Computational framework for extreme responses of dynamical systems

Abstract

A computational framework is developed for estimating the distribution of extreme responses of dynamical systems subjected to random inputs, referred to as target responses. Generally, these problems do not admit analytical solutions and cannot be solved numerically since they have infinite stochastic dimensions. We construct finite-dimensional (FD) versions of the posed problems, i.e., problems depending on finite sets of random variables, and establish conditions under which extreme of their FD responses can be used as surrogates for extremes of target responses. Under these conditions, extreme of target responses can be characterized from paths of FD responses, which can be generated by standard numerical algorithms. Statical methods based on level crossing and conditional probabilities are employed to improve the estimates of the distributions of extremes of FD responses. Two dynamical systems are used to illustrate the implementation and the performance of our computational framework.