NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS

International audienceWe consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The overspecified Cauchy data are split between two players: the first player solves the elliptic equation 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 nonused Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a difference term. We prove that there always exists a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion Nash game has a stable solution with respect to noisy data. Some numerical two- and three-dimensional experiments are provided to illustrate the efficiency and stability of our algorithm

Nash strategies for the inverse inclusion Cauchy-Stokes problem

International audienceWe introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundarymeasurements. 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

Determination of point-forces via extended boundary measurements using a game strategy approach

International audienceIn this work, we consider a game theory approach to deal with an inverse problem related to the Stokes system. The problem consists in detecting the unknown point-forces acting on the fluid from incomplete measurements on the boundary of a domain. The approach that we propose deals simultaneously with the reconstruction of the missing data and the determination of the unknown point-forces. The solution is interpreted in terms of Nash equilibrium between both problems. We develop a new point-force detection algorithm, and we present numerical results to illustrate the efficiency and robustness of the method

A Three-player Nash game for point-wise source identification in Cauchy-Stokes problems

International audienceWe consider linear steady Stokes flow under the action of a finite number of particles located inside the flow domain. The particles exert point-wise forces on the fluid, and are unknown in number, location and magnitude. We are interested in the determination of these point-wise forces, using only a single pair of partially available Cauchy boundary measurements. The inverse problem then couples two harsh problems : identification of point-wise sources and recovery of missing boundary data. We reformulate it as a threeplayer Nash game. The first two players aim at recovering the Dirichlet and Neumann missing data, while the third one aims at the point-forces reconstruction of the number, location and magnitude of the point-forces. To illustrate the efficiency and robustness of the proposed algorithm, we finally present several numerical experiments for different geometries and source distribution, including the case of noisy measurements

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

International audienceWe 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

A nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology

International audienceWe consider the identification problem of the conductivity coefficient for an elliptic operator using an incomplete over specified measures on the surface. Our purpose is to introduce an original method based on a game theory approach, and design a new algorithm for the simultaneous identification of conductivity coefficient and data completion process. We define three players with three corresponding criteria. The two first players use Dirichlet and Neumann strategies to solve the completion problem, while the third one uses the conductivity coefficient as strategy, and uses a cost which basically relies on an identifiability theorem. In our work, the numerical experiments seek the development of this algorithm for the electrocardiography imaging inverse problem, dealing with in-homogeneities in the torso domain. Furthermore, in our approach, the conductivity coefficients are known only by an approximate values. we conduct numerical experiments on a 2D torso case including noisy measurements. Results illustrate the ability of our computational approach to tackle the difficult problem of joint identification and data completion. Mathematics Subject Classification. 35J25, 35N05, 91A80. The dates will be set by the publisher

A Nash-game approach to joint image restoration and segmentation

International audienceWe propose a game theory approach to simultaneously restore and segment noisy images. We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm

Identification de fissures droites depuis des mesures frontière incomplètes

Nous nous intéressons à la détection et à l'identification de fissures rectilignes à l'intérieur d'un conducteur isotrope plan (2D) à partir de mesures de la solution du problème de Neumann pour le Laplacien effectuées sur une partie seulement de la frontière extérieure du domaine. Nous procédons d'abord à l'extension à toute la frontière d'une estimation de la trace de la solution, au moyen de méthodes constructives d'analyse complexe et d'approximation issues de [13], puis à la localisation de la fissure, en utilisant les algorithmes proposés dans [17]

A Nash-game approach to solve the Cauchy problem for elliptic equations

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A Nash Game Approach to Solve Elliptic Data Completion Problems

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