Faculty Publications

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    A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations
    (2012) George, S.; Elmahdy, A.I.
    In this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y; where the right hand side is replaced by noisy data y? ? X with ?y - y ?? ? ? and T : D(T) ? X ? X is a nonlinear monotone operator defined on a Hilbert space X: The iteration x ?n,? converges quadratically to the unique solution x?? of the equation T(x) + ?(x - x0) = y? (x0 := x 0,??). It is known that (Tautanhahn (2002)) x?? converges to the solution x? of Tx = y: The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. Under a general source condition on x 0 - x? we proved that the error ?x? - x n, ??;? is of optimal order. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate. © 2012 Institute of Mathematics, NAS of Belarus.
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    Expanding the applicability of a modified Gauss-Newton method for solving nonlinear ill-posed problems
    (2013) Argyros, I.K.; George, S.
    We expand the applicability of a modified Gauss-Newton method recently presented in George (2013) [19] for approximate solution of a nonlinear ill-posed operator equation between two Hilbert spaces. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in earlier studies such as George (2013, 2010) [19,18]. 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. Numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations. © 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.
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    Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions
    (Springer-Verlag Italia s.r.l., 2016) Argyros, I.K.; George, S.
    We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111–115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study. © 2015, Springer-Verlag Italia.
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    On the convergence of Broyden's method with regularity continuous divided differences and restricted convergence domains
    (Mathematical Research Press, 2017) Argyros, I.K.; George, S.
    We present a semilocal convergence analysis for Broyden's method with regularly continuous divided differences in a Banach/Hilbert space setting. By using: center-Lipschitz-type conditions defining restricted convergence domains, at least as weak hypotheses and the same computational cost as in [6] we provide a new convergence analysis for Broyden's method with the following advantages: larger convergence domain; finer error bounds on the distances involved, and at least as precise information on the location of the solution. © 2017 Journal of Nonlinear Functional Analysis.
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    A derivative-free iterative method for nonlinear ill-posed equations with monotone operators
    (Walter de Gruyter GmbH info@degruyter.com, 2017) George, S.; Nair, M.T.
    Recently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova's method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11]. © 2017 Walter de Gruyter GmbH, Berlin/Boston.
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    Unified convergence analysis of frozen Newton-like methods under generalized conditions
    (Elsevier B.V., 2019) Argyros, I.K.; George, S.
    The objective in this article is to present a unified convergence analysis of frozen Newton-like methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations. Numerical examples complete the presentation of this article. © 2018 Elsevier B.V.
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    On the complexity of choosing majorizing sequences for iterative procedures
    (Springer-Verlag Italia s.r.l., 2019) Argyros, I.K.; George, S.
    The aim of this paper is to introduce general majorizing sequences for iterative procedures which may contain a non-differentiable operator in order to solve nonlinear equations involving Banach valued operators. A general semi-local convergence analysis is presented based on majorizing sequences. The convergence criteria, if specialized can be used to study the convergence of numerous procedures such as Picard’s, Newton’s, Newton-type, Stirling’s, Secant, Secant-type, Steffensen’s, Aitken’s, Kurchatov’s and other procedures. The convergence criteria are flexible enough, so if specialized are weaker than the criteria given by the aforementioned procedures. Moreover, the convergence analysis is at least as tight. Furthermore, these advantages are obtained using Lipschitz constants that are least as tight as the ones already used in the literature. Consequently, no additional hypotheses are needed, since the new constants are special cases of the old constants. These ideas can be used to study, the local convergence, multi-step multi-point procedures along the same lines. Some applications are also provided in this study. © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature.
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    Local convergence analysis of jarratt-type schemes for solving equations
    (Biemdas Academic Publishers, 2019) Argyros, I.K.; George, S.
    The aim of this paper is to present a local convergence analysis for Jarratt-type schemes to solve equations with Banach space valued operators. The convergence conditions are are weaker than the corresponding ones in earlier results and some numerical examples are provided. © 2019 Applied Set-Valued Analysis and Optimization.
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    Expanding the applicability of an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems
    (Springer-Verlag Italia s.r.l., 2019) Argyros, I.K.; Cho, Y.J.; George, S.; Xiao, Y.
    We expand the applicability of an a posteriori parameter choice strategy for Tikhonov regularization of the nonlinear ill-posed problem presented in Jin and Hou (Numer Math 83:139–159, 1999) by weakening the conditions needed in Jin and Hou [13]. Using a center-type Lipschitz condition instead of a Lipschitz-type condition used in Jin and Hou [13], Scherzer et al. (SIAM J Numer Anal 30:1796–1838, 1993), we obtain a tighter convergence analysis. Numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations. © 2019, The Royal Academy of Sciences, Madrid.