Digital inversive vectors can achieve strong polynomial tractability for the weighted star discrepancy and for multivariate integration

TL;DR

This paper demonstrates that digital inverse vectors can achieve strong polynomial tractability in high-dimensional numerical integration, especially for weighted star discrepancy and certain Fourier series, by replacing random nodes with pseudorandom vectors.

Contribution

The paper introduces the use of digital inverse vectors as pseudorandom nodes to establish tractability in high-dimensional integration problems, improving upon traditional random sampling methods.

Findings

Digital inverse vectors achieve strong polynomial tractability.

Method effective for weighted star discrepancy.

Applicable to Fourier and cosine series integration.

Abstract

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not explode exponentially with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of H\"older continuous, absolute convergent Fourier- and cosine series, where the proposed method is successful.