This paper is dedicated to enhancing the computational efficiency of traditional parallel-in-time methods for solving stochastic initial-value problems. The standard parareal algorithm often suffers from slow convergence when applied to problems with stochastic inputs, primarily due to the poor quality of the initial guess. To address this issue, we propose a hybrid parallel algorithm, termed KLE-CGC, which integrates the Karhunen-Lo\`{e}ve (KL) expansion with the coarse grid correction (CGC). The method first employs the KL expansion to achieve a low-dimensional parameterization of high-dimensional stochastic parameter fields. Subsequently, a generalized Polynomial Chaos (gPC) spectral surrogate model is constructed to enable rapid prediction of the solution field. Utilizing this prediction as the initial value significantly improves the initial accuracy for the parareal iterations. A rigorous convergence analysis is provided, establishing that the proposed framework retains the same theoretical convergence rate as the standard parareal algorithm. Numerical experiments demonstrate that KLE-CGC maintains the same convergence order as the original algorithm while substantially reducing the number of iterations and improving parallel scalability.

翻译：本文致力于提升传统并行时间方法求解随机初值问题的计算效率。标准parareal算法在应用于随机输入问题时，常因初始猜测质量不佳而导致收敛缓慢。为解决此问题，我们提出一种混合并行算法，称为KLE-CGC，该算法将Karhunen-Loève（KL）展开与粗网格修正（CGC）相结合。该方法首先利用KL展开实现高维随机参数场的低维参数化，随后构建广义多项式混沌（gPC）谱代理模型以实现解场的快速预测。以此预测值作为初始值，可显著提升parareal迭代的初始精度。本文提供了严格的收敛性分析，证明所提框架保持了与标准parareal算法相同的理论收敛速率。数值实验表明，KLE-CGC在保持与原算法相同收敛阶数的同时，大幅减少了迭代次数并提升了并行可扩展性。