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Uncertainty Quantification"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>This paper considers the problem of optimal recovery of an element $u$ of a Hilbert space ${\\cal H}$ from measurements of the form $\\ell_j(u)$, $j=1,\\dots,m$, where the $\\ell_j$ are known linear functionals on ${\\cal H}$. Problems of this type are well studied [C. A. Micchelli, T. J. Rivlin, and S. Winograd, Numer. Math., 26 (1976), pp. 191--200] and usually are carried out under an assumption that $u$ belongs to a prescribed model class, typically a known compact subset of ${\\cal H}$. Motivated by reduced modeling for solving parametric partial differential equations, this paper considers another setting, where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of ${\\cal H}$ of dimension $n$, or more generally, in the form of how well $u$ can be approximated by each subspace from a sequence of nested subspaces $V_0\\subset V_1\\cdots\\subset V_n$ with each $V_k$ of dimension $k$. A recovery algorithm for the one-space formulation was proposed in [Y. Maday, A. T. Patera, J. D. Penn and M. Yano, Internat. J. Numer. Methods Engrg., 102 (2015), pp. 933--965]. Their algorithm is proved, in the present paper, to be optimal. It is also shown how the recovery problem for the one-space problem has a simple formulation if certain favorable bases are chosen to represent $V_n$ and the measurements. The major contribution of the present paper is to analyze the multispace case by exploiting additional information derived from the whole hierarchy of spaces $V_j$ rather than only from the largest space $V_n$. It is shown that in this multispace case, the set of all $u$ that satisfy the given information can be described as the intersection of a family of known ellipsoids in ${\\cal H}$. It follows that a near optimal recovery algorithm in the multispace problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of $u$ in the multispace setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multispace problem, and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$.<\/jats:p>","DOI":"10.1137\/15m1025384","type":"journal-article","created":{"date-parts":[[2017,1,3]],"date-time":"2017-01-03T15:16:27Z","timestamp":1483456587000},"page":"1-29","source":"Crossref","is-referenced-by-count":61,"title":["Data Assimilation in Reduced Modeling"],"prefix":"10.1137","volume":"5","author":[{"given":"Peter","family":"Binev","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Albert","family":"Cohen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wolfgang","family":"Dahmen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ronald","family":"DeVore","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guergana","family":"Petrova","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Przemyslaw","family":"Wojtaszczyk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,1,3]]},"reference":[{"key":"atypb1","first-page":"1010","author":"Bashir O.","year":"2007","journal-title":"New York"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/100795772"},{"key":"atypb3","first-page":"371","author":"Bojanov B.","year":"1994","journal-title":"Basel"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(67)90040-7"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2011056"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492915000033"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530511001728"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2012027"},{"key":"atypb9","first-page":"155","author":"Combettes P.L.","year":"1996","journal-title":"New York"},{"key":"atypb10","first-page":"185","author":"Combettes P.L.","year":"2011","journal-title":"New York"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-013-9186-2"},{"key":"atypb12","unstructured":"R. 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