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

TL;DR

The paper addresses efficient global sensitivity analysis when gradient information is available, by introducing gradient-enhanced chaos expansions based on a Poincaré basis. It proves that the derivative-stable property (orthogonal derivatives) uniquely identifies a Poincaré basis tied to weighted Poincaré inequalities and Sturm-Liouville problems. A full framework combines sparse gradient-enhanced regression with flexible derivative weighting (including the Stein kernel weight $w_{\mathrm{lin}}$) to estimate Sobol' indices and DGSM for broad distributions. Numerical experiments on toy and flood models show consistent improvements in Sobol' index estimation and surrogate accuracy, especially with weighting and derivative augmentation. The approach broadens applicability of gradient-enhanced GSA to non-classical distributions and offers practical, open-source tooling.

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.

Figures (8)

• Figure 1: First Poincaré basis functions (omitting the constant one) for $\mathcal{U}(0,1)$ and $\mathcal{E}(1)$ truncated on $[0,3]$, and $w \equiv 1$. Solid line: the basis function computed from the analytic expression; Dotted line: the basis function estimated by finite elements.
• Figure 2: First Poincaré basis functions (omitting the constant one) for $\mathcal{U}(0,1)$ and $\mathcal{E}(1)$ truncated on $[0,3]$, and $w = w_{\mathrm{lin}}$. The dotted lines represent the basis functions estimated by finite elements. On the left plot, the solid lines are the theoretical functions associated to $\mathcal{U}(0,1)$ (i.e. the Legendre polynomials)
• Figure 3: Results for the toy model in the unweighted case. Dotted lines represent the true total Sobol' indices.
• Figure 4: Results for the toy model in the weighted case. Dotted lines represent the true total Sobol' indices.
• Figure 5: $L_2(\mu)$ error. Results for the toy model, both unweighted and weighted cases.

Theorems & Definitions (10)

• Definition 1: Poincaré basis
• Proposition 1: Existence of the Poincaré basis
• Proposition 2: Poincaré basis and stability by differentiation
• Proposition 3
• Remark 1
• Lemma 2
• proof
• proof : Proof of Proposition \ref{['prop:sufficient_conditions_for_existence']}
• proof : Proof of Proposition \ref{['prop:PoincareBasis_and_Derivation']}
• proof : Proof of Proposition \ref{['prop:DGSMfromPoinCE']}