A hybrid method for the parallel computation of Green's functions

  • Authors:

  • Dan Erik Petersen;Song Li;Kurt Stokbro;Hans Henrik B. SøRensen;Per Christian Hansen;Stig Skelboe;Eric Darve

  • Affiliations:

  • Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark;Institute for Computational and Mathematical Engineering, Stanford University, 496 Lomita Mall, Durand Building, Stanford, CA 94305-4042, USA;Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark;Department of Computer Science, University of Aarhus, IT-Parken, Aabogade 34, DK-8200 Aarhus N, Denmark;Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads, Bldg. 321, DK-2800 Lyngby, Denmark;Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark;Department of Mechanical Engineering, Stanford University, 496 Lomita Mall, Durand Building, Room 209, Stanford, CA 94305-4040, USA and Institute for Computational and Mathematical Engineering, St ...

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

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Abstract

Quantum transport models for nanodevices using the non-equilibrium Green's function method require the repeated calculation of the block tridiagonal part of the Green's and lesser Green's function matrices. This problem is related to the calculation of the inverse of a sparse matrix. Because of the large number of times this calculation needs to be performed, this is computationally very expensive even on supercomputers. The classical approach is based on recurrence formulas which cannot be efficiently parallelized. This practically prevents the solution of large problems with hundreds of thousands of atoms. We propose new recurrences for a general class of sparse matrices to calculate Green's and lesser Green's function matrices which extend formulas derived by Takahashi and others. We show that these recurrences may lead to a dramatically reduced computational cost because they only require computing a small number of entries of the inverse matrix. Then, we propose a parallelization strategy for block tridiagonal matrices which involves a combination of Schur complement calculations and cyclic reduction. It achieves good scalability even on problems of modest size.