A decomposition-based approach to uncertainty quantification of multicomponent systems

Abstract

To support effective decision making, engineers should comprehend and manage various uncertainties throughout the design process. In today's modern systems, quantifying uncertainty can become cumbersome and computationally intractable for one individual or group to manage. This is particularly true for systems comprised of a large number of components. In many cases, these components may be developed by different groups and even run on different computational platforms, making it challenging or even impossible to achieve tight integration of the various models. This thesis presents an approach for overcoming this challenge by establishing a divide-and-conquer methodology, inspired by the decomposition-based approaches used in multidisciplinary analysis and optimization. Specifically, this research focuses on uncertainty analysis, also known as forward propagation of uncertainties, and sensitivity analysis. We present an approach for decomposing the uncertainty analysis task amongst the various components comprising a feed-forward system and synthesizing the local uncertainty analyses into a system uncertainty analysis. Our proposed decomposition-based multicomponent uncertainty analysis approach is shown to converge in distribution to the traditional all-at-once Monte Carlo uncertainty analysis under certain conditions. Our decomposition-based sensitivity analysis approach, which is founded on our decomposition-based uncertainty analysis algorithm, apportions the system output variance among the system inputs. The proposed decomposition-based uncertainty quantification approach is demonstrated on a multidisciplinary gas turbine system and is compared to the traditional all-at-once Monte Carlo uncertainty quantification approach. To extend the decomposition-based uncertainty quantification approach to high dimensions, this thesis proposes a novel optimization formulation to estimate statistics from a target distribution using random samples generated from a (different) proposal distribution. The proposed approach employs the well-defined and determinable empirical distribution function associated with the available samples. The resulting optimization problem is shown to be a single linear equality and box-constrained quadratic program and can be solved efficiently using optimization algorithms that scale well to high dimensions. Under some conditions restricting the class of distribution functions, the solution of the optimization problem yields importance weights that are shown to result in convergence in the Ll-norm of the weighted proposal empirical distribution function to the target distribution function, as the number of samples tends to infinity. Results on a variety of test cases show that the proposed approach performs well in comparison with other well-known approaches. The proposed approaches presented herein are demonstrated on a realistic application; environmental impacts of aviation technologies and operations. The results demonstrate that the decomposition-based uncertainty quantification approach can effectively quantify the uncertainty of a multicomponent system for which the models are housed in different locations and owned by different groups.