The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 13
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 88961)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 13
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
text
string | meta
dict |
|---|---|
\section{\label{sec:level1}Introduction}
Scalar transport in turbulence plays significant roles in various practical applications, ranging from geophysical and environmental problems such as the advection and dispersion of pollutants in the atmosphere \citep{nironi2015dispersion,mazzitelli2012active}, mixing of nutrients in the ocean \citep{bhamidipati2020turbulent, chor2020mixing}, and chemical reactions in industrial flows \citep{dimotakis2005turbulent, hill1976homogeneous}. Scalar transport is also important from the perspective of fundamental turbulence research, with previous studies showing that the scalar field is a sensitive detector of the structures in turbulent flows, such that their study has yielded insights into the physics of turbulent flows themselves \citep{tong1994passive}. The scalar field may feedback on the velocity field under certain conditions, such as for stratified turbulence or thermally driven turbulence \citep{zhang2022analysis,lohse2010small}. The focus of this paper is, however, on the case of passive scalars.
The transport of scalars in turbulent flows is challenging to understand both because of the complexity of the underlying turbulent flow that advects, stretches and compresses the scalar field, and also because of the molecular diffusion to which it is subject that leads to non-trivial differences compared to the transport of fluid particles \citep{ottino1989kinematics,warhaft2000passive}. Indeed, while it has often been assumed that the statistical properties of passive scalars in turbulent flows should reflect the analogous properties of the underlying velocity field, e.g. a similarity in the statistical distribution of the turbulent kinetic energy and scalar dissipation rates, this is not in general the case. For example, it has been found that Kolmogorov's hypothesis of local isotropy of the small scales of a turbulent flow is strongly violated when applied to passive scalars, with both experiments and direct numerical simulations (DNS) finding that the skewness of the scalar derivative remains of the order of unity when a large-scale mean scalar gradient is imposed, while it would be zero for a locally isotropic flow \citep{sreenivasan1977skewness,sreenivasan1991local,pumir1994numerical,mestayer1976local}. This strong violation of small-scale isotropy of the scalar field is often attributed to the presence of ramp-cliff structures in the scalar field through which large and small scales of the scalar field are directly connected \citep{buaria2021small,shraiman2000scalar}. It has also been found that intermittency in the scalar field is even stronger than that for the velocity field \citep{watanabe2004statistics}. Indeed, even for a stochastic model where the velocity field has Gaussian statistics, the scalar field has non-Gaussian statistics \citep{tong1994passive, kraichnan1994anomalous,falkovich01}. Another profound difference is that while the velocity field exhibits a dissipation anomaly (i.e.~the averaged turbulent kinetic energy dissipation rate is independent of viscosity for high Reynolds numbers), the scalar field does not, with the scalar dissipation rate decreasing as $\sim 1/\log(Sc)$ as the Schmidt number $Sc$ is increased \citep{buaria2021turbulence}. Interestingly, however, recent DNS results have shown that while the expected correspondence between the analogous velocity and scalar statistics is not observed for $Sc\leq 1$, it is recovered for sufficiently large $Sc$ (even for $Sc=7$) where a viscous-convective sub-range emerges in the scalar field \citep{Shete22}.
In general, the properties of a passive scalar field depend upon both the Reynolds number $\Rey$ and Schmidt number $Sc$, and in many practical problems, both $\Rey$ and $Sc$ are large. For $Sc>1$, the smallest scale (in a mean-field sense) in the scalar field is thought to be the Batchelor scale \citep{batchelor1959small} $\eta_B=Sc^{-1/2}\eta$, where $\eta$ is the Kolmogorov length scale, and therefore resolving flows with high $\Rey$ and $Sc$ is very challenging using DNS (as well as experiments) due to the spatial and temporal resolution constraints. For problems where the small-scale properties of the scalar field are important, large eddy simulations are not helpful. It is therefore highly desirable to develop models for the small scales of the scalar field that are capable of handling large ranges of $\Rey$ and $Sc$, as well as being computationally efficient. One possibility is to develop Lagrangian models for the scalar gradients in turbulent flows, inspired by the corresponding models for the velocity gradient that have been highly successful both in terms of making predictions and leading to new insights into the small-scale dynamics of turbulent flows \citep{meneveau2011lagrangian}. Analogous models for scalars could be used to explore and understand the small-scale dynamics of scalar fields at $\Rey$ and $Sc$ that are currently far out of reach using either DNS or experiments.
Lagrangian models for the velocity gradients are derived from the Navier-Stokes equations, but they require modelling/approximations for the pressure Hessian and viscous terms which are unclosed in the reference frame of a single fluid particle trajectory. Various models have been proposed, including the restricted Euler model \citep{vieillefosse1982local}, the tetrad model \citep{chertkov1999lagrangian}, the recent fluid deformation model \citep{chevillard2006lagrangian}, as well as closures based on random Gaussian fields \citep{wilczek2014pressure, johnson2016closure} and, more generally, on tensor function representation of the unclosed terms \citep{Leppin2020}. With the exception of the restricted Euler model, these models predict the steady-state, non-trivial properties of the velocity gradients in turbulent flows, including the preferential alignment of the vorticity with the intermediate strain-rate eigenvector, intermittency, and the multifractal scaling of the moments of the velocity gradients. Unfortunately, most of the models are only capable of predicting flows with low to moderate $\Rey$.
Recently, the multi-level recent deformation of Gaussian fields (ML-RDGF) model has been developed, which was shown to predict DNS data accurately for arbitrary Reynolds numbers \citep{johnson2017turbulence}.
In the equation for a passive scalar gradient along a fluid particle trajectory, the scalar gradient diffusion term is unclosed. A closed model for scalar gradients in turbulent flows was previously derived based on a simple linear relaxation model for the scalar gradient diffusion \citep{martin2005joint}, with the velocity gradient in the equation specified using a restricted Euler model that was modified to include a linear damping term to model viscous effects \citep{martin1998dynamics}. Although the model showed general qualitative agreement with DNS data, there were significant quantitative inaccuracies for a number of key quantities, including significant errors in the predictions for the alignments of the scalar gradients with the strain-rate eigenvectors, and significant underprediction of large fluctuations of the scalar gradients. More advanced models also have been proposed that use the recent fluid deformation approximation \citep{chevillard2006lagrangian} to model the scalar gradient diffusion term \citep{hater2011lagrangian, gonzalez2009kinematic}. These models also yielded qualitatively reasonable predictions, but suffered from quantitative inaccuracies and can only make predictions for relatively low Reynolds numbers due to their use of the recent fluid deformation approximation \citep{chevillard2006lagrangian}. Our work significantly advances these models in two ways.
First, the velocity gradients will be specified using the much more sophisticated ML-RDGF model that also allows for predictions to be made at arbitrarily large $\Rey$. Second, the recent deformation of Gaussian fields closure \citep{johnson2016closure} will be used to provide a more sophisticated closure for the scalar gradient diffusion term. This closure leads to nonlinear terms that can regulate the growth of the scalar gradients in regions of intense stretching, which can be vital to preventing blow-ups from occurring in the model. The closure also incorporates the effect of $Sc$ on the scalar gradient dynamics. In this way, the model is capable of predicting the effect of both $\Rey$ and $Sc$ on the scalar gradient dynamics.
\section{\label{sec:level2_mod_scal}Model for scalar gradients}
\subsection{\label{sec:level3_eq}Governing equations for instantaneous and characteristic variables}
For an incompressible, Newtonian fluid, the equations governing the evolution of the fluid velocity gradient $\boldsymbol{A}\equiv\boldsymbol{\nabla u}$ and scalar gradient $\boldsymbol{B}\equiv\boldsymbol{\nabla}\phi$ are
\begin{align}
D_t\boldsymbol{A}&=-\boldsymbol{A\cdot A}-\boldsymbol{H}+\nu\nabla^2\boldsymbol{A}+\boldsymbol{F_A},\label{Aeq}\\
D_t\boldsymbol{B}&=-\boldsymbol{A}^\top\boldsymbol{\cdot B}+\kappa\nabla^2\boldsymbol{B}+\boldsymbol{F_B},\label{Beq}
\end{align}
where $D_t\equiv \partial_t +\boldsymbol{u\cdot \nabla}$ is the Lagrangian derivative, $\boldsymbol{H}\equiv \boldsymbol{\nabla\nabla}p$ is the pressure Hessian, $p$ is the fluid pressure (normalized by the fluid density), $\nu$ is the fluid kinematic viscosity, $\kappa$ is the scalar diffusivity, and $\boldsymbol{F_A}, \boldsymbol{F_B}$ are forcing terms (assumed to be known/prescribed). In the Lagrangian frame, the spatial derivative terms in \eqref{Aeq} and \eqref{Beq} are unknown, and therefore these terms must be modeled in terms of functionals of the associated quantity, e.g $\nu\nabla^2\boldsymbol{A}$ must be modelled as some functional of $\boldsymbol{A}$.
Concerning \eqref{Aeq}, various closure approaches have been developed, including the Recent Fluid Deformation Approximation (RFDA) \citep{chertkov1999lagrangian}, Random Gaussian Fields Closure (RGFC) \citep{wilczek2014pressure}, and the Recent Deformation of Gaussian Fields (RDGF) closure \citep{johnson2016closure}. More recently, a multi-level version of the RDGF model has emerged (referred to as ML-RDGF) that provides a model for $D_t\boldsymbol{A}$ that is valid for arbitrary Reynolds numbers. These closures for \eqref{Aeq} all lead to a model for $\boldsymbol{A}$ that can generate steady-state statistics. This is in contrast to the Restricted Euler (RE) model \citep{vieillefosse1982local,meneveau2011lagrangian} that ignores the anisotropic contribution to $\boldsymbol{H}$ and sets $\nu=0$, leading to a model for $\boldsymbol{A}$ that exhibits a finite-time singularity. Subsequent studies found that the addition of viscous effects to the RE model is not sufficient to prevent a finite-time singularity; the anisotropic pressure Hessian must also be accounted for. This could point to a potential challenge in closing \eqref{Beq}; if the closure approximation for $\kappa\nabla^2\boldsymbol{B}$ is not sufficiently accurate then the predictions from the resulting model could also generate finite-time singular solutions for $\boldsymbol{B}$ since the equation for $\boldsymbol{B}$ contains no pressure Hessian to regulate the amplification term $\boldsymbol{B\cdot A}$ that causes $\|\boldsymbol{B}\|$ to grow when $\boldsymbol{B\cdot}(\boldsymbol{B\cdot A})<0$. In this sense, developing a suitable closure for \eqref{Beq} may be more challenging than that for \eqref{Aeq}.
The RGFC model (and also the RDGF and ML-RDGF models, since they are extensions of the RGFC model) does not directly approximate the unclosed terms in \eqref{Aeq} (unlike the RFDA model) but instead closes the equation for $d_t\boldsymbol{\mathcal{A}}$, which is defined as the solution to
\eqref{Aeq}
averaged over the subset of fluid particles that experience the same value of $\boldsymbol{A}$. The statistics of $\boldsymbol{\mathcal{A}}$ correspond to the statistics of $\boldsymbol{A}$, however, the advantage is that the terms requiring closure in the equation for $d_t\boldsymbol{\mathcal{A}}$ are conditionally averaged quantities, and therefore statistical approaches based on the properties of random Gaussian fields may be employed to close the terms in the equation. Such a procedure cannot be done when directly closing the terms in \eqref{Aeq} since these involve instantaneous quantities.
Mathematically, this procedure that replaces $D_t\boldsymbol{A}$ with $d_t\boldsymbol{\mathcal{A}}$ may be formalized as follows. Suppose that $\boldsymbol{A}$ evolves in a phase-space with time-independent coordinates $\boldsymbol{a}\in\mathbb{R}^{3\times 3}$. The PDF of $\boldsymbol{A}$ is then defined as $\mathcal{P}(\boldsymbol{a},t)\equiv \langle\delta(\boldsymbol{A}-\boldsymbol{a})\rangle$ which solves the Liouville equation
\begin{align}
\partial_t\mathcal{P}=-\boldsymbol{\nabla_a\cdot}\Big(\mathcal{P}\Big\langle D_t\boldsymbol{A}\Big\rangle_{\boldsymbol{A}=\boldsymbol{a}} \Big),
\end{align}
where $\langle\cdot\rangle_{\boldsymbol{A}=\boldsymbol{a}} $ denotes an ensemble average conditioned on $\boldsymbol{A}=\boldsymbol{a}$. We now introduce the time-dependent characteristic variable $\boldsymbol{\mathcal{A}}$ defined via
\begin{align}
d_t\boldsymbol{\mathcal{A}}&\equiv\boldsymbol{G}(\boldsymbol{\mathcal{A}},t),\label{calAdt}\\
\boldsymbol{G}(\boldsymbol{a},t)&\equiv\Big\langle D_t\boldsymbol{A}\Big\rangle_{\boldsymbol{A}=\boldsymbol{a}}.
\end{align}
Whereas $\boldsymbol{A}$ evolves according to the instantaneous Navier-Stokes equation, $\boldsymbol{\mathcal{A}}$ evolves according to the conditionally averaged Navier-Stokes equation. Nevertheless, since $\langle D_t\boldsymbol{A}\rangle_{\boldsymbol{A}=\boldsymbol{a}}$ only involves a partial average over the flow, the field $\boldsymbol{G}$ will in general exhibit nonlinear dependence on $\boldsymbol{a}$ and $t$, and hence the trajectories $\boldsymbol{\mathcal{A}}$ generated by \eqref{calAdt} will still vary chaotically in time for a turbulent flow.
We may then define the PDF $\varrho(\boldsymbol{a},t)\equiv \langle\delta(\boldsymbol{\mathcal{A}}-\boldsymbol{a})\rangle$ which solves
\begin{align}
\partial_t\varrho&=-\boldsymbol{\nabla_a\cdot}\Big(\varrho\Big\langle d_t\boldsymbol{\mathcal{A}}\Big\rangle_{\boldsymbol{\mathcal{A}}=\boldsymbol{a}} \Big).
\end{align}
From the definition of the characteristic variable we have
\begin{align}
\Big\langle d_t\boldsymbol{\mathcal{A}}\Big\rangle_{\boldsymbol{\mathcal{A}}=\boldsymbol{a}}=\Big\langle \boldsymbol{G}(\boldsymbol{\mathcal{A}},t)\Big\rangle_{\boldsymbol{\mathcal{A}}=\boldsymbol{a}}=\boldsymbol{G}(\boldsymbol{a},t)=\Big\langle D_t\boldsymbol{A}\Big\rangle_{\boldsymbol{A}=\boldsymbol{a}},
\end{align}
where the second equality follows since $\boldsymbol{G}(\boldsymbol{a},t)$ is not random. In view of this, if $\varrho(\boldsymbol{a},0)=\mathcal{P}(\boldsymbol{a},0)$ then it follows that $\varrho(\boldsymbol{a},t)=\mathcal{P}(\boldsymbol{a},t)\,\forall t$, and hence the statistics of $\boldsymbol{A}$ may be obtained via solutions to
\eqref{calAdt}.
Based on the definitions above we have
\begin{align}
d_t\boldsymbol{\mathcal{A}}&=-\boldsymbol{\mathcal{A}\cdot \mathcal{A}}-\langle\boldsymbol{H}\rangle_{\boldsymbol{\mathcal{A}}}+\nu\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}+\langle\boldsymbol{F_A}\rangle_{\boldsymbol{\mathcal{A}}},\label{Acaleq}
\end{align}
where we have used the short-hand notation
\begin{align}
\langle\cdot\rangle_{\boldsymbol{\mathcal{A}}}=\langle\cdot\rangle_{\boldsymbol{A}=\boldsymbol{a}}\Big\vert_{\boldsymbol{a}=\boldsymbol{\mathcal{A}}}.
\end{align}
While closing the equation for $D_t\boldsymbol{A}$ requires approximating instantaneous quantities, closing $d_t\boldsymbol{\mathcal{A}}$ requires approximating conditionally averaged quantities. This is a major advantage since it means that powerful statistical approaches can be used to develop systematic closures for $d_t\boldsymbol{\mathcal{A}}$. This then is the approach adopted in the RGFC and RDGF models, where $\langle\boldsymbol{H}\rangle_{\boldsymbol{\mathcal{A}}}$ and $\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}$ are closed under the approximation that the velocity field $\boldsymbol{u}$ has Gaussian statistics.
A similar approach to this may be adopted for \eqref{Beq}, but now the averaging must be conditional on both $\boldsymbol{A}=\boldsymbol{a}$ and $\boldsymbol{B}=\boldsymbol{b}$, where $(\boldsymbol{a},\boldsymbol{b})\in (\mathbb{R}^{3\times 3},\mathbb{R}^3)$ are the time-independent coordinates of the phase-space in which $\boldsymbol{A}$ and $\boldsymbol{B}$ evolve. We define
\begin{align}
d_t\boldsymbol{\mathcal{B}}&\equiv\boldsymbol{J}(\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}},t),\label{calBdt}\\
\boldsymbol{J}(\boldsymbol{a},\boldsymbol{b},t)&\equiv\Big\langle D_t\boldsymbol{B}\Big\rangle_{\boldsymbol{A}=\boldsymbol{a},\boldsymbol{B}=\boldsymbol{b}},
\end{align}
and then based on \eqref{Beq} this leads to
\begin{align}
d_t\boldsymbol{\mathcal{B}}&=-\boldsymbol{\mathcal{A}}^\top\boldsymbol{\cdot\mathcal{B}}+\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}+\langle\boldsymbol{F_B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}},\label{Bcaleq}\\
\langle\cdot\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}&=\langle\cdot\rangle_{\boldsymbol{A}=\boldsymbol{a},\boldsymbol{B}=\boldsymbol{b}}\Big\vert_{\boldsymbol{a}=\boldsymbol{\mathcal{A}},\boldsymbol{b}=\boldsymbol{\mathcal{B}}}.
\end{align}
An approach based on the RGFC or RDGF models may then be used to close the term $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ under the assumption that the scalar field $\phi$ has Gaussian statistics. Just as the solutions to
\eqref{calAdt} can be used to construct the statistics of $\boldsymbol{A}$, so also can the solutions to
\eqref{calBdt} be used to construct the statistics of $\boldsymbol{B}$.
\subsection{\label{sec:level2_mod}Model for the velocity gradients}
In our model for $\boldsymbol{\mathcal{B}}$, the ML-RDGF model will be used to prescribe $\boldsymbol{\mathcal{A}}$, and since our model for $\boldsymbol{\mathcal{B}}$ will be based on the RDGF approach, we summarize the key ideas in this modeling approach before applying them to deriving a closed equation for $d_t\boldsymbol{\mathcal{B}}$
\subsubsection{\label{sec:level3_RDGF}RDGF closure}
The deformation tensor $\boldsymbol{D}(t,s)$ for a fluid particle with position $\boldsymbol{x}^f(t\vert\boldsymbol{X},s)$ that satisfies $\boldsymbol{x}^f(s\vert\boldsymbol{X},s)=\boldsymbol{X}$ is defined as
\begin{align}
\boldsymbol{D}(t,s)\equiv\frac{\partial}{\partial\boldsymbol{X}}\boldsymbol{x}^f(t\vert\boldsymbol{X},s),\quad s\in[0,t],
\end{align}
and evolves according to
\begin{align}
\partial_t\boldsymbol{D}=\boldsymbol{A\cdot D},\quad \boldsymbol{D}(0,0)=\mathbf{I},
\end{align}
where $\mathbf{I}$ is the identity matrix. While the solution to this is given by a time-ordered exponential, the RFDA approximates the solution by assuming that $\boldsymbol{A}$ is constant over a recent-deformation timescale $\tau$, with the deformation at times $s<t-\tau$ ignored. In this case, the approximate solution is
\begin{align}
\boldsymbol{D}(t,s)&\approx \exp\Big((t-s)\boldsymbol{A} \Big),\quad t-s\in[0,\tau].\label{Dappx}
\end{align}
The idea then is as follows: the quantities $\boldsymbol{H}\equiv\boldsymbol{\nabla\nabla}p$ and $\nabla^2\boldsymbol{A}$ are evaluated along the fluid particle trajectory $\boldsymbol{x}^f(t\vert\boldsymbol{X},s)$, and they may be related to their corresponding values at the reference configuration $\boldsymbol{x}^f(s\vert\boldsymbol{X},s)=\boldsymbol{X}$ using $\boldsymbol{D}$. Using the approximate solution for $\boldsymbol{D}$ in \eqref{Dappx} and setting $s=t-\tau$ leads to
\begin{align}
\langle\boldsymbol{H}\rangle_{\boldsymbol{\mathcal{A}}}&\approx \Big\langle\boldsymbol{D}^{-\top}\boldsymbol{\cdot}\Big(\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{p}\Big)\boldsymbol{\cdot}\boldsymbol{D}^{-1}\Big\rangle_{\boldsymbol{\mathcal{A}}}=\boldsymbol{\mathcal{D}}^{-\top}\boldsymbol{\cdot}\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{p}\Big\rangle_{\boldsymbol{\mathcal{A}}}\boldsymbol{\cdot}\boldsymbol{\mathcal{D}}^{-1},\label{Defappx1}\\
\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}&\approx \Big\langle\boldsymbol{D}^{-\top}\boldsymbol{\cdot}\Big(\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{A}}\Big)\boldsymbol{\cdot}\boldsymbol{D}^{-1}\Big\rangle_{\boldsymbol{\mathcal{A}}}=\boldsymbol{\mathcal{D}}^{-\top}\boldsymbol{\cdot}\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{A}}\Big\rangle_{\boldsymbol{\mathcal{A}}}\boldsymbol{\cdot}\boldsymbol{\mathcal{D}}^{-1},\label{Defappx2}
\end{align}
where $\boldsymbol{\mathcal{D}}= \exp(\tau\boldsymbol{\mathcal{A}})$, and $(\cdot)^{-\top}$ denotes the transpose of the inverse of a tensor. The RDGF model then uses the RGFC approach \citep{wilczek2014pressure} to approximate the conditional averages in \eqref{Defappx1} and \eqref{Defappx2}.
The basic motivation for the RDGF model is that the closure approximations for $\langle\boldsymbol{H}\rangle_{\boldsymbol{\mathcal{A}}}$ and $\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}$ can be improved by applying the RGFC at $\boldsymbol{X},t-\tau$ rather than $\boldsymbol{x},t$. This is because when the RGFC is applied at $\boldsymbol{X},t-\tau$ it is then transformed under the flow map into something more realistic. For example, while the conditional averages in \eqref{Defappx1} and \eqref{Defappx2} are approximated assuming that $\boldsymbol{u}$ has Gaussian statistics, due to the Lagrangian transformation described by $\boldsymbol{\mathcal{D}}$, the resulting approximations for $\langle\boldsymbol{H}\rangle_{\boldsymbol{\mathcal{A}}}$ and $\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}$ obtained through \eqref{Defappx1} and \eqref{Defappx2} will, in general, correspond to those for a non-Gaussian field $\boldsymbol{u}$.
The final closed equation obtained using the RDGF closure has the form \citep{johnson2016closure}
\begin{align}
d\boldsymbol{\mathcal{A}}=\boldsymbol{\mathcal{N}}_{\mathcal{A}}\{\boldsymbol{\mathcal{A}},\tau,\tau_\eta\}dt+\boldsymbol{\Sigma\cdot}d\boldsymbol{\mathcal{W}},\label{RDGF_equation}
\end{align}
where the forcing term has been chosen to be $\langle\boldsymbol{F_A}\rangle_{\boldsymbol{\mathcal{A}}}dt=\boldsymbol{\Sigma\cdot}d\boldsymbol{\mathcal{W}}$, with $d\boldsymbol{\mathcal{W}}$ denoting a tensor-valued Wiener process, and $\boldsymbol{\Sigma}$ denoting a diffusion tensor that depends on the coefficients $D_s$ and $D_a$ which determine the growth rate of the mean-square values of the strain-rate $\boldsymbol{\mathcal{S}}\equiv (\boldsymbol{\mathcal{A}}+\boldsymbol{\mathcal{A}}^\top)/2$ and rotation-rate $\boldsymbol{\mathcal{R}}\equiv (\boldsymbol{\mathcal{A}}-\boldsymbol{\mathcal{A}}^\top)/2$ tensors. For brevity, we do not include the details of the nonlinear operator $\boldsymbol{\mathcal{N}}_{\mathcal{A}}\{\cdot \}$ which may be found in previous works \citep{johnson2016closure}.
The three unknown parameters $\tau, D_a, D_s$ are obtained by an optimization procedure that seeks those values for which the model satisfies known constraints for isotropic turbulence, namely $2\langle\|\boldsymbol{\mathcal{S}}\|^2\rangle =1/\tau_\eta^2$, and the two homogeneity relations due to \cite{betchov1956inequality}, $\langle \boldsymbol{\mathcal{A}}\boldsymbol{:\mathcal{A}}\rangle=0$, $\langle (\boldsymbol{\mathcal{A}}\boldsymbol{\cdot\mathcal{A}})\boldsymbol{:\mathcal{A}}\rangle =0$. The values obtained by this procedure are $\tau=0.1302\tau_\eta$, $D_s=0.1014/\tau_\eta^3$, and $D_a=0.0505/\tau_\eta^3$.
\subsubsection{\label{sec:level3}ML-RDGF closure}
The RDGF model described by \eqref{RDGF_equation} does not contain any dependence on $\Rey_\lambda$. To address this, a multi-level version of the RDGF closure (called ML-RDGF) was developed \citep{johnson2017turbulence}. The key idea behind this model is that in a turbulent flow, there exist velocity gradients at different scales in the flow, and the velocity gradient dynamics at different scales are coupled because of the energy cascade. Moreover, this coupling will be influenced by the fact that the energy flux through the cascade is not constant, but fluctuates in time and space. The ML-RDGF model extends the RDGF model to take this into account by replacing \eqref{RDGF_equation} with
\begin{align}
d\boldsymbol{\mathcal{A}}^{[n]}=\boldsymbol{\mathcal{N}}_{\mathcal{A}}\{\boldsymbol{\mathcal{A}}^{[n]},\tau,\tau_n\}dt -d_t(\ln\tau_n)\boldsymbol{\mathcal{A}}^{[n]} +\boldsymbol{\Sigma}^{[n]}\boldsymbol{\cdot}d\boldsymbol{\mathcal{W}}^{[n]},\quad n=1,2,...N,\label{MLRDGF_equation}
\end{align}
in which the $\tau_\eta$ appearing in \eqref{RDGF_equation} has been replaced by the time-dependent timescale $\tau_n(t)$, so that now $\tau=0.1302\tau_n$, $D_s=0.1014/\tau_n^3$, and $D_a=0.0505/\tau_n^3$.
Equation \eqref{MLRDGF_equation} is actually a system of $N$ coupled equations, and $\boldsymbol{\mathcal{A}}^{[n]}$ represents the velocity gradient at the $n^{th}$ level, corresponding to the velocity gradient filtered on some scale. For the first level $n=1$, the timescale is fixed $\tau_1=\beta^{N-1}\tau_\eta$, where $\beta=10$ is chosen in the model. For $n\geq 2$, $\tau_n(t)\equiv \beta^{-1}\|\boldsymbol{\mathcal{S}}^{[n-1]}\|^{-1}$ (where $\boldsymbol{\mathcal{S}}^{[n-1]}$ is the strain-rate associated with $\boldsymbol{\mathcal{A}}^{[n-1]}$), such that the time evolution of $\boldsymbol{\mathcal{A}}^{[n]}$ is coupled to the evolution at the larger scale where the velocity gradient is $\boldsymbol{\mathcal{A}}^{[n-1]}$. The solution at level $n=N$ then corresponds to the full (unfiltered) velocity gradient, i.e.~$\boldsymbol{\mathcal{A}}^{[N]}=\boldsymbol{\mathcal{A}}$. This multi-level model is capable of predicting $\boldsymbol{\mathcal{A}}$ for the discrete Reynolds numbers $\Rey_\lambda=\Rey_\lambda^{[n=1]}\beta^{N-1}$, where $\Rey_\lambda^{[n=1]}$ is the Taylor Reynolds number corresponding to the first level $n=1$, which was chosen in previous studies \citep{johnson2017turbulence} to be $\Rey_\lambda^{[n=1]}=60$. Then a modified timescale for the second level, $\tau_2$, enables the model to predict flows at arbitrary $\Rey_\lambda$.
The predictions for $\boldsymbol{\mathcal{A}}^{[N]}=\boldsymbol{\mathcal{A}}$ were shown to be in excellent agreement with DNS and experimental data \citep{johnson2017turbulence}, and revealed that the model makes robust predictions for the intermittency of $\boldsymbol{\mathcal{A}}$ up to the highest Reynolds number considered, $\Rey_\lambda= \textit{O}(10^6)$.
\subsection{\label{sec:level2}Closure for scalar gradient equation based on RDGF}
Based on its excellent performance, the ML-RDGF model will be used to specify $\boldsymbol{\mathcal{A}}$ in the equation for the scalar gradient $\boldsymbol{\mathcal{B}}$. The RDGF closure scheme will be used to close the scalar gradient diffusion term $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$. A multi-level version is not required since the effect of $\Rey_\lambda$ on the model for $\boldsymbol{\mathcal{B}}$ will already be accounted for through the use of the ML-RDGF to specify $\boldsymbol{\mathcal{A}}$ in the equation for $\boldsymbol{\mathcal{B}}$.
Analogous to \eqref{Defappx1} and \eqref{Defappx2}, under the recent fluid deformation approximation, the scalar gradient diffusion term may be expressed as
\begin{align}
\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}\approx\boldsymbol{\mathcal{D}}^{-\top}\boldsymbol{\cdot}\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}\Big\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}\boldsymbol{\cdot}\boldsymbol{\mathcal{D}}^{-1}.\label{Defappx3}
\end{align}
The Random Gaussian Fields Closure (RGFC) can be used to derive a closed expression for the conditional average appearing in this expression, leading through \eqref{Defappx3} to an RDGF closure for $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$.
In the present context of the scalar field, the RGFC begins with the assumption that $\phi$ has Gaussian statistics defined in terms of the characteristic functional
\begin{align}
\Sigma^{\phi}[\lambda(\boldsymbol{x})]=\exp\Bigg[-\frac{1}{2}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\lambda(\boldsymbol{x})R^{\phi}(\boldsymbol{x},\boldsymbol{x}')\lambda(\boldsymbol{x}')\,d\boldsymbol{x}\, d\boldsymbol{x}' \Bigg],\label{phi_char}
\end{align}
(time label is suppressed here and in what follows for simplicity) where $\lambda(\boldsymbol{x})$ is the Fourier variable conjugate to $\phi(\boldsymbol{x})$, and $R^{\phi}(\boldsymbol{x},\boldsymbol{x}')\equiv\langle \phi(\boldsymbol{x})\phi(\boldsymbol{x}')\rangle$, which for isotropic turbulence has the form
\begin{align}
R^{\phi}(\boldsymbol{x},\boldsymbol{x}')=R^{\phi}(r)=\langle\phi^2\rangle f_\phi(r),\quad \boldsymbol{r}\equiv \boldsymbol{x}-\boldsymbol{x}',\quad r\equiv\|\boldsymbol{r}\|,\label{iso_Rphi}
\end{align}
where $f_\phi(r)$ is the scalar spatial correlation function. Following the work of \cite{wilczek2014pressure}, the characteristic function for the scalar gradient $\boldsymbol{B}\equiv\boldsymbol{\nabla}\phi$ is obtained from \eqref{phi_char} as
\begin{align}
\Sigma^{\boldsymbol{B}}[\boldsymbol{\mu}(\boldsymbol{x})]=\exp\Bigg[-\frac{1}{2}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\boldsymbol{\mu}(\boldsymbol{x})\boldsymbol{\cdot}\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{x},\boldsymbol{x}')\boldsymbol{\cdot}\boldsymbol{\mu}(\boldsymbol{x}')\,d\boldsymbol{x}\, d\boldsymbol{x}' \Bigg],\label{B_char}
\end{align}
where $\boldsymbol{\mu}$ is the Fourier variable conjugate to $\boldsymbol{B}$, and $\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{x},\boldsymbol{x}')\equiv\langle\boldsymbol{B}(\boldsymbol{x})\boldsymbol{B}'(\boldsymbol{x}')\rangle$, which is related to $R^{\phi}(\boldsymbol{x},\boldsymbol{x}')$ through
\begin{align}
\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{x},\boldsymbol{x}')=\langle\boldsymbol{\nabla} \phi(\boldsymbol{x})\boldsymbol{\nabla}'\phi(\boldsymbol{x}')\rangle = \boldsymbol{\nabla}\boldsymbol{\nabla}'R^{\phi}(\boldsymbol{x},\boldsymbol{x}').
\end{align}
For an isotropic scalar field where \eqref{iso_Rphi} applies, we then have
\begin{align}
\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{x},\boldsymbol{x}')=\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{r})=\langle\phi^2\rangle\Bigg[\Bigg(\frac{f_\phi'(r)}{r}-f_\phi^{''}(r)\Bigg)\frac{\boldsymbol{rr}}{r^2}-\frac{f_\phi'(r)}{r}\mathbf{I} \Bigg],\label{Bcov}
\end{align}
where prime superscripts denote differentiation with respect to $r$.
According to \eqref{B_char}, the statistics of $\boldsymbol{B}$ are described by a Gaussian characteristic functional when $\phi$ is assumed to be Gaussian. Just as in previous works \citep{wilczek2014pressure,johnson2016closure} when deriving a closure model for $D_t\boldsymbol{\mathcal{A}}$, it is acknowledged that the statistics of $\boldsymbol{B}$ are not Gaussian in a real turbulent flow (nor are they Gaussian even when $\boldsymbol{u}$ is Gaussian \citep{falkovich01}). The motivation for this choice is simply that it is the only option when deriving a statistical field closure. It must be appreciated, however, that only with respect to the closure of $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$, are the statistics of $\boldsymbol{B}$ approximated as being Gaussian; the model that follows from this choice nevertheless generates statistics for $\boldsymbol{\mathcal{B}}$ that are highly non-Gaussian (as will be shown later).
The specific term to be closed using RGFC is
\begin{align}
\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}\Big\rangle_{\boldsymbol{a},\boldsymbol{b}}=\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{x}^f(t\vert\boldsymbol{X},s),t)\Big\rangle_{{\boldsymbol{A}}(\boldsymbol{x}^f(t\vert\boldsymbol{X},s),t)=\boldsymbol{a},{\boldsymbol{B}}(\boldsymbol{x}^f(t\vert\boldsymbol{X},s),t)=\boldsymbol{b}},
\end{align}
which is evaluated at $\boldsymbol{a}=\boldsymbol{\mathcal{A}}, \boldsymbol{b}=\boldsymbol{\mathcal{B}}$ in \eqref{Defappx3}. Just as the RFDA assumes ${\boldsymbol{A}}(\boldsymbol{x}^f(t\vert\boldsymbol{X},s),t)\approx {\boldsymbol{A}}(\boldsymbol{X},s)$ for $t-s\in[0,\tau]$, we also assume ${\boldsymbol{B}}(\boldsymbol{x}^f(t\vert\boldsymbol{X},s),t)\approx {\boldsymbol{B}}(\boldsymbol{X},s)$ for $t-s\in[0,\tau]$, and inserting this yields
\begin{align}
\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}\Big\rangle_{\boldsymbol{a},\boldsymbol{b}}\approx\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{X},s)\Big\rangle_{{\boldsymbol{A}}(\boldsymbol{X},s)=\boldsymbol{a},{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}},
\end{align}
which puts the conditional average into a form to which the RGFC procedure \citep{wilczek2014pressure} can be applied.
Before proceeding, we note that although the argument of the conditional average\[\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{X},s)\Big\rangle_{{\boldsymbol{A}}(\boldsymbol{X},s)=\boldsymbol{a},{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}\]does not contain $\boldsymbol{A}$ but only $\boldsymbol{B}$, the conditionality on $\boldsymbol{A}(\boldsymbol{X},s)=\boldsymbol{a}$ cannot formally be removed. This is because since the evolution of $\boldsymbol{B}$ depends upon $\boldsymbol{A}$ (see \eqref{Beq}), then the value of \[\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}\]will not be uncorrelated from $\boldsymbol{A}$, in general. While a closure for the full conditional average can be obtained using the RDGF approach, in order to simplify the closure analysis the following approximation is made
\begin{align}
\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{X},s)\Big\rangle_{{\boldsymbol{A}}(\boldsymbol{X},s)=\boldsymbol{a},{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}\approx \Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{X},s)\Big\rangle_{{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}.\label{to_close}
\end{align}
It is crucial to note, however, that the overall closure for $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ does partially include the effect of the conditioning upon $\boldsymbol{\mathcal{A}}$, since the right hand side of \eqref{Defappx3} contains $\boldsymbol{\mathcal{D}}$ and not $\boldsymbol{D}$ precisely because of the conditionality in the averaging operator on the left hand side of \eqref{Defappx3}. Therefore, the closure for $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ captures some of the dependency of the scalar diffusion on the local velocity gradients in the flow.
With the statistics of $\boldsymbol{B}$ prescribed using \eqref{B_char} and the covariance tensor for $\boldsymbol{B}$ prescribed using \eqref{Bcov}, a closure for \eqref{to_close}, and hence $\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$, may be obtained following the same steps as in the work of \cite{wilczek2014pressure}. The basic steps are as follows. First, using the approach described in Appendix A of the work of \cite{wilczek2014pressure}, the unclosed quantity is re-written in terms of a two-point quantity
\begin{align}
\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{X},s)\Big\rangle_{{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}=\lim_{r\to 0}\frac{\partial^2}{\partial\boldsymbol{r}\partial\boldsymbol{r}}\Big\langle \boldsymbol{B}(\boldsymbol{X}+\boldsymbol{r},s)\Big\rangle_{{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}.
\end{align}
Applying the steps outlined in Appendix B of \cite{wilczek2014pressure} to the scalar field we obtain
\begin{align}
\Big\langle \boldsymbol{B}(\boldsymbol{X}+\boldsymbol{r},s)\Big\rangle_{{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}=\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{r})\boldsymbol{\cdot}[\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{0})]^{-1}\boldsymbol{\cdot b},
\end{align}
where
\begin{align}
[\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{0})]^{-1}=\frac{\mathbf{I}}{\langle\phi^2\rangle f_\phi^{''}(0)}.
\end{align}
The resulting expression
\begin{align}
\Big\langle\frac{\partial^2}{\partial\boldsymbol{X}\partial\boldsymbol{X}}{\boldsymbol{B}}(\boldsymbol{X},s)\Big\rangle_{{\boldsymbol{B}}(\boldsymbol{X},s)=\boldsymbol{b}}=\lim_{r\to 0}\Bigg(\frac{\partial^2}{\partial\boldsymbol{r}\partial\boldsymbol{r}}\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{r})\Bigg)\boldsymbol{\cdot}[\boldsymbol{R}^{\boldsymbol{B}}(\boldsymbol{0})]^{-1}\boldsymbol{\cdot b},
\end{align}
may then be computed using \eqref{Bcov}, and when the result is evaluated at $\boldsymbol{b}=\boldsymbol{\mathcal{B}}$ and substituted into \eqref{Defappx3}, the closed expression obtained is
\begin{align}
\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}&\approx \delta_\mathcal{B}\Big(\boldsymbol{\mathcal{C}}_R^{-1}\boldsymbol{\cdot\mathcal{B}}+\boldsymbol{\mathcal{C}}_R^{-\top}\boldsymbol{\cdot\mathcal{B}} +\mathrm{tr}(\boldsymbol{\mathcal{C}}_R^{-1})\boldsymbol{\mathcal{B}}\Big),\label{closed_B_term}\\
\delta_\mathcal{B}&\equiv\frac{\kappa}{3}\frac{f_\phi^{''''}(0)}{f_\phi^{''}(0)},
\end{align}
where $\boldsymbol{\mathcal{C}}_R^{-1}\equiv\boldsymbol{\mathcal{D}}^{-1}\boldsymbol{\cdot}\boldsymbol{\mathcal{D}}^{-\top}$ is the inverse of the right Cauchy-Green tensor.
In the work of \cite{johnson2016closure}, the coefficient analogous to $\delta_\mathcal{B}$ in the closure for $\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}$, namely $\delta_\mathcal{A}$, was estimated based on the enstrophy production-dissipation balance at steady-state. The same procedure can be applied to approximate $\delta_\mathcal{B}$ based on the steady-state production-dissipation balance $\langle\boldsymbol{S:BB}\rangle=-\kappa\langle\|\boldsymbol{\nabla B}\|^2 \rangle $, leading to
\begin{align}
\delta_\mathcal{B}&\approx \frac{1}{9\tau_\eta}\mathrm{tr}(\boldsymbol{\mathcal{C}}_L)\gamma_\mathcal{B},\label{deltaBappx}\\
\gamma_\mathcal{B}&\equiv \tau_\eta\frac{\langle\boldsymbol{S:BB}\rangle}{\langle \|\boldsymbol{B}\|^2\rangle},
\end{align}
where $\boldsymbol{\mathcal{C}}_L\equiv\boldsymbol{\mathcal{D}}\boldsymbol{\cdot}\boldsymbol{\mathcal{D}}^\top$ is the left Cauchy-Green tensor.
Whereas the RDGF model for $\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}$ is nonlinear in $\boldsymbol{\mathcal{A}}$ due to the contributions from $\boldsymbol{\mathcal{C}}_L$ and $\boldsymbol{\mathcal{C}}_R$, the closure for $ \langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ in \eqref{closed_B_term} is linear in $\boldsymbol{\mathcal{B}}$. This could potentially suggest an issue with \eqref{closed_B_term} since it is known that models for $\langle\nabla^2\boldsymbol{A}\rangle_{\boldsymbol{\mathcal{A}}}$ that are linear in $\boldsymbol{\mathcal{A}}$ can suffer from finite-time singularities, depending on the initial conditions \citep{martin1998dynamics}. However, given that the production term in the equation for $\boldsymbol{\mathcal{B}}$ is $-\boldsymbol{\mathcal{A}}^\top\boldsymbol{\cdot\mathcal{B}}$, then the dependence of the closure in \eqref{closed_B_term} on $\boldsymbol{\mathcal{C}}_L$ and $\boldsymbol{\mathcal{C}}_R$ may indirectly prevent singular growth of $\|\boldsymbol{\mathcal{B}}\|$, at least in regions where this growth is associated with large values of $\|\boldsymbol{\mathcal{A}}\|$, because in those regions the growth of $\boldsymbol{\mathcal{B}}$ could be modulated through $\boldsymbol{\mathcal{C}}_L$ and $\boldsymbol{\mathcal{C}}_R$. Moreover, we performed tests where in the closure for $ \langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$, we set $\boldsymbol{\mathcal{D}}=\mathbf{I}$, i.e.~so that the recent deformation mapping was removed. Simulations of the model using this blew up, showing that the contributions in \eqref{closed_B_term} involving $\boldsymbol{\mathcal{D}}$ do indeed play a key indirect role in preventing singular growth of $ \boldsymbol{\mathcal{B}}$.
In view of \eqref{deltaBappx}, any dependence of the closed expression for $\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ on $\kappa$ is contained within $\gamma_\mathcal{B}$, and this term must be specified using data. However, in order for the model to depend upon $\kappa$ and hence $Sc$, then the data must specify $\gamma_\mathcal{B}$ as a function of $Sc$, which is not desirable. An alternative approach is as follows: The closure approximation in \eqref{closed_B_term} is linear in $\boldsymbol{\mathcal{B}}$, and $1/\delta_\mathcal{B}$ may be regarded as a linear relaxation timescale for $\boldsymbol{\mathcal{B}}$, describing how the diffusion term $\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ causes $\boldsymbol{\mathcal{B}}$ to relax to its equilibrium value. In view of this, $1/\delta_\mathcal{B}$ should scale with the small-scale scalar timescale $\tau_\phi$, and for $Sc\geq 1$ this timescale can be estimated using the Batchelor scale \citep{batchelor1959small,donzis2005scalar} as $\tau_\phi\sim \tau_\eta Sc^{-1/3}$, while for $Sc<1$ the Corrsin scale \citep{corrsin1951spectrum} may be used to obtain $\tau_\phi\sim \tau_\eta Sc^{-1/2}$ . In view of this, the $Sc$ dependence may be explicitly accounted for in $\delta_\mathcal{B}$ by using
\begin{align}
\delta_\mathcal{B}&\approx \frac{Sc^{-\xi}}{9\tau_\eta}\mathrm{tr}(\boldsymbol{\mathcal{C}}_L)\alpha_\mathcal{B},\label{deltaBappx2}\\
\alpha_\mathcal{B}&\equiv \tau_\eta\frac{\langle\boldsymbol{S:BB}\rangle}{\langle \|\boldsymbol{B}\|^2\rangle}\Bigg\vert_{Sc=1},\label{alphaB}
\end{align}
where $\xi=1/2$ for $Sc<1$, and $\xi=1/3$ for $Sc\geq 1$. From our DNS (see \S\ref{NSims}) we obtain the value $\alpha_\mathcal{B}\approx -0.32$. However, in anticipation of results to be shown later, we note that using this fixed value for $\alpha_B$ yields a model whose predictions are not accurate (see figure \ref{PDF_B2}). Therefore, a modification will be introduced wherein $\alpha_\mathcal{B}$ is specified based on the local scalar gradient production along the trajectory history of the particle, as in equation \eqref{alphaBnew}, which dramatically improves the predictions from the model (cf.~figure \ref{PDF_B2_new}).
Analogous to the equation for $\boldsymbol{\mathcal{A}}$, the forcing term in the scalar gradient equation is chosen to be $\langle\boldsymbol{F_B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}dt=\sigma d\boldsymbol{\mathcal{W}}$, where now $d\boldsymbol{\mathcal{W}}$ is a vector-valued Wiener process with increments defined by
\begin{align}
\langle d\boldsymbol{\mathcal{W}}\rangle&=\boldsymbol{0},\\
\langle d\boldsymbol{\mathcal{W}}d\boldsymbol{\mathcal{W}}\rangle&=\mathbf{I}dt.
\end{align}
Since the equation for $\boldsymbol{\mathcal{B}}$ is linear, the statistics scale with the forcing amplitude $\sigma$, and therefore $\sigma$ may be chosen arbitrarily if the results generated by the model are suitably normalized. Using this forcing term, the final form of the model equation is written as a stochastic differential equation
\begin{align}
d\boldsymbol{\mathcal{B}}&\approx-\boldsymbol{\mathcal{A}}^\top\boldsymbol{\cdot\mathcal{B}}dt+\delta_\mathcal{B}\Big(\boldsymbol{\mathcal{C}}_R^{-1}\boldsymbol{\cdot\mathcal{B}}+\boldsymbol{\mathcal{C}}_R^{-\top}\boldsymbol{\cdot\mathcal{B}} +\mathrm{tr}(\boldsymbol{\mathcal{C}}_R^{-1})\boldsymbol{\mathcal{B}}\Big)dt+\sigma d\boldsymbol{\mathcal{W}}.\label{Beq_closed_forcing}
\end{align}
For applications where there is a mean scalar gradient, the forcing term could be replaced by the term describing the production of fluctuating scalar gradients due to the imposed mean scalar gradient.
Finally, just as the ML-RDGF model replaces the constant timescale for $\boldsymbol{\mathcal{A}}^{[N]}$ with the fluctuating timescale $\tau_N(t)$, the Kolmogorov timescale $\tau_\eta$ that appears in \eqref{deltaBappx2} should also for consistency be replaced by $\tau_N(t)$ (since the ML-RDGF model is being used to specify $\boldsymbol{\mathcal{A}}$ in \eqref{Beq_closed_forcing}). In other words \eqref{deltaBappx2} is to be replaced by
\begin{align}
\delta_\mathcal{B}&\approx \frac{Sc^{-\xi}}{9\tau_N(t)}\mathrm{tr}(\boldsymbol{\mathcal{C}}_L)\alpha_\mathcal{B}.\label{deltaBappx3}
\end{align}
This ensures that for $Sc=1$, the local timescale on which both $\boldsymbol{\mathcal{B}}$ and $\boldsymbol{\mathcal{A}}$ fluctuate is $ O(\tau_N(t))$.
\section{Numerical Simulations}\label{NSims}
We test the model predictions against data from Direct Numerical Simulation (DNS) of a passive scalar field advected by an incompressible, three-dimensional, statistically steady and isotropic turbulent velocity field.
The DNS code uses a standard Fourier pseudo-spectral method \citep{canuto1988spectral} to solve the discretized Navier-Stokes and passive scalar equations on a triply-periodic cubic domain. The required Fourier transforms are executed in parallel using the P3DFFT library \citep{pekurovsky2012p3dfft} and the aliasing error is removed via the $3/2$ rule \citep{canuto1988spectral}.
The code is described in further detail in \cite{Carbone2019}.
The non-dimensional governing equation for the passive scalar in Fourier space reads
\begin{align}
\partial_t \hat{\phi} + \textrm{i} \boldsymbol{k}\bcdot\widehat{\boldsymbol{u}\phi} =
-\kappa\|\boldsymbol{k}\|^2\hat{\phi}
+ \hat{F},
\label{eq_scal_field}
\end{align}
where the hat indicates a Fourier transform, ``$\textrm{i}$'' is the imaginary unit, $\boldsymbol{u}(\boldsymbol{x},t)$ is the turbulent velocity field, $\phi(\boldsymbol{x},t)$ is the passive scalar field and $\hat{\phi}(\boldsymbol{k},t)$ denotes its spatial Fourier transform.
For consistency with the stochastic model, the passive scalar field is driven by a stochastic forcing, such that the complex forcing $\hat{F}$ is Gaussian and white in time, with correlation
\begin{align}
\left\langle \hat{F}(\boldsymbol{k},t)\hat{F}(\boldsymbol{k}',t') \right\rangle = 2\sigma_0^2 \|\boldsymbol{k}\|^2 \delta(\boldsymbol{k}+\boldsymbol{k}')\delta(t-t'), \; 0<\|\boldsymbol{k}\|<k_f.
\label{eq_forcing_correlations}
\end{align}
The forcing is confined to the wavevectors $\boldsymbol{k}$ within a sphere of radius $k_f$, and we choose $k_f=\sqrt{7}$ since it yields large-scale statistics that are close to being isotropic.
Finally, the spectrum of the forcing $|\hat{F}|^2(k)$ scales as $\|\boldsymbol{k}\|^2$, compatible with energy equipartition among the smallest Fourier modes.
The constant parameter $\sigma_0$ in \eqref{eq_forcing_correlations} regulates the dissipation rate and it is adjusted to yield an appropriate Kolmogorov scale $\eta$ prescribed by the spatial resolution requirements.
We simulate a flow with $\Rey_\lambda=100$ and $Sc=1$. The spatial resolution is $k_{\max}\eta\simeq 3$, with $k_{\max}=N/2$ being the maximum resolved wavenumber, for all three simulations.
The time integration of equation \eqref{eq_scal_field} is performed by means of a second-order Runge-Kutta scheme designed for stochastic differential equations \citep{Honeycutt1992} and the CFL number stays below 0.3.
Regarding the numerical simulations of the model, we solve the scalar gradient equation \eqref{Beq_closed_forcing} through a second-order predictor-corrector method \citep{kloeden2018numerical} with a time step $dt=0.05\tau_{\eta}$. Each level of the ML-RDGF model equation was solved using its own appropriate time step, namely, level $n$ was solved using $dt=0.05\langle \tau_n(t)\rangle$. Tests were performed using smaller time steps and these tests indicated that the aforementioned time steps were small enough to achieve convergence of the results.
We numerically solve the model equation \eqref{Beq_closed_forcing} for $\Rey_\lambda=100, 300, 500$ and the results from these will be referred to as M1, M2, and M3 in the results section. While M1 is designed to match the DNS, M2 and M3 will be used to explore the model's ability to capture the effect of $\Rey_\lambda$ on the scalar gradient statistics.
\section{Results and Discussion}
\subsection{\label{sec:level2_A} Velocity Gradients}
In our scalar gradient model, the velocity gradients are specified using the ML-RDGF model\citep{johnson2017turbulence}. We therefore begin by comparing the predictions of this model against the DNS data in order to assess its accuracy in predicting the statistics $\boldsymbol{\mathcal{A}}$, since any inaccuracies in this model will in turn lead to inaccuracies in our model for the scalar gradients.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim = 5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/A11_PDF_all_Re_lambda_withDNS.pdf}
\put(70,0){$a_{11}/a_{11,rms}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\centering
\subfloat[]
{\hspace{2mm}\begin{overpic}
[trim = 5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/A12_PDF_all_Re_lambda_withDNS.pdf}
\put(70,0){$a_{12}/a_{12,rms}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\caption{PDFs of (a) longitudinal $a_{11}/a_{11,rms}$ and (b) transverse $a_{12}/a_{12,rms}$ velocity gradients.}
\label{PDF_aij}
\end{figure}}
In Figure \ref{PDF_aij} we compare the PDFs of the longitudinal and transverse components of the velocity gradient predicted by the ML-RDGF model with the DNS results. (Recall that $\boldsymbol{a}$ and $\boldsymbol{b}$ are the phase-space variables conjugate to $\boldsymbol{\mathcal{A}}$ and $\boldsymbol{\mathcal{B}}$, respectively, and we use the subscripts ``$rms$'' and ``$av$'' to denote the root-mean-square and averaged values of the variable under consideration). One can see that the PDFs for the longitudinal gradients (Figure \ref{PDF_aij}(a)) are negatively skewed while the PDFs for the transverse gradients are symmetric (Figure \ref{PDF_aij}(b)). The former is associated with the self-amplification of the velocity gradients \citep{tsinober}, while the latter is a constraint due to isotropy. The predictions from M1 are in excellent agreement with the DNS data, and the model makes realistic predictions at the higher $\Rey_\lambda$ to which M2 and M3 correspond. The ability of the ML-RDGF model to accurately predict the components of the velocity gradient, and realistic intermittency trends with increasing $\Rey_\lambda$ (and over a much larger range of $\Rey_\lambda$ than considered here) were previously demonstrated in detail in the original ML-RDGF paper of \cite{johnson2017turbulence}.
Further insight into the ability of the ML-RDGF model to predict the velocity gradient dynamics can be obtained by considering its predictions for the velocity gradient invariants $Q\equiv-\boldsymbol{a}\boldsymbol{:a}/2$ and $R\equiv-(\boldsymbol{a \cdot} \boldsymbol{a}) \boldsymbol{:a}/3$. The invariant $Q$ measures the relative strength of the local strain rate and vorticity in the flow, with $Q>0$ denoting vorticity-dominated regions of the flow, while $R$ measures the relative importance of the local strain self-amplification and enstrophy production, with $R<0$ denoting regions dominated by enstrophy production \citep{tsinober}. The joint PDFs of $Q$ and $R$ from the ML-RDGF model for M1, M2, M3 are presented in Figure \ref{PDF_QR_A}, along with the DNS results for comparison. As observed previously \citep{johnson2017turbulence}, the ML-RDGF captures the main features of the $Q,R$ joint-PDF well, including the signature sheared-drop shape of the joint-PDF, which is preserved by the model as $\Rey_\lambda$ is increased. The joint-PDF contours extend along the right Viellefosse tail, and the PDF is concentrated in the quadrants $Q>0, R<0$ and $Q<0, R>0$, consistent with previous studies \citep{johnson2016closure, johnson2017turbulence}. For M1, the joint-PDF is more compact compared to the DNS, indicating that the model underpredicts the probability of large values of $Q,R$. The agreement between the ML-RDGF model and DNS data was shown to be better in \cite{johnson2017turbulence}, however, their DNS data was for $\Rey_\lambda=430$, indicating that the quantitive accuracy of the model is better at higher $\Rey_\lambda$. Considering the results for M2 and M3 shows that as $\Rey_\lambda$ increases, the contours of the joint-PDF spread out in the $Q,R$ phase-space, showing that the model captures the effect of $\Rey_\lambda$ on the intermittency in the flow.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_2_JPDF_DNS.pdf}
\put(75,-1){$R/\tilde{Q}_{av}^{3/2}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim =0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_2_JPDF_Re_lambda_100.pdf}
\put(75,-1){$R/\tilde{Q}_{av}^{3/2}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}\\
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_2_JPDF_Re_lambda_300.pdf}
\put(75,-1){$R/\tilde{Q}_{av}^{3/2}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_2_JPDF_Re_lambda_500.pdf}
\put(75,-1){$R/\tilde{Q}_{av}^{3/2}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\caption{Logarithm of joint-PDFs of $Q/\tilde{Q}_{av}$ and $R/\tilde{Q}_{av}^{3/2}$ (where $\tilde{Q}_{av}\equiv\langle\|\mathcal{A}\|^2\rangle$) from (a) DNS, (b) M1, (c) M2, (d) M3. Colours indicate the values of the logarithm of the PDF.}
\label{PDF_QR_A}
\end{figure}}
A more careful and quantitative comparison between the model predictions and DNS data for $Q,R$ can be made by comparing the PDFs of $Q$ and $R$ separately. In Figure \ref{PDF_QR}(a), one can see that in the DNS, the PDF of $Q$ is strongly positively skewed, which is associated with the fact that the vorticity is more intermittent than the strain-rate in turbulent flows \citep{yeung18}. By contrast, the ML-RDGF model predicts PDFs for $Q$ that are much more symmetric, with the model significantly underpredicting regions of intense vorticity, and slightly underpredicting regions of intense straining. These discrepancies are, however, mainly in the tails of the PDF, with the model predictions in good agreement with the DNS for values of the PDF that are $\geq O(10^{-2})$.
In Figure \ref{PDF_QR}(b) it is seen that the model also predicts PDFs of $R$ that are relatively symmetric compared with the DNS data for which the PDF is negatively skewed. The model prediction for M2, which corresponds to $\Rey_\lambda=300$, is much closer to the DNS data than those of M1 which has the same $\Rey_\lambda=100$ as the DNS.
In view of these results, it is seen that while the ML-RDGF model predicts the components of the velocity gradients very accurately (as well as the alignments between the vorticity and strain-rate eigenvectors\citep{johnson2017turbulence}), it does not predict the invariants of the velocity gradients accurately when compared with the DNS, at least not for the $\Rey_\lambda$ considered here. This could in turn lead to inaccuracies in the scalar gradient model, above and beyond any arising from the closure approximations for the scalar gradient diffusion term. These results point to the need for further refinements in the ML-RDGF model.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim = 5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_PDF_all_Re_lambda.pdf}
\put(80,-2){$Q/\tilde{Q}_{av}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\centering
\subfloat[]
{\hspace{2mm}\begin{overpic}
[trim = 5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/x_2_PDF_all_Re_lambda.pdf}
\put(80,-2){$R/\tilde{Q}_{av}^{3/2}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\caption{PDFs of (a) $Q/\tilde{Q}_{av}$ and (b) $R/\tilde{Q}_{av}^{3/2}$, where $\tilde{Q}_{av}\equiv\langle\|\boldsymbol{\mathcal{A}}\|^2\rangle$.}
\label{PDF_QR}
\end{figure}}
\subsection{\label{sec:level2_SG} Scalar Gradients}
We now turn to consider the predictions from our new model for the scalar gradients. In Figure \ref{PDF_B2}, we plot the PDFs of $Q_b/Q_{b,av}$ (where $Q_b\equiv\|\boldsymbol{b}\|^2$) which is proportional to the scalar dissipation rate $\epsilon_\phi\equiv \kappa\|\boldsymbol{b}\|^2$, as well as the PDF of $b_1$, the scalar gradient component in one of the (arbitrary, due to isotropy) directions. Concerning the PDF of $Q_b/Q_{b,av}$, the results show that while the model is in good qualitative agreement with the DNS data, capturing the slowly decaying tail of the PDF, it significantly underpredicts the values of the PDF. The model predictions for the PDF of $b_1$ are also in significant error, underpredicting small to intermediate values of $b_1/b_{1,rms}$, and significantly overpredicting large values of $b_1/b_{1,rms}$, such that the overall shape of the PDF is not well captured by the model.
An investigation into the cause of these significant underpredictions revealed that the problem is due to the model generating extremely large values of $\|\boldsymbol{\mathcal{B}}(t)\|^2$. An example of the time series of $\|\boldsymbol{\mathcal{B}}(t)\|^2/\langle \|\boldsymbol{\mathcal{B}}(t)\|^2\rangle$ generated by the model at $\Rey_\lambda=100$ is shown in Figure \ref{time_series}, together with the time series of $\|\boldsymbol{\mathcal{S}}(t)\|^2/\langle \|\boldsymbol{\mathcal{S}}(t)\|^2\rangle$ for comparison. Although the signal $\|\boldsymbol{\mathcal{S}}(t)\|^2$ exhibits significant fluctuations about the mean, $\|\boldsymbol{\mathcal{B}}(t)\|^2$ exhibits infrequent but enormous fluctuations about the mean, which only get stronger as $\Rey_\lambda$ is increased. Although $\|\boldsymbol{\mathcal{B}}(t)\|^2$ would be expected to be more intermittent than $\|\boldsymbol{\mathcal{S}}(t)\|^2$, one would not anticipate intermittent fluctuations in $\|\boldsymbol{\mathcal{B}}(t)\|^2$ as large as these, nor are they manifested in the DNS data, and therefore they seem to indicate an issue with the model. Since the integral of the PDF of $Q_b$ over its sample-space is one, then because the model vastly overpredicts the probability of extremely large values of $\|\boldsymbol{\mathcal{B}}(t)\|^2$, it underpredicts the probability of values in the sample-space range shown in figure \ref{PDF_B2}.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/x_3_PDF_all_Re_lambda_Fixed_alpha.pdf}
\put(80,-2){$Q_b/Q_{b,av}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\subfloat[]
{\hspace{2mm}\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/b1_PDF_all_Re_lambda_withDNS_FIXED_alpha.pdf}
\put(80,-2){$b_1/b_{1,rms}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\caption{PDFs of (a) $Q_b/Q_{b,av}$, (b) $b_1/b_{1,rms}$. The results from the model are obtained with the (uncorrected) model coefficient $\alpha_\mathcal{B}$ specified by equation \eqref{alphaB}.}
\label{PDF_B2}
\end{figure}}
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/time_series_B2.pdf}
\put(80,2){$t/\tau_\eta$}
\put(-15,40){\rotatebox{90}{$\|\boldsymbol{\mathcal{B}}(t)\|^2/\langle \|\boldsymbol{\mathcal{B}}(t)\|^2\rangle$}}
\end{overpic}}
\subfloat[]
{\hspace{2mm}\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/time_series_S2.pdf}
\put(80,2){$t/\tau_\eta$}
\put(-5,40){\rotatebox{90}{$\|\boldsymbol{\mathcal{S}}(t)\|^2/\langle \|\boldsymbol{\mathcal{S}}(t)\|^2\rangle$}}
\end{overpic}}
\caption{Time series of $\|\boldsymbol{\mathcal{B}}(t)\|^2$ and $\|\boldsymbol{\mathcal{S}}(t)\|^2$, normalized by their mean values, generated from the model with $\Rey_\lambda=100$ and the (uncorrected) coefficient $\alpha_\mathcal{B}$ specified by equation \eqref{alphaB}.}
\label{time_series}
\end{figure}}
That the model vastly overpredicts the probability of extremely large values of $\|\boldsymbol{\mathcal{B}}(t)\|^2/\langle \|\boldsymbol{\mathcal{B}}(t)\|^2\rangle$ must be due to deficiencies in the closure for $\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$. In particular, the values of $\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ predicted by the closure approximation when $\|\boldsymbol{\mathcal{B}}\|$ is large are too small to sufficiently counteract the scalar production term (although apparently, they are large enough to prevent singularities in the model since simulations of the model do not blow up). A relatively simple modification to the model to address this deficiency is to modify its specification of the coefficient $\alpha_\mathcal{B}$ in \eqref{alphaB} so that it includes information on the locally averaged scalar production, rather than simply the global mean value. This is achieved by replacing \eqref{alphaB} with
\begin{align}
\alpha_\mathcal{B}(t)&= \tau_\eta\int^t_{t-\tau_1} \boldsymbol{\mathcal{S}}(t')\boldsymbol{:}\boldsymbol{\mathcal{B}}(t')\boldsymbol{\mathcal{B}}(t')\,dt'\Bigg/\int^t_{t-\tau_1} \|\boldsymbol{\mathcal{B}}(t')\|^2\,dt',\label{alphaBnew}
\end{align}
which can be computed when solving the model since it only depends on $\boldsymbol{\mathcal{S}}$ and $\boldsymbol{\mathcal{B}}$ at previous times, which are known. With this, the global average involved in \eqref{alphaB} is replaced with a local time average over the trajectory history of the particle. The time integral is chosen to span $[t-\tau_1,t]$ in view of the fact that $\boldsymbol{\mathcal{S}}\boldsymbol{:}\boldsymbol{\mathcal{B}}\boldsymbol{\mathcal{B}}$ and $\|\boldsymbol{\mathcal{B}}\|^2$ have timescales on the order of the integral timescale, which in the ML-RDGF is specified by $\tau_1$.
The advantage of using \eqref{alphaBnew} is that the coefficient $\delta_\mathcal{B}$ in \eqref{Beq_closed_forcing} will then depend upon the local scalar gradient dynamics, and in regions where the production of $\|\boldsymbol{\mathcal{B}}\|^2$ is large, $\delta_\mathcal{B}$ will also be large relative to its value in regions where the production of $\|\boldsymbol{\mathcal{B}}\|^2$ is small. In other words, using \eqref{alphaBnew} introduces nonlinearity into the closure for $\kappa\langle\nabla^2\boldsymbol{B}\rangle_{\boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{B}}}$ with respect to its dependence on $\boldsymbol{\mathcal{B}}$, and this may help oppose the extremely large fluctuations predicted by the original form of the model. In practice, since \eqref{alphaBnew} requires time-history information, the model is solved for $t\leq \tau_1$ using \eqref{alphaB}, and then for $t>\tau_1$, \eqref{alphaBnew} is used.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/x_3_PDF_all_Re_lambda.pdf}
\put(80,-2){$Q_b/Q_{b,av}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\subfloat[]
{\hspace{2mm}\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/b1_PDF_all_Re_lambda_withDNS.pdf}
\put(80,-2){$b_1/b_{1,rms}$}
\put(-13,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\caption{PDFs of (a) $Q_b/Q_{b,av}$, (b) $b_1/b_{1,rms}$, based on using \eqref{alphaBnew} instead of \eqref{alphaB} to specify $\alpha_\mathcal{B}$ in the model.}
\label{PDF_B2_new}
\end{figure}}
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/x_4_PDF_all_Re_lambda.pdf}
\put(73,-2){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(-10,60){\rotatebox{90}{PDF}}
\put(122,115){DNS}
\put(122,107){M1}
\put(122,98){M2}
\put(122,89){M3}
\end{overpic}}
\subfloat[]
{\hspace{2mm}\begin{overpic}
[trim =5mm 60mm 0mm 60mm,scale=0.3,clip,tics=20]{Figures/CosineAngles_PDF_Re_lambda_100_withDNS.pdf}
\put(80,-2){$\boldsymbol{e}_b\boldsymbol{\cdot}\boldsymbol{e}_i$}
\put(-10,60){\rotatebox{90}{PDF}}
\put(40,116){DNS, $i=1$}
\put(40,107){DNS, $i=2$}
\put(40,98){DNS, $i=3$}
\put(40,90){M1, $i=1$}
\put(40,81){M1, $i=2$}
\put(40,72){M1, $i=3$}
\end{overpic}}
\caption{PDFs of (a) $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$, (b) the inner product between the unit vector $\boldsymbol{e}_b\equiv\boldsymbol{b}/\|\boldsymbol{b}\|$ and the eigenvectors $\boldsymbol{e}_i$ of $\boldsymbol{\mathcal{S}}$.}
\label{PDF_x4_align}
\end{figure}}
Figure \ref{PDF_B2_new} once again shows the PDFs of $Q_b$ and $b_1$, but this time using \eqref{alphaBnew} instead of \eqref{alphaB} to specify $\alpha_\mathcal{B}$ in the model. Comparing the results to those in figure \ref{PDF_B2}, it can be seen that the new specification of $\alpha_\mathcal{B}$ dramatically improves the predictions from the model, being now in excellent agreement with the DNS data. The results also show the impact of $\Rey_\lambda$ on $Q_b$ as predicted by the model, with the probability of intermediate values of $Q_b/Q_{b,av}$ and $b_1/b_{1,rms}$ predicted to decrease as $\Rey_\lambda$ increases, while the probability of large $Q_b/Q_{b,av}$ and $b_1/b_{1,rms}$ increases as $\Rey_\lambda$ increases. However, further tests of the model revealed that when $\Rey_\lambda$ is increased much beyond $\Rey_\lambda=500$ the predictions of the model become unrealistic, with extremely large values of the scalar gradient occurring in the model, and the model can even blow up. Therefore, although the use of \eqref{alphaBnew} dramatically improves the performance of the model over the range of $\Rey_\lambda$ considered, it is not sufficient to guarantee that the model makes reasonable predictions for arbitrarily large $\Rey_\lambda$. An investigation into the causes of the failure of the model at high $\Rey_\lambda$ and possible remedies for this are left to future work.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_3_JPDF_DNS.pdf}
\put(75,-1){$Q_b/Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_3_JPDF_Re_lambda_100.pdf}
\put(75,-1){$Q_b/Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}\\
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_3_JPDF_Re_lambda_300.pdf}
\put(75,-1){$Q_b/Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_3_JPDF_Re_lambda_500.pdf}
\put(75,-1){$Q_b/Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\caption{Logarithm of joint-PDFs of $Q/\tilde{Q}_{av}$ and $Q_b/Q_{b,av}$ from (a) DNS, (b) M1, (c) M2, (d) M3. Colors indicate the values of the logarithm of the PDF.}
\label{PDF_Q_Qb}
\end{figure}}
In figure \ref{PDF_x4_align}(a) we show the PDF of the scalar gradient production $R_b\equiv \boldsymbol{a:bb}$. The results show that the model predictions are in good agreement with the DNS data, with some underpredictions for the largest fluctuations. The model captures the strong negative skewness of the PDF that is associated with the predominance of scalar gradient production over destruction. The model also predicts the largest fluctuations in $R_b$ become more probable as $\Rey_\lambda$ is increased due to intermittency in the flow. In figure \ref{PDF_x4_align}(b) we show the PDF of the inner product between the unit vector $\boldsymbol{e}_b\equiv\boldsymbol{b}/\|\boldsymbol{b}\|$ and the eigenvectors $\boldsymbol{e}_i$ (corresponding to the ordered eigenvalues) of $\boldsymbol{\mathcal{S}}$. The model predicts these non-trivial alignments very well, capturing the strong preferential alignment with the compressional eigendirection $\boldsymbol{e}_3$, and misalignment with the intermediate eigendirection $\boldsymbol{e}_2$ and extensional eigendirection $\boldsymbol{e}_1$. However, the model predicts a misalignment with $\boldsymbol{e}_1$ that is a little too strong, and a misalignment with $\boldsymbol{e}_2$ that is a little too weak. Only the results for M1 are shown, as the results from M2 and M3 are almost identical. The current model predictions for these alignment PDFs are in much better agreement with the DNS data than those of the model of \cite{martin2005joint} which uses a much more simplistic closure approximation for the scalar diffusion term.
\subsection{\label{sec:level2_JPDF}Joint-PDFs of Velocity and Scalar Gradients}
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_4_JPDF_DNS.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_4_JPDF_Re_lambda_100.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}\\
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_4_JPDF_Re_lambda_300.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_1_4_JPDF_Re_lambda_500.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,55){\rotatebox{90}{$Q/\tilde{Q}_{av}$}}
\end{overpic}}
\caption{Logarithm of joint-PDFs of $Q/\tilde{Q}_{av}$ and $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$ from (a) DNS, (b) M1, (c) M2, (d) M3. Colors indicate the values of the logarithm of the PDF.}
\label{PDF_Q_Rb}
\end{figure}}
We now turn to consider the relationship between the velocity and scalar gradients predicted by the model. In Figure \ref{PDF_Q_Qb} we show the joint-PDFs of the velocity gradient invariant $Q/\tilde{Q}_{av}$ and scalar invariant $Q_b/Q_{b,av}$ from the DNS and model. For M1, there is an excellent qualitative agreement with the DNS data, with the model capturing the elongation of the PDF (note however that the appearance of the elongation is somewhat exaggerated due to the different axis ranges) along the horizontal axis toward regions of large $Q_b/Q_{b,av}$, indicating that large fluctuations in the scalar gradients are much more probable than they are for the velocity gradients. The model also captures the exponential-like behaviour of the isocontours of the PDF, whose shape indicates that large values of $Q_b/Q_{b,av}$ tend to occur in regions where $Q/\tilde{Q}_{av}$ is small, and vice-versa. The quantitative errors in M1 are mainly associated with the variation of the joint-PDF along the $Q/\tilde{Q}_{av}$ axis, which can be understood in terms of the ML-RGDF's underprediction of large fluctuations of $Q/\tilde{Q}_{av}$ at $\Rey_\lambda=100$, as already observed when considering the PDF of $Q/\tilde{Q}_{av}$ in figure \ref{PDF_QR}. Comparing the results from M1, M2, and M3 shows that the model predicts that as $\Rey_\lambda$ is increased the shape of the joint-PDF is preserved, but becomes stretched along the two axes. Correspondingly the probability of regions with comparable values of $Q/\tilde{Q}_{av}$ and $Q_b/Q_{b,av}$ is predicted to decrease as $\Rey_\lambda$ increases.
{\vspace{0mm}\begin{figure}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_3_4_JPDF_DNS.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,50){\rotatebox{90}{$Q_b/Q_{b,av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_3_4_JPDF_Re_lambda_100.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,50){\rotatebox{90}{$Q_b/Q_{b,av}$}}
\end{overpic}}\\
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_3_4_JPDF_Re_lambda_300.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,50){\rotatebox{90}{$Q_b/Q_{b,av}$}}
\end{overpic}}
\centering
\subfloat[]
{\begin{overpic}
[trim = 0mm 60mm 10mm 60mm,scale=0.3,clip,tics=20]{Figures/x_3_4_JPDF_Re_lambda_500.pdf}
\put(70,-1){$R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$}
\put(8,50){\rotatebox{90}{$Q_b/Q_{b,av}$}}
\end{overpic}}
\caption{Logarithm of joint-PDFs of $Q_b/Q_{b,av}$ and $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$ from (a) DNS, (b) M1, (c) M2, (d) M3. Colors indicate the values of the logarithm of the PDF.}
\label{PDF_Qb_Rb}
\end{figure}}
In Figure \ref{PDF_Q_Rb} we show the joint-PDFs of the velocity gradient invariant $Q/\tilde{Q}_{av}$ and scalar production invariant $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$ from the DNS and model. This joint PDF provides insights into how straining and vortical regions of the flow might contribute differently to the scalar gradient production. The model accurately reproduces the qualitative behaviour of the PDF seen in the DNS data, including the elongation of the PDF along the $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}<0$ direction, associated with the predominance of scalar gradient production over destruction. The results also show that events with strong scalar gradient production are more probable in regions where $Q/\tilde{Q}_{av}<0$, despite the fact that $Q/\tilde{Q}_{av}$ itself has a positively skewed PDF. This is because the antisymmetric part of the velocity gradient, and therefore the vorticity, does not directly contribute to the invariant $R_b$, but only contributes indirectly through its impact on the local alignment of $\boldsymbol{\mathcal{B}}(t)$ with $\boldsymbol{\mathcal{S}}(t)$, whereas the strain-rate directly affects $R_b$. The model underpredicts the probability of the largest fluctuations, which again is principally due to the ML-RDGF underpredicting the intermittency of $Q/\tilde{Q}_{av}$. Comparing the results from M1, M2, and M3 shows that the model predicts that as $\Rey_\lambda$ is increased the shape of the joint-PDF is largely preserved, except for being stretched along the axes due to the increased intermittency of the flow. The model does seem to predict, however, that the overall probability of the system being in the quadrant $Q<0,R_b>0$ reduces as $\Rey_\lambda$ increases. Future comparisons with DNS at higher $\Rey_\lambda$ will be needed to assess the accuracy of this prediction. It is possible that this is a defect in the model that is in some way related to the failure of the model at higher $\Rey_\lambda$.
Finally, in Figure \ref{PDF_Qb_Rb} we show the joint-PDFs of the scalar gradient invariant $Q_b/Q_{b,av}$ and scalar production invariant $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$. The prediction of the model for M1 is in very good agreement with the DNS, both qualitatively and quantitatively, with only small deviations. As with the other joint-PDFs, comparing the results from M1, M2, and M3 shows that the model predicts that as $\Rey_\lambda$ is increased the shape of the joint-PDF of $Q_b/Q_{b,av}$ and $R_b/\tilde{Q}_{av}^{1/2}Q_{b,av}$ is preserved, except for being stretched along the axes due to the increased intermittency of the flow.
\FloatBarrier
\section{Conclusions}
A Lagrangian model for passive scalar gradients in isotropic turbulence has been developed, with the scalar gradient diffusion term closed using the RDGF approach, which has recently been very successfully applied to close the equation for the fluid velocity gradients along fluid particle trajectories. This closure yields a diffusion term that is nonlinear in the velocity gradients, but linear in the scalar gradients, and comparisons of the statistics generated by the closed model with DNS data revealed large errors. An investigation revealed that these large errors were due to the scalar gradient model generating erroneously large fluctuations, possibly due to the diffusion term being linear in the scalar gradient under the RDGF closure. This defect was addressed by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. With this modification, the closed form of the diffusion term is now a nonlinear functional of the scalar gradients, and the resulting model is in very good agreement with the DNS data.
Since the ML-RDGF model of \cite{johnson2017turbulence} is used to specify the velocity gradients in the scalar gradient equation, we begin by comparing its predictions with DNS data. In agreement with the results of \cite{johnson2017turbulence}, the model very accurately predicts the longitudinal and transverse components of the velocity gradients, and reproduces the key features of the joint-PDF of $Q$ and $R$, the second and third invariants of the velocity gradient tensor. However, a more quantitative test of the model predictions for the PDFs of $Q$ and $R$ individually against DNS data revealed that at least for $\Rey_\lambda=100$, the ML-RDGF significantly underpredicts the probability of large positive values of $Q$ and large negative values of $R$, which are primarily associated with regions of intense enstrophy and enstrophy production, respectively. These inaccuracies could then impact the accuracy of the scalar gradient model, and highlight the need for further improvements in the ML-RDGF model (although the model may predict the statistics of $Q$ and $R$ more accurately at higher $\Rey_\lambda$).
Comparisons between the scalar gradient model and DNS data for $\Rey_\lambda=100$ showed very good agreement. In particular, the model accurately predicts the squared magnitude of the scalar gradients (which are proportional to the scalar dissipation rates), as well as the individual components of the scalar gradients. The model also captures well the PDF of the scalar production, including its strong negative skewess that is associated with the predominance of scalar gradient production of destruction, but slightly underpredicts the most extreme fluctuations of the scalar gradient production and destruction. Next, the PDFs of the inner product between the scalar gradient direction and the strain-rate eigendirections were considered, which provide insights into the nontrivial statistical geometry of the passive scalar and velocity gradient dynamics. The model is in excellent agreement with the DNS data regarding the strong preferential alignment between the scalar gradient and the compressional eigendirection. However, the model predicts a misalignment with the extensional eigendirection that is a little too strong, and a misalignment with the intermediate eigendirection that is a little too weak.
The ability of the model to capture the statistical relationship between the velocity and scalar dynamics was considered next, by considering various joint-PDFs of the velocity and scalar gradient tensors. The results showed excellent qualitative agreement with the DNS data, with some quantitative errors that seem to be rooted in the ML-RDGF model underpredicting the probability of extreme fluctuations in the $Q$ and $R$ invariants. The joint-PDF of the squared magnitude of the scalar gradient with the scalar gradient production term predicted by the model was in excellent qualitative as well as quantitative agreement with the DNS.
The predictions of the model at $\Rey_\lambda$ greater than that of the DNS were also considered, and the predictions are reasonable up to around $\Rey_\lambda\approx 500$. However, beyond this, the model breaks down and leads to extremely large scalar gradients that can even cause the numerical simulations of the model to blow up. Therefore, while the modification to the scalar gradient diffusion term that incorporates the scalar gradient production along the local trajectory history of the particle leads to excellent predictions from the model at lower $\Rey_\lambda$, it is not sufficient to prevent the model from generating extremely large fluctuations at high $\Rey_\lambda$ where intermittency in the velocity gradients can lead to very large local scalar gradient production events. Therefore, as anticipated earlier, developing an accurate model for scalar gradients in turbulence is in some ways more complicated than that for velocity gradients, because scalar gradient dynamics lack a mechanism similar to the pressure Hessian that controls the growth of the velocity gradients. For scalar gradients, the closure for the diffusion term is a delicate matter since this term alone is dynamically responsible for preventing finite-time singularities of the scalar gradients. A crucial point for future work is therefore to understand in more detail how the diffusion term regulates the growth of the scalar gradients, and developing a closure model that is sufficiently sophisticated to capture this.
Another aspect to be explored is the influence of the Schmidt number $Sc$ on the scalar gradients. While the model does capture a $Sc$ dependence in the scalar gradient diffusion term, given that the model breaks down for $Sc=1$ when $\Rey_\lambda>500$, it is likely that the model will also break down for $\Rey_\lambda<500$ when $Sc$ becomes sufficiently large, since both regimes promote the intensification of the scalar gradients.
\backsection[Acknowledgements]{This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant number ACI-1548562 \citep{xsede}. Specifically, the Expanse cluster was used under allocation CTS170009.}
\backsection[Funding]{A.D. Bragg and X. Zhang acknowledge support through a National Science Foundation (NSF) CAREER award \# 2042346}
\backsection[Declaration of interests]{The authors report no conflict of interest.}
\bibliographystyle{jfm}
|
{
"timestamp": "2022-11-01T01:27:09",
"yymm": "2210",
"arxiv_id": "2210.17336",
"language": "en",
"url": "https://arxiv.org/abs/2210.17336"
}
|
\section{Background and notation}
\label{sec:background}
\subsection{Discrete DPPs}
\label{sec:discrete-dpps}
For more background on discrete DPPs, we refer readers to
\cite{kulesza2012determinantal} and \cite{tremblay2022extended}. Discrete DPPs
are a specific instance of a \emph{discrete point process}. We say $\mathcal{X}$ is a
discrete point process on a ground set $\Omega$, if it is a random subset of
$\Omega$. Without loss of generality, we let $\Omega = \{1, \ldots,n \}$ so that
$\mathcal{X}$ is a random subset of indices.
\begin{definition}[DPP]
$\mathcal{X}$ is a DPP on $\Omega$
with marginal kernel $\bK \in \R^{n \times n}$ such that $\mathbf{0}\leq \bK \leq \bI$, noted $\mathcal{X} \sim DPP(\bK)$, if for all fixed subsets
$S \subseteq \Omega$, we have
\begin{equation}
\label{eq:incl-prob}
p(S \subseteq \mathcal{X}) = \det \bK_{S}
\end{equation}
\end{definition}
Here $\bK_{S}$ is the principal submatrix of $\bK$ with indices given by $S$.
We shall use ``Matlab'' notation, where $\bK_{A,B}$ denotes the submatrix with
row indices $A$ and column indices $B$, $\bK_{A,:}$ means all columns and
$\bK_{:,B}$ all rows.
\begin{definition}[Projection DPP]
A projection DPP is a DPP whose kernel is a projection matrix (ie.
$\bK^2 = \bK$).
\end{definition}
If $\bK$ is a projection matrix, it can be written as $\bQ\bQ^\top$ where $\bQ \in
\R^{n \times m}$ is an orthonormal basis for $\mspan \bK$ ($\bQ^\top\bQ = \bI$
and $\mspan \bQ = \mspan \bK$). Note that any orthonormal basis for $\bK$ is
enough, $\bQ$ need not be a basis of eigenvectors.
Projection DPPs are important because of the following mixture decomposition,
due to \cite{Hough:DPPandIndep}.
\begin{theorem}[mixture representation]\label{houghs_thm}
Let $\mathcal{X} \sim DPP(\bK)$, and $\bK = \bU \matr{\Lambda} \bU^\top$ the
eigendecomposition of $\bK$, with $\matr{\Lambda}$ the diagonal matrix of eigenvalues $\{\lambda_j\}_{j=1,\ldots,n}$ and $\bU=\left(\bu_1|\bu_2|\ldots|\bu_n\right)$ the matrix of eigenvectors. Then the following process produces a sample from
$\mathcal{X}$.
\begin{enumerate}
\item Sample a subset $\mathcal{Y}$ of eigenvectors by including each eigenvector
$\bu_j$ with probability $\lambda_j$.
\item Form the projection kernel $\bK_{\mathcal{Y}} = \bU_{:,\mathcal{Y}}(\bU_{:,\mathcal{Y}})^\top$
\item Sample $\mathcal{X} \sim DPP(\bK_{\mathcal{Y}}) $
\end{enumerate}
\end{theorem}
The cost of sampling a DPP when following this recipe equals the cost of
computing the eigendecomposition of $\bK$ ($\O(np^2)$ with $p$ the rank of
$\bK$) followed by the cost of sampling a projection DPP (step 3). It is the latter step that we focus on here.
\subsection{Fixed-size DPPs}
The cardinal of a DPP is in general random. Such varying-sized samples are
not practical in many applications,
which led authors in~\cite{kulesza2011k} to define fixed-size DPPs\footnote{They are often called k-DPPs in the literature, but we prefer ``fixed-size DPPs'' in order not to overload the symbol $k$ too much.}
\begin{definition}[Fixed-size DPP]
\label{def:fsDPP}
A fixed size DPP of size $m$ is a DPP $\mathcal{X}$ conditioned on $|\mathcal{X}|=m$.
\end{definition}
To sample a fixed-size DPP with kernel $\bK$, one follows the same recipe as in Theorem~\ref{houghs_thm} except for the first step that is replaced by:
\begin{enumerate}
\item \emph{Sample a subset $\mathcal{Y}$ of eigenvectors by including each eigenvector
$\bu_j$ with probability $\lambda_j$; \emph{conditioned on $|\mathcal{Y}|=m$.}}
\end{enumerate}
Performing such a conditioned sampling can be done by Algorithm 8
of~\cite{kulesza2011k}, which works by explicitly computing elementary polynomials.
\subsection{L-ensembles and fixed-size L-ensembles}
\label{sec:ell-ensembles}
L-ensembles are a subclass of DPPs popular in Machine Learning applications
because of their intuitive definition:
\begin{definition}[L-ensemble]
Let $\matr{L}$ be a positive semi-definite matrix.
$\mathcal{X}$ is a L-ensemble on $\Omega$ if for all $X \subseteq \Omega$
\begin{equation}
\label{eq:ell-ensemble}
p(\mathcal{X} = X) = \frac{1}{\det (\bI + \matr{L})} \det(\matr{L}_X)
\end{equation}
\end{definition}
L-ensembles are specified via their likelihood function, Eq.~\eqref{eq:ell-ensemble},
which states that those subsets of $\Omega$ where the submatrix $\matr{L}_X$ is
well-conditioned, are preferred. Intuitively, if $L_{i,j}$ represents a
similarity between items $i$ and $j$ of $\Omega$, then the L-ensemble favours
subsets of $\Omega$ that hold dissimilar items.
As with DPPs, one defines fixed-size L-ensembles as:
\begin{definition}[Fixed-size L-ensemble]
\label{def:fsLens}
A fixed size L-ensemble of size $m$ is a L-ensemble $\mathcal{X}$ conditioned on $|\mathcal{X}|=m$.
\end{definition}
(Fixed-size) L-ensembles are (fixed-size) DPPs with kernel $\bK =(\bI+\matr{L})^{-1} \matr{L}$~\cite{kulesza2012determinantal, tremblay2022extended}, and the mixture
representation thus applies.
To conclude this section, we have seen that all (fixed-size) L-ensembles and more generally all (fixed-size) DPPs have a mixture representation that divides the sampling algorithm in two steps: i/ a diagonalisation step that costs $\mathcal{O}(np^2)$, ii/ a step consisting of sampling a projection DPP. Step ii/ is known\footnote{This is an average (resp. deterministic) cost for DPPs (resp. fixed-size DPPs) for which $m$ refers to the average (resp. desired) size of the sample.} to cost $\mathcal{O}(nm^2)$ in the literature. The purpose of this paper is to show that the cost of this second step can always be reduced to $\O(nm + m^3 \log m)$.
\subsection{State-of-the-art}
\label{sec:SOTA}
Various directions have been explored when designing fast samplers for discrete
DPPs. Some have focused on bypassing the eigendecomposition of $\bK$ or $\matr{L}$
\cite{poulson2020high,launay2020exact,derezinski2019exact}.
If $\bK$ is a sparse matrix, then the algorithms in \cite{poulson2020high} can
be quite advantageous compared to standard $\O(n^3)$ algorithms. These
algorithms are difficult to adapt to L-ensembles in an efficient manner (for
instance, they cannot take advantage of sparsity in $\matr{L}$). Random spanning
forests \cite{avena_two_2017} are an example of a discrete DPP where a fast
sampler (Wilson's algorithm, \cite{wilson_generating_1996}) is available, and
sparsity in $\matr{L}$ seems to play a role. However, Wilson's algorithm does not
extend readily to L-ensembles with arbitrary structures
Another direction for generic DPP samplers is to give up on exactness.
Approximate samplers are available, based on Markov Chain Monte Carlo methods.
The most recent results in that direction are in \cite{anari2022optimal}, where
the authors show that given some preprocessing there are MCMC samplers that run
in $\tilde{\O}(m^\omega)$, where the $\tilde{\O}$ is shorthand for $\O(.)$ ``up to
logarithmic factors'', and $\omega$ is the exponent of matrix-multiplication
time, which for practical values of $m$ is effectively 3. The pre-processing
consists essentially in estimating the inclusion probabilities, also known as
the leverage scores, and its runtime is given by \cite{anari2022optimal} as
$\tilde{\O}(nm^{\omega - 1})$. Our results are essentially the same (preprocessing
in $\O(nm^2)$, sampling in $\tilde{\O}(m^3)$), but our sampler is exact.
Finally, two recent papers \cite{han2022mcmc,han2022scalable} extend the tree-based sampling approach of \cite{gillenwater2019tree} to obtain both approximate and exact samplers with complexity slightly larger than our proposal. There are also more complicated to implement. However, these can handle non-symmetric DPPs, which the algorithm given here cannot do.
\section{Sampling via accept-reject}
\label{sec:accept-reject}
In this section the goal is to formulate and analyse an algorithm for sampling a
projection DPP $\mathcal{X} \sim DPP(\bQ\bQ^\top)$ with $\bQ \in \R^{n \times m}$, verifying $\bQ^\top\bQ = \bI$. The
first exact such algorithm was described by \cite{Hough:DPPandIndep}, and
adapted in \cite{kulesza2012determinantal} to the discrete case. The first
efficient version appeared in \cite{gillenwater2014approximate}.
It is effectively a variant of the Gram-Schmidt algorithm.
For completeness we give an easy derivation in the next section (Section~\ref{subsec:SOTA_alg}), and the notation will serve to
describe our own variant, in Section~\ref{subsec:rejection-sampling}.
\subsection{State-of-the-art algorithm}
\label{subsec:SOTA_alg}
A projection DPP has size $m$ almost surely (see \cite{kulesza2012determinantal}). We shall sample the $m$
elements successively. Let $\vec{\mathcal{X}} = (x_1,\ldots,x_m)$ be an ordered version of
$\mathcal{X}$; we can go from $\vec{\mathcal{X}}$ to $\mathcal{X}$ by forgetting the order and from $\mathcal{X}$ to
$\vec{\mathcal{X}}$ by ordering randomly. The latter can be achieved by picking a first
item uniformly from $\mathcal{X}$, then a second, then a third etc.
The sampling algorithm proceeds via the following decomposition:
\begin{equation}
\label{eq:chain-rule}
p(\vec{\mathcal{X}}) = p(x_1)p(x_2|x_1)p(x_3|x_1,x_2)\dots p(x_m|x_1 \dots x_{m-1})
\end{equation}
The algorithm samples $x_1$ first, then $x_2$ given $x_1$ has been selected, etc.
The law of $x_1$ is the law of the first element of $\vec{\mathcal{X}}$, a randomly
ordered version of $\mathcal{X}$. That is equivalent to $x_1$ being sampled uniformly at
random from $\mathcal{X}$, and so:
\[ p(x_1 = i) = \frac{1}{m}p(i \in \mathcal{X}) = \frac{K_{i,i}}{m} \]
We can similarly obtain the law of $x_2$ given $x_1$, as two elements drawn
randomly (without replacement) from $\mathcal{X}$:
\begin{align*}
p(x_2 = i|x_1=j) &= \frac{p(x_1 = j, x_2=i)}{p(x_1=j)}=
\frac{p(i \in \mathcal{X} , j \in \mathcal{X}) }{m(m-1)} \times \frac{m}{p(j \in \mathcal{X}) } \\
&=
\frac{1}{(m-1)} \frac{\det \bK_{\{i,j\}}}{K_{j,j}}
\end{align*}
The formula for determinants of block matrices yields:
\[ \frac{\det \bK_{\{i,j\}}}{K_{j,j}} = K_{i,i}- \frac{K_{i,j}^2}{K_{j,j}} \]
and so:
\[ p(x_2 = i|x_1=j) = \frac{1}{(m-1)} \left(K_{i,i}- \frac{K_{i,j}^2}{K_{j,j}} \right)\]
For the general term in the chain rule decomposition (eq.
(\ref{eq:chain-rule})), the same reasoning applies. We obtain:
\begin{align}
\label{eq:chain-rule-gen-term}
p \left(x_t = i| \vec{\mathcal{X}}_{1:(t-1)} = X_t \right) = \frac{1}{(m-t)} \left( K_{i,i} - \bK_{i,X_t}(\bK_{X_t})^{-1}\bK_{X_t,i} \right)
\end{align}
Eq. (\ref{eq:chain-rule-gen-term}) is enough to give us a sampling algorithm,
since at each step we have an explicit (discrete) probability distribution to
sample from. However, implementing eq. (\ref{eq:chain-rule-gen-term}) naïvely, we would be computing a matrix
inverse at each step, which would turn out to be quite expensive for large $m$. To get an efficient algorithm a bit more work is needed.
Let us reexpress eq. (\ref{eq:chain-rule-gen-term}) in terms of $\bQ$. Recalling
$\bK = \bQ\bQ^\top$, we obtain:
\begin{align}
\label{eq:chain-rule-Q}
p \left(x_t = i| \vec{\mathcal{X}}_{1:(t-1)} = X_t \right)= \frac{1}{ (m-t)}
\left( K_{ii} - \bQ_{i,:}\bM_t(\bQ_{i,:})^\top \right)
\end{align}
where $\bM_t = (\bQ_{X_t,:})^\top \left(
\bQ_{X_t,:}(\bQ_{X_t,:})^\top \right)^{-1} \bQ_{X_t,:}$
is a projection matrix ($\bM_t^2 = \bM_t$) of size $m$ and rank $|X_t| = t-1$, and so can be
rewritten
\[ \bM_t = \sum_{i=1}^{t-1} \vect{s}_{i}\vect{s}_{i}^\top \]
where $\vect{s}_1 \dots \vect{s}_t$ form an orthonormal basis for $\mspan \bM_t = \mspan
\bQ_{X_t,:}^\top$, the linear subspace spanned by the rows of $\bQ$ selected so
far.
A first source of computational savings comes from realising that $\bM_t$ can be
computed iteratively via the Gram-Schmidt process.
Notice that $\bM_t = \bM_{t-1} + \vect{s}_{t-1}\vect{s}_{t-1}^\top$, and $\bM_{t-1}$ spans
$\mspan \bQ_{X_{t-1},:}^\top$. We obtain $\vect{s}_{t-1}$ via Gram-Schmidt: first we
compute the residual
\[ \vect{z}_{t-1} = (\bI - \bM_{t-1}) \bQ_{x_{t-1},:}^\top\]
and then we normalise:
\[ \vect{s}_{t-1} = \frac{\vect{z}_{t-1}}{\norm{ \vect{z}_{t-1}} }\]
At each step $t$ this costs $\O(m(t-1))$ operations, and we will need to do this
$m-1$ times at a total cost of $\O(m^3)$.
Next, we can show that the probability distribution we sample from at step $t$
can be easily obtained from the one we had at step $t-1$. It is more convenient
to write this using unnormalised versions of the densities.
Let $\pi^{(1)}(i) = m p(x_1 = i) = K_{i,i}$. Next, we define:
\begin{align*}
\pi^{(2)}(i) = (m-1)p(x_2 = i | x_1 = j)
= K_{i,i} -
\frac{K_{i,j}^2}{K_{i,i}} = \pi^{(1)}(i) - \frac{K_{i,j}^2}{K_{i,i}} \end{align*}
Note that we have suppressed the dependency on the past in the notation
$\pi^{(2)}(i)$: it is to be understood as the (unnormalised) density we draw
from at the second step of the algorithm.
In the general case, we define:
\begin{equation}
\label{eq:unnormalised-general}
\pi^{(t)}(i) = (m-t+1) \;p\left(x_t = i | \vec{\mathcal{X}}_{1:(t-1)}=X_t \right)
\end{equation}
Injecting eq. (\ref{eq:chain-rule-gen-term}) and eq. (\ref{eq:chain-rule-Q}), we find
\begin{align}
\label{eq:unnormalised-recursion}
\pi^{(t)}(i) &= \pi^{(1)}(i) - \bQ_{i,:} \bM_t (\bQ_{i,:})^\top
= \pi^{(1)}(i) - \bQ_{i,:} (\bM_{t-1} + \vect{s}_{t-1}\vect{s}_{t-1}^\top) (\bQ_{i,:})^\top \nonumber\\
&= \pi^{(t-1)}(i) - (\bQ_{i,:}\vect{s}_{t-1})^2
\end{align}
All we need to do at each step of the algorithm is to
\begin{enumerate}
\item pick an item $i$ according to $\pi^{(t)}$
\item perform a step of the Gram-Schmidt algorithm to update $\bM_t$ based on
the new vector $\bQ_{i,:}$
\item Update the probability distribution to $\pi^{(t+1)}$ according to eq. (\ref{eq:unnormalised-recursion})
\end{enumerate}
Sampling from a discrete distribution (step 1 above) can be done at cost
$\O(n)$, and is needed $m$ times, for a total cost of $\O(nm)$. We have already
established that the cost of the Gram-Schmidt algorithm is $\O(m^3)$. It is the
update to the probability distribution that is the most costly, with each step
costing $\O(nm)$ ($n$ dot products in $\R^m$) for a total of $\O(nm^2)$. Since
$n>m$ the cost of the algorithm scales as $\O(nm^2)$. We show pseudo-code for this standard
algorithm as alg. \ref{alg:sampling-proj}.
\begin{algorithm}[tb]
\caption{Sampling from a projection DPP $\mathcal{X} \sim DPP(\bK=\bQ\bQ^\top)$, standard algorithm}
Initialise $\pi(x) \gets \sum_{j=1}^m Q_{x,j}^2$, $\mathcal{X} = \emptyset$, Gram-Schmidt
basis $\bS=[]$ \;
\ForEach{$t \in 1\ldots m$}
{
Sample $x$ from $ \frac{\pi(x)}{m-(t-1)}$, add to $\mathcal{X}$ \;
Compute residual $\vect{z}_t = (\bI - \bS\bS^\top)(\bQ_{x,:})^\top$ \;
Add column $\vect{s}_t = \frac{\vect{z}_t}{\norm{\vect{z}_t}}$ to $\bS$ \;
Compute $\vect{v} = \bQ\vect{s}_{t} \in \R^n$ \;
Update probabilities $\pi(x) \gets \pi(x) - (v_{x})^2$\;
}
\label{alg:sampling-proj}
\end{algorithm}
In the next section, we move on to the core of our contribution, showing that this $\O(nm^2)$ cost can be reduced to $\O(nm + m^3 \log m)$ via rejection sampling.
\subsection{Using rejection sampling}
\label{subsec:rejection-sampling}
The use of rejection sampling is not exactly new in this context, since
algorithms for sampling continuous DPPs use this strategy out of necessity \cite{lavancier2015determinantal}.
In the discrete case, we believe it has not been used before because it looked
like an unnecessary complication. In fact, it leads to an algorithm that is
faster but no more complicated to implement than the traditional discrete
sampler described in alg.~\ref{alg:sampling-proj}.
As we discussed above, the most expensive part of alg.
\ref{alg:sampling-proj}, lies in updating the probability distribution to sample
from (last step of the \emph{for} loop). It turns out that
the accept-reject method lets us bypass this step.
To briefly recall the rejection sampling idea, suppose we have an unnormalised
density $\pi(x)$ (the target) we wish to draw from, and a proposal $q(x)$, also
unnormalised, but that we know how to draw from, and is not too far from $\pi$.
Further, $q(x)$ has support at least as wide as $\pi(x)$, and upper bounds it
($\pi(x) \leq q(x)$ over the support). We may then draw $x$ from $q(x)$ and
accept the sample with probability $\frac{\pi(x)}{q(x)}$. The accepted samples then have
density $\pi(x)$. If $q(x)$ is a good bound for $\pi(x)$, then the rejection
sampler will be quite efficent. In the limit where $\pi = q$, the acceptance
probability goes to 1. If on the other hand $q(x)$ is quite loose, the
acceptance probability may be bad.
In the DPP sampling algorithm, we need to sample from $\pi^{(1)}$, then
$\pi^{(2)}$, etc. up to $\pi^{(m)}$. Our proposal is to compute $\pi^{(1)}$
exactly for all $n$ entries, then use $\pi^{(1)}$ as proposal distribution for
the rest of the sequence.
Recall that $\pi^{(t)}$ is the unnormalised density defined in eq.
(\ref{eq:unnormalised-general}). From the recursion in eq.
(\ref{eq:unnormalised-recursion}), we have that
\[ \pi^{(t)}(x) \geq \pi^{(t+1)}(x) \]
for all $x$ and $1\leq t \leq m-1$. This is true in particular for $\pi^{(1)}$, which
can therefore be used as a proposal distribution for all subsequent $\pi^{(t)}$.
We can directly compute the probability of accepting a proposed sample. At step $t$, we
sample $x$ from the normalised density $\frac{1}{m}\pi^{(1)}(x)$ and accept it
with probability $\frac{\pi^{(t)}(x)}{\pi^{(1)}(x)}$. The acceptance probability
equals:
\begin{equation}
\label{eq:acceptance-prob}
\rho_t = \sum_{i=1}^n \frac{\pi^{(t)}(i)}{\pi^{(1)}(i)}\frac{\pi^{(1)}(i)}{m} = \frac{1}{m} \sum_{i=1}^n \pi^{(t)}(i) = \frac{(m-t+1)}{m}
\end{equation}
This probability decreases at each step of the algorithm, but at
the final step it is still positive and equals $\frac{1}{m}$.
Let us outline the proposed algorithm. First, one computes every entry of
$\pi^{(1)}$. For this we use the following formula
\begin{equation}
\label{eq:leverage}
\pi^{(1)}(i) = K_{i,i} = \sum_{i=1}^n Q_{ij}^2
\end{equation}
The computation is equivalent to computing the norm of each row of $\bQ$, at
cost $\O(nm)$. Because we need to sample from $\pi^{(1)}$ repeatedly, it pays to
use Walker's alias method \cite{walker1977efficient} (see also Chapter III.4 of~\cite{devroye86nonuniformrandom}). Given a preprocessing cost
of $\O(n)$, the alias method gives us all subsequent samples at cost
$\O(1)$ instead of $\O(n)$.
At the first step we sample our first item from $\pi^{(1)}$. At step 2, and all
subsequent steps, we use rejection sampling, which involves computing the ratio
\begin{equation}
\label{eq:acc-ratio}
\frac{\pi^{(t)}(x)}{\pi^{(1)}(x)}= 1 - \frac{1}{\pi^{(1)}(x)}\sum_{j=1}^{t-1} (\bQ_{x,:} \vect{s}_j )^2
\end{equation}
by eq. (\ref{eq:unnormalised-recursion}). Computing the acceptance ratio has
cost $\O(m(t-1))$ at step $t$, the cost of $t-1$ dot products in $\R^m$. We do
this repeatedly until a proposal is accepted, at which point we need to perform
a Gram-Schmidt step to update $\bM_t$ to $\bM_{t+1}$. We then move on to the
next iteration, or stop if $t=m$.
We summarise the whole process as alg. \ref{alg:sampling-proj-rejection}.
To recapitulate the different computational costs:
\begin{itemize}
\item Preprocessing cost: computing $\pi^{(1)}$ for all
entries comes at cost $\O(nm)$ and setting up Walker's alias method at cost $\O(n)$
\item The Gram-Schmidt process (computing $\vect{z}_t$ then $\vect{s}_t$) costs $\O(tm)$ at step $t$. Summing this figure for $t=1$ to $m$ gives a cost of $\O(m^3)$
\item We now need to compute the average cost of the \emph{while} loop at step $t$.
The rejection sampler has probability
$\rho_t$ of succeeding, given by eq. (\ref{eq:acceptance-prob}). The number of
proposals $R_t$ that are required until acceptance is thus a random variable that follows a geometric
distribution with success probability $\rho_t$: the expected number of proposals at iteration $t$ is
therefore
\begin{equation}
\label{eq:expected-number-of-props}
\E(R_t) = \frac{1}{\rho_t} = \frac{m}{m-t+1}.
\end{equation}
Since computing the acceptance ratio has cost $\O(m(t-1))$ for each trial, the expected cost of the \emph{while} loop at
step $t$ scales as $\O\left(\frac{m^2(t-1)}{m-t+1}\right)$. Summing this
figure for $t=1$ to $m$, one has $\sum_{t=1}^m\frac{m^2(t-1)}{m-t+1}\leq
m^3\sum_{t=1}^m\frac{1}{m-t+1}$ which scales as \footnote{$\sum_{t=1}^m\frac{1}{m-t+1}=\sum_{t=1}^m\frac{1}{t}$ scales as $\O(\log m)$: see, \emph{e.g.}, Chapter 6 of~\cite{abramowitz1964handbook}} $\O(m^3\log m)$.
\end{itemize}
Tallying everything we obtain the following theorem:
\begin{theorem}
\label{thm:runtime}
Alg. \ref{alg:sampling-proj-rejection} samples a projection DPP, with an
expected runtime scaling as $\O(nm +m^3
\log m)$. Also, any additional sample from the same DPP can be obtained in an extra $\O(m^3\log m)$ expected runtime. \\
Moreover, these expected runtimes are representative. Indeed,
$\forall\delta\in\left(0,\frac{1}{2}\right)$, the total number of proposals $R=\sum_{t=1}^m R_t$ satisfies, with probability greater than $1-\delta$:
$$R \leq 2 m \log m + 3m\log\frac{1}{\delta}$$
\end{theorem}
\begin{proof}
The fact that Alg.~\ref{alg:sampling-proj-rejection} samples a projection DPP in $\O(nm +m^3
\log m)$ expected runtime is proven above the Theorem's statement.
The fact that any additional sample from the same DPP only costs an extra $\O(m^3\log m)$ expected runtime comes from the fact that all initialisation steps (the computation of $\pi^{(1)}$ and the setting-up cost of Walkers' algorithm) have already been computed for the first sample. To obtain any extra sample, one only needs to run the \emph{for} loop once more, costing $\O(m^3\log m)$. \\
The final statement relates to concentration properties of $R$. For $m=1$, the statement is trivial. We now show the result for $m\geq 2$.
At step $t$ of Alg.~\ref{alg:sampling-proj-rejection} the number of proposals of the sampler is
a random geometric variable $R_t$ with parameter $(m-t+1)/m$. Note that all the $R_t$'s are independent.
We study here the behavior of $R=\sum_{t=1}^{m} R_t$ which is the total number of rejection sampling steps used during the whole course of Alg.~\ref{alg:sampling-proj-rejection}.
In \cite{Jans18}, the following one-tailed bound for $R$ is given in Thm 2.3. $\forall\lambda\geq1$:
\begin{equation*}
p\left( R \geq \lambda E[R]\right) \leq \frac{1}{\lambda} \left( 1-\frac{1}{m} \right)^{(\lambda- 1-\log\lambda)\E[R]}
\end{equation*}
We look for $\lambda$ large enough s.t. $p\left( R \geq \lambda E[R]\right)\leq \delta$, \emph{i.e.}:
\begin{align}
\label{eq:to_verify}
\log\left(\frac{1}{\delta}\right)\leq-(\lambda-1-\log\lambda)E[R]\log\left(1-\frac{1}{m}\right)
\end{align}
One has\footnote{In fact, the function $\lambda-1-\log\lambda$ is convex for all $\lambda\geq 1$ and thus lower-bounded by all its tangents. The one we use is the tangent in $3/2$. Other choices lead to other constants in the result.} $\forall\lambda\geq 1, \frac{1}{3}\lambda-\log \frac{3}{2} \leq \lambda-1-\log\lambda$
such that Eq.~\eqref{eq:to_verify} is verified provided that:
$$\lambda\geq 3\left(\log \frac{3}{2} - \frac{\log\delta^{-1}}{E[R]\log\left(1-\frac{1}{m}\right)}\right)$$
Stated differently, setting $\lambda$ to this lower bound implies $p(R \leq \lambda E[R])\leq 1-\delta$.
All is left to show is that $\forall m\geq2$:
\begin{align*}
\forall\delta\in\left(0,\frac{1}{2}\right),\quad 3\left(\log\frac{3}{2} - \frac{\log\delta^{-1}}{E[R]\log\left(1-\frac{1}{m}\right)}\right)E[R] \leq
2 m \log m + 3m\log\delta^{-1}
\end{align*}
For this, we use two upper-bounds. The first one is $\forall m>1, -\frac{1}{\log\left(1-\frac{1}{m}\right)}\leq m-1/2$. The second one is the following bound on $E[R]$. As $E[R_t]=m/(m-t+1)$, one has (see, \emph{e.g.}, Chapter 6 of~\cite{abramowitz1964handbook}): $\frac{\E[R]}{m}= \sum_{t=1}^m \frac{1}{m-t+1}=\sum_{t=1}^{m}\frac{1}{t}=\gamma+\psi(m)+\frac{1}{m}$
where $\gamma\approx 0.577$ is Euler's constant and $\psi(m)$ the digamma function. Now, a known bound is $\psi(m)\leq \log m - \frac{1}{2m}$, which yields $E[R]\leq m(\log m +\gamma) + \frac{1}{2}$.
\end{proof}
\begin{algorithm}[tb]
\caption{Sampling from a projection DPP $\mathcal{X} \sim DPP(\bK=\bQ\bQ^\top)$ with
rejection sampling }
Initialise $\pi^{(1)}(x) \gets \sum_{j=1}^m Q_{x,j}^2$, $\mathcal{X} \gets \emptyset$,
Gram-Schmidt basis $\bS \gets []$ \;
Initialise alias table for Walker's alias algorithm to sample from $\pi^{(1)}$.\\
\ForEach{$t \in 1\ldots m$}
{
accept $\gets$ false \;
\While{not accept}
{
Draw $x$ from $\pi^{(1)}$ using the alias method\;
Compute acceptance ratio $r = 1 - \frac{1}{\pi^{(1)}(x)}\sum_{j=1}^{t-1}
(\bQ_{x,:} \vect{s}_t )^2$ \;
\If{rand() $< r$}{accept $\gets$ true}
}
Add $x$ to $\mathcal{X}$ \;
Compute residual $\vect{z}_t = (\bI - \bS\bS^\top)(\bQ_{x,:})^\top$ \;
Add column $\vect{s}_t = \frac{\vect{z}_t}{\norm{\vect{z}_t}}$ to $\bS$ \;
}
\label{alg:sampling-proj-rejection}
\end{algorithm}
\subsection{Some refinements}
\label{sec:refinements}
Algorithm \ref{alg:sampling-proj-rejection} works well enough as is but there
are some refinements that can reduce the computational cost further (at least
theoretically).
\subsubsection{Preprocessing for general DPPs}
\label{sec:preprocessing}
In some applications we require several samples from the same DPP, and
algorithms have been described that trade higher set-up cost for a lower cost
per sample (see \cite{gillenwater2019tree}).
If the target DPP is a projection DPP, then setting up alg.
\ref{alg:sampling-proj-rejection} for repeated sampling could not be easier.
In practice a substantial amount of time is spent computing $\pi^{(1)}$, and a
smaller amount setting up the alias table. All of this can be done as part of
preprocessing, so the first sample from the DPP costs $\O(nm+m^3\log m)$ but
after that the cost is just $\O(m^3\log m)$ per sample.
If the target DPP is not a projection DPP, then one has to use the mixture
representation (Thm~\ref{houghs_thm}): draw a random set of eigenvectors
$\mathcal{Y}$ and run alg. \ref{alg:sampling-proj-rejection} with $\bQ = \bU_{\mathcal{Y},:}$.
Since the kernel changes every time, so does $\pi^{(1)}$ and it cannot be computed
as part of pre-processing. However, the kernels encountered in practice tend to
have rapidly decreasing eigenvalues, so that the variance in $\mathcal{Y}$ is quite small
and the DPP is close to a projection DPP. Without getting into too much detail,
it is possible to pre-compute the partial sum
\[ \widehat{\pi^{(1)}}(i) = \sum_{j \in \hat{\mathcal{Y}}} U_{i,j}^2 \]
for some highly likely subset of $\mathcal{Y}$, denoted here $\hat{\mathcal{Y}}$. $\pi^{(1)}$ for the
actual sampled $\mathcal{Y}$ can be obtained efficiently by removing the extra entries
and adding the missing ones. The alias table can be computed from scratch.
This type of preprocessing brings down the cost to $\O(ns + m^3\log
m)$ per sample, where $m = \E|\mathcal{Y}|$ and $s$ is the expected size of the symmetric
difference between $\hat{\mathcal{Y}}$ and $\mathcal{Y}$.
\subsubsection{Caching computations and updating the proposal distribution}
\label{sec:updating-prop}
Clearly, Alg.~\ref{alg:sampling-proj-rejection} has some wasted
computation, since all the computations done when a proposal $x$ is rejected are
performed again should $x$ come up a second time. When $n$ is small, or when
$\pi^{(1)}$ has low entropy, this may indeed happen several times.
Caching is one way of reducing the amount of redundant computations that are
performed. Going back to the recursive formula for $\pi^{(t)}$ (eq.
(\ref{eq:unnormalised-recursion})), we see that it is cheaper to compute $\pi^{(t)}$
from $\pi^{(t-1)}$ then it is to compute it from scratch. A reasonable caching
strategy is then to keep track for every point $x$ of the last density
evaluation performed for that point. If $x$ comes up again, the evaluation of
the acceptance ratio is simplified.
Another natural idea is to update the proposal distribution over the course of
the algorithm (instead of sticking with $\pi^{(1)}$ throughout). The most basic
version is that any $x$ that has already been selected has an acceptance
probability of 0, so we may as well not suggest them. Another is that points
similar to a selected point are quite unlikely to come up further down, and
so it may be worth computing $\pi^{(t)}$ for these neighbours to tighten the bound.
Finally, we may combine this idea with the caching idea, which provides a better
bound for every point that has ever been suggested. Unfortunately this runs
against the difficulty of updating the alias table in Walker's algorithm, which
one would have to compute from scratch at every update (at cost
$\O(n)$). A better way would be to use a binary tree representation
\cite{devroye86nonuniformrandom}, which can be updated at cost $\O(\log n)$ and
provides samples also at cost $\O(\log n)$. The implementation complexity
increases a lot however, and we have not pursued this further. As we shall see
below, Alg.~\ref{alg:sampling-proj-rejection} is quite fast in practice,
and implementation time may be better invested in feature computation and
orthogonalisation.
\section{Empirical results}
\label{sec:empirical-results}
We compare the Accept/Reject algorithm (Alg.~\ref{alg:sampling-proj-rejection}) to its classical counterpart (Alg.~\ref{alg:sampling-proj}) for
different values of $n$ and $m$. Both algorithms are implemented in the Julia
language and are publicly available~\footnote{we've added a folder in our DPP.jl repository containing the code necessary to reproduce the figures, available here: \url{https://github.com/dahtah/DPP.jl/tree/main/misc/sampling_paper}}. For each value of $n$, we
sample a random projection matrix of size $n \times m$ (via QR decomposition of
a matrix with Gaussian entries). We then pre-compute the leverage scores, and
run each algorithm 100 times. Fig. \ref{fig:results_runtime} a) and b) show the measured
median runtimes. All tests are run on a 2017 Linux
laptop with i7-8550U Intel CPU.
For very small values of $n$, the classical algorithm is faster, which can be
explained by the efficiency of BLAS calls. At each step the whole conditional
distribution is computed, the main cost being a matrix multiplication (i.e., a
BLAS call), which benefits from efficient multithreaded code. However, the
different asymptotic scalings ($\O(nm^2)$ vs. $\O(nm)$) soon makes the classical
algorithm uncompetitive. The cross-over point in our simulations is at around
$n=1,000$, and by $n=10,000$ the difference is stark.
In many cases, sampling a projection DPP is only one of the steps in a process
that involves feature computation and orthogonalisation. For instance, in
\cite{Tremblay:DPPforCoresets}, a DPP based on the Gaussian kernel is used to
produce a subset of the data suitable for running clustering algorithms.
Starting from $n$ points, $\vect{x}_1,\ldots,\vect{x}_n$ in $\R^d$ they use a DPP with
L-ensemble given by $L_{ij} = \gamma \kappa(\vect{x},\vect{y}) = \gamma \exp \left( - \frac{1}{\sigma^2} \norm{\vect{x}-\vect{y}}^2 \right)$,
where $\gamma$ is a tuning parameter that determines the expected size.
Producing a sample from this exact DPP requires the eigendecomposition of $\matr{L}$
which is impractical; however, $\matr{L}$ is numerically low-rank for relevant values
of $\sigma$ and this can be exploited. In \cite{Tremblay:DPPforCoresets} the
authors use a low-rank approximation of $\matr{L}$ from Random Fourier Features
\cite{rahimi_random_2008} followed by a SVD. This brings down the total cost to
$\O(nm^2)$. We now sketch (without any formal justification) another procedure
which gives comparable results at lower cost.
\begin{figure}
\includegraphics[width=\linewidth]{runtime_aistats.pdf}
\caption{\emph{Left panels}: median time needed to sample a projection DPP,
using the standard approach (Alg.~\ref{alg:sampling-proj}) vs. A/R (Alg.~\ref{alg:sampling-proj-rejection}). For two values of $m$: a) $m=30$, b) $m=60$. Note
that here the time taken to compute an orthogonal basis is not taken into
account (see text). \emph{Right panels}: simulation of a full workload that includes feature
generation and orthogonalisation (see text). c) time taken by each step in
the computation as a function of $n$. Solid lines are runtimes when sampling
using the classical method, dashed, when using A/R. Note that the first two
steps (labeled ``kernel'' and ``RRQR'', in red and green respectively) are identical. A/R sampling becomes
beneficial at around $n=1,000$ and is orders of magnitude faster at
$n=100,000$. d) Sampling time as a function of total computation time. }
\label{fig:results_runtime}
\end{figure}
First, Gaussian kernel matrices have rapidly decaying spectra (see e.g.
\cite{wathen2015eigenvalues}), which implies in particular that DPPs
sampled from a Gaussian L-ensemble are well approximated by projection DPPs with
kernels $\bP_m= \bU_m\bU_m^t$ where $\bP_m$ projects onto the dominant
eigenspace of $\matr{L}$ of order $m$. Therefore, all we need is a good basis for
the dominant eigenspace. Methods from randomised linear algebra offer good
practical tools (``range finders'') to obtain a basis for such a space, see
\cite{martinsson2020randomized}. For these simulations, we used the following
approximation:
\begin{enumerate}
\item First, select $5m$ columns uniformly from $\matr{L}$. Call this matrix $\bA$.
We call this the ``kernel step''.
\item Second, use Rank-Revealing QR (RRQR, \cite{chan1987rank}) and random
projections, as implemented in the Julia package LowRankApprox.jl, to produce
$\bQ$, an orthonormal matrix of size $n \times m$ that approximates the image
of $\bA$. We call this the ``RRQR'' step.
\item Third, sample a DPP with projection kernel $\bQ\bQ^t$ using either the
classical or the A/R algorithm.
\end{enumerate}
We set $m=100$ and time each step. This results in a total
runtime of around 1.2 sec. at $n=10^5$ with the A/R sampler, which challenges
the notion that DPPs are very slow to sample from. With this procedure, the time
spent sampling the actual DPP goes up to ~20\% of total time for the classical
algorithm at $n=10^5$, but using the A/R sampler sampling time becomes
negligible. See Fig.~\ref{fig:results_runtime} c) and d) to see how these times vary with $n$. \\
This indicates that for some computations the implementation effort
may be better allocated to speeding up the (deterministic) linear algebra and
feature computation part rather than the sampling part. In this particular
instance, step (1) at least could be sped up relatively easily by exploiting
parallelism, or the GPU, which we did not attempt.
\section{Discussion and perspectives}
\label{sec:m-vs-mlogm}
On top of the improvement on the sampling time of DPPs, our results imply the following intriguing by-product. Let $\mathcal{X}$ be a projection DPP of size $m$. A set of $\O(m \log m)$ points sampled i.i.d. from the inclusion probability distribution $p(i \in \mathcal{X})=\pi^{(1)}(i)/m$ (also known as \emph{leverage scores}) contains with high probability a realisation from the DPP.
The consequences of this fact are worth discussing.
First, let us put the result a bit more formally.
\begin{definition}
Let $\mathcal{X}$ be a DPP on $\Omega$ and $\mathcal{Y}\subseteq\Omega$. We call $\Phi(\mathcal{Y})$ a \emph{thinning algorithm} if it returns a subset of $\mathcal{Y}$. Moreover,
we say $\Phi(\mathcal{Y})$ is \emph{successful} when it returns a realisation from $\mathcal{X}$.
\end{definition}
\begin{corollary}
\label{cor:guaranteed}
Let $\mathcal{X}$ be a projection DPP of size $m$, and $\mathcal{Y}$ be a set of i.i.d. points sampled with replacement with probability proportional to the leverage scores: $p(i \in \mathcal{X})=\pi^{(1)}(i)/m$. Let $\delta\in(0,1/2)$. A simple modification of Alg.~\ref{alg:sampling-proj-rejection} gives a thinning algorithm $\Phi$ that verifies: $\Phi(\mathcal{Y})$ is successful with probability greater than $1-\delta$ provided that $|\mathcal{Y}|\geq 2 m \log m + 3m\log\left(\frac{1}{\delta}\right)$.
\end{corollary}
\begin{proof}
Let $\mathcal{Y}$ be drawn i.i.d. with replacement from the leverage score distribution $\pi^{(1)}$.
$\Phi$ is the following simple modification of Alg.~\ref{alg:sampling-proj-rejection}. Instead of drawing a new proposal $x$ using the alias method at the beginning of the \emph{while} loop as in Alg.~\ref{alg:sampling-proj-rejection}, draw uniformly and without replacement from $\mathcal{Y}$. If $\Phi$ finishes before emptying $\mathcal{Y}$, then it is successful. If there is no more item in $\mathcal{Y}$ to draw from and $\Phi$ is not terminated, then it fails.
The probability that $\Phi$ succeeds is thus equal to the probability that $|\mathcal{Y}|$
is larger than the number of proposals $R$ of Alg.~\eqref{alg:sampling-proj-rejection}.
As shown in Theorem~\ref{thm:runtime}, setting $|\mathcal{Y}|= 2 m \log m + 3m\log\left(\frac{1}{\delta}\right)$ yields: $p(|\mathcal{Y}|\geq R)\leq 1-\delta$, finishing the proof.
\end{proof}
Note that this is a substantial improvement over the work of \cite{derezinski2019exact}, which gives this result only for $|\mathcal{Y}| \geq \O(m^2)$ i.i.d. points.
A natural question is then to ask if the result is optimal. Can we find a thinning algorithm that succeeds with high probability for even smaller i.i.d sets? The answer is no in general (proved in the supplementary material):
\begin{proposition}
Corollary~\ref{cor:guaranteed} is optimal in the following sense.
Let $\mathcal{X}$ and $\mathcal{Y}$ be as previously. There does not exist a generic thinning algorithm able to succeed with fixed non-null probability if $|\mathcal{Y}| = o(m \log m)$.
\end{proposition}
\begin{proof}
The proposition states that there does not in general exist a thinning algorithm that succeeds with a non-null probability if $|\mathcal{Y}|$ is asymptotically smaller than $m\log m$. We shall prove this by exhibiting a type of DPP $\mathcal{X}$ for which this is verified.
It is well-known in the folklore that a form of \emph{stratified sampling} is a special case of projection DPPs. In stratified sampling, we partition the ground set $\Omega$ into $m$ classes, and sample an item uniformly from each segment of the partition. To simplify the argument, assume $\Omega$ can be cut into $m$ subsets of equal size, and define vector $\vect{e}_j$ as the (normalised) indicator of segment $j$, i.e. $\vect{e}_j(i) = \sqrt{\frac{n}{m}}$ if item $i$ is in segment $j$ and $0$ otherwise. Let $\matr{E} = \left[ \vect{e}_1 \ldots \vect{e}_m \right]$. Then it is easy to show that stratified sampling is equivalent to a DPP $\mathcal{X}$ with marginal kernel $\bK = \matr{E}\matr{E}^t$. Since $\matr{E}^t\matr{E} = \bI$, the DPP in question is a projection DPP.
Because of the nature of stratified sampling, we know that $\mathcal{X}$ contains a point from each one of the segments, and that $p(i \in \mathcal{X}) = \frac{m}{n}$ for all $i$. Now in order for any thinning algorithm to produce a stratified sample, the i.i.d. sample $\mathcal{Y}$ needs to contain at least one point from each segment. Let $l \stackrel{\textrm{def}}{=} |\mathcal{Y}|$: how large does $l$ need to be so that $\mathcal{Y}$ contains at least one point from each segment with probability at least $\alpha$ ($\alpha>0$ fixed)? This is an instance of the coupon collector's problem. Assume that at each time $t$ we add a ball to one of $m$ urns with equal probability, and call $T$ the smallest $t$ such all urns have at least one ball. We will show that for any $l(m) = o(m\log m)$, $\lim_{m \to \infty} p(T < l(m) ) = 0$.
To do this, we need an upper bound for $p(T < l(m))$. Many bounds exist in the literature for large values of $T$, but we have not been able to find one for the left tail in the published literature. There is, however, one on the stackexchange website by a user called ``cardinal'' \footnote{see \url{https://stats.stackexchange.com/q/7774}}, which we draw from. $T$ can be viewed as a sum of $m$ geometric variables: $T= \sum_{i=1}^m T_i$, where $T_i$ is the time at which $i$ urns have at least one ball.
All the $T_i$'s are independent geometric random variables with success probability $p_i = 1-\frac{i-1}{m}$. Indeed, the same representation is obtained by considering alg. 2 in the special case of stratified sampling (each iteration fills one urn).
Next, we use, for any $s>0$
\begin{equation}
\label{eq:chernoff}
p(T < l) = p(\exp(-sT) > \exp(-sl)) \leq \exp(sl) \E \left(\exp(-sT)\right)
\end{equation}
where we used Markov's inequality. Since $T$ is a sum of independent geometric variables, $\E(\exp(-sT))$ is easy to compute\footnote{Using $E\left(\exp^{-sT}\right) = \prod_{j=1}^m E\left(\exp^{-sT_j}\right) = \prod_{j=1}^m \frac{p_j\exp^{-s}}{1-(1-p_j)\exp^{-s}}$ and changing variable $i \leftarrow m-j+1$ in the product}:
\[ \E(\exp(-sT)) = \prod_{i=1}^m \frac{i}{m(e^s - 1) + i}\]
Picking $s = \frac{1}{m}$, we obtain:
\[ p(T < l) \leq \exp^\frac{l}{m} \prod_{i=1}^m \frac{i}{m(e^{1/m} - 1) + i} \]
Since $e^{\frac{1}{m}} \geq 1+\frac{1}{m}$, we can upper bound the right-hand side to:
\[ p(T < l) \leq \exp^\frac{l}{m} \prod_{i=1}^m \frac{i}{1 + i} = \frac{\exp(\frac{l}{m})}{m+1} \]
Therefore, any choice of sample size $l(m)$ such that $\frac{\exp(\frac{l(m)}{m})}{m+1}$ goes to 0 in the limit is asymptotically too small (the probability of success goes to 0). One can check that $\frac{\exp(\frac{l}{m})}{m+1} = o(1)$ is equivalent to $l(m) = o(m \log m)$, which proves our claim.
\end{proof}
\bigskip
This transition occuring at $\O(m\log m)$ calls for discussion, and paves the way to future interesting lines of research.
First of all, Corollary~\ref{cor:guaranteed} shows, from an original angle, that
the repulsiveness of DPPs is weak. Indeed, other repulsive processes such as hard-core processes cannot verify such property in all generality. For instance, in the high density limit of a hard-sphere model, the probability that the position of $m$ non-overlapping spheres can be found within a set of only $\O(m \log m)$ iid points drawn uniformly, tends to $0$.
In addition, these results ask the following question: in what cases should one pay the extra cost of sampling $m$ elements from a DPP, rather than simply sampling $\O(m\log m)$ elements i.i.d. ? Of course, when the objective is to sample a diverse set, such as in search engines, then it is always worthwhile to sample the DPP. However, in the case of integration~\cite{bardenet2016monte, coeurjolly2020monte} for instance; or in the case of coresets~\cite{Tremblay:DPPforCoresets}, the answer is not so clear and requires investigation.
\subsubsection*{Acknowledgements}
This work was supported by the ANR project GRANOLA (ANR-21-CE48-0009), as well as the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01), the Grenoble Data Institute (ANR-15-IDEX- 02), MIAI@Grenoble Alpes (ANR-19-P3IA-0003), the LIA CNRS/Melbourne Univ Geodesic, and the IRS (Initiatives de Recherche Stratégiques) of the IDEX Université Grenoble Alpes.
|
{
"timestamp": "2022-11-01T01:28:00",
"yymm": "2210",
"arxiv_id": "2210.17358",
"language": "en",
"url": "https://arxiv.org/abs/2210.17358"
}
|
"\\chapter*{Abstract}\n\\addcontentsline{toc}{chapter}{Abstract}\n\nReinforcement learning (rl) is a(...TRUNCATED)
| {"timestamp":"2022-11-01T01:28:11","yymm":"2210","arxiv_id":"2210.17368","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nStudying causality in real-world environments is often challenging because(...TRUNCATED)
| {"timestamp":"2022-11-01T01:26:03","yymm":"2210","arxiv_id":"2210.17283","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\n\tA classical result of Hajnal \\cite{hajnal1983partition} and Thomassen(...TRUNCATED)
| {"timestamp":"2022-11-15T02:28:43","yymm":"2210","arxiv_id":"2210.17371","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nThere are lots of interests in studying supersymmetric theory with eight s(...TRUNCATED)
| {"timestamp":"2022-11-01T01:26:49","yymm":"2210","arxiv_id":"2210.17324","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\label{sec:intro}\n\nThese guidelines include complete descriptions of th(...TRUNCATED)
| {"timestamp":"2022-11-01T01:26:36","yymm":"2210","arxiv_id":"2210.17310","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\subsection{Overview}\n\\subsubsection{Traces and characters}\nIn linear (...TRUNCATED)
| {"timestamp":"2022-11-01T01:28:07","yymm":"2210","arxiv_id":"2210.17364","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\label{sec:intro}\n\nMultimodal data can provide complementary informatio(...TRUNCATED)
| {"timestamp":"2022-11-01T01:26:12","yymm":"2210","arxiv_id":"2210.17292","language":"en","url":"http(...TRUNCATED)
|
"\\section{}} \\textcolor{PineGreen} \\textcolor{BrickRed}\n\n\n\n \\def\\mathfrak sp{\\mathfrak sp(...TRUNCATED)
| {"timestamp":"2022-11-30T02:17:55","yymm":"2210","arxiv_id":"2210.17306","language":"en","url":"http(...TRUNCATED)
|
End of preview.