From b8a8579c0d9587a071e34c19f8f12348ea8cd88c Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Mon, 23 Jan 2023 08:52:32 +0100
Subject: Minor changes

---
 main.tex | 340 +++++++++++++++++++++++++++++++--------------------------------
 1 file changed, 165 insertions(+), 175 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 6d005a9..b27a8ff 100755
--- a/main.tex
+++ b/main.tex
@@ -23,8 +23,8 @@
 \AtEveryCitekey{\clearfield{month}}
 \addbibresource{main.bib}
 
-\title{Mobility estimation for Langevin dynamics using control variates\\[.3cm]
-  \small \textcolor{yellow}{AMMP Seminar}
+\title{Variance reduction for applications in computational statistical physics\\[.3cm]
+  \small \textcolor{yellow}{IRMAR -- Séminaire de probabilités}%
 }
 
 \author{%
@@ -37,10 +37,9 @@
     École des Ponts ParisTech
 }
 
-\date{October 2022}
+\date{27 October 2022}
 \begin{document}
 
-
 \begin{frame}[plain]
   \begin{figure}[ht]
     \centering
@@ -116,6 +115,7 @@
       \color{blue}
       Part I: Mobility estimation for Langevin dynamics
     \end{center}
+    \medskip
     \begin{figure}
         \centering
         \begin{minipage}[t]{.2\linewidth}
@@ -172,7 +172,7 @@
 }
 
 \begin{frame}
-  {Goals of molecular dynamics}
+  {Goals of computational statistical physics}
 
   {\large $\bullet$} Computation of \emph{macroscopic properties} from Newtonians atomistic models:
   \vspace{-.1cm}
@@ -322,12 +322,12 @@
 
 \begin{frame}
   {Effective diffusion}
-  It is possible to show a \emph{functional central limit theorem} for the Langevin dynamics\footfullcite{MR663900}:
+  It is possible to show a \emph{functional central limit theorem} for the Langevin dynamics:
   \begin{equation*}
-    \varepsilon  \vect q_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, \vect W_s
-    \qquad  \text{weakly on } C([0, \infty)).
+    \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, $\vect q_t/\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly.
+  In particular, $\widetilde {\vect q}_t /\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly.
 
   \vspace{-.25cm}
   \begin{figure}[ht]
@@ -356,7 +356,7 @@
   \end{align*}
   and then rearrange:
   \begin{align*}
-    \alert\varepsilon (q_{t/\alert\varepsilon^2} - q_{0}) &= \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} p_s \, \d s \\
+    \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 $L^p(\Omega, C([0, T], \real))$}
                                                           }
@@ -373,39 +373,8 @@
   \textbf{In the multidimensional setting}, $D_{\vect e} = \ip{\phi_{\vect e}}{\vect e^\t \vect p}$ with $- \mathcal L \phi_{\vect e} = \vect e^\t \vect p$
 \end{frame}
 
-\begin{frame}
-  {Open question: surface diffusion when $\gamma \ll 1$\footnote{Source of the video: \url{https://en.wikipedia.org/wiki/Surface_diffusion}}}
-    \begin{figure}[ht]
-        \centering
-        \href{run:videos/surface_diffusion.webm?autostart&loop}%
-        {\includegraphics[width=0.4\linewidth]{videos/surface_diffusion.png}}
-        \hspace{1cm}
-        % \href{run:videos/diffusion.webm?autostart&loop}%
-        % {\includegraphics[width=0.4\linewidth]{figures/mean_square.pdf}}
-    \end{figure}
-
-    \vspace{-.3cm}
-    Applications:
-    \begin{itemize}
-      \item integrated circuits;
-      \item catalysis.
-    \end{itemize}
-
-    \textbf{Open question}: behavior of the effective diffusion coefficient when $\gamma \ll 1$?
-    \[
-      D = \lim_{t \to \infty} \frac{\langle \abs{\vect q(t)}^2 \rangle}{4 t} \sim \gamma^{-\alert{\sigma}}, \qquad \alert{\sigma} =\, ???
-    \]
-    % \vspace{-.3cm}
-
-    % \textbf{Difficulty}: slow convergence of Monte Carlo methods when $\gamma$ is small.
-    % \vspace{.3cm}
-\end{frame}
-
-
-% \subsection{Some background material on the Langevin equation}
-
-
 \begin{frame}{Langevin dynamics: \textcolor{yellow}{underdamped} and \textcolor{yellow}{overdamped} regimes\footfullcite{MR2394704}}
+  \vspace{-.3cm}
   \begin{figure}[ht]
     \centering
     \href{run:videos/particles_underdamped.webm?autostart&loop}%
@@ -415,108 +384,78 @@
     \caption{Langevin dynamics with friction $\gamma = 0.1$ (left) and $\gamma = 10$ (right)}
   \end{figure}
 
-  \vspace{-.3cm}
+  \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}
-  {The \textcolor{yellow}{underdamped} limit in \textcolor{yellow}{dimension 1}}
-  As \emph{$\gamma \to 0$},
-  the Hamiltonian of the rescaled process
-  \begin{equation*}
-    \left\{
-    \begin{aligned}
-    q_{\gamma}(t) = q(t/\gamma), \\
-    p_{\gamma}(t) = p(t/\gamma),
-    \end{aligned}
-    \right.
-  \end{equation*}
-  converges weakly to a diffusion process on a graph.
-  \vspace{-.6cm}
-
-  \begin{figure}[ht!]
-      % \centering
-      % #1f77b4', u'#ff7f0e', u'#2ca02c
-      \definecolor{c1}{RGB}{31,119,180}
-      \definecolor{c2}{RGB}{255,127,14}
-      \definecolor{c3}{RGB}{44,160,44}
-      \begin{tikzpicture}%
-      \node[anchor=south west,inner sep=0] at (0,0) {%
-          \includegraphics[width=.7\textwidth]{figures/separatrix.eps}
-      };
-      \coordinate (origin) at (10,0);
-      \coordinate (Emin) at ($ (origin) + (0,.5) $);
-      \coordinate (E0) at ($ (origin) + (0,2) $);
-      \coordinate (E1) at ($ (origin) + (-1,4) $);
-      \coordinate (E2) at ($ (origin) + (1,4) $);
-      \node at ($ (Emin) + (.7,0) $) {$E_{\min}$};
-      \node[color=red] at ($ (E0) + (.5,0) $) {$E_{0}$};
-      \node at ($ (E1) + (0,.3) $) {$p < 0$};
-      \node at ($ (E2) + (0,.3) $) {$p > 0$};
-      \draw[thick,color=c2] (Emin) -- (E0) node [color=black, midway, right] {};
-      \draw[thick,color=c1] (E0) -- (E1) node [color=black, midway, left] {};
-      \draw[thick,color=c3] (E0) -- (E2) node [color=black, midway, right] {};
-      \node at (E0) [circle,fill,inner sep=1.5pt,color=red]{};
-      \node at (Emin) [circle,fill,inner sep=1.5pt]{};
-      \end{tikzpicture}%
-  \end{figure}
-  \vspace{-.5cm}
-  In this limit, it holds that
-  \[
-    % \norm{\mathcal L^{-1}}_{\mathcal B\left(L^2_0(\mu)\right)} = \mathcal O \left( \alert{\gamma^{-1}} \right),
-    % \qquad
-    \phi = - \mathcal L^{-1} p = \alert{\gamma^{-1}} \phi_{\rm und} + \mathcal O(\gamma^{-1/2}).
-  \]
-  % The limiting function $\phi_{\rm und}$ is continuous but \alert{not in $H^1(\mu)$}.
-\end{frame}
-
-
 \begin{frame}
   {Scaling of the effective diffusion coefficient for \textcolor{yellow}{Langevin} dynamics\footfullcite{MR2427108}}
   In \alert{dimension 1},
-  \( \lim_{\gamma \to 0} \gamma D^{\gamma} = D_{\rm und} \) and \( \lim_{\gamma \to \infty} \gamma D^{\gamma} = D_{\rm ovd}. \)
+  \( \lim_{\gamma \to 0} \gamma D^{\gamma} = D_{\rm und} := \ip{\phi_{\rm und}}{p} \) and \( \lim_{\gamma \to \infty} \gamma D^{\gamma} = D_{\rm ovd}. \)
   \begin{figure}[ht]
       \centering
-      \includegraphics[width=0.5\linewidth,height=0.33\linewidth]{figures/scaling_diffusion_langevin.png}
+      \includegraphics[width=0.7\linewidth,height=0.45\linewidth]{figures/scaling_diffusion_langevin.png}
   \end{figure}
 
-  \textbf{\emph{Our aims in this work:}}
-  \begin{itemize}
-    \item How can we efficiently estimate the effective diffusion coefficient when \alert{$\gamma \ll 1$}?
-    \item How does the mobility scale as \alert{$\gamma \to 0$} in the multidimensional setting?
-  \end{itemize}
+  % \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}
 
-
-\subsection{Efficient mobility estimation}%
-
 \begin{frame}
-  {Brief literature review}
-  % Consider the Langevin dynamics with $(\vect q_t, \vect p_t) \in (\real^{\alert{d}} \times \real^{\alert{d}})$:
-  % \begin{equation*}
-  %     \left\{
-  %       \begin{aligned}
-  %       & \d \vect q_t = \vect p_t \,\d t, \\
-  %       & \d \vect p_t = - \grad V (\vect q_t)  \, \d t - \gamma \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \, \d \vect W_t.
-  %       \end{aligned}
-  %     \right.
-  % \end{equation*}
+  {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.
+  $\gamma D^{\gamma}_{\vect e} \xrightarrow[\gamma \to 0]{} D_{\rm und}$ when $V$ is \alert{non-separable}, e.g.
   \[
     V(\vect q) = - \frac{1}{2} \Big( \cos(q_1) + \cos(q_2) \Big) - \alert{\delta} \cos(q_1) \cos(q_2)
   \]
 
-  \textbf{Open question:}
-  how does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}?
-  % \begin{block}
+  \textbf{Open question}: behavior of the effective diffusion coefficient when $\gamma \ll 1$?
+    \[
+      D^{\gamma}_{\vect e} = \lim_{t \to \infty} \frac{\expect \Bigl[ \abs{\vect e^\t \vect q_t}^2 \Bigr]}{2 t} \sim \gamma^{-\alert{\sigma}}, \qquad \alert{\sigma} =\, ???
+    \]
+\end{frame}
+
+\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{$d = 2$}?}
 
   Various answers are given in the literature:
@@ -530,10 +469,21 @@
     \item
       $D^{\gamma}_{\vect e} \propto \gamma^{-\sigma}$ with $\sigma$ depending on the potential~\footfullcite{roussel_thesis}.
   \end{itemize}
+
+  \vspace{.5cm}
+  \textbf{Difficulty with $\gamma \ll 1$}:
+  \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.
+  \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}
@@ -544,57 +494,29 @@
         \end{equation*}
         where $\mu_{\eta}$ is the invariant distribution of the system with external forcing.
 
-    \item \emph{Green--Kubo formula}: Since $-\mathcal L^{-1} = \int_{0}^{\infty} \e^{t \mathcal L} \, \d t$,
-      \begin{align*}
-        D_{\vect e} &= \int - \mathcal L^{-1}(\vect e^\t \vect p) \, (\vect \e^\t \vect p) \, \d \mu = \int_{0}^{\infty} \! \! \! \int \e^{t \mathcal L} (\vect e^\t \vect p) (\vect e^\t \vect p) \, \d \mu \, \d t  \\
-          &= \int_{0}^{\infty} \expect_{\mu}\bigl((\vect e^\t \vect p_0) (\vect e^\t \vect p_t)\bigr) \, \d t.
-      \end{align*}
-
     \item \emph{Einstein's relation}:
       \[
-        D_{\vect e} = \lim_{t \to \infty}  \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl|\vect e^\t (\vect q_t - \vect q_0)\bigr|^2 \Bigr].
+        D_{\vect e} = \lim_{t \to \infty}  \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl|\vect e^\t (\widetilde {\vect q}_t - \vect q_0)\bigr|^2 \Bigr].
       \]
 
     \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}
-%   {Fourier/Hermite Galerkin method for one-dimensional Langevin dynamics}
-%
-%   Saddle-point formulation\footfullcite{roussel2018spectral}:
-%   find $(\Phi_N, \alpha_N) \in V_N \times \real$ such that
-%   \begin{align}
-%        \notag
-%        - \Pi_N \, \mathcal L \, \Pi_N \alert{\Phi_N} + \alert{\alpha_N} u_N &= \Pi_N p, \\
-%        \label{eq:constraint}
-%         \ip{\Phi_N}{u_N} &= 0,
-%   \end{align}
-%   where
-%   \begin{itemize}
-%     \item $\Pi_N$ is the $L^2(\mu)$ projection operator on a finite-dimensional subspace $V_N$,
-%     \item $u_N = \Pi_N 1 / \norm{\Pi_N 1}$.
-%       Eq.~\eqref{eq:constraint} ensures that the system is \emph{well-conditioned}.
-%   \end{itemize}
-%
-%   \vspace{.2cm}
-%   For $V_N$, we use the following basis functions:
-%   \[
-%     e_{i,j} = {\left( Z \, \e^{\beta \left( H(q,p) + |z|^2 \right)} \right)}^{\frac{1}{2}} \, G_i(q) \, H_j(p), \qquad 0 \leq i,j \leq N,
-%   \]
-%   where $(G_i)_{i \geq 0}$ are \emph{trigonometric functions} and $(H_j)_{i \geq 0}$ are \emph{Hermite polynomials}.
-%
-%   $\rightarrow$ \alert{Impractical} in two or more spatial dimensions.
-% \end{frame}
-
 \begin{frame}
   {Estimation of the effective diffusion coefficient from Einstein's relation}
   Consider the following estimator of the effective diffusion coefficient $D_{\vect e}$:
   \[
-    \emph{u(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2T}}, \qquad (\vect q_0, \vect p_0) \sim \mu.
+    \emph{u(T) = \frac{\abs{\vect e^\t (\widetilde{\vect q}_T - \widetilde {\vect q}_0)}^2}{2T}}, \qquad (\vect q_0, \vect p_0) \sim \mu.
   \]
 
   \textbf{Bias of this estimator:}
@@ -620,9 +542,9 @@
   For $T \gg 1$,
   it holds approximately that
   \[
-    \frac{\vect e^\t (\vect q_T - \vect q_0)}{\sqrt{2T}} \sim \mathcal N(0, D_{\vect e})
+    \frac{\vect e^\t (\widetilde {\vect q}_T - \widetilde {\vect q}_0)}{\sqrt{2T}} \sim \mathcal N(0, D_{\vect e})
     \qquad \leadsto \qquad
-    u(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2 D_{\vect e} T} \sim \chi^2 (1).
+    \frac{u(T)}{D_{\vect e}} = \frac{\abs{\vect e^\t (\widetilde {\vect q}_T - \widetilde {\vect q}_0)}^2}{2 D_{\vect e} T} \sim \chi^2 (1).
   \]
   Therefore, we deduce
   \[
@@ -646,11 +568,10 @@
 % \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}
+  Let $\phi_{\vect e}$ denote the solution to the \emph{Poisson equation},
   \[
-    - \mathcal L \phi_{\vect e}(\vect q, \vect p) = \vect e^\t \vect p, \qquad \phi_{\vect e} \in L^2_0(\mu).
+    - \mathcal L \phi_{\vect e}(\vect q, \vect p) = \vect e^\t \vect p, \qquad \phi_{\vect e} \in L^2_0(\mu)
   \]
-  and let $\psi_{\vect e}$ denote an approximation of $\phi_{\vect e}$.
   By It\^o's formula,
   we obtain
   \[
@@ -658,15 +579,16 @@
     = - \int_{0}^{T} \vect e^\t \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \int_{0}^{T} \grad_{\vect p} \phi_{\vect e}(\vect q_t, \vect p_t) \cdot \d \vect W_t.
   \]
   Therefore
+  if $\psi_{\vect e}$ denote an approximation of $\phi_{\vect e}$, then
   \begin{align*}
-    \vect e^\t (\vect q_T - \vect q_0)
+    \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}
   \[
-    v(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2T} - \left( \frac{\abs{\xi_T}^2}{2T} - \lim_{T\to \infty}\expect \left[ \frac{\abs{\xi_T}^2}{2T} \right]  \right).
+    v(T) = \frac{\abs{\vect e^\t (\widetilde {\vect q}_T - \widetilde {\vect q}_0)}^2}{2T} - \left( \frac{\abs{\xi_T}^2}{2T} - \lim_{T\to \infty}\expect \left[ \frac{\abs{\xi_T}^2}{2T} \right]  \right).
   \]
 \end{frame}
 
@@ -757,7 +679,7 @@
   \end{figure}
   \begin{itemize}
     \item Variance reduction is possible if $\abs{\delta}/\gamma \ll 1$;
-    \item Control variates are \alert{not very useful} when $\gamma \ll 1$ and $\delta$ is fixed.
+    \item Control variates are \alert{not very useful} as $\gamma \to 0$ and $\delta$ is fixed…
   \end{itemize}
 \end{frame}
 
@@ -780,7 +702,7 @@
   \textbf{Perspectives for future work:}
   \begin{itemize}
     \item Use alternative methods (PINNs, Gaussian processes) to solve the Poisson equation;
-    \item Improve and study variance reduction approaches for other transport coefficients.
+    \item Study and improve variance reduction approaches for other transport coefficients.
   \end{itemize}
 \end{frame}
 
@@ -853,12 +775,15 @@
 \begin{frame}
   {The sampling problem}
 
+  \vspace{.2cm}
+
   \begin{block}
     {Objective of the sampling problem}
-    Calculate averages with respect to
+    Calculate averages of the form
     \[
-      \mu = \frac{\e^{-V}}{Z},
-      \qquad Z = \int_{\torus^d} \e^{-V}.
+      I := \mu(f) := \int_{\torus^d} f \, \d \mu,
+      \qquad \mu := \frac{\e^{-V}}{Z},
+      \qquad Z := \int_{\torus^d} \e^{-V}.
     \]
     \vspace{-.4cm}
   \end{block}
@@ -937,7 +862,7 @@
   \textbf{Numerator:} by the \emph{central limit theorem},
   \[
     \frac{1}{\sqrt{N}} \sum_{n=1}^{N} \left((f-I) \e^U\right) (X^{n})
-    \xrightarrow[N \to \infty]{\rm Law} \mathcal N\left(0, \int_{\torus^d} \abs*{(f-I) \e^U}^2 \, \d \mu_{U}\right)
+    \xrightarrow[N \to \infty]{\rm Law} \mathcal N\left(0, \int_{\torus^d} \Bigl\lvert (f-I) \e^U \Bigr\rvert^2 \, \d \mu_{U}\right)
   \]
 
   \textbf{Denominator:} by the strong law of large numbers,
@@ -951,7 +876,7 @@
       \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{2 Z_U^2}{Z^2} \int_{\torus^n} \bigl\lvert (f-I) \e^U \bigr\rvert^2 \, \d \mu_{U}.
+      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}
 
@@ -961,7 +886,8 @@
   it holds that
   \[
       s^2_f[U]
-      \geq \frac{2Z_U^2}{Z^2} \left( \int_{\torus^d} \abs{f-I} \e^U \, \d \mu_{U} \right)^2,
+      \geq \frac{Z_U^2}{Z^2} \left( \int_{\torus^d} \abs{f-I} \e^U \, \d \mu_{U} \right)^2
+      = \frac{1}{Z^2} \left( \int_{\torus^d} \abs{f-I} \e^{-V}  \right)^2,
   \]
   with equality when $\abs{f-I} \e^U$ is constant.
 
@@ -1100,9 +1026,9 @@
   \]
   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 $\mathcal L_{U_*} \phi_U = 0$.
+    \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}
@@ -1110,7 +1036,7 @@
 \end{frame}
 
 \begin{frame}
-  {Example (1/2)}
+  {Example (1/3)}
   Assume that $V = 0$ and $f(x) = \sin(x_1) + \sin(x_2)$.
   \begin{figure}[ht]
       \centering
@@ -1126,7 +1052,7 @@
 \end{frame}
 
 \begin{frame}
-  {Example (2/2): multimodal target $\e^{-V}$}
+  {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
@@ -1137,6 +1063,25 @@
   \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}}
@@ -1204,6 +1149,51 @@
 
 \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 $\varphi \in L^1(\mu)$,
-- 
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