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@INPROCEEDINGS{DiNapoli:155016,
      author       = {Di Napoli, Edoardo and Berljafa},
      title        = {{A}n optimized subspace iteration eigensolver applied to
                      sequences of dense eigenproblems in ab initio simulations},
      reportid     = {FZJ-2014-04209},
      year         = {2014},
      abstract     = {Sequences of eigenvalue problems consistently appear in a
                      large class of applications based on the iterative solution
                      of a non-linear eigenvalue problem. A typical example is
                      given by the chemistry and materials science ab initio
                      simulations relying on computational methods developed
                      within the framework of Density Functional Theory (DFT). DFT
                      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 several sequences of
                      generalized eigenproblems $P: Ax=\lambda Bx$. Each sequence,
                      $P^{(1)}, \dots P^{(\ell)} \dots P^{(N)}$, groups together
                      eigenproblems with increasing outer-iteration index $\ell$.
                      In more general terms a set of eigenvalue problems
                      $\{P^{(1)}, \dots P^{(N)}\}$ is said to be a ``sequence'' if
                      the solution of the $\ell$-th eigenproblem determines, in an
                      application-specific manner, the initialization of the
                      $(\ell+1)$-th eigenproblem. For instance at the beginning of
                      each DFT cycle an initial function $\rho^{(\ell)}({\bf r})$
                      determines the initialization of the $\ell$-th eigenproblem.
                      A large fraction of $P^{(\ell)}$ eigenpairsare then use to
                      compute a new $\rho^{(\ell+1)}({\bf r})$ which, in turn,
                      leads to the initialization of a new eigenvalue problem
                      $P^{(\ell+1)}$. In addition to be part of a sequence,
                      successive eigenproblems might possess a certain degree of
                      correlation connecting their respective eigenpairs. In DFT
                      sequences, correlation becomes manifest in the way
                      eigenvectors of successive eigenproblems become
                      progressively more collinear to each other as the
                      $\ell$-indexincreases. We developed a subspace iteration
                      method (ChFSI) specifically tailored for sequences of
                      eigenproblems whose correlation appears in the form of
                      increasingly collinear eigenvectors. Our strategy is to take
                      the maximal advantage possible from the information that the
                      solution of the $P^{(\ell)}$ eigenproblem is providing when
                      solving for the successive $P^{(\ell+1)}$ problem. As a
                      consequence the subspace iteration was augmented with a
                      Chebyshevpolynomial filter whose degree gets dynamically
                      optimized so as to minimize the number of matrix-vector
                      multiplications. The effectiveness of the Chebyshev filter
                      is substantially increased when inputed the approximate
                      eigenvectors $\{ x_1^{(\ell)}, \dots x_{\rm
                      nev}^{(\ell)}\}$, as well as very reliable estimates, namely
                      $[\lambda_1^{(\ell)},\ \lambda_{\rm nev}^{(\ell)}]$, for the
                      limits of the eigenspectrum interval $[a,\ b]$ to be
                      filtered in. In additionthe degree of the polynomial filter
                      is adjusted so as to be minimal with respect to the required
                      tolerance for the eigenpairs residual. This result is
                      achieved by exploiting the dependence each eigenpair
                      residual have with respect to its convergence ratio as
                      determined by the rescaled Chebyshev polynomial and its
                      degree. The solver is complemented with an efficient
                      mechanism which locks and deflates the converged
                      eigenpairs.The resulting eigensolver was implemented in C
                      language and parallelized for both shared and distributed
                      architectures. Numerical tests show that, when the
                      eigensolver is inputed approximate solutions instead of
                      random vectors, it achieves up to a 5X speedup. Moreover
                      ChFSI takes great advantage of computational resources by
                      scaling over a large range of cores commensurate with the
                      size of the eigenproblems. Specifically, by making better
                      use of massively parallel architectures, the distributed
                      memory version will allow DFT users to simulate physical
                      systems quite larger than are currently accessible.},
      month         = {Jul},
      date          = {2014-07-01},
      organization  = {6th International Workshop on Parallel
                       Matrix Algorithms and Applications,
                       Lugano (Switzerland), 1 Jul 2014 - 4
                       Jul 2014},
      subtyp        = {After Call},
      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/155016},
}