  Three algorithms are developed for uncertainty quantification in modeling
coupled Stokes and Darcy flows. The porous media may consist of multiple
regions with different properties. The permeability is modeled as a
non-stationary stochastic variable, with its log represented as a sum of local
Karhunen-Lo\`eve (KL) expansions. The problem is approximated by stochastic
collocation on either tensor-product or sparse grids, coupled with a multiscale
mortar mixed finite element method for the spatial discretization. A
non-overlapping domain decomposition algorithm reduces the global problem to a
coarse scale mortar interface problem, which is solved by an iterative solver,
for each stochastic realization. In the traditional implementation, each
subdomain solves a local Dirichlet or Neumann problem in every interface
iteration. To reduce this cost, two additional algorithms based on
deterministic or stochastic multiscale flux basis are introduced. The basis
consists of the local flux (or velocity trace) responses from each mortar
degree of freedom. It is computed by each subdomain independently before the
interface iteration begins. The use of the multiscale flux basis avoids the
need for subdomain solves on each iteration. The deterministic basis is
computed at each stochastic collocation and used only at this realization. The
stochastic basis is formed by further looping over all local realizations of a
subdomain's KL region before the stochastic collocation begins. It is reused
over multiple realizations. Numerical tests are presented to illustrate the
performance of the three algorithms, with the stochastic multiscale flux basis
showing significant savings in computational cost.
