It is known that standard stochastic Galerkin methods face challenges when solving partial differential equations (PDEs) with random inputs. These challenges are typically attributed to the large number of required physical basis functions and stochastic basis functions. Therefore, it becomes crucial to select effective basis functions to properly reduce the dimensionality of both the physical and stochastic approximation spaces. In this study, our focus is on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC). We delve into the low-rank approximation of the quasimatrix, whose columns represent the coefficients in the gPC expansions of the solution. We conduct an investigation into the singular value decomposition (SVD) of this quasimatrix, proposing a strategy to identify the rank required for a desired accuracy. Subsequently, we introduce both a simultaneous low-rank projection approach and an alternating low-rank projection approach to compute the low-rank approximation of the solution for PDEs with random inputs. Numerical results demonstrate the efficiency of our proposed methods for both diffusion and Helmholtz problems.

翻译：众所周知，标准随机伽辽金方法在求解含随机输入的偏微分方程（PDE）时面临挑战。这些挑战通常归因于所需物理基函数和随机基函数的数量庞大。因此，选择有效的基函数以适当降低物理近似空间和随机近似空间的维度变得至关重要。在本研究中，我们重点关注与广义多项式混沌（gPC）相关的随机伽辽金近似。我们深入研究了拟矩阵的低秩近似，其列向量表示解在gPC展开中的系数。我们对该拟矩阵的奇异值分解（SVD）进行了研究，提出了一种确定达到期望精度所需秩的策略。随后，我们引入了同时低秩投影方法和交替低秩投影方法，以计算含随机输入PDE解的低秩近似。数值结果表明，我们提出的方法对于扩散问题和亥姆霍兹问题均具有高效性。