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When solving partial differential equations with random fields ascoefficients the efficient sampling of random field realisations can bechallenging. In this paper we focus on the fast sampling of Gaussian fieldsusing quasi-random points in a finite element and multilevel quasi Monte Carlo(MLQMC) setting. Our method uses the SPDE approach of Lindgren et al.~combinedwith 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 intwo parts. The first part is sampled using quasi-random points and contains afinite number of terms in order of decaying importance to ensure good QMCconvergence. The second part is a correction term which is sampled usingstandard pseudo-random numbers. We show how the sampling of both terms can beperformed in linear time and memory complexity in the number of mesh cells viaa supermesh construction, yielding an overall linear cost. Furthermore, ourtechnique can be used to enforce the MLQMC coupling even in the case ofnon-nested mesh hierarchies. We demonstrate the efficacy of our method withnumerical experiments.

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