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Column reduced digital nets

Vishnupriya Anupindi, Peter Kritzer

TL;DR

The paper develops column reduced digital nets as a structure to accelerate quasi-Monte Carlo matrix-matrix products and provides rigorous $t$-value bounds for the reduced nets. It introduces a fast algorithm that exploits the repetitive structure arising from zeroed columns, along with an error-analysis framework based on weighted star discrepancy and projection bounds. The results demonstrate both computational speedups and theoretical guarantees, with numerical experiments validating the practical benefits over row-reduced nets. This work enhances the toolkit for efficient QMC-based integration, especially in high-dimensional settings common in uncertainty quantification.

Abstract

Digital nets provide an efficient way to generate integration nodes of quasi-Monte Carlo (QMC) rules. For certain applications, as e.g. in Uncertainty Quantification, we are interested in obtaining a speed-up in computing products of a matrix with the vectors corresponding to the nodes of a QMC rule. In the recent paper "The fast reduced QMC matrix-vector product" (J. Comput. Appl. Math. 440, 115642, 2024), a speed up was obtained by using so-called reduced lattices and row reduced digital nets. In this work, we propose a different multiplication algorithm where we exploit the repetitive structure of column reduced digital nets instead of row reduced digital nets. This method has advantages over the previous one, as it facilitates the error analysis when using the integration nodes in a QMC rule. We also provide an upper bound for the quality parameter of column reduced digital nets, and numerical tests to illustrate the efficiency of the new algorithm.

Column reduced digital nets

TL;DR

The paper develops column reduced digital nets as a structure to accelerate quasi-Monte Carlo matrix-matrix products and provides rigorous -value bounds for the reduced nets. It introduces a fast algorithm that exploits the repetitive structure arising from zeroed columns, along with an error-analysis framework based on weighted star discrepancy and projection bounds. The results demonstrate both computational speedups and theoretical guarantees, with numerical experiments validating the practical benefits over row-reduced nets. This work enhances the toolkit for efficient QMC-based integration, especially in high-dimensional settings common in uncertainty quantification.

Abstract

Digital nets provide an efficient way to generate integration nodes of quasi-Monte Carlo (QMC) rules. For certain applications, as e.g. in Uncertainty Quantification, we are interested in obtaining a speed-up in computing products of a matrix with the vectors corresponding to the nodes of a QMC rule. In the recent paper "The fast reduced QMC matrix-vector product" (J. Comput. Appl. Math. 440, 115642, 2024), a speed up was obtained by using so-called reduced lattices and row reduced digital nets. In this work, we propose a different multiplication algorithm where we exploit the repetitive structure of column reduced digital nets instead of row reduced digital nets. This method has advantages over the previous one, as it facilitates the error analysis when using the integration nodes in a QMC rule. We also provide an upper bound for the quality parameter of column reduced digital nets, and numerical tests to illustrate the efficiency of the new algorithm.
Paper Structure (14 sections, 2 theorems, 58 equations, 2 figures, 1 algorithm)

This paper contains 14 sections, 2 theorems, 58 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The problem setting
  3. Digital nets and sequences
  4. The $t$-values of column reduced digital nets
  5. Column reduction for $(t,m,s)$-nets
  6. Case 1: $\max\{t,w_2\}=w_2$.
  7. Case 2: $\max\{t,w_2\}=t$.
  8. Column reduction for $(t,m,{\bf{\textit{e}}},s)$-nets
  9. Projections of column reduced digital nets
  10. Applications of column reduced digital nets
  11. A reduced matrix product algorithm
  12. Error analysis
  13. Numerical experiments
  14. Conclusion

Key Result

Theorem 1

Let ${\mathcal{P}}$ be a digital $(t,m,s)$-net over $\mathbb{F}_b$ with generating matrices $C_1^{(m)}, \dots ,C_s^{(m)}$ derived from a digital $(t,s)$-sequence over $\mathbb{F}_b$, where we assume that $m\ge t$. Let $\widetilde{C}_1^{(m)},\ldots,\widetilde{C}_s^{(m)}$ be as defined in eq:def_Ci_re and $\widetilde{t} \le \min\{m,w_s +t\}$. Furthermore, if ${\mathcal{P}}$ is a strict digital $(t,m

Figures (2)

  • Figure 1: $m=12,\tau = 20$, varying $w_j$
  • Figure 2: $s=800, \tau = 20$, varying $m$

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Example 1
  • ...and 5 more