Analysis of the discrepancy principle for Tikhonov regularization under low order source conditions

TL;DR

This paper investigates the convergence rates of Tikhonov regularization using the discrepancy principle for ill-posed nonlinear problems, especially in oversmoothing cases where solutions are outside the penalty domain.

Contribution

It extends existing convergence rate results to low order source conditions and oversmoothing scenarios within Hilbert scales framework.

Findings

Established low order convergence rates under logarithmic source conditions

Extended analysis to oversmoothing cases where solutions are outside the penalty domain

Provided theoretical insights into Tikhonov regularization behavior in complex scenarios

Abstract

We study the application of Tikhonov regularization to ill-posed nonlinear operator equations. The objective of this work is to prove low order convergence rates for the discrepancy principle under low order source conditions of logarithmic type. We work within the framework of Hilbert scales and extend existing studies on this subject to the oversmoothing case. The latter means that the exact solution of the treated operator equation does not belong to the domain of definition of the penalty term. As a consequence, the Tikhonov functional fails to have a finite value.