Inverse problems for linear parabolic equations using mixed formulations -Part 1 : Theoretical analysis

Arnaud Münch (1) , Diego Araujo de Souza (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 EDAN - Departamento de Ecuaciones Differenciales y Analysis numérico [Sevilla]

We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in Ω × (0, T)-Ω a bounded subset of R N-from a partial distributed observation. We employ a least-squares technique and minimize the L 2-norm of the distance from the observation to any solution. Taking the parabolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. The well-posedness of this mixed formulation-in particular the inf-sup property-is a consequence of classical energy estimates. We then reproduce the arguments to a linear first order system, involving the normal flux, equivalent to the linear parabolic equation. The method, valid in any dimension spatial dimension N , may also be employed to reconstruct solution for boundary observations. With respect to the hyperbolic situation considered in [10] by the first author, the parabolic situation requires-due to regularization properties-the introduction of appropriate weights function so as to make the problem numerically stable.

Mots clés: inverse problems; Parabolic equations

Domaines: Optimisation et contrôle [math.OC]

Arnaud Münch, Diego Araujo de Souza. Inverse problems for linear parabolic equations using mixed formulations -Part 1 : Theoretical analysis. 2015. ⟨hal-01188851⟩