  Some classical uncertainty quantification problems require the estimation of
multiple expectations. Estimating all of them accurately is crucial and can
have a major impact on the analysis to perform, and standard existing Monte
Carlo methods can be costly to do so. We propose here a new procedure based on
importance sampling and control variates for estimating more efficiently
multiple expectations with the same sample. We first show that there exists a
family of optimal estimators combining both importance sampling and control
variates, which however cannot be used in practice because they require the
knowledge of the values of the expectations to estimate. Motivated by the form
of these optimal estimators and some interesting properties, we therefore
propose an adaptive algorithm. The general idea is to adaptively update the
parameters of the estimators for approaching the optimal ones. We suggest then
a quantitative stopping criterion that exploits the trade-off between
approaching these optimal parameters and having a sufficient budget left. This
left budget is then used to draw a new independent sample from the final
sampling distribution, allowing to get unbiased estimators of the expectations.
We show how to apply our procedure to sensitivity analysis, by estimating
Sobol' indices and quantifying the impact of the input distributions. Finally,
realistic test cases show the practical interest of the proposed algorithm, and
its significant improvement over estimating the expectations separately.
