Poincar\'e inequalities on intervals -- application to sensitivity analysis

TL;DR

This paper advances the understanding of Poincaré inequalities on intervals, providing new theoretical results and computational methods to improve sensitivity analysis in models with truncated distributions.

Contribution

It introduces new bounds and spectral methods for Poincaré constants on intervals, enhancing sensitivity analysis accuracy for truncated distributions.

Findings

Exact Poincaré constants for various distributions are computed.

Semi-analytical formulas are provided for common distributions.

Application demonstrates improved sensitivity analysis in hydrology.

Abstract

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincar\'e inequalities. Typically two kind of sensitivity indices are linked by a Poincar\'e type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper to compute. This allows performing a low-cost screening of unessential variables. The efficiency of this screening then highly depends on the accuracy of the upper bounds in Poincar\'e inequalities. The novelty in the questions concern the wide range of probability distributions involved, which are often truncated on intervals. After providing an overview of the existing knowledge and techniques, we add some theory about Poincar\'e constants on intervals, with improvements for symmetric intervals. Then we exploit the spectral interpretation for computing exact value of Poincar\'e constants of any admissible distribution on a given interval. We give semi-analytical results for some frequent distributions (truncated exponential, triangular, truncated normal), and present a numerical method in the general case. Finally, an application is made to a hydrological problem, showing the benefits of the new results in Poincar\'e inequalities to sensitivity analysis.