Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space.

In this paper we consider the iteratively regularized Gauss-Newton method (IRGNM) in its classical Tikhonov version as well as two further-Ivanov type and Morozov type-versions. In these two alternative versions, regularization is achieved by imposing bounds on the solution or by minimizing some regularization functional under a constraint on the data misfit, respectively. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces. Moreover, we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators. The results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem, for the cases of a measure valued and of an source. For the latter, we also provide numerical results with the Ivanov type IRGNM.