Abstract
We discuss the numerical realization of an a-posteriori parameter choice for Tikhonov regularization of linear integral equations of the first kind in Hilbert scales that leads to optimal convergence rates and that does not need any information about the exact solution. Numerical examples confirm the theoretical results.
Zusammenfassung
Wir diskutieren die numerische Realisierung einer a-posteriori Parameterwahl für die Tikhonov Regularisierung von linearen Integralgleichungen erster Art in Hilbertskalen, die optimale Konvergenzraten liefert und die keine Information über die exakte Lösung benötigt. Numerische Beispiele bestätigen die theoretischen Ergebnisse.
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References
Adams, R.: Sobolev Spaces. New York: Academic Press 1975.
Eldén, L.: Algorithms for the regularization of ill-conditioned least squares problems. BIT17, 134–145 (1977).
Eldén, L.: A program for interactive regularization. Part I: Numerical Algorithms. Linköping University, Department of Mathematics, Report Lith-Mat-R-79-25.
Engl, H. W., Neubauer, A.: Convergence rates for Tikhonov-Regularization in finite-dimensional subspaces of Hilbert scales, to appear in AMS Proceedings.
Gferer, H.: An a-posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, to appear in Math. Comp.
Groetsch, C. W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Boston: Pitman 1984.
Krein, S. G., Petunin, J. I.: Scales of Banach spaces. Russian Math. Surveys21, 85–160 (1966).
Natterer, F.: Error bounds for Tikhonov regularization in Hilbert scales. Applic. Analysis18, 29–37 (1984).
Natterer, F.: On the order of regularization methods. In: Improperly Posed Problems and Their Numerical Treatment (Hämmerlin, G., Hoffmann, K. H., eds.), pp. 189–203 Basel: Birkhäuser 1983.
Neubauer, A.: An a-posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates, submitted.
Neubauer, A.: Zur Tikhonov-Regularisierung von linearen Operatorgleichungen. Diplomarbeit, Universität Linz, 1984.
Prenter, P. M.: Splines and Variational Methods, Pure and Appl. Math. New York: J. Wiley & Sons 1975.
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. New York: Springer 1980.
Swartz, B. K., Varga, R. S.: Error bounds for spline andL-spline interpolation. J. Appr. Theory6, 6–49 (1972).
Wilkinson, J. H., Reinsch, C. H.: Linear Algebra, Vol. II. Berlin: Springer 1971.
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The research that lead to this paper was partially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (project S32/03). The author is on leave from Universität Linz, Austria; the travel support from the Fulbright Commission is gratefully acknowledged.
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Neubauer, A. Numerical realization of an optimal discrepancy principle for Tikhonov-regularization in Hilbert scales. Computing 39, 43–55 (1987). https://doi.org/10.1007/BF02307712
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DOI: https://doi.org/10.1007/BF02307712