Tractability of non-homogeneous tensor product problems in the worst case setting

TL;DR

This paper investigates the computational complexity of multivariate tensor product problems in the worst case setting, providing comprehensive criteria for various tractability notions and applying to multiple kernel-based approximation problems.

Contribution

It introduces a unified method to establish necessary and sufficient conditions for different tractability types in tensor product problems, covering a broad class of kernels.

Findings

Derived conditions for strong polynomial tractability

Established criteria for polynomial and quasi-polynomial tractability

Applied results to problems with Euler, Wiener, Korobov, Gaussian, and analytic kernels

Abstract

We study multivariate linear tensor product problems with some special properties in the worst case setting. We consider algorithms that use finitely many continuous linear functionals. We use a unified method to investigate tractability of the above multivariate problems, and obtain necessary and sufficient conditions for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability, $(s,t)$-weak tractability, and weak tractability. Our results can apply to multivariate approximation problems with kernels corresponding to Euler kernels, Wiener kernels, Korobov kernels, Gaussian kernels, and analytic Korobov kernels.