This paper presents two novel ensemble domain decomposition methods for fast-solving the Stokes-Darcy coupled models with random hydraulic conductivity and body force. To address such random systems, we employ the Monte Carlo (MC) method to generate a set of independent and identically distributed deterministic model samples. To facilitate the fast calculation of these samples, we adroitly integrate the ensemble idea with the domain decomposition method (DDM). This approach not only allows multiple linear problems to share a standard coefficient matrix but also enables easy-to-use and convenient parallel computing. By selecting appropriate Robin parameters, we rigorously prove that the proposed algorithm has mesh-dependent and mesh-independent convergence rates. For cases that require mesh-independent convergence, we additionally provide optimized Robin parameters to achieve optimal convergence rates. We further adopt the multi-level Monte Carlo (MLMC) method to significantly lower the computational cost in the probability space, as the number of samples drops quickly when the mesh becomes finer. Building on our findings, we propose two novel algorithms: MC ensemble DDM and MLMC ensemble DDM, specifically for random models. Furthermore, we strictly give the optimal convergence order for both algorithms. Finally, we present several sets of numerical experiments to showcase the efficiency of our algorithm.

翻译：本文针对具有随机水力传导系数与体积力的Stokes-Darcy耦合模型，提出了两种新型集成区域分解快速求解方法。为处理此类随机系统，我们采用蒙特卡洛（MC）方法生成一组独立同分布的确定性模型样本。为实现这些样本的快速计算，我们巧妙地将集成思想与区域分解方法（DDM）相结合。该方法不仅允许多个线性问题共享标准系数矩阵，还能实现易于使用的便捷并行计算。通过选取合适的Robin参数，我们严格证明了所提算法具有网格依赖与网格无关的收敛速率。对于需要网格无关收敛的情形，我们还提供了优化的Robin参数以实现最优收敛速率。我们进一步采用多级蒙特卡洛（MLMC）方法以显著降低概率空间的计算成本，因为当网格加密时所需样本数量会快速下降。基于以上研究，我们针对随机模型提出了两种新型算法：MC集成DDM与MLMC集成DDM，并严格给出了两种算法的最优收敛阶。最后，我们通过多组数值实验展示了所提算法的高效性。