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@INPROCEEDINGS{Schelthoff:200820,
      author       = {Schelthoff, Christof and Basermann, Achim},
      title        = {{P}olynomial {P}reconditioning for the {C}onjugate
                      {G}radient {M}ethod on {M}assively {P}arallel {S}ystems},
      volume       = {95/1},
      address      = {Clausthal-Zellerfeld},
      publisher    = {Institut für Informatik},
      reportid     = {FZJ-2015-03205},
      series       = {Informatik-Bericht},
      pages        = {150-167},
      year         = {1995},
      comment      = {Proceedings des 13. Workshops über Parallelverarbeitung},
      booktitle     = {Proceedings des 13. Workshops über
                       Parallelverarbeitung},
      abstract     = {A frequently used iterative algorithm for solving large,
                      sparse, symmetric and positiv definite systems of linear
                      equations is the method of conjugate gradients (CG).This
                      method requires one matrix-vector product and some dot
                      products in each iteration. Convergence is dependent on the
                      condition number of the coefficient matrix. So
                      preconditioning techniques are used to reduce the number of
                      iterations.In this context, polynomial preconditioning was
                      developed. This method decreases the total number of dot
                      products by reducing the total number of iterations. Of
                      course, some additional work has to be done for the
                      preconditioning. When a polynomial of degree k is used, k
                      matrix-vector products per iteration have to be calculated
                      rather than one. On scalar machines, this shift between
                      matrix-vector products and dot products influences the
                      performance of the algorithm only slightly. On massively
                      parallel systems, dot products require global
                      synchronization, while the calculation of matrix-vector
                      products merely results in communication with a small number
                      of processors. Hence, polynomial preconditioned CG seems to
                      scale better than CG without preconditioning. Of course,
                      this is not the case in general. The performance of this
                      preconditioner depends on several issues, e.g., the sparsity
                      pattern and the eigenvalue distribution of the matrix, an
                      efficient communication scheme for the matrix-vector
                      products and the time needed for global synchronization of
                      the specific parallel machine. The actual implementation
                      used here is based on Chebyshev polynomials. Performance
                      tests were carried out on the Intel Paragon XP/S 10 with 140
                      nodes at the Research Centre Jülich (KFA). The CG method
                      with polynomial preconditioning shows better performance and
                      scalability than the basic method on massively parallel
                      machines. Additionally there are some numerical advantages
                      like a higher accuracy and an increased stability.},
      month         = {Sep},
      date          = {1994-09-25},
      organization  = {13. Workshop über
                       Parallelverarbeitung, Lessach
                       (Austria), 25 Sep 1994 - 1 Oct 1994},
      cin          = {JSC / ZAM},
      ddc          = {004},
      cid          = {I:(DE-Juel1)JSC-20090406 / I:(DE-Juel1)VDB62},
      pnm          = {899 - ohne Topic (POF2-899)},
      pid          = {G:(DE-HGF)POF2-899},
      typ          = {PUB:(DE-HGF)8},
      url          = {https://juser.fz-juelich.de/record/200820},
}