论文标题
错误分析具有近似模型的罕见事件的概率
Error analysis for probabilities of rare events with approximate models
论文作者
论文摘要
对罕见事件的可能性的估计是可靠性和风险评估中的重要任务。我们考虑以极限状态函数表示的故障事件，这取决于偏微分方程（PDE）的解决方案。在许多应用中，PDE无法分析解决。我们只能评估精确的PDE溶液的近似值。因此，相对于极限状态函数的近似值估计了罕见事件的概率。这导致了罕见事件概率估计的近似误差。实际上，我们证明了失败概率的近似误差限制的误差，该误差的行为就像PDE的离散精度乘以失败概率的近似值，第一阶可靠性方法（形式）估计值。该边界需要破坏域的凸度。对于非Convex故障域，我们证明了对形式估计的相对误差绑定的误差。因此，我们得出了罕见事件估计的概率所需准确性与PDE离散水平之间的关系。这种关系可用于指导可行的可靠性分析，例如多级方法。
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.
