A Model Reduction Method for Elliptic PDEs with Random Input Using the Heterogeneous Stochastic FEM Framework
by Thomas Y. Hou   Pengfei Liu   Zhiwen Zhang  

Vol. 11 No. 1 (2016) P.179~P.216

ABSTRACT

We introduce a model reduction method for elliptic PDEs with random input, which follows the heterogeneous stochastic finite element method framework and exploits the compactness of the solution operator in the stochastic direction on local regions of the spatial domain. This method consists of two stages and suits the multi-query setting. In the offline stage, we adaptively construct local stochastic basis functions that can capture the stochastic structure of the solution space in local regions of the domain. This is achieved through local Hilbert-Karhunen-Loève expansions of sampled stochastic solutions with randomly chosen forcing functions. In the online stage, for given forcing functions, we discretize the equation using the heterogeneous coupling of spatial basis with the constructed local stochastic basis, and obtain the numerical solutions through Galerkin projection. Convergence of the online numerical solutions is proved based on the thresholding in the offline stage. Numerical results are presented to demonstrate the effectiveness of this model reduction method.

KEYWORDS
Model reduction. Local stochastic basis. Hilbert-Karhunen-Loève expansion.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 65N30.

MILESTONES

Received: 2015-04-20
Revised : 2015-09-16
Accepted: 2015-09-16

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