Faculty Publications
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Item Finite dimensional realization of a Guass-Newton method for ill-posed hammerstein type operator equations(2012) Erappa, M.E.; George, S.Finite dimensional realization of an iterative regularization method for approximately solving the non-linear ill-posed Hammerstein type operator equations KF(x) = f, is considered. The proposed method is a combination of the Tikhonov regularization and Guass-Newton method. The advantage of the proposed method is that, we use the Fr chet derivative of F only at one point in each iteration. We derive the error estimate under a general source condition and the regularization parameter is chosen according to balancing principle of Pereverzev and Schock (2005). The derived error estimate is of optimal order and the numerical example provided proves the efficiency of the proposed method. © 2012 Springer-Verlag.Item Some aspects of location identification of PD source using AE signals by an iterative method(2012) Punekar, G.S.; Jadhav, P.; Bhavani, S.T.; Nagamani, H.N.An acoustic Partial Discharge (PD) location problem modeled mathematically, gives system of sphere equations, which are non-linear. These equations are formed with known acoustic emission (AE) sensors co-ordinates, with PD locations co-ordinates as unknowns. Newton's method is implemented to locate the PD activity using the AE signals. This is an iterative method and the convergence depends on the initial guess. Different aspects such as initial guess, location of sensors (sensor co-ordinates) and tank orientation in space are studied in this paper by numerical experiments on the algorithm implemented using the experimental data published (available) in a literature. The published data considered for the study here uses 8 number of sensors (4 on the front and 4 on the back wall of the transformer tank; laboratory model). The method of locating acoustic emission partial discharge (AEPD) requires at most 4 sensors (three to identify the coordinates of the location and one for arrival time of AE signal). Hence, results of such 70 combinations (i.e. 8C4) are studied using the algorithm implemented. The numerical test runs indicate that some combinations either do not lead to convergence or yield results with high errors. At least such 10 combinations (out of 70) are identified and analyzed. © 2012 IEEE.Item Improvements in AEPD location identification by removing outliers and post processing(Institute of Electrical and Electronics Engineers Inc., 2016) Antony, D.; Punekar, G.S.The mathematical model of an Acoustic Emission Partial Discharge (AEPD) system is solved in the literature using Newton's method with redundant number of sensors (more than 4; eight in this case). The system for numerical experiments consists of eight sensors. The algorithm is implemented using three different initial guesses. For the calculated PD source coordinates, histograms are plotted. After finding the mean and standard deviation, coordinate values which are lying outside different fractions of sigma are removed. The average of remaining set is calculated and it is found that, the accuracy of location identification can be greatly improved. © 2015 IEEE.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 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 A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales(Rocky Mountain Mathematics Consortium PO Box 871904 Tempe AZ 85287-1804, 2014) Erappa, M.E.; George, S.; Kunhanandan, M.In this paper regularized solutions of ill-posed Hammerstein type operator equation KF(x) = y, where K : X ? Y is a bounded linear operator with non-closed range and F : X ? X is non-linear, are obtained by a two step Newton type iterative method in Hilbert scales, where the available data is y? in place of actual data y withItem 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 Ball convergence for a Newton-steffensen-type third-order method(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for a composite Newton-Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [1], [5]-[28] using hypotheses up to the second derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.Item Ball convergence of some fourth and sixth-order iterative methods(World Scientific Publishing Co. Pte Ltd wspc@wspc.com.sg, 2016) Argyros, I.K.; George, S.We present a local convergence analysis for some families of fourth and sixth-order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies [V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990) 355-367; C. Chun, P. Stanica and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011) 1665-1675; J. M. Gutiérrez and M. A. Hernández, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998) 1-8; M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000) 131-143; M. A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433-455; M. A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl. 104(3) (2000) 501-515; J. L. Hueso, E. Martinez and C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth-order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015) 412-420; Á. A. Magre nán, Estudio de la dinámica del método de Newton amortiguado, Ph.D. Thesis, Servicio de Publicaciones, Universidad de La Rioja (2013), http://dialnet.unirioja.es/servlet/tesis?codigo=38821; J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970); M. S. Petkovic, B. Neta, L. Petkovic and J. Džunic, Multi-Point Methods for Solving Nonlinear Equations (Elsevier, 2013); J. F. Traub, Iterative Methods for the Solution of Equations, Automatic Computation (Prentice-Hall, Englewood Cliffs, NJ, 1964); X. Wang and J. Kou, Semilocal convergence and R-order for modified Chebyshev-Halley methods, Numer. Algorithms 64(1) (2013) 105-126] have used hypotheses on the fourth Fréchet derivative of the operator involved. We use hypotheses only on the first Fréchet derivative in our local convergence analysis. This way, the applicability of these methods is extended. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples illustrating the theoretical results are also presented in this study. © 2016 World Scientific Publishing Company.