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Convergence rates and finite-dimensional approximation for a class of ill-posed variational inequalities

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Abstract

The purpose of this paper is to investigate an operator version of Tikhonov regularization for a class of ill-posed variational inequalities under arbitrary perturbation operators. Aspects of convergence rate and finite-dimensional approximations are considered. An example in the theory of generalized eigenvectors is given for illustration.

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References

  1. J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Wunod Gauthier-Villars, Paris (1969).

    MATH  Google Scholar 

  2. M. Sibony, “Sur l’approximation d’équation et inéquations aux derivées partieles nonlinéaires de type monotone,” J. Math. Anal. Appl., 34, 502–564 (1992).

    Article  MathSciNet  Google Scholar 

  3. M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations (in Russian), Nauka, Moscow (1972).

    Google Scholar 

  4. G. Bruckner, “On the speed of c 0-sequences in connection with a lemma of Toeplitz,” Abh. Akad. Wiss. DDR, Nonlinear Anal.: Theory Appl., No. 2, 53–76 (1981).

    MathSciNet  Google Scholar 

  5. K. Kluge, “Optimal control with minimum problems and variational inequalities,” Lect. Notes Comput. Sci., 27, 377–382 (1975).

    Google Scholar 

  6. K. Kluge, “Approximation methods for nonlinear problems with constrain in form of variational inequalities,” Banach Center Publ., 1, 131–138 (1976).

    Google Scholar 

  7. I. P. Ryazantseva, “Operator method for regularization of problems of optimal programming with monotone maps,” Sib. Mat. Zh., 24, No. 6, 214 (1983).

    Google Scholar 

  8. I. J. Alber, “On the solution of nonlinear equations involving monotone operators in Banach spaces,” Sib. Mat. Zh., 26, 3–11 (1975).

    Google Scholar 

  9. I. P. Ryazantseva, “On the choice of a regularization parameter for equations under monotone perturbations,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 8, 39–43 (1981).

    MathSciNet  Google Scholar 

  10. O. A. Liskovets, “Regularization for problems involving discontinuous monotone operators under arbitrary perturbations,” Dokl. Akad. Nauk SSSR, 272, No. 1, 30–34 (1983).

    MathSciNet  Google Scholar 

  11. I. J. Alber and A. I. Notik, “Geometrical properties of Banach spaces and approximate methods for the solution of nonlinear operator equations,” Dokl. Akad. Nauk SSSR, 276, 1033–1037 (1984).

    MathSciNet  Google Scholar 

  12. A. Neubauer, “An a posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates,” SIAM J. Numer. Math., 25, 1313–1326 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  13. H. W. Engl and C. W. Groetsch, “Projection-regularization for linear operator equations of the first kind,” in: Special Programs on Inverse Problems. Proc. Center Math. Anal. Australian Nat. Univ. (1988), pp. 17–31.

  14. H. W. Engl, K. Kunisch, and A. Neubauer, “Convergence rates for Tikhonov regularization of nonlinear ill-posed problems,” Inverse Probl., 5, 523–540 (1989).

    Article  MATH  MathSciNet  Google Scholar 

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Institute of Information Technology, Vietnam. Published in Ukrainskii Matematicheskii Zhunal, Vol. 49, No. 5, pp. 629–637, May, 1997.

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Buong, N. Convergence rates and finite-dimensional approximation for a class of ill-posed variational inequalities. Ukr Math J 49, 697–707 (1997). https://doi.org/10.1007/BF02486451

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  • DOI: https://doi.org/10.1007/BF02486451

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