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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 pbkp_redict the mean of original system output. We also explore additional capabilities of the multivariate PCM, in terms of cross-statistics pbkp_rediction, 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.