Taking into account uncertainties in computational models requires the construction of suitable probabilistic models. On the one hand, these models must ensure some fundamental properties for the operators involved in the associated stochastic boundary value problem (SBVP): such properties allow one to prove, in particular, the existence and uniqueness of the solution to the SBVP. On the other hand, the stochastic representations must faithfully reproduce the underlying physical phenomena, without introducing an arbitrary modeling bias.
In this context, we have proposed, identified and validated new classes of tensor-valued random fields. Additionally, new generators relying on families of stochastic differential equations were constructed in order to sample fields with values on highly-constrained matrix sets. Such models are useful for the modeling of stochastic elliptic operators, such as those entering formulations in linear elasticity or in the simulation of Darcean flows.
Various applications were proposed, including the modeling of elasticity and permeability random fields, or the modeling of random interphases in nanocomposites. An illustration for elasticity random fields that are orthotropic almost surely can be found in the figure below.
Realizations of the random fields modeling the components (11) and (44) of the elasticity tensor