Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data
Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data
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DOI：
10.1137/20m1372214
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
Yifan Chen;T. Hou
中科院分区：
文献类型：
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作者：
Yifan Chen;T. Hou
.多尺度偏微分方程的数值放大与非均匀函数的离散数据近似之间有着密切的联系：用于推导放大方程（前者）的粗变量对应于用于近似（后者）的采样信息。因此，这两个问题都可以被认为是基于一些粗略数据来恢复目标函数，这些粗略数据要么是通过升级算法人为选择的，要么是通过一些物理测量过程确定的。然后，本文的目的是研究，在这样的设置和一个特定的椭圆问题，如何的粗数据，我们称之为二次采样的lengthscale，影响恢复的准确性，有限的计算预算。我们的分析和实验表明，减少子采样长度尺度可能会提高精度，这意味着在这种计算约束的情况下，粗粒度或数据采集的指导标准，特别是导致直接的见解，在数值均匀化文献中的Gamblets方法的实施。此外，如果目标函数不具有足够的规律性，则将长度尺度减小到零可能导致近似误差的放大，这表明需要对要近似的目标函数进行更强的先验假设。我们引入了一个奇异权函数来处理它，无论是理论上还是数值上。这项工作揭示了粗糙数据的长度尺度，计算成本，目标函数的规律性，以及近似和数值模拟的准确性的相互作用。
. There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm or determined by some physical measurement process. The purpose of this paper is then to study, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations.