Computing Interior Eigenvalues of Large Matrices by Preconditioned Krylov Subspace Methods

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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.