Hein, Torsten : Regularization of ill-posed problems in Banach spaces - approximative source conditions and convergence rates results

Author(s):

Hein, Torsten

Title:

Regularization of ill-posed problems in Banach spaces - approximative source conditions and convergence rates results

Electronic source:

application/postscript

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 28, 2007

Mathematics Subject Classification:

47A52			[ Ill-posed problems, regularization ]
47J06			[ Nonlinear ill-posed problems ]
49N45			[ Inverse problems ]

Abstract:

In this preprint we deal with convergence rates for regularizing linear and nonlinear ill-posed problems with operators mapping from a Hilbert space into a Banach space. Thereby we deal with so-called distance functions which quantify the violation of a reference source condition. With the aid of these functions we present error bounds and convergence rates for regularized solutions of linear and non-linear problems when the reference source condition is not satisfied. The way of applying distance functions transfers the idea of considering generalized source conditions in Hilbert spaces to inverse problems in Banach spaces in a natural way. Introducing this topic for linear ill-posed problems we additionally show that this theory can be easily extended to nonlinear problems as well as to more general penalty terms using Bregman distances. Moreover, the application of the discrepancy principle as a posteriori choice strategy of the regularization parameter is discussed for both linear and nonlinear problems.

Keywords:

ill-posed problem, regularization, distance function, convergence rates, Bregman distance

Language:

English

Publication time:

12 / 2007