Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling

TL;DR

This paper introduces a fast, linear-time sampling method for Gaussian random fields in PDEs using multilevel quasi-Monte Carlo, combining wavelet expansions and supermesh techniques for efficient, scalable simulations.

Contribution

It presents a novel fast white noise sampling algorithm tailored for MLQMC, enabling efficient PDE solutions with random coefficients and non-nested meshes.

Findings

Sampling complexity is linear in the number of mesh cells.

The method achieves good QMC convergence with finite importance terms.

Numerical experiments demonstrate efficiency and accuracy.

Abstract

When solving partial differential equations with random fields as coefficients the efficient sampling of random field realisations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the SPDE approach of Lindgren et al.~combined with a new fast algorithm for white noise sampling which is taylored to (ML)QMC. We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.