A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction
A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction
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DOI：
10.1137/19m1277485
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
Sijing Li;Zhiwen Zhang;Hongkai Zhao
中科院分区：
文献类型：
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作者：
Sijing Li;Zhiwen Zhang;Hongkai Zhao
我们提出了一个数据驱动的方法来解决多尺度椭圆偏微分方程的随机系数的基础上内在的低维结构的基础上的椭圆微分算子。我们的方法包括离线和在线阶段。在离线阶段，从数据中提取低维空间及其基，以实现解空间的显著降维。在在线阶段，提取的基将被用于有效地解决一个新的多尺度椭圆偏微分方程。低维结构的存在性是通过显示底层绿色函数的高度可分性来建立的。根据问题设置的不同，提出了不同的在线构造方法。在建立数据驱动基的过程中，给出了基于采样误差和截断阈值的误差分析。最后，我们提出的数值例子来证明所提出的方法的准确性和效率。
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the underlying Green's functions. Different online construction methods are proposed depending on the problem setup. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method.