A Randomised Lattice Rule Algorithm with Pre-determined Generating Vector and Random Number of Points for Korobov Spaces with  \(0 < \alpha \le 1/2\)

In previous work [12], we showed that a lattice rule with a pre-determined generating vector but random number of points can achieve the near optimal convergence of \(O(n^{-\alpha -1/2+\epsilon })\) , \(\epsilon > 0\) , for the worst case expected error, commonly referred to as the randomised error, for numerical integration of high-dimensional functions in the Korobov space with smoothness \(\alpha > 1/2\) . Compared to the optimal deterministic rate of \(O(n^{-\alpha +\epsilon })\) , \(\epsilon > 0\) , such a randomised algorithm is capable of an extra half in the rate of convergence. In this paper, we show that a pre-determined generating vector also exists in the case of \(0 < \alpha \le 1/2\) . Also here we obtain the near optimal convergence of \(O(n^{-\alpha -1/2+\epsilon })\) , \(\epsilon > 0\) ; or in more detail, we obtain \(O(\sqrt{r} \, n^{-\alpha -1/2+1/(2r)+\epsilon '})\) which holds for any choices of \(\epsilon ' > 0\) and \(r \in {\mathbb {N}}\) with \(r > 1/(2\alpha )\) .