Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability. (English) Zbl 1503.35288
Summary: In this paper, we study the nonstationary iterated Tikhonov regularization method (NITRM) proposed by Q. Jin and M. Zhong [Numer. Math. 127, No. 3, 485–513 (2014; Zbl 1297.65062)] to solve the inverse problems, where the inverse mapping fulfills a Hölder 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 Hölder 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.
MSC:
| 35R30 | Inverse problems for PDEs |
| 47A52 | Linear operators and ill-posed problems, regularization |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
Keywords:
regularization; nonlinear ill-posed operator equations; Tikhonov regularization; convergence analysis; iterative regularization methodsCitations:
Zbl 1297.65062References:
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