Faculty Publications
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Item 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.Item 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.Item 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 Unified semi-local convergence for k-Step iterative methods with flexible and frozen linear operator(MDPI AG indexing@mdpi.com Postfach Basel CH-4005, 2018) Argyros, I.K.; George, S.The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton's, or Stirling's, or Steffensen's, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis. © 2018 by the authors.Item Local convergence analysis of two competing two-step iterative methods free of derivatives for solving equations and systems of equations(Udruga Matematicara Osijek oml@mathos.hr, 2019) Argyros, I.K.; George, S.We present the local convergence analysis of two-step iterative methods free of derivatives for solving equations and systems of equations under similar hypotheses based on Lipschitz-type conditions. The methods are in particular useful for solving equations or systems involving non-differentiable terms. A comparison is also provided using suitable numerical examples. © 2019 Department of Mathematics, University of Osijek.Item Local convergence for an eighth order method for solving equations and systems of equations(De Gruyter Open Ltd, 2019) Argyros, I.K.; George, S.The aim of this study is to extend the applicability of an eighth convergence order method from the k-dimensional Euclidean space to a Banach space setting. We use hypotheses only on the first derivative to show the local convergence of the method. Earlier studies use hypotheses up to the eighth derivative although only the first derivative and a divided difference of order one appear in the method. Moreover, we provide computable error bounds based on Lipschitz-type functions. © 2019 I.K Argyros and S. George.Item 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.Item 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.Item Derivative Free Iterative Scheme for Monotone Nonlinear Ill-posed Hammerstein-Type Equations(International Association of Engineers, 2021) Erappa, S.M.; George, S.An iterative scheme which is free of derivative is employed to approximately solve nonlinear ill-posed Hammer-stein type operator equations )TG(x) = Y, where G is a non-linear monotone operator and ) is a bounded linear operator defined on Hilbert spaces X,Y,Z. The convergence analysis adapted in the paper includes weaker Lipschitz condition and adaptive choice of Perverzev and Schock(2005) is employed to choose the regularization parameter U. Furthermore, order optimal error bounds are obtained and the method is validated by a numerical example. © 2021, IAENG International Journal of Applied Mathematics. All Rights Reserved.Item EXTENDING THE SOLVABILITY OF EQUATIONS USING SECANT-TYPE METHODS IN BANACH SPACE(Publishing House of the Romanian Academy, 2021) Argyros, I.K.; George, S.We extend the solvability of equations defined on a Banach space using numerically efficient secant-type methods.The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sufficient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational effort, since the new Lipschitz constants are special cases of the old ones. © 2021, Publishing House of the Romanian Academy. All rights reserved.