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Multigrid solution techniques for anisotropic structured linear systems

机译：各向异性结构化线性系统的多网格求解技术

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摘要

Multigrid methods are highly efficient solution techniques for large sparse structured linear systems which are positive definite and ill-conditioned. In a recent paper [R. Fischer, T. Huckle, Multigrid methods for anisotropic BTTB systems, Linear Algebra Appl. (2005), submitted for publication], multigrid methods have been developed which are especially designed for anisotropic matrices belonging to the two-level Toeplitz class. These methods are primarily based on the use of a suitable combination of semicoarsening and full coarsening steps. In this paper the main focus is on the design of efficient smoothing techniques. Moreover, we are not only interested in two-level Toeplitz matrices, but also in matrices of two-level trigonometric matrix algebras. First, we describe methods for systems with anisotropy along coordinate axes. Although some of the ideas are known from the solution of partial differential equations, we present them here in a more formal way using generating functions and their level curves. This allows us not only to obtain theoretical results on convergence and reduction of anisotropy, but also to carry over the results to systems with anisotropy in other directions. We introduce new coordinates in order to describe these more complicated systems in terms of generating functions. This enables us to develop smoothers which are especially suitable for these more complicated systems.
机译：多网格方法是正定和病态的大型稀疏结构线性系统的高效求解技术。在最近的一篇论文中[R. Fischer，T. Huckle，各向异性BTTB系统的Multigrid方法，线性代数应用。 （2005年，已提交出版），已经开发了多网格方法，这些方法是专门为属于两级Toeplitz类的各向异性矩阵设计的。这些方法主要基于半粗化和完全粗化步骤的适当组合。本文主要关注高效平滑技术的设计。此外，我们不仅对两层Toeplitz矩阵感兴趣，而且对两层三角矩阵代数矩阵感兴趣。首先，我们描述了沿坐标轴具有各向异性的系统的方法。尽管从偏微分方程的解决方案中可以了解到一些思想，但是我们在这里使用生成函数和它们的水平曲线以更正式的方式介绍它们。这不仅使我们可以获得关于各向异性的收敛和减小的理论结果，而且可以将结果推广到在其他方向上具有各向异性的系统。我们引入新的坐标，以便根据生成函数来描述这些更复杂的系统。这使我们能够开发出特别适用于这些更复杂系统的平滑器。

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