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Reduced digital nets

Vishnupriya Anupindi, Peter Kritzer

TL;DR

The paper investigates using row, column, and column-row reduced digital nets to accelerate QMC vector-matrix products $XA$ in integration rules, by inducing repetitive structure in the quadrature points. It derives explicit bounds on the quality parameters $\rho_m$ and the minimal $t$-value for row-reduced and column-row reduced nets, enabling error analysis via the weighted star discrepancy. It provides concrete computational cost reductions for point generation and the matrix-matrix product, and it extends the framework to projected nets with performance guarantees. Numerical experiments in Julia demonstrate practical speed-ups for column-row and column reductions relative to row reductions and standard methods, validating the proposed approach for high-dimensional QMC integration. These results offer a principled route to faster QMC-based integration when the vector-matrix product is the bottleneck, accompanied by rigorous error control through projected net parameters and discrepancy bounds.

Abstract

In the recent papers ``The fast reduced QMC matrix-vector product'' (J. Comput. Appl. Math. 440, 115642, 2024) and ``Column reduced digital nets'' (submitted), it was proposed to use QMC rules based on reduced digital nets which provide a speed-up in the computation of QMC vector-matrix products that may occur in practical applications. In this paper, we provide upper bounds on the quality parameter of row reduced and column-row reduced digital nets, which are helpful for the error analysis of using reduced point sets as integration nodes in a QMC rule. We also give remarks on further aspects and comparisons of row reduced, column reduced, and column-row reduced digital net.

Reduced digital nets

TL;DR

The paper investigates using row, column, and column-row reduced digital nets to accelerate QMC vector-matrix products in integration rules, by inducing repetitive structure in the quadrature points. It derives explicit bounds on the quality parameters and the minimal -value for row-reduced and column-row reduced nets, enabling error analysis via the weighted star discrepancy. It provides concrete computational cost reductions for point generation and the matrix-matrix product, and it extends the framework to projected nets with performance guarantees. Numerical experiments in Julia demonstrate practical speed-ups for column-row and column reductions relative to row reductions and standard methods, validating the proposed approach for high-dimensional QMC integration. These results offer a principled route to faster QMC-based integration when the vector-matrix product is the bottleneck, accompanied by rigorous error control through projected net parameters and discrepancy bounds.

Abstract

In the recent papers ``The fast reduced QMC matrix-vector product'' (J. Comput. Appl. Math. 440, 115642, 2024) and ``Column reduced digital nets'' (submitted), it was proposed to use QMC rules based on reduced digital nets which provide a speed-up in the computation of QMC vector-matrix products that may occur in practical applications. In this paper, we provide upper bounds on the quality parameter of row reduced and column-row reduced digital nets, which are helpful for the error analysis of using reduced point sets as integration nodes in a QMC rule. We also give remarks on further aspects and comparisons of row reduced, column reduced, and column-row reduced digital net.
Paper Structure (9 sections, 3 theorems, 44 equations, 2 figures)

This paper contains 9 sections, 3 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The problem setting
  3. Digital nets and sequences
  4. Row reduced digital nets
  5. The advantage of using reduced digital nets
  6. Projections of digital nets
  7. Error analysis
  8. Combination of column and row reduced digital nets
  9. Numerical experiments

Key Result

Theorem 1

Let ${\mathcal{P}}$ be a digital $(t,m,s)$-net over ${\mathbb{F}}_b$ with generating matrices $C_1^{(m)}, \dots ,C_s^{(m)}$. Let $\widetilde{C}_1^{(m)},\ldots,\widetilde{C}_s^{(m)}$ be row reduced with respect to reduction indices $0= w_1 \leq \dots \leq w_s$, and let $\widetilde{t}$ be the minimal and $\widetilde{t} \leq \min \{m, \max \{t,w_s \} \}$.

Figures (2)

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

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Remark 1: Row reduced nets
  • Remark 2: Column reduced nets
  • Corollary 1
  • Lemma 1
  • proof
  • ...and 2 more