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Optimal quadrature for Haar wavelet spaces

Authors: Stefan Heinrich, Fred J. Hickernell and Rong-Xian Yue
Journal: Math. Comp. 73 (2004), 259-277
MSC (2000): Primary 65C05, 65D30
Published electronically: April 28, 2003
MathSciNet review: 2034121
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Abstract: This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, $\mathcal {H}_{\text {wav}}$. The asymptotic orders of the errors are derived for the case of the scrambled $(\lambda ,t,m,s)$-nets and $(t,s)$-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands $\mathcal {H}_{\text {wav}}$.

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Additional Information

Stefan Heinrich
Affiliation: FB Informatik, Universität Kaiserslautern, PF 3049, D-67653 Kaiserslautern, Germany

Fred J. Hickernell
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
ORCID: 0000-0001-6677-1324

Rong-Xian Yue
Affiliation: College of Mathematical Science, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China

Keywords: Quasi-Monte Carlo methods, Monte Carlo methods, high dimensional integration, lower bounds
Received by editor(s): July 9, 2001
Received by editor(s) in revised form: May 13, 2002
Published electronically: April 28, 2003
Additional Notes: This work was partially supported by a Hong Kong Research Grants Council grant HKBU/2030/99P, by Hong Kong Baptist University grant FRG/97-98/II-99, by Shanghai NSF Grant 00JC14057, and by Shanghai Higher Education STF grant 01D01-1.
Article copyright: © Copyright 2003 American Mathematical Society