Random Matrix Models and Nonparametric Method for Uncertainty Quantification

Abstract

This paper deals with the fundamental mathematical tools and the associated computational aspects for constructing the stochastic models of random matrices that appear in the nonparametric method of uncertainties and in the random constitutive equations for multiscale stochastic modeling of heterogeneous materials. The explicit construction of ensembles of random matrices, but also the presentation of numerical tools for constructing general ensembles of random matrices are presented and can be used for high stochastic dimension. The developments presented are illustrated for the nonparametric method for multiscale stochastic modeling of heterogeneous linear elastic materials and for the nonparametric stochastic models of uncertainties in computational structural dynamics.

Uncertainty Quantification

“…This algorithm, introduced for the high dimensions, is based on the use of an ISDE associated with a stochastic dissipative Hamiltonian dynamical system, driven by a stochastic Gaussian white noise [192,100,206,207], and for which a damping parameter allows for controlling the obtention of the stationary solution. The developments are based on the use of the theoretical results that have been presented in Section 3.6.…”

“…The volume element dG on Euclidean space M n (R) and the volume element d S G on Euclidean space M S n (R) are defined [185,207] by The volume element dG on Euclidean space M n (R) and the volume element d S G on Euclidean space M S n (R) are defined [185,207] by…”

“…Such ensembles of random matrices have been introduced in [140,184,185,190,203,207] and are presented in Sections 5.4.7 and 5.4.8. New ensembles of random matrices are necessary for the development of the nonparametric probabilistic approach of the model uncertainties induced by the modeling errors in computational solid mechanics and in computational fluid mechanics, which differ from the GOE and from the other known ensembles from the random matrix theory.…”

“…Such advanced numerical tools for the high dimensions have been introduced in [98,192,207] and are presented in Section 5.4.10. In addition to the fundamental ensembles of positive-definite symmetric real random matrices for which the probability distribution and the associated generator can explicitly be constructed (see Sections 5.4.7 and 5.4.8), we are in need of advanced numerical tools for construct-ing any ensemble of random matrices for the high dimensions by using the Max-Ent principle, for which the explicit calculation of the Lagrange multipliers cannot be performed.…”

“…Mean Value [190,207] Decomposition of the mean value of a random matrix belonging to SE rect . Any random rectangular matrix [A] in SE rect is a second-order random matrix with values in M m,n (R), whose mean value…”