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Sparse Hierarchical Preconditioners Using Piecewise Smooth Approximations of Eigenvectors
arXiv - CS - Numerical Analysis Pub Date : 2019-07-08 , DOI: arxiv-1907.03406
Bazyli Klockiewicz, Eric Darve

When solving linear systems arising from PDE discretizations, iterative methods (such as Conjugate Gradient, GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. The efficiency of the preconditioner depends largely on its accuracy on the eigenvectors corresponding to small eigenvalues, and unfortunately, black-box methods typically cannot guarantee sufficient accuracy on these eigenvectors. Thus, constructing the preconditioner becomes a problem-dependent task. However, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with $O(n)$ or $O(n \log{n})$ construction complexities. Our methods exhibit the optimal $O(n)$ solution times in benchmarks run on large elliptic problems of different types, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes.

中文翻译:

使用特征向量的分段平滑逼近的稀疏分层预处理器

在求解由 PDE 离散化产生的线性系统时,迭代方法(例如共轭梯度、GMRES 或 MINRES)通常是唯一实用的选择。然而,为了在少量迭代中收敛,它们必须与有效的预处理器相结合。预处理器的效率很大程度上取决于它在对应于小特征值的特征向量上的准确性,不幸的是,黑盒方法通常无法保证这些特征向量的足够准确性。因此,构建预处理器成为一个依赖于问题的任务。然而,对于一大类问题,包括许多椭圆方程,小特征值对应的特征向量是偏微分方程网格的光滑函数。在本文中,我们描述了一种分层近似分解方法,该方法侧重于提高平滑特征向量的准确性。通过保留因式分解矩阵对网格的分段多项式函数的作用来提高精度。基于分解,我们提出了一系列具有 $O(n)$ 或 $O(n \log{n})$ 构造复杂度的稀疏预处理器。我们的方法在不同类型的大型椭圆问题上运行的基准测试中表现出最佳的 $O(n)$ 求解时间,例如在流动或机械模拟中出现。在线性弹性方程的情况下,预处理器在近核刚体模式上是精确的。基于分解,我们提出了一系列具有 $O(n)$ 或 $O(n \log{n})$ 构造复杂度的稀疏预处理器。我们的方法在不同类型的大型椭圆问题上运行的基准测试中表现出最佳的 $O(n)$ 求解时间,例如在流动或机械模拟中出现。在线性弹性方程的情况下,预处理器在近核刚体模式上是精确的。基于分解,我们提出了一系列具有 $O(n)$ 或 $O(n \log{n})$ 构造复杂度的稀疏预处理器。我们的方法在不同类型的大型椭圆问题上运行的基准测试中表现出最佳的 $O(n)$ 求解时间,例如在流动或机械模拟中出现。在线性弹性方程的情况下,预处理器在近核刚体模式上是精确的。
更新日期:2020-02-25
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