Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
6
3
2016
BAYESIAN NONPARAMETRIC GENERAL REGRESSION
195-213
Ka Veng
Yuen
University of Macau
Gilberto A.
Ortiz
Faculty of Science and Technology, University of Macau, 999078, Macao, China
Bayesian identification has attracted considerable interest in various research areas for the determination of the mathematical model with suitable complexity based on input-output measurements. Regression analysis is an important tool in which Bayesian inference and Bayesian model selection have been applied. However, it has been noted that there is a subjectivity problem of model selection results due to the assignment of the prior distribution of the regression coefficients. Since regression coefficients are not physical parameters, assignment of their prior distribution is nontrivial. To resolve this problem, we propose a novel nonparametric regression method using Bayesian model selection in conjunction with general regression. In order to achieve this goal, we also reformulate the general regression under the Bayesian framework. There are two attractive features of the proposed method. First, it eliminates the subjectivity of model selection results due to the prior distribution of the regression coefficients. Second, the number of model candidates is drastically reduced, compared with traditional regression using the same number of design/input variables. Therefore, this allows for the consideration of a much larger number of potential design variables. The proposed method will be assessed and validated through two simulated examples and two real applications.
EMPIRICAL ACCELERATION FUNCTIONS AND FUZZY INFORMATION
215-228
Muhammad
Shafiq
Kohat University of Science and Technology
Alamgir
Khalil
Department of Statistics, University of Peshawar, KP, Pakistan
Muhammad
Atif
Department of Statistics, University of Peshawar, KP, Pakistan
Qamruz
Zaman
Department of Statistics, University of Peshawar, KP, Pakistan
In accelerated life testing approaches lifetime data are obtained under various conditions which are considered more severe than the usual condition. Classical analysis techniques are based on obtained precise measurements, and used to model variation among the observations. In fact, data have two types of uncertainty: variation among the observations and fuzziness of the single observation. Analysis techniques, which do not consider fuzziness and are only based on precise lifetime observations, lead to pseudo results. The aim of this study was to examine the behavior of empirical acceleration functions using fuzzy lifetime data. Furthermore the results showed an increased fuzziness in the transformed lifetimes compared to the input data.
STOCHASTIC SAMPLING BASED BAYESIAN MODEL UPDATING WITH INCOMPLETE MODAL DATA
229-244
Sahil
Bansal
Indian Institute of Technology Roorkee
Sai Hung
Cheung
School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore
In this paper, we are interested in model updating of a linear dynamic system based on incomplete modal data including modal frequencies, damping ratios, and partial mode shapes of some of the dominant modes. To quantify the uncertainties and plausibility of the model parameters, a Bayesian approach is developed. The mass and stiffness matrices in the identification model are represented as a linear sum of the contribution of the corresponding mass and stiffness matrices from the individual prescribed substructures. The damping matrix is represented as a sum of the contribution from individual substructures in the case of viscous damping, in terms of mass and stiffness matrices in the case of classical damping (Caughey damping), or a combination of the viscous and classical damping. A Metropolis-within-Gibbs sampling based algorithm is proposed that allows for an efficient sampling from the posterior probability distribution. The effectiveness and efficiency of the proposed method are illustrated by numerical examples with complex modes.
A STOPPING CRITERION FOR ITERATIVE SOLUTION OF STOCHASTIC GALERKIN MATRIX EQUATIONS
245-269
Christophe
Audouze
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, Canada M3H 5T6
Pär
Håkansson
School of Chemistry, University of Southampton, Highfield, SO171BJ, United Kingdom
Prasanth B.
Nair
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, Canada M3H 5T6
In this paper we consider generalized polynomial chaos (gPC) based stochastic Galerkin approximations of linear random algebraic equations where the coefficient matrix and the right-hand side are parametrized in terms of a finite number of i.i.d random variables. We show that the standard stopping criterion used in Krylov methods for solving the stochastic Galerkin matrix equations resulting from gPC projection schemes leads to a substantial number of unnecessary and computationally expensive iterations which do not improve the solution accuracy. This trend is demonstrated by means of detailed numerical studies on symmetric and nonsymmetric linear random algebraic equations. We present some theoretical analysis for the special case of linear random algebraic equations with a symmetric positive definite coefficient matrix to gain more detailed insight into this behavior. Finally, we propose a new stopping criterion for iterative Krylov solvers to avoid unnecessary iterations while solving stochastic Galerkin matrix equations. Our numerical studies suggest that the proposed stopping criterion can provide up to a threefold reduction in the computational cost.
FAST AND ACCURATE MODEL REDUCTION FOR SPECTRAL METHODS IN UNCERTAINTY QUANTIFICATION
271-286
Francisco Damascene
Freitas
Department of Electrical Engineering, University of Brasilia, CEP: 70910-900, Brasilia, DF, Brazil
Roland
Pulch
Institute of Mathematics and Computer Science, University of Greifswald,
Walther-Rathenau-Straße 47, D-17489 Greifswald, Germany
Joost
Rommes
Mentor Graphics, 110 rue Blaise Pascal, Montbonnot, France
A fast and accurate model order reduction procedure is presented that can successfully be applied to spectral methods for uncertainty quantification problems. The main novelties include (1) the application of model order reduction to uncertainty quantification problems; (2) the improvement of existing model order reduction methods in order to meet the accuracy and performance requirements; and (3) an efficient approach for systems with many outputs. Numerical experiments for large-scale realistic systems illustrate the suitability and performance (50× speedup while preserving accuracy) for uncertainty quantification problems.