Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. The problem may appear in the form of a low-degree polynomial (quartic or quintic, say) expressed in the monomial basis, or a high-degree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a nonmonomial basis. We have implemented a parallel solver in SLEPc covering both cases that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart and illustrate the robustness and performance of the solver with some numerical experiments.

[1]  Karl Meerbergen,et al.  The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem , 2008, SIAM J. Matrix Anal. Appl..

[2]  Timo Betcke,et al.  Optimal Scaling of Generalized and Polynomial Eigenvalue Problems , 2008, SIAM J. Matrix Anal. Appl..

[3]  Zhaojun Bai,et al.  SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem , 2005, SIAM J. Matrix Anal. Appl..

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

[5]  Nicholas J. Higham,et al.  Backward Error of Polynomial Eigenproblems Solved by Linearization , 2007, SIAM J. Matrix Anal. Appl..

[6]  Nicholas J. Higham,et al.  NLEVP: A Collection of Nonlinear Eigenvalue Problems , 2013, TOMS.

[7]  Andrés Tomás,et al.  Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement , 2007, Parallel Comput..

[8]  Paul Van Dooren,et al.  Normwise Scaling of Second Order Polynomial Matrices , 2004, SIAM J. Matrix Anal. Appl..

[9]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

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

[11]  P. Lancaster,et al.  Factorization of selfadjoint matrix polynomials with constant signature , 1982 .

[12]  Frann Coise Tisseur.  Backward Error and Condition of Polynomial Eigenvalue Problems , 1999 .

[13]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[14]  Weichung Wang,et al.  A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation , 2010, J. Comput. Phys..

[15]  Karl Meerbergen,et al.  The Quadratic Eigenvalue Problem , 2001, SIAM Rev..

[16]  Volker Mehrmann,et al.  Vector Spaces of Linearizations for Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[17]  Sven Hammarling,et al.  An algorithm for the complete solution of quadratic eigenvalue problems , 2013, TOMS.

[18]  José E. Román,et al.  Parallel iterative refinement in polynomial eigenvalue problems , 2016, Numer. Linear Algebra Appl..

[19]  Daniel Kressner,et al.  Perturbation, extraction and refinement of invariant pairs for matrix polynomials , 2011 .

[20]  Daniel Kressner,et al.  A block Newton method for nonlinear eigenvalue problems , 2009, Numerische Mathematik.

[21]  D. Kressner,et al.  Chebyshev interpolation for nonlinear eigenvalue problems , 2012 .

[22]  P. Lancaster,et al.  Linearization of matrix polynomials expressed in polynomial bases , 2008 .

[23]  Marlis Hochbruck,et al.  A Multilevel Jacobi--Davidson Method for Polynomial PDE Eigenvalue Problems Arising in Plasma Physics , 2010, SIAM J. Sci. Comput..

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

[25]  Robert M. Corless,et al.  Block LU factors of generalized companion matrix pencils , 2007, Theor. Comput. Sci..

[26]  Daniel Kressner,et al.  Memory‐efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis , 2014, Numer. Linear Algebra Appl..