Multi-GPU Scalable Implementation of a Contour-Integral-Based Eigensolver for Real Symmetric Dense Generalized Eigenvalue Problems

We consider a parallel eigensolver for generalized eigenvalue problems for distributed GPU systems. In this paper, we propose a distributed parallel implementation of the Sakurai-Sugiura (SS) eigenvalue solver for solving generalized eigenvalue problems with real symmetric matrices using GPU linear algebra libraries. In the SS method, the target subspace is constructed from solutions of linear systems. The dominant part of this method is calculating solutions of linear equations. By assigning the solution of independent linear systems to each GPU, a coarse-grained parallelism can be obtained, and high scalability is expected. We also proposed the performance model of this implementation. We evaluate its parallel performance using numerical examples that involve medium-size dense matrices.

[1]  Hiroto Tadano,et al.  A numerical method for nonlinear eigenvalue problems using contour integrals , 2009, JSIAM Lett..

[2]  Toshiyuki Imamura,et al.  Development of a High-Performance Eigensolver on a Peta-Scale Next-Generation Supercomputer System (Selected Papers of the Joint International Conference of Supercomputing in Nuclear Applications and Monte Carlo : SNA + MC 2010) , 2011 .

[3]  Tetsuya Sakurai,et al.  Efficient Parameter Estimation and Implementation of a Contour Integral-Based Eigensolver , 2013 .

[4]  Eric Polizzi,et al.  A Density Matrix-based Algorithm for Solving Eigenvalue Problems , 2009, ArXiv.

[5]  Tetsuya Sakurai,et al.  A projection method for nonlinear eigenvalue problems using contour integrals , 2013, JSIAM Lett..

[6]  Tetsuya Sakurai,et al.  A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method , 2008, J. Comput. Appl. Math..

[7]  Wolf-Jurgen Beyn,et al.  An integral method for solving nonlinear eigenvalue problems , 2010, 1003.1580.

[8]  T. Sakurai,et al.  A projection method for generalized eigenvalue problems , 2002 .

[9]  Lukas Krämer,et al.  Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations , 2011, Parallel Comput..

[10]  T. Sakurai,et al.  A projection method for generalized eigenvalue problems using numerical integration , 2003 .

[11]  T. Fujiwara,et al.  An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system , 2012, Journal of physics. Condensed matter : an Institute of Physics journal.

[12]  Tetsuya Sakurai,et al.  Efficient Algorithm for Linear Systems Arising in Solutions of Eigenproblems and Its Application to Electronic-Structure Calculations , 2012, VECPAR.

[13]  Toshiyuki Imamura,et al.  Development of a High-Performance Eigensolver on a Peta-Scale Next-Generation Supercomputer System , 2011 .

[14]  T. Sakurai,et al.  CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems , 2007 .

[15]  H. V. D. Vorst,et al.  A Petrov-Galerkin type method for solving Axk=b, where A is symmetric complex , 1990 .

[16]  Eric Polizzi,et al.  Density-Matrix-Based Algorithms for Solving Eingenvalue Problems , 2009 .

[17]  Tetsuya Sakurai,et al.  Performance comparison of parallel eigensolvers based on a contour integral method and a Lanczos method , 2013, Parallel Comput..

[18]  T. Sakurai,et al.  A quadrature-based eigensolver with a Krylov subspace method for shifted linear systems for Hermitian eigenproblems in lattice QCD , 2010, JSIAM Lett..

[19]  T. Fujiwara,et al.  Domain boundary formation in helical multishell gold nanowires , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[20]  Mitsuhisa Sato,et al.  A parallel method for large sparse generalized eigenvalue problems using a GridRPC system , 2008, Future Gener. Comput. Syst..