Discretized Newton-Tikhonov Method for Ill-posed Hammerstein Type Equations

Argyros, Ioannis K and George, Santhosh and Shobha, Monnanda Erappa (2016) Discretized Newton-Tikhonov Method for Ill-posed Hammerstein Type Equations. Communications on Applied Nonlinear Analysis, 23 (1). pp. 34-55. ISSN 1074-133X

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Abstract

George and Shobha (2012) considered the finite dimensional realization of an iterative method for non-linear ill-posed Hammerstein type operator equation KF(x) = f, when the Fr´echet derivative F′(.) of the non-linear operator F is not invertible. In this pa- per we consider the special case i.e., F′(.)−1 exists and is bounded. We analyze the convergence using Lipschitz-type conditions used in [10], [13], [22] and also analyze the convergence using a center type Lipschitz condition. The center type Lipschitz con- dition provides a tighter error estimate and expands the applicability of the method. Using a logarithmic-type source condition on F(x0)−F(ˆx) (here ˆx is the actual solution of KF(x) = f) we obtain an optimal order convergence rate. Regularization param- eter is chosen according to the balancing principle of Pereverzev and Schock (2005). Numerical illustrations are given to prove the reliability of our approach.

Item Type: Article
Uncontrolled Keywords: Newton Tikhonov method, Ill-posed Hammerstein operator, Balancing principle, Regularization method.
Subjects: Engineering > MIT Manipal > Mathematics
Depositing User: MIT Library
Date Deposited: 16 Jul 2016 11:43
Last Modified: 16 Jul 2016 11:43
URI: http://eprints.manipal.edu/id/eprint/146627

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