Hot new directions for quasi-Monte Carlo research in step with applications

TL;DR

This paper reviews recent advances in quasi-Monte Carlo methods, highlighting their theoretical foundations, applications, and strategies for cost efficiency, emphasizing their suitability for high-dimensional problems.

Contribution

It provides a comprehensive overview of new theoretical developments and practical strategies in QMC, connecting theory with diverse applications and emphasizing dimension-independent error bounds.

Findings

Error bounds can be independent of dimension s under certain conditions

Efficient algorithms can find good parameters even for large s

QMC methods are closely linked with practical applications and cost-saving strategies

Abstract

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications.