粗糙系数多尺度偏微分方程的广义粗糙多调和样条
Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients
摘要：
在本文中，我们证明了广义粗糙多谐样条（GRPS）的贝叶斯框架内的建设，特别是，粗糙系数的多尺度偏微分方程。该方法通过对原始偏微分方程进行随机化处理，在给定边缘或导数测量的部分信息的情况下，利用适当的先验分布和条件期望，自动得到最优粗基。我们证明了所得到的基的（准）最优局部化和逼近性质，并通过数值实验验证了理论结果。
英文摘要：
In this paper, we demonstrate the construction of generalized Rough Polyhamronic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on edge or derivative measurements. We prove the (quasi)-optimal localization and approximation properties of the obtained bases, and justify the theoretical results with numerical experiments.
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