M. Croci, M. B. Giles, M. E. Rognes, and P. E. Farrell, Multilevel quasi Monte Carlo methods for elliptic partial differential equations driven by spatial white noise, submitted to SISC. Preprint, Talk.
When solving partial differential equations driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this paper we focus on the efficient sampling of white noise using quasi-random points in a finite element method and multilevel Quasi Monte Carlo (MLQMC) setting. This work is an extension of previous research on white noise sampling for MLMC.
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. 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.