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

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 $\Omega\times (0,T)$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - 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 \cite{NC-AM-InverseProblems} 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.