pith. sign in

arxiv: 1409.5894 · v2 · pith:I45RVFQDnew · submitted 2014-09-20 · 🧮 math.NA · cs.NA

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

classification 🧮 math.NA cs.NA
keywords integrationoptimalperiodicpointbivariatecarlocaseerror
0
0 comments X
read the original 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 $\mathcal{O}(N^{-1}\log(N)^{\frac{1}{2}})$, 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 $H^1_\text{mix}$ for small $N$. Moreover, we investigate the situation for pointsets 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.

This paper has not been read by Pith yet.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.