One-dimensional Poincare inequalities are used in Global Sensitivity Analysis (GSA) to provide derivative-based upper bounds and approximations of Sobol indices. We add new perspectives by investigating weighted Poincare inequalities. Our contributions are twofold. In a first part, we provide new theoretical results for weighted Poincare inequalities, guided by GSA needs. We revisit the construction of weights from monotonic functions, providing a new proof from a spectral point of view. In this approach, given a monotonic function g, the weight is built such that g is the first non-trivial eigenfunction of a convenient diffusion operator. This allows us to reconsider the linear standard, i.e. the weight associated to a linear g. In particular, we construct weights that guarantee the existence of an orthonormal basis of eigenfunctions, leading to approximation of Sobol indices with Parseval formulas. In a second part, we develop specific methods for GSA. We study the equality case of the upper bound of a total Sobol index, and link the sharpness of the inequality to the proximity of the main effect to the eigenfunction. This leads us to theoretically investigate the construction of data-driven weights from estimators of the main effects when they are monotonic, another extension of the linear standard. Finally, we illustrate the benefits of using weights on a GSA study of two toy models and a real flooding application, involving the Poincare constant and/or the whole eigenbasis.

翻译：一维庞加莱不等式在全局敏感性分析中用于提供基于导数的Sobol指数上界及其近似估计。本文通过研究加权庞加莱不等式为该领域带来新的视角。我们的贡献主要体现在两个方面：在第一部分中，我们基于全局敏感性分析的需求，提出了加权庞加莱不等式的新理论结果。我们从谱视角出发重新审视基于单调函数构造权重的方法，并给出了新的证明。该方法针对单调函数g构造权重，使得g成为适当扩散算子的第一非平凡特征函数。基于此框架，我们重新考察了线性标准情形（即与线性函数g相关联的权重），特别构造了能保证特征函数正交基存在的权重，从而可通过帕塞瓦尔公式实现Sobol指数的近似计算。在第二部分中，我们针对全局敏感性分析开发了具体方法：研究总Sobol指数上界的等式成立条件，并将不等式的紧致性与主效应函数接近特征函数的程度建立联系。这促使我们从理论上探索当主效应函数具有单调性时，如何基于其估计量构造数据驱动权重——这是对线性标准情形的另一拓展。最后，我们通过两个玩具模型和一个真实洪水模拟案例，分别从庞加莱常数和完整特征基两个维度，具体展示了加权方法在全局敏感性分析中的优势。