Convergence rates for inverse Problems in Hilbert spaces: A Comparative Study

TL;DR

This paper investigates convergence rates for inverse problems in Hilbert spaces using H"older stability estimates, providing new insights for both linear and non-linear cases without relying on classical spectral theory.

Contribution

It introduces a novel approach using H"older stability estimates and variational inequalities to determine convergence rates without classical spectral theory or additional non-linearity estimates.

Findings

Convergence rates for linear inverse problems derived without spectral theory.

Convergence rates for non-linear inverse problems obtained without extra non-linearity estimates.

Comparison of Tikhonov and Lavrentiev regularization methods using new stability concepts.

Abstract

In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates for the determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without incorporating the classical concept of spectral theory and for non-linear inverse problems, we obtain the convergence rates without incorporating any additional non-linearity estimate. Further, we employ the smoothness concept of inhomogeneous variational inequalities to deduce the convergence rates for non-linear inverse problems. In addition to Tikhonov regularization, we also consider Lavrentiev's regularization method for non-linear inverse problems and determine its convergence rates by incorporating the H\"older stability estimates as well as inhomogeneous variational inequalities. And finally, we discuss the co-action between the variational inequalities and the H\"older stability estimates.