# Mathematics Colloquium

#### Design of Monte Carlo when dealing with metastable processes: rates of convergence and optimization for parallel methods

**Speaker:**
Paul Dupuis, Brown University

**Location:**
TBA

**Videoconference link:** https://nyu.zoom.us/j/99223527647

**Date:**
Monday, November 16, 2020, 3:45 p.m.

**Synopsis:**

A challenging problem for Markov chain Monte Carlo methods occurs when different parts of the state

space communicate poorly under the process being simulated. For example, if the process is a small

noise diffusion then times between visits to neighborhoods of attractors of the noiseless system scale

exponentially in the inverse of the noise strength, and a very long time is required for good

approximations. This talk will be in three parts: (1) a discussion on useful properties for any rate of

convergence in this setting, and selection of such a rate; (2) presentation of a large deviation theory for

first and second moments of the empirical measure of a small noise diffusion as time gets large and the

noise gets small, and (3) its application to the problem of optimal temperature selection for certain

parallel Monte Carlo methods in the (computationally demanding) small noise limit. Although the

results are asymptotic as the size of the noise tends to zero, the optimal placement of temperatures is

explicit and independent of the particular functional being integrated or (with mild restrictions) on the

potential. Formulas for the resulting variances are phrased in terms of graph optimization problems

with nodes corresponding to metastable states of the single temperature dynamics.