Radu Ioan Bot Bernd Hofmann, Peter Mathé: Regularizability of ill-posed problems and the modulus of continuity

Author(s):

Radu Ioan Bot
Bernd Hofmann
Peter Mathé

Title:

Regularizability of ill-posed problems and the modulus of continuity

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 17, 2011

Mathematics Subject Classification:

47A52	[]
65J20	[]

Abstract:

The regularization of linear ill-posed problems is based on theirconditional well-posedness when restricting the problem to certainclasses of solutions. Given such class one may considerseveral related real-valued functions, which measure thewell-posedness of the problem on such class. Among those functionsthe modulus of continuity is best studied. For solution classes whichenjoy the additional feature of being star-shaped at zero, the authorsdevelop a series of results with focus on continuity properties ofthe modulus of continuity. In particular it is highlighted that the problem is conditionally well-posed if and only if the modulus ofcontinuity is right-continuous at zero. Those results are then applied to smoothness classes in Hilbert space.This study concludes with a new perspective on a concavity problem forthe modulus of continuity, recently addressed by two of the authors in the paper "Some note on the modulus of continuity for ill-posed problems in Hilbert space", 2011.

Keywords:

Linear ill-posed problems, modulus of continuity, conditional stability

Language:

English

Publication time:

09/2011