Component-by-component digit-by-digit construction of good polynomial lattice rules in weighted Walsh spaces

TL;DR

This paper presents a digit-by-digit construction method for polynomial lattice rules in weighted Walsh spaces, enabling efficient high-dimensional integration with excellent convergence and overcoming the curse of dimensionality.

Contribution

It introduces a novel component-by-component construction approach for polynomial lattice rules that is fast, smoothness-independent, and effective in high-dimensional settings.

Findings

Achieves excellent convergence order in integration error

Can overcome the curse of dimensionality with coordinate weights

Provides extensive numerical validation

Abstract

We consider the efficient construction of polynomial lattice rules, which are special cases of so-called quasi-Monte Carlo (QMC) rules. These are of particular interest for the approximate computation of multivariate integrals where the dimension $d$ may be in the hundreds or thousands. We study a construction method that assembles the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a digit-by-digit (or, equivalently, coefficient-by-coefficient) fashion. As we will show, the integration error of the corresponding QMC rules achieves excellent convergence order, and, under suitable conditions, we can vanquish the curse of dimensionality by considering function spaces equipped with coordinate weights. The construction algorithm is based on a quality measure that is independent of the underlying smoothness of the function space and can be implemented in a fast manner (without the use of fast Fourier transformations). Furthermore, we illustrate our findings with extensive numerical results.