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 An application of newton type iterative method for lavrentiev regularization for ill-posed equations: Finite dimensional realization(2012) George, S.; Pareth, S.In this paper, we consider, a finite dimensional realization of Newton type iterative method for Lavrentiev regularization of ill-posed equations. Precisely we consider the ill-posed equation F(x) = f when the available data is f ? withItem Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scales(2013) George, S.; Pareth, S.; Kunhanandan, M.In this paper we consider the two step method for approximately solving the ill-posed operator equation F(x)=f, where F:D(F) ⊆X?X, is a nonlinear monotone operator defined on a real Hilbert space X, in the setting of Hilbert scales. We derive the error estimates by selecting the regularization parameter ? according to the adaptive method considered by Pereverzev and Schock in (2005), when the available data is f? with ?-f-f??- ??. The error estimate obtained in the setting of Hilbert scales { Xr}r?R generated by a densely defined, linear, unbounded, strictly positive self adjoint operator L:D(L)X?X is of optimal order. © 2013 Elsevier Inc. All rights reserved.Item On improving the semilocal convergence of newton-type iterative method for ill-posed Hammerstein type operator equations(2013) Erappa, M.E.; George, S.George and Pareth( 2012), presented a quartically convergent Two Step Newton type method for approximately solving an ill-posed operator equation in the finite dimensional setting of Hilbert spaces. In this paper we use the analogous Two Step Newton type method to approximate a solution of ill-posed Hammerstein type operator equation.Item Finite dimensional realization of a quadratic convergence yielding iterative regularization method for ill-posed equations with monotone operators(Elsevier Inc. usjcs@elsevier.com, 2016) Shubha, V.S.; George, S.; Padikkal, P.; Erappa, M.E.Recently Jidesh et al. (2015), considered a quadratic convergence yielding iterative method for obtaining approximate solution to nonlinear ill-posed operator equation F(x)=y, where F: D(F) ? X ? X is a monotone operator and X is a real Hilbert space. In this paper we consider the finite dimensional realization of the method considered in Jidesh et al. (2015). Numerical example justifies our theoretical results. © 2015 Elsevier Inc. All rights reserved.Item A derivative-free iterative method for nonlinear ill-posed equations with monotone operators(Walter de Gruyter GmbH info@degruyter.com, 2017) George, S.; Nair, M.T.Recently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova's method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11]. © 2017 Walter de Gruyter GmbH, Berlin/Boston.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.Item Expanding the applicability of an iterative regularization method for ill-posed problems(Biemdas Academic Publishers, 2019) Argyros, I.K.; George, S.An iteratively regularized projection method, which converges quadratically, is considered for stable approximate solutions to a nonlinear ill-posed operator equation F(x) = y, where F : D(F) ? X ? X is a nonlinear monotone operator defined on the real Hilbert space X. We assume that only a noisy data y? with ky? y? k ? ? are available. Under the assumption that the Fréchet derivative F0 of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that, under a general source condition on x0 ? x, the error kxn h ? ? ? xk between the regularized approximation xn h ? ? , (x0 h ? ? := Phx0, where Ph is an orthogonal projection on to a finite dimensional subspace Xh of X) and the solution x is of optimal order. © 2019 Journal of Nonlinear and Variational AnalysisItem Derivative Free Iterative Scheme for Monotone Nonlinear Ill-posed Hammerstein-Type Equations(International Association of Engineers, 2021) Erappa, S.M.; George, S.An iterative scheme which is free of derivative is employed to approximately solve nonlinear ill-posed Hammer-stein type operator equations )TG(x) = Y, where G is a non-linear monotone operator and ) is a bounded linear operator defined on Hilbert spaces X,Y,Z. The convergence analysis adapted in the paper includes weaker Lipschitz condition and adaptive choice of Perverzev and Schock(2005) is employed to choose the regularization parameter U. Furthermore, order optimal error bounds are obtained and the method is validated by a numerical example. © 2021, IAENG International Journal of Applied Mathematics. All Rights Reserved.