### Abstract

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 language | English (US) |
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Pages (from-to) | 375-381 |

Number of pages | 7 |

Journal | Structural and Multidisciplinary Optimization |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - Apr 25 2015 |

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### Keywords

- Decision variables
- Probability of failure
- Sensitivity

### ASJC Scopus subject areas

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

### Cite this

*Structural and Multidisciplinary Optimization*,

*52*(2), 375-381. https://doi.org/10.1007/s00158-015-1232-1