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High-dimensional integration: The quasi-Monte Carlo way*†

Published online by Cambridge University Press:  02 April 2013

Josef Dick ,
Frances Y. Kuo  and
Ian H. Sloan
Josef Dick
Affiliation:
University of New South Wales, Sydney, NSW 2052, Australia E-mail: josef.dick@unsw.edu.au
Frances Y. Kuo
Affiliation:
>University of New South Wales, Sydney, NSW 2052, Australia E-mail: f.kuo@unsw.edu.au
Ian H. Sloan
Affiliation:
University of New South Wales, Sydney, NSW 2052, Australia E-mail: i.sloan@unsw.edu.au King Fahd University of Petroleum and Minerals, Dhahran 34463, Saudi Arabia
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Abstract

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

Type
Research Article
Information
Acta Numerica , Volume 22 , May 2013 , pp. 133 - 288
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

†

The URLs cited in this work were correct at the time of going to press, but the publisher and the authors make no undertaking that the citations remain live or are accurate or appropriate.

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