Tikhonov functionals with a tolerance measure introduced in the regularization

TL;DR

This paper introduces a modified Tikhonov functional incorporating a tolerance measure to better handle small deviations in solutions of ill-posed nonlinear inverse problems, with theoretical guarantees and practical insights.

Contribution

It proposes a new Tikhonov regularization approach with tolerance measures, providing theoretical analysis and exploring sparse solution reconstruction.

Findings

Existence, stability, and convergence of minimizers proven.

Effect of tolerances on solutions demonstrated through examples.

Parameter choice rules for optimal regularization identified.

Abstract

We consider a modified Tikhonov-type functional for the solution of ill-posed nonlinear inverse problems. Motivated by applications in the field of production engineering, we allow small deviations in the solution, which are modeled through a tolerance measure in the regularization term of the functional. The existence, stability, and weak convergence of minimizers are proved for such a functional, as well as the convergence rates in the Bregman distance. We present an example for illustrating the effect of tolerances on the regularized solution and examine parameter choice rules for finding the optimal regularization parameter for the assumed tolerance value. In addition, we discuss the prospect of reconstructing sparse solutions when tolerances are incorporated in the regularization functional.