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Nash strategies for the inverse inclusion Cauchy-Stokes problem

Abstract : We introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms is not straightforward. We reformulate the problem as a three-player Nash game. Thanks to an identifiability result derived for the Cauchy-Stokes inclusion problem, it is enough to set up two Stokes boundary value problems, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. We used a level-set approach to get rid of the tricky control dependence of functional spaces, and we provided the third player with the level-set function as strategy, with a cost functional of Kohn-Vogelius type. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for three different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems.
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https://hal.inria.fr/hal-01945094
Contributor : Abderrahmane Habbal <>
Submitted on : Wednesday, December 5, 2018 - 10:23:16 AM
Last modification on : Wednesday, October 28, 2020 - 12:10:04 PM
Long-term archiving on: : Wednesday, March 6, 2019 - 12:55:22 PM

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Abderrahmane Habbal, Moez Kallel, Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem. Inverse Problems and Imaging , AIMS American Institute of Mathematical Sciences, 2019, 13 (4), pp.36. ⟨10.3934/ipi.2019038⟩. ⟨hal-01945094⟩

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