The multi-level Monte Carlo method for computing quantities of interest

The Monte Carlo (MC) method is the foundation for most ensemble simulations. It is well known that the MC converges to the theoretical expectation as fast as the inverse of the square root of the number of samples, which is considered slow when the variation is large. To obtain an accurate estimate of the expectation, a large number of simulations have to be carried, and the total cost for the ensemble simulations is N square, when N represents the number of degrees of freedom of the discrete system. The multi-level Monte Carlo (MMC) method, which is essentially a variation reduction method, was first proposed by Giles for stochastic differential equations, and then extended to stochastic partial differential equations by other authors. Through an approach not unlike the multigrid method, the MMC utilize a hierarchy of grids from high to low resolutions. The total computational cost is reduced to the order of N times the alpha power of log of N, where alpha depends on the dimensions. This talk consists of a brief introduction to the MMC method, a new analytical approach for addressing the challenges in applying the method to long-term climate modelings, and recent results from applying the method to quantify the volume transport of an idealized channel model for the Antarctic Circumpolar Current.