A double regularization approach for inverse problems with noisy data and inexact operator

In standard inverse problems, the task is to solve an operator equation from given noisy data. However, sometimes also the operator is not known exactly. Therefore we propose a method that allows errors both in the operator and the data. In particular, we consider operator equations where the operator can be characterized by a function. For the stable reconstruction we propose the use of a Tikhonov-type functional with a generalized misfit term and an additional penalty term which promotes sparsity. Using an appropriate parameter choice rule for the two regularization parameters, we prove convergence and convergence rates for the method, and provide a first numerical example.