  Monte Carlo (MC) sampling is a popular method for estimating the statistics
(e.g. expectation and variance) of a random variable. Its slow convergence has
led to the emergence of advanced techniques to reduce the variance of the MC
estimator for the outputs of computationally expensive solvers. The control
variates (CV) method corrects the MC estimator with a term derived from
auxiliary random variables that are highly correlated with the original random
variable. These auxiliary variables may come from surrogate models. Such a
surrogate-based CV strategy is extended here to the multilevel Monte Carlo
(MLMC) framework, which relies on a sequence of levels corresponding to
numerical simulators with increasing accuracy and computational cost. MLMC
combines output samples obtained across levels, into a telescopic sum of
differences between MC estimators for successive fidelities. In this paper, we
introduce three multilevel variance reduction strategies that rely on
surrogate-based CV and MLMC. MLCV is presented as an extension of CV where the
correction terms devised from surrogate models for simulators of different
levels add up. MLMC-CV improves the MLMC estimator by using a CV based on a
surrogate of the correction term at each level. Further variance reduction is
achieved by using the surrogate-based CVs of all the levels in the MLMC-MLCV
strategy. Alternative solutions that reduce the subset of surrogates used for
the multilevel estimation are also introduced. The proposed methods are tested
on a test case from the literature consisting of a spectral discretization of
an uncertain 1D heat equation, where the statistic of interest is the expected
value of the integrated temperature along the domain at a given time. The
results are assessed in terms of the accuracy and computational cost of the
multilevel estimators, depending on whether the construction of the surrogates,
and the associated computational cost, precede the evaluation of the estimator.
It was shown that when the lower fidelity outputs are strongly correlated with
the high-fidelity outputs, a significant variance reduction is obtained when
using surrogate models for the coarser levels only. It was also shown that
taking advantage of pre-existing surrogate models proves to be an even more
efficient strategy.
