Abstract
In this paper the stability of the solutions of parameter estimation problems in their output least squares formulation is analyzed. The concepts of output least squares stability (OLS stability) is defined and sufficient conditions for this property are proved for abstract elliptic equations. These results are applied to the estimation of the diffusion, convection, and friction coefficient in second-order elliptic equations inℝ n,n=2, 3. Results on Tikhonov regularization in a nonlinear setting are also given.
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References
R. A. Adams: Sobolev Spaces, Academic Press, New York, 1975
G. Alessandrini: An identification problem for an elliptic equation in two variables, Ann Mat Pura Appl, to appear
W. Alt: Lipschitzian perturbations of infinite optimization problems, in: Mathematical Programming with Data Pertubations, II, ed. by A. V. Fiacco, Lecture Notes in Pure and Applied Mathematics, 85, Marcel Dekker, New York, 1984, 7–21
M. S. Berger: Nonlinearity and Functional Analysis, Academic Press, New York, 1977
H. Brezis: Analyse Fonctionelle, Théorie et Applications, Masson, Paris, 1983
G. Chavent: Local stability of the output least square parameter estimation technique, Mat Apl Comput 2 (1983), 3–22
G. Chavent: On parameter identifiability, Proceedings of the IFAC Symposium on Identification and Parameter Estimation, York, July 1985, 531–536
F. Colonius and K. Kunisch: Stability for parameter estimation in two-point boundary-value problems, J Reine Angew Math 370 (1986), 1–29
A. V. Fiacco: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, 1983
C. W. Groetsch: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics, 105, Pitman, Boston, 1984
T. Kato: Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, Berlin, 1980
C. Kravaris and J. H. Seinfeld: Identification of parameters in distributed systems by regularization, SIAM J Control Optim 23 (1985), 217–241
K. Kunisch and L. W. White: Regularity properties in parameter estimation of diffusion coefficients in elliptic boundary-value problems, Appl Anal 21 (1986), 71–88
O. A. Ladyzhenskaya and N. N. Ural'tseva: Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968
H. Maurer and J. Zowe: First- and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math Programming, 16 (1979), 98–110
G. R. Richter: An inverse problem for the steady state diffusion equation, SIAM J Appl Math 41 (1981), 210–221
D. L. Russell: Some remarks on numerical aspects of coefficient identification in elliptic systems, in: Optimal Control of Partial Differential Equations, ed. by K. H. Hoffmann and W. Krabs, Birkhäuser, Basel, 1984, 210–228
M. H. Schultz: Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ
A. N. Tikhonov and V. Y. Arsenin: Solutions of Ill-Posed Problems, Winston, Washington, 1977
M. M. Vainberg: Variational Methods for the Study of Nonlinear Operators, Holden Day, San Francisco, 1964
J. Wloka: Partielle Differentialgleichungen, Teubner, Stuttgart, 1982
A. Wouk: A Course of Applied Functional Analysis, Wiley, New York, 1979
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Communicated by I. Lasiecka
Part of this research was carried out while both authors were visitors at the Division of Applied Mathematics at Brown University. The research of F. Colonius was supported in part by the Deutsche Forschungsgemeinschaft. Both authors also acknowledge support from the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under Grant No. S3206.
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Colonius, F., Kunisch, K. Output least squares stability in elliptic systems. Appl Math Optim 19, 33–63 (1989). https://doi.org/10.1007/BF01448191
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DOI: https://doi.org/10.1007/BF01448191