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Quasi-Monte Carlo methods for high dimensional integration - the standard (weighted Hilbert space) setting and beyond

by F.Y. Kuo and Ch. Schwab and I. H. Sloan

(Report number 2012-01)

Abstract
This paper is a contemporary review of QMC ("quasi-Monte Carlo") methods, i.e., 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 CBC ("component-by-component") construction of lattice rules that achieve the optimal convergence order (i.e., a rate of almost 1=N, where N is the number of points, independently of dimension) to so-called POD ("product-and-order-dependent") weights, as seen in some recent applications.

Keywords:

@Techreport{KSS12_153,
  author = {F.Y. Kuo and Ch. Schwab and I. H. Sloan},
  title = {Quasi-Monte Carlo methods for high dimensional integration - the standard (weighted Hilbert space) setting and beyond},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-01.pdf },
  year = {2012}
}

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