Study of micro-macro acceleration schemes for linear slow-fast stochastic differential equations with additive noise

TL;DR

This paper analyzes a micro-macro acceleration method for linear stochastic differential equations with multiple time scales, demonstrating its stability and efficiency in simulating slow dynamics over long periods.

Contribution

It provides explicit stability analysis and numerical validation of a micro-macro acceleration scheme for linear slow-fast stochastic systems, highlighting its advantages over direct discretization.

Findings

Stability threshold largely independent of time-scale separation.

Method allows larger time steps than direct discretization.

Numerical experiments confirm theoretical predictions.

Abstract

Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro-macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages forward in time. To have explicit derivations, we elicit an amenable linear test equation containing multiple time scales. We make derivations and perform numerical experiments in the Gaussian setting, where only the evolution of mean and variance matters. The analysis shows that, for this test model, the stability threshold on the extrapolation step is largely independent of the time-scale separation. In consequence, the micro-macro acceleration method increases the admissible time steps far beyond those for which a direct time discretization becomes unstable.