Identification of the diffusion in a nonlinear parabolic problem and numerical resolution

Abstract

This paper presents an iterative method to identify the diffusion in a semi-linear parabolic problem. This method can be generalized to other kind of problems, elliptic, parabolic and hyperbolic in two-dimensional and three-dimensional case. The diffusion is obtained by solving an optimal control problem. By imposing specific conditions to the data, we build a sequence of linear problems which converge to the exact solution. We discretize our problem by a finite element method in the first case and a spectral method in the second case, using the sensibility method for approximating the gradient of the functional. Some numerical experiments prove the efficiency of this method. References

• P. A. Raviart, D. J. M. Thomas, Introduction a l'analyse numerique des equations aux derivees partielles, Masson (1983).
• J.L. Lions, E. Magenes, Problemes aux limites non homogenes et applications, Vol. 1, Dunod (1968).
• M. Bouchiba, S. Abidi, Identification of the Diffusion in a lineair Parabolic problem, International Journal of Applied Mathematics, Vol. 23, No. 3, 2010, 491–501.
• M. Kern, Problemes Inverses Aspects Numeriques, INRIA (2003).
• J. E. Rakotoson, J. M. Rakotoson, Analyse Fonctionnelle appliquee aux equations aux derivees partielles, Presse Universitaires de France, (1999).
• C. Bernardi, Y. Maday, Spectral Methods, in the Handbook of Numerical Analysis V, P. G. Ciarlet and J. L. Lions eds., North-Holland, (1997), 209–485.
• V. Girault, P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms, Springer-Verlag (1986).
• C. Bernardi, Y. Maday, F. Rapetti, Discretisation Variationnelles de problemes aux limites elliptiques, Springer-Verlag (2004).
• C. Bernardi, Y. Maday, Approximations spectrales de problemes aux limites elliptiques, Springer-Verlag (1992).
• I. Ekeland, R. Teman, Analyse Convexe et problemes Variationnels, Bordas (19674).
• J. L. Lions, Controle Optimal de systemes gouvernes par des equations aux derivees partielles, Dunod, Paris, (1968).
• J. Henry, Etude de la controlabilite de certaines equations paraboliques, These d'etat, Universite Paris VI, (1978).
• J. Satouri, Methodes delements spectraux avec joints pour des geometries axisymetriques, These de l'Universite Pierre et Marie Curie, Paris VI (2010).
• K. Van de Doel, U. M. Ascher, On level set regularization for highly ill-posed distributed parameter systems, Journal of Computational Physics, 2006, 216: 707–723. http://people.cs.ubc.ca/ ascher/papers/da.pdf
• M. Bohm, M. A. Demetriou, S. Reich, and I. G. Rosen, Model Reference Adaptive Control of Distributed Parameter Systems, SIAM J. Control Optim., 36(1), 33–81; doi:10.1002/rnc.1098
• C. Jia, A note on the model reference adaptive control of linear parabolic systems with constant coefficients. Journal of Systems Science and Complexity 24, 1110–1117; doi:10.1007/s11424-011-0042-9
• D. B. Pietria, M. Krstic, Output-feedback adaptive control of a wave PDE with boundary anti-damping. Automatica 50, 1407–1415; doi:10.1016/j.automatica.2014.02.040