Tikhonov and Landweber convergence rates: characterization by interpolation spaces

TL;DR

This paper characterizes the convergence rates of Tikhonov and Landweber regularization methods in linear inverse problems using the framework of real interpolation spaces, providing a unified theoretical understanding.

Contribution

It introduces a novel characterization of convergence rates via intermediate interpolation spaces, linking regularization performance to functional analysis properties.

Findings

Convergence rates are characterized by solution membership in interpolation spaces.

Results unify Tikhonov and Landweber convergence analysis.

Provides a functional analytic framework for regularization rate estimation.

Abstract

Algebraic convergences rates of (iterated) Tikhonov regularization for linear inverse problems in Hilbert spaces are characterized by the membership of the exact solution to intermediate spaces produced by the K-method of real interpolation. Similar results are obtained for the Landweber iteration.