Summary
The paper describes a numerical strategy for the approximate solution of nonlinear, discretized, inverse problems by regularization. It is assumed that the solution of the associated direct problems and the computation of Fréchet derivatives are expensive. In order to minimize the amount of work, a predictor-corrector type algorithm is proposed. From a series of solutions to problems with a coarse discretization one obtains a starting approximation for a problem with a fine discretization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Björck, Å., Eldén, L.: Methods in numerical algebra for ill-posed problems. Preprint LiTH-MAT-R-33-1979, Linköping University, Dept. of Mathematics, Sweden
Collatz, L.: Funktionalanalysis und numerische Mathematik. Berlin, Heidelberg, New York: Springer 1968
Eriksson, G., Dahlquist, G.: On an inverse non-linear diffusion problem. In: Deufelhard, P., Hairer, E. (eds.), Numerical Treatment of Inverse Problems in Differential and Integral Equations, pp. 238–245. Boston: Birkhäuser 1983
Friedrich, V., Hofmann, B., Tautenhahn, U.: Möglichkeiten der Regularisierung bei der Auswertung von Meßdaten. Wiss. Schriftenreihe der Techn. Hochschule Karl-Marx-Stadt 10/1979
Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics21, 215–223 (1979)
Groetsch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind. Boston: Pitman 1984
Grossmann, Ch., Kaplan, A.A.: Strafmethoden und modifizierte Lagrange-Funktionen in der nichtlinearen Optimierung. Leipzig: Teubner 1979
Hoffmann, K.-H.: Identifizierungsprobleme bei partiellen Differentialgleichungen. In: Numerische Behandlung von Differentialgleichungen, vol. 3, pp. 97–116. Basel: Birkhäuser 1981
Hofmann, B.: Optimization aspects of the generalized discrepancy principle in regularization. Optimization17, 305–316 (1986)
Hofmann, B.: Regularization for applied inverse and ill-posed problems. Leipzig: Teubner 1986
Mitchell, T., Draper, N.R.: Using external information in linear regression, with a commentary on ridge regression. TR no. 2327, Madison Research Centre 1982, USA
Morozov, V.A.: Methods for solving incorrectly posed problems. Berlin, Heidelberg, New York, Tokyo: Springer 1984
Schwetlick, H.: Numerische Lösung nichtlinearer Gleichungen. Berlin: Verlag der Wissenschaften 1979
Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. New York: John Wiley 1977
Wahba, G.: Numerical and statistical methods for mildly, moderately and severely ill-posed problems with noisy data. TR no. 595, Univ. of Wisconsin Madison 1980, USA
Wahba, G., Wold, S.: A complete automatic French curve: Fitting spline functions by crossvalidation. Commun. Stat4, 1–17 (1975)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Friedrich, V., Hofmann, B. A predictor-corrector technique for constrained least-squares regularization. Numer. Math. 51, 353–367 (1987). https://doi.org/10.1007/BF01400119
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01400119