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

TL;DR

This work extends randomised lattice rules for Korobov spaces to the low-smoothness regime $0<\alpha\le 1/2$, where points are shifted to accommodate noncontinuous functions. It introduces a pre-determined generating vector alongside a random number of points (and random shifts for $\alpha\le1/2$) and proves the existence of a vector $\boldsymbol{z}$ that achieves near-optimal randomised error, with explicit bounds shrinking as $n$ grows. The key innovation is the construction of good sets $G_{n,\lambda}$ via primes in $\mathcal{P}_n$ and the Chinese Remainder Theorem, enabling rigorous control over the RMS error and yielding $e^{\text{rms}}=O(n^{-\alpha-1/2+\epsilon})$ for any $\epsilon>0$ by suitable parameter choices. A dimension-independent bound is obtained under a tractability-type condition on the weights, and lower bounds are provided to validate the near-optimality of the method. Overall, the paper generalises prior results for $\alpha>1/2$ to the challenging $0<\alpha\le 1/2$ regime, offering a practical path to improved randomised integration in high dimensions.

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 $O(n^{-α-1/2+ε})$, $ε> 0$, 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 $α> 1/2$. Compared to the optimal deterministic rate of $O(n^{-α+ε})$, $ε> 0$, 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 $0 < α\le 1/2$. Also here we obtain the near optimal convergence of $O(n^{-α-1/2+ε})$, $ε> 0$; or in more detail, we obtain $O(\sqrt{r} \, n^{-α-1/2+1/(2r)+ε'})$ which holds for any choices of $ε' > 0$ and $r \in \mathbb{N}$ with $r > 1/(2α)$.

Theorems & Definitions (10)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Theorem 2
• proof