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International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2018024436
pages 193-210

AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES

Heyrim Cho
Department of Mathematics, University of Maryland, College Park, MD 20742
Howard C. Elman
Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, USA

ABSTRACT

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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