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 iter...

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Bibliographic Details
Published inESAIM. Control, optimisation and calculus of variations Vol. 27; p. S27
Main Authors Boulakia, Muriel, de Buhan, Maya, Schwindt, Erica L.
Format Journal Article
LanguageEnglish
Published Les Ulis EDP Sciences 2021
Subjects
Analysis of PDEs
Convergence
Iterative algorithms
Iterative methods
Mathematics
Reaction-diffusion equations
Reconstruction
Numerical reconstruction
Inverse problems
93B07
Carleman estimates
numerical reconstruction AMS subject classifications: 35R30
35K57
35K55
Nonlinear parabolic equations
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Summary: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.
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content type line 14
ISSN:1292-8119
1262-3377
DOI:10.1051/cocv/2020086