数据稀缺与更新条件下基于概率密度演化-测度变换的认知不确定性量化分析

工程设计中往往需要同时处理固有不确定性与认知不确定性。对于固有不确定性分析与量化，国内外已有诸多研究，例如Monte Carlo方法、正交多项式展开理论和概率密度演化理论等。而对认知不确定性、特别是固有不确定性与认知不确定性耦合情况下的研究，则还相对缺乏。该文中，针对数据稀缺与数据更新导致的认知不确定性，首先分别引入Bootstrap方法和Bayes更新方法进行不确定性表征。在此基础上，结合基于概率密度演化-测度变换的两类不确定性量化统一理论新框架，提出了存在认知不确定性情况下的不确定性传播与可靠性分析高效方法及其具体数值算法。由此，给出了基于数据进行工程系统不确定性量化、传播与可靠性分析的基本途径。通过具有工程实际数据的3个工程实例分析，包括无限边坡稳定性分析、挡土墙稳定性分析和屋面桁架结构可靠性分析，验证了该文方法的精度和效率。

QUANTIFICATION OF EPISTEMIC UNCERTAINTY DUE TO DATA SPARSITY AND UPDATING BASED ON THE FRAMEWORK VIA SYNTHESIZING PROBABILITY DENSITY EVOLUTION METHOD AND CHANGE OF PROBABILITY MEASURE

Aleatory uncertainty and epistemic uncertainty generally exist simultaneously in engineering design.A variety of studies, including, e.g., the Monte Carlo simulation, orthogonal polynomial expansion and probabilitydensity evolution method, have been carried out in terms of the aleatory uncertainty. However, rather limitedattention has been paid to the epistemic uncertainty, especially the coupling of aleatory and epistemic uncertainty.In the present paper, in order to represent the epistemic uncertainty due to data sparsity or data updating, theBootstrap method and Bayesian update method are introduced, respectively. Further, the newly developedcompatible framework via synthesizing the probability density evolution method (PDEM) and the change ofprobability measure (COM) is incorporated to develop a highly efficient approach for the quantification andpropagation of uncertainty and reliability evaluation of systems involving not only aleatory but also epistemicuncertainty. Numerical algorithms are elaborated. Therefore, a path from observed data to quantification andpropagation of uncertainty and reliability evaluation is shaped. Three engineering cases with real data, including astability analysis of infinite slope model, a stability analysis of retaining wall model, and a reliability analysis ofroof truss structure, are illustrated, demonstrating the accuracy and efficiency of the proposed method.