On the root mean square weighted \(L_{2}\) discrepancy of scrambled nets
type: Item
instance of: scholarly article
title: On the root mean square weighted \(L_{2}\) discrepancy of scrambled nets
zbMATH Open document ID: 1073.65005
author: Friedrich Pillichshammer
published in: Journal of Complexity
publication date: 2004-11-23T00:00:00Z
review text: The author proves an estimate for the root mean square weighted \(L_{2}\) discrepancy of scrambled \((0,m,s)\)-nets in base \(b\) which are frequently used quasi-Monte Carlo (QMC) point sets. For a net of size \(N\), the bound is essentially (i.e. up to logarithmic factors) proportional to \(N^{-1}\) which is known to be the optimal order. In contrast to this, Monte-Carlo points exhibit a weighted \(L_{2}\) discrepancy proportional to \(N^{-1/2}\). For certain sets of weights, the bound is explicitly computed. Unfortunately the dependence on the dimension is exponentially. Nevertheless, the bound can serve as substitute for explicit expressions for the \(L_{2}\) discrepancy which are only computable for fairly low dimensions.
reviewed by: Michael Junk
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 11K45
zbMATH DE Number: 2117865
zbMATH Keywords: quasi-Monte Carlo methods
zbMATH Keywords: scrampled nets
zbMATH Keywords: weighted \(L_2\) discrepancy
MaRDI profile type: Publication
full work available at URL: j.jco.2003.08.007
OpenAlex ID: W2092904482
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DOI: 10.1016/J.JCO.2003.08.007
description: scientific article; zbMATH DE number 2117865