  We introduce in this document a direct method allowing to solve numerically
inverse type problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave 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
hyperbolic 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. Under usual geometric optic conditions,
we show the well-posedness of this mixed formulation (in particular the inf-sup
condition) and then introduce a numerical approximation based on space-time
finite elements discretization. We prove the strong convergence of the
approximation and then discussed several examples for $N=1$ and $N=2$. The
problem of the reconstruction of both the state and the source term is also
addressed.
