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Numerical Integration III pp 157–171Cite as

Birkhäuser

Quasi-Monte Carlo Methods for Multidimensional Numerical Integration

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Abstract

The Monte Carlo approximation to an integral is obtained by averaging values of the integrand at randomly selected nodes. Concretely, if we normalize the integration domain to be \({\bar I^S}\) = [0,1]s, s ≥ 1, then

$$\int\limits_{{{\overline I }^s}} {f\left( {\underline t } \right)} d\underline t \approx \frac{1}{N}\sum\limits_{n = 1}^N {f\left( {{{\underline x }_n}} \right),}$$
((1))

where x 1,..., x N are N independent random samples from the uniform distribution on \({\bar I^S}\) . The expected value of the integration error is O(N-½). The basic idea of a quasi-Monte Carlo method is to replace random nodes by well-chosen deterministic nodes, with the aim of getting a deterministic error bound that is smaller than the stochastic error bound under weak regularity conditions on the integrand.

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Authors and Affiliations

  1. Mathematical Institute, Austrian Academy of Sciences, Dr.-Ignaz-Seipel-Platz 2, A-1010, Vienna, Austria

    Prof. Dr. Harald Niederreiter

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  1. Prof. Dr. Harald Niederreiter
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Editors and Affiliations

  1. Institut für angewandte Mathematik, TU Braunschweig, Pockelsstrasse 14, D-3300, Braunschweig, Germany

    H. Braß

  2. Mathematisches Institut, Ludwig-Maximilian-Universität, Theresienstrasse 39, D-8000, München 2, Germany

    G. Hämmerlin

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Niederreiter, H. (1988). Quasi-Monte Carlo Methods for Multidimensional Numerical Integration. In: Braß, H., Hämmerlin, G. (eds) Numerical Integration III. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 85. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6398-8_15

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  • DOI: https://doi.org/10.1007/978-3-0348-6398-8_15

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  • Print ISBN: 978-3-7643-2205-2

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