Abstract
In this paper, we investigate the numerical identification of the diffusion parameters in a linear parabolic problem. The identification is formulated as a constrained minimization problem. By using the augmented Lagrangian method, the inverse problem is reduced to a coupled nonlinear algebraic system, which can be solved efficiently with the preconditioned conjugate gradient method. Finally, we present some numerical experiments to show the efficiency of the proposed methods, even for identifying highly discontinuous parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. L. Keung J. Zou (1998) ArticleTitleNumerical Identifications of Parameters in Parabolic Systems Inverse Problems 14 83–100
H. W. Engl J. Zou (2000) ArticleTitleA New Approach to Convergence Rate Analysis of Tikhonov Regularization for Parameter Identification in Heat Conduction Inverse Problems 16 1907–1923
B. Gou J. Zou (2001) ArticleTitleAn Augmented Lagrangian Method for Parameter Identification in Parabolic Systems Journal of Mathematical Analysis and Applications 263 49–68
T. Kärkkäinen (1997) ArticleTitleError Estimates for Distributed Parameter Identification in Parabolic Problems with Output Least Squares and Crank-Nicolson Method Applications of Mathematics 42 259–277
T. F. Chan X. C. Tai (2003) ArticleTitleIdentification of Discontinuous Coefficients from Elliptic Problems Using Total Variation Regularization SIAM Journal on Scientific Computing 25 881–904
K. Ito K. Kunisch (1990) ArticleTitleThe Augmented Lagrangian Method for Parameter Estimation in Elliptic Systems SIAM Journal on Control and Optimization 28 113–136
Y. L. Keung J. Zou (2000) ArticleTitleAn Efficient Linear Solver for Nonlinear Parameter Identification Problems SIAM Journal on Scientific Computing 22 1511–1526
K. Kunisch G. Peichl (1991) ArticleTitleEstimation of a Temporally and Spatially Varying Diffusion Coefficient in a Parabolic System by an Augmented Lagrangian Technique Numerische Mathematik 59 473–509
K. Kunisch X. C. Tai (1997) ArticleTitleSequential and Parallel Splitting Methods for Bilinear Control Problems in Hilbert Spaces SIAM Journal on Numerical Analysis 34 91–118
P. G. Ciarlet (1978) The Finite-Element Method for Elliptic Problems North-Holland Publishing Company Amsterdam, Holland
Tai, X.-C., Frø yen, J., Espedal, M. S., and Chan, T. F., Overlapping Domain Decomposition and Multigrid Methods for Inverse Problems, Domain Decomposition Methods, 10th International Conference on Domain Decomposition Methods, Contemporary Mathematics, Vol. 218, pp. 523–529, 1998; see also URL: http//www.mi.uib.no/~tai/.
Z. Chen J. Zou (1999) ArticleTitleAn Augmented Lagrangian Methods for Identifying Discontinuous Parameters in Elliptic Systems SIAM Journal on Control and Optimization 37 892–910
W. Hackbusch (1994) Iterative Solution of Large Sparse Systems of Equations Applied Mathematical Sciences Springer Verlag New York, NY
Author information
Authors and Affiliations
Additional information
This work was partially supported by the Research Council of Norway, Grant NFR-128224/431.
Rights and permissions
About this article
Cite this article
Nilssen, T.K., Tai, X.C. Parameter Estimation with the Augmented Lagrangian Method for a Parabolic Equation. J Optim Theory Appl 124, 435–453 (2005). https://doi.org/10.1007/s10957-004-0944-y
Issue Date:
DOI: https://doi.org/10.1007/s10957-004-0944-y