Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part I

TL;DR

This paper establishes a lower error bound for multivariate operator approximation in Hilbert spaces, revealing that using function values leads to the curse of dimensionality, unlike linear functionals.

Contribution

It introduces a new lower error bound linking operator approximation to linear functional bounds, and applies it to show the curse of dimensionality for standard Sobolev spaces with function values.

Findings

Error bounds relate operator approximation to linear functional bounds

Curse of dimensionality occurs with function values in Sobolev spaces

Linear functionals avoid the curse in multivariate approximation

Abstract

We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms use function values. Furthermore, some of these linear functionals have the same norm as the linear operators. We then apply this error bound for linear (unweighted) tensor products. In this way we use negative tractability results known for linear functionals to conclude the same negative results for linear operators. In particular, we prove that $L_2$-multivariate approximation defined for standard Sobolev space suffers the curse of dimensionality if function values are used although the curse is not present if linear functionals are allowed.