# 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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