Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients - a survey of analysis and implementation

TL;DR

This survey reviews recent advances in applying quasi-Monte Carlo methods to elliptic PDEs with random coefficients, comparing different approaches, analyzing errors, and providing practical software guidance.

Contribution

It offers a comprehensive overview of QMC techniques for elliptic PDEs with random coefficients, including analysis, comparisons, and implementation guidance.

Findings

QMC methods effectively handle high-dimensional PDE problems.

Error analysis techniques unify various QMC approaches.

Practical software tools facilitate QMC implementation for PDEs.

Abstract

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers, and contrasts, the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first order QMC rules versus higher order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.