Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces

TL;DR

This paper introduces interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces, achieving optimal error rates with reduced construction costs by leveraging digit interlacing and component-by-component methods.

Contribution

It develops a new class of interlaced polynomial lattice rules for smooth functions, providing explicit constructions with optimal error bounds and improved efficiency over existing higher order rules.

Findings

Achieves optimal worst-case error rates in weighted Walsh spaces.

Provides explicit construction methods with reduced computational cost.

Demonstrates good dimension dependence under certain weight conditions.

Abstract

Quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions. A construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in [J. Dick, SIAM J. Numer. Anal., 45 (2007) pp.~2141--2176], which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension. In this paper, we study the use of polynomial lattice point sets for interlaced components. We call quadrature rules using such point sets {\em interlaced polynomial lattice rules}. We consider weighted Walsh spaces containing smooth functions and derive two upper bounds on the worst-case error for interlaced polynomial lattice rules, both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules. We investigate the component-by-component construction and the Korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst-case error. Through this approach we are able to obtain a good dependence of the worst-case error bounds on the dimension under certain conditions on the weights, while significantly reducing the construction cost as compared to higher order polynomial lattice rules.