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@INPROCEEDINGS{diNapoli:15613,
      author       = {di Napoli, E. and Bientinesi, Paolo},
      title        = {{S}equences of generalized eigenproblems in {DFT}},
      reportid     = {PreJuSER-15613},
      year         = {2011},
      note         = {Record converted from VDB: 12.11.2012},
      comment      = {10th IMACS International Symposium on Iterative Methods in
                      Scientific Computing},
      booktitle     = {10th IMACS International Symposium on
                       Iterative Methods in Scientific
                       Computing},
      abstract     = {Research in several branches of chemistry and material
                      science relies on large numerical simulations. Many of these
                      simulations are based on Density Functional Theory (DFT)
                      models that lead to sequences of generalized eigenproblems
                      $\{P^{(i)}\}$. Every simulation normally requires the
                      solution of hundreds of sequences, each comprising dozens of
                      large and dense eigenproblems; in addition, the problems at
                      iteration $i+1$ of $\{P^{(i)}\}$ are constructed
                      manipulating the solution of the problems at iteration $i$.
                      The size of each problem ranges from 10 to 40 thousand and
                      the interest lays in the eigenpairs corresponding to the
                      lower 10-30\\% part of the spectrum. Due to the dense nature
                      of the eigenproblems and the large portion of the spectrum
                      requested, iterative solvers are not competitive; as a
                      consequence, current simulation codes uniquely use direct
                      methods. In this talk we present a study that highlights how
                      eigenproblems in successive iterations are strongly
                      correlated to one another. In order to understand this
                      result, we need to stress the importance of the basis wave
                      functions, which constitute the building blocks of any DFT
                      scheme. Indeed, the matrix entries of each problem in
                      $\{P^{(i)}\}$ are calculated through superposition integrals
                      of a set of basis wave functions. Moreover, the state wave
                      functions---describing the quantum states of the
                      material---are linear combinations of basis wave functions
                      with coefficients given by the eigenvectors of the problem.
                      Since a new set of basis wave functions is determined at
                      each iteration of the simulation, the eigenvectors between
                      adjacent iterations are only loosely linked with one
                      another. In light of these considerations it is surprising
                      to find such a deep correlation between the eigenvectors of
                      successive problems. We set up a mechanism to track the
                      evolution over iterations $i=1,\dots,n$ of the angle between
                      eigenvectors $x^{(i)}$ and $x^{(i+1)}$ corresponding to the
                      $j^{th}$ eigenvalue. In all cases the angles decrease
                      noticeably after the first few iterations and become almost
                      negligible, even though the overall simulation is not close
                      to convergence. Even the state of the art direct
                      eigensolvers cannot exploit this behavior in the solutions.
                      In contrast, we propose a 2-step approach in which the use
                      of direct methods is limited to the first few iterations,
                      while iterative methods are employed for the rest of the
                      sequence. The choice of the iterative solver is dictated by
                      the large number of eigenpairs required in the simulation.
                      For this reason we envision the Subspace Iterations
                      Method---despite its slow convergence rate---to be the
                      method of choice. Nested at the core of the method lays an
                      inner loop $V \leftarrow A V$; due to the observed
                      correlation between eigenvectors, the convergence is reached
                      in a limited number of steps. In summary, we propose
                      evidence in favor of a mixed solver in which direct and
                      iterative methods are combined together.},
      month         = {May},
      date          = {2011-05-18},
      organization  = {Marrakech, Morocco, 18 May 2011},
      cin          = {JSC},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {Scientific Computing / Simulation and Data Laboratory
                      Quantum Materials (SDLQM) (SDLQM)},
      pid          = {G:(DE-Juel1)FUEK411 / G:(DE-Juel1)SDLQM},
      typ          = {PUB:(DE-HGF)6},
      url          = {https://juser.fz-juelich.de/record/15613},
}