Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes

TL;DR

This paper introduces an efficient white noise sampling method for SPDEs within finite element and multilevel Monte Carlo frameworks, enabling better coupling across non-nested meshes and achieving optimal convergence.

Contribution

A novel sampling technique that exploits local matrix factorization to efficiently generate white noise samples and enforce coupling in MLMC with non-nested meshes.

Findings

Achieves optimal convergence rates in 2D and 3D SPDE solutions

Ensures proper coupling of samples across MLMC levels

Demonstrates convergence of sampled field covariances

Abstract

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.