Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics
Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics
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DOI：
10.1137/10082364x
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发表时间：
2012-04
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影响因子：
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通讯作者：
H. Calandra;S. Gratton;J. Langou;X. Pinel;X. Vasseur
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文献类型：
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作者：
H. Calandra;S. Gratton;J. Langou;X. Pinel;X. Vasseur
在计算科学和工程的许多应用中，要求同时给出多个右端的大型线性方程组的解。已知基于高斯消去法的直接方法在该设置中特别有吸引力。然而，当问题的维数非常大时，预条件块Krylov空间求解器通常被认为是选择的方法。因此，本文的目的是提出基于块重新启动GMRES的迭代方法，允许变量预处理的多右端线性系统的解决方案。使用灵活的方法是特别感兴趣的近似可能迭代求解器被认为是在预处理阶段。首先，我们介绍了一个新的变种块灵活重新启动GMRES，其中包括一个策略，用于检测系统的线性组合时，近似收敛。这种显式的块大小减小通常称为紧缩。我们分析了这种基于收缩的灵活方法的主要性质，并证明了块残差的Frobenius范数总是非增的。我们还提出了一个灵活的变体，特别是在有限的内存的情况下使用的基础上放气和截断。最后，我们说明了这些灵活的块方法的数值行为产生的大规模工业模拟中的物理学，不确定的线性系统的大小高达10亿个未知数与多个右手边已成功地解决了在并行分布式存储器环境。
In a wide number of applications in computational science and engineering the solution of large linear systems of equations with several right-hand sides given at once is required. Direct methods based on Gaussian elimination are known to be especially appealing in that setting. Nevertheless, when the dimension of the problem is very large, preconditioned block Krylov space solvers are often considered as the method of choice. The purpose of this paper is thus to present iterative methods based on block restarted GMRES that allow variable preconditioning for the solution of linear systems with multiple right-hand sides. The use of flexible methods is especially of interest when approximate possibly iterative solvers are considered in the preconditioning phase. First we introduce a new variant of block flexible restarted GMRES that includes a strategy for detecting when a linear combination of the systems has approximately converged. This explicit block size reduction is often called deflation. We analyze the main properties of this flexible method based on deflation and notably prove that the Frobenius norm of the block residual is always nonincreasing. We also present a flexible variant based on both deflation and truncation to especially be used in case of limited memory. Finally we illustrate the numerical behavior of these flexible block methods for large industrial simulations arising in geophysics, where indefinite linear systems of size up to 1 billion unknowns with multiple right-hand sides have been successfully solved in a parallel distributed memory environment.