On the regularization of nonlinear ill-posed problems via inexact Newton iterations

Inexact Newton methods for the stable solution of nonlinear ill-posed problems are considered. The corresponding inner scheme can be chosen to be any linear regularization with a sufficient modulus of convergence. The regularization property of these Newton-type algorithms is verified, that is, the iterates converge to a solution of the nonlinear problem with exact data when the noise level tends to zero. Moreover, convergence rates are given. Finally, implementation issues are discussed and the algorithm is applied to a parameter identification problem for an elliptic PDE. The numerical results reproduce nicely theoretical predictions and show the efficiency of the proposed method.