Tractability results for the weighted star-discrepancy

TL;DR

This paper establishes conditions under which the weighted star-discrepancy, a measure of integration error in high-dimensional spaces, is strongly tractable, using probabilistic methods and empirical process theory.

Contribution

It provides new tractability results and sharp sufficient conditions for the weighted star-discrepancy, enhancing understanding of high-dimensional numerical integration.

Findings

Identifies conditions for strong tractability of weighted star-discrepancy

Uses probabilistic proofs and empirical process theory

Provides sharp bounds for error in weighted Sobolev spaces

Abstract

The weighted star-discrepancy has been introduced by Sloan and Wo{\'z}niakowski to reflect the fact that in multidimensional integration problems some coordinates of a function may be more important than others. It provides upper bounds for the error of multidimensional numerical integration algorithms for functions belonging to weighted function spaces of Sobolev type. In the present paper, we prove several tractability results for the weighted star-discrepancy. In particular, we obtain rather sharp sufficient conditions under which the weighted star-discrepancy is strongly tractable. The proofs are probabilistic, and use empirical process theory.