Constitutive law gap functionals to solve Cauchy problem for a linear elliptic PDE: a review

Thouraya Baranger (1) , Stéphane Andrieux (2)

1 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]
2 LAMSID - UMR 8193 - Laboratoire de Mécanique des Structures Industrielles Durables

This paper describes a general method based on minimizing constitutive law gap functional in order to solve the Cauchy problem for a linear elliptic PDE. This functional measures the gap between the solutions of two well-posed problems. Each of these problems has one of the Cauchy data as known boundary condition: Dirichlet or Neumann, and on the boundary where the data is lacking, unknown Robin boundary conditions are imposed. These latter are controlled two by positives scalars. This approach generalizes that presented in Andrieux et al. and encompasses various methods proposed in the literature. According to the values of Robin's parameters when they tend to zero or infinity, there are two groups of methods: the first group includes those which depends on only one unknown data Dirichlet, Neumann or Robin. The second group includes those which depend on two unknown data Dirichlet and Neumann. Then, the equivalence between Euler-Lagrange conditions for the constitutive law functionals and interfacial operators usually used in the Domain Decomposition field is shown. Using the Hadamard example we analyse analytically the behavior of these operators as functions of the parameters. Then, the derivatives of the functional are given using adjoint fields which are parametrized by the same parameters. Finally, numerical examples are given to illustrate the behavior of these methods, which are not function of the parameters but also of the regularity of the Cauchy data and the overall geometry of the domain.

Mots clés: Inverse problem; Control; Cauchy problem; Domain Decomposition; Identification

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Biomécanique [physics.med-ph]; Biomécanique [physics.med-ph]; Thermique [physics.class-ph]; Thermique [physics.class-ph]

Thouraya Baranger, Stéphane Andrieux. Constitutive law gap functionals to solve Cauchy problem for a linear elliptic PDE: a review. 2010. ⟨hal-00489572v2⟩