Fast construction of higher order digital nets for numerical integration in weighted Sobolev spaces

TL;DR

This paper introduces a fast algorithm for constructing higher order digital nets using polynomial lattice point sets, improving efficiency and performance for numerical integration in weighted Sobolev spaces.

Contribution

It presents a new fast search algorithm for classical digital nets suitable for interlacing, leveraging polynomial lattice point sets and component-by-component construction.

Findings

The algorithm achieves $O(dsN \log N)$ construction complexity.

Constructed point sets often outperform Sobol' and Niederreiter-Xing sequences.

The method provides tractability results under certain weight conditions.

Abstract

Higher order digital nets are special classes of point sets for quasi-Monte Carlo rules which achieve the optimal convergence rate for numerical integration of smooth functions. An explicit construction of higher order digital nets was proposed by Dick, which is based on digitally interlacing in a certain way the components of classical digital nets whose number of components is a multiple $ds$ of the dimension $s$. In this paper we give a fast computer search algorithm to find good classical digital nets suitable for interlaced components by using polynomial lattice point sets. We consider certain weighted Sobolev spaces of smoothness of arbitrarily high order, and derive an upper bound on the mean square worst-case error for digitally shifted higher order digital nets. Employing this upper bound as a quality criterion, we prove that the component-by-component construction can be used efficiently to find good polynomial lattice point sets suitable for interlaced components. Through this approach we are able to get some tractability results under certain conditions on the weights. Fast construction using the fast Fourier transform requires the construction cost of $O(dsN \log N)$ operations using $O(N)$ memory, where $N$ is the number of points and $s$ is the dimension. This implies a significant reduction in the construction cost as compared to higher order polynomial lattice point sets. Numerical experiments confirm that the performance of our constructed point sets often outperforms those of higher order digital nets with Sobol' sequences and Niederreiter-Xing sequences used for interlaced components, indicating the usefulness of our algorithm.