Multi-level Monte Carlo ensemble domain decomposition method for the random Stokes-Darcy models with uncertain parameters

This paper presents a novel multi-level Monte Carlo ensemble domain decomposition method for efficiently solving Stokes-Darcy models characterized by random hydraulic conductivity and external forces. The multi-level Monte Carlo method is employed to significantly reduce computational cost in the probability space, as the required number of samples decreases substantially with spatial mesh refinement. By generating a set of independent and identically distributed deterministic model samples in different spatial meshes, we integrate the ensemble idea with the domain decomposition method to enable rapid computation. This integration not only allows multiple linear problems to share a common coefficient matrix, but also facilitates efficient parallel computations. Through a judicious selection of Robin parameters, we rigorously prove that the proposed algorithm exhibits both mesh-dependent and mesh-independent convergence rates. Furthermore, optimized Robin parameters are provided to achieve optimal convergence rates. Moreover, we rigorously establish the optimal convergence order for the proposed algorithm, demonstrating the superiority of the multi-level Monte Carlo method over traditional Monte Carlo. Finally, numerical experiments are presented to validate the efficiency of our proposed algorithm.