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Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

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Date
2018
Dewey
Analyse
Sujet
greedy algorithms; data assimilation; reduced models; sensor placement; inf-sup stability
Journal issue
SIAM/ASA Journal on Uncertainty Quantification
Volume
6
Number
3
Publication date
08-2018
Article pages
1101-1126
Publisher
SIAM - Society for Industrial and Applied Mathematics
DOI
http://dx.doi.org/10.1137/17M1157635
URI
https://basepub.dauphine.fr/handle/123456789/19808
Collections
  • CEREMADE : Publications
Metadata
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Author
Binev, Peter
106904 University of South Carolina
Cohen, Albert
25 Laboratoire Jacques-Louis Lions [LJLL]
Mula, Olga
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Nichols, James
25 Laboratoire Jacques-Louis Lions [LJLL]
Type
Article accepté pour publication ou publié
Abstract (EN)
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.

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