Discrepancy Driven Adaptive Monte Carlo for Forward Uncertainty Forecasting in Nonlinear Dynamical Systems

Two serious limitations of the widely popular Monte Carlo simulations (MCS) for uncertainty forecasting are that: i) its rate of convergence is slow, especially when applied to complex systems where each simulation is computationally expensive, and ii) it is almost impossible to hold the quality of MC ensemble over time using a fixed number of simulations, given the fact that the propagated state uncertainty is time-varying and normally unknown in advance. In this paper, an adaptive MCS framework is developed that aims to control its performance within the desired upper and lower accuracy bounds on-the-fly, while using a "minimum" number of simulations. When the accuracy of MCS falls below the lower accuracy bound (i.e. error exceeds the prescribed threshold), additional "optimally" selected particles are sequentially introduced at the initial time, and then forward propagated to join the current ensemble until reaching the required level of accuracy. This is done by following a two layer approach targeted at improving its rate of convergence via both non-collapsing (projective distance) and space-filling criteria (discrepancy metric). On the contrary, when MCS outperforms the required accuracy level (i.e. error is lower than needed), particles are removed from the current ensemble in the interest of reducing computational load. Particle removal is based on their relative weightage obtained by numerically solving the associated stochastic Liouville equation via the method of characteristics. Numerical simulations illustrate the benefits of proposed methodology.