The tail wags the distribution: Only sample the tails for efficient reliability analysis

TL;DR

The paper introduces Tail Stratified Sampling (TSS), a novel method for efficiently estimating rare failure probabilities by focusing sampling efforts on the distribution tails, improving reliability analysis accuracy and computational efficiency.

Contribution

It presents the TSS estimator, a versatile stratified sampling approach that directly targets distribution tails, enhancing failure probability estimation in complex systems.

Findings

TSS outperforms traditional methods in various analytical examples.

TSS effectively estimates small failure probabilities with fewer evaluations.

The method is applicable to high-dimensional reliability problems.

Abstract

To ensure that real-world infrastructure is safe and durable, systems are designed to not fail for any but the most rarely occurring parameter values. By only happening deep in the tails of the parameter distribution, failure probabilities are kept small. At the same time, it is essential to understand the risk associated with the failure of a system, no matter how unlikely. However, estimating such small failure probabilities is challenging; numerous system performance evaluations are necessary to produce even a single system state corresponding to failure, and each such evaluation is usually significantly computationally expensive. To alleviate this difficulty, we propose the Tail Stratified Sampling (TSS) estimator - an intuitive stratified sampling estimator for the failure probability that successively refines the tails of the system parameter distribution, enabling direct sampling of the tails, where failure is expected to occur. The most general construction of TSS is presented, highlighting its versatility and robustness for a variety of applications. The intuitions behind the formulation are explained, followed by a discussion of the theoretical and practical benefits of the method. Various details of the implementation are presented. The performance of the algorithm is then showcased through a host of analytical examples with varying failure domain geometries and failure probabilities as well as multiple numerical case studies of moderate and high dimensionality. To conclude, a qualitative comparison of TSS against the existing foundational variance-reduction methods for reliability analysis is presented, along with suggestions for future developments.