Reliability Analysis Combining the Method of Moments with Control Variates

Abstract

Estimating failure probabilities is a critical challenge in practice, due to high-dimensional parameter spaces and small failure probability levels. Existing sample-based methods are dimensionally robust but face efficiency challenges when estimating small failure probabilities. Approximate methods provide a balance between accuracy and computational efficiency; however, their performance is often sensitive to the dimensionality of the parameter spaces.Among existing approximate methods, the Method of Moments (MoM) estimates failure probabilities by utilizing the higher-order moments of the performance function. While it provides analytical efficiency, it faces challenges in high-dimensional problems due to the difficulties in efficient moment estimation. Control Variates (CV), a variance reduction technique based on sampling, enhances moment estimation with efficiency independent of dimensionality by leveraging numerical models of different fidelities. However, it is rarely applied to the estimation of higher-order moments.This paper introduces an approach for reliability analysis that combines the MoM with CV, proposing estimators for the third and fourth raw moments of the performance function based on CV. The approach achieves significant computational savings compared to traditional sample-based approaches and demonstrates strong potential for high-dimensional applications.The effectiveness of the proposed approach is validated through three numerical examples, including non-Gaussian problems, computationally intensive finite-element models, and nonlinear dynamic systems. The results highlight its accuracy and efficiency.

Keywords: Failure probability, Method of Moments, Control Variates, High dimension, Small failure probability