A note about exponential tractability of linear weighted tensor product problems in the worst-case setting

TL;DR

This paper investigates the conditions under which weighted linear tensor product problems are exponentially tractable in the worst-case setting, providing necessary and sufficient criteria for various exponential weak tractability notions.

Contribution

It fills the gaps in understanding exponential tractability by establishing necessary and sufficient conditions for EXP-$(s,t)$-WT and EXP-UWT in weighted tensor product problems.

Findings

Derived necessary and sufficient conditions for EXP-$(s,t)$-WT with $ ext{max}(s,t)<1$

Established criteria for exponential uniform weak tractability (EXP-UWT)

Extended the theoretical understanding of tractability in weighted tensor product problems

Abstract

This paper is devoted to discussing the weighted linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential $(s, t)$-weak tractability (EXP-$(s, t)$-WT) with $\max(s,t)<1$ and exponential uniform weak tractability (EXP-UWT) under the absolute or normalized error criterion. We solve the problem by filling the remaining gaps left open on EXP-tractability. That is, we obtain necessary and sufficient conditions for EXP-$(s, t)$-WT with $\max(s, t) < 1$ and for EXP-UWT.