Faculty Publications
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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 "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.Item Local convergence for some third-order iterative methods under weak conditions(Korean Mathematical Society kms@kms.or.kr, 2016) Argyros, I.K.; Cho, Y.J.; George, S.The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot. © 2016 Korean Mathematical Society.Item Extended convergence of gauss-newton’s method and uniqueness of the solution(SINUS Association Office_CJEES@yahoo.ro, 2018) Argyros, I.K.; Cho, Y.J.; George, S.The aim of this paper is to extend the applicability of the Gauss-Newton’s method for solving nonlinear least squares problems using our new idea of restricted convergence domains. The new technique uses tighter Lipschitz functions than in earlier papers leading to a tighter ball convergence analysis. © 2018, SINUS Association. All rights reserved.Item Iterative Roots of Non-PM Functions and Denseness(Birkhauser Verlag AG, 2018) Cho, Y.J.; Suresh Kumar, S.K.; Murugan, M.The characteristic interval plays a vital role on the existence of iterative roots of PM functions with height less than or equal to one. In this paper, we define the characteristic interval for continuous functions and prove theorems on extension and nonexistence of iterative roots for a class of continuous non-PM functions on a closed and bounded interval I. Also, we prove that a class of continuous non-PM functions, which do not possess any iterative roots, is dense in C(I, I). © 2018, Springer International Publishing AG, part of Springer Nature.Item Improved local convergence analysis for a three point method of convergence order 1.839…(Korean Mathematical Society kms@kms.or.kr, 2019) Argyros, I.K.; Cho, Y.J.; George, S.In this paper, we present a local convergence analysis of a three point method with convergence order 1.839… for approximating a locally unique solution of a nonlinear operator equation in setting of Banach spaces. Using weaker hypotheses than in earlier studies, we obtain: larger radius of convergence and more precise error estimates on the distances involved. Finally, numerical examples are used to show the advantages of the main results over earlier results. © 2019 Korean Mathematical Society.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 Local Convergence of Inexact Newton-Like Method under Weak Lipschitz Conditions(Springer, 2020) Argyros, I.K.; Cho, Y.J.; George, S.; Xiao, Y.The paper develops the local convergence of Inexact Newton-Like Method (INLM) for approximating solutions of nonlinear equations in Banach space setting. We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis. The obtained results compare favorably with earlier ones such as [7, 13, 14, 18, 19]. A numerical example is also provided. © 2020, Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences.Item Extending the Convexity of Nonlinear Image of a Ball Appearing in Optimization(Universal Wiser Publisher, 2020) Argyros, I.K.; Cho, Y.J.; George, S.Let X, Y be Hilbert spaces and F: X ? Y be Fréchet differentiable. Suppose that F? is center-Lipschitz on U(w, r) and F?(w) be a surjection. Then, S1 = F(U(w, ?1)) is convex where ?1 ? r. The set S1 contains the corresponding set given in [18] under the Lipschitz condition. Numerical examples where the old conditions are not satisfied but the new conditions are satisfied are provided in this paper. © 2020 Ioannis K. Argyros, et al. DOI: https://doi.o.Item Projection method for Fractional Lavrentiev Regularisation method in Hilbert scales(Springer Science and Business Media B.V., 2022) Mekoth, C.; George, S.; Padikkal, P.; Cho, Y.J.We study finite dimensional Fractional Lavrentiev Regularization (FLR) method for linear ill-posed operator equations in the Hilbert scales. We obtain an optimal order error estimate under Hölder type source condition and under a parameter choice strategy. Numerical experiments confirming the theoretical results are also given. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.