Closed-Loop Adaptive Monte Carlo Framework for Uncertainty Forecasting in Nonlinear Dynamic Systems

A novel adaptive Monte Carlo simulation (MCS) framework for uncertainty forecasting in nonlinear dynamical systems is presented. A closed-loop architecture is created that controls transient forecasting performance as well as associated computational burden. Performance is quantified in terms of estimation accuracy of application-dependent quantities of interest (QoI), bounds on which are prescribed by the user. When the QoI estimation error, measured periodically via bootstrap sampling, exceeds the prescribed upper threshold, optimally selected particles are sequentially introduced into the initial ensemble and then forward propagated to join the current ensemble. Optimality of new particles is defined in terms of ensemble efficiency, quantified in turn by space-filling and noncollapsing properties. On the other hand, when the QoI estimation error is less than the prescribed lower threshold, particles are removed in the interest of alleviating computational load. Probability of particle retention is proportional to the state probability density function value at its location, computed numerically by solving the associated stochastic Liouville equation via the method of characteristics. This approach creates a minimal particle representation of state uncertainty while maintaining guaranteed performance of MCS within user-defined accuracy bounds. Numerical simulations demonstrate the effectiveness of the proposed adaptive forecasting method.