Multiscale Modeling &Simulation, Volume 18, Issue 3, Page 1242-1271, January 2020.
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic approximate low-dimensional structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low-dimensional space and its basis are extracted from solution samples to achieve significant dimension reduction in the solution space. At the online stage, the extracted data-driven basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of approximate low-dimensional structure is established in two scenarios based on (1) high separability of the underlying Green's functions, and (2) smooth dependence of the parameters in the random coefficients. Various online construction methods are proposed for different problem setups. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present extensive numerical examples to demonstrate the accuracy and efficiency of the proposed method.

中文翻译：

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基于内在维数的随机系数多尺度椭圆PDE的数据驱动方法

2020年1月，多尺度建模与仿真，第18卷，第3期，第1242-1271页。
我们提出了一种数据驱动的方法，基于底层椭圆微分算子的内在近似低维结构，解决了具有随机系数的多尺度椭圆PDE。我们的方法包括离线阶段和在线阶段。在离线阶段，从解决方案样本中提取低维空间及其基础，以实现解决方案空间中的显着降维。在在线阶段，提取的数据驱动基础将用于有效地求解新的多尺度椭圆PDE。基于以下两种情况建立了近似低维结构的存在：（1）基础格林函数的高可分离性，以及（2）随机系数中参数的平滑依赖性。针对不同的问题设置，提出了各种在线构建方法。我们在建立数据驱动的基础上，基于采样误差和截断阈值提供误差分析。最后，我们提供大量的数值示例来证明所提方法的准确性和效率。