  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.
