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Mathematics of Computation

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

Authors: Pallavi Mahale and M. Thamban Nair
Journal: Math. Comp. 78 (2009), 171-184
MSC (2000): Primary 65J20
Published electronically: June 10, 2008
MathSciNet review: 2448702
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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.

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Additional Information

Pallavi Mahale
Affiliation: Department of Mathematics, IIT Madras, Chennai 600036, India

M. Thamban Nair
Affiliation: Department of Mathematics, IIT Madras, Chennai 600036, India

Received by editor(s): July 2, 2007
Received by editor(s) in revised form: January 13, 2008
Published electronically: June 10, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.