B.Hofmann; O. Scherzer : Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert Spaces

Author(s) :

B.Hofmann; O. Scherzer

Title :

Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert Spaces

Electronic source :

[gzipped dvi-file] 28 kB
[gzipped ps-file] 75 kB

Preprint series

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 98-2, 1998

Mathematics Subject Classification :

65J15 [ Equations with nonlinear operators (numerical methods) ]

65J20 [ Improperly posed problems (numerical methods in abstract spaces) ]

47H15 [ Equations involving nonlinear operators ]

47H17 [ Methods for solving equations involving nonlinear operators ]

Abstract :

The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results.

Keywords :

nonlinear operator equations, ill-posedness, Frechet derivative, source conditions, stability estimates, ill-posedness measurres, a posteriori estimates

Language :

english

Publication time :

1/1998

Notes :

The work of B.H. is supported in part by the Alexander von Humboldt Foundation Bonn (Germany) and by the Johannes-Kepler-University Linz (Austria), the work of O.S. is supported in part by the Christian Doppler Society (Austria) and by the Fonds zur Förderung der Wissenschaftlichen Forschung (Austria) , SFB F1310