Preconditioning Chebyshev subspace iteration applied to sequences of dense eigenproblems in ab initio simulations

Research in several branches of chemistry and materials science relies on large ab initio numerical simulations. The majority of these simulations are based on computational methods developed within the framework of Density Functional Theory (DFT) [1]. Among all the DFT-based methods the Full-potential Linearized Augmented Plane Wave (FLAPW) [2, 3] method constitutes the most precise computational framework to calculate ground state energy of periodic and crystalline materials. FLAPW provides the means to solve a high-dimensional quantum mechanical problem by representing it as a non-linear generalized eigenvalue problem which is solved self-consistently through a series of successive outer-iteration cycles. As a consequence each self-consistent simulation is made of dozens of sequences of dense generalized eigenproblems P : Ax = λBx. Each sequence, P1 , . . . Pi . . . PN , groups together eigenproblems with increasing outer-iteration index i. Successive eigenproblems in a FLAPW-generated sequence possess a high degree of correlation. In particular it has been demonstrated that eigenvectors of adjacent eigenproblems become progressively more collinear to each other as the outer-iteration index increases [4]. This result suggests one could use eigenvectors, computed at a certain outer-iteration, as approximate solutions to improve the performance of the eigensolver at the next one. In order to maximally exploit the approximate solution, we developed a subspace iteration method augmented with an optimized Chebyshev polynomial accelerator together with an efficient locking mechanism (ChFSI). The resulting eigensolver was implemented in C language and parallelized for both shared and distributed architectures. Numerical tests show that, when the eigensolver is preconditioned with approximate solutions instead of random vectors, it achieves up to a 5X speedup. Moreover ChFSI takes great advantage of computational resources by obtaining levels of efficiency up to 80 % of the theoretical peak performance. In particular, by making better use of massively parallel architectures, the distributed memory version will allow the FLAPW method users to simulate larger physical systems than are currently accessible. Additionally, despite the eigenproblems in the sequence being relatively large and dense, the parallel ChFSI preconditioned with approximate solutions performs substantially better than the corresponding direct eigensolvers, even for a significant portion of the sought-after spectrum. [1] R. M. Dreizler, and E. K. U. Gross, Density Functional Theory (Springer-Verlag, 1990) [2] A. J. Freeman, H. Krakauer, M. Weinert, and E. Wimmer, Phys. Rev. B 24 (1981) 864. [3] A. J. Freeman, and H. J. F. Jansen, Phys. Rev. B 30 (1984) 561 [4] E. Di Napoli, S. Blu ̈gel, and P. Bientinesi, Comp. Phys. Comm. 183 (2012), pp. 1674- 1682, [arXiv:1108.2594