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
DOI:
https://doi.org/10.1090/S0025-5718-02-01420-5
Published electronically:
March 20, 2002
MathSciNet review:
1933047
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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:
https://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