Faculty Publications
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Item A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations(2012) George, S.; Elmahdy, A.I.In this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y; where the right hand side is replaced by noisy data y? ? X with ?y - y ?? ? ? and T : D(T) ? X ? X is a nonlinear monotone operator defined on a Hilbert space X: The iteration x ?n,? converges quadratically to the unique solution x?? of the equation T(x) + ?(x - x0) = y? (x0 := x 0,??). It is known that (Tautanhahn (2002)) x?? converges to the solution x? of Tx = y: The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. Under a general source condition on x 0 - x? we proved that the error ?x? - x n, ??;? is of optimal order. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate. © 2012 Institute of Mathematics, NAS of Belarus.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 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.