  In this paper, we study the development of efficient multiscale methods for
flows in heterogeneous media. Our approach uses the Generalized Multiscale
Finite Element (GMsFEM) framework. The main idea of GMsFEM is to approximate
the solution space locally using a few multiscale basis functions. This is
typically achieved by selecting an appropriate snapshot space and a local
spectral decomposition, e.g., the use of oversampled regions in order to
achieve an efficient model reduction. However, the successful construction of
snapshot spaces may be costly if too many local problems need to be solved in
order to obtain these spaces. In this paper, we show that this efficiency can
be achieved using a moderate quantity of local solutions (or snapshot vectors)
with random boundary conditions on oversampled regions with zero forcing.
Motivated by the randomized algorithm presented in [19], we consider a snapshot
space which consists of harmonic extensions of random boundary conditions
defined in a domain larger than the target region. Furthermore, we perform an
eigenvalue decomposition in this small space. We study the application of
randomized sampling for GMsFEM in conjunction with adaptivity, where local
multiscale spaces are adaptively enriched. Convergence analysis is provided. We
present representative numerical results to validate the method proposed.
