论文标题
在大型Lyapunov方程的集成Krylov-Adi求解器上
On an integrated Krylov-ADI solver for large-scale Lyapunov equations
论文作者
论文摘要
大规模Lyapunov方程的低级别ADI方法中最昂贵的步骤之一是在每次迭代时移动的线性系统的解决方案。我们建议将扩展的Krylov子空间方法用于此任务。特别是，我们说明了如何构建一个单个近似空间来求解从Lyapunov剩余规范方面实现规定准确性所需的所有移动线性系统。此外，我们展示了如何完全合并这两个迭代程序，以便为重要的方程式获得新颖，有效地实现低级ADI方法。许多用于换档计算的最新算法也可以很容易地纳入我们的新方案中。与基于偏移的线性系统的稀疏直接求解器的实现相比，几个数值结果说明了我们新过程的潜力。
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems.
