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@INPROCEEDINGS{DiNapoli:150535,
      author       = {Di Napoli, Edoardo and Berljafa, Mario},
      title        = {{P}reconditioning {C}hebyshev subspace iteration applied to
                      sequences of dense eigenproblems in ab initio simulations},
      reportid     = {FZJ-2014-00589},
      year         = {2013},
      abstract     = {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]},
      month         = {Jun},
      date          = {2013-06-24},
      organization  = {Numerical Analysis and Scientific
                       Computation with Applications, Calais
                       (France), 24 Jun 2013 - 26 Jun 2013},
      subtyp        = {Invited},
      cin          = {JSC},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {411 - Computational Science and Mathematical Methods
                      (POF2-411) / Simulation and Data Laboratory Quantum
                      Materials (SDLQM) (SDLQM)},
      pid          = {G:(DE-HGF)POF2-411 / G:(DE-Juel1)SDLQM},
      typ          = {PUB:(DE-HGF)6},
      url          = {https://juser.fz-juelich.de/record/150535},
}