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@@ -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 $\varphi \in L^1(\mu)$, |