Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints

TL;DR

This paper develops a probabilistic learning inference method using Kullback-Leibler divergence to estimate posterior models for stochastic boundary value problems with uncertainties, demonstrated through a complex application in stochastic homogenization.

Contribution

It introduces a novel methodology combining KL divergence minimization with implicit constraints and surrogate modeling for stochastic boundary value problems.

Findings

Effective estimation of posterior probability measures for stochastic BVPs.

Application to 3D stochastic homogenization of elastic media.

Inclusion of residual constraints improves model accuracy.

Abstract

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.