An asymmetric dimension-adaptive tensor-product method for reliability analysis

Reliability analysis plays an essential role in the development of structural systems. However, commonly used reliability analysis methods suffer from either the curse of dimensionality or the lack of accuracy in many structural problems. This paper presents an asymmetric dimension-adaptive tensor-product (ADATP) method to resolve the difficulties of existing reliability analysis methods. The proposed method leverages three ideas: (i) an asymmetric dimension-adaptive scheme to efficiently build the tensor-product interpolation considering both directional and dimensional importance, (ii) a hierarchical interpolation scheme using either piecewise multi-linear basis functions or cubic Lagrange splines, (iii) a hierarchical surplus as an error indicator to automatically detect the highly nonlinear regions in a random space and adaptively refine the collocation points in these regions. The proposed method has three distinct features for reliability analysis: (a) automatically detecting and adaptively reproducing tri- and higher-variate interactions, (b) greatly alleviating the curse of dimensionality, and (c) no need of response sensitivities. Several mathematical and engineering problems involving high nonlinearity are used to demonstrate the effectiveness of the ADATP method.