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URN: urn:nbn:de:0030-drops-1486
URL: http://drops.dagstuhl.de/opus/volltexte/2005/148/
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Nuyens, Dirk ; Cools, Ronald

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

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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)).

BibTeX - Entry

@InProceedings{nuyens_et_al:DSP:2005:148,
  author =	{Dirk Nuyens and Ronald Cools},
  title =	{Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  year =	{2005},
  editor =	{Thomas M{\"u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  number =	{04401},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2005/148},
  annote =	{Keywords: numerical integration , cubature/quadrature , rank-1 lattice , component-by-component construction , fast algorithm}
}

Keywords: numerical integration , cubature/quadrature , rank-1 lattice , component-by-component construction , fast algorithm
Seminar: 04401 - Algorithms and Complexity for Continuous Problems
Issue Date: 2005
Date of publication: 19.04.2005

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