Computing Interior Eigenvalues of Large Matrices by Preconditioned Krylov Subspace Methods

研究者将发展预条件Krylov子空间方法并分析计算大规模矩阵特征值问题的几个内部特征值。它还将为公共分发开发黑盒实现。在PI以前的工作中，这种类型的方法已经被开发用于计算对称问题的一些极端（最小或最大）特征值，这也被实现在一个图书馆质量的软件称为EIGIFP。研究者提出了一个推广现有的方法计算内部特征值的对称和非对称矩阵问题。该算法不仅继承了已有Krylov子空间方法的优良特性，而且还允许通过使用预处理器来加速收敛（或近似逆）大矩阵的内部特征值的计算出现在许多重要的应用中，例如粒子加速器模型的腔中的电磁场模拟和局部化的安德森模型。量子力学尽管在开发大规模特征值问题的迭代方法和软件包方面取得了巨大的进展，但这些应用仍构成了重大挑战。所提出的工作将推进内部特征值计算的理论、算法和软件工具箱。它的目的是通过开发一个非专业用户可以方便使用的高效和公开访问的软件包，在应用程序中产生最大的影响。

The investigator will develop preconditioned Krylov subspace methods with analysis for computing a few interior eigenvalues of large scale matrix eigenvalue problems. It will also develop black-box implementations for public distributions. In a previous work of PI, a method of this type has been developed for computing some extreme (smallest or largest) eigenvalue of the symmetric problems, which has also been implemented in a library-quality software called EIGIFP. The investigator proposes to develop a generalization of the existing method for computing interior eigenvalues for symmetric and nonsymmetric matrix problems. The resulting algorithms not only inherit desirable characteristics of the existing Krylov subspace methods, but also allow convergence acceleration through the use of a preconditioner (or approximate inverse) rather than the inverse of a shifted matrix.Computations of interior eigenvalues for large matrices arise in many important applications such as electromagnetic field simulations in cavities for particle accelerator models and the Anderson model of localization in quantum mechanics. In spite of tremendous progresses made in developing iterative methods and software packages for large-scale eigenvalue problems, these applications pose a significant challenge. The proposed work shall advance the theory, algorithms and software toolboxes for the computation of interior eigenvalues. It aims to generate maximal impact in applications by developing an efficient and publicly accessible software package that can be easily used by non-expert users.