Neumann-Dirichlet Nash Strategies for the Solution of Elliptic Cauchy Problems

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 ...

[1]  Vladimir Maz’ya,et al.  An iterative method for solving the Cauchy problem for elliptic equations , 1991 .

[2]  Patrick Redont,et al.  A New Class of Alternating Proximal Minimization Algorithms with Costs-to-Move , 2007, SIAM J. Optim..

[3]  Sergei V. Pereverzev,et al.  The balancing principle for the regularization of elliptic Cauchy problems , 2007 .

[4]  J. Leblond,et al.  VSP 2003 Parameter identi ̄ cation for Laplace equation and approximation in Hardy classes , 2003 .

[5]  Jérémi Dardé,et al.  About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains , 2010 .

[6]  Amel Ben Abda,et al.  Missing boundary data reconstruction via an approximate optimal control , 2008 .

[7]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[8]  Christian Kanzow,et al.  Formulation and Numerical Solution of Nash Equilibrium Multiobjective Elliptic Control Problems , 2013, SIAM J. Control. Optim..

[9]  Laurent Bourgeois.  About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains , 2010 .

[10]  Jean-Antoine Désidéri.  Cooperation and competition in multidisciplinary optimization , 2012, Comput. Optim. Appl..

[11]  Faker Ben Belgacem,et al.  Why is the Cauchy problem severely ill-posed? , 2007 .

[12]  Fahmi Ben Hassen,et al.  Sources recovery from boundary data: A model related to electroencephalography , 2009, Math. Comput. Model..

[13]  Amel Ben Abda,et al.  Line segment crack recovery from incomplete boundary data , 2002 .

[14]  Moez Kallel,et al.  Recovery of Cracks from Incomplete Boundary Data , 2002 .

[15]  Luca Rondi,et al.  The stability for the Cauchy problem for elliptic equations , 2009, 0907.2882.

[16]  Tamer Basar,et al.  Distributed algorithms for the computation of noncooperative equilibria , 1987, Autom..

[17]  Daniel Lesnic,et al.  The Cauchy problem for Laplace’s equation via the conjugate gradient method , 2000 .

[18]  Thouraya Baranger,et al.  Solving Cauchy problems by minimizing an energy-like functional , 2006 .

[19]  R. Rubinstein,et al.  On relaxation algorithms in computation of noncooperative equilibria , 1994, IEEE Trans. Autom. Control..

[20]  Amel Ben Abda,et al.  A control type method for solving the Cauchy–Stokes problem , 2013 .

[21]  A. Habbal,et al.  A topology Nash game for tumoral antiangiogenesis , 2005 .

[22]  Alfio Quarteroni,et al.  Theory and Application of Steklov-Poincaré Operators for Boundary-Value Problems , 1991 .

[23]  J. Bolte,et al.  Alternating Proximal Algorithms for Weakly Coupled Minimization Problems. Applications to Dynamical Games and PDE’s , 2008 .

[24]  Mohamed Jaoua,et al.  Solution of the Cauchy problem using iterated Tikhonov regularization , 2001 .

[25]  A. Nachaoui,et al.  Convergence analysis for finite element approximation to an inverse Cauchy problem , 2006 .

[26]  T. Basar,et al.  Relaxation techniques and asynchronous algorithms for on-line computation of noncooperative equilibria , 1987, 26th IEEE Conference on Decision and Control.

[27]  Abderrahmane Habbal,et al.  Multidisciplinary topology optimization solved as a Nash game , 2004 .