Computing Partial Spectra with Least-Squares Rational Filters

We present a method for computing partial spectra of Hermitian matrices, based on a combination of subspace iteration with rational filtering. In contrast with classical rational filters derived from Cauchy integrals or from uniform approximations to a step function, we adopt a least-squares (LS) viewpoint for designing filters. One of the goals of the proposed approach is to build a filter that will lead to linear systems that are easier to solve by iterative methods. Among the main advantages of the proposed LS filters is their flexibility. Thus, we can place poles in more general locations than with the formulations mentioned above, and we can also repeat these poles a few times for better efficiency. This leads to a smaller number of required poles than in existing methods. As a consequence, factorization costs are reduced when direct solvers are used and the scheme is also beneficial for iterative solvers. The paper discusses iterative schemes to solve the linear systems resulting from the filtered s...

[1]  Tetsuya Sakurai,et al.  Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems , 2015, Numerical Algorithms.

[2]  Y. Saad.  Numerical Methods for Large Eigenvalue Problems , 2011 .

[3]  Fei Xue,et al.  Krylov Subspace Recycling for Sequences of Shifted Linear Systems , 2013, ArXiv.

[4]  Qiang Ye,et al.  A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares Problems , 2003, SIAM J. Matrix Anal. Appl..

[5]  T. Rose,et al.  Substructuring in MSC/NASTRAN for large scale parallel applications , 1991 .

[6]  Lloyd N. Trefethen,et al.  Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic , 2015, SIAM J. Sci. Comput..

[7]  Y. Saad,et al.  Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[9]  Marc Van Barel,et al.  Designing rational filter functions for solving eigenvalue problems by contour integration , 2015 .

[10]  Marc Van Barel,et al.  Nonlinear eigenvalue problems and contour integrals , 2016, J. Comput. Appl. Math..

[11]  Y. Saad,et al.  Experimental study of ILU preconditioners for indefinite matrices , 1997 .

[12]  Ping Tak Peter Tang,et al.  Feast Eigensolver for Non-Hermitian Problems , 2015, SIAM J. Sci. Comput..

[13]  Y. Saad,et al.  PARSEC – the pseudopotential algorithm for real‐space electronic structure calculations: recent advances and novel applications to nano‐structures , 2006 .

[14]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

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

[16]  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..

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

[18]  Ping Tak Peter Tang,et al.  FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection , 2013, SIAM J. Matrix Anal. Appl..

[19]  Ren-Cang Li,et al.  Krylov type subspace methods for matrix polynomials , 2006 .

[20]  Yousef Saad,et al.  Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems , 2018, Numer. Linear Algebra Appl..

[21]  Y. Saad,et al.  Iterative Solution of Indefinite Symmetric Linear Systems by Methods Using Orthogonal Polynomials over Two Disjoint Intervals , 1983 .

[22]  D. Sorensen,et al.  4. The Implicitly Restarted Arnoldi Method , 1998 .

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

[24]  Youcef Saad,et al.  Overview of Krylov subspace methods with applications to control problems , 1989 .

[25]  Andreas Stathopoulos,et al.  Iterative Validation of Eigensolvers: A Scheme for Improving the Reliability of Hermitian Eigenvalue Solvers , 2006, SIAM J. Sci. Comput..

[26]  Ping Tak Peter Tang,et al.  Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver , 2014, SIAM J. Sci. Comput..

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

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

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

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

[31]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[32]  Andreas Stathopoulos,et al.  PRIMME: preconditioned iterative multimethod eigensolver—methods and software description , 2010, TOMS.

[33]  Yousef Saad,et al.  A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems , 2012, SIAM J. Sci. Comput..