The Anderson Model of Localization: A Challenge for Modern Eigenvalue Methods
SIAM Journal on Scientific Computing
Thick-restart Lanczos method for electronic structure calculations
Journal of Computational Physics
Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Filtered Conjugate Residual-type Algorithms with Applications
SIAM Journal on Matrix Analysis and Applications
Self-consistent-field calculations using Chebyshev-filtered subspace iteration
Journal of Computational Physics
Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis
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A central goal in computational materials science is to find efficient methods for solving the Kohn-Sham equation. The realization of this goal would allow one to predict properties such as phase stability, structure and optical and dielectric properties for a wide variety of materials. Typically, a solution of the Kohn-Sham equation requires computing a set of low-lying eigenpairs. Standard methods for computing such eigenpairs require two procedures: (a) maintaining the orthogonality of an approximation space, and (b) forming approximate eigenpairs with the Rayleigh-Ritz method. These two procedures scale cubically with the number of desired eigenpairs. Recently, we presented a method, applicable to any large Hermitian eigenproblem, by which the spectrum is partitioned among distinct groups of processors. This "divide and conquer" approach serves as a parallelization scheme at the level of the solver, making it compatible with existing schemes that parallelize at a physical level and at the level of primitive operations, e.g., matrix-vector multiplication. In addition, among all processor sets, the size of any approximation subspace is reduced, thereby reducing the cost of orthogonalization and the Rayleigh-Ritz method. We will address the key aspects of the algorithm, its implementation in real space, and demonstrate the nature of the algorithm by computing the electronic structure of a metal-semiconductor interface.