Variance-based global sensitivity analysis, in particular Sobol' analysis, is widely used for determining the importance of input variables to a computational model. Sobol' indices can be computed cheaply based on spectral methods like polynomial chaos expansions (PCE). Another choice are the recently developed Poincar\'e chaos expansions (PoinCE), whose orthonormal tensor-product basis is generated from the eigenfunctions of one-dimensional Poincar\'e differential operators. In this paper, we show that the Poincar\'e basis is the unique orthonormal basis with the property that partial derivatives of the basis form again an orthogonal basis with respect to the same measure as the original basis. This special property makes PoinCE ideally suited for incorporating derivative information into the surrogate modelling process. Assuming that partial derivative evaluations of the computational model are available, we compute spectral expansions in terms of Poincar\'e basis functions or basis partial derivatives, respectively, by sparse regression. We show on two numerical examples that the derivative-based expansions provide accurate estimates for Sobol' indices, even outperforming PCE in terms of bias and variance. In addition, we derive an analytical expression based on the PoinCE coefficients for a second popular sensitivity index, the derivative-based sensitivity measure (DGSM), and explore its performance as upper bound to the corresponding total Sobol' indices.

翻译：基于方差的全局灵敏度分析，特别是Sobol分析，广泛应用于确定输入变量对计算模型的重要性。Sobol指数可通过谱方法（如多项式混沌展开PCE）低成本计算。另一选择是近期发展的庞加莱混沌展开（PoinCE），其正交张量积基由一维庞加莱微分算子的特征函数生成。本文证明：庞加莱基具有唯一性，其偏导数构成与原始基同测度的正交基。这一特性使PoinCE特别适合在代理模型过程中整合导数信息。假设计算模型的偏导数可求，我们分别基于庞加莱基函数或基偏导数通过稀疏回归计算谱展开。通过两个数值案例证明，基于导数的展开能为Sobol指数提供精确估计，在偏差和方差上甚至优于PCE。此外，基于庞加莱系数推导出另一常用灵敏度指数——基于导数的灵敏度度量（DGSM）的解析表达式，并探究其作为相应总Sobol指数上界的性能。