A Nash-game approach to solve the Coupled problem of conductivity identification and data completion

Abstract : We consider the identification problem of the conductivity coefficient for an elliptic operator using an incomplete over-specified measurements on the surface (Cauchy data). Data completion problems are widely discussed in literature by several methods (see, e.g., for control and game oriented approaches [1, 2], and references therein). The identification of conductivity and permit-tivity parameters has also been investigated in many studies (see, e.g., [3, 4]). In this work, our purpose is to extend the method introduced in [1], based on a game theory approach, to develop a new algorithm for the simultaneous identification of conductivity coefficient and missing boundary data. We shall say that there are three players and we define three objective functions. Each player controls one variable and minimizes his own cost function in order to seek a Nash equilibrium which is expected to approximate the inverse problem solution. The first player solves the elliptic equation (div(k.(u)) = 0) with the Dirichlet part of the Cauchy data prescribed over the accessible boundary and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the non used Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two players consider a response of the unknown conductivity of the third player. The third player controls the conductivity coefficient, and uses the over specified Dirichlet condition as well as the second's player Dirichlet condition strategy prescribed over the inaccessible part of the boundary.
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Communication dans un congrès
PICOF (Problèmes Inverses, Contrôle et Optimisation de Formes), Jun 2016, Autrans, France
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  • HAL Id : hal-01402976, version 1

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Rabeb Chamekh, Abderrahmane Habbal, Moez Kallel, Nejib Zemzemi. A Nash-game approach to solve the Coupled problem of conductivity identification and data completion. PICOF (Problèmes Inverses, Contrôle et Optimisation de Formes), Jun 2016, Autrans, France. 〈hal-01402976〉

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