Tikhonov Regularization of Nonlinear Differential-Algebraic Equations

  • H. W. Engl
  • M. Hanke
  • A. Neubauer
Part of the Inverse Problems and Theoretical Imaging book series (IPTI)

Abstract

Using recent results about Tikhonov regularization of nonlinear ill-posed problems, we show that nonlinear index-2 differential-algebraic equations can be solved in a stable way via Tikhonov regularization in various ways. Convergence rates for the regularized solutions are derived.

Keywords

Assure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Adams, Sobolev Spaces; Academic Press, New York 1975MATHGoogle Scholar
  2. [2]
    K. E. Brenan, L. R. Engquist, Backward differentiation approximations of nonlinear differential algebraic systems; Math. Comp. 51 (1988), pp. 659–676CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    H. W. Engl, C. W. Groetsch (eds.), Inverse and Ill-Posed Problems, Academic Press, Orlando 1987MATHGoogle Scholar
  4. [4]
    H. W. Engl, K. Klinisch, A. Neubauer, Convergence rates for Tikhonov regularization of non-linear ill-posed problems; Inverse Problems 5 (1989), pp. 523–540MATHGoogle Scholar
  5. [5]
    C. W. Gear, Maintaining solution invariants in the numerical solution of ODE’s, SIAM J. Sci. Comput. 7 (1986), pp. 734–743MATHMathSciNetGoogle Scholar
  6. [6]
    C. W. Gear, G. K. Gupta, B. Leimkuhler, Automatic integration of Euler-Lagrange equations with constraints; J. Comp. Appl. Math. 12 and 13, (1985), pp. 77–90CrossRefMathSciNetGoogle Scholar
  7. [7]
    E. Griepentrog, R. März, Differential-Algebraic Equations and their Numerical Treatment, Teubner-Texte zur Math. 88, Teubner, Leipzig 1986Google Scholar
  8. [8]
    C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston 1984MATHGoogle Scholar
  9. [9]
    E. Hairer, CH. Lubich, M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York 1989Google Scholar
  10. [10]
    M. Hanke, Linear differential-algebraic equations in spaces of integrable functions; J. Differ. Equations 79 (1989), pp. 14–30CrossRefMATHADSMathSciNetGoogle Scholar
  11. [11]
    M.Hanke, R.März, A.Neubauer, On the regularization of linear differential-algebraic equations; in [3], pp. 523–540Google Scholar
  12. [12]
    M. Hanke, R. März, A. Neubauer, On the regularization of a class of nontransferable differential-algebraic equations; J. Differ. Equations 73 (1988), pp. 119–132CrossRefMATHGoogle Scholar
  13. [13]
    E. Kamke, Differentialgleichungen: Lösungsmethode und Lösungen I, 8.Auflage; Akad. Verlagsgesellschaft Geest and Portig, Leipzig 1967Google Scholar
  14. [14]
    P. Lötstedt, L. R. Petzold, Numerical solution of nonlinear differential equations with algebraic constraints I, SIAM J.Sci.Stat. Comput. 7 (1986), pp. 720–733MATHGoogle Scholar
  15. [15]
    R. März, Higher Index Differential-Algebraic Equations: Analysis and Numerical Treatment, Banach Center Publ. XIV, Warszawa, Polish Scientific Publishers 1989Google Scholar
  16. [16]
    V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York 1976Google Scholar
  17. [17]
    M. Z. Nashed (ed.), Generalized Inverses and Applications, Academic Press, New York 1976MATHGoogle Scholar
  18. [18]
    A.Neubauer, When do Sobolev spaces form a Hilbert scale? Proc.Amer.Math.Soc. 103 (1988), pp. 557–562CrossRefMathSciNetGoogle Scholar
  19. [19]
    A. Neubauer, Tikhonov regularization for nonlinear ill-posed problems: optimal convergence rates and finite dimensional approximation, Inverse Problems 5 (1989), pp. 541–557CrossRefMATHADSMathSciNetGoogle Scholar
  20. [20]
    T. I. Seidman, C. R. Vogel, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems 5 (1989), pp. 227–238CrossRefMATHADSMathSciNetGoogle Scholar
  21. [21]
    A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems, Wiley, New York 1977Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • H. W. Engl
    • 1
  • M. Hanke
    • 2
  • A. Neubauer
    • 1
  1. 1.Institut für MathematikJohannes-Kepler-UniversitätLinzAustria
  2. 2.Sektion MathematikHumboldt-UniversitätGerman Democratic Republic

Personalised recommendations