  In this paper we investigate adaptive discretization of the iteratively
regularized Gauss- Newton method IRGNM. All-at-once formulations considering
the PDE and the measurement equation simultaneously allow to avoid
(approximate) solution of a potentially nonlinear PDE in each Newton step as
compared to the reduced form [22]. We analyze a least squares and a generalized
Gauss-Newton formulation and in both cases prove convergence and convergence
rates with a posteriori choice of the regularization parameters in each Newton
step and of the stopping index under certain accuracy requirements on four
quantities of interest. Estimation of the error in these quantities by means of
a weighted dual residual method is discussed, which leads to an algorithm for
adaptive mesh refinement. Numerical experiments with an implementation of this
algorithm show the numerical efficiency of this approach, which especially for
strongly nonlinear PDEs outperforms the nonlinear Tikhonov regularization
considered in [21].
