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A Generalized Tikhonov Regularization for Nonlinear Inverse Ill-Posed Problems

Michael Ulbrich

Technische Universität München

Zentrum Mathematik

Lehrstuhl für Angewandte Mathematik und Mathematische Statistik

July 1998

Technical Report

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We propose and analyze a generalization of the Tikhonov regularization for nonlinear ill-posed operator equations F(y)=z. Hereby, regularization functionals of the form ||R(y)||_W^2 with linear or nonlinear operator R are considered. This class contains both regularization in state space and the usual Tikhonov regularization as special cases. We develop existence, stability, and convergence results of approximate solutions to the regularized problem in a Banach space setting. Under the additional assumption that the regularization space W is embedded in a Hilbert scale, we present error estimates if the regularization parameter is chosen either according to the discrepancy principle of Morozov or by the usual a-priori strategy. Moreover, we propose a generalization of Neubauer's Tikhonov-type regularization to our class of regularization functionals and derive error bounds. Compared to the usual assumptions, we prove all our error estimates under substantially weakened requirements on the injectivity of F. This is particularly valuable for parameter identification problems with scattered observations. The developed theory is applied to a simple nonlinear parameter identification problem to illustrate our findings.