Unified reliability assessment for problems with low- to high-dimensional random inputs using the Laplace transform and a mixture distribution

Efficient evaluation of the failure probability of systems with low- to high-dimensional random inputs in a unified way is still a challenging task since the methods that work in low dimensions usually become inefficient in high dimensions and vice versa. In this paper, a unified method is proposed to address this challenge. First, the Laplace transform (LT) is introduced to characterize the output variable of the limit state function (LSF). Two fifth-degree cubature formulae are employed to numerically approximate the LT when the input parameter space is low/moderate-dimensional, whereas a low-discrepancy sampling technique is adopted for high-dimensional problems. A mixture of skew normal distributions, is then developed to recover the probability distribution of the LSF from the knowledge of its LT. By matching with discrete values of the LT, the parameters of the mixture distribution are identified and the probability distribution of the LSF can be reconstructed. Five numerical examples are investigated to verify and exemplify the proposed method, where some standard reliability analysis methods are also conducted for comparison. The results indicate that the proposed method can efficiently recover the probability distribution of the LSF and estimate the failure probability for problems with low- to high-dimensional random inputs within a unified framework. The source code is readily available at: http://github.com/Chao-Dang/Reliability-Analysis-Using-Laplace-Transform-and-Mixture-Distribtution.