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

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.