Computing Interior Eigenvalues of Large Matrices by Preconditioned Krylov Subspace Methods

研究人员将发展预条件的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。它还将为公共发行版开发黑盒实现。在PI之前的一项工作中，已经开发了这种类型的方法来计算对称问题的一些极端（最小或最大）特征值，该方法也已在称为EIGIFP的库质量软件中实现。研究者建议对现有的计算对称和非对称矩阵问题的内部特征值的方法进行推广。所得到的算法不仅继承了现有Krylov子空间方法的理想特性，而且还允许通过使用预条件（或近似逆）而不是移位矩阵的逆来加速收敛。大矩阵的内部特征值计算在许多重要的应用中出现，如粒子加速器模型的腔内电磁场模拟和量子力学中局域化的安德森模型。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进展，但这些应用提出了重大挑战。所提出的工作将推进计算内部特征值的理论、算法和软件工具箱。它的目标是通过开发一个非专业用户可以轻松使用的高效且可公开访问的软件包，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.