Accuracy improvement of the most probable point‐based dimension reduction method using the hessian matrix

Abstract

Summary This paper proposes a most probable point (MPP)‐based dimension reduction method (DRM) using the Hessian matrix called HeDRM to improve accuracy of reliability analysis in existing MPP‐based DRM methods. Conventional MPP‐based DRMs contain two types of errors: (1) error due to eliminating cross‐terms of a performance function by using the univariate DRM; (2) error because of dependency of an axis direction after a rotational transformation. The proposed method minimizes the aforementioned errors by utilizing the Hessian matrix of a performance function. By performing an orthogonal transformation using the eigenvectors of the Hessian matrix, the cross‐term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the Gaussian quadrature points for numerical integration are arranged along the eigenvector directions. In this way, the error incurred by exiting MPP‐based DRMs can be reduced that leads to more accurate probability of failure estimation. In addition, this paper proposes to allocate the Gaussian quadrature points using the magnitude of the eigenvalues of the Hessian matrix. This allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently. Copyright © 2016 John Wiley & Sons, Ltd.

Second-order reliability methods: a review and comparative study

“…Thereafter, system reliability analysis (Hao et al 2020;Song and Der Kiureghian 2006;Radhika et al 2008;Yu et al 2018;Hu and Du 2018b;Cheng and Du 2016) for series and parallel systems can be used to compute the system level reliability. Furthermore, sensitivity analysis (Song et al 2009;Hu and Mahadevan 2016b;Xiao et al 2011;Jacques et al 2006;Mara and Tarantola 2012;McRae et al 1982;Sudret 2008;Hu and Xiong 2015) and dimension reduction techniques (Li et al 2019;Hawchar et al 2017;Hu and Youn 2011;Sargsyan et al 2014;Guo et al 2019;Yadav and Rahman 2012;Huang and Du 2006;Yadav and Rahman 2014;Bin et al 2017) can also be adopted to reduce the complexity of large-scale engineering system before using SORM. To the best of the author's knowledge, the use of SORM for very large number of random variables, i.e., thousands, has been limited.…”