  This paper deals with model-order reduction of parametric partial
differential equations (PPDE). More specifically, we consider the problem of
finding a good approximation subspace of the solution manifold of the PPDE when
only partial information on the latter is available. We assume that two sources
of information are available: i) a "rough" prior knowledge, taking the form of
a manifold containing the target solution manifold, ii) partial linear
measurements of the solutions of the PPDE (the term partial refers to the fact
that observation operator cannot be inverted). We provide and study several
tools to derive good approximation subspaces from these two sources of
information. We first identify the best worst-case performance achievable in
this setup and propose simple procedures to approximate the corresponding
optimal approximation subspace. We then provide, in a simplified setup, a
theoretical analysis relating the achievable reduction performance to the
choice of the observation operator and the prior knowledge available on the
solution manifold.
