Constructions of general polynomial lattice rules based on the weighted star discrepancy |
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Authors: | Josef Dick Peter Kritzer Gunther Leobacher Friedrich Pillichshammer |
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Institution: | aSchool of Mathematics, University of New South Wales, Sydney 2052, Australia;bFachbereich Mathematik, Universität Salzburg, Hellbrunnerstraße 34, A-5020 Salzburg, Austria;cInstitut für Finanzmathematik, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria |
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Abstract: | In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration.Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy.In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy. |
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Keywords: | Weighted star discrepancy Digital nets Polynomial lattice rule |
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