Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients

TL;DR

This paper develops multi-level quasi-Monte Carlo finite element methods to efficiently estimate expected values of solutions to elliptic PDEs with random coefficients, achieving high accuracy with minimal computational effort.

Contribution

It introduces a multi-level QMC FE approach with level-dependent dimension truncation, providing error bounds and demonstrating near-optimal computational complexity for elliptic PDEs with randomness.

Findings

Error of order h^2 or N^{-1+δ} achieved

Total work comparable to a single PDE solve at finest discretization

QMC rules with POD weights effectively handle high-dimensional integrals

Abstract

Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient, to estimate expected values of linear functionals of the solution. The expected value is considered as an infinite-dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. We use a multi-level algorithm, with the number of QMC points depending on the discretization level, and with a level-dependent dimension truncation strategy. In some scenarios, we show that the overall error is $\mathcal{O}(h^2)$, where $h$ is the finest FE mesh width, or $\mathcal{O}(N^{-1+\delta})$ for arbitrary $\delta>0$, where $N$ is the maximal number of QMC sampling points. For these scenarios, the total work is essentially of the order of one single PDE solve at the finest FE discretization level. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). Families of QMC rules with "POD weights" ("product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.