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The Output Least Squares Identifiability of the Diffusion Coefficient from an H1–Observation in a 2–D Elliptic Equation

Published online by Cambridge University Press:  15 August 2002

Guy Chavent  and
Karl Kunisch
Guy Chavent
Affiliation:
Ceremade, Université Paris–Dauphine, Paris Cedex 16, France; guy.chavent@inria.fr.
Karl Kunisch
Affiliation:
Institute of Mathematics, University of Graz, Austria.
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Abstract

Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.

Keywords

Parameter estimationdiffusion coefficientinverse problemidentifiabilityleast squares.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 423 - 440
Copyright
© EDP Sciences, SMAI, 2002

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References

Alessandrini, G. and Magnanini, R., Elliptic Equations in Divergence Form, Geometric Critical Points of Solutions, and Stekloff Eigenfunctions. SIAM J. Math. Anal . 25-5 (1994) 1259-1268. CrossRef
G. Chavent, Identification of distributed parameter systems: About the output least square method, its implementation and identifiability, in Proc. IFAC Symposium on Identification. Pergamon (1979) 85-97.
Chavent, G., Quasi convex sets and size x curvature condition, application to nonlinear inversion. J. Appl. Math. Optim. 24 (1991) 129-169. CrossRef
Chavent, G., New size x curvature conditions for strict quasi convexity of sets. SIAM J. Control Optim. 29-6 (1991) 1348-1372. CrossRef
Chavent, G. and Kunisch, K., A geometric theory for the L 2-stability of the inverse problem in a 1-D elliptic equation from an H 1-observation. Appl. Math. Optim. 27 (1993) 231-260. CrossRef
Chavent, G. and Kunisch, K., Weakly Nonlinear Inverse Problems, On. SIAM J. Appl. Math. 56-2 (1996) 542-572. CrossRef
Chavent, G. and Kunisch, K., State-space regularization: Geometric theory. Appl. Math. Optim. 37 (1998) 243-267. CrossRef
G. Chavent and K. Kunisch, The Output Least Squares Identifiability of the Diffusion Coefficient from an H 1 -Observation in a 2D Elliptic Equation. INRIA Report 4067 (2000).
G. Chavent and J. Liu, Multiscale parametrization for the estimation of a diffusion coefficient in elliptic and parabolic problems, in Fifth IFAC Symposium on Control of Distributed Parameter Systems. Perpignan, France (1989).
Chicone, C. and Gerlach, J., A note on the identifiability of distributed parameters in elliptic systems. SIAM J. Math. Anal. 18 (1987) 13781-384. CrossRef
V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1979).
A. Grimstad, K. Kolltveit, T. Mannseth and J. Nordtvedt, Assessing the validity of a linearized error analysis for a nonlinear parameter estimation problem. Preprint.
A. Grimstad and T. Mannseth, Nonlinearity, scale, and sensitivity for parameter estimation problems. Preprint.
V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998).
Ito, K. and Kunisch, K., On the injectivity and linearization of the coefficient to solution mapping for elliptic boundary value problems. J. Math. Anal. Appl. 188 (1994) 1040-1066. CrossRef
Liu, J., A multiresolution method for distributed parameter estimation. SIAM J. Sci. Stat. Comp. 14 (1993) 389-405. CrossRef
G.R. Richter, An inverse problem for the steady state diffusion equation. SIAM J. Appl. Math. 4 (1981), 210-221.
G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987).