In this article a new family of preconditioners is introduced for symmetric positive definite linear systems. The new preconditioners, called the AWG preconditioners (for Algebraic-Woodbury-GenEO) are constructed algebraically. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable non-assembled form is known by construction. The extra cost is a second (and to this day rather expensive) coarse solve in the preconditioner. One of the AWG preconditioners has already been presented in the short preprint [38]. This article is the first full presentation of the larger family of AWG preconditioners. It includes proofs of the spectral bounds as well as numerical illustrations.
翻译:在本条中,为对称正确定线性系统引入了一套新的先决条件。 新的先决条件( 被称为AWG 先决条件者( 为 Algebraic- Woodbury- GenEO ) ), 以代数制建。 我们的意思是, 只需要对正在解决线性系统所需的矩阵A A 的了解。 由于GENEO光谱粗略的空间技术, 先决条件操作者的条件号从理论上从上面捆绑起来。 通过以更光谱模式丰富粗糙的空间, 这个上限可以更小一些。 与以前GENEO 粗糙空间的工作不同, 新的是不需要对部分非混合的A型形式有所了解。 事实上, 光谱粗空间技术不是直接适用于A,而是对A 的低级修改, 其适当的非混合形式是建筑学所知道的。 额外成本是先决条件的第二个( 到今天成本相当昂贵) 粗糙的解决方案。 特设工作组的一个先决条件已经在短的预印页中作了介绍。 它的完整地展示了AWG 。