  Partial differential equations (PDEs) are fundamental for theoretically
describing numerous physical processes that are based on some input fields in
spatial configurations. Understanding the physical process, in general,
requires computational modeling of the PDE. Uncertainty in the computational
model manifests through lack of precise knowledge of the input field or
configuration. Uncertainty quantification (UQ) in the output physical process
is typically carried out by modeling the uncertainty using a random field,
governed by an appropriate covariance function. This leads to solving
high-dimensional stochastic counterparts of the PDE computational models. Such
UQ-PDE models require a large number of simulations of the PDE in conjunction
with samples in the high-dimensional probability space, with probability
distribution associated with the covariance function. Those UQ computational
models having explicit knowledge of the covariance function are known as
aleatoric UQ (AUQ) models. The lack of such explicit knowledge leads to
epistemic UQ (EUQ) models, which typically require solution of a large number
of AUQ models. In this article, using a surrogate, post-processing, and domain
decomposition framework with coarse stochastic solution adaptation, we develop
an offline/online algorithm for efficiently simulating a class of EUQ-PDE
models.
