Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T02:02:57.639Z Has data issue: false hasContentIssue false
  • English
  • Français

A Galerkin strategy with Proper Orthogonal Decomposition forparameter-dependent problems – Analysis, assessments and applications to parameterestimation

Published online by Cambridge University Press:  07 October 2013

D. Chapelle ,
A. Gariah ,
P. Moireau  and
J. Sainte-Marie
D. Chapelle
Affiliation:
Inria Saclay Ile-de-France, Palaiseau, France.. [email protected]
A. Gariah
Affiliation:
Inria Paris-Rocquencourt, Le Chesnay, France.
P. Moireau
Affiliation:
Inria Saclay Ile-de-France, Palaiseau, France.. [email protected]
J. Sainte-Marie
Affiliation:
Inria Paris-Rocquencourt, Le Chesnay, France. Université Paris 6, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Paris, France. CETMEF, Margny-les-Compiègne, France.
Get access

Abstract

We address the issue of parameter variations in POD approximations of time-dependentproblems, without any specific restriction on the form of parameter dependence.Considering a parabolic model problem, we propose a POD construction strategy allowing usto obtain some a priori error estimates controlled by the POD remainder –in the construction procedure – and some parameter-wise interpolation errors for the modelsolutions. We provide a thorough numerical assessment of this strategy with theFitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in twoparameter estimation applications, the first in a variational estimation framework withthe FitzHugh − Nagumo model, and the second with a beating heart mechanical model forwhich we employ a sequential estimation method to characterize model parameters using realimage data in a clinical case.

Keywords

Proper Orthogonal Decompositionparameter variationsestimationFitzHugh − Nagumo equationscardiac modeling
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1821 - 1843
Copyright
© EDP Sciences, SMAI 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amsallem, D. and Farhat, C., An online method for interpolating linear parametric reduced-order models. SIAM J. Sci. Comput. 33 (2011) 2169. Google Scholar
Banks, H.T., Joyner, M.L., Winchesky, B. and Winfree, W.P., Nondestructive evaluation using a reduced-order computational methodology. Inverse Problems 16 (2000) 1–17. Google Scholar
Berkooz, G., Holmes, P. and Lumley, J.L., The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1993) 539–575. Google Scholar
Buffa, A., Maday, Y., Patera, A.T., Prud’homme, C. and Turinici, G., A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: M2AN 46 (2012) 595–603.Google Scholar
Chabiniok, R., Moireau, P., Lesault, P.-F., Rahmouni, A., Deux, J.-F. and Chapelle, D., Estimation of tissue contractility from cardiac cine-MRI using a biomechanical heart model. Biomech. Model. Mechanobiol. 11 (2012) 609–630. Google ScholarPubMed
Chapelle, D. and Bathe, K.J., The inf-sup test. Comput. Struct. 47 (1993) 537–545. Google Scholar
Chapelle, D., Gariah, A. and Sainte-Marie, J., Galerkin approximation with Proper Orthogonal Decomposition: new error estimates and illustrative examples. ESAIM: M2AN 46 (2012) 731–757. Google Scholar
Chapelle, D., Le Tallec, P., Moireau, P. and Sorine, M., An energy-preserving muscle tissue model: formulation and compatible discretizations. J. Multiscale Comput. Engrg. 10 (2012) 189–211. Google Scholar
G. Chavent, Nonlinear Least Squares for Inverse Problems: Theoretical foundations and step-by-step guide for applications. Scientific Computation. Springer, New York (2009).
Ciarlet, P.G. and Raviart, P.A., General Lagrange and Hermite interpolation in R with applications to finite element methods. Arch. Rational Mech. Anal. 46 (1972) 177–199. Google Scholar
FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445–466. Google ScholarPubMed
Galbally, D., Fidkowski, K., Willcox, K. and Ghattas, O., Non-linear model reduction for uncertainty quantification in large-scale inverse problems. International J. Numer. Methods Engrg. 81 (2010) 1581—1608. Google Scholar
Haasdonk, B., Convergence rates of the POD-greedy method. ESAIM: M2AN 47 (2012) 859–873. Google Scholar
Julier, S., Uhlmann, J. and Durrant-Whyte, H., A new method for the nonlinear transformation of means and covariances in filter and estimators. IEEE Trans. Automat. Contr. 45 (2000) 447–482. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492–515. Google Scholar
Manzoni, A., Quarteroni, A. and Rozza, G., Shape optimization for viscous flows by reduced basis methods and free form deformation. Int. J. Numer. Methods in Fluids 70 (2012) 646–670. Google Scholar
Moireau, P. and Chapelle, D., Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems. ESAIM: COCV 17 (2011) 380–405. Google Scholar
Moireau, P., Chapelle, D. and Le Tallec, P., Joint state and parameter estimation for distributed mechanical systems. Comput. Methods Appl. Mechanics Engrg. 197 (2008) 659–677. Google Scholar
Moireau, P., Chapelle, D. and Le Tallec, P., Filtering for distributed mechanical systems using position measurements: Perspectives in medical imaging. Inverse Problems 25 (2009) 035010. Google Scholar
Nagumo, J., Arimoto, S. and Yoshizawa, S., An active pulse transmission line simulating nerve axon. Proc. of IRE 50 (1962) 2061–2070. Google Scholar
Pham, D.-T., Verron, J. and Gourdeau, L., Filtres de Kalman singuliers évolutifs pour l’assimilation de données en océanographie. C.R. l’Acad. Sci. – Series IIA 326 (1998) 255–260. Google Scholar
Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T. and Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engrg. 124 (2002) 70–80. Google Scholar
Sainte-Marie, J., Chapelle, D., Cimrman, R. and Sorine, M., Modeling and estimation of the cardiac electromechanical activity. Comput. Struct. 84 (2006) 1743–1759. Google Scholar
D. Simon, Optimal State Estimation: Kalman, H ∞, and Nonlinear Approaches. Wiley-Interscience (2006).
Smolyak, S.A.. Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4 (1963) 240–243. Google Scholar
Veroy, K. and Patera, A.T., Certified real-time solution of the parametrized steady incompressible navier-stokes equations. Internat. J. Numer. Methods Fluids 47 (2004) 773–788. Google Scholar