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

TL;DR

This paper introduces a filtered multilevel Monte Carlo method that uses grid transfer operators and filtering techniques from multigrid methods to improve the estimation of expectations of discretized random fields, especially in complex hierarchical settings.

Contribution

The paper develops a novel filtered MLMC estimator incorporating multigrid-inspired filtering, enhancing variance reduction and estimator accuracy for discretized random fields.

Findings

Filtering improves variance estimation for small- and large-scale components.

The F-MLMC estimator outperforms crude MC and unfiltered MLMC in experiments.

Theoretical spectral analysis aligns with numerical results.

Abstract

In this paper, we investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of numerical simulators have inconsistent dimensions across the multilevel hierarchy. This motivates the introduction of grid transfer operators borrowed from multigrid methods. By adapting mathematical tools from multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. We then propose filtered MLMC (F-MLMC) estimators based on a filtering mechanism similar to the smoothing process of multigrid methods, and we show that the filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. Next, the conclusions of the spectral analysis are experimentally verified with a one-dimensional illustration. Finally, the proposed F-MLMC estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.