Multilevel Monte Carlo method based on multigrid discretizations for stochastic eigenvalue problem

In this paper, for a stochastic eigenvalue problem, we establish a new multilevel Monte Carlo (MLMC) method based on multigrid discretization to calculate the expectation of the minimum eigenvalue. We use the MLMC method to disperse the samples into the discrete eigenvalue problem at each level, and use the multigrid discretization based on the shifted-inverse iteration to solve the discrete eigenvalue problem at each level. By making full use of the characteristics of these two methods, the computational cost is reduced, meanwhile the accuracy of the approximate solution is maintained. We present ample numerical experiments to verify the complexity theorem and show the efficiency of the new algorithms in terms of the mean, variance, computation time, and sample size.