Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.