On the effectiveness of Monte Carlo for initial uncertainty forecasting in nonlinear dynamical systems

This paper considers the problem of initial uncertainty forecasting in deterministic nonlinear continuous-time dynamical systems via particle ensembles. The popular Monte Carlo method, which while simple to implement, faces fundamental issues. In particular, it is not clear how well the propagated particles continue to represent the true state-uncertainty at future times. This paper evaluates the performance of Monte Carlo forecasting by analyzing it in the context of Markov chain Monte Carlo (MCMC) theory. The propagated ensemble is viewed as the realization of a Markov chain at each time instant, generated by an associated instantaneous transition kernel. It is shown that for a special class of nonlinear systems that have zero divergence, the propagated kernel is in detailed balance with the true state probability density function. This guarantees statistical consistency of the Monte Carlo ensemble with the truth at all times for such systems. On the other hand, no such guarantee is possible for systems with non-zero divergence.