Abstract Sensitivity analysis plays an important role in reliability evaluation, structural optimization and structural design, etc. The local sensitivity, i.e., the partial derivative of the quantity of interest in terms of parameters or basic variables, is inadequate when the basic variables are random in nature. Therefore, global sensitivity such as the Sobol’ indices based on the decomposition of variance and the moment-independent importance measure, among others, have been extensively studied. However, these indices are usually computationally expensive, and the information provided by them has some limitations for decision making. Specifically, all these indices are positive, and therefore they cannot reveal whether the effects of a basic variable on the quantity of interest are positive or adverse. In the present paper, a novel global sensitivity index is proposed when randomness is involved in structural parameters. Specifically, a functional perspective is firstly advocated, where the probability density function (PDF) of the output quantity of interest is regarded as the output of an operator on the PDF of the source basic random variables. The Frechet derivative is then naturally taken as a measure for the global sensitivity. In some sense such functional perspective provides a unified perspective on the concepts of global sensitivity and local sensitivity. In the case the change of the PDF of a basic random variable is due to the change of parameters of the PDF of the basic random variable, the computation of the Frechet-derivative-based global sensitivity index can be implemented with high efficiency by incorporating the probability density evolution method (PDEM) and change of probability measure (COM). The numerical algorithms are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method.

中文翻译：

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基于Fréchet导数的全局敏感度指数及其高效数值分析

摘要 灵敏度分析在可靠性评价、结构优化和结构设计等方面具有重要作用。自然。因此，已广泛研究了全局敏感性，例如基于方差分解的 Sobol 指数和与矩无关的重要性度量等。然而，这些指标通常计算量很大，而且它们提供的信息对决策有一定的局限性。具体来说，所有这些指标都是正的，因此它们不能揭示一个基本变量对感兴趣数量的影响是正面的还是负面的。在本文中，当结构参数中涉及随机性时，提出了一种新的全局灵敏度指数。具体而言，首先提倡函数视角，将感兴趣的输出量的概率密度函数（PDF）视为算子在源基本随机变量的PDF上的输出。然后自然地将 Frechet 导数作为全局灵敏度的度量。从某种意义上说，这种功能视角为全局敏感性和局部敏感性的概念提供了统一的视角。如果基本随机变量的PDF的变化是由于基本随机变量的PDF参数的变化，通过结合概率密度演化方法（PDEM）和概率测度变化（COM），可以高效地实现基于 Frechet 导数的全局灵敏度指数的计算。详细阐述了数值算法。说明了几个例子，证明了所提出方法的有效性。