The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit-state function, which depends on the solution of a partial differential equation (PDE). In many applications, the PDE cannot be solved analytically. We can only evaluate an approximation of the exact PDE solution. Therefore, the probability of rare events is estimated with respect to an approximation of the limit-state function. This leads to an approximation error in the estimate of the probability of rare events. Indeed, we prove an error bound for the approximation error of the probability of failure, which behaves like the discretization accuracy of the PDE multiplied by an approximation of the probability of failure, the first order reliability method (FORM) estimate. This bound requires convexity of the failure domain. For non-convex failure domains, we prove an error bound for the relative error of the FORM estimate. Hence, we derive a relationship between the required accuracy of the probability of rare events estimate and the PDE discretization level. This relationship can be used to guide practicable reliability analyses and, for instance, multilevel methods.
翻译:对稀有事件概率的估计是可靠性和风险评估方面的一项重要任务。我们考虑的是以限值函数表示的故障事件,这取决于部分差分方程(PDE)的解决方案。在许多应用中,PDE无法通过分析解决。我们只能评估精确的PDE解决方案的近似值。因此,稀有事件的概率是按限值函数的近似值估算的。这导致稀有事件概率估计的近似误差。事实上,我们证明发生故障概率近似差错是注定的,这表现为PDE的离散精确度乘以故障概率的近似值,第一顺序可靠性方法(FORM)的估算。这一约束要求故障域的一致性。对于非convex故障域,我们证明与FORM估算的相对错误有关。因此,我们得出了稀有事件估计概率所要求的准确度与PDE离散程度之间的关系。这种关系可以用来指导实用的可靠性分析,例如多层次的方法。