ON QUASI-MONTE CARLO METHODS IN WEIGHTED ANOVA SPACES

P. Kritzer, F. Pillichshammer, G. W. Wasilkowski

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

In the present paper we study quasi-Monte Carlo rules for approximating integrals over the d-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasiMonte Carlo rules. Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number n of used integration nodes. For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.

Original languageEnglish
Pages (from-to)1381-1406
Number of pages26
JournalMathematics of Computation
Volume90
Issue number329
DOIs
StatePublished - 2021

Bibliographical note

Publisher Copyright:
© 2021 American Mathematical Society. All Rights Reserved.

Keywords

  • ANOVA space
  • Quasi-Monte Carlo integration
  • weighted discrepancy
  • worst case error

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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