A computational framework for the quantification of rare events in systems with instabilities

We consider the problem of probabilistic quantification of dynamical systems exhibiting heavy tailed distributions. These heavy tail features are associated with rare transient re- sponses due to the occurrence of internal instabilities. Systems with these characteristics can be found in a variety of areas including mechanics, fluids, and waves. Here we are interested for the development of a computational approach, a probabilistic-decomposition-synthesis method that will take into account the nature of these internal instabilities and will inexpensively provide the non-Gaussian probability density function for the quantities of interest. Our approach relies on the decomposition of the statistics to a stable Gaussian core and a heavy-tailed distribution. Statistics in the stable region are analytically characterized using a Gaussian approximation approach, while the non-Gaussian distributions associated with the intermittently unstable region of the phase space, are inexpen- sively quantified through reduced-order methods. Applications are presented for nonlinear water waves as well as subjected mechanical systems.