Quasi-Monte carlo integration

William J. Morokoff, Russel Caflisch

Research output: Contribution to journalArticle

Abstract

The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)-random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo-random sequences with more uniformly distributed sequences known as quasi-random. In this paper quasi-random (Halton, Sobol', and Faure) and pseudo-random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand, including variance, variation, smoothness, and dimension. The results show that variation, which plays an important role in the theoretical upper bound given by the Koksma-Hlawka inequality, does not affect convergence, while variance, the determining factor in random Monte Carlo, is shown to provide a rough upper bound, but does not accurately predict performance. In general, quasi-Monte Carlo methods are superior to random Monte Carlo, but the advantage may be slight, particularly in high dimensions or for integrands that are not smooth. For discontinuous integrands, we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi-Monte Carlo is only slightly larger than 1/2 in high dimensions.

Original languageEnglish (US)
Pages (from-to)218-230
Number of pages13
JournalJournal of Computational Physics
Volume122
Issue number2
DOIs
StatePublished - 1995

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pseudorandom sequences
Monte Carlo methods
quadratures
Monte Carlo method
exponents
Experiments
decay

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)
  • Computer Science Applications

Cite this

Quasi-Monte carlo integration. / Morokoff, William J.; Caflisch, Russel.

In: Journal of Computational Physics, Vol. 122, No. 2, 1995, p. 218-230.

Research output: Contribution to journalArticle

Morokoff, WJ & Caflisch, R 1995, 'Quasi-Monte carlo integration', Journal of Computational Physics, vol. 122, no. 2, pp. 218-230. https://doi.org/10.1006/jcph.1995.1209
Morokoff, William J. ; Caflisch, Russel. / Quasi-Monte carlo integration. In: Journal of Computational Physics. 1995 ; Vol. 122, No. 2. pp. 218-230.
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