A reduced fast component-by-component construction of lattice points for integration in weighted spaces with fast decreasing weights

TL;DR

This paper introduces a modified fast component-by-component algorithm that reduces construction costs for lattice rules in weighted spaces by focusing on more important coordinates, improving efficiency in high-dimensional integration.

Contribution

It presents a novel modification to the fast component-by-component algorithm that decreases computational effort by reducing the search space based on coordinate importance.

Findings

Significant reduction in construction cost for lattice rules.

Effective in high-dimensional weighted integration spaces.

Numerical results confirm improved efficiency.

Abstract

Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the $s$-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions $s > 2$, one usually has to resort to computer search algorithms. The fast component-by-component approach is a useful algorithm for finding suitable quadrature rules. We present a modification of the fast component-by-component algorithm which yields savings of the construction cost for (polynomial) lattice rules in weighted function spaces. The idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large weights. We analyze tractability conditions of the resulting QMC rules. Numerical results demonstrate the effectiveness of our method.