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Shift-and-Invert Parallel Spectral Transformation Eigensolver: Massively Parallel Performance for Density-Functional Based Tight-Binding
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DOI：10.1002/jcc.24254.png)
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The Shift-and-invert parallel spectral transformations (SIPs), a computational approach to solve sparse eigenvalue problems, is developed for massively parallel architectures with exceptional parallel scalability and robustness. The capabilities of SIPs are demonstrated by diagnalization of density-functional based tight-binding (DFTB) Hamiltonian and overlap matrices for single-wall matallic carbon nanotubes, diamond nanowires, and bulk diamond crystals. The largest (smallest) example studies is a 128,000 (2000) atom nanotube for which similar to 330,000 (similar to 5600) eigenvalues and eigenfunctions are obtained in similar to 190 (similar to 5) seconds when parallelized over 266,144 (16,384) Blue Gene/Q cores. Weak scaling and strong scaling of SIPs are analyzed and the performance of SIPs is compared with other novel methods. Different matrix ordering methods are investigated to reduce the cost of the factorization step, which dominates the time to solution at the strong scaling limit. A parallel implementation of assembling the density matrix from the distributed eigenvectors is demonstrated. (C) 2015 Wiley Periodicals, Inc.
Keyword:
sparse matrices
spectrum slicing
generalized eigenvalue problem
semi-empirical methods
tight-binding DFT
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