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We prove lower bounds for the error of optimal cubature formulae for$d$-variate functions from Besov spaces of mixed smoothness$B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 < p, \theta \le \infty$and $\alpha > 1/p$, where ${\mathbb G}^d$ is either the $d$-dimensional torus${\mathbb T}^d$ or the $d$-dimensional unit cube ${\mathbb I}^d$. We proveupper bounds for QMC methods of integration on the Fibonacci lattice forbivariate periodic functions from $B^{\alpha}_{p,\theta}({\mathbb T}^2)$ in thecase $1\leq p \leq \infty$, $0 < \theta \leq \infty$, $\alpha>1/p$. Anon-periodic modification of this classical formula yields upper bounds for$B^{\alpha}_{p,\theta}({\mathbb I}^2)$ if $1/p<\alpha<1+1/p$. In combinationthese results yield the correct asymptotic error of optimal cubature formulaefor functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ and indicate that acorresponding result is most likely also true in case $d>2$. This is comparedto the correct asymptotic of optimal cubature formulae on Smolyak grids whichresults in the observation that any cubature formula on Smolyak grids is neveroptimal for the general setting.

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square