  In this paper, we develop a Bayesian multiscale approach based on a
multiscale finite element method. Because of scale disparity in many multiscale
applications, computational models can not resolve all scales. Various subgrid
models are proposed to represent un-resolved scales. Here, we consider a
probabilistic approach for modeling un-resolved scales using the Multiscale
Finite Element Method (cf., [1, 2]). By representing dominant modes using the
Generalized Multiscale Finite Element, we propose a Bayesian framework, which
provides multiple inexpensive (computable) solutions for a deterministic
problem. These approximate probabilistic solutions may not be very close to the
exact solutions and, thus, many realizations are needed. In this way, we obtain
a rigorous probabilistic description of approximate solutions. In the paper, we
consider parabolic and wave equations in heterogeneous media. In each time
interval, the domain is divided into subregions. Using residual information, we
design appropriate prior and posterior distributions. The likelihood consists
of the residual minimization. To sample from the resulting posterior
distribution, we consider several sampling strategies. The sampling involves
identifying important regions and important degrees of freedom beyond permanent
basis functions, which are used in residual computation. Numerical results are
presented. We consider two sampling algorithms. The first algorithm uses
sequential sampling and is inexpensive. In the second algorithm, we perform
full sampling using the Gibbs sampling algorithm, which is more accurate
compared to the sequential sampling. The main novel ingredients of our approach
consist of: defining appropriate permanent basis functions and the
corresponding residual; setting up a proper posterior distribution; and
sampling the posteriors.
