Quasi-Monte Carlo Methods for Numerical Integration: Comparison of Different Low Discrepancy Sequences

Radović, Igor; Sobol, I. M.; Tichy, Robert F.

doi:10.1515/mcma.1996.2.1.1

Cited by 61 publications

(22 citation statements)

References 7 publications

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“…stream 0 -leap 5 and leap 33). These effects (and differences to earlier findings) may be due to the fact that we use a test function with singularity located in the origin and that leaping in general reduces the share of points taken from the start of the original sequence (which is known to be of lower quality [39,40]). It is also interesting to note that there are significant differences among the results of the different streams.…”

Section: Single Substream Resultscontrasting

confidence: 69%

Quasi Monte Carlo Integration in Grid Environments: Further Leaping Effects

Hofbauer

Uhl

Zinterhof

2006

Parallel Process. Lett.

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The splitting of Quasi-Monte Carlo (QMC) point sequences into interleaved substreams has been suggested to raise the speed of distributed numerical integration and to lower the traffic on the network. The usefulness of this approach in GRID environments is discussed. After specifying requirements for using QMC techniques in GRID environments in general we review and evaluate the proposals made in literature so far. In numerical integration experiments we investigate the quality of single leaped QMC point sequence substreams, comparing the respective properties of Sobol', Halton, Faure, Niederreiter-Xing, and Zinterhof sequences in detail. Numerical integration results obtained on a distributed system show that leaping sensitivity varies tremendously among the different sequences and we provide examples of deteriorated results caused by leaping effects, especially in heterogeneous settings which would be expected in GRID environments.

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Section: Single Substream Resultscontrasting

confidence: 69%

Quasi Monte Carlo Integration in Grid Environments: Further Leaping Effects

Hofbauer

Uhl

Zinterhof

2006

Parallel Process. Lett.

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“…For g 3 , the constant α s is determined numerically so that g 3 also integrates to 1. The first function was proposed in Radovic et al [1996] and is also used, for example, in Wang and Hickernell [2000]. Here we consider the same choices that have been studied in these two papers, that is (i) α j = 0.01, (ii) α j = 1; (iii) α j = j , and (iv) α j = j 2 , for 1 ≤ j ≤ s.…”

Section: Test-functionsmentioning

confidence: 99%

Generalized Halton sequences in 2008

Faure

Lemieux

2009

ACM Trans. Model. Comput. Simul.

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Halton sequences have always been quite popular with practitioners, in part because of their intuitive definition and ease of implementation. However, in their original form, these sequences have also been known for their inadequacy to integrate functions in moderate to large dimensions, in which case (t, s)-sequences such as the Sobol' sequence are usually preferred. To overcome this problem, one possible approach is to include permutations in the definition of Halton sequencesthereby obtaining generalized Halton sequences-an idea that goes back to almost thirty years ago, and that has been studied by many researchers in the last few years. In parallel to these efforts, an important improvement in the upper bounds for the discrepancy of Halton sequences has been made by Atanassov in 2004. Together, these two lines of research have revived the interest in Halton sequences. In this article, we review different generalized Halton sequences that have been proposed recently, and compare them by means of numerical experiments. We also propose a new generalized Halton sequence which, we believe, offers a practical advantage over the surveyed constructions, and that should be of interest to practitioners.

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“…This function was …rst considered by Radović, Sobol', Tichy [14], and used subsequently by several authors. The exact value of the integral is one, and the sensitivity of the function to x i quickly decreases as a i increases.…”

Section: Methodsmentioning

confidence: 99%

Random and Deterministic Digit Permutations of the Halton Sequence

Ökten

Shah

Goncharov

2012

Springer Proceedings in Mathematics &Amp; Statistics

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The Halton sequence is one of the classical low-discrepancy sequences. It is e¤ec-tively used in numerical integration when the dimension is small, however, for larger dimensions, the uniformity of the sequence quickly degrades. As a remedy, generalized (scrambled) Halton sequences have been introduced by several researchers since the 1970s. In a generalized Halton sequence, the digits of the original Halton sequence are permuted using a carefully selected permutation. Some of the permutations in the literature are designed to minimize some measure of discrepancy, and some are obtained heuristically.In this paper, we investigate how these carefully selected permutations di¤er from a permutation simply generated at random. We use a recent genetic algorithm, test problems from numerical integration and computational …nance, and a recent randomized quasi-Monte Carlo method, to compare generalized Halton sequences with randomly chosen permutations, with the traditional generalized Halton sequences. Numerical results suggest that the random permutation approach is as good as, or better than, the "best" deterministic permutations.

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Quasi-Monte Carlo Methods for Numerical Integration: Comparison of Different Low Discrepancy Sequences

Abstract: Further numerical experience on quasi-Monte Carlo integration is established. Different kinds of low discrepancy point sequences including Halton, Sobol', Faure, and Niederreiter sequences are considered and a new family of test functions is used.

Cited by 61 publications

References 7 publications

Quasi Monte Carlo Integration in Grid Environments: Further Leaping Effects

Quasi Monte Carlo Integration in Grid Environments: Further Leaping Effects

Generalized Halton sequences in 2008

Random and Deterministic Digit Permutations of the Halton Sequence

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