Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration
type: Item
instance of: scholarly article
title: Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration
zbMATH Open document ID: 0876.11042
DOI: 10.1007/BF01298952
author: Gerhard Larcher
author: Harald Niederreiter
author: Wolfgang Ch. Schmid
published in: Monatshefte für Mathematik
publication date: 1996-06-04T00:00:00Z
full work available at URL: https://eudml.org/doc/178723
review text: The concept of a \((t,m,s)\) net (for this notion see the excellent monograph [\textit{H. Niederreiter}, Random number generation and quasi-Monte Carlo methods, SIAM (1992; Zbl 0761.65002)]) and of \((t,s)\)-sequences is of fundamental importance for high-dimensional quasi-Monte Carlo integration, in theory as well as in practice. It allows for the construction of so-called ``low discrepancy'' point sets, which serve as the nodes at which the integrand is computed. Until now, all construction methods for \((t,m,s)\) nets of practical relevance are digital methods over certain rings \(R\). That means that digit expansions in some integer base \(b\geq 2\) are used to define the net. In this paper, the authors present a much more general version of this approach to numerical integration based on digital nets over arbitrary finite rings. They provide an error bound for integrands with rapidly converging Walsh series. Further important results of this paper concern the existence of digital nets and of digital sequences constructed over arbitrary rings. In particular, necessary and sufficient conditions for the existence of digital \((0,m,s)\)-nets and of digital \((0,s)\)-sequences are exhibited. In the final section, the authors give explicit constructions of digital \((t,m,s)\)-nets and digital \((t,s)\)-sequences in the case \(R=\mathbb{Z}_b\).
reviewed by: Peter Hellekalek
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 11K45
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 874082
zbMATH Keywords: \((t,m,s)\) net
zbMATH Keywords: low discrepancy point sets
zbMATH Keywords: high-dimensional quasi-Monte Carlo integration
zbMATH Keywords: error bound
zbMATH Keywords: rapidly converging Walsh series
zbMATH Keywords: digital nets
zbMATH Keywords: digital sequences
MaRDI profile type: Publication
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OpenAlex ID: W2086236484
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description: scientific article; zbMATH DE number 874082