Hofmann, Bernd : A note on convergence rates for variational regularization with non-convex residual term

Author(s):

Hofmann, Bernd

Title:

A note on convergence rates for variational regularization with non-convex residual term

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 11, 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:

This note formulates some assertions and conjectures concerning an up to now missing case of convergence rates results for variational regularization of nonlinear ill-posed problems in Banach spaces. If the residual term is the $p$-th power of a Banach space norm, then the use of powers $0<p<1$ instead of the common values $1 \le p <\infty$ leads to an artificial limitation of convergence rates. This effect also occurs for general residual terms when they represent concave monomials of that distance which is bounded by the noise level and expresses some kind of qualification for the regularization method.

Keywords:

Nonlinear ill-posed problems, variational regularization, variational inequalities, non-convex residual term, convergence rates, qualification, Banach space

Language:

English

Publication time:

5 / 2009