  In this paper we study adaptive discretization of the iteratively regularized
Gauss-Newton method IRGNM with an a posteriori (discrepancy principle) choice
of the regularization parameter in each Newton step and of the stopping index.
We first of all prove convergence and convergence rates under some accuracy
requirements formulated in terms of four quantities of interest. Then
computation of error estimators for these quantities based on a weighted dual
residual method is discussed, which results in an algorithm for adaptive
refinement.
  Finally we extend the results from the Hilbert space setting with quadratic
penalty to Banach spaces and general
  Tikhonov functionals for the regularization of each Newton step.
