Existence, uniquennes and stability of minimizers of generalized Tikhonov-Phillips functionals

Abstract

The Tikhonov-Phillips method is widely used for regularizing ill-posed inverse problemsmainly due to the simplicity of its formulation as an optimization problem. The useof different penalizers in the functionals associated to the corresponding optimizationproblems has originated a variety of other methods which can be considered as"variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case forinstance of the Tikhonov-Phillips method of order one, the total variation regularizationmethod, etc. In this article we find sufficient conditions on the penalizers in generalizedTikhonov-Phillips functionals which guarantee existence, uniqueness and stability of theminimizers. The particular cases in which the penalizers are given by the bounded variationnorm, by powers of seminorms and by linear combinations of powers of seminormsassociated to closed operators, are studied. Several examples are presented and a fewresults on image restoration are shown.