Poincaré Bases Extend Gradient Sensitivity Analysis Beyond Polynomials

Global sensitivity analysis often has to work in the regime where model evaluations are expensive and the input distribution is not one of the few cases handled cleanly by classical polynomial chaos. Derivatives can help, but they create a compatibility problem: a basis that is orthogonal for function approximation need not have derivatives that are also orthogonal in the right weighted space. This paper argues that the compatible choice is not an arbitrary polynomial family but the basis induced by weighted Poincaré inequalities and one-dimensional Sturm--Liouville eigenproblems.

Core Contribution

The central result is a characterization: under the paper's assumptions, the only basis whose derivatives also form an orthogonal basis is the Poincaré basis associated with a weighted Poincaré inequality. That gives the authors a principled replacement for standard polynomial chaos when the measure is nonstandard or when derivative-based sensitivity information is available.

The contribution is both theoretical and algorithmic. The paper defines Poincaré chaos expansions, or PoinCE, and a weighted variant, wPoinCE. It then connects them to two gradient-enhanced estimation schemes: an averaged derivative expansion and a combined regression using both model values and gradients. The point is not merely to add gradients to a regression design, but to choose basis functions whose derivative structure matches the sensitivity quantities being estimated.

Figure 2 shows the construction at the basis-function level: for uniform and truncated exponential measures, the finite-element estimates track the analytic Poincaré functions, while the weighted choice changes the shapes relative to the unweighted case.

Technical Approach

For each input variable, the method solves a one-dimensional weighted Sturm--Liouville problem to obtain basis functions adapted to the marginal distribution and derivative weight. Tensor-product chaos expansions are then built from these one-dimensional bases, with total-degree truncation used in the experiments. The implementation described in the paper uses UQLab, finite-element approximations for the spectral problem, Latin hypercube experimental designs, and LARS sparse regression with leave-one-out model selection.

The paper compares several estimators. Plain PoinCE and wPoinCE use only model evaluations. PoinCE-der-aggr and wPoinCE-der-aggr use averaged derivative expansions. PoinCE-comb-regr and wPoinCE-comb-regr solve a combined regression problem using both model values and gradients. The evaluation tracks three quantities: an $H^1(\mu,w)$ validation error combining function and derivative mismatch, an $L^2(\mu)$ validation error, and total Sobol indices computed from the expansion.

A useful detail is the treatment of gradient cost. If gradients are available at no extra cost, the derivative-enhanced methods get a direct advantage. If gradients are approximated by finite differences, one gradient evaluation costs roughly $d$ model evaluations, so the fair comparison shifts to larger model-only designs. The authors explicitly discuss both views rather than reporting a single favorable setting.

Results and Analysis

The first experiment is a four-dimensional interaction toy model with independent uniform inputs, total degree $p=8$, and experimental design sizes from 25 to 200. In the unweighted case, both derivative-based expansions reduce the $H^1(\mu,w)$ error relative to value-only PoinCE, and their total Sobol index estimates for $X_1$ and $X_4$ converge toward the reference dotted lines as the design grows. In the weighted case, wPoinCE-comb-regr is visibly stronger in $H^1(\mu,w)$ error, while the total Sobol estimates are similar across the weighted variants once the design is large enough.

The paper's more informative comparison is the $L^2(\mu)$ error, because $H^1(\mu,w)$ changes with the weight. Figure 5 shows a clear advantage for the weighted construction on the toy model, especially for the combined gradient regression at larger design sizes.

The flood-cost model tests the method on a less polynomial-friendly setting. The model has eight independent inputs and a cost function based on maximal annual overflow and dyke maintenance cost. Several marginals are nonstandard for classical polynomial chaos, including truncated and triangular distributions, so the Poincaré basis is a better fit to the problem statement than a fixed Wiener--Askey family. The authors report the same pattern: derivative-based methods outperform value-only PoinCE in the unweighted case, and in the weighted case wPoinCE-comb-regr gives the best reported behavior. When gradient cost is counted as $d=8$ model evaluations, the advantage narrows, which is the right caveat for practitioners whose simulators do not provide adjoints or analytic gradients.

Limitations

The evidence supports the paper's claim that Poincaré bases are a coherent way to combine chaos expansions, derivatives, and Sobol index estimation. The strongest part is the mathematical characterization of the basis. The empirical case is more modest: two numerical studies, finite-element basis approximations, and comparisons under assumptions about gradient availability. For users with cheap gradients and nonstandard input measures, the method is attractive. For users who must estimate gradients by finite differences in high dimension, the cost accounting may erase much of the gain.