Journal Articles
Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/123456789/19884
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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.