Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives
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 , 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 for small . Moreover, we investigate the situation for pointsets whose cardinality is not a Fibonacci number. It turns out that for 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