19th GAMM-Seminar Leipzig on
High-dimensional problems - Numerical treatment and applications

Max-Planck-Institute for Mathematics in the Sciences
Inselstr. 22-26, D-04103 [O->]Leipzig
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  19th GAMM-Seminar
January, 23th-25th, 2003
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  Abstract Emanouil I. Atanassov, Fri, 17.30-17.55 Previous Contents Next  
  Optimal Generation of Scrambled Low-Discrepancy Sequences
Emanouil I. Atanassov (Uni Saarbrücken)

Several families of sequences have upper bounds on their discrepancy of order O(N-1log1/2N), and are the most popular candidates for usage in quasi-Monte Carlo methods. They are called low-discrepancy sequences, and the Sobol and the Halton family are two of the most popular among them. Two problems exist before the widespread adoption of quasi-Monte Carlo methods. The first one is that the quality of distribution of the quasi-random sequence usually depends on the choice of a certain set of generation parameters, and finding the optimal choice is not easy. The second one is that the generation of these sequences requires thorough understanding of their theory, and thus the low-level computer hardware issues are usually neglected. The idea of scrambling the low-discrepancy sequences was introduced with the purpose to add some automatic statistical error estimation, and even to improve the equidistribution of the sequence, in the case of Owen type scrambling.

Our results in selecting the best parameters for the modified Halton sequences so that the upper limits on their discrepancy is reduced, will be shown. We will also show how our improved generation algorithms can substantially reduce the generation time, even when Owen-type scrambling is applied.

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