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We investigate the use of multilevel Monte Carlo (MLMC) methods forestimating the expectation of discretized random fields. Specifically, weconsider a setting in which the input and output vectors of the numericalsimulators have inconsistent dimensions across the multilevel hierarchy. Thisrequires the introduction of grid transfer operators borrowed from multigridmethods. Starting from a simple 1D illustration, we demonstrate numericallythat the resulting MLMC estimator deteriorates the estimation of high-frequencycomponents of the discretized expectation field compared to a Monte Carlo (MC)estimator. By adapting mathematical tools initially developed for multigridmethods, we perform a theoretical spectral analysis of the MLMC estimator ofthe expectation of discretized random fields, in the specific case of linear,symmetric and circulant simulators. This analysis provides a spectraldecomposition of the variance into contributions associated with each scalecomponent of the discretized field. We then propose improved MLMC estimatorsusing a filtering mechanism similar to the smoothing process of multigridmethods. The filtering operators improve the estimation of both the small- andlarge-scale components of the variance, resulting in a reduction of the totalvariance of the estimator. These improvements are quantified for the specificclass of simulators considered in our spectral analysis. The resulting filteredMLMC (F-MLMC) estimator is applied to the problem of estimating the discretizedvariance field of a diffusion-based covariance operator, which amounts toestimating the expectation of a discretized random field. The numericalexperiments support the conclusions of the theoretical analysis even withnon-linear simulators, and demonstrate the improvements brought by the proposedF-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

A filtered multilevel Monte Carlo method for estimating the expectation of discretized random fields