Error analysis for probabilities of rare events with approximate models

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. Hence, we derive a relationship between the required accuracy of the probability of rare events estimate and the PDE discretization level. As our error bound depends on a computable approximation of the failure probability, it is applicable in practicable reliability analyses and, for instance, in multilevel methods.