RT Journal Article ID 659db390164d3751 A1 Cho, Heyrim A1 Elman, Howard C. T1 AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES JF International Journal for Uncertainty Quantification JO IJUQ YR 2018 FD 2018-05-11 VO 8 IS 3 SP 193 OP 210 K1 reduced basis method K1 ANOVA decomposition K1 probabilistic collocation method K1 anisotropic stochasticity K1 high-dimensionality AB The combination of reduced basis and collocation methods enables efficient and accurate evaluation of the solutions to parametrized partial differential equations (PDEs). In this paper, we study the stochastic collocation methods that can be combined with reduced basis methods to solve high-dimensional parametrized 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. PB Begell House LK https://www.dl.begellhouse.com/journals/52034eb04b657aea,2a63c994718e44bd,659db390164d3751.html