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The construction of good extensible rank-1 lattices


Authors: Josef Dick, Friedrich Pillichshammer and Benjamin J. Waterhouse
Journal: Math. Comp. 77 (2008), 2345-2373
MSC (2000): Primary 11K45, 65C05, 65D30
DOI: https://doi.org/10.1090/S0025-5718-08-02009-7
Published electronically: May 1, 2008
MathSciNet review: 2429889
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Abstract: It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield small integration errors for $ n = p, p^2, \ldots$ for all integers $ p \ge 2$. This paper provides algorithms for the construction of generating vectors which are finitely extensible for $ n = p, p^2, \ldots$ for all integers $ p \ge 2$. The proofs which show that our algorithms yield good extensible rank-1 lattices are based on a sieve principle. Particularly fast algorithms are obtained by using the fast component-by-component construction of Nuyens and Cools. Analogous results are presented for generating vectors with small weighted star discrepancy.


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Additional Information

Josef Dick
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
Email: josi@maths.unsw.edu.au

Friedrich Pillichshammer
Affiliation: Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria
Email: friedrich.pillichshammer@jku.at

Benjamin J. Waterhouse
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
Email: benjw@maths.unsw.edu.au

DOI: https://doi.org/10.1090/S0025-5718-08-02009-7
Keywords: Quasi-Monte Carlo, numerical integration, extensible lattice rule, component-by-component construction
Received by editor(s): June 13, 2006
Received by editor(s) in revised form: December 8, 2006
Published electronically: May 1, 2008
Additional Notes: This work was supported by the Austrian Research Foundation (FWF), Project S 9609, that is part of the Austrian Research Network “Analytic Combinatorics and Probabilistic Number Theory”.
The support of the Australian Research Council under its Centre of Excellence program is gratefully acknowledged.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.