Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

Abstract

In this paper, we study the nonstationary iterated Tikhonov regularization method (NITRM) proposed by Jin et al. [19] to solve the inverse problems, where the inverse mapping fulfills a Holder stability estimate. The iterates of NITRM are defined through certain minimization problems in the settings of Banach spaces. In order to study the various important characteristics of the sought solution, we consider the non-smooth uniformly convex penalty terms in the minimization problems. In the case of noisy data, we terminate the method via a discrepancy principle and show the strong convergence of the iterates as well as the convergence with respect to the Bregman distance. For noise free data, we show the convergence of the iterates to the sought solution. Additionally, we derive the convergence rates of NITRM method for both the noisy and noise free data that are missing from the literature. In order to derive the convergence rates, we solely utilize the Holder stability of the inverse mapping that opposes the standard analysis which requires a source condition as well as a nonlinearity estimate to be satisfied by the inverse mapping. Finally, we discuss three numerical examples to show the validity of our results.

Covengence analysis of a generalized Levenberg-Marquardt method for possibly non-smooth inverse problems

“…It has been highly focused on recently and some interesting results has been given, see [26,27,28] for example. Moreover, Mittal et al showed in [35] that the assumption 3 holds for many inverse problems.…”