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Publication Date:
May 2012
ISSN:
1569-3961
DOI:
10.1515/mcma-2012-0006

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Monte Carlo Methods and Applications

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Probabilistic error bounds for the discrepancy of mixed sequences

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1Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

2Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

Citation Information: Monte Carlo Methods and Applications. Volume 18, Issue 2, Pages 181–200, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: 10.1515/mcma-2012-0006, May 2012

Publication History:
Received:
2011-03-31
Accepted:
2012-02-23
Published Online:
2012-05-15

Abstract.

In many applications Monte Carlo (MC) sequences or Quasi-Monte Carlo (QMC) sequences are used for numerical integration. In moderate dimensions the QMC method typically yield better results, but its performance significantly falls off in quality if the dimension increases. One class of randomized QMC sequences, which try to combine the advantages of MC and QMC, are so-called mixed sequences, which are constructed by concatenating a d-dimensional QMC sequence and an ()-dimensional MC sequence to obtain a sequence in dimension s. Ökten, Tuffin and Burago proved probabilistic asymptotic bounds for the discrepancy of mixed sequences, which were refined by Gnewuch. In this paper we use an interval partitioning technique to obtain improved probabilistic bounds for the discrepancy of mixed sequences. By comparing them with lower bounds we show that our results are almost optimal.

Keywords: Monte Carlo; Quasi-Monte Carlo; discrepancy; hybrid sequences; mixed sequences; probabilistic methods

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