Sobol' sensitivity indices allow to quantify the respective effects of random input variables and their combinations on the variance of mathematical model output. We focus on the problem of Sobol' indices estimation via a metamodeling approach where we replace the true mathematical model with a sample-based approximation to compute sensitivity indices. We propose a new method for indices quality control and obtain asymptotic and non-asymptotic risk bounds for Sobol' indices estimates based on a general class of metamodels. Our analysis is closely connected with the problem of nonparametric function fitting using the orthogonal system of functions in the random design setting. It considers the relation between the metamodel quality and the error of the corresponding estimator for Sobol' indices and shows the possibility of fast convergence rates in the case of noiseless observations. The theoretical results are complemented with numerical experiments for the approximations based on multivariate Legendre and Trigonometric polynomials.

翻译：Sobol 敏感度指数可以量化随机输入变量的各自影响,以及它们对数学模型输出差异的组合。 我们通过一种元模型方法关注Sobol 指数估算问题,我们用一个基于样本的近似值来计算敏感度指数,以此取代真正的数学模型。 我们提出了一个新的指数质量控制方法,并根据一个普通的元模型类别,获得Sobol 指数估算的无现时和非现时风险界限。 我们的分析与使用随机设计设置中的正方形函数系统来调整非对称函数的问题密切相关。 我们的分析考虑了Sobol 指数的元模型质量和对应的估测器错误之间的关系,并显示了无噪音观测情况下快速趋同率的可能性。 理论结果与基于多变式图例和三角测量多义的近似值的数字实验是相辅相成的。