🤖 AI Summary
Gradient-enhanced global sensitivity analysis (GSA) faces significant challenges under data-scarce conditions, where accurate estimation of Sobol′ indices requires both high-fidelity gradient information and efficient sample utilization.
Method: This paper proposes a surrogate modeling framework based on the Poincaré chaos expansion. Unlike conventional orthogonal bases, the Poincaré basis is uniquely characterized by simultaneous orthogonality of both basis functions and their first-order derivatives, derived from a Sturm–Liouville eigenvalue problem and compatible with arbitrary probability measures and weighting schemes. The method integrates a weighted Poincaré inequality, sparse gradient-enhanced regression, and a derivative-driven sensitivity weighting mechanism.
Contribution/Results: It enables highly accurate and computationally efficient estimation of Sobol′ indices. Evaluated on a flood modeling case study, the approach achieves reliable sensitivity ranking using only a small number of samples, demonstrating strong adaptability to complex real-world problems and significant computational advantages over existing methods.
📝 Abstract
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{é} inequalities and Sturm-Liouville eigenvalue problems, which we refer to as the Poincar{é} 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.