Greedy algorithms for optimal measurements selection in state estimation using reduced models
| Autor: | Peter Binev, Albert Cohen, James A. Nichols, Olga Mula |
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| Přispěvatelé: | University of South Carolina [Columbia], Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Discrete mathematics Basis (linear algebra) Applied Mathematics Hilbert space 010103 numerical & computational mathematics Function (mathematics) State (functional analysis) Space (mathematics) 01 natural sciences Manifold 010101 applied mathematics symbols.namesake [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] Position (vector) Modeling and Simulation symbols Discrete Mathematics and Combinatorics 0101 mathematics Statistics Probability and Uncertainty Greedy algorithm [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] Mathematics |
| Zdroj: | SIAM/ASA Journal on Uncertainty Quantification SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2018, ⟨10.1137/17M1157635⟩ |
| ISSN: | 2166-2525 |
| DOI: | 10.1137/17M1157635⟩ |
| Popis: | International audience; We consider the problem of optimal recovery of an unknown function u in a Hilbert space V from measurements of the form j (u), j = 1,. .. , m, where the j are known linear functionals on V. We are motivated by the setting where u is a solution to a PDE with some unknown parameters , therefore lying on a certain manifold contained in V. Following the approach adopted in [9, 3], the prior on the unknown function can be described in terms of its approximability by finite-dimensional reduced model spaces (V n) n≥1 where dim(V n) = n. Examples of such spaces include classical approximation spaces, e.g. finite elements or trigonometric polynomials, as well as reduced basis spaces which are designed to match the solution manifold more closely. The error bounds for optimal recovery under such priors are of the form µ(V n , W m)ε n , where ε n is the accuracy of the reduced model V n and µ(V n , W m) is the inverse of an inf-sup constant that describe the angle between V n and the space W m spanned by the Riesz representers of (1 ,. .. , m). This paper addresses the problem of properly selecting the measurement func-tionals, in order to control at best the stability constant µ(V n , W m), for a given reduced model space V n. Assuming that the j can be picked from a given dictionary D we introduce and analyze greedy algorithms that perform a sub-optimal selection in reasonable computational time. We study the particular case of dictionaries that consist either of point value evaluations or local averages, as idealized models for sensors in physical systems. Our theoretical analysis and greedy algorithms may therefore be used in order to optimize the position of such sensors. |
| Databáze: | OpenAIRE |
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