A convergence rates result for an iteratively regularized Gauss-Newton-Halley method in Banach space

TL;DR

This paper extends a second order iterative regularization method for inverse problems from Hilbert to Banach spaces, providing convergence rates and demonstrating advantages through numerical simulations.

Contribution

It introduces a convergence analysis of a second order Gauss-Newton-Halley method in Banach spaces, including new rate results and comparison with first order methods.

Findings

Second order information improves convergence rates.

Advantages are most significant in exact penalization cases.

Numerical simulations confirm theoretical benefits.

Abstract

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.