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Optimal Point Sets for Quasi-Monte Carlo Integration of Bivariate Periodic Functions with Bounded Mixed Derivatives

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Monte Carlo and Quasi-Monte Carlo Methods

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 163))

Abstract

We investigate quasi-Monte Carlo (QMC) integration of bivariate periodic functions with dominating mixed smoothness of order one. While there exist several QMC constructions which asymptotically yield the optimal rate of convergence of \(\mathscr {O}(N^{-1}\log (N)^{\frac{1}{2}})\), it is yet unknown which point set is optimal in the sense that it is a global minimizer of the worst case integration error. We will present a computer-assisted proof by exhaustion that the Fibonacci lattice is the unique minimizer of the QMC worst case error in periodic \(H^1_\text {mix}\) for small Fibonacci numbers N. Moreover, we investigate the situation for point sets whose cardinality N is not a Fibonacci number. It turns out that for \(N=1,2,3,5,7,8,12,13\) the optimal point sets are integration lattices.

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Acknowledgments

The authors thank Christian Kuske and André Uschmajew for valuable hints and discussions. Jens Oettershagen was supported by the Sonderforschungsbereich 1060 The Mathematics of Emergent Effects of the DFG.

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Authors and Affiliations

  1. Institut für Analysis, Johannes-Kepler-Universität Linz, Altenberger Straße 69, 4040, Linz, Austria

    Aicke Hinrichs

  2. Institute for Numerical Simulation, Wegelerstraße 6, 53115, Bonn, Germany

    Jens Oettershagen

Authors
  1. Aicke Hinrichs
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  2. Jens Oettershagen
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Corresponding author

Correspondence to Jens Oettershagen .

Editor information

Editors and Affiliations

  1. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Ronald Cools

  2. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Dirk Nuyens

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Hinrichs, A., Oettershagen, J. (2016). Optimal Point Sets for Quasi-Monte Carlo Integration of Bivariate Periodic Functions with Bounded Mixed Derivatives. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_19

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