Generalized sentinels defined via least squares |
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Authors: | Chavent G |
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Institution: | (1) Domaine de Volucau-Rocquencourt, INRIA, BP 105, 78153 Le Chesnay Cédex, France;(2) CEREMADE, Université Paris Dauphine, 75775 Paris Cedex 16, France |
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Abstract: | We address the problem of monitoring a linear functional (c, x)Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorA![epsi](/content/vg442355g10g1808/xxlarge949.gif) (E; F). WhenE=E
1×E
2,c=(c
1 0), andA is injective and defined through the solution of a partial differential equation, Lions (6]–8]) introduced sentinelss F such that (s, Ax)Fis sensitive to x1 E
1 but insensitive to x2 E2. In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, ) ×E, where F withF dense in 80, such that for anya priori guess x0 ofx, we have s, Ax![rang](/content/vg442355g10g1808/xxlarge9002.gif) ![Fscr](/content/vg442355g10g1808/xxlarge8497.gif) + ( , x0)E=(c, x)E, where x is the least-squares estimate ofx closest tox
0, and (ii) a family of regularized sentinels (s
n
,
n
) F×E which converge to (s, ). Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of sentinels with special sensitivity in the sense of Lions 8]). |
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Keywords: | Least squares Sentinels Optimal control Regularization Duality |
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