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

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.