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Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales

Bernd Hofmann, Peter Mathé
2017 Inverse Problems  
We study the Tikhonov regularization for ill-posed non-linear operator equations in Hilbert scales. Our focus is on the interplay between the smoothness-promoting properties of the penalty and the smoothness inherent in the solution. The objective is to study the situation when the unknown solution fails to have a finite penalty value, hence when the penalty is oversmoothing. By now this case was only studied for linear operator equations in Hilbert scales. We extend those results to certain
more » ... sults to certain classes of non-linear problems. The main result asserts that under appropriate assumptions order optimal reconstruction is still possible. In an appendix we highlight that the non-linearity assumption underlying the present analysis is met for specific applications.
doi:10.1088/1361-6420/aa9b59 fatcat:64cyc63unrabro7itkebpg4utu