Time-Adaptive FEM for Distributed Parameter Identification in Biological Models

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 48)


We propose a time-adaptive finite element method for the solution of a parameter identification problem for ODE which describes dynamical systems of biological models. We present framework of a posteriori error estimate in the Tikhonov functional, in Lagrangian, and in the reconstructed function. We also present time-mesh relaxation property in the adaptivity and formulate the time-mesh refinement recommendation and an adaptive algorithm which can be used to find optimal values of the distributed parameters in biological models.


Human Immunodeficiency Virus Type Mesh Refinement Tikhonov Regularization Posteriori Error Estimate Relaxation Property 
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.



This research was supported by the Swedish Research Council, the Swedish Foundation for Strategic Research (SSF) through the Gothenburg Mathematical Modelling Centre (GMMC), the Swedish Institute, Visby Program, the program of the Russian Academy of Sciences “Basic Research for Medicine,” and the Russian Foundation for Basic Research (Grant 11-01-00117).


  1. 1.
    B.M. Adams, H.T. Banks, M. Davidian, H.-D. Kwon, H.T. Tran, S.N. Wynne and E.S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols, Journal of Computational and Applied Mathematics, 184, 10–49, 2005.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    M. Asadzadeh and L. Beilina, A posteriori error analysis in a globally convergent numerical method for a hyperbolic coefficient inverse problem, Inverse Problems, 26, 115007, 2010.MathSciNetCrossRefGoogle Scholar
  3. 3.
    A.B. Bakushinskii and M.Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Springer, New York, 2004.Google Scholar
  4. 4.
    A.B. Bakushinsky, M.Y. Kokurin, A. Smirnova, Iterative Methods for Ill-posed Problems, Walter de Gruyter GmbH&Co., Berlin, 2011.Google Scholar
  5. 5.
    W. Bangerth and A. Joshi, Adaptive finite element methods for the solution of inverse problems in optical tomography, Inverse Problems, 24, 034011, 2008.MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element method, Acta Numerica, 10, 1–102, 2001.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    L. Beilina, Adaptive finite element/difference method for inverse elastic scattering waves, Applied and Computational Mathematics, 1, 158–174, 2002.MathSciNetGoogle Scholar
  8. 8.
    L. Beilina, Adaptive finite element method for a coefficient inverse problem for the Maxwell’s system, Applicable Analysis, 90, 1461–1479, 2011.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    L. Beilina and C. Clason, An adaptive hybrid FEM/FDM method for an inverse scattering problem in scanning acoustic microscopy,  SIAM Journal on Scientific Computing, 28, 382–402, 2006.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    L. Beilina and C. Johnson, A hybrid FEM/FDM method for an inverse scattering problem. In  Numerical Mathematics and Advanced Applications - ENUMATH 2001, Springer-Verlag, Berlin, 2001.Google Scholar
  11. 11.
    L. Beilina and C. Johnson, A posteriori error estimation in computational inverse scattering, Mathematical Models and Methods in Applied Sciences, 15, 23–37, 2005.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    L. Beilina and M.V. Klibanov, A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem, Inverse Problems, 26, 045012, 2010.MathSciNetCrossRefGoogle Scholar
  13. 13.
    L. Beilina and M.V. Klibanov, Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inverse Problems, 26, 125009, 2010.MathSciNetCrossRefGoogle Scholar
  14. 14.
    L. Beilina and M.V. Klibanov, Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D, Journal of Inverse and Ill-posed Problems, 18, 85–132, 2010.MathSciNetCrossRefGoogle Scholar
  15. 15.
    L. Beilina and M.V. Klibanov, Relaxation property of the adaptivity technique for some ill-posed problems, preprint, Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, ISSN 1652–9715; nr 2012:4.Google Scholar
  16. 16.
    L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012.MATHCrossRefGoogle Scholar
  17. 17.
    L. Beilina, M.V. Klibanov and M.Yu Kokurin, Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem, Journal of Mathematical Sciences, 167, 279–325, 2010.MathSciNetCrossRefGoogle Scholar
  18. 18.
    V.A. Chereshnev, S.I. Bazhan, B.A. Bakhmetev, I.A. Gainova and G.A. Bocharov, Systems analysis of HIV infection pathogenesis, Biology Bulletin Reviews, 132(2), 115–140, 2012.Google Scholar
  19. 19.
    K. Eriksson, D. Estep and C. Johnson,Calculus in Several Dimensions, Springer, Berlin, 2004.MATHGoogle Scholar
  20. 20.
    T. Feng, N. Yan and W. Liu, Adaptive finite element methods for the identification of distributed parameters in elliptic equation, Advances in Computational Mathematics, 29, 27–53, 2008.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, Ltd, New York, 1986.Google Scholar
  22. 22.
    A. Griesbaum, B. Kaltenbacher and B. Vexler, Efficient computation of the Tikhonov regularization parameter by goal-oriented adaptive discretization, Inverse Problems, 24, 025025, 2008.MathSciNetCrossRefGoogle Scholar
  23. 23.
    B. Kaltenbacher, A. Krichner and B. Vexler, Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems, Inverse Problems, 27, 125008, 2011.MathSciNetCrossRefGoogle Scholar
  24. 24.
    B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, de Gruyter, New York, 2008.Google Scholar
  25. 25.
    N.A. Koshev and L. Beilina, A posteriori error estimates for the Fredholm integral equation of the first kind, accepted to book series Springer Proceedings in Mathematics, 2012.Google Scholar
  26. 26.
    O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Springer Verlag, Berlin, 1985.CrossRefGoogle Scholar
  27. 27.
    R. Medzhitov and D. Littman, HIV immunology needs a new direction: Commentary, Nature, 455(7213), 591, 2008.Google Scholar
  28. 28.
    A.N. Tikhonov and V.Y. Arsenin,  Solutions of Ill-Posed Problems, Winston and Sons, Washington, DC, 1977.MATHGoogle Scholar
  29. 29.
    A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov and A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, London, 1995.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and Gothenburg UniversityGothenburgSweden
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia

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