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For the Tikhonov regularization of ill-posed nonlinear operator equations in a Hilbert scale setting, the convergence of regularized solutions is studied. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Holder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies, without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low order convergence rates under associated (for example logarithmic) source conditions.

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term