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On the quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences

Josef Dick, Takashi Goda, Kosuke Suzuki

TL;DR

This work investigates whether common low-discrepancy digital nets and sequences are quasi-uniform by introducing well-separated point sets and an algebraic criterion to certify quasi-uniformity. It provides a concrete two-dimensional, low-discrepancy digital net that is well-separated (hence quasi-uniform) and documents several counterexamples where low-discrepancy does not imply quasi-uniformity, including Sobol', Hammersley, Faure, and Fibonacci polynomial lattices. The results delineate the limitations of quasi-uniformity in higher dimensions and establish a framework to assess quasi-uniformity via algebraic conditions on generating matrices, with implications for space-filling designs and scattered-data approximation. The paper also notes that lattice-based quasi-Monte Carlo nets will be addressed in a forthcoming work, suggesting a broader program to align discrepancy with space-filling properties in QMC constructions.

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 \emph{well-separated} point sets and provide an algebraic criterion to determine whether a given digital net 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. The quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences will be studied in a forthcoming paper.

On the quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences

TL;DR

This work investigates whether common low-discrepancy digital nets and sequences are quasi-uniform by introducing well-separated point sets and an algebraic criterion to certify quasi-uniformity. It provides a concrete two-dimensional, low-discrepancy digital net that is well-separated (hence quasi-uniform) and documents several counterexamples where low-discrepancy does not imply quasi-uniformity, including Sobol', Hammersley, Faure, and Fibonacci polynomial lattices. The results delineate the limitations of quasi-uniformity in higher dimensions and establish a framework to assess quasi-uniformity via algebraic conditions on generating matrices, with implications for space-filling designs and scattered-data approximation. The paper also notes that lattice-based quasi-Monte Carlo nets will be addressed in a forthcoming work, suggesting a broader program to align discrepancy with space-filling properties in QMC constructions.

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 \emph{well-separated} point sets and provide an algebraic criterion to determine whether a given digital net 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. The quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences will be studied in a forthcoming paper.

Paper Structure

This paper contains 14 sections, 17 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Digital nets and sequences
  4. Quasi-Uniformity
  5. Some known results
  6. Well-separated nets
  7. Basic concepts and properties
  8. An algebraic criterion for well-separated digital nets
  9. An example of two-dimensional well-separated digital nets
  10. Not well-separated digital nets
  11. The Hammersley $(0,m,2)$-net
  12. Some $(0,2)$-sequences in base $2$
  13. Faure sequence
  14. Some Fibonacci polynomial lattice

Key Result

Lemma 2.8

Let $\mathcal{S} = \{\boldsymbol{x}_0, \boldsymbol{x}_1, \ldots \} \subset [0,1]^d$ be a sequence. Let $P_i = \{ \boldsymbol{x}_0, \boldsymbol{x}_1, \ldots, \boldsymbol{x}_{i-1}\}$ denote the first $i$ points of $\mathcal{S}$. Let $1 \le i_1 < i_2 < i_3 < \cdots$ be an increasing sequence.

Theorems & Definitions (40)

  • Definition 2.1: Digital net
  • Definition 2.2: Digitally shifted digital net
  • Definition 2.3: Digital sequence
  • Remark 2.4: Lower bound on the covering radius
  • Remark 2.5: Upper bound on the separation radius
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Definition 3.1
  • ...and 30 more