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

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.2014.02.04