Rational Krylov decompositions : theory and applications
Rational Krylov decompositions : theory and applications
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有理 Krylov 分解：理论与应用
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Mario Berljafa
中科院分区：
文献类型：
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作者：
Mario Berljafa
基于有理Krylov空间的数值方法已成为科学计算中不可缺少的工具。在这篇论文中，我们研究理性Krylov空间考虑理性Krylov分解，矩阵关系，在一定条件下，与这些空间。我们研究了这种分解的代数性质，并给出了有理Krylov空间的一个隐式Q定理。
我们推导出标准的和调和的里兹提取策略，用于近似矩阵的特征对和近似矩阵函数的作用
一个向量。虽然这些主题已经考虑到以前，我们的方法不需要最后一个极点是无限的，这使得提取过程
计算效率更高。
通常，用于计算有理Krylov基的有理Arnoldi算法的计算上最昂贵的组件是在每次迭代时求解大型线性方程组。我们探讨的选择，同时解决几个线性系统，从而并行构建合理的Krylov基础。如果不小心这样做，正交化的基础可能会变得条件不良，导致
正交化过程中的数值不稳定性。我们引入了新的概念，连续对产生一个接近最佳的并行化策略，允许控制增长的条件数的非正交基础。因此，我们得到了一个更准确和可靠的并行有理Arnoldi算法。使用我们的高性能C++实现的计算的好处。
本文提出了一种求解非线性有理最小二乘问题的迭代算法。困难在于寻找有理函数的极点。为此，在每次迭代的合理Krylov分解构造和修改的线性问题，以重新定位的极点到新的。我们的数值结果表明，该算法，称为RKFIT，是非常适合的线性时不变动力系统的模型降阶和指数积分相关的优化问题。此外，我们推导出一个策略的RKFIT得到的近似降阶。通过RKFIT得到的有理函数表示与援助的标量有理Krylov分解和一个额外的系数向量。以这种形式表示的函数称为RKFUN。我们开发了有效的方法进行评估，极点和根的发现，并执行基本的算术运算与RKFUNs。
最后，我们讨论了RKALK，一个MATLAB的rational Krylov工具箱，它实现了我们所有的算法，可以从http://rktoolbox.org免费获得。RKALK还提供了一个广泛的指南和越来越多的例子。特别是，我们的大多数数值实验很容易通过下载来重现
工具箱并在MATLAB中运行相应的示例文件。
Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. In this thesis we study rational Krylov spaces by considering rational Krylov decompositions; matrix relations which, under certain conditions, are associated with these spaces. We investigate the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces.
We derive standard and harmonic Ritz extraction strategies for approximating the eigenpairs of a matrix and for approximating the action of a matrix function
onto a vector. While these topics have been considered previously, our approach does not require the last pole to be infinite, which makes the extraction procedure
computationally more efficient.
Typically, the computationally most expensive component of the rational Arnoldi algorithm for computing a rational Krylov basis is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become poorly conditioned, leading
to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using our high performance C++ implementation.
We develop an iterative algorithm for solving nonlinear rational least squares problems. The difficulty is in finding the poles of a rational function. For this purpose, at each iteration a rational Krylov decomposition is constructed and a modified linear problem is solved in order to relocate the poles to new ones. Our numerical results indicate that the algorithm, called RKFIT, is well suited for model order reduction of linear time invariant dynamical systems and for optimisation problems related to exponential integration. Furthermore, we derive a strategy for the degree reduction of the approximant obtained by RKFIT. The rational function obtained by RKFIT is represented with the aid of a scalar rational Krylov decomposition and an additional coefficient vector. A function represented in this form is called an RKFUN. We develop efficient methods for the evaluation, pole and root finding, and for performing basic arithmetic operations with RKFUNs.
Lastly, we discuss RKToolbox, a rational Krylov toolbox for MATLAB, which implements all our algorithms and is freely available from http://rktoolbox.org. RKToolbox also features an extensive guide and a growing number of examples. In particular, most of our numerical experiments are easily reproducible by downloading
the toolbox and running the corresponding example files in MATLAB.
影响因子：
2.1
作者：
Schröder C;Taslaman L
通讯作者：
Taslaman L
影响因子：
1.5
作者：
S. Güttel;L. Knizhnerman
通讯作者：
S. Güttel;L. Knizhnerman