Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

TL;DR

This paper proposes a novel recycling approach for samples in the multigrid multilevel Monte Carlo method, combining Quasi-Monte Carlo points and an improved variance estimate to enhance efficiency in PDE uncertainty quantification.

Contribution

It introduces an alternative sample recycling method using Quasi-Monte Carlo points and provides a new variance estimate, improving convergence speed and efficiency over traditional Monte Carlo approaches.

Findings

Sample recycling is more effective with nonsmooth random fields.

The method accelerates convergence in elliptic PDEs with lognormal coefficients.

Numerical results demonstrate improved efficiency across various covariance smoothness.

Abstract

The Multilevel Monte Carlo method is an efficient variance reduction technique. It uses a sequence of coarse approximations to reduce the computational cost in uncertainty quantification applications. The method is nowadays often considered to be the method of choice for solving PDEs with random coefficients when many uncertainties are involved. When using Full Multigrid to solve the deterministic problem, coarse solutions obtained by the solver can be recycled as samples in the Multilevel Monte Carlo method, as was pointed out by Kumar, Oosterlee and Dwight [Int. J. Uncertain. Quantif., 7 (2017), pp. 57--81]. In this article, an alternative approach is considered, using Quasi-Monte Carlo points, to speed up convergence. Additionally, our method comes with an improved variance estimate which is also valid in case of the Monte Carlo based approach. The new method is illustrated on the example of an elliptic PDE with lognormal diffusion coefficient. Numerical results for a variety of random fields with different smoothness parameters in the Mat\'ern covariance function show that sample recycling is more efficient when the input random field is nonsmooth.