Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial

TL;DR

This paper offers an overview and tutorial on applying quasi-Monte Carlo methods to PDEs with random coefficients, aiming to bridge QMC experts and PDE practitioners with practical guidance and recent research insights.

Contribution

It provides a comprehensive overview, a step-by-step tutorial, and discusses software tools for applying QMC methods to PDEs with random coefficients.

Findings

Effective QMC strategies for PDEs with random coefficients

Guidelines for implementing first-order QMC rules in uniform cases

Accessible entry point for researchers in QMC and PDE analysis

Abstract

This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tutorial of the required analysis for the setting known as the uniform case with first order QMC rules. The aim of this article is to provide an easy entry point for QMC experts wanting to start research in this direction and for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods.