Harmonic transform-based density estimation method for uncertainty propagation and reliability analysis involving multi-modal distributions

Abstract

Uncertainty propagation and reliability analysis involving random variables following multi-modal distributions are still challenging, where conventional methods could not be applicable. In this paper, a novel harmonic transform-based density estimation method is proposed to efficiently and accurately deal with this problem in a non-parametric manner, where the input random variables involving multi-modal distributions are modeled by Gaussian mixture model. First, the kernel density estimation is performed to obtain a rough estimation about the unknown distribution, where the requirement of the proposed method is checked. Then, the harmonic transform-based density estimation method is presented, where the formulation of harmonic transform as well as the analytical expression of unknown distribution are put forward accordingly. Numerical estimation scheme of the proposed method is also established accordingly, where some computational issues are discussed. The proposed method is first validated via recovering some analytical distributions with different multi-modal characteristics. Then, numerical examples with static and dynamic analysis are investigated to verify the efficacy of the proposed method, where both explicit and implicit limit state functions with different shaped multi-modal distributions are involved. The results reveal that the proposed method can effectively obtain the whole distribution region, especially for the distribution tail, for both uncertainty propagation and reliability analysis with multi-modally distributed random variables.