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    Unified semi-local convergence for k-Step iterative methods with flexible and frozen linear operator
    (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.
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    Unified convergence analysis of frozen Newton-like methods under generalized conditions
    (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.
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    Unified Convergence for Multi-Point Super Halley-Type Methods with Parameters in Banach Space
    (2019) Argyros, I.K.; George, S.
    We present a local convergence analysis of a multi-point super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier works was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the Super-Halley-like method by using hypotheses only on the first derivative. We also provide: A computable error on the distances involved and a uniqueness result based on Lipschitz constants. The convergence order is also provided for these methods. Numerical examples are also presented in this study. � 2019, Indian National Science Academy.
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    Unified convergence domains of Newton-like methods for solving operator equations
    (2016) Argyros, I.K.; George, S.
    We present a unified semilocal convergence analysis in order to approximate a locally unique zero of an operator equation in a Banach space setting. Using our new idea of restricted convergence domains we generate smaller Lipschitz constants than in earlier studies leading to the following advantages: weaker sufficient convergence criteria, tighter error estimates on the distances involved and an at least as precise information on the location of the zero. Hence, the applicability of these methods is extended. These advantages are obtained under the same cost on the parameters involved. Numerical examples where the old sufficient convergence criteria cannot apply to solve equations but the new criteria can apply are also provided in this study. � 2016 Elsevier Inc. All rights reserved.
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    Third-order derivative-free methods in Banach spaces for nonlinear ill-posed equations
    (2019) Shubha, V.S.; George, S.; Jidesh, P.
    We develop three third order derivative-free iterative methods to solve the nonlinear ill-posed oprerator equation F(x) = f approximately. The methods involve two steps and are free of derivatives. Convergence analysis shows that these methods converge cubically. The adaptive scheme introduced in Pereverzyev and Schock (SIAM J Numer Anal 43(5):2060 2076, 2005) has been employed to choose regularization parameter. These methods are applied to the inverse gravimetry problem to validate our developed results. 2019, Korean Society for Computational and Applied Mathematics.
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    The asymptotic mesh independence principle of Newton's method under weaker conditions
    (2016) Argyros, I.K.; Sheth, S.M.; Younis, R.M.; George, S.
    We present a new asymptotic mesh independence principle of Newton's method for discretized nonlinear operator equations. Our hypotheses are weaker than in earlier studies such as [1], [8]-[12]. This way we extend the applicability of the mesh independence principle which asserts that the behavior of the discretized version is asymptotically the same as that of the original iteration and consequently, the number of steps required by the two processes to converge within a given tolerance is essentially the same. The results apply to solve a boundary value problem that cannot be solved with the earlier hypotheses given in [12]. 2016 International Publications. All rights reserved.
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    Shock coupled fourth-order diffusion for image enhancement
    (2012) Jidesh, P.; George, S.
    In this paper a shock coupled fourth-order diffusion filter is proposed for image enhancement. This filter converges at a faster rate while preserving and enhancing edges, ramps and textures present in the images. The proposed filter diffuses with varying magnitudes in the directions normal to the level-curve and along it. The magnitude of the directional diffusion is controlled by a diffusion function, meant to provide a good response in the direction along the level-curves, than across them. The proposed filter can still preserve the planar approximation of the image, thereby avoiding the discrepancy caused due to the staircase effect, as in the second-order counterparts. The anisotropic property of the filter is thoroughly studied, analyzed and demonstrated with perspective and quantitative results. The performance of the proposed filter is compared with the state-of-the-art methods for image enhancement. The quantitative and perspective measures provided endorse the capability of the method to enhance various kinds of images. 2012 Elsevier Ltd. All rights reserved.
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    Reconstruction of signals by standard Tikhonov method
    (2011) George, S.; Jidesh, P.
    In this work we propose a standard Tikhonov regularization approach for obtaining the signal f from the observed signal ye. The observed signal is distorted by an additive noise or error e. Deviating from the usual assumption on the bound on
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    Proximal methods with invexity and fractional calculus
    (2017) Anastassiou, G.A.; Argyros, I.K.; George, S.
    We present some proximal methods with invexity results involving fractional calculus.
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    Projection method for newton-tikhonov regularization for non-linear ill-posed hammerstein type operator equations
    (2013) Shobha, M.E.; George, S.
    An iteratively regularized projection scheme for the ill-posed Hammerstein type operator equation KF(x) = f has been considered. Here F : D(F)X X is a non-linear operator, K : X ? Y is a bounded linear operator, X and Y are Hilbert spaces. The method is a combination of dis- cretized Tikhonov regularization and modified Newton's method. It is assumed that the F?echet derivative of F at x0 is invertible i.e., the ill-posedness of the operator KF is due to the ill-posedness of the linear operator K. Here x0 is an initial approximation to the solution x of KF(x) = f. Adaptive choice of the parameter suggested by Perverzev and Schock(2005) is employed in select- ing the regularization parameter ?. A numerical example is given to test the reliability of the method. 2013 Academic Publications, Ltd.