Posterior probabilistic learning constrained by stochastic PDE and experimental statistical moments of physics observations

Invited LectureInternational audienceWe consider a stochastic boundary value problem (SBVP) on a bounded domain of the three-dimensional space for which the unknown state is the random field Y. The partial differential equation of this SBVP depends on an uncontrolled non-Gaussian tensor-valued random field, G, and on a controlled vector-valued random parameter, W. We also consider a random vector O = obs(Y,G,W) of physics observations. It is assumed that are given the prior non-Gaussian probability model of G and W and experimental targets represented by a vector b of statistical moments E{h(O)} of random vector O (such as the mean vector and a global dispersion parameter). We are interested in estimating the posterior probability model of G and W given the target b. Such a problem belongs to the class of statistical inverse problems and is generally solved by using the maximum likelihood or Bayesian approaches. In the present framework, such statistics methodologies cannot be used for the following reasons. (1) The numerical cost of one evaluation of the stochastic computational model (SCM) that discretizes the SBVP is very high. Consequently, the SCM can only be called a small number of times. (2) The problem is non-Gaussian and is in very high dimension. (3) The experimental realizations of random observation O are unknown; only statistical moments of O, represented by vector b = E{h(O)}, are given as targets. Consequently maximum likelihood and Bayesian methodologies cannot be used. For solving this challenging problem, we propose an approach that can be briefly summarized as follows. (i) A training set having a small number of points is constructed using the SCM: given a realization (g,w) of (G,W) using the prior probability model, the SBVP is solved for computing the corresponding realization y of Y and then the realization o = obs(y,g,w) of O is deduced. (ii) A formulation of an optimization problem based on the Kullback-Leibler minimum cross-entropy principle is proposed for constructing the posterior probability model under the constraint defined by the targets E{h(O)} = b and by the constraint consisting in minimizing the norm of the random residue of the SBVP. (iii) This optimization problem is solved by using a probabilistic learning method from the training set under the introduced constraints. As an application, we present the stochastic homogenization of a 3D random anisotropic elastic medium with uncertain spectrum for which observation O is the reshaping of the fourth-order effective elasticity tensor, C, at macroscale. We consider the case for which there is no scale separation, yielding C as a random tensor