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.

翻译：标准随机伽辽金方法在求解具有高维随机输入的偏微分方程时面临挑战，这通常源于所需随机基函数数量庞大。合理选择有效基函数以降低随机逼近空间维数变得至关重要。本文聚焦于基于广义多项式混沌（gPC）的随机伽辽金逼近，通过方差分析（ANOVA）分解探索gPC展开。针对ANOVA展开的每个分量函数给出了简洁形式的gPC展开，并提出自适应ANOVA流程以构建整体随机伽辽金系统。数值结果表明，本文提出的自适应ANOVA随机伽辽金方法在扩散问题和亥姆霍兹问题中均具有高效性。