Faculty Publications
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Item 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.Item An application of newton type iterative method for lavrentiev regularization for ill-posed equations: Finite dimensional realization(2012) George, S.; Pareth, S.In this paper, we consider, a finite dimensional realization of Newton type iterative method for Lavrentiev regularization of ill-posed equations. Precisely we consider the ill-posed equation F(x) = f when the available data is f ? withItem 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.Item A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations(Elsevier Inc. usjcs@elsevier.com, 2015) Padikkal, P.; Shubha, V.S.; George, S.George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution x?? of the method of Lavrentiev regularization, i.e., F(x) + ?(x - x0) = y?, approximating the solution x of the ill-posed problem F(x) = y where F:D(F)?X?X is a nonlinear monotone operator defined on a real Hilbert space X. The convergence analysis of the method was based on a majorizing sequence. In this paper we are concerned with the problem of expanding the applicability of the method considered by George and Elmahdy (2012) by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as George and Elmahdy (2012), Mahale and Nair (2009), Mathe and Perverzev (2003), Nair and Ravishankar (2008), Semenova (2010) and Tautanhahn (2002). We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem. © 2014 Elsevier Inc. All rights reserved.Item Ball convergence comparison between three iterative methods in Banach space under hypothese only on the first derivative(Elsevier Inc. usjcs@elsevier.com, 2015) Argyros, I.K.; George, S.Abstract We present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for these methods under hypotheses only on the first Fréchet derivative in contrast to earlier studies such as Adomian (1994) [1], Babajee et al. (2008) [13], Cordero and Torregrosa (2007) [17], Cordero et al. [18], Darvishi and Barati (2007) [19], using hypotheses reaching up to the fourth Fréchet derivative although only the first derivative appears in these methods. This way we expand the applicability of these methods. Numerical examples are also presented in this study. © 2015 Elsevier Inc.Item Finite dimensional realization of a quadratic convergence yielding iterative regularization method for ill-posed equations with monotone operators(Elsevier Inc. usjcs@elsevier.com, 2016) Shubha, V.S.; George, S.; Padikkal, P.; Erappa, M.E.Recently Jidesh et al. (2015), considered a quadratic convergence yielding iterative method for obtaining approximate solution to nonlinear ill-posed operator equation F(x)=y, where F: D(F) ? X ? X is a monotone operator and X is a real Hilbert space. In this paper we consider the finite dimensional realization of the method considered in Jidesh et al. (2015). Numerical example justifies our theoretical results. © 2015 Elsevier Inc. All rights reserved.Item 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.Item Local convergence of a fast Steffensen-type method on Banach space under weak conditions(Inderscience Publishers, 2017) Argyros, I.K.; George, S.This paper is devoted to the study of the seventh-order Steffensen-type methods for solving nonlinear equations in Banach spaces. Using the idea of a restricted convergence domain, we extended the applicability of the seventh-order Steffensen-type methods. Our convergence conditions are weaker than the conditions used in the earlier studies. Numerical examples are also given in this study. © © 2017 Inderscience Enterprises Ltd.Item Convergence rate results for steepest descent type method for nonlinear ill-posed equations(Elsevier Inc. usjcs@elsevier.com, 2017) George, S.; Sabari, M.Convergence rate result for a modified steepest descent method and a modified minimal error method for the solution of nonlinear ill-posed operator equation have been proved with noisy data. To our knowledge, convergence rate result for the steepest descent method and minimal error method with noisy data are not known. We provide a convergence rate results for these methods with noisy data. The result in this paper are obtained under less computational cost when compared to the steepest descent method and minimal error method. We present an academic example which satisfies the assumptions of this paper. © 2016 Elsevier Inc.Item Local Convergence of Jarratt-Type Methods with Less Computation of Inversion Under Weak Conditions(Taylor and Francis Ltd., 2017) Argyros, I.K.; George, S.We present a local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Earlier studies cannot be used to solve equations using such methods. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. © 2017, © Vilnius Gediminas Technical University, 2017.