Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators

Abstract

In this paper we deal with Morozov’s discrepancy principle as an a-posteriori parameter choice rule for Tikhonov regularization with general convex penalty terms Ψ for non-linear inverse problems. It is shown that a regularization parameter α fulfilling the discprepancy principle exists, whenever the operator F satisfies some basic conditions, and that for this parameter choice rule holds α → 0, δ q /α → 0 as the noise level δ goes to 0. It is illustrated that for suitable penalty terms this yields convergence of the regularized solutions to the true solution in the topology induced by Ψ. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.