  We consider the problem of modeling heterogeneous materials where micro-scale
dynamics and interactions affect global behavior. In the presence of
heterogeneities in material microstructure it is often impractical, if not
impossible, to provide quantitative characterization of material response. The
goal of this work is to develop a Bayesian framework for uncertainty
quantification (UQ) in material response prediction when using nonlocal models.
Our approach combines the nonlocal operator regression (NOR) technique and
Bayesian inference. Specifically, we use a Markov chain Monte Carlo (MCMC)
method to sample the posterior probability distribution on parameters involved
in the nonlocal constitutive law, and associated modeling discrepancies
relative to higher fidelity computations. As an application, we consider the
propagation of stress waves through a one-dimensional heterogeneous bar with
randomly generated microstructure. Several numerical tests illustrate the
construction, enabling UQ in nonlocal model predictions. Although nonlocal
models have become popular means for homogenization, their statistical
calibration with respect to high-fidelity models has not been presented before.
This work is a first step towards statistical characterization of nonlocal
model discrepancy in the context of homogenization.
