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

Zirong Liu, Heping Wang, Kai Wang

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.

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

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 -weak tractability (EXP--WT) with 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--WT with and for EXP-UWT.
Paper Structure (6 sections, 11 theorems, 141 equations)

This paper contains 6 sections, 11 theorems, 141 equations.

Table of Contents

  1. Introduction and main results
  2. Weighted linear tensor product problems.
  3. Various notions of EXP-tractability
  4. Main results
  5. Premilinaries
  6. Proofs of Theorems \ref{['thm1']} and \ref{['thm2']}

Key Result

Theorem 1.1

Let $0<s<1$ and $0<t\leq1$. EXP-$(s,t)$-WT holds if and only if and

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 7 more