Comptes Rendus
no. 23-24
Numerical analysis
A model-data weak formulation for simultaneous estimation of state and model bias
[Estimation de la variable dʼétat et du biais de modèle par une formulation faible incorporant les données]

Présenté par : Olivier Pironneau

Masayuki Yano1 ; James D. Penn1 ; Anthony T. Patera1
1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 937-941.

We introduce a Petrov–Galerkin regularized saddle approximation which incorporates a “model” (partial differential equation) and “data” (M experimental observations) to yield estimates for both state and model bias. We provide an a priori theory that identifies two distinct contributions to the reduction in the error in state as a function of the number of observations, M: the stability constant increases with M; the model-bias best-fit error decreases with M. We present results for a synthetic Helmholtz problem and an actual acoustics system.

Nous présentons une approximation de Petrov–Galerkin pour un problème de point selle incorporant un « modèle » (équation aux dérivées partielles) et des « données » (M observations expérimentales) afin dʼobtenir une estimation conjointe de la variable dʼétat et du biais de modèle. Notre théorie a priori identifie deux contributions à la décroissance de lʼerreur sur lʼétat en fonction du nombre dʼobservations expérimentales, M : la croissance de la constante stabilité avec M ; la décroissance de lʼestimation par moindre carré du biais de modèle avec M. Nous présentons des résultats pour un problème de Helmholtz synthétique ainsi que pour un système acoustique réel.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.10.034

Masayuki Yano 1 ; James D. Penn 1 ; Anthony T. Patera 1

1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 
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Masayuki Yano; James D. Penn; Anthony T. Patera. A model-data weak formulation for simultaneous estimation of state and model bias. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 937-941. doi : 10.1016/j.crma.2013.10.034. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.034/

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