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.

DOI: 10.1051/cocv/2014042

Keywords: Data assimilation, time under-sampled measurements, Luenberger observers, numerical analysis, interpolation

^{1}; Imperiale, Alexandre

^{2}; Moireau, Philippe

^{2}

@article{COCV_2015__21_3_635_0, author = {C{\^\i}ndea, Nicolae and Imperiale, Alexandre and Moireau, Philippe}, 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}, publisher = {EDP-Sciences}, volume = {21}, number = {3}, year = {2015}, doi = {10.1051/cocv/2014042}, mrnumber = {3358625}, zbl = {1405.93045}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2014042/} }

TY - JOUR AU - Cîndea, Nicolae AU - Imperiale, Alexandre AU - Moireau, Philippe TI - Data assimilation of time under-sampled measurements using observers, the wave-like equation example JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 635 EP - 669 VL - 21 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014042/ DO - 10.1051/cocv/2014042 LA - en ID - COCV_2015__21_3_635_0 ER -

%0 Journal Article %A Cîndea, Nicolae %A Imperiale, Alexandre %A Moireau, Philippe %T Data assimilation of time under-sampled measurements using observers, the wave-like equation example %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 635-669 %V 21 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014042/ %R 10.1051/cocv/2014042 %G en %F COCV_2015__21_3_635_0

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, Volume 21 (2015) no. 3, pp. 635-669. doi : 10.1051/cocv/2014042. http://www.numdam.org/articles/10.1051/cocv/2014042/

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