标题： 具有预确定生成向量和随机点数的随机化格子规则算法，用于 Korobov 空间中的$0 < α\le 1/2$ 显示英文标题

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

摘要： 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 $0 < \alpha \le 1/2$. 在这里，我们同样得到$O(n^{-\alpha-1/2+\epsilon})$和$\epsilon > 0$的近最优收敛性；或者更详细地说，我们得到$O(\sqrt{r} \, n^{-\alpha-1/2+1/(2r)+\epsilon'})$，该式对于任何选择的$\epsilon' > 0$和$r \in \mathbb{N}$以及$r > 1/(2\alpha)$都成立。 显示英文摘要

摘要： 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 $O(\sqrt{r} \, n^{-\alpha-1/2+1/(2r)+\epsilon'})$ which holds for any choices of $\epsilon' > 0$ and $r \in \mathbb{N}$ with $r > 1/(2\alpha)$.