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

Abstract

Iterative regularization methods for nonlinear ill-posed equations of the form $F(x)= y$, where $F: D(F) \subset X \to Y$ 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 $x^\dagger$. 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 $x_0$ of the solution $x^\dagger$. The error analysis for this method is done under a general source condition which also involves the Fréchet derivative only at $x_0$. 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.