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Accelerating the Arnoldi iteration: Theory and practice.

机译：加速Arnoldi迭代：理论与实践。

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The Arnoldi iteration is widely used to compute a few eigenvalues of a large sparse or structured matrix. However, the method may suffer from slow convergence when the desired eigenvalues are not dominant or well separated. A systematic approach is taken in this dissertation to address the issue of how to accelerate the convergence of the Arnoldi algorithm within a subspace of limited size.; The acceleration strategies presented here are grouped into three categories. They are the method of restarting, the method of spectral transformation and the Newton-like acceleration.; Simply put, the method of restarting repeats a k-step Arnoldi iteration after improving the starting vector. The method is further divided into polynomial and rational restarting based on the way the starting vector is modified. We show that both mechanisms can be implemented in an implicit fashion by relating the restarted Arnoldi to a truncated QR or the RQ iteration. The rational restarting via a Truncated RQ (TRQ) iteration converges extremely fast. However, a linear system must be solved for each restarting. The possibility of replacing a direct linear solver with a preconditioned iterative solver while maintaining the rapid convergence of TRQ is explored in this thesis.; The method of spectral transformation is based on the idea of transforming the original eigenvalue problem into one that is easier to solve. Again, both polynomial and rational transformations are possible. Practical issues regarding the design and implementation of effective spectral transformations are discussed.; Finally, one can treat the eigenvalue problem as a nonlinear equation upon which Newton-like methods can be applied. The Jacobi-Davidson (JD) algorithm proposed by Sleijpen and Van der Vorst takes this approach. In JD, the Arnoldi iteration is merely used as a global searching tool to provide a good starting point for the Newton iteration. This algorithm shares many similar properties with the TRQ iteration. Numerical comparisons between these two methods are made in this thesis.
机译：Arnoldi迭代被广泛用于计算大型稀疏或结构化矩阵的一些特征值。但是，当所需的特征值不是主要的或没有很好地分离时，该方法可能会收敛缓慢。本文采用系统的方法解决了在有限大小的子空间内如何加快Arnoldi算法收敛的问题。此处介绍的加速策略分为三类。它们是重新启动的方法，频谱变换的方法和类似牛顿的加速度。简而言之，重新启动的方法在改善了起始向量后重复了k步Arnoldi迭代。根据修改起始向量的方式，该方法又分为多项式和有理重启。我们表明，通过将重新启动的Arnoldi与截断的QR或RQ迭代相关联，可以以隐式方式实现这两种机制。通过截断的RQ（TRQ）迭代进行的合理重启非常快速地收敛。但是，每次重启都必须解决一个线性系统。本文探讨了在保持TRQ快速收敛的同时，用预处理的迭代求解器代替直接线性求解器的可能性。频谱变换方法基于将原始特征值问题变换为更易于解决的思想。同样，多项式和有理变换都是可能的。讨论了有关有效频谱变换的设计和实现的实际问题。最后，可以将特征值问题视为可以应用牛顿法的非线性方程。 Sleijpen和Van der Vorst提出的Jacobi-Davidson（JD）算法采用了这种方法。在JD中，Arnoldi迭代仅用作全局搜索工具，为Newton迭代提供了良好的起点。该算法与TRQ迭代具有许多相似的属性。本文对这两种方法进行了数值比较。

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