Flexible Krylov Methods and Schwarz Preconditioners

标题：灵活的Krylov方法和Schwarz预调节器。krylov子空间方法是目前求解线性代数方程组，特别是微分方程离散化问题的首选迭代方法。这些方法的优点部分来源于预条件的使用，预条件是改变线性系统的谱性质的矩阵（或算子）。近年来，一些研究人员（包括PI）提出并分析了Krylov方法，其中允许前置条件从一个（外部）步骤变化到下一个步骤。特别地，前置条件可以是克雷洛夫方法本身。这些由内而外的方法中有一些已经被实验证明是有效的。作为该项目的一部分，建议对这类方法的一般内外方法进行详细分析。这样的分析应该能让我们对这些方法有一个了解，同时也说明了如何去思考新的内外方法。所有这些方法都将进行实验，并与重新启动的方法进行比较。不精确克雷洛夫子空间方法是指在每一步不精确地执行矩阵向量乘法的情况。这种情况出现在许多应用中，包括块矩阵和每一步的舒尔补近似。一些研究人员通过实验表明，随着迭代的进行，不精确的数量可以增加。我们建议详细研究这一现象，以提供对这一现象的理解，并设计计算中使用的不精确量的界限。在过去的几年中，我们开发了一种新的加法和乘法Schwarz方法的代数公式。这些方法作为微分方程（并联）解的固定预调节器，广泛应用于工业、科学和工程应用。这个新的公式使我们能够利用丰富的线性代数理论来研究这些方法。这个新理论补充了通常用于这些方法的分析理论。例如，我们最近完成了限制加性Schwarz（限制性加性Schwarz）预条件的分析，其中没有解析收敛结果。在该项目的第二部分中，提出进一步使用该新公式来分析其他Schwarz变体。

title: Flexible Krylov methods and Schwarz preconditioners.Proposal Number: 0207525Krylov subspace methods are nowadays the premier iterative methods for the solution of linear algebraic systems of equations, especially for those which arise from the discretization of differential equations. The strength of these methods derives in part by the use of preconditioners, which are matrices (or operators) changing the spectral properties of the linear system. In recent years, several researchers (including the PI), have proposed and analyzed Krylov methods in which the preconditioner is allowed to change from one (outer) step to the next. In particular, the preconditioner can be a Krylov method itself. Some of these inner-outer methods have been shown experimentally to work well. As part of this project, it is proposed to undertake a detailed analysis of general inner-outer methods of this kind. This analysis should give us an understanding of these methods, and also indicate how to think of new inner-outer methods. Experiments will be conducted with all these methods together with comparison with restarted ones. Inexact Krylov subspace methods refer to the situation where the matrix-vector multiplication at each step is not performed exactly. This situation appears in numerous applications, including block matrices and the approximation of Schur complements at each step. It was shown experimentally by some researchers that the amount of inexactness can be allowed to grow as the iterations progress. We propose to study this phenomenon in detail, both to provide an understanding of this phenomenon, and to devise bounds on the amount of inexactness to be used computationally. During the last few years we have developed a new algebraic formulation of additive and multiplicative Schwarz methods. These methods, which are used as fixed) preconditioners for the (parallel) solution of differential equations, are extensively used in industry, science and engineering applications. This new formulation allow us to study these methods using the rich theory of linear algebra. This new theory complements the analytical theory usually used for these methods. For example, we have recently completed the analysis of the Restrictive Additive Schwarz (RAS) preconditioner, for which there is no analytical convergence results.In the second part of this project, it is proposed to further use this new formulation to analyze other Schwarz variants.