Tractability of multivariate problems for standard and linear information in the worst case setting: part II

TL;DR

This paper investigates the conditions under which multivariate linear tensor product problems over Hilbert spaces are quasi-polynomially tractable in the worst case, focusing on function value algorithms and specific spectral assumptions.

Contribution

It establishes necessary and sufficient conditions for quasi-polynomial tractability using only function values in the worst case setting for Hilbert space problems.

Findings

QPT holds if univariate errors decay polynomially

QPT requires the largest singular value to be simple

Eigenfunction must be a multiple of a point evaluation function

Abstract

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.