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QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND

Part of: Numerical approximation and computational geometry

Published online by Cambridge University Press:  29 May 2012

F. Y. KUO ,
CH. SCHWAB  and
I. H. SLOAN
F. Y. KUO
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia (email: f.kuo@unsw.edu.au, i.sloan@unsw.edu.au)
CH. SCHWAB
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, ETH Zentrum, HG G57.1, CH8092 Zürich, Switzerland (email: christoph.schwab@sam.math.ethz.ch)
I. H. SLOAN*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia (email: f.kuo@unsw.edu.au, i.sloan@unsw.edu.au)
*
✉For correspondence; e-mail: i.sloan@unsw.edu.au
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Abstract

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This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

Keywords

quasi-Monte Carlo methodsQMChigh-dimensional integrationweighted spacesreproducing kernel Hilbert spacesBanach space settingsworst-case errordiscrepancyweighted Koksma–Hlawka inequalitylattice ruleslow-discrepancy sequencesfast CBC constructionPOD weights

MSC classification

Secondary: 65D30: Numerical integration
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 1 , July 2011 , pp. 1 - 37
Copyright
Copyright © Australian Mathematical Society 2012

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