Reduced digital nets

TL;DR

This paper analyzes the quality of reduced digital nets used in quasi-Monte Carlo methods, providing bounds on their parameters to facilitate error analysis and compare different reduction techniques.

Contribution

It offers upper bounds on the quality parameters of various reduced digital nets, advancing understanding of their effectiveness in QMC integration.

Findings

Upper bounds on quality parameters for row reduced digital nets

Upper bounds on quality parameters for column-row reduced digital nets

Comparative remarks on different reduction methods

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.