English

Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives

Numerical Analysis 2015-12-11 v2

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 O(N1log(N)12)\mathcal{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 Hmix1H^1_\text{mix} for small NN. Moreover, we investigate the situation for pointsets whose cardinality NN is not a Fibonacci number. It turns out that for N=1,2,3,5,7,8,12,13N=1,2,3,5,7,8,12,13 the optimal point sets are integration lattices.

Keywords

Cite

@article{arxiv.1409.5894,
  title  = {Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives},
  author = {Aicke Hinrichs and Jens Oettershagen},
  journal= {arXiv preprint arXiv:1409.5894},
  year   = {2015}
}

Comments

20 pages, version 2: several minor changes incorporating referee's suggestions