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

An adaptive reduced basis collocation method based on PCM ANOVA decomposition for anisotropic stochastic PDEs