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Conditional stability estimates are a popular tool for the regularization ofill-posed problems. A drawback in particular under nonlinear operators is thatadditional regularization is needed for obtaining stable approximate solutionsif the validity area of such estimates is not completely known. In this paperwe consider Tikhonov regularization under conditional stability estimates fornonlinear ill-posed operator equations in Hilbert scales. We summarizeassertions on convergence and convergence rate in three cases describing therelative smoothness of the penalty in the Tikhonov functional and of the exactsolution. For oversmoothing penalties, for which the rue solution no longerattains a finite value, we present a result with modified assumptions for apriori choices of the regularization parameter yielding convergence rates ofoptimal order for noisy data. We strongly highlight the local character of theconditional stability estimate and demonstrate that pitfalls may occur throughincorrect stability estimates. Then convergence can completely fail and thestabilizing effect of conditional stability may be lost. Comprehensivenumerical case studies for some nonlinear examples illustrate such effects.

Case studies and a pitfall for nonlinear variational regularization under conditional stability