Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

TL;DR

This paper introduces a component-by-component construction algorithm for randomized rank-1 lattice rules that achieves near-optimal error rates in high-dimensional integration, without needing to verify error bounds during construction.

Contribution

It presents a new algorithm for constructing randomized lattice rules that guarantees near-optimal error bounds without prior checking, extending to various function spaces.

Findings

Achieves almost optimal randomized error rates in high-dimensional spaces.

Error bounds are independent of dimension under certain summability conditions.

Extends results to tent-transformed lattice rules and polynomial lattice rules.

Abstract

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters, $\gamma_j$, satisfy the summability condition $\sum_{j=1}^{\infty}\gamma_j^{1/\alpha}<\infty$, where $\alpha$ is a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces. In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition $\sum_{j=1}^{\infty}\gamma_j^{1/\alpha}<\infty$ holds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.