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Licensed Unlicensed Requires Authentication Published by De Gruyter May 16, 2018

Remarks on randomization of quasi-random numbers

  • Sergej M. Ermakov and Svetlana N. Leora ORCID logo EMAIL logo

Abstract

In this paper we discuss estimation of the quasi-Monte Carlo methods error in the case of calculation of high-order integrals. Quasi-random Halton sequences are considered as a special case. Randomization of these sequences by the random shift method turns out to lead to well-known random quadrature formulas with one free node. Some new properties of such formulas are pointed out. The subject is illustrated by a number of numerical examples.

MSC 2010: 11K45; 65C05

Award Identifier / Grant number: 17-01-00267-a

Funding statement: This work was partially supported by the Russian Foundation for Basic Research, Project No. 17-01-00267-a.

References

[1] S. M. Ermakov, Die Monte Carlo Methode und verwandte Fragen, Deutscher Verlag der Wissenschaften, Berlin, 1975. Search in Google Scholar

[2] J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 84–90. 10.1007/BF01386213Search in Google Scholar

[3] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974. Search in Google Scholar

[4] C. Schlier, Discrepancy behaviour in the non-asymptotic regime, Appl. Numer. Math. 50 (2004), 227–276. 10.1016/j.apnum.2003.12.004Search in Google Scholar

Received: 2017-12-24
Accepted: 2018-4-24
Published Online: 2018-5-16
Published in Print: 2018-6-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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