Generalized Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

TL;DR

This paper investigates convergence rate conditions for regularization in Hilbert spaces, revealing that some recent generalized conditions are not equivalent to classical ones, but are optimal within the standard setting, and introduces a new source condition.

Contribution

It demonstrates the non-equivalence of variational and standard source conditions in Hilbert spaces and introduces a novel source condition that simplifies convergence rate proofs.

Findings

Novel source condition relates to existing conditions

Generalized convergence rates are optimal in Hilbert spaces

Spectral theory can be bypassed in proofs

Abstract

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting. We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer, and generalize existing convergence rates results. The key tool in our analysis is a novel source condition, which we put into relation to the existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces.