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

Abstract

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 (c) 2016 John Wiley Sons, Ltd.