# Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales

## TL;DR

This paper extends the analysis of Tikhonov regularization with oversmoothing penalties from linear to certain non-linear ill-posed problems in Hilbert scales, demonstrating order optimal reconstruction under specific conditions.

## Contribution

It provides the first extension of oversmoothing Tikhonov regularization results to non-linear operator equations in Hilbert scales, under appropriate assumptions.

## Key findings

- Order optimal reconstruction is achievable for certain non-linear problems.
- The non-linearity assumption is verified for specific applications.
- The study broadens the applicability of oversmoothing regularization techniques.

## Abstract

We study the Tikhonov regularization for ill-posed non-linear operator equations in Hilbert scales. Our focus is on the interplay between the smoothness-promoting properties of the penalty and the smoothness inherent in the solution. The objective is to study the situation when the unknown solution fails to have a finite penalty value, hence when the penalty is oversmoothing. By now this case was only studied for linear operator equations in Hilbert scales. We extend those results to certain classes of non-linear problems. The main result asserts that under appropriate assumptions order optimal reconstruction is still possible. In an appendix we highlight that the non-linearity assumption underlying the present analysis is met for specific applications.