We consider the problem of optimal recovery of an element u of a Hilbert spac e H from measurements of the form l_j(u)\, j = 1\, ... \, m\, where the l_j are known linear functio nals on H. Motivated by reduced modeling for solvi ng parametric partial diff\;erential equations \, we investigate a setting where the additional i nformation about the solution u is in the form of how well u can be approximated by a certain known subspace V_n of H of dimension n\, or more general ly\, in the form of how well u can be approximated by each of a sequence of nested subspaces V_0\, V _1\, ... \, V_n with each V_k of dimension k. The goal is to exploit additional information derived from the whole hierarchy of spaces rather than onl y from the largest space V_n. It is shown that\, i n this multispace case\, the set of all u that sat isfy the given information can be described as the intersection of a family of known ellipsoidal cyl inders in H and that a near optimal recovery algor ithm in the multi-space pr oblem is provided by id entifying any point in this intersection. LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR