Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

Abstract

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.