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Author: search.filters.author.Pareth, S.Subject: search.filters.subject.balancing principle

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    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(x̂) = 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-x̂ (here x 0 is the initial approximation). The test example provided endorses the reliability and effectiveness of our method. © 2012 Springer-Verlag.
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    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.