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We analyze a new random algorithm for numerical integration of $d$-variatefunctions over $[0,1]^d$ from a weighted Sobolev space with dominating mixedsmoothness $\alpha\ge 0$ and product weights$1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where the functions are continuous andperiodic when $\alpha>1/2$. The algorithm is based on rank-$1$ lattice ruleswith a random number of points~$n$. For the case $\alpha>1/2$, we prove thatthe algorithm achieves almost the optimal order of convergence of$\mathcal{O}(n^{-\alpha-1/2})$, where the implied constant is independent ofthe dimension~$d$ if the weights satisfy $\sum_{j=1}^\infty\gamma_j^{1/\alpha}<\infty$. The same rate of convergence holds for the moregeneral case $\alpha>0$ by adding a random shift to the lattice rule withrandom $n$. This shows, in particular, that the exponent of strong tractabilityin the randomized setting equals $1/(\alpha+1/2)$, if the weights decay fastenough. We obtain a lower bound to indicate that our results are essentiallyoptimal. This paper is a significant advancement over previous related workswith respect to the potential for implementation and the independence of errorbounds on the problem dimension. Other known algorithms which achieve theoptimal error bounds, such as those based on Frolov's method, are verydifficult to implement especially in high dimensions. Here we adapt alesser-known randomization technique introduced by Bakhvalov in 1961. Thisalgorithm is based on rank-$1$ lattice rules which are very easy to implementgiven the integer generating vectors. A simple probabilistic approach can beused to obtain suitable generating vectors.

Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-\alpha-1/2})$ error independently of the dimension