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An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization

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Języki publikacji
EN
Abstrakty
EN
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 fδ with ||f- fδ||≤δ. We prove that the TSNLM converges cubically to a solution of the equation F(x)+α(x-x0)= fδ (such solution is an approximation of 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 α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.
Wydawca
Rocznik
Strony
181--196
Opis fizyczny
Bibliogr. 20 poz.
Twórcy
autor
  • Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, 575 025, India
autor
  • Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, 575 025, India
Bibliografia
  • [1] I. K. Argyros and S. Hilout, A convergence analysis for directional two-step Newton methods, Numer. Algor. 55 (2010), 503-528.
  • [2] B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal. 17 (1997), 421-436.
  • [3] P. Deuflhard, H. W. Engl and O. Scherzer, A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse Probl. 14 (1998), 1081-1106.
  • [4] S. George, Newton-Lavrentiev regularization of ill-posed Hammerstein type operator equation, J. Inverse Ill-Posed Probl. 14 (2006), no. 6, 573-582.
  • [5] S. George and A. I. Elmahdy, A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations, Comput. Methods Appl. Math. 12 (2012), no. 1, 32-45.
  • [6] S. George and M. T. Nair, A modified Newton-Lavrentiev regularization for nonlinear ill-posed Hammerstein-Type operator equation, J. Complexity 1A (2008), 228-240.
  • [7] C. W. Groetsch, J. T. King and D. Murio, Asymptotic analysis of a finite element method for Fredholm equations of the first kind, in: Treatment of integral Equations by Numerical Methods, Academic Press, London (1982), 1-11.
  • [8] J. Jaan and U. Tautenhahn, On Lavrentiev regularization for ill-posed problems in Hilbert scales, Numer. Funct. Anal. Optim. 2A (2003), no. 5-6, 531-555.
  • [9] Q. Jin, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales, Inverse Probl. 16 (2000), 187-197.
  • [10] Q. Jin, On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems, Math. Comp. 69 (2000), no. 232, 1603-1623.
  • [11] B. Kaltenbacher, A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems, Numer. Math. 79 (1998), 501-528.
  • [12] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadel-phia, 1995.
  • [13] P. Mahale and M. T. Nair, Iterated Lavrentiev regularization for nonlinear ill-posed problems, ANZIAM J. 51 (2003), 191-217.
  • [14] P. Mathe and S. V. Perverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inrerse Probl. 19 (2003), no. 3, 789-803.
  • [15] M. T. Nair and P. Ravishankar, Regularized versions of continuous Newton’s method and continuous modified Newton’s method under general source conditions, Numer. Funct. Anal. Optim. 29 (2008), no. 9-10, 1140-1165.
  • [16] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in General Variables, Academic Press, New York, 1970.
  • [17] S. V. Perverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), 2060-2076.
  • [18] A. G. Ramm, Dynamical System Method for Sohing Operator Equations, Elsevier, Amsterdam, 2007.
  • [19] E. V. Semenova, Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators, Comput. Methods Appl. Math. 10 (2010), no. 4, 444-454.
  • [20] U. Tautanhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Probl. 18 (2002), 191-207.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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