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Articles written by Red-Horse, J.R

  1. Ghanem, R.G. and Red-Horse, J.R. "Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach" Physica D. 133 (1-4). 1999. pp. 137--144.

    This paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients. These typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory. The concepts of projection, orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that, in the stochastic case, inner products refer to expectation operations. Specifically, the Karhunen–Loève expansion is used to discretize these processes into a denumerable set of random variables, thus providing a denumerable function space in which the problem is cast. The polynomial chaos expansion is then used to represent the solution in this space, and the coefficients in the expansion are evaluated as generalized Fourier coefficients via a Galerkin procedure in the Hilbert space of random variables.


  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.