Saddlepoint Approximation Method in Reliability Analysis: A Review

Abstract

The escalating need for reliability analysis (RA) and reliability-based design optimization (RBDO) within engineering challenges has prompted the advancement of saddlepoint approximation methods (SAM) tailored for such problems. This article offers a detailed overview of the general SAM and summarizes the method characteristics first. Subsequently, recent enhancements in the SAM theoretical framework are assessed. Notably, the mean value first-order saddlepoint approximation (MVFOSA) bears resemblance to the conceptual framework of the mean value second-order saddlepoint approximation (MVSOSA); the latter serves as an auxiliary approach to the former. Their distinction is rooted in the varying expansion orders of the performance function as implemented through the Taylor method. Both the saddlepoint approximation and third-moment (SATM) and saddlepoint approximation and fourth-moment (SAFM) strategies model the cumulant generating function (CGF) by leveraging the initial random moments of the function. Although their optimal application domains diverge, each method consistently ensures superior relative precision, enhanced efficiency, and sustained stability. Every method elucidated is exemplified through pertinent RA or RBDO scenarios. By juxtaposing them against alternative strategies, the efficacy of these methods becomes evident. The outcomes proffered are subsequently employed as a foundation for contemplating prospective theoretical and practical research endeavors concerning SAMs. The main purpose and value of this article is to review the SAM and reliability-related issues, which can provide some reference and inspiration for future research scholars in this field.

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