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diff --git a/main.tex b/main.tex new file mode 100755 index 0000000..fb194c7 --- /dev/null +++ b/main.tex @@ -0,0 +1,929 @@ +\documentclass[9pt]{beamer} +\renewcommand{\emph}[1]{\textcolor{blue}{#1}} +\newif\iflong +\longfalse +\setbeamerfont{footnote}{size=\scriptsize} + +\input{header} +\input{macros} + +\newcommand{\highlight}[2]{% + \colorbox{#1!20}{$\displaystyle#2$}} + +\newcommand{\hiat}[4]{% + \only<#1>{\highlight{#3}{#4}}% + \only<#2>{\highlight{white}{#4}}% +} + +\graphicspath{{figures/}} +\AtEveryCitekey{\clearfield{pages}} +\AtEveryCitekey{\clearfield{eprint}} +\AtEveryCitekey{\clearfield{volume}} +\AtEveryCitekey{\clearfield{number}} +\AtEveryCitekey{\clearfield{month}} +\addbibresource{main.bib} + +\title{Mobility estimation for Langevin dynamics using control variates\\[.3cm] + \small \textcolor{yellow}{AMMP Seminar} +} + +\author{% + Urbain Vaes \texorpdfstring{\\\texttt{urbain.vaes@inria.fr}}{} +} + +\institute{% + MATHERIALS -- Inria Paris + \textcolor{blue}{\&} CERMICS -- + École des Ponts ParisTech +} + +\date{October 2022} +\begin{document} + + +\begin{frame}[plain] + \begin{figure}[ht] + \centering + % \includegraphics[height=1.5cm]{figures/logo_matherials.png} + % \hspace{.5cm} + \includegraphics[height=1.2cm]{figures/logo_inria.png} + \hspace{.5cm} + \includegraphics[height=1.5cm]{figures/logo_ponts.png} + \hspace{.5cm} + \includegraphics[height=1.5cm]{figures/logo_ERC.jpg} + \hspace{.5cm} + \includegraphics[height=1.5cm]{figures/logo_EMC2.png} + \end{figure} + \titlepage +\end{frame} + +\begin{frame} + {Outline} + \tableofcontents +\end{frame} + +% \section{Some background material on fast/slow systems of SDEs}% +% \label{sec:numerical_solution_of_multiscale_sdes} + + +% \begin{frame} +% {Homogenization result} +% \begin{itemize} +% \item Effective drift: +% \[ +% \vect F(x) = \int_{\torus^n} \left(\vect f \, \cdot \, \grad_x \right) \vect \Phi(x,y) \, \rho^{\infty}(y;x) \, \d y. +% \] +% \item Effective diffusion: +% \begin{align*} +% & \mat A(x) \, \mat A(x)^T = \frac12 \left(\mat A_0(x) + \mat A_0(x)^T\right), \\ +% & \text{with } \mat A_0(x) := 2 \int_{\real^n} \vect f(x,y) \, \otimes \, \vect \Phi(x,y) \, \rho^{\infty}(y;x) \, \d y. +% \end{align*} +% \end{itemize} +% \begin{example} +% Multiscale system: +% \begin{alignat*}{2} +% & \d X^{\varepsilon}_t = \frac{1}{\varepsilon} X^{\varepsilon}_t \, Y^{\varepsilon}_t \, \d t, \quad & X^{\varepsilon}_0 = 1, \\ +% & \d Y^{\varepsilon}_t = - \frac{1}{\varepsilon^2} \, Y_t^{\varepsilon} \, \d t +% + \frac{\sqrt 2}{\varepsilon} \,\d W_{y}(t), \quad & Y^{\varepsilon}_0 = 0. +% \end{alignat*} +% Effective equation: +% \[ +% \d X_t = X_t \, \d t + \, \sqrt{2} \, X_t \, \d W_{y} (t). +% \] +% \end{example} +% \end{frame} + +% \begin{frame} +% {Example: Stratonovich correction} +% \begin{figure}[ht] +% \centering +% \href{run:videos/spectral/slow.avi?autostart&loop}% +% {\includegraphics[width=0.8\textwidth]{videos/spectral/slow.png}}% + +% \href{run:videos/spectral/fast.avi?autostart&loop}% +% {\includegraphics[width=0.8\textwidth]{videos/spectral/fast.png}}% +% \caption{% +% Convergence to the solution of the effective equation as $\varepsilon \to 0$. +% } +% \end{figure} +% \end{frame} + +\section{Mobility estimation for Langevin dynamics using control variates} +\begin{frame} + {Collaborators and reference} + \begin{figure} + \centering + \begin{minipage}[t]{.2\linewidth} + \centering + \raisebox{\dimexpr-\height+\ht\strutbox}{% + \includegraphics[height=\linewidth]{figures/collaborators/greg.jpg} + } + \end{minipage}\hspace{.01\linewidth}% + \begin{minipage}[t]{.24\linewidth} + Grigorios Pavliotis + \vspace{0.2cm} + + \includegraphics[height=1cm,width=\linewidth,keepaspectratio]{figures/collaborators/imperial.pdf} + \flushleft \scriptsize + Department of Mathematics + \end{minipage}\hspace{.1\linewidth}%% + \begin{minipage}[t]{.2\linewidth} + \centering + \raisebox{\dimexpr-\height+\ht\strutbox}{% + \includegraphics[height=\linewidth]{figures/collaborators/gabriel.jpg} + } + \end{minipage}\hspace{.01\linewidth}% + \begin{minipage}[t]{.24\linewidth} + Gabriel Stoltz + \vspace{0.2cm} + + \includegraphics[height=1.2cm,width=\linewidth,keepaspectratio]{figures/collaborators/enpc.png} + \flushleft \scriptsize + CERMICS + \end{minipage} + \end{figure} + + \vspace{.7cm} + \textbf{Reference:} + \fullcite{2022arXiv220609781P} +\end{frame} + + +% \begin{frame}[plain] +% \frametitle{Outline} +% \tableofcontents[subsectionstyle=show] +% \end{frame} + +\subsection{Background and problem statement}% + +\AtBeginSubsection[] +{ + \begin{frame}<beamer> + % \frametitle{Outline for section \thesection} + \frametitle{Outline} + \tableofcontents[currentsubsection,sectionstyle=show/shaded,subsectionstyle=show/shaded/hide] + % \tableofcontents[currentsubsection] + \end{frame} +} + +\begin{frame} + {Goals of molecular dynamics} + + {\large $\bullet$} Computation of \emph{macroscopic properties} from Newtonians atomistic models: + \vspace{-.1cm} + \begin{minipage}{.51\textwidth} + \vspace{-.7cm} + \begin{itemize} + \item Static properties, such as + \begin{itemize} + \item the heat capacity and + \item the equations of state $P = P(\rho, T)$. + \end{itemize} + + \vspace{.2cm} + \item Dynamical properties, such as \emph{transport coefficients}: + % mobilité, + % viscosité de cisaillement; + % conductivité thermique. + \begin{itemize} + \item the viscosity; + \item the thermal conductivity; + \item the \emph{mobility} of ions in solution. + \end{itemize} + \end{itemize} + \end{minipage} + \hspace{.5cm} + \begin{minipage}{.4\textwidth} + \begin{figure}[ht] + \centering + \includegraphics[width=.8\linewidth, angle=270]{figures/loi_argon-crop.pdf} + \caption*{\hspace{1.2cm}% + Equation of state of argon at 300K. + + \tiny\hspace{1.2cm}$\bullet$ `+': molecular simulation; + + \hspace{1.2cm}$\bullet$ Solid line: experimental measurements\footnotemark. + } + \end{figure} + \end{minipage} + \footnotetext{\url{https://webbook.nist.gov/chemistry/fluid/}} + + \vspace{.2cm} + {\large $\bullet$} \emph{Numerical microscope}: + used in physics, biology, chemistry. +\end{frame} + +\begin{frame} + {Some background material on the Langevin equation} + Consider the (one-particle) Langevin equation + \[ + \left\{ + \begin{aligned} + & \d \vect q_t = \textcolor{blue}{\vect p_t \, \d t}, \\ + & \d \vect p_t = \textcolor{blue}{- \grad V(\vect q_t) \, \d t} \, \textcolor{red}{- {\color{black}\gamma} \vect p_t \, \d t + \sqrt{2 {\color{black}\gamma} \beta^{-1}} \, \d \vect W_t}, + \end{aligned} + \right. + \qquad (\vect q_0, \vect p_0) \sim \mu, + \] + where $\gamma$ is the friction, $V$ is a \emph{periodic} potential, and $\beta = \frac{1}{k_{\rm B} T}$. + \begin{itemize} + % \item The dynamics is composed of a \textcolor{blue}{Hamiltonian} part and a \textcolor{red}{fluctuation/dissipation} part; + \item The invariant probability measure is + \[ + \mu(\vect q, \vect p) = \frac{1}{Z} \e^{-\beta H(\vect q, \vect p)} = \frac{1}{Z} \e^{-\beta \left(V(\vect q) + \frac{\abs{\vect p}^2}{2}\right)}, \quad \text{on}~ \emph{\torus^d} \times \real^d. + \] + \item The generator of the associated Markov semigroup + \[ + \left (\e^{\mathcal L t} \varphi\right) (\vect q, \vect p) = \expect \bigl(\varphi(\vect q_t, \vect p_t) \big| (\vect q_0, \vect p_0) = (\vect q, \vect p) \bigr) + \] + is the following operator: + \begin{align*} + \mathcal L &= \textcolor{blue}{\left(\vect p \cdot \grad_{\vect q} - \grad V(q) \cdot \grad_{\vect p} \right)} + + \gamma \, \textcolor{red}{\left( - \vect p \grad_{\vect p} + \beta^{-1} \laplacian_{\vect p} \right)} + =: \textcolor{blue}{\mathcal L_{\textrm{ham}}} + \gamma \, \textcolor{red}{\mathcal L_{\textrm{FD}}}. + \end{align*} + \end{itemize} + We denote by $\norm{\cdot}$ and $\ip{\cdot}{\cdot}$ the norm and inner product of~$L^2(\mu)$, and + \[ + L^2_0(\mu) = \Bigl\{\varphi \in L^2(\mu) : \ip{\varphi}{1} = \expect_{\mu} \varphi = 0 \Bigr\}. + \] +\end{frame} + + +% \begin{frame} +% {Common models in molecular simulation} +% We consider the following hierarchy of models: +% \begin{align} +% \label{eq:gle:model:overdamped} \tag{OL} +% \dot {\vect q} &= - \grad V(\vect q) + \sqrt{2 \, \beta^{-1}} \, \dot {\vect W}, \\ +% \label{eq:gle:model:langevin} \tag{L} +% \ddot {\vect q} &= - \grad V(\vect q) - \gamma \, \dot {\vect q} + \sqrt{2 \gamma \, \beta^{-1}} \, \dot {\vect W}, \\ +% \label{eq:gle:model:generalized} \tag{GLE} +% \ddot {\vect q} &= -\grad V(\vect q) - \int_{0}^{t} \widehat \gamma(t-s) \, \dot {\vect q}(s) \, \d s + \vect F(t). +% \end{align} +% where +% \begin{itemize} +% \item $V$ is a potential, in this talk \emph{periodic}; +% \item $\gamma$ is the friction coefficient; +% \item $\widehat \gamma(\cdot)$ is the memory kernel; +% \item $\vect F$ is a stationary non-Markovian noise process. +% \end{itemize} +% \vspace{.2cm} + +% The kernel $\widehat \gamma(\cdot)$ and the noise $F$ are related by the \emph{fluctuation/dissipation} relation: +% \[ +% \expect\bigl[\vect F(t) \otimes \vect F(s)\bigr] = \beta^{-1} \, \widehat \gamma(t-s) \mat I_d. +% \] +% \end{frame} + +% \subsection{Mobility and effective diffusion} +\begin{frame} + {Definition of the mobility} + Consider Langevin dynamics with additional forcing in a direction $\vect e$: + % \[ + % \ddot {\vect q} = - \grad V(\vect q) + \alert{\eta \vect e} - \gamma \, \dot {\vect q} + \sqrt{2 \, \gamma} \, \beta^{-1} \, \dot {\vect W}. + % \] + % This equation may be rewritten as a system for the position and momentum: + \[ + \left\{ + \begin{aligned} + & \d \vect q_t = \vect p_t \, \d t, \\ + & \d \vect p_t = - \grad V(\vect q_t) \, \d t + \alert{\eta \vect e} \, \d t - \gamma \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \, \d \vect W_t. + \end{aligned} + \right. + \] + This dynamics admits a unique invariant probability distribution $\mu_{\alert{\eta}} \in \mathcal P(\emph{\torus^d} \times \real^d)$. + + \begin{definition} + [Mobility] + The mobility in direction $\vect e$ is defined mathematically as + \[ + M_{\vect e} = + \lim_{\alert{\eta} \to 0} \frac{1}{\alert{\eta}}\expect_{\mu_{\alert{\eta}}} [\vect e^\t \vect p] + \] + $\approx $ factor relating the mean momentum to the strength of the inducing force. + \end{definition} + + \begin{itemize} + \item There is a symmetric mobility tensor $\mat M$ such that $M_{\vect e} = \vect e^\t \mat M \vect e$. + + \item + \textbf{Einstein's relation:} + \( + \mat D = \beta^{-1} \mat M, + \) with $\mat D$ the \emph{effective diffusion coefficient}. + \end{itemize} +\end{frame} + +\begin{frame} + {Effective diffusion} + It is possible to show a \emph{functional central limit theorem} for the Langevin dynamics\footfullcite{MR663900}: + \begin{equation*} + \varepsilon \vect q_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, \vect W_s + \qquad \text{weakly on } C([0, \infty)). + \end{equation*} + In particular, $\vect q_t/\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly. + + \vspace{-.25cm} + \begin{figure}[ht] + \centering + \href{run:videos/gle/effective-diffusion.webm?autostart&loop}% + {\includegraphics[width=0.75\textwidth]{videos/gle/effective-diffusion.png}}% + \caption{Histogram of $q_t/\sqrt{t}$. The potential $V(q) = - \cos(q) / 2$ is illustrated in the background.} + \end{figure} +\end{frame} + +\begin{frame} + {Mathematical expression for the effective diffusion (dimension 1)} + \vspace{.2cm} + \begin{exampleblock}{Expression of $D$ in terms of the solution to a Poisson equation} + The effective diffusion coefficient is given by where $D = \emph{ \ip{\phi}{p}}$ and $\phi$ is the solution to + \[ + \emph{- \mathcal L \phi = p}, + \qquad \phi \in L^2_0(\mu) := \bigl\{ u \in L^2(\mu): \ip{u}{1} = 0 \bigr\}. + \] + \end{exampleblock} + \textbf{Key idea of the proof:} Apply It\^o's formula to $\phi$ + \begin{align*} + \d \phi(q_s, p_s) + % &= \frac{1}{\varepsilon^2} \mathcal L_{L} \phi (q_t, p_t) + \frac{1}{\varepsilon} \, \sqrt{2 \gamma \beta^{-1}} \, \derivative{1}[\phi]{p}(q_t, p_t) \, \d W_t, \\ + &= - p_s \, \d s + \sqrt{2 \gamma \beta^{-1}} \, \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s + \end{align*} + and then rearrange: + \begin{align*} + \alert\varepsilon (q_{t/\alert\varepsilon^2} - q_{0}) &= \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} p_s \, \d s \\ + &= \underbrace{\alert\varepsilon \bigl(\phi(q_0, p_0) - \phi(q_{t/\alert\varepsilon^2}, p_{t/\alert\varepsilon^2})\bigr)}_{\to 0 + % ~\text{in $L^p(\Omega, C([0, T], \real))$} + } + + \underbrace{\sqrt{2 \gamma \beta^{-1}} \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s}_{\to \sqrt{2 D} W_t~\text{weakly by MCLT}}. + \end{align*} + % where + % \begin{align*} + % D &= \gamma \beta^{-1} \, \int \abs{\textstyle \derivative{1}[\phi]{p}(q, p)}^2 \, \mu(\d q \, \d p) + % = - \int \phi (\mathcal L \phi) \, \d \mu + % = \ip{\phi}{p}. + % \end{align*} + + \vspace{.3cm} + \textbf{In the multidimensional setting}, $D_{\vect e} = \ip{\phi_{\vect e}}{\vect e^\t \vect p}$ with $- \mathcal L \phi_{\vect e} = \vect e^\t \vect p$ +\end{frame} + +\begin{frame} + {Open question: surface diffusion when $\gamma \ll 1$\footnote{Source of the video: \url{https://en.wikipedia.org/wiki/Surface_diffusion}}} + \begin{figure}[ht] + \centering + \href{run:videos/surface_diffusion.webm?autostart&loop}% + {\includegraphics[width=0.4\linewidth]{videos/surface_diffusion.png}} + \hspace{1cm} + % \href{run:videos/diffusion.webm?autostart&loop}% + % {\includegraphics[width=0.4\linewidth]{figures/mean_square.pdf}} + \end{figure} + + \vspace{-.3cm} + Applications: + \begin{itemize} + \item integrated circuits; + \item catalysis. + \end{itemize} + + \textbf{Open question}: behavior of the effective diffusion coefficient when $\gamma \ll 1$? + \[ + D = \lim_{t \to \infty} \frac{\langle \abs{\vect q(t)}^2 \rangle}{4 t} \sim \gamma^{-\alert{\sigma}}, \qquad \alert{\sigma} =\, ??? + \] + % \vspace{-.3cm} + + % \textbf{Difficulty}: slow convergence of Monte Carlo methods when $\gamma$ is small. + % \vspace{.3cm} +\end{frame} + + +% \subsection{Some background material on the Langevin equation} + + +\begin{frame}{Langevin dynamics: \textcolor{yellow}{underdamped} and \textcolor{yellow}{overdamped} regimes\footfullcite{MR2394704}} + \begin{figure}[ht] + \centering + \href{run:videos/particles_underdamped.webm?autostart&loop}% + {\includegraphics[width=0.49\textwidth]{videos/particles_underdamped.png}}% + \href{run:videos/particles_overdamped.webm?autostart&loop}% + {\includegraphics[width=0.49\textwidth]{videos/particles_overdamped.png}}% + \caption{Langevin dynamics with friction $\gamma = 0.1$ (left) and $\gamma = 10$ (right)} + \end{figure} + + \vspace{-.3cm} + \begin{itemize} + \item The \alert{underdamped} limit as $\gamma \to 0$ is well understood in dimension 1 but not in the \alert{multi-dimensional setting}. + \item \emph{Overdamped} limit: + as $\gamma \to \infty$, the rescaled process $t \mapsto q_{\gamma t}$ converges weakly to the solution of the \emph{overdamped Langevin equation}: + \[ + \dot {\vect q} = - \grad V(q) + \sqrt{2 \, \beta^{-1}} \, \dot {\vect W}. + \] + \end{itemize} +\end{frame} + +\begin{frame} + {The \textcolor{yellow}{underdamped} limit in \textcolor{yellow}{dimension 1}} + As \emph{$\gamma \to 0$}, + the Hamiltonian of the rescaled process + \begin{equation*} + \left\{ + \begin{aligned} + q_{\gamma}(t) = q(t/\gamma), \\ + p_{\gamma}(t) = p(t/\gamma), + \end{aligned} + \right. + \end{equation*} + converges weakly to a diffusion process on a graph. + \vspace{-.6cm} + + \begin{figure}[ht!] + % \centering + % #1f77b4', u'#ff7f0e', u'#2ca02c + \definecolor{c1}{RGB}{31,119,180} + \definecolor{c2}{RGB}{255,127,14} + \definecolor{c3}{RGB}{44,160,44} + \begin{tikzpicture}% + \node[anchor=south west,inner sep=0] at (0,0) {% + \includegraphics[width=.7\textwidth]{figures/separatrix.eps} + }; + \coordinate (origin) at (10,0); + \coordinate (Emin) at ($ (origin) + (0,.5) $); + \coordinate (E0) at ($ (origin) + (0,2) $); + \coordinate (E1) at ($ (origin) + (-1,4) $); + \coordinate (E2) at ($ (origin) + (1,4) $); + \node at ($ (Emin) + (.7,0) $) {$E_{\min}$}; + \node[color=red] at ($ (E0) + (.5,0) $) {$E_{0}$}; + \node at ($ (E1) + (0,.3) $) {$p < 0$}; + \node at ($ (E2) + (0,.3) $) {$p > 0$}; + \draw[thick,color=c2] (Emin) -- (E0) node [color=black, midway, right] {}; + \draw[thick,color=c1] (E0) -- (E1) node [color=black, midway, left] {}; + \draw[thick,color=c3] (E0) -- (E2) node [color=black, midway, right] {}; + \node at (E0) [circle,fill,inner sep=1.5pt,color=red]{}; + \node at (Emin) [circle,fill,inner sep=1.5pt]{}; + \end{tikzpicture}% + \end{figure} + \vspace{-.5cm} + In this limit, it holds that + \[ + % \norm{\mathcal L^{-1}}_{\mathcal B\left(L^2_0(\mu)\right)} = \mathcal O \left( \alert{\gamma^{-1}} \right), + % \qquad + \phi = - \mathcal L^{-1} p = \alert{\gamma^{-1}} \phi_{\rm und} + \mathcal O(\gamma^{-1/2}). + \] + % The limiting function $\phi_{\rm und}$ is continuous but \alert{not in $H^1(\mu)$}. +\end{frame} + + +\begin{frame} + {Scaling of the effective diffusion coefficient for \textcolor{yellow}{Langevin} dynamics\footfullcite{MR2427108}} + In \alert{dimension 1}, + \( \lim_{\gamma \to 0} \gamma D^{\gamma} = D_{\rm und} \) and \( \lim_{\gamma \to \infty} \gamma D^{\gamma} = D_{\rm ovd}. \) + \begin{figure}[ht] + \centering + \includegraphics[width=0.5\linewidth,height=0.33\linewidth]{figures/scaling_diffusion_langevin.png} + \end{figure} + + \textbf{\emph{Our aims in this work:}} + \begin{itemize} + \item How can we efficiently estimate the effective diffusion coefficient when \alert{$\gamma \ll 1$}? + \item How does the mobility scale as \alert{$\gamma \to 0$} in the multidimensional setting? + \end{itemize} +\end{frame} + + +\subsection{Efficient mobility estimation}% + +\begin{frame} + {Brief literature review} + % Consider the Langevin dynamics with $(\vect q_t, \vect p_t) \in (\real^{\alert{d}} \times \real^{\alert{d}})$: + % \begin{equation*} + % \left\{ + % \begin{aligned} + % & \d \vect q_t = \vect p_t \,\d t, \\ + % & \d \vect p_t = - \grad V (\vect q_t) \, \d t - \gamma \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \, \d \vect W_t. + % \end{aligned} + % \right. + % \end{equation*} + In dimension $> 1$, it \alert{does not hold} that + $\gamma D^{\gamma}_{\vect e} \xrightarrow[\gamma \to 0]{} D_{\rm und}$ when $V$ is \alert{non separable}, e.g. + \[ + V(\vect q) = - \frac{1}{2} \Big( \cos(q_1) + \cos(q_2) \Big) - \alert{\delta} \cos(q_1) \cos(q_2) + \] + + \textbf{Open question:} + how does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}? + % \begin{block} + % {Open question: how does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}?} + + Various answers are given in the literature: + \begin{itemize} + \item + $D^{\gamma}_{\vect e} \propto \gamma^{-1/2}$ for specific potentials\footfullcite{chen1996surface}; + + \item + $D^{\gamma}_{\vect e} \propto \gamma^{-1/3}$ for specific potentials~\footfullcite{Braun02}; + + \item + $D^{\gamma}_{\vect e} \propto \gamma^{-\sigma}$ with $\sigma$ depending on the potential~\footfullcite{roussel_thesis}. + \end{itemize} + % \end{block} + \vspace{.5cm} +\end{frame} + +\begin{frame}[label=continue] + {Numerical approaches for calculating the effective diffusion coefficient} + \begin{itemize} + \itemsep.5cm + \item \emph{Linear response approach}: + \begin{equation*} + D_{\vect e} = \lim_{\eta \to 0} \frac{1}{\beta \alert{\eta}} \expect_{\alert{\mu_\eta}} \, (\vect e^\t \vect p). + \end{equation*} + where $\mu_{\eta}$ is the invariant distribution of the system with external forcing. + + \item \emph{Green--Kubo formula}: Since $-\mathcal L^{-1} = \int_{0}^{\infty} \e^{t \mathcal L} \, \d t$, + \begin{align*} + D_{\vect e} &= \int - \mathcal L^{-1}(\vect e^\t \vect p) \, (\vect \e^\t \vect p) \, \d \mu = \int_{0}^{\infty} \! \! \! \int \e^{t \mathcal L} (\vect e^\t \vect p) (\vect e^\t \vect p) \, \d \mu \, \d t \\ + &= \int_{0}^{\infty} \expect_{\mu}\bigl((\vect e^\t \vect p_0) (\vect e^\t \vect p_t)\bigr) \, \d t. + \end{align*} + + \item \emph{Einstein's relation}: + \[ + D_{\vect e} = \lim_{t \to \infty} \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl|\vect e^\t (\vect q_t - \vect q_0)\bigr|^2 \Bigr]. + \] + + \item Deterministic method, e.g. \emph{Fourier/Hermite Galerkin}, for the Poisson equation + \[ + - \mathcal L \phi_{\vect e} = \vect e^\t \vect p, \qquad D_{\vect e} = \ip{\phi_{\vect e}}{p}. + \] + \end{itemize} +\end{frame} + +% \begin{frame} +% {Fourier/Hermite Galerkin method for one-dimensional Langevin dynamics} +% +% Saddle-point formulation\footfullcite{roussel2018spectral}: +% find $(\Phi_N, \alpha_N) \in V_N \times \real$ such that +% \begin{align} +% \notag +% - \Pi_N \, \mathcal L \, \Pi_N \alert{\Phi_N} + \alert{\alpha_N} u_N &= \Pi_N p, \\ +% \label{eq:constraint} +% \ip{\Phi_N}{u_N} &= 0, +% \end{align} +% where +% \begin{itemize} +% \item $\Pi_N$ is the $L^2(\mu)$ projection operator on a finite-dimensional subspace $V_N$, +% \item $u_N = \Pi_N 1 / \norm{\Pi_N 1}$. +% Eq.~\eqref{eq:constraint} ensures that the system is \emph{well-conditioned}. +% \end{itemize} +% +% \vspace{.2cm} +% For $V_N$, we use the following basis functions: +% \[ +% e_{i,j} = {\left( Z \, \e^{\beta \left( H(q,p) + |z|^2 \right)} \right)}^{\frac{1}{2}} \, G_i(q) \, H_j(p), \qquad 0 \leq i,j \leq N, +% \] +% where $(G_i)_{i \geq 0}$ are \emph{trigonometric functions} and $(H_j)_{i \geq 0}$ are \emph{Hermite polynomials}. +% +% $\rightarrow$ \alert{Impractical} in two or more spatial dimensions. +% \end{frame} + +\begin{frame} + {Estimation of the effective diffusion coefficient from Einstein's relation} + Consider the following estimator of the effective diffusion coefficient $D_{\vect e}$: + \[ + \emph{u(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2T}}, \qquad (\vect q_0, \vect p_0) \sim \mu. + \] + + \textbf{Bias of this estimator:} + \begin{align*} + \notag + \expect \bigl[u(T)\bigr] + % &= \int_{0}^{\infty} \ip{\e^{t \mathcal L}(\vect e^\t \vect p)}{\vect e^\t \vect p} \d t + % - \int_{0}^{\infty} \ip{\e^{t \mathcal L} (\vect e^\t \vect p)}{\vect e^\t \vect p} \min\left\{1, \frac{t}{T}\right\} \, \d t \\ + &= D_{\vect e} - \int_{0}^{\infty} \ip{\e^{t \mathcal L} (\vect e^\t \vect p)}{\vect e^\t \vect p} \min\left\{1, \frac{t}{T}\right\} \, \d t. + \end{align*} + Using the decay estimate for the semigroup\footfullcite{roussel2018spectral} + \[ + \norm{\e^{t \mathcal L}}_{\mathcal B\left(L^2_0(\mu)\right)} \leq L \e^{- \ell \min\{\gamma, \gamma^{-1}\}t}, + \] + we deduce + \[ + \left\lvert \expect[u(T)] - D_{\vect e} \right\rvert \leq \frac{C \textcolor{red}{\max\{\gamma^2, \gamma^{-2}\}}}{T}. + \] +\end{frame} + +\begin{frame} + {Variance of the estimator $u(T)$ for large $T$} + For $T \gg 1$, + it holds approximately that + \[ + \frac{\vect e^\t (\vect q_T - \vect q_0)}{\sqrt{2T}} \sim \mathcal N(0, D_{\vect e}) + \qquad \leadsto \qquad + u(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2 D_{\vect e} T} \sim \chi^2 (1). + \] + Therefore, we deduce + \[ + \lim_{T \to \infty} \var \bigl[u(T)\bigr] = 2 D_{\vect e}^2. + \] + The relative standard deviation (asymptotically as $T \to \infty$) is therefore + \[ + \lim_{T \to \infty} \frac{\sqrt{\var \bigl[u(T)\bigr]}}{\expect \bigl[u(T)\bigr]} = \sqrt{2} + \qquad \leadsto \text{\emph{independent} of $\gamma$}. + \] + + \begin{exampleblock}{Scaling of the mean square error when using $J$ realizations} + Assuming an asymptotic scaling as $\gamma^{-\sigma}$ of $D_{\vect e}$, we have + \[ + \forall \gamma \in (0, 1), \qquad + \frac{\rm MSE}{D_{\vect e}^2} \leq \frac{C}{\gamma^{4-2 \sigma} T^2} + \frac{2}{J} + \] + \end{exampleblock} +\end{frame} + +% \subsection{Variance reduction using control variates} +\begin{frame} + {Variance reduction using \textcolor{yellow}{control variates}} + Let $\phi_{\vect e}$ denote the solution to the \emph{Poisson equation} + \[ + - \mathcal L \phi_{\vect e}(\vect q, \vect p) = \vect e^\t \vect p, \qquad \phi_{\vect e} \in L^2_0(\mu). + \] + and let $\psi_{\vect e}$ denote an approximation of $\phi_{\vect e}$. + By It\^o's formula, + we obtain + \[ + \phi_{\vect e}(\vect q_T, \vect p_T) - \phi_{\vect e}(\vect q_0, \vect p_0) + = - \int_{0}^{T} \vect e^\t \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \int_{0}^{T} \grad_{\vect p} \phi_{\vect e}(\vect q_t, \vect p_t) \cdot \d \vect W_t. + \] + Therefore + \begin{align*} + \vect e^\t (\vect q_T - \vect q_0) + &= \int_{0}^{T} \vect e^\t \vect p_t \, \d t \\ + &\approx - \psi_{\vect e}(\vect q_T, \vect p_T) + \psi_{\vect e}(\vect q_0, \vect p_0) + \sqrt{2 \gamma \beta^{-1}} \int_{0}^{T} \grad_{\vect p} \psi_{\vect e}(\vect q_t, \vect p_t) \cdot \d \vect W_t + =: \emph{\xi_T}. + \end{align*} + which suggests the \emph{improved estimator} + \[ + v(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2T} - \left( \frac{\abs{\xi_T}^2}{2T} - \lim_{T\to \infty}\expect \left[ \frac{\abs{\xi_T}^2}{2T} \right] \right). + \] +\end{frame} + +\begin{frame} + {Properties of the improved estimator} + \textbf{Smaller bias} if $-\mathcal L \psi_{\vect e} \approx \vect e^\t \vect p$: + \begin{align*} + \label{eq:basic_bound_bias} + \abs{\expect \bigl[ v(T) \bigr] - D^{\gamma}_{\vect e}} + &\leq \frac{L \max\{\gamma^2, \gamma^{-2}\}}{T \ell^2 } \, \emph{\norm{\vect e^\t \vect p + \mathcal L \psi_{\vect e}}} \left(\beta^{-1/2} + \norm{\mathcal L \psi_{\vect e}} \right). + \end{align*} + + \textbf{Smaller variance}: + \begin{equation*} + \begin{aligned}[b] + \var \bigl[v(T)\bigr] + \leq + C &\left( T^{-1} \emph{\norm{\phi_{\vect e} - \psi_{\vect e}}[L^4(\mu)]}^2 + \gamma \emph{\norm{\grad_{\vect p} \phi_{\vect e} - \grad_{\vect p} \psi_{\vect e}}[L^4(\mu)]}^2 \right) \\ + &\quad \times \left( T^{-1} \norm{\phi_{\vect e} + \psi_{\vect e}}[L^4(\mu)]^2 + \gamma \norm{\grad_{\vect p} \phi_{\vect e} + \grad_{\vect p} \psi_{\vect e}}[L^4(\mu)]^2 \right). + \end{aligned} + \end{equation*} + + + \textbf{Construction of $\psi_{\vect e}$ in the \alert{one-dimensional setting}}. We consider two approaches: + \begin{itemize} + \item Approximate the solution to the Poisson equation by a Galerkin method. + \item Use asymptotic result for the Poisson equation: + \[ + \gamma \phi \xrightarrow[\gamma \to 0]{L^{2}(\mu)} \phi_{\rm und}, + \] + which suggests letting $\psi = \phi_{\rm und} / \gamma$. + \end{itemize} +\end{frame} + +\begin{frame} + {Construction of the approximate solution $\psi_{\vect e}$ \textcolor{yellow}{in dimension 2}} + We consider the potential + \[ + V(\vect q) = - \frac{1}{2} \Big( \cos(q_1) + \cos(q_2) \Big) - \alert{\delta} \cos(q_1) \cos(q_2). + \] + \begin{itemize} + \item + For this potential, $\mat D$ is isotropic + $\leadsto$ sufficient to consider $\vect e = (1, 0)$, + \[ + D_{(1,0)} = \ip{\phi_{(1, 0)}}{p_1}, + \qquad - \mathcal L \phi_{(1,0)}(\vect q, \vect p) = p_1. + \] + + \item + If \emph{$\delta = 0$}, then the solution is $\phi_{(1, 0)}(\vect q, \vect p) = \phi_{\rm 1D} (q_1, p_1)$, + where $\phi_{\rm 1D}$ solves + \[ + - \mathcal L_{\rm 1D} \phi_{\rm 1D}(q, p) = p, \qquad V_{\rm 1D}(q) = \frac{1}{2} \cos (q). + \] + + \item + We take $\emph{\psi_{(1,0)}(\vect q, \vect p) = \psi_{\rm 1D}(q_1, p_1)}$, + where $\psi_{\rm 1D} \approx \phi_{\rm 1D}$. + \end{itemize} +\end{frame} + +\subsection{Numerical experiments}% +\begin{frame} + {Numerical experiments for the one-dimensional case (1/2)} + \begin{figure}[ht] + \centering + \includegraphics[width=0.99\linewidth]{figures/underdamped_1d.pdf} + \end{figure} +\end{frame} + +\begin{frame} + {Numerical experiments for the one-dimensional case (2/2)} + \begin{figure}[ht] + \centering + \includegraphics[width=0.99\linewidth]{figures/time.pdf} + \caption{Evolution of the sample mean and standard deviation, estimated from $J = 5000$ realizations for $\gamma = 10^{-3}$.} + \end{figure} +\end{frame} + +\begin{frame} + {Performance of the control variates approach in dimension 2} + \begin{figure}[ht] + \centering + \includegraphics[width=0.49\linewidth]{figures/var-delta-galerkin.pdf} + \includegraphics[width=0.49\linewidth]{figures/var-delta-underdamped.pdf} + \label{fig:time_bias_deviation_2d} + \end{figure} + \begin{itemize} + \item Variance reduction is possible if $\abs{\delta}/\gamma \ll 1$; + \item Control variates are \alert{not very useful} when $\gamma \ll 1$ and $\delta$ is fixed. + \end{itemize} +\end{frame} + +\begin{frame} + {Scaling of the mobility in dimension 2} + \begin{figure}[ht] + \centering + \includegraphics[width=0.9\linewidth]{figures/diffusion.pdf} + \label{fig:time_bias_variance_2d} + \end{figure} +\end{frame} + +\begin{frame}{Summary and perspectives for future work} + In this talk, we presented + \begin{itemize} + \item a variance reduction approach for efficiently estimating the mobility; + \item numerical results showing that the scaling of the mobility is \emph{not universal}. + \end{itemize} + + \textbf{Perspectives for future work:} + \begin{itemize} + \item Use alternative methods (PINNs, Gaussian processes) to solve the Poisson equation; + \item Improve and study variance reduction approaches for other transport coefficients. + \end{itemize} + + \vspace{1cm} + \begin{center} + Thank you for your attention! + \end{center} +\end{frame} + +\section{Optimal importance sampling for overdamped Langevin dynamics} + +\begin{frame} + {Collaborators} + \begin{figure} + \centering + \begin{minipage}[t]{.2\linewidth} + \centering + \raisebox{\dimexpr-\height+\ht\strutbox}{% + \includegraphics[height=\linewidth]{figures/collaborators/tony.jpg} + } + \end{minipage}\hspace{.03\linewidth}% + \begin{minipage}[t]{.21\linewidth} + Tony Lelièvre + \vspace{0.2cm} + + \includegraphics[height=1.2cm,width=\linewidth,keepaspectratio]{figures/collaborators/enpc.png} + \flushleft \scriptsize + CERMICS \& Inria + \end{minipage}\hspace{.1\linewidth}%% + \begin{minipage}[t]{.2\linewidth} + \centering + \raisebox{\dimexpr-\height+\ht\strutbox}{% + \includegraphics[height=\linewidth]{figures/collaborators/gabriel.jpg} + } + \end{minipage}\hspace{.01\linewidth}% + \begin{minipage}[t]{.24\linewidth} + Gabriel Stoltz + \vspace{0.2cm} + + \includegraphics[height=1.2cm,width=\linewidth,keepaspectratio]{figures/collaborators/enpc.png} + \flushleft \scriptsize + CERMICS \& Inria + \end{minipage} + \end{figure} + + \vspace{.7cm} + \textbf{Outline:} + \vspace{.2cm} + \tableofcontents +\end{frame} + +\subsection{Background and problem statement} + +\begin{frame} + {The sampling problem} + + \begin{exampleblock} + {Objective of the sampling problem} + Calculate averages with respect to + \[ + \mu = \frac{\e^{-V}}{Z}, + \qquad Z = \int_{\torus^d} \e^{-V}. + \] + \vspace{-.4cm} + \end{exampleblock} + + \vspace{-.2cm} + \textbf{Often in applications}: + \begin{itemize} + \item The dimension $d$ is large; + \item The normalization constant $Z$ is unknown; + \item We cannot generate i.i.d.\ samples from~$\mu$. + \end{itemize} + + \textbf{Markov chain Monte Carlo (MCMC) approach}: + \[ + \mu(f) \approx \mu^T (f) := \frac{1}{T} \int_{0}^{T} f(Y_t) \, \d t + \] + for a Markov process $(Y_t)_{t\geq 0}$ that is \emph{ergodic} with respect to~$\mu$. + + \textbf{Example}: \emph{overdamped Langevin} dynamics + \[ + \d Y_t = -\nabla V(Y_t) \, \d t + \sqrt{2} \, \d W_t, + \qquad Y_0 = y_0. + \] +\end{frame} + +\begin{frame} + {Importance sampling in the MCMC context} + If $(X_t)_{t \geq 0}$ is a Markov process ergodic with respect to + \[ + \mu_{U} = \frac{\e^{-V - U}}{Z_U}, + \qquad Z_U = \int_{\torus^d} \e^{-V-U}, + \] + then $\mu(f)$ may be approximated by + \begin{equation*} + \label{eq:estimator} + \mu^T_U(f) := + \frac + {\displaystyle \frac{1}{T} \int_0^T (f \e^U)(X_t) \, \d t} + {\displaystyle \frac{1}{T} \int_0^T(\e^U)(X_t) \, \d t}. + \end{equation*} + + \textbf{Asymptotic variance}: + Under appropriate conditions, + it holds that + \[ + \sqrt{T} \bigl( \mu^T_U(f) - \mu(f)\bigr) + \xrightarrow[T \to \infty]{\rm Law} \mathcal N\bigl(0, \sigma^2_f[U]\bigr). + \] + + \begin{exampleblock} + {Objective} + Find $U$ such that the asymptotic variance $\sigma^2_f[U]$ is minimized. + \end{exampleblock} +\end{frame} + +\begin{frame} + {Background} +\end{frame} + + +\appendix + +\begin{frame}[noframenumbering,plain] + {Connection with the asymptotic variance of MCMC estimators} + \textbf{Ergodic theorem\footfullcite{MR885138}}: for an observable $\varphi \in L^1(\mu)$, + \[ + \widehat \varphi_t = \frac{1}{t} \int_{0}^{t} \varphi(\vect q_s, \vect p_s) \, \d s + \xrightarrow[t \to \infty]{a.s.} \expect_{\mu} \varphi. + \] + + \textbf{Central limit theorem\footfullcite{MR663900}}: + If the following \emph{Poisson equation} has a solution $\phi \in L^2(\mu)$, + \[ + - \mathcal L \phi = \varphi - \expect_{\mu} \varphi, + \] + then a central limit theorem holds: + \[ + \sqrt{t} \bigl(\widehat \varphi_t - \expect_{\mu}\varphi\bigr) + \xrightarrow[t \to \infty]{\rm Law} \mathcal N(0, \sigma^2_{\varphi}), + \qquad + \sigma^2_{\varphi} + = \ip{\phi}{\varphi - \expect_{\mu} \varphi}. + \] + + \textbf{Connection with effective diffusion}: Apply this result with $\varphi(\vect q, \vect p) = \vect e^\t \vect p$. +\end{frame} + +\end{document} + +% vim: ts=2 sw=2 |