论文标题
通过子空间分解的空间网络模型的迭代解决方案
Iterative solution of spatial network models by subspace decomposition
论文作者
论文摘要
我们介绍并分析了用于解决空间网络问题的预处理共轭梯度方法（PCG）。首先，我们考虑基于纤维材料的扩散和结构力学模拟，但是该方法可以应用于各种模型，从而实现了一组抽象假设。所提出的方法基于经典的子空间分解为粗糙的子空间，这是将有限元空间限制到空间网络节点的限制，以及在网格恒星上具有支持的局部化子空间。这项工作的主要贡献是对拟议方法的收敛分析。该分析将有限元理论（包括插值边界）转化为空间网络设置。 PCG算法的收敛速率仅取决于运算符的全局界限以及网络的均匀性，连通性和局部性常数。理论结果通过几个数值实验证实。
We present and analyze a preconditioned conjugate gradient method (PCG) for solving spatial network problems. Primarily, we consider diffusion and structural mechanics simulations for fiber based materials, but the methodology can be applied to a wide range of models, fulfilling a set of abstract assumptions. The proposed method builds on a classical subspace decomposition into a coarse subspace, realized as the restriction of a finite element space to the nodes of the spatial network, and localized subspaces with support on mesh stars. The main contribution of this work is the convergence analysis of the proposed method. The analysis translates results from finite element theory, including interpolation bounds, to the spatial network setting. A convergence rate of the PCG algorithm, only depending on global bounds of the operator and homogeneity, connectivity and locality constants of the network, is established. The theoretical results are confirmed by several numerical experiments.
