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In this paper we consider the Iteratively Regularized Gauss-Newton Method(IRGNM) in its classical Tikhonov version and in an Ivanov type version, whereregularization is achieved by imposing bounds on the solution. We do so in ageneral Banach space setting and under a tangential cone condition, whileconvergence (without source conditions, thus without rates) has so far onlybeen proven under stronger restrictions on the nonlinearity of the operatorand/or on the spaces. Moreover, we provide a convergence result for thediscretized problem with an appropriate control on the error and show how toprovide the required error bounds by goal oriented weighted dual residualestimators. The results are illustrated for an inverse source problem for anonlinear elliptic boundary value problem, for the cases of a measure valuedand of an $L^\infty$ source. For the latter, we also provide numerical resultswith the Ivanov type IRGNM.

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