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Strong tractability of integration using scrambled Niederreiter points

Authors: Rong-Xian Yue and Fred J. Hickernell
Journal: Math. Comp. 74 (2005), 1871-1893
MSC (2000): Primary 65C05, 65D30
Published electronically: March 3, 2005
MathSciNet review: 2164101
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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 $\varepsilon$-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.

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Additional Information

Rong-Xian Yue
Affiliation: Division of Scientific Computation, E-Institute of Shanghai Universities, 100 Guilin Road, Shanghai 200234, People’s Republic of China

Fred J. Hickernell
Affiliation: Department of Applied Mathematics, Shanghai Normal University, Shanghai, People’s Republic of China
Address at time of publication: Department of Applied Mathematics, Illinois Institute of Technology, 10 West 32nd Street, E1 Building, Room 208, Chicago, Illinois 60616-3793
ORCID: 0000-0001-6677-1324

Keywords: Multivariate integration, quasi–Monte Carlo methods, nets and sequences, scrambling
Received by editor(s): November 24, 2003
Received by editor(s) in revised form: July 6, 2004
Published electronically: March 3, 2005
Additional Notes: 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
Article copyright: © Copyright 2005 American Mathematical Society