An Adaptive Split-Merge Scheme for Uncertainty Propagation using Gaussian Mixture Models

A novel adaptive scheme is presented in order to refine and to coarse the number of Gaussian components in a Gaussian mixture and it is integrated with a previous adaptive Gaussian sum for uncertainty propagation through dynamical systems. The previously presented adaptive Gaussian sum for uncertainty propagation adapts the weights of different components under the assumption that the number of Gaussian components remains constant during propagation. In this work, this assumption is relaxed using a component refining and coarsening scheme. The scheme is based on an iterative split-merge procedure which selects the mixand with the largest contribution to the uncertainty propagation error and splits it along the direction with most nonlinear dynamics. Merging is driven by the approximation error of two Gaussian components with a single Gaussian density function. A numerical example is presented to illustrate the concept of adapting the number of Gaussian components during propagation.