Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data

Romain Rischette (1) , Thouraya Baranger (1) , Naima Debit (2)

1 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]
2 ICJ - Institut Camille Jordan

This paper is concerned with solving Cauchy problem for elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, Cauchy problem is presented as an optimal control problem. Numerical convergence analysis is carried out and leads to an adapted stopping criteria for the minimization process depending on noise rate. Numerical examples involving smooth and singular data are presented.

Mots clés: Inverse problem; ill-posed problem; Cauchy problem; data completion; boundary condition identification; optimal control; finite element; noise; a priori error estimates

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

Romain Rischette, Thouraya Baranger, Naima Debit. Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data. Journal of Computational and Applied Mathematics, 2011, 235, pp.3257-3269. ⟨10.1016/j.cam.2010.12.019⟩. ⟨hal-00426772v3⟩