Smoothed Circulant Embedding with Applications to Multilevel Monte Carlo Methods for PDEs with Random Coefficients

TL;DR

This paper enhances the efficiency of Monte Carlo methods for PDEs with random coefficients by integrating a smoothing technique into circulant embedding, significantly reducing computational costs in practical scenarios.

Contribution

It introduces a novel smoothing approach within circulant embedding to decouple mesh coarseness from correlation length, improving MLMC efficiency for oscillatory random fields.

Findings

Achieved 5-10 times reduction in computational cost.

Enabled coarser initial meshes regardless of correlation length.

Validated improvements through numerical experiments.

Abstract

We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We make use of the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields, we devise and implement a smoothing technique integrated into the circulant embedding method. This allows to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of factor 5-10 in computational cost for accuracies of practical interest.