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arXiv:1311.1563

Lower and upper bounds for the error of optimal cubature in Besov spaces of mixed smoothness

Ðinh Dũng, Tino Ullrich

Published Nov 7, 2013• 0 citations
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Abstract

We prove lower bounds for the error of optimal cubature formulas for $d$-variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 1/p$, and upper bounds for QMC methods on the Fibonacci lattice for two-variate periodic functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ in the case $1\leq p \leq \infty$, $0 1/p$, where ${\mathbb G}^d$ is either the $d$-dimensional torus ${\mathbb T}^d$ or the $d$-dimensional cube $[0,1]^d$. In combination these results yield the correct asymptotic of the error of optimal cubature formulas for functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ and indicate that a corresponding results is most likely also true in case $d>2$. This is compared to the obtained correct asymptotic of optimal cubature formulas on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.

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