Faculty Publications

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Subject: search.filters.subject.Discrepancy principle

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    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.
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    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.
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    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.
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    Discrepancy principles for fractional Tikhonov regularization method leading to optimal convergence rates
    (Springer, 2020) Kanagaraj, K.; Reddy, G.D.; George, S.
    Fractional Tikhonov regularization (FTR) method was studied in the last few years for approximately solving ill-posed problems. In this study we consider the Schock-type discrepancy principle for choosing the regularization parameter in FTR and obtained the order optimal convergence rate. Numerical examples are provided in this study. © 2019, Korean Society for Informatics and Computational Applied Mathematics.
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    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.
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    Finite dimensional realization of the FTR method with Raus and Gfrerer type discrepancy principle
    (Springer-Verlag Italia s.r.l., 2023) George, S.; Padikkal, J.; Krishnendu, R.
    It is known that the standard Tikhonov regularization methods oversmoothen the solution x^ of the ill-posed equation T(x) = y, so the computed approximate solution lacks many inherent details that are expected in the desired solution. To rectify this problem, Fractional Tikhonov Regularization (FTR) method have been introduced. Kanagaraj et al. (J Appl Math Comput 63(1):87–105, 2020), studied FTR method for solving ill-posed problems. Techniques are developed to study the Finite Dimensional FTR (FDFTR) method. We also study Raus and Gfrerer type discrepancy principle for FDFTR method and compare the numerical results with other discrepancy principles of the same type. © 2023, The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature.