Faculty Publications
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Item Projection scheme for newton-type iterative method for Lavrentiev regularization(2012) Pareth, S.; George, S.In this paper we consider the finite dimensional realization of a Newton-type iterative method for obtaining an approximate solution to the nonlinear 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. It is assumed that F(xÌ‚) = f and that the only available data are f δ with ∥f - f δ∥ ≤ δ. It is proved that the proposed method has a local convergence of order three. The regularization parameter α is chosen according to the balancing principle considered by Perverzev and Schock (2005) and obtained an optimal order error bounds under a general source condition on x 0-xÌ‚ (here x 0 is the initial approximation). The test example provided endorses the reliability and effectiveness of our 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 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 An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization(2013) George, S.; Pareth, S.Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero x* of a differentiable function F defined on a convex set D of a Hilbert space H, we consider a two-step Newton-Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear 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. It is assumed that F(x) = f and that the only available data are f? with ¶f - f? ¶= ? ?. We prove that the TSNLM converges cubically to a solution of the equation F(x)+?(x-xo) = f? (such solution is an approximation of O x) where x0 is the initial guess. Under a general source condition on x0- x?, we derive order optimal error bounds by choosing the regularization parameter ? according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method. © 2013 by Walter de Gruyter Berlin Boston.