Multivariate Probabilistic Collocation Method for Effective Uncertainty Evaluation With Application to Air Traffic Flow Management

Abstract

Modern large-scale infrastructure systems have typical complicated structure and dynamics, and extensive simulations are required to evaluate their performance. The probabilistic collocation method (PCM) has been developed to effectively simulate a system's performance under parametric uncertainty. In particular, it allows reduced-order representation of the mapping between uncertain parameters and system performance measures/outputs, using only a limited number of simulations; the resultant representation of the original system is provably accurate over the likely range of parameter values. In this paper, we extend the formal analysis of single-variable PCM to the multivariate case, where multiple uncertain parameters may or may not be independent. Specifically, we provide conditions that permit multivariate PCM to precisely predict the mean of original system output. We also explore additional capabilities of the multivariate PCM, in terms of cross-statistics prediction, relation to the minimum mean-square estimator, computational feasibility for large dimensional parameter sets, and sample-based approximation of the solution. At the end of the paper, we demonstrate the application of multivariate PCM in evaluating air traffic system performance under weather uncertainties.