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Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)

Authors: Dirk Nuyens and Ronald Cools

Published in: Dagstuhl Seminar Proceedings, Volume 4401, Algorithms and Complexity for Continuous Problems (2005)

Abstract

Part I: (this part of the talk by Ronald Cools) We restate our previous result which showed that it is possible to construct the generating vector of a rank-1 lattice rule in a fast way, i.e. O(s n log(n)), with s the number of dimensions and n the number of points assumed to be prime. Here we explicitly use basic facts from algebra to exploit the structure of a matrix – which introduces the crucial cost in the construction – to get a matrix-vector multiplication in time O(n log(n)) instead of O(n^2). We again stress the fact that the algorithm works for any tensor product reproducing kernel Hilbert space. Part II: (this part of the talk by Dirk Nuyens) In the second part we generalize the tricks used for primes to non-primes, by basically falling back to algebraic group theory. In this way it can be shown that also for a non-prime number of points, this crucial matrix-vector multiplication can be done in time O(n log(n)). We conclude that the construction of rank-1 lattice rules in an arbitrary r.k.h.s. for an arbitrary amount of points can be done in a fast way of O(s n log(n)).

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Dirk Nuyens and Ronald Cools. Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II). In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{nuyens_et_al:DagSemProc.04401.3,
  author =	{Nuyens, Dirk and Cools, Ronald},
  title =	{{Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--26},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4401},
  editor =	{Thomas M\"{u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.3},
  URN =		{urn:nbn:de:0030-drops-1486},
  doi =		{10.4230/DagSemProc.04401.3},
  annote =	{Keywords: numerical integration , cubature/quadrature , rank-1 lattice , component-by-component construction , fast algorithm}
}

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Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)

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