  Conditional stability estimates are a popular tool for the regularization of
ill-posed problems. A drawback in particular under nonlinear operators is that
additional regularization is needed for obtaining stable approximate solutions
if the validity area of such estimates is not completely known. In this paper
we consider Tikhonov regularization under conditional stability estimates for
nonlinear ill-posed operator equations in Hilbert scales. We summarize
assertions on convergence and convergence rate in three cases describing the
relative smoothness of the penalty in the Tikhonov functional and of the exact
solution. For oversmoothing penalties, for which the rue solution no longer
attains a finite value, we present a result with modified assumptions for a
priori choices of the regularization parameter yielding convergence rates of
optimal order for noisy data. We strongly highlight the local character of the
conditional stability estimate and demonstrate that pitfalls may occur through
incorrect stability estimates. Then convergence can completely fail and the
stabilizing effect of conditional stability may be lost. Comprehensive
numerical case studies for some nonlinear examples illustrate such effects.

Numerical Analysis