An adaptive dimension decomposition and reselection method for reliability analysis

Recently, the research community in reliability analysis has seen a strong surge of interest in the dimension decomposition approach, which typically decomposes a multi-dimensional system response into a finite set of low-order component functions for more efficient reliability analysis. However, commonly used dimension decomposition methods suffer from two limitations. Firstly, it is often difficult or impractical to predetermine the decomposition level to achieve sufficient accuracy. Secondly, without an adaptive decomposition scheme, these methods may unnecessarily assign sample points to unimportant component functions. This paper presents an adaptive dimension decomposition and reselection (ADDR) method to resolve the difficulties of existing dimension decomposition methods for reliability analysis. The proposed method consists of three major components: (i) an adaptive dimension decomposition and reselection scheme to automatically detect the potentially important component functions and adaptively reselect the truly important ones, (ii) a test error indicator to quantify the importance of potentially important component functions for dimension reselection, and (iii) an integration of the newly developed asymmetric dimension-adaptive tensor-product (ADATP) method into the adaptive scheme to approximate the reselected component functions. The merits of the proposed method for reliability analysis are three-fold: (a) automatically detecting and adaptively representing important component functions, (b) greatly alleviating the curse of dimensionality, and (c) no need of response sensitivities. Several mathematical and engineering high-dimensional problems are used to demonstrate the effectiveness of the ADDR method.