A simplified generalized Gauss-Newton method for nonlinear ill-posed problems

Iterative regularization methods for nonlinear ill-posed equations of the form F (x) = y, where F : D(F ) ⊂ X → Y is an operator between Hilbert spaces X and Y , usually involve calculation of the Fréchet derivatives of F at each iterate and at the unknown solution x†. In this paper, we suggest a modified form of the generalized Gauss-Newton method which requires the Fréchet derivative of F only at an initial approximation x0 of the solution x†. The error analysis for this method is done under a general source condition which also involves the Fréchet derivative only at x0. The conditions under which the results of this paper hold are weaker than those considered by Kaltenbacher (1998) for an analogous situation for a special case of the source condition.