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Optimal Inequalities to Bound a Performance Probability

机译：绑定性能概率的最佳不等式

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A challenging problem encountered in engineering applications is the estimation of a probability-of-failure based on incomplete knowledge of the sources of uncertainty and/or limited sampling. Theories formulated to derive upper probability bounds offer an attractive alternative because first, they avoid postulating the probability laws that are often unknown and second, they substitute numerical optimization for statistical sampling. A critical assessment of one such technique is presented. It derives upper probability bounds from the McDiarmid concentration-of-measure theory, which postulates that fluctuations of a function are more-or-less concentrated about its mean value. Two applications of this theory are presented. The first application analyzes a "toy" polynomial function defined in two dimensions. The upper bounds of probability are calculated and compared to sampling-based estimates of the true-but-unknown probabilities. For this function, the upper bounds obtained are too broad to be useful. These results are confirmed by conducting a similar analysis on a real engineering system, where upper bounds of probability associated with resonant frequencies of a structural system are estimated. A high-fidelity finite element model, previously validated using vibration measurements, is used to predict the frequencies. In this application, the uncertainty is introduced by way of material properties and the effective preload of a beam-to-column connection, modeled explicitly. These applications suggest that the theory not only leads to upper bounds that are inefficient but that can also be sub-optimal if their numerical estimation is based on too few model runs. It is concluded that this particular theory, while mathematically attractive, may not be well suited for engineering applications.
机译：工程应用中遇到的具有挑战性的问题是基于对不确定性和/或有限的采样源的不完全知识来估计失败概率。制定衍生上概率界限的理论提供了一个有吸引力的替代方案，因为首先，他们避免假设通常未知的概率法，其次，它们替代统计采样的数值优化。提出了对一种这种技术的关键评估。它来自McDiarmid的测量浓度理论的上概率界限，其假设函数的波动更加或更少集中于其平均值。提出了这一理论的两个应用。第一个应用分析了两个维度定义的“玩具”多项式函数。计算概率的上限并将其与基于采样的基于采样的估计进行了比较的真实但未知的概率。对于此功能，所获得的上限太广泛而是有用的。通过对实际工程系统进行类似的分析来确认这些结果，其中估计与结构系统的谐振频率相关的概率的上限。以前使用振动测量验证的高保真有限元模型用于预测频率。在本申请中，不确定性通过材料特性和梁到列连接的有效预载引入，明确建模。这些应用程序表明，该理论不仅导致上限，效率低，但如果它们的数值估计基于太少的模型运行，则也可以是子最佳的。得出结论，这种特殊的理论在数学上有吸引力，不适合工程应用。

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