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Item Expanding the applicability of a Newton-Lavrentiev regularization method for ill-posed problems(Publishing House of the Romanian Academy Calea 13 Septembrie nr. 13, Sector 5, 050711. P.O. Box 5-42, Bucuresti, 2013) Argyros, I.K.; George, S.We present a semilocal convergence analysis for a simplified Newton-Lavrentiev regularization method for solving ill-posed problems in a Hilbert space setting. We use a center-Lipschitz instead of a Lipschitz condition in our conver-gence analysis. This way we obtain: weaker convergence criteria, tighter error bounds and more precise information on the location of the solution than in earlier studies (such as [13]).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 Extending the applicability of newton's method on riemannian manifolds with values in a cone(2013) Argyros, I.K.; George, S.We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math. 13(2B) (2009) 633-656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton-Kantorovich theorem. © 2013 World Scientific Publishing Company.Item Modification of the kantorovich-type conditions for newton's method involving twice frechet differentiable operators(2013) Argyros, I.K.; George, S.We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal. 11 (2004) 103-119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math. 79 (1997) 131-145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217]). This way, our convergence criteria can be weaker; the error estimates tighter and the information on the location of the solution more precise. Numerical examples are presented to show that our results apply in cases not covered before such as [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal. 11 (2004) 103-119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math. 79 (1997) 131-145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217]. © 2013 World Scientific Publishing Company.Item Expanding the applicability of Lavrentiev regularization methods for ill-posed problems(2013) Argyros, I.K.; Cho, Y.J.; George, S.In this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). © 2013 Argyros et al.; licensee Springer.Item On the semilocal convergence of newton's method for sections on riemannian manifolds(World Scientific Publishing Co. Pte. Ltd. wspc@wspc.com.sg, 2014) Argyros, I.K.; George, S.; Dass, B.K.We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant ?-theory, J. Complexity 24 (2008) 423-451] in combination with the weaker center 2-piece L 1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant ?-theory, J. Complexity 24 (2008) 423-451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal. 22 (2002) 359-390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. 8 (2008) 197-226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant ?-theory, IMA J. Numer. Anal. 23 (2003) 395-419]. © World Scientific Publishing Company.Item Inverse free iterative methods for nonlinear ill-posed operator equations(Hindawi Publishing Corporation 410 Park Avenue, 15th Floor, 287 pmb New York NY 10022, 2014) Argyros, I.K.; George, S.; Padikkal, P.We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equation F (x) = y. The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper. © 2014 Ioannis K. Argyros et al.Item On extended convergence domains for the newton-kantorovich method(Publishing House of the Romanian Academy Calea 13 Septembrie nr. 13, Sector 5, 050711. P.O. Box 5-42, Bucuresti, 2014) Argyros, I.K.; George, S.We present results on extended convergence domains and their applications for the Newton-Kantorovich method (NKM), using the same information as in previous papers. Numerical examples are provided to emphasize that our results can be applied to solve nonlinear equations using (NKM), in contrast with earlier results which are not applicable in these cases. © 2014, Publishing House of the Romanian Academy. All rights reserved.Item Expanding the applicability of a two step Newton-type projection method for ILL-posed problems(Adam Mickiewicz University Press Nowowiejskiego 55 Poznan 61-734, 2014) Argyros, I.K.; Erappa, M.E.; George, S.There are many classes of ill-posed problems that cannot be solved with existing iterative methods, since the usual Lipschitz-type assumptions are not satisfied. In this study, we expand the applicability of a two step Newton-type projection method considered in [10], [11], using weaker assumptions. Numerical examples for the method and examples where the old assumptions are not satisfied but the new assumptions are satisfied are provided at the end of this study.Item On the "terra incognita" for the newton-kantrovich method with applications(2014) Argyros, I.K.; Cho, Y.J.; George, S.In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fréchet-derivative of the operator involved is p-Hölder continuous (p ?(0, 1]). Numerical examples involving two boundary value problems are also provided. © 2014 Korean Mathematical Society.