  Multilevel Monte Carlo (MLMC) is a recently proposed variation of Monte Carlo
(MC) simulation that achieves variance reduction by simulating the governing
equations on a series of spatial (or temporal) grids with increasing
resolution. Instead of directly employing the fine grid solutions, MLMC
estimates the expectation of the quantity of interest from the coarsest grid
solutions as well as differences between each two consecutive grid solutions.
When the differences corresponding to finer grids become smaller, hence less
variable, fewer MC realizations of finer grid solutions are needed to compute
the difference expectations, thus leading to a reduction in the overall work.
This paper presents an extension of MLMC, referred to as multilevel control
variates (MLCV), where a low-rank approximation to the solution on each grid,
obtained primarily based on coarser grid solutions, is used as a control
variate for estimating the expectations involved in MLMC. Cost estimates as
well as numerical examples are presented to demonstrate the advantage of this
new MLCV approach over the standard MLMC when the solution of interest admits a
low-rank approximation and the cost of simulating finer grids grows fast.
