High-Discrepancy Sequences for High-Dimensional Numerical Integration

Shu Tezuka

doi:10.1007/978-3-642-27440-4_40

High-Discrepancy Sequences for High-Dimensional Numerical Integration

Shu Tezuka

Division of Applied Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

In this paper, we consider a sequence of points in [0, 1]^d, which are distributed only on the diagonal line between (0,...,0) and (1,...,1). The sequence is constructed based on a one-dimensional low-discrepancy sequence. We apply such sequences to d-dimensional numerical integration for two classes of integrals. The first class includes isotropic integrals. Under a certain condition, we prove that the integration error for this class is O(√logN/N), where N is the number of points. The second class is called as Kolmogorov superposition integrals for which, under a certain condition, we prove that the integration error for this class is O((logN)/N).

Original language	English
Title of host publication	Monte Carlo and Quasi-Monte Carlo Methods 2010
Publisher	Springer New York LLC
Pages	685-694
Number of pages	10
ISBN (Print)	9783642274398
DOIs	https://doi.org/10.1007/978-3-642-27440-4_40
Publication status	Published - Jan 1 2012
Event	9th International Conference on Monte Carlo and Quasi Monte Carlo Methods in Scientific Computing, MCQMC 2010 - Warsaw, Poland Duration: Aug 15 2010 → Aug 20 2010

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	23
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	9th International Conference on Monte Carlo and Quasi Monte Carlo Methods in Scientific Computing, MCQMC 2010
Country/Territory	Poland
City	Warsaw
Period	8/15/10 → 8/20/10

All Science Journal Classification (ASJC) codes

Mathematics(all)

Access to Document

10.1007/978-3-642-27440-4_40

Cite this

Tezuka, S 2012, High-Discrepancy Sequences for High-Dimensional Numerical Integration. in Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics and Statistics, vol. 23, Springer New York LLC, pp. 685-694, 9th International Conference on Monte Carlo and Quasi Monte Carlo Methods in Scientific Computing, MCQMC 2010, Warsaw, Poland, 8/15/10. https://doi.org/10.1007/978-3-642-27440-4_40

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abstract = "In this paper, we consider a sequence of points in [0, 1]d, which are distributed only on the diagonal line between (0,...,0) and (1,...,1). The sequence is constructed based on a one-dimensional low-discrepancy sequence. We apply such sequences to d-dimensional numerical integration for two classes of integrals. The first class includes isotropic integrals. Under a certain condition, we prove that the integration error for this class is O(√logN/N), where N is the number of points. The second class is called as Kolmogorov superposition integrals for which, under a certain condition, we prove that the integration error for this class is O((logN)/N).",

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doi = "10.1007/978-3-642-27440-4_40",

language = "English",

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AB - In this paper, we consider a sequence of points in [0, 1]d, which are distributed only on the diagonal line between (0,...,0) and (1,...,1). The sequence is constructed based on a one-dimensional low-discrepancy sequence. We apply such sequences to d-dimensional numerical integration for two classes of integrals. The first class includes isotropic integrals. Under a certain condition, we prove that the integration error for this class is O(√logN/N), where N is the number of points. The second class is called as Kolmogorov superposition integrals for which, under a certain condition, we prove that the integration error for this class is O((logN)/N).

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High-Discrepancy Sequences for High-Dimensional Numerical Integration

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