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 Avtomat. i Telemekh., 2012, Issue 3, Pages 28–38 (Mi at3775)

Applications of Mathematical Programming

The Levenberg–Marquardt method for approximation of solutions of irregular operator equations

V. V. Vasin

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

Abstract: An ill-posed problem is considered in the form of a nonlinear operator equation with a discontinuous inverse operator. It is known that in investigating a high convergence of the methods of the type of Levenberg–Marquardt (LM) method, one is forced to impose very severe constraints on the problem operator. In the suggested article the LM method convergence is set up not for the initial problem, but for the Tikhonov-regularized equation. This makes it possible to construct a stable Fejer algorithm for approximation of the solution of the initial irregular problem at the conventional, comparatively nonburdensome conditions on the operator. The developed method is tested on the solution of an inverse problem of geophysics.

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English version:
Automation and Remote Control, 2012, 73:3, 440–449

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Presented by the member of Editorial Board: À. È. Êèáçóí

Citation: V. V. Vasin, “The Levenberg–Marquardt method for approximation of solutions of irregular operator equations”, Avtomat. i Telemekh., 2012, no. 3, 28–38; Autom. Remote Control, 73:3 (2012), 440–449

Citation in format AMSBIB
\Bibitem{Vas12} \by V.~V.~Vasin \paper The Levenberg--Marquardt method for approximation of solutions of irregular operator equations \jour Avtomat. i Telemekh. \yr 2012 \issue 3 \pages 28--38 \mathnet{http://mi.mathnet.ru/at3775} \transl \jour Autom. Remote Control \yr 2012 \vol 73 \issue 3 \pages 440--449 \crossref{https://doi.org/10.1134/S0005117912030034} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000301791500003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84862148863} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Vasin V., “Irregular Nonlinear Operator Equations: Tikhonov's Regularization and Iterative Approximation”, J. Inverse Ill-Posed Probl., 21:1 (2013), 109–123
2. V. V. Vasin, “Modified Newton-type processes generating Fejér approximations of regularized solutions to nonlinear equations”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 145–158
3. Boeckmann C., Osterloh L., “Runge-Kutta Type Regularization Method for Inversion of Spheroidal Particle Distribution From Limited Optical Data”, Inverse Probl. Sci. Eng., 22:1, SI (2014), 150–165
4. A. F. Skurydina, “A regularized Levenberg–Marquardt type method applied to the structural inverse gravity problem in a multilayer medium and its parallel realization”, Vestn. YuUrGU. Ser. Vych. matem. inform., 6:3 (2017), 5–15
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