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\title{Mobility estimation for Langevin dynamics using control variates\\[.3cm]
  \small \textcolor{yellow}{AMMP Seminar}
}

\author{%
    Urbain Vaes \texorpdfstring{\\\texttt{urbain.vaes@inria.fr}}{}
}

\institute{%
    MATHERIALS -- Inria Paris
    \textcolor{blue}{\&} CERMICS --
    École des Ponts ParisTech
}

\date{October 2022}
\begin{document}


\begin{frame}[plain]
  \begin{figure}[ht]
    \centering
    % \includegraphics[height=1.5cm]{figures/logo_matherials.png}
    % \hspace{.5cm}
    \includegraphics[height=1.2cm]{figures/logo_inria.png}
    \hspace{.5cm}
    \includegraphics[height=1.5cm]{figures/logo_ponts.png}
    \hspace{.5cm}
    \includegraphics[height=1.5cm]{figures/logo_ERC.jpg}
    \hspace{.5cm}
    \includegraphics[height=1.5cm]{figures/logo_EMC2.png}
  \end{figure}
  \titlepage
\end{frame}

\begin{frame}
  {Outline}
  \tableofcontents
\end{frame}

% \section{Some background material on fast/slow systems of SDEs}%
% \label{sec:numerical_solution_of_multiscale_sdes}


% \begin{frame}
%   {Homogenization result}
%   \begin{itemize}
%     \item Effective drift:
%     \[
%       \vect F(x) = \int_{\torus^n} \left(\vect f \, \cdot \, \grad_x \right) \vect \Phi(x,y) \, \rho^{\infty}(y;x) \, \d y.
%     \]
%     \item Effective diffusion:
%       \begin{align*}
%         & \mat A(x) \, \mat A(x)^T = \frac12 \left(\mat A_0(x) + \mat A_0(x)^T\right), \\
%         & \text{with } \mat A_0(x) :=  2 \int_{\real^n} \vect f(x,y) \, \otimes \, \vect \Phi(x,y) \, \rho^{\infty}(y;x) \, \d y.
%       \end{align*}
%   \end{itemize}
%   \begin{example}
%     Multiscale system:
%     \begin{alignat*}{2}
%       & \d X^{\varepsilon}_t = \frac{1}{\varepsilon} X^{\varepsilon}_t \, Y^{\varepsilon}_t \, \d t, \quad & X^{\varepsilon}_0 = 1, \\
%       & \d Y^{\varepsilon}_t = - \frac{1}{\varepsilon^2} \, Y_t^{\varepsilon} \, \d t
%         + \frac{\sqrt 2}{\varepsilon}  \,\d W_{y}(t), \quad & Y^{\varepsilon}_0 = 0.
%     \end{alignat*}
%     Effective equation:
%     \[
%       \d X_t = X_t \, \d t + \, \sqrt{2} \, X_t \, \d W_{y} (t).
%     \]
%   \end{example}
% \end{frame}

% \begin{frame}
%   {Example: Stratonovich correction}
%   \begin{figure}[ht]
%   \centering
%   \href{run:videos/spectral/slow.avi?autostart&loop}%
%   {\includegraphics[width=0.8\textwidth]{videos/spectral/slow.png}}%

%   \href{run:videos/spectral/fast.avi?autostart&loop}%
%   {\includegraphics[width=0.8\textwidth]{videos/spectral/fast.png}}%
%   \caption{%
%     Convergence to the solution of the effective equation as $\varepsilon \to 0$.
%   }
%   \end{figure}
% \end{frame}

\section{Mobility estimation for Langevin dynamics using control variates}
\begin{frame}
    % {Part I:  Mobility estimation for Langevin dynamics using control variates}
    \begin{center}
      \Large
      \color{blue}
      Part I: Mobility estimation for Langevin dynamics
    \end{center}
    \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[]
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    % \frametitle{Outline for section \thesection}
    \frametitle{Outline}
    % \tableofcontents[currentsubsection,sectionstyle=show/shaded,subsectionstyle=show/shaded/hide]
    \tableofcontents[currentsubsection]
  \end{frame}
}

\begin{frame}
  {Goals of molecular dynamics}

  {\large $\bullet$} Computation of \emph{macroscopic properties} from Newtonians atomistic models:
  \vspace{-.1cm}
  \begin{minipage}{.51\textwidth}
    \vspace{-.7cm}
    \begin{itemize}
      \item Static properties, such as
        \begin{itemize}
          \item the heat capacity and
          \item the equations of state $P = P(\rho, T)$.
        \end{itemize}

        \vspace{.2cm}
      \item Dynamical properties, such as \emph{transport coefficients}:
        % mobilité,
        % viscosité de cisaillement;
        % conductivité thermique.
        \begin{itemize}
          \item the viscosity;
          \item the thermal conductivity;
          \item the \emph{mobility} of ions in solution.
        \end{itemize}
    \end{itemize}
  \end{minipage}
  \hspace{.5cm}
  \begin{minipage}{.4\textwidth}
    \begin{figure}[ht]
      \centering
      \includegraphics[width=.8\linewidth, angle=270]{figures/loi_argon-crop.pdf}
      \caption*{\hspace{1.2cm}%
        Equation of state of argon at 300K.

        \tiny\hspace{1.2cm}$\bullet$ `+': molecular simulation;

        \hspace{1.2cm}$\bullet$ Solid line: experimental measurements\footnotemark.
      }
    \end{figure}
  \end{minipage}
  \footnotetext{\url{https://webbook.nist.gov/chemistry/fluid/}}

  \vspace{.2cm}
  {\large $\bullet$} \emph{Numerical microscope}:
  used in physics, biology, chemistry.
\end{frame}

\begin{frame}
  {Some background material on the Langevin equation}
  Consider the (one-particle) Langevin equation
  \[
    \left\{
    \begin{aligned}
      & \d \vect q_t = \textcolor{blue}{\vect p_t \, \d t}, \\
      & \d \vect p_t = \textcolor{blue}{- \grad V(\vect q_t) \, \d t} \, \textcolor{red}{- {\color{black}\gamma} \vect p_t \, \d t + \sqrt{2 {\color{black}\gamma} \beta^{-1}} \, \d \vect W_t},
    \end{aligned}
    \right.
    \qquad (\vect q_0, \vect p_0) \sim \mu,
  \]
  where $\gamma$ is the friction, $V$ is a \emph{periodic} potential, and $\beta = \frac{1}{k_{\rm B} T}$.
  \begin{itemize}
    % \item The dynamics is composed of a \textcolor{blue}{Hamiltonian} part and a \textcolor{red}{fluctuation/dissipation} part;
    \item The invariant probability measure is
      \[
        \mu(\vect q, \vect p) = \frac{1}{Z} \e^{-\beta H(\vect q, \vect p)} = \frac{1}{Z} \e^{-\beta \left(V(\vect q) + \frac{\abs{\vect p}^2}{2}\right)}, \quad \text{on}~ \emph{\torus^d} \times \real^d.
      \]
    \item The generator of the associated Markov semigroup
      \[
        \left (\e^{\mathcal L t} \varphi\right) (\vect q, \vect p) = \expect \bigl(\varphi(\vect q_t, \vect p_t) \big| (\vect q_0, \vect p_0) = (\vect q, \vect p) \bigr)
      \]
      is the following operator:
      \begin{align*}
        \mathcal L &= \textcolor{blue}{\left(\vect p \cdot \grad_{\vect q} - \grad V(q) \cdot \grad_{\vect p} \right)}
        + \gamma \, \textcolor{red}{\left( - \vect p \grad_{\vect p} + \beta^{-1} \laplacian_{\vect p} \right)}
        =: \textcolor{blue}{\mathcal L_{\textrm{ham}}} + \gamma \, \textcolor{red}{\mathcal L_{\textrm{FD}}}.
      \end{align*}
  \end{itemize}
  We denote by $\norm{\cdot}$ and $\ip{\cdot}{\cdot}$ the norm and inner product of~$L^2(\mu)$, and
  \[
    L^2_0(\mu) = \Bigl\{\varphi \in L^2(\mu) : \ip{\varphi}{1} = \expect_{\mu} \varphi =  0 \Bigr\}.
  \]
\end{frame}


% \begin{frame}
%   {Common models in molecular simulation}
%   We consider the following hierarchy of models:
%   \begin{align}
%     \label{eq:gle:model:overdamped} \tag{OL}
%     \dot {\vect q} &= - \grad V(\vect q) + \sqrt{2 \, \beta^{-1}} \, \dot {\vect W}, \\
%     \label{eq:gle:model:langevin} \tag{L}
%     \ddot {\vect q} &= - \grad V(\vect q) - \gamma \, \dot {\vect q} + \sqrt{2 \gamma \, \beta^{-1}} \, \dot {\vect W}, \\
%     \label{eq:gle:model:generalized} \tag{GLE}
%     \ddot {\vect q} &= -\grad V(\vect q) - \int_{0}^{t} \widehat \gamma(t-s) \, \dot {\vect q}(s) \, \d s + \vect F(t).
%   \end{align}
%   where
%   \begin{itemize}
%     \item $V$ is a potential, in this talk \emph{periodic};
%     \item $\gamma$ is the friction coefficient;
%     \item $\widehat \gamma(\cdot)$ is the memory kernel;
%     \item $\vect F$ is a stationary non-Markovian noise process.
%   \end{itemize}
%   \vspace{.2cm}

%   The kernel $\widehat \gamma(\cdot)$ and the noise $F$ are related by the \emph{fluctuation/dissipation} relation:
%   \[
%       \expect\bigl[\vect F(t) \otimes \vect F(s)\bigr] = \beta^{-1} \, \widehat \gamma(t-s) \mat I_d.
%   \]
% \end{frame}

% \subsection{Mobility and effective diffusion}
\begin{frame}
  {Definition of the mobility}
  Consider Langevin dynamics with additional forcing in a direction $\vect e$:
  % \[
  %   \ddot {\vect q} = - \grad V(\vect q) + \alert{\eta \vect e} - \gamma \, \dot {\vect q} + \sqrt{2 \, \gamma} \, \beta^{-1} \, \dot {\vect W}.
  % \]
  % This equation may be rewritten as a system for the position and momentum:
  \[
    \left\{
    \begin{aligned}
      & \d \vect q_t = \vect p_t \, \d t, \\
      & \d \vect p_t = - \grad V(\vect q_t) \, \d t + \alert{\eta \vect e} \, \d t - \gamma \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \, \d \vect W_t.
    \end{aligned}
    \right.
  \]
  This dynamics admits a unique invariant probability distribution $\mu_{\alert{\eta}} \in \mathcal P(\emph{\torus^d} \times \real^d)$.

  \begin{definition}
    [Mobility]
    The mobility in direction $\vect e$ is defined mathematically as
    \[
      M_{\vect e} =
      \lim_{\alert{\eta} \to 0} \frac{1}{\alert{\eta}}\expect_{\mu_{\alert{\eta}}} [\vect e^\t \vect p]
    \]
    $\approx $ factor relating the mean momentum to the strength of the inducing force.
  \end{definition}

  \begin{itemize}
    \item There is a symmetric mobility tensor $\mat M$ such that $M_{\vect e} = \vect e^\t \mat M \vect e$.

    \item
      \textbf{Einstein's relation:}
      \(
      \mat D = \beta^{-1} \mat M,
      \) with $\mat D$ the \emph{effective diffusion coefficient}.
  \end{itemize}
\end{frame}

\begin{frame}
  {Effective diffusion}
  It is possible to show a \emph{functional central limit theorem} for the Langevin dynamics\footfullcite{MR663900}:
  \begin{equation*}
    \varepsilon  \vect q_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, \vect W_s
    \qquad  \text{weakly on } C([0, \infty)).
  \end{equation*}
  In particular, $\vect q_t/\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly.

  \vspace{-.25cm}
  \begin{figure}[ht]
  \centering
  \href{run:videos/gle/effective-diffusion.webm?autostart&loop}%
  {\includegraphics[width=0.75\textwidth]{videos/gle/effective-diffusion.png}}%
  \caption{Histogram of $q_t/\sqrt{t}$. The potential $V(q) = - \cos(q) / 2$ is illustrated in the background.}
  \end{figure}
\end{frame}

\begin{frame}
  {Mathematical expression for the effective diffusion (dimension 1)}
  \vspace{.2cm}
  \begin{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 (q_{t/\alert\varepsilon^2} - q_{0}) &= \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} p_s \, \d s \\
                                                            &= \underbrace{\alert\varepsilon \bigl(\phi(q_0, p_0) - \phi(q_{t/\alert\varepsilon^2}, p_{t/\alert\varepsilon^2})\bigr)}_{\to 0
                                                            % ~\text{in $L^p(\Omega, C([0, T], \real))$}
                                                          }
    + \underbrace{\sqrt{2 \gamma \beta^{-1}} \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s}_{\to \sqrt{2 D} W_t~\text{weakly by MCLT}}.
  \end{align*}
  % where
  % \begin{align*}
  %   D &= \gamma \beta^{-1} \, \int \abs{\textstyle \derivative{1}[\phi]{p}(q, p)}^2 \, \mu(\d q \, \d p)
  %   = - \int \phi (\mathcal L \phi) \, \d \mu
  %   = \ip{\phi}{p}.
  % \end{align*}

  \vspace{.3cm}
  \textbf{In the multidimensional setting}, $D_{\vect e} = \ip{\phi_{\vect e}}{\vect e^\t \vect p}$ with $- \mathcal L \phi_{\vect e} = \vect e^\t \vect p$
\end{frame}

\begin{frame}
  {Open question: surface diffusion when $\gamma \ll 1$\footnote{Source of the video: \url{https://en.wikipedia.org/wiki/Surface_diffusion}}}
    \begin{figure}[ht]
        \centering
        \href{run:videos/surface_diffusion.webm?autostart&loop}%
        {\includegraphics[width=0.4\linewidth]{videos/surface_diffusion.png}}
        \hspace{1cm}
        % \href{run:videos/diffusion.webm?autostart&loop}%
        % {\includegraphics[width=0.4\linewidth]{figures/mean_square.pdf}}
    \end{figure}

    \vspace{-.3cm}
    Applications:
    \begin{itemize}
      \item integrated circuits;
      \item catalysis.
    \end{itemize}

    \textbf{Open question}: behavior of the effective diffusion coefficient when $\gamma \ll 1$?
    \[
      D = \lim_{t \to \infty} \frac{\langle \abs{\vect q(t)}^2 \rangle}{4 t} \sim \gamma^{-\alert{\sigma}}, \qquad \alert{\sigma} =\, ???
    \]
    % \vspace{-.3cm}

    % \textbf{Difficulty}: slow convergence of Monte Carlo methods when $\gamma$ is small.
    % \vspace{.3cm}
\end{frame}


% \subsection{Some background material on the Langevin equation}


\begin{frame}{Langevin dynamics: \textcolor{yellow}{underdamped} and \textcolor{yellow}{overdamped} regimes\footfullcite{MR2394704}}
  \begin{figure}[ht]
    \centering
    \href{run:videos/particles_underdamped.webm?autostart&loop}%
    {\includegraphics[width=0.49\textwidth]{videos/particles_underdamped.png}}%
    \href{run:videos/particles_overdamped.webm?autostart&loop}%
    {\includegraphics[width=0.49\textwidth]{videos/particles_overdamped.png}}%
    \caption{Langevin dynamics with friction $\gamma = 0.1$ (left) and $\gamma = 10$ (right)}
  \end{figure}

  \vspace{-.3cm}
  \begin{itemize}
    \item The \alert{underdamped} limit as $\gamma \to 0$ is well understood in dimension 1 but not in the \alert{multi-dimensional setting}.
    \item \emph{Overdamped} limit:
      as $\gamma \to \infty$, the rescaled process $t \mapsto q_{\gamma t}$ converges weakly to the solution of the \emph{overdamped Langevin equation}:
      \[
        \dot {\vect q} = - \grad V(q) + \sqrt{2 \, \beta^{-1}} \, \dot {\vect W}.
      \]
  \end{itemize}
\end{frame}

\begin{frame}
  {The \textcolor{yellow}{underdamped} limit in \textcolor{yellow}{dimension 1}}
  As \emph{$\gamma \to 0$},
  the Hamiltonian of the rescaled process
  \begin{equation*}
    \left\{
    \begin{aligned}
    q_{\gamma}(t) = q(t/\gamma), \\
    p_{\gamma}(t) = p(t/\gamma),
    \end{aligned}
    \right.
  \end{equation*}
  converges weakly to a diffusion process on a graph.
  \vspace{-.6cm}

  \begin{figure}[ht!]
      % \centering
      % #1f77b4', u'#ff7f0e', u'#2ca02c
      \definecolor{c1}{RGB}{31,119,180}
      \definecolor{c2}{RGB}{255,127,14}
      \definecolor{c3}{RGB}{44,160,44}
      \begin{tikzpicture}%
      \node[anchor=south west,inner sep=0] at (0,0) {%
          \includegraphics[width=.7\textwidth]{figures/separatrix.eps}
      };
      \coordinate (origin) at (10,0);
      \coordinate (Emin) at ($ (origin) + (0,.5) $);
      \coordinate (E0) at ($ (origin) + (0,2) $);
      \coordinate (E1) at ($ (origin) + (-1,4) $);
      \coordinate (E2) at ($ (origin) + (1,4) $);
      \node at ($ (Emin) + (.7,0) $) {$E_{\min}$};
      \node[color=red] at ($ (E0) + (.5,0) $) {$E_{0}$};
      \node at ($ (E1) + (0,.3) $) {$p < 0$};
      \node at ($ (E2) + (0,.3) $) {$p > 0$};
      \draw[thick,color=c2] (Emin) -- (E0) node [color=black, midway, right] {};
      \draw[thick,color=c1] (E0) -- (E1) node [color=black, midway, left] {};
      \draw[thick,color=c3] (E0) -- (E2) node [color=black, midway, right] {};
      \node at (E0) [circle,fill,inner sep=1.5pt,color=red]{};
      \node at (Emin) [circle,fill,inner sep=1.5pt]{};
      \end{tikzpicture}%
  \end{figure}
  \vspace{-.5cm}
  In this limit, it holds that
  \[
    % \norm{\mathcal L^{-1}}_{\mathcal B\left(L^2_0(\mu)\right)} = \mathcal O \left( \alert{\gamma^{-1}} \right),
    % \qquad
    \phi = - \mathcal L^{-1} p = \alert{\gamma^{-1}} \phi_{\rm und} + \mathcal O(\gamma^{-1/2}).
  \]
  % The limiting function $\phi_{\rm und}$ is continuous but \alert{not in $H^1(\mu)$}.
\end{frame}


\begin{frame}
  {Scaling of the effective diffusion coefficient for \textcolor{yellow}{Langevin} dynamics\footfullcite{MR2427108}}
  In \alert{dimension 1},
  \( \lim_{\gamma \to 0} \gamma D^{\gamma} = D_{\rm und} \) and \( \lim_{\gamma \to \infty} \gamma D^{\gamma} = D_{\rm ovd}. \)
  \begin{figure}[ht]
      \centering
      \includegraphics[width=0.5\linewidth,height=0.33\linewidth]{figures/scaling_diffusion_langevin.png}
  \end{figure}

  \textbf{\emph{Our aims in this work:}}
  \begin{itemize}
    \item How can we efficiently estimate the effective diffusion coefficient when \alert{$\gamma \ll 1$}?
    \item How does the mobility scale as \alert{$\gamma \to 0$} in the multidimensional setting?
  \end{itemize}
\end{frame}


\subsection{Efficient mobility estimation}%

\begin{frame}
  {Brief literature review}
  % Consider the Langevin dynamics with $(\vect q_t, \vect p_t) \in (\real^{\alert{d}} \times \real^{\alert{d}})$:
  % \begin{equation*}
  %     \left\{
  %       \begin{aligned}
  %       & \d \vect q_t = \vect p_t \,\d t, \\
  %       & \d \vect p_t = - \grad V (\vect q_t)  \, \d t - \gamma \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \, \d \vect W_t.
  %       \end{aligned}
  %     \right.
  % \end{equation*}
  In dimension $> 1$, it \alert{does not hold} that
  $\gamma D^{\gamma}_{\vect e} \xrightarrow[\gamma \to 0]{} D_{\rm und}$ when $V$ is \alert{non separable}, e.g.
  \[
    V(\vect q) = - \frac{1}{2} \Big( \cos(q_1) + \cos(q_2) \Big) - \alert{\delta} \cos(q_1) \cos(q_2)
  \]

  \textbf{Open question:}
  how does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}?
  % \begin{block}
    % {Open question: how does $D^\gamma_{\vect e}$ behave when $\gamma \ll 1$ and \alert{$d = 2$}?}

  Various answers are given in the literature:
  \begin{itemize}
    \item
      $D^{\gamma}_{\vect e} \propto \gamma^{-1/2}$ for specific potentials\footfullcite{chen1996surface};

    \item
      $D^{\gamma}_{\vect e} \propto \gamma^{-1/3}$ for specific potentials~\footfullcite{Braun02};

    \item
      $D^{\gamma}_{\vect e} \propto \gamma^{-\sigma}$ with $\sigma$ depending on the potential~\footfullcite{roussel_thesis}.
  \end{itemize}
  % \end{block}
  \vspace{.5cm}
\end{frame}

\begin{frame}[label=continue]
  {Numerical approaches for calculating the effective diffusion coefficient}
  \begin{itemize}
    \itemsep.5cm
    \item \emph{Linear response approach}:
        \begin{equation*}
          D_{\vect e} = \lim_{\eta \to 0} \frac{1}{\beta \alert{\eta}} \expect_{\alert{\mu_\eta}} \, (\vect e^\t \vect p).
        \end{equation*}
        where $\mu_{\eta}$ is the invariant distribution of the system with external forcing.

    \item \emph{Green--Kubo formula}: Since $-\mathcal L^{-1} = \int_{0}^{\infty} \e^{t \mathcal L} \, \d t$,
      \begin{align*}
        D_{\vect e} &= \int - \mathcal L^{-1}(\vect e^\t \vect p) \, (\vect \e^\t \vect p) \, \d \mu = \int_{0}^{\infty} \! \! \! \int \e^{t \mathcal L} (\vect e^\t \vect p) (\vect e^\t \vect p) \, \d \mu \, \d t  \\
          &= \int_{0}^{\infty} \expect_{\mu}\bigl((\vect e^\t \vect p_0) (\vect e^\t \vect p_t)\bigr) \, \d t.
      \end{align*}

    \item \emph{Einstein's relation}:
      \[
        D_{\vect e} = \lim_{t \to \infty}  \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl|\vect e^\t (\vect q_t - \vect q_0)\bigr|^2 \Bigr].
      \]

    \item Deterministic method, e.g. \emph{Fourier/Hermite Galerkin}, for the Poisson equation
      \[
        - \mathcal L \phi_{\vect e} = \vect e^\t \vect p, \qquad D_{\vect e} = \ip{\phi_{\vect e}}{p}.
      \]
  \end{itemize}
\end{frame}

% \begin{frame}
%   {Fourier/Hermite Galerkin method for one-dimensional Langevin dynamics}
%
%   Saddle-point formulation\footfullcite{roussel2018spectral}:
%   find $(\Phi_N, \alpha_N) \in V_N \times \real$ such that
%   \begin{align}
%        \notag
%        - \Pi_N \, \mathcal L \, \Pi_N \alert{\Phi_N} + \alert{\alpha_N} u_N &= \Pi_N p, \\
%        \label{eq:constraint}
%         \ip{\Phi_N}{u_N} &= 0,
%   \end{align}
%   where
%   \begin{itemize}
%     \item $\Pi_N$ is the $L^2(\mu)$ projection operator on a finite-dimensional subspace $V_N$,
%     \item $u_N = \Pi_N 1 / \norm{\Pi_N 1}$.
%       Eq.~\eqref{eq:constraint} ensures that the system is \emph{well-conditioned}.
%   \end{itemize}
%
%   \vspace{.2cm}
%   For $V_N$, we use the following basis functions:
%   \[
%     e_{i,j} = {\left( Z \, \e^{\beta \left( H(q,p) + |z|^2 \right)} \right)}^{\frac{1}{2}} \, G_i(q) \, H_j(p), \qquad 0 \leq i,j \leq N,
%   \]
%   where $(G_i)_{i \geq 0}$ are \emph{trigonometric functions} and $(H_j)_{i \geq 0}$ are \emph{Hermite polynomials}.
%
%   $\rightarrow$ \alert{Impractical} in two or more spatial dimensions.
% \end{frame}

\begin{frame}
  {Estimation of the effective diffusion coefficient from Einstein's relation}
  Consider the following estimator of the effective diffusion coefficient $D_{\vect e}$:
  \[
    \emph{u(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2T}}, \qquad (\vect q_0, \vect p_0) \sim \mu.
  \]

  \textbf{Bias of this estimator:}
  \begin{align*}
      \notag
      \expect \bigl[u(T)\bigr]
      % &= \int_{0}^{\infty} \ip{\e^{t \mathcal L}(\vect e^\t \vect p)}{\vect e^\t \vect p}  \d t
      % - \int_{0}^{\infty} \ip{\e^{t \mathcal L} (\vect e^\t \vect p)}{\vect e^\t \vect p} \min\left\{1, \frac{t}{T}\right\} \, \d t \\
      &= D_{\vect e} - \int_{0}^{\infty} \ip{\e^{t \mathcal L} (\vect e^\t \vect p)}{\vect e^\t \vect p} \min\left\{1, \frac{t}{T}\right\} \, \d t.
  \end{align*}
  Using the decay estimate for the semigroup\footfullcite{roussel2018spectral}
  \[
    \norm{\e^{t \mathcal L}}_{\mathcal B\left(L^2_0(\mu)\right)} \leq L \e^{- \ell \min\{\gamma, \gamma^{-1}\}t},
  \]
  we deduce
  \[
    \left\lvert \expect[u(T)] - D_{\vect e} \right\rvert \leq \frac{C \textcolor{red}{\max\{\gamma^2, \gamma^{-2}\}}}{T}.
  \]
\end{frame}

\begin{frame}
  {Variance of the estimator $u(T)$ for large $T$}
  For $T \gg 1$,
  it holds approximately that
  \[
    \frac{\vect e^\t (\vect q_T - \vect q_0)}{\sqrt{2T}} \sim \mathcal N(0, D_{\vect e})
    \qquad \leadsto \qquad
    u(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2 D_{\vect e} T} \sim \chi^2 (1).
  \]
  Therefore, we deduce
  \[
    \lim_{T \to \infty} \var \bigl[u(T)\bigr] = 2 D_{\vect e}^2.
  \]
  The relative standard deviation (asymptotically as $T \to \infty$) is therefore
  \[
    \lim_{T \to \infty} \frac{\sqrt{\var \bigl[u(T)\bigr]}}{\expect \bigl[u(T)\bigr]} = \sqrt{2}
    \qquad \leadsto \text{\emph{independent} of $\gamma$}.
  \]

  \begin{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}
  \[
    - \mathcal L \phi_{\vect e}(\vect q, \vect p) = \vect e^\t \vect p, \qquad \phi_{\vect e} \in L^2_0(\mu).
  \]
  and let $\psi_{\vect e}$ denote an approximation of $\phi_{\vect e}$.
  By It\^o's formula,
  we obtain
  \[
    \phi_{\vect e}(\vect q_T, \vect p_T) - \phi_{\vect e}(\vect q_0, \vect p_0)
    = - \int_{0}^{T} \vect e^\t \vect p_t \, \d t + \sqrt{2 \gamma \beta^{-1}} \int_{0}^{T} \grad_{\vect p} \phi_{\vect e}(\vect q_t, \vect p_t) \cdot \d \vect W_t.
  \]
  Therefore
  \begin{align*}
    \vect e^\t (\vect q_T - \vect q_0)
    &= \int_{0}^{T} \vect e^\t \vect p_t \, \d t \\
    &\approx  - \psi_{\vect e}(\vect q_T, \vect p_T) + \psi_{\vect e}(\vect q_0, \vect p_0) + \sqrt{2 \gamma \beta^{-1}} \int_{0}^{T} \grad_{\vect p} \psi_{\vect e}(\vect q_t, \vect p_t) \cdot \d \vect W_t
    =: \emph{\xi_T}.
  \end{align*}
  which suggests the \emph{improved estimator}
  \[
    v(T) = \frac{\abs{\vect e^\t (\vect q_T - \vect q_0)}^2}{2T} - \left( \frac{\abs{\xi_T}^2}{2T} - \lim_{T\to \infty}\expect \left[ \frac{\abs{\xi_T}^2}{2T} \right]  \right).
  \]
\end{frame}

\begin{frame}
  {Properties of the improved estimator}
  \textbf{Smaller bias} if $-\mathcal L \psi_{\vect e} \approx \vect e^\t \vect p$:
    \begin{align*}
        \label{eq:basic_bound_bias}
        \abs{\expect \bigl[ v(T) \bigr] - D^{\gamma}_{\vect e}}
                &\leq  \frac{L \max\{\gamma^2, \gamma^{-2}\}}{T \ell^2 }  \,  \emph{\norm{\vect e^\t \vect p + \mathcal L \psi_{\vect e}}}  \left(\beta^{-1/2} + \norm{\mathcal L \psi_{\vect e}} \right).
    \end{align*}

  \textbf{Smaller variance}:
    \begin{equation*}
        \begin{aligned}[b]
            \var \bigl[v(T)\bigr]
            \leq
            C &\left( T^{-1} \emph{\norm{\phi_{\vect e} - \psi_{\vect e}}[L^4(\mu)]}^2  + \gamma \emph{\norm{\grad_{\vect p} \phi_{\vect e} - \grad_{\vect p} \psi_{\vect e}}[L^4(\mu)]}^2 \right) \\
              &\quad \times \left( T^{-1} \norm{\phi_{\vect e} + \psi_{\vect e}}[L^4(\mu)]^2  + \gamma \norm{\grad_{\vect p} \phi_{\vect e} + \grad_{\vect p} \psi_{\vect e}}[L^4(\mu)]^2 \right).
        \end{aligned}
    \end{equation*}


    \textbf{Construction of $\psi_{\vect e}$ in the \alert{one-dimensional setting}}. We consider two approaches:
    \begin{itemize}
      \item Approximate the solution to the Poisson equation by a Galerkin method.
      \item Use asymptotic result for the Poisson equation:
        \[
          \gamma \phi \xrightarrow[\gamma \to 0]{L^{2}(\mu)} \phi_{\rm und},
        \]
        which suggests letting $\psi = \phi_{\rm und} / \gamma$.
    \end{itemize}
\end{frame}

\begin{frame}
  {Construction of the approximate solution $\psi_{\vect e}$ \textcolor{yellow}{in dimension 2}}
  We consider the potential
  \[
    V(\vect q) = - \frac{1}{2} \Big( \cos(q_1) + \cos(q_2) \Big) - \alert{\delta} \cos(q_1) \cos(q_2).
  \]
  \begin{itemize}
    \item
      For this potential, $\mat D$ is isotropic
      $\leadsto$ sufficient to consider $\vect e = (1, 0)$,
      \[
        D_{(1,0)} = \ip{\phi_{(1, 0)}}{p_1},
        \qquad - \mathcal L \phi_{(1,0)}(\vect q, \vect p) = p_1.
      \]

    \item
      If \emph{$\delta = 0$}, then the solution is $\phi_{(1, 0)}(\vect q, \vect p) = \phi_{\rm 1D} (q_1, p_1)$,
      where $\phi_{\rm 1D}$ solves
      \[
        - \mathcal L_{\rm 1D} \phi_{\rm 1D}(q, p) = p, \qquad V_{\rm 1D}(q) = \frac{1}{2} \cos (q).
      \]

    \item
      We take $\emph{\psi_{(1,0)}(\vect q, \vect p) = \psi_{\rm 1D}(q_1, p_1)}$,
      where $\psi_{\rm 1D} \approx \phi_{\rm 1D}$.
  \end{itemize}
\end{frame}

\subsection{Numerical experiments}%
\begin{frame}
  {Numerical experiments for the one-dimensional case (1/2)}
  \begin{figure}[ht]
      \centering
      \includegraphics[width=0.99\linewidth]{figures/underdamped_1d.pdf}
  \end{figure}
\end{frame}

\begin{frame}
  {Numerical experiments for the one-dimensional case (2/2)}
  \begin{figure}[ht]
    \centering
    \includegraphics[width=0.99\linewidth]{figures/time.pdf}
    \caption{Evolution of the sample mean and standard deviation, estimated from $J = 5000$ realizations for $\gamma = 10^{-3}$.}
  \end{figure}
\end{frame}

\begin{frame}
  {Performance of the control variates approach in dimension 2}
  \begin{figure}[ht]
    \centering
    \includegraphics[width=0.49\linewidth]{figures/var-delta-galerkin.pdf}
    \includegraphics[width=0.49\linewidth]{figures/var-delta-underdamped.pdf}
    \label{fig:time_bias_deviation_2d}
  \end{figure}
  \begin{itemize}
    \item Variance reduction is possible if $\abs{\delta}/\gamma \ll 1$;
    \item Control variates are \alert{not very useful} when $\gamma \ll 1$ and $\delta$ is fixed.
  \end{itemize}
\end{frame}

\begin{frame}
  {Scaling of the mobility in dimension 2}
  \begin{figure}[ht]
    \centering
    \includegraphics[width=0.9\linewidth]{figures/diffusion.pdf}
    \label{fig:time_bias_variance_2d}
  \end{figure}
\end{frame}

\begin{frame}{Summary and perspectives for future work}
  In this talk, we presented
  \begin{itemize}
    \item a variance reduction approach for efficiently estimating the mobility;
    \item numerical results showing that the scaling of the mobility is \emph{not universal}.
  \end{itemize}

  \textbf{Perspectives for future work:}
  \begin{itemize}
    \item Use alternative methods (PINNs, Gaussian processes) to solve the Poisson equation;
    \item Improve and study variance reduction approaches for other transport coefficients.
  \end{itemize}

  \vspace{1cm}
  \begin{center}
    Thank you for your attention!
  \end{center}
\end{frame}

\section{Optimal importance sampling for overdamped Langevin dynamics}

% \begin{frame}
%   \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}

  \begin{block}
    {Objective of the sampling problem}
    Calculate averages with respect to
    \[
      \mu = \frac{\e^{-V}}{Z},
      \qquad Z = \int_{\torus^d} \e^{-V}.
    \]
    \vspace{-.4cm}
  \end{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},
  \[
    \frac{1}{\sqrt{N}} \sum_{n=1}^{N} \left((f-I) \e^U\right) (X^{n})
    \xrightarrow[N \to \infty]{\rm Law} \mathcal N\left(0, \int_{\torus^d} \abs*{(f-I) \e^U}^2 \, \d \mu_{U}\right)
  \]

  \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,
  \[
      \sqrt{N} \bigl( \mu^N_U(f) - I\bigr)
      \xrightarrow[T \to \infty]{\rm Law} \mathcal N\bigl(0, s^2_f[U]\bigr),
      \qquad
      s^2_f[U] := \frac{2 Z_U^2}{Z^2} \int_{\torus^n} \bigl\lvert (f-I) \e^U \bigr\rvert^2 \, \d \mu_{U}.
  \]
\end{frame}

\begin{frame}
  {Background: importance sampling in the i.i.d.\ setting (2/2)}
  By the Cauchy--Schwarz inequality,
  it holds that
  \[
      s^2_f[U]
      \geq \frac{2Z_U^2}{Z^2} \left( \int_{\torus^d} \abs{f-I} \e^U \, \d \mu_{U} \right)^2,
  \]
  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}
  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}

\appendix

\begin{frame}[noframenumbering,plain]
  {Connection with the asymptotic variance of MCMC estimators}
  \textbf{Ergodic theorem\footfullcite{MR885138}}:  for an observable $\varphi \in L^1(\mu)$,
  \[
    \widehat \varphi_t = \frac{1}{t} \int_{0}^{t} \varphi(\vect q_s, \vect p_s) \, \d s
    \xrightarrow[t \to \infty]{a.s.} \expect_{\mu} \varphi.
  \]

  \textbf{Central limit theorem\footfullcite{MR663900}}:
  If the following \emph{Poisson equation} has a solution $\phi \in L^2(\mu)$,
  \[
      - \mathcal L \phi = \varphi - \expect_{\mu} \varphi,
  \]
  then a central limit theorem holds:
  \[
    \sqrt{t} \bigl(\widehat \varphi_t - \expect_{\mu}\varphi\bigr)
    \xrightarrow[t \to \infty]{\rm Law} \mathcal N(0, \sigma^2_{\varphi}),
    \qquad
    \sigma^2_{\varphi}
    = \ip{\phi}{\varphi - \expect_{\mu} \varphi}.
  \]

  \textbf{Connection with effective diffusion}: Apply this result with $\varphi(\vect q, \vect p) = \vect e^\t \vect p$.
\end{frame}

\end{document}

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