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Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations

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Abstract

For the solution of nonlinear ill-posed problems, a Two Step Newton-Tikhonov methodology is proposed. Two implementations are discussed and applied to nonlinear ill-posed Hammerstein type operator equations \(KF(x)=y\), where K defines the integral operator and F the function of the solution x on which K operates. In the first case, the Fr\(\acute{e}\)chet derivative of F is invertible in a neighbourhood which includes the initial guess \(x_0\) and the solution \({\hat{x}}\). In the second case, F is monotone. For both cases, local cubic convergence is established and order optimal error bounds are obtained by choosing the regularization parameter according to the the balancing principle of Pereverzev and Schock (2005).We also present the results of computational experiments giving the evidence of the reliability of our approach.

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Acknowledgments

The authors thank Prof. Bob Anderssen for his detailed suggestions and comments. Ms. Shobha, thanks National Institute of Technology Karnataka, India, for the financial support.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

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

    Santhosh George

  3. Department of Mathematics, Manipal Institute of Technology, Manipal, 576104, India

    Shobha Monnanda Erappa

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  1. Ioannis K. Argyros
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  2. Santhosh George
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  3. Shobha Monnanda Erappa
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Correspondence to Shobha Monnanda Erappa.

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Argyros, I.K., George, S. & Monnanda Erappa, S. Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations. Rend. Circ. Mat. Palermo, II. Ser 66, 303–323 (2017). https://doi.org/10.1007/s12215-016-0254-x

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