Quasi-Monte Carlo numerical integration on $\mathbb{R}^s$: digital nets and worst-case error

TL;DR

This paper extends quasi-Monte Carlo methods to integrate functions on  space using transformed digital nets, achieving optimality in fractional Besov spaces through Walsh phase plane analysis.

Contribution

It introduces a novel transformation of digital nets for  integration, enabling quasi-Monte Carlo rules to operate effectively on  spaces with proven optimality.

Findings

Rules are optimal for fractional Besov spaces.

Transformation maintains local qMC properties and global distribution.

Analysis uses Walsh phase plane tilings.

Abstract

Quasi-Monte Carlo rules are equal weight quadrature rules defined over the domain $[0,1]^s$. Here we introduce quasi-Monte Carlo type rules for numerical integration of functions defined on $\mathbb{R}^s$. These rules are obtained by way of some transformation of digital nets such that locally one obtains qMC rules, but at the same time, globally one also has the required distribution. We prove that these rules are optimal for numerical integration in fractional Besov type spaces. The analysis is based on certain tilings of the Walsh phase plane.