Beyond Monte Carlo for the initial uncertainty propagation problem

This paper revisits the problem of particle uncertainty characterization and propagation for deterministic nonlinear continuous time dynamical systems. A revealing comparison is made between the traditional Monte Carlo method and a recently developed particle framework based on Markov chain Monte Carlo and the method of characteristics (MCMC-MOC). The focus is on dynamical systems that do not admit stationary solutions that often pose bigger problems for the Liouville equation than systems with fixed points. We demonstrate through dispersion analysis that while traditional Monte Carlo is unable to propagate the state density function in a statistically consistent manner and suffers from degeneracy, the MCMC-MOC approach performs very well. The new approach automatically extracts the domain of significance (support) of the state density function by constructing a Markov Chain guided by the solution of the stochastic Liouville equation. Being equivalent in measure to the true state probability density, the particles in the Markov chain can be used directly to compute desired expectations. Simple examples are used to illustrate the potential pitfalls of using traditional Monte Carlo and the corresponding advantages of the MCMC-MOC technique.