Geodesic
Data assimilation of time under-sampled measurements using observers, the wave-like equation example
Cîndea, Nicolae1  ; Imperiale, Alexandre2  ; Moireau, Philippe2 
1 Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), Campus des Cézeaux –, BP 80026, 63171, Aubière cedex, France
2 Project-Team M Ξ DISIM, Inria Saclay-Île de France, France
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 635-669

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We propose a sequential data assimilation scheme using Luenberger type observers when only some space restricted time under-sampled measurements are available. More precisely, we consider a wave-like equation for which we assume known the restriction of the solution to an open non-empty subset of the spatial domain and for some time samples (typically the sampling step in time is much larger than the time discretization step). To assimilate the available data, two strategies are proposed and analyzed. The first strategy consists in assimilating data only if they are available and the second one in assimilating interpolation of the available data at all the discretization times. In order to tackle the spurious high frequencies which appear when we discretize the wave equation, for both strategies, we introduce a numerical viscous term. In this case, we prove some error estimates between the exact solution and our observers. Numerical simulations illustrate the theoretical results in the case of the one dimensional wave equation.

Reçu le :
DOI : 10.1051/cocv/2014042
Classification : 35L05, 35L20, 65D05, 65M12, 93B07, 93B70, 93D15
Keywords: Data assimilation, time under-sampled measurements, Luenberger observers, numerical analysis, interpolation

Cîndea, Nicolae 1 ; Imperiale, Alexandre 2 ; Moireau, Philippe 2

1 Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), Campus des Cézeaux –, BP 80026, 63171, Aubière cedex, France
2 Project-Team M Ξ DISIM, Inria Saclay-Île de France, France 
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     title = {Data assimilation of time under-sampled measurements using observers, the wave-like equation example},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {635--669},
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Cîndea, Nicolae; Imperiale, Alexandre; Moireau, Philippe. Data assimilation of time under-sampled measurements using observers, the wave-like equation example. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 635-669. doi: 10.1051/cocv/2014042

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