Conditions for tractability of the weighted $L_p$-discrepancy and integration in non-homogeneous tensor product spaces

TL;DR

This paper investigates the conditions under which the weighted $L_p$-discrepancy and integration in non-homogeneous tensor product spaces are tractable, providing comprehensive bounds for all $p \, \in\, (1, \infty)$ and linking discrepancy to integration complexity.

Contribution

It establishes matching sufficient and necessary conditions for polynomial and weak tractability of weighted $L_p$-discrepancy for all $p$ in the range, extending previous results beyond even integers.

Findings

Derived upper bounds for tractability conditions.

Proved lower bounds using general results on integration complexity.

Applied results to tensor products of degree 2 polynomials.

Abstract

We study tractability properties of the weighted $L_p$-discrepancy. The concept of {\it weighted} discrepancy was introduced by Sloan and Wo\'{z}\-nia\-kowski in 1998 in order to prove a weighted version of the Koksma-Hlawka inequality for the error of quasi-Monte Carlo integration rules. The weights have the aim to model the influence of different coordinates of integrands on the error. A discrepancy is said to be tractable if the information complexity, i.e., the minimal number $N$ of points such that the discrepancy is less than the initial discrepancy times an error threshold $\varepsilon$, does not grow exponentially fast with the dimension. In this case there are various notions of tractabilities used in order to classify the exact rate. For even integer parameters $p$ there are sufficient conditions on the weights available in literature, which guarantee the one or other notion of tractability. In the present paper we prove matching sufficient conditions (upper bounds) and neccessary conditions (lower bounds) for polynomial and weak tractability for all $p \in (1, \infty)$. The proofs of the lower bounds are based on a general result for the information complexity of integration with positive quadrature formulas for tensor product spaces. In order to demonstrate this lower bound we consider as a second application the integration of tensor products of polynomials of degree at most 2.