  In this paper we propose an extension of the iteratively regularized
Gauss--Newton method to the Banach space setting by defining the iterates via
convex optimization problems. We consider some a posteriori stopping rules to
terminate the iteration and present the detailed convergence analysis. The
remarkable point is that in each convex optimization problem we allow
non-smooth penalty terms including $L^1$ and total variation (TV) like penalty
functionals. This enables us to reconstruct special features of solutions such
as sparsity and discontinuities in practical applications. Some numerical
experiments on parameter identification in partial differential equations are
reported to test the performance of our method.
