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@@ -770,8 +770,8 @@ \end{figure} \end{frame} -\begin{frame}{Summary and perspectives for future work} - In this talk, we presented +\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}. @@ -782,11 +782,6 @@ \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} @@ -1020,7 +1015,7 @@ \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 + where \[ F(x) := \int_0^x \bigl( f(\xi)-I \bigr) \e^{-V(\xi)}\d \xi. \] @@ -1061,27 +1056,152 @@ \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 + + \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 \[ - F(x) := \int_0^x \bigl( f(\xi)-I \bigr) \e^{-V(\xi)}\d \xi. + 0 = \frac{1}{2} \d \sigma^2_f[U_*] \cdot \delta U + = \frac{Z_U^2}{Z^2} \int_{\torus^d} \delta U \bigg( \abs*{\nabla{\phi_{U_*}}}^2 - \int_{\torus^d} \abs*{\nabla {\phi_{U_*}}}^2 \, \d \mu_{U_*} \bigg) \, \d \mu_{U_*}, \] - This inequality~\eqref{eq:lower_bound_asymvar} is an equality for + 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 (1/2)} + 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 (2/2): 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} + +\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 \[ - U(x) = U_*(x) = - V(x) -\ln\abs*{F(x) + A_*}, + f = \sum_{j=1}^{J} \sqrt{\lambda_j} u_j f_j, + \qquad u_j \sim \mathcal N(0, 1), + \qquad \lambda_j \in (0, \infty). \] - where $A_*$ is the constant achieving the infimum in~\eqref{eq:lower_bound_asymvar}. + \textbf{Question:} can we find~$U$ such that $\sigma^2[U] := \expect \bigl( \sigma^2_f[U] \bigr)$ is minimized? \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~$U$. + \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}[noframenumbering,plain] |