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Hofmann, Bernd : Interplay of source conditions and variational inequalities for nonlinear ill-posed problems

Hofmann, Bernd : Interplay of source conditions and variational inequalities for nonlinear ill-posed problems

Author(s):
Hofmann, Bernd
Title:
Interplay of source conditions and variational inequalities for nonlinear ill-posed problems
Electronic source:
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 6, 2009
Mathematics Subject Classification:
47J06 [ Nonlinear ill-posed problems ]
65J20 [ Improperly posed problems; regularization ]
47J20 [ Variational and other types of inequalities involving nonlinear operators ]
47A52 [ Ill-posed problems, regularization ]
Abstract:
In the last years convergence rates results for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces have been published, where the classical concept of source conditions was replaced with variational inequalities holding on some level sets. Also this advanced essentially the analysis of non-smooth situations with respect to forward operators and solutions. In fact, such variational inequalities combine both structural conditions on the nonlinearity of the operator and smoothness properties of the solution. Varying exponents in the variational inequalities correspond to different levels of convergence rates. In this paper, we discuss the range of occurring exponents in the Banach space setting. To lighten the cross-connections between generalized source conditions, degree of nonlinearity of the forward operator and associated variational inequalities we study the Hilbert space situation and even prove some converse result for linear operators.
Keywords:
Nonlinear ill-posed problems, regularization, variational inequalities, source conditions, degree of nonlinearity, convergence rates
Language:
English
Publication time:
3 / 2009