  It is known that standard stochastic Galerkin methods encounter challenges
when solving partial differential equations with high-dimensional random
inputs, which are typically caused by the large number of stochastic basis
functions required. It becomes crucial to properly choose effective basis
functions, such that the dimension of the stochastic approximation space can be
reduced. In this work, we focus on the stochastic Galerkin approximation
associated with generalized polynomial chaos (gPC), and explore the gPC
expansion based on the analysis of variance (ANOVA) decomposition. A concise
form of the gPC expansion is presented for each component function of the ANOVA
expansion, and an adaptive ANOVA procedure is proposed to construct the overall
stochastic Galerkin system. Numerical results demonstrate the efficiency of our
proposed adaptive ANOVA stochastic Galerkin method for both diffusion and
Helmholtz problems.
