Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||34|
|Published online||01 March 2021|
Numerical reconstruction based on Carleman estimates of a source term in a reaction–diffusion equation*
Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions, équipe COMMEDIA,
2 Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France.
3 Inria Nancy Grand-Est and CNRS UMR 7503 LORIA, Villers-lès-Nancy, France.
** Corresponding author: firstname.lastname@example.org
Accepted: 30 November 2020
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.
Mathematics Subject Classification: 35R30 / 35K55 / 35K57 / 93B07
Key words: Inverse problems / nonlinear parabolic equations / Carleman estimates / numerical reconstruction
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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