  We consider the problem of numerically approximating the solutions to a
partial differential equation (PDE) when there is insufficient information to
determine a unique solution. Our main example is the Poisson boundary value
problem, when the boundary data is unknown and instead one observes finitely
many linear measurements of the solution. We view this setting as an optimal
recovery problem and develop theory and numerical algorithms for its solution.
The main vehicle employed is the derivation and approximation of the Riesz
representers of these functionals with respect to relevant Hilbert spaces of
harmonic functions.
