Large Deviation Methods for the Analysis and Design of Accelerated Monte Carlo Schemes

Brown University, Providence RI

Investigators

Abstract

The main focus of the research program is the use of large deviation ideas for the design and analysis of accelerated Monte Carlo methods. In addition, the underlying large deviation theory will be developed where needed. The PIs will emphasize two classes of problems, which are (i) accelerating the convergence of an empirical measure to a target stationary distribution and (ii) the estimation of the probability of a single rare event. In prior work the PIs have initiated use of the large deviation rate for the empirical measure of a Markov process as a tool for algorithm analysis. By considering suitable limits of the already effective parallel tempering algorithm, they developed a new algorithm called infinite swapping, as well as computationally tractable variants called partial infinite swapping. The research program involves further theoretical development of the algorithm and the associated large deviation tools, as well as applications to challenging problems from materials science. With regard to the estimation of the probability of a rare event, the PIs will continue to develop an approach based on large deviations and subsolutions to an associated Hamilton-Jacobi-Bellman equation for the design and analysis of importance sampling and certain types of branching algorithms. The primary applications focus of the research program is the study of nanoscale materials. Because their basic physical and chemical properties can differ significantly from those of their bulk counterparts, such materials are of substantial practical and theoretical interest. Understanding how these basic properties are determined by the atomic-level details involved, an essential element in realizing the ultimate design potential of these systems, involves overcoming a number of rare event challenges. Monte Carlo algorithms are one of the most flexible and useful numerical tools in the applied sciences and engineering. They are used to solve a broad range of problems, such as determining thermodynamic properties in models from chemistry and material science, evaluating equilibrium properties in large scale networks, and in problems of classification and estimation in Bayesian statistics. However, with many Monte Carlo algorithms, rare events play a dominant role in determining the quality and hence usefulness of the resulting numerical approximation. An important example is the approximation of integrals via Markov chain Monte Carlo. These applications are very challenging when the distribution has pockets of significant probability and transitions between these pockets are rare. Another example is estimation of the probability of a single rare event, such as unexpectedly large payouts in insurance claims or the transition between metastable wells in a model from chemical physics. The PIs will develop and apply methods from large deviation theory, which is the branch of probability that analyzes and characterizes rare events, to the problem of efficient algorithm design. They will apply these algorithms to problems of materials science.