  In this paper, we propose a Bayesian approach for multiscale problems with
the availability of dynamic observational data. Our method selects important
degrees of freedom probabilistically in a Generalized multiscale finite element
method framework. Due to scale disparity in many multiscale applications,
computational models can not resolve all scales. Dominant modes in the
Generalized Multiscale Finite Element Method are used as "permanent" basis
functions, which we use to compute an inexpensive multiscale solution and the
associated uncertainties. Through our Bayesian framework, we can model
approximate solutions by selecting the unresolved scales probabilistically. We
consider parabolic equations in heterogeneous media. The temporal domain is
partitioned into subintervals. Using residual information and given dynamic
data, we design appropriate prior distribution for modeling missing subgrid
information. The likelihood is designed to minimize the residual in the
underlying PDE problem and the mismatch of observational data. Using the
resultant posterior distribution, the sampling process identifies important
degrees of freedom beyond permanent basis functions. The method adds important
degrees of freedom in resolving subgrid information and ensuring the accuracy
of the observations.
