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On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

Authors: I. H. Sloan, F. Y. Kuo and S. Joe
Journal: Math. Comp. 71 (2002), 1609-1640
MSC (2000): Primary 65D30, 65D32; Secondary 68Q25
Published electronically: March 20, 2002
MathSciNet review: 1933047
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Abstract: We develop and justify an algorithm for the construction of quasi-Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found ``component-by-component'': the ($d+1$)-th component of the generator vector and the shift are obtained by successive $1$-dimensional searches, with the previous $d$ components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for $n$-point rules with $n$ prime and all dimensions 1 to $d$ requires a total cost of $O(n^3d^2)$ operations. This may be reduced to $O(n^3d)$ operations at the expense of $O(n^2)$ storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and $n$ up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.

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Additional Information

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

F. Y. Kuo
Affiliation: Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

S. Joe
Affiliation: Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

Received by editor(s): October 30, 2000
Published electronically: March 20, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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