Abstract
In this work, we consider the identification problem of the diffusion coef-ficient in two-dimensional elliptic equations. For parameterization, we use the zonation method: the diffusion coefficient is assumed to be a piecewise constant space function and unknowns are both the diffusion coefficient values and the geometry of the zones. An algorithm based on geometric principles is developed in order to determine the boundaries between the zones. This algorithm uses the refinement indicators which are easily computed from the gradient of the objective function. The efficiency of the algorithm is proved by testing it in some simple cases with and without noise on the data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ackerer, Ph., Siegel, P., Blaschke, A.: Inverse problem applied to groundwater flow and transport equations using a downscaling parametrization. Comput. Mech. Pub. 1, 321–328 (1996)
Ben Ameur, H., Chavent, G., Jaffré, J.: Refinement and coarsening indicators for adaptive parameterization: application of the estimation of hydraulic transmissivities. Inverse Probl. 18, 775–794 (2002)
Bonnans, J., Gilbert, J.-C., Lemaréchal, C., Sagastizabal, C.: Optimisation Numérique Aspects Théoriques et Pratiques. Springer, Berlin Heidelberg New York (1997)
Chavent, G.: Identification of functional parameter in partial differential equations. In: Goodson, R.E., Polis, M. (eds.) Identifications of Parameters in Distributed Systems, pp. 31–48. ASME, New York (1974)
Chavent, G. Non linear Least Squares for Inverse Problems. (Book in press)
Chavent, G., Bissel, R.: Identification of functional parameter in partial differential equations. In: Tanaka, M., Dulikravich, G.S. (eds.) Inverse Problems in Engineering Mechanics, pp. 309–314. Elsevier, Amsterdam, The Netherlands (1998)
Eppstein, M., Dougherty, D.: Simultaneous estimation of transmissivity values and zonation. Water Resour. Res. 32, 3321–3336 (1996)
Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: 1. Maximum likehood method incorporating prior information. Water Resour. Res. 22(2), 199–210 (1986)
Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: 2. Uniqueness, stability and solution algorithms. Water Resour. Res. 22(2), 211–227 (1986)
Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: 3. Application to synthetic and field data. Water Resour. Res. 22(2), 228–242 (1986)
Liu, J.: A multiresolution method for distributed parameter estimation. SIAM J. Sci. Comput. 14(2), 389–405 (1993)
Siegel, P.: Transfert de masse en milieux poreux fortement hétérogènes: modélisation et estimation de paramètres par éléments finis mixtes hybrides et discontinus. Thèse Université Louis Pasteur – Institut de Mécanique des Fluides, URA CNRS 854, Strasbourg, France, p. 185. (1997)
Sun, N.-Z.: Inverse Problems in Groundwater Modeling. Kluwer, Massachusets (1994)
Tsai, F.T.-C., Sun, N.-Z., Yeh, W.W.-G.: A combinatorial optimization scheme for parameter structure identification in groundwater modeling. Ground Water. 41(2), 156–169 (2003)
Yeh, W.W.-G.: Review of parameter identification procedures in groundwater hydrology: the inverse problem. Water Resour. Res. 22, 95–108 (1986)
Yeh, W.W.-G., Yoon, Y.-S.: Parameter identification with optimum dimension in parametrization. Water Resour. Res. 17(3), 664–672 (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hayek, M., Ackerer, P. An Adaptive Subdivision Algorithm for the Identification of the Diffusion Coefficient in Two-dimensional Elliptic Problems. J Math Model Algor 6, 529–545 (2007). https://doi.org/10.1007/s10852-006-9046-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10852-006-9046-1