A Short-Time Non-Gaussian Probability Approximation Based Method for Response Transitions of a Lévy-Driven Nonsmooth System

This paper proposes an efficient short-time probability approximation with Lévy excitation to capture the transient probability distribution and its evolving path. Using principal component analysis (PCA), the method constructs a probability core to exclude outliers beyond it. The statistics of samples that fall inside the core are treated, with a prescribed fiducial probability, as an easy-to-estimate Gaussian type. The idea is verified numerically by compared with Monte-Carlo results. Then, it is integrated into the path integral (PI) method, combined with evolving probabilistic vector (EPV) techniques, to efficiently obtain probability distributions in each time step of PI. This scheme is semianalytical, only dependent on a relatively small amount of response samples to form the probability core; thus, it can have very computational advantages over full Monte-Carlo simulation to capture transient responses and probability distributions. The application to investigating response transitions of a nonsmooth system driven by Lévy shock and jump has revealed the performance of the proposed method. Also, the exit times of stochastic response are characterized quantitatively from the perspective of global dynamic transition. These investigations will be helpful to achieve the efficient probability estimation for nonlinear system with non-Gaussian inputs and quantify the reliability of the mechanical system.