  In this paper, we consider the implementation of multi-level Monte Carlo
method to a stochastic optimal control problem with log-normal coefficients and
its surrogate model problem. From the perspective of two optimization problems,
i.e., minimizing the accuracy using a fixed computational cost and minimizing
the total computational cost to attain a given accuracy, we derive formulas to
determine the optimal sample sizes for each level of multi-level Monte Carlo
method. Furthermore, we put forward the multi-level Monte Carlo algorithm for
our stochastic optimal control problem and some tricks to deal with the
multi-level log-normal coefficients. Finally, we present the numerical results
of both the elliptic SPDEs and our control problem to validate the
effectiveness over the traditional Monte Carlo method.
