A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations

TL;DR

This paper introduces a micro-macro acceleration method that enhances Monte Carlo simulations of stochastic differential equations by combining short path simulations with macroscopic extrapolation, improving efficiency and accuracy.

Contribution

The paper proposes a novel micro-macro acceleration algorithm with a matching operator, providing convergence proof and analysis of different matching strategies for stochastic differential equations.

Findings

Convergence of the method is proven without statistical error.

Different matching strategies impact the accuracy of macroscopic predictions.

Numerical experiments demonstrate the effectiveness of the approach.

Abstract

We present and analyse a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) time-scale of individual trajectories and the (slow) time-scale of the macroscopic function of interest. The algorithm combines short bursts of path simulations with extrapolation of a number of macroscopic state variables forward in time. The new microscopic state, consistent with the extrapolated variables, is obtained by a matching operator that minimises the perturbation caused by the extrapolation. We provide a proof of the convergence of this method, in the absence of statistical error, and we analyse various strategies for matching, as an operator on probability measures. Finally, we present numerical experiments that illustrate the effects of the different approximations on the resulting error in macroscopic predictions.