Discretized Newton-Tikhonov method for ill-posed hammerstein type equations

Ioannis K. Argyros, Santhosh George, Monnanda Erappa Shobha

Research output: Contribution to journalArticle

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 Fréchet 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(◯) (here ◯ 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.

Original languageEnglish
Pages (from-to)34-55
Number of pages22
JournalCommunications on Applied Nonlinear Analysis
Volume23
Issue number1
Publication statusPublished - 01-01-2016
Externally publishedYes

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Lipschitz condition
Newton-Raphson method
Iterative methods
Derivatives
Regularization Parameter
Nonlinear Operator
Operator Equation
Balancing
Invertible
Expand
Lipschitz
Convergence Rate
Error Estimates
Logarithmic
Iteration
Derivative

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

Argyros, Ioannis K. ; George, Santhosh ; Shobha, Monnanda Erappa. / Discretized Newton-Tikhonov method for ill-posed hammerstein type equations. In: Communications on Applied Nonlinear Analysis. 2016 ; Vol. 23, No. 1. pp. 34-55.
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Discretized Newton-Tikhonov method for ill-posed hammerstein type equations. / Argyros, Ioannis K.; George, Santhosh; Shobha, Monnanda Erappa.

In: Communications on Applied Nonlinear Analysis, Vol. 23, No. 1, 01.01.2016, p. 34-55.

Research output: Contribution to journalArticle

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