  The combination of reduced basis and collocation methods enables efficient
and accurate evaluation of the solutions to parameterized PDEs. In this paper,
we study the stochastic collocation methods that can be combined with reduced
basis methods to solve high-dimensional parameterized stochastic PDEs. We also
propose an adaptive algorithm using a probabilistic collocation method (PCM)
and ANOVA decomposition. This procedure involves two stages. First, the method
employs an ANOVA decomposition to identify the effective dimensions, i.e.,
subspaces of the parameter space in which the contributions to the solution are
larger, and sort the reduced basis solution in a descending order of error.
Then, the adaptive search refines the parametric space by increasing the order
of polynomials until the algorithm is terminated by a saturation constraint. We
demonstrate the effectiveness of the proposed algorithm for solving a
stationary stochastic convection-diffusion equation, a benchmark problem chosen
because solutions contain steep boundary layers and anisotropic features. We
show that two stages of adaptivity are critical in a benchmark problem with
anisotropic stochasticity.
