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

作者：Ma Xiang; Zabaras Nicholas^*

来源：Journal of Computational Physics, 2010, 229(10): 3884-3915.

DOI：10.1016/j.jcp.2010.01.033

• Stochastic partial differential equations
• High-dimensional model representation
• Stochastic collocation method
• Sparse grids
• Random heterogeneous media

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 d

• 出版日期2010-5-20