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

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.

I. Introduction

There is a growing need to effectively and strategically manage large-scale infrastructure systems, such as air traffic systems, power grids, and environmental systems. These systems are typically subject to a wide range of uncertainties, which significantly complicate the evaluation and management of system performance. To give some examples, flow contingency management solutions are being developed for air traffic systems, which seek to automatically generate management plans over a 2–15 h lookahead time that are robust to weather uncertainties [26], [27]. In analogy, strategic resource scheduling and real-time surveillance/control algorithms are sought for the power grid, that are flexible to uncertainties in renewable generation and load, and robust to complex and uncertain fault events [12], [29]. As a step toward real-time management, techniques are needed for accurate yet computationally efficient evaluation/prediction of system performance over a range of parametric uncertainties. To address this need in broad infrastructure system applications, this paper develops a systematic method to effectively evaluate output statistics for systems with multiple uncertain input parameters.