Probability of failure sensitivity with respect to decision variables

Sylvain Lacaze, Loïc Brevault, Samy Missoum, Mathieu Balesdent

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


This note introduces a derivation of the sensitivities of a probability of failure with respect to decision variables. For instance, the gradient of the probability of failure with respect to deterministic design variables might be needed in RBDO. These sensitivities might also be useful for Uncertainty-based Multidisciplinary Design Optimization. The difficulty stems from the dependence of the failure domain on variations of the decision variables. This dependence leads to a derivative of the indicator function in the form of a Dirac distribution in the expression of the sensitivities. Based on an approximation of the Dirac, an estimator of the sensitivities is analytically derived in the case of Crude Monte Carlo first and Subset Simulation. The choice of the Dirac approximation is discussed.

Original languageEnglish (US)
Pages (from-to)375-381
Number of pages7
JournalStructural and Multidisciplinary Optimization
Issue number2
StatePublished - Apr 25 2015


  • Decision variables
  • Probability of failure
  • Sensitivity

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Computer Science Applications
  • Control and Optimization
  • Computer Graphics and Computer-Aided Design


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