On one dimensional weighted Poincaré inequalities for Global Sensitivity Analysis

Global Sensitivity Analysis (GSA) is an active field of mathematics that aims at quantifying the influence of input parameters on complex systems arising in engineering. In this paper, we provide new perspectives on one-dimensional weighted Poincaré inequalities and apply them in GSA to establish derivative-based upper bounds and approximations of Sobol indices. In a first part, we provide new theoretical results for weighted Poincaré inequalities. Based on spectral properties of the associated diffusion operator, we study the construction of weights from monotonic functions, extending the classical case of linear functions. We then construct non-vanishing weights that guarantee the existence of an orthonormal basis of eigenfunctions. This allows us to approximate Sobol indices using Parseval formulas. In a second part we develop specific methods for GSA. We investigate the construction of data-driven weights from estimators of the main effects when they are monotonic. Finally, we illustrate numerically our results on two toy models and a real flooding application.