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A simplified generalized Gauss-Newton method for nonlinear ill-posed problems. (English) Zbl 1198.65101
Summary: 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 {\it B. Kaltenbacher} [Numer. Math. 79, No. 4, 501--528 (1998; Zbl 0908.65042)] for an analogous situation for a special case of the source condition.

65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
35R30Inverse problems for PDE
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