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

Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variates