Quantifying Uncertainty
Ghanem, R.G. and Pellissetti, M.. "Adaptive data refinement in the spectral stochastic finite element method" Communications in Numerical Methods in Engineering. 18
(2).
FEB 2002.
pp. 141--151.
One version of the stochastic finite element method involves representing the solution with respect to a basis in the space of random variables and evaluating the co-ordinates of the solution with respect to this basis by relying on Hilbert space projections. The approach results in an explicit dependence of the solution on certain statistics of the data. The error in evaluating these statistics, which is directly related to the amount of available data, can be propagated into errors in computing probabilistic measures of the solution. This provides the possibility of controlling the approximation error, due to limitations in the data, in probabilistic statements regarding the performance of the system under consideration. In addition to this error associated with data resolution, is added the more traditional error, associated with mesh resolution. This latter also contributes to polluting the estimated probabilities associated with the problem. The present paper will develop the above concepts and indicate how they can be coupled in order to yield a more meaningful and useful measure of approximation error in a given problem.
Ghanem, R.G., Masri, S., Pellissetti, M., and Wolfe, R.. "Identification and prediction of stochastic dynamical systems in a polynomial chaos basis" Computer Methods in Applied Mechanics and Engineering. 194
(12-16).
2005.
pp. 1641--1654.
Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust. (C) 2004 Elsevier B.V. All rights reserved.