Add some slides

1 files changed, 50 insertions, 1133 deletions

diff --git a/main.tex b/main.tex
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--- a/main.tex
+++ b/main.tex
@@ -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}