Analysis of a micro-macro acceleration method with minimum relative entropy moment matching

TL;DR

This paper analyzes the convergence and stability of a micro-macro acceleration Monte Carlo method for stochastic differential equations, focusing on a relative entropy-based approach to minimize perturbations during extrapolation.

Contribution

It provides a rigorous analysis of a specific relative entropy minimization technique within micro-macro acceleration methods for stochastic simulations.

Findings

The method is stable under certain conditions.

Local errors depend on extrapolation step size and number of macroscopic variables.

Convergence to microscopic dynamics is established as parameters tend to their limits.

Abstract

We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of the macroscopic function of interest. We consider a class of methods, presented in [Debrabant, K., Samaey, G., Zieli\'nski, P. A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs short bursts of path simulations, combined with the extrapolation of a few macroscopic state variables forward in time. After extrapolation, a new microscopic state is then constructed, consistent with the extrapolated variable and minimising the perturbation caused by the extrapolation. In the present paper, we study a specific method in which this perturbation is minimised in a relative entropy sense. We discuss why relative entropy is a useful metric, both from a theoretical and practical point of view, and rigorously study local errors and numerical stability of the resulting method as a function of the extrapolation time step and the number of macroscopic state variables. Using these results, we discuss convergence to the full microscopic dynamics, in the limit when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.