Improved Efficiency of Multilevel Monte Carlo for Stochastic PDE through Strong Pairwise Coupling

TL;DR

This paper demonstrates that strong pairwise coupling in multilevel Monte Carlo methods significantly enhances efficiency when applied to stochastic reaction-diffusion PDEs, improving convergence rates and computational tractability.

Contribution

It proves that strong pairwise coupling increases the efficiency of MLMC for SPDEs and provides numerical evidence comparing different coupling strategies.

Findings

Strong pairwise coupling improves convergence rates in MLMC for SPDEs.

Theoretical proof of higher efficiency rates with strong coupling.

Numerical comparisons validate the importance of coupling strategies.

Abstract

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when (MLMC) is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the (MLMC) method with strong pairwise coupling that was developed and studied numerically on filtering problems in [{\it Chernov et al., Numer. Math., 147 (2021), 71-125}], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this feature.