# Recent advances in higher order quasi-Monte Carlo methods

## TL;DR

This paper reviews recent theoretical and practical advances in higher order quasi-Monte Carlo methods, highlighting their development, analysis, and applications in complex computational problems.

## Contribution

It provides a unified overview of how Walsh analysis has advanced the theory and applications of HoQMC methods since their inception.

## Key findings

- Significant progress in discrepancy and multivariate integration theory.
- Successful applications to PDEs with random coefficients.
- Enhanced understanding of Walsh analysis in HoQMC development.

## Abstract

In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007, 2008) who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration.