论文标题
块预处理梯度型eigensolvers中单个丽兹值的收敛速率
Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers
论文作者
论文摘要
许多流行的大型和稀疏遗产矩阵或矩阵对的经济植物可以解释为加速块预处理梯度（BPG）迭代，以通过组成已知估计值来分析其收敛行为。 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确保计算聚类特征值的合理性能。可以轻松地解释此功能，以通过对非经过专门的Eigensolvers的经典估计进行调整，以进行精确的分离（确切的移位式）预处理，而现有的结果仍然可以改善更一般的预处理。我们期望将相应的矢量迭代的某些尖锐估计扩展到BPG，其中将得出单个丽思族值的融合速率的适当界限。在[数学中，BPG都实现了固定步骤的BPG。 comp。 88（2019），2737--2765]。本文涉及的更实际的案例是，台阶大小通过Rayleigh-Ritz方法隐式优化。鉴于简洁和更灵活的界限，我们的新估计提高了一些以前的估计。
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.
