Regression on parametric manifolds: Estimation of spatial fields, functional outputs, and parameters from noisy data

Anthony T. Patera; Einar M. Rønquist

doi:10.1016/j.crma.2012.05.002

Numerical Analysis

Regression on parametric manifolds: Estimation of spatial fields, functional outputs, and parameters from noisy data
[Régression sur des variétés paramétriques : estimation de champs spatiaux, sorties fonctionnelles, et paramètres à partir de données bruitées]

Présenté par : Philippe G. Ciarlet

Anthony T. Patera ¹ ; Einar M. Rønquist ²

¹ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
² Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 543-547.

Résumé
Abstract

Nous étendons la méthode dʼinterpolation empirique, EIM en abrégé (pour Empirical Interpolation Method), au contexte de la régression en présence de données bruitées sur une variété paramétrique. Les fonctions de bases sont calculées hors-ligne sur la base de la variété sans bruit ; les coefficients EIM dʼune fonction quelconque sur la variété sont calculés en-ligne sur la base des observations expérimentales à travers une formulation moindres carrés. Les erreurs induites par les données bruitées dans les coefficients EIM aussi bien que les sorties fonctionelle-linéaire associées sont quantifiées en intervalles de confiance et sans connaissance ni de la valeur du paramètre ni de la variance du bruit. Nous proposons aussi, dans le même esprit, une procédure dʼestimation de paramètre.

In this Note we extend the Empirical Interpolation Method (EIM) to a regression context which accommodates noisy (experimental) data on an underlying parametric manifold. The EIM basis functions are computed Offline from the noise-free manifold; the EIM coefficients for any function on the manifold are computed Online from experimental observations through a least-squares formulation. Noise-induced errors in the EIM coefficients and in linear-functional outputs are assessed through standard confidence intervals and without knowledge of the parameter value or the noise level. We also propose an associated procedure for parameter estimation from noisy data.

Reçu le : 2012-03-30
Accepté le : 2012-05-03
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.05.002

Affiliations des auteurs :

Anthony T. Patera ¹ ; Einar M. Rønquist ²

¹ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
² Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

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     author = {Anthony T. Patera and Einar M. R{\o}nquist},
     title = {Regression on parametric manifolds: {Estimation} of spatial fields, functional outputs, and parameters from noisy data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {543--547},
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     volume = {350},
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     year = {2012},
     doi = {10.1016/j.crma.2012.05.002},
     language = {en},
}

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Anthony T. Patera; Einar M. Rønquist. Regression on parametric manifolds: Estimation of spatial fields, functional outputs, and parameters from noisy data. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 543-547. doi : 10.1016/j.crma.2012.05.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.002/

Bibliographie
Cité par

[1] H.T. Banks; K. Kunisch Estimation Techniques for Distributed Parameter Systems, Birkhäuser, 1989

[2] M. Barrault; Y. Maday; N.C. Nguyen; A.T. Patera An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 667-672

[3] S. Chaturantabut, D.C. Sorensen, Discrete empirical interpolation for nonlinear model reduction, TR09-05, CAAM, Rice University, 2009.

[4] N.R. Draper; H. Smith Applied Regression Analysis, Wiley, 1998

[5] N.C. Nguyen; A.T. Patera; J. Peraire A ‘best points’ interpolation method for efficient approximation of parametrized functions, Int. J. Numer. Methods Eng., Volume 73 (2008), pp. 521-543

[6] N.C. Nguyen, J. Peraire, An interpolation method for the reconstruction and recognition of face images, in: A. Ranchordas, H. Araújo, J. Vitrià (Eds.), VISAPP 2007 Proceedings Second International Conference on Computer Vision Theory and Applications, vol. 2, Barcelona, Spain, March 8–11, 2007, pp. 91–96.

Ngoc Cuong Nguyen Model reduction techniques for parametrized nonlinear partial differential equations, Error Control, Adaptive Discretizations, and Applications, Part 1, Volume 58 (2024), p. 149 | DOI:10.1016/bs.aams.2024.03.005
Michael A. Demetriou Adaptive Techniques for the Online Estimation of Spatial Fields With Mobile Sensors, IEEE Transactions on Automatic Control, Volume 68 (2023) no. 9, p. 5669 | DOI:10.1109/tac.2022.3222752
Markus Dihlmann; Bernard Haasdonk A reduced basis Kalman filter for parametrized partial differential equations, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 22 (2016) no. 3, pp. 625-669 | DOI:10.1051/cocv/2015019 | Zbl:1346.35245
N. C. Nguyen; H. Men; R. M. Freund; J. Peraire Functional regression for state prediction using linear PDE models and observations, SIAM Journal on Scientific Computing, Volume 38 (2016) no. 2, p. b247-b271 | DOI:10.1137/14100275x | Zbl:1350.60038
Y. Maday; O. Mula; A. T. Patera; M. Yano The generalized empirical interpolation method: stability theory on Hilbert spaces with an application to the Stokes equation, Computer Methods in Applied Mechanics and Engineering, Volume 287 (2015), pp. 310-334 | DOI:10.1016/j.cma.2015.01.018 | Zbl:1423.76346
N. C. Nguyen; J. Peraire Gaussian functional regression for linear partial differential equations, Computer Methods in Applied Mechanics and Engineering, Volume 287 (2015), pp. 69-89 | DOI:10.1016/j.cma.2015.01.008 | Zbl:1423.35362
Yvon Maday; Anthony T. Patera; James D. Penn; Masayuki Yano A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics, International Journal for Numerical Methods in Engineering, Volume 102 (2015) no. 5, pp. 933-965 | DOI:10.1002/nme.4747 | Zbl:1352.65529
Yvon Maday; Olga Mula A Generalized Empirical Interpolation Method: Application of Reduced Basis Techniques to Data Assimilation, Analysis and Numerics of Partial Differential Equations, Volume 4 (2013), p. 221 | DOI:10.1007/978-88-470-2592-9_13
Masayuki Yano; James D. Penn; Anthony T. Patera A model-data weak formulation for simultaneous estimation of state and model bias, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 351 (2013) no. 23-24, pp. 937-941 | DOI:10.1016/j.crma.2013.10.034 | Zbl:1281.65148

Cité par 9 documents. Sources : Crossref, zbMATH

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