Skip to main content
SpringerLink
Log in

Modified Newton-type processes generating Fejér approximations of regularized solutions to nonlinear equations

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We investigate a two-stage algorithm for the construction of a regularizing algorithm that solves approximately a nonlinear irregular operator equation. First, the initial equation is regularized by a shift (Lavrent’ev’s scheme). To approximate the solution of the regularized equation, we apply modified Newton and Gauss-Newton type methods, in which the derivative of the operator is calculated at a fixed point for all iterations. Convergence theorems for the processes, error estimates, and the Fejér property of iterations are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. B. Bakushinskii, “A regularizing algorithm based on the Newton-Kantorovich method for solving variational inequalities,” USSR Comput. Math. Math. Phys. 16(6), 16–23 (1976).

    Article  MathSciNet  Google Scholar 

  2. A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

  3. A. B. Bakushinskii, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Comput. Math. Math. Phys. 32(9), 1353–1359 (1992).

    MathSciNet  Google Scholar 

  4. M. Hanke, “The regularizing Levenberg-Marquardt scheme is of optimal order,” J. Integral Equations Appls. 22(2), 259–282 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  5. V. V. Vasin, “The Levenberg-Marquardt method for approximation of solutions of irregular operator equations,” Autom. Remote Control 73(3), 440–449 (2012).

    Article  MathSciNet  Google Scholar 

  6. V. V. Vasin, “Irregular nonlinear operator equations: Tikhonov’s regularization and iterative approximation,” J. Inv. Ill-Posed Problems 21(1), 109–123 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  7. S. George, “On convergence of regularized modified Newton’s method for nonlinear ill-posed problems,” J. Inv. Ill-Posed Problems 18(2), 133–146 (2010).

    Article  MATH  Google Scholar 

  8. V. V. Vasin, “Iterative methods for solving ill-posed problems with a priori information in Hilbert spaces,” USSR Comput. Math. Math. Phys. 28(4), 6–13 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  9. V. V. Vasin and A. L. Ageev, Ill-Posed Problems with A Priori Information (Nauka, Yekaterinburg, 1993; VSP, Utrecht, 1995).

    MATH  Google Scholar 

  10. V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type: Theory and Applications (Regulyarn. Khaotich. Dinamika, Izhevsk, 2005; Gruyter, Berlin, 2009).

    Google Scholar 

  11. B. Martinet, “Determination approachee d’un point fixe d’une application pseudo-contractante. Cas de l’application prox,” C. R. Acad. Sci. Paris, Ser. A-B 274, A163–A165 (1972).

    MathSciNet  Google Scholar 

  12. I. I. Eremin, “Methods of Fejér’s approximations in convex programming,” Math. Notes 3(2), 143–149 (1968).

    Article  MathSciNet  Google Scholar 

  13. I. I. Eremin, “On the general theory of Fejér mappings,” Mat. Zap. Ural. Gos. Univ. 7(2), 50–58 (1969).

    MATH  MathSciNet  Google Scholar 

  14. I. I. Eremin, “The application of the method of Fejér approximations to the solution of problems of convex programming with non-smooth constraints,” USSR Comput. Math. Math. Phys. 9(5), 225–235 (1969).

    Article  MathSciNet  Google Scholar 

  15. E. M. Mukhamadiev, V. A. Morozov, and A. B. Nazimov, “On the regularization of singular systems of linear algebraic equations by shifts,” Dokl. Math. 77(2), 254–257 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  16. E. M. Mukhamadiev, V. A. Morozov, and A. B. Nazimov, “Regularization of singular systems of linear algebraic equations by shifts,” Comput. Math. Math. Phys. 47(12), 1885–1892 (2007).

    Article  MathSciNet  Google Scholar 

  17. A. B. Nazimov, E. M. Mukhamadiev, V. A. Morozov, and M. Mullodzhanov, The Method of Regularization by Shifts. Theory and Appllications (Izd. Vologodsk. Gos. Tekhn. Univ., Vologda, 2012) [in Russian].

    Google Scholar 

  18. A. Kufner and S. Fučik, Nonlinear Differential Equations (Elsevier, New York, 1980; Nauka, Moscow, 1988).

    MATH  Google Scholar 

  19. L. Fejér, “Über die Lag der Nullstellen von Polynomen, die aus Minimumforderung gewisser Art entspringen,” Math. Ann. 85(1), 41–48 (1992)

    Article  Google Scholar 

  20. T. S. Motzkin and J. J. Schoenberg, “The relaxation method for linear inequalities,” Canad. J. Math. 6(3), 393–404 (1954).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    V. V. Vasin

  2. Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    V. V. Vasin

Authors
  1. V. V. Vasin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Vasin.

Additional information

Dedicated to I.I. Eremin’s 80th birthday.

Original Russian Text © V.V. Vasin, 2013, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vasin, V.V. Modified Newton-type processes generating Fejér approximations of regularized solutions to nonlinear equations. Proc. Steklov Inst. Math. 284 (Suppl 1), 145–158 (2014). https://doi.org/10.1134/S0081543814020138

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543814020138

Keywords

Search

Navigation