  The Stokes-Brinkman equations model fluid flow in highly heterogeneous porous
media. In this paper, we consider the numerical solution of the Stokes-Brinkman
equations with stochastic permeabilities, where the permeabilities in
subdomains are assumed to be independent and uniformly distributed within a
known interval. We employ a truncated anchored ANOVA decomposition alongside
stochastic collocation to estimate the moments of the velocity and pressure
solutions. Through an adaptive procedure selecting only the most important
ANOVA directions, we reduce the number of collocation points needed for
accurate estimation of the statistical moments. However, for even modest
stochastic dimensions, the number of collocation points remains too large to
perform high-fidelity solves at each point. We use reduced basis methods to
alleviate the computational burden by approximating the expensive high-fidelity
solves with inexpensive approximate solutions on a low-dimensional space. We
furthermore develop and analyze rigorous a posteriori error estimates for the
reduced basis approximation. We apply these methods to 2D problems considering
both isotropic and anisotropic permeabilities.
