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The construction of good extensible rank-1 lattices

Authors:	Josef Dick Friedrich Pillichshammer Benjamin J Waterhouse

Institution:	School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia ; Institut für Finanzmathematik, Universität Linz, Altenbergstrasse 69, A-4040 Linz, Austria ; School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

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.

Keywords:	Quasi-Monte Carlo numerical integration extensible lattice rule component-by-component construction

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