  The recent growth in multi-fidelity uncertainty quantification has given rise
to a large set of variance reduction techniques that leverage information from
model ensembles to provide variance reduction for estimates of the statistics
of a high-fidelity model. In this paper we provide two contributions: (1) we
utilize an ensemble estimator to account for uncertainties in the optimal
weights of approximate control variate (ACV) approaches and derive lower bounds
on the number of samples required to guarantee variance reduction; and (2) we
extend an existing multi-fidelity importance sampling (MFIS) scheme to leverage
control variates. As such we make significant progress towards both increasing
the practicality of approximate control variates$-$for instance, by accounting
for the effect of pilot samples$-$and using multi-fidelity approaches more
effectively for estimating low-probability events. The numerical results
indicate our hybrid MFIS-ACV estimator achieves up to 50% improvement in
variance reduction over the existing state-of-the-art MFIS estimator, which had
already shown outstanding convergence rate compared to the Monte Carlo method,
on several problems of computational mechanics.
