Numerische Mathematik

, Volume 134, Issue 1, pp 163–196 | Cite as

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

  • Aicke Hinrichs
  • Lev Markhasin
  • Jens Oettershagen
  • Tino Ullrich
Article

Abstract

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces \(S^r_{p,q}B(\mathbb {T}^d)\) with dominating mixed smoothness \(1/p<r<2\). We show that order 2 digital nets achieve the optimal rate of convergence \(N^{-r} (\log N)^{(d-1)(1-1/q)}\). The logarithmic term does not depend on r and hence improves the known bound of Dick (SIAM J Numer Anal 45:2141–2176, 2007) for the special case of Sobolev spaces \(H^r_{\text {mix}}(\mathbb {T}^d)\). Secondly, the rate of convergence is independent of the integrability p of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of order 2 digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case \(r=2\).

Mathematics Subject Classification

11K06 11J71 42C10 46E35 65C05 65D30 65D32 91G60 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Aicke Hinrichs
    • 1
  • Lev Markhasin
    • 2
  • Jens Oettershagen
    • 3
  • Tino Ullrich
    • 3
  1. 1.Institute for AnalysisJohannes Kepler UniversityLinzAustria
  2. 2.Institute for Stochastics and ApplicationsUniversity of StuttgartStuttgartGermany
  3. 3.Institute for Numerical SimulationUniversity of BonnBonnGermany

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