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

TL;DR

This paper introduces an adaptive importance sampling and control variates method to efficiently estimate multiple expectations from the same sample, significantly reducing computational costs in uncertainty quantification tasks.

Contribution

It develops an adaptive algorithm that approaches optimal estimators for multiple expectations, with a stopping criterion and application to sensitivity analysis.

Findings

Significant reduction in estimation cost compared to separate methods

Effective application to Sobol' indices and input impact quantification

Demonstrated practical benefits on realistic test cases

Abstract

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.