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

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 illposedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity combined 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 and 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. 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