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DOI: 10.4230/DagSemProc.04401.3
URN: urn:nbn:de:0030-drops-1486
URL: https://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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04401.NuyensDirk.Paper.148.pdf (0.3 MB)


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: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/opus/volltexte/2005/148},
  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}
}

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


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