Goal oriented adaptivity in the IRGNM for parameter identification in PDEs I:\\ reduced formulation

Abstract

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.