Uncertainty estimation using the moments method facilitated by automatic differentiation in Matlab

摘要

Computational models have long been used to predict the performance of some baselinedesign given its design parameters. Given inconsistencies in manufacturing,the manufactured product always deviates from the baseline design. There is currentlymuch interest in both evaluating the effects of variability in design parameterson a design’s performance (uncertainty estimation), and robust optimization of thebaseline design such that near optimal performance is obtained despite variabilityin design parameters. Traditionally, uncertainty analysis is performed by expensiveMonte-Carlo methods. This work considers the alternative moments method for uncertaintypropagation and its implementation in Matlab.In computational design it is assumed a computational model gives a sufficientlyaccurate approximation to a design’s performance. As such it can be used for estimatingstatistical moments (expectation, variance, etc.) of the design due to knownstatistical variation of the model’s parameters, e.g., by the Monte Carlo approach. Inthe moments method we further assume the model is sufficiently differentiable thata Taylor series approximation to a model may be constructed, and the moments ofthe Taylor series may be taken analytically to yield approximations to the model’smoments.In this thesis we generalise techniques considered within the engineering communityand design and document associated software to generate arbitrary order Taylorseries approximations to arbitrary order statistical moments of computational modelsimplemented in Matlab; Taylor series coefficients are calculated using automatic differentiation.This approach is found to be more efficient than a standard Monte Carlomethod for the small-scale model test problems we consider. Previously Christiansonand Cox (2005) have indicated that the moments method will be non-convergent inthe presence of complex poles of the computational model and suggested a partitioningmethod to overcome this problem. We implement a version of the partitioningmethod and demonstrate that it does result in convergence of the moments method.Additionally, we consider, what we term, the branch detection problem in order toascertain if our Taylor series approximation might only be valid piecewise.