Faculty Publications
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Item On convergence of regularized modified Newton's method for nonlinear ill-posed problems(Walter de Gruyter GmbH and Co. KG, 2010) George, S.In this paper we consider regularized modified Newton's method for approximately solving the nonlinear ill-posed problem F(x) = y, where the right hand side is replaced by noisy data y?? Y with y - y?? ? and F : D(F) ? X ? Y is a nonlinear operator between Hilbert spaces X and Y. Under the assumption that Fréchet derivative F? of F is Lipschitz continuous, a choice of the regularization parameter and a stopping rule based on a majorizing sequence are presented. We prove that under a general source condition on , the error between the regularized approximation and the solution of optimal order. © de Gruyter 2010.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 Newton type iteration for Tikhonov regularization of non-linear ill-posed Hammerstein type equations(2014) George, S.; Erappa, M.E.An iterative method is investigated for a nonlinear ill-posed Hammerstein type operator equation KF(x)=f. We use a center-type Lipschitz condition in our convergence analysis instead of the usual Lipschitz condition. The adaptive method of Pereverzev and Schock (SIAM J. Numer. Anal. 43(5):2060-2076, 2005) is used for choosing the regularization parameter. The optimality of this method is proved under a general source condition involving the Fréchet derivative of F at some initial guess x 0. A numerical example of nonlinear integral equation shows the efficiency of this procedure. © 2013 Korean Society for Computational and Applied Mathematics.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 Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method(Elsevier B.V., 2018) George, S.; Sabari, M.We present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill-posed operator equation F(x)=y. The proposed method is a modified form of the method considered by Argyros et al. (2014). The balancing principle considered by Pereverzev and Schock (2005) is used for choosing the regularization parameter. The error estimate is derived under a general source condition and is of optimal order. The provided numerical example proves the efficiency of the proposed method. © 2017 Elsevier B.V.Item Modified Minimal Error Method for Nonlinear Ill-Posed Problems(Walter de Gruyter GmbH cmam@cmam.info, 2018) Sabari, M.; George, S.An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate. © 2018 Walter de Gruyter GmbH, Berlin/Boston.Item Extended Newton-type iteration for nonlinear ill-posed equations in Banach space(Springer Verlag service@springer.de, 2019) Sreedeep, C.D.; George, S.; Argyros, I.K.In this paper, we study nonlinear ill-posed equations involving m-accretive mappings in Banach spaces. We produce an extended Newton-type iterative scheme that converges cubically to the solution which uses assumptions only on the first Fréchet derivative of the operator. Using general Hölder type source condition we obtain an error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) for choosing the regularization parameter. © 2018, Korean Society for Computational and Applied Mathematics.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 regularization method for nonlinear ill-posed equations in Hilbert scales(De Gruyter Open Ltd, 2019) George, S.; Kanagaraj, K.In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space. © 2019 De Gruyter. All rights reserved.