On the Randomization of Frolov's Algorithm for Multivariate Integration
  We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ 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 worst case error has the order $n^{-r} \, (\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant 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 $n^{-r-1/2}$ is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich 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 thesis is a review of the mentioned upper bounds and their proofs. 
David Krieg
math.NA