Regularization of inverse problems with large noise

Regularization of ill-posed problems is only possible if certain bounds on the data noise level are available. We consider here the case of large, possibly unbounded noise, and propose a class of modified regularization methods that are capable of dealing with that case. After some modification, these methods can be analyzed by standard regularization theory, and optimal convergence rates are obtained. An analysis in the spirit of regularization in Hilbert scales allows to relate the results obtained to other approaches dealing with large noise, and to clarify the influence of the relaxed assumptions regarding the noise on the convergence rates. Finally, the theoretical results are illustrated by examples and numerical tests are presented