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

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Title: NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS.
Authors: HABBAL, A.1 habbal@polytech.unice.fr, KALLEL, M.2 moez.kallel@ipeit.rnu.tn
Source: SIAM Journal on Control & Optimization. 2013, Vol. 51 Issue 5, p4066-4083. 18p.
Subjects: Cauchy problem, Elliptic operators, Nash equilibrium, Dirichlet problem, Neumann boundary conditions
Abstract: We 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. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Control & Optimization is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS.
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  Data: <searchLink fieldCode="AR" term="%22HABBAL%2C+A%2E%22">HABBAL, A.</searchLink><relatesTo>1</relatesTo><i> habbal@polytech.unice.fr</i><br /><searchLink fieldCode="AR" term="%22KALLEL%2C+M%2E%22">KALLEL, M.</searchLink><relatesTo>2</relatesTo><i> moez.kallel@ipeit.rnu.tn</i>
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  Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Control+%26+Optimization%22">SIAM Journal on Control & Optimization</searchLink>. 2013, Vol. 51 Issue 5, p4066-4083. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Cauchy+problem%22">Cauchy problem</searchLink><br /><searchLink fieldCode="DE" term="%22Elliptic+operators%22">Elliptic operators</searchLink><br /><searchLink fieldCode="DE" term="%22Nash+equilibrium%22">Nash equilibrium</searchLink><br /><searchLink fieldCode="DE" term="%22Dirichlet+problem%22">Dirichlet problem</searchLink><br /><searchLink fieldCode="DE" term="%22Neumann+boundary+conditions%22">Neumann boundary conditions</searchLink>
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  Data: We 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. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Control & Optimization is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/120869808
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      – Code: eng
        Text: English
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        PageCount: 18
        StartPage: 4066
    Subjects:
      – SubjectFull: Cauchy problem
        Type: general
      – SubjectFull: Elliptic operators
        Type: general
      – SubjectFull: Nash equilibrium
        Type: general
      – SubjectFull: Dirichlet problem
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      – SubjectFull: Neumann boundary conditions
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      – TitleFull: NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS.
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              Text: 2013
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