Approximation to Multivariate Normal Integral and its Application in Time-Dependent Reliability Analysis


It is common to evaluate high-dimensional normal probabilities in many uncertainty-related applications such as system and time-dependent reliability analysis. An accurate method is proposed to evaluate high-dimensional normal probabilities, especially when they reside in tail areas. The normal probability is at first converted into the cumulative distribution function of the extreme value of the involved normal variables. Then the series expansion method is employed to approximate the extreme value with respect to a smaller number of mutually independent standard normal variables. The moment generating function of the extreme value is obtained using the Gauss-Hermite quadrature method. The saddlepoint approximation method is finally used to estimate the cumulative distribution function of the extreme value, thereby the desired normal probability. The proposed method is then applied to time-dependent reliability analysis where a large number of dependent normal variables are involved with the use of the First Order Reliability Method. Examples show that the proposed method is generally more accurate and robust than the widely used randomized quasi Monte Carlo method and equivalent component method.


Mechanical and Aerospace Engineering

Research Center/Lab(s)

Center for High Performance Computing Research

Second Research Center/Lab

Intelligent Systems Center

Keywords and Phrases

Dimension reduction; Extreme value distribution; Gauss-Hermite quadrature; Multivariate normal distribution; Reliability; Saddlepoint approximation

International Standard Serial Number (ISSN)


Document Type

Article - Journal

Document Version


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© 2021 Elsevier, All rights reserved.

Publication Date

01 Jan 2021