  The use of second order information on the forward operator often comes at a
very moderate additional computational price in the context of parameter
identification probems for differential equation models. On the other hand the
use of general (non-Hilbert) Banach spaces has recently found much interest due
to its usefulness in many applications. This motivates us to extend the second
order method previously considered by the author in a Hilbert space setting,
(see also Hettlich and Rundell 2000) to a Banach space setting and analyze its
convergence. We here show rates results for a particular source condition and
different exponents in the formulation of Tikhonov regularization in each step.
This includes a complementary result on the (first order) iteratively
regularized Gauss-Newton method (IRGNM) in case of a one-homogeneous data
misfit term, which corresponds to exact penalization. The results clearly show
the possible advantages of using second order information, which get most
pronounced in this exact penalization case. Numerical simulations for a
coefficient identification problem in an elliptic PDE illustrate the
theoretical findings.
