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Journal of Mathematics
Volume 2013 (2013), Article ID 439316, 9 pages
http://dx.doi.org/10.1155/2013/439316
Research Article

Newton-Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems

Department of Mathematical and Computational Sciences, National Institute of Technology, Mangalore, Karnataka 575 025, India

Received 14 August 2012; Accepted 15 October 2012

Academic Editor: De-Xing Kong

Copyright © 2013 Santhosh George. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equation , where is a nonlinear operator between the Hilbert spaces and . The analysis in George, 2010 was carried out using a majorizing sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure.