Identification of Bayesian posteriors for coefficients of chaos expansions

Handbook of Uncertainty Quantification

2015

The statistical inverse problem for the experimental identification of a non-Gaussian matrix-valued random field that is the model parameter of a boundary value problem, using some partial and limited experimental data related to a model observation, is a very difficult and challenging problem. A complete advanced methodology and the associated tools are presented for solving such a problem in the following framework: the random field that must be identified is a nonGaussian matrix-valued random field and is not simply a real-valued random field; this non-Gaussian random field is in high stochastic dimension and is identified in a general class of random fields; some fundamental algebraic properties of this non-Gaussian random field must be satisfied such as symmetry, positiveness, invertibility in mean square, boundedness, symmetry class, spatial-correlation lengths, etc; the available experimental data sets correspond only to partial and limited data for a model observation of the boundary value problem. The developments presented are mainly related to the elasticity framework, but the methodology is general and can be used in many areas of computational sciences and engineering. The developments are organized as follows. The first part is devoted to the definition of the statistical inverse problem that has to be solved in high stochastic dimension, and is focussed on stochastic elliptic operators such that the ones that are encountered in the boundary value problems of the linear elasticity. The second one deals with the construction of two possible parameterized representations for a non-Gaussian positive-definite matrix-valued random field that models the model parameter of a boundary value problem. A 1 Corresponding author, christian.soize@univ-paris-est.fr Quantification, Springer, 2016. parametric model-based representation is then constructed in introducing a statistical reduced model and a polynomial chaos expansion, first with deterministic coefficients and after with random coefficients. This parametric model-based representation is directly used for solving the statistical inverse problem. The third part is devoted to the description of all the steps of the methodology allowing the statistical inverse problem to be solved in high stochastic dimension. These steps are based on the identification of a prior stochastic model of the Non-Gaussian random field by using the maximum likelihood method and then, on the identification of a posterior stochastic model of the Non-Gaussian random field by using the Bayes method.The fourth part presents the construction of an algebraic prior stochastic model of the model parameter of the boundary value problem, for a non-Gaussian matrix-valued random field. The generator of realizations for such an algebraic prior stochastic model for a non-Gaussian matrix-valued random field is presented. Preprint submitted to Handbook for Uncertainty

Section: Parameterization Of the Non-gaussian Second-order Random Vecmentioning

confidence: 99%

Random Vectors and Random Fields in High Dimension: Parametric Model-Based Representation, Identification from Data, and Inverse Problems

Handbook of Uncertainty Quantification

2015

“…In the last decades, this very promising method has therefore been applied in many works (see [34,35,36,37,38,39,40,41,42] for more details about the inverse PCE identification from experimental data). For practical purposes, the sum that is defined by Eq.…”

Section: Kl and Pce Expansionsmentioning

confidence: 99%

Track irregularities stochastic modeling

Perrin

Probabilistic Engineering Mechanics

Duhamel

et al. 2013

Self Cite

High speed trains are currently meant to run faster and to carry heavier loads, while being less energy consuming and still respecting the security and comfort certification criteria. To face these challenges, a better understanding of the interaction between the dynamic train behavior and the track geometry is needed. As during its lifecycle, the train faces a great variability of track conditions, this dynamic behavior has indeed to be characterized on track portions sets that are representative of the whole railway network. This paper is thus devoted to the development of a stochastic modeling of the track geometry and its identification with experimental measurements. Based on a spatial and statistical decomposition, this model allows the spatial and statistical variability and dependency of the track geometry to be taken into account. Moreover, it allows the generation of realistic track geometries that are representative of a whole railway network. First, this paper describes a practical implementation of the proposed method and then applies this method to the modeling of a particular French high speed line, for which experimental data are available.

“…(24), had to be rejected. The identification of the vector-valued coefficients y 1 ,...,y N for a fixed value of N is done using the maximum likelihood method [36,42] as performed in [2,43,10]. For a given value of y 1 ,...,y N , the estimation of…”

Section: Summarizing the Identification Of A High-dimension Polynomiamentioning

confidence: 99%

“…However, taking into account a high number of random parameters is of first importance in a stochastic multiscale analysis and we thus propose a different methodology based on polynomial chaos representations. Initiated in [14], the methodology to construct a polynomial chaos expansion of random fields has been intensely developed to solve stochastic partial differential equations [3,15,16,13,24,22,29,32,31,35,38,11] but also for the identification of random fields using experimental data and classical inference techniques [17,2] or maximum likelihood estimation [9,10,43,18]. A new methodology has been recently introduced to deal with the identification of polynomial chaos representations in high-dimension [39,41].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials

Clément

Computer Methods in Applied Mechanics and Engineering

Yvonnet

2013

Self Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. AbstractThis paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials. At the microscale level, the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used. The geometry of the microstructure is random and characterized by a high number of random parameters. The method is based on a deterministic non-concurrent multiscale approach devoted to micro-macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale Cauchy Green strain states. To overcome the curse of dimensionality, due to the high number of involved random variables, the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion. Several strategies, based on novel algorithms dedicated to high stochastic dimension, are used and adapted for the class of multi-modal random variables which may characterize the potential. Numerical examples, at both small and large scales, allow analyzing the efficiency of the approach through comparisons with classical methods.