Construction of good polynomial lattice rules in weighted Walsh spaces by an alternative component-by-component construction

TL;DR

This paper presents an efficient, dimension-independent component-by-component construction method for polynomial lattice rules in weighted Walsh spaces, achieving near-optimal error convergence for multivariate integration.

Contribution

It introduces an alternative, easy-to-implement component-by-component algorithm that produces high-quality polynomial lattice rules with error bounds independent of dimension.

Findings

Achieves almost optimal error convergence order.

Error bounds can be made independent of dimension under certain weights.

Numerical experiments confirm theoretical results.

Abstract

We study the efficient construction of good polynomial lattice rules, which are special instances of quasi-Monte Carlo (QMC) methods. The integration rules obtained are of particular interest for the approximation of multivariate integrals in weighted Walsh spaces. In particular, we study a construction algorithm which assembles the components of the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a component-wise fashion. We show that the constructed QMC rules achieve the almost optimal error convergence order in the function spaces under consideration and prove that the obtained error bounds can, under certain conditions on the involved weights, be made independent of the dimension. We also demonstrate that our alternative component-by-component construction, which is independent of the underlying smoothness of the function space, can be implemented relatively easily in a fast manner. Numerical experiments confirm our theoretical findings.