Preconditioning complex symmetric linear systems

A new preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to conjugate orthogonal conjugate gradient (COCG) or conjugate orthogonal conjugate residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. To reduce the computational cost the preconditioner is approximated with an inexact variant based on incomplete Cholesky decomposition or on orthogonal polynomials. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the inexact polynomial version completes the description of the preconditioner.

[1]  Richard F. Barrett,et al.  Matrix Market: a web resource for test matrix collections , 1996, Quality of Numerical Software.

[2]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[3]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[4]  R. P. Kendall,et al.  An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations , 1968 .

[5]  R.D. Falgout,et al.  An Introduction to Algebraic Multigrid Computing , 2006, Computing in Science & Engineering.

[6]  T. Sogabe,et al.  A COCR method for solving complex symmetric linear systems , 2007 .

[7]  Jennifer A. Scott,et al.  On Positive Semidefinite Modification Schemes for Incomplete Cholesky Factorization , 2014, SIAM J. Sci. Comput..

[8]  Artem Napov,et al.  An Algebraic Multigrid Method with Guaranteed Convergence Rate , 2012, SIAM J. Sci. Comput..

[9]  A. Cicone.  A note on the Joint Spectral Radius , 2015, 1502.01506.

[10]  C. Micchelli,et al.  Polynomial Preconditioners for Conjugate Gradient Calculations , 1983 .

[11]  S. Ashby.  Minimax polynomial preconditioning for Hermitian linear systems , 1991 .

[12]  A. Bunse-Gerstner,et al.  On a conjugate gradient-type method for solving complex symmetric linear systems , 1999 .

[13]  Amin Rafiei.  Left-looking version of AINV preconditioner with complete pivoting strategy , 2014 .

[14]  D. K. Salkuyeh,et al.  A sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of an SPD matrix , 2008, 0807.3644.

[15]  R. Freund.  On conjugate gradient type methods and polynomial preconditioners for a class of complex non-hermitian matrices , 1990 .

[16]  Gerard L. G. Sleijpen,et al.  BiCR variants of the hybrid BiCG methods for solving linear systems with nonsymmetric matrices , 2010, J. Comput. Appl. Math..

[17]  Yousef Saad,et al.  ILUM: A Multi-Elimination ILU Preconditioner for General Sparse Matrices , 1996, SIAM J. Sci. Comput..

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

[19]  Yong Zhang,et al.  Lanczos-type variants of the COCR method for complex nonsymmetric linear systems , 2009, J. Comput. Phys..

[20]  Owe Axelsson,et al.  Iteration number for the conjugate gradient method , 2003, Math. Comput. Simul..

[21]  Carlo Janna,et al.  Adaptive Pattern Research for Block FSAI Preconditioning , 2011, SIAM J. Sci. Comput..

[22]  Luca Bergamaschi,et al.  Approximate inverse preconditioning in the parallel solution of sparse eigenproblems , 2000, Numer. Linear Algebra Appl..

[23]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[24]  Michele Benzi,et al.  A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method , 1996, SIAM J. Sci. Comput..

[25]  B. Fischer.  Polynomial Based Iteration Methods for Symmetric Linear Systems , 1996 .

[26]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

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

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

[29]  M. Sugihara,et al.  An extension of the conjugate residual method to nonsymmetric linear systems , 2009 .

[30]  Lorenzo Codecasa,et al.  Base functions and discrete constitutive relations for staggered polyhedral grids , 2009 .

[31]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[32]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[33]  M. Ng,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[34]  Fang Chen,et al.  Modified HSS iteration methods for a class of complex symmetric linear systems , 2010, Computing.

[35]  Y. Saad,et al.  Practical Use of Polynomial Preconditionings for the Conjugate Gradient Method , 1985 .

[36]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[37]  Thomas A. Manteuffel,et al.  A Comparison of Adaptive Chebyshev and Least Squares Polynomial Preconditioning for Hermitian Positive Definite Linear Systems , 1992, SIAM J. Sci. Comput..

[38]  J. Scott,et al.  HSL_MI20: An efficient AMG preconditioner for finite element problems in 3D , 2010 .

[39]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[40]  Owe Axelsson,et al.  A survey of preconditioned iterative methods for linear systems of algebraic equations , 1985 .

[41]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[42]  Anne Greenbaum,et al.  Approximating the inverse of a matrix for use in iterative algorithms on vector processors , 1979, Computing.

[43]  Chih-Jen Lin,et al.  Incomplete Cholesky Factorizations with Limited Memory , 1999, SIAM J. Sci. Comput..

[44]  S. Ashby.  Polynomial Preconditioning for Conjugate Gradient Methods , 1988 .

[45]  Roland W. Freund,et al.  Conjugate Gradient-Type Methods for Linear Systems with Complex Symmetric Coefficient Matrices , 1992, SIAM J. Sci. Comput..

[46]  Hassane Sadok,et al.  CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm , 1999, Numerical Algorithms.

[47]  V. Kozyakin.  On accuracy of approximation of the spectral radius by the Gelfand formula , 2008, 0810.2856.

[48]  Edmond Chow,et al.  Parallel Implementation and Practical Use of Sparse Approximate Inverse Preconditioners with a Priori Sparsity Patterns , 2001, Int. J. High Perform. Comput. Appl..

[49]  Gerard L. G. Sleijpen,et al.  Maintaining convergence properties of BiCGstab methods in finite precision arithmetic , 1995, Numerical Algorithms.

[50]  Antonello Tamburrino,et al.  Multi-frequency identification of defects in conducting media , 2008 .

[51]  Fang Chen,et al.  On preconditioned MHSS iteration methods for complex symmetric linear systems , 2011, Numerical Algorithms.

[52]  Karin Schwab.  The Joint Spectral Radius Theory And Applications , 2016 .

[54]  Anshul Gupta,et al.  Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning , 2010, SIAM J. Sci. Comput..

[55]  Timothy A. Davis,et al.  Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2) , 2006 .

[56]  D FalgoutRobert.  An Introduction to Algebraic Multigrid , 2006 .

[57]  D. R. Fokkema,et al.  BICGSTAB( L ) FOR LINEAR EQUATIONS INVOLVING UNSYMMETRIC MATRICES WITH COMPLEX , 1993 .

[58]  Ursula van Rienen,et al.  Numerical Methods in Computational Electrodynamics - Linear Systems in Practical Applications , 2001, Lecture Notes in Computational Science and Engineering.

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