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 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.

Automated summary

In this work, a reaction-diffusion equation with a cubic reaction term is considered, and a iterative algorithm based on Carleman estimates is proposed for the numerical reconstruction of the time-independent part of the source term. The algorithm minimizes cost functionals which are strongly convex on bounded sets at each iteration, demonstrating global convergence and linear convergence speed in weighted norm. Numerical reconstructions in one or two dimensions are used to illustrate the effectiveness of the proposed method.