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Mathematics of Computation

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On strong tractability of weighted multivariate integration

Authors: Fred J. Hickernell, Ian H. Sloan and Grzegorz W. Wasilkowski
Journal: Math. Comp. 73 (2004), 1903-1911
MSC (2000): Primary 65D30, 65D32, 65Y20, 11K38
Published electronically: April 22, 2004
MathSciNet review: 2059742
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Abstract: We prove that for every dimension $s$ and every number $n$ of points, there exists a point-set $\mathcal{P}_{n,s}$ whose $\boldsymbol \gamma$-weighted unanchored $L_{\infty}$ discrepancy is bounded from above by $C(b)/n^{1/2-b}$ independently of $s$ provided that the sequence $\boldsymbol\gamma=\{\gamma_k\}$ has $\sum_{k=1}^\infty\gamma_k^a<\infty$ for some (even arbitrarily large) $a$. Here $b$ is a positive number that could be chosen arbitrarily close to zero and $C(b)$ depends on $b$ but not on $s$ or $n$. This result yields strong tractability of the corresponding integration problems including approximation of weighted integrals $\int_Df(\mathbf{x})\,\rho(\mathbf{x})\,d\mathbf{x}$over unbounded domains such as $D=\mathbb{R}^s$. It also supplements the results that provide an upper bound of the form $C\sqrt{s/n}$ when $\gamma_k\equiv1$.

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Additional Information

Fred J. Hickernell
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Ian H. Sloan
Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia

Grzegorz W. Wasilkowski
Affiliation: Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, Kentucky 40506-0046

Keywords: Weighted integration, quasi--Monte Carlo methods, low discrepancy points, tractability
Received by editor(s): December 16, 2002
Received by editor(s) in revised form: April 30, 2003
Published electronically: April 22, 2004
Article copyright: © Copyright 2004 American Mathematical Society