Approximate solution of nonlinear inverse problems by fixed-point iteration

In this paper we propose a derivative-free iterative method for the approximate solution of a nonlinear inverse problem Fx = y. In this method the iterations are defined as Gxk+1 = Gxk +(Sy−SFxk), where G is an easily invertible operator and S is an operator from a data space to a solution space. We give general suggestions for the choice of operators G and S and show a practically relevant example of an inverse problem where such a method is succesfully applied. We carry out analysis of the proposed method for linear inverse problems. Using the recently introduced balancing principle we construct a stopping rule. Under reasonable assumptions, we show that this stopping rule leads to the regularization algorithm. Numerical results for a test example show its satisfactory behavior. ∗ Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; and Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, 4040 Linz, Austria; e-mail: [email protected] † Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; e-mail: [email protected] ‡ Fraunhofer-Institut für Technound Wirtschaftsmathematik, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany; e-mail: [email protected]