Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE
Grenier, E. ; Louvet, V. ; Vigneaux, P.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014), p. 1303-1329 / Harvested from Numdam
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Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases impossible to do within a reasonable time. We propose here a very simple methodology which may accelerate population parametrization of complex models, including partial differential equations models. We illustrate our method on the classical KPP equation.

Publié le : 2014-01-01
DOI : https://doi.org/10.1051/m2an/2013140
Classification:  35Q62,  35Q92,  35R30,  65C40,  35K57 
@article{M2AN_2014__48_5_1303_0,
     author = {Grenier, Emmanuel and Louvet, V. and Vigneaux, P.},
     title = {Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {48},
     year = {2014},
     pages = {1303-1329},
     doi = {10.1051/m2an/2013140},
     mrnumber = {3264355},
     zbl = {1301.35177},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2014__48_5_1303_0}
}

Grenier, E.; Louvet, V.; Vigneaux, P. Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 1303-1329. doi : 10.1051/m2an/2013140. http://gdmltest.u-ga.fr/item/M2AN_2014__48_5_1303_0/

[1] H.T. Banks and K. Kunisch, Estimation techniques for distributed parameter systems, vol. 1. Systems & Control: Foundations & Appl. Birkhäuser Boston Inc., Boston, MA (1989). | MR 1045629 | Zbl 0695.93020

[2] G.T. Brauns, R.J. Bishop, M.B. Steer, J.J. Paulos and S.H. Ardalan, Table-based modeling of delta-sigma modulators using Zsim. IEEE Trans. Computer-Aided Design Integr. Circuits Syst. 9 (1990) 142-150.

[3] B. Delyon, M. Lavielle and E. Moulines, Convergence of a stochastic approximation version of the EM algorithm. Ann. Statis. 27 94-128 (1999). | MR 1701103 | Zbl 0932.62094

[4] S. Donnet, J.-L. Foulley and A. Samson, Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations. Biometrics 66 (2010) 733-741. | MR 2758209 | Zbl 1203.62187

[5] S. Donnet and A. Samson, Parametric inference for mixed models defined by stochastic differential equations. ESAIM: PS 12 (2008) 196-218. | Numdam | MR 2374638 | Zbl 1182.62164

[6] R.A. Fisher, The Genetical Theory of Natural Selection. Oxford University Press (1930). | JFM 56.1106.13 | MR 1785121

[7] M.B. Giles and N.A. Pierce, An introduction to the adjoint approach to design. Flow, Turbulence and Combustion 65 (2000) 393-415. | Zbl 0996.76023

[8] W.M. Goughran, E. Grosse and D.J. Rose, CAzM: A circuit analyzer with macromodeling. IEEE Trans. Electron. Devices 30 (1983) 1207-1213.

[9] V. Isakov, Inverse problems for partial differential equations, vol. 127. Appl. Math. Sci., 2nd edition. Springer, New York (2006). | MR 2193218 | Zbl 1092.35001

[10] J. Kaipio and E. Somersalo, Statistical and computational inverse problems, vol. 160. Appl. Math. Sci. Springer-Verlag, New York (2005). | MR 2102218 | Zbl 1068.65022

[11] A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Étude de l'equation de la diffusion avec croissance de la quantite de matiere et son application a un problème biologique. Bulletin de l'université d'État à Moscou, Section A I (1937) 1-26. | Zbl 0018.32106

[12] E. Kuhn and M. Lavielle, Maximum likelihood estimation in nonlinear mixed effects models. Comput. Statis. Data Anal. 49 (2005) 1020-1038. | MR 2143055 | Zbl pre05374202

[13] M. Lavielle, Private Communication (2012).

[14] M. Lavielle and K. Bleakley, Population Approach & Mixed Effects Models - Models, Tasks, Tools & Methods. Avalaible at http://popix.lixoft.net/ INRIA (2013).

[15] M. Lavielle and F. Mentré, Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the monolix software. J. Pharmacokinetics and Pharmacodynamics 34 (2007) 229-249.

[16] J.-L. Lions, Optimal control of systems governed by partial differential equations. Translated from the French by S.K. Mitter. Springer-Verlag, New York (1971). | Zbl 0203.09001

[17] L. Nie, Strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models. Metrika 63 (2006) 123-143. | Zbl 1095.62028

[18] L. Nie and M. Yang, Strong consistency of mle in nonlinear mixed-effects models with large cluster size. Sankhya: Indian J. Statis. 67 (2005) 736-763. | Zbl 1193.62028

[19] E. Snoeck, P. Chanu, M. Lavielle, P. Jacqmin, E.N. Jonsson, K. Jorga, T. Goggin, J. Grippo, N.L. Jumbe and N. Frey, A Comprehensive Hepatitis C Viral Kinetic Model Explaining Cure. Clinical Pharmacology & Therapeutics 87 (2010) 706-713.

[20] A. Tarantola, Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005). | MR 2130010 | Zbl 1074.65013

[21] M. Team, The Monolix software, Version 4.1.2. Analysis of mixed effects models. Available at http://www.lixoft.com/ LIXOFT and INRIA (2012).

[22] G. Yu and P. Li, Efficient look-up-table-based modeling for robust design of sigma-delta ADCs. IEEE Trans. Circuits Syst. - I 54 (2007) 1513-1528.