Parallel Symmetric Eigenvalue Problem Solvers

Abstract : Sparse symmetric eigenvalue problems arise in many computational science and engineering applications: in structural mechanics, nanoelectronics, and spectral reordering, for example. Often, the large size of these problems requires the development of eigensolvers that scale well on parallel computing platforms. In this dissertation, I describe two such eigensolvers, TraceMin and TraceMin-Davidson. These methods are different from many other eigensolvers in that they do not require accurate linear solves to be performed at each iteration in order to find the smallest eigenvalues and their associated eigenvectors. After introducing these closely related eigensolvers, I discuss alternative methods for solving saddle point problems arising in each iteration, which can improve the overall running time. Additionally, I present TraceMin-Multisectioning, a new TraceMin implementation geared towards finding large numbers of eigenpairs in any interval of the spectrum. I conclude with numerical experiments comparing my trace-minimization solvers to other popular eigensolvers (such as Krylov-Schur, LOBPCG, Jacobi- Davidson, and FEAST), establishing the competitiveness of my methods.

[1]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[2]  F. L. Bauer.  Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischer Eigenwertprobleme , 1957 .

[3]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .

[4]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[5]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[6]  Ahmed H. Sameh,et al.  Trace Minimization Algorithm for the Generalized Eigenvalue Problem , 1982, PPSC.

[7]  Robert B. Wilhelmson.  High-speed computing: scientific applications and algorithm design , 1988 .

[8]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.

[9]  Rudolf A. Römer,et al.  The Anderson Model of Localization: A Challenge for Modern Eigenvalue Methods , 1999, SIAM J. Sci. Comput..

[10]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[11]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM Rev..

[12]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[13]  A. Sameh,et al.  The trace minimization method for the symmetric generalized eigenvalue problem , 2000 .

[14]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[15]  Andrew V. Knyazev,et al.  Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method , 2001, SIAM J. Sci. Comput..

[16]  Olaf Schenk,et al.  Solving unsymmetric sparse systems of linear equations with PARDISO , 2002, Future Gener. Comput. Syst..

[17]  Vicente Hernández,et al.  SLEPc: Scalable Library for Eigenvalue Problem Computations , 2002, VECPAR.

[18]  G. W. Stewart,et al.  A Krylov-Schur Algorithm for Large Eigenproblems , 2001, SIAM J. Matrix Anal. Appl..

[19]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[20]  Michele Benzi,et al.  Spectral Properties of the Hermitian and Skew-Hermitian Splitting Preconditioner for Saddle Point Problems , 2005, SIAM J. Matrix Anal. Appl..

[21]  Ahmed Sameh,et al.  A nested iterative scheme for indefinite linear systems in particulate flows , 2004 .

[22]  Pierre-Antoine Absil,et al.  Adaptive Model Trust Region Methods for Generalized Eigenvalue Problems , 2005, International Conference on Computational Science.

[23]  Vicente Hernández,et al.  SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems , 2005, TOMS.

[24]  Tamara G. Kolda,et al.  An overview of the Trilinos project , 2005, TOMS.

[25]  Rudolf A. Römer,et al.  On Large-Scale Diagonalization Techniques for the Anderson Model of Localization , 2006, SIAM J. Sci. Comput..

[26]  O. Schenk,et al.  ON FAST FACTORIZATION PIVOTING METHODS FOR SPARSE SYMMETRI C INDEFINITE SYSTEMS , 2006 .

[27]  Patrick Amestoy,et al.  Hybrid scheduling for the parallel solution of linear systems , 2006, Parallel Comput..

[28]  Merico E. Argentati,et al.  Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc , 2007, SIAM J. Sci. Comput..

[29]  Pierre-Antoine Absil,et al.  Trust-Region Methods on Riemannian Manifolds , 2007, Found. Comput. Math..

[30]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[31]  Yousef Saad,et al.  Block Krylov–Schur method for large symmetric eigenvalue problems , 2008, Numerical Algorithms.

[32]  Richard B. Lehoucq,et al.  Anasazi software for the numerical solution of large-scale eigenvalue problems , 2009, TOMS.

[33]  D. Wiersma,et al.  Fifty years of Anderson localization , 2009 .

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

[35]  Murat Manguoglu,et al.  PSPIKE: A Parallel Hybrid Sparse Linear System Solver , 2009, Euro-Par.

[36]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[37]  Volker Mehrmann,et al.  Nonlinear eigenvalue and frequency response problems in industrial practice , 2011 .

[38]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[39]  James Demmel,et al.  Communication-optimal Parallel and Sequential QR and LU Factorizations , 2008, SIAM J. Sci. Comput..

[40]  Eric Polizzi,et al.  A High-Performance Numerical Library for Solving Eigenvalue Problems: FEAST Solver v2.0 User's Guide , 2012, ArXiv.

[41]  Edoardo Di Napoli,et al.  Efficient estimation of eigenvalue counts in an interval , 2013, Numer. Linear Algebra Appl..

[42]  Anshul Gupta,et al.  WSMP : Watson Sparse Matrix Package Part I – direct solution of symmetric systems Version 20 . 12 , 2017 .