In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability of our method, we apply it to address two crucial uncertainty quantification problems: stochastic elliptic equations and optimal control problems governed by stochastic elliptic PDE constraints. Based on varying dimension reduction ratios, our algorithm exhibits the capability to yield a high precision numerical solution for stochastic partial differential equations, or provides a rough representation of the exact solutions as a pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal control problems allows a diverse range of gradient-based unconstrained optimization methods, rendering it particularly appealing for computationally intensive large-scale problems. Numerical experiments are conducted and the results provide strong validation of the feasibility and effectiveness of our algorithm.

翻译：本文提出了一种低秩逼近方法，用于高效求解随机偏微分方程。该方法通过对刚度矩阵进行新颖的低秩逼近，在保持精度的同时显著降低了矩阵求逆的计算负载和存储需求。为展示所提方法的通用性与适用性，我们将其应用于两类关键的不确定性量化问题：随机椭圆方程和受随机椭圆型PDE约束的最优控制问题。基于不同的维度缩减比例，所提算法既能以高精度数值解求解随机偏微分方程，也可作为预处理阶段提供精确解的粗略表示。同时，该算法在求解随机最优控制问题时支持多种基于梯度的无约束优化方法，尤其适用于计算密集型大规模问题。数值实验结果有力验证了所提算法的可行性和有效性。