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@@ -1,8 +1,7 @@ -\documentclass[9pt]{beamer} +\documentclass[10pt]{beamer} \renewcommand{\emph}[1]{\textcolor{blue}{#1}} \newif\iflong \longfalse -\setbeamerfont{footnote}{size=\scriptsize} \input{header} \input{macros} @@ -23,8 +22,8 @@ \AtEveryCitekey{\clearfield{month}} \addbibresource{main.bib} -\title{Variance reduction for applications in computational statistical physics\\[.3cm] - \small \textcolor{yellow}{IRMAR -- Séminaire de probabilités}% +\title{Nonequilibrium systems and computation of transport coefficients\\[.3cm] + \small \textcolor{yellow}{SINEQ Summer school}% } \author{% @@ -37,7 +36,7 @@ École des Ponts ParisTech } -\date{27 October 2022} +\date{\today} \begin{document} \begin{frame}[plain] @@ -57,1166 +56,84 @@ \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} - % {Part I: Mobility estimation for Langevin dynamics using control variates} - \begin{center} - \Large - \color{blue} - Part I: Mobility estimation for Langevin dynamics - \end{center} - \medskip - \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 computational statistical physics} - - {\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 $\beta = \frac{1}{k_{\rm B} T}$. + {Some references} \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*} + \item \fullcite{MR3509213} + \item \fullcite{pavliotis2011applied} + \item Lecture notes by Gabriel Stoltz on computational statistical physics: + \url{http://cermics.enpc.fr/~stoltz/Cours/intro_phys_stat.pdf} \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: - \begin{equation*} - \varepsilon \widetilde {\vect q}_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, \vect W_s - \qquad \text{weakly on } C([0, \infty)), \qquad \widetilde {\vect q}_t := \vect q_0 + \int_{0}^{t} \vect p_s \, \d s \in \emph{\real^{d}}. - \end{equation*} - In particular, $\widetilde {\vect q}_t /\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ 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{block}{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{block} - \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 (\widetilde q_{t/\alert\varepsilon^2} - \widetilde 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}{Langevin dynamics: \textcolor{yellow}{underdamped} and \textcolor{yellow}{overdamped} regimes\footfullcite{MR2394704}} - \vspace{-.3cm} - \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{-.4cm} - \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}. - In dimension 1, it holds that - \[ - \phi = - \mathcal L^{-1} p = \alert{\gamma^{-1}} \phi_{\rm und} + \mathcal O(\gamma^{-1/2}). - \] - \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} - \vspace{.2cm} -\end{frame} - -\begin{frame} - {Scaling of the effective diffusion coefficient for \textcolor{yellow}{Langevin} dynamics\footfullcite{MR2427108}} - In \alert{dimension 1}, - \( \lim_{\gamma \to 0} \gamma D^{\gamma} = D_{\rm und} := \ip{\phi_{\rm und}}{p} \) and \( \lim_{\gamma \to \infty} \gamma D^{\gamma} = D_{\rm ovd}. \) - \begin{figure}[ht] - \centering - \includegraphics[width=0.7\linewidth,height=0.45\linewidth]{figures/scaling_diffusion_langevin.png} - \end{figure} - - % \textbf{\emph{Our aims in this part:}} - % \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} - -\begin{frame} - {Open question: surface diffusion when $\gamma \ll 1$\footnote{Source of the video: \url{https://en.wikipedia.org/wiki/Surface_diffusion}}} - \vspace{-.1cm} - \begin{minipage}[t]{.49\linewidth} - Applications: - \begin{itemize} - \item integrated circuits; - \item catalysis. - \end{itemize} - \end{minipage} - \begin{minipage}[t]{.49\linewidth} - \vspace{-.3cm} - \begin{figure}[ht] - \centering - \href{run:videos/surface_diffusion.webm?autostart&loop}% - {\includegraphics[width=0.8\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} - \end{minipage} - - 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}: behavior of the effective diffusion coefficient when $\gamma \ll 1$? - \[ - D^{\gamma}_{\vect e} = \lim_{t \to \infty} \frac{\expect \Bigl[ \abs{\vect e^\t \vect q_t}^2 \Bigr]}{2 t} \sim \gamma^{-\alert{\sigma}}, \qquad \alert{\sigma} =\, ??? - \] + {Outline} + \tableofcontents \end{frame} +\section{Introduction} \begin{frame} - {Brief literature review} - % \textbf{Open question:} - \begin{block}{Open question:} - How does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}? - \end{block} - % {Open question: how does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}?} + {Transport coefficients} + At the \alert{macroscopic} level, + transport coefficients relate an external forcing to an average response expressed through some steady-state flux. - Various answers are given in the literature: + \textbf{Examples:} \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}. + \item The \emph{mobility} relates an external force to a velocity; + \item The \emph{heat conductivity} relates a temperature difference to a heat flux; + \item The \emph{shear viscosity} relates a shear velocity to a shear stress; \end{itemize} - \vspace{.5cm} - \textbf{Difficulty with $\gamma \ll 1$}: + \vspace{.3cm} + \textbf{Challenges we do not address:} \begin{itemize} - \item Deterministic methods for the Poisson equation $-\mathcal L \phi_{\vect e} = \vect e^\t \vect p$ are ill-conditioned. - \item Probabilistic methods are very slow to converge. + \item Choose thermodynamical ensemble; + \item Prescribe microscopic dynamics; \end{itemize} - % \end{block} - \vspace{.5cm} -\end{frame} - - -\subsection{Efficient mobility estimation}% - - -\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{Einstein's relation}: - \[ - D_{\vect e} = \lim_{t \to \infty} \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl|\vect e^\t (\widetilde {\vect q}_t - \vect q_0)\bigr|^2 \Bigr]. - \] - - \item Deterministic method, e.g. \emph{Fourier/Hermite Galerkin}, for the Poisson equation - \[ - - \mathcal L \phi_{\vect e} = \vect e^\t \vect p, \qquad D_{\vect e} = \ip{\phi_{\vect e}}{p}. - \] - - \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*} - \end{itemize} -\end{frame} - -\begin{frame} - {Estimation of the effective diffusion coefficient from Einstein's relation} - Consider the following estimator of the effective diffusion coefficient $D_{\vect e}$: - \[ - \emph{u(T) = \frac{\abs{\vect e^\t (\widetilde{\vect q}_T - \widetilde {\vect q}_0)}^2}{2T}}, \qquad (\vect q_0, \vect p_0) \sim \mu. - \] - - \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 - \[ - \frac{\vect e^\t (\widetilde {\vect q}_T - \widetilde {\vect q}_0)}{\sqrt{2T}} \sim \mathcal N(0, D_{\vect e}) - \qquad \leadsto \qquad - \frac{u(T)}{D_{\vect e}} = \frac{\abs{\vect e^\t (\widetilde {\vect q}_T - \widetilde {\vect q}_0)}^2}{2 D_{\vect e} T} \sim \chi^2 (1). - \] - 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{block}{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{block} -\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}, - \[ - - \mathcal L \phi_{\vect e}(\vect q, \vect p) = \vect e^\t \vect p, \qquad \phi_{\vect e} \in L^2_0(\mu) - \] - 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 - if $\psi_{\vect e}$ denote an approximation of $\phi_{\vect e}$, then - \begin{align*} - \vect e^\t (\widetilde {\vect q}_T - \widetilde {\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} - \[ - v(T) = \frac{\abs{\vect e^\t (\widetilde {\vect q}_T - \widetilde {\vect q}_0)}^2}{2T} - \left( \frac{\abs{\xi_T}^2}{2T} - \lim_{T\to \infty}\expect \left[ \frac{\abs{\xi_T}^2}{2T} \right] \right). - \] -\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 - \[ - V(\vect q) = - \frac{1}{2} \Big( \cos(q_1) + \cos(q_2) \Big) - \alert{\delta} \cos(q_1) \cos(q_2). - \] + {Computation of transport coefficients} + Three main classes of methods: \begin{itemize} + \itemsep.2cm \item - For this potential, $\mat D$ is isotropic - $\leadsto$ sufficient to consider $\vect e = (1, 0)$, - \[ - D_{(1,0)} = \ip{\phi_{(1, 0)}}{p_1}, - \qquad - \mathcal L \phi_{(1,0)}(\vect q, \vect p) = p_1. - \] + Non-equilibrium techniques + \begin{itemize} + \item Calculations from the steady state of a system out of equilibrium. + \item Comprises bulk-driven and boundary-driven approaches. + \end{itemize} \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 + Equilibrium techniques based on the Green--Kubo formula \[ - - \mathcal L_{\rm 1D} \phi_{\rm 1D}(q, p) = p, \qquad V_{\rm 1D}(q) = \frac{1}{2} \cos (q). + \rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t. \] - \item - We take $\emph{\psi_{(1,0)}(\vect q, \vect p) = \psi_{\rm 1D}(q_1, p_1)}$, - where $\psi_{\rm 1D} \approx \phi_{\rm 1D}$. + Transient techniques: \end{itemize} \end{frame} -\subsection{Numerical experiments}% +\section{Equilibrium and nonequilibrium dynamics} \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} as $\gamma \to 0$ 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 of part I and perspectives for future work} - In this part, 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 Study and improve variance reduction approaches for other transport coefficients. - \end{itemize} -\end{frame} - -\section{Optimal importance sampling for overdamped Langevin dynamics} - -% \begin{frame} -% \begin{center} -% \huge Part II: Optimal importance sampling for overdamped Langevin dynamics -% \end{center} -% \end{frame} - -\begin{frame} - \begin{center} - \Large - \color{blue} - Part II: importance sampling for overdamped Langevin dynamics - \end{center} - - \begin{figure} - \centering - \begin{minipage}[t]{.2\linewidth} - \centering - \raisebox{\dimexpr-\height+\ht\strutbox}{% - \includegraphics[height=\linewidth]{figures/collaborators/martin.jpg} - } - \end{minipage}\hspace{.03\linewidth}% - \begin{minipage}[t]{.21\linewidth} - Martin Chak - \vspace{0.2cm} - - \includegraphics[height=1.2cm,width=\linewidth,keepaspectratio]{figures/collaborators/sorbonne.png} - \flushleft \scriptsize - Sorbonne Université - \end{minipage}\hspace{.1\linewidth}%% - \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}%% - - \vspace{.5cm} - \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} -\end{frame} - -\subsection{Background and problem statement} - -\begin{frame} - {The sampling problem} - - \vspace{.2cm} - - \begin{block} - {Objective of the sampling problem} - Calculate averages of the form - \[ - I := \mu(f) := \int_{\torus^d} f \, \d \mu, - \qquad \mu := \frac{\e^{-V}}{Z}, - \qquad Z := \int_{\torus^d} \e^{-V}. - \] - \vspace{-.4cm} - \end{block} - - \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}: - \[ - I := \mu(f) \approx \mu^T (f) := \frac{1}{T} \int_{0}^{T} f(Y_t) \, \d t - \] - 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 $I = \mu(f)$ 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{Markov process}: \emph{overdamped Langevin} dynamics - \[ - \d X_t = -\nabla (V+U)(X_t) \, \d t + \sqrt{2} \, \d W_t, - \qquad X_0 = x_0. - \] - - \textbf{Asymptotic variance}: - Under appropriate conditions, - it holds that - \[ - \sqrt{T} \bigl( \mu^T_U(f) - I \bigr) - \xrightarrow[T \to \infty]{\rm Law} \mathcal N\bigl(0, \sigma^2_f[U]\bigr). - \] - - \begin{block} - {Objective} - Find $U$ such that the asymptotic variance $\sigma^2_f[U]$ is minimized. - \end{block} -\end{frame} - -\begin{frame} - {Background: importance sampling in the i.i.d.\ setting (1/2)} - Given i.i.d.\ samples $\{X^1, X^2, \dotsc\}$ from $\mu_U$, - we define - \[ - \mu_U^N(f) := - \displaystyle \frac - {\sum_{n=1}^{N} (f \e^U)(X^{n})} - {\sum_{n=1}^{N} (\e^U)(X^{n})} - = I + \displaystyle \frac - {\frac{1}{N} \sum_{n=1}^{N} \left((f-I) \e^U\right)(X^{n})} - {\frac{1}{N} \sum_{n=1}^{N} (\e^U)(X^{n})}, - \] - - \textbf{Numerator:} by the \emph{central limit theorem}, - \[ - \frac{1}{\sqrt{N}} \sum_{n=1}^{N} \left((f-I) \e^U\right) (X^{n}) - \xrightarrow[N \to \infty]{\rm Law} \mathcal N\left(0, \int_{\torus^d} \Bigl\lvert (f-I) \e^U \Bigr\rvert^2 \, \d \mu_{U}\right) - \] - - \textbf{Denominator:} by the strong law of large numbers, - \[ - \frac{1}{N} \sum_{n=1}^{N} \left(\e^U\right)\left(X^{n}\right) \xrightarrow[N \to \infty]{\rm a.s.} - \frac{Z}{Z_U}. - \] - - \textbf{Therefore}, by Slutsky's theorem, - \[ - \sqrt{N} \bigl( \mu^N_U(f) - I\bigr) - \xrightarrow[T \to \infty]{\rm Law} \mathcal N\bigl(0, s^2_f[U]\bigr), - \qquad - s^2_f[U] := \frac{Z_U^2}{Z^2} \int_{\torus^n} \bigl\lvert (f-I) \e^U \bigr\rvert^2 \, \d \mu_{U}. - \] -\end{frame} - -\begin{frame} - {Background: importance sampling in the i.i.d.\ setting (2/2)} - By the Cauchy--Schwarz inequality, - it holds that - \[ - s^2_f[U] - \geq \frac{Z_U^2}{Z^2} \left( \int_{\torus^d} \abs{f-I} \e^U \, \d \mu_{U} \right)^2 - = \frac{1}{Z^2} \left( \int_{\torus^d} \abs{f-I} \e^{-V} \right)^2, - \] - with equality when $\abs{f-I} \e^U$ is constant. - - \begin{block} - {Optimal importance distribution} - The \emph{optimal $\mu_U$} in the i.i.d.\ setting is - \[ - \mu_{U} \propto \abs{f-I} \e^{-V} - \] - \end{block} - - \textbf{Objectives}: - \begin{itemize} - \item Is there a counterpart of this formula in the \emph{MCMC setting}? - \item If not, can we approximate the optimal distribution numerically? - \end{itemize} -\end{frame} - -\subsection{Minimizing the asymptotic variance for one observable} -\begin{frame} - {Formula for the asymptotic variance} - Let $\mathcal L_U$ denote the generator of the Markov semigroup associated to the modified potential; - \[ - \mathcal L_U = - \nabla (V + U) \cdot \nabla + \Delta. - \] - \begin{block} - {Limit theorem} - Under appropriate conditions, - it holds that - \[ - \sqrt{T} \bigl( \mu^T_U(f) - I\bigr) - \xrightarrow[T \to \infty]{\rm Law} \mathcal N\bigl(0, \sigma^2_f[U]\bigr). - \] - The \emph{asymptotic variance} is given by - \[ - \sigma^2_f[U] - = \frac{2Z_U^2}{Z^2}\int_{\torus^d} \phi_U (f-I) \, \e^U \, \d\mu_{U}, - \] - where $\phi_U$ is the unique solution in~$H^1(\mu_{U}) \cap L^2_0(\mu_{U})$ to - \[ - -\mathcal L_U \phi_{U} = (f- I) \e^U. - \] - \end{block} - \textbf{Main ideas of the proof:} central limit theorem for martingales, Slutsky's theorem. -\end{frame} - -\begin{frame} - {Explicit optimal $U$ in dimension 1} - In \emph{dimension one}, it holds that - \begin{equation} - \label{eq:lower_bound_asymvar} - \sigma^2_f[U] \geq \frac{2}{Z^2} \inf_{A \in \real} \bigg(\int_{\torus} \bigl\lvert F(x) + A \bigr\rvert \d x \bigg)^2. - \end{equation} - where - \[ - F(x) := \int_0^x \bigl( f(\xi)-I \bigr) \e^{-V(\xi)}\d \xi. - \] - This inequality~\eqref{eq:lower_bound_asymvar} is an equality for - \[ - U(x) = U_*(x) = - V(x) -\ln\abs*{F(x) + A_*}, - \] - where $A_*$ is the constant achieving the infimum in~\eqref{eq:lower_bound_asymvar}. - - \begin{itemize} - \item The potential $U_*$ is generally \alert{singular}: impractical for numerics\dots - \item The lower bound in~\eqref{eq:lower_bound_asymvar} can be approached by a smooth~$U$. - \end{itemize} -\end{frame} - -\begin{frame} - {Example (1/2)} - Assume that $V = 0$ and $f(x) = \cos(x)$. - \begin{figure}[ht] - \centering - \includegraphics[width=0.8\linewidth]{figures/driftopt/1d_optimal_cosine.pdf} - \label{fig:optimal_perturbation_potential} - \end{figure} - $\rightsquigarrow$ The optimal potential ``divides'' the domain into two parts. -\end{frame} - -\begin{frame} - {Example (2/2)} - Assume that $V(x) = 5\cos(2 x)$ and~$f(x) = \sin(x)$. - The target measure is \alert{multimodal}. - \begin{figure}[ht] - \centering - \includegraphics[width=0.8\linewidth]{figures/driftopt/1d_optimal_metastable.pdf} - \label{fig:optimal_perturbation_potential_1d_metastable} - \end{figure} - \emph{Variance reduction} by a factor $> 1000!$ -\end{frame} - -\begin{frame} - {Finding the optimal $U$ in the multidimensional setting} - - \begin{proposition} - [Functional derivative of the asymptotic variance] - Let $\phi_U$ denote the solution to - \begin{equation} - \label{eq:poisson} - -\mathcal L_U \phi_{U} = (f- I) \e^U. - \end{equation} - Under appropriate conditions, - it holds for all $\delta U \in C^{\infty}(\torus^d)$ that - \begin{align} - \notag - \frac{1}{2} \d \sigma^2_f[U] \cdot \delta U - &:= \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} \bigl(\sigma^2_f[U + \varepsilon \delta U] - \sigma^2_f[U]\bigr) \\ - \label{eq:funcder} - &= \frac{Z_U^2}{Z^2} \int_{\torus^d} \delta U \bigg( \abs*{\nabla{\phi_{U}}}^2 - \int_{\torus^d} \abs*{\nabla {\phi_{U}}}^2 \, \d \mu_{U} \bigg) \, \d \mu_{U}. - \end{align} - \end{proposition} - \textbf{Steepest descent approach}: - \begin{itemize} - \item Solve the Poisson equation~\eqref{eq:poisson} numerically; - \item Construct an ascent direction $G$ for $\sigma^2_f$ using~\eqref{eq:funcder}, e.g.\ $\delta U = \abs*{\nabla{\phi_{U}}}^2$; - \item Perform a step in this direction: $U \leftarrow U - \eta G$; - \item Repeat until convergence. - \end{itemize} -\end{frame} - -\begin{frame} - {No smooth minimizers} - \begin{corollary} - [No smooth minimizer] - \label{corollary:no_smooth_minimizer} - Unless~$f$ is constant, - there is no perturbation potential~$U \in C^\infty(\torus^n)$ that is a critical point of $\sigma^2_f[U]$. - \end{corollary} - \textbf{Proof.} - Assume by contradiction that $U_*$ is smooth critical point. - Then - \[ - 0 = \frac{1}{2} \d \sigma^2_f[U_*] \cdot \delta U - = \frac{Z_U^2}{Z^2} \int_{\torus^d} \delta U \bigg( \abs*{\nabla{\phi_{U_*}}}^2 - \int_{\torus^d} \abs*{\nabla {\phi_{U_*}}}^2 \, \d \mu_{U_*} \bigg) \, \d \mu_{U_*}, - \] - for all $\delta U \in C^{\infty}(\torus^d)$. - \begin{itemize} - \item Therefore, it must hold that $\abs*{\nabla {\phi_{U_*}}}^2 = C$ is constant. - \item Since $\phi_{U_*}$ is a smooth function, there is $x \in \torus^d$ such that $\nabla \phi_{U_*}(x) = 0$. - \item Consequently $C = 0$ and so $\nabla \phi_{U_*} = 0$: \alert{contradiction} because then $\mathcal L_{U_*} \phi_{U_*} = 0$. - \end{itemize} - - \vspace{.5cm} - $\rightsquigarrow$ The optimal perturbation potential is \alert{not convenient} in practice\dots -\end{frame} - -\begin{frame} - {Example (1/3)} - Assume that $V = 0$ and $f(x) = \sin(x_1) + \sin(x_2)$. - \begin{figure}[ht] - \centering - \includegraphics[width=0.49\linewidth]{figures/driftopt/2d_optimal.pdf} - \includegraphics[width=0.49\linewidth]{figures/driftopt/2d_optimal_poisson.pdf} - \caption{% - Optimal total potential (left) - together with the solution to the associated Poisson equation (right). - } - \label{fig:2d_first_example} - \end{figure} - $\rightsquigarrow$ The domain is again divided into subdomains that suffice for estimating~$I$. -\end{frame} - -\begin{frame} - {Example (2/3): multimodal target $\e^{-V}$} - Assume that $V(x) = 2\cos(x_1) - \cos(x_2)$ and~$f(x) = \sin(x_1)$. - \begin{figure}[ht] - \centering - \includegraphics[width=0.49\linewidth]{figures/driftopt/2d_optimal_multimodal.pdf} - \includegraphics[width=0.49\linewidth]{figures/driftopt/2d_optimal_multimodal_heatmap.pdf} - \label{fig:2d_metastable} - \end{figure} - \emph{Variance reduction} by a factor $\approx 6$! -\end{frame} - -\begin{frame} - {Example (2/3): A more complicated example} - In this case we consider that - \[ - V(x) = \exp\left(\cos(x_1) \sin(x_2) + \frac{1}{5} \cos(3x_1)\right), - \qquad - f(x) = \sin\Bigl(x_1 + \cos(x_2)\Bigr)^3. - \] -\begin{figure}[ht] - \centering - \includegraphics[width=0.49\linewidth]{figures/2d_difficult_V.pdf} - \includegraphics[width=0.49\linewidth]{figures/2d_difficult_optimal} - \caption{% - Unperturbed potential~$V$ (left) and optimal potential~$V+U$ (right). - } - \label{fig:2d_difficult} -\end{figure} -\end{frame} - -\subsection{Minimizing the asymptotic variance for a class of observables} -\begin{frame} - {Alternative: minimizing the expected variance over \textcolor{yellow}{a class of observables}} - Assume that the observables are well described by a Gaussian random field - \[ - f = \sum_{j=1}^{J} \sqrt{\lambda_j} u_j f_j, - \qquad u_j \sim \mathcal N(0, 1), - \qquad \lambda_j \in (0, \infty). - \] - \textbf{Question:} can we find~$U$ such that $\sigma^2[U] := \expect \bigl( \sigma^2_f[U] \bigr)$ is minimized? - - \begin{itemize} - \item It holds that - \[ - \sigma^2[U] = \sum_{j=1}^{J} \lambda_j \sigma^2_{f_j}. - \] - - \item - The functional derivative of $\sigma^2[U]$ is given by - \[ - \frac{1}{2} \d\sigma^2[U] \cdot \delta U - = \frac{Z_U^2}{Z^2} \int_{\torus^d} \left( \delta U - \int_{\torus^d} \delta U \, \d \mu_U \right) \left( \sum_{j=1}^{J} \lambda_j \abs*{\nabla{\phi_j}}^2 \right) \, \d \mu_{U}. - \] - - \item - The steepest descent approach can be employed in this case too! - \end{itemize} -\end{frame} - -\begin{frame} - {Example} - Here $V(x) = 2 \cos(2 x_1) - \cos(x_2)$ and $f \sim \mathcal N\bigl(0, (\laplacian + \mathcal I)^{-1}\bigr)$. - \begin{figure}[ht] - \centering - \includegraphics[width=0.49\linewidth]{figures/driftopt/2d_initial_class_metastable.pdf} - \includegraphics[width=0.49\linewidth]{figures/driftopt/2d_optimal_class_metastable.pdf} - \caption{% - Potential~$V$ (left) and optimal potential~$V+U$ (right). - } - \end{figure} -\end{frame} - -\begin{frame} - {Summary of part II and perspectives for future work} - In this part, - \begin{itemize} - \item We studied an importance sampling approach for the overdamped Langevin dynamics. - \item We proposed an approach for calculating the optimal perturbation potential. - \end{itemize} - - \textbf{Perspectives}: - \begin{itemize} - \item Solving the Poisson equation accurately is not possible in high dimension. - \item Application to high-dimensional systems: - \[ - U(x) = U\bigl(\xi(x)\bigr), \qquad \xi \text{ reaction coordinate}. - \] - \end{itemize} - \vspace{1cm} - \begin{center} - \Large - \emph{Thank you for your attention!} - \end{center} -\end{frame} - -\appendix - -\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} - % The limiting function $\phi_{\rm und}$ is continuous but \alert{not in $H^1(\mu)$}. -\end{frame} - - -\begin{frame}[noframenumbering,plain] - {Connection with the asymptotic variance of MCMC estimators} - \textbf{Ergodic theorem\footfullcite{MR885138}}: for an observable $\varphi \in L^1(\mu)$, + {Equilibrium and nonequilibrium dynamics} + Consider a general diffusion process of the form \[ - \widehat \varphi_t = \frac{1}{t} \int_{0}^{t} \varphi(\vect q_s, \vect p_s) \, \d s - \xrightarrow[t \to \infty]{a.s.} \expect_{\mu} \varphi. + \d x_t = b(x_t) \, \d t + \sigma(x_t) \, \d W_t, \] + and assume that it admits an invariant distribution $\mu$. - \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: - \[ - \sqrt{t} \bigl(\widehat \varphi_t - \expect_{\mu}\varphi\bigr) - \xrightarrow[t \to \infty]{\rm Law} \mathcal N(0, \sigma^2_{\varphi}), - \qquad - \sigma^2_{\varphi} - = \ip{\phi}{\varphi - \expect_{\mu} \varphi}. - \] + \vspace{.2cm} + \begin{definition} + [Time-reversibility] + A stationary ($x_0 \sim \mu$) stochastic process $(x_t)$ is time-reversible if its law is invariant under time reversal: + the law of the \emph{forward paths} $(x_s)_{0 \leq s \leq t}$ + coincides with the law of the \emph{backward paths} $(x_{t-s})_{0 \leq s \leq t}$. + \end{definition} - \textbf{Connection with effective diffusion}: Apply this result with $\varphi(\vect q, \vect p) = \vect e^\t \vect p$. + \vspace{.2cm} + \begin{theorem} + A stationary diffusion processes $x_t$ in $\real^d$ with generator $\mathcal L$ and invariant measure~$\mu$ is reversible if and only if $\mathcal L$ is self-adjoint in~$L^2(\mu)$. + \end{theorem} \end{frame} \end{document} |