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 Finite-dimensional realization of lavrentiev regularization for nonlinear III-posed equations(Springer Verlag service@springer.de, 2014) Pareth, S.A finite-dimensional realization of the two-step Newton method is considered for obtaining an approximate solution (reconstructed signals) for the nonlinear ill-posed equation when the available data (noisy signal) is with and the operator F is monotone. We derived an optimal-order error estimate under a general source condition on, where is the initial approximation to the actual solution (signal) The choice of the regularization parameter is made according to the adaptive method considered by Pereverzev and Schock (2005). 2D visualization shows the effectiveness of the proposed method. © 2014 Springer India.Item Iterative regularization methods for ill-posed hammerstein type operator equation with monotone nonlinear part(2010) George, S.; Kunhanandan, M.We considered a procedure for solving an ill-posed Hammerstein type operator equation KF (x) = y, by solving the linear equation Kz = y first for z and then solving the nonlinear equation F (x) = z. Convergence analysis is carried out by means of suitably constructed majorizing sequences. The derived error estimate using an adaptive method proposed by Perverzev and Schock (2005) in relation to the noise level and a stopping rule based on the majorizing sequences are shown to be of optimal order with respect to certain assumptions on F (x?), where x? is the solution of KF (x) = y.Item An iteratively regularized projection method with quadratic convergence for nonlinear Ill-posed problems(2010) George, S.; Elmahdy, A.I.An iteratively regularized projection method, which converges quadratically, has been considered for obtaining stable approximate solution to nonlinear ill-posed operator equations 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 y-y? ? ? are available. Under the assumption that the Fréchet derivative F? 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 between the regularized approximation where Ph is an orthog-onal projection on to a nite dimensional subspace Xh of X) and the solution x? is of optimal order.Item Dynamical system method for ill-posed Hammerstein type operator equations with monotone operators(2012) Erappa, M.E.; George, S.The problem of approximately solving an ill-posed Hammerstein type operator equation KF(x) = y in a Hilbert space is considered, where K is a bounded linear operator and F is a non-linear monotone operator. The method involves the Dynamical System Method (DSM) - both continuous and iterative schemes, studied by Ramm (2005), and known as Tikhonov regularization. By choosing the regularization parameter according to an adaptive scheme considered by Pereverzev and Schock (2005) an order optimal error estimate has been obtained. © 2012 Academic Publications, Ltd.Item Expanding the applicability of a two step Newton-type projection method for ILL-posed problems(Adam Mickiewicz University Press Nowowiejskiego 55 Poznan 61-734, 2014) Argyros, I.K.; Erappa, M.E.; George, S.There are many classes of ill-posed problems that cannot be solved with existing iterative methods, since the usual Lipschitz-type assumptions are not satisfied. In this study, we expand the applicability of a two step Newton-type projection method considered in [10], [11], using weaker assumptions. Numerical examples for the method and examples where the old assumptions are not satisfied but the new assumptions are satisfied are provided at the end of this study.Item Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations(Springer-Verlag Italia s.r.l. springer@springer.it, 2017) Argyros, I.K.; George, S.; Erappa, S.M.For the solution of nonlinear ill-posed problems, a Two Step Newton-Tikhonov methodology is proposed. Two implementations are discussed and applied to nonlinear ill-posed Hammerstein type operator equations KF(x) = y, where K defines the integral operator and F the function of the solution x on which K operates. In the first case, the Fre´ chet derivative of F is invertible in a neighbourhood which includes the initial guess x0 and the solution x^. In the second case, F is monotone. For both cases, local cubic convergence is established and order optimal error bounds are obtained by choosing the regularization parameter according to the the balancing principle of Pereverzev and Schock (2005).We also present the results of computational experiments giving the evidence of the reliability of our approach. © 2016, Springer-Verlag Italia.