On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences

TL;DR

This paper investigates the quasi-uniformity of digital nets used in quasi-Monte Carlo methods, introducing criteria for well-separated sets and providing examples and counterexamples of low-discrepancy digital nets that are or are not quasi-uniform.

Contribution

It introduces an algebraic criterion for quasi-uniformity in digital nets and constructs examples of low-discrepancy digital nets that are quasi-uniform.

Findings

A criterion for well-separated digital nets

Existence of low-discrepancy quasi-uniform digital nets

Counterexamples of non-quasi-uniform digital nets

Abstract

We study the quasi-uniformity properties of digital nets, a class of quasi-Monte Carlo point sets. Quasi-uniformity is a space-filling property used for instance in experimental designs and radial basis function approximation. However, it has not been investigated so far whether common low-discrepancy digital nets are quasi-uniform, with the exception of the two-dimensional Sobol' sequence, which has recently been shown not to be quasi-uniform. In this paper, with the goal of constructing quasi-uniform low-discrepancy digital nets, we introduce the notion of well-separated point sets and provide an algebraic criterion to determine whether a given sequence of digital nets is well-separated. Using this criterion, we present an example of a two-dimensional digital net which has low-discrepancy and is quasi-uniform. Additionally, we provide several counterexamples of low-discrepancy digital nets that are not quasi-uniform.