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 ()-th component of the generator vector and the shift are obtained by successive -dimensional searches, with the previous 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 -point rules with prime and all dimensions 1 to requires a total cost of operations. This may be reduced to operations at the expense of storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and 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

Email:
sloan@maths.unsw.edu.au

**F. Y. Kuo**

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

Email:
f.kuo@math.waikato.ac.nz

**S. Joe**

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

Email:
stephenj@math.waikato.ac.nz

DOI:
http://dx.doi.org/10.1090/S0025-5718-02-01420-5

Received by editor(s):
October 30, 2000

Published electronically:
March 20, 2002

Article copyright:
© Copyright 2002
American Mathematical Society