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

TL;DR

This paper extends previous results by demonstrating that a pre-determined generating vector for lattice rules can achieve near-optimal convergence rates in Korobov spaces with smoothness $0 < \alpha extless= 1/2$, enhancing high-dimensional numerical integration.

Contribution

It proves the existence of a pre-determined generating vector for lattice rules in the case $0 < \alpha extless= 1/2$, achieving near-optimal convergence rates similar to the case $ extgreater 1/2$.

Findings

Achieves near-optimal convergence rate of $O(n^{-\alpha - 1/2 + \epsilon})$ for $0 < \alpha extless= 1/2$.

Provides explicit bounds involving the parameter $r$ and $\epsilon'$ for the convergence rate.

Extends previous work to cover the case of lower smoothness in Korobov spaces.

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