title: Shift-and-invert parallel spectral transformation eigensolver: Massively parallel performance for density-functional based tight-binding.
abstract: 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 diagonalization of density-functional based tight-binding (DFTB) Hamiltonian and overlap matrices for single-wall metallic carbon nanotubes, diamond nanowires, and bulk diamond crystals. The largest (smallest) example studied is a 128,000 (2000) atom nanotube for which ∼330,000 (∼5600) eigenvalues and eigenfunctions are obtained in ∼190 (∼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.
authors.affiliation: Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois, 60439.
authors.forename: Murat
authors.initials: M
authors.lastname: Keçeli
authors.affiliation: Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 60439.
authors.forename: Hong
authors.initials: H
authors.lastname: Zhang
authors.affiliation: Argonne National Laboratory, Materials Science Division, Argonne, Illinois, 60439.
authors.forename: Peter
authors.initials: P
authors.lastname: Zapol
authors.affiliation: Department of Chemistry, the University of Alabama, Tuscaloosa, Alabama, 35487.
authors.forename: David A
authors.initials: DA
authors.lastname: Dixon
authors.affiliation: Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois, 60439.
authors.forename: Albert F
authors.initials: AF
authors.lastname: Wagner
keywords: generalized eigenvalue problem; semi-empirical methods; sparse matrices; spectrum slicing; tight-binding DFT