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Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

CiteScore 2017: 0.67

SCImago Journal Rank (SJR) 2017: 0.417
Source Normalized Impact per Paper (SNIP) 2017: 0.860

Mathematical Citation Quotient (MCQ) 2017: 0.25

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Volume 16, Issue 3-4


A genetic algorithm approach to estimate lower bounds of the star discrepancy

Manan Shah
Published Online: 2010-10-20 | DOI: https://doi.org/10.1515/mcma.2010.014


We consider a new method using genetic algorithms to obtain lower bounds for the star discrepancy for any number of points in [0, 1]s. We compute lower bounds for the star discrepancy of samples of a number of sequences in several dimensions and successfully compare with existing results from the literature. Despite statements in the quasi-Monte Carlo literature stating that computing the star discrepancy is either intractable or requires a lot of computational work for s ≥ 3, we show that it is possible to compute the star discrepancy exactly or at the very least obtain reasonable lower bounds without a huge computational burden. Our method is fast and consistent and can be easily extended to estimate lower bounds of other discrepancy measures. Our method can be used by researchers to measure the uniformity quality of point sets as given by the star discrepancy rather than having to rely on the L 2 discrepancy, which is easy to compute, but is flawed (and it is well known that the L 2 discrepancy is flawed).

Keywords.: Star discrepancy; genetic algorithm; Thiémard; Halton; Faure

About the article

Received: 2009-12-17

Revised: 2010-09-13

Published Online: 2010-10-20

Published in Print: 2010-12-01

Citation Information: Monte Carlo Methods and Applications, Volume 16, Issue 3-4, Pages 379–398, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma.2010.014.

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