We present a reduced basis stochastic Galerkin method for partial differential equations with random inputs. In this method, the reduced basis methodology is integrated into the stochastic Galerkin method, resulting in a significant reduction in the cost of solving the Galerkin system. To reduce the main cost of matrix-vector manipulation involved in our reduced basis stochastic Galerkin approach, the secant method is applied to identify the number of reduced basis functions. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

翻译：我们提出了一种适用于随机输入偏微分方程的约化基随机伽辽金方法。该方法将约化基方法论融入随机伽辽金方法中，显著降低了求解伽辽金系统的计算成本。为减少约化基随机伽辽金方法中矩阵向量操作的主要开销，我们采用割线法确定约化基函数的数目。本文建立了该方法的通用数学框架，通过数值实验验证了其精度，并展示了其计算效率。