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In this article we develop a multi-grid multi-level Monte Carlo (MGMLMC)method for the stochastic Stokes-Darcy interface model with random hydraulicconductivity both in the porous media domain and on the interface. Because therandomness through the interface affects the flow in the Stokes domain, weinvestigate the coupled stochastic Stokes-Darcy model to improve the fidelityas this model also considers the second and third porosity of the free flow.Then we prove the existence and uniqueness of the weak solution of thevariational form. For the numerical solution, we adopt the Monte Carlo (MC)method and finite element method (FEM), for the discrete form in theprobability space and physical space, respectively. In the traditionalsingle-level Monte Carlo (SLMC) method, more accurate numerical approximaterequires both larger number of samples in probability space and smaller meshsize in the physical space. Then the computational cost increase significantlyas the mesh size becomes smaller for the more accurate numerical approximate.Therefore we adopt the multi-level Monte Carlo (MLMC) method to dramaticallyreduce the computational cost in the probability space, because the number ofsamples decays fast while the mesh size decreases. We also develop a strategyto calculate the number of samples needed in MLMC method for the stochasticStokes-Darcy model. Furthermore MLMC naturally provides the hierarchial gridsand sufficient information on these grids for multi-grid (MG) method, which canin turn improve the efficiency of MLMC. In order to fully make use of thedynamical interaction between this two methods, we propose the multi-gridmulti-level Monte Carlo method for more efficiently solving the stochasticmodel. Numerical examples are provided to verify and illustrate the proposedmethod and the theoretical conclusions.

Multi-grid Multi-Level Monte Carlo Method for Stokes-Darcy interface Model with Random Hydraulic Conductivity