  In this paper a novel modification of the multilevel Monte Carlo approach,
allowing for further significant complexity reduction, is proposed. The idea of
the modification is to use the method of control variates to reduce variance at
level zero. We show that, under a proper choice of control variates, one can
reduce the complexity order of the modified MLMC algorithm down to
$\varepsilon^{-2+\delta}$ for any $\delta\in [0,1)$ with $\varepsilon$ being
the precision to be achieved. These theoretical results are illustrated by
several numerical examples.
