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

Abstract

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

Uncertainty Quantification

“…, w νexp exp relative to the random observation vector W of the stochastic computational model (see also [200,201,153,208]). In this section, this methodology is detailed for the random fields and allows for identifying the non-Gaussian matrix-valued random field {[K(x)], x ∈ Ω} by using the partial and limited experimental data w 1 exp , .…”

“…This ensemble will be useful for stochastic modeling of elliptic operators [152,203,208] (see also point 3 of Comment-2 in Section 5.4.7.1). This ensemble will be useful for stochastic modeling of elliptic operators [152,203,208] (see also point 3 of Comment-2 in Section 5.4.7.1).…”

“…In this section, we summarize the advanced prior stochastic model introduced and developed in [98] and more detailed in [208].…”

Computational stochastic homogenization of heterogeneous media from an elasticity random field having an uncertain spectral measure

“…The apparent elasticity field at mesoscale is modeled by a non-Gaussian positive-definite fourth-order tensor-valued homogeneous random field. This paper present an extension of the works [47,48,56,57,61] devoted to random field representations for stochastic elliptic boundary value problems and computational stochastic homogenization. We propose a novel probabilistic modeling to take into account uncertainties in the spectral measure of the random apparent elasticity field and we solve a stochastic elliptic boundary value problem (BVP) to perform the computational stochastic homogenization.…”

A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models

“…It consists in constructing prior and posterior stochastic models of uncertain model parameters pertaining, for example, to geometry, boundary conditions, and material properties . This approach was shown to be computationally efficient for both a μ ‐parametric HDM and its associated μ ‐parametric ROM (for example, see ) and for large‐scale statistical inverse problems . However, it does not take into account neither the model uncertainties induced by modeling errors introduced during the construction of a μ ‐parametric HDM (model form uncertainties) nor those due to model reduction .…”