A Markov Chain Monte Carlo Particle Solution of the Initial Uncertainty Propagation Problem

Abstract

This paper presents a particle approach based on Markov chain Monte Carlo and the method of characteristics to numerically solve the stochastic Liouville equation for nonlinear dynamical systems. Markov chain theory helps accomplish two important objectives: (i) it provides a viable approach for systems with high dimensional state space by using a compact particle representation that is equivalent in measure to the time varying probability density function of the state, and (ii) it automatically extracts the domain of significance of the state uncertainty (i.e. support of the density function) by constructing a Markov chain guided by the solution of the stochastic Liouville equation. Being equivalent in measure to state probability density, the particles obtained can be used directly to compute desired expectations of the system state.