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

Authors Dirk Nuyens, Ronald Cools



PDF
Thumbnail PDF

File

DagSemProc.04401.3.pdf
  • Filesize: 332 kB
  • 26 pages

Document Identifiers

Author Details

Dirk Nuyens
Ronald Cools

Cite AsGet BibTex

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)
https://doi.org/10.4230/DagSemProc.04401.3

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)).
Keywords
  • numerical integration
  • cubature/quadrature
  • rank-1 lattice
  • component-by-component construction
  • fast algorithm

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail