2019/11/27 by Matieyendou Lamboni, Lamboni, Matieyendou
Decision Sciences · Engineering · #FOS: Mathematics #Fatigue and fracture mechanics #Probabilistic and Robust Engineering Design #Probability (math.PR) #Soil, Finite Element Methods
paper · pdf · doi:10.48550/arxiv.1911.12444
openalex publication_date 2019/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Weighted Poincaré-type and related inequalities provide upper bounds of the variance of functions. Their application in sensitivity analysis allows for quickly identifying the active inputs. Although the efficiency in prioritizing inputs depends on the upper bounds, the latter can be big, and therefore useless in practice. In this paper, an optimal weighted Poincaré-type inequality and gradient-based expression of the variance (integral equality) are studied for a wide class of probability measures. For a function f : ℝ → ℝn, we show that \rm Varμ(f) = ∫Ω× Ω ∇ f (x) ∇ f (x')T (F(min(x, x') - F(x)F(x'))/(ρ(x) ρ(x')) dμ(x) dμ(x') , and \rm Varμ(f) \preceq (1)/(2)∫Ω∇ f(x) ∇ f(x)T (F(x) (1-F(x)))/([ρ(x)]2) dμ(x) , with \rm Varμ(f) = ∫ΩffT dμ-∫Ωf dμ∫ΩfT dμ, F and ρ the distribution and the density functions, respectively. These results are generalized to cope with multivariate functions by making use of cross-partial derivatives, and they allow for proposing a new proxy-measure for multivariate sensitivity analysis, including Sobol' indices. Finally, the analytical and numerical tests show the relevance of our proxy-measure for identifying important inputs by improving the upper bounds from Poincaré inequalities.