Time-Adaptive FEM for Distributed Parameter Identification in Biological Models
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.
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).
- 3.A.B. Bakushinskii and M.Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Springer, New York, 2004.Google Scholar
- 4.A.B. Bakushinsky, M.Y. Kokurin, A. Smirnova, Iterative Methods for Ill-posed Problems, Walter de Gruyter GmbH&Co., Berlin, 2011.Google Scholar
- 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
- 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
- 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
- 21.R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, Ltd, New York, 1986.Google Scholar
- 24.B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, de Gruyter, New York, 2008.Google Scholar
- 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
- 27.R. Medzhitov and D. Littman, HIV immunology needs a new direction: Commentary, Nature, 455(7213), 591, 2008.Google Scholar