The presentation is devoted to Tikhonov regularization under conditional stability
estimates for nonlinear ill-posed operator equations in Hilbert scales. Our focus
is on the case of oversmoothing penalties, for which the true solution no longer
attains a finite value. In this context, we present some new results on convergence
and recall assertions on rates. We strongly highlight the local character of
conditional stability estimates for nonlinear problems and demonstrate that
pitfalls may occur. Then convergence can completely fail and the stabilizing
effect of conditional stability may be lost. Numerical case studies for some
nonlinear examples illustrate such effects.
This talk presents joint work with Peter Mathé (Berlin), Robert Plato (Siegen),
Daniel Gerth and Christopher Hofmann (Chemnitz). Research is supported by the
Deutsche Forschungsgemeinschaft (DFG) under grant HO 1454/12-1.