A micro/macro algorithm to accelerate Monte Carlo simulation of stochastic differential equations

TL;DR

This paper introduces a micro/macro acceleration method for Monte Carlo simulation of SDEs that efficiently handles different time-scales by combining microscopic simulations with macroscopic extrapolation, improving computational efficiency.

Contribution

The paper proposes a novel micro/macro acceleration technique for SDE Monte Carlo simulations, analyzing its convergence and demonstrating its effectiveness through numerical experiments.

Findings

The method accelerates SDE simulations by combining short microscopic bursts with macroscopic extrapolation.

Convergence analysis shows the method's accuracy depends on extrapolation step and macroscopic variables.

Numerical experiments validate the method's efficiency and error behavior.

Abstract

We present and analyze a micro/macro acceleration technique for the Monte Carlo simulation of stochastic differential equations (SDEs) in which there is a separation between the (fast) time-scale on which individual trajectories of the SDE need to be simulated and the (slow) time-scale on which we want to observe the (macroscopic) function of interest. The method performs short bursts of microscopic simulation using an ensemble of SDE realizations, after which the ensemble is restricted to a number of macroscopic state variables. The resulting macroscopic state is then extrapolated forward in time and the ensemble is projected onto the extrapolated macroscopic state. We provide a first analysis of its convergence in terms of extrapolation time step and number of macroscopic state variables. The effects of the different approximations on the resulting error are illustrated via numerical experiments.