On k-step CSCS-based polynomial preconditioners for Toeplitz linear systems with application to fractional diffusion equations

Abstract The implicit finite difference scheme with the shifted Gruwald formula for discretizing the fractional diffusion equations (FDEs) often results in the ill-conditioned non-Hermitian Toeplitz systems. In the present paper, we consider to solve such Toeplitz systems by exploiting the preconditioned GMRES method. A k -step polynomial preconditioner is designed based on the circulant and skew-circulant splitting (CSCS) iteration method proposed by Ng (2003). Theoretical and experimental results involving numerical solutions of FDEs demonstrate that the proposed k -step preconditioner is efficient to accelerate the GMRES solver for non-Hermitian Toeplitz systems.

[1]  A. Wathen,et al.  Iterative Methods for Toeplitz Systems , 2005 .

[2]  Raymond H. Chan,et al.  Preconditioners for non-Hermitian Toeplitz systems , 2001, Numer. Linear Algebra Appl..

[3]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[4]  Siu-Long Lei,et al.  Circulant and skew-circulant splitting iteration for fractional advection–diffusion equations , 2014, Int. J. Comput. Math..

[5]  Siu-Long Lei,et al.  A circulant preconditioner for fractional diffusion equations , 2013, J. Comput. Phys..

[6]  Faezeh Toutounian,et al.  Recursive self preconditioning method based on Schur complement for Toeplitz matrices , 2012, Numerical Algorithms.

[7]  Yu-Mei Huang,et al.  On $${\varvec{m}}$$m-step Hermitian and skew-Hermitian splitting preconditioning methods , 2015 .

[8]  O. Axelsson,et al.  Preconditioned Conjugate Gradient Methods , 1990 .

[9]  Loyce M. Adams,et al.  Additive polynomial preconditioners for parallel computers , 1989, Parallel Comput..

[10]  C.-C. Jay Kuo,et al.  Spectral properties of preconditioned rational Toeplitz matrices: the nonsymmetric case , 1993 .

[11]  Yang Wang,et al.  On validity of m-step multisplitting preconditioners for linear systems , 2002, Appl. Math. Comput..

[12]  Apostolos Hadjidimos,et al.  Some notes on multisplltting methods and m-step preconditioners for linear systems , 1996 .

[13]  L. Adams.  m-Step Preconditioned Conjugate Gradient Methods , 1985 .

[14]  Ting-Zhu Huang,et al.  Strang-type preconditioners for solving fractional diffusion equations by boundary value methods , 2013, J. Comput. Appl. Math..

[15]  Ting-Zhu Huang,et al.  On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors , 2009, Math. Comput. Simul..

[16]  Stefano Serra Capizzano,et al.  Preconditioning strategies for non‐Hermitian Toeplitz linear systems , 2005, Numer. Linear Algebra Appl..

[17]  Chen Fang,et al.  On structured variants of modified HSS Iteration methods for complex Toeplitz linear systems , 2013 .

[18]  Gabriele Steidl,et al.  Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems , 1998 .

[19]  M. Ng.  Circulant and skew-circulant splitting methods for Toeplitz systems , 2003 .

[20]  Raymond H. Chan,et al.  Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems , 1991 .

[21]  Michael K. Ng,et al.  Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems , 2009, Numerical Algorithms.

[22]  Raymond H. Chan,et al.  Toeplitz-Circulant Preconditioners for Toeplitz Systems and their Applications to Queueing Networks with Batch Arrivals , 1996, SIAM J. Sci. Comput..