Error analysis for probabilities of rare events with approximate models

TL;DR

This paper develops error bounds for estimating rare event probabilities using approximate PDE solutions, linking PDE discretization accuracy to reliability estimate precision, aiding practical risk assessment.

Contribution

It introduces new error bounds for probability estimates based on approximate PDE solutions, applicable to convex and non-convex failure domains, guiding reliability analysis.

Findings

Error bounds depend on PDE discretization accuracy and FORM estimates.

Bounds are applicable to convex failure domains and relative errors for non-convex domains.

Provides a relationship to guide PDE discretization for reliable rare event probability estimation.

Abstract

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.