Variational regularization with oversmoothing penalty term in Banach spaces

TL;DR

This paper extends variational regularization techniques with oversmoothing penalties from Hilbert to Banach spaces, providing convergence rates and tools for handling low order smoothness in nonlinear ill-posed problems.

Contribution

It introduces a framework for oversmoothing regularization in Banach spaces and derives convergence rates for different smoothness types, expanding previous Hilbert space results.

Findings

Convergence rates established for a priori regularization parameters.

Extension of oversmoothing regularization to Banach spaces.

Tools developed for low order smoothness analysis.

Abstract

In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present study tries to continue and to extend those investigations to Banach scales. This new study includes convergence rates results for a priori choices of the regularization parameter, both for H\"older-type smoothness and low order-type smoothness. The necessary tools for low order smoothness in the Banach space setting are provided.