超越周期设置的均匀化降基方法
Reduced-Basis approach for homogenization beyond the periodic setting
AI总结：
针对椭圆偏微分方程均匀化中大量宏观参数化微观问题，提出降基方法加速计算，通过后验误差估计和离线/在线两阶段实现高精度加速。
摘要：
我们考虑椭圆偏微分方程均匀化中平均系数的计算。在此问题中，如同许多多尺度问题一样，在微观尺度上需要大量由宏观尺度参数化的类似计算。这是一个非常适合模型降阶尝试的框架。本文旨在展示降基方法如何在不损失精度的情况下加速大量单元问题的计算。该降基方法的基本组成部分是{\it 后验}误差估计，它为感兴趣的输出提供尖锐的误差界，以及一个分为离线阶段和在线阶段的逼近过程，该过程将逼近空间的生成与其在Galerkin投影中的使用解耦。
英文摘要：
We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the macroscopic scale is required at the microscopic scale. This is a framework very much adapted to model order reduction attempts. The purpose of this work is to show how the reduced-basis approach allows to speed up the computation of a large number of cell problems without any loss of precision. The essential components of this reduced-basis approach are the {\it a posteriori} error estimation, which provides sharp error bounds for the outputs of interest, and an approximation process divided into offline and online stages, which decouples the generation of the approximation space and its use for Galerkin projections.
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