Strong tractability of integration using scrambled niederreiter points

Rong Xian Yue; Fred J. Hickernell

doi:10.1090/S0025-5718-05-01755-2

Strong tractability of integration using scrambled niederreiter points

Rong Xian Yue^*, Fred J. Hickernell

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › peer-review

9 Citations (Scopus)

Abstract

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The ε-exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

Original language	English
Pages (from-to)	1871-1893
Number of pages	23
Journal	Mathematics of Computation
Volume	74
Issue number	252
Early online date	3 Mar 2005
DOIs	https://doi.org/10.1090/S0025-5718-05-01755-2
Publication status	Published - Oct 2005
Externally published	Yes

Scopus Subject Areas

Algebra and Number Theory
Computational Mathematics
Applied Mathematics

User-Defined Keywords

Multivariate integration
Nets and sequences
Quasi-Monte Carlo methods
Scrambling

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10.1090/S0025-5718-05-01755-2

Cite this

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title = "Strong tractability of integration using scrambled niederreiter points",

abstract = "We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The ε-exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.",

keywords = "Multivariate integration, Nets and sequences, Quasi-Monte Carlo methods, Scrambling",

author = "Yue, {Rong Xian} and Hickernell, {Fred J.}",

note = "This work was partially supported by Hong Kong Research Grants Council grant HKBU/2020/02P, National Science Foundation of China grant 10271078, E-Institute of Shanghai Municipal Education Commission (E03004), and the Special Funds for Major Specialties of the Shanghai Education Committee.",

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T1 - Strong tractability of integration using scrambled niederreiter points

AU - Yue, Rong Xian

AU - Hickernell, Fred J.

N1 - This work was partially supported by Hong Kong Research Grants Council grant HKBU/2020/02P, National Science Foundation of China grant 10271078, E-Institute of Shanghai Municipal Education Commission (E03004), and the Special Funds for Major Specialties of the Shanghai Education Committee.

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N2 - We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The ε-exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

AB - We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The ε-exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

KW - Multivariate integration

KW - Nets and sequences

KW - Quasi-Monte Carlo methods

KW - Scrambling

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JO - Mathematics of Computation

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ER -

Strong tractability of integration using scrambled niederreiter points

Abstract

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