A note on $\Gamma$-convergence of Tikhonov functionals for nonlinear inverse problems

TL;DR

This paper investigates the conditions under which Tikhonov functionals for nonlinear inverse problems in Banach spaces exhibit $\Gamma$-convergence, ensuring that approximate solutions converge to the true minimizer despite model uncertainties.

Contribution

It provides new criteria for $\Gamma$-convergence of Tikhonov functionals in nonlinear inverse problems, facilitating reliable approximations under uncertainties.

Findings

Established criteria for $\Gamma$-convergence in various topologies.

Demonstrated convergence of minimizers under certain conditions.

Addressed stability issues in variational regularization of inverse problems.

Abstract

We consider variational regularization of nonlinear inverse problems in Banach spaces using Tikhonov functionals. This article addresses the problem of $\Gamma$-convergence of a family of Tikhonov functionals and assertions of the convergence of their respective infima. Such questions arise, if model uncertainties, inaccurate forward operators, finite dimensional approximations of the forward solutions and / or data, etc. make the evaluation of the original functional impossible and, thus, its minimizer not computable. But for applications it is of utmost importance that the minimizer of the replacement functional approximates the original minimizer. Under certain additional conditions this is satisfied if the approximated functionals converge to the original functional in the sense of $\Gamma$-convergence. We deduce simple criteria in different topologies which guarantee $\Gamma$-convergence as well as convergence of minimizing sequences.