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Author: search.filters.author.Vasin, V.Subject: search.filters.subject.Newton-Raphson method

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    An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems
    (2014) Vasin, V.; George, S.
    In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x)=y where F:D(F)?X?X is a nonlinear monotone operator or F?(x0) is nonnegative selfadjoint operator defined on a real Hilbert space X. We assume that only a noisy data y??X with ?y- y???? are available. Further we assume that Fréchet derivative F? of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x0-x?, the error ?x?-xn,??? between the regularized approximation xn,??(x0,??;=x0) and the solution x? is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem. © 2013 Elsevier Inc. All rights reserved.
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    Expanding the applicability of Tikhonov's regularization and iterative approximation for ill-posed problems
    (Walter de Gruyter GmbH, 2014) Vasin, V.; George, S.
    Recently, Vasin [J. Inverse Ill-Posed Probl. 21 (2013), 109-123] considered a new iterative method for approximately solving nonlinear ill-posed operator equation in Hilbert spaces. In this paper we introduce a modified form of the method considered by Vasin. This paper weakens the conditions needed in the existing results. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in [J. Inverse Ill-Posed Probl. 21 (2013), 109-123]. This way a tighter convergence analysis is obtained and under less computational cost, since the more precise and easier to compute center-Lipschitz instead of the Lipschitz constant is used in the convergence analysis. Order optimal error bounds are given in case the regularization parameter is chosen a priori and by the adaptive method of Pereverzev and Schock [SIAM J. Numer. Anal. 43 (2005), 2060-2076]. A numerical example of a nonlinear integral equation proves the efficiency of the proposed method. © 2014 by De Gruyter.