Population parametrization of costly black box models using iterations between SAEM algorithm and kriging
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Population parametrization of costly black box models using iterations between SAEM algorithm and kriging Emmanuel Grenier1,6 · Celine Helbert2 · Violaine Louvet2,3,6 · Adeline Samson4,5 · Paul Vigneaux1,6
Received: 5 November 2015 / Revised: 22 February 2016 / Accepted: 24 March 2016 / Published online: 16 April 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In this article we focus on parametrization of black box models from repeated measurements among several individuals (population parametrization). We introduce a variant of the SAEM algorithm, called KSAEM algorithm, which couples the standard SAEM algorithm with the dynamic construction of an approximate metamodel. The costly evaluation of the genuine black box is replaced by a kriging step, using a basis of precomputed values, basis which is enlarged during SAEM algorithm to improve the accuracy of the metamodel in regions of interest. Keywords Parameters estimation · SAEM algorithm · Kriging · Non-linear mixed effect models · Partial differential equations · KPP equation
Communicated by Florence Hubert.
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Paul Vigneaux [email protected] Emmanuel Grenier [email protected] Celine Helbert [email protected] Violaine Louvet [email protected] Adeline Samson [email protected]
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Univ Lyon, ENS de Lyon, CNRS UMR 5669, UMPA, 69364 Lyon, France
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Univ Lyon, CNRS UMR 5208, Ecole Centrale de Lyon, ICJ, 69622 Villeurbanne, France
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Present Address: GRICAD, Universite Grenoble Alpes, Grenoble, France
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Universite Grenoble Alpes, LJK, 38000 Grenoble, France
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CNRS, LJK, 38000 Grenoble, France
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INRIA Grenoble Rhone-Alpes, Numed Team, 69364 Lyon, France
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E. Grenier et al.
Mathematics Subject Classification 65C60 · 65C40 · 62M05 · 60G15 · 65M32 · 65N21 · 35K57
1 Introduction In this article, we are concerned with the parametrization of models of the form y = f (t, Z ) + ε where y is the observable, t is the time of observation, Z the individual parameters and ε is a measurement error term. The model f is referred to as a “black box” model. It may be a system of ordinary differential equations, of partial differential equations, or a multi-agents system, or any combination of these model types. We will assume that it is costly, namely that its evaluation is very long. For instance one single evaluation of a reaction-diffusion equation in a complex geometry may last a few minutes or even a few hours if the coefficients are large or small, leading to a stiff behavior. In this paper, we focus on population parametrization from observations of f along time among N individuals. From these repeated longitudinal data, we search the distribution of the parameters Z in that given population of individuals. To take into account the various sources of variabilities (inter-individual and intra-individual variabilities), we use a non-linear mixed effect model. The non-linear mixed effect model links the jth measure, j = 1, . . . , Ni , yi j at times ti j for
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