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Articles written by Doostan, A.

  1. Ghanem, R.G. and Doostan, A.. "On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data" Journal of Computational Phyics. 217 (1). 2006. pp. 63--81.

    This paper investigates the predictive accuracy of stochastic models. In particular, a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models, on their overall predictive accuracy. In the course of this development, a new method for the characterization of stochastic processes from corresponding experimental observations is obtained. Specifically, polynomial chaos representations of these processes are estimated that are consistent, in some useful sense, with the data. The estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function, thus permitting the analysis of the dependence of their values on further experimental evidence. Moreover, the error in these coefficients, associated with limited data, is propagated through a physical system characterized by a stochastic partial differential equation (SPDE). This formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model. A Bayesian inference scheme is relied upon as the logic for parameter estimation, with its computational engine provided by a Metropolis-Hastings Markov chain Monte Carlo procedure.


  2. Ghanem, R.G., Doostan, A., and Red-Horse, J.R. "A probabilistic construction of model validation" Computer Methods in Applied Mechanics and Engineering. 197 (29-32). 2008. pp. 2585--2595.

    We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise. (C) 2007 Elsevier B.V. All rights reserved.


  3. Doostan, A., Ghanem, R.G., and Red-Horse, J.R. "Stochastic model reduction for chaos representations" Computer Methods in Applied Mechanics and Engineering. 196 (37-40). 2007. pp. 3951--3966.

    This paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic Galerkin schemes for the solution of partial differential equations with stochastic coefficients. In particular, an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation (SPDE) in the Hilbert space spanned by Wiener chaos. A description of the stochastic aspect of the problem on two well-separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale. With such algorithms at hand, the solution of SPDE's becomes both computationally manageable and efficient. Moreover, a solid foundation is thus provided for the adaptive error control in stochastic Galerkin procedures. Different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics.


  4. Ghanem, R.G, Saad, G., and Doostan, A.. "Efficient solution of stochastic systems: Application to the embankment dam problem" Structural Safety. 29 (3). 2007. pp. 238--251.

    The embankment dam problem of the benchmark study is treated using the newly developed Stochastic Model Reduction for Polynomial Chaos Representations method. The elastic and shear moduli of the material, in the present problem, are modeled as two stochastic processes that are explicit functions of the same process possessing a relatively low correlation length. The state of the system can thus be viewed as a function defined on a high dimensional space, associated with the fluctuations of the underlying process. In such a setting, the spectral stochastic finite element method (SSFEM) for the specified spatial discretization is computationally prohibitive. The approach adopted in this paper enables the stochastic characterization of a fine mesh problem based on the high dimensional polynomial chaos solution of a coarse mesh analysis. After relatively reducing the dimensionality of the problem through a Karhunen-Loeve representation of the stochastic variables, the SSFEM solution consisting of a high dimensional polynomial in Gaussian independent variables is obtained for the coarse mesh problem. Then the attained solution is used to define a new basis for solving the fine mesh problem. The paper presents some new algorithms for the estimation of chaos coefficients in the presence of complex non-Gaussian dependencies. A numerical convergence study is presented together with a discussion of the results.