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Acta Applicandae Mathematicae

, Volume 106, Issue 3, pp 473–499 | Cite as

Dynamical Systems Gradient Method for Solving Nonlinear Equations with Monotone Operators

  • N. S. Hoang
  • A. G. RammEmail author
Article

Abstract

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.

Keywords

Dynamical systems method (DSM) Nonlinear operator equations Monotone operators Discrepancy principle 

Mathematics Subject Classification (2000)

47J05 47J06 47J35 65R30 

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References

  1. 1.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) zbMATHGoogle Scholar
  2. 2.
    Hoang, N.S., Ramm, A.G.: Solving ill-conditioned linear algebraic systems by the dynamical systems method. Inverse Probl. Sci. Eng. 16(N5), 617–630 (2008) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Hoang, N.S., Ramm, A.G.: An iterative scheme for solving nonlinear equations with monotone operators (2008, submitted) Google Scholar
  4. 4.
    Ivanov, V., Tanana, V., Vasin, V.: Theory of Ill-posed Problems. VSP, Utrecht (2002) zbMATHGoogle Scholar
  5. 5.
    Lions, J.L.: Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Dunod, Gauthier-Villars, Paris (1969) zbMATHGoogle Scholar
  6. 6.
    Morozov, V.A.: Methods of Solving Incorrectly Posed Problems. Springer, New York (1984) Google Scholar
  7. 7.
    Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Noordhoff, Leyden (1978) zbMATHGoogle Scholar
  8. 8.
    Ramm, A.G.: Theory and Applications of Some New Classes of Integral Equations. Springer, New York (1980) zbMATHGoogle Scholar
  9. 9.
    Ramm, A.G.: Stationary regimes in passive nonlinear networks. In: Uslenghi, P. (ed.) Nonlinear Electromagnetics, pp. 263–302. Acad. Press, New York (1980) Google Scholar
  10. 10.
    Ramm, A.G.: Dynamical Systems Method for Solving Operator Equations. Elsevier, Amsterdam (2007) zbMATHGoogle Scholar
  11. 11.
    Ramm, A.G.: Global convergence for ill-posed equations with monotone operators: the dynamical systems method. J. Phys. A 36, L249–L254 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ramm, A.G.: Dynamical systems method for solving nonlinear operator equations. Int. J. Appl. Math. Sci. 1(N1), 97–110 (2004) zbMATHGoogle Scholar
  13. 13.
    Ramm, A.G.: Dynamical systems method for solving operator equations. Commun. Nonlinear Sci. Numer. Simul. 9(N2), 383–402 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ramm, A.G.: DSM for ill-posed equations with monotone operators. Commun. Nonlinear Sci. Numer. Simul. 10(N8), 935–940 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ramm, A.G.: Discrepancy principle for the dynamical systems method. I, II. Commun. Nonlinear Sci. Numer. Simul. 10, 95–101 (2005); 13, 1256–1263 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ramm, A.G.: Dynamical systems method (DSM) and nonlinear problems. In: Lopez-Gomez, J. (ed.) Spectral Theory and Nonlinear Analysis, pp. 201–228. World Scientific, Singapore (2005) Google Scholar
  17. 17.
    Ramm, A.G.: Dynamical systems method (DSM) for unbounded operators. Proc. Am. Math. Soc. 134(N4), 1059–1063 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ramm, A.G.: Random Fields Estimation. World Sci., Singapore (2005) zbMATHGoogle Scholar
  19. 19.
    Ramm, A.G.: Iterative solution of linear equations with unbounded operators. J. Math. Anal. Appl. 1338–1346 Google Scholar
  20. 20.
    Ramm, A.G.: On unbounded operators and applications. Appl. Math. Lett. 21, 377–382 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. American Mathematical Society, Providence (1994) zbMATHGoogle Scholar
  22. 22.
    Tautenhahn, U.: On the asymptotical regularization method for nonlinear ill-posed problems. Inverse Probl. 10, 1405–1418 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Tautenhahn, U.: On the method of Lavrentiev regularization for nonlinear ill-posed problems. Inverse Probl. 18, 191–207 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Vainberg, M.M.: Variational Methods and Method of Monotone Operators in the Theory of Nonlinear Equations. Wiley, London (1973) Google Scholar
  25. 25.
    Zeidler, E.: Nonlinear Functional Analysis. Springer, New York (1985) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentKansas State UniversityManhattanUSA

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