Variance-reduced multiscale simulation of slow-fast stochastic differential equations

TL;DR

This paper introduces a variance reduction method for efficiently simulating the averaged behavior of stochastic slow-fast systems, leveraging control variables and MCMC techniques to improve accuracy and computational efficiency.

Contribution

It proposes a novel variance reduction scheme using control variables for multiscale stochastic simulations, with analysis and validation on linear and nonlinear models.

Findings

Significant variance reduction achieved in simulations.

Improved accuracy in estimating averaged dynamics.

Method effective for both linear and nonlinear systems.

Abstract

We study a variance reduction strategy based on control variables for simulating the averaged macroscopic behavior of a stochastic slow-fast system. We assume that this averaged behavior can be written in terms of a few slow degrees of freedom, and that the fast dynamics is ergodic for every fixed value of the slow variable. The time derivative for the averaged dynamics can then be approximated by a Markov chain Monte Carlo method. The variance-reduced scheme that is introduced here uses the previous time instant as a control variable. We analyze the variance and bias of the proposed estimator and illustrate its performance when applied to a linear and nonlinear model problem.