Advertisement

Computing

, Volume 40, Issue 2, pp 91–109 | Cite as

A variant of finite-dimensional Tikhonov regularization with a-posteriori parameter choice

  • J. T. King
  • A. Neubauer
Article

Abstract

In this paper we consider a particular variant of finite-dimensional Tikhonov regularization for ill-posed operator equations. Convergence rates are established and an a-posteriori parameter choice method is derived that leads to optimal convergence rates with respect to data errors and with respect to the finite-dimensional subspace, without using any information about the exact solution. Finally, using linear splines we present several numerical examples that confirm the theoretical results.

AMS Subject Classifications

65R20 45L10 

Key words

Integral equations Tikhonov regularization parameter selection method 

Eine Variante endlich-dimensionaler Tikhonov-Regularisierung mit a-posteriori Parameterwahl

Zusammenfassung

In dieser Arbeit betrachten wir eine besondere Variante endlich-dimensionaler Tikhonov-Regularisierung schlechtgestellter Operatorgleichungen. Konvergenzraten werden nachgewiesen und eine a-posteriori Parameterwahl wird hergeleitet, die optimale Konvergenzraten bezüglich des Datenfehlers und der endlich-dimensionalen Approximation liefert, ohne Information über die exakte Lösung zu benötigen. Schließlich präsentieren wir auch einige numerische Beispiele, bei denen lineare Splinefunktionen verwendet werden, die die theoretischen Ergebnisse bestätigen.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Beck, J. V., Murio, D. A.: Combined function specification — regularization procedure for solution of inverse heat conduction problem. AIAA Journal24/1, 180–185 (1984).Google Scholar
  2. [2]
    Elden, L.: Algorithms for the regularization of ill-conditioned least squares problems. BIT17, 134–145 (1977).Google Scholar
  3. [3]
    Elden, L.: The numerical solution of a noncharacteristic Cauchy problem for a parabolic equation. In: Numerical Treatment of Inverse Problems in Differential and Integral Equations (Deuflhard, P., Hairer, E., eds.), pp. 246–268. Basel: Birkhäuser 1983.Google Scholar
  4. [4]
    Elden, L.: An efficient algorithm for the regularization of ill-conditioned least squares problems with triangular Toeplitz matrix. Siam J. Sci. Stat. Comp.5, 237–254 (1984).Google Scholar
  5. [5]
    Engl, H. W., Neubauer, A.: An improved version of Marti's method for solving ill-posed linear integral equations. Math. Comp.45, 405–416 (1985).Google Scholar
  6. [6]
    Engl, H. W.: Regularization and least squares collocation. In: Numerical Treatment of Inverse Problems in Differential and Integral Equations (Deuflhard, P., Hairer, E., eds.), pp. 345–354. Basel: Birkhäuser 1983.Google Scholar
  7. [7]
    Garbor, B. S., et al.: Matrix Eigensystems Routines — EISPACK Guide, Lecture Notes in Computer Science No. 51 New York: Springer 1976.Google Scholar
  8. [8]
    Gfrerer, H.: An a-posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math. Comp.49, 507–522, S5–S12 (1987).Google Scholar
  9. [9]
    Groetsch, C. W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Boston: Pitman 1984.Google Scholar
  10. [10]
    Groetsch, C. W.: Comments on Morozov's discrepancy principle. In: Improperly Posed Problems and Their Numerical Treatment (Hämmerlin, G., Hoffmann, K. H., eds.), pp. 97–104. Basel: Birkhäuser 1983.Google Scholar
  11. [11]
    Groetsch, C. W., King, J. T., Murio, D.: Asymptotic analysis of a finite element method for Fredholm equations of the first kind. In: Treatment of Integral Equations by Numerical Methods (Baker, C. T. H., Miller, G. F., eds.), pp. 1–11. London: Academic Press 1982.Google Scholar
  12. [12]
    Krasnoselskii, M. A., et al.: Integral operators in spaces of summable functions. Leyden: Noordhoff Int. Publ. 1976.Google Scholar
  13. [13]
    Morozov, A.: On the solution of functional equations by the method of regularization. Soviet Math. Dokl.7, 414–417 (1966).Google Scholar
  14. [14]
    Neubauer, A.: An a-posteriori parameter choice for Tikhonov-regularization in Hilbert scales leading to optimal convergence rates. Siam J. Num. Anal., to appear.Google Scholar
  15. [15]
    Neubauer, A.: Numerical realization of an optimal discrepancy principle for Tikhonov regularization in Hilbert scales. Computing39, 43–55 (1987).Google Scholar
  16. [16]
    Neubauer, A.: Zur Tikhonov-Regularisierung von linearen Operatorgleichungen. Diplomarbeit, Univ. Linz, Austria, 1984.Google Scholar
  17. [17]
    Prenter, P. M.: Splines and Variational Methods, Pure and Appl. Math. New York: J. Wiley and Sons 1975.Google Scholar
  18. [18]
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. New York: Springer 1980.Google Scholar
  19. [19]
    Tikhonov, A. N.: Solution of incorrectly formulated problems and the regularization method. Soviet Math. Doklady4, 1035–1038 (1963).Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • J. T. King
    • 1
  • A. Neubauer
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.Institut für MathematikJohannes Kepler UniversitätLinzAustria

Personalised recommendations