每年出版 6 期
ISSN 打印: 2152-5080
ISSN 在线: 2152-5099
Indexed in
AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES
摘要
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.
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