Numerical reconstruction based on Carleman estimates of a source term in a reaction–diffusion equation

Abstract

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 cost functionals which are strongly convex on bounded sets. Despite the nonlinear nature of the problem, we prove that our method 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 method with several numerical reconstructions in dimension one or two.