Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves
Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves
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DOI：
10.1137/080716281
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发表时间：
2009-08
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影响因子：
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通讯作者：
M. Freitag;A. Spence
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文献类型：
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作者：
M. Freitag;A. Spence
我们考虑了用移位-逆Arnoldi方法计算大型稀疏非对称矩阵的几个特征向量和相应的特征值，并考虑了隐式重启和不隐式重启。对于内迭代，我们使用预条件GMRES作为非精确迭代求解器。通过迭代求解器在算法的每个外部步骤所需的内部迭代次数来衡量求解的成本。我们首先推广了Simoncini[SIAM J.Numer.Anal.，43(2005)，pp.1155-1174]隐含地重新启动了Arnoldi的方法，这产生了该方法的总体成本的改进。其次，我们将一种新的预处理策略应用到内解算器中。我们表明，对预处理器进行小的秩值改变可以显著节省总迭代次数。在这里考虑的例子中，新的预处理器与隐式重新启动的Arnoldi中的放松策略的组合产生了大约50%的总成本增加。数值实验说明了整个论文的理论。
We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. The costs of the solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini [SIAM J. Numer. Anal., 43 (2005), pp. 1155-1174] to implicitly restarted Arnoldi's method, which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show that small rank changes to the preconditioner can produce significant savings in the total number of iterations. The combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 percent in the examples considered here. Numerical experiments illustrate the theory throughout the paper.