Solving Cauchy problems by minimizing an energy-like functional

Stéphane Andrieux (1) , Thouraya Baranger (2) , Amel Ben Abda (3)

1 LAMSID - UMR 8193 - Laboratoire de Mécanique des Structures Industrielles Durables
2 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]
3 LR-LAMSIN-ENIT - Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis]

An energy-like error functional is introduced in the context of the ill-posed problem of boundary data recovering, which is well known as a Cauchy problem. Links with existing methods for data completion are detailed. Here the problem is converted into an optimization problem; the computation of the gradients of the energy-like functional is given for both the continuous and the discrete problems. Numerical experiments highlight the efficiency of the proposed method as well as its robustness in the model context of Laplace's equation, but also for anisotropic conductivity problems.

Mots clés: data completion; Inverse problem; Cauchy problem; thermostatics; boundary condition identification

Domaines: Mécanique [physics.med-ph]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Stéphane Andrieux, Thouraya Baranger, Amel Ben Abda. Solving Cauchy problems by minimizing an energy-like functional. Inverse Problems, 2006, 22, pp.115-133. ⟨hal-00139569⟩