From 7d6969615c87706c45cac0fa2e6396c50db65abb Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Thu, 13 Oct 2022 14:52:58 +0200
Subject: Improve presentation

---
 main.tex | 160 +++++++++++++++++++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 140 insertions(+), 20 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index e1fbc9e..35354b9 100755
--- a/main.tex
+++ b/main.tex
@@ -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]
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