Improving MLMC for SDEs with application to the Langevin equation

TL;DR

This paper enhances the efficiency of the Multilevel Monte Carlo method for stochastic differential equations, especially the Langevin equation, by applying numerical tricks, operator splitting, and discrete random variables, achieving significant computational gains.

Contribution

It introduces novel MLMC techniques for Langevin equations using operator splitting and discrete random variables, improving efficiency without requiring strong-approximation theory.

Findings

Discrete random variables increase efficiency by nearly two orders of magnitude for small-noise problems.

Modified equations analysis allows for improved MLMC without strong-approximation theory.

Combining operator splitting and extrapolation enhances MLMC performance for SDEs.

Abstract

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the Multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis to circumvent the need for a strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.