Finite Element Methods for Solving Parabolic Inverse Problems

  • Yee Lo Keung
  • Jun Zou
Part of the Applied Optimization book series (APOP, volume 33)


In this paper, we apply the finite element method to identify physical parameters in parabolic initial-boundary value problems. The identifying problem is formulated as a constrained minimization of the L 2-norm error between the observation data and the physical solution to the original system, with the H 1-regularization or BV-regularization. Then the finite element method is used to approximate the constrained minimization problem, and the resulting discrete system is further reduced to a sequence of unconstrained minimizations. Numerical experiments are presented to show the efficiency of the proposed method, for continuous and discontinuous parameters with noised observations.


Finite Element Method Parabolic System Time Step Size Unconstrained Minimization Good Initial Guess 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bank, H. T., and K. Kunisch. Estimation techniques for distributed parameter system. Birkhäuser, Boston, 1989.CrossRefGoogle Scholar
  2. [2]
    Chan, T., B. Smith, and J. Zou. Overlapping schwarz methods on unstructured meshes using non-matching coarse grids. Numer. Math., 73:149–167, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Chan, T., and J. Zou. Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes. Numerical Algorithms, 8:329–346, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Chen, Z., and J. Zou. An augmented lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control Optim., 1998. Accepted for publication.Google Scholar
  5. [5]
    Engl, H. W., M. Hanke, and A. Neubauer. Regularization of inverse problems. Kluwer Academic Publishers, Dordrecht, 1996.zbMATHCrossRefGoogle Scholar
  6. [6]
    Gutman S. Identification of discontinuous parameters in flow equations. SIAM J. Control Optim., 28:1049–1060, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Ito, K., and K. Kunisch. Augmented Lagrangian-SQP-methods in Hilbert spaces and applications to control in the coefficients problems. SIAM J. Optim., 6:96–125, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Keung, Y., and J. Zou. Identifying parameters in parabolic systems by finite element methods. Technical Report 97–10 (126), Dept of Math, The Chinese University of Hong Kong, 1997.Google Scholar
  9. [9]
    Keung, Y., and J. Zou. Numerical identifications of parameters in parabolic systems. Inverse Problems, 14:1–18, 1998.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Kunisch, K., and L. White. The parameter estimation problem for parabolic equations and discontinuous observation operators. SIAM J. Control Optim., 23:900–927, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Seid, Y. F., and J. Zou. Finite element simulations in parameter identifications. Technical Report 97–10 (120), Dept of Math, The Chinese University of Hong Kong, 1997.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Yee Lo Keung
  • Jun Zou

There are no affiliations available

Personalised recommendations