Reliability Sensitivity with Response Gradient

TL;DR

This work develops a general theory for reliability sensitivity by expressing the derivative of the failure probability with respect to a sensitivity parameter as the product of the failure density and the conditional expectation of the response gradient given a threshold. A kernel-smoothed Monte Carlo approach is built to estimate this sensitivity for all thresholds in a single Subset Simulation run, enabling efficient handling of rare events and black-box responses. The methodology is demonstrated across multiple examples (normal response, buckling, SDOF first passage, pile design) to show robustness to different sensitivity parameter types and to illustrate bias-variance trade-offs and practical performance. The results highlight the potential to integrate gradient information directly into reliability analyses, improving risk-informed decision making and design under uncertainty.

Abstract

Engineering risk is concerned with the likelihood of failure and the scenarios when it occurs. The sensitivity of failure probability to change in system parameters is relevant to risk-informed decision making. Computing sensitivity is at least one level more difficult than the probability itself, which is already challenged by a large number of input random variables, rare events and implicit nonlinear `black-box' response. Finite difference with Monte Carlo probability estimates is spurious, requiring the number of samples to grow with the reciprocal of step size to suppress estimation variance. Many existing works gain efficiency by exploiting a specific class of input variables, sensitivity parameters, or response in its exact or surrogate form. For general systems, this work presents a theory and associated Monte Carlo strategy for computing sensitivity using response values and gradients with respect to sensitivity parameters. It is shown that the sensitivity at a given response threshold can be expressed via the expectation of response gradient conditional on the threshold. Determining the expectation requires conditioning on the threshold that is a zero-probability event, but it can be resolved by the concept of kernel smoothing. The proposed method offers sensitivity estimates for all response thresholds generated in a single Monte Carlo run. It is investigated in a number of examples featuring sensitivity parameters of different nature. As response gradient becomes increasingly available, it is hoped that this work can provide the basis for embedding sensitivity calculations with reliability in the same Monte Carlo run.

Figures (8)

• Figure 1: Example 1 (Normal response). Results of a typical SS run with $m = 3$ levels, $p_0 = 0.1$ and $N=1000$ samples per level. Blue = SS, red = Benchmark, vertical green lines = thresholds at $\{10^{-1}, 10^{-2}, 10^{-3}\}$. The right column shows the fractional change of $F$ per fractional change of sensitivity parameter, the recommended sensitivity measure.
• Figure 2: Example 1 (Normal response). Sample mean (blue solid line) and $\pm 1\sigma$ bounds (blue dashed line) from 1000 SS runs. Red = Benchmark. Vertical green lines = thresholds at $\{10^{-1}, 10^{-2}, 10^{-3}\}$ of SS sample mean. Same layout as Figure \ref{['fig:eg01_single']} except that scatter plots in the left column are omitted.
• Figure 3: Example 2 (Shear building buckling). Results of a typical run. Same legend as Figure \ref{['fig:eg01_single']}
• Figure 4: Example 2 (Shear building buckling). Sample mean (blue solid line) and $\pm 1\sigma$ bounds (blue dashed line) from 1000 SS runs. Same legend as Figure \ref{['fig:eg01_stats']}
• Figure 5: Example 3 (SDOF first passage). Results of a typical run. Same legend as Figure \ref{['fig:eg01_single']}