  This paper presents new results allowing an unknown non-Gaussian positive
matrix-valued random field to be identified through a stochastic elliptic
boundary value problem, solving a statistical inverse problem. A new general
class of non-Gaussian positive-definite matrix-valued random fields, adapted to
the statistical inverse problems in high stochastic dimension for their
experimental identification, is introduced and its properties are analyzed. A
minimal parametrization of discretized random fields belonging to this general
class is proposed. Using this parametrization of the general class, a complete
identification procedure is proposed. New results of the mathematical and
numerical analyzes of the parameterized stochastic elliptic boundary value
problem are presented. The numerical solution of this parametric stochastic
problem provides an explicit approximation of the application that maps the
parameterized general class of random fields to the corresponding set of random
solutions. This approximation can be used during the identification procedure
in order to avoid the solution of multiple forward stochastic problems. Since
the proposed general class of random fields possibly contains random fields
which are not uniformly bounded, a particular mathematical analysis is
developed and dedicated approximation methods are introduced. In order to
obtain an algorithm for constructing the approximation of a very
high-dimensional map, complexity reduction methods are introduced and are based
on the use of low-rank approximation methods that exploit the tensor structure
of the solution which results from the parametrization of the general class of
random fields.
