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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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Discrepancy principle for generalized GN iterations combined with the reverse connection control

Anatoly Bakushinsky1 / Alexandra Smirnova2

1Institute of System Analysis RAS, Moscow 117312, Russia. E-mail:

2Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA. E-mail:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 18, Issue 4, Pages 421–431, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jiip.2010.019, October 2010

Publication History

Received:
2010-04-01
Published Online:
2010-10-20

Abstract

In this paper we investigate the generalized Gauss–Newton method in the following form

x n+1 = ξnθ(F′*(xn)F′(xn), τn)F′*(xn){(F(xn) – ƒδ) – F′(xn)(xnξn)}, x 0, ξn ∈ 𝒟 ⊂ H 1.

The modified source condition

which depends on the current iteration point xn, is used. We call this inclusion the undetermined reverse connection. The new source condition leads to a much larger set of admissible control elements ξn as compared to the previously studied versions, where ξn = ξ. The process is combined with a novel a posteriori stopping rule, where is the number of the first transition of ‖F(xn) – ƒδ‖ through the given level δω, 0, < ω < 1, i.e.,

The convergence analysis of the proposed algorithm is given.

Keywords.: Stopping rule; discrepancy principle; ill-posed problem; iterative regularization; Fréchet derivative; Gauss–Newton's method

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[1]
A. Smirnova, A. Bakushinsky, and L. DeCamp
Applicable Analysis, 2015, Volume 94, Number 4, Page 672
[2]
Alexandra Smirnova
Inverse Problems, 2012, Volume 28, Number 8, Page 085005

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