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The geometry of limit state function graphs and subset simulation: Counterexamples

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  • Breitung, Karl

Abstract

In the last fifteen years the subset sampling method has often been used in reliability problems as a tool for calculating small probabilities. This method is extrapolating from an initial Monte Carlo estimate, for which the probability content of a failure domain found by a suitable higher level of the original limit state function. Then iteratively conditional probabilities are estimated for failures domains decreasing to the original failure domain. However, there are implied premises, regarding the structure of the failure domains, which must be fulfilled for the method to work properly. The examples studied in this paper demonstrate that inaccurate results might be obtained if the said premises are not fulfilled. This demonstrates that there are limitations for the application of this method.

Suggested Citation

  • Breitung, Karl, 2019. "The geometry of limit state function graphs and subset simulation: Counterexamples," Reliability Engineering and System Safety, Elsevier, vol. 182(C), pages 98-106.
    • Handle: RePEc:eee:reensy:v:182:y:2019:i:c:p:98-106
    DOI: 10.1016/j.ress.2018.10.008
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    1. Alexander Shapiro & Jos Berge, 2002. "Statistical inference of minimum rank factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 67(1), pages 79-94, March.
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    2. Ma, Yuan-Zhuo & Zhu, Yi-Chen & Li, Hong-Shuang & Nan, Hang & Zhao, Zhen-Zhou & Jin, Xiang-Xiang, 2022. "Adaptive Kriging-based failure probability estimation for multiple responses," Reliability Engineering and System Safety, Elsevier, vol. 228(C).
    3. Yu, Quanfu & Xu, Jun, 2025. "Distribution reconstruction and reliability assessment of complex LSFs via an adaptive Non-parametric Density Estimation Method," Reliability Engineering and System Safety, Elsevier, vol. 254(PB).
    4. Millar, Robert & Li, Hui & Li, Jinglai, 2023. "Multicanonical sequential Monte Carlo sampler for uncertainty quantification," Reliability Engineering and System Safety, Elsevier, vol. 237(C).
    5. Zhang, Tianyu & Zhang, Jize, 2025. "iCE-NGM: Improved cross-entropy importance sampling with non-parametric adaptive Gaussian mixtures and budget-informed stopping criterion," Reliability Engineering and System Safety, Elsevier, vol. 264(PA).
    6. Eshra, Elsayed & Papakonstantinou, Konstantinos G. & Nikbakht, Hamed, 2025. "A direct importance sampling-based framework for rare event uncertainty quantification in non-Gaussian spaces," Reliability Engineering and System Safety, Elsevier, vol. 264(PA).
    7. Naderi, Leila & Jia, Gaofeng, 2026. "Multi-fidelity Subset Simulation for rare event simulation," Reliability Engineering and System Safety, Elsevier, vol. 266(PB).
    8. Dehghani, Nariman L. & Zamanian, Soroush & Shafieezadeh, Abdollah, 2021. "Adaptive network reliability analysis: Methodology and applications to power grid," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    9. Li, Binbin & Xia, Weili & Liao, Zihan, 2025. "Relaxed subset simulation for reliability estimation," Reliability Engineering and System Safety, Elsevier, vol. 264(PA).
    10. Xin, Fukang & Wang, Pan & Chen, Yi & Yang, Rong & Hong, Fangqi, 2025. "A new uncertainty reduction-guided single-loop Kriging coupled with subset simulation for time-dependent reliability analysis," Reliability Engineering and System Safety, Elsevier, vol. 261(C).
    11. Chen, Zhenzhong & Huang, Dongyu & Li, Xiaoke & Qiu, Guiming & Zhao, Pengcheng, 2024. "A reliability analysis method based on the intersection area division of hypersphere and paraboloid," Reliability Engineering and System Safety, Elsevier, vol. 252(C).
    12. Xu, Chunlong & Yang, Ya & Wu, Huajun & Zhou, Jianping, 2023. "A new active learning method for system reliability analysis with multiple failure modes," Reliability Engineering and System Safety, Elsevier, vol. 240(C).
    13. Dasgupta, Agnimitra & Johnson, Erik A., 2024. "REIN: Reliability Estimation via Importance sampling with Normalizing flows," Reliability Engineering and System Safety, Elsevier, vol. 242(C).
    14. Ameryan, Ala & Ghalehnovi, Mansour & Rashki, Mohsen, 2022. "AK-SESC: a novel reliability procedure based on the integration of active learning kriging and sequential space conversion method," Reliability Engineering and System Safety, Elsevier, vol. 217(C).
    15. Rashki, Mohsen & Faes, Matthias G.R. & Wei, Pengfei & Song, Jingwen, 2025. "Asymptotic subset simulation: An efficient extrapolation tool for small probabilities approximation," Reliability Engineering and System Safety, Elsevier, vol. 260(C).
    16. Wang, Hao & Li, Luyi & Liu, Junchao & Yuan, Xiukai, 2025. "An efficient method for solving system failure probability functions based on subset simulation and probability reanalysis techniques," Reliability Engineering and System Safety, Elsevier, vol. 262(C).
    17. Dong, Sichen & Zhong, Anbiao & Li, Lei & Li, Honglin & Yuan, Tianyu, 2026. "Hybrid reliability analysis based on an active learning method considering the coupling effects of random-interval uncertainty," Reliability Engineering and System Safety, Elsevier, vol. 265(PA).
    18. Wang, Lei & Wang, Tao & Dong, You & Frangopol, Dan M. & Li, Zhengliang, 2025. "Structural reliability analysis based on fractional moments-based iterative maximum entropy method and multiplicative exact dimension reduction integration method," Reliability Engineering and System Safety, Elsevier, vol. 264(PA).

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