The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets
type: Item
instance of: scholarly article
title: The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets
zbMATH Open document ID: 1111.65002
author: N. E. Zubov
published in: Numerische Mathematik
publication date: 2007-01-30T00:00:00Z
review text: The authors investigate the worst-case error of the multivariate integration in reproducing kernel Sobolev spaces. These spaces are composed of functions whose second partial derivatives are square integrable. For an approximation of the \(s\)-dimensional integrals of the functions of these spaces a quasi-Monte Carlo (QMC) algorithm is used. For a deterministically chosen point set in \([0,1)^{s}\) two component-by-component transformations are used. The first one is a digital shift with a random vector in \([0,1)^{s}\) and the second one is a fold with tent transformation. The well-known digital \((t,m,s)\)-nets over \({\mathbb Z}_{2}\) and nets, constructed by polynomial lattice rules are used in the paper. In section 2 some preliminary definitions and results are presented. In section 3 the notions: a worst-case error of a QMC algorithm of the integration in reproducing kernel Hilbert space, a mean square worst-case error for the integration, a digitally shifted and folded kernel are reminded. Theorem 1 represents the relation beween the mean square worst-case error of the integration in a reproducing kernel Hilbert space, generated by an ordinary kernel and the worst-case error of the integration in a reproducing kernel Hilbert space, generated by a digitally shifted and then folded initial kernel. Theorem 2 gives a formula for the digitally shifted and then folded kernel in the terms of the Walsh functions in base 2. Theorem 3 gives a formula for the mean square worst-case error of the integration using an arbitrary \(s\)-dimensional net and the form of this formula in the case when the net is chosen to be a digital \((t,m,s)\)-net over \({\mathbb Z}_{2}.\) In Theorem 4 the formula for the mean square worst-case error of the integration using an arbitrary \(s\)-dimensional net in the terms of the Walsh functions in base 2 is shown. In the cases when the net is chosen to be a digital \((t,m,s)\)-net over \({\mathbb Z}_{2}\) and a net, constructed by polynomial l…
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K45
zbMATH DE Number: 5119708
zbMATH Keywords: reproducing kernel Sobolev space
zbMATH Keywords: worst-case error
zbMATH Keywords: mean square worst-case error
zbMATH Keywords: digital shift
zbMATH Keywords: folded point set
zbMATH Keywords: Walsh functions
zbMATH Keywords: (t,m,s)-nets, polynomial lattice rules
zbMATH Keywords: Korobov polynomial lattice rules
reviewed by: Vassil Grozdanov
MaRDI profile type: Publication
full work available at URL: s00211-006-0046-x
OpenAlex ID: W1976420124
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DOI: 10.1007/S00211-006-0046-X
description: scientific article; zbMATH DE number 5119708