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

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

Abstract

We investigatequasi-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.

Keywords

Multivariate integration Quasi-Monte Carlo Optimal quadrature points Fibonacci lattice 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für AnalysisJohannes-Kepler-Universität LinzLinzAustria
  2. 2.Institute for Numerical SimulationBonnGermany

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