Notes on $(s,t)$-weak tractability: A refined classification of problems with (sub)exponential information complexity

TL;DR

This paper introduces the concept of $(s,t)$-weak tractability to better classify the complexity of multivariate problems, providing quantitative measures for problems that are not polynomially tractable, with applications to Sobolev space approximation and integration.

Contribution

It proposes a new $(s,t)$-weak tractability framework that refines existing classifications and characterizes it for linear compact operators using singular values.

Findings

Characterization of $(s,t)$-weak tractability for Hilbert space operators

Application to Sobolev space approximation problems

Application to integration problems for smooth functions

Abstract

In the last 20 years a whole hierarchy of notions of tractability was proposed and analyzed by several authors. These notions are used to classify the computational hardness of continuous numerical problems $S=(S_d)_{d\in\mathbb{N}}$ in terms of the behavior of their information complexity $n(\epsilon,S_d)$ as a function of the accuracy $\epsilon$ and the dimension $d$. By now a lot of effort was spend on either proving quantitative positive results (such as, e.g., the concrete dependence on $\epsilon$ and $d$ within the well-established framework of polynomial tractability), or on qualitative negative results (which, e.g., state that a given problem suffers from the so-called curse of dimensionality). Although several weaker types of tractability were introduced recently, the theory of information-based complexity still lacks a notion which allows to quantify the exact (sub-/super-)exponential dependence of $n(\epsilon,S_d)$ on both parameters $\epsilon$ and $d$. In this paper we present the notion of $(s,t)$-weak tractability which attempts to fill this gap. Within this new framework the parameters $s$ and $t$ are used to quantitatively refine the huge class of polynomially intractable problems. For linear, compact operators between Hilbert spaces we provide characterizations of $(s,t)$-weak tractability w.r.t. the worst case setting in terms of singular values. In addition, our new notion is illustrated by classical examples which recently attracted some attention. In detail, we study approximation problems between periodic Sobolev spaces and integration problems for classes of smooth functions. Keywords: Information-based complexity, Multivariate numerical problems, Hilbert spaces, Tractablity, Approximation, Integration.