Domain decomposition based $${\mathcal H}$$ -LU preconditioning
Domain decomposition based $${\mathcal H}$$ -LU preconditioning
复制标题
基于域分解 $${mathcal H}$$ -LU 预处理
DOI：
10.1007/s00211-009-0218-6
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发表时间：
2009
影响因子：
2.1
通讯作者：
S. Borne
中科院分区：
文献类型：
--
作者：
L. Grasedyck;Ronald Kriemann;S. Borne
层次矩阵提供了一种数据稀疏的方法来近似完全填充的矩阵。构造$${{\mathcal H}}$$ -矩阵的两个基本步骤是（a）矩阵分块的分层构造，和（B）用低秩矩阵对矩阵数据进行分块逼近。在本文中，我们开发了一种新的方法来构建必要的分区的区域分解的基础上。与标准的基于几何二分的$${\mathcal H}}$$ -矩阵相比，该方法可以得到有限元刚度矩阵的$${\mathcal H}$$ -LU分解，并显著提高了存储和计算复杂度。这些严格证明和数值验证的改进结果从一个$${\mathcal H}$$ -矩阵块结构，这是自然适合并行化，其中大型子块的刚度矩阵保持为零的LU分解。我们提供的数值结果中，区域分解为基础的$${{\mathcal H}}$$ -LU分解被用作一个预条件的离散（三维）对流扩散方程的迭代解。
Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an $${{\mathcal H}}$$ -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based $${{\mathcal H}}$$ -matrices, this new approach yields $${\mathcal H}$$ -LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an $${\mathcal H}$$ -matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based $${{\mathcal H}}$$ -LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.