Tractability properties of the weighted star discrepancy of regular grids

TL;DR

This paper characterizes when regular grids with varying mesh sizes achieve different levels of tractability for the weighted star discrepancy, based on the decay of the weight sequence.

Contribution

It provides exact conditions on weight sequences for various tractability properties of regular grids in discrepancy theory.

Findings

Weak tractability occurs if and only if lim_{j→∞} j γ_j=0.

Conditions for polynomial and strong polynomial tractability are explicitly characterized.

The results offer precise criteria linking weight decay to discrepancy tractability.

Abstract

In this paper we study tractability properties of the weighted star discrepancy with general coefficients of centered regular grids with different mesh-sizes. We give exact characterizations of the weight sequences $(\gamma_j)_{j \ge 1}$ such that the regular grid with different mesh-sizes achieves weak, uniform weak, quasi polynomial, polynomial or strong polynomial tractability for the $\boldsymbol{\gamma}$-weighted star discrepancy. For example, a necessary and sufficient condition such that the regular grid with different mesh-sizes achieves weak tractability for the $\boldsymbol{\gamma}$-weighted star discrepancy is $\lim_{j \rightarrow \infty}j \gamma_j=0$.