A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems - Analysis, assessments and applications to parameter estimation

Chapelle, D.; Gariah, A.; Moireau, P.; Sainte-Marie, J.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1821-1843

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

Classification : 65M60, 35A35, 35B45, 93E10
Keywords: proper orthogonal decomposition, parameter variations, estimation, Fitzhugh − Nagumo equations, cardiac modeling

[1] D. Amsallem and C. Farhat, An online method for interpolating linear parametric reduced-order models. SIAM J. Sci. Comput. 33 (2011) 2169. | Zbl | MR

[2] H.T. Banks, M.L. Joyner, B. Winchesky and W.P. Winfree, Nondestructive evaluation using a reduced-order computational methodology. Inverse Problems 16 (2000) 1-17. | Zbl | MR

[3] G. Berkooz, P. Holmes and J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1993) 539-575. | MR

[4] A. Buffa, Y. Maday, A.T. Patera, C. Prud'Homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: M2AN 46 (2012) 595-603. | Zbl | Numdam

[5] R. Chabiniok, P. Moireau, P.-F. Lesault, A. Rahmouni, J.-F. Deux and D. Chapelle, Estimation of tissue contractility from cardiac cine-MRI using a biomechanical heart model. Biomech. Model. Mechanobiol. 11 (2012) 609-630.

[6] D. Chapelle and K.J. Bathe, The inf-sup test. Comput. Struct. 47 (1993) 537-545. | Zbl | MR

[7] D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with Proper Orthogonal Decomposition: new error estimates and illustrative examples. ESAIM: M2AN 46 (2012) 731-757. | Zbl | MR | Numdam

[8] D. Chapelle, P. Le Tallec, P. Moireau and M. Sorine, An energy-preserving muscle tissue model: formulation and compatible discretizations. J. Multiscale Comput. Engrg. 10 (2012) 189-211.

[9] G. Chavent, Nonlinear Least Squares for Inverse Problems: Theoretical foundations and step-by-step guide for applications. Scientific Computation. Springer, New York (2009). | Zbl | MR

[10] P.G. Ciarlet and P.A. Raviart, General Lagrange and Hermite interpolation in R with applications to finite element methods. Arch. Rational Mech. Anal. 46 (1972) 177-199. | Zbl | MR

[11] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445-466. | MR

[12] D. Galbally, K. Fidkowski, K. Willcox and O. Ghattas, Non-linear model reduction for uncertainty quantification in large-scale inverse problems. International J. Numer. Methods Engrg. 81 (2010) 1581-1608. | Zbl | MR

[13] B. Haasdonk, Convergence rates of the POD-greedy method. ESAIM: M2AN 47 (2012) 859-873. | Zbl | MR | Numdam

[14] S. Julier, J. Uhlmann and H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filter and estimators. IEEE Trans. Automat. Contr. 45 (2000) 447-482. | Zbl | MR

[15] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492-515. | Zbl | MR

[16] A. Manzoni, A. Quarteroni and G. Rozza, Shape optimization for viscous flows by reduced basis methods and free form deformation. Int. J. Numer. Methods in Fluids 70 (2012) 646-670. | MR

[17] P. Moireau and D. Chapelle, Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems. ESAIM: COCV 17 (2011) 380-405. | Zbl | MR | Numdam

[18] P. Moireau, D. Chapelle and P. Le Tallec, Joint state and parameter estimation for distributed mechanical systems. Comput. Methods Appl. Mechanics Engrg. 197 (2008) 659-677. | Zbl | MR

[19] P. Moireau, D. Chapelle and P. Le Tallec, Filtering for distributed mechanical systems using position measurements: Perspectives in medical imaging. Inverse Problems 25 (2009) 035010. | Zbl | MR

[20] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon. Proc. of IRE 50 (1962) 2061-2070.

[21] D.-T. Pham, J. Verron and L. Gourdeau, Filtres de Kalman singuliers évolutifs pour l'assimilation de données en océanographie. C.R. l'Acad. Sci. - Series IIA 326 (1998) 255-260.

[22] C. Prud'Homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engrg. 124 (2002) 70-80.

[23] J. Sainte-Marie, D. Chapelle, R. Cimrman and M. Sorine, Modeling and estimation of the cardiac electromechanical activity. Comput. Struct. 84 (2006) 1743-1759. | MR

[24] D. Simon, Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches. Wiley-Interscience (2006).

[25] S.A. Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4 (1963) 240-243. | Zbl

[26] K. Veroy and A.T. Patera, Certified real-time solution of the parametrized steady incompressible navier-stokes equations. Internat. J. Numer. Methods Fluids 47 (2004) 773-788. | Zbl | MR