We consider a family of parametric elliptic pdes \u2212div(a\u2207u)) = f on a domain D with zero boundary conditions and diffusion coefficient a satisfying the ellipticity condition r \u2264 a \u2264 R on D with r > 0, If a varies continuously over a parameter set, then the solution ua describes a solution manifold. We study the question of whether we can determine a when ua is known and the related question of how well we can approximate a when we have partial information of ua given by data observations. In the case that a is uniquely determined by ua, we study the related question of how smooth is the inverse map ua \u2192 a. Concerning the recovery of a from partial measurements of the state, we discuss possible strategies and their cost in terms of the entropy of the solution manifold.<\/p>\n