# A Review on Local Failure Probability Sensitivity Analysis
## 1 Introduction  
In order to derive the reliability of a system, a limit state function \(g\) is often defined depending on different identified inputs of the system. The scalar output of the limit state function is then called the response of the system. This limit state function makes it possible to assess the state of the system thanks to the value of the response: if the value is positive, the system is safe, otherwise it is failing. We assume here that the inputs of the system are divided into two categories: deterministic inputs \(\mathbf{s}\in\mathbb{R}^{p}\) and random inputs \(\mathbf{X}\in\mathbb{R}^{d}\). Consequently, the response of the system is random and is denoted \(Y=g(\mathbf{s},\mathbf{X})\). The deterministic inputs \(\mathbf{s}\) of the system can be of different nature and, for the rest of this paper, depending on their nature, a specific notation will be used. Physical variables like design parameters of the system are thus denoted \(\delta\in\mathbb{R}^{n}\). Whereas distribution parameters of the probabilistic framework selected to model the random inputs \(\mathbf{X}\) are denoted \(\boldsymbol{\theta}\in\mathbb{R}^{m}\). Therefore, the deterministic input vector \(\mathbf{s}\) is written \(\mathbf{s}=[\boldsymbol{\theta},\delta]\), and \(m+n=p\). It is assumed that the limit state function \(g\) is numerically expensive (it can require the resolution of partial derivative equations for instance). Accordingly, the reliability analysis of the system must require a number of evaluations of \(g\) as small as possible and the use of advanced reliability techniques [1]. Understanding how the deterministic parameters \(\boldsymbol{s}\) impact the system's failure is highly valuable, especially in the context of Reliability-Based Design Optimization (RBDO), as discussed in [2]. RBDO is a field of research for engineers tasked with designing structures in the face of uncertainties. The principle of the RBDO problem usually revolves around striking a balance between the structure's cost and its reliability. Therefore, the optimization problem can be written with the minimization of an objective function describing the cost of the structure [3], under some probabilistic reliability constraints. Thoseconstraints typically involve the estimation of the failure probability \(P_{f}=\mathbb{P}(g(\mathbf{s},\mathbf{X})<0)\). The derivatives of \(P_{f}\) with respect to \(\mathbf{s}\) are then needed for gradient-based algorithms used to solve RBDO problems [4]. By integrating these derivatives with gradient-based optimization techniques, reliability-based design optimization becomes a data-driven and systematic approach to ensuring the safety and performance of critical systems. The estimation of these derivatives with respect to \(\mathbf{s}\) is part of the local reliability sensitivity analysis (RSA) [5] of the system. The derivatives, also called sensitivities, make it possible to optimize the reliability of the system. Indeed, these sensitivities provide valuable insights into how specific model parameters affect the failure probability, allowing engineers and designers to make informed decisions about adjusting those parameters to enhance reliability [6]. RSA [5] is in fact a specific form of sensitivity analysis (SA) [7] where instead of directly studying the influence of the system's inputs on the output \(Y\), one examines their influence on other quantities such as the failure probability. In this article, we propose to review different approaches for local RSA with regard to deterministic inputs. The quantity of interest is thus the vector of derivatives of \(P_{f}(\mathbf{s})\) according to \(\mathbf{s}\) for a fixed value of \(\mathbf{s}\). In practice, the estimation of these derivatives usually follows the computation of \(P_{f}(\mathbf{s})\). Therefore, the evaluations of \(g\) needed to estimate \(P_{f}(\mathbf{s})\) are repurposed for the computation of those derivatives to reduce the additional numerical cost [8].