Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation. *

Muriel Boulakia (1, 2) , Maya de Buhan (3) , Erica Schwindt (4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 COMMEDIA - COmputational Mathematics for bio-MEDIcal Applications
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
4 LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications

In this article, we consider a reaction-diffusion equation where the reaction term is given by a cubic function and we are interested in the numerical reconstruction of the time-independent part of the source term from measurements of the solution. For this identification problem, we present an iterative algorithm based on Carleman estimates which consists of minimizing at each iteration strongly convex cost functionals. Despite the nonlinear nature of the problem, we prove that our algorithm globally converges and the convergence speed evaluated in weighted norm is linear. In the last part of the paper, we illustrate the effectiveness of our algorithm with several numerical reconstructions in dimension one or two.

Mots clés: Nonlinear parabolic equations; Inverse problems; Carleman estimates; Numerical reconstruction

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Muriel Boulakia, Maya de Buhan, Erica Schwindt. Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation. *. 2019. ⟨hal-02185889v1⟩