Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation
| Autor: | Erica L. Schwindt, Maya de Buhan, Muriel Boulakia |
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| Přispěvatelé: | Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), COmputational Mathematics for bio-MEDIcal Applications (COMMEDIA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), This work was partially funded by the DGA 2014-91-00-79 project. Maya de Buhan was partially supported by the Project 'Analysis and simulation of optimal shapes - application to life sciences' of the Paris City Hall., Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Géometrie et Lumière (ALICE-POST), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, This work was partially funded by the DGA 2014-91-00-79 project. The second author was partially supported by the Project 'Analysis and simulation of optimal shapes - application to life sciences' of the Paris City Hall. |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Inverse problems
Control and Optimization Iterative method 010103 numerical & computational mathematics AMS subject classifications: 35R30 35K55 35K57 93B07 Mathematics Subject Classification.35R30 35K55 35K57 93B07 01 natural sciences Reaction–diffusion system Applied mathematics numerical reconstruction AMS subject classifications: 35R30 [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics [MATH]Mathematics [math] Mathematics nonlinear parabolic equations Inverse problem Carleman estimates 93B07 010101 applied mathematics Parameter identification problem Computational Mathematics Nonlinear system Control and Systems Engineering Numerical reconstruction 35K57 Norm (mathematics) Bounded function 35K55 Convex function |
| Zdroj: | ESAIM: Control, Optimisation and Calculus of Variations ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, In press, ⟨10.1051/cocv/2020086⟩ ESAIM: Control, Optimisation and Calculus of Variations, 2021, 27, pp.34. ⟨10.1051/cocv/2020086⟩ ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2021, 27 (S27), pp.34. ⟨10.1051/cocv/2020086⟩ |
| ISSN: | 1292-8119 1262-3377 |
| DOI: | 10.1051/cocv/2020086⟩ |
| Popis: | International audience; 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. |
| Databáze: | OpenAIRE |
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