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Articles Published in 1998

  1. Ghanem, R.G. and Dham, S.. "Stochastic finite element analysis for multiphase flow in heterogeneous porous media" Transport in Porous Media. vol. 32. 1998. pp. 239--262.


  2. Ghanem, R.G.. "Probabilistic characterization of transport in heterogeneous media" Computer Methods in Applied Mechanics and Engineering. 158 (3-4). JUN 8 1998. pp. 199--220.

    The mechanics of transport and flow in a random porous medium are addressed in this paper. The hydraulic properties of the porous medium are modeled as spatial random processes. The random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented. Thus, the hydraulic processes are discretized using the spectral Karhunen - Loeve expansion. This expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes. These amplitudes form a basis in the manifold associated with the random processes. The concentrations over the whole domain are also random processes, with unknown probabilistic structure. These processes are represented using the Polynomial Chaos basis. This is a basis in the functional space described by all second order random variables. The deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator. Once the spatio-temporal variation of the concentrations has been specified in terms of the Polynomial Chaos expansion, individual realizations can be readily computed. (C) 1998 Elsevier Science S.A.


  3. Li, R. and Ghanem, R.G.. "Adaptive Polynomial Chaos expansions applied to statistics of extremes in nonlinear random vibration" Prob. Engrg. Mech.. 13 (2). 1998. pp. 125--136.

    This paper presents a new module towards the development of efficient computational stochastic mechanics. Specifically, the possibility of an adaptive polynomial chaos expansion is investigated. Adaptivity in this context refers to retaining, through an iterative procedure, only those terms in a representation of the solution process that are significant to the numerical evaluation of the solution. The technique can be applied to the calculation of statistics of extremes for nongaussian processes. The only assumption involved is that these processes be the response of a nonlinear oscillator excited by a general stochastic process. The proposed technique is an extension of a technique developed by the second author for the solution of general nonlinear random vibration problems. Accordingly, the response process is represented using its Karhunen-Loeve expansion. This expansion allows for the optimal encapsulation of the information contained in the stochastic process into a set of discrete random variables. The response process is then expanded using the polynomial chaos basis, which is a complete orthogonal set in the space of second-order random variables. The time dependent coefficients in this expansion are then computed by using a Galerkin projection scheme which minimizes the approximation error involved in using a finite-dimensional subspace. These coefficients completely characterize the solution process, and the accuracy of the approximation can be assessed by comparing the contribution of successive coefficients. A significant contribution of this paper is the development and implimentation of adaptive schemes for the polynomial chaos expansion. These schemes permit the inclusion of only those terms in the expansion that have a significant contribution. (C) 1997 Elsevier Science Ltd.


  4. Griebel, M.. "Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences" Computing. 61 (2). 1998. pp. 151--179.


  5. Gerstner, T. and Griebel, M.. "Numerical integration using sparse grids" Numer. Alg.. 18 (3-4). 1998. pp. 209--232.

    We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one-dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw-Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction.