Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers
Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers
复制标题
块预条件梯度型特征求解器中各个 Ritz 值的收敛率
DOI：
--
复制
发表时间：
2022
影响因子：
1.3
通讯作者：
K. Neymeyr
中科院分区：
文献类型：
--
作者：
M. Zhou;K. Neymeyr
许多流行的大型稀疏埃尔米特矩阵或矩阵对的特征求解器可以解释为加速块预处理梯度（BPG）迭代，以便通过组合已知的估计来分析其收敛行为。 BPG的一个重要特征是集群鲁棒性，即通过足够大的块大小来保证计算集群特征值的合理性能。通过采用非预处理特征值求解器的经典估计，可以很容易地解释精确逆（精确移位逆）预处理的这一特征，而更一般预处理的现有结果仍然可以改进。我们期望将对相应向量迭代的某些尖锐估计扩展到 BPG，其中将导出各个 Ritz 值的收敛速度的适当范围。这种扩展已经在[Math.比较。 88（2019），2737--2765]。本文讨论了更实际的情况，即通过 Rayleigh-Ritz 方法隐式优化步长。鉴于简洁和更灵活的界限，我们的新估计改进了之前的一些估计。
Many popular eigensolvers for large and sparse Hermitian matrices or matrix pairs can be interpreted as accelerated block preconditioned gradient (BPG) iterations in order to analyze their convergence behavior by composing known estimates. An important feature of BPG is the cluster robustness, i.e., reasonable performance for computing clustered eigenvalues is ensured by a sufficiently large block size. This feature can easily be explained for exact-inverse (exact shift-inverse) preconditioning by adapting classical estimates on nonpreconditioned eigensolvers, whereas the existing results for more general preconditioning are still improvable. We expect to extend certain sharp estimates for the corresponding vector iterations to BPG where proper bounds of convergence rates of individual Ritz values are to be derived. Such an extension has been achieved for BPG with fixed step sizes in [Math. Comp. 88 (2019), 2737--2765]. The present paper deals with the more practical case that the step sizes are implicitly optimized by the Rayleigh-Ritz method. Our new estimates improve some previous ones in view of concise and more flexible bounds.
DOI：
10.1137/17m1157568
发表时间：
2017-11
期刊：
ArXiv
影响因子：
--
作者：
Lingfei Wu;Fei Xue;A. Stathopoulos
通讯作者：
Lingfei Wu;Fei Xue;A. Stathopoulos