标题： 带Poincar{é}混沌展开的梯度增强全局敏感性分析 显示英文标题

标题： Gradient-enhanced global sensitivity analysis with Poincar{é} chaos expansions

摘要： 混沌展开在全局敏感性分析（GSA）中被广泛使用，因为它们利用L2空间的正交基来高效计算Sobol'指数，特别是在数据稀缺的情况下。 当存在导数时，我们认为一个理想的特性是基函数的导数也应形成正交基。 我们证明，满足这一特性的唯一基与加权Poincaré不等式和Sturm-Liouville本征值问题相关，我们将其称为Poincaré基。 然后，我们引入了一个全面的梯度增强GSA框架，该框架结合了最近在稀疏、梯度增强回归用于代理建模方面的进展，以及基于导数的敏感性分析的加权方案构建。 所提出的方法适用于一大类概率测度，并支持各种权重的选择。 我们在一个具有挑战性的洪水建模案例研究中展示了该方法的有效性，在此案例中，使用有限的数据准确估计了Sobol'指数。 显示英文摘要

摘要： Chaos expansions are widely used in global sensitivity analysis (GSA), as they leverage orthogonal bases of L2 spaces to efficiently compute Sobol' indices, particularly in data-scarce settings. When derivatives are available, we argue that a desirable property is for the derivatives of the basis functions to also form an orthogonal basis. We demonstrate that the only basis satisfying this property is the one associated with weighted Poincar{\'e} inequalities and Sturm-Liouville eigenvalue problems, which we refer to as the Poincar{\'e} basis. We then introduce a comprehensive framework for gradient-enhanced GSA that integrates recent advances in sparse, gradient-enhanced regression for surrogate modeling with the construction of weighting schemes for derivative-based sensitivity analysis. The proposed methodology is applicable to a broad class of probability measures and supports various choices of weights. We illustrate the effectiveness of the approach on a challenging flood modeling case study, where Sobol' indices are accurately estimated using limited data.