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Fast convergence of quasi-Monte Carlo for a class of isotropic integrals

Author: A. Papageorgiou
Journal: Math. Comp. 70 (2001), 297-306
MSC (2000): Primary 65D30, 65D32
Published electronically: February 23, 2000
MathSciNet review: 1709157
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Abstract: We consider the approximation of $d$-dimensional weighted integrals of certain isotropic functions. We are mainly interested in cases where $d$ is large. We show that the convergence rate of quasi-Monte Carlo for the approximation of these integrals is $O(\sqrt{\log n}/n)$. Since this is a worst case result, compared to the expected convergence rate $O(n^{-1/2})$ of Monte Carlo, it shows the superiority of quasi-Monte Carlo for this type of integral. This is much faster than the worst case convergence, $O(\log^d n/n)$, of quasi-Monte Carlo.

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Additional Information

A. Papageorgiou
Affiliation: Department of Computer Science, Columbia University, New York, NY 10027

Keywords: Multidimensional integration, quadrature, Monte Carlo methods, low discrepancy sequences, quasi-Monte Carlo methods
Received by editor(s): March 2, 1999
Published electronically: February 23, 2000
Additional Notes: This research has been supported in part by the NSF
Article copyright: © Copyright 2000 American Mathematical Society