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

Yano, Masayuki1 ; Penn, James D.1 ; Patera, Anthony T.1
1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Comptes Rendus. Mathématique, Tome 351 (2013) no. 23-24, pp. 937-941.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.10.034
Yano, Masayuki 1 ; Penn, James D. 1 ; Patera, Anthony T. 1

1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 
@article{CRMATH_2013__351_23-24_937_0,
     author = {Yano, Masayuki and Penn, James D. and Patera, Anthony T.},
     title = {A model-data weak formulation for simultaneous estimation of state and model bias},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {937--941},
     publisher = {Elsevier},
     volume = {351},
     number = {23-24},
     year = {2013},
     doi = {10.1016/j.crma.2013.10.034},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.10.034/}
}
TY  - JOUR
AU  - Yano, Masayuki
AU  - Penn, James D.
AU  - Patera, Anthony T.
TI  - A model-data weak formulation for simultaneous estimation of state and model bias
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 937
EP  - 941
VL  - 351
IS  - 23-24
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.10.034/
DO  - 10.1016/j.crma.2013.10.034
LA  - en
ID  - CRMATH_2013__351_23-24_937_0
ER  - 
%0 Journal Article
%A Yano, Masayuki
%A Penn, James D.
%A Patera, Anthony T.
%T A model-data weak formulation for simultaneous estimation of state and model bias
%J Comptes Rendus. Mathématique
%D 2013
%P 937-941
%V 351
%N 23-24
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.10.034/
%R 10.1016/j.crma.2013.10.034
%G en
%F CRMATH_2013__351_23-24_937_0
Yano, Masayuki; Penn, James D.; Patera, Anthony T. A model-data weak formulation for simultaneous estimation of state and model bias. Comptes Rendus. Mathématique, Tome 351 (2013) no. 23-24, pp. 937-941. doi : 10.1016/j.crma.2013.10.034. http://www.numdam.org/articles/10.1016/j.crma.2013.10.034/

[1] Antoni, J. A Bayesian approach to sound source reconstruction: optimal basis, regularization, and focusing, J. Acoust. Soc. Amer., Volume 131 (2012), pp. 2873-2890

[2] Dahmen, W.; Plesken, C.; Welper, G. Double greedy algorithms: reduced basis methods for transport dominated problems, Math. Model. Numer. Anal. (2013) (in press) | DOI

[3] Demkowicz, L.F.; Gopalakrishnan, J. A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation, Comput. Methods Appl. Mech. Eng., Volume 23–24 (2010), pp. 1558-1572

[4] Franceschini, G.; Macchietto, S. Model-based design of experiments for parameter precision: state of the art, Chem. Eng. Sci., Volume 63 (2008), pp. 4846-4872

[5] Kalman, R.E. A new approach to linear filtering and prediction problems, Trans. ASME – J. Basic Eng., Ser. D, Volume 82 (1960), pp. 35-45

[6] Li, Z.L.; Navon, I.M. Optimality of variational data assimilation and its relationship with the Kalman filter and smoother, Q. J. Roy. Meteor. Soc., Volume 127 (2001), pp. 661-683

[7] Lorenc, A.C. A global three-dimensional multivariate statistical interpolation scheme, Mon. Weather Rev., Volume 109 (1981), pp. 701-721

[8] Maday, Y.; Nguyen, N.C.; Patera, A.T.; Pau, G.S.H. A general multipurpose interpolation procedure: the magic points, Commun. Pure Appl. Math., Volume 8 (2009) no. 1, pp. 383-404

[9] Patera, A.T.; Rønquist, E.M. Regression on parametric manifolds: estimation of spatial field, functional outputs, and parameters from noisy data, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012), pp. 543-547

[10] Quarteroni, A.; Valli, A. Numerical Approximation of Partial Differential Equations, Springer, New York, 1997

[11] Yano, M.; Penn, J.D.; Patera, A.T. A model-data weak formulation for estimation of state and model bias; application to acoustics, Math. Model. Numer. Anal. (2013) (in preparation)

Cité par Sources :