%0 Journal Article
%A Cho, Heyrim
%A Elman, Howard C.
%D 2018
%I Begell House
%K reduced basis method, ANOVA decomposition, probabilistic collocation method, anisotropic stochasticity, high-dimensionality
%N 3
%P 193-210
%R 10.1615/Int.J.UncertaintyQuantification.2018024436
%T AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES
%U http://dl.begellhouse.com/journals/52034eb04b657aea,2a63c994718e44bd,659db390164d3751.html
%V 8
%X 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.
%8 2018-05-11