Circulant embedding with QMC -- analysis for elliptic PDE with lognormal coefficients

TL;DR

This paper analyzes the convergence of a combined circulant embedding and quasi-Monte Carlo method for solving elliptic PDEs with lognormal coefficients, providing theoretical guarantees and numerical validation.

Contribution

It offers the first convergence analysis for the method using QMC with circulant embedding in the context of elliptic PDEs with random coefficients.

Findings

Convergence depends on eigenvalues of the block circulant matrix.

Analysis applies to general covariance matrix factorizations.

Numerical results confirm theoretical predictions on complex domains.

Abstract

In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but a convergence theory was missing. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.