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An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations,10.1016/j.jcp.201

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations   (Citations: 3)
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A computational methodology is developed to address the solution of high-dimensional stochastic problems. It utilizes high-dimensional model representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. HDMR is efficient at capturing the high-dimensional input–output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms using only the important dimensions. The newly developed adaptive sparse grid collocation (ASGC) method is incorporated into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output. Several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.
Journal: Journal of Computational Physics - J COMPUT PHYS , vol. 229, no. 10, pp. 3884-3915, 2010
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    • ...The HDMR expansion is built iteratively by including only the “most significant” component functions to minimize the computational cost of building the surrogate model [5]...
    • ...On the other hand, one can apply the MC method to an approximation of ( ) V x , namely its surrogate model, a compact polynomial representation of ( ) V x . In this case, the difficulty lies in obtaining an accurate but cheap to evaluate surrogate model of ( ) V x . This can be achieved using the iterative HDMR method proposed in [5]...
    • ...Representation (1) can be constructed using the cut-HDMR method [5], which expresses the component functions in terms of observable values on lines, planes, and hyperplanes (i.e...
    • ...iD ∈ but S i ∉ u , are set to their corresponding mean values (see [4, 5] for details)...
    • ...[5]. This limits the direct application of cut-HDMR in realistic large-scale...
    • ...EMC/EMI problems for large dof N . This high cost can be reduced considerably by integrating an iterative scheme to the hierarchical cut-HDMR method, which automatically selects random variables that significantly contribute to () V x and iteratively includes these variables’ higher-order component functions in the cut-HDMR expansion [5]...
    • ...Iterative cut-HDMR Construction: The iterative cut-HDMR scheme [5] first constructs the first-order component functions by setting 1 S = . Then, it computes...

    Abdulkadir C. Yucelet al. Efficient stochastic EMC/EMI analysis using HDMR-generated surrogate m...

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