Article: An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations Journal: International Journal of Computing Science and Mathematics (IJCSM) 2017 Vol.8 No.4 pp.342 - 352 Abstract: Study of inverse problems are interesting and mathematically challenging due to the fact that in most of the situation they are unstable with respect to perturbations of the data. In this paper to solve such operator equations, we propose a modified form of Gauss-Newton method combined with an a posteriori parameter choice strategy with the inexact data. Convergence and the convergence rate results are proven. We consider both a-priori and a-posteriori choice rule of parameter that guarantees the scheme converges to the exact solution. The theoretical results are illustrated through numerical examples and compared with the standard scheme to demonstrate that the scheme is stable and achieves good computational output. The salient features of our proposed scheme are: 1) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature; 2) consideration of an adaptive and numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; 3) computation of an optimal order regularisation parameter of the order O(δ<SUP align="right"><SMALL>2/3</SMALL></SUP>) using a discrepancy principle. Inderscience Publishers - linking academia, business and industry through research

Title: An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations

Authors: D. Pradeep; M.P. Rajan

Addresses: School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, CET Campus, Thiruvananthapuram 695-016, Kerala, India ' School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, CET Campus, Thiruvananthapuram 695-016, Kerala, India

Abstract: Study of inverse problems are interesting and mathematically challenging due to the fact that in most of the situation they are unstable with respect to perturbations of the data. In this paper to solve such operator equations, we propose a modified form of Gauss-Newton method combined with an a posteriori parameter choice strategy with the inexact data. Convergence and the convergence rate results are proven. We consider both a-priori and a-posteriori choice rule of parameter that guarantees the scheme converges to the exact solution. The theoretical results are illustrated through numerical examples and compared with the standard scheme to demonstrate that the scheme is stable and achieves good computational output. The salient features of our proposed scheme are: 1) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature; 2) consideration of an adaptive and numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; 3) computation of an optimal order regularisation parameter of the order O(δ^2/3) using a discrepancy principle.

Keywords: nonlinear ill-posed problems; regularisation; iterative method.

DOI: 10.1504/IJCSM.2017.085852

International Journal of Computing Science and Mathematics, 2017 Vol.8 No.4, pp.342 - 352

Received: 18 Jun 2015
Accepted: 24 Sep 2016
Published online: 16 Aug 2017 *

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Title: An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations

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