On one dimensional weighted Poincare inequalities for Global Sensitivity Analysis

TL;DR

This paper advances the theoretical understanding of weighted one-dimensional Poincare inequalities in Global Sensitivity Analysis, introducing new spectral methods, data-driven weight construction, and demonstrating practical benefits in toy and real-world models.

Contribution

It provides new spectral-based theoretical results for weighted Poincare inequalities, constructs weights from monotonic functions, and develops methods for data-driven weight estimation in GSA.

Findings

New spectral proof for weighted Poincare inequalities.

Construction of weights ensuring eigenfunction basis for Sobol index approximation.

Demonstrated benefits of weighted inequalities in toy and flooding models.

Abstract

One-dimensional Poincare inequalities are used in Global Sensitivity Analysis (GSA) to provide derivative-based upper bounds and approximations of Sobol indices. We add new perspectives by investigating weighted Poincare inequalities. Our contributions are twofold. In a first part, we provide new theoretical results for weighted Poincare inequalities, guided by GSA needs. We revisit the construction of weights from monotonic functions, providing a new proof from a spectral point of view. In this approach, given a monotonic function g, the weight is built such that g is the first non-trivial eigenfunction of a convenient diffusion operator. This allows us to reconsider the linear standard, i.e. the weight associated to a linear g. In particular, we construct weights that guarantee the existence of an orthonormal basis of eigenfunctions, leading to approximation of Sobol indices with Parseval formulas. In a second part, we develop specific methods for GSA. We study the equality case of the upper bound of a total Sobol index, and link the sharpness of the inequality to the proximity of the main effect to the eigenfunction. This leads us to theoretically investigate the construction of data-driven weights from estimators of the main effects when they are monotonic, another extension of the linear standard. Finally, we illustrate the benefits of using weights on a GSA study of two toy models and a real flooding application, involving the Poincare constant and/or the whole eigenbasis.