Quasi-Monte Carlo integration
type: Item
instance of: scholarly article
title: Quasi-Monte Carlo integration
zbMATH Open document ID: 0863.65005
author: William J. Morokoff
author: Russel Caflisch
published in: Journal of Computational Physics
publication date: 1997-06-08T00:00:00Z
review text: Monte Carlo methods for multidimensional integration using random (pseudo-random) and quasi-random nodes are compared both through error analysis and extensive numerical computations. Known error expressions in terms of variance, discrepancy and variation are reviewed, and the expected advantages of some quasi-random nodes (Halton, Sobol', Faure) of low discrepancy are investigated numerically. Convergence like \(N^{-\alpha}({1\over 2}\leq \alpha\leq 1)\) for \(N\) points is found empirically for problems with widely differing characteristics and dimension \(s\). The authors find that the integrand's variation is not a useful indicator of error in these practical problems, but that its variance, the determining factor for random Monte Carlo analysis, can often provide a rough bound. The general conclusion is that quasi-Monte Carlo methods are superior to random Monte Carlo, but the advantages can sometimes be slight, especially for higher dimensions or non-smooth integrands. For discontinuous integrands an error bound is found suggesting that behaviour better than \(\alpha=s/(2s-1)\) cannot be counted on even when using nodes of low discrepancy.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 851733
zbMATH Keywords: random Monte Carlo methods
zbMATH Keywords: convergence
zbMATH Keywords: multidimensional integration
zbMATH Keywords: quasi-random
zbMATH Keywords: error analysis
zbMATH Keywords: low discrepancy
zbMATH Keywords: quasi-Monte Carlo methods
zbMATH Keywords: discontinuous integrands
reviewed by: William E. Smith
MaRDI profile type: Publication
OpenAlex ID: W2079884250
DOI: 10.1006/JCPH.1995.1209
description: scientific article; zbMATH DE number 851733