Efficient Computation of Extreme Excursion Probabilities for Dynamical Systems through Rice's Formula

Abstract

We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses a Markov chain Monte Carlo or a Laplace approximation approach to construct a biasing distribution that in turn is used in an importance sampling procedure to estimate the extreme excursion probabilities. The prior and likelihood of the biasing distribution are obtained by using Rice's formula from excursion probability theory. We use Gaussian mixture biasing distributions and approximate the non-Gaussian initial state distribution by the method of moments to circumvent the linearity and Gaussianity assumptions needed by excursion probability theory. The method requires the tangent linear model of the nonlinear dynamical system to be bounded. Additionally, our method works best when the uncertainty-induced trajectories of the nonlinear dynamical system are small perturbations around a mean linear trajectory and thus when the target system exhibits only mild nonlinearity. We demonstrate the effectiveness of this computational framework for nonlinear dynamical systems of up to 100 dimensions.