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 nonlinear 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 suitable penalty terms the regularized solutions converge to the true solution in the topology induced by Ψ. It is illustrated that for this parameter choice rule it holds α → 0, δq/α → 0 as the noise level δ goes to 0. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.