发表机构
Heidelberg University; RWTH Aachen University（海德堡大学; 亚琛工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于Grassmann流形插值的局部模型降阶方法，用于高效求解多尺度参数化PDE，通过分解局部子问题并插值最优子空间，兼顾近指数空间收敛与参数收敛，数值实验验证了理论。
AI 中文摘要
多尺度、参数相关的偏微分方程（PDEs）由于强烈的系数非均匀性和高维参数空间，带来了严峻的计算挑战。我们在多尺度广义有限元方法（MS-GFEM）框架内发展了一种几何插值方法，针对其最昂贵的组成部分：计算参数相关的最优局部逼近空间。利用MS-GFEM的空间局部性并假设局部参数依赖性，我们将全局问题分解为参数低维的局部子问题。每个参数的最优子空间被识别为Grassmann流形上的点，并通过稀疏网格上的Grassmann插值进行逼近，这保留了这些空间的几何结构，同时高效处理高维参数空间。所得到的局部模型降阶方法继承了MS-GFEM的近指数空间收敛性和稀疏网格的参数收敛速率。针对椭圆问题的数值实验证实了理论收敛结果。
英文摘要
Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter spaces. We develop a geometric interpolation approach within the multiscale generalized finite element method (MS-GFEM) that targets the most expensive component: computing parameter-dependent optimal local approximation spaces. Leveraging the spatial localization of MS-GFEM and assuming local parameter dependence, we decompose the global problem into parametrically low-dimensional local subproblems. The optimal subspaces for each parameter are identified as points on a Grassmann manifold and approximated via Grassmann interpolation on sparse grids, which preserves the geometric structure of these spaces while efficiently handling high-dimensional parameter spaces. The resulting localized model reduction method inherits the nearly exponential spatial convergence of MS-GFEM and the parametric convergence rates of sparse grids. Numerical experiments for elliptic problems confirm the theoretical convergence results.
