Gradient-enhanced global sensitivity analysis with Poincar'e chaos expansions

Spectral methods, also known as 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 of the model are available, a desirable property, both for modeling and GSA purposes, 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 call Poincar'e basis. We also show that under certain conditions the Poincar'e basis achieves the same convergence rate as the best polynomial approximation for classes of smooth functions. We then introduce a comprehensive framework for gradient-enhanced GSA that integrates recent advances both in the construction of the expansion - with gradient-enhanced regression - and in the construction of weights for derivative-based sensitivity analysis. Furthermore, the proposed methodology is applicable to a broad class of probability measures and various choices of weights. We illustrate its efficiency on a challenging flood modeling case study, where Sobol' indices are accurately estimated using limited data.