On strong tractability of weighted multivariate integration期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On strong tractability of weighted multivariate integration

Authors:	Fred J Hickernell Ian H Sloan Grzegorz W Wasilkowski

Institution:	Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong ; School of Mathematics, University of New South Wales, Sydney 2052, Australia ; Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, Kentucky 40506-0046

Abstract:	We prove that for every dimension and every number 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 provided that the sequence $\boldsymbol\gamma=\{\gamma_k\}$ has $\sum_{k=1}^\infty\gamma_k^a<\infty$ for some (even arbitrarily large) . Here is a positive number that could be chosen arbitrarily close to zero and depends on but not on or . 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 $\gamma_k\equiv1$ .

Keywords:	Weighted integration quasi--Monte Carlo methods low discrepancy points tractability

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