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

Guy Chavent; Karl Kunisch

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 423-440
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Chavent, Guy, and Kunisch, Karl. "The Output Least Squares Identifiability of the Diffusion Coefficient from an H1–Observation in a 2–D Elliptic Equation." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 423-440. <http://eudml.org/doc/90655>.

@article{Chavent2010,
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. },
author = {Chavent, Guy, Kunisch, Karl},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Parameter estimation; diffusion coefficient; inverse problem; identifiability; least squares.; parameter estimation; inverse problem; least squares},
language = {eng},
month = {3},
pages = {423-440},
publisher = {EDP Sciences},
title = {The Output Least Squares Identifiability of the Diffusion Coefficient from an H1–Observation in a 2–D Elliptic Equation},
url = {http://eudml.org/doc/90655},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Chavent, Guy
AU - Kunisch, Karl
TI - The Output Least Squares Identifiability of the Diffusion Coefficient from an H1–Observation in a 2–D Elliptic Equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 423
EP - 440
AB - 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.
LA - eng
KW - Parameter estimation; diffusion coefficient; inverse problem; identifiability; least squares.; parameter estimation; inverse problem; least squares
UR - http://eudml.org/doc/90655
ER -

References

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  9. 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).  
  10. C. Chicone and J. Gerlach, A note on the identifiability of distributed parameters in elliptic systems. SIAM J. Math. Anal.18 (1987) 13781-384.  Zbl0644.35092
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  13. A. Grimstad and T. Mannseth, Nonlinearity, scale, and sensitivity for parameter estimation problems. Preprint.  Zbl0952.35153
  14. V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998).  Zbl0908.35134
  15. K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient to solution mapping for elliptic boundary value problems. J. Math. Anal. Appl.188 (1994) 1040-1066.  Zbl0817.35021
  16. J. Liu, A multiresolution method for distributed parameter estimation. SIAM J. Sci. Stat. Comp.14 (1993) 389-405.  Zbl0773.65059
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