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It is known that standard stochastic Galerkin methods encounter challengeswhen solving partial differential equations with high-dimensional randominputs, which are typically caused by the large number of stochastic basisfunctions required. It becomes crucial to properly choose effective basisfunctions, such that the dimension of the stochastic approximation space can bereduced. In this work, we focus on the stochastic Galerkin approximationassociated with generalized polynomial chaos (gPC), and explore the gPCexpansion based on the analysis of variance (ANOVA) decomposition. A conciseform of the gPC expansion is presented for each component function of the ANOVAexpansion, and an adaptive ANOVA procedure is proposed to construct the overallstochastic Galerkin system. Numerical results demonstrate the efficiency of ourproposed adaptive ANOVA stochastic Galerkin method for both diffusion andHelmholtz problems.

An adaptive ANOVA stochastic Galerkin method for partial differential equations with high-dimensional random inputs