss
events and clarify the kinematics of the residues.
While there is much to be discovered about this type of reaction, it is
evident that this presents a new opportunity in the fast-beam regime which
uniquely complements classic low-energy reactions, such as multi-step Coulomb
excitation and multi-nucleon transfer. Fast beams allow for the use thick
targets and capitalize on an increase in $\gamma$-ray yield by a factor of
about 4300 for the specific example of a 188-mg/cm2 9Be target used here vs. a
1-mg/cm2 Pb target often employed for multi-step Coulomb excitation, for
example. Also, strong forward focusing enhances the collection efficiency as
compared to low-energy reactions that fill a larger phase space.
Multi-step Coulomb excitation studies with low-energy rare-isotope beams have
been performed at beam intensities similar to those used here, but have been
limited to a complementary level scheme selectively comprising cascades
connected by strong $E2$ transitions, with at most the first $3^{-}$ state
[26, 7]. We illustrate this with the example of the state-of-the art low-
energy Coulomb excitation of the neighboring Ca isotope 42Ca on Pb [27]. The
measurement was performed at 1 pnA stable-beam intensity for 5 days (resulting
in more than 110,000 times the number of Ca projectiles on target as in the
present measurement) – excited states up to the $4^{+}_{1,2}$ states were
reported with no evidence for any of the negative-parity cross-shell
excitations observed here.
Multi-nucleon transfer, largely limited to stable beams at pnA beam
intensities, is known to populate complex-structure states, however, without
efficiently reaching 38Ca in spite of 40Ca being an often-used beam (see [28]
and references within). When low-energy neutron-rich beams become available at
near stable-beam intensities, multi-nucleon transfer may become an alternative
to access such states in selected neutron-rich nuclei [29]. While it is
interesting to also extend our approach to collective nuclei, it already
promises to be a unique method to probe cross-shell excitations near magic
numbers, elucidating shell evolution in rare isotopes and exploring the
necessary model spaces for a region’s description on the quest for a
predictive model of nuclei.
In conclusion, the in-beam $\gamma$-ray spectroscopy is reported of higher-
spin, complex-structure negative-parity states in 38Ca populated in highly-
dissipative processes induced by a fast 38Ca projectile beam reacting with a
9Be target. This work constitutes the first high-resolution $\gamma$-ray
spectroscopy of 38Ca with a modern HPGe $\gamma$-ray tracking array. The final
states observed in the inelastic scattering,
9Be($\hbox{${}^{38}$Ca},\hbox{${}^{38}$Ca}+\gamma$)X, at large momentum loss
are characterized through their particle-hole character relative to the 40Ca
closed-shell ground state. Excellent agreement is obtained with shell-model
calculations using the FSU cross-shell effective interaction. Based on the
strongly populated negative-parity states and the non-observation of the first
excited $0^{+}_{2}$ state, we propose a consistent picture in which these
multi-particle multi-hole states are formed by simultaneous rearrangement of
multiple nucleons in a single, highly-dissipative collision. These reaction
processes, seen here in the extreme low-momentum tail of 38Ca+9Be inelastic
scattering, identify a new pathway to gain access to excited states not
usually observed in fast-beam induced reactions and likely out of reach for
low-energy reactions.
This work was supported by the U.S. National Science Foundation (NSF) under
Grants No. PHY-1565546 and PHY-2110365, by the DOE National Nuclear Security
Administration through the Nuclear Science and Security Consortium, under
Award No. DE-NA0003180, and by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under Grants No. DE-SC0020451 (MSU) and
DE-FG02-87ER-40316 (WashU) and under Contract No. DE-AC02-06CH11357 (ANL).
GRETINA was funded by the DOE, Office of Science. Operation of the array at
NSCL was supported by the DOE under Grant No. DE-SC0019034. J.A.T.
acknowledges support from the Science and Technology Facilities Council (U.K.)
Grant No. ST/V001108/1.
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|
# Modelling of Spintronic Terahertz Emitters as a function of spin generation
and diffusion geometry
Yingshu Yang School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore, Singapore Stefano Dal Forno School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore, Singapore Marco Battiato<EMAIL_ADDRESS>School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore, Singapore
###### Abstract
Spintronic THz emitters (STE) are efficient THz sources constructed using thin
heavy-metal (HM) and ferromagnetic-metal (FM) layers. To improve the
performance of the STE, different structuring methods (trilayers, stacked
bilayers) have been experimentally applied. A theoretical description of the
overall THz emission process is necessary to optimize the efficiency of STE.
In particular, geometry, composition, pump laser frequency, and spin diffusion
will be significant in understanding the pathways for further research
developments. This work will apply a generalized model based on a modified
Transfer Matrix Method (TMM). We will consider the spin generation and
diffusion in the FM and HM layers and explain the spintronic THz emission
process. This model is suitable for calculating emitted THz signal as a
function of FM and HM thicknesses for different geometrical configurations. We
will investigate a bilayer geometry as a test case, but the extension to a
multi-layer configuration is straightforward. We will show how the different
configurations of the sample will influence the THz emission amplitude.
††preprint: AIP/123-QED
THz (0.1 -30 THz) is a frequency range that is gaining popularity because of
its enormous potential in basic process studies of materials as it is able to
resonantly couple to conduction-electron transport, plasmons, excitons,
phonons, or magnons Kampfrath, Tanaka, and Nelson (2013); Seifert _et al._
(2022). It is also an effective tool in imaging, sensing for biomedical
purposes, and security applications Seifert _et al._ (2022); Feng _et al._
(2021); Agarwal _et al._ (2022a). THz radiation can be generated using
various methods, including photoconductive antennas and nonlinear optical
crystals Seifert _et al._ (2022). Spintronics THz emitters (STE), a new type
of THz source, based on spin-to-charge conversion, have gained popularity in
recent years due to their high efficiency, broad bandwidth, and ease of
manufacture. STE is usually constructed using two simple layers of
ferromagnetic-metal(FM), and heavy-metal(HM)Kampfrath _et al._ (2013);
Seifert _et al._ (2016); Wu _et al._ (2017). The mechanism at the core of
THz emission first involves the photoexcitation of the FM layer. This induces
a spin population that diffuses to the HM layer in a process called
superdiffusive spin transport Battiato, Carva, and Oppeneer (2010, 2012);
Battiato, Maldonado, and Oppeneer (2014); Battiato and Held (2016); Battiato
(2017); Carva, Battiato, and Oppeneer (2011); Carva _et al._ (2013). Here,
due to the high spin-orbit-coupling (SOC) of the HM, inverse Spin Hall effect
(ISHE) Seifert _et al._ (2016) will take place and produce a transverse
charge current responsible for the THz emission (as shown in Fig. 1(a)).
Figure 1: (a) Schematic of a spintronic THz emitter and the THz emission
process. (b) and (c) Schematics of a bilayer and trilayer systems with the
spin diffusion profile in the FM layer (purple line), the spin current
diffusion and reflection in the HM layer (blue line), and the energy
distribution profile of the laser pump along the system (red line and shaded
area).
The study of STE optimization has recently received much attention in order to
broaden their applications. As a result, there have been different
experimental studies carried out to test the THz generation efficiency based
on different materials of the structureSeifert _et al._ (2016, 2018); Qiu
_et al._ (2018); Liu _et al._ (2022), different thicknesses of the layersZhou
_et al._ (2018); Torosyan _et al._ (2018); Qiu _et al._ (2018); Wu _et al._
(2017), and even different stacks of layersFeng _et al._ (2018); Cheng _et
al._ (2021); Seifert _et al._ (2016); Cheng _et al._ (2019); Herapath _et
al._ (2019). To analyze the performance of the STE based on the above
approaches, a theoretical model to describe the STE geometry and THz emission
process is needed. Models based on basic THz emission processes involving the
pump-pulse absorption, spin generation, and spin to charge conversion have
been developed and used in different analysis Seifert _et al._ (2016);
Torosyan _et al._ (2018). These models usually calculate the THz emission
amplitude assuming the spin current amplitude to be proportional to the
excitation energy deposition Seifert _et al._ (2022) and at the same time
assuming the FM/HM interface is transparent to spins and electrons Seifert
_et al._ (2022, 2016). However, no general model has considered the detailed
geometry of the excitation laser and spin current with specific STE structures
in the THz emission process.
In a typical experimental setup, STEs are constructed by combining substrate,
FM, and HM layers in a number of different arrangements. The ordering of the
layers, their thicknesses, and the side illuminated by the laser pump will
impact the geometry of the spin generation, the spin diffusion, and the energy
profile of the laser pump. The model we applied here addresses all of the
above properties on the same footing. Specifically, we want to describe the
three fundamental processes that characterize any STE: the propagation of the
laser excitation pump through the system, the generation and diffusion of
spins in the HM and FM layers, and the generation and propagation of the THz
pulse throughout the system. These processes are controlled by a number of
parameters, namely, the type of materials, their thicknesses, the arrangement
of the layers, the frequency of the pump, the spin diffusion length, and the
degree of spin reflections between interfaces. Fig.1(b) and (c) show
schematics of the processes mentioned above in both bilayer and trilayer
systems. The dependence of the outgoing THz pulse strongly depends on the
choice of these parameters. Our findings show that properly fitting the
material properties in optical and THz regions is necessary to achieve
quantitative predictions.
## I Methods
To properly model STEs, we have to describe the three sub-processes that occur
in the THz emission: (1) Propagation and absorption of the optical pump into
the STE’s layers; (2) Production of the spin current, its propagation through
the STE’s layers and the eventual conversion into a transversal charge
current; and (3) Production, propagation and extraction from the STE of the
THz electromagnetic radiation. The following sections will address each of
these tasks individually.
### I.1 Pump laser absorption profile
We describe the optical EM waves propagation and absorption using a Transfer
Matrix Method (TMM) Coutaz, Garet, and Wallace (2018); Born and Wolf
(2013).The system is composed by three layers: a Substrate, a FM layer and a
HM layer, in different stacking orders. According to the standard Transfer
Matrix Method, the transmission and reflection of the waves throughout an
layer sample can be expressed as,
$\begin{bmatrix}f_{\infty}^{>}\\\
f_{\infty}^{<}\end{bmatrix}=\bar{\bar{T}}_{[0,\infty]}\begin{bmatrix}f_{0}^{>}\\\
f_{0}^{<}\end{bmatrix},$ (1)
where $\bar{\bar{T}}_{[0,\infty]}$ is the frequency-dependent $2\times 2$
transfer matrix that propagates the fields from the beginning to the end of
the multilayer, and $f_{0}$, $f_{\infty}$ represent the field amplitudes at
the beginning and the end of the multilayer. The superscripts > and <
represent the right and left propagating waves respectively. Assuming the pump
pulse is impinging on the sample from the left, $f_{0}^{>}$ is the time
profile of the pump (which we assume known). Generally, there will be no
second pump incoming from the right, so $f_{\infty}^{<}=0$. The two remaining
field amplitudes $f_{\infty}^{>}$ and $f_{0}^{<}$, which represent the
transmitted and reflected waves respectively, are the unknowns in the system
in Eq. 1. We remind that the system of two equations in Eq. 1 is to be solved
for every frequency independently.
The generation of the spin currents and the subsequent diffusion of the spins
in the FM and HM layers strongly depends on how the energy deposited by the
laser pump is partitioned through the system Zhou _et al._ (2018); Torosyan
_et al._ (2018). According to the Poynting theorem, the total energy loss due
to Joule effects can be expressed as,
$Q_{loss}=-\int_{-\infty}^{+\infty}\mathrm{d}t\oint_{S}(\mathbf{E}\times\mathbf{H})\cdot\mathrm{d}\bm{S}.$
(2)
Here, $Q_{loss}$ is the total energy dissipated by a system enclosed by the
closed surface $S$. Because of the planar symmetry of STE, the surface $S$ can
be chosen to be a parallelepiped enclosing the layer. At normal incidence,
only faces of $S$ parallel to the interfaces between layers contribute to the
integral in Eq. 2. Hence, we can define the energy per unit area that crossed
a surface $S_{z}$ at position $z$ as,
$\Phi(z)=-\int_{-\infty}^{+\infty}E[t,z]H[t,z]\mathrm{d}t,$ (3)
where $z$ is the position of the surface $S_{z}$. Thus, if we assume the two
interface position of a layer with thickness $d$ in a multilayer system as
$z_{0}$ and $z_{0}+d$, and assuming the initial input of the pump laser as
$Q_{in}$, the net absorption for this layer will be:
$A_{layer}=\frac{Q_{loss}}{Q_{in}}=\frac{\Phi(z_{0})-\Phi(z_{0}+d)}{Q_{in}}.$
(4)
where $Q_{in}$ can also be calculated using Eq. 3 with fields at the initial
surface of the system by ignoring the reflected fields ($f_{0}^{<}$ in Eq. 1).
This net absorption can be used when simulating STEs with fixed FM
thicknesses. Similar ideas can be used to obtain the energy distribution
profile. If we assume a local axis within one single layer, the energy
distribution profile can be expressed as,
$D(z)=-\frac{d(\Phi(z)/Q_{in})}{dz}$ (5)
The energy distribution profile of Eq.5 is schematically displayed at the
bottom of Figs. 1(b) and (c) with a red solid line.
### I.2 Spin generation in the FM layer
We now address how the absorbed energy in the FM layer produces a spin current
profile over the whole sample. We focus first on how energy deposited between
$z$ and $z+dz$ within the FM layer propagates through the whole sample. We
assume that the spin current spatial distribution generated by an
infinitesimally thin layer and in the absence of interfaces is in the form of
an exponential decay, with an effective spin diffusion length $\lambda$, both
towards the left and the right of the emission plane (See Fig. 1). However due
to the finite thickness of the FM layer, as well as the presence of the other
layers, we must explicitly consider the reflections and the transmissions of
the spins at the layers’ boundaries.
We call $\alpha$ and $\beta$ the probabilities for a spin to be transmitted
over interfaces to the right and the left of the considered spin current
emission point $z$. Such probabilities depend on the materials and the quality
of the interfaces. We assume that the probability of transmission at the
interface with air or with the substrate (assumed insulating, with
sufficiently large bandgap to prevent injection of the excited spin currents)
to be 0. The rate at which the spin population is transmitted to the left or
to the right can be calculated by exploiting the properties of geometric
series (see Suppl. Info. for the detailed derivation). By summing up the
contributions of all the multiple reflections at the boundaries, we obtain the
following expressions
$\displaystyle\zeta_{R}(z)$
$\displaystyle=\frac{\alpha}{N}\left[e^{-\frac{d-z}{\lambda}}+\bar{\beta}e^{-\frac{d+z}{\lambda}}\right],$
(6) $\displaystyle\zeta_{L}(z)$
$\displaystyle=\frac{\beta}{N}\left[e^{-\frac{z}{\lambda}}+\bar{\alpha}e^{-\frac{2d-z}{\lambda}}\right],$
(7)
where $N={1-\bar{\alpha}\bar{\beta}e^{-\frac{2d}{\lambda}}}$,
$\bar{\alpha}=1-\alpha$, $\bar{\beta}=1-\beta$. Eqs. 6 and 7 measure the
number of spins (generated at position $z$) that have crossed (transmitted)
the right or left interfaces of the FM layer, respectively. We call $\zeta(z)$
the spin generation efficiency profile function (purple solid lines in Fig.
1).
Finally, to compute the effective number of spins that is injected from FM to
HM, we must consider the effects of the laser pump. We assume that the spin
density is proportional to the laser fluence at the point $z$ times the left
or right spin generation efficiency profile. The total spin population becomes
$S(d)=\int_{z_{i}}^{z_{i}+d}D(z)\zeta(z)dz,$ (8)
where $D(z)$ is the energy (per unit area) deposited by the laser pump (Eq.
5), $z_{i}$ is the position of the FM layer and $\zeta(z)$ must be chosen
either to be right or left, depending on the geometrical configuration of the
system. For example, if we consider the bilayer THz emitter shown in Fig.1(b),
we need to calculate only the transmittance to the right interface. However,
if we want to model the trilayer case shown in Fig.1(c), then the
transmittance for both the left and right interfaces have to be considered.
### I.3 Spin current profile in the HM layer
After the laser pump has excited the spin population in the FM layer, spins
start diffusing and are eventually injected at the edge of the HM layer. We
again assume an exponentially decaying spatial distribution with an effective
spin diffusion length $\lambda$. We call $\gamma$ and $\mu$ the probabilities
for a spin to be transmitted over the right and the left interfaces,
respectively. We want to calculate the average number of spins at a given
position $z$ inside the HM layer. Again, by exploiting the properties of
geometric series (see Supp. Info.)Agarwal _et al._ (2022b), we can sum up the
contributions of the multiple reflections and obtain the following
$\sigma(z,d)=\frac{1}{N}\left[e^{-\frac{z}{\lambda}}+\bar{\gamma}e^{-\frac{2d-z}{\lambda}}\right],$
(9)
where $N=1-\bar{\gamma}\bar{\mu}e^{-\frac{2d}{\lambda}}$, $d$ is the thickness
of the HM layer and $\bar{\gamma}=1-\gamma$, $\bar{\mu}=1-\mu$. The above
equation measures the spin current density at position $z$ for a given HM
layer of thickness $d$ (see Fig. 1 light blue curves). The transversal charge
current density is finally obtained by multiplying by the inverse spin Hall
coefficient of the HM, similarly to our previous work. Yang, Dal Forno, and
Battiato (2021a, b)
Two other mechanisms of generation of transversal charge curret are also
possible. The first one is caused by spin-to-charge current conversion in the
FM layer. This contribution is usually much smaller than the HM layer
contribution due to the low spin to charge conversion efficiency of FM
layerSeifert _et al._ (2022) and will be neglected in this work. A second
mechanisms involves the creation of hot electrons in the HM layer. These hot
electron will diffuse back to the FM layer and act as a secondary excitation
of the FM layer. This enhancement will become large when HM layer is thick
Agarwal _et al._ (2022b). We stress that the above contributions, even if
ignored in most cases, can be described by our model (but it is not included
in this work for simplicity). The FM contribution can be included by adding a
source layer to the modified TMM model developed in Sec.I.4. The secondary
enhancement from the HM layer can also be added by specifically calculating
the absorption of the pump-pulse in the HM layer, which can be calculated
using Sec.I.1. As we are more interested in the performance of the main
contribution, we show results of the HM layer emission only in the following
section. We stress that the full model of including all three contributions is
straightforward. However, specifying the percentage of each contribution in
experiment requires detailed material data fitting (both THz frequencies and
pump laser frequencies) to increase the accuracy.
### I.4 THz radiation production and propagation
The final step is computing the THz radiation extracted from the STE. This
requires the computation of the production of the THz within the HM layer as
well as its propagation though the multilayer. In this case, standard TMM
cannot be used. In the case one of the layers act as a source of
electromagnetic radiation (by mean of a time and position dependent volume
current) we use the modified TMM, which we call TMM-with-source, that we
developed in Ref. Yang, Dal Forno, and Battiato, 2021b. The expression that we
obtained maintains the structure of the TMM, but includes a source term
$\begin{bmatrix}f_{\infty}^{>}\\\
f_{\infty}^{<}\end{bmatrix}=\bar{\bar{T}}_{[0,\infty]}\begin{bmatrix}f_{0}^{>}\\\
f_{0}^{<}\end{bmatrix}+\begin{bmatrix}J^{>}\\\ J^{<}\end{bmatrix}.$ (10)
where $J^{>}$ and $J^{<}$ are the amplitudes of the right and left propagating
fields generated in the source layer and account for time and position
dependent charge currents within the multilayer. In this work, the position
dependent profile used for the construction of the source term is taken as Eq.
9 and the time dependent profile is taken as a similar shape as used in Ref.
Yang, Dal Forno, and Battiato, 2021b.
It should be noticed that, while in the case of TMM applied to the absorption
of the optical pump laser, in the case of TMM-with-source for the emitted THz
radiation the constraints on the field amplitudes on the right and left of the
sample are different. In this case, no external THz pulse is sent to the
multilayer, and therefore $f_{\infty}^{<}$ and $f_{0}^{>}$ are to be set to
zero. This means that in a THz emission process the only source is the THz
generated from the HM layer and Eq. 10 can be solved for the $f_{\infty}^{>}$
and $f_{0}^{<}$ amplitudes. To be noticed that if we want to describe a
trilayer system with two HM layers acting as sources (case shown in Fig.1(c)),
the calculation will be straightforward with two additional terms describing
fields coming from two different layers.
Figure 2: The schematic of the four different configurations of STE and its
corresponding emission map. (a-1)–(d-1) Different configurations of STE
structure and a schematic show of the energy distribution profile (red shaded,
Eq.5), overall spin generation efficiency profile (blue shaded, Eq.7), and
spin current diffusion profile (green shaded, Eq.9) for each configuration.
(a-2)–(d-2)The THz emission map with changing Pt(HM) (0-15nm) and
Co(FM)(0-15nm) thicknesses for four different configurations.The yellow dashed
line represents the peak position of the THz emission profile as a function of
HM thickness, while the blue dashed line represents the peak position of the
THz emission profile as a function of HM thickness. Figure 3: The linear fit
of THz emission profile peak positions with changing theoretical Spin
diffusion length of FM and HM under different conditions. The first row,
(a)-(c), shows the peak position change of the THz emission profile as a
function of HM thickness when the theoretical spin diffusion length for the HM
layer changes from 1-6nm with the plasma frequency of HM layer for THz range
material properties is changing from 3-7eV, and the extinction coefficient for
optical range material properties is changing from 4-12. The second row,
(d)-(f), shows the peak position change of the THz emission profile as a
function of FM thickness when the theoretical spin diffusion length for the FM
layer is changing from 1-6nm with the plasma frequency of FM layer for THz
range material properties is changing from 3-7eV, and the extinction
coefficient for optical range material properties is changing from 4-12.
## II Results
In the following, we present the calculated THz amplitudes as a function of
geometry, layer arrangements, materials, and laser pump frequency. In Fig.
2(a-1)–(d-1), we show the four different layer arrangements we considered in
this work and label them C1, C2, C3, and C4, respectively.
We start with a simple STE structure constructed with quartz (substrate, 1mm),
Co (FM), and Pt (HM, 0–15nm). The material properties of Co and Pt at THz
frequencies are taken from Ref. Ordal _et al._ , 1985, while at optical
frequencies taken from Ref. Werner, Glantschnig, and Ambrosch-Draxl, 2009. The
optical properties for quartz are taken from Ref. Gao, Lemarchand, and
Lequime, 2012 and at the THz range are taken as the experimentally measured
dielectric constants in Ref. Agarwal _et al._ , 2022b. We take the Pt and Co
spin diffusion length to be 1.1 nm and 1nm, respectively,Seifert _et al._
(2016); Torosyan _et al._ (2018); Zhou _et al._ (2018); Zhang _et al._
(2013) as a test case.
Figs. 2(a-2)-(d-2) show the calculated THz emission peak amplitude for
different HM and FM thicknesses for the C1, C2, C3, and C4 configurations. One
can notice that at parity of FM thickness, the emitted THz intensity increases
at first, as more HM thickness allows for more efficient spin-to-charge
conversion. However, eventually, further increases do not provide further gain
in the THz intensity but become detrimental as larger metallic regions lead to
absorption of the generated THz radiation within the sample itself. This
reproduces the known fact that the THz intensity peaks for relatively thin
layers for both changing HM thickness and FM thickness.Seifert _et al._
(2016, 2022); Cheng _et al._ (2021); Torosyan _et al._ (2018)
The THz emission amplitude depends on the arrangement of the layers. We
observe that exciting from the substrate side produces stronger THz emission
compared to exiting from the active bilayer side (C1 and C2 $>$ C3 and C4).
The reason is that in the second cases, the produces THz radiation has to
traverse the quartz substrate, which absorbs in that frequency range. This
reproduces experimental findings on quartz substrates Agarwal _et al._
(2022b). The situation is reversed in the case of a sapphire substrate
(material properties taken from Ref. Sanjuan and Tocho, 2012 for THz range and
Ref. Kelly, 1972 for optical range), where instead absorption of the pumping
radiation in the substrate becomes more important. Experiments confirm this
scenario.Liu _et al._ (2021)
Apart from intensity, we can also extract the behaviour of the emitted THz
with changing layers’ thicknesses. Although configurations C1 and C2 display
similar maximum THz intensities, their behaviour with layers’ thicknesses is
different. We can observe that the most relevant characteristic controlling
these dependences is which one of the two active layers (HM or FM) faces the
pumping laser (see Fig. 2). Each layer has three key effects in the THz
production process. The FM acts as the generator of spin current, as absorber
of optical photons, and absorber of THz radiation, while the HM as spin-to-
charge converter and, again, as absorber of both optical and THz frequencies.
The thickness dependence of all this processes depends on the material
properties. Yet the relative weights of each effect in each layer depends on
the relative positions. For instance the role of optical photons absorber is
higher in the first layer traversed by the pump pulse.
One further interesting finding is that the peak positions of the THz emission
with changing FM (or HM) layer thickness are not fixed (yellow dashed line for
HM peak position, blue dashed line for FM peak position) and they depend on
the thickness of the adjacent layer. However, the optimal HM thickness at
which the THz emission is maximal is generally used in experiments as a quick
and quantitative estimation of the spin diffusion length and vice
versaPapaioannou and Beigang (2021); Seifert _et al._ (2022); Zhou _et al._
(2018); Torosyan _et al._ (2018); Seifert _et al._ (2016). For that to be a
meaningful estimation, the peak position should only depend on the spin
diffusion length and not be affected by other characteristics of the sample.
However, this contradicts our findings.
To understand this better, we performed a larger set of calculations where we
compare how those peak positions compare with the actual spin diffusion length
when other parameters are changed. In Figs. 3(a)-(c) we show on the y axis the
thickness of the HM layer for which we obtain the strongest THz emission,
while on the x coordinates the spin diffusion length in the HM used in the
calculations. To really claim that the peak HM thickness can be used as an
estimation of the spin diffusion length, one should require the points to be
over, or close enough to, the $y=x$ line (blue dashed line in Figs), or more
generally have a stable functional dependence unaffected by other layer’s
properties. However, while a sufficiently linear relationship can be found
between the two quantities, important deviations can be observed from the
desired correlation.
Yet even more crucially, the relationship between the two quantities is very
strongly dependent on the other parameters of the system. In Fig. 3(a) the
thickness of the FM layer is shown to impact the peak position. For instance
if the measured HM peak thickness were $4$nm, the extrapolated spin diffusion
should be $3$nm if the FM is $10$nm thick, or twice as large for a sample with
a $1$nm thick FM. One could still argue that the FM thickness is generally
known and one could couple experimental results to theory to do more reliable
estimations of the spin diffusion length. However Figs. 3(b) and (c) show that
that will require a very careful characterisation of the sample. A change in
the dielectric properties of the HM (THz plasma frequency $\omega_{p}$ and
optical extinction coefficient $k$) can, in fact, strongly impact the
relationship (in particular Fig. 3(c) shows the huge change in the dependence,
with increasing extinction coefficient).
Similarly, we compute the FM layer thickness at which one obtains the highest
THz emission at parity of HM thickness. It might be tempting to use that to
estimate the spin current diffusion length in the FM. However Figs. 3(d)-(f)
again show that the correlation between the two quantities is far too strongly
dependent on other properties of the multilayer.
Figure 4: Three different tests of THz emission. (a) THz emission profile for
the same Sub/HM/FM configuration with different FM material. (b) THz emission
profile for the same Sub/HM/FM configuration with different HM material. (c)
THz emission profile for the same Sub/Pt/Fe configuration with different
excitation lasers.
We now move onto analysing the dependence of the THz emission on the used
materials and pump laser frequency. We choose configuration C1 as the test
case and describe three material combinations. We set the HM material as Pt
and Pd and the FM material as Co and Fe (material properties for pump laser
frequencies taken from Ref. Werner, Glantschnig, and Ambrosch-Draxl, 2009 and
for THz frequencies from Ref. Ordal _et al._ , 1985.) We set the FM
thicknesses to 3nm, the spin diffusion length $\lambda=1.1$nm for the HM, and
$\lambda=1$nm for the FM cases. Fig. 4(a) and (b) show the calculated THz
amplitudes. Changes are direct consequence of the different optical properties
of the materials. Fig. 4(c) shows the dependence of the THz pulse for two
different pump laser wavelengths (400nm and 800nm). We see that, for the given
thicknesses and $\lambda$, the THz emission profile changes. Specifically, the
400nm emitted THz amplitude is higher than 800nm one. This is in qualitative
agreement with experiments Adam _et al._ (2019).
Figure 5: The comparison of THz emission profile with changing thicknesses of
HM between experimental data and theoretical calculation when spin reflection
at FM/HM interface is 20% and 100%. (a) Comparison for sapphire/Co/Pt
structure with data taken from Zhou _et al._ (2018). (b) Comparison for
MgO/Fe/Pt structure with data taken from Torosyan _et al._ (2018).
Finally, we show another interesting finding for the THz emission profile at
low HM layer thicknesses. From the generalized spin diffusion model (Eq. 9) in
the HM layer, we know that the spin reflections at the interfaces will play a
more significant role at lower compared to higher thicknesses. To see the
influence of the spin reflections, we calculated two sets of THz emission
profiles as a function of HM layer thicknesses. Then, we compared them to two
different sets of normalized experimental data for
Sapphire(1mm)/Co(3nm)/Pt(xnm) and MgO(0.5mm)/Fe(12nm)/Pt(xnm) samples taken
from Refs. Torosyan _et al._ , 2018; Zhou _et al._ , 2018.
In these two sets of calculations, we took the spin reflection percentage as
20% and 100% for the FM/HM interface and compared them to the experimental
data ($\bar{\mu}=20\%$ and $\bar{\mu}=100\%$ in Eq. 9). We can see that when
20% of spins are reflected (80% spins transmitted), the profile shows a much
better fit to the experimental data at lower thicknesses. This can be evidence
to show that one possible reason for the appearance of a positive second
derivative at low thicknesses in the experiment comes from the spin
reflections at the interfaces in the HM layer. Hence, it is crucial to
consider the spin reflections when dealing with low HM thicknesses.
## III Conclusion
We built a theoretical model that included a spin diffusion profile in HM, a
spin generation efficiency profile in FM, and an excitation laser energy
profile in FM with the modified TMM to describe the spintronic THz emission.
The thicknesses of the layers and the substrate, the material choice, the
layer arrangement, and the pump laser frequency all affect the THz emission
profile. Using this model, we showed that the peak position of the THz pulse
as a function of the HM thickness is dominated by the spin diffusion length.
However, other contributions, such as laser absorption and layer arrangements,
play a non-negligible role. We also showed that an accurate description of the
spin current reflection at interfaces is important to achieve accurate
predictions at low thicknesses ($d_{HM}<4nm$). This is important because the
THz signal usually peaks within this thickness range . Finally, we showed that
a proper experimental fitting needs to be done to extract reliable information
about the STE, such as spin diffusion length for both HM and FM, spin
transmission and reflection percentage at each interface, and THz emission
efficiency.
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|
# Solution to the Number Rotation Puzzle
Thomas Lam Carnegie Mellon University<EMAIL_ADDRESS>
###### Abstract.
The Number Rotation Puzzle (NRP) is a combination puzzle in which the goal is
to rearrange a scrambled rectangular grid of numbers back into order via moves
that consist of rotating square blocks of numbers of fixed size. Over all
possible boards and rotating block sizes, we find all solvable initial
configurations and provide algorithms to solve such configurations. For
sufficiently large board and rotating block sizes, solvability conditions
depend only on parity restrictions, with special additional conditions for
smaller sizes. One special case leads to a novel construction of the exotic
outer automorphism on $S_{6}$.
2010 Mathematics Subject Classification. Primary 05E99; Secondary 00A08
Key Words. combination puzzle, three-cycle, exotic automorphism
## 1\. Introduction
The Number Rotation Puzzle (NRP) is a combination puzzle in which the goal is
to rearrange a scrambled square grid of integers back into ascending order.
Numbers can be moved by rotating square blocks of numbers by some multiple of
$90^{\circ}$. The game is most commonly played on a $3\times 3$ board of the
integers $1$ to $9$, with $2\times 2$ rotating blocks. We call this the
standard NRP. See Figure 1 for an example of a game played on the standard
NRP.
Figure 1. A two move solution to a sample standard NRP.
In addition to the standard NRP, other variations exist, such as those with
large board size and rotating block size, e.g. a $5\times 5$ grid with
$3\times 3$ rotating blocks, and those with rectangular boards.
The NRP has been implemented on Nokia phones and a few puzzle game websites,
most notably Simon Tatham’s Twiddle (see [4]). There has been very little
research done on the NRP. In fact, it has no “official" name. Most researchers
focus on computational methods for solving the standard NRP. For example, Lee
programmed an AI using decision matrices to solve the standard NRP [2], and
Wang and Song developed an efficient brute-force algorithm to find solutions
to the standard NRP [5]. However, one of the more outstanding feats in the
research of the NRP is Fernando’s solution to the $n\times n$ NRP with
$(n-1)\times(n-1)$ rotating blocks, which was in turn used to bound God’s
Number — the least number of moves required to solve a combination puzzle — of
the NRP [1].
The focus of this paper will be solving the NRP on rectangular boards, and
thus generalizing the work of Fernando. By solve, it is meant that all
possible solvable initial configurations are identified, and a logical solving
algorithm is developed that will solve all such configurations to prove that
they are indeed solvable.
Particularly, our main result is as follows:
###### Theorem 1.1.
Consider the $n\times m$ board with $b\times b$ rotating blocks, where $n$ is
the number of rows and $m$ is the number of columns.
* •
If $b=n=m$, then there are exactly four solvable initial configurations, each
a rotation of the solved board. Assume in the following criteria that this
case does not occur.
* •
If $b=1$, then the only solvable initial configuration is the solved board.
* •
If $b=2$, then there are two further cases.
* –
If $(m,n)=(2,3),(3,2)$, then there are only 120 solvable configurations.
* –
Otherwise, all initial configurations are solvable.
* •
If $b=3$, then there are two further cases.
* –
If $(m,n)=(3,4),(4,3)$, then there are only $6!$ solvable initial
configurations. In fact, solving one parity will solve the other.
* –
Otherwise, all initial configurations whose numbers lie in their correct
parity set are solvable.
* •
If $b\geq 4$, then case on the residue of $b$ modulo 8.
* –
If $b\equiv 2,6\pmod{8}$, all initial configurations are solvable.
* –
If $b\equiv 0,4\pmod{8}$, the solvable boards are those that are even
permutations of the solved boards.
* –
If $b\equiv 3,\ 5\pmod{8}$, the solvable boards are those that are even
permutations of the solved boards such that each number lies in its correct
parity.
* –
If $b\equiv 1,\ 7\pmod{8}$, the solvable boards are such that each parity is
an even permutation of their respective solved states.
We will also reveal a surprising construction of the exotic outer automorphism
on $S_{6}$ in the special case $(n,m,b)=(3,4,3)$.
Through this research, it is hoped that this puzzle be popularized in light of
its intuitive mechanics.
## 2\. Preliminaries
We denote by $(n,m,b)$ the NRP on an $n\times m$ board with $b\times b$
rotating blocks, where $n$ denotes the number of rows and $m$ denotes the
number of columns.
We denote moves by capital letters, and typically such a move is a
$90^{\circ}$ counter-clockwise rotation of a block. If $X$ is such a move,
then $X^{-1}$ is the $90^{\circ}$ clockwise rotation of said block, and
$X^{2}$ is the $180^{\circ}$ rotation of said block. We read algorithms from
left to right, so that e.g. $XYZ$ denotes executing $X$, $Y$, then $Z$. For an
algorithm $A$ we write $A^{n}$ to denote $A$ repeated $n$ times, and write
$A^{-1}$ to denote the inverse algorithm of $A$.
We also impose a coordinate system with $(i,j)$ denoting the square in the
$i$th row from the top and the $j$th column from the left.
## 3\. Parity Restrictions
We let the case in which $m,n,b\geq 5$ with $m\neq n$ be the general case. We
start by proving that our claimed conditions for solvability are necessary in
the general case. These conditions come from three parity restrictions (PRs):
1. (1)
Parity of a square
2. (2)
Even permutations
3. (3)
Even permutations within a parity of squares
The “parity of a square" PR simply refers to the invariance of the color of a
square in which a number lies if the board is colored like a checkerboard.
Clearly this invariance occurs exactly when $b$ is odd, hence for odd $b$ we
have that each of the numbers in the board must lie in the correct square
parity in order to have solvability. This gives us the first PR.
For some values of $b$, moves will execute even permutations, which indicates
that solvable boards must be even permutations of the solved state. To
identify such $b$, we take cases:
* •
If $b$ is even, then a move will move $b^{2}$ numbers in $b^{2}/4$ 4-cycles.
Since 4-cycles are odd, we have that the permutation introduced by a move is
even iff $b^{2}/4$ is even, and this occurs exactly when $b\equiv 0\pmod{4}$.
* •
If $b$ is odd, then each move will move $b^{2}-1$ numbers because the center
of rotation stays in place. Again, it follows that moves execute even
permutations iff $\frac{b^{2}-1}{4}$ is even, and some simple casework shows
that this occurs for all odd $b$.
Hence for $b\equiv 0,1,3\pmod{4}$, solvable boards must be even permutations,
giving the second PR.
The third PR is a combination of the first two. For odd $b$, each move
executes a permutation that can be decomposed into a permutation of one parity
and a permutation of the other parity. If these permutations are even, then
solvable boards must have each of their parities be even permutations of their
respective solved states.
Since a $b\times b$ rotating block moves $\frac{b^{2}-1}{2}$ numbers within
each parity, we see as before that a move is an even permutation within a
parity iff $\frac{b^{2}-1}{8}$ is even. It is not hard to show that this
occurs exactly when $b\equiv 1,7\pmod{8}$, giving the third PR and completing
the proof of the necessary direction in the general case.
## 4\. General Solving Algorithm
We now show that these conditions are sufficient for solvability in the
general case by constructing a solving algorithm. We claim that it is
sufficient to prove that
* •
for even $n$, we may 3-cycle any three numbers, and
* •
for odd $n$, we may 3-cycle any three numbers lying in a common parity.
To see this, let us consider each of the cases described in the PRs.
* •
If $n\equiv 0\pmod{4}$, then the board must be an even permutation, and since
we can execute any 3-cycle, we clearly can solve the board.
* •
For $n\equiv 2\pmod{4}$, the board may be an odd permutation. In this case,
simply rotating any block $90^{\circ}$ will bring the board to an even
permutation, in which case we can solve the board since we can execute any
3-cycle.
* •
For $n\equiv 1,7\pmod{8}$, each parity must be an even permutation of their
respective solved states. Since we can compute any 3-cycle of numbers of the
same number, it follows that we can execute any even permutation of a parity,
hence the solvability.
* •
For $n\equiv 3,5\pmod{8}$, the board must be an even permutation. Looking at
each parity, they are either both even permutations or both odd permutations
of their respective solved states. If the latter, then rotating any block
$90^{\circ}$ must bring both parities to even permutations, from which we
deduce solvability as in the previous case.
In practice, a solving algorithm would involve first executing a move if
needed to bring the board to an even permutation, and then using 3-cycles to
solve one number at a time until the entire board is inevitably solved.
Next, we claim that it is sufficient to solve the case in which $m=n+1$ and
$b=n$, i.e. the $(n,n+1,n)$ NRP for $n\geq 5$. One way to argue this is as
follows: For the $(M,N,n)$ NRP with $M\geq n$ and $N\geq n+1$, we may 3-cycle
any three numbers for $n$ odd, and 3-cycle any three numbers on the same
parity for $n$ even, by executing 3-cycles within the $n\times(n+1)$ sub-
boards in which each of the three numbers lie until they lie in a common
$n\times(n+1)$ sub-board. Then we may 3-cycle them and undo the previous
intermediate moves, resulting in the desired 3-cycle.
For the $(n,n+1,b)$ NRP, there are only two rotating blocks that may be used.
We denote rotation of the left and right rotating blocks counter-clockwise
$90^{\circ}$ by $X$ and $Y$, respectively. To prove that any three numbers may
be 3-cycled, subject to parity restrictions, we will first prove that there
exists an algorithm cycling some three numbers.
### 4.1. Existence of a 3-cycle Algorithm
We define an extremely useful algorithm, called cycle and denoted by $C$, as
$C:=(YX)^{4}=YXYXYXYX.$
The importance of $C$ is that it does not move many numbers. Imagine that the
numbers along the leftmost column, the bottom-most row, and the rightmost
column form a sort of looped “conveyor belt", with $(1,n+1)$ looping back to
$(1,1)$. Then $C$ will “cycle" the numbers on the conveyor belt counter-
clockwise a total of $n-3$ squares along the belt (See Figure 2).
Figure 2. The algorithm cycle in action on the $6\times 7$ board. Left:
Initial board. Right: The board after execution of $(YX)^{4}$. Since $n=6$,
all numbers in the belt (outlined in bold) are cycled $6-3=3$ squares counter-
clockwise.
To see that $C$ indeed executes this sort of behavior, first note that the
numbers not in the belt form a square which is rotated $90^{\circ}$ with every
execution of $YX$, hence these numbers’ positions are unaffected after cycle.
Then, observe that the cyclic order of the numbers on the belt is invariant.
After a $YX$, the number at $(1,1)$ will move to $(n,1)$. Thus, the belt is
advanced $n-1$ squares counter-clockwise after each $YX$. It follows that
$(YX)^{4}$ advances the belt a total of $4n-4$ squares counter-clockwise.
Since there are only $3n-1$ squares on the belt, this is a net advancement of
$n-3$ squares counter-clockwise.
We are now ready to define an algorithm $\varphi$ that executes a 3-cycle. If
we let $A=XYYXY^{-1}X^{-1}$, then we take
$\varphi:=C(ACA^{-1})C^{-1}(AC^{-1}A^{-1}).$
The motivation is that the sequence $ACA^{-1}$ essentially “applies cycle to a
different set of squares", so in some sense $\varphi$ is simply commuting two
different instances of cycle. If the squares moved by $C$ and $ACA^{-1}$
intersect at exactly one square, then it is not hard to see that $\varphi$
must be a 3-cycle.
To prove this, it is sufficient to show that after $A$, only one number that
was in the belt will remain in the belt. This can be visually seen in Figure
3. See Section A.1 for a complete proof.
Figure 3. The cycle “belt", after executing $A$.
### 4.2. The Spiral Algorithm
To build up our plan for sending the three numbers that we wish to 3-cycle to
the squares cycled by $\varphi$, we first develop a methodology for sending
one number to any desired square (while obeying PRs in the sense that the
number’s square and the goal square lie in the same parity when $n$ is odd).
To do this, it is sufficient to be able to send any number to a “center
square" of the board. In the discussion that follows, we will assume that if
$n$ is odd then the numbers we are maneuvering lie in the “even parity" in the
sense that their coordinates have even sum. The case where $n$ is even and the
numbers in question have odd parity is handled by reflecting our arguments.
Our specific goal is to send a number, which we will call $a$, to the square
$u=(n/2,n/2+1)$ if $n$ is even (the “upper center square"), and
$u=\left(\frac{n+1}{2},\frac{n+1}{2}\right)$ if $n$ is odd (the center of the
$X$-rotating block). Note that these squares are of minimum positive distance
to the center of the $Y$-rotating block in either case. The procedure for
accomplishing this is the spiral algorithm, described as follows.
1. (1)
If $a$ is located in the leftmost column, execute $X^{2}$ so that it is not.
2. (2)
If either $a$ is at $u$, or is at any of the other three squares of minimum
positive distance to the center of rotation of $Y$, then execute $Y$ as needed
to move $a$ to $u$ and terminate.
3. (3)
Execute $Y$ until the following inequalities hold, where $(i,j)$ are the
coordinates of $a$:
$\frac{n+1}{2}\leq i\leq n,\qquad 2\leq j\leq\frac{n+3}{2}.$
Visually, this condition entails that $a$ is located in the lower-left
quadrant of the square rotated by $Y$, so this step is necessarily possible.
4. (4)
Execute $X$.
5. (5)
Loop back to Step 2.
Figure 4. Geometrical interpretation of the spiral distance monovariant.
$c_{x}$ is the center of rotation of the move $X$, and similarly for $c_{y}$.
We claim that this algorithm must terminate. Let $c_{y}$ be the point at the
center of rotation of $Y$, which may or may not be the center of a square. It
suffices to show that the straightline distance from $a$ to $c$ is a strictly
decreasing monovariant over each loop of the algorithm, since there are only a
finite number of possible distances between the centers of two squares. Note
that this distance is invariant as $Y$ is executed, so it suffices to observe
a decrease in the distance after $X$ is executed.
Right before $X$ is executed, let the distance from $a$ to $c_{y}$ be $d$.
Then, since the coordinates of $a$ will satisfy the inequalities specified in
step 3, $a$ will lie on the lower-left quarter arc of the circle centered at
$c$ with radius $d$ (See Figure 4).
Now, let us calculate a lower bound for $d$. If $n$ odd, we see that the
coordinates of $c_{y}$ sum to $n+1$, which is odd, so the premise that $a$ is
at $c_{y}$ contradicts the assumption that $a$ lies on an even-parity square.
Moreover if $d=1$ then $a$ is already as close as it can be to $c_{y}$, which
would have terminated the algorithm. Hence we may assume $d>1$ for odd $n$.
For even $n$, we see that the closest possible straightline distance to
$c_{y}$ is $\frac{\sqrt{2}}{2}$, and if this distance is obtained then the
algorithm would have terminated. Otherwise, the next closest distance to
$c_{y}$ would be $\frac{\sqrt{10}}{2}$, so we may assume that
$d\geq\frac{\sqrt{10}}{2}$ for even $n$. In either case, we may assume that
$d>1$.
Now, executing $X$ will rotate the aforementioned arc $90^{\circ}$ counter-
clockwise about the center of rotation of $x$, denoted by $c_{x}$. This will
bring the arc closer to $c_{y}$. Since only the distances between this arc and
$c_{y}$ are of concern here, it is equivalent to rotate $c_{y}$ $90^{\circ}$
clockwise about $c_{x}$ to $c^{\prime}_{y}$. The new distance between $a$ and
$c_{y}$ after $X$ will simply be the distance between $a$ and
$c_{y}^{\prime}$. Let this new distance be $d^{\prime}$.
Now let the angle formed by $a$, $c_{y}$, $c^{\prime}_{y}$ be $\theta$. Since
$a$ must lie on the lower-left quarter arc, we see that
$0^{\circ}\leq\theta\leq 45^{\circ}$. The distance from $a$ to $c_{y}$ is the
radius $d$, and since the distances between $c_{x}$ to $c_{y}$ and
$c^{\prime}_{y}$ are both 1, the distance from $c_{y}$ to $c_{y}^{\prime}$ is
$\sqrt{2}$. By Law of Cosines we have that
$d^{\prime}=\sqrt{d^{2}+2-2\sqrt{2}d\cos{\theta}}.$
We want to show that $d^{\prime}<d$, or that
$\sqrt{d^{2}+2-2\sqrt{2}d\cos{\theta}}<d.$
After algebraic manipulations, we see that this holds if and only if
$d\cos{\theta}>\frac{1}{\sqrt{2}}$. This follows because we may multiply the
assumption $d>1$ with the inequality $\cos\theta\geq\frac{1}{\sqrt{2}}$ which
holds for $0\leq\theta\leq 45$. This completes the proof that spiral
terminates. We conclude that any number $a$ may be sent to any target square,
provided that the square in which $a$ lies shares the same parity as the
target square in the case that $n$ is odd.
We note that in spiral, there was nothing special about using the lower-left
quadrant of the $Y$-rotating block and then executing $X$. It would have been
just as effective to instead move $a$ to the upper-left quadrant of the
$Y$-rotating block and then executing $X^{-1}$. Denote this variant as
spiral*.
### 4.3. The Spiral-Cycle Algorithm
We are now ready to describe the procedure for sending any three numbers to
any three desired squares. As before, if $n$ is odd then we will require that
the squares occupied by the three numbers, as well as the three goal squares,
all lie on the same parity. In this case we will assume without loss of
generality that these six squares lie on the even parity, as before.
It suffices to choose three fixed squares $u_{1},u_{2},u_{3}$ and show that we
can move any three numbers $a_{1},a_{2},a_{3}$ to these squares, since we can
find the moves $B_{1}$ that send the three numbers we need to cycle to
$u_{1},u_{2},u_{3}$, find the moves $B_{2}$ that send the goal squares to
$u_{1},u_{2},u_{3}$, and then execute $B_{1}B_{2}^{-1}$.
Our choice for $u_{1},u_{2},u_{3}$ is as follows:
* •
If $n$ is even, we choose $u_{1}=(1,1)$, $u_{2}=(2,1)$, and
$u_{3}=(\frac{n}{2},\frac{n}{2}+1)$.
* •
If $n$ is odd, we choose $u_{1}=(1,1)$, $u_{2}=(3,1)$, and
$u_{3}=(\frac{n+1}{2},\frac{n+1}{2})$. Note that these squares are chosen to
lie on the even parity.
The procedure for sending $a_{1},a_{2},a_{3}$ to $u_{1},u_{2},u_{3}$ is the
spiral-cycle algorithm, described as follows for all $n\geq 6$:
1. (1)
Use spiral to move $a_{1}$ to $u_{1}$.
2. (2)
If $a_{2}$ is now in the first column, we move it out via $XYX^{-1}$.
3. (3)
Use a modified spiral* to bring $a_{2}$ to $(n,n+1)$ for even $n$ or
$(n-1,n+1)$ for odd $n$, with the following caveats:
* •
If we require usage of the $X$ rotating block in the spiral* algorithm, but
rotating the $X$ block will displace the current position of $a_{1}$ (i.e.
$a_{1}$ lies in the first column), then we execute the inverse cycle $C^{-1}$.
This “hides" $a_{1}$ in the last column. Note that $C^{-1}$ cannot displace
$a_{2}$, which would have to lie in the upper-left quadrant of the $Y$
rotating block when spiral* requires the use of the $X$ block, so $a_{2}$
cannot lie in the belt at this time.
* •
If we require usage of the $Y$ rotating block in the spiral* algorithm, but
rotating the $Y$ block will displace the current position of $a_{1}$ (i.e.
$a_{1}$ lies in the last column), then we execute $C$. This cannot displace
$a_{2}$ either, since at this stage it must be the case that at least one
$X^{-1}$ move of the spiral* algorithm was executed, so $a_{2}$ is too close
to the center of the $Y$ rotating block to lie in the belt at this time.
* •
Once spiral* is terminated in this way, we execute $C$ is needed to restore
$a_{1}$ to its position at $u_{1}=(1,1)$.
4. (4)
Execute $XY^{-1}X^{-1}$. This will bring $a_{2}$ to $(2,1)$ for even $n$, or
$(3,1)$ for odd $n$. Now $a_{1},a_{2}$ are at $u_{1},u_{2}$.
5. (5)
If $a_{3}$ is now in the first column, then move it out using
$XY^{2}XYXY^{-1}X^{2}Y^{-1}XY^{-1}X^{-1}$ for odd $n$, or
$XY^{2}X^{-1}Y^{-1}XY^{-1}X^{-1}$ for even $n$. See Section A.2 for a proof
that this works for $n\geq 5$.
6. (6)
Mimic Step 3 in order to get $a_{3}$ to $u_{3}$. This will still work because
$C^{-1}$ always moves the numbers on $(1,1),(2,1),$ and $(3,1)$ in the last
column.
If $n\geq 5$, replace each instance of $C$ with $C^{2}$.
To see that the application of cycle always “hides" the numbers on
$(1,1),(2,1),$ and $(3,1)$ as described, recall that $C^{-1}$ moves the
numbers along the belt $n-3$ squares clockwise. For $n\geq 6$ we see that
$3\leq n-3\leq n$, so it must move the first three numbers on the belt into
the last $n$, i.e. the last column. For $n=5$, we use two applications of
cycle instead so that the numbers move $2(5)-6=4$ squares clockwise along the
belt. Since $3\leq 4\leq 5$, this gives the same result of moving
$(1,1),(2,1),$ and $(3,1)$ into the last column. This justifies the
correctness of spiral-cycle. We conclude that
* •
for $n$ even, we may send any three numbers to any three squares, and
* •
for $n$ odd, we may send any three numbers lying in a common parity to any
three squares of that parity.
Consequently,
* •
for $n$ even, we may 3-cycle any three numbers and
* •
for $n$ odd, we may 3-cycle any three numbers lying in a common parity,
which is what we wanted to prove.
This completes the proof of the theorem in the general case.
## 5\. Special Cases
Our general solution fails for some small cases. This is due to the fact that
the general 3-cycle algorithms described do not have enough space to execute
the permutation that contains the 3-cycle. As a result, we have some special
cases for smaller values of $m$, $n$, and $b$, that will introduce further
solvability conditions.
### 5.1. $(2,n,2)$
$n$ is not large enough for any of the general algorithms to work. In fact,
for $n=3$, general 3-cycle algorithms do not exist. It was proven by Jaap
Scherphuis that out of all $6!$ theoretically reachable permutations in the
$(2,3,2)$ variation, only $5!$ are achievable [3].
However, when $n\geq 4$, there exist algorithms that switch two numbers. Take
any $4\times 2$ sub-board, and label the possible moves from left to right
$X$, $Y$, and $Z$. Then the following algorithm will switch two numbers:
$XYZ^{-1}Y^{2}X^{-1}Z^{-1}YZ^{2}Y^{-1}$
### 5.2. $(m,n,2)$ where $m,n\geq 3$
Although this is classified as a special case, this is the most common version
of the NRP, especially $(3,3,2)$, which is the standard NRP. Thus, many
different resolutions to this variation are well-known.
There are no restrictions on solvability. Consider a $3\times 3$ sub-board,
and let $X$ rotate the lower-left $2\times 2$ block, $Y$ rotate the upper-
right block, and $Z$ rotate the lower-right block. The following algorithm
switches two numbers:
$XY^{-1}X^{-1}YZ$
From here, it is easy to prove that all initial configurations are solvable.
### 5.3. $(3,4,3)$
We claim that out of all $12!$ theoretically reachable permutations, only $6!$
are achievable. Moreover, we claim that solving one parity will solve the
other.
Figure 5. The $3\times 4$ NRP with $3\times 3$ rotating blocks. The two
possible moves are notated as shown.
To show this, first observe that since $b$ is odd, the parity in which each
number lies is preserved by each move. Hence we may view the puzzle as two
separate but superimposed puzzles formed by by the 6 numbers in each parity.
Now, let the two counter-clockwise moves be $X$ and $Y$, as in Figure 5. We
let $P_{0}$ and $P_{1}$ be the set of coordinates with even coordinate sum and
odd coordinate sum, respectively. To prove our claim, we need to show that, if
the initial configuration is solvable, then
1. (1)
we can obtain any permutation of $P_{0}$ (without any regard as to what
happens to $P_{1}$), and
2. (2)
$P_{0}$ is solved if and only if $P_{1}$ is solved.
To prove (1), note that $XYX^{-1}Y^{-1}XY^{-1}XYXYX$ swaps $(2,4)$ and
$(3,3)$, and fixes all other elements of $P_{0}$. It follows easily that any
two numbers in $P_{0}$ may be swapped. As for (2), it suffices to prove that
$P_{1}$ being solved implies that $P_{0}$ is solved by symmetry. In
particular, it is sufficient to prove that any sequence of moves that fixes
all elements of $P_{1}$ must also fix all elements of $P_{0}$. This is because
if for some solvable board we have that all elements of $P_{1}$ are solved but
$P_{0}$ is not, then by virtue of solvability there exists an algorithm $A$
that solves the $P_{0}$ parity but fixes $P_{1}$. What we wish to prove will
contradict this.
We define a pair to be a set of two elements. Let $S_{0}$ be the set of all
pairs with elements in $P_{0}$, and let $S_{1}$ be the set of partitions of
$P_{1}$ into three pairs. For a sequence of moves $A$, we let $A$ “act" on
$S_{0}$ and $S_{1}$ in an element-wise sense. For example:
* •
We have the pair $\\{(1,1),(2,4)\\}\in S_{0}$, and after executing the $Y^{2}$
this pair becomes $\\{(1,1),(2,2)\\}$.
* •
We have the partition
$\\{\\{(1,2),(3,4)\\},\\{(1,4),(3,2)\\},\\{(2,1),(2,3)\\}\\}\in S_{1}$, and
after executing $X$ this becomes the partition
$\\{\\{(2,1),(3,4)\\},\\{(1,4),(2,3)\\},\\{(3,2),(1,2)\\}\\}$.
We shall denote such actions by $\cdot$, so e.g.
$Y^{2}\cdot\\{(1,1),(2,4)\\}=\\{(1,1),(2,2)\\}$.
The remainder of the proof relies in the following remarkably strong property.
###### Lemma 5.4.
There exists a bijection $\phi:S_{0}\to S_{1}$ that preserves structure in the
following sense: For all pairs $p\in S_{0}$, we have that $\phi(A\cdot
p)=A\cdot\phi(p)$ for all $A$.
Figure 6. Movement graphs on the sets $S_{1}$ and $S_{2}$. Top: Movement graph
on the set of pairs on the parity set $P_{0}$. Bottom: Movement graph on the
set of partitions into pairs on the parity set $P_{1}$. Each edge represents a
move. It can be seen that the structures of these graphs are identical.
###### Proof.
We list out all 15 elements of $S_{0}$ in a graph with two types of directed
edges: an “$X$ edge" and a “$Y$ edge", where each edge represents the actions
of $X$ and $Y$. That is, we place an $X$ edge between the pairs $p_{1}$ and
$p_{2}$ if and only if $X\cdot p_{1}=p_{2}$, and similarly for the $Y$ edges.
We may construct the same sort of graph for $S_{1}$. In doing so, we see that
the two graphs are isomorphic! See Figure 6 for a visual of these graphs. The
proof of our claim follows from this isomorphic structure by taking $\phi$ to
be the natural correspondence that arises.
∎
To finish, consider a sequence of moves $A$ that fixes the elements of
$P_{1}$. Take any pair $\\{a,b\\}\in S_{0}$. We claim that $A$ fixes this
pair, i.e. $A\cdot\\{a,b\\}=\\{a,b\\}$. Indeed, using the lemma and the fact
that $A$ fixes the elements of $P_{1}$, we have that
$A\cdot\\{a,b\\}=\phi^{-1}(A\cdot\phi(\\{a,b\\}))=\phi^{-1}(\phi(\\{a,b\\}))=\\{a,b\\}.$
Since $A$ fixes all pairs, it must fix all elements of $P_{0}$. This is
because for any $a,b,c\in P_{0}$, we have $A\cdot\\{a,b\\}=\\{a,b\\}$ and
$A\cdot\\{a,c\\}=\\{a,c\\}$, so $A\cdot a\in\\{a,b\\}\cap\\{a,c\\}=\\{a\\}$,
forcing $A\cdot a=a$ for all $a\in P_{0}$, as needed.
As a remark, we note the following fascinating connection to group theory: Let
$\sigma_{P_{0}}$ and $\sigma_{P_{1}}$ denote that permutation groups on
$P_{0}$ and $P_{1}$, respectively. Construct a map
$\psi:\sigma_{P_{0}}\to\sigma_{P_{1}}$ as follows: If a sequence of moves
executes a permutation $\phi_{0}\in\sigma_{P_{0}}$ on $P_{0}$ and a
permutation $\phi_{1}\in\sigma_{P_{1}}$ on $P_{1}$, then
$\psi:\phi_{0}\mapsto\phi_{1}$. Then by the work we have done, $\psi$ is a
well-defined isomorphism, and since $\sigma_{P_{0}}\cong\sigma_{p_{1}}\cong
S_{6}$, we may view $\psi$ as an automorphism on $S_{6}$. In fact, a quick
check reveals that $\psi$ is the exotic outer automorphism on $S_{6}$!
### 5.5. $(3,n,3)$ for $n\geq 5$
Consider some $3\times 5$ sub-board. Let $P_{0}$ and $P_{1}$ denote the
squares in this sub-board of even and odd parity respectively, so that $P_{0}$
contains the upper-left corner. Label, from left to right, the three possible
moves, $X$, $Y$, and $Z$. Then the algorithm
$YZY^{-1}Z^{-1}YZ^{2}Y^{-1}Z^{-1}$ switches two numbers in $P_{0}$ (with other
side effects in $P_{1}$), hence we can solve all numbers in $P_{0}$.
Since each move is a rotation of a $3\times 3$ block, each move enacts an even
permutation, so the resulting configuration of the numbers in the other
parity, $P_{1}$, must be an even permutation of the solved order. Thus, it
suffices to find an arbitrary 3-cycle algorithm on numbers in $P_{1}$ that
does not displace the numbers in $P_{0}$. Consider $(XZX^{-1}Z^{-1})^{2}$,
which is a 3-cycle. Then, one can demonstrate that any three numbers can be
sent anywhere by adapting the transposition algorithm on $P_{0}$ to work on
$P_{1}$ instead by "translating" it as such:
$XYX^{-1}Y^{-1}XY^{2}X^{-1}Y^{-1}$. This will act as a transposition on
$P_{1}$, and applying this repeatedly to move the three desired numbers to be
cycled to the correct squares will serve as an intermediary algorithm.
### 5.6. $(m,n,3)$ where $m,n\geq 4$
The solvability condition here is the same as in the general case: the initial
configuration must be an even permutation of the solved board, and every
number must lie in its correct parity. Fernando’s general 3-cycle algorithm on
square boards is enough to resolve this case [1]. Nevertheless, we present a
short 3-cycle algorithm. Let $A$ rotate the upper-left $3\times 3$ block, $B$
rotate the upper-right block, $C$ rotate the lower-left block, and $D$ rotate
the lower-right block. Then
$ADA^{-1}D^{-1}C^{-1}DA^{-1}D^{-1}AC$
is a 3-cycle.
### 5.7. $(m,n,4)$ where $m,n\geq 4$ with $m\neq n$
The solvability condition here is the same as in the general case: the initial
configuration must be an even permutation of the solved board. As in the proof
of the general case, it suffices to consider the case $(4,5,4)$ and find a
3-cycle. An “easy" 3-cycle is given by
$(XY^{2}X^{-1}YX^{-1}Y^{2})^{35}.$
All remaining special cases are completely trivial.
## 6\. Future Research
There are still some open problems concerning the NRP. For example, all
rotations were assumed to preserve the orientations of the numbers, so they
always stay upright. It would be interesting to determine the additional
solvability conditions if orientation of numbers are not preserved with
rotation. Additionally, all moves were assumed to be rotations of square
rotating blocks. Thus, a variation of the NRP can be made by letting the
rotating blocks be rectangular, with moves consisting of rotating said blocks
$180^{\circ}$. This would add an exciting dimension to an already convoluted
puzzle.
## 7\. Acknowledgments
I would like to thank Ravi Fernando for revealing to me the connection between
my result on the $(3,4,3)$ NRP and the exotic automorphism on $S_{6}$.
## References
* [1] R. Fernando. Nxn corner rotation puzzle. https://www.speedsolving.com/forum/threads/nxn- corner-rotation-puzzle.15472/, 2009.
* [2] Y. Lee. Solving the “rotation” puzzle in stages. https://codemyroad.wordpress.com/2015/04/13 /solving-the-rotation-puzzle-in-stages/, 2015.
* [3] J. Scherphuis. Two-generator corners group. URL:https://www.jaapsch.net/puzzles/pgl25.htm.
* [4] S. Tatham. Chapter 7: Twiddle, 2018.
* [5] G. Wang and J. Song. Bbfs-stt: An efficient algorithm for number rotation puzzle. Entertainment Computing, 12:1–7, November 2015.
## 8\. Appendix
Here we include the unsightly proofs necessary to justify some of the claims
we have made.
We recall the notations of $X$ and $Y$ denoting the $90^{\circ}$ counter-
clockwise rotations of the left and right blocks respectively on the
$(n,n+1,n)$ NRP. We may view $X$ and $Y$ as functions on the board
$\\{1,\cdots,n\\}\times\\{1,\cdots,n+1\\}$ defined via:
$\displaystyle X(x,y)$ $\displaystyle=\begin{cases}(n+1-y,x),&1\leq y\leq n\\\
(x,y),&\text{otherwise}\end{cases}$ $\displaystyle Y(x,y)$
$\displaystyle=\begin{cases}(n+2-y,x+1),&2\leq y\leq n+1\\\
(x,y),&\text{otherwise}\end{cases}$
It will also be useful to state the behavior of $X^{-1},Y^{-1},X^{2},$ and
$Y^{2}$:
$\displaystyle X^{-1}(x,y)$ $\displaystyle=\begin{cases}(y,n+1-x),&1\leq y\leq
n\\\ (x,y),&\text{otherwise}\end{cases}$ $\displaystyle Y^{-1}(x,y)$
$\displaystyle=\begin{cases}(y-1,n+2-x),&2\leq y\leq n+1\\\
(x,y),&\text{otherwise}\end{cases}$ $\displaystyle X^{2}(x,y)$
$\displaystyle=\begin{cases}(n+1-x,n+1-y),&1\leq y\leq n\\\
(x,y),&\text{otherwise}\end{cases}$ $\displaystyle Y^{2}(x,y)$
$\displaystyle=\begin{cases}(n+1-x,n+3-y),&2\leq y\leq n+1\\\
(x,y),&\text{otherwise}\end{cases}$
For ease, let us define the abuse of notation
$([a,b],c):=\\{(a,c),(a+1,c),\cdots,(b,c)\\}$, and likewise for $(a,[b,c])$.
For further ease and abuse, we view $\\{(a,b)\\}$ and $(a,b)$ as the same.
### 8.1. Image of the belt under the algorithm $A$
Let $A=XYYXY^{-1}X^{-1}$ be an algorithm, and let
$E=([1,n],1)\cup(n,[2,n])\cup([1,n],n+1)$ denote the “belt". We show that the
image of $E$ under the algorithm $A$ intersects with $E$ at exactly one
square, provided that $n\geq 5$.
We recall that our algorithms are read left-to-right, so eg.
$(XY)(E)=Y(X(E))$. Our proof strategy simply involves computing the image of
$E$ after every move, and splitting said image to prepare for the next move.
* •
After $X$:
$\displaystyle X(E)$ $\displaystyle=X([1,n],1)\cup X(n,[2,n])\cup([1,n],n+1)$
$\displaystyle=(n,[1,n])\cup([1,n-1],n)\cup([1,n],n+1)$
$\displaystyle=(n,1)\cup(n,[2,n])\cup([1,n-1],n)\cup([1,n],n+1)$
* •
After $Y^{2}$:
$\displaystyle(XY^{2})(E)$ $\displaystyle=(n,1)\cup Y^{2}(n,[2,n])\cup
Y^{2}([1,n-1],n)\cup Y^{2}([1,n],n+1)$
$\displaystyle=(n,1)\cup(1,[3,n+1])\cup([2,n],3)\cup([1,n],2)$
$\displaystyle=(n,1)\cup(1,n+1)\cup(1,[3,n])\cup([2,n],3)\cup([1,n],2)$
* •
After $X$:
$\displaystyle(XY^{2}X)(E)$ $\displaystyle=X(n,1)\cup(1,n+1)\cup
X(1,[3,n])\cup X([2,n],3)\cup X([1,n],2)$
$\displaystyle=(n,n)\cup(1,n+1)\cup([1,n-2],1)\cup(n-2,[2,n])\cup(n-1,[1,n])$
$\displaystyle=(n,n)\cup(1,n+1)\cup([1,n-1],1)\cup(n-2,[2,n])\cup(n-1,[2,n])$
* •
After $Y^{-1}$:
$\displaystyle\phantom{{}={}}(XY^{2}XY^{-1})(E)$
$\displaystyle=Y^{-1}(n,n)\cup Y^{-1}(1,n+1)\cup([1,n-1],1)\cup
Y^{-1}(n-2,[2,n])\cup Y^{-1}(n-1,[2,n])$
$\displaystyle=(n-1,2)\cup(n,n+1)\cup([1,n-1],1)\cup([1,n-1],4)\cup([1,n-1],3)$
* •
After $X^{-1}$:
$\displaystyle\phantom{{}={}}(XY^{2}XY^{-1}X^{-1})(E)$
$\displaystyle=X^{-1}(n-1,2)\cup X^{-1}([1,n-1],1)\cup X^{-1}([1,n-1],3)\cup
X([1,n-1],4)\cup(n,n+1)$
$\displaystyle=(2,2)\cup(1,[2,n])\cup(3,[2,n])\cup(4,[2,n])\cup(n,n+1)$
From this we see that, under the key assumption that $n\geq 4$, we have that
$(n,n+1)$ is the sole element in the intersection of $E$ and $A(E)$.
### 8.2. Moving $a_{3}$ out of the first column
After executing the first four steps of the Spiral-Cycle algorithm, it may be
the case that $a_{3}$ lies in the first column, in which case we cannot
proceed. If so, then it must be moved out of this column without disturbing
the positions of $a_{1}$ and $a_{2}$.
Let us first consider the case in which $n$ is even. We claim that after
$XY^{2}X^{-1}Y^{-1}XY^{-1}X^{-1}$, all numbers in the first column are moved
out of it, without displacing $a_{1}$ and $a_{2}$, which are are $(1,1)$ and
$(2,1)$ respectively.
It is easy to verify that $a_{1}$ and $a_{2}$ are fixed under this algorithm.
$\displaystyle(1,1)$ $\displaystyle\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle
Y^{2}}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle X^{-1}}}{{\mapsto}}(1,1)$
$\displaystyle\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(1,1)\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle X^{-1}}}{{\mapsto}}(1,1)$
$\displaystyle(2,1)$ $\displaystyle\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,2)\stackrel{{\scriptstyle
Y^{2}}}{{\mapsto}}(1,n+1)\stackrel{{\scriptstyle X^{-1}}}{{\mapsto}}(1,n+1)$
$\displaystyle\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,n+1)\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,n+1)\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,2)\stackrel{{\scriptstyle X^{-1}}}{{\mapsto}}(2,1)$
Now let $E=([3,n],1)$ be the rest of the first column. We show that the image
of $E$ under the algorithm is disjoint from $E$, which suffices. Here we must
assume $n\geq 4$.
* •
After $X$:
$X(E)=(n,[3,n])$
* •
After $Y^{2}$:
$(XY^{2})(E)=Y^{2}(n,[3,n])=(1,[3,n])$
* •
After $X^{-1}$:
$(XY^{2}X^{-1})(E)=X^{-1}(1,[3,n])=([3,n],n)$
* •
After $Y^{-1}$:
$(XY^{2}X^{-1}Y^{-1})(E)=Y^{-1}([3,n],n)=(n-1,[2,n-1])$
* •
After $X$:
$(XY^{2}X^{-1}Y^{-1}X)(E)=X(n-1,[2,n-1])=([2,n-1],n-1)$
* •
After $Y^{-1}$:
$(XY^{2}X^{-1}Y^{-1}XY^{-1})(E)=Y^{-1}([2,n-1],n-1)=(n-2,[3,n])$
* •
After $X^{-1}$:
$(XY^{2}X^{-1}Y^{-1}XY^{-1}X^{-1})(E)=X^{-1}(n-2,[3,n])=([3,n],3)$
Evidently $([3,n],3)$ is disjoint from $([3,n],1)$, as needed.
Now we consider the case in which $n$ is odd. Then we claim that
$XY^{2}XYXY^{-1}X^{2}Y^{-1}XY^{-1}X^{-1}$ moves out all numbers in the first
column without displacing $a_{1}$ and $a_{2}$, which are at $(1,1)$ and
$(3,1)$ respectively.
This algorithm is more difficult to motivate, but nevertheless we may manually
verify that $a_{1}$ and $a_{2}$ are fixed under this algorithm.
$\displaystyle(1,1)$ $\displaystyle\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle
Y^{2}}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,n)\stackrel{{\scriptstyle
Y}}{{\mapsto}}(2,n+1)\stackrel{{\scriptstyle X}}{{\mapsto}}(2,n+1)$
$\displaystyle\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,n)\stackrel{{\scriptstyle
X^{2}}}{{\mapsto}}(1,1)\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(1,1)\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle X^{-1}}}{{\mapsto}}(1,1)$
$\displaystyle(3,1)$ $\displaystyle\stackrel{{\scriptstyle
X}}{{\mapsto}}(n,3)\stackrel{{\scriptstyle
Y^{2}}}{{\mapsto}}(1,n)\stackrel{{\scriptstyle
X}}{{\mapsto}}(1,1)\stackrel{{\scriptstyle
Y}}{{\mapsto}}(1,1)\stackrel{{\scriptstyle X}}{{\mapsto}}(n,1)$
$\displaystyle\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,1)\stackrel{{\scriptstyle
X^{2}}}{{\mapsto}}(1,n)\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n-1,n+1)\stackrel{{\scriptstyle
X}}{{\mapsto}}(n-1,n+1)\stackrel{{\scriptstyle
Y^{-1}}}{{\mapsto}}(n,3)\stackrel{{\scriptstyle X^{-1}}}{{\mapsto}}(3,1)$
The difficulty is showing that the rest of the column, $E=(2,1)\cup([4,n],1)$,
is moved out. We will need the assumpton that $n\geq 5$.
* •
After $X$:
$X(E)=X(2,1)\cup X([4,n],1)=(n,2)\cup(n,[4,n])$
* •
After $Y^{2}$:
$(XY^{2})(E)=Y^{2}(n,2)\cup Y^{2}(n,[4,n])=(1,n+1)\cup(1,[3,n-1])$
* •
After $X$:
$(XY^{2}X)(E)=X(1,n+1)\cup X(1,[3,n-1])=(1,n+1)\cup([2,n-2],1)$
* •
After $Y$:
$(XY^{2}XY)(E)=Y(1,n+1)\cup Y([2,n-2],1)=(1,2)\cup([2,n-2],1)$
* •
After $X$:
$(XY^{2}XYX)(E)=X(1,2)\cup X([2,n-2],1)=(n-1,1)\cup(n,[2,n-2])$
* •
After $Y^{-1}$:
$(XY^{2}XYXY^{-1})(E)=Y^{-1}(n-1,1)\cup
Y^{-1}(n,[2,n-2])=(n-1,1)\cup([1,n-3],2)$
* •
After $X^{2}$:
$\displaystyle(XY^{2}XYXY^{-1}X^{2})(E)$ $\displaystyle=X^{2}(n-1,1)\cup
X^{2}([1,n-3],2)$ $\displaystyle=(2,n)\cup([4,n],n-1)$
* •
After $Y^{-1}$:
$\displaystyle(XY^{2}XYXY^{-1}X^{2}Y^{-1})(E)$ $\displaystyle=Y^{-1}(2,n)\cup
Y^{-1}([4,n],n-1)$ $\displaystyle=(n-1,n)\cup(n-2,[2,n-2])$
* •
After $X$:
$\displaystyle(XY^{2}XYXY^{-1}X^{2}Y^{-1}X)(E)$ $\displaystyle=X(n-1,n)\cup
X(n-2,[2,n-2])$ $\displaystyle=(1,n-1)\cup([3,n-1],n-2)$
* •
After $Y^{-1}$:
$\displaystyle(XY^{2}XYXY^{-1}X^{2}Y^{-1}XY^{-1})(E)$
$\displaystyle=Y^{-1}(1,n-1)\cup Y^{-1}([3,n-1],n-2)$
$\displaystyle=(n-2,n+1)\cup(n-3,[3,n-1])$
* •
After $X^{-1}$:
$\displaystyle(XY^{2}XYXY^{-1}X^{2}Y^{-1}XY^{-1}X^{-1})(E)$
$\displaystyle=X^{-1}(n-2,n+1)\cup X^{-1}(n-3,[3,n-1])$
$\displaystyle=(n-2,n+1)\cup([3,n-1],4)$
We see that $(n-2,n+1)\cup([3,n-1],4)$ does not intersect the first column, as
needed.
|
# Anomalous relaxation of density waves in a ring-exchange system
Pranay Patil<EMAIL_ADDRESS>Max-Planck-Institut für Physik komplexer
Systeme, Nöthnitzer Straße 38, Dresden 01187, Germany Laboratoire de Physique
Théorique, Université de Toulouse, CNRS, UPS, France Markus Heyl Max-Planck-
Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, Dresden 01187,
Germany Theoretical Physics III, Center for Electronic Correlations and
Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg,
Germany Fabien Alet Laboratoire de Physique Théorique, Université de
Toulouse, CNRS, UPS, France
###### Abstract
We present the analysis of the slowing down exhibited by stochastic dynamics
of a ring-exchange model on a square lattice, by means of numerical
simulations. We find the preservation of coarse-grained memory of initial
state of density-wave types for unexpectedly long times. This behavior is
inconsistent with the prediction from a low frequency continuum theory
developed by assuming a mean field solution. Through a detailed analysis of
correlation functions of the dynamically active regions, we exhibit an
unconventional transient long ranged structure formation in a direction which
is featureless for the initial condition, and argue that its slow melting
plays a crucial role in the slowing-down mechanism. We expect our results to
be relevant also for the dynamics of quantum ring-exchange dynamics of hard-
core bosons.
## I Introduction
The field of dynamics in isolated quantum systems has recently received an
increasing amount of attention thanks to the discoveries of a plethora of
interesting non-equilibrium behaviors Calabrese (2018); Garrahan (2018);
Gogolin and Eisert (2016), and of versatile experimental platforms to realize
the same Browaeys and Lahaye (2020); Gross and Bloch (2017). These studies
have been partially motivated by the desire to achieve the protection of
quantum information from scrambling caused by Hamiltonian dynamics or
environmental noise. This has lead to the rapid development of the field of
many body localization Abanin _et al._ (2019); Alet and Laflorencie (2018);
Abanin and Papić (2017); Luitz _et al._ (2015), which relies on strong
disorder to provide a safeguard against scrambling many body dynamics, and the
nascent field of Hilbert space fragmentation Sala _et al._ (2020); Khemani
_et al._ (2020); Moudgalya _et al._ (2021); Yang _et al._ (2020); Mukherjee
_et al._ (2021); Karpov _et al._ (2021), which results from the highly
constrained configuration space of Hamiltonians with a large number of strong
local constraints and/or highly frustrated interactions Zhang and Røising
(2022); Sikora _et al._ (2011).
A milder version of the total arresting of dynamics generated by the phenomena
mentioned above is realized by systems which approach equilibrium in a manner
which is qualitatively different from standard diffusion. An example of this
has recently been explored for the spin-1/2 Heisenberg chain as well as its
classical version at intermediate energies where evidence of super-diffusion
has recently been seen both theoretically Ilievski _et al._ (2018); McRoberts
_et al._ (2022) and experimentally Scheie _et al._ (2021), leading to
connections to surface growth dynamics studied by Kardar-Parisi-Zhang Weiner
_et al._ (2020); Kardar _et al._ (1986). An approach to equilibrium which is
slower than that expected from diffusion has also been realized in systems
which conserve higher moments (such as dipolar and octupolar) of the spin
configuration Iaconis _et al._ (2019); Feldmeier _et al._ (2020);
Morningstar _et al._ (2020); Iaconis _et al._ (2021); Richter and Pal
(2022). For two- and higher-dimensional systems, this has also lead to the
realization of exotic fractonic phases Gromov _et al._ (2020); Nandkishore
and Hermele (2019); Pretko _et al._ (2020). Remarkable advances have also
been made on probing experimental realizations of Hilbert space fragmentation
and/or higher-moment conservation and associated sub-diffusion Guardado-
Sanchez _et al._ (2020); Scherg _et al._ (2021); Kohlert _et al._ (2021).
While progress has been made on the analytical description and experimental
detection of the phenomena mentioned above, there has been a dire need for
numerical simulations on microscopic models to lend support to many of the
predictions. This is in general a difficult task as powerful methods to
simulate large-scale quantum dynamics on the needed long-time scales are
relatively few and capable of handling only specific regimes Baiardi and
Reiher (2019); Schmitt and Heyl (2020). It was found insightful Iaconis _et
al._ (2019); Iadecola and Vijay (2020) to adopt methods from stochastic
classical mechanics in the framework of cellular automata Iaconis _et al._
(2019); Gopalakrishnan and Zakirov (2018); Iaconis (2021), which ignore part
of the quantum phase fluctuations and have been able to successfully describe
the long-term dynamical behavior of strongly interacting quantum systems. This
intuition arises from the expectation that for generic systems with
sufficiently large Hilbert spaces and for times long compared to the
microscopic energy scales of the Hamiltonian, the dynamics does not show
quantum coherence, thus reducing to a classical dynamics problem. As mentioned
above, well-known exceptions to this exist, but as simulation of exact quantum
dynamics is out of reach using current methods, a study of the classical
equivalent becomes of particular interest. This can also serve as a natural
starting point to understand the complete quantum dynamics.
Following up on the studies of constrained systems, we consider in this work
the case of a simple hard-core bosonic model living on a square lattice,
undergoing ring-exchange dynamics. This model has already been studied from
the perspective of cuprates, as they serve as promising candidates to realize
high-Tc superconductivity Paramekanti _et al._ (2002); Rousseau _et al._
(2005); Schaffer _et al._ (2009). Although traditionally most studies have
focused on the possible exotic ground state features of this model Tay and
Motrunich (2010); Tay _et al._ (2011), some recent works Iaconis _et al._
(2019); Khudorozhkov _et al._ (2021) have considered the constrained dynamics
generated by the ring-exchange mechanism, including starting from random
configurations Iaconis _et al._ (2019). However, the relevance of
fragmentation to generic low-momentum states which are only described by
macroscopic patterns has not been investigated.
In this work we address this question using a classical approach based on
stochastic dynamics. We find that structured initial configurations in the
form of a boson density wave retain their coarse grained structure for a time
which grows as a tunable power of the wavelength, with an eventual melting
which is approximately described by a continuum model derived from a simple
Taylor expansion. We study the detailed structure of the melting process via
spatial correlation functions and find that the dynamics proceed through the
development of strongly correlated large active regions which merge and
destroy the initial modulated pattern.
The detailed plan of the paper is as follows. In Sec.II, we present the model,
discuss the quantities conserved under the dynamics, and elucidate the general
profile of stripe-like configurations which are perfectly frozen under the
dynamics. The bulk of this section discusses the effects of small
perturbations on these exactly frozen patterns, and the preservation of the
memory of the initial state to infinite times as illustrated by simulation of
exact quantum and stochastic classical dynamics on a small system size. The
close agreement between exact quantum dynamics and its stochastic classical
equivalent seen in this section motivates our approximation and we focus
purely on the classical system for the following sections.
Sec.III recalls the expected continuum field theory based on Taylor expansions
by first considering the simpler case of correlated random walkers, along with
numerical checks of the equations developed. This is followed by a treatment
of the hard-core model using a mean-field assumption.
For Sec.IV, we move to more general configurations which take the form of
boson density waves, and show the persistence of the memory of the initial
state for unexpectedly long periods of time. We also compare the prediction of
the continuum field theory developed in Sec.III with the initial dynamics.
We follow up in Sec.V with a detailed analysis of the evolution of the
dynamical active regions, and present a phenomenological picture of the
mechanism leading up to the melting of the initial density wave configuration.
We summarize our results in Sec.VI and present possibilities for future follow
ups via direct simulation of the quantum many body dynamics.
In a following appendix A, we briefly discuss the long-time momentum space
profile of correlation functions, show that it is consistent with a mean-field
treatment obtained in a previous work, and that it cannot be a basis of the
anomalous scaling observed in the present work (as could have been deduced
from a recent analysis Sala _et al._ (2021) of a related model).
## II Ring-exchange boson dynamics, conserved quantities and frozen patterns
We consider a system of hard-core bosons living on a square lattice, which
evolve stochastically in time using only ring-exchange dynamics where bosons
hop by pairs around a 4-sites plaquette of the lattice if and only if a single
diagonal of the plaquette is occupied by two bosons (“flippable” plaquettes).
This only allowed dynamics is shown in Fig. 1. This dynamic rule trivially
conserves the total particle number as well as the number of particles in each
individual column and in each individual row Tay and Motrunich (2010); Tay
_et al._ (2011); Iaconis _et al._ (2019); Khudorozhkov _et al._ (2021). For
the rest of this work, we restrict ourselves to half-filling, i.e. $L^{2}/2$
sites occupied by bosons, on a periodic $L\times L$ lattice, and to the sector
where each column and each row has exactly $L/2$ occupied sites. One expects
it to be the sector with the largest number of configurations, as it is
maximally symmetric. The total number of configurations in this sector can be
computed on large lattices using combinatorial techniques Canfield and McKay
(2005).
We find that the conserved quantities discussed above do not in themselves
completely describe the dynamically connected sets of configurations. This can
be seen by considering a “perfect” stripe configuration as shown in Fig. 1,
where an alternating pattern of filled and empty sites does not leave a single
flippable plaquette, making the configuration frozen under ring-exchange
dynamics. By varying the widths and locations of the filled and empty stripes,
we can create many similar frozen configurations. One may also naively expect
that this restriction on the number of accessible configurations extends to
the case where we do not have perfect stripes. To see this, we consider a
configuration generated by interchanging the two neighboring diagonals on the
edge between a filled and an empty region. This creates a diagonal made of
flippable plaquettes, where the influence of this active region may be
expected to only extend to a few lattice spacings around this diagonal.
Figure 1: (a) The ring-exchange rule that is the only allowed dynamics in our
system. (b) A perfect stripe configuration which is frozen under the dynamic
rule above. (c) Initial configuration with mixed filled-staggered pattern on a
$L=16$ sample used to seed Hilbert space fragment. The dynamics of the (d)
overlap $O(t)$ and (e) Fourier ratio $R(t)$ with this initial state for both
exact dynamics (ED) in the quantum case and stochastic classical automaton
(SCA) dynamics, which quickly saturate to the exact values calculated from
enumerating all connected configurations.
To test this intuition and illustrate this effect, we exactly enumerate all
the configurations connected to an initial condition of the form described
above for an $16\times 16$ lattice (shown in Fig. 1). We find that the total
number of configurations, which we call $N_{c}$ which belong to this
“fragment” is $27,990$. To show that they retain some structure of the
representative which we have chosen, we compute an overlap as defined below,
$\bar{O}=\frac{4}{N_{c}L^{2}}\sum_{c}\sum_{x,y}n^{\rm seed}_{x,y}n^{c}_{x,y}.$
(1)
where $n_{x,y}^{c/{\rm seed}}=0,1$ is the number of bosons at site of
coordinates $(x,y)$ in the ($c$-th/initial) configuration. If we use all
possible configurations (in which case $N_{c}$ would be the total Hilbert
space size), we would expect $\bar{O}=0$ due to the symmetry under $n\to 1-n$.
For the restricted Hilbert space belonging to the fragment being considered we
find $\bar{0}=0.844003...$, showing that for a dynamic simulation restricted
to this sector, the initial and late time states would still retain
significant overlap. To confirm this, we also run a stochastic classical
automaton (SCA) simulation, where at each time step, we propose $L^{2}$ random
plaquette flips, and if the chosen plaquette is flippable, we flip it with
probability $1/2$. The resulting overlap with the initial state is shown in
Fig. 1, and we see that it quickly approaches the value expected from exact
enumeration, and retains it indefinitely. To study the accuracy with which the
SCA reproduces the exact quantum dynamics, we also perform an exact
diagonalization for this Hilbert space fragment, and compute
$\braket{\psi(t)}{O}{\psi(t)}$ for $\ket{\psi(t)}=e^{-iHt}\ket{\psi_{0}}$
using
$H=\sum_{x,y}b^{\dagger}_{x,y}b_{x+1,y}b_{x,y+1}b^{\dagger}_{x+1,y+1}+h.c.$,
where the initial condition is the same occupation-basis state as we
initialized the SCA with and $O$ now denotes an operator which is diagonal in
the occupation-basis and measures the overlap with the initial state. This
operator can be generated directly from the expression above for the classical
case by promoting $n^{c}_{x,y}$ to the number operator, while retaining the
integer status of $n^{\rm seed}_{x,y}$. As shown in Fig. 1, we find that the
overlap in the quantum dynamics closely traces the SCA during the initial
decay away from an overlap of $1$. In the following sections, we quantify the
relaxation to equilibrium using the Fourier transform $n_{k}(t)$ with
frequency $k$ along the $\hat{x}+\hat{y}$ direction, at time $t$. To this end,
we measure the Fourier ratio $R(t)=n_{k}(t)/n_{k}(0)$, and make a similar
comparison as done for the spatial overlap above. This is presented in Fig. 1
as well, and once again we find a close agreement between the exact quantum
dynamics and it’s classical equivalent in the way they approach the
equilibrium value of $R(t)$ within the sector of interest. This result
suggests that the quantum dynamics within sectors matches the classical
automaton upon coarse graining past time scales of $O$(1), suggesting that
quantum phases do not play a substantial role in the aspects of dynamics which
we want to study, and that a classical automaton approach could be sufficient
to study the effect of the kinetic constraints on large scale features.
Following the arguments above and noting that exact quantum dynamics for
larger sizes is not possible with current computational capabilities, we
directly study the long time behavior of our stochastic dynamics simulations
to gather information about large system sizes in the following sections.
## III Expected hydrodynamic description
Here we discuss the coarse grained description in continuous space, of the
microscopic dynamics which we have introduced in the previous section. We
begin by relaxing the hard-core constraint and considering the limit of
average number of particles per site being $\gg 1$. This allows us to reduce
the problem to non-interacting correlated random walkers, and leads to an
analogue of the diffusion equation which encodes vanilla sub-diffusion. We
show numerical evidence for the validity of the same. The expected continuum
dynamical behavior of this type of ring-exchange in 2d was first presented in
Ref. Iaconis _et al._ (2019), but we recall it for completeness as well as to
understand how it should be perturbed to take into account the hard-core
nature. Next we move to an equivalent model of a real-valued field on the
lattice which makes the connection to the hydrodynamic limit more transparent,
and we again provide support with numerical simulations.
Lastly, we return to the hard-core model presented in the previous section,
and show that the continuum theory describing the model must include non-
linear terms in addition to the sub-diffusive term, and we show the
possibility for a quantitative change in the behavior of the system due to the
non-linearities.
### III.1 Large particle number limit
We relax the hard-core constraint of the stripe configurations described in
the previous section and assume the pattern to exist over a featureless
background of an average density of $n_{d}\gg 1$ particles per site. The
dynamics can now be understood in terms of correlated random walkers in the
following way. First, we label each particle in the system as an independent
walker. A move is defined now as first picking a walker (called walker $a$) at
random, picking one of its four next nearest neighboring sites with
probability $1/4$, and moving one of the walkers on the chosen site (called
walker $b$) in tandem with walker $a$ in a ring exchange type manner. Due to
the large number of walkers per site, we expect to always be able to find
walker $b$.
To write down the number of particles $n$ at a particular position $(x,y)$ at
time $t+1$ as a function of the values at time $t$, we must consider all
processes which can change $n_{x,y}$. These processes are (1) choosing a
walker at $(x,y)$ and moving it away ($\propto n_{x,y}$); (2) choosing a
walker at one of the four nearest neighbor sites and moving it or its partner
to $(x,y)$ ($\propto 1/2(n_{x\pm 1,y})$ or $1/2(n_{x,y\pm 1})$); (3) choosing
one of the walkers at one of the next nearest neighbors and moving it in
tandem with a walker at $(x,y)$, thus reducing the number of walkers at
$(x,y)$ by one ($\propto 1/4(n_{x\pm 1,y\pm 1})$). Putting these terms
together, we can write the change in $n_{x,y}$, which is an whole number, from
time $t$ to time $t+1$ as (note that all terms in the expression below are at
time $t$, and we do not explicitly mention it for ease of representation)
$\displaystyle\Delta_{t}$ $\displaystyle n_{x,y}\propto-n_{x,y}$
$\displaystyle+\frac{1}{2}[n_{x+1,y}+n_{x-1,y}+n_{x,y+1}+n_{x,y-1}]$
$\displaystyle-\frac{1}{4}[n_{x+1,y+1}+n_{x+1,y-1}+n_{x-1,y+1}+n_{x-1,y-1}].$
(2)
This equation was obtained using similar arguments in Ref. Iaconis _et al._
(2019). If the system is initialized over a background of $n_{d}$ particles
per site using the stripe configuration described in the previous section
(shown in Fig. 1), $n_{(x,y)}$ can be seen as a step function switching
periodically between $n_{d}$ and $n_{d}+1$. We would naively expect that the
correlated dynamics discussed above would quickly eliminate the sharp
boundaries of the $n_{(x,y)}$ texture and lead to a smooth function once we
average over many realizations of the stochastic dynamics.
For a function which varies slowly as a function of $(x,y)$ (note that this
implies that the stripes in the initial condition should be wide compared to
lattice spacing), we can perform a Taylor expansion of the expression above.
We find that all terms to fourth order cancel, and the only term at fourth
order yields
${\partial_{t}}n_{(x,y)}=-c{\partial_{x}^{2}}{\partial_{y}^{2}}n_{(x,y)}$ (3)
where $c=1$ if following the treatment above. For further convenient reporting
of the wave-vector $k$ in units of $\pi$, and taking into account in addition
a factor of $4$ coming from the acceptance probabilities in our numerical
implementation, we consider instead a different normalization with
$c=\pi^{4}/4$. Rescaling the x-axis as in Fig. 2, this allows to recover a
match to $e^{-x}$ for the fit in Fig. 2. For the rest of this manuscript, we
maintain this convention for $c$.
As the stripe initial condition is a periodic square wave in $(x+y)$, it is
convenient to rewrite the above equation in the Fourier basis, and consider
only the lowest harmonic (largest wavelength) of the square wave transform.
This reduces the dynamical equation to
${\partial_{t}}n_{k_{x},k_{y}}=-ck_{x}^{2}k_{y}^{2}n_{k_{x},k_{y}}$. For
diagonal stripes, $k_{x}=k_{y}=k$, and $n_{k}$ can be exactly reduced to
$n_{k}(0)\exp{(-ck^{4}t)}$, where $n_{k}(0)$ is the value at $t=0$. We can now
numerically verify this behavior by calculating the Fourier ratio
$R_{k}(t)=n_{k}(t)/n_{k}(0)$, and looking for a data collapse onto a single
exponential for various values of $k$. We find that $n_{d}=3$ provides a
sufficiently large background for a good data collapse, and show the same for
a $128\times 128$ lattice for various stripe widths (encoded in $k$) in Fig. 2
and averaged over $80$ realizations of the stochastic dynamics for each $k$.
For smaller values of $n_{d}$, we find an increasing discrepancy between
different values of $k$, with the divergence growing with decreasing $n_{d}$.
Figure 2: Dynamics of the ratio $R_{k}(t)=n_{k}(t)/n_{k}(0)$, against $k^{4}t$
for various values of $k$ and on a log-linear scale, for a $128\times 128$
system of correlated walkers, with a background base walker density per site
$n_{d}=3$. The solid line is a fit to a single exponential.
### III.2 Discretization of continuum equation
Here we study a discretization of Eq. (3) on the lattice by defining the real
valued field $n_{x,y}$, and attempt to recover the limit of the hard-core
model. From the analysis of the correlated random walkers above, we expect
that a ring exchange dynamic on a plaquette should lead to the
${\partial_{x}^{2}}{\partial_{y}^{2}}$ form. To this end, we define the
“activity” of a plaquette whose left bottom site is $(x,y)$ to be
$a=[n_{x,y+1}-n_{x,y}]-[n_{x+1,y+1}+n_{x+1,y}]$. This is evaluated at time
$t$, and the fields living on the plaquette are updated at time $t+1$ by
adding $\epsilon a$ to sites $(x,y)$ and $(x+1,y+1)$ and subtracting $\epsilon
a$ from the other two sites. The constant $\epsilon$ is included to ensure
that the field remains $\in[-1,1]$, and plays only a quantitative role in the
scaling study. To deduce the dynamic rule for the activity at a single lattice
site, we must consider the four neighboring plaquettes around it which can
affect the field on the chosen site via plaquette updates. By summing $a$ for
the four plaquettes with equal weight, and carrying out a careful grouping of
terms, we see that the equation reduces exactly to the dynamical equation
discussed in the previous subsection. Using a Taylor series expansion once
again leads to a dynamical equation of the form given in Eq. (3). We attempt a
data collapse for the decay of $R(t)$ as defined in the previous subsection
for stripe configurations using the dynamic rule mentioned above. As shown in
Fig. 3, we once again find a satisfactory data collapse to a single
exponential for a large range of $k$ values.
Before we turn to the case of the hard-core model, we must note that the
dynamics described in this subsection differ in one crucial way from the hard-
core model. We can see this by considering all the plaquette configurations
which yield a non-zero value of $a$ (these are listed in Fig. 4), and
observing that only the two completely “flippable” (or equivalently with the
largest magnitude of $a$) contribute to dynamics in the hard-core case. This
is not the case if one considers the dynamical rule used here and in the
previous subsection (all plaquettes with all values of $a$ are updated). We
argue in the next subsection that this leads to a strong violation of Eq. (3).
Figure 3: Dynamics of $R(t)$ for a $128\times 128$ system with continuum
fields at each lattice site, against $k^{4}t$ for various values of $k$ and on
a log-linear scale. The solid line is a fit to a single exponential.
### III.3 Continuum theory under the hard-core constraint
Now we turn to the model described in the previous section, and restrict
particle number to at most one per site, and dynamics only to proceed via
exactly flippable plaquettes. Note that now we cannot define an updated state
using just the definition of $a$ as in the previous subsection. We require a
function which evaluates to unity only for the two flippable plaquette
configurations and zero for all others. For the purpose of this subsection, it
is more convenient to set $n_{(x,y)}=1$ for an occupied site and $-1$ for an
unoccupied site. It is not apparent if there is a unique function which
achieves this, and one of the simplest functions which we were able to find on
the plaquette to satisfy these constraints is
$\displaystyle h=\frac{1}{4}\big{[}$
$\displaystyle(n_{x,y+1}-n_{x,y})-(n_{x+1,y+1}-n_{x+1,y})$
$\displaystyle-\frac{1}{4}(\Sigma_{p}n)(n_{x,y}n_{x+1,y}-n_{x+1,y+1}n_{x,y+1})$
$\displaystyle\ \ \ \ \ \ \
\times(n_{x,y}n_{x,y+1}-n_{x+1,y}n_{x+1,y+1})\big{]},$ (4)
where $\Sigma_{p}n=n_{x,y}+n_{x,y+1}+n_{x+1,y+1}+n_{x+1,y}$ is the sum of all
$n$ belonging to the plaquette. A careful consideration of the expression
above reveals that it generates a value of $\pm 1$ for the flippable
plaquettes shown in Fig. 4a, while returning zero for all other configurations
(including those in Fig. 4b), thus satisfying our requirements for a hard-core
ring exchange. The second term in the above expression is formed by noticing
that the only configurations which violated the assignment of values we
desired had a difference in the types of pair arrangements on opposite edges.
Before performing approximations on this expression to derive a continuum
theory, we find that it is convenient to expand the product of the last two
bracketed terms discussed above as
$(2n_{x+1,y}n_{x,y+1}-2n_{x,y}n_{x+1,y+1})$, where we have used
$n_{x,y}^{2}=1$ for all $(x,y)$.
Figure 4: Plaquettes which have a non-zero contribution to dynamics in Sec.
III.2. Plaquettes (a) are flippable, and (b) do not contribute under hard-core
discrete dynamics of Sec. III.3. For (b), we did not include particle-hole
symmetric partners (with black and white dots interchanged) for simplicity.
For the evolution $\Delta_{t}n_{x,y}$ of density of a single site, we must
once again consider the four plaquettes in which it participates. The
resulting expression has a term identical to the dynamic equation in the
previous subsections, but has an important addition in the form of the sum of
$\frac{1}{2}(\Sigma_{p}n)(n_{{x^{\prime}},y}n_{x,{y^{\prime}}}-n_{x,y}n_{{x^{\prime}},{y^{\prime}}})$
over the four plaquettes.
To get the continuum limit as done previously, we assume initial conditions
with several flippable plaquettes and a smooth density profile which varies
slowly spatially. An average over stochastic dynamics and initial conditions
thus allows us to replace the terms linear in $n$ with the differential form
$-{\partial_{x}^{2}}{\partial_{y}^{2}}g$, where $g$ is a real valued field
living in continuous space-time aiming at replacing $n$ (we explicitly
distinguish it from $n$ for this subsection due to the correlations possibly
generated by the products of $n$).
To understand the behavior of the averaged value of terms such as
$n_{x+1,y}n_{x,y+1}$, an important assumption about the correlations has to be
made. In the following of this subsection, we work in a mean-field picture
ignoring correlation effects and assume that such terms can be rewritten as
the product of $g$ at the two points. In doing so, we will be able to derive
the complete dynamics only in terms of the field $g$. This mean-field like
assumption is true for the initial condition we work with, as it is drawn from
an uncorrelated ensemble, but it is not a priori evident if the dynamics
maintains this uncorrelated nature or rapidly builds up correlations. The non-
linear term $f$ is expressed as
$\displaystyle(\sum_{x,y}n)(n_{x+1,y}n_{x,y+1}-n_{x,y}n_{x+1,y+1})$
$\displaystyle+$
$\displaystyle(\sum_{x-1,y-1}n)(n_{x,y-1}n_{x-1,y}-n_{x-1,y-1}n_{x,y})$
$\displaystyle-$
$\displaystyle(\sum_{x-1,y}n)(n_{x,y}n_{x-1,y+1}-n_{x-1,y}n_{x,y+1})$
$\displaystyle-$
$\displaystyle(\sum_{x,y-1}n)(n_{x+1,y-1}n_{x,y}-n_{x,y-1}n_{x+1,y}),$
where the subscript below the sum indicates the plaquette over which the sum
is performed indexed by its left bottom site. Once again we expand the
products and ensure that we replace all occurrences of $n_{x,y}^{2}$ with
unity for all $(x,y)$. The equation above is thus reduced to a linear
combination of single body and three body terms. This can be further reduced
by performing the mean field decoupling
$\braket{n_{x,y}n_{{x^{\prime}},{y^{\prime}}}n_{{x^{\prime\prime}},{y^{\prime\prime}}}}=\braket{n_{x,y}}\braket{n_{{x^{\prime}},{y^{\prime}}}}\braket{n_{{x^{\prime\prime}},{y^{\prime\prime}}}}=g_{x,y}g_{{x^{\prime}},{y^{\prime}}}g_{{x^{\prime\prime}},{y^{\prime\prime}}}$,
followed by a Taylor expansion around the relevant site in the derivatives of
$g$. As we began our analysis by considering a sum over the four plaquettes
surrounding the site $(x,y)$, symmetry restrictions apply to the terms
generated by the Taylor series, which require that only terms with non-zero
coefficients have an even number of derivatives in $x$ and $y$, and are
symmetric with respect to the $x\to y$ transformation. Using this constraint
and after some algebra, we find that the only surviving terms arise at fourth
order in the derivatives and that the complete dynamical equation reduces to
the following expression (ignoring a global factor of $1/4$), given by
$\displaystyle\partial_{t}g=-\Big{(}\frac{1}{2}-g^{2}\Big{)}$
$\displaystyle{\partial_{x}^{2}}{\partial_{y}^{2}}g-{\partial_{y}^{2}}g({\partial_{x}}g)^{2}-{\partial_{x}^{2}}g({\partial_{y}}g)^{2}$
$\displaystyle-2({\partial_{x}}g)({\partial_{y}}g)({\partial_{x}}{\partial_{y}}g)-g{\partial_{x}^{2}}g{\partial_{y}^{2}}g.$
(5)
The presence of non-linearities, derived even under a crude mean-field
approximation, suggests that at leading order in the dynamics, the hard core
constraint indeed plays an important role, and may invalidate the expectation
that the coarse grained dynamics are equivalent to those of vanilla sub-
diffusion.
## IV Long time persistence and eventual melt of approximate stripe
configurations
We have seen in the Sec. II that configurations which can be viewed as small
perturbations around a perfect stripe configuration may maintain the memory of
the initial state indefinitely. As these configurations are highly specific,
it would seem unrealistic to choose one of these as the initial state for the
dynamics of large systems. This motivated us to study “approximate” stripe
patterns, which are chosen to be boson density waves with a wave-vector
$\vec{k}=(k_{x},k_{y})$.
Figure 5: A sample boson configuration created by annealing to the target
function $f_{\vec{k}}$ Eq. 6 with $k_{x}^{t}=k_{y}^{t}=0.1875$ and $A=1.0$ for
a $96\times 96$ lattice, along with Fourier components of random initial
states for larger sizes generated using the annealing method for
$k_{x}^{t}=k_{y}^{t}=0.1875$, plotted against the nearest distance $k_{d}$,
shows a clear peak for $k_{d}=0$.
### IV.1 Initial state preparation
To prepare such configurations, we first generate a target distribution using
the function
$f_{\vec{k^{t}}}(\vec{r})=A\sin(\vec{k^{t}}\cdot\vec{r}),$ (6)
where $A$ takes continuous values between $0$ and $1$. We cannot generate a
configuration which has exactly this pattern as a boson configuration can take
only $\pm 1$ (filled/empty) at each site. Thus we perform a Monte Carlo
simulated annealing starting from a charge-density wave state with
$\vec{k^{t}}=k^{t}\hat{e}_{x}+k^{t}\hat{e}_{y}$ using an energy defined as
$E=\sum_{l}(D_{l}-Lf_{\vec{k^{t}}}(\vec{r}))^{2}$, where the sum is over all
diagonals and $D_{l}=\sum_{(i,j)\in l}\sigma_{i,j}$, is the occupancy in
diagonal number $l$. This favors an exact match between the current and target
configurations. By tuning the inverse temperature for the annealing from
$\beta=0.01$ to $\beta=20.48$ in a geometric progression consisting of
multiplying by $2$ every $10L^{2}$ steps, we achieve a stripe configuration
which has a single Fourier component at the target $\vec{k^{t}}$. We ensure
that the proposed configuration changes are long-ranged and respect the
conservation laws discussed in the previous section, thus staying in the
sector where every column and every row has exactly $L/2$ bosons. An example
of a configuration created by this procedure is shown in Fig. 5. To check the
effectiveness of the annealing procedure, we record the final energy and find
it to be $O(L)$, implying that each $D_{l}$ is within an $O(1)$ value of its
target value, which is expected as $D_{l}$ is an integer and the target value
is in general a real number, and not necessarily close to an integer value. We
find that this procedure generates a sharp peak for the desired $\vec{k^{t}}$
amid a weak background which decays with increasing system size. This can be
seen in Fig. 5, where we plot the Fourier component as a function of the
distance from the target peak location in Fourier space
$k_{d}^{2}=(k_{x}-k^{t})^{2}+(k_{y}-k^{t})^{2}$ for various sizes. Due to the
periodic boundary conditions in Fourier space, given by
$(k_{x},k_{y})\leftrightarrow(1-k_{x},1-k_{y})$, we consider the shortest
distance to the expected peak over the naive distance for open boundary
conditions. Note that such a configuration still contains a large number of
flippable plaquettes due to the smooth nature of the target pattern.
Figure 6: Plots for $A=0.7$ : (a) Comparison of average scaled profile
$B_{x}(t)$ from microscopic hard-core dynamics compared to continuum
expression (Eq. (III.3)) for wavelength $\lambda=16$ for $t_{0}=16$. (b)
$R(t)=n_{k}(t)/n_{k}(0)$ against $k^{4}t$ on log-linear scale, with long times
fit to a single exponential.
### IV.2 Comparing with continuum field theory
In the context of the continuum field theory discussed in Sec.IIIC, the field
$g$ at time zero would now simply be equal to $A\sin(\vec{k}\cdot\vec{r})$
with $\vec{k}=\vec{k^{t}}$. It is essential to numerically verify the validity
of the assumption used in developing our continuum theory, namely that spatial
correlations do not play a role in the initial dynamics.
Using this as the initial source field in Eq. (III.3), we find that
${\partial_{t}}g$ reduces to
$Ak^{4}\sin(\vec{k}\cdot\vec{r})\Big{[}-\frac{1}{2}+4A^{2}\cos^{2}(\vec{k}\cdot\vec{r})-\frac{A^{2}}{2}\sin^{2}(\vec{k}\cdot\vec{r})\Big{]}.$
(7)
This immediately indicates that we have lost linearity (which would imply no
dependence on amplitude other than a global proportionality), and that the
theory is no longer exactly separable in Fourier space. Note here that the
terms proportional to $A^{2}$ within the brackets oppose the decay generated
by the simple sub-diffusion, leading to a possibility of further slowing down
the dynamics. The above equation also suggests that for $A\ll 1$, we should
expect to recover vanilla sub-diffusive dynamics. For
$\vec{k}\cdot\vec{r}\in(0,\pi)$, the sign is controlled by the expression
within the brackets, and we can see that a growth of the function can be
achieved for $\sin(\vec{k}\cdot\vec{r})<((8A^{2}-1)/(9A^{2}))^{1/2}$, provided
$8A^{2}>1$ (that is large enough amplitude). This condition is satisfied for
$\vec{k}\cdot\vec{r}$ in the vicinity of $0$ and $\pi$. This effect is rather
non-intuitive as it implies that the local density tends away (towards $\pm
1$) from its equilibrium value (0) under non-equilibrium dynamics for a
tunable range of $\vec{r}$, leading to a local reduction of entropy as the
number of states available locally reduces if we require their average to be
closer to the most extreme values which it can take. To ascertain the extent
to which our microscopic dynamics is consistent with the continuum field
theory developed under the mean field assumption, we can now compare the
averaged value of $n_{x,y}$ against a numerical evolution of Eq. (III.3).
Looking specifically for the feature described above, we plot
$B_{x}(t)=\langle n_{x,x}(t)\rangle/\langle n_{x,x}(0)\rangle$ in Fig. 6a as a
function of $x$ for the minimal period. Our initial condition already sets
$n_{x,x}(0)=A\sin(2kx)$. We see a qualitative match to the prediction from the
continuum equation (in the sense of the sign of the difference to unity of the
ratio is well captured), but a quantitative disagreement (in the amplitude of
this difference) potentially due to the build up of correlation effects beyond
mean-field.
Before we move to the study of the microscopic dynamics, it is worth
considering the time scales over which the initial pattern melts under the
continuum dynamics, as done in Sec. II. We show in Fig. 6b that the dynamics
is still consistent with a single scaling variable, given by $k^{4}t$,
although the relaxation deviates from the single exponential seen in the
simpler cases considered in Sec. II. This implies that the additional non-
linear terms in the continuum dynamics do not alter the scaling of space-time,
and this can be intuitively understood by observing that all non-linear terms
have the same order in derivatives.
Figure 7: For “weak” amplitude $A=0.2$. (a) Dynamics of $R(t)$ as a function
of $k^{4}t$, in a log-linear scale. (b) Zoom on initial dynamics, that shows a
consistent scaling with $k^{4}t$ for sample initial conditions. We do not
present an average over samples as it leads to large fluctuations over the
time axis.
Figure 8: For “large” values of amplitude $A$ : (a) Slow initial behavior of
$R(t)$ as a function of time, for $A=0.5$, $k=0.125$ and a $L=256$ sample. To
detect the beginning of the melting process, we define a threshold of 0.99
(shown by dotted line), and define $t_{0}$ as the intercept. (b) Average
melting starting time $t_{0}$ as a function of $1/k$, for different values of
$A$. The log-log scale emphasizes the power-law dependence. The black solid
line corresponds to a $1/k^{4}$ power-law.
### IV.3 Exact numerical evolution: Beginning of melting
We now study the dynamical behavior starting from a configuration generated by
the method discussed above. To carry out an analysis of the evolution of the
coarse-grained structure of the initial configuration, we choose to study the
decay of the dominant Fourier component via the already defined ratio
$R(t)=\frac{n_{\vec{k}}(t)}{n_{\vec{k}}(0)}$. To get an intuition about the
effect of lattice spacing, we first consider the decay of $R(t)$ over a range
of decreasing values of $k$, starting from $k=1/2$. As shown in Fig. 7a, for
$A=0.2$, there are clear deviations away from a single exponential for large
values of $k$, whereas intermediate values of $k$ appear to agree partially,
and for small values of $k$, we once again deviate from the expected $k^{-4}$
scaling. To study the initial stages of the decay of the wave pattern, we can
look at $R(t)$ in the regime where it is close to unity. Here we find a
collapse consistent with a time scaling with $k^{-4}$ for a large range of
$k$, including large $k$ values (as shown in Fig. 7b). This is quite
surprising, because the regime of short-time scales and large-wave vectors is
not the one where we expect the hydrodynamic prediction of Sec. III to hold.
We have no simple explanation for this observation.
For larger values of $A$, we observe a completely different behavior. We find
that $R(t)\approx 1$ for a non-trivial amount of time after the initialization
of the dynamics, an illustration of this behavior of $R(t)$ is given in Fig. 8
for $A=0.5$. We define the beginning of the melting process by the first time
$t_{0}$ at which $R(t)$ crosses $0.99$. This threshold is chosen arbitrarily
and a different threshold does not change the result qualitatively (as shown
using a threshold of $0.9$ in the inset of Fig. 8b). We study this for a few
values of $k$, and for $20$ realizations of the initial condition for each
$k$. In addition to this, for each realization, we run sufficient number of
realization of the stochastic dynamics to ensure that we get a good estimate
for $t_{0}$. We have taken systems of linear size in the range of $150-250$,
as the property of self-averaging allows us to narrow the spread in the values
of $t_{0}$. We find a strong dependence of the averaged $t_{0}$ on the inverse
wave-vector $1/k$. A linear fit on a log-log scale of $t_{0}$ vs $1/k$ reveals
various power law regimes (see Fig. 8b) that depend on the amplitude $A$, the
most extreme of which is achieved for $A=1$, where the dependence is
$t_{0}\sim(1/k)^{18.1(3)}$. Such a strong dependence on the initial pattern
suggests that the mechanism for melting is initiated by a coordinated
rearrangement of bosons which is favorable for dynamics. Note that for all
values of $A$, there exists a window of $1/2>k>k^{0}(A)$, where we find good
agreement with $(1/k)^{4}$, as predicted from the continuum theory, with
$k^{0}(A)$ reducing with reducing $A$.
### IV.4 Exact numerical evolution: Post-melt scaling
The continuum theory developed in Sec. III, which can be valid only for small
wave-vectors and long times, cannot describe the results in Fig. 8b and the
period for which we observed $R(t)\approx 1$, and this suggests that it is not
the appropriate theory to understand this “prethermal” behavior.
We can check however if the continuum prediction is upheld after the melting
process with a simple scaling collapse. We first define a time
${t^{\prime}}=0$, which is taken to be the well after the beginning of the
melting process, as the time at which $R(t)$ drops below a threshold of $0.01$
(chosen just for the convenience of analysis, once again we checked that this
value only plays a quantitative role). We can now attempt a parameter free
scaling collapse of $R({t^{\prime}})$ and would expect to get a linear profile
on a log-linear plot for a range of $k$ by scaling the ${t^{\prime}}\to
k^{4}{t^{\prime}}$. We present this analysis in Fig. 9 for $A=1.0$ and find a
reasonable agreement with the expectation developed above, even if not
entirely satisfactory. To further check the applicability of the continuum
theory, we also consider ring exchange dynamics over $2\times 1$ plaquettes,
which we expect to the same Eq. (3), but have much short pre-thermal times (as
naturally be expected from a longer-range type of exchange). This allows to
probe a larger range of wave-vectors $k$ values, as presented in Fig. 9 where
the adherence to a $k^{4}{t^{\prime}}$ is clearly improved.
Figure 9: $R(t)$ scaled by the value closest to $0.01$ ($R(t_{b})$) in our
dataset vs $k^{4}t^{\prime}$ where $t^{\prime}=t-t_{b}$ is the time shifted by
the reference time $t_{b}$,for different values of $k$. A log-linear scale is
chosen to emphasize the expected scaling in $k$ for different values of $k$.
(a) : For the main ring-exchange model discussed in the manuscript. (b)
Similar scaling for an extended model where we allow ring-exchanges models
around $2\times 1$ and $1\times 2$ plaquettes.
## V Pattern of dynamical activity in prethermal and melting process
Figure 10: (a) A sample initial configuration for an $80\times 80$ lattice
with $k=0.05$ and $A=0.6$. Intersection of axes indicates origin. (b-d)
Flippability correlator $C(x,y,t)$ for the same system size, averaged over
$80$ initial conditions and $128$ realizations of the dynamics for each, at
times (left to right) $t=2^{17},2^{19},2^{21}$.
The previous section shows that a mean field treatment is unable to recreate
the slow dynamics seen numerically, thus hinting at the presence of
correlations which are neglected at a mean field level. To understand the
large-scale mechanisms involved in the prolongation of the prethermal lifetime
and the onset of melting, we consider the build up and correlations of
flippable plaquettes. This serves as a proxy for identifying dynamically
active regions and their evolution. We study real-space flippability
correlations through the normalized connected correlator
$C(x,y,t)=\frac{\braket{P(0,0,t)P(x,y,t)}-\braket{P(0,0,t)}\braket{P(x,y,t)}}{\braket{P(t)}^{2}},$
where $P(x,y,t)=1$ if the plaquette whose left bottom site is $(x,y)$ is
flippable and zero otherwise, and $P(t)$ is the spatially averaged density of
flippable plaquettes. Large (low) values of $C(x,y)$ at a given time will thus
indicate strong (weak) correlations of flippability with the initial point.
Recall that we have chosen starting configurations from annealing to a
potential which is only a function of the tilted coordinate $x+y$. This means
that we have a freedom of choosing the origin for our correlation function at
any $x-y$ for fixed $x+y$. We take advantage of this symmetry by averaging
over all equivalent positions of the origin. For the $x+y$ position of the
origin we use our potential function as defined in Eq. (6) and set our origin
to be the point satisfying $\vec{k}\cdot\vec{r}=\pi/4$. We make this choice as
it lies at the threshold between highly active and inactive regions, defined
by $\braket{n}=1/2$ (medium boson density, high density of flippable
plaquettes) and $\braket{n}=1$ (high boson density, low density of flippable
plaquettes).
Figure 11: (a) A sample filled-staggered-unfilled-staggered type initial
configuration, as described in the main text, with $W=13$ consecutive
diagonals filled. (b) $R(t)$ at the dominant wave-vector for various values of
$W$, against rescaled time $k^{4}t$. (c) Flippability correlator $C(x,y,t)$
for configuration in (1) at $t=2^{23}$ (d) $C(x,y,t)$ for $W=27$ at $t=2^{21}$
with the same conditions. (e) Same as plot for $W=27$, scaled instead by
$\langle P_{d_{i}}\rangle\langle P_{d_{j}}\rangle$.
### V.1 Density wave patterns
We begin by considering the initial conditions studied in the previous
sections. To ensure that we are able to observe large scale (slowly varying
spatial) features in the correlation function, we choose a system size of
$80\times 80$ and $k=0.05$ at $A=0.6$ (a particular realization of this is
shown in Fig. 10). To gather high quality statistics we average over 80
realizations for the initial condition and for each realization we run 128
realizations of the stochastic dynamics. The time $t_{0}$, which denotes the
beginning of the melting process, is $\approx 10^{5}$ or $2^{17}$ for
$k=0.05$, as seen in Fig. 8. We study $C(x,y,t)$ up to time scales of $2^{21}$
and find that strong anti-correlations develop at short range for the density
of flippable plaquettes at early times, and sustain until the intermediate
stages of melting. This is shown in Fig. 10, where snapshots of $C(x,y,t)$ at
$t=2^{17}\to 2^{21}$ are presented.
This profile shows the development of active regions (given in red) surrounded
by inactive regions (blue), and suggests that a mechanism of excluded
dynamical regions may be the source of dynamical slowing down. Note also that
Fig. 10 suggests a bias of the dynamics towards the $x$ and $y$ axes, which
may be expected from the continuum theory as well, due to the lack of radial
symmetry. We see that even before the melting time, weak correlations are
already built up along the $x$ and $y$ axes, but in a manner which is anti-
correlated and modulated with the approximate stripe width. We find that this
pattern begins to appear at times as short as $t=2^{4}=16$, but we present
data at the last possible time value before melting which we have recorded as
the pattern is visible with significantly more clarity.
Another important aspect of the correlation pattern to note here is the
dependence on the $x-y$ coordinate (“perpendicular” to the initial wave-
pattern), which is not built in in the initial condition as the energy used in
the annealing process used to generate the configuration has no $x-y$
dependence. This behavior cannot also be expected from the continuum theory
since we start of with a single wave like configuration with $k_{x}=k_{y}$.
We find that an important condition for the existence of such correlation
patterns is that they are specific values of $A$ where the dynamics is not
described by simple sub-diffusion. For example, we studied the case of
$k=0.05$ at $A=0.2$, which shows conventional dynamics (as far as we can see
from Fig. 8), and did not find any non-trivial correlations down to a
precision of $10^{-4}$ in $C(x,y,t)$. This is as expected from a mean-field
treatment, as it forbids correlations by definition.
The evolution of the correlation landscape for the duration of the melting
process presented in Fig. 10 suggests that it proceeds through a merging of
dynamically active regions. This will be discussed in more detail for step-
like initial conditions, where a complete melting process can be observed with
higher clarity and on longer times.
### V.2 Maximally active square wave initial conditions
To gain a better understanding of the melting process, we begin with a more
artificial initial condition chosen to have regions of flippable plaquettes
with maximal and minimal density. We consider alternating patterns such as the
one shown in Fig. 11, with $W$ (necessarily odd) consecutive diagonals filled,
followed by $W+1$ in a perfect staggered (“Néel”) pattern, followed by $W$
empty diagonals, and finally by another $W+1$ in a Néel pattern to close out
one period. These constraints are chosen to ensure that a filled region is
bounded by empty diagonals and vice versa. The lattice size is then
$N(2W+2(W+1))$, where $N$ is the number of periods. Although these
configurations show melting times (quantified by the decay of the appropriate
Fourier component) which are faster than those studied above, they show a
scaling fore pre-melting which is slower than the conventional $k^{-4}$, as
seen in Fig.11. We find that the flippability correlator $C(x,y,t)$ for these
unconventional initial conditions shows an extension of the anti-correlated
pattern in the $x-y$ direction with a periodic modulation across distances
large compared to lattice spacing. This is seen clearly for $W=13$ with $N=2$
at time of $t=2^{23}=8\times 10^{6}$ in Fig. 11.
Figure 12: (a) $R(t)$ for $W=7$ and 2 stripes with vertical lines marking the
times (b-f) $[4,6,8,11,14]\times 2^{20}$ for which $C(x,y,t)$ profiles are
collected.
An important characteristic of the pattern described above is the value of the
wave-vector associated to this pattern. We find that this is determined by the
approximate stripe width, i.e. the wavelength $1/k$. Our data suggest that
this occurs due to a development of the correlation along the $x$ and $y$
axes, which is limited by the size of active regions (high density of
flippable plaquettes). Growth of correlation within an active region is
mediated by alternating patterns of correlation and anti-correlation, which
can be thought of as being generated by reflections off of the boundary
between active and inactive regions, as shown in Fig. 11. Repeated reflections
create the observed periodic pattern, thus linking the periodicity and the
width of the initial stripes. This is clearly seen in the approximate
periodicities of the pattern for $W=13$ and $27$ in Fig. 11 where the profiles
look similar even though the stripe width is changed by a factor of 2. Strong
signatures of the periodic pattern for stripe widths (or alternatively
wavelengths) of $13$ and $27$ at approximately the same time indicates the
emergence of this pattern at a time scale which only weakly depends on the
wave-vector $k$, and that it develops at a time scale parametrically much
smaller than the melting time. Note that an extrapolation based on Fig.11b
indicates the beginning of melting to occur earliest at $t\approx 2^{27}$,
assuming a lower bound of the scaling behavior as $k^{-4}$ (which is clearly
slower than the scaling our numerics show in Fig.11b).
To develop an estimate of the relevance of this patterning to the process of
melting, we first remove the normalization of $\braket{P(t)}$ which we have
absorbed into the definition and instead look at $C(x,y,t)$ scaled by $\langle
P_{d_{i}}(t)\rangle\langle P_{d_{j}}(t)\rangle$, where $\braket{P_{d_{i}}(t)}$
is the average number of flippable plaquettes in diagonal number $i$ at time
$t$. This makes the correlation function effectively lie between $-1$ and $1$,
with either limit describing a saturation to the largest (smallest) values. In
particular, a value of $-1$ implies that the density of flippable plaquettes
is zero, leading to a complete arrest of the dynamics. This version of
$C(x,y,t)$ is plotted in Fig. 11e and we see clearly that close to the
frontier of dynamical activity, we do get a complete anti-correlation around
the reference region. This suggests that the dynamical activity is likely
strongly controlled by the unconventional pattern in the $x-y$ direction.
Now we turn to an investigation of the role of the periodic patterns observed
above in the melting process. We can study this by choosing a small enough $W$
so that the melting process is captured within the time scale of $10^{9}$
which we can simulate. We find that for $W=7$, the ratio $R(t)$ begins to
deviate from unity at a time $t_{0}\sim 2^{21}=2\times 10^{6}$, and reaches a
value of 0.01 by $t_{b}\sim 2^{24}=1.6\times 10^{7}$ (data plotted in Fig.
12). In this interval, we find at time $t=2^{20}$ a formation of stable
periodic structures in the $x-y$ direction, and see a connection of the
dynamically active regions for $t=2^{22}$. Finally by $t=2^{24}$, the
correlation pattern has relaxed into a single wave in the $x+y$ direction,
restoring the symmetry in the $x-y$ direction expected from the continuum
prediction. The role of correlations is expected to be insignificant past this
time point.
## VI Summary and outlook
We studied a system of hard-core bosons on a square lattice evolving under
classical stochastic dynamics using ring exchanges. We find that boson density
waves motivated by the patterns present in the Hilbert space fragmentation of
this model approach equilibrium on an extremely long time scale, which
diverges with the inverse wave-vector as $k^{-\alpha}$ for small $k$, where
$\alpha$ depends on the amplitude $A$ of the initial density wave
configuration. For decreasing values of $A$, we find increasingly larger
windows in the large $k$ regime where we observe a scaling of $k^{-4}$. Our
numerical studies and mean-field treatments of simplified models show the
emergence of sub-diffusion on the other-hand in the hydrodynamic low $k$
limit. For the hard-core case, we derive a dynamical equation using a mean-
field like assumption which is able to capture partially the unconventional
feature of an increase in slope around the nodes of the pattern driven by the
non-linear terms in the dynamical equation but which also predicts a time
scaling of $k^{-4}$. This leads us to the conclusion that the microscopic
dynamics are controlled by strong correlations which are built up in the
prethermal regime, which cannot be accounted for by this mean-field treatment.
To understand the mechanism driving the long prethermal regime, we study the
correlation function of flippable plaquette density for different instances in
time. We find that a strong anti-correlation is built up in the direction
orthogonal to the modulation direction for the initial condition, even
extending to the entire length of the diagonals for intermediate system sizes.
We thus observe the formation of isolated dynamical puddles and find that the
mechanism leading to the melting is characterized by a merger of these
puddles, which relax into a profile which is consistent with what we expect
from the continuum theory. Thus, we have found an example where correlations
control crucial aspects of the dynamical behavior, and a simple mean-field
like hydrodynamic description is insufficient. An important question remains:
how generic are the long melting processes that we observe, in particular to
which additional classes of initial conditions do they apply?
We have shown that the decay of the overlap with the initial state with time
shows a similar behavior upon comparing the exact quantum dynamics and our
classical automaton for a small system size. This leads us to speculate that
the behavior studied in the automaton language might inherit properties which
can also be relevant for the exact quantum evolution for long times, as
already seen for the study of sub-diffusive behavior in 1D Feldmeier _et al._
(2020). First of all, we expect that our hydrodynamic description might be
equally valid also for the quantum dynamics, in case the quantum system
exhibits a hydrodynamic long-time behavior. This would imply sub-diffusion
also in the quantum case. Further, it might be possible that the onset of
hydrodynamic behavior also in the quantum model might be delayed to very long
times. This would have the consequence that a large intermediate time window
exists with unconventionally slow dynamics. As this model is related to the
quantum link model Karpov _et al._ (2021) via a relaxation of charge
conservationKhudorozhkov _et al._ (2021), similar physics may be expected to
emerge there in a more restricted setting.
The crossover between Hilbert space fragmentation, which dictates preservation
of the memory of the initial state to arbitrarily long times, and the long
prethermal plateaus we see here, is another promising direction in which
investigations can be carried out to better our understanding of the processes
involved. Finally, the rapid increase of the lifetime of the prethermal
behavior with inverse wave-vector suggests that rare events, which create
configurations allowing the applicability of the hydrodynamic theory, play a
key role. An improved understanding of the potentially large deviations which
lead to the above mentioned phenomenon would help greatly in developing a
coarse-grained description of the dynamics at the threshold between the
prethermal and equilibrium regimes.
###### Acknowledgements.
We would like to thank Clément Sire, Kedar Damle and Frank Pollmann for
fruitful discussions. Computational resources for this project were provided
by LPT Toulouse and MPIPKS. This project has received funding from the
European Research Council (ERC) under the European Unions Horizon 2020
research and innovation program (grant agreement No. 853443).
## Appendix A Low energy behavior of random configurations
Figure 13: $C(\vec{k},t)$ normalized by $C(\vec{0},t)$ for times $t=2^{12}$
(a) and $2^{16}$ (b).
Here we follow the analysis developed in Ref. Sala _et al._ (2021) for a
similar stochastic model. The authors consider a classical automaton on a
square lattice which also conserves the dipole moment similar to the model we
have considered. The microscopic dynamics is considered from the perspective
of the long time correlation function, defined as
$C(\vec{r},t)=\braket{n_{\vec{r}}(t)n_{0}(0)}$, where the expectation value is
over random initial conditions and realizations of stochastic dynamics. The
authors of Ref. Sala _et al._ (2021) estimate the correlation function in
momentum space ($C(\vec{k},t)$) for a time much larger than the scale of the
microscopic dynamics (using both automaton dynamics and an effective
analytical ansatz) and find sub-diffusive features with “hidden” modulated
symmetries corresponding to certain patterns in the Brillouin zone (see Fig.2b
of Ref. Sala _et al._ (2021)).
In this appendix, we would like to check if this analysis can help identify
the slow dynamics we observe for large scale modulated patterns. The energy
spectrum as a function of $\vec{k}$ has already been identified in a mean-
field framework by Paramekanti et al. in Ref. Paramekanti _et al._ (2002) for
the form of ring-exchange used in our work, to be given by
$E_{\vec{k}}\propto|\sin(k_{x}/2)\sin(k_{y}/2)|$. This trivially implies that
the lines $k_{x}=0$ and $k_{y}=0$ host zero modes, and that this should be
visible in $C(\vec{k},t)$ in the long time limit. Note that this profile does
not explain the slow dynamics which we have explored in this work, as that
would at the minimum require the presence of slow modes along tilted
directions such as $k_{x}=k_{y}$. To check if this expectation is borne out by
ring-exchange dynamics, we perform numerical evolutions of random
configurations of a $200\times 200$ lattice for $t=2^{12}$ and $2^{16}$, shown
in Fig. 13. We can clearly see that the profile is consistent with non-zero
values being present only along the lines $k_{x}=0$ and $k_{y}=0$ at long
times. This aspect of the dynamics is consistent with the mean-field treatment
developed in the body of the manuscript. This is to be expected since random
configurations correspond to the $A\to 0$ limit of our study of modulated
patterns, which was shown to be well-described by the “vanilla” mean-field
dynamics.
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|
# NOPE-SAC: Neural One-Plane RANSAC for Sparse-View Planar 3D Reconstruction
Bin Tan, Nan Xue, Tianfu Wu, Gui-Song Xia B. Tan, N. Xue and G.-S. Xia are
with the School of Computer Science, Wuhan University, Wuhan, China, 430072.
T. Wu is with the Department of Electrical and Computer Engineering, North
Carolina State University, Raleigh, NC, USA, 27606.(Corresponding author: Nan
Xue.)
###### Abstract
This paper studies the challenging two-view 3D reconstruction in a rigorous
sparse-view configuration, which is suffering from insufficient
correspondences in the input image pairs for camera pose estimation. We
present a novel Neural One-PlanE RANSAC framework (termed NOPE-SAC in short)
that exerts excellent capability to learn one-plane pose hypotheses from 3D
plane correspondences. Building on the top of a siamese plane detection
network, our NOPE-SAC first generates putative plane correspondences with a
coarse initial pose. It then feeds the learned 3D plane parameters of
correspondences into shared MLPs to estimate the one-plane camera pose
hypotheses, which are subsequently reweighed in a RANSAC manner to obtain the
final camera pose. Because the neural one-plane pose minimizes the number of
plane correspondences for adaptive pose hypotheses generation, it enables
stable pose voting and reliable pose refinement in a few plane correspondences
for the sparse-view inputs. In the experiments, we demonstrate that our NOPE-
SAC significantly improves the camera pose estimation for the two-view inputs
with severe viewpoint changes, setting several new state-of-the-art
performances on two challenging benchmarks, i.e., MatterPort3D and ScanNet,
for sparse-view 3D reconstruction. The source code is released at
https://github.com/IceTTTb/NopeSAC for reproducible research.
###### Index Terms:
Planar 3D Reconstruction, Two-view Camera Pose Estimation, Sparse-view 3D
Reconstruction, Deep Learning.
## 1 Introduction
Two-view 3D reconstruction is a fundamental and longstanding problem in
computer vision, which is usually formulated to recover the relative camera
pose and the scene geometry by feature correspondences [1]. Although such a
classic formulation has been well established with keypoints in conventional
Structure-from-Motion systems [2, 3, 4, 5], they are challenged by severe
viewpoint changes and low-texture appearance of the captured images in the
indoor scenes as shown in Fig. 1(a). In this paper, we are interested in
addressing such a challenging configuration for two-view pose estimation and
scene reconstruction, which is named as _sparse-view_ 3D reconstruction in
literature [6, 7, 8].
(a) SuperGlue Keypoint Correspondences[9]
(b) Plane Correspondences
(c) Camera Poses and Planar Reconstruction
Figure 1: An illustration of the indoor sparse-view 3D reconstruction. (a)
shows the keypoint correspondences of SuperGlue [9] including only 4 inliers
(Green lines) and 8 outliers (Blue lines). (b) and (c) present the plane
correspondences and reconstructed 3D planar scene of our NOPE-SAC from two
sparse-view images. The Black frustum shows the ground truth camera of the
second image. The Pink, Gray, and Blue frustums show the initial, the one-
plane hypothetical, and the final cameras of the first image estimated by our
NOPE-SAC.
The _sparse-view images_ are often resulted by large viewpoint changes up to
scene scale in image capturing, which manifests as a low overlapping rate
between the images. For instance, as shown in Fig. 1(a), even though adopting
the state-of-the-art SuperGlue [9] for keypoint correspondences in the sparse-
view image pair, it is infeasible for the well-known 5-point algorithm [10]
because there are only $4$ inliers among the $12$ putative correspondences.
However, when we carefully check the input sparse-view images from pixel level
to object level with a focus on the wall, floor, and door, in our mind, these
two images can be aligned in a unified 3D space. The pioneering works [7, 8]
justified such an observation in learning-based reconstruction systems by
relying on 3D plane correspondences, in which they formulate the challenging
relative pose estimation problem in a voting form in accordance with the
geometric correctness of plane correspondences.
Voting the pose hypotheses by putative 3D plane correspondences in [7, 8]
shares a similar spirit with the keypoint-based RANSAC framework [11] for pose
estimation, but they use precomputed pose hypotheses instead of computing them
from the inlier correspondences. To ensure that the precomputed pose
hypotheses can cope with complicated pose distribution, the number of
hypotheses would be large (_e.g., 1024 in SparsePlane[7]_), which induced a
contradiction between a large number of hypotheses and the small number of
plane correspondences. For example, there are only 4 plane correspondences in
Fig. 1(b) for hypotheses voting, thus being unstable and unreliable to
distinguish the best hypothesis from the 1024 ones. To this end, a time-
consuming continuous optimization with the assistance of keypoint
correspondences is required [7] for accurate sparse-view 3D reconstruction.
Although they achieved impressive reconstruction results, from the perspective
of RANSAC [11], they ignored a key problem of pose hypotheses generation and
are thus inflexible and computationally expensive for sparse-view 3D
reconstruction.
In this paper, we pursue to formulate the camera pose estimation with pure
plane correspondences in RANSAC to get rid of the fixed pose hypotheses.
Considering the fact that explicitly modeling the relationship between the
plane correspondences and the relative camera pose in the sparse-view setting
would lead to ill-posed minimal problems [12], we leverage neural networks to
encode the parameters of plane correspondences into a pose embedding space and
learn to generate the pose hypotheses. We minimize the number of plane
correspondences to 1 for the pose hypothesis generation, namely one-plane
pose, and thus propose our Neural One-PlanE RANSAC (NOPE-SAC) framework. With
the one-plane pose generation, the number of pose hypotheses is significantly
reduced to be the same as the number of plane correspondences. Consequently,
the contradiction between the number of hypotheses and those of the supported
inlier plane correspondences is largely alleviated, which makes it possible to
fully utilize the information of plane correspondences for recovering accurate
camera pose from sparse-view images without using post-processing steps such
as the continuous optimization in [7].
We build our NOPE-SAC together with the plane detection and matching modules
to establish a complete planar reconstruction system. More precisely, given
two sparse-view images, our NOPE-SAC first detects 3D planes on each image and
makes the plane correspondences by solving an optimal transport problem as in
[9]. To start the pose estimation in RANSAC with a coarse initial camera pose,
we leverage a convolutional neural network (ConvNet) for the learning of pose
embedding which is then used to decode the initial pose. Then, the parameters
of each plane correspondence are encoded together with the initial pose
embedding to form a new pose embedding for the generation of one-plane pose
hypotheses. Due to the flexibility issue of pose initialization from ConvNets,
we present an arbitrary pose initialization module (AIM) that follows the
auto-encoder structure to encode poses from other methods into embeddings.
Next, with a coarse initial camera pose and $N$ putative plane
correspondences, our NOPE-SAC generates $N$ one-plane pose hypotheses, which
are then voted by the geometric cost of the plane correspondences. Finally,
the refined pose is estimated by fusing all hypotheses together with voting
scores, and the holistic planar reconstruction is approached as the final
output. Fig. 1 presents an illustrative example of our NOPE-SAC for the
sparse-view input images. For the four plane correspondences predicted by the
siamese network, our NOPE-SAC yields an initial pose (with the pink frustum)
and four one-plane pose hypotheses (with the gray frustums), and then use the
plane correspondences to vote and fuse the hypotheses (including the initial
pose) for achieving a holistic 3D reconstruction.
In the experiments, we evaluate our NOPE-SAC on two indoor benchmark datasets,
i.e., Matterport3D [13] with sparse-view splits created by SparsePlanes [7]
and ScanNet [14] with a more challenging split created by ourselves (see Sec.
5.1 for details). On both benchmarks, our NOPE-SAC achieves state-of-the-art
performance in terms of pose estimation accuracy and planar 3D reconstruction
precision. Compared to the prior arts (_e.g.,_ PlaneFormers [8]), our NOPE-SAC
pushed the accuracy of pose estimation on the Matterport3D dataset to $73.2\%$
and $89.0\%$ for translation and rotation ($6.4\%$ and $5.2\%$ absolute
improvements), the accuracy of pose estimation on the ScanNet dataset to
$82.0\%$ and $82.6\%$ for translation and rotation ($6.7\%$ and $9.4\%$
absolute improvements), and the Average Precision (AP) of 3D plane
reconstruction to $43.29\%$ on the Matterport3D and $39.39\%$ on the ScanNet
($5.76\%$ and $4.75\%$ absolute improvements). The comprehensive ablation
studies further verified the design rationales of the proposed NOPE-SAC.
In summary, in this article, we present a novel approach, i.e., NOPE-SAC, to
address the challenging problem of sparse-view planar 3D reconstruction in a
RANSAC framework, which fully takes the advantage of end-to-end learning of
deep neural networks. By our NOPE-SAC, the neural one-plane pose hypotheses
alleviate the contradiction between the number of the pose hypotheses and the
3D plane correspondences, thus improving the accuracy of camera pose
estimation without incurring any offline optimization procedures. In
performance, our method sets several new state-of-the-art performances on both
the Matterport3D [13] and the ScanNet [14] datasets for pose estimation and
holistic planar reconstruction. The comprehensive experimental analyzes also
aligned benchmarking protocols for sparse-view 3D reconstruction.
## 2 Related Work
### 2.1 Single-View 3D Reconstruction
One relevant task to indoor 3D reconstruction is to recover 3D scenes from
single images. As one of the most widely used solutions, single-view depth
estimation has been extensively studied [15, 16, 17, 18, 19] to pixel-wisely
predict the depth values of the input images. Benefitting from the advances of
deep learning and the richness of the training data for depth estimation [20,
21], we have witnessed their significant improvements in estimation accuracy
and generalization ability.
Because the single-view depth estimation can only yield unstructured 3D point
clouds, they would bring structural distortions into predictions for
structured scenes, _e.g.,_ the indoor scenes. To this end, some researchers
propose to directly predict 3D planes from a single image to reconstruct the
geometry of structured scene [22, 23, 24, 25]. For example, Liu et al. [23]
applied a two-stage instance segmentation framework to jointly detect plane
instance masks and estimate 3D plane parameters for the single view planar
reconstruction. Although these insightful methods work well for single-view
indoor plane reconstruction, they can not recover the holistic scene because
of the limited field of view in every single-view image.
In this paper, we take the merits of single-view planar 3D reconstruction and
go further to the challenging sparse two-view configurations. We show that the
estimated 3D planes from single-view images are favorable for both camera pose
estimation and planar 3D reconstruction from the low-overlapping two-view
images.
### 2.2 Two-View Camera Pose Estimation
Two-view 3D reconstruction is the most fundamental task in computer vision,
which was formulated to solve the relative camera poses between the input
images and estimate the scene geometry from the camera motions. The problem of
camera pose estimation is the core of this task.
A common solution is to estimate camera poses from keypoint correspondences
[26, 27, 28, 29, 30] relying on a typical 5-point solver [10] with a RANSAC
[11] framework. Following this paradigm, there have been tremendous efforts on
improving the performance of keypoint detection and matching by neural
networks [31, 32, 33, 34, 9, 35]. As our study mainly focuses on the sparse-
view configuration for two-view 3D reconstruction, such a common solution
would be infeasible due to the fact of low-overlapping rate between the
sparse-view inputs for feature correspondences.
Recently, some approaches leverage neural networks to directly estimate camera
poses from feature cost volumes built up on dense pixel correspondences [36,
37, 38, 39]. Besides, Wei et al. [36] combined the traditional and learning-
based methods by estimating dense pixel correspondences with a neural network
and retrieving camera poses with the 5-point solver [10]. Although these
learning-based methods have shown their excellent performances, they largely
depend on sufficient image overlap to achieve motion cues from correspondences
and thus are challenged by sparse-view indoor images with small overlapping
rates.
Most recently, the problem of pose estimation from sparse-view indoor images
has been extensively studied [6, 7, 8]. Due to the challenges of obtaining a
sufficient number of inlier correspondences, these approaches use a large
number of a-priori poses (by clustering the ground-truth camera poses) as the
label space and formulate the problem of pose estimation to label scoring with
respect to classification likelihoods and plane correspondences. Because a
large number of a-priori poses would affect the flexibility and their
estimation accuracy, these approaches have to adopt an optimization scheme for
estimation refinement. Compared to those approaches, we show that although the
number of 3D plane correspondences is very limited in sparse-view
configuration, it works very well in a consensus sampling pipeline by
generating the one-plane pose hypotheses and voting to obtain the accurate
estimation of camera poses in end-to-end learning.
Figure 2: Overview of the proposed NOPE-SAC. Our network first detects 3D
planes (Sec. 3.1) and estimates plane correspondences (Sec. 3.2) from the
input sparse-view images. Then, the final relative camera pose of the first
image (Blue frustum) is voted from a coarse initial pose (Pink frustum) and a
few one-plane poses (Gray frustum) according to the geometric cost of plane
correspondences (Sec. 4). At last, the planar 3D reconstruction is achieved as
the final output. The Black frustum shows the camera of the second image.
## 3 Planar Correspondences Preparation
The overview of our proposed Nope-SAC for planar 3D reconstruction is
illustrated in Fig. 2, which detects 3D planes from the two-view images and
establishes the plane correspondences between views for camera pose estimation
and planar 3D reconstruction. We focus on 3D plane detection and matching in
this section and leave the key components for pose estimation in Sec. 4. With
the estimated relative camera pose, the scene reconstruction is finally
achieved by aligning the 3D plane correspondences between viewpoints.
### 3.1 3D Plane Instance Detection
Similar to [7], we define a plane instance as
$\pi=\left\\{\mathbf{n},d,\mathcal{M},\text{emb}\right\\}$, where
$\mathbf{n}\in\mathbb{R}^{3}$ and $d$ are the plane normal and the offset from
the plane to the camera center, $\mathcal{M}\in\mathbb{R}^{H\times W}$ is the
plane segmentation mask ($H$ and $W$ are the image height and width,
respectively.) and $\text{emb}\in\mathbb{R}^{256}$ is a plane appearance
embedding.
We use the recent PlaneTR [25] as the 3D plane detection module for each input
view with two main modifications: (1) the line segmentation branch in PlaneTR
[25] is excluded to keep the simplicity; (2) the backbone network of PlaneTR
[25] is replaced to ResNet-50 [40] for efficiency.
In computation, we use the output feature of the Transformer decoder in [25]
as the plane appearance embedding and predict the normal and offset of each
plane on top of the plane appearance embedding with a linear layer. The 3D
plane detection module is supervised as:
$\mathcal{L}_{\text{plane}}=\mathcal{L}_{\text{cls}}+20\mathcal{L}_{\text{mask}}+\mathcal{L}_{\text{dice}}+\mathcal{L}_{\text{parm}}+0.5\mathcal{L}_{\text{center}},$
(1)
where $\mathcal{L}_{cls}$, $\mathcal{L}_{mask}$, $\mathcal{L}_{dice}$,
$\mathcal{L}_{parm}$ and $\mathcal{L}_{center}$ are the classification loss,
the mask loss, the dice loss [41], the plane parameter loss and the plane
center loss as used in [25, 42]. We refer readers to [25, 42] for more
details.
### 3.2 Plane Matching
Denoted by $\Pi_{1}={\\{\pi_{i}^{(1)}\\}}_{i=1}^{K_{1}}$ and
$\Pi_{2}={\\{\pi_{j}^{(2)}\\}}_{j=1}^{K_{2}}$ the plane sets from two view
images, we aim to find an optimal partial assignment $A\in[0,1]^{K_{1}\times
K_{2}}$ from $\Pi_{1}$ to $\Pi_{2}$ for plane matching, where $K_{1}$ and
$K_{2}$ are the number of plane instances in two images. The optimal partial
assignment $A$ is solved as an Optimal Transport (OT) problem based on the
scoring matrix $S\in\mathbb{R}^{K_{1}\times K_{2}}$.
#### Scoring Matrix.
Given the plane sets $\Pi_{1}$ and $\Pi_{2}$, we compute the affinity matrices
in terms of appearance and geometry and linearly add them as the final scoring
matrix. For the affinity of plane appearance, we apply an Attentional Graph
Neural Network used in [9] on the plane appearance embeddings to obtain two
encoded sequence ${E_{1}}\in\mathbb{R}^{K_{1}\times 256}$ and
${E_{2}}\in\mathbb{R}^{K_{2}\times 256}$. Then, the appearance score matrix is
computed by
$S_{e}={E_{1}}{E_{2}}^{T}.$ (2)
The geometric affinity is defined by the plane parameters between two 3D
planes. Given an initial relative camera pose $R,\mathbf{t}$ (see Sec. 4.1 for
details), we first convert the 3D plane parameters of $\Pi_{1}$ to
${\widetilde{\Pi}}_{1}$, the warped counterpart of $\Pi_{1}$ under the
coordinate frame of $\Pi_{2}$, and then calculate the geometric affinity
matrix by:
$S_{g}(i,j)=-\lambda_{1}\text{acos}(\tilde{n}_{i}^{(1)},{n}_{j}^{(2)})-\lambda_{2}|\tilde{d}_{i}^{(1)}-d_{j}^{(2)}|,$
(3)
where $\tilde{n}_{i}^{(1)}$ and $\tilde{d}_{i}^{(1)}$ are the $i^{th}$ plane
normal and offset in ${\widetilde{\Pi}}_{1}$. ${n}_{j}^{(2)}$ and
$d_{j}^{(2)}$ are the $j^{th}$ plane parameters in $\Pi_{2}$. $\lambda_{1}$
and $\lambda_{2}$ are set to 0.125 and 0.25 to balance the magnitude of the
two terms.
Finally, the scoring matrix $S$ can be calculated as:
$S=S_{e}+S_{g}$ (4)
#### Optimal Matching.
Given a scoring matrix $S$, we apply a differentiable Sinkhorn Algorithm [43,
44] to output a soft assignment matrix $A$. Then, for a pair of planes
$\\{\pi_{i}^{(1)},\pi_{j}^{(2)}\\}$, they can be regarded as matched planes if
$A(i,j)$ is the maximal score both in the $i^{th}$ row and the $j^{th}$ column
of $A$, and $A(i,j)$ is larger than a fixed matching threshold (0.2 in this
paper). We supervise the plane matching module with the loss defined in Eqn.
(5). More precisely, let $\bar{A}\in\mathbb{R}^{(K_{1}+1)\times(K_{2}+1)}$ to
be the soft assignment matrix augmented with dustbins and the expected soft
assignment matrix $A=\bar{A}_{1:K_{1},1:K_{2}}$, the plane matching loss can
be calculated as:
$\displaystyle\mathcal{L}_{\text{match}}=$
$\displaystyle-\sum_{(i,j)\in\mathcal{A}}\text{log}\bar{A}_{i,j}-\sum_{i\in\mathcal{I}}\text{log}\bar{A}_{i,K_{2}+1}$
(5) $\displaystyle-\sum_{j\in\mathcal{J}}\text{log}\bar{A}_{K_{1}+1,j},$
where, $\mathcal{A}=\\{(i,j)\\}$ are the indices of ground truth matches.
$\mathcal{I}$ and $\mathcal{J}$ are the indices of unmatched planes in two
images respectively.
#### Correspondences Preparation for Pose Estimation.
After the computation of optimal matching by the Sinkhorn layer, we extract a
total number of $M$ plane correspondences, denoted in the set
$\mathbb{P}=\left\\{(\mathcal{P}_{m}^{(1)},\mathcal{P}_{m}^{(2)})\right\\}_{m=1}^{M}$,
where the superscript (i) indicates the viewpoint index. For each plane
$\mathcal{P}_{m}^{(i)}$, we denote the normal and offset in
$\mathbf{n}_{m}^{(i)}\in\mathbb{R}^{3}$ and $d_{m}^{(i)}\in\mathbb{R}_{+}$,
respectively. With the plane correspondence set $\mathbb{P}$, we aim at
estimating the relative camera pose in Sec. 4.
## 4 NOPE-SAC Pose Estimation
Given the initial camera pose $\xi_{0}=(R_{0},\mathbf{t}_{0})$ and a pair of
matched planes $\mathcal{P}_{m}^{(1)}$ in the first-view image
$\mathcal{I}_{1}$ and $\mathcal{P}_{m}^{(2)}$ in the second-view image
$\mathcal{I}_{2}$, the proposed NOPE-SAC learns the one plane pose in the
embedding space of the initial camera pose $\xi_{0}$ and the plane
correspondence $(\mathcal{P}_{m}^{(1)},\mathcal{P}_{m}^{(2)})$. For a total of
$M$ putative plane matches, there will be $M$ one-plane poses and we then
learn to vote those one-plane pose hypotheses according to the geometric cost
in the planar consensus set. Thanks to the pose hypotheses by NOPE-SAC only
requiring one plane correspondence to yield a pose hypothesis, the imbalance
issue between the number of hypotheses and the cardinal of the consensus set
is largely alleviated for accurate pose estimation.
### 4.1 Camera Pose Initialization
In the consensus sampling pipelines for camera pose estimation, a coarse
estimation is required to kick off the estimation of pose parameters.
Different from the keypoint-based RANSAC (and its variants) that did the
coarse estimation by randomly sampling a small subset of keypoint
correspondences, it would be more challenging to obtain the coarse estimation
from plane correspondences by an explicit mathematical model. To this end, we
resort to the convolutional neural network to learn the initial pose $\xi_{0}$
from the feature volume of image $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$.
#### Pose Regression Network.
We take ResNet-50 [40] as our backbone network for the initial pose
regression. Given two input images $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ of
which they are with the spatial size of $H\times W$, the output backbone
features $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are taken to build a 4D
affinity volume $V\in\mathbb{R}^{(h\times w)\times(h\times w)}$ by
$V(\mathbf{p},\mathbf{p}^{\prime})=\frac{\exp(\mathcal{F}_{2}^{T}(\mathbf{p})\mathcal{F}_{1}^{T}(\mathbf{p}^{\prime}))}{\sum_{\mathbf{p}}\exp(\mathcal{F}_{2}^{T}(\mathbf{p})\mathcal{F}_{1}^{T}(\mathbf{p}^{\prime}))},$
(6)
for any pair of pixels $\mathbf{p}$ and $\mathbf{p}^{\prime}$ in the
coordinates of $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, where $h=H/s$,
$w=W/s$, and $s=32$ is the stride of backbone network. The computation of $V$
is similar to [7], however, we choose to directly predict the pose parameters
instead of learning the classification likelihood with respect to a set of
discrete pose templates as the formulation of classification would hurt the
end-to-end learning for camera pose estimation.
To predict the initial camera pose
$\xi_{0}=(R_{0},\mathbf{t}_{0})\in\text{SE}(3)$, the affinity volume is
rearranged to $\mathcal{V}\in\mathbb{R}^{hw\times h\times w}$, and is then
transformed respectively by 6 convolutional layers (including BatchNorm and
LeakyReLU) and a linear layer to yield two feature embeddings,
$f_{r}^{(0)},f_{t}^{(0)}\in\mathbb{R}^{256}$ for the rotation $R_{0}$ and
translation $\mathbf{t}_{0}$. Here, the stride of each even-numbered
convolution operation is set to 2 to reduce the feature size. With
$f_{r}^{(0)},f_{t}^{(0)}$, it is straightforward to predict the rotation and
translation by linear layers $\text{LinearRot}(\cdot)$ and
$\text{LinearTrans}(\cdot)$ with
$\mathbf{q}_{0}=\text{LinearRot}(f_{r}^{(0)}),\mathbf{t}_{0}=\text{LinearTrans}(f_{t}^{(0)}),$
(7)
where $\mathbf{q}_{0}\in\mathbb{R}^{4}$ is the unit quaternion of the rotation
matrix that satisfies $\left\|\mathbf{q}_{0}\right\|_{2}=1$. The weights of
$\text{LinearRot}(\cdot)$ and $\text{LinearTrans}(\cdot)$ are optimized by the
MSE (Mean-Square Error) loss
$\mathcal{L}_{\text{cam}}^{\text{init}}=\|\mathbf{q}_{0}-\mathbf{q}^{gt}\|_{2}^{2}+\|\mathbf{t}_{0}-\mathbf{t}^{gt}\|_{2}^{2},$
(8)
where $\mathbf{q}^{gt}$ and $\mathbf{t}^{gt}$ are the ground truths of the
rotation (in unit quaternions) and translation between the input images
$(\mathcal{I}_{1},\mathcal{I}_{2})$.
Besides predicting the initial camera poses, the embedding $f_{r}^{(0)}$ and
$f_{t}^{(0)}$ plays a vital role in our proposed NOPE-SAC because it generates
the pose hypotheses by the embedding vectors of initial poses and plane
correspondences. Thus, there would be an issue of inflexibility for the
initial poses that do not come from the convolutional pose estimation network.
We present an Arbitrary Initialization Module (AIM) that reconstructs a given
camera pose via auto-encoding to cope with this situation.
#### Arbitrary Initialization Module.
We featurize any given quaternion $\mathbf{q}^{*}$ (of rotation
$R^{*}\in\text{SO}(3)$) and translation $\mathbf{t}^{*}\in\mathbb{R}^{3}$ in
an auto-encoding manner by:
$\begin{split}f_{r}^{*}=\text{RotEnc}(\mathbf{q}^{*}),\,f_{t}^{*}=\text{TransEnc}(\mathbf{t}^{*}),\\\
\tilde{\mathbf{q}}^{*}=\text{LinearRot}(f_{r}^{*}),\,\tilde{\mathbf{t}}^{*}=\text{LinearTrans}(f_{t}^{*}),\end{split}$
(9)
where $\text{RotEnc}(\cdot)$ and $\text{TransEnc}(\cdot)$ are the encoders for
rotation and translation by MLPs, $\text{LinearRot}(\cdot)$ and
$\text{LinearTrans}(\cdot)$ are the shared layers defined in Eqn. (7) to
reconstruct the camera poses by
$\mathcal{L}_{\text{rec}}=\left\|\tilde{\mathbf{q}}^{*}-\mathbf{q}^{*}\right\|_{2}^{2}+\left\|\tilde{\mathbf{t}}^{*}-\mathbf{t}^{*}\right\|_{2}^{2}.$
(10)
For the training of AIM, we randomly sample the rotations and translations
from the uniform distribution, please move to Section 5.3 for the details.
TABLE I: Detailed architecture of MLPs used in NOPE-SAC Pose Estimation
Module. For each MLP, the corresponding equation, notation, the number of
linear layers, the channel dimensions of inputs and outputs, and the
activation functions are listed.
Equation | Notation | $\\#$ Layers | $\\#$ Channels | Activation
---|---|---|---|---
In | Out
Eqn. (9) | RotEnc / TransEnc | 6 | 256 | 256 | ReLU
Eqn. (11) | $\mathcal{G}$ | 1 | 8 | 1024 | ReLU
4 | 1024 | 1024 | ReLU
1 | 1024 | 1024 | -
2 | 1024 | 1024 | ReLU
1 | 1024 | 1024 | -
Eqn. (12) | $\mathcal{E}_{r}$ | 1 | 1024 | 512 | ReLU
4 | 512 | 512 | ReLU
1 | 512 | 256 | -
Eqn. (12) | $\mathcal{E}_{t}$ | 1 | 1280 | 1024 | ReLU
1 | 1024 | 1024 | ReLU
1 | 1024 | 1024 | -
1 | 1024 | 512 | ReLU
4 | 512 | 512 | ReLU
1 | 512 | 256 | -
Eqn. (13) | $\mathcal{D}_{r}$ / $\mathcal{D}_{t}$ | 2 | 512 | 512 | ReLU
1 | 512 | 256 | ReLU
### 4.2 Camera Pose Refinement
With the initial camera pose $\xi_{0}=(R_{0},\mathbf{t}_{0})$ and its
corresponding feature embedding $f_{r}^{(0)}$ and $f_{t}^{(0)}$, we present
the core of NOPE-SAC that consists of the following steps: (1) One-plane Pose
Hypotheses Generation, (2) Hypotheses Scoring and (3) Pose Refinement based on
the $M$ 3D plane correspondences $\mathbb{P}=\\{\mathcal{P}_{m}\\}_{m=1}^{M}$.
For each plane correspondence $\mathcal{P}_{m}$, the normal vector (3D real
vectors) and the offset (real scalars) from the plane to the camera center in
the input images $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are denoted by
$(\mathbf{n}_{m}^{(1)},d_{m}^{(1)})$ and $(\mathbf{n}_{m}^{(2)},d_{m}^{(2)})$,
respectively.
#### One-plane Pose Hypotheses Generation.
For the $m$-th 3D plane correspondence
$\mathcal{P}_{m}^{(1)}=(\mathbf{n}_{m}^{(1)},{d}_{m}^{(1)})$ in
$\mathcal{I}_{1}$ and
$\mathcal{P}_{m}^{(2)}=(\mathbf{n}_{m}^{(2)},{d}_{m}^{(2)})$ in
$\mathcal{I}_{2}$, we leverage an MLP layer
$\mathcal{G}(\cdot):\mathbb{R}^{8}\mapsto\mathbb{R}^{1024}$ to embed the
correspondence by
$g_{m}=\mathcal{G}(\tilde{\mathbf{n}}_{m}^{(1)}\oplus{\tilde{\mathbf{d}}}_{m}^{(1)}\oplus\mathbf{n}_{m}^{(2)}\oplus\mathbf{d}_{m}^{(2)}),$
(11)
where $\tilde{\mathbf{n}}_{m}^{(1)}$ and $\tilde{d}_{m}^{(1)}$ are the warped
normal of $\mathbf{n}_{m}^{(1)}$ and offset of $d_{m}^{(1)}$ by the relative
camera pose $(R_{0},\mathbf{t}_{0})$, $\oplus$ is the concatenation operation.
Here, the warped plane parameters by the initial camera pose could be regarded
as a kind of normalization to facilitate the learning of plane correspondence
embedding in neural networks.
With the correspondence embedding $g_{m}$, we further transform it to the
rotation feature $e_{m}^{r}\in\mathbb{R}^{256}$ and the translation feature
$e_{m}^{t}\in\mathbb{R}^{256}$ by two MLPs $\mathcal{E}_{r}$ and
$\mathcal{E}_{t}$ as
$\begin{split}e_{m}^{r}&=\mathcal{E}_{r}(g_{m}),\\\
e_{m}^{t}&=\mathcal{E}_{t}(g_{m}\oplus e_{m}^{r}).\end{split}$ (12)
Then, we concatenate $e_{m}^{r}$ and $e_{m}^{t}$ with the embedding
$f_{r}^{(0)}$ and $f_{t}^{(0)}$ of the initial camera pose, and transform the
concatenated embedding by two MLP layers
$\mathcal{D}_{r}(\cdot):\mathbb{R}^{512}\mapsto\mathbb{R}^{256}$ and
$\mathcal{D}_{t}(\cdot):\mathbb{R}^{512}\mapsto\mathbb{R}^{256}$ to yield the
one-plane pose embedding $f_{m}^{r}$ and $f_{m}^{t}$ as
$\begin{split}f_{m}^{r}&=\mathcal{D}_{r}(f_{r}^{(0)}\oplus e_{m}^{r}),\\\
f_{m}^{t}&=\mathcal{D}_{t}(f_{t}^{(0)}\oplus e_{m}^{t}).\end{split}$ (13)
Finally, to obtain the one-plane rotation and translation for the $m$-th plane
correspondence, we leverage the linear layers $\text{LinearRot}(\cdot)$ and
$\text{LinearTrans}(\cdot)$ defined in Eqn. (9) as the shared headnets by
$\begin{split}\mathbf{q}^{(m)}&=\text{LinearRot}(f_{m}^{r}),\\\
\mathbf{t}^{(m)}&=\text{LinearTrans}(f_{m}^{r}),\end{split}$ (14)
where $\mathbf{q}^{(m)}$ and $\mathbf{t}^{(m)}$ are the predicted quaternion
of rotation and the translation.
#### Hypotheses Scoring Layer.
Next to the pose hypotheses generation, we compute the geometric cost for each
predicted rotation and translation. For each pose hypothesis $R^{(m)}$ and
$\mathbf{t}^{(m)}$, the rotation cost $c_{r}^{(m)}\in\mathbb{R}^{M}$ and
translation cost $c_{t}^{(m)}\in\mathbb{R}^{M}$ throughout all $M$ plane
correspondences are calculated by
$\begin{split}c_{r}^{(m)}&=\left(\|\mathbf{n}_{i,m}^{(1)}-\mathbf{n}_{i}^{(2)}\|_{2}\right)_{i=1}^{M}\\\
c_{t}^{(m)}&=\left(\|d_{i,m}^{(1)}\mathbf{n}_{i,m}^{(1)}-d_{i}^{(2)}\mathbf{n}_{i}^{(2)}\|_{2}\right)_{i=1}^{M},\end{split}$
(15)
where $\mathbf{n}_{i,m}^{(1)}$ and $d_{i,m}^{(1)}$ are the warped normal and
offset of the plane $\mathcal{P}_{i}^{(1)}$ by the $m$-th pose hypothesis. By
including the costs of initial pose $(R_{0},\mathbf{t}_{0})$, there will be
two cost matrices $C_{r}\in\mathbb{R}^{(M+1)\times M}$ and
$C_{t}\in\mathbb{R}^{(M+1)\times M}$, which are then fed into two parallel
Hypotheses Scoring Layers to yield the rotation scores
$\mathbf{w}_{r}=(w_{r}^{(0)},\ldots,w_{r}^{(M)})$ and the translation scores
$\mathbf{w}_{t}=(w_{t}^{(0)},\ldots,w_{t}^{(M)})$, respectively. Here, the sum
of scores is equal to $1$ and we use the Softmax operation in our
implementation. As shown in Fig. 3, each Hypotheses Scoring Layer consists of
an MLP layer (including 3 linear layers with ReLU activations and 64 channels
per layer) and one linear layer for score prediction.
To train the scoring layer, we design a scoring loss to supervise the learning
of pose scores dynamically. Let $i$ and $j$ be the indices of rotation and
translation in all pose hypotheses $\\{R^{(m)}\\}_{m=0}^{M}$ and
$\\{t^{(m)}\\}_{m=0}^{M}$ that are closest to the ground truth camera poses.
Then, the heuristic scoring loss can be calculated as:
$\displaystyle\mathcal{L}_{\text{score}}=\|1-w_{r}^{i}\|_{1}+2\|1-w_{t}^{j}\|_{1}+\frac{10}{M}\sum_{m=1}^{M}c_{t}^{(m)}(m),$
(16)
where $w_{r}^{i}$ and $w_{t}^{j}$ are the predicted rotation and translation
scores, and $c_{t}^{(m)}(m)$ means the translation cost of plane
correspondence $\mathcal{P}_{m}$ with the $m$-th pose hypothesis.
Figure 3: An illustration of the Hypotheses Scoring Layer. The input cost
matrix is first passed through an MLP layer and then converted to the scores
with a linear layer and Softmax operation. The top line shows the output
dimensions of each step.
#### Final Pose Estimation.
Following the hypotheses scoring layer, we present a _Soft_ fusion strategy by
default to fuse all the pose hypotheses to obtain the final pose. In detail,
we use the predicted scores $\mathbf{w}_{r}$, $\mathbf{w}_{t}$ to obtain the
embedding of the final pose by
$f_{r}=\sum_{i=0}^{M}w_{r}^{i}f_{r}^{(i)},\,f_{t}=\sum_{i=0}^{M}w_{t}^{i}f_{t}^{(i)},$
(17)
and leverages the layers $\text{LinearRot}(\cdot)$ and
$\text{LinearTrans}(\cdot)$ again as the shared pose predictors to obtain the
final pose $\xi_{\rm ref}=(\mathbf{q}_{\rm ref},\mathbf{t}_{\rm ref})$. We
denote the computation of the camera pose refinement module by
$\xi_{\rm ref}={\rm PoseRef}(\xi_{0},\mathbb{P};\Theta),$ (18)
where $\xi_{0}$ and $\mathbb{P}$ are the required input of the initial pose
and the plane correspondences, and $\Theta$ indicates the parameters of the
neural networks.
Apart from the _Soft_ fusion scheme, there are three representative
alternatives to get the final refined pose from pose hypotheses, which are
summarized as follows:
1. -
Avg: It treats all pose hypotheses equally and sets all scores used in Eqn.
(17) to be $\frac{1}{M+1}$.
2. -
Min-Cost: It selects the minimal-cost rotation/translation hypothesis
according to Eqn. (15) and discard all the remaining hypotheses.
3. -
Max-Score: It takes the maximal-score rotation/translation hypothesis
according to the hypotheses scoring layer as the final pose prediction.
If not explicitly stated, we use the Soft fusion strategy in all experiments,
and we analyze these fusion strategies in the ablation study.
We supervise the final pose prediction with the MSE loss. Because there are
two initial pose predictions $\xi_{0}^{\rm reg}$ (from the regression branch)
and $\xi_{0}^{\rm rec}$ (from the Arbitrary Initialization Module), by taking
the predicted plane correspondences $\mathbb{P}_{\rm pred}$ and the ground-
truth $\mathbb{P}_{gt}$ into account during training, there will be four
predictions of the refined camera pose for the _Soft_ fusion scheme. We
compute the loss functions $\mathcal{L}_{\rm cam}^{\rm soft}$ and
$\mathcal{L}_{\rm score}$ for the refinement module similar to Eqn. (8) and
Eqn. (16), but the four predictions are all taken in the computation. To avoid
instability during training, the Avg fusion strategy is also involved to
compute a loss function $\mathcal{L}_{\rm cam}^{\rm avg}$. Finally, the total
loss function of the refinement module is denoted by
$\mathcal{L}_{\rm cam}^{\rm ref}=\mathcal{L}_{\rm cam}^{\rm
soft}+\mathcal{L}_{\rm cam}^{\rm avg}+0.01\mathcal{L}_{\rm score}.$ (19)
To train the NOPE-SAC for pose estimation, we linearly add the loss functions
$\mathcal{L}_{\text{cam}}^{\text{init}}$, $\mathcal{L}_{\text{rec}}$ and
$\mathcal{L}_{\text{cam}}^{\text{ref}}$ by
$\mathcal{L}_{\text{pose}}=\mathcal{L}_{\text{cam}}^{\text{init}}+\mathcal{L}_{\text{rec}}+\mathcal{L}_{\text{cam}}^{\text{ref}}$
(20)
as the total loss for the camera pose estimation.
## 5 Experiments and Analysis
This section presents the experiments of our NOPE-SAC.
(a) Matterport3D [13] train
(b) Matterport3D [13] test
(c) ScanNet [14] train
(d) ScanNet [14] test
(e) Rotation train
(f) Rotation test
(g) Translation train
(h) Translation test
Figure 4: Dataset analysis of the sparse-view split on the Matterport3D [13]
and the ScanNet [14] datasets. The top row shows the image overlap ratio on
the training and test set of two datasets (from left to right, the average
percentages of image overlap are $20.9\%$, $21.0\%$, $20.6\%$ and $18.6\%$).
At the bottom, we visualize the rotation and translation distributions on the
ScanNet (Blue dot) and the Matterport3D (Red dot) datasets.
### 5.1 Datasets
#### Matterport3D Dataset.
We use the sparse view dataset based on Matterport3D [13] with ground truth
camera poses and plane annotations generated by [7], which contains 31932,
4707, and 7996 image pairs for training, validation, and testing. The size of
each image is $480\times 640$.
#### ScanNet Dataset.
We create a new sparse view split on the indoor ScanNet [14] video dataset
with plane annotations generated by [23]. The image size is $480\times 640$.
We randomly sample 17237/4051 image pairs from 1210/303 non-overlapping scenes
for training/testing. The frame interval within a sampled image pair is at
least 20 and 40 frames in the training and test sets, respectively.
#### Dataset Analysis.
We first analyze the image overlap on the Matterport3D [13] and the ScanNet
[14] datasets. As shown in the top row of Fig. 4, our split on the ScanNet
dataset contains more low-overlap image pairs than the split on the
Matterport3D dataset created by [7]. Further, we visualize the distributions
of rotations (represented as rotation vectors) and translations on the
training and test set of these two datasets. As shown in the bottom row of
Fig. 4, although the translation range on the ScanNet dataset is smaller than
the Matterport3D dataset, the rotation distribution on the ScanNet dataset is
much more complex than the Matterport3D dataset, which makes our split on the
ScanNet dataset more challenging.
### 5.2 Evaluation Metrics
#### Metrics for Camera Pose Estimation.
We evaluate camera poses with the rotation angle error and the translation
distance error as used in [7, 37, 36]. The reported metrics include the mean
and median errors, and the percentage of errors lower than a threshold.
#### Metrics for 3D Plane Reconstruction.
According to SparsePlanes [7], the matched planes are merged together and all
3D planes from two views are converted to the same coordinate system. Then,
the reconstructed full scene is evaluated like a detection problem with the
metric of average precision (AP). A reconstructed 3D plane is regarded as a
true positive if the following conditions of (1) its mask IoU $\geq$ 0.5, (2)
its plane normal angle error in degree is less than $\alpha^{\circ}$, and (3)
its plane offset error in meters is less than $\beta$ are all satisfied.
TABLE II: Comparison of camera poses on the Matterport3D [13] dataset and the
ScanNet [14] dataset.
Method | Translation | Rotation
---|---|---
Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$1m) $\uparrow$ | ($\leq$0.5m) $\uparrow$ | ($\leq$0.2m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 30^{\circ}$) $\uparrow$ | ($\leq 15^{\circ}$) $\uparrow$ | ($\leq 10^{\circ}$) $\uparrow$
Matterport3D dataset |
SuperGlue [9] | - | - | - | - | - | 3.88 | 24.17 | 77.8% | 71.0% | 65.7%
SparsePlanes [7] | 0.63 | 1.15 | 66.6% | 40.4% | 11.9% | 7.33 | 22.78 | 83.4% | 72.9% | 61.2%
PlaneFormers [8] | 0.66 | 1.19 | 66.8% | 36.7% | 8.7% | 5.96 | 22.20 | 83.8% | 77.6% | 68.0%
SparsePlanes-TR [7, 25] | 0.61 | 1.13 | 67.3% | 41.7% | 12.2% | 6.87 | 22.17 | 83.8% | 74.5% | 63.3%
PlaneFormers-TR [8, 25] | 0.64 | 1.17 | 67.9% | 38.7% | 8.9% | 5.28 | 21.90 | 83.9% | 79.0% | 70.8%
NOPE-SAC-Cls (ours) | 0.66 | 1.20 | 65.6% | 37.9% | 9.8% | 2.98 | 19.68 | 84.9% | 83.1% | 80.2%
NOPE-SAC-Reg (ours) | 0.52 | 0.94 | 73.2% | 48.3% | 16.2% | 2.77 | 14.37 | 89.0% | 86.9% | 84.0%
ScanNet dataset |
SuperGlue [9] | - | - | - | - | - | 10.90 | 31.00 | 67.8% | 56.0% | 48.4%
SparsePlanes [7] | 0.56 | 0.81 | 73.7% | 44.6% | 10.7% | 15.46 | 33.38 | 70.5% | 48.7% | 28.0%
PlaneFormers [8] | 0.55 | 0.81 | 75.3% | 45.5% | 11.3% | 14.34 | 32.08 | 73.2% | 52.1% | 32.3%
SparsePlanes-TR [7, 25] | 0.57 | 0.82 | 73.4% | 43.6% | 10.1% | 14.57 | 32.36 | 72.8% | 51.2% | 30.1%
PlaneFormers-TR [8, 25] | 0.53 | 0.79 | 76.2% | 47.0% | 11.4% | 13.81 | 31.58 | 74.5% | 54.1% | 33.6%
NOPE-SAC-Cls (ours) | 0.49 | 0.76 | 77.5% | 50.9% | 14.1% | 9.01 | 27.84 | 77.9% | 69.5% | 55.1%
NOPE-SAC-Reg (ours) | 0.41 | 0.65 | 82.0% | 59.1% | 21.2% | 8.27 | 22.12 | 82.6% | 73.2% | 59.5%
### 5.3 Implementation Details
Our NOPE-SAC is implemented with Detectron2 [45] and the AdamW optimizer [46]
is used for training with a batch size of 16. Because the sparse-view planar
3D reconstruction consists of multiple neural modules for the 3D plane
detection and matching, the camera pose initialization (including the initial
pose regression and the arbitrary pose initialization), and the pose
refinement with one-plane pose hypotheses, we train the whole network in three
stages on the Matterport3D [13] training set. In the first stage, only the 3D
plane detection module is optimized for 12k iterations with the learning rate
of $10^{-4}$. Then, we add the plane matching module, the initial pose
regression network, and the arbitrary initialization module for the second
stage of training in 50K iterations. The initial learning rate in this
training stage is also $10^{-4}$, and we use the multi-step learning rate
schedule to adjust the learning rate at the milestones of 34k and 44k
iterations with the decay factor of 0.1. In the last stage, we add the camera
pose refinement module in training for a total of 14k iterations. The learning
rate in this stage is initialized to $10^{-4}$ for the first 6k iterations and
is decayed to $10^{-5}$ for the last 8k iterations.
On the ScanNet [14] dataset, we finetune the model trained on the Matterport3D
dataset with two additional stages. In the first stage, we train the 3D plane
detection module, the plane matching module, the initial pose regression
network, and the arbitrary initialization module for 20K iterations. The
initial learning rate is set to $10^{-4}$ and divided by 10 after 4.4K
iterations. In the second stage, we add the camera pose refinement module and
train the whole network for 15K iterations. The initial learning rate is set
to $10^{-4}$ and divided by 10 after 2.2K iterations.
When training the arbitrary initialization module (AIM), we first represent
the input rotation as a rotation vector $\textbf{v}^{*}\in\mathbb{R}^{3}$.
Then, we randomly sample each axis of $\textbf{v}^{*}$ and the input
translation $\textbf{t}^{*}\in\mathbb{R}^{3}$ from the uniform distribution
$U(-2.5,2.5)$. At last, the sampled rotation vector $\textbf{v}^{*}$ is
converted to a unit quaternion $\textbf{q}^{*}$ as used in Eqn. 9.
TABLE III: Detailed camera pose comparison with SparsePlanes-TR [7, 25] on the
Matterport3D [13] dataset and the ScanNet [14] dataset. ‘Cls. Top-1’ means the
Top-1 classification pose of SparsePlanes [7]. ‘Con.’ means the continuous
optimization proposed by SparsePlanes.
Method | Translation | Rotation
---|---|---
Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$0.5m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 15^{\circ}$) $\uparrow$
Matterport3D dataset
Cls. Top-1 | 0.90 | 1.40 | 21.1% | 7.65 | 24.57 | 71.7%
[7, 25] w/o Con. | 0.88 | 1.35 | 21.9% | 7.17 | 22.36 | 74.6%
[7, 25] | 0.61 | 1.13 | 41.7% | 6.87 | 22.17 | 74.5%
NOPE-SAC-Cls | 0.66 | 1.20 | 37.9% | 2.98 | 19.68 | 83.1%
ScanNet dataset
Cls. Top-1 | 0.56 | 0.83 | 43.9% | 15.21 | 33.16 | 49.5%
[7, 25] w/o Con. | 0.55 | 0.83 | 44.2% | 14.55 | 32.36 | 51.3%
[7, 25] | 0.57 | 0.82 | 43.6% | 14.57 | 32.36 | 51.2%
NOPE-SAC-Cls | 0.49 | 0.76 | 50.9% | 9.01 | 27.84 | 69.5%
### 5.4 Baseline Configurations
We compare our method with the state-of-the-art learning solutions including
SparsePlanes [7] and PlaneFormers [8] for pose estimation and planar 3D
reconstruction, as well as a keypoint-based solution built on the top-
performing keypoint matcher, SuperGlue [9]. For the SparsePlanes [7] and
PlaneFormers [8], we additionally set two variant baselines to compare the
performance gain with the same 3D plane detection module.
#### SparsePlanes and SparsePlanes-TR
. SparsePlanes [7] is the first sparse-view planar 3D reconstruction approach.
Given a set of precompute pose hypotheses, it detects 3D planes in each view
and predicts the likelihood of the precomputed set of camera poses by deep
neural networks. Then, a discrete optimization problem is solved to obtain the
optimal plane correspondences and the camera pose by taking the geometric
correctness and embedding consistency between 3D planes across views, as well
as the classification likelihood into account. Following the discrete
optimization, a continuous optimization problem is formulated to refine the
camera pose with the assistance of SIFT [26] keypoints. Since the 3D plane
detection modules between SparsePlanes [7] and NOPE-SAC are different, we
build a variant baseline SparsePlanes-TR to ensure the fairness of
comparisons, in which we replace the plane detection module to PlaneTR [25] as
our network used. Because the official implementation of SparsePlanes [7] did
not provide results on the ScanNet [14] dataset, we train these baselines on
the ScanNet [14] dataset by ourselves.
#### PlaneFormers and PlaneFormers-TR
. PlaneFormers [8] is the prior art developed based on SparsePlanes [7], which
selects Top-9 classification poses from [7] as hypotheses. For each pose
hypothesis, PlaneFormers jointly estimates the pose score and plane matching
cost with a neural network. Finally, the pose with the best score is selected
and refined with an extra estimated pose residual, and the plane
correspondences are achieved by conducting an offline Hungarian algorithm on
the matching cost. Like SparsePlane [7], we implemented the results of
PlaneFormers on the ScanNet [14] dataset by ourselves. In PlaneFormers-TR, we
use the planes detected by PlaneTR [25] as our network used.
#### SuperGlue
. It is a strong baseline for keypoints detection and matching with neural
networks. The camera poses are then calculated from matched keypoints with
‘5-point solver [10] \+ RANSAC [11]’. Since SuperGlue [9] lacks scale
information in translations, we do not report its translation errors. In case
SuperGlue fails to estimate camera poses, we set its rotation results as
identity matrices like [7]. For the results of plane reconstruction, we use
the ground truth scales in translations and match planes detected by PlaneTR
[25] with our plane matching module.
(a) Image 1
(b) Image 2
(c) Plane Correspondences
(d) Poses (view 1)
(e) Poses (view 2)
Figure 5: Qualitative results of the refined poses and plane correspondences
on the Matterport3D [13] dataset (first three rows) and the ScanNet [14]
dataset (last three rows). The Pink, Blue and Red frustums show the initial,
the refined, and the ground truth cameras of the first image respectively.
Gray frustums show the one-plane pose hypotheses of the first image generated
from plane correspondences. Black frustums show the camera of the second
image.
### 5.5 Comparison of Camera Pose Estimation
We evaluate our NOPE-SAC with the regressed initial poses, namely NOPE-SAC-
Reg, which is the main version of our proposed method. We also evaluate a
variant of our NOPE-SAC which directly uses the Top-1 a-pripori pose with a
classification module of SparsePlanes [7] as initial poses, namely NOPE-SAC-
Cls.
#### Quantitative Results.
We first compare the performance of rotation estimation. As shown in Tab. II,
both our NOPE-SAC-Reg and Nope-SAC-Cls significantly outperform all baselines
on both two datasets especially when the rotation threshold is small, _e.g.,_
$84.0\%$ (NOPE-SAC-Reg) v.s. $63.3\%$ (SparsePlanes-TR) with the threshold of
$10^{\circ}$ on the Matterport3D [13] dataset. Then, we further evaluate the
translation results. When compared with SparsePlanes-TR [7, 25] and
PlaneFormers-TR [8, 25] which use the a-pripori classification poses, our
NOPE-SAC-Cls performs slightly worse than them on the Matterport3D [13]
dataset but outperforms them on the more challenging ScanNet [14] dataset.
With a further comparison to SparsePlanes-TR [7, 25] as shown in Tab. III, we
find that the improvement of translation estimation for SparsePlanes-TR [7,
25] largely depends on its keypoint-based continuous optimization. Although
they are well-performing on the MatterPort3D [13] dataset, the performance of
continuous optimization dramatically degenerates on the ScanNet [14] dataset
because it is much more difficult to achieve good keypoint matches on the
ScanNet dataset (as shown in Fig. 1(a)). By contrast, our NOPE-SAC-Cls
consistently improves the initial poses with only the plane correspondences on
both datasets.
#### Qualitative Results.
Fig. 5 shows the pose estimation results of our NOPE-SAC from two different
viewpoints (last two columns) on the Matterport3D [13] and the ScanNet [14]
datasets. As described in Sec. 4, we compute one-plane pose hypotheses (Gray
frustums) from estimated plane correspondences (the third column). Despite the
outlier poses caused by incorrect correspondences and inaccurate plane
parameters, NOPE-SAC effectively achieved the final refined pose (Blue
frustum) from all one-plane pose hypotheses (Gray frustums) and the initial
pose (Pink frustum).
TABLE IV: Average Precision (AP) of 3D plane reconstruction conditioned with
mask IoU, normal angle error, and offset distance error. The threshold of mask
IoU is fixed to 0.5. ‘All’ means we consider all three conditions. ‘-Offset’
and ‘-Normal’ mean we ignore the offset and the normal conditions
respectively. ‘Con.’ means the continuous optimization proposed by
SparsePlanes [7].
Method | Offset$\leq$1m, Normal$\leq 30^{\circ}$ | Offset$\leq$0.5m, Normal$\leq 15^{\circ}$ | Offset$\leq$0.2m, Normal$\leq 5^{\circ}$
---|---|---|---
All | -Offset | -Normal | All | -Offset | -Normal | All | -Offset | -Normal
Matterport3D dataset
SuperGlue-TR [9, 25] | 39.51 | 44.11 | 44.08 | 28.98 | 37.53 | 34.40 | 11.29 | 21.87 | 17.27
SparsePlanes [7] w/o Con. | 35.87 | 42.13 | 38.80 | 23.36 | 35.34 | 27.48 | 8.07 | 17.28 | 12.99
SparsePlanes [7] | 36.02 | 42.01 | 39.04 | 23.53 | 35.25 | 27.64 | 6.76 | 17.18 | 11.52
PlaneFormers [8] | 37.62 | 43.19 | 40.36 | 26.10 | 36.88 | 29.99 | 9.44 | 18.82 | 14.78
SparsePlanes-TR [7, 25] w/o Con. | 39.91 | 46.50 | 42.53 | 27.37 | 40.79 | 31.03 | 9.99 | 22.80 | 14.64
SparsePlanes-TR [7, 25] | 40.35 | 46.39 | 43.03 | 27.81 | 40.65 | 31.38 | 9.02 | 22.80 | 13.66
PlaneFormers-TR [8, 25] | 41.87 | 47.50 | 44.43 | 30.78 | 42.82 | 34.03 | 12.45 | 25.98 | 17.34
NOPE-SAC-Cls Init. (ours) | 38.94 | 46.60 | 41.96 | 26.17 | 40.48 | 29.89 | 9.89 | 22.55 | 14.29
NOPE-SAC-Cls Ref. (ours) | 41.92 | 48.18 | 44.01 | 31.36 | 44.24 | 34.01 | 13.59 | 30.05 | 17.45
NOPE-SAC-Reg Init. (ours) | 40.07 | 46.03 | 43.59 | 26.78 | 36.76 | 31.95 | 10.09 | 19.09 | 15.55
NOPE-SAC-Reg Ref. (ours) | 43.29 | 49.00 | 45.32 | 32.61 | 44.94 | 35.36 | 14.25 | 30.39 | 18.37
ScanNet dataset
SuperGlue-TR [9, 25] | 33.20 | 33.77 | 43.40 | 22.78 | 24.94 | 36.72 | 4.35 | 6.19 | 19.33
SparsePlanes [7] w/o Con. | 33.20 | 34.12 | 40.74 | 22.89 | 25.62 | 33.67 | 3.03 | 4.52 | 17.17
SparsePlanes [7] | 33.08 | 34.12 | 40.51 | 21.69 | 25.59 | 32.20 | 2.52 | 4.50 | 14.85
PlaneFormers [8] | 34.64 | 35.47 | 41.37 | 24.48 | 27.19 | 34.69 | 3.93 | 5.52 | 18.58
SparsePlanes-TR [7, 25] w/o Con. | 35.56 | 36.51 | 42.14 | 26.01 | 29.61 | 35.12 | 3.96 | 6.10 | 18.59
SparsePlanes-TR [7, 25] | 35.32 | 36.50 | 41.92 | 24.71 | 29.55 | 33.50 | 3.21 | 6.07 | 15.32
PlaneFormers-TR [8, 25] | 36.82 | 37.87 | 43.01 | 27.41 | 30.72 | 36.31 | 4.83 | 7.02 | 19.94
NOPE-SAC-Cls Init. (ours) | 35.41 | 36.68 | 42.44 | 25.21 | 28.78 | 34.96 | 3.84 | 5.74 | 18.66
NOPE-SAC-Cls Ref. (ours) | 38.23 | 39.36 | 43.27 | 30.25 | 34.15 | 36.93 | 6.23 | 9.57 | 20.56
NOPE-SAC-Reg Init. (ours) | 36.39 | 37.35 | 43.15 | 25.59 | 28.54 | 36.06 | 4.59 | 6.41 | 19.92
NOPE-SAC-Reg Ref. (ours) | 39.39 | 40.30 | 43.88 | 31.21 | 34.89 | 37.88 | 6.74 | 10.10 | 21.41
(a) Image 1
(b) Image 2
(c) SparsePlanes-TR
(d) PlaneFormers-TR
(e) Ours
(f) Ground Truth
Figure 6: Comparison of 3D plane reconstruction results on the Matterport3D
[13] dataset (first four rows) and the ScanNet [14] dataset (last three rows).
Blue and Black frustums show cameras of the first and the second images
respectively. TABLE V: Ablation studies for NOPE-SAC camera pose refinement.
Method | Translation | Rotation
---|---|---
Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$1m) $\uparrow$ | ($\leq$0.5m) $\uparrow$ | ($\leq$0.2m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 30^{\circ}$) $\uparrow$ | ($\leq 15^{\circ}$) $\uparrow$ | ($\leq 10^{\circ}$) $\uparrow$
Matterport3D dataset |
Initial Pose | 0.69 | 1.08 | 65.0% | 37.0% | 10.1% | 11.16 | 21.49 | 81.3% | 60.5% | 46.5%
Nume-Ref-I | 0.52 | 1.06 | 69.2% | 48.5% | 19.1% | 7.17 | 21.96 | 79.8% | 67.4% | 58.3%
Nume-Ref-II | 0.68 | 1.07 | 65.5% | 38.2% | 10.7% | 8.38 | 19.97 | 82.2% | 65.7% | 54.9%
Homo-Ref | 0.61 | 1.05 | 68.5% | 42.9% | 14.9% | 5.03 | 17.47 | 85.1% | 74.5% | 66.3%
PlaneFormers-TR [8, 25] | 0.66 | 1.06 | 66.8% | 37.5% | 9.8% | 8.96 | 20.01 | 83.0% | 65.9% | 53.5%
NOPE-SAC (ours) | 0.52 | 0.94 | 73.2% | 48.3% | 16.2% | 2.77 | 14.37 | 89.0% | 86.9% | 84.0%
ScanNet dataset |
Initial Pose | 0.48 | 0.72 | 77.7% | 51.9% | 16.5% | 14.68 | 26.75 | 73.7% | 51.0% | 34.4%
Nume-Ref-I | 0.54 | 0.80 | 73.2% | 47.2% | 13.7% | 16.57 | 29.47 | 68.7% | 47.0% | 32.3%
Nume-Ref-II | 0.48 | 0.72 | 77.7% | 51.8% | 16.5% | 14.64 | 26.72 | 73.7% | 51.1% | 34.6%
Homo-Ref | 0.49 | 0.73 | 77.5% | 51.3% | 16.5% | 14.03 | 26.27 | 74.2% | 52.7% | 36.8%
PlaneFormers-TR [8, 25] | 0.48 | 0.72 | 78.3% | 52.2% | 14.2% | 14.30 | 26.51 | 74.5% | 52.2% | 33.7%
NOPE-SAC (ours) | 0.41 | 0.65 | 82.0% | 59.1% | 21.2% | 8.27 | 22.12 | 82.6% | 73.2% | 59.5%
### 5.6 Comparison of 3D Planar Reconstruction
We evaluate two versions of our method named NOPE-SAC-Reg and NOPE-SAC-Cls as
described in Sec. 5.5 on the Matterport3D [13] and ScanNet [14] datasets.
#### Quantitative Results.
We evaluate our method with initial (Init.) and refined (Ref.) poses, and
compare with baselines under various plane offset and normal error thresholds
from loose to strict. As shown in Tab. IV, Our NOPE-SAC-Reg achieves state-of-
the-art performance especially under the strictest settings with
‘Offset$\leq$0.2m’ and ‘Normal$\leq 5^{\circ}$’ on both Matterport3D [13] and
ScanNet[14] datasets. When compared with SparsePlanes-TR [7, 25] which also
uses classification poses, our NOPE-SAC-Cls effectively improves the
reconstruction performance (_e.g.,_ from 25.21 to 30.25 with offset$\leq$0.5m
and normal$\leq 10^{\circ}$ on the ScanNet dataset). Those performance gains
confirmed the superiority of our proposed NOPE-SAC.
#### Qualitative Results.
Fig. 6 visualizes the 3D plane reconstruction results by different approaches
on the Matterport3D [13] and the ScanNet [14] datasets. As it is shown, our
method successfully reconstructs the scenes from sparse views even when the
image overlap is very small (_e.g.,_ the third row in Fig. 6) and the
viewpoint change is very large (_e.g.,_ the fourth row in Fig. 6).
### 5.7 Ablation for NOPE-SAC Pose Estimation
This section presents a series of ablation study for our NOPE-SAC pose
estimation as it plays the most important role for the end task. We use the
initial pose achieved by our pose regression network by default.
#### NOPE-SAC VS. Other Pose Refinement Methods.
Here, we first compare our NOPE-SAC with three traditional methods and one
learning based method for pose refinement, including:
1. (1)
Nume-Ref-I: We use the numerical optimization like SparsePlanes [7] but only
optimizes the initial camera poses $R^{(0)}$, $t^{(0)}$ as follows:
$\begin{split}\mathop{\min}\limits_{R,t}\sum_{\mathcal{P}\in\mathbb{P}}&L(d_{\text{par}}(\mathcal{P},R,t))+d_{\text{pix}}(\mathcal{P},R,t)\\\
&+d_{\text{cam}}(R,R^{(0)}),\end{split}$ (21)
where $\mathcal{P}$ is a pair of matched planes in the predicted plane
correspondence set $\mathbb{P}$, $L(\cdot)$ is the Huber loss,
$d_{\text{par}}$ calculates the euclidean distance between plane parameters,
$d_{\text{pix}}$ is the reprojection error of matched SIFT [26] points on
plane regions, and $d_{\text{cam}}$ is a regularization term which restricts
the geodesic distance of rotations.
2. (2)
Nume-Ref-II: A variant of Nume-Ref-I which excludes $d_{\text{pix}}$ in Eq.
21.
3. (3)
Homo-Ref: We use homographies to estimate refined poses from predicted plane
correspondences $\mathbb{P}$. Specifically, to each predicted plane
correspondence $\mathcal{P}\in\mathbb{P}$, matched SIFT points are found from
plane regions and then used to calculate the homography matrix $H$. Then a
refined pose hypothesis can be decomposed from $H$. We use the scale of the
initial translation and select the refined pose from all hypotheses which
minimizes $\sum_{\mathcal{P}\in\mathbb{P}}d_{\text{par}}(\mathcal{P},R,t)$.
4. (4)
PlaneFormers-TR [8, 25]: We estimate the refined pose by giving planes
detected by PlaneTR [25] and our regressed initial pose as inputs to the
PlaneFormers [8].
As shown in Tab. V, compared to the learning based PlaneFormers-TR [8, 25]
which directly estimates a pose residual from features of all plane
correspondences, our method improves the initial poses more effectively than
PlaneFormers-TR [8, 25] on both two datasets with our accurate one-plane pose
hypotheses. Benefiting from matched keypoints, the traditional Nume-Ref-I
achieves the results which are most close to our NOPE-SAC on the Matterport3D
[13] dataset. However, due to the difficulty to find sufficient good keypoint
matches, Nume-Ref-I performs even worse than the initial pose on the ScanNet
[14] dataset. Similarly, Homo-Ref also suffers from unsatisfied keypoint
matches on the ScanNet dataset. In contrast, benefiting from directly learning
pose refinement with plane parameters in embedding space, our NOPE-SAC avoids
the problem of keypoints detection and matching and achieves state-of-the-art
performance.
TABLE VI: Ablation studies of the arbitrary initialization module and plane
warping in NOPE-SAC pose estimation on the Matterport3D [13] and the ScanNet
[14] datasets.
Settings | Trans. | Rot.
---|---|---
AIM | Warp Plane | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$0.5m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 15^{\circ}$) $\uparrow$
Matterport3D dataset
$\checkmark$ | | 0.61 | 1.00 | 42.3% | 3.19 | 14.10 | 85.7%
| $\checkmark$ | 0.51 | 0.92 | 49.3% | 2.97 | 14.34 | 86.8%
$\checkmark$ | $\checkmark$ | 0.52 | 0.94 | 48.3% | 2.77 | 14.37 | 86.9%
ScanNet dataset
$\checkmark$ | | 0.42 | 0.66 | 57.4% | 8.17 | 22.05 | 73.2%
| $\checkmark$ | 0.40 | 0.64 | 59.9% | 8.21 | 21.95 | 73.0%
$\checkmark$ | $\checkmark$ | 0.41 | 0.65 | 59.1% | 8.27 | 22.12 | 73.2%
#### Arbitrary Initialization Module.
In this part, we analyze the necessity of our arbitrary initialization module
(AIM). As introduced in Sec. 4.1, the initial pose embeddings can be achieved
from (1) the pose regression network or (2) the AIM. Thus, we first compare
the influence of these two methods to pose refinement. As shown in Tab. VI,
using initial pose embeddings from AIM achieves similar results to embeddings
from the regression network, which demonstrates the effectiveness of our AIM.
Furthermore, with the usage of AIM, our NOPE-SAC is flexible to refine initial
poses that do not come from the convolutional pose regression network.
#### Plane Warping for Correspondence Embedding.
Here, we discuss the influence of warping plane parameters before calculating
correspondence embeddings as described in Eqn. 11. As shown in Tab. VI,
warping plane parameters effectively improves the translation results,
especially on the Matterport3D [13] dataset (_e.g.,_ from $42.3\%$ to
$48.3\%$).
#### Pose Hypotheses Fusion.
We then discuss the fusion strategies of one-plane pose hypotheses for pose
refinement in Sec. 4.2, including (1)Soft, (2)Avg, (3)Min-Cost, (4)Max-Score.
As shown in Tab. VII, all strategies, even selecting only one pose hypothesis
(Min-Cost and Max-Score), can improve the initial pose, which demonstrates the
effectiveness and the reasonability of our one-plane RANSAC. Specifically,
when evaluating translations, Avg performs closely to Soft, while Min-Cost and
Max-Score degenerate significantly on both two datasets. It indicates that it
is necessary to leverage more than one hypothesis to get better translation
refinement results. When evaluating rotations, both Min-Cost and Max-Score
perform closely to or slightly better than Soft, while Avg degenerates
significantly. It indicates that rotations can be refined more easily than
translations with one rotation hypothesis. However, because of the influence
of matching outliers and the plane parameter errors, the strategies of Avg,
Min-Cost, and Max-Score are not stable for camera pose estimation. By
contrast, Soft achieves the best overall performance benefiting from the
learned pose scores.
TABLE VII: Ablation studies of NOPE-SAC pose estimation with different one
plane pose fusion strategies on the Matterport3D [13] and the ScanNet [14]
datasets.
Method | Trans. | Rot.
---|---|---
Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$0.5m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 15^{\circ}$) $\uparrow$
Matterport3D dataset
Initial Pose | 0.69 | 1.08 | 37.0% | 11.16 | 21.49 | 60.5%
Avg | 0.56 | 0.96 | 45.3% | 5.03 | 17.21 | 77.1%
Min-Cost | 0.56 | 1.00 | 45.4% | 2.85 | 14.47 | 86.8%
Max-Score | 0.59 | 1.03 | 43.1% | 2.84 | 14.38 | 87.0%
Soft | 0.52 | 0.94 | 48.3% | 2.77 | 14.37 | 86.9%
ScanNet dataset
Initial Pose | 0.48 | 0.72 | 51.9% | 14.68 | 26.75 | 51.0%
Avg | 0.42 | 0.65 | 57.8% | 10.28 | 23.33 | 63.2%
Min-Cost | 0.46 | 0.73 | 53.6% | 8.75 | 22.68 | 72.1%
Max-Score | 0.46 | 0.72 | 53.0% | 8.77 | 22.91 | 72.4%
Soft | 0.41 | 0.65 | 59.1% | 8.27 | 22.12 | 73.2%
TABLE VIII: Influence of plane parameter accuracy to NOPE-SAC pose estimation
on the Matterport3D [13] and the ScanNet [14] datasets.
Setting | Trans. | Rot.
---|---|---
Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$0.5m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 15^{\circ}$) $\uparrow$
Matterport3D dataset
Initial Pose | 0.69 | 1.08 | 37.0% | 11.16 | 21.49 | 60.5%
w/ GT | 0.32 | 0.70 | 63.5% | 0.34 | 3.85 | 96.2%
(0.1m, $5^{\circ}$) | 0.39 | 0.76 | 58.6% | 3.27 | 6.85 | 95.4%
(0.2m, $10^{\circ}$) | 0.48 | 0.83 | 51.5% | 6.75 | 10.76 | 83.6%
(0.3m, $15^{\circ}$) | 0.56 | 0.89 | 45.3% | 10.40 | 15.17 | 66.0%
ScanNet dataset
Initial Pose | 0.48 | 0.72 | 51.9% | 14.68 | 26.75 | 51.0%
w/ GT | 0.26 | 0.47 | 72.1% | 3.48 | 8.42 | 90.7%
(0.1m, $5^{\circ}$) | 0.29 | 0.49 | 70.4% | 7.22 | 11.63 | 88.2%
(0.2m, $10^{\circ}$) | 0.35 | 0.55 | 65.4% | 11.95 | 16.35 | 65.4%
(0.3m, $15^{\circ}$) | 0.40 | 0.60 | 59.9% | 16.35 | 21.11 | 44.3%
TABLE IX: Influence of the plane matching precision (P) to NOPE-SAC pose
estimation on the Matterport3D [13] and the ScanNet [14] datasets.
Threshold | P | Translation | Rotation
---|---|---|---
Med. $\downarrow$ | Mean $\downarrow$ | ($\leq$0.5m) $\uparrow$ | Med. $\downarrow$ | Mean $\downarrow$ | ($\leq 15^{\circ}$) $\uparrow$
Matterport3D dataset |
0.2 | 49.9 | 0.52 | 0.94 | 48.3% | 2.77 | 14.37 | 86.9%
0.1 | 48.8 | 0.52 | 0.94 | 48.4% | 2.75 | 14.40 | 86.9%
0.01 | 46.4 | 0.53 | 0.94 | 47.9% | 2.76 | 14.63 | 86.7%
0.001 | 44.7 | 0.53 | 0.94 | 47.7% | 2.77 | 14.96 | 86.3%
ScanNet dataset |
0.2 | 44.3 | 0.41 | 0.66 | 59.1% | 8.27 | 22.12 | 73.2%
0.1 | 43.0 | 0.41 | 0.65 | 58.9% | 8.26 | 22.11 | 73.5%
0.01 | 40.8 | 0.41 | 0.65 | 58.6% | 8.21 | 22.36 | 73.2%
0.001 | 39.8 | 0.41 | 0.65 | 58.5% | 8.22 | 22.56 | 73.2%
#### Influence of Plane Parameters.
In this part, we study the influence of plane parameter accuracy on our NOPE-
SAC pose estimation on the Matterport3D [13] and the ScanNet [14] datasets.
The upper bound of our method is achieved by using ground truth plane
correspondences and ground truth plane parameters in our pose refinement,
defined as ‘w/ GT’ in Tab. VIII. Then, we add various Gaussian noises to the
offset and normal of ground truth plane parameters in both two image views.
The mean of the Gaussian noises is set to zero. The standard deviation of
plane offset increases from 0.1m to 0.3m, and the standard deviation of plane
normal increases from $5^{\circ}$ to $15^{\circ}$. As shown in Tab. VIII, our
method effectively improves the initial pose even in the challenging setting
of (0.2m, $10^{\circ}$), but fails to improve rotations in the setting of
(0.3m, $15^{\circ}$) on the ScanNet dataset because of too large noises on
plane parameters. It demonstrates that our method is robust to the accuracy of
plane parameters.
#### Influence of Matching Precision.
Besides plane parameters, we also evaluate the influence of plane matching
precision to our NOPE-SAC on the Matterport3D [13] and ScanNet [14] datasets.
Here, a predicted plane correspondence is regarded as a true positive if it
can be matched to a ground truth plane correspondence with mask IoU $\geq$ 0.5
in both two images. We conduct the experiments by gradually reducing the
threshold of plane matching from 0.2 to 0.001. As shown in Tab. IX, with the
reduction of the matching threshold, the plane matching precision reduces from
49.9 to 44.7 while the pose metrics only change slightly on the Matterport3D
dataset. Similar results can be found on the ScanNet dataset. It demonstrates
that our NOPE-SAC is able to cope with incorrect plane correspondences.
## 6 Conclusion
This paper studies the challenging two-view 3D reconstruction in a rigorous
sparse-view configuration. At the core of estimating the camera poses from
one-plane pose hypotheses generated from plane correspondences with neural
networks, our proposed NOPE-SAC formulates the problem in a consensus sampling
paradigm while enjoying end-to-end learning without incurring any offline
optimization schemes. In the experiments, our NOPE-SAC achieves new state-of-
the-art performances for sparse-view camera pose estimation and planar 3D
reconstruction in indoor scenes on the challenging Matterport3D [13] and
ScanNet [14] datasets. The comprehensive analyzes also align benchmarking
protocols for sparse-view 3D reconstruction.
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|
Probing New Physics in the Vector-like Lepton
Model by Lepton Electric Dipole Moments
Koichi Hamaguchia,b***<EMAIL_ADDRESS>Natsumi Nagataa†††
<EMAIL_ADDRESS>Genta Osakia‡‡‡ osaki@hep-
th.phys.s.u-tokyo.ac.jp , and Shih-Yen Tsenga§§§ shihyen@hep-
th.phys.s.u-tokyo.ac.jp
aDepartment of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan
bKavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277–8583, Japan
We examine the lepton dipole moments in an extension of the Standard Model
(SM), which contains vector-like leptons that couple only to the second-
generation SM leptons. The model naturally leads to sizable contributions to
the muon $g-2$ and the muon electric dipole moment (EDM). One feature of this
model is that a sizable electron EDM is also induced at the two-loop level due
to the existence of new vector-like leptons in the loops. We find parameter
regions that can explain the muon $g-2$ anomaly and are also consistent with
the experimental constraints coming from the electron EDM and the Higgs decay
$h\rightarrow\mu^{+}\mu^{-}$. The generated EDMs can be as large as
$\mathcal{O}(10^{-22})~{}e\cdot\mathrm{cm}$ for the muon and
$\mathcal{O}(10^{-30})~{}e\cdot\mathrm{cm}$ for the electron, respectively,
which can be probed in future experiments for the EDM measurements.
## 1 Introduction
Last year, Fermilab published their first result [1] on the measurement of the
muon anomalous magnetic moment
$a_{\mu}\equiv\frac{g_{\mu}-2}{2}~{},$ (1)
which gives a value of
$a_{\mu}(\mathrm{exp})=116592061(41)\times 10^{-11}~{},$ (2)
while the Standard Model (SM) prediction is [2]
$a_{\mu}(\mathrm{SM})=116591810(43)\times 10^{-11}~{}.$ (3)
There is a $4.2\sigma$ tension between the experiment and theory,
$\Delta a_{\mu}=a_{\mu}(\mathrm{exp})-a_{\mu}(\mathrm{SM})=251(59)\times
10^{-11}~{},$ (4)
which may indicate the existence of physics beyond the Standard Model
(BSM)111The hadronic vacuum polarization (HVP) contribution to the muon $g-2$
has been a challenge for theoretical calculations. The value obtained in the
data-driven method, which is adopted in Ref. [2], is
$a^{\mathrm{HVP}}_{\mu}=6845(40)\times 10^{-11}$. A discrepancy exists between
this value and the lattice QCD calculations performed by the Budapest-
Marseille-Wuppertal (BMW) group [3], which gives
$a^{\mathrm{HVP,LO}}_{\mu,\mathrm{BMW}}=7075(55)\times 10^{-11}$. This is
$2.1\sigma$ larger than the recommended data-driven result. Naturally, the two
values should be compatible with each other because they correspond to the
same physical processes in the SM. However, the current situation is that
there is a significant difference between the two approaches, and the reason
is not yet clear. More intriguingly, recent results from other lattice QCD
groups [4, 5] support the result obtained by the BMW group. For the time
being, we simply fix on the value given in Eq. (3)..
Various new physics models have been proposed to explain this deviation. In
general, these models contain hypothetical new particles and couplings, whose
corresponding parameters are complex, and thus contain complex phases that
break the $CP$ symmetry. It is well-known that the flavor-conserving $CP$
violation in the SM is very small, such that the induced particle electric
dipole moments (EDMs) are vanishingly small. The non-zero SM contributions to
lepton EDMs appear at the four-loop level and are thus strongly suppressed.
For example, the electron EDM is estimated to be
$\left|d^{\mathrm{SM}}_{e}\right|\leq 10^{-38}~{}e\cdot\mathrm{cm}$ [6]. Since
it is far below the sensitivity of the current experimental techniques, any
observation of a particle EDM will be an unambiguous sign of the new physics
beyond the SM.
Currently, the upper bound on the muon EDM is
$\left|d_{\mu}\right|<1.8\times
10^{-19}~{}e\cdot\mathrm{cm}~{}\left(95\%~{}\mathrm{C.L.}\right)$ (5)
set by the Muon $\left(g-2\right)$ Collaboration at Brookhaven National
Laboratory [7], which is about ten orders of magnitude weaker than the one on
the electron EDM,
$\left|d_{e}\right|<1.1\times
10^{-29}~{}e\cdot\mathrm{cm}~{}\left(90\%~{}\mathrm{C.L.}\right)$ (6)
set by the ACME Collaboration [8]. In order to improve the sensitivity on the
muon EDM, there are several future experiments proposed to measure the muon
EDM. For example, J-PARC Muon $g-2$/EDM experiment [9] and the one using the
frozen-spin technique at the Paul Scherrer Institute (PSI) [10] will have
sensitivities of $\sigma\left(d_{\mu}\right)\leq 1.5\times
10^{-21}~{}e\cdot\mathrm{cm}$ and $\sigma\left(d_{\mu}\right)\leq 6\times
10^{-23}~{}e\cdot\mathrm{cm}$, respectively.
In this paper, we consider a model with extra vector-like leptons (VLLs) as a
possible explanation of the muon $g-2$ deviation, and investigate the EDMs of
the muon and electron in the model. Models with VLLs have been discussed
previously as solutions to the muon $g-2$ anomaly [11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. See also Refs. [29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] for the works that
studied the muon EDMs in the models motivated by the muon $g-2$ anomaly. We
consider a simple extension to the SM with two vector-like leptons, one SU(2)L
doublet and one SU(2)L singlet. We show that the model can naturally induce
sizable EDMs of the muon and the electron at the one-loop and two-loop levels,
respectively, in the parameter regions motivated by the muon $g-2$; the
predicted EDMs can be as large as $|d_{\mu}|\sim 10^{-22}~{}e\cdot\mathrm{cm}$
and $|d_{e}|\sim 10^{-30}~{}e\cdot\mathrm{cm}$, which are within the reach of
the proposed EDM experiments. There are also researches discussing the
indirect constraints on the muon EDM extracted from the EDM measurements of
heavy atoms and molecules; for example, see Ref. [47].
This paper is structured as follows. In section 2, we describe the model used
in the analysis. In section 3, we summarize the calculation of the one-loop
contributions to the muon dipole moments. In section 4, the induced electron
EDM in this model is discussed. The experimental constraints on this model are
presented in section 5, and the results are given in section 6. Finally, we
summarize the study in section 7.
## 2 Model
| $\ell_{L}$ | $\mu_{R}$ | $H$ | $L_{L,R}$ | $E_{L,R}$
---|---|---|---|---|---
SU(3)C | 1 | 1 | 1 | 1 | 1
SU(2)L | 2 | 1 | 2 | 2 | 1
U(1)Y | $-\frac{1}{2}$ | $-1$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$
Table 1: The quantum numbers of the SM and extra vector-like leptons in our
model.
We consider an extension of the SM with one SU(2)L doublet ($L$) and one
SU(2)L singlet ($E$) vector-like leptons. For simplicity, we consider the
minimal scenario where the vector-like leptons couple only to the second-
generation lepton, not to the first- and third-generation leptons.222Such kind
of structure may be realized by imposing flavor symmetries on the model. We
discuss a specific example in appendix B. Therefore, the electron and tau do
not mix with extra vector-like leptons, and their masses totally originate
from the Higgs Yukawa couplings. The quantum numbers of the fields necessary
for the analysis are listed in Table 1, where $\ell_{L}$ and $\mu_{R}$ are the
second-generation leptons in the SM and $L_{L,R}$ and $E_{L,R}$ denote the
vector-like leptons, which respectively have the same quantum numbers as can
be seen from the table. In the discussion below, our notation basically
follows the one used in Ref. [13].
The components of the doublet fields are labeled as
$\displaystyle\ell_{L}=\begin{pmatrix}\nu_{\mu}\\\
\mu_{L}\end{pmatrix},~{}L_{L,R}=\begin{pmatrix}L^{0}_{L,R}\\\
L^{-}_{L,R}\end{pmatrix},~{}H=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v+h\end{pmatrix},$ (7)
where $v=246.22$ GeV is the vacuum expectation value of the Higgs field. In
the rest of this paper, we denote the indices of muon and two vector-like
leptons in the mass basis as $2,4,5$, respectively, and for the muon, we use
the symbol $\mu$ and $2$ interchangeably.
Without loss of generality, we work in the basis where the Yukawa matrix of
the leptons in the SM sector is already diagonal. The most relevant part of
the Lagrangian is the Yukawa interactions among the muon and the vector-like
leptons and the mass term of the vector-like leptons, which are given by333In
general, the other mass terms $M^{\prime}_{L}\bar{\ell}_{L}L_{R}$ and
$M^{\prime}_{R}\bar{E}_{L}\mu_{R}$ are also allowed, but they can be removed
by field redefinitions.
$\displaystyle\begin{split}\mathcal{L}\supset&-y_{\mu}\bar{\ell}_{L}\mu_{R}H-\lambda_{E}\bar{\ell}_{L}E_{R}H-\lambda_{L}\bar{L}_{L}\mu_{R}H-\lambda\bar{L}_{L}E_{R}H-\bar{\lambda}H^{{\dagger}}\bar{E}_{L}L_{R}\\\
&-M_{L}\bar{L}_{L}L_{R}-M_{E}\bar{E}_{L}E_{R}+\mathrm{h.c.}.\end{split}$ (8)
The parameters $y_{\mu},\lambda_{E},\lambda_{L},\lambda,\bar{\lambda},M_{L}$
and $M_{E}$ are in general complex. However, most of the complex phases are
not physical since they can be removed via field redefinitions. It turns out
that in the model there are two independent complex phases
$\displaystyle\phi_{\lambda}$
$\displaystyle=\arg\left(y_{\mu}\lambda^{\ast}_{L}\lambda^{\ast}_{E}\lambda\right)~{},$
(9) $\displaystyle\phi_{\bar{\lambda}}$
$\displaystyle=\arg\left(y_{\mu}\lambda_{L}^{*}\lambda_{E}^{*}M_{L}M_{E}\bar{\lambda}^{*}\right)~{},$
(10)
which are invariant under the phase rotation of the fields and therefore can
serve as the sources of new $CP$ violation. In this work, we take the two $CP$
phases to be the phases of $\lambda$ and $\bar{\lambda}$ and set the other
parameters to be real.
After the Higgs field develops a vacuum expectation value, the leptons acquire
masses and the mass matrix of charged leptons is given by
$\displaystyle\bar{f}_{L}Mf_{R}=(\bar{\mu}_{L},\bar{L}^{-}_{L},\bar{E}_{L})\begin{pmatrix}y_{\mu}v/\sqrt{2}&0&\lambda_{E}v/\sqrt{2}\\\
\lambda_{L}v/\sqrt{2}&M_{L}&\lambda v/\sqrt{2}\\\
0&\bar{\lambda}v/\sqrt{2}&M_{E}\end{pmatrix}\begin{pmatrix}\mu_{R}\\\
L^{-}_{R}\\\ E_{R}\end{pmatrix},$ (11)
where the leptons in the flavor eigenbasis are denoted collectively as
$f_{L}=(\mu_{L},L^{-}_{L},E_{L})^{T}$ and
$f_{R}=(\mu_{R},L^{-}_{R},E_{R})^{T}$.
We can diagonalize the mass matrix in Eq. (11) by a bi-unitary transformation
with two unitary matrices $U_{L}$ and $U_{R}$:
$\displaystyle U^{\dagger}_{L}\begin{pmatrix}y_{\mu}v&0&\lambda_{E}v\\\
\lambda_{L}v&M_{L}&\lambda v\\\
0&\bar{\lambda}v&M_{E}\end{pmatrix}U_{R}=\begin{pmatrix}m_{2}&0&0\\\
0&m_{4}&0\\\ 0&0&m_{5}\\\ \end{pmatrix},$ (12)
where the eigenmass $m_{2}$ is set to be the mass of muon, $m_{2}=m_{\mu}$,
and the mass ordering is fixed as $m_{2}<m_{4}<m_{5}$. We also note that the
mass of the neutral component of the doublet $L$, $L^{0}\equiv\nu_{4}$, solely
originates from the mass term $-M_{L}\bar{L}_{L}L_{R}$ and hence its mass is
determined by the value of $M_{L}$.
## 3 Muon dipole moments
In this section, we summarize the one-loop contributions to the dipole moments
of the muon, which are induced by the diagrams of the Higgs, $Z$ boson and $W$
boson mediations, as shown in Fig. 1. For the relevant interactions, see
appendix A.
Figure 1: One-loop contributions to the muon dipole moments. $\ell_{2,4,5}$
are the muon and two vector-like leptons in the mass basis, while $\nu_{2,4}$
are the muon neutrino and the heavy neutrino in the vector-like doublet,
respectively.
### 3.1 Higgs boson mediation
The contributions to the moun dipole moments from the Higgs boson mediation
are given by
$\displaystyle\Delta a_{\mu}^{h}$
$\displaystyle=\frac{m_{\mu}}{8\pi^{2}m_{h}^{2}}\sum_{i=2,4,5}\left[(|\lambda_{i2}|^{2}+|\lambda_{2i}|^{2})\,m_{\mu}f_{h}(r_{i})+{\rm
Re}\,(\lambda_{i2}\lambda_{2i})\,m_{i}g_{h}(r_{i})\right]-a^{h,\mathrm{SM}}_{\mu},$
(13) $\displaystyle d_{\mu}^{h}$
$\displaystyle=-\frac{e}{16\pi^{2}m_{h}^{2}}\sum_{i=2,4,5}{\rm
Im}(\lambda_{i2}\lambda_{2i})m_{i}g_{h}(r_{i}),$ (14)
where $a^{h,\mathrm{SM}}_{\mu}$ is the SM contribution from the diagram with
muons in the loop. The loop functions are
$\displaystyle g_{h}(r_{i})$
$\displaystyle=-\frac{r_{i}^{2}-4r_{i}+3+2\,\mathrm{ln}(r_{i})}{2(1-r_{i})^{3}},$
(15) $\displaystyle f_{h}(r_{i})$
$\displaystyle=\frac{r_{i}^{3}-6r_{i}^{2}+3r_{i}+2+6r_{i}\,\mathrm{ln}(r_{i})}{12(1-r_{i})^{4}},$
(16)
with $r_{i}=m^{2}_{i}/m^{2}_{h}$ and $i=2,4,5$.
### 3.2 $Z$ boson mediation
The contributions to the moun dipole moments from the $Z$ boson mediation are
given by
$\displaystyle\Delta a_{\mu}^{Z}$
$\displaystyle=\frac{m_{\mu}}{8\pi^{2}m_{Z}^{2}}\sum_{i=2,4,5}\big{\\{}\left[|(g^{Z}_{L})_{i2}|^{2}+|(g^{Z}_{R})_{i2}|^{2}\right]\,m_{\mu}f_{Z}(r_{i})$
$\displaystyle\qquad\qquad\qquad\qquad+{\rm
Re}\,\left[(g^{Z}_{L})_{i2}(g^{Z\ast}_{R})_{i2}\right]\,m_{i}g_{Z}(r_{i})\big{\\}}-a^{Z,\mathrm{SM}}_{\mu},$
(17) $\displaystyle d_{\mu}^{Z}$
$\displaystyle=\frac{e}{16\pi^{2}m^{2}_{Z}}\sum_{i=2,4,5}{\rm
Im}\,\left[(g^{Z}_{L})_{i2}(g^{Z\ast}_{R})_{i2}\right]\,m_{i}g_{Z}(r_{i}),$
(18)
where $a^{Z,\mathrm{SM}}_{\mu}$ is the SM contribution from the diagram with
muons in the loop. The loop functions are then given by
$\displaystyle g_{Z}(r_{i})$
$\displaystyle=-\frac{r_{i}^{3}+3r_{i}-4-6r_{i}\,\mathrm{ln}(r_{i})}{2(1-r_{i})^{3}},$
(19) $\displaystyle f_{Z}(r_{i})$
$\displaystyle=-\frac{5r_{i}^{4}-14r_{i}^{3}+39r^{2}_{i}-38r_{i}+8-18r^{2}_{i}\,\mathrm{ln}(r_{i})}{12(1-r_{i})^{4}},$
(20)
with $r_{i}=m^{2}_{i}/m^{2}_{Z}$ and $i=2,4,5$.
### 3.3 $W$ boson mediation
The contributions to the muon dipole moments from the $W$ boson mediation are
given by
$\displaystyle\Delta a^{W}_{\mu}$
$\displaystyle=\frac{m_{\mu}}{8\pi^{2}m_{W}^{2}}\big{\\{}\sum_{i=2,4}\left[|(g^{W}_{L})_{i2}|^{2}+|(g^{W}_{R})_{i2}|^{2}\right]m_{\mu}f_{W}(r_{i})$
$\displaystyle\quad\qquad\qquad\qquad+{\rm
Re}\,\left[(g^{W}_{L})_{42}(g^{W\ast}_{R})_{42}\right]\,M_{L}g_{W}(r_{4})\big{\\}}-a^{W,\mathrm{SM}}_{\mu},$
(21) $\displaystyle d^{W}_{\mu}$
$\displaystyle=\frac{e}{16\pi^{2}m^{2}_{W}}{\rm
Im}\,\left[(g^{W}_{L})_{42}(g^{W\ast}_{R})_{42}\right]\,M_{L}g_{W}(r_{4}),$
(22)
where $a^{W,\mathrm{SM}}_{\mu}$ is the SM contribution from the diagram with
muon neutrino in the loop. The loop functions are given by
$\displaystyle g_{W}(r_{i})$
$\displaystyle=\frac{r^{3}_{i}-12r^{2}_{i}+15r_{i}-4+6r^{2}_{i}\,\mathrm{ln}(r_{i})}{2(1-r_{i})^{3}},$
(23) $\displaystyle f_{W}(r_{i})$
$\displaystyle=\frac{4r^{4}_{i}-49r^{3}_{i}+78r^{2}_{i}-43r_{i}+10+18r^{3}_{i}\,\mathrm{ln}(r_{i})}{12(1-r_{i})^{4}},$
(24)
where $r_{2}=m^{2}_{\nu_{\mu}}/m^{2}_{W}$ assuming $m_{\nu_{\mu}}\simeq 0$,
and $r_{4}=M^{2}_{L}/m^{2}_{W}$.
## 4 Electron EDM
In this model, the mixings among the muon and the extra vector-like leptons
also contribute to the electron EDM. These contributions appear in the
internal lepton loop of the two-loop Barr-Zee diagrams [48] shown in Fig. 2,
where all possible combinations of muon and vector-like leptons contribute to
the inner loop, as shown in Fig. 3.
Figure 2: Contributions to the electron EDM from the Barr-Zee diagram induced
by the vector-like leptons.
Figure 3: The inner loop inside the blob of Fig. 2.
To obtain the value of the electron EDM, let us first discuss the left diagram
in Fig. 2. The first step is to extract the effective vertex
$\Gamma^{\mu\nu}_{hV\gamma}$ from the inner loop. By the gauge invariance of
the on-shell photon, $q_{\mu}\Gamma^{\mu\nu}=0$, the effective vertex must
have the form of [49]
$\displaystyle\Gamma^{\mu\nu,ij}_{hV\gamma}(q,k)=\int^{1}_{0}dx\frac{eQ}{4\pi^{2}D^{ij}}\left[c_{E}^{Vij}(q^{\nu}k^{\mu}-q\cdot
k\,g^{\mu\nu})+ic_{O}^{Vij}\epsilon^{\mu\nu\alpha\beta}q_{\alpha}k_{\beta}\right],$
(25)
where $V=\gamma$, $Z$, and $Q$ is the charge of the lepton coupled to the
external photon which is $-1$ for all cases considered here. The coefficients
$c_{E}^{Vij}$, $c_{O}^{Vij}$, and the $\Delta^{ij}$ in the denominator are
given by
$\displaystyle c^{Vij}_{E}$
$\displaystyle=m_{i}x^{2}(1-x)(g^{ij\ast}_{s}g^{Vij}_{v}+g^{ij\ast}_{p}g^{Vij}_{a})+m_{j}(1-x)^{3}(g^{ij\ast}_{s}g^{Vij}_{v}-g^{ij\ast}_{p}g^{Vij}_{a}),$
(26) $\displaystyle c^{Vij}_{O}$
$\displaystyle=-m_{i}x(1-x)(g^{ij\ast}_{s}g^{Vij}_{a}+g^{ij\ast}_{p}g^{Vij}_{v})+m_{j}(1-x)^{2}(g^{ij\ast}_{s}g^{Vij}_{a}-g^{ij\ast}_{p}g^{Vij}_{v}),$
(27) $\displaystyle D^{ij}$
$\displaystyle=x(1-x)k^{2}-xm^{2}_{i}-(1-x)m^{2}_{j}.$ (28)
Here, $i,j=2,4,5$, and $g_{s},g_{p},g_{v}^{V},g_{a}^{V}$ are the couplings of
scalar, pseudoscalar, vector, and axial-vector types, respectively.444 The
details of interaction couplings can be found in appendix A. The particle $j$
is defined as the one who interacts with the external photon, accompanied by
particle $i$ in the inner loop of the Barr-Zee diagram. The expression of the
electron EDM from the $hV\gamma$ diagrams is given by
$\displaystyle
d^{hV\gamma}_{e}=\frac{eg^{Vee}_{v}g^{ee}_{s}}{32\pi^{4}}\sum_{i,j=2,4,5}\int^{1}_{0}dx\,\mathrm{Im}(c^{Vij}_{O})\,I^{ij}_{hV},$
(29)
where $g^{Vee}_{v}$ and $g^{ee}_{s}$ are the SM couplings of the electron to
vector bosons, $V=\gamma$, $Z$, and Higgs boson. The momentum integration over
$k$ in the outer loop, $I^{ij}_{hV}$, is given by
$\displaystyle
I^{ij}_{hV}=\frac{1}{m^{2}_{h}}\left[F\left(x,\frac{m^{2}_{V}}{m^{2}_{h}},\frac{\Delta_{ij}}{m^{2}_{h}}\right)-F\left(x,\frac{m^{2}_{V}}{m^{2}_{h}},\frac{\Delta_{ij}}{m^{2}_{V}}\right)\right],$
(30)
where
$\displaystyle F(x,y,z)$
$\displaystyle=\frac{1}{(1-y)[z-x(1-x)]}\mathrm{ln}\frac{z}{x(1-x)},$ (31)
$\displaystyle\Delta_{ij}$ $\displaystyle=xm^{2}_{i}+(1-x)m^{2}_{j}.$ (32)
Another class of diagrams is the ones with two $W$ bosons as shown in the
right diagram of Fig. 2. In this case, the contribution to the electron EDM
originates from the interaction of the heavy neutrino $\nu_{4}$ with the
charged leptons, and is given by
$\displaystyle
d^{WW\gamma}_{e}=\frac{eg^{2}}{256\pi^{4}}\sum_{i=2,4,5}\frac{m_{e}m_{i}M_{L}}{m^{2}_{W}}\mathrm{Im}\left[(g^{W}_{L})_{4i}(g^{W\ast}_{R})_{4i}\right]\int^{1}_{0}dxI^{4i}_{WW},$
(33)
where
$\displaystyle
I^{4i}_{WW}=\frac{(1-x)}{-x(1-x)m^{2}_{W}+(1-x)m^{2}_{i}+xM^{2}_{L}}\mathrm{ln}\frac{(1-x)m^{2}_{i}+xM^{2}_{L}}{x(1-x)m^{2}_{W}}.$
(34)
## 5 Constraints
In this section, we discuss the constraints on the model of vector-like
leptons coming from the precision electroweak measurements, the electron EDM,
the Higgs decay $h\rightarrow\mu^{+}\mu^{-}$, and the collider constraints on
the heavy charged leptons.
### 5.1 Precision electroweak measurements
Since the muon and the muon neutrino mix with extra vector-like leptons in
this model, the gauge couplings in the mass basis are modified (for details,
see appendix A). Therefore, several electroweak observables are affected
correspondingly, such as the muon lifetime, decay asymmetries involving muons,
partial widths of $W$ and $Z$ bosons, etc. These constraints have been
considered in the previous analysis [26, 11] and can be translated into upper
bounds of the couplings $\lambda_{L}$ and $\lambda_{E}$ given by
$\displaystyle\frac{\lambda_{L}v}{\sqrt{2}M_{L}}\lesssim
0.04~{},~{}\frac{\lambda_{E}v}{\sqrt{2}M_{E}}\lesssim 0.03.$ (35)
### 5.2 $h\rightarrow\mu^{+}\mu^{-}$
Since the Yukawa coupling of the muon is also modified from its SM value, the
deviation from the SM prediction is expected in the decay channel of the Higgs
boson to a muon pair, $h\rightarrow\mu^{+}\mu^{-}$. In the latest search of
this Higgs decay channel, the CMS group found the first evidence of the Higgs-
to-dimuon decay channel [50] with a significance of three standard deviations.
Currently, the branching fraction of $h\rightarrow\mu^{+}\mu^{-}$ is
constrained to be within the range $0.8\times
10^{-4}<\mathcal{B}(h\rightarrow\mu^{+}\mu^{-})<4.5\times 10^{-4}$ at 95$\%$
confidence level. This can be transformed into
$\displaystyle
0.37<R(h\rightarrow\mu^{+}\mu^{-})\equiv\frac{\Gamma(h\rightarrow\mu^{+}\mu^{-})}{\Gamma(h\rightarrow\mu^{+}\mu^{-})_{\mathrm{SM}}}<2.1,$
(36)
where we have used
$\Gamma(h\rightarrow\mu^{+}\mu^{-})_{\mathrm{SM}}=2.16\times 10^{-4}$ [51] for
$m_{H}=125.25$ GeV [52]. In our setup, the ratio is given by
$R(h\rightarrow\mu^{+}\mu^{-})=|\lambda_{22}|^{2}/(m_{\mu}/v)^{2}$.
### 5.3 Electron EDM
The latest measurement of the electron EDM is performed by the ACME
collaboration [8], which measured the precession of the electron spin in a
superposition of the quantum states of an electron inside a strong
intramolecular electric field. The group obtained an upper limit on the value
of the electron EDM,
$\displaystyle|d_{e}|<1.1\times 10^{-29}~{}e\cdot\mathrm{cm}$ (37)
at 90$\%$ confidence level. We note that in obtaining the above limit,
possible contributions to the spin precession frequency from the $CP$-odd
electron-nucleon scalar coupling are set to zero. In the case of the model of
vector-like leptons, such kinds of coupling exist as higher-order quantum
effects and, therefore, can be safely ignored in our analysis.555The size of
these higher-order effects can be estimated from the results in Ref. [47],
where the contribution of the muon EDM to the electron EDM $d_{e}$ and the CP-
odd electron-nucleon coupling $C_{S}$ is evaluated. This is expressed in terms
of the equivalent electron EDM, $d_{e}^{\rm equiv}$, a linear combination of
$d_{e}$ and $C_{S}$ that is constrained by experiments: $d_{e}^{\rm
equiv}\simeq 5.8\times 10^{-10}d_{\mu}$. As we see in the subsequent section,
in our model we have $|d_{\mu}|\lesssim 1.5\times
10^{-22}~{}e\cdot\mathrm{cm}$, and thus the size of the equivalent electron
EDM induced radiatively by the muon EDM is $\lesssim 8.7\times
10^{-32}~{}e\cdot\mathrm{cm}$, which is smaller than the Barr-Zee contribution
for most of the parameter points shown in Fig. 5.
### 5.4 Direct search of heavy leptons
The extra vector-like leptons decay via charged or neutral weak currents
through the mixing with the SM second generation leptons. The tree-level decay
modes include $\ell_{i}\to\ell_{j}+Z/h$, and $\ell_{i}\to\nu_{j}+W$, with
$i>j$. However, there are not so many experimental searches concentrating on
vector-like leptons compared to the searches on vector-like quarks.
The LEP experiment searched for such kinds of decays and set a lower bound on
the mass of these new leptons [53]. The lower bound was set to be around 100
GeV. Recently, the CMS group reported the lower bound on the mass of vector-
like leptons coupled to the third generation of SM lepton, $\tau$ [54]. The
doublet type is constrained to be heavier than 1045 GeV, and for the singlet
type, the mass range 125–150 GeV is excluded. Expecting that the constraints
on the vector-like leptons coupled to the muons are comparable, in this study,
we assume that the vector-like leptons have masses of $\mathcal{O}(1)$ TeV.
## 6 Results
In this section, we show the results of our analysis on the model of vector-
like leptons. We randomly choose the sampling points and scan over the
parameters in the model with the ranges of parameters listed in Table 2. We
note that the masses of the vector-like leptons are chosen to be at the TeV
scale in this analysis so that the constraints from the direct search of heavy
leptons are avoided. The muon Yukawa coupling $y_{\mu}$ is solved for the
correct muon mass with all other 8 parameters fixed randomly in the range
indicated by Table 2. We note that in solving the Yukawa coupling, it can be
written in the form of a quadratic equation, therefore there are in general
two solutions of Yukawa coupling for each randomly chosen parameter set. We
include both of them in the result.
Parameter | Value
---|---
$M_{L},M_{E}$ | $1-5$ TeV
$|\lambda_{L}|,|\lambda_{E}|$ | $\leq$ EW constraints in Eq. (35)
$|\lambda|,|\bar{\lambda}|$ | $0-1$
$\phi_{\lambda},\phi_{\bar{\lambda}}$ | $0-2\pi$
$y_{\mu}$ | solved for the correct $m_{\mu}$
Table 2: Ranges of parameters randomly chosen for the sampling points. Around
$4\times 10^{5}$ sampling points for the results are shown in the scattering
plots.
In the left plot of Fig. 4, the correlation between the muon $g-2$ and the
muon EDM is shown. The horizontal red line and the (light) red bands
correspond to the central value and (2$\sigma$) 1$\sigma$ regions of the
observed muon $g-2$ in the experiment, respectively. The gray dots are the
ones excluded by the $h\rightarrow\mu^{+}\mu^{-}$ constraint. After this
selection, we further exclude the sampling points that predict the electron
EDM to be too large and exceed the current upper bound given in Eq. (37).
These are represented by the blue dots. The black dots are the sampling points
that satisfy both the constraints from $h\rightarrow\mu^{+}\mu^{-}$ and
electron EDM. We can see that there are predictions consistent with the
observed muon $g-2$, and the largest muon EDM is around $|d_{\mu}|\simeq
1.5\times 10^{-22}~{}e\cdot\mathrm{cm}$. The PSI experiment [10] can examine
some regions of the parameters, and its sensitivity to the muon EDM is shown
by the vertical dotted line. In this result, we confirm the elliptical
correlation between muon $g-2$ and muon EDM obtained from the EFT approach
[46].
In the right plot of Fig. 4, the correlation between the muon $g-2$ and the
electron EDM is shown. The gray dots show the sampling points excluded by the
$h\rightarrow\mu^{+}\mu^{-}$ constraint, and the region with the blue band is
excluded by the current limit on the electron EDM from the ACME experiment.
From the plot, we see that most of the sampling points that are consistent
with both the $h\rightarrow\mu^{+}\mu^{-}$ and muon $g-2$ predict
$|d_{e}|\gtrsim 10^{-32}~{}e\cdot\mathrm{cm}$. The vertical dashed line,
$|d_{e}|=10^{-30}~{}e\cdot\mathrm{cm}$, indicates a representative sensitivity
of near future experiments. The next-generation ACME experiment (ACME III) is
expected to improve the current sensitivity by an order of magnitude [55], and
there are also proposals to improve the sensitivity by several orders of
magnitude in the next 10-20 years [56].
Figure 4: The correlations of muon $g-2$ with muon EDM (left plot) and
electron EDM (right plot). The horizontal red line and (light) red bands
correspond to the central value and (2$\sigma$) 1$\sigma$ regions of the
observed muon $g-2$, respectively. The $h\rightarrow\mu^{+}\mu^{-}$ constraint
is first applied in both plots and the sampling points which do not satisfy it
are shown as the gray dots. In the left plot, we also show the sampling points
further excluded by the electron EDM constraint in blue. The black dots are
the points that satisfy both the constraints from $h\rightarrow\mu^{+}\mu^{-}$
and electron EDM. The vertical dotted line represents the PSI experiment. In
the right plot, the blue band shows the constraint from electron EDM
measurement and the vertical dashed line indicates the prospect of the near-
future electron EDM experiments.
In Fig. 5, we show the correlation between the muon and the electron EDMs.
Again, we first excluded the sampling points that do not satisfy the
$h\rightarrow\mu^{+}\mu^{-}$ constraint. These are the gray dots in the
figure. The remaining sampling points are categorized into two classes, with
the (dark) red dots representing the points satisfying the (1$\sigma$)
2$\sigma$ region of the muon $g-2$, respectively. The horizontal blue band is
the constraint from the ACME experiment for the electron EDM measurement, and
the prospects of the future muon/electron EDM experiments are also given in
the plot. We find that these proposed experiments can investigate a
significant fraction of the parameter space of this model, as they can exclude
most of the available sampling points in the figure if no signals of EDMs are
discovered. Furthermore, all of the allowed parameter space in Fig. 5 can be
covered by the electron EDM experiments offering sensitivities better than the
current value by several orders of magnitude [56].
Figure 5: The correlations between the muon and electron EDMs. The gray dots
show the points excluded by the $h\rightarrow\mu^{+}\mu^{-}$ constraint. The
(dark) red dots represent the points satisfying the (1$\sigma$) 2$\sigma$
region of the muon $g-2$. The blue band is the constraint from the ACME
experiment for the electron EDM measurement. The prospect of the PSI
experiment for the muon EDM measurement and the near-future electron EDM
experiments are shown as the vertical dotted line and horizontal dashed line,
respectively.
Finally, we show the dependence of the muon and electron EDMs on the two
$CP$-violating phases, $\phi_{\bar{\lambda}}$ and $\phi_{\lambda}$, in Fig. 6.
Here, we fix the other parameters to be
$|\lambda|=0.2,|\bar{\lambda}|=0.6,M_{E}=2.5~{}\mathrm{TeV},M_{L}=5~{}\mathrm{TeV}$,
$\lambda_{L}=-0.04\times\sqrt{2}M_{L}/v\sim-0.57$, and
$\lambda_{E}=0.03\times\sqrt{2}M_{E}/v\sim 0.86$. The magnitude of the EDMs is
indicated by the black contours. The gray bands are the regions where no
solution exists for the $y_{\mu}$ to give the correct muon mass after the mass
diagonalization. The (2$\sigma$) 1$\sigma$ level of muon $g-2$ are also given
in the left figure as the (light) red bands. As we can see from the plots, the
muon EDM only depends on one of the two $CP$-violating phases,
$\phi_{\bar{\lambda}}$, while in the case of the electron EDM, it depends on
both of the phases. If the muon EDM is indeed nonzero and the vector-like
leptons exist, we may be able to determine the masses and couplings of the
vector-like leptons from the measurement of their decays, and the value of
$\phi_{\bar{\lambda}}$ can then be fixed by the result of the muon EDM
measurement. We can input these parameters to the electron EDM and find the
value of the remaining phase $\phi_{\lambda}$ according to the result of the
electron EDM measurement.
Figure 6: The dependence of muon EDM (left) and electron EDM (right) on the
two $CP$-violating phases in the model. The other parameters are chosen to be
$|\lambda|=0.2,|\bar{\lambda}|=0.6,M_{E}=2.5~{}\mathrm{TeV},M_{L}=5~{}\mathrm{TeV}$,
$\lambda_{L}=-0.04\times\sqrt{2}M_{L}/v\sim-0.57$, and
$\lambda_{E}=0.03\times\sqrt{2}M_{E}/v\sim 0.86$. The size of the EDMs are
given by the black contours. The gray bands represent the regions where no
solution exists for the $y_{\mu}$ to give the correct muon mass after the mass
diagonalization. The region consistent with muon $g-2$ at (2$\sigma$)
1$\sigma$ level is also indicated in the left figure by the (light) red bands.
## 7 Summary
In this work, we have investigated a simple extension of the Standard Model,
with the addition of one SU(2)L doublet and one SU(2)L singlet vector-like
leptons, which are coupled to the second-generation SM leptons only. In this
framework, the muon dipole moments are generated through the mediation of the
Higgs, $W$, and $Z$ bosons. Furthermore, one interesting feature of this model
is that a sizable electron EDM can also be induced at the two-loop level.
Besides the latest value of the muon $g-2$ published by the Muon $g-2$
collaboration at the Fermilab, we have also considered the recent constraint
on the Higgs decay channel $h\rightarrow\mu^{+}\mu^{-}$ reported by the CMS
group and the constraint on the electron EDM from the ACME experiment. We have
found that there are parameter regions consistent with all these constraints.
Because of the chirality flip from the heavy vector-like leptons, the sizes of
muon dipole moments are enhanced. The muon EDM can be as large as
$10^{-22}~{}e\cdot\mathrm{cm}$, which can be probed in the future muon EDM
measurement such as the PSI experiment, whose sensitivity is estimated to be
around $6\times 10^{-23}~{}e\cdot\mathrm{cm}$ [10]. An electron EDM of
$\mathcal{O}(10^{-30})~{}e\cdot\mathrm{cm}$ can be generated in this model,
and future electron EDM measurement like the ACME III experiment, whose
sensitivity is expected to be one order of magnitude better than the previous
ACME II [55], is able to examine the prediction.
## Acknowledgments
This work is supported in part by the Grant-in-Aid for Innovative Areas
(No.19H05810 [KH], No.19H05802 [KH], No.18H05542 [NN]), Scientific Research B
(No.20H01897 [KH and NN]), Young Scientists (No.21K13916 [NN]), and JSPS
KAKENHI Grant (No.20J22214 [SYT]).
## Appendix A Interactions
In this appendix, we briefly summarize the interactions in the model we used
in the analysis. Concerning the index notation, the quantities related to the
flavor basis are indexed by Greek alphabets ($\alpha,\beta,\gamma,\cdots$),
while the quantities related to the mass basis are indexed by Latin alphabets
($i,j,k,\cdots$). For the flavor indices, we have $2=\mu,4=L$ and $5=E$.
### A.1 Higgs boson couplings
Since the couplings related to electron and tau are not modified by the
vector-like leptons, their couplings with the Higgs boson are the SM values
$\lambda_{e,\tau}=-m_{e,\tau}/v$. Other couplings of charged leptons to the
Higgs boson are modified by the mixings. The Yukawa interaction in the flavor
basis is given by
$\displaystyle{\cal
L}_{Y}\supset-\frac{1}{\sqrt{2}}\,\bar{f}_{L\rho}\,Y_{\rho\sigma}\,f_{R\sigma}\,h+\mathrm{h.c.}.$
(38)
This can be transformed to the the mass basis given by
$\displaystyle{\cal
L}_{Y}\supset-\frac{1}{\sqrt{2}}\sum_{\rho,\sigma=2,4,5}\bar{f}_{Li}(U^{{\dagger}}_{L})_{i\rho}\,Y_{\rho\sigma}\,(U_{R})_{\sigma
j}\,f_{Rj}\,h+\mathrm{h.c.}\equiv\bar{f}_{Li}\,\lambda_{ij}\,f_{Rj}\,h+\mathrm{h.c.},$
(39)
where the Yukawa matrix is written as
$\displaystyle Y=\begin{pmatrix}y_{\mu}&0&\lambda_{E}\\\
\lambda_{L}&0&\lambda\\\ 0&\bar{\lambda}&0\end{pmatrix},$ (40)
and the Yukawa couplings in the mass basis is
$\displaystyle\lambda_{ij}=-\frac{1}{\sqrt{2}}\sum_{\rho,\sigma=2,4,5}(U^{\dagger}_{L})_{i\rho}Y_{\rho\sigma}\left(U_{R}\right)_{\sigma
j}.$ (41)
We notice that the Yukawa coupling in the flavor basis is similar to the mass
matrix of the charged leptons with the absence of the masses of vector-like
leptons, that is, $Yv/\sqrt{2}=M-\mathrm{diag}(0,M_{L},M_{E})$. With this
relation, the Higgs boson couplings in the mass basis can be written as:
$\displaystyle-\lambda v$
$\displaystyle=U^{\dagger}_{L}\begin{pmatrix}y_{\mu}v/\sqrt{2}&0&\lambda_{E}v/\sqrt{2}\\\
\lambda_{L}v/\sqrt{2}&M_{L}&\lambda v/\sqrt{2}\\\
0&\bar{\lambda}v/\sqrt{2}&M_{E}\end{pmatrix}U_{R}-U^{\dagger}_{L}\begin{pmatrix}0&0&0\\\
0&M_{L}&0\\\ 0&0&M_{E}\end{pmatrix}U_{R}$ (42)
$\displaystyle=\begin{pmatrix}m_{\mu}&0&0\\\ 0&m_{4}&0\\\
0&0&m_{5}\end{pmatrix}-U^{\dagger}_{L}\begin{pmatrix}0&0&0\\\ 0&M_{L}&0\\\
0&0&M_{E}\end{pmatrix}U_{R}.$ (43)
In this expression, we can see clearly that the first term has a simple form
which is the same as the Yukawa couplings in the SM. The second term
corresponds solely to the contributions from the mass term of the vector-like
leptons.
### A.2 $Z$ boson couplings
The couplings of leptons to the $Z$ boson come from the kinetic term of the
leptons. Due to the mixing among muon and vector-like leptons, the kinetic
term is modified to
$\displaystyle{\cal
L}_{\mathrm{kin}}\supset\bar{f}_{L\sigma}i\not{D}_{\sigma}f_{L\sigma}+\bar{f}_{R\sigma}i\not{D}_{\sigma}f_{R\sigma}=\bar{f}_{Li}(U^{{\dagger}}_{L})_{i\sigma}i\not{D}_{\sigma}(U_{L})_{\sigma
j}f_{Lj}+\bar{f}_{Ri}(U^{{\dagger}}_{R})_{i\sigma}i\not{D}_{\sigma}(U_{R})_{\sigma
j}f_{Rj}.$ (44)
The covariant derivative $D_{\mu\sigma}$ is defined as
$\displaystyle
D_{\mu\sigma}=\partial_{\mu}+i\,\frac{g}{\mathrm{cos}\,\theta_{W}}(T^{3}_{\sigma}-Q_{\sigma}\mathrm{sin}^{2}\theta_{W})Z_{\mu}+ieQ_{\sigma}A_{\mu},$
(45)
where $e$ and $g$ are the electromagnetic and the SU(2) couplings of the
Standard Model, and $T^{3}$ and $Q$ are the weak isospin and electric charge
of leptons obtained from the quantum numbers listed in Table 1.
The couplings of the $Z$ boson to charged leptons $\ell_{i}$ and $\ell_{j}$
are defined in the Lagrangian
$\displaystyle{\cal
L}_{Z}\supset\left[\bar{f}_{Li}\gamma^{\mu}(g^{Z}_{L})_{ij}f_{Lj}+\bar{f}_{Ri}\gamma^{\mu}(g^{Z}_{R})_{ij}f_{Rj}\right]Z_{\mu}.$
(46)
The couplings of left-handed and right-handed charged leptons with $Z$ boson
are given by
$\displaystyle(g^{Z}_{L})_{ij}=-\frac{g}{\cos\theta_{W}}\displaystyle\sum\limits_{\sigma=2,4,5}(T^{3}_{L\sigma}-\sin^{2}\theta_{W}Q_{\sigma})(U^{\dagger}_{L})_{i\sigma}(U_{L})_{\sigma
j},$ (47)
$\displaystyle(g^{Z}_{R})_{ij}=-\frac{g}{\cos\theta_{W}}\displaystyle\sum\limits_{\sigma=2,4,5}(T^{3}_{R\sigma}-\sin^{2}\theta_{W}Q_{\sigma})(U^{\dagger}_{R})_{i\sigma}(U_{R})_{\sigma
j},$ (48)
where the electric charge $Q_{\sigma}$ and the third component of weak
isospins $T^{3}_{L\sigma}$ and $T^{3}_{R\sigma}$ for each flavor are
summarized in Table 3.
Flavor | 2 | 4 | 5
---|---|---|---
$Q_{\sigma}$ | $-$1 | $-$1 | $-$1
$T^{3}_{L\sigma}$ | $-$1/2 | $-$1/2 | 0
$T^{3}_{R\sigma}$ | 0 | $-$1/2 | 0
Table 3: This table shows the electric charge $Q_{\sigma}$ and the third
component of weak isospins $T^{3}_{L\sigma}$ and $T^{3}_{R\sigma}$ for each
flavor considered in the mixing of charged leptons. Recall that in the flavor
basis, we have $2=\mu,4=L$, and $5=E$.
Since the electric charge $Q_{\sigma}$ is the same for all the charged leptons
considered, no modification is made on the couplings of photon with charged
leptons. On the other hand, because of the different weak isospins of the
charged leptons in the mixing, the couplings of the $Z$ boson in Eq. (47) and
Eq. (48) are modified from their SM values. In $(g^{Z}_{L})_{ij}$, we have
$\displaystyle-\frac{g}{\mathrm{cos}\,\theta_{W}}\left\\{\left(-\frac{1}{2}+\mathrm{sin}^{2}\theta_{W}\right)[(U^{\dagger}_{L})_{i2}(U_{L})_{2j}+(U^{\dagger}_{L})_{i4}(U_{L})_{4j}]+\mathrm{sin}^{2}\theta_{W}(U^{\dagger}_{L})_{i5}(U_{L})_{5j}\right\\}.$
(49)
To see the modification to the SM coupling, we can arrange and separate Eq.
(47) in two parts. The first one is
$\displaystyle-\frac{g}{\mathrm{cos}\,\theta_{W}}\left\\{\left(-\frac{1}{2}+\mathrm{sin}^{2}\theta_{W}\right)[(U^{\dagger}_{L})_{i2}(U_{L})_{2j}+(U^{\dagger}_{L})_{i4}(U_{L})_{4j}+(U^{\dagger}_{L})_{i5}(U_{L})_{5j}]\right\\}.$
(50)
By the unitarity of the matrix $U_{L}$, we have
$\sum_{\sigma=2,4,5}(U^{\dagger}_{L})_{i\sigma}(U_{L})_{\sigma
j}=\delta_{ij}$. This gives the form of the SM coupling of $Z$ boson with the
left-handed leptons
$\displaystyle(g^{Z}_{L})^{\mathrm{SM}}_{ij}=-\frac{g}{\mathrm{cos}\,\theta_{W}}\left(-\frac{1}{2}+\mathrm{sin}^{2}\theta_{W}\right)\delta_{ij}.$
(51)
The second one corresponds to the modification of the SM coupling,
$\displaystyle(\delta
g^{Z}_{L})_{ij}=-\frac{g}{2\,\mathrm{cos}\,\theta_{W}}(U^{\dagger}_{L})_{i5}(U_{L})_{5j}.$
(52)
Similarly, we can separate Eq. (48) in parts of the SM coupling
$\displaystyle(g^{Z}_{R})^{\mathrm{SM}}_{ij}=-\frac{g}{\mathrm{cos}\,\theta_{W}}\sin^{2}\theta_{W}\,\delta_{ij}$
(53)
and the modification
$\displaystyle(\delta
g^{Z}_{R})_{ij}=\frac{g}{2\mathrm{cos}\,\theta_{W}}(U^{\dagger}_{R})_{i4}(U_{R})_{4j}.$
(54)
### A.3 $W$ boson couplings
The couplings of $W$ with leptons are also derived from the kinetic term.
Since the charged lepton $E$ is an SU(2)L singlet, it decouples from the
interaction with $W$ boson. The kinetic term is given by
$\displaystyle{\cal L}_{\mathrm{kin}}$
$\displaystyle\supset-\frac{g}{\sqrt{2}}\left(\bar{\nu}_{\mu}\gamma^{\mu}\mu_{L}+\bar{L}_{L}^{0}\gamma^{\mu}L_{L}^{-}+\bar{L}_{R}^{0}\gamma^{\mu}L_{R}^{-}\right)W^{+}_{\mu}+h.c.$
(55)
$\displaystyle=-\frac{g}{\sqrt{2}}\left(\bar{\nu}_{2}\gamma^{\mu}(U_{L})_{2j}f_{Lj}+\bar{\nu}_{L4}\gamma^{\mu}(U_{L})_{4j}f_{Lj}+\bar{\nu}_{R4}\gamma^{\mu}(U_{R})_{4j}f_{Rj}\right)W^{+}_{\mu}+h.c..$
(56)
This can be written in a more compact form
$\displaystyle{\cal
L}_{W}\supset\left[\bar{\nu}_{Li}\gamma^{\mu}(g^{W}_{L})_{ij}f_{Lj}+\bar{\nu}_{Ri}\gamma^{\mu}(g^{W}_{R})_{ij}f_{Rj}\right]W^{+}_{\mu}+h.c.,$
(57)
and the couplings of $W$ boson with leptons are defined as
$\displaystyle(g^{W}_{L})_{2j}=-\frac{g}{\sqrt{2}}(U_{L})_{2j},\qquad(g^{W}_{L})_{4j}=-\frac{g}{\sqrt{2}}(U_{L})_{4j},\qquad(g^{W}_{R})_{4j}=-\frac{g}{\sqrt{2}}(U_{R})_{4j}~{},$
(58)
with $(g^{W}_{R})_{2j}=(g^{W}_{L})_{5j}=(g^{W}_{R})_{5j}=0$.
### A.4 Couplings in the electron EDM formulas
The interaction of the Higgs boson with the charged leptons is
$\displaystyle\bar{f}_{Li}\,\lambda_{ij}\,f_{Rj}\,h+\bar{f}_{Ri}\,\lambda^{\ast}_{ji}\,f_{Lj}\,h=\bar{f}_{i}(\lambda_{ij}P_{R}+\lambda^{\ast}_{ji}P_{L})f_{j}\,h,$
(59)
where $P_{L,R}$ are the projection operators for the left-handed and right-
handed components of the leptons. This can be rearranged in the form of the
scalar and pseudoscalar couplings
$\displaystyle\bar{f}_{i}(g_{s}^{ij}+g_{p}^{ij}\gamma_{5})f_{j}\,h,$ (60)
where
$\displaystyle
g_{s}^{ij}=\frac{1}{2}(\lambda_{ij}+\lambda^{\ast}_{ji})~{},~{}g_{p}^{ij}=\frac{1}{2}(\lambda_{ij}-\lambda^{\ast}_{ji})$
(61)
are the scalar and pseudoscalar types of coupling.
The interaction of charged leptons with $Z$ boson can be written as
$\displaystyle\bar{f}_{i}\gamma^{\mu}\left(g^{Z,ij}_{v}+g^{Z,ij}_{a}\gamma_{5}\right)f_{j}Z_{\mu},$
(62)
where
$\displaystyle
g^{Z,ij}_{v}=\frac{1}{2}\left[(g^{Z}_{L})_{ij}+(g^{Z}_{R})_{ij}\right]~{},~{}g^{Z,ij}_{a}=\frac{1}{2}\left[(-g^{Z}_{L})_{ij}+(g^{Z}_{R})_{ij}\right]$
(63)
are the vector and axial-vector types of coupling.
The electron couplings in Eq. (29) are given by
$\displaystyle g^{ee}_{s}$ $\displaystyle=-m_{e}/v,$ (64) $\displaystyle
g^{\gamma ee}_{v}$ $\displaystyle=e,$ (65) $\displaystyle g^{Zee}_{v}$
$\displaystyle=-\frac{g}{4\,\mathrm{cos}\,\theta_{W}}\left(-1+4\,\mathrm{sin}^{2}\theta_{W}\right),g^{Zee}_{a}=-\frac{g}{4\,\mathrm{cos}\,\theta_{W}}.$
(66)
## Appendix B An example for the muon-only coupling
In the model discussed in our work, we just assume that the vector-like
leptons couple only to the second-generation leptons, not to the first- and
third-generation ones. In this appendix, we briefly discuss a model where this
setup is realized as a consequence of U(1) gauge symmetries.
As discussed in Refs. [57, 58], there are two types of lepton flavor-dependent
U(1) symmetries which can be gauged and introduced to a model without
anomalies. One is the linear combination of the lepton numbers
U(1)${}_{\alpha_{e}L_{e}+\alpha_{\mu}L_{\mu}-(\alpha_{e}+\alpha_{\mu})L_{\tau}}$
and the other is a linear combination of baryon and lepton numbers
U(1)${}_{B+\beta_{e}L_{e}+\beta_{\mu}L_{\mu}-(3+\beta_{e}+\beta_{\mu})L_{\tau}}$.
As an example, we choose them to be U(1)${}_{L_{\mu}-L_{\tau}}$ and
U(1)${}_{B+3L_{e}-L_{\mu}-5L_{\tau}}$. The charge assignment of the fields is
summarized in Table 4. Note that we introduce three right-handed neutrinos,
$N_{e},N_{\mu},N_{\tau}$. To break the new U(1) gauge symmetries, we also
introduce two scalar fields that are singlet under the SM gauge interactions,
$\sigma$ and $\sigma^{\prime}$. We choose the charges of the vector-like
leptons to be the same as the ones of muon so that the interaction structure
of coupling solely to the muon in Eq. (8) can be realized. The charges of the
new scalars are chosen for the reason explained in the following.
| $e$ | $\mu$ | $\tau$ | $H$ | $L_{L,R}$ | $E_{L,R}$ | $\sigma$ | $\sigma^{\prime}$ | $N_{e}$ | $N_{\mu}$ | $N_{\tau}$
---|---|---|---|---|---|---|---|---|---|---|---
U(1)${}_{L_{\mu}-L_{\tau}}$ | 0 | 1 | $-$1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | $-$1
U(1)${}_{B+3L_{e}-L_{\mu}-5L_{\tau}}$ | 3 | $-$1 | $-$5 | 0 | $-$1 | $-$1 | 2 | 6 | 3 | $-$1 | $-$5
Table 4: Charge assignment of gauged U(1)${}_{L_{\mu}-L_{\tau}}$ and
U(1)${}_{B+3L_{e}-L_{\mu}-5L_{\tau}}$ symmetries to the fields considered in
the model in appendix B.
After the spontaneous symmetry breaking of scalar fields, we obtain a Majorana
mass matrix of the right-handed neutrinos in the form of
$\displaystyle\begin{pmatrix}\lambda_{ee}\langle\sigma^{\prime\ast}\rangle&\lambda_{e\mu}\langle\sigma^{\ast}\rangle&\lambda_{e\tau}\langle\sigma\rangle\\\
\lambda_{e\mu}\langle\sigma^{\ast}\rangle&0&\lambda_{\mu\tau}\langle\sigma^{\prime}\rangle\\\
\lambda_{e\tau}\langle\sigma\rangle&\lambda_{\mu\tau}\langle\sigma^{\prime}\rangle&0\end{pmatrix},$
(67)
while the neutrino Dirac mass matrix is diagonal. This implies that after
applying the seesaw mechanism [59, 60, 61, 62], we obtain an active neutrino
mass matrix with the so-called two-zero minor structure [63, 64]. We can then
follow the same analysis performed in [65, 66, 67] to extract the predictions
of the sum of three active neutrino masses and the Dirac $CP$-violating phase
$\delta$.
In Fig. 7, we show these predictions as a function of the neutrino mixing
angle $\theta_{23}$, with other two mixing angles and the two squared mass
differences fixed by the values provided in [68]. The neutrino masses are
taken to be in the normal ordering, $m_{1}<m_{2}<m_{3}$, as the inverse
ordering turns out to be incompatible with the observed neutrino oscillation
data in this model [65]. In both plots, the boundary of $x$ axis corresponds
to the 3$\sigma$ range of $\theta_{23}$, while the vertical dashed line shows
the central value of $\theta_{23}$ and the 1$\sigma$ range of $\theta_{23}$
are enclosed by the vertical dotted lines. The horizon dashed line in the left
plot represents the upper bound on the sum of active neutrino masses given in
[69], which is $\sum m_{\nu}<0.13$ eV at 95$\%$ confidence level. The (light)
green band in the right plot shows the (3) 1$\sigma$ range of the Dirac
$CP$-violating phase obtained in [68]. The predictions from the two-zero minor
structure are plotted as red curves, dark (light) red bands corresponding to
the parameters fixed to their central value and 1 (3) sigma value,
respectively, given in [68]. As we can see from the plots, although a large
fraction of the parameter region are excluded by the constraint on the sum of
the neutrino masses, there is still a possibility around the corner close to
the 3$\sigma$ boundary of $\theta_{23}$, which can be examined in the future
neutrino experiments.
Figure 7: Left plot: predictions of the sum of the three active neutrino
masses as a function of $\theta_{23}$. Right plot: predictions of the Dirac
$CP$ phase as a function of $\theta_{23}$. The predictions are plotted as red
curves, dark (light) red bands corresponding to the parameters fixed to their
central value and 1 (3) sigma value given in [68]. In both plots, the boundary
of $x$ axis corresponds to the 3$\sigma$ range of $\theta_{23}$, while the
vertical dashed line shows the central value of $\theta_{23}$ and the
1$\sigma$ range of $\theta_{23}$ are enclosed by the vertical dotted lines.
The horizon dashed line in the left plot represents the upper bound on the sum
of the active neutrino masses given in [69], which is $\sum m_{\nu}<0.13$ eV
at 95$\%$ confidence level. The (light) green band in the right plot shows the
(3) 1$\sigma$ range of the Dirac $CP$-violating phase obtained in [68].
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# Generalised Spherical Text Embedding
Souvik Banerjee1, Bamdev Mishra2, Pratik Jawanpuria2 Manish Shrivastava1
1 International Institute of Information Technology, Hyderabad
2 Microsoft India
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
This paper aims to provide an unsupervised modelling approach that allows for
a more flexible representation of text embeddings. It jointly encodes the
words and the paragraphs as individual matrices of arbitrary column dimension
with unit Frobenius norm. The representation is also linguistically motivated
with the introduction of a novel similarity metric. The proposed modelling and
the novel similarity metric exploits the matrix structure of embeddings. We
then go on to show that the same matrices can be reshaped into vectors of unit
norm and transform our problem into an optimization problem over the spherical
manifold. We exploit manifold optimization to efficiently train the matrix
embeddings. We also quantitatively verify the quality of our text embeddings
by showing that they demonstrate improved results in document classification,
document clustering, and semantic textual similarity benchmark
tests.111https://github.com/SouvikBan/matrix_rep.
## 1 Introduction
Most unsupervised text embedding models are trained by encoding the words or
paragraphs acquired from the training data as a feature length vector, with
the assumption that they reside in the Euclidean space. Such models are
ubiquitous for good reason. Aside from their efficiency, they have also proven
to be very effective providing us with state of the art results in various
intrinsic and extrinsic embedding evaluation tasks. Word2vecMikolov et al.
(2013b, a), and GLoVE Pennington et al. (2014) are two notable examples where
word embeddings are learned in the Euclidean space and are trained to be
oriented such that word vectors that appear in the same context have higher
cosine similarity. Some of the most common methods of intrinsic evaluation of
word embeddings include word similarity, word analogy, and compositionality.
Doc2vec Le and Mikolov (2014), an unsupervised document embedding model
generalises the training method introduced in Word2vec to documents and
achieves improved results in various downstream tasks like sentiment analysis,
information retrieval and multi-class classification. There are other document
embedding models like skip-thought Kiros et al. (2015) and infersent Conneau
et al. (2017); Moghadasi and Zhuang (2020).
The joint spherical embedding model, JoSE as proposed in Meng et al. (2019),
shows that directional similarity is often more effective in tasks such as
word similarity and document clustering. They show that when embeddings are
trained in the Euclidean space, there is a performance gap between the
training stage and usage stage of text embeddings. To bridge that gap, they
propose a model which trains both words and paragraphs on a spherical space
with tools from Riemannian optimization methods. The resulting embeddings are
also shown to give considerably better results in word similarity, document
clustering, and document classification tasks when compared with other
standard models. Such application of manifold geometry has also been explored
in substantial depth in works like Batmanghelich et al. (2016); Reisinger et
al. (2010); Gopal and Yang (2014). There are also other notable Riemannian
optimization based embedding training models like Tifrea et al. (2018); Nickel
and Kiela (2017) which train embeddings on the hyperbolic manifold space and
uses its tree like property for better hierarchical representation of data.
Hyperbolic word embeddings are also intrinsically linked with Gaussian word
embeddings Vilnis and McCallum (2014) which gives a lot more insight into the
geometry of word embeddings.
However, most of these text embedding models like JoSE, Word2vec, Doc2vec, and
fastText Bojanowski et al. (2017); Meng et al. (2020) are trained with the
goal of getting a single dense vector representation per word or document.
These models treat both polysemous and monosemous words in the same way
resulting in the most frequent meaning of the word dominating the others or
the meanings getting mixed in the case of former. It is especially detrimental
for documents where we use a single dense vector representation to encode
information which span over several sentences, often involving multiple
topics.
This paper aims to address this problem by using matrices as the mode of
representation instead of vectors. Our model is the joint word and document
training generative model proposed in JoSE where we replace the cosine
similarity metric with a novel metric that exploits the matrix structure of
the embeddings. This robust metric takes word or document matrices of
arbitrary number of columns and calculates the similarity between them. We
also show that a few reshape operations allow us to reformulate the
optimization problem of our model in terms of the spherical manifold
optimization problem. Thus, we offer more flexibility in the way of matrix
dimensions while retaining efficiency. Our choice of metric also suggests that
the word, sentence, paragraph/document embeddings do not need to have the same
number of columns, which has linguistic validation.
## 2 Matrix Representation of Texts and Optimization Problem
The text embeddings are represented as elements of the following set
$\mathcal{S}(p,r)=\\{\textbf{X}\in\mathbb{R}^{p\times
r}:||\textbf{X}||_{F}=1\\},$
where $r\leq p$ and $\|\cdot\|_{F}$ denotes the Frobenius norm. The Frobenius
norm is the matrix norm of a $p\times r$ matrix X defined as the square root
of the sum of the absolute squares of its elements, i.e.,
$||\textbf{X}||_{F}=\sqrt{\sum_{i=1}^{p}\sum_{j=1}^{r}x_{ij}^{2}}.$
Our model design is consistent with JoSE where it is assumed that text
generation is a two-step process: a center word is first generated according
to the semantics of the paragraph, and then the surrounding words are
generated based on the center word’s semantics. Consider a positive tuple
$(\mathcal{U},\mathcal{V},\mathcal{D})$ where word $\mathcal{V}$ appears in
the local context window of word $\mathcal{U}$ in paragraph $\mathcal{D}$ and
negative tuple $(\mathcal{V},\mathcal{U}^{\prime},\mathcal{D})$ where
$\mathcal{U}^{\prime}$ is a randomly sampled word from the vocabulary serving
as a negative sample. We represent words
$\mathcal{V},\mathcal{U},\mathcal{U}^{\prime}$ as matrices
$\textbf{V},\textbf{U},\textbf{N}$ which are elements of the set
$\mathcal{S}(p,r_{1})$ and paragraph $\mathcal{D}$ as matrix D which is an
element of the set $\mathcal{S}(p,r_{2})$, where $p,r_{1},r_{2}>0$. From a
linguistic perspective, these matrices can be considered as a set of latent
variables that govern the semantics of a word or a document. Each column is
given some arbitrary unit of linguistic information to encode, a latent
variable which contributes to the mathematical representation of a word or a
document. For example, the columns of a matrix D that represent the document
$\mathcal{D}$ might encode latent variables that contain information about
some topic contained in that document. Similarly, the columns of the word
matrix U might encode information about a specific context in which a
polysemous word $\mathcal{U}$ appears. We also keep the number of columns for
word matrices less than or equal to the number of columns for
sentence/document matrices, i.e., $r_{1}\leq r_{2}$, so that the number of
latent variables governing a word should not be more than the ones that govern
a sentence or paragraph.
Novel metric. To model the above mentioned linguistic representation
mathematically, we define a novel similarity metric for the ambient space in
which we train our matrix embeddings. The proposed metric function is a
measure of similarity between two sets of latent variables (matrices) - a
function analogous to the cosine similarity measure for vectors in the
Euclidean space. Given two arbitrary matrices
$\textbf{A}\in\mathcal{S}(p,r_{1}),\textbf{B}\in\mathcal{S}(p,r_{2})$, we
propose the similarity metric
$\textbf{g}:\mathcal{S}(p,r_{1})\times\mathcal{S}(p,r_{2})\rightarrow\mathbb{R}$
as
$\textbf{g}(\textbf{A},\textbf{B})=\frac{\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}a_{i}^{\top}b_{j}}{r_{1}r_{2}},$
(1)
where $\textbf{A}=[a_{1}\,a_{2}\,\cdots\,a_{r_{1}}]$ ,
$\textbf{B}=[b_{1}\,b_{2}\,\cdots\,b_{r_{2}}]$ ,
$a_{i},b_{j}\in\mathbb{R}^{p}\,\forall i\,\in[1,2,\cdots r_{1}]$ ,
$\forall\,j\in[1,2,\cdots r_{2}]$.
Motivation for our similarity metric. The metric g (1) calculates the average
of all the entries in the matrix $\textbf{A}^{\top}\textbf{B}$. The linguistic
intuition behind the choice of this metric is that we want to define a metric
that takes the average of dot products between all possible pairs of latent
variables (columns) from each matrix. In the case of $r_{1}=r_{2}=1$,
$\textbf{g}(\textbf{A},\textbf{B})$ reduces to the cosine similarity metric
between unit norm vectors A and B which is the metric used in the spherical
space model of JoSE. However, in the case of higher values of $r_{1},r_{2}$,
For example, let two words $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ be
represented by proposed $p\times r_{1}$ matrix embeddings -
$\textbf{V}_{1}=[\text{a}_{1},\text{a}_{2}]$ and
$\textbf{V}_{2}=[\text{b}_{1},\text{b}_{2}]$, where $p=1$ and $r_{1}=2$. The
proposed similarity metric $\textbf{g}(\textbf{V}_{1},\textbf{V}_{2})$ is
computed as ${(a_{1}b_{1}+a_{1}b_{2}+a_{2}b_{1}+a_{2}b_{2}})/{4}$. Note that
this is different from computing the cosine similarity which gives
$(a_{1}b1+a_{2}b_{2})$. Moreover, the regular cosine similarity between word
and paragraph embedding matrices with unequal dimensions $(p\times r_{1}$ and
$p\times r_{2}$ respectively) is not defined. On the other hand, our proposed
similarity metric is still applicable.
Modelling. As our model has the same generative process as JoSE, we take the
same max-margin loss function and substitute the cosine similarity metric with
our new similarity metric g where the word matrices
$\textbf{U},\textbf{V},\textbf{N}\in\mathcal{S}(p,r_{1})$ and paragraph matrix
$\textbf{D}\in\mathcal{S}(p,r_{2})$ with $r_{1}\leq r_{2}$. We get the
following loss, i.e.,
$\begin{array}[]{lll}\mathcal{L}(\textbf{V},\textbf{U},\textbf{N},\textbf{D})\\\
\quad\quad=\max\,(0,m-\textbf{g}(\textbf{V},\textbf{U})-\textbf{g}(\textbf{U},\textbf{D})\\\
\qquad\qquad\qquad\quad+\
\textbf{g}(\textbf{V},\textbf{N})+\textbf{g}(\textbf{N},\textbf{D})).\end{array}$
(2)
where $m>0$ is the margin.
Optimization. For the purpose of optimization, matrices of different
dimensions are reshaped and embedded into Riemannian spherical manifolds of
different dimensions. Overall, they are combined using the Riemannian product
manifold structure. Therefore, the optimization of $\mathcal{L}$ (2) is done
by performing two reshape operations per iteration while training. For
example, the unit Frobenius norm matrices of dimension $\mathbb{R}^{p\times
r}$ can be reshaped into vectors of dimension $\mathbb{R}^{pr}$ with the unit
norm. To calculate the value of our loss function (2) at every iteration and
the Euclidean gradient (partial derivatives), the vectors in question are
reshaped into matrices for calculating the g values and their gradients.
Subsequently, the matrices are reshaped back into vectors. We then apply the
Riemannian gradient descent update rule to update the parameters Meng et al.
(2019); Absil et al. (2008); Smith (2014); Edelman et al. (1998). Note that
our proposed modelling and optimization are different from just training on
the spherical manifold with unit vectors and using the cosine similarity
metric (which is the case in JoSE).
## 3 Experiments
To highlight the quality of our obtained matrix representations, we run the
same set of evaluations as JoSE with a relatively lower number of columns,
i.e., $1\leq r_{1}\leq r_{2}\leq 6$. We notice that for even higher values,
the quality of our embeddings gradually decrease. Moreover, the word
similarity experiment results are not added as words seemingly do not benefit
from our representation directly. Indeed, the best word similarity score are
obtained for $r_{1}=r_{2}=1$. Instead, we add semantic textual similarity
benchmark tests to show that sentences can benefit from this matrix
representation model. Unless otherwise stated, our model and JoSE are trained
for 35 iterations on the respective corpora; the local context window size is
5; the embedding dimension is kept at 100; the number of negative samples are
2. Other hyperparameters in our model are kept the same as JoSE.
### 3.1 Document Clustering
We perform document clustering on the 20
Newsgroup222http://qwone.com/~jason/20Newsgroups/ dataset using spectral
clustering. Each paragraph in the dataset is separated by a new line and is
considered a separate document while training. JoSE uses K-Means and SK-Means
as the clustering algorithm that assume the ambient space to be the Euclidean
and the spherical space, respectively. Our non-Euclidean space with its custom
metric requires a clustering algorithm that allows the freedom of using custom
metric, i.e., the algorithm should be space agnostic. We found spectral
clustering to suit those requirements perfectly. The four external measures
used for validating the results are kept unchanged from JoSE Banerjee et al.
(2005); Manning et al. (2008); Steinley (2004). These measures are Mutual
Information (MI), Normalized Mutual Information (NMI), Adjusted Rand Index
(ARI), and Purity. We run the clustering algorithm with our custom similarity
metric as written in (1) with kernel coefficient, $\gamma=0.001$. Table 1
shows quantitatively how matrix representations benefit document embeddings
for clustering tasks. Keeping $r_{1}=1$ fixed, we see a steady increase in
performance as $r_{2}$ is increased from 1 (the score of our baseline model -
JoSE ) to 6.
Table 1: Evaluation results for spectral clustering of document embeddings on
the 20 Newsgroup dataset for kernel coefficient, $\gamma=0.001$
($r_{1}=1,r_{2}=1$ is JoSE score). Document embeddings benefit from matrix
representations as demonstrated by better scores for higher values of $r_{2}$.
Here, $r_{1}=1$.
$r_{2}$, $r_{1}=1$ | MI | NMI | ARI | Purity
---|---|---|---|---
$r_{2}=1$ | 1.73 | 0.58 | 0.45 | 0.64
$r_{2}=2$ | 1.75 | 0.59 | 0.46 | 0.63
$r_{2}=3$ | 1.75 | 0.59 | 0.46 | 0.62
$r_{2}=4$ | 1.77 | 0.60 | 0.46 | 0.63
$r_{2}=5$ | 1.84 | 0.62 | 0.49 | 0.65
$r_{2}=6$ | 1.85 | 0.62 | 0.49 | 0.67
### 3.2 Document Classification
Following Meng et al. (2019), the document classification evaluations are ran
on the following two datasets: the topic classification 20 Newsgroup dataset
(which we used for document clustering as well) and a binary sentiment
classification dataset consisting of 1 000 positive and 1 000 negative movie
reviews333http://www.cs.cornell.edu/people/pabo/movie-review-data/. The
train/test split is the original split for 20 Newsgroup while for the movie
review datasets, the splitting is done by randomly selecting 80% of the data
as training and 20% as testing. The classification algorithm we use is K-NN
with $k=3$ and a custom distance metric that is suitable for our space. The
custom distance metric for two paragraph matrices
$\textbf{U}=[u_{1}\,u_{2}\,\cdots\,u_{r_{2}}]$ and
$\textbf{V}=[v_{1}\,v_{2}\,\cdots\,v_{r_{2}}]$ where
$\textbf{U},\textbf{V}\in\mathcal{S}(p,r_{2})$ and
$u_{i},v_{i}\in\mathbb{R}^{p}\,\forall i\in[1,2,\cdots,r_{2}]$ is defined as
$\begin{array}[]{lll}\text{dist}^{2}(\textbf{U},\textbf{V})=\frac{{\sum_{k=1}^{r_{2}}\sum_{l=1}^{r_{2}}(u_{k}-v_{l})^{\top}(u_{k}-v_{l})}}{r_{2}^{2}}.\end{array}$
(3)
The intuition for the distance metric in (3) comes from our interpretation of
each individual column as encoding a latent variable governing the semantics
of that specific document. A quick look at (3) tells us that the distance
metric takes the square root of the average of the squared Euclidean distances
between all pairs of columns formed from one matrix with another. Tables 2 and
3 list the Macro-F1 and Micro-F1 scores for 20 Newsgroup dataset and Movie
Reviews dataset respectively for increasing values of $r_{1}$ and $r_{2}$. We
again see an increase in scores for higher values of both $r_{2}$ and $r_{1}$
compared to JoSE ($r_{1}$=1, $r_{2}$=1).
Table 2: F1-macro, F1-micro for 20 Newsgroup dataset classification using K-NN
with K=3 ($r_{1}=1,r_{2}=1$ is JoSE score). Increasing the value $r_{2}$
benefits documents embeddings in classification tasks.
$r_{1}$\$r_{2}$ | $r_{2}$=1 | $r_{2}$=2 | $r_{2}$=3 | $r_{2}$=4
---|---|---|---|---
$r_{1}$=1 | 0.74, 0.74 | 0.77, 0.77 | 0.78, 0.78 | 0.78, 0.78
$r_{1}$=2 | – | 0.76, 0.76 | 0.77, 0.77 | 0.76, 0.77
$r_{1}$=3 | – | – | 0.76, 0.76 | 0.73, 0.74
$r_{1}$=4 | – | – | – | 0.72, 0.72
Table 3: F1-macro, F1-micro for movie review dataset classification using K-NN
with K=3 ($r_{1}=1,r_{2}=1$ is the JoSE score).
$r_{1}$\$r_{2}$ | $r_{2}$=1 | $r_{2}$=2 | $r_{2}$=3 | $r_{2}$=4
---|---|---|---|---
$r_{1}$=1 | 0.74, 0.74 | 0.75, 0.75 | 0.76, 0.76 | 0.75, 0.76
$r_{1}$=2 | – | 0.75, 0.75 | 0.75, 0.75 | 0.74, 0.74
$r_{1}$=3 | – | – | 0.74, 0.74 | 0.76, 0.76
$r_{1}$=4 | – | – | – | 0.74, 0.74
### 3.3 Semantic Textual Similarity Task
Semantic Textual Similarity Benchmark comprises a selection of the English
datasets used in the STS tasks organized in the context of SemEval Cer et al.
(2017) between 2012 and
2017444http://ixa2.si.ehu.eus/stswiki/index.php/STSbenchmark. We perform
semantic textual similarity tasks on the sts-benchmark dataset to show that
even sentences can benefit from being represented as matrices. The benchmark
comprises of 8 628 sentence pairs split into 3 partitions: train, development
and test. The results are reported on both the test and dev sets. Each
sentence in the dataset is treated as a separate document by our model and we
use all the sentences in the train, development and test set to train. The
rationale for this is that the model is completely unsupervised, i.e., it
takes only the raw text and uses no supervised or annotated information, and
thus there is no need to hold out the test data as it is unlabelled. We train
for 1 000 iters with window size 15 and negative samples 5 while the rest of
the hyperparameters were kept at their default values. To score a sentence
pair representation, similarity was computed between them using our custom
metric described in 1 for our model. We report the dev and test Pearson
correlation score for $r_{1},r_{2}=1,2,3,4,\,r_{1}\leq r_{2}$. As Table 4
reports, higher values of $r_{2}$ give better scores compared to our baseline
model JoSE ($r_{1}=r_{2}=1$).
Table 4: Pearson Correlation for STS Benchmark on dev and test data
($r_{1}=1,r_{2}=1$ is the JoSE score). Even sentences can benefit from our
matrix representation as demonstrated by better scores with higher values of
$r_{2}$.
$r_{1}$\$r_{2}$ | $r_{2}$=1 | $r_{2}$=2 | $r_{2}$=3 | $r_{2}$=4
---|---|---|---|---
$r_{1}$=1 | 0.51, 0.40 | 0.51, 0.39 | 0.52, 0.40 | 0.53, 0.40
$r_{1}$=2 | – | 0.53, 0.40 | 0.53, 0.40 | 0.53, 0.40
$r_{1}$=3 | – | – | 0.53, 0.40 | 0.54, 0.40
$r_{1}$=4 | – | – | – | 0.53, 0.40
## 4 Conclusion
In this paper, we extend the joint modelling idea used for training text
embeddings from vectors with unit norm to matrices with unit Frobenius norm.
Each word/sentence/document matrix is made to encode information in a way that
each column of the matrix represents some latent topic, context, or discourse.
Since the standard vector dot product can no longer be applied, we introduce a
novel similarity metric that allows the measurement of similarity between
matrices of arbitrary number of columns. For optimization simplicity, we
reshape our matrices to vectors of unit norm that allows using the Riemannian
gradient descent optimization algorithm on the spherical manifold. Our theory
is validated quantitatively by the results which shows that our text
embeddings outperform or produce similar results when compared with JoSE in
document classification, clustering, and semantic textual similarity tasks.
This paper is meant to serve as a ground work for more involved research
topics which integrate concepts of differential geometry and NLP. Future
directions could include qualitative analysis on the columns of the matrices
to see what tangible information they encode which will allow for better
modelling. Another direction would be to exploit other matrix structures on
the embeddings, e.g., treating each word embedding as a symmetric positive
definite matrix and to study whether they can be beneficial.
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|
[title=Index of definitions and notations,intoc]
# Existence obstructions for separating morphisms and totally real pencils
Matilde Manzaroli
###### Abstract.
It goes back to Ahlfors that a real algebraic curve $C$ admits a separating
morphism $f$ to the complex projective line if and only if the real part of
the curve disconnects its complex part, i.e. the curve is separating. The
degree of such $f$ is bounded from below by the number $l$ of real connected
components of $\mathbb{R}C$. The sharpness of this bound is not a priori
clear. We prove that real algebraic separating curves, embedded in some
ambient surface and with $l$ bounded in a certain way, do not admit separating
morphisms of lowest possible degree. Moreover, this result of non-existence
can be applied to show that certain real separating plane curves of degree
$d$, do not admit totally real pencils of curves of degree $k$ such that
$kd\leq l$. Finally, we construct an explicit family of real separating plane
curves of increasing degree $2s+1$ such that each member of the family does
not admit totally real pencils of curves of degree lower or equal to
$\frac{s^{2}+1}{2s+1}$.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
3. 3 Theorem 1.13 and applications
## 1\. Introduction
A real algebraic variety is a compact complex algebraic variety $X$ equipped
with an anti-holomorphic involution $\sigma:X\rightarrow X$, called real
structure. The real part $\mathbb{R}X$ of $X$ is the set of points fixed by
the involution $\sigma$.
Let $C$ be any non-singular real algebraic curve. Denote by
$C_{1},\dots,C_{l}$ the connected components of $\mathbb{R}C$. Harnack-Klein’s
inequality [Har76, Kle73] bounds $l$ by the genus $g$ of $C$ plus one. If
$C\setminus\mathbb{R}C$ is connected, we say that $C$ is of type II or non-
separating, otherwise of type I or separating [Kle73]. Looking at the real
part of the curve and its position with respect to its complexification gives
information about $l$ and viceversa. For example, we know that if $l$ equals
$g+1$, then $C$ is of type I. Or, if $C$ is of type I then $l$ has the parity
of $g+1$. Rokhlin promoted and contributed to this new point of view; an
example of Rokhlin’s contribution is the introduction and the study of the
complex orientations of a separating real algebraic curve [Rok72]. Namely, if
$C$ is of type I, the two halves of $C\setminus\mathbb{R}C$ induce two
opposite orientations on $\mathbb{R}C$ called complex orientations of the
curve. Looking at complex orientations of separating real curves has allowed a
change of prospective and a remarkable progress in the study of real algebraic
curves, their topology and a refinement of their classifications.
###### Definition 1.1.
We say that a real morphism $f$ from a real algebraic curve $C$ to the complex
projective line $\mathbb{C}\mathbb{P}^{1}$ is separating if
$f^{-1}(\mathbb{R}\mathbb{P}^{1})=\mathbb{R}C$.
According to Ahlfors [Ahl50, §4.2], there exists a separating morphism
$f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ if and only if $C$ is separating. In
this paper, we focus on the relation of separating curves embedded in some
ambient surface and their separating morphisms; see Definition 1.1. Before
stating the main result of this paper, Theorem 1.13, in its generality, we
present it restricted to the case of real algebraic plane curves.
### 1.1. Real plane curves
Let us consider separating real algebraic plane curves $C$. Let $l$ denote the
number of the connected components of $\mathbb{R}C$. The following proposition
show that there is a relation between $l$ and the existence of separating
morphisms $f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ of degree $r(l)$.
###### Proposition 1.2 (Particular case of Theorem 1.13).
Let $C$ be a non-singular real algebraic plane curve of type I and with $l$
real connected components. Assume one of the following:
1. (1)
The degree of $C$ is $2s+1$, for some $s\in\mathbb{Z}_{>1}$ and
$l+\varepsilon\leq s^{2}+s-2,$ where $\varepsilon\in\\{0,1\\}$ such that
$l+\varepsilon$ is even.
2. (2)
The degree of $C$ is $2s$, for some $s\in\mathbb{Z}_{\geq 4}$ and
$l+\varepsilon\leq s^{2}-2s+2,$ where $\varepsilon\in\\{0,1\\}$ such that
$l-s+\varepsilon$ is even.
Then $C$ admits no separating morphisms of degree $l$.
Proposition 1.2 falls within the line of results relating topology, complex
orientations and properties of separating plane curves, such as Rokhlin’s
complex orientation formulas (2.1), (2.2), and such as [Ore21, Theorem 1.1],
where Orevkov shows that there are finer relations for the numbers which
intervene in (2.2).
As corollaries of Proposition 1.2, one can find obstructions for the
existences of totally real pencils of curves of a certain degree for given
separating plane curves; see Section 1.3.
### 1.2. Generalities of separating morphisms
A separating morphism $f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ is always
unramified once restricted to $\mathbb{R}C$; see [KS20a, Theorem 2.19].
Therefore, the restriction of $f$ to each connected component of $\mathbb{R}C$
is a covering map of $\mathbb{R}\mathbb{P}^{1}$. This implies that the degree
of a separating morphism is at least as big as the number of connected
components of $\mathbb{R}C$. Actually, the definition of separating morphism
is more general and includes real morphisms between any real algebraic
varieties of same dimension. In order to a have a general idea of the subject,
we refer the interested reader to [KS20a] and [KTM22]. In the context of this
paper, we only need Definition 1.1.
In [Hui01], [Gab06],[CH13], [Cop13], [KS20b], [Ore18] properties of separating
morphisms and their existence are treated. For example, [Gab06, Theorem 7.1]
states that a genus $g$ real separating curve with $l$ real connected
components admits a separating morphism of degree at most $\frac{g+l+1}{2}$.
Later, Coppens, in [Cop13], constructed, for every value $h$ between
$\tilde{l}$, the minimum for a separating morphism degree, and
$\frac{\tilde{g}+\tilde{l}+1}{2}$, a separating curve $\tilde{C}$ of genus
$\tilde{g}$ and with $\tilde{l}$ real connected components such that $h$ is
the smallest possible degree of a separating morphism of $\tilde{C}$. In
[KS20b], the authors fix a separating curve $C$ and study all separating
morphisms of $C$ as follows. Set $d_{i}(f)\in\mathbb{N}$ the degree of the
covering map $f:C_{i}\rightarrow\mathbb{R}\mathbb{P}^{1}$ and set
$d(f):=(d_{1}(f),\dots,d_{l}(f))$. Let us denote by
$t:=\sum_{i=1}^{l}d_{i}(f)$ the degree of $f$. The set $Sep(C)$ of all such
degree partitions forms a semigroup, called separating semigroup, and for all
element $d\in Sep(C)$, it is shown that $d+\mathbb{Z}^{l}_{\geq 0}$ is also
contained in $Sep(C)$. So, in order to understand $Sep(C)$ for a given
separating curve $C$, it is important to understand which minimal possible
element $Sep(C)$ contains, where with minimal we mean that $t$ is of minimal
possible value.
###### Remark 1.3.
Thanks to Harnack-Klein inequality, any real curve of genus $g$ cannot have
more than $g+1$ real connected components. By Riemann-Roch theorem, all non-
singular real curves with $l=g+1$ real connected components admits a
separating morphism of degree $l$. But, whenever $l<g+1$, it is not a priori
clear whether a separating morphism of degree $l$ exists.
### 1.3. Obstruction of existence of totally real pencils
Kummer and Shaw, in [KS20b], also prove that for all real separating curves
embedded in the complex projective plane, there exist infinitely many totally
real pencils of curves.
###### Theorem 1.4.
([KS20b, Theorem 1.6]) Let $C$ be a non-singular real algebraic plane curve of
type I. Then there exists an integer $k$ such that $\forall k^{\prime}\geq k$,
there are $f,g\in\mathbb{R}[x,y,z]_{k^{\prime}}$ such that $V(f,g)\cap
C=\emptyset$ and $V(\lambda f+\mu g)\cap C$ consists of real points only for
all $\lambda,\mu\in\mathbb{R}$ not both zero.
###### Definition 1.5.
Let $C$ be a non-singular real algebraic plane curve of type I. We say that
$C$ admits a totally real pencil of curves of degree $k$ if there exists an
integer $k$ such that there are $f,g\in\mathbb{R}[x,y,z]_{k}$ such that
$V(f,g)\cap C=\emptyset$ and $V(\lambda f+\mu g)\cap C$ consists of real
points only for all $\lambda,\mu\in\mathbb{R}$ not both zero.
In [KS20b, Question 1], the authors wonder which may be the minimal possible
value of $k$ in Theorem 1.4. An immediate corollary of Proposition 1.2 gives a
lower bound for $k$, as follows.
###### Corollary 1.6.
Let $C$ be a non-singular degree $d$ real algebraic plane curve of type I and
with $l$ real connected components. Assume that $C$ respects either $(1)$ or
$(2)$ of Proposition 1.2. Then $C$ admits no totally real pencil of curves of
degree $k$ such that $kd\leq l$.
###### Remark 1.7.
As remarked in [KS20b, Remark 3.5], according to [Ahl50, §4.2], real
separating plane curves admit separating morphisms (whose degree is a priori
unknown). Hence, there exists an integer $k$ such that there are
$f,g\in\mathbb{R}[x,y,z]_{k}$ such that $V(\lambda f+\mu g)\cap C$ consists of
real points only for all $\lambda,\mu\in\mathbb{R}$ not both zero. The
difference with Definition 1.5 is that $V(f,g)\cap C$ may be non-empty.
Analogously, [Gab06, Theorem 1.7] implies the existence of separating
morphisms of degree between $l$ and $\frac{g+l+1}{2}$ and, once again, this
does not imply that a given real separating plane curve of genus $g$ and with
$l$ real connected components, admits totally real pencils of curves of degree
$k(l,g)$ in the sense of Definition 1.5.
In order to have examples of separating real curves respecting the hypothesis
of Proposition 1.2, we construct an explicit family of real separating plane
curves of increasing degree $2s+1$ such that each member of the family admits
no totally real pencils of curves of degree lower than $\frac{s^{2}+1}{2s+1}$;
see Proposition 1.8.
In particular, Proposition 1.8 presents some explicit examples of real curves
in which the bound for the minimal degree of separating morphisms is higher
than the Brill-Noether bound. Namely, the Brill-Noether theorem states that a
generic curve of genus $g$ has gonality $\lfloor\frac{g+3}{2}\rfloor$. This
implies in particular that for a generic real curve of genus $g$, there cannot
be any separating morphism of degree less than $\frac{g+3}{2}$. For the real
separating plane curves $C$ of odd degree $2s+1$ constructed to prove
Proposition 1.8, this gonality $\displaystyle{\frac{2s^{2}-s+3}{2}}$ is
strictly smaller than $s^{2}+1$, which is the number of connected components
of $\mathbb{R}C$.
###### Proposition 1.8.
For all $s\in\mathbb{N}_{\geq 2}$, there exists a non-singular real plane
curve $B_{d}$ of degree $d=2s+1$, of type I and with $s^{2}+1$ real connected
components, which admit no totally real pencils of curves of degree $k$ such
that $k\leq\frac{s^{2}+1}{2s+1}$.
###### Remark 1.9.
We do not know yet whether the curves $B_{2s+1}$, constructed (see Lemma 3.2)
to prove Proposition 1.8, admit totally real pencils of degree
$k^{\prime}=\lceil\frac{s^{2}+1}{2s+1}\rceil$.
### 1.4. Statement of the main result and strategy
In this paper, we consider real separating curves embedded in some ambient
real surface and focus on their separating morphisms. The key tools of our
approach are the fact that all separating curves come equipped with two
possible complex orientations and the use of [Ore21, Theorem 3.2] as shown in
[Ore21, Example 3.3]. Before recalling [Ore21, Theorem 3.2] and state the main
result of this paper (Theorem 1.13), let us introduce some notation.
###### Notation 1.10.
Let $X$ be a smooth real algebraic surface and $C\subset X$ a non-singular
real algebraic curve of class $d$ in $H_{2}(X;\mathbb{Z})$. Let us consider a
real divisor $D$ belonging to the linear system $|C+K_{X}|$. Assume that $D$
has not $C$ as a component. This divisor can always be written as
$2D_{0}+D_{1}$ with the following properties. The divisor $D_{0}$ is an
effective divisor in $H_{2}(X;\mathbb{Z})$ and $D_{1}$ is a reduced curve
realising some class $\beta\in H_{2}(X;\mathbb{Z})$.
###### Example 1.11.
Let $C$ be a non-singular real algebraic curve of degree $2k$ in
$\mathbb{C}\mathbb{P}^{2}$. Then $D_{0}$ can be chosen as the divisor
realising the class of a plane curve of degree $k-2$ and $D_{1}$ as a line in
$\mathbb{C}\mathbb{P}^{2}$.
###### Theorem 1.12.
([Ore21, Theorem 3.2]) Let $X$ be a smooth real algebraic surface, let $C$ be
a smooth irreducible real separating curve on $X$ and $D$ be a real divisor on
$X$ as in Notation 1.10. Let us fix a complex orientation on $\mathbb{R}C$ and
an orientation $\mathcal{O}$ on
$\mathbb{R}X\setminus(\mathbb{R}C\cup\mathbb{R}D_{1})$ which changes each time
we cross $\mathbb{R}C\cup\mathbb{R}D_{1}$ at its smooth points. The latter
orientation induces a boundary orientation on
$\mathbb{R}C\setminus(\mathbb{R}C\cap D_{1})$. Let
$f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ be a separating morphism. Then it is
impossible that, for some $p\in\mathbb{R}\mathbb{P}^{1}$, the set
$f^{-1}(p)\setminus supp(D)$ is non-empty and the two orientations coincide at
each point of the set.
In [Ore21], Orevkov shows powerful applications of Theorem 1.12, such
as[Ore21, Theorem 1.1] and the construction of complex schemes (Definition
2.7) in the real projective plane which are realisable by real
pseudoholomorphic plane curves of odd degree and not realisable by real
algebraic plane curves of same degree. Moreover, other applications of [Ore21,
Theorem 3.2] are presented via examples. Therefore, the possible directions of
use of the content of [Ore21, Theorem 3.2] are still wide open. Now, let us
present the main result of this paper. We refer the reader to Notation 1.10.
Unless differently stated, we choose $\beta\in H_{2}(X;\mathbb{Z})$ in order
to maximise $\frac{D_{0}^{2}-D_{0}K_{X}}{2}\in\mathbb{Z}_{>0}$.
###### Theorem 1.13.
Let $X$ be a smooth real algebraic surface with holomorphic Euler
characteristic of the trivial bundle $\chi(0)\geq 1$ and $C\subset X$ a non-
singular real algebraic curve of class $d$ and of type I. Denote by $l$ the
number of connected components of $\mathbb{R}C$. Assume one of the following:
1. (1)
$D_{1}$ can be chosen empty and
$2\lceil\frac{l}{2}\rceil\leq\frac{d^{2}-k_{X}^{2}}{4}$;
2. (2)
$D_{1}$ cannot be chosen empty and $2\lceil\frac{l-m-\theta}{2}\rceil\leq
D_{0}^{2}-D_{0}K_{X}-2\theta$, where $\theta+m$ is the number of connected
components of $\mathbb{R}C$ which have non-empty intersection with
$\mathbb{R}D_{1}$, and $m=min(\frac{\beta^{2}-\beta K_{X}}{2},l)$;
Then there are no separating morphisms
$f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ of degree $l$.
###### Remark 1.14.
For example, Theorem 1.13 holds for separating real curves in any smooth
rational surface $X$, because the holomorphic Euler characteristic of the
trivial bundle equals one. In particular, Theorem 1.13 directly implies
Proposition 1.2. In fact, for plane curves of odd degree $2s+1$, the curve
$D_{1}$ can be chosen empty and
$2\lceil\frac{l}{2}\rceil\leq\frac{(2s+1)^{2}-9}{4}=s^{2}+s-2$. On the other
hand, for plane curves of even degree $2s$, the curve $D_{1}$ can be chosen to
be a real line so that $m+\theta\leq s$ with $m=2$, and $D_{0}=(s-2)[D_{1}]\in
H_{2}(\mathbb{C}\mathbb{P}^{2};\mathbb{Z})$. Hence
$2\Bigl{\lceil}\frac{l-s}{2}\Bigl{\rceil}\leq
l-2-\theta\leq(s-2)^{2}+3(s-2)-2\theta$
and one gets $l+\varepsilon\leq s^{2}-s-\theta$, where $\varepsilon=1$ if
$l-s$ is odd and $0$ otherwise. At worst, the number $\theta$ equals $s-2$ and
we obtain the bound given in the hypothesis of Proposition 1.2.
### 1.5. Acknowledgements
I would like to thank Mario Kummer and Kris Shaw for useful discussions. A
special thanks to Stepan Orevkov for addressing my attention to [Vir86,
(4.8)]. Moreover I would like to thank Erwan Brugallé and Hannah Markwig for
suggestions and remarks on a preliminary version of the paper. Thanks to
Athene Grant.
### Organisation of the paper
We start presenting some tools and terminology in Section 2. Then, in Section
3.1, we give a proof of Theorem 1.13. Finally, in Section 3.2, we present
examples and applications of Theorem 1.13, and we prove Proposition 1.8.
## 2\. Preliminaries
Let us consider non-singular real algebraic curves embedded in the complex
projective plane. The real part of real plane curves is homeomorphic to a
disjoint union of $S^{1}$ embedded in $\mathbb{R}\mathbb{P}^{2}$. Each $S^{1}$
can be embedded in $\mathbb{R}\mathbb{P}^{2}$ in two different ways: if it
realises the trivial-class in
$H_{1}(\mathbb{R}\mathbb{P}^{2};\mathbb{Z}/2\mathbb{Z})$, it is called oval,
otherwise it is called pseudo-line and denoted by $J$. If a non-singular real
plane curve has even degree then its real part consists of ovals only;
otherwise of exactly one pseudo-line and ovals.
An oval in $\mathbb{R}P^{2}$ separates two disjoint non-homeomorphic connected
components: the connected component homeomorphic to a disk is called interior
of the oval; the other one is called exterior of the oval. For each pair of
ovals, if one is in the interior of the other we speak about an injective
pair, otherwise a non-injective pair. For a separating real plane curve,
equipped with one of its two complex orientations, an injective pair of ovals
is said to be positive if the complex orientations of the ovals are induced by
an orientation of the annulus bounded by the ovals; otherwise negative. An
oval $\mathcal{O}$ of an odd degree separating curve is said positive if
$[\mathcal{O}]=-2[J]$ in $H_{1}(M)$, where $M$ is the closure of the non-
orientable component of $\mathbb{R}\mathbb{P}^{2}\setminus\mathcal{O}$; and
negative otherwise.
### 2.1. Rokhlin’s complex orientation formula
For any separating plane curve of even degree $2s$ and with $l$ real connected
components, Rokhlin in [Rok74] proved that
(2.1) $2(\Pi_{+}-\Pi_{-})=l-s^{2},$
where $\Pi_{+}$ and $\Pi_{-}$ denote the number of respectively positive and
negative injective pairs of ovals of the curve.
Later, for separating plane curves of odd degree $2s+1$ and with $l$ real
connected components, Mishachev [Mis75] proved that
(2.2) $\Lambda_{+}-\Lambda_{-}+2(\Pi_{+}-\Pi_{-})=l-1-s(s+1),$
where $\Lambda_{+}$ and $\Lambda_{-}$ denote respectively positive and
negative ovals of the curve.
In order to give examples of separating real curves respecting the hypothesis
of Proposition 1.2 and to prove Proposition 1.8, we need the following result.
###### Proposition 2.3.
[Vir86, Section (4.8)] Let $A_{1}$ and $A_{2}$ be separating real plane curves
of degree $d_{1}$ and $d_{2}$, transversal to each other and intersecting in
$r$ real points. Let $B$ be a real curve of degree $d=d_{1}+d_{2}$ obtained
from $A_{1}\cup A_{2}$ by a small perturbation such that some complex
orientations of $A_{1}$ and $A_{2}$ determine an orientation $\mathfrak{O}$ of
$\mathbb{R}B$. Then $B$ is of type I if and only if the pair
$(\mathbb{R}B,\mathfrak{O})$ respects Rokhlin’s complex orientation formula.
### 2.2. Encoding real schemes
In this section, it is introduced some notation and definitions used Section
3.2.
###### Definition 2.4.
Let $B_{1},\dots,B_{n}$ be any collection of disjoint ovals (and of a pseudo-
line) in $\mathbb{R}\mathbb{P}^{2}$. We call topological type the arrangement
$\mathcal{S}$ realised by the pair
$(\mathbb{R}\mathbb{P}^{2},\bigsqcup_{i=1}^{n}B_{i})$.
###### Notation 2.5 (Encoding topological types).
Let us consider collections of disjoint ovals in $\mathbb{R}\mathbb{P}^{2}$.
An empty union of ovals is denoted by $0$. We say that a union of $l$ ovals
realises $l$ if there are no injective pairs. The symbol
$\langle\mathcal{S}\rangle$ denotes the disjoint union of a non-empty
collection of ovals realising $\mathcal{S}$, and an oval forming an injective
pair with each oval of the collection. A collection of $h$ ovals in
$\mathbb{R}\mathbb{P}^{2}$ is called a nest of depth $h$ if any two ovals of
the collection form an injective pair. The disjoint union of any two
collections of ovals, realising respectively $\mathcal{S}^{\prime}$ and
$\mathcal{S}^{\prime\prime}$ in $\mathbb{R}\mathbb{P}^{2}$, is denoted by
$\mathcal{S}^{\prime}\sqcup\mathcal{S}^{\prime\prime}$ if none of the ovals of
one collection forms an injective pair with the ovals of the other one and
they are both non-empty collections. A disjoint union of the form
$\mathcal{S}^{\prime}\sqcup 0$ is still denoted $\mathcal{S}^{\prime}$.
Moreover, as previously mentioned, a pseudo-line in $\mathbb{R}\mathbb{P}^{2}$
is denoted by $J$.
###### Definition 2.6.
A topological type $\mathcal{S}$, up to homeomorphism of
$\mathbb{R}\mathbb{P}^{2}$, is called real scheme. Let
$A\subset\mathbb{C}\mathbb{P}^{2}$ be a non-singular real curve. We say that
$A$ has real scheme $\mathcal{S}$ if the pair
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}A)$ realises $\mathcal{S}$.
###### Definition 2.7.
A real scheme $\mathcal{S}$ in $\mathbb{R}\mathbb{P}^{2}$, where each
connected component is endowed with an orientation is called complex scheme.
Let $A\subset\mathbb{C}\mathbb{P}^{2}$ be a non-singular real separating
curve. We say that $A$ has complex scheme $\mathcal{S}$ if the pair
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}A)$ realises $\mathcal{S}$ as real scheme
and one of the two complex orientations of $\mathbb{R}A$ agrees with that of
$\mathcal{S}$.
## 3\. Theorem 1.13 and applications
Section 3.1 is uniquely devoted to the proof of Theorem 1.13. Then Section 3.2
presents the proof of Proposition 1.8 and some examples.
### 3.1. Proof of Theorem 1.13
###### Proof of Theorem 1.13.
Denote by $C_{i}$ the connected components of $\mathbb{R}C$ where $i=1,\dots
l$. Let us consider a real divisor $D$ belonging to the linear system
$|C+K_{X}|$. For the sake of contradiction, assume that there exists a
separating morphism $f:C\rightarrow\mathbb{P}^{1}$ of degree $l$. Then, for
all $p\in\mathbb{R}\mathbb{P}^{1}$, the set
$f^{-1}(p)=\\{p_{1},\dots,p_{l}\\}$ is a collection of $l$ real points such
that every $C_{i}$ contains exactly one $p_{i}$. Let us choose a curve $D_{1}$
and a curve of class $D_{0}$; see Notation 1.10. For our purposes, the choice
of $\beta\in H_{2}(X;\mathbb{Z})$, and consequently of $D_{1}$, is done in
order to maximise $\frac{D_{0}^{2}-D_{0}K_{X}}{2}\in\mathbb{Z}_{>0}$; where,
thanks to Riemann-Roch theorem and the hypothesis that the holomorphic Euler
characteristic of the trivial bundle of $X$ is bigger or equal to one, the
positive integer $\frac{D_{0}^{2}-D_{0}K_{X}}{2}$ is always less or equal than
the dimension of the linear system of curves in $X$ of class $D_{0}$.
Therefore, fixed a configuration of $\frac{D_{0}^{2}-D_{0}K_{X}}{2}$ points,
there always exists at least one curve of class $D_{0}$ passing through the
configuration.
Once $D_{1}$ is chosen, let us fix an orientation on
$\mathbb{R}X\setminus(\mathbb{R}C\cup\mathbb{R}D_{1})$ which changes each time
we cross $\mathbb{R}C\cup\mathbb{R}D_{1}$ at its smooth points; see [Ore21,
Proof of Theorem 3.2] for details. The latter orientation induces a boundary
orientation $\mathfrak{O}$ on $\mathbb{R}C$. Fix some $p$ in
$\mathbb{R}\mathbb{P}^{1}$.
Before treating cases $(1)$ and $(2)$ and prove the statement. Let us give an
idea of the strategy. In order to get a contradiction, we use Theorem 1.12.
Namely, define $Y_{p}$ (resp. $NY_{p}$) as the set of points in
$f^{-1}(p)\setminus supp(D)$ on which the complex orientation and the
orientation $\mathfrak{O}$ agree (resp. do not agree). Then, Theorem 1.12 tell
us that if the set $f^{-1}(p)\setminus supp(D)$ is non-empty, one has that
$Y_{p}$ and $NY_{p}$ are both non-empty.
First, let us treat case $(1)$ in which $D_{1}$ can be chosen empty. Fix one
of the two complex orientations on $\mathbb{R}C$. Then, remark that either the
number $V^{1}_{p}$ of $p_{i}$’s where the complex orientation and the
orientation $\mathfrak{O}$ agree, or the number $V^{2}_{p}$ of the $p_{i}$’s
where the two orientations do not agree, does not exceed
$\lceil\frac{l}{2}\rceil$. Given a collection $\mathcal{P}$ of
$\frac{d^{2}-K_{X}^{2}}{8}$ real points in $X$, there exists a real curve $A$
of class $D_{0}$ passing through $\mathcal{P}$. By hypothesis
$2\lceil\frac{l}{2}\rceil\leq\frac{d^{2}-K_{X}^{2}}{4}$, therefore it is
possible to choose such a collection $\mathcal{P}$ so that it contains at
least $\lceil\frac{l}{2}\rceil$ points of $f^{-1}(p)$ belonging to
$V^{j}_{p}$, for an opportune $j\in\\{1,2\\}$. It follows that the set
$f^{-1}(p)\setminus supp(D)$ is non-empty and either $NY_{p}$ or $Y_{p}$ is
empty, which contradicts Theorem 1.12.
Now, let us consider case $(2)$ in which $D_{1}$ is not empty. Let us pick
$D_{1}$ as the real curve passing through a given collection
$\tilde{\mathcal{P}}$ of $\frac{\beta^{2}-\beta K_{X}}{2}$ real points
contained in $m=min(\frac{\beta^{2}-\beta K_{X}}{2},l)$ connected components
$C_{j_{1}},\dots,C_{j_{m}}$. Moreover, choose the collection
$\tilde{\mathcal{P}}$ such that it contains the points
$p_{j_{1}},\dots,p_{j_{m}}$ of $f^{-1}(p)$. Now, we proceed as in the previous
case with the difference that one must take $D_{1}$ into account when looking
at the orientation $\mathfrak{O}$. Let $C_{j_{m+1}},\dots,C_{j_{m+\theta}}$ be
the connected components of
$\mathbb{R}C\setminus\bigsqcup_{i=j_{1}}^{j_{m}}C_{i}$ intersecting
$\mathbb{R}D_{1}$. Each connected components $C_{j_{m+i}}$, with
$i=1,\dots,\theta$, may contain, regardless of the chosen complex orientation
on $\mathbb{R}C$, a point of $f^{-1}(p)$ where the orientation $\mathfrak{O}$
and the complex orientation do not coincide. Fix one of the two complex
orientations on $\mathbb{R}C$. Then, remark that either the number $V^{1}_{p}$
of $p_{i}$’s contained in
$\mathbb{R}C\setminus\bigsqcup_{i=1}^{m+\theta}C_{j_{i}}$, where the complex
orientation and the orientation $\mathfrak{O}$ agree, or the number
$V^{2}_{p}$ of the $p_{i}$’s in
$\mathbb{R}C\setminus\bigsqcup_{i=1}^{m+\theta}C_{j_{i}}$, where the two
orientations do not agree, does not exceed $\lceil\frac{l-m-\theta}{2}\rceil$.
Therefore, we pick a collection $\mathcal{P}$ of
$\frac{D_{0}^{2}-D_{0}K_{X}}{2}$ real points such that $\theta$ of those are
the points of $f^{-1}(p)$ contained in the $C_{j_{m+i}}$’s, where
$i=1,\dots,\theta$; and the others include the points of $V^{j}_{p}$, for an
opportune $j\in\\{1,2\\}$. Once again, the set $f^{-1}(p)\setminus supp(D)$ is
non-empty and either $NY_{p}$ or $Y_{p}$ is empty, which contradicts Theorem
1.12. ∎
### 3.2. Examples and Applications
Let us consider real separating curves in the complex projective plane and
focus on [KS20b, Question 1]. A first known example [KS20b, Example 2.8]
concerns real separating plane curves of degree $2s$ or $2s+1$ having a a nest
of maximal depth $s$. Such curves admit a totally real pencil of lines. Some
more examples can be found in [Tou13], where for some pairs $(A,\mathcal{S})$,
where $A$ is a real separating plane sextic curve with $9$ real connected
components and $\mathcal{S}$ its real scheme in $\mathbb{R}\mathbb{P}^{2}$,
the minimal value for $k$ of Theorem 1.4, is shown to be equal to $3$.
$q$ Figure 1. $(\mathbb{R}\mathbb{P}^{2},\mathbb{R}A,\mathbb{R}L)$ of Example
3.1. Double arrows denote $\mathfrak{O}$, simple arrows the fixed complex
orientation of $\mathbb{R}A$ and $\bullet$ the points in $f^{-1}(p)$.
###### Example 3.1 (Application of Theorem 1.12).
Let $A$ be a non-singular plane quintic of type I realising the real scheme
$J\quad\sqcup\quad 4$. We use Theorem 1.12 in order to show that there exists
no separating morphism $f:A\rightarrow\mathbb{C}\mathbb{P}^{1}$ of degree $5$.
For the sake of contradiction assume that such a $f$ exists. By Rokhlin’s
complex orientation formula (2.2), among the ovals of $\mathbb{R}A$, three are
negative and one is positive. Fix $p\in\mathbb{R}\mathbb{P}^{1}$. Denote $q$
the point lying on the positive oval of $\mathbb{R}A$ and contained in
$f^{-1}(p)$. Let us consider Notation 1.10. Choose $D_{1}$ to be empty.
Therefore $D_{0}$ realises the class of a line. Fix an orientation
$\mathfrak{O}$ that changes every time time we cross $\mathbb{R}A$. Up to
choose one of the two complex orientations of $\mathbb{R}A$, one gets that the
fixed complex orientation and $\mathfrak{O}$ agree on the pseudo-line and the
three negative ovals of $\mathbb{R}A$. Hence, if we pick any line $L$ passing
through the point $q$, we get a contradiction thanks to Theorem 1.12; see Fig.
1,
A key information used to deal with the examples above is the fact that their
real schemes are given. In this paper, on the other hand, we focus on real
separating curves $C$ about which only the number of real connected components
is set. We manage to obstruct the existence of certain totally real pencils of
curves with respect to $C$, but we do not know yet how to compute the minimal
value for $k$ of Theorem 1.4. The rest of the section is dedicated to the
proof of Proposition 1.8. We start with a lemma, where a recursive
construction of curves produces odd degree plane curves respecting the
hypothesis of Proposition 1.8.
###### Lemma 3.2.
For all positive integer $s$ and $\varepsilon\in\\{0,1\\}$, there exist non-
singular real plane projective curves of degree $2s+\varepsilon$ of type I and
with $s^{2}+\varepsilon$ real connected components.
###### Proof.
For small positive integers $d\leq 4$, real maximal, therefore separating,
plane curves of degree $d$ have the required properties of the statement.
Therefore, we present a recursive construction of real plane curves $B_{d}$
with the wanted features via a variation of Harnack construction method
[Har76] combined with Proposition 2.3, which allows us to check, at every step
of the construction, that the realised curves are of type I. Set $l_{d}$ as
the number of connected components of $\mathbb{R}B_{d}$. Moreover if $d=2s+1$,
denote $\Lambda_{\pm}$ with $\Lambda^{s}_{\pm}$ for $B_{d}$. In the following,
whenever we consider a plane curve $A$, we denote its polynomial by
$a(x,y,z)$.
Let us fix a real line $L$.
1. (i)
Step $0$ induction: In order to construct a curve with the wanted properties
for any given degree, we start from the following. One can construct a non-
singular real maximal cubic $B_{3}$ whose pseudo-line $J_{3}$ intersects
$\mathbb{R}L$ in $3$ distinct points; see Fig 2.1. Equip $\mathbb{R}B_{3}$
with one of its two complex orientations and $\mathbb{R}L$ with the complex
orientation opposite to that of $J_{3}$. Let $\mathcal{H}_{1},\mathcal{H}_{2}$
and $\mathcal{H}_{3}$ be the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{3}$. Up to fix $\mathcal{H}_{1}$, label the
$\mathcal{H}_{i}$’s following the order given by the fixed complex orientation
on $J_{3}$. Pick four real lines $L_{1},\dots,L_{4}$ intersecting pairwise
respectively $\mathcal{H}_{1},\mathcal{H}_{2}$. Now, we apply Brusotti’s
perturbation method [Bru21] to $B_{3}\cup L$ in a compatible way with the
fixed complex orientations of $B_{3}$ and $L$. Namely, we take a small
perturbation of $L\cup B_{3}$ replacing $b_{3}(x,y,z)l(x,y,z)$ with
$b_{4}(x,y,z):=b_{3}(x,y,z)l(x,y,z)+\varepsilon
l_{1}(x,y,z)\dots,l_{4}(x,y,z)$, where $\varepsilon\not=0$ is a sufficient
small real number. Up to a choice of the sign of $\varepsilon$, the real
curve, defined by $b_{4}(x,y,z)=0$, is a non-singular real curve $B_{4}$ of
degree $4$. In particular $B_{4}$ is maximal, therefore of type I, equipped
with a complex orientation $\mathfrak{O}_{4}$ coming form the perturbation;
see Fig. 2.2.
Take for $L$ the opposite complex orientation to the previously chosen one.
Set $\mathcal{H}_{1},\dots,\mathcal{H}_{4}$ for the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{4}$. Label the $\mathcal{H}_{i}$’s following
the order given by the fixed complex orientation on $\mathbb{R}L$. Without
loss of generality, we can say that $\mathcal{H}_{1}$ and $\mathcal{H}_{3}$
are in the interior of the ovals of $\mathbb{R}B_{4}$ intersecting
$\mathbb{R}L$. Pick five real lines $L_{1},\dots,L_{5}$ such that $L_{1}$
intersects $\mathcal{H}_{2}$ and the others $\mathcal{H}_{4}$. Now, we apply
Brusotti’s perturbation method to $L\cup B_{4}$ with respect to
$\bigcup_{i=1}^{5}L_{i}$, in a compatible way with the complex orientation
$\mathfrak{O}_{4}$ and the fixed complex orientation of $L$, and obtain a non-
singular real curve $B_{5}$ of degree $5$ whose real part is equipped with an
orientation $\mathfrak{O}_{5}$ coming from the perturbation. In order to
assure that $\mathfrak{O}_{5}$ is a complex orientation of $B_{5}$ and, hence
$B_{5}$ is of type I, it is enough to check that $B_{5}$ respects (2.2); see
Proposition 2.3. Because of the absence of nests in $\mathbb{R}B_{5}$ and the
number of positive and negative ovals of $\mathbb{R}B_{5}$ differ by $-2$, it
follows that $(B_{5},\mathfrak{O}_{5})$ respect Rokhlin’s complex orientation
formula.
Fig. 2.1: $(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{3},\mathbb{R}L)$Fig. 2.3:
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{5},\mathbb{R}L)$Fig. 2.2:
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{4},\mathbb{R}L)$Fig. 2.4:
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{6},\mathbb{R}L)$ Figure 2.
$\mathbb{R}L$ in red and $\mathbb{R}B_{d}$ in black, where, from top to bottom
and from left to right, $d=3,4,5,6$.
2. (ii)
Inductive Step: Assume that there exists a non-singular real plane curve
$B_{2s-1}$ of degree $2s-1$ and of type I. Equip $\mathbb{R}B_{2s-1}$ with one
of its two complex orientations and $\mathbb{R}L$ with the complex orientation
opposite to that of $J_{2s-1}$. Moreover, assume that
* •
the pseudo-line $J_{2s-1}$ intersects $\mathbb{R}L$ in $2s-1$ distinct points
such that, if one labels the intersections points $2s-1,\dots,1$ with respect
to the complex orientation on $J_{2s-1}$, the order of the labeled points
agrees with the complex orientation of $\mathbb{R}L$;
* •
the number $l_{2s-1}$ equals $(s-1)^{2}+1$;
* •
the curve $B_{2s-1}$ has no nests and
$\Lambda^{(s-1)}_{+}=\frac{(s-1)(s-2)}{2}\quad\text{and}\quad\Lambda^{(s-1)}_{-}=\frac{s(s-1)}{2}.$
Let $\mathcal{H}_{1},\dots,\mathcal{H}_{2s-1}$ be the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{2s-1}$. Up to fix $\mathcal{H}_{1}$, label
the $\mathcal{H}_{i}$’s following the order given by the fixed complex
orientation on $J_{2s-1}$. Pick $2s$ real lines $L_{1},\dots,L_{2s}$
intersecting pairwise
$\mathcal{H}_{j},\mathcal{H}_{j+1},\dots,\mathcal{H}_{j+s}$, for some
$j\in\\{1,\dots,2s+1\\}$. Now, we apply Brusotti’s perturbation method to
$L\cup B_{2s-1}$ with respect to $\bigcup_{i=1}^{2s}L_{i}$, in a compatible
way with the fixed complex orientations of $B_{2s-1}$ and $L$, and obtain a
degree $2s$ non-singular real algebraic curve $B_{2s}$, equipped with an
orientation $\mathfrak{O}_{2s}$. Let us check that (2.1) applies to the pair
$(B_{2s},\mathfrak{O}_{2s})$. The curve $B_{2s}$ has no nests and
$l_{2s}=l_{2s-1}-1+2s-1=s^{2}$. Hence, Proposition 2.3 implies that $B_{2s}$
is of type I. In addition, remark that $s$ ovals of $\mathbb{R}B_{2s}$
intersect $\mathbb{R}L$.
Take for $L$ the opposite complex orientation to the previously chosen one.
Set $\mathcal{H}_{1},\dots,\mathcal{H}_{2s}$ for the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{2s}$. Label the $\mathcal{H}_{i}$’s
following the order given by the fixed complex orientation on $\mathbb{R}L$.
Without loss of generality, we can say that $\mathcal{H}_{2h+1}$, for $0\leq
h\leq s$ are in the interior of the ovals of $\mathbb{R}B_{2s}$ intersecting
$\mathbb{R}L$. Pick $2s+1$ real lines $L_{1},\dots,L_{2s+1}$ such that $L_{h}$
intersects $\mathcal{H}_{2h}$, for all $1\leq h\leq s-1$ and all others lines
intersect $\mathcal{H}_{2s}$. Once again, apply Brusotti’s perturbation method
to $L\cup B_{2s}$ with respect to $\bigcup_{i=1}^{2s+1}L_{i}$, in a compatible
way with the fixed complex orientations of $B_{2s}$ and $L$. We get a degree
$2s+1$ non-singular real curve $B_{2s+1}$ whose real part is equipped with an
orientation $\mathfrak{O}_{2s+1}$ coming from the perturbation, and
$l_{2s+1}=l_{2s}+1=s^{2}+1$. Let us check that the pair
$(B_{2s+1},\mathfrak{O}_{2s+1})$ respects (2.2). First, observe that
$\Lambda^{s}_{+}=\Lambda^{(s-1)}_{+}+s-1\quad\text{and}\quad\Lambda^{s}_{-}=\Lambda^{(s-1)}_{-}+s.$
It follows that, because of the absence of nests in $\mathbb{R}B_{2s+1}$ and
the number of positive and negative ovals of $\mathbb{R}B_{2s+1}$ differ by
$-s$, the pair $(B_{2s+1},\mathfrak{O}_{2s+1})$ satisfies Rokhlin’s complex
orientation formula. In conclusion, one obtains a real curve of degree $2s+1$
which is separating and with $s^{2}+1$ real connected components.
∎
###### Proof of Proposition 1.8.
The non-singular real algebraic plane curves $B_{2s+1}$ of odd degree
$2s+1\geq 5$ and of type I, constructed in Lemma 3.2, satisfy the hypothesis
of Corollary 1.6. Therefore $B_{2s+1}$ admit no totally real pencils of curves
of degree lower or equal to $\frac{s^{2}+1}{2s+1}$. ∎
###### Remark 3.3.
Notice that the non-singular real algebraic plane curves $B_{2s}$ of even
degree $2s\geq 6$ and of type I constructed in Lemma 3.2, does not satisfy
hypothesis of Corollary 1.6. We do not know yet whether there is a way to
apply [Ore21, Theorem 3.2] to this particular case in order to show that
$B_{2s}$ admit no totally real pencils of curves of degree lower or equal to
$\frac{s}{2}$.
Mimic the constructions adopted in the proof of Lemma 3.2, one may also
construct an explicit family of real separating plane curves of increasing
even degree, which respect the hypothesis of Proposition 1.2 or even those in
Remark 1.14. Instead of presenting to the reader such a construction, that
would be a modified version of proof of Lemma 3.2, we conclude with an example
of separating real curves in a different ambient surface, respecting
hypothesis of Theorem 1.13 and, therefore, non admitting separating morphisms
of degree equal to the number of their real connected components.
###### Example 3.4.
Let $Q$ be $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$, equipped with the anti-
holomorphic involution $\sigma:Q\rightarrow Q$ sending $(x,y)$ to
$(\overline{y},\overline{x})$, where $x=[x_{0}:x_{1}]$ and $y=[y_{0}:y_{1}]$
are in $\mathbb{C}P^{1}$ and
$\overline{x}=[\overline{x_{0}}:\overline{x_{1}}]$ and
$\overline{y}=[\overline{y_{0}}:\overline{y_{1}}]$ are respectively the images
of $x$ and $y$ via the standard complex conjugation on $\mathbb{C}P^{1}$. The
real part of $Q$ is homeomorphic to a $2$-sphere $S^{2}$ and $Q$ is called
quadric ellipsoid. Moreover, the Euler characteristic of the trivial bundle of
$Q$ equals one; see Remark 1.14. A non-singular real algebraic curve A on $Q$
is defined by a bi-homogeneous polynomial of bidegree $(d,d)$
$P(x,y)=\sum\limits_{0\leq i,j\leq
d}a_{i,j}x_{0}^{i}x_{1}^{d-i}y_{0}^{j}y_{1}^{d-j}=0,$
where $d$ is a positive integer and the coefficients satisfy
$a_{i,j}=\overline{a_{j,i}}$. Real schemes realised by non-singular real
algebraic curves of bidegree $(d,d)$ on $Q$ are completely classified for
$d\leq 5$; see [GS80], [Mik94] and [Man21]. Moreover such classifications
distinguish the cases in which a given topological type may or may not be
realised by a real algebraic curve of type I.
For $d\leq 5$, we report in Table 1 a list of real schemes realised by type I
real algebraic curves of bidegree $(d,d)$, with $l$ real connected components,
where $l$ undergoes the hypothesis of Theorem 1.13. Hence, the curves do not
admit separating morphisms of degree $l$. In order to understand Table 1, let
us recall some notation. Let $A$ be a non-singular bidegree $(d,d)$ real curve
on $Q$. The connected components of $A$ are called ovals. An oval in $S^{2}$
bounds two disks; therefore, on $S^{2}$ interior and exterior of an oval are
not well defined. It follows that the encoding of real schemes on $S^{2}$ is
not well defined either and it depends on the choice of a point $p$ on
$S^{2}\setminus\mathbb{R}A$. Let us take $S^{2}$ deprived of $p$, which is
homeomorphic to $\mathbb{R}^{2}$ and let us call oval any circle embedded in
$\mathbb{R}^{2}$. Analogously to the case of $\mathbb{R}P^{2}$, in
$\mathbb{R}^{2}$ one define interior and exterior of an oval and
(non-)injective pairs for each pair of ovals. Up to choose a point, we can now
adopt the same notation introduced in Section 2.2.
d | Real schemes realised by type I curves | $l\leq$ | $r(D_{0},D_{1})$ | $D_{0}$ | $D_{1}$ | $m$ | $\theta$
---|---|---|---|---|---|---|---
$3$ | $\langle\langle 1\rangle\rangle$ | $3\leq$ | 3 | $-$ | $(1,1)$ | 3 | 0
$4$ | $\langle\langle\langle 1\rangle\rangle\rangle$ | $4\leq$ | 6 | $(1,1)$ | $-$ | $-$ | $-$
$4$ | $4\quad\sqcup\quad\langle 1\rangle$ | $6\leq$ | 6 | $(1,1)$ | $-$ | $-$ | $-$
$4$ | $3\quad\sqcup\quad\langle 2\rangle$ | $6\leq$ | 6 | $(1,1)$ | $-$ | $-$ | $-$
$5$ | $\langle\langle\langle\langle 1\rangle\rangle\rangle\rangle$ | $5\leq$ | 7 | $(1,1)$ | $(1,1)$ | 3 | 2
$5$ | $\alpha\quad\sqcup\quad\langle\beta\rangle\quad\sqcup\quad\langle\gamma\rangle,$ for all $\alpha,\beta,\gamma$ such | $7\leq$ | 7 | $(1,1)$ | $(1,1)$ | 3 | 2
| that $\alpha=0\pmod{2}$ and $\alpha+\beta+\gamma=5$ | | | | | |
Table 1. For all $d\leq 5$, this is a list of real schemes realised by type I
real algebraic curves of bidegree $(d,d)$, with $l$ real connected components,
where $l$ undergoes the hypothesis of Theorem 1.13. Hence, the curves do not
admit separating morphisms of degree $l$. The symbol $r(D_{0},D_{1})$
represents the bound for the number $l$, depending on choices of
$D_{0},D_{1}$, for which the hypothesis of Theorem 1.13 are satisfied and such
that one maximises the number
$\frac{D_{0}^{2}-D_{0}K_{S}}{2}\in\mathbb{Z}_{>0}$.
###### Remark 3.5.
In the context of Example 3.4, one can construct recursively, in a way similar
to the proof of Lemma 3.2, for all $d\geq 6$, bidegree $(d,d)$ separating real
algebraic curves on the quadric ellipsoid not admitting separating morphism of
degree equal to the number of their real connected components. In fact,
variations of Harnack’s construction method, Brusotti’s perturbation method
and Rokhlin’s complex orientation formula [Zvo83, Ore07] are also available on
the quadric ellipsoid.
## References
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* [Gab06] A. Gabard. Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes. Comment. Math. Helv., 81(4):945–964, 2006.
* [GS80] D. A. Gudkov and E. I. Shustin. Classification of non-singular 8th order curves on ellipsoid. In ”Methods of Qualitative Theory”. Gorky university Press, pages 104–107 (in Russian), 1980.
* [Har76] A. Harnack. Über vieltheiligkeit der ebenen algebraischen Curven. In Math. Ann., 10: 189-199,, volume 1060 of Lecture Notes in Math., pages 187–200. 1876.
* [Hui01] J. Huisman. On the geometry of algebraic curves having many real components. Rev. Mat. Complut., 14(1):83–92, 2001.
* [Kle73] F. Klein. Über Flächen dritter Ordnung. Math. Ann. 6, pag. 551–581, 1873.
* [KS20a] M. Kummer and E. Shamovich. Real fibered morphisms and Ulrich sheaves. J. Algebraic Geom., 29(1):167 – 198, 2020.
* [KS20b] M. Kummer and K. Shaw. The separating semigroup of a real curve. Annales de la Faculté des sciences de Toulouse: Mathématiques, 29(1):79–96, 2020.
* [KTM22] M. Kummer, C. Le Texier, and M. Manzaroli. Real-Fibered Morphisms of del Pezzo Surfaces and Conic Bundles. Discrete & Computational Geometry, https://doi.org/10.1007/s00454-022-00427-3, 2022.
* [Man21] M. Manzaroli. Real algebraic curves of bidegree (5,5) on the quadric ellipsoid. St. Petersburg Math. J., 32(2):279–306, 2021.
* [Mik94] G. Mikhalkin. Congruences for real algebraic curves on an ellipsoid. In Topology of manifolds and varieties, volume 18 of Adv. Soviet Math., pages 223–233. Amer. Math. Soc., Providence, RI, 1994.
* [Mis75] N. M. Mishachev. Complex orientations of plane M-curves of odd degree. Funktsional Anal. i Prilozhen., 9:77–78, 1975.
* [Ore07] S. Y. Orevkov. Positions of an $M$-quintic with respect to a conic that maximally intersect the odd branch of the quintic. Algebra i Analiz, 19(4):174–242, 2007.
* [Ore18] S. Y. Orevkov. On the hyperbolicity locus of a real curve. Funk. anal. i ego prilozhenia, 52(2):86–89 (Russian), 2018.
* [Ore21] S. Y. Orevkov. Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves. Geom. and Funct. Anal., (31):930–947, 2021.
* [Rok72] V. A. Rokhlin. Congruences modulo 16 in Hilbert’s sixteenth problem. Funktsional. Anal. i Prilozhen. 6, 58–64 (Russian), 1972.
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Matilde Manzaroli, Universität Tübingen, Germany
E-mail address: matilde.manzaroli’at’uni-tuebingen.de
|
# Camelira: An Arabic Multi-Dialect Morphological Disambiguator
Ossama Obeid1, Go Inoue1,2, Nizar Habash1
1Computational Approaches to Modeling Language (CAMeL) Lab
New York University Abu Dhabi
2Mohamed bin Zayed University of Artificial Intelligence
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
## 1 Introduction
The last two decades have witnessed remarkable progress in Natural Language
Processing (NLP) for Arabic and its dialects despite many challenges such as
its diglossic nature, morphological complexity, and orthographic ambiguity
Darwish et al. (2021). These efforts have led to many practical applications
for various NLP tasks including tokenization, part-of-speech (POS) tagging,
morphological disambiguation, named entity recognition, dialect identification
(DID), and sentiment analysis (Pasha et al., 2014; Abdelali et al., 2016;
Obeid et al., 2019; Abdul-Mageed et al., 2020b, inter alia). Tools for core
technologies like POS tagging and morphological disambiguation are primary
examples of such successful applications, e.g., MADAMIRA Pasha et al. (2014),
Farasa Abdelali et al. (2016), UDPipe Straka et al. (2016), and Stanza Qi et
al. (2020). However, there are still gaps to be filled in terms of coverage
and usability. For example, these systems only support Modern Standard Arabic
(MSA) and Egyptian Arabic, but not other widely spoken dialects such as Gulf
and Levantine. In addition, these web interfaces only present the top
prediction, although the alternative readings could provide valuable
information for analyzing the models’ behavior. In contrast, morphological
analyzers such as ElixirFM Smrž (2007), CALIMAStar Taji et al. (2018b), CALIMA
Egyptian Habash et al. (2012) show all the different readings for a given word
out of context but without disambiguated analyses in context. These tools
assume that users already know the input DID; however, this is not necessarily
the case for second language learners. To address these limitations, we
present Camelira,111http://camelira.camel-lab.com,222Camelira is named after
CAMeL Tools Obeid et al. (2020), and in homage to MADAMIRA Pasha et al.
(2014). a web interface for Arabic multi-dialect morphological disambiguation
that covers four major variants of Arabic: MSA, Egyptian, Gulf, and Levantine.
Our system takes an input sentence and provides automatically disambiguated
readings for each word in context, as well as its alternative out-of-context
readings. We also showcase the integration of a state-of-the-art morphological
disambiguator Inoue et al. (2022) with the highest performing fine-grained
Arabic DID system Salameh et al. (2018) on the MADAR DID shared task Bouamor
et al. (2019). Camelira provides an option to automatically choose a dialect-
specific disambiguator based on the prediction of the DID component. To the
best of our knowledge, our work is the first to demonstrate an integrated web
application that leverages both Arabic morphological disambiguation and DID
systems. Our contributions are as follows: (a) We present a user-friendly web
interface that allows researchers and language learners to explore the
detailed linguistic analysis of a given Arabic sentence. (b) We include three
major Arabic dialects (Egyptian, Gulf, and Levantine) in addition to MSA, to
make our tool more accessible to a wider audience. (c) We integrate DID to
automatically select the appropriate disambiguator; a feature that helps users
with limited knowledge of Arabic dialects.
## 2 Arabic Linguistic Facts
The Arabic language poses a number of challenges for NLP Habash (2010). We
highlight three aspects that are most relevant to multi-dialectal
morphological modeling: dialectal variations, morphological richness, and
orthographic ambiguity. First, Arabic is characterized with diglossia and its
large number of dialects Ferguson (1959); Holes (2004). MSA is the shared
standard variant used in official contexts, while the dialects are the
varieties of daily use. MSA and the dialects vary among themselves in
different aspects, such as lexicons, morphology, and syntax. Second, Arabic is
a morphologically rich and complex language. It employs a combination of
templatic, affixational, and cliticization morphological operations to
represent numerous grammatical features such as gender, number, person, case,
state, mood, aspect, and voice, in addition to a number of attachable
pronominal, preposition, and determiner clitics. Third, Arabic is
orthographically highly ambiguous. This is due to its orthographic conventions
where diacritical marks are often omitted, leading to a high degree of
ambiguity. For example, MSA can have 12 different morphological analyses per
word on average Pasha et al. (2014).
## 3 Related Work
#### Morphological Analysis and Disambiguation
Morphological analysis is the task of producing a complete list of readings
(analyses) for a given word out of context. Morphological analysis has a wide
range of applications, including treebank annotation Maamouri et al. (2003,
2011, 2009) and improving morphological modeling Habash et al. (2005); Inoue
et al. (2017); Zalmout and Habash (2017); Khalifa et al. (2020). Over the past
two decades, there have been numerous efforts in building morphological
analyzers for Arabic, e.g. BAMA Buckwalter (2002), MAGEAD Habash and Rambow
(2006); Altantawy et al. (2010), ALMORGEANA Habash (2007), ElixirFM Smrž
(2007), SAMA Graff et al. (2009), CALIMA Egyptian Habash et al. (2012), CALIMA
Gulf Khalifa et al. (2017), AlKhalil Morpho Sys Boudlal et al. (2010);
Boudchiche et al. (2017) and CALIMAStar Taji et al. (2018b). Among these
efforts, ElixirFM333http://quest.ms.mff.cuni.cz/elixir and
CALIMAStar444http://calimastar.camel-lab.com/ provide easy-to-use web
interfaces, allowing the user to explore all the possible morphological
analyses for a given word. In addition to these rule-based approaches,
Eskander et al. (2016) used a corpus-based paradigm completion technique
Eskander et al. (2013) to develop a morphological analyzer for Levantine; and
Khalifa et al. (2020) used the same technique to develop a morphological
analyzer for Gulf. Morphological disambiguation is the subsequent process of
identifying the correct analysis in context from the list of different
analyses produced by a morphological analyzer. Examples of this in Arabic
start with MADA Habash et al. (2005) and many following efforts Pasha et al.
(2014); Khalifa et al. (2016); Zalmout and Habash (2017, 2020); Khalifa et al.
(2020); Inoue et al. (2022), where they rank the analyses based on the
predictions of morphological taggers. While these models have achieved
significant improvement over time, only MADAMIRA Pasha et al. (2014) offers a
web interface555http://madamira.camel-lab.com/ that’s accessible to a general
audience. In this work, we present a user-friendly web interface for state-of-
the-art morphological disambiguation models to make these recent advances more
accessible to a wider audience, such as linguists and language learners. Our
interface also provides all the alternative readings of each input word with
the associated prediction scores, allowing researchers to investigate the
model’s behavior.
Figure 1: The Camelira interface with an MSA example sentence celebrating the
winning of a racehorse named “Dream.” In this example, the automatically
diacritized forms of the words are presented together with their POS. The
first word (on the right), which is highlighted, is selected by the user. The
two lower boxes show all the possible out-of-context analyses (on the right)
and the detailed features and gloss for the top in-context analysis (on the
left).
Figure 2: The Camelira interface presenting the same example in Figure 1
using the Arabic user interface.
#### Dialect Identification
Dialect identification (DID) is the task of automatically identifying the
language variety of a given text. DID for Arabic and its variants has
attracted increasing attention in recent years. A number of shared tasks have
been organized, including VarDial Malmasi et al. (2016); Zampieri et al.
(2017, 2018), MADAR Bouamor et al. (2019), and NADI Abdul-Mageed et al.
(2020a, 2021, 2022), along with continuous efforts in dataset creation (Zaidan
and Callison-Burch, 2011; Mubarak and Darwish, 2014; Zaghouani and Charfi,
2018; Baimukan et al., 2022, inter alia). These evaluation campaigns have led
to the development of practical applications, such as
ADIDA666http://adida.camel-lab.com/ Obeid et al. (2019), a web interface for
fine-grained Arabic DID based on the highest performing system in the MADAR
shared task Salameh et al. (2018). In this work, we employ one of the DID
systems described by Salameh et al. (2018);777We use regional level
classification instead of fine-grained city-level classification because the
morphological analyzers are designed at the regional level. however, we differ
from their work in that we combine DID with multi-dialect morphological
disambiguation to allow users to easily select an appropriate dialect-specific
Arabic disambiguator based on the DID prediction.
## 4 System Design and Implementation
### 4.1 Design Considerations
We want an easy-to-use one-stop online-accessible user interface that supports
the analysis of Arabic sentences from different dialects, and with access to
under-the-hood decisions about disambiguation. To that end, we are inspired by
three web interfaces: MADAMIRA Pasha et al. (2014) for in-context
disambiguation, CALIMAStar Taji et al. (2018a) for out-of-context analysis,
and ADIDA Obeid et al. (2019) for dialect identification. Furthermore, we
would like the web interface to have a responsive design with streamlined user
experiences across a range of devices from mobile to desktops.
### 4.2 Implementation
#### Back-end
The back-end is implemented in Python using
Flask888https://flask.palletsprojects.com/ to serve a REST API. We implemented
the MODEL-6 DID system described by Salameh et al. (2018) for automatic
dialect identification and the morphological disambiguation system described
by Inoue et al. (2022). The implementation of the morphological disambiguator
was provided by the CAMeL Tools999https://github.com/CAMeL-Lab/camel_tools
Python API Obeid et al. (2020). We plan to add our MODEL-6 implementation to
CAMeL Tools. For morphological disambiguation, we use the unfactored model
with a morphological analyzer for all variants. We chose the unfactored models
because they are faster than the factored models and only slightly lower in
performance. Table 1 shows the performance accuracy of Camelira’s
morphological disambiguation models. We report numbers on DEV as presented in
Inoue et al. (2022). For DID, we train our MODEL-6 using the TRAIN split and
evaluate using the DEV and TEST splits following Salameh et al. (2018). Table
2 compares the performance of our implementation with that of Salameh et al.
(2018). Our results are slightly lower due to implementation differences.
| ALL TAGS | POS
---|---|---
MSA | 95.9 | 98.7
EGY | 90.5 | 94.0
GLF | 93.8 | 96.6
LEV | 85.5 | 92.7
Table 1: Accuracy of Camelira’s morphological disambiguation models based on Inoue et al. (2022)’s unfactored+Morph models. ALL TAGS is the accuracy of the combined morphosyntactic features. | DEV | TEST
---|---|---
Camelira | 92.8 | 93.5
Salameh et al. | 93.1 | 93.6
Table 2: Accuracy of Camelira’s implementation of the MODEL-6 DID model
compared with Salameh et al. (2018)’s implementation of the same model.
#### Front-end
The front-end was implemented using Vue.js101010https://vuejs.org/ for model
view control and Bulma111111https://bulma.io/ for styling and creating a
responsive design that works well across devices.
Figure 3: The Camelira interface with an MSA example sentence and “Tokenized”
display tab. This is an exact replica of the input and output choices as in
Figure 1 except that the word forms are presented in full tokenization.
### 4.3 The Camelira Interface
The Camelira interface is divided into three main areas, the Input Area, Text
Output Area, and Morphological Analysis Area. Figure 1 shows an example of a
disambiguated MSA sentence in the Camelira web interface. We also provide the
option of viewing the interface in Arabic as seen in Figure 2.
#### Input Area
At first, only the Input Area is displayed which provides users with an input
box where they can enter the sentence they wish to disambiguate. Users are
also presented with a drop-down menu where they can select whether to
disambiguate the input sentence as a particular dialect (MSA, Egyptian, Gulf,
or Levantine) or to have the dialect be automatically selected.
#### Text Output Area
Once the submit button is clicked and the sentence has been disambiguated, the
Text Output Area is displayed. First, the dialect indicator displays which
dialect was used to analyze the provided input. Then, an output box displays
the disambiguated sentence in three different views: (a) the Diacritized/POS
view which displays the diacritized text (if supported by the selected
dialect’s resources) along with the POS tag of each word, (b) the Tokenized
view which displays each disambiguated word in its tokenized form where tokens
are delimited by a ‘+’ character, and (c) the Lemmatized view where each word
is displayed in its lemmatized form. Figure 3 is the same as Figure 1 except
that the text output is in Tokenized mode.
#### Morphological Analysis Area
Below the Text Output Area, the Morphological Analysis Area consists of the
Analysis List box (on the right), which displays all analyses of a given word
sorted by their disambiguation ranking order, and the Analysis Viewer box (on
the left), which displays a selected analysis in an easy-to-read form with
more morphological feature details. The analysis list displays the
disambiguation score of each analysis as well as the values for a reduced set
of features. Clicking on a word in the Text Output Area selects that word,
displaying its analyses in the analysis list and analysis viewer boxes.
Clicking on an analysis in the Analysis List will display its user-friendly
form in the Analysis Viewer. By default, the top analysis is selected.
#### Dialect Identification and Morphological Disambiguation
Figures 4 and 5 present Egyptian and Gulf Arabic examples, respectively. Both
are presented in a mobile setting to demonstrate our responsive design. In the
case of Figure 4, the user selected Auto-Detect for dialect identification. In
the Gulf example, the user selected Gulf Arabic directly. Note that the Gulf
Arabic does not show diacritizations since its training data did not include
diacritized forms Khalifa et al. (2020).
Figure 4: The Camelira interface with an Egyptian example sentence: _"A very
cool song [video clip], you’ll regret it if you don’t watch it."_ In this
example, the input text is automatically correctly detected as Egyptian.
Figure 5: The Camelira interface with a Gulf example sentence: _"Go up to
your room, I don’t want to hear you talking about this subject again."_ In
this example, the user specified the input dialect as Gulf.
## 5 Conclusion and Future Work
We presented Camelira, a user-friendly web interface for Arabic multi-dialect
morphological disambiguation that covers four major variants of Arabic. The
system takes a sentence as input and provides an automatically disambiguated
reading for each word, as well as its alternative readings, allowing users to
explore various linguistic information, such as part-of-speech, morphological
features, and lemmas. Camelira also provides an option to automatically choose
an appropriate dialect-specific disambiguator based on the prediction of its
dialect identification component. In the future, we plan to extend our
disambiguation system to cover other Arabic dialects such as Maghrebi and
Yemeni Arabic. We also plan to continue to update the system using future
improvements in terms of efficiency and accuracy in CAMeL Tools Obeid et al.
(2020).
## Limitations and Ethical Considerations
We acknowledge that our system is currently limited to specific variants of
Arabic and it can produce erroneous predictions especially on different
dialects, genres, and styles that are not covered in the current system’s
training data. We also acknowledge that our work on core and generic NLP
technologies can be used as part of the pipeline of other systems with
malicious intents.
## Acknowledgements
Some of this work was carried out on the High Performance Computing resources
at New York University Abu Dhabi. We thank Salam Khalifa and Bashar Alhafni
for their insightful comments and helpful discussions. We also thank anonymous
reviewers for their helpful comments.
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|
# Efficient Adversarial Input Generation via Neural Net Patching
Tooba Khan, Kumar Madhukar, Subodh Vishnu Sharma
###### Abstract
The adversarial input generation problem has become central in establishing
the robustness and trustworthiness of deep neural nets, especially when they
are used in safety-critical application domains such as autonomous vehicles
and precision medicine. This is also practically challenging for multiple
reasons – scalability is a common issue owing to large-sized networks, and the
generated adversarial inputs often lack important qualities such as
naturalness and output-impartiality. We relate this problem to the task of
patching neural nets, i.e. applying small changes in some of the network’s
weights so that the modified net satisfies a given property. Intuitively, a
patch can be used to produce an adversarial input because the effect of
changing the weights can also be brought about by changing the inputs instead.
This work presents a novel technique to patch neural networks, and an
innovative approach of using it to produce perturbations of inputs which are
adversarial for the original net. We note that the proposed solution is
significantly more effective than the prior state-of-the-art techniques.
## Introduction
Deep Neural Networks (DNNs) today are omnipresent. An important reason behind
their widespread use is their ability to generalize and thereby perform well
even on previously unseen inputs. While this is a great practical advantage,
it may sometimes make DNNs unreliable. In safety- or business-critical
applications this lack of reliability can indeed have dreadful costs.
Evidently, central to a trained network’s unreliability lies the lack of
robustness against input perturbations, i.e., small changes to some inputs
cause a substantial change in the network’s output. This is undesirable in
many application domains. For example, consider a network that has been
trained to issue advisories to aircrafts to alter their paths based on
approaching intruder aircrafts. It is natural to expect such a network to be
robust in its decision-making, i.e. the advisory issued for two very similar
situations should not be vastly different. At the same time, if that is not
the case, then demonstrating the lack of robustness through _adversarial
inputs_ can help not only in improving the network but also in deciding when
the network should relinquish control to a more dependable entity.
Given a network and an input, an adversarial input is one which is _very
close_ to the given input and yet the network’s outputs for the two inputs are
quite different. In the last several years, there has been much work on
finding adversarial inputs (Pei et al. 2017; Guo et al. 2018; Goodfellow,
Shlens, and Szegedy 2015; Kurakin, Goodfellow, and Bengio 2017; Alparslan et
al. 2021). These approaches can be divided into black-box and white-box
methods based on whether they consider the network’s architecture during the
analysis or not. A variety of techniques have been developed in both these
classes, ranging from generation of random attacks (Goodfellow, Shlens, and
Szegedy 2015; Kurakin, Goodfellow, and Bengio 2017) and gradient-based methods
(Alparslan et al. 2021) to symbolic execution (Wang et al. 2018; Yang et al.
2021; Li et al. 2019), fault localization (Eniser, Gerasimou, and Sen 2019),
coverage-guided testing (Pei et al. 2017; Guo et al. 2018), and SMT and ILP
solving (Gopinath et al. 2018). However, there are several issues that limit
the practicability of these techniques – poor success rate, large distance
between the adversarial and the original input (both in terms of the number of
input values changed, and the degree of the change), unnatural or perceivably
different inputs, and output partiality (techniques’ bias to produce
adversarial examples for just one of the network’s outputs). This paper
presents a useful approach to generate adversarial inputs in a way that
addresses these issues.
We relate the problem of finding adversarial inputs to the task of patching
neural nets, i.e. applying small changes in some of the network’s weights so
that the modified net satisfies a given property. Patching DNNs is a topic of
general interest to the Machine Learning community because of its many
applications, which include bug-fixing, watermark resilience, and fine-tuning
of DNNs, among others (Goldberger et al. 2020). Intuitively, the relation
between these two problems is based on the observation that a patch can be
translated into an adversarial input because the effect of changing the
weights may be brought about by changing the inputs instead. In fact, a patch
in the very first _edge-layer_ of a network can very easily be transformed
into a corresponding change in the input by just solving linear equations.
While there are techniques to solve the patching problem (Goldberger et al.
2020; Refaeli and Katz 2021), it is in general a difficult task, particularly
for layers close to the input layer. Informally, this is because the
computation of the entire network needs to be encoded and passed to a
constraint solver in order to obtain a patch. For large-sized networks, this
gives rise to a big monolithic constraint leading to scalability issues for
the solver. We address this by proposing an improvement in the technique of
(Goldberger et al. 2020), and then using it repeatedly to find a middle-layer
patch and chop off the latter half of the network, till a first-layer patch
has been obtained. Our experiments on three popular image-dataset benchmarks
show that our approach does significantly better than other state-of-the-art
techniques, in terms of the number of pixels modified as well as the magnitude
of the change. This reflects in the quality of the adversarial images, both
visually and in several qualitative metrics that we present later.
The core contributions of this paper are:
* •
A novel technique to patch neural networks, and an innovative approach of
using it to produce small perturbations of inputs which are adversarial for
the original net.
* •
An extensive experimental evaluation using CIFAR-10 (Krizhevsky and Hinton
2009), MNIST (Deng et al. 2009a), and ImageNet (Deng et al. 2009b) datasets,
and a number of qualitative parameters, to show the efficacy of our approach
over the state-of-the-art.
## Illustrative Example
Let us consider a toy DNN, $\mathcal{N}$, shown in Fig. 1. It has two neurons
in each of its four layers – the input layer first, followed by two hidden
layers, and then the output layer. We assume that the hidden layers have
ReLU111ReLU($x$) = $\mathit{max}(0,x)$ as the activation function. For neurons
without an activation function, the value of a neuron is computed by summing
up, for each incoming edge to the neuron, the product of the edge weight and
the value of the neuron at the edge’s source. In presence of an activation
function, the value is computed by applying the function on this sum. For
example, for the input $\langle 0.5,0.5\rangle$, the values at the next three
layers are $\langle 1,0.5\rangle$, $\langle 3.5,1.5\rangle$, and
$\langle-5,-6.5\rangle$ respectively.
1$\rightarrow$-3/42$\rightarrow$7/8 1/12$\leftarrow$1/2 1/24$\leftarrow$1/2
1/2$\rightarrow$0 3/2$\rightarrow$1/8 -6.5$\rightarrow$-0.375
1$\rightarrow$1/8 7/2$\rightarrow$1/4 -5$\rightarrow$-0.5
$i_{1}$$i_{2}$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$o_{1}$$o_{2}$$1$$-1$$2$$1$$3$$1$$2$$5$$-4$$-8$
Figure 1: Adversarial inputs from first-layer modification. The adversarial
input and the corresponding values of each neuron are written in red. The
modification required in first layer weights are shown in black boxes.
Finding an adversarial input $\langle i_{1},i_{2}\rangle$ for the input
$\langle 0.5,0.5\rangle$ amounts to finding a value for each $i_{k}$
($k\in\\{1,2\\}$) such that $|i_{k}-0.5|\leq\delta$ and the corresponding
output $o_{2}>o_{1}$, for a given small $\delta$. It is noteworthy that if we
want the second output to become bigger than the first one, this can be
achieved by modifying the weights instead of the inputs. For example, if the
edges connecting the second input neuron to the first hidden layer had the
weights $\langle-0.75,0.875\rangle$ instead of $\langle 1,2\rangle$, then the
next three layers would have the values $\langle 0.125,0\rangle$, $\langle
0.25,0.125\rangle$, and $\langle-0.5,-0.375\rangle$, and our goal would be met
by modifying the weights while keeping the inputs unchanged. We will come to
the question of how to find the changed first-layer weights in a bit, but let
us first see how the changed weights can help us obtain an adversarial input.
This is a rather simple exercise. Note that with the changed weights, the
values of the neurons in the first hidden layer were $\langle 0.125,0\rangle$.
So, our task is simply to find inputs for which the first hidden layer values
stay as $\langle 0.125,0\rangle$, but with the original weights $\langle
1,2\rangle$. This can be done by solving the following equations, where
$\delta_{1}$ and $\delta_{2}$ ($\delta_{1},\delta_{2}$ $\leq\delta\leq 0.5$,
say) are the changes in the two inputs respectively.
$(1/2+\delta_{1})+(1/2+\delta_{2})=1/8$ (1)
$-(1/2+\delta_{1})+2*(1/2+\delta_{2})=0$ (2)
We get $\delta_{1}=-5/12,\delta_{2}=-11/24$ and, thus, the adversarial input
as $\langle 1/12,1/24\rangle$. The dotted rectangles in Fig. 1 contain the
adversarial inputs and the corresponding values at each layer.
The first thing to notice here is that Eqn. 2 could have been relaxed as
$-(1/2+\delta_{1})+2*(1/2+\delta_{2})\leq 0$ because of the ReLU activation
function. This may give us smaller values of $\delta_{i}$’s. Moreover, along
with minimizing the change in each input pixel, we can also minimize the
number of pixels that are modified, as shown here.
$(1/2+\delta_{1}*M_{1})+(1/2+\delta_{2}*M_{2})=1/8$ (3)
$-(1/2+\delta_{1}*M_{1})+2*(1/2+\delta_{2}*M_{2})<=0$ (4)
$M_{1},M_{2}\in\\{0,1\\};~{}~{}~{}\mathit{minimize}~{}~{}\sum M_{i}$ (5)
In pratice, we solve these constraints in place of Eqns. 1-2; this gives us
better adversarial inputs.
There are a few more points to note before we proceed. First, it is only the
first-layer modification that may be easily translated into an adversarial
input as illustrated. Modification in deeper layers are not immediately
helpful; they cannot be translated easily into an adversarial input because of
the non-linear activation functions. Second, we need to find _small_
modifications in the weights, so that the corresponding $\delta_{i}$’s in the
inputs fall within the allowed $\delta$. And, lastly, while there are ways to
compute a first-layer change directly using an SMT or an ILP solver
(Goldberger et al. 2020), this approach is not very scalable in practice as
large-sized networks give rise to big monolithic formulas that may be
difficult for the solver. Instead, we propose an iterative approach that finds
a middle-layer modification using (Goldberger et al. 2020)222In fact, we use
an improved version of this which is described in the next section. and chops
off the latter half of the network, repeatedly till a first-layer patch is
found.
Since our network has three edge-layers, we start by finding a small
modification of the weights in the second edge-layer with which we can achieve
our target of making $o_{2}$ bigger than $o_{1}$ for the input $\langle
0.5,0.5\rangle$. We propose an $\epsilon_{i,j}$ change in the weight of the
$j^{th}$ edge in $i^{th}$ edge-layer, and encode the constraints for
$o_{2}>o_{1}$ as follows:
$x_{3}=\mathit{max}(0,1*(2+\epsilon_{2,1})+1/2*(3+\epsilon_{2,2}))$ (6)
$x_{4}=\mathit{max}(0,1*(1+\epsilon_{2,3})+1/2*(1+\epsilon_{2,4}))$ (7)
$-4*x_{3}+5*x_{4}>2*x_{3}-8*x_{4}$ (8)
The range of each $\epsilon_{i,j}$ is [$-\alpha,\alpha$] if $\alpha$ is the
biggest permissible modification for an edge-weight. We minimize the magnitude
of the total change using Gurobi (Gurobi Optimization, LLC 2022). For the
equations above, we get
$\langle\epsilon_{2,1},\epsilon_{2,2},\epsilon_{2,3},\epsilon_{2,4}\rangle=\langle-9/8,-17/4,-5/4,-1/4\rangle$.
These changes indeed make the second output bigger (see Fig. 2).
Extracted Network 1/2 3/2$\rightarrow$1/8 -6.5$\rightarrow$-0.375 1
7/2$\rightarrow$1/4 -5$\rightarrow$-0.5
$i_{1}$$1/2$$i_{2}$$1/2$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$o_{1}$$o_{2}$$1$$2$$-1$$1$2$\rightarrow$7/81$\rightarrow$3/43$\rightarrow$-5/41$\rightarrow$-1/4$2$$5$$-4$$-8$
Figure 2: Middle-layer modification and sub-net extraction
Our next step is to extract a sub-network (see Fig. 2) and look for a
modification in its middle layer. Since the extracted network has only two
edge-layers, this step will give us a first-layer modification. The equations,
subject to the constraint that $x_{3}$ and $x_{4}$ get the values 1/4 and 1/8
respectively, are shown below.
$x_{1}=\mathit{max}(0,1/2*(1+\epsilon_{1,1})+1/2*(1+\epsilon_{1,2}))$ (9)
$x_{2}=\mathit{max}(0,1/2*(-1+\epsilon_{1,3})+1/2*(2+\epsilon_{1,4}))$ (10)
$2*x_{1}+3*x_{2}=1/4;~{}~{}~{}1*x_{1}+1*x_{2}=1/8$ (11)
Gurobi gives us the solution
$\langle\epsilon_{1,1},\epsilon_{1,2},\epsilon_{1,3},\epsilon_{1,4}\rangle=\langle
0,-7/4,0,-9/8\rangle$, from which we can obtain the adversarial input $\langle
1/12,1/24\rangle$ as already shown above.
## Methodology
We now describe the technical details of our approach and present our
algorithm. We begin with the notation that we use. Let $\mathcal{N}$ denote
the DNN that we have with $n$ inputs ($i_{1},i_{2},\ldots,i_{n}$), $m$ outputs
($o_{1},o_{2},\ldots,o_{m}$), and $k$ layers ($l_{1},l_{2},\ldots,l_{k}$). We
use $v_{p,q}$ to denote the value of the $q^{th}$ neuron in $l_{p}$. We assume
that $\mathcal{N}$ is feed-forward, i.e., the (weighted) edges connect neurons
in adjacent layers only, and point in the direction of the output layer. We
use the term _edge-layer_ to refer to all the edges between any two adjacent
layers of $\mathcal{N}$, and denote the edge layers as
$el_{1},el_{2},\ldots,el_{k-1}$. We also assume the hidden layers
($l_{2},l_{3},\ldots,l_{k-1}$) in $\mathcal{N}$ have ReLU activation function,
and that there is no activation function on any output neuron. For simplicity,
we assume that the neurons do not have any biases. This is not a limitation in
any sense; our implementation handles them directly. Moreover, a DNN with
biases can be converted into an equivalent one without any biases. Like in the
previous section, we use $\delta_{p}$ to denote the change in the $p^{th}$
input, and $\epsilon_{q,s}$ to denote the change in the weight of the $q^{th}$
edge in $el_{s}$. The $\delta_{p}$’s are constrained to be $\leq\delta$, which
is the biggest perturbation allowed in any pixel to find an adversarial input.
Algorithm 1 presents a pseudocode of our algorithm. The inputs to the
algorithm are: $\mathcal{N},\delta$, and the values for the input neurons
$v_{1,1},v_{1,2},\ldots,v_{1,n}$, for which the corresponding output does not
satisfy a given adversarial property $\phi(o_{1},o_{2},\ldots,o_{m})$ (denoted
simply as $\phi$, henceforth). The aim of our algorithm is to find a new set
of input values $v^{\prime}_{1,1},v^{\prime}_{1,2},\ldots,v^{\prime}_{1,n}$
such that $\mathcal{N}$’s output corresponding to these new inputs satisfies
$\phi$. The algorithm works in the following two steps. First, it finds a
_small_ modification in the weights of $el_{1}$ to derive $\mathcal{N}_{mod}$
(which is essentially $\mathcal{N}$ with the modified weights in $el_{1}$),
such that the output of $\mathcal{N}_{mod}$ for the input
$v_{1,1},v_{1,2},\ldots,v_{1,n}$ satisfies $\phi$. Then, the algorithm
translates this first-layer modification into adversarial inputs
$v^{\prime}_{1,1},v^{\prime}_{1,2},\ldots,v^{\prime}_{1,n}$, subject to the
constraint that $|v^{\prime}_{1,p}-v_{1,p}|\leq\delta$, for every $p\in[1,n]$.
This second step is shown in the algorithm as the function $\mathit{mod2adv}$,
the pseudocode of which has been omitted as this is a simple call to Gurobi as
illustrated in the previous section.
Let us assume for the time being that we have a sub-routine
$\mathit{modifyEdgeLayer}$ that takes as input a DNN $\mathcal{N}$, one of its
edge-layers $el_{k}$, values of the input neurons
$v_{1,1},v_{1,2},\ldots,v_{1,n}$, and a property $\phi$ on the output layer
neurons, and returns a new network $\mathcal{N}_{mod}$ with the constraints
that:
* •
$\mathcal{N}_{mod}$ is same as $\mathcal{N}$ except for the weights in
$el_{k}$, and
* •
the output of $\mathcal{N}_{mod}$ on $v_{1,1},v_{1,2},\ldots,v_{1,n}$
satisfies $\phi$.
Clearly, with such a sub-routine, the first step of our algorithm becomes
trivial. We would simply call $\mathit{modifyEdgeLayer}$ with the given input,
$\mathcal{N}$, $\phi$, and $el_{1}$. We refer to the work of Goldberger et al.
(Goldberger et al. 2020) which gives us exactly this. However, we do not use
it directly to find our first-layer modification. Informally, the technique of
(Goldberger et al. 2020) uses variables $\epsilon_{q,s}$ to denote the changes
in the weights (in a given edge-layer $s$) and encodes the computation of the
entire network, and then adds the constraint that the output must satisfy
$\phi$. It then uses Gurobi on these constraints, to solve for (and optimize)
the values of $\epsilon_{q,s}$. Consider an example (reproduced from
(Goldberger et al. 2020)) shown in Fig. 3 with the output property
$\phi:=(v_{3,1}\geq v_{3,2})$. Let us ignore the color of the output neurons
for now. If the input neurons are given values $v_{1,1}=3$ and $v_{1,2}=4$,
the output neurons get the value $v_{3,1}=-2$ and $v_{3,2}=2$, which does not
satisfy $\phi$. Note that the hidden layer neurons have ReLU activation
function. In order to obtain a second edge-layer modification such that $\phi$
holds for the input $\langle 3,4\rangle$, the technique of (Goldberger et al.
2020) generates the following constraints.
$v_{1,1}$$v_{1,2}$$v_{2,1}$$v_{2,2}$$v_{3,1}$$v_{3,2}$1-12-2 1 1 -1 -1 Figure
3: Example illustrating DNN modification, from (Goldberger et al. 2020)
$\mathit{minimize}~{}~{}M$ (12)
$M\geq 0$ (13)
$-M\leq\epsilon_{1,2}\leq M$ (14)
$-M\leq\epsilon_{2,2}\leq M$ (15)
$-M\leq\epsilon_{3,2}\leq M$ (16)
$-M\leq\epsilon_{4,2}\leq M$ (17)
$v_{3,1}=0.(1+\epsilon_{1,2})+2.(-1+\epsilon_{2,2})$ (18)
$v_{3,2}=0.(-1+\epsilon_{3,2})+2.(1+\epsilon_{4,2})$ (19)
$v_{3,1}\geq v_{3,2}$ (20)
Figure 4: DNN modification constraints for Fig.3
In a similar way, the constraints can be generated for modification in any
layer, by propagating the input values up to that layer, encoding the
computation from there onward, and adding the output property $\phi$. If a
first-layer modification has to be found, this gives rise to a big monolithic
constraint, particularly for large-sized networks. This does not scale in
practice, and therefore we propose an iterative approach of doing this in our
algorithm. The iterative approach uses the above technique to find a middle-
layer modification, derive a new output property $\phi^{\prime}$ for the first
half of the network, and does this repeatedly till a first-layer modification
is found. This has been illustrated in the function
$\mathit{findFirstLayerMod}$ in Alg. 1, which calls $\mathit{modifyEdgeLayer}$
in a loop. The last bit here is to understand how the modified output property
$\phi^{\prime}$ may be derived in each iteration. The way this works is as
follows. Let us say that the last call to $\mathit{modifyEdgeLayer}$ was made
on edge-layer $el_{(j-1)}$, which connects the layers $l_{(j-1)}$ and $l_{j}$.
We can use the modified weights to propagate the inputs all the way to layer
$l_{j}$, by simply simulating the network on the inputs. This gives us values
for all the neurons in layer $l_{j}$, say $c_{1},c_{2},\ldots$ and so on. Now,
the modified weights are replaced by the original weights, and all the layers
after $l_{j}$ are dropped off from the network. This reduced network
$\mathcal{N}^{\prime}$ has exactly the layers $l_{1}$ to $l_{j}$ of
$\mathcal{N}$. We denote this reduction as
$\mathcal{N}\downarrow_{(l_{1}\ldots l_{j})}$. Ideally, we would want to find
a middle-layer modification in $\mathcal{N}^{\prime}$ under the new output
constraint as
$\phi^{\prime}:=(v_{j,1}=c_{1})\wedge(v_{j,2}=c_{2})\wedge\ldots$ and so on.
The updated $\phi^{\prime}$ is correct because we know that $\phi$ gets
satisfied when these values are propagated further to the output layer. But,
we can relax the strict equalities of $\phi^{\prime}$ into inequalities as
discussed in the next subsection.
Algorithm 1 Adversarial Inputs via Network Patching
Input: $\mathcal{N},l,\delta,\phi,$ and input $\langle
v_{1,1},\ldots,v_{1,n}\rangle$
Output: Adversarial input $\langle
v^{\prime}_{1,1},\ldots,v^{\prime}_{1,n}\rangle$
findFirstLayerMod($\mathcal{N},l,\mathit{in},\phi$):
1: while true do
2: $p\leftarrow\lceil(l/2)\rceil$
3:
$\mathcal{N}_{mod}\leftarrow\mathit{modifyEdgeLayer}(\mathcal{N},\mathit{in},\phi,el_{p})$
4: return $\mathcal{N}_{mod}$ if$(p=1)$
5: $\langle
c_{1},c_{2},\ldots\rangle\leftarrow\mathit{simulate}(\mathcal{N}_{mod},\mathit{in})$
6: $\phi^{\prime}=(v_{(p+1),1}=c_{1})\wedge(v_{(p+1),2}=c_{2})\wedge\ldots$
7: $\mathcal{N}^{\prime}=\mathcal{N}\downarrow_{(l_{1}\ldots l_{(p+1)})}$
8: $\mathcal{N}\leftarrow\mathcal{N}^{\prime}$; $l\leftarrow(p+1)$;
$\phi\leftarrow\phi^{\prime}$
9: end while
main():
1: $\mathit{in}\leftarrow\langle v_{1,1},\ldots,v_{1,n}\rangle$
2:
$\mathcal{N}_{mod}\leftarrow\mathit{findFirstLayerMod}$($\mathcal{N},l,\mathit{in},\phi$)
3: $\langle\delta_{1},\delta_{2},\ldots,\delta_{n}\rangle$ =
$\mathit{mod2adv}$($\mathcal{N}_{mod},\mathit{in},\delta$)
4: $\langle v^{\prime}_{1,1},\ldots,v^{\prime}_{1,n}\rangle$ = $\langle
v_{1,1},\ldots,v_{1,n}\rangle$ \+ $\langle\delta_{1},\ldots,\delta_{n}\rangle$
5: return $\langle v^{\prime}_{1,1},\ldots,v^{\prime}_{1,n}\rangle$
### Simplifying DNN Modification Constraints
Let us revisit the example of Fig. 3, and the constraints corresponding to the
modification problem shown in Fig. 4. Recollect that the modification problem
in this example was to find small changes in the weights of the second edge-
layer, such that $v_{3,1}\geq v_{3,2}$ for the input $\langle 3,4\rangle$.
This is not true for the DNN in this example as $v_{3,1}$ gets the value -2,
whereas $v_{3,2}$ gets the value 2\. A possible way of satisfying $v_{3,1}\geq
v_{3,2}$ is to change weights in such a way that $v_{3,2}$ _decreases_ and
$v_{3,2}$ _increases_. This can give us a marking of the final layer neurons
as decrement and increment, which has been indicated by the neuron colors red
and green in Fig. 3. The useful thing about such a marking is that it can be
propagated backward to other layers. For instance, in the same example,
$v_{2,1}$ and $v_{2,2}$ can also be colored green and red, resp. This works by
looking at the edge weights. Since $v_{2,1}$ is connected to $v_{3,1}$ with a
positive-weight edge, an increase in $v_{3,1}$ can be brought about by
_increasing_ $v_{2,1}$. If we look at $v_{2,2}$, since it connected to
$v_{3,1}$ with a negative-weight edge, a _decrease_ in $v_{2,2}$ would result
in an increase in $v_{3,1}$. This marking was proposed by Elboher et al.
(Elboher, Gottschlich, and Katz 2020), although it was in the context of
abstraction-refinement of neural networks. We refer the interested readers to
(Elboher, Gottschlich, and Katz 2020) for more details about this marking
scheme. In what follows, we explain how this marking can be useful in
simplifying the modification constraints.
Since we are interested in modifying weights in the second edge-layer
($el_{2}$), we propagate the inputs to the second layer of neurons. This gives
us the values $\langle 0,2\rangle$ for $\langle v_{2,1},v_{2,2}\rangle$.
Having propagated the input to the source neurons of $el_{2}$, and the
increment-decrement marking at the target neurons of $el_{2}$, we claim that
we can identify whether a given edge-weight should be increased, or decreased.
Let us consider the edge between $v_{2,2}$, which has a value of 2, and
$v_{3,1}$, which has an increment marking. We claim that the change in this
edge, $\epsilon_{2,2}$ should be positive. Naturally, since the value is
positive, we should multiply with a _bigger_ weight to get an increased
output. Instead, if a positive value was connected to a decrement neuron, we
should decrease the weight (for example, for the edge between $v_{2,2}$ and
$v_{3,2}$). In case of negative values, just the opposite needs to be done.
And if the value is zero, no change needs to be made at all. With this, the
constraints in Fig. 4 get simplified as
$\epsilon_{1,2}=\epsilon_{2,2}=0,0\leq\epsilon_{3,2}\leq M$ and
$-M\leq\epsilon_{3,2}\leq 0$. We have implemented this on top of the tool
corresponding to (Goldberger et al. 2020) and used it in our call to
$\mathit{modifyEdgeLayer}$.
We end this section with a brief note on how this increment decrement marking
may help us relax the modified output property $\phi^{\prime}$. Recollect that
$\phi^{\prime}$ was derived as a conjunction of equality constraints, where
each conjunct was equating a last-layer neuron of the reduced network with the
values obtained by simulating the input on $\mathcal{N}_{mod}$. Since we know
the increment-decrement marking of last layer neurons, we can relax each
conjunct into an inequality by replacing the equality sign with $\leq$
($\geq$) for decrement (increment) neurons. This allows us to obtain, possibly
better, solutions more often.
## Related Work
Robustness of DNNs has gained a lot of attention in the last several years as
DNNs are permeating our lives, even safety- and business-critical aspects of
it. A number of techniques have been developed to establish robustness or
demonstrate the lack of it through adversarial examples. These techniques may
be broadly classified as black box and white box approaches.
Black box methods do not consider the architecture of DNNs in trying to argue
about its robustness. Attacks based on the L2 and L0 norms were introduced in
(Alparslan et al. 2021). These attacks change the pixel values by some fixed
amount, and measure their effectiveness by the decrease in confidence of the
original label. Fast Gradient sign method (Goodfellow, Shlens, and Szegedy
2015) is a gradient based method, which uses gradient of the loss of neural
network when the original and modified images are fed to it. These methods
modify a large number of pixels in the original image, to induce a
misclassification. Changing more pixels makes the adversarial image visibly
different from the original image.
White box methods, on the other hand, involve looking at the complete
architecture of the DNNs to obtain adversarial inputs or argue that none
exists. Verification of neural networks using Symbolic execution is one such
white box approach. As discussed in (Gopinath et al. 2018), it translates the
neural network into an imperative program, and uses SMT (Symbolic Modulo
Theory) based solver to prove given properties. However, such techniques are
not scalable due to exponential time complexity. Symbolic propagation is
another white box method which converts inputs to symbolic representations and
propagates them through the hidden layers and the output layer. (Li et al.
2019)] uses symbolic propagation along with abstract interpretation to compute
bounds on the neurons. (Wang et al. 2018) has used integer arithmetic to find
bounds on the outputs. (Yang et al. 2021) verifies local robustness properties
using symbolic propagation and global robustness properties using Lipschitz
constant based framework. However, these techniques often given loose bounds
and lack precision.
Another white box technique is to find flaws in the training phase of the
neural network such as the use of a inappropriate loss function. The most
popularly used loss function while training classification networks is softmax
cross entropy loss. The authors of (Pang et al. 2020) claim that the presence
of adversarial behavior in a deep neural network can be attributed to its loss
function. They propose Max-Mahalanobis center (MMC) loss which learns from
dense feature regions in the input images. (Raj Dhamija, Gunther, and Boult
2019) has proposed two loss functions: Entropic Open-Set Loss and
Objectoshpere Loss. Entropic Open-Set Loss makes the softmax values of all
classes uniform when the input is unknown. Objectoshpere Loss increases the
boundary of feature magnitudes of known and unknown samples. Loss function
replacement techniques are still vulnerable to adversarial attacks. They can
not be transferred either. These methods cannot be used to test the robustness
of DNNs that are already in use.
DeepFault (Eniser, Gerasimou, and Sen 2019) is another white-box approach for
testing DNNs. It uses fault localization, i.e, finding the areas of the
network that are mainly responsible for incorrect behaviors. It analyses a
trained DNN to find faulty neurons from its hit spectrum analysis using
suspiciousness measures. Adversarial inputs are those inputs that achieve high
activation values for suspicious neurons. Another popular category of white
box methods depends on coverage based techniques. These methods use structural
coverage metrics such as neuron coverage and modified condition/decision
coverage (MC/DC). These metrics are claimed to be related to faults in neural
networks. (Sun et al. 2019) developed a tool which uses MC/DC variants for
neural networks and neuron coverage. This tool is used to verify neural
networks based on the results of the specified coverage. (Xie et al. 2019)
uses mutation based strategies to generate test cases which can maximize
neuron coverage. Similarly, (Gao et al. 2020) uses a genetic algorithm based
approach which can generate adversarial inputs to maximize neuron coverage and
they retrain the DNN simultaneously while generating inputs. (Pei et al. 2017)
implements a white-box approach that maximises neuron coverage and
differential behaviour of a group of DNNs, implying that different DNNs will
produce different output classes for the adversarial input while maximising
neuron coverage. (Guo et al. 2018) also focuses on maximising neuron coverage
in order to generate adversarial inputs.
Even though the code coverage criteria of software engineering test
methodologies corresponds to neuron coverage, it is not a useful indicator of
the production of adversarial inputs. Neuron coverage statistics, according to
authors in (Harel-Canada et al. 2020), lead to the detection of fewer flaws,
making them inappropriate for proving the robustness of DNNs. They also
present three new standards – defect detection, naturalness, and output
impartiality – that can be used to gauge the quality of adversarial inputs
produced, as alternatives to the L2 and L-inf norms. Their findings establish
that the adversarial image set generated by using neuron coverage measures did
not perform well on these three standards. Thus, neuron coverage is not a good
measure for testing neural networks and generating adversarial images. We have
explained these metrics in the following section.
Since our technique relies on finding modifications in DNNs, we also discuss a
couple of recent work that is related in this respect. (Goldberger et al.
2020) proposes a technique to find minimal modifications in a single-layer in
a DNN, in order to meet a given outcome. This work has been extended further
in (Refaeli and Katz 2021) to come up with multi-layer modifications by
dividing the problem into multiple sub problems and applying the idea of
(Goldberger et al. 2020) on each one of them. Our work proposes an improvement
over their idea and uses the improved technique to find small modifications
which are then translated into adversarial inputs.
## Experimental Setup
We implemented our approach in a tool called Aigent (Adversarial Input
Generator), using the Tensorflow and Keras libraries for working with the
DNNs. Aigent uses Gurobi for constrained optimization.
### Benchmark datasets
We conducted our experiments on three popular datasets: MNIST, CIFAR-10, and
ImageNet. We chose these benchmarks because they are readily available and are
acceptable as inputs by a number of tools, making it easier to compare
different techniques in a fair way. MNIST consists of 60,000 black and white
images of handwritten digits for the purpose of training and 10,000 for
testing. Each image is of 28x28 size. It has 10 classes with labels
corresponding to each digit. CIFAR-10 consists of 60000 32x32 colour images in
10 classes. ImageNet is a large dataset which consists of images in 1000
classes.
### Metrics of Evaluation
In addition to the usual metrics like L2 and L-$\infty$ distance, and the time
taken, we have used the following metrics to compare our results with the
results of existing approaches.
1. 1.
Defect detection: to highlight the attack success rate, or the number of
benchmarks for which our tool could successfully produce an adversarial image.
2. 2.
Naturalness: to score the adversarial images for being admissible, i.e.
visibly not very different from the original image. We use Frechet Incéption
Distance (FID) (Harel-Canada et al. 2020) to measure naturalness. Values of
FID close to 0 indicate that the adversarial images are natural, and are
therefore desirable.
3. 3.
Output impartiality: to reflect whether the adversarial image generation is
biased towards any one of the output classes or not. We measure this using the
Pielou score (Harel-Canada et al. 2020), which can range from 0 (biased,
undesirable) to 1 (unbiased, desirable).
(Harel-Canada et al. 2020) have observed that neuron coverage was negatively
correlated with defect detection, naturalness, and output impartiality.
Naturalness is considered an essential metric while assessing the quality of
adversarial images. Fig. 5 shows how some of the existing methods generate
images that perform well on L2 and L-$\infty$ distance metric, but appear very
different from the original image.
Figure 5: Examples of adversarial images generated by other techniques lacking
originality.
## Results
Table 1 presents a comparison of Aigent with other state-of-the-art approaches
of generating adversarial images, on all the three benchmarks datasets, for
several important metrics including FID, Pielou score, defect detection rate,
L-2 and L-$\infty$ distance, and the number of pixels modified. The results
demonstrate that Aigent performs better than all the other approaches in terms
of FID, which indicates that the adversarial images generated by our method
are natural and visibly quite similar to the corresponding original images.
This is further reinforced by the fact that Aigent modifies far fewer pixels
as compared to the other approaches.
S. No. Technique FID Pielou score L-2 L-$\infty$ Time (seconds) Pixels
modified % of pixels modified Defect detection Benchmark dataset: MNIST 1
Aigent 0.001 0.725 1.82 0.66 1.726 24 3.06% 72.00% 2 Aigent (high defect)1
0.03 0.74 4.1 0.80 1.799 24 3.06% 91.40% 3 FGSM2 1.73 0.95 2.8 0.1 0.069 784
100.00% 99.00% 4 Black Box3 1.98 0.14 6.58 0.23 0.065 784 100.00% 88.40% 5
DeepXplore 0.02 0.47 5.16 1 11.74 60 7.65% 45.66% 6 DLFuzz4 0.17 0.88 2.29
0.39 30 586 74.74% 92.36% Benchmark dataset: CIFAR-10 1 Aigent 0.00009 0.927
1.6 0.5 12.01 12 0.39% 100.0% 2 FGSM2 0.071 0.92 5.5 0.1 0.079 3072 100.00%
100.0% 3 Black Box3 0.44 0.703 13.04 0.23 0.082 3072 100.00% 76.20% Benchmark
dataset: ImageNet 1 Aigent 0.00011 0.75 6.81 0.73 35 300 0.61% 98.60% 2 FGSM2
0.43 0.87 22 0.1 0.4 16384 100.00% 97.00% 3 Black Box3 0.05 0.8 52 0.4 0.3
16384 100.00% 90.00% 4 DeepXplore 0.032 N.A 58.04 0.51 84 15658 95.57% 59.13%
5 DLFuzz4 0.11 N.A 61.1 0.6 57 16102 98.28% 92.00%
Table 1: Comparison of Aigent with other state-of-the-art techniques on MNIST,
CIFAR-10 and ImageNet datasets. Bold values indicate the best figure for each
metric. DeepXplore and DLFuzz did not work on CIFAR-10. 1: Tuned to achieve
higher defect detection. 2:Gradient Based technique. 3:(Goodfellow, Shlens,
and Szegedy 2015). 4: We were getting a few compilation errors in the DLFuzz
code (https://github.com/turned2670/DLFuzz) which we fixed for this
comparison.
Our method could achieve 72% defect detection for MNIST when constraints were
stricter. When we allowed the quality of generated adversarial images to
degrade slightly in order to achieve a higher defect detection, shown as
Aigent (high defect) in the table, we were able to generate adversarial images
for 91.4% of the original images. Our method performs comparable to white box
methods. Although black box methods achieve higher defect detection, they
modify 100% pixels which leads to visibly distinguishable images. Thus, our
method performs well in terms of defect detection, while keeping the
modification quite small. Default detection on CIFAR-10 and ImageNet
benchmarks for Aigent was better than all other approaches.
For measuring the Pielou score, we took 50 original images of each class and
then calculated the frequency distribution on the classes of adversarial
images generated. Aigent was able to achieve a good Pielou score on all the
benchmark datasets. While techniques such as FGSM and DLFuzz have a better
Pielou score, it comes at the expense of other metrics such as FID and the
percentage of pixels modified. Our method aims at striking a good balance
between these metrics as it is crucial for the quality of adversarial images.
Figure 6: Adversarial images produced by Aigent (bottom row), and the
corresponding original images (top row).
Fig. 6 shows sample adversarial images produced for MNIST, CIFAR-10 and
ImageNet datasets. The first row contains original images and the second row
contains their corresponding adversarial images. Fig. 7 shows a comparison of
Aigent with black-box, gradient-based, and coverage-guided approaches, on the
same input image.
Figure 7: Sample adversarial images from different tools.
## Conclusion
Finding adversarial inputs for DNNs is not just useful for identifying
situations when a network may behave unexpectedly, but also for adversarial
training, which can make the network robust. We have proposed a technique to
generate adversarial inputs via patching of neural networks. In our
experiments over three benchmark image datasets, we observed that the proposed
method is significantly more effective than the existing state-of-the-art, in
terms of the naturalness of the adversarial images as well as the fraction of
pixels that were modified.
There are several interesting directions for future work. Since the proposed
method works by finding a patch repeatedly, better algorithms for DNN patching
would also make our technique more efficient. Another useful direction would
be to find a _minimal_ patch in order to get the _closest_ adversarial
example. And even though the technique can, in principle, be applied to DNNs
with any activation function, it would be worthwhile to engineer our approach
to handle activations other than ReLU efficiently.
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|
# A note on additive complements of the squares
Yuchen Ding, Yu–Chen Sun, Li–Yuan Wang and Yutong Xia (Yuchen Ding) School of
Mathematical Science, Yangzhou University, Yangzhou 225002, People’s Republic
of China<EMAIL_ADDRESS>(Yu–Chen Sun) Department of Mathematics and
Statistics, University of Turku, Turku 20014 , Finland<EMAIL_ADDRESS>(Li-Yuan Wang) School of Physical and Mathematical Sciences, Nanjing Tech
University, Nanjing 211816, People’s Republic of China<EMAIL_ADDRESS>(Yutong Xia) School of Mathematical Science, Yangzhou University, Yangzhou
225002, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.
Let $\mathcal{S}=\\{1^{2},2^{2},3^{2},...\\}$ be the set of squares and
$\mathcal{W}\subset\mathbb{N}$ be an additive complement of $\mathcal{S}$ so
that $\mathcal{S}+\mathcal{W}\supset\\{n\in\mathbb{N}:n\geq N_{0}\\}$ for some
$N_{0}$. Let also
$\mathcal{R}_{\mathcal{S},\mathcal{W}}(n)=\\#\\{(s,w):n=s+w,s\in\mathcal{S},w\in\mathcal{W}\\}$.
Chen and Fang [5] proved that
$\sum_{n=1}^{N}R_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/4}\log N.$
In this note, we improve their result and obtain that
$\sum_{n=1}^{N}R_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/3}.$
2020 Mathematics Subject Classification. Primary 11B13; Secondary 11B75.
Keywords. Additive complements, Squares.
## 1\. Introduction
Let $\mathbb{N}$ be the set of all non-negative integers. We say that
$A,B\subset\mathbb{N}$ are additive complements if
$A+B\supset\\{n\in\mathbb{N}:n\geq N_{0}\\}$ for some fixed $N_{0}$. If $A,B$
are additive complements, we say that $A$ is an additive complement of $B$.
Let $\mathcal{S}=\\{1^{2},2^{2},3^{2},...\\}$ the set of squares and
$\mathcal{W}\subset\mathbb{N}$ be an additive complement of $\mathcal{S}$. Let
$N$ be a large integer and $\mathcal{W}(N)$ be the number of elements of
$\mathcal{W}$ not exceeding $N$. Clearly, we have $\mathcal{W}(N)\sqrt{N}\geq
N-N_{0}$ for some given integer $N_{0}$, from which we deduce that
$\liminf_{N\rightarrow\infty}\mathcal{W}(N)/\sqrt{N}\geq 1.$
It is of great interest to ask whether
$\liminf_{N\rightarrow\infty}\mathcal{W}(N)/\sqrt{N}$ is strictly larger than
$1$.
In 1956, Erdős [8] posed this problem, which was later settled affirmatively
by Moser [10] with an accurate lower bound $1.06$. The number was then
improved in a few articles [1, 2, 4, 7, 11, 12]. Up to now the best result is
$\displaystyle\liminf_{N\rightarrow\infty}\mathcal{W}(N)/\sqrt{N}\geq 4/\pi,$
(1.1)
given by Cilleruelo [6], Habsieger [9], Balasubramanian and Ramana [3].
Note that $\mathcal{W}(N)/\sqrt{N}>1$ implies that there are some integers
$n\in\mathbb{N}$ which do not have an unique representation. Thus it is
worthwhile to study the number of representations for writing $n$ as a sum
$s+w$ with $s\in\mathcal{S}$ and $w\in\mathcal{W}$.
For any $A,B\subset\mathbb{N}$, let
$R_{A,B}(n)=\\#\\{(a,b):n=a+b,a\in A,b\in B\\}.$
Obviously, if $A,B$ are complements, then
$\liminf_{N\rightarrow\infty}\frac{1}{N}\sum_{n\leq N}R_{A,B}(n)\geq 1.$
Ben Green had a nice observation that for $B=\\{b_{1},b_{2},...\\}$ with
$b_{n}=\frac{\pi^{2}}{16}n^{2}+o(n^{2})$,
$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}R_{S,B}(n)=1.$
Later, he asked Fang (private communication, see [5]) whether there is an
additive complement of $\mathcal{S}$ has the same property as $B$. Namely, can
we find an additive complement $\mathcal{W}=\\{w_{n}\\}_{n=1}^{\infty}$ of
$\mathcal{S}$ satisfying the asymptotic condition
$w_{n}=\frac{\pi^{2}}{16}n^{2}+o(n^{2})$?
Motivated by Ben Green’s problem, Chen and Fang [5, Theorem 1.1] proved the
following result.
Let $N$ be a sufficiently large integer. For any additive complement
$\mathcal{W}$ of $\mathcal{S}$, we have
$\displaystyle\sum_{n=1}^{N}R_{\mathcal{S},\mathcal{W}}(n)-N\gg\mathcal{W}(2\sqrt{N})\log\mathcal{W}(2\sqrt{N})$
(1.2)
From (1.1), one can immediately obtain the following corollary. If
$\mathcal{W}$ is any complement of $\mathcal{S}$, then
$\sum_{n=1}^{N}R_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/4}\log N.$ (1.3)
Chen and Fang [5, Remark 1] also gave a counterexample to show that (1.3) does
not always hold, if $\mathcal{S}$ and $\mathcal{W}$ are substituted with
general additive complements $A,B$.
In this note, we will give an improvement of the result of Chen and Fang and
get the following theorem.
###### Theorem 1.1.
Let $N$ be a sufficiently large integer. For any additive complement
$\mathcal{W}$ of $\mathcal{S}$, we have
$\sum_{n=1}^{N}R_{\mathcal{S},\mathcal{W}}(n)-N\gg\mathcal{W}\left(\frac{1}{5}N^{2/3}\right).$
###### Remark 1.1.
The coefficient $1/5$ can be further optimized by carefully calculating.
From Theorem 1.1 and (1.1), one can immediately obtain the following
corollary.
###### Corollary 1.1.
For any additive complement $\mathcal{W}$ of $\mathcal{S}$, we have
$\sum_{n=1}^{N}R_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/3}.$
The proof of Theorem 1.1 is based on the structure of the proof in Chen–Fang
[5]. In Chen–Fang’s proof, for $d_{1},d_{2}\in\mathcal{W}$, they considered
the equation $x^{2}+d_{1}=y^{2}+d_{2}<N$ and required that this equation has
$\gg\log N$ solutions (see [5, Lemma 2.1]). Thus the $\log$-factor, in (1.3),
comes from the number of solutions. However, in Chen-Fang’s arguments, the
above equation has some solutions $(x,y)$ such that $|x-y|$ is very small,
namely $O(1)$, and thus $x+y$ is very large, namely $O(|d_{2}-d_{1}|)$. Since
$\max\\{x^{2},y^{2}\\}<N$, we must have $|d_{2}-d_{1}|\ll N^{1/2}$. For this
reason, in their arguments, they restricted
$d_{1},d_{2}\in\mathcal{W}\cap[cN^{1/2}]$ for some $c>0$. Here $[N]$ denotes
the set $\\{n\in\mathbb{N}:n\leq N\\}$.
Our new idea is that we exclude those solutions $(x,y)$ such that $|x-y|$ is
small. Namely, we require solutions $(x,y)$ to satisfy that $|x-y|\gg
N^{1/6}$, so $x+y$ is not too large, namely, $x+y\ll|d_{2}-d_{1}|/N^{1/6}$.
Hence, in our case, $\max\\{x^{2},y^{2}\\}<N$ follows from
$|d_{2}-d_{1}|/N^{1/6}\ll N^{1/2}$. Thus, one can allow $d_{1}$ and $d_{2}$ to
come from the larger set $\mathcal{W}\cap[cN^{1/6+1/2}]$ for some $c>0$ and
$c$ will be determined in our proof. Here the threshold $N^{1/6}$ is chosen to
balance certain terms in the proof.
Consequently, we can improve the lower bound $\mathcal{W}(2N^{1/2})$ in (1.2)
to $\mathcal{W}(cN^{2/3})$ for some explicit $c>0$, but the payoff is that
since we just pick solutions $(x,y)$ such that $|x-y|$ is large, we can only
show that the number of available solutions in our case is $\gg 1$ instead of
$\gg\log N$.
###### Acknowledgment.
The authors would like to thank Kaisa Matomäki for her helpful comments.
The first author was supported by National Natural Science Foundation of China
(Grant No. 12201544), Natural Science Foundation of Jiangsu Province of China
(Grant No. BK20210784) and China Postdoctoral Science Foundation (Grant No.
2022M710121). He was also supported by foundation numbers JSSCBS20211023 and
YZLYJF2020PHD051. The second author was supported by UTUGS funding and was
working in the Academy of Finland project No. $333707$. The third author was
supported by the National Natural Science Foundation of China (Grant No.
12201291) and the Natural Science Foundation of the Higher Education
Institutions of Jiangsu Province (21KJB110001).
## 2\. Proof of Theorem 1.1
The proof of our theorem relies on the following lemma, which is a refinement
of the original argument of Chen and Fang [5, Theorem 2.1].
###### Lemma 2.1.
Let $K$ be a positive integer and $\mathcal{D}\subseteq\mathbb{N}$ be such
that $4K^{2}|d-d^{\prime}$ for any $d,d^{\prime}\in\mathcal{D}$. Then for all
positive numbers $x$, we have
$\sum_{\begin{subarray}{c}n\leq x\\\ R_{\mathcal{S},\mathcal{D}}(n)\geq
1\end{subarray}}\left(R_{\mathcal{S},\mathcal{D}}(n)-1\right)\geq\mathcal{D}(2K\sqrt{x})-2.$
###### Proof.
The lemma is trivial if $\mathcal{D}(2K\sqrt{x})\leq 2$. So we only need to
consider the case $\mathcal{D}(2K\sqrt{x})>2$. In this case, we assume
$\mathcal{D}(2K\sqrt{x})=\ell$ and
$\mathcal{D}\cap\left[0,2K\sqrt{x}\right]=\\{d_{1}<d_{2}<\cdot\cdot\cdot<d_{\ell}\\}.$
For any $2<s\leq\ell$, we have $4K^{2}|d_{s}-d_{1}$. So we can assume
$d_{s}-d_{1}=4K^{2}n_{s}$ for some positive integer $n_{s}>1$. Then one of the
solutions of the equation
$x^{2}-y^{2}=d_{s}-d_{1}=4K^{2}n_{s}$
has the form
$\begin{cases}x_{s}=Kn_{s}+K,\\\ y_{s}=Kn_{s}-K.\end{cases}$ (2.1)
Moreover, the solution given by (2.1) satisfies
$\displaystyle x_{s}^{2}+d_{1}=y_{s}^{2}+d_{s}<x,$ (2.2)
because
$x_{s}^{2}+d_{1}<\left(\frac{d_{s}-d_{1}}{2K}\right)^{2}+2\frac{d_{1}(d_{s}-d_{1})}{(2K)^{2}}<\left(\frac{d_{s}}{2K}\right)^{2}\leq\frac{(2K\sqrt{x})^{2}}{4K^{2}}=x.$
If an integer $n\leq x$ can be written as the sum of $d_{1}$ and a square,
then
$\displaystyle
R_{\mathcal{S},\mathcal{D}}(n)-1\geq\sum_{\begin{subarray}{c}d_{s}>d_{1}\\\
n-d_{s}\in\mathcal{S}\end{subarray}}1,$
from which we deduce that
$\sum_{\begin{subarray}{c}n\leq x\\\ R_{\mathcal{S},\mathcal{D}}(n)\geq
1\end{subarray}}\\!\\!\\!\left(R_{\mathcal{S},\mathcal{D}}(n)-1\right)\geq\\!\\!\\!\sum_{\begin{subarray}{c}n\leq
x\\\
n-d_{1}\in\mathcal{S}\end{subarray}}\\!\\!\\!\left(R_{\mathcal{S},\mathcal{D}}(n)-1\right)\geq\\!\\!\\!\sum_{\begin{subarray}{c}n\leq
x\\\
n-d_{1}\in\mathcal{S}\end{subarray}}\sum_{\begin{subarray}{c}d_{s}>d_{1}\\\
n-d_{s}\in\mathcal{S}\end{subarray}}1=\sum_{\begin{subarray}{c}1<s\leq\ell\end{subarray}}\sum_{\begin{subarray}{c}n\leq
x\\\ n-d_{s}\in\mathcal{S}\\\ n-d_{1}\in\mathcal{S}\end{subarray}}1.$ (2.3)
For any $2<s\leq\ell$, from (2.1) and (2.2) with $n=x_{s}^{2}+d_{1}$, we have
$\sum_{\begin{subarray}{c}n\leq x\\\ n-d_{s}\in\mathcal{S}\\\
n-d_{1}\in\mathcal{S}\end{subarray}}1\geq 1$
It follows that by (2.3),
$\sum_{\begin{subarray}{c}n\leq x\\\ R_{\mathcal{S},\mathcal{D}}(n)\geq
1\end{subarray}}\left(R_{\mathcal{S},\mathcal{D}}(n)-1\right)\geq\sum_{\begin{subarray}{c}2<s\leq\ell\end{subarray}}1=\ell-2=\mathcal{D}(2K\sqrt{x})-2.$
∎
Now, we turn to prove Theorem 1.1.
###### Proof of Theorem 1.1.
Let $N$ be a sufficiently large integer. It suffices to prove
$\displaystyle\sum_{n=1}^{N}\left(R_{\mathcal{S},\mathcal{W}}(n)-1\right)\gg\mathcal{W}\left(\frac{1}{5}N^{2/3}\right).$
Let $K<N^{1/2}$ be a large number to be chosen later. We follow the proof of
Chen and Fang [5]. For any $1\leq j\leq 4K^{2}$, let
$\mathcal{W}_{j}=\\{w\in\mathcal{W}:w\equiv j\ ({\rm{mod}}\ 4K^{2})\\}.$
Then we have
$\mathcal{W}=\bigcup_{j=1}^{4K^{2}}\mathcal{W}_{j}$
with $\mathcal{W}_{i}\bigcap\mathcal{W}_{j}=\varnothing$ for $i\neq j$. Thus,
$\displaystyle\sum_{n=1}^{N}\left(R_{\mathcal{S},\mathcal{W}}(n)-1\right)=\sum_{n=1}^{N}\left(\sum_{j=1}^{4K^{2}}R_{\mathcal{S},\mathcal{W}_{j}}(n)-1\right)$
$\displaystyle\geq$
$\displaystyle\sum_{n=1}^{N}\sum_{\begin{subarray}{c}j=1\\\
R_{\mathcal{S},\mathcal{W}_{j}}(n)\geq
1\end{subarray}}^{4K^{2}}\left(R_{\mathcal{S},\mathcal{W}_{j}}(n)-1\right)=\sum_{j=1}^{4K^{2}}\sum_{\begin{subarray}{c}n=1\\\
R_{\mathcal{S},\mathcal{W}_{j}}(n)\geq
1\end{subarray}}^{N}\left(R_{\mathcal{S},\mathcal{W}_{j}}(n)-1\right).$ (2.4)
By Lemma 2.1 with $\mathcal{D}=\mathcal{W}_{j}$, we have
$\displaystyle\sum_{\begin{subarray}{c}n=1\\\
R_{\mathcal{S},\mathcal{W}_{j}}(n)\geq
1\end{subarray}}^{N}\left(R_{\mathcal{S},\mathcal{W}_{j}}(n)-1\right)\geq\mathcal{W}_{j}(2K\sqrt{N})-2$
(2.5)
for any $1\leq j\leq 4K^{2}$. Combining (2) and (2.5), we obtain
$\displaystyle\sum_{n=1}^{N}\left(R_{\mathcal{S},\mathcal{W}}(n)-1\right)\geq\sum_{j=1}^{4K^{2}}(\mathcal{W}_{j}(2K\sqrt{N})-2)=\mathcal{W}(2K\sqrt{N})-8K^{2}.$
(2.6)
We choose $K=\frac{1}{10}N^{1/6}$, then our claims follows from (1.1). ∎
## References
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* [2] R. Balasubramanian,On the additive completion of squares, J. Number Theory, 29 (1988), 10–12.
* [3] R. Balasubramanian, D.S. Ramana, Additive complements of the squares, C. R. Math. Acad. Sci. Soc. R. Can., 23 (2001), 6–11.
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