A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with $0<\alpha \le 1/2$

Abstract

In previous work (Kuo, Nuyens, Wilkes, 2023), 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 $O(\sqrt{r}{n}^{-\alpha -1/2+1/(2r)+{\epsilon }^{\prime }})$ which holds for any choices of ${\epsilon }^{\prime }>0$ and $r\in \mathbb{N}$ with $r>1/(2\alpha )$.