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We are concerned with the numerical integration of functions from the Sobolevspace $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness$r\in\mathbb{N}$ over the $d$-dimensional unit cube.  In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worstcase error has the order $n^{-r} \, (\log n)^{(d-1)/2}$ with respect to thenumber $n$ of function evaluations. This is known to be optimal. 39 yearslater, Erich Novak and me introduced a randomized version of this algorithmusing $d$ random dilations. We showed that its error is bounded above by aconstant multiple of $n^{-r-1/2} \, (\log n)^{(d-1)/2}$ in expectation and by$n^{-r} \, (\log n)^{(d-1)/2}$ almost surely. The main term $n^{-r-1/2}$ isagain optimal and it turns out that the very same algorithm is also optimal forthe isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also addeda random shift to this algorithm to make it unbiased. Just recently, MarioUllrich proved that the expected error of the resulting algorithm on$H^{r,\text{mix}}([0,1]^d)$ is even bounded above by $n^{-r-1/2}$. This thesisis a review of the mentioned upper bounds and their proofs.

On the Randomization of Frolov's Algorithm for Multivariate Integration