  We consider the problem of numerically approximating the solutions to an
elliptic partial differential equation (PDE) for which the boundary conditions
are lacking. To alleviate this missing information, we assume to be given
measurement functionals of the solution. In this context, a near optimal
recovery algorithm based on the approximation of the Riesz representers of
these functionals in some intermediate Hilbert spaces is proposed and analyzed
in [Binev et al. 2024]. Inherent to this algorithm is the computation of $H^s$,
$s>1/2$, inner products on the boundary of the computational domain. We take
advantage of techniques borrowed from the analysis of fractional diffusion
problems to design and analyze a fully practical near optimal algorithm not
relying on the challenging computation of $H^s$ inner products.
