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

Abstract EN

Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equation F(x)=y, where F:D(F)⊆X→Y is a nonlinear operator between the Hilbert spaces X and Y.

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.