This paper is dedicated to exploring the NASA Langley Challenge on Optimization under Uncertainty by proposing a series of approaches for both forward and inverse treatment of uncertainty propagation and quantification. The primary effort is placed on the categorization of the subproblems as to be forward or inverse procedures, such that dedicated techniques are proposed for the two directions, respectively. The sensitivity analysis and reliability analysis are categorized as forward procedures, while modal calibration & uncertainty reduction, reliability-based optimization, and risk-based design are regarded as inverse procedures. For both directions, the overall approach is based on imprecise probability characterization where both aleatory and epistemic uncertainties are investigated for the inputs, and consequently, the output is described as the probability-box (P-box). Theoretic development is focused on the definition of comprehensive uncertainty quantification criteria from limited and irregular time-domain observations to extract as much as possible uncertainty information, which will be significant for the inverse procedure to refine uncertainty models. Furthermore, a decoupling approach is proposed to investigate the P-box along two directions such that the epistemic and aleatory uncertainties are decoupled, and thus a two-loop procedure is designed to propagate both epistemic and aleatory uncertainties through the systematic model. The key for successfully addressing this challenge is in obtaining on the balance among an appropriate hypothesis of the input uncertainty model, a comprehensive criterion of output uncertainty quantification, and a computational viable approach for both forward and inverse uncertainty treatment.

本文致力于通过提出一系列对不确定性传播和量化进行正向和逆向处理的方法来探索 NASA Langley 在不确定性下的优化挑战。主要工作是将子问题分类为正向或逆向过程，从而分别针对两个方向提出专用技术。敏感性分析和可靠性分析被归类为正向程序，而模态校准和不确定性降低、基于可靠性的优化和基于风险的设计被视为逆向程序。对于两个方向，总体方法都基于不精确的概率表征，其中对输入的偶然和认知不确定性进行了调查，因此，输出被描述为概率盒（P-box）。理论发展的重点是从有限的和不规则的时域观测中定义综合不确定性量化标准，以尽可能多地提取不确定性信息，这对于改进不确定性模型的逆过程具有重要意义。此外，提出了一种解耦方法来研究沿两个方向的 P-box，从而使认知不确定性和随机不确定性解耦，因此设计了一个双循环程序以通过系统模型传播认知不确定性和随机不确定性。成功应对这一挑战的关键是在输入不确定性模型的适当假设、输出不确定性量化的综合标准、